AVERTISSEMENT. D'autre part, toute contrefaçon, plagiat, reproduction encourt une poursuite pénale. LIENS

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1 AVERTISSEMENT Ce document est le frut d'un long traval approuvé par le jury de soutenance et ms à dsposton de l'ensemble de la communauté unverstare élarge. Il est soums à la proprété ntellectuelle de l'auteur. Cec mplque une oblgaton de ctaton et de référencement lors de l utlsaton de ce document. D'autre part, toute contrefaçon, plagat, reproducton encourt une poursute pénale. llcte Contact : ddoc-theses-contact@unv-lorrane.fr LIENS Code de la Proprété Intellectuelle. artcles L Code de la Proprété Intellectuelle. artcles L L

2 Thèse présentée en vue de l obtenton du ttre de Docteur de l Unversté de Lorrane en Physque par Perre Wendenbaum Intrcaton et dynamque de trempe dans les chaînes de spns quantques Entanglement and quench dynamcs n quantum spn chans Soutenance publque devant la commsson d examen le 8 décembre 2014 Composton du jury: Présdent: M. Bertrand Berche Professeur, Unversté de Lorrane Rapporteurs: M. Stéphane Attal Professeur, Unversté Claude Bernard, Lyon I M. Gunter Schütz Professeur, Forschungszentrum, Jülch, Allemagne Examnateurs: M. Drag Karevsk Professeur, Unversté de Lorrane (drecteur de thèse) M. Therry Platn Senor Lecturer, Coventry Unversty, Royaume-Un M. Gullaume Roux Maître de Conférences, Unversté Pars-Sud Insttut Jean Lamour - Département de Physque de la Matère et des Matéraux Unversté de Lorrane - U.F.R Faculté des Scences et Technologes - Vandœuvre-lès-Nancy Secteur Physque Géoscences Chme Mécanque École doctorale EMMA

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4 "-Bonjour, dt le Pett Prnce. - Bonjour dt le marchand." C état un marchand de plules perfectonnées qu apasent la sof. On en avale une par semane et l on n éprouve plus le beson de bore. "-Pourquo vends tu ça? dt le Pett Prnce. - C est une grosse économe de temps, dt le marchand. Les experts ont fat des calculs. On épargne cnquante-tros mnutes par semane. - Et que fat on des cnquante-tros mnutes? - On en fat ce qu on veut... - Mo, dt le Pett Prnce, s j avas cnquante-tros mnutes à dépenser, je marcheras tout doucement vers une fontane..." Antone de Sant Exupéry Le Pett Prnce

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6 Remercements/Acknowledgements Un seul mot, usé, mas qu brlle comme une velle pèce de monnae: merc! Pablo Neruda Le moment est mantenant venu de s asseor et de regarder en arrère vers ces dernères années. En se retournant, on vot un long chemn. Ce long chemn de tros ans que consttue une thèse de doctorat, je ne l a, heureusement, pas arpenté seul. Peu mporte la sute donnée à cette thèse, les rencontres effectuées ont été source d un grand enrchssement, tant scentfque qu human. Un grand nombre de personnes, scentfques ou non, a contrbué de près ou de lon à la réusste de cette entreprse. Je doute pouvor nommément lster toutes ces personnes, auss, que celles qu ne trouvent pas leur nom c-dessous soent gratfées de ma reconnassance. Je tens en tout premer leu à remercer Stéphane Attal et Gunter Schütz d avor accepté de prendre un peu de leur temps pour être rapporteurs de ce manuscrt. Merc également à Bertrand Berche d avor présdé mon jury, et à Therry Platn et Gullaume Roux d avor accepté d en fare parte. Cette thèse s est déroulée au sen du département de Physque de la Matère et des Matéraux de l Insttut Jean Lamour. Je remerce donc son drecteur Mchel Vergnat pour son accuel. Je tens évdemment à exprmer ma profonde et sncère grattude à Drag Karevsk, mon drecteur de thèse. Merc pour la confance que tu m as témognée, notamment à l ssue de mon stage de master en cherchant un moyen pour que je pusse contnuer en thèse avec to, alors que ce n état pas ce qu état ntalement prévu. Tes connassances sur les systèmes quantques hors équlbre, ton expérence de la recherche, ton souten et la grande autonome que tu m as lassée m ont été préceux et ont été l un des moteurs de ce traval. C est avec un grand plasr que je remerce chaleureusement tous les membres de l équpe de physque statstque pour les bons moments que j a passés avec chacun d entre eux, et pour l envronnement scentfque très stmulant qu ls ont crée. Les pauses autour d un café ont été des moments de détente apprécables, où j a pu en apprendre ben plus sur le fonctonnement en "off" d nsttutons comme l unversté ou le laboratore. Ce sera sûrement avec une ponte de nostalge que je contnurea de suvre vos recherches et la ve du groupe. J a une pensée toute partculère pour v

7 touts les thésards et thésardes passés et présents de la 106 que j a eu le bonheur de côtoyer durant ces tros ans et dem: Maro Collura, Xaver Durang, Sophe Mantell, Nelson Bolvar, Ncolas Allegra, Dmtrs Volots, Marana Krasnytska, Emlo Flores, Sascha Wald et Hugo Tschrhart. Partager son bureau avec des gens venus de contrées lontanes pour certans apporte un certan enrchssement culturel, merc pour cela. Je vous souhate à tous bonheur et épanoussement dans vos projets et ve futurs. Je remerce partculèrement Bertand, alors chef du département de physque, pour avor démêlé une stuaton ben mal embarquée concernant mon montorat. Je remerce également toutes les personnes membres d autres équpes de l jl, ans que tous les autres thésards que j a pu cotoyer. I spent a non neglgble part of my Ph.D n the Theoretcal Quantum Physcs Group of the Saarland Unversty. I would lke to thank ts leader Govanna Morg to have welcomed me n the group, and to have proposed me an nterestng research subject. Even f the orgnal plan has not been fulflled, I really apprecated the human and scentfc envronment of the team. There are two peoples that I would lke to thank specally. The frst one s Endre Kajar. Thanks for havng taken care of me when I arrved n these unknown country and group. Your enthusasm for my work, your dsponblty to answer all my stupd questons and your great pedagogcal sklls helped me to start my work on the rght track. I wsh you the best n your new lfe as a physcs teacher. The second person s of course Bruno Taketan. More than a smple collaborator, your became over the months a frend. You mplcaton and your advses, on both scentfc and poltcal levels, have been very precous to me. I m very grateful to you to have shared wth me your bg knowledge on quantum entanglement and quantum nformaton processng durng our "ten mnutes meetngs", whch became most of the tme "one hour and a half meetngs". Of course I also acknowledge every people I met durng my stay n Saarbrücken, and n partcular my two offce roommates, Susanne for her help on admnstratve stuff, and the crazy Irshman Mossy for s daly good mood. Le derner projet de ma thèse a été réalsé en collaboraton avec Therry Platn. Je le (re)remerce donc pour son mplcaton dans ce traval, pour sa dsponblté et pour avor répondu à mes questons concernant les systèmes en nteractons répétées. Merc également de m avor accuell lors de mon séjour à Coventry. J a eu la chance d avor pu ensegner durant cette thèse. Je remerce donc mes collègues de mécanque du pont, Sacha Ourjoumtsev, Xaver Glad et Therry Revellé, ce derner m ayant partculèrement adé en répondant à chacune de mes questons concernant l ensegnement. Merc également à Vrgne Pchon avec qu j a ensegné en PACES durant ma dernère année. Qu dt contrat DCCE dt forcément formatons et stages résdentels... Je remerce donc mes compagnons d nfortune Thomas Drouot, Lucas Echenberger, Marjore Etque et Mylène Rchard. Ces journées auraent été beaucoup, beaucoup plus longues sans votre présence. Parce que pour arrver à l étape "thèse" l faut d abord passer par l étape "cnq années de cours à l unversté", je remerce tous mes camarades de promoton, en v

8 partculer Gullaume, P-A et Pnpn, avec qu j a passé d agréables moments. Comme dt plus haut, le bon déroulement d une thèse résde auss dans le contact avec des personnes extéreures au monde la recherche unverstare. Je voudras remercer deux secrétares dont les compétences ne sont plus à prouver. Merc donc à Martne Gauler pour son mmense gentllesse et à Chrstne Sartor pour son dynamsme, son mplcaton auprès des doctorants, et pour m avor sort plus d une fos d mbroglos comme seule l admnstraton françase sat en créer. Merc aux gérants de la cafet, Phlppe, Mare et leur flle Améle pour leur bonne humeur quotdenne, pour m avor nourr toutes ces années et pour avor accepté de mettre de la sauce salade dans mes sandwchs. Je voudras ben sûr remercer tous mes ams hors unversté, passés et présents. En partculer, je remerce Émle pour son amté de presque deux décennes mantenant. Je ne compte plus le nombre de bons souvenrs en ta compagne. J a également une pensée amcale pour Jessca et Jérémy Greco (l tenat au nom!). Merc pour toutes les nvtatons à Aumetz et pour contnuer de jouer à Ffa avec mo malgré toutes tes défates :). Je tens évdemment à remercer mes grands-parents, oncles, tantes et toute ma famlle. En partculer ma sœur Sabrna, mon beau-frère Martal et surtout mes parents. Merc à eux pour leur souten moral et fnancer 1, et pour m avor toujours lassé chosr mon chemn malgré le flou professonnel que peut représenter une thèse en physque théorque. Ces remercements ne sauraent être complets sans quelques lgnes pour Auréle. Merc de m avor accompagné pendant ces tros ans, cette thèse n aurat défntvement pas été la même sans to. Dans les moments dffcles, je savas que, même séparés par la dstance, tu étas toujours un peu avec mo quand même. Merc pour ton souten et tes consels "d ancenne thésarde". Ta présence m aura perms de relatvser les dffcultés, et de ne pas oubler l essentel. Merc à to. 1. On ouble souvent de le précser, mas ne pas avor à se soucer de trouver de quo payer son logement ou sa nourrture et un plus ndénable dans la réusste de ses études. Merc pour cela. v

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10 Contents Remercements/Acknowledgements Contents Résumé détallé (en franças) v x x Introducton 1 1 Quantum Entanglement and Models Quantum entanglement Defntons and separablty crtera Entanglement measures Entanglement and quantum phase transtons Quantum teleportaton Quantum decoherence The XY model Presentaton and canoncal dagonalzaton Phase dagram Dynamcs Entanglement entropy of the XX chan The Bose-Hubbard model Self-trappng of Hard Core bosons on optcal lattce Bose-Hubbard model and Hard-Core lmt Contnuum lmt and local equlbrum hypothess Intal states Dynamcs after the sudden quench Free expanson of the condensate Self Trappng wth a lnear potental Concluson Entanglement creaton between two spns embedded n an Isng chan Model and theoretcal treatment Hamltonans and ntal states Hgh magnetzaton lmt Characterzaton of the bath Full Hamltonan n normal coordnates x

11 3.2 Tme evoluton of the spn defects Entanglement dynamcs Spns coupled to the same pont Spns coupled at two dfferent ponts Spectral densty theory Concluson Dsentanglement of Bell state by nteracton wth a non equlbrum envronment Hamltonan and dynamcs Loschmdt Echo n the fermonc representaton Quench dynamcs Weak and strong couplng regmes Effect of the quench on the dsentanglement dynamcs Short tmes dynamcs Revval tme Independent dynamcs Concluson Non Equlbrum and Equlbrum Steady State entanglement drven by quantum repeated nteractons Quantum repeated nteractons Descrpton of the repeated nteractons process Tme evoluton of the system XY model Intal states Dynamcs of the Clfford operators Tme evoluton of the reduced densty matrx Tme evoluton of the two-pont correlaton matrx Contnuous lmt of the evoluton equaton Toy model Model and shape of the reduced densty matrx Intal state Tme evoluton of the correlaton matrx Spn-spn correlaton functons and evoluton of the concurrence Loss of entanglement n the envronment General case: System of two chans of sze N Model an ntal states Study for tmes t < 2N: NESS Steady state Convergence toward the steady state Concluson Concluson 135 A Tme evoluton of the reduced densty matrx 137 B Gaussan character of the Bell state 143 x

12 C Statonary spn-spn correlaton functons 145 Bblography 147 x

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14 Résumé détallé (en franças) Le concept d ntrcaton trouve ses orgnes dès les premers temps du développement de la théore quantque. Schrödnger est le premer à mettre un nom sur ce concept ("Verschränkung" en allemand), tands qu Ensten, Podolsk et Rosen mentonnaent, dans leur célèbre artcle de 1935 [EPR35], a spooky acton at a dstance que l on pourrat tradure par une acton à dstance fantomatque et effrayante. S ce concept a, d un pont de vue théorque et phlosophque, perms une melleure compréhenson des fondements de la mécanque quantque, l ntrcaton est devenue depus les années quatre-vngt un ngrédent clé dans le développement de l nformaton quantque [NC00]. Ce domane de recherche récent tente de surpasser les lmtes mposées par les concepts d nformaton classque. Une des premères applcatons modernes de l ntrcaton en tant que ressource technologque est la "quantum key dstrbuton", premer protocole de cryptographe quantque permettant des communcatons sécursées. L ntrcaton quantque est également pressente pour être d une grande mportance dans le développement d ordnateurs quantques, dans lesquels l nformaton n est plus stockée dans des bts classques, mas dans des bts quantques, ou "qubts" [DV95]. Toutes ces nouvelles applcatons ont beson d un solde cadre théorque, c est pourquo de nombreux efforts ont été menés pour développer des défntons rgoureuses d un état ntrqué, ntrodure des crtères de séparablté [HHH96], ou des mesures d ntrcaton [PV07]. Ces méthodes développées en nformaton quantque ont attré la curosté d autres domanes de la physque, comme la physque de la matère condensée [Pre00, Sch13]. En effet, l ntrcaton d un état quantque peut ader à la caractérsaton de nouveaux états de la matère avec des corrélatons à longue portée, comme les supraconducteurs à haute température [Ver04] ou l effet Hall quantque [LH08]. L étude de l ntrcaton est également un outl pussant dans la détecton de transtons de phase quantques à température nulle. Au pont crtque, le système présente des corrélatons à longue portée à cause de l ntrcaton de l état fondamental. Par exemple, les proprétés d échelle d états fondamentaux ntrqués de systèmes crtques ont été analysées au travers du comportement de l entrope d ntrcaton, une mesure utlsée lorsque le système est pur. Il a notamment été montré que l entrope d ntrcaton de certanes théores crtques peut être relée à la charge centrale de la théore conforme correspondante [CC09, VLRK03, PS05]. Le concept de spectre d ntrcaton a été également ntrodut comme un outl pussant dans l étude des phénomènes crtques. L analyse du spectre de la matrce densté rédute d une bpartton d un système quantque et du gap de Schmdt (la dfférence entre les deux valeurs propres les plus élevées) a, par exemple, parfatement décrt le comportement du paramètre d ordre dans la régon crtque [DCLLS12]. S tous ces travaux tratent de l ntrcaton de l état fondamental, on peut se poser x

15 la queston quant au comportement de l ntrcaton dans des systèmes hors de l équlbre. En effet, grâce aux nombreux progrès expérmentaux ces dernères années, la physque des systèmes hors de l équlbre a connu un regan d ntérêt, spécalement dans des domanes comme la matère condensée, ou l optque quantque. Malgré ces progrès expérmentaux, une théore générale décrvant les systèmes quantques hors équlbre est toujours manquante. La compréhenson du comportement de ce genre de système est donc d une grande mportance dans la physque théorque moderne. Les travaux exposés dans ce manuscrt sont consacrés à l étude du comportement hors de l équlbre de systèmes quantques, et plus partculèrement de leurs proprétés d ntrcaton. Pluseurs protocoles peuvent être ms en œuvre pour amener un système hors de l équlbre. On peut par exemple réalser une trempe quantque [CG07, PSSV11] en varant plus ou mons rapdement un paramètre de contrôle dans l hamltonen. Le couplage d un système à un envronnement mène également à de très ntéressantes dynamques et proprétés d ntrcaton. La chaîne de spns quantque consttue un très bon canddat pour cette étude. En effet, l avantage de ce type de modèle résde dans sa fable dmensonnalté, nous autorsant la plupart du temps le développement de méthodes analytques et numérques permettant de dédure leur prncpales proprétés [LSM61, BMD70, BM71a, BM71b, MBA71]. Cette thèse est dvsée en cnq chaptres. Le chaptre 1 est consacré à la présentaton du cadre théorque et mathématque de cette thèse. On se consacre dans un premer temps à la descrpton du phénomène d ntrcaton quantque. Un état quantque pur Ψ dans un espace de Hlbert composte H = H 1 H 2 H N est dt séparable s l peut s écrre sous la forme Ψ = φ 1 φ 2... φ N, (1) où φ H. S l ne peut pas s écrre sous la forme précédente, cet état est dt ntrqué. Dans la sute de ce traval, nous nous lmterons au cas N = 2, et nous parlerons donc d ntrcaton bpartte. S l on consdère deux spns 1/2, un état pur ntrqué ben connu est par exemple l un des états de Bell Ψ = 1 2 ( + ), où l état (resp. ) est état propre de l opérateur σ z avec la valeur propre 1 (resp. 1). Cet état possède une proprété ntéressante lorsque l on effectue une mesure sur l un des deux spns, par exemple une mesure de sa polarsaton. En effet, dû au prncpe de réducton du paquet d onde, s on trouve que l état du premer spn est (resp. ), cela veut dre que le deuxème spn a été nstantanément projeté dans l état (resp. ) ben qu aucune mesure n at été effectuée sur ce spn. Lorsque l état consdéré est un mélange statstque (cas des états thermques par exemple), l ne peut être représenté par un vecteur de l espace de Hlbert, et l on dot fare appel au formalsme de la matrce densté. La noton d ntrcaton est alors généralsée de la manère suvante: un état ρ est dt séparable s l peut s écrre comme ρ = p ρ A ρ B, avec p = 1, p 0, (2) et est ntrqué autrement. Ic, ρ A et ρ B sont des matrces densté rédutes assocées aux xv

16 parttons A et B. Nous ntrodusons ensute la noton de crtère de séparablté, qu nous permet de connaître le caractère ntrqué ou non d un état. S cet état est pur, un de ces crtères est basé sur la décomposton de Schmdt [Sch06, EK95]. S un seul coeffcent apparaît dans cette décomposton, alors l état consdéré est séparable. Pour un mélange statstque, l un des crtères les plus fréquemment utlsé est le crtère PPT [Per96], pour postve partal transpose (transposée partelle postve en franças). S l on obtent un spectre postf après avor effectué la transposée partelle de la matrce de densté, alors l état est séparable. Dans la pratque, la seule connassance de l ntrcaton ou non d un état quantque est souvent nsuffsante. En effet, ces crtères ne nous donnent pas accès à la quantté d ntrcaton présente, et l est alors judceux de défnr une mesure d ntrcaton [PV07]. Une mesure d ntrcaton est une applcaton de la matrce densté vers un nombre réel qu nous ndque s l état est plus ou mons ntrqué. Dans ce traval, nous avons utlsé deux mesures dfférentes. La premère de ces mesures est l entrope d ntrcaton, utle quand l état est pur. Elle est défne comme l entrope de von Neumann de la matrce densté rédute d une des deux parttons E( ψ ) = S(ρ A ) = Tr{ρ A ln ρ A } = S(ρ B ). (3) Cette mesure étant ncapable de dscerner les corrélatons classques des corrélatons quantques dans le cas d un mélange statstque, nous ntrodusons également l ntrcaton de formaton [HW97, Woo98, Woo01, PV07], défne comme l ntrcaton moyenne mnmum sur toutes les décompostons possbles de l état sur une base d états purs: E f (ρ) = mn p E( ψ ). (4) { ψ } Dans la pratque, cette mnmsaton est très dffcle, mas l a cependant été montré que dans le cas où le système consdéré est consttué de deux spns 1/2 (ce qu sera majortarement notre cas dans la sute), l ntrcaton de formaton peut s écrre en foncton d un nombre C [0 : 1], appelé la concurrence, et défn comme C(ρ) = max{0, λ 1 λ 2 λ 3 λ 4 }. (5) Ic, les λ sont les valeurs propres en ordre décrossant de la matrce R = ρ ρ ρ où ρ est défne par ρ = (σ y σ y )ρ (σ y σ y ). L ntrcaton de formaton étant une foncton monotone crossante de C, la concurrence peut être utlsée comme mesure en elle-même. Plus ce nombre est grand, plus les deux spns sont ntrqués. Après cette brève présentaton de l ntrcaton quantque, nous nous focalsons sur les deux modèles utlsés, le modèle XY et le modèle de Bose-Hubbard. Le premer de ces modèles décrt une assemblée de spns en nteracton. Son hamltonen est H XY = J ( 2 1+κ n 2 σx n σn+1 x + 1 κ ) 2 σy nσ y n+1 h 2 σ z, (6) n où les σ α n sont les matrces de Paul dans la drecton α (α = x, y, z) assocées au ste n, h est un champ magnétque transverse, J est le couplage entre deux spns premers vosns, et κ est un paramètre d ansotrope. Le cas sotrope (κ = 0) correspond au modèle XX, tands que la lmte ansotrope (κ = 1) correspond au modèle d Isng. Cet hamltonen prend une forme fermonque quadratque après ntroducton de la xv

17 transformaton de Jordan-Wgner qu assoce aux opérateurs de montée et descente de spn des opérateurs de Clfford Γ 1 n = n 1 ( σ z ) σx n, Γ 2 n 1 n = =1 =1 ( σ z ) σy n, (7) et l devent H = 1 4 Γ TΓ, (8) où T est la matrce hamltonenne et Γ un vecteur contenant les opérateurs de Clfford. L ntroducton des opérateurs fermonques dagonaux η q et η q, relés aux opérateur de Clfford par une transformaton de Bogolubov permet l écrture de l hamltonen sous une forme dagonale ( H = ε q η q η q 1 ), (9) q 2 où ε q est l énerge de l exctaton fermonque q. L analyse de ε q nous permet d avor accès au dagramme de phase du modèle XY [Kar06] qu est dscuté dans la secton Après cela, nous nous ntéressons à la dynamque des opérateurs de Clfford dans la représentaton de Hesenberg, qu s écrt Γ(t) = R(t)Γ(0), (10) avec R(t) = e tt, et nous dérvons une formule pour l entrope d ntrcaton d une partton A d un système dans le cas du modèle XX. Nous rappelons qu elle peut être relée aux valeurs propres ξ q de la matrce de corrélaton, C A = c c j où les ndces et j sont restrents au sous-espace A. Fnalement, le deuxème modèle que nous avons consdéré est le modèle de Bose- Hubbard. Ce modèle décrt de façon smple la dynamque de bosons sur un réseau undmensonnel. Son hamltonen est c H BH = t ( b b +1+ b b +1 ) µ n + U 2 n (n 1), (11) ou les b et b sont des opérateurs bosonques, et n = b b est un opérateur de densté locale. Le premer terme de l hamltonen est un terme cnétque décrvant le saut d un boson sur le ste vosn. Le deuxème terme est proportonnel au potentel chmque qu fxe le nombre de bosons du système, et le derner terme est un terme d nteracton locale entre bosons sur le même ste. Cette nteracton est répulsve pour U > 0 et attractve s U < 0. L hamltonen devent soluble dans les deux lmtes de paramètres t/u 0 et t/u. Dans la lmte où t 0, le terme d nteracton domne et le système est caractérsé par une densté n 0 constante sur tous les stes, fxée par la mnmsaton de l énerge. L état fondamental est dt état de Mott et est donné par le produt tensorel Ψ = (b )n 0 0, (12) xv

18 où 0 est l état vde de boson. Dans la lmte opposée, le terme cnétque est domnant, et le système est dans une phase superflude où les bosons sont délocalsés sur tout le réseau. L état fondamental est dans ce cas donné par Ψ = ( N b ) 0. (13) Le dagramme de phase de ce modèle est présenté sur la fgure 1.2. Les quatre chaptres suvants contennent les prncpaux résultats de la thèse. Le chaptre 2 est consacré à l étude de la dynamque de trempe d une condensat de bosons ntalement pégé sur un réseau optque undmensonnel par un potentel de confnement harmonque. Le condensat de partcules est ntalement pégé entre les postons x = A < 0 et x = 0. En foncton de la valeur du paramètre de contrôle du potentel, nous avons travallé avec deux types d états ntaux. Dans le premer cas, tout le condensat est dans une phase superflude, avec une varaton spatale de la densté. Dans le deuxème cas, deux phases superfludes entourent une phase de Mott, centrée en x = A/2, où la densté locale reste constante et égale à un. Dans la lmte d une répulson nfne entre bosons sur un même ste (lmte bosons de cœur dur), nous avons développé une théore hydrodynamque qu reprodut parfatement le comportement des bosons. Cette théore repose essentellement sur une hypothèse d équlbre local avec le potentel. Dans le cadre de cette théore, les partcules sont émses vers la gauche et vers la drote le long de trajectores d énerge ε q = V(x) cos(q) constante. Les denstés de partcules dans l espace des phases (x, q) se déplaçant vers la gauche (left movers, sgne ) et vers la drote (rght movers, sgne +) sont données par ρ ± (x, q, t) = 1 2 dx 0 dq 0 ρ 0 (x 0, q 0 )δ(x x ± (x 0, q 0, t))δ(q q ± (x 0, q 0, t)), (14) et la densté totale par ρ(x, q, t) = ρ + (x, q, t)+ρ (x, q, t). Dans l expresson précedente, x ± (x 0, q 0, t) et q ± (x 0, q 0, t) sont les équatons d évoluton d une partcule ntalement stuée en (x 0, q 0 ). Nous avons consdéré deux types de trempe dfférents. Le premer consste à soudanement supprmer le potentel de confnement. Après la trempe, l équaton d évoluton d une partcule ntalement en (x 0, q 0 ) est celle d une partcule lbre x ± (x 0, q 0 ) = x 0 ± v q0 t où v q0 est la vtesse de la partcule ne dépendant que de son énerge ntale. Comme le potentel est nul, les partcules sont lbres d aller explorer tout l espace, et le condensat s étale vers la drote et la gauche jusqu à ce que les partcules attegnent les deux bords du système. Les prédctons données par la théore hydrodynamque permettent de parfatement décrre l évoluton temporelle du profl de densté, ans que l évoluton du nombre de partcules présentes dans la régon ntale du pège. Le second protocole de trempe consdéré consste au passage d un potentel harmonque à un potentel lnéare d équaton V(x) = Fx Θ( x) où la foncton de Heavsde assure que le potentel n est dfférent de zéro que dans la parte négatve de l axe. Le condensat présente un comportement plus rche que dans le cas précédent. En effet, on observe dans ce cas une séparaton du condensat en deux xv

19 partes: les partcules d énerge ntale comprse entre 1 et 1 quttent la rampe et se propagent vers la drote de l axe, tands que les autres restent pégées par le potentel en effectuant des oscllatons de Bloch. En foncton de l état ntal et de la force F, pluseurs régmes d oscllatons sont observés: Pour un état ntal complètement superflude, toutes les partcules quttent la rampe pour une force F < Fesc, S f où Fesc S f est la force telle que le pont le plus énergétque du condensat ntal sot égale à 1. Pour des valeurs de forces ntermédares, on observe un régme d oscllatons où le condensat pégé se sépare en deux partes, une parte correspondant aux left movers, qu commencent leur mouvement vers la gauche, et une parte correspondant aux rght movers, qu commencent leur mouvement vers la drote. Pour des forces élevées, tout le condensat commence son mouvement dans la même drecton, et l osclle comme un tout. Lorsque l état ntal est un mélange Mott/superflude, la seule force qu permet à toutes les partcules de s échapper du pège est F = 0. On observe pour cet état ntal des oscllatons de part et d autre d un plateau où la densté reste constante au cours du temps, et égale à sa valeur ntale. Tros régmes d oscllatons dfférents sont observés: Pour des fables valeurs de force, les deux côtés oscllants du plateau commencent leurs oscllatons dans des drectons opposées (vers la gauche sur le côté gauche, et vers la drote sur le côté drot), menant à un breathng regme. Pour des valeurs de force ntermédares, le côté gauche du plateau se sépare en deux partes, correspondant aux left et rght movers, oscllant en sens contrare, alors que le mouvement du côté drot est nchangé. Fnalement, pour des valeurs de force mportantes, les deux côtés du condensat oscllent en phase, c est à dre dans la même drecton à un nstant donné. Un dagramme de phase décrvant les dfférents régmes d oscllatons est présenté fgure Fnalement, on mentonne que nous avons également étudé la densté de partcules s échappant du pège, le courant de partcules, ans que l évoluton temporelle de l entrope d ntrcaton entre les partcules pégées et celles s échappant du pège. Les tros chaptre suvant sont consacrés à l étude de systèmes quantques ouverts sur un envronnement (ou ban). Dans le chaptre 3, nous focalsons notre attenton sur la dynamque de deux spns "défauts", localement couplés de manère symétrque à un envronnement modélsé par une chaîne de 2N spns avec nteracton d Isng, et dans un état thermque. Les deux défauts sont ntalement préparés dans un état séparable (c est à dre désntrqués), et l on se pose la queston de savor s de l ntrcaton peut être créée entre ces deux spns par l ntermédare du couplage à un envronnement commun. Dans la sute, la chaîne de spns a été transformée en une chaîne de bosons en nteractons par la transformaton d Holsten-Prmakoff. Nous avons chos de travaller dans la lmte où le champ magnétque transverse du ban est très supéreur au couplage ntra-chaîne, rendant tous les spns de l envronnement presque parfatement polarsés dans la drecton du champ, et de plus, la tempéraxv

20 ture de l envronnement est supposée très fable. Ces approxmatons nous amènent fnalement à une chaîne d Isng transformée en une assemblée d oscllateurs harmonques en nteracton. La symétre d échange des bosons n et n dans la chaîne nous pousse à l ntroducton de nouvelles coordonnées pour la descrpton du ban: les coordonnées de centre de masse (symétrques) et les coordonnées relatves (antsymétrques). L ntroducton de ces nouvelles varables a pour effet de découpler la chaîne de talle 2N en deux chaînes de N oscllateurs contenant respectvement les coordonnées symétrques et antsymétrques. De la même manère, nous ntrodusons une base d états non locaux symétrques et antsymétrques pour la descrpton des deux spns défauts. Cette nouvelle base mène à nouveau à un découplage: les états symétrques ne sont couplés qu au ban composé des varables symétrques, et de même pour les états antsymétrques. Ce découplage est brsé par l ntroducton de l hamltonen Zeeman des deux défauts. En effet, cet hamltonen, une fos écrt dans la nouvelle base, couple les sous-espaces symétrque et antsymétrque, et l devent donc mpossble d écrre l opérateur d évoluton total comme un produt de deux opérateurs agssant dans les espaces symétrque et antsymétrque respectvement. Pour pouvor écrre l opérateur d évoluton de cette manère (et donc découpler la dynamque), nous avons fat appel à l approxmaton suvante: la dynamque est analysée pour des temps très nféreurs au temps caractérstque mcroscopque des défauts donné par l nverse de leur gap en énerge. Dans ce régme temporel, l hamltonen Zeeman n nfluence pas la dynamque des défauts, et l peut être en conséquence néglgé dans l hamltonen total. Dans la pratque, nous avons consdéré des spns défauts dégénérés (c est à dre que nous avons posé le champ transverse des défauts égal à zéro), ce qu nous autorse à étuder la dynamque pour des temps arbtrarement longs. Grâce à l approxmaton précédente, nous avons pu établr l évoluton temporelle des éléments de la matrce densté rédute des spns défauts (équaton (3.52)), ce qu nous a perms de suvre l évoluton temporelle de la concurrence. Nous avons dans un premer temps analysé le cas de deux défauts couplés à la même poston dans la chaîne, pus le cas de deux spns défauts couplés à deux endrots dfférents. Dans les deux cas, nous avons observé une créaton d ntrcaton entre les deux spns défauts. La concurrence assocée osclle au cours du temps, avec des oscllatons dépendant fortement des paramètres du système. La dfférence prncpale entre les deux cas précédemment mentonnés est que lorsque les spns sont couplés au même endrot, l ntrcaton est créée nstantanément alors qu un temps d établssement est nécessare lorsqu ls sont séparés par une dstance non nulle. Ce temps d établssement d ntrcaton croît exponentellement avec la dstance entre les spns défauts. La pérode d oscllaton augmente lorsque les couplages ntra-chaîne et défauts-chaîne sont augmentés, et le maxmum attent par la concurrence ne dépend lu que du couplage entre le défauts et la chaîne, ans que de l état ntal des défauts. En effet, l ntrcaton est créée pour tous les état ntaux, sauf lorsque l un ou les deux défauts sont préparés ntalement dans un état propre de l hamltonen d nteracton. Dans le chaptre 4, nous étudons un modèle smlare au modèle étudé dans le chaptre précédent, à savor deux spns défauts localement couplés aux postons 0 et d d une chaîne d Isng en champ transverse avec des condtons de bord pérxx

21 odques. La dfférence vent du fat que les défauts sont c ntalement préparés dans un état de Bell, c est à dre que l ntrcaton est maxmum. De plus, le champ transverse de l envronnement est soudanement trempé d une valeur ntale h à une valeur fnale h f, le forçant à évoluer dans un régme hors de l équlbre. Le but est c d étuder l nfluence de la trempe de l envronnement sur l ntrcaton ntale des défauts par rapport à la stuaton d équlbre (c est à dre lorsqu aucune trempe n est effectuée) déjà tratée dans la lttérature [CP08b]. La concurrence s exprme dans ce cas de manère très smple en foncton de l écho de Loschmdt L(t), qu est lu même relé à l évoluton temporelle de la matrce de corrélaton fermonque ntale des deux défauts. Cette évoluton temporelle a été étudée numérquement par dagonalsaton exacte. Nous nous sommes dans un premer temps penchés sur l nfluence de la valeur du couplage ε entre les défauts et l envronnement. Nous avons dentfé deux régmes dfférents, correspondant à un couplage fable et à un couplage fort. Dans le régme de couplage fable l écho décrot lentement et de manère monotone. Lorsque le couplage est augmenté, la décrossance de l écho est plus rapde et, pour de fortes valeurs de ε, l écho développe des oscllatons dont la fréquence est foncton de la valeur du couplage. Ces oscllatons sont contenues dans une enveloppe ndépendante de ε. Nous avons ensute étudé plus attentvement l effet de la trempe sur les proprétés de désntrcaton. Il apparaît que la trempe est toujours néfaste du pont de vue de la cohérence du système, dans le sens où elle accélère toujours la désntrcaton. Nous avons également ms en évdence que l écho reflète les proprétés crtques de l envronnement. En effet, la dérvé de l écho par rapport au champ ntal à un temps fxé présente une anomale lorsque le champ ntal est proche de la valeur h = 1. Le système étant de talle fn dans nos smulatons numérques, l anomale observée est un pc et non une dvergence totale de la dérvée. Une analyse d effets de talle fne nous a perms de trouver la lo gouvernant la poston du pc ans que sa valeur maxmale en foncton de la talle de l envronnement: h c h max N γ, d h L hmax ln N, (15) où γ est un exposant qu peut être relé à l exposant crtque de la longueur de corrélaton ν de la classe d unversalté du modèle d Isng (ν = 1). Nous avons ensute consdéré la dynamque aux temps courts. Dans ce régme temporel, la décrossance de l écho est gaussenne L(t) exp( αt 2 ), et on trouve fnalement: L(t) = 1 t 2[ H 2 I H I 2] +O(t 3 ), (16) où H I est la valeur moyenne de l hamltonen d nteracton dans l état ntal de l envronnement. Il apparaît donc que la trempe n nfluence pas le début de l évoluton de l écho, pusque c est la varance de l hamltonen d nteracton dans l état ntal qu gouverne la dynamque de l echo aux temps courts. En terme de foncton de corrélaton spn-spn, le paramètre gaussen α est donné par α = 2ε 2( 1+ σ z 0 σz d c σ z 0 2). (17) Lon du pont crtque, ce paramètre sature lorsque la dstance devent grande. Au contrare, lorsque le champ transverse de l envronnement est proche du pont crtque h = 1, cette saturaton n exste pas. En effet, le corrélateur connecté σ z 0 σz d c xx

22 décrot exponentellement hors du pont crtque, menant à la saturaton à grandes dstances, alors que la décrossance est algébrque au pont crtque. La talle fne de l envronnement dans nos smulatons numérques ndut un changement sgnfcatf de l écho pour des temps de l ordre de N. Ce phénomène de revval est dû au transport des quas-partcules le long de la chaîne. Une des dfférences majeures par rapport à la stuaton d équlbre résde dans le fat que ce revval apparaît à des temps deux fos plus courts que lorsque l envronnement n est pas trempé. En effet, dans le cas de la trempe globale consdérée c, tous les stes de la chaîne sont émetteurs d exctatons, ces dernères n ont donc beson de parcourr que la moté de la chaîne pour reconstrure l état ntal. Au contrare, dans la stuaton d équlbre, ces exctatons ne sont émses qu aux postons où les défauts sont couplés. Il leur est donc nécessare de parcourr la chaîne entère afn de reconstrure la cohérence ntale. Enfn, la dernère parte de cette étude tente d dentfer la part de la désntrcaton venant drectement du couplage avec l envronnement, et celle venant de l nteracton mutuelle des deux défauts au travers de la chaîne. Pour cela, nous avons étudé la dfférence entre l écho dans la stuaton où les défauts sont couplés à un envronnement commun et l écho dans la stuaton lmte où les deux défauts sont couplés à des envronnements dfférents. Lorsque le champ magnétque ntal est lon de la valeur crtque, l évoluton de l écho est la même dans les deux stuatons jusqu à ce que les spns soent corrélés par l ntermédare des quas-partcules parcourant l envronnement. Au contrare, pour une chaîne crtque, la dfférence entre les échos est non nulle à t = 0 +, reflétant la corrélaton ntale des deux défauts au travers de la longueur de corrélaton crtque. Fnalement, le chaptre 5, derner chaptre de cette thèse, est consacré à la dynamque d un système quantque couplé à un envronnement décrt par un processus d nteractons répétées [AP06, AJ07, AD10]. L envronnement joue le rôle de réservor d ntrcaton, et nous étudons le transfert de cette ntrcaton au système consdéré. Nous commençons ce chaptre par une ntroducton détallée du processus d nteractons répétées. L envronnement est, dans ce contexte, modélsé par une assemblée de "copes" dentques et ndépendantes et l nteracton avec le système se fat de manère répétée, c est à dre que le système nteragt avec toutes les copes, l une après les autres, pendant un temps caractérstque τ. Notons que cela est équvalent à rafraîchr l état ntal de l envronnement après chaque temps t = nτ. S l on ne s ntéresse qu à l évoluton temporelle du système, on peut montrer que sa matrce densté rédute après la nème nteracton est donnée par ρ s (nτ) = Tr n { U (n) [ ρ s ((n 1)τ) ρ b n ] } U (n), (18) où U (n) est l opérateur d évoluton couplant le système à la nème cope de l envronnement. Dans la lmte contnue où le temps d nteracton τ tend vers zéro, ρ s obét à l équaton de Lndblad avec L{X} = [H s, X] 1 2 t ρ s (t) = L{ρ s }, (19) } ) ({L L, X 2L XL. (20) xx

23 Lorsque les hamltonens consdérés peuvent s écrre sous une forme quadratque d opérateurs fermonques, et que l état ntal total est Gaussen, l est possble de montrer que l évoluton temporelle préserve le caractère gaussen de l état [Pes03]. Il s en sut donc, d après le théorème de Wck, que toutes les observables décrvant le système peuvent s écrre en foncton des corrélateurs fermonques à deux ponts. Dans la lmte contnue, la matrce de corrélaton fermonque évolue selon l équaton t G S = [T S, G S ] 1 2 ({ G S, ΘΘ } 2ΘG B Θ ) (21) où G S et G B sont les matrces de corrélaton restrentes au système et à une cope du ban respectvement, et Θ est une matrce contenant le couplage système-ban. Dans le reste de cette étude, les copes consttuant le ban sont supposées être formées d une pare de spns 1/2, non couplés entre eux, et préparés dans un état de Bell maxmalement ntrqué. Nous avons dans un premer temps étudé un modèle trval, pour lequel la dynamque a pu être détermnée analytquement. Il consste en deux spns 1/2, chacun de ces spns étant couplé à un des consttuants de la pare de Bell formant une cope de l envronnement (vor la fgure 5.2 pour une représentaton du modèle). Ces deux spns sont préparés dans un état thermque séparable paramétré par leur amantaton. Afn d évaluer la concurrence dans le système, la matrce densté rédute est reconstrute à l ade des fonctons de corrélaton spn-spn, elles mêmes détermnées grâce à l évoluton temporelle de la matrce de corrélaton fermonque. Nous trouvons que la concurrence évolue en suvant { C(t) = max 0, 1 e γ2t 1 ( [ ] 2e γ2t +(m m0 4 2 t 1)e 2γ2, (m m0 4 )2 e 2γ2 t ) 1/2 }. (22) où m 0 1 and m0 4 sont les amantatons ntales des spns 1 et 4 respctvement. Dans l état statonnare, la concurrence attent la valeur C 14 (t ) = 1, ndquant une ntrcaton maxmale entre les spns du système, ben qu ls n aent jamas nterag drectement. Après ce modèle trval, nous avons consdéré un système plus complqué, où les deux spns sont remplacés par deux chaînes dentques de spns avec nteractons de type XX. Chacune des chaînes est couplée sur un bord à un des spns consttuant la pare de Bell (vor fgure 5.7). Nous avons étudé la dynamque du système sur deux régmes temporels dfférents: un régme de temps courts par rapport à la longueur des chaînes 1 < t N, et un régme de temps longs t N. La vtesse des exctatons étant normalsée à v = 1, le temps t = N correspond au temps nécessare à la premère exctaton njectée pour attendre le bord opposé du pont d nteracton avec l envronnement. L étude du premer de ces régmes nous rensegne sur le comportement d un système sem-nfn. Dans ce cas, le système attent un état statonnare hors équlbre, ou NESS (pour non-equlbrum-steady-state en anglas) caractérsé par un courant statonnare traversant les deux chaînes. On étude ce NESS au travers du comportement d observables telles que l amantaton locale et le courant. On s ntéresse également à deux types d ntrcaton: l ntrcaton longtudnale, mesurée entre deux spns consécutfs dans une chaîne, et la concurrence crosée, mesurée xx

24 entre un spn de la premère chaîne et son équvalent dans la deuxème (vor les doubles flèches de la fgure 5.7). En partculer, nous avons ms en évdence le comportement d echelle de l amantaton, du courant et de la concurrence longtudnale. La concurrence crosée, quant à elle, décrot exponentellement avec la dstance au pont d nteracton de la pare de spns p consdérée, C (p) exp( p/ξ ent ). On peut extrare de cette décrossance une longueur d ntrcaton typque ξ ent, qu semble se comporter proportonnellement à l nverse du courant statonnare s établssant dans les chaînes. Fnalement, pour des temps t N, le système attent un état statonnare d équlbre où le courant s annule. Cet état est caractérsé par un produt d états de Bell sur les spns se fasant face dans les deux chaînes, décrvant une ntrcaton crosée maxmale. On peut fnalement noter que cet état statonnare d équlbre est ndépendant de l état ntal du système. Enfn, tros appendces vennent conclure ce manuscrt. Dans l appendce A, nous présentons en détal l établssement de la formule d évoluton temporelle des éléments de la matrce densté rédute utlsée dans le chaptre 3. Nous montrons dans l appendce B que la matrce densté assocée à chacune des pares de Bell peut s écrre comme la lmte basse température d une matrce de densté thermale. Enfn, dans l appendce C, nous dédusons la foncton de corrélaton spn-spn statonnare entre spns se fasant face dans le cadre des nteractons répétées. Deux artcles scentfques drectement en len avec cette thèse ont été publés, et deux autres sont actuellement en préparaton: Hydrodynamc descrpton of Hard-core Bosons on a Galleo ramp P. Wendenbaum, M. Collura and D. Karevsk Physcal Revew A (2013). Decoherence of Bell states by local nteractons wth a suddenly quenched spn envronment P. Wendenbaum, B.G. Taketan and D. Karevsk Physcal Revew A (2014). Entanglement creaton between two spns embedded n a sngle spn chan P. Wendenbaum, B.G. Taketan, E. Kajar, G. Morg and D. Karevsk En préparaton. Entanglement replcaton va quantum repeated nteractons P. Wendenbaum, T. Platn and D. Karevsk En préparaton. xx

25

26 General ntroducton The concept of entanglement takes ts orgn n the early tmes of the development of quantum theory. Schrödnger s the frst one to gve t ts name ("Verschränkung" n German), whereas, n ther famous artcle of 1935 [EPR35], Ensten, Podolsy and Rosen mentoned t as a "spooky acton at a dstance". If t has, from a purely theoretcal and phlosophcal pont of vew, allowed a better understandng of the foundatons of quantum mechancs, quantum entanglement became, snce the eghtes, a technologcal key ngredent wth the development of quantum nformaton processng [NC00]. Ths recent feld of research tres to surpass the lmts mposed by the concepts of classcal nformaton usng quantum physcs. One of the frst concrete modern applcaton of entanglement used as a resource s the quantum key dstrbuton [BB84], frst protocol of quantum cryptography, allowng secure communcatons. Quantum entanglement s also supposed to be of prmary mportance for the buldng of quantum computers, where nformaton s no longer stored nto classcal bts, but nto quantum bts, or "qubts" [DV95]. All these applcatons need a strong theoretcal background, that s why a lots of efforts have been made to develop rgorous defntons of an entangled state, and ntroduce separablty crtera [HHH96] or entanglement measures [PV07]. These methods developed n quantum nformaton processng have attracted the curosty of other areas of physcs lke condensed matter theory [Pre00, Sch13]. Indeed, the entanglement present n a quantum many-body state can help to the characterzaton of new states of matter wth strong long-range correlatons, lke superconductvty [Ver04] or quantum Hall effect [LH08]. The study of entanglement s also a very powerful tool for the detecton of quantum phase transtons at zero temperature. At the quantum crtcal pont, the system exhbts long-range correlatons due to the entanglement present n the many-body ground state. For example, scalng propertes of ground state entanglement of quantum many-body crtcal systems have been analyzed through the scalng of entanglement entropy [VLRK03], an entanglement measure used when the state of the system s pure. For example, the entanglement entropy of certan crtcal theores can be related to the central charge of the correspondng conformal feld theory [CC09, VLRK03, PS05]. The concept of entanglement spectrum has also been ntroduced as a powerful tool for the study of crtcal phenomena. The analyze of the spectrum of the reduced densty matrx of a bpartton of a quantum system, together wth the Schmdt gap (the dfference between the two largest egenvalues) has, for nstance, perfectly reproduced the crtcal scalng behavor of order parameter close to crtcalty [DCLLS12]. If all these works deal wth ground state entanglement, one may ask the queston of entanglement behavor of system out of equlbrum. Indeed, thanks to expermental progresses durng the few past decades, the physcs of non equlbrum systems knows an mpulse of nterest, especally n the area of condensed matter physcs 1

27 2 General ntroducton and quantum optcs. Despte these expermental progresses, there s stll a lack of a good general theoretcal framework for the descrpton of non-equlbrum quantum phenomena. Understandng the behavor of such phenomena s then one of the most challengng ssues of modern theoretcal physcs. The work presented n ths thess s devoted to the study of the non equlbrum behavor of one-dmensonal strongly nteractng quantum systems, wth a specal focus on ther entanglement dynamcs. Several protocols can drve the system under consderaton nto an out of equlbrum regme. For example, on can more or less suddenly vary one control parameter of the Hamltonan, realzng the so-called quantum quench [CG07, PSSV11, MDDS07, RSMS09, CEF12a, CEF12b, Rou09, BRK11]. The couplng of the system to a bggest envronment leads also to very nterestng dynamcs [KPS13, PKS13] and entanglement propertes. The quantum spn chans are a very good canddate for ths study. Indeed, the bg advantage of such models s ther low dmensonalty, allowng most of the tme the development of analytcal or numercal methods to extract ther man propertes [LSM61, BMD70, BM71a, BM71b, MBA71]. Ths thess s dvded n fve chapters. The frst one s devoted to the ntroducton of the theoretcal and mathematcal framework of the work. We gve a general descrpton of the concept of entanglement, and we turn to the presentaton of the models used, namely the quantum XY and the Bose-Hubbard models. The second chapter treats the effects of a sudden quench on the potental of a Bose cloud ntally trapped n a certan regon of an optcal lattce. Two dfferent ntal states are consdered, dependng on the ntal trappng potental. In the nfnte repulson regme (hard-core regme), we develop an hydrodynamcal theory whch perfectly catch the man features of the dynamcs. The results obtaned wth ths theory are confronted wth numercal data gven by exact dagonalzaton. The two followng chapters deal wth the study of the dynamcs of a small system (typcally two spns 1/2) coupled to an envronment modeled by a chan of nteractng spns. In the thrd chapter, the two "defect" spns are ntally prepared nto a separable state and we ask f the couplng wth the envronment can create an effectve nteracton leadng to entanglement. The reduced dynamcs of the two defects s analyzed after performng a bosonzaton of the envronment, and we look n partcular to the dependence of the entanglement on the parameters of the system. In chapter four, we start n the opposte stuaton where the two defect spns are ntally maxmally entangled, and we look at the effects of the couplng wth the envronment on ths ntal entangled state. In addton to the couplng, we also perform a sudden quench n the envronment, forcng t to evolve n an out-of-equlbrum regme. We focus here on the nfluence of the degree of out-of-equlbrum on the dsentanglement dynamcs. Fnally, n a last chapter of ths thess, we analyze the dynamcs of an open quantum system coupled to an envronment by means of the repeated nteractons process [AP06, AJ07, AD10]. The envronment plays the role of a reservor of entanglement, and we analyze f ths entanglement can be transfered to the system. After a general descrpton of the repeated nteractons process, we study a smple toy model for whch the entanglement propertes can be completely determned analytcally, and

28 General ntroducton 3 we move afterward to a more general case analyzed for tmes shorter than the system sze t < N, and tmes much larger than N. Two artcles have been publshed n lnk wth ths thess, and two others are currently n preparaton: Hydrodynamc descrpton of Hard-core Bosons on a Galleo ramp P. Wendenbaum, M. Collura and D. Karevsk Physcal Revew A (2013). Decoherence of Bell states by local nteractons wth a suddenly quenched spn envronment P. Wendenbaum, B.G. Taketan and D. Karevsk Physcal Revew A (2014). Entanglement creaton between two spns embedded n a sngle spn chan P. Wendenbaum, B.G. Taketan, E. Kajar, G. Morg and D. Karevsk In preparaton. Entanglement replcaton va quantum repeated nteractons P. Wendenbaum, T. Platn and D. Karevsk In preparaton.

29

30 Quantum Entanglement and Models 1 In ths chapter, we ntroduce the basc concepts used n ths thess. The frst secton s devoted to the presentaton of a key ngredent of ths work, namely quantum entanglement. We frst gve the basc concepts and ntroduce separablty crtera for both pure and mxed states. Afterward, we ntroduce two measures of entanglement, necessary to know the amount of entanglement present n a quantum state. We also present the close relaton between entanglement and quantum phase transtons, and the quantum teleportaton s ntroduced as a drect applcaton where entanglement s used as a resource. Afterward, we move to the descrpton of the two models of quantum systems used n ths work, namely the quantum XY and the Bose-Hubbard models. For the frst of these models, we sketch n detals the canoncal dagonalzaton procedure by means of the Jordan Wgner transformaton, and brefly dscuss ts phase dagram. The phase dagram of the Bose-Hubbard model s presented through two lmts of the system s parameters. 1.1 Quantum entanglement Defntons and separablty crtera by Consder a quantum pure state Ψ lvng n a composte Hlbert space H gven H = H 1 H 2... H N, (1.1) where the H ( = 1,..., N) are the N parttons of the total space. If the state Ψ can be wrtten lke Ψ = φ 1 φ 2... φ N, (1.2) where φ H, then ths state s sad to be separable. On the contrary, f t can not be wrtten n the prevous form, the state s sad to be entangled. If the full Hlbert space H s compose by only two parttons (.e N = 2 n the prevous expresson), we wll then speak about bpartte entanglement. As an example, let us consder the quantum state of two spns 1/2 labeled A and B. The state Ψ 1 = 1 2 ( + ) can be wrtten lke Ψ 1 = 5

31 6 Chapter 1. Quantum Entanglement and Models A 1 2 ( + ) B = φ A φ B and s, as a consequence, separable. On the other hand, for the state Ψ 2 = 1 2 ( + ), t s mpossble to fnd two states belongng to the subspaces A and B respectvely such that t can be wrtten n the form (1.2), ths state s then entangled. The entanglement has an nterestng consequence when one makes a measurement on state Ψ 2. Suppose that an expermentalst wants to know the state of the spn A by measurng ts polarzaton. He wll fnd the spn n one of the two states A or A wth the same probablty 1/2. If he fnds the spn A n the state A (resp. A ), t means that the spn B, whch can be arbtrary far from A, wll nstantaneously collapse nto the state B (resp. B ) although no measurement has been done on t. Gven a certan quantum state, t s not always an easy task to know f ths state s entangled or not. One crteron for separablty s based on the Schmdt decomposton [Sch06, EK95]. Suppose a bpartte quantum state ψ leavng on a Hlbert space H = H A H B where we set, wthout loss of generalty, dm(h A ) = dm(h B ) = N. Then, the state ψ AB can be decomposed lke ψ AB = N p u A v B, (1.3) =1 where the { u A } and { v B } are the egenvectors of the reduced densty matrces ρ A = Tr B {ρ} and ρ B = Tr A {ρ} respectvely, wth the same egenvalues p ρ A u A =p u A, (1.4) ρ B v B =p v B. (1.5) To prove ths, we expend the state ψ AB on the base formed by the egenstates of the reduced densty matrces ρ A and ρ B ψ AB = d j u A v B j, wth ua v B j ua v B j, (1.6) j and where the coeffcents d j are gven by the overlap d j = u A v B j ψ AB. Now we wrte down the reduced densty matrces of the part A by tracng the part B from the full densty matrx ρ AB = ψ AB ψ AB : ρ A = Tr B {ρ AB } { = Tr B d j d j ua v B j ua vb j } j j = j = j j k d j d j vb k ua v B j ua vb j vb k d j d j u A u A. (1.7) Ths expresson (1.7) can be compared to the decomposton of ρ A nto ts egenstates ρ A = p u A u A, (1.8)

32 1.1. Quantum entanglement 7 gvng the condton d j d j = δ p (1.9) j such that (1.6) reduces to (1.3). The non negatve numbers p are called the Schmdt coeffcents, and the number of p = 0 s the Schmdt rank. The expresson (1.3) together wth the defnton of an entangled state gve us a crteron for the separablty of a pure state: the state ψ AB s separable f only one of ts Schmdt coeffcent s dfferent from zero. Indeed, the only way to wrte (1.3) n the form (1.2) s to have only one non zero p. Let us suppose that more than one p are dfferent from zero. Because p < 1, we have p 2 < p, and then t follows Tr ρ 2 A = Tr ρ2 B = p 2 < 1. (1.10) It follows that the reduced densty matrx of one part of a pure bpartte entangled state s a mxed state. In other words, when a pure state s entangled, we do not have all the nformaton about one part by tracng out the full state over the second part, reflectng quantum correlatons exstng between parts A and B. The prevous defnton of an entangled state and the separablty crteron based on Schmdt decomposton held when the state of the quantum system s pure. What happens f the system s n a statstcal mxture? The noton of entanglement s generalzed to mxed state n the followng way: a mxed bpartte quantum state ρ s separable f t can be wrtten n the form ρ = p ρ A ρ B, wth p = 1, p 0, (1.11) and s entangled otherwse. In other words, ρ s separable f t s a convex sum of separable states belongng to subspaces A and B. The crteron based on the Schmdt decomposton does not hold anymore n ths case. For mxed state, t exsts an other crteron, the Postve Partal Transpose (PPT), also called Peres-Horodeck crteron [Per96]. Suppose a generc bpartte quantum state ρ actng on the Hlbert spaceh A H B ρ = ρ jmn j m n, (1.12) jmn a necessary condton for ts separablty s that the partal transpose taken wth respect to one part only, let s say the part B (ρ) T B = ρ jnm j n m, (1.13) jmn has only non negatve egenvalues. More formally, the partal transposton of a separable state ρ has to be postve (ρ) T B = (1 A T)ρ > 0, (1.14) where T s the transposton operator. Indeed, f the state ρ s separable, t can be wrtten n the form (1.11). After the partal transposton, t becomes (ρ) T B = p ρ A (ρ B )T B. (1.15)

33 8 Chapter 1. Quantum Entanglement and Models The partal transposton conservng the egenvalues, t follows that the spectra of (ρ) T B and ρ are dentcal. The latter operator beng postve semdefnte, ths proves the necessty of the crteron. The subsystem transposed s not mportant here snce (ρ) T A = (ρ T B ) T. Ths crteron s necessary when we deal wth arbtrary sze of Hlbert space H A and H B, but t has been shown n [HHH96] that t becomes suffcent when the full Hlbert space has the structure H = H 2 H 2, H = H 3 H 2 or H = H 2 H 3. To llustrate ths crteron, let us consder Werner state [Wer89]. A Werner state s a N N state nvarant under any untary transformaton of the form U U. Consderng two spns 1/2, a Werner state can be defned assumng the spns n a fracton f of the entangled snglet state ψ = 1 2 ( ) wth an mpurty fracton 1 4 (1 f)1. In the canoncal base, the densty matrx of such a state s ρ W = f f 2 f f 1+ f f Its partal transpose wth respect to the second spn becomes (ρ W ) T B = f f 0 1+ f f 0 2 f f. (1.16), (1.17) and the dagonalzaton of the last matrx gves three egenvalues equals to (1+ f)/4 and one equal to (1 3 f)/4. Ths last egenvalue s the smallest one and gves a threshold value of the fracton f = 1 3 such that the state s separable f f > 1 3 and entangled f f Entanglement measures So far, we have ntroduced entanglement crtera whch tell us f a quantum state s entangled or not. But these crtera are unable to gve how strong are these quantum correlatons. Then, we need to ntroduce the noton of entanglement measure whch gves us access to the amount of entanglement present n a gven quantum state. It does not exst a unque defnton of a measure of entanglement, but t s generally admtted that a measure E(ρ) has to fulfll some requrements [Hor01, PV07]: A measure s a mappng between the densty matrx and a real number. E(ρ) has to be zero for separable state. ρ E(ρ) R. (1.18) ρ separable E(ρ) = 0. (1.19) E(ρ) does not ncrease under LOCC (Local Operaton and Classcal Communcaton).

34 1.1. Quantum entanglement 9 E(ρ) has to reduce to the entanglement entropy, that we wll defne below, when the state s pure. In the rest of ths work, we wll use two dfferent measures of entanglement, namely the entanglement entropy and the concurrence Entanglement entropy The entanglement entropy s an entanglement measure used when the system s pure. It s based on the von Neumann entropy defned by S(ρ) = Tr{ρ ln ρ}, (1.20) and quantfyng the classcal mxng degree of a quantum state. Obvously, the von Neumann entropy of a pure state vanshes S( ψ ψ ) = 0. As we saw n the prevous secton, the reduced densty matrx of a bpartte entangled pure state ψ s a mxed state. Therefor, a way to quantfy the entanglement present n ψ s to measure how ths reduced densty matrx s mxed by the ntroducton of the entanglement entropy, whch s nothng else but the von Neumann entropy of one of the reduced densty matrces ρ A or ρ B E( ψ ) = S(ρA) = Tr{ρA ln ρa} = S(ρB), (1.21) that can be rewrtten usng the spectral decomposton (1.8) of ρ A lke S(ρ A ) = N =1 p ln p. (1.22) The value of E( ψ ) s equal to zero when the state s separable and reaches ts maxmum value E( ψ ) = ln N when p = 1/N, where N s the dmenson of ρ A. To prove ths last feature, we are lookng for the extremum of the functon S usng the constrant p = 1. Imposng ths last constrant, we have The dervatve S p j N 1 S(ρ A ) = p ln p =1 gves S p j = ln p j + ln ( 1 ( 1 N 1 =1 N 1 =1 p ) Settng ths dervatve to zero s equvalent to have p j = 1 N 1 =1 p ) ln ( 1 N 1 =1 p ). (1.23) j = 1,, N 1. (1.24) p = p N j = 1,, N 1, (1.25) that s fulfll only when p j = 1/N j. Now we know the probablty dstrbuton leadng to the maxmum entropy, we can easly compute S for ths dstrbuton S max = N =1 1 N ln 1 N = ln N. (1.26)

35 10 Chapter 1. Quantum Entanglement and Models When the entanglement entropy s maxmum, the state s sad to be maxmally entangled. To llustrate ths, consder two spns n the pure state ψ AB = 1 2 ( + ). The reduced densty matrx ρ A of the spn A obtaned by tracng ρ = ψ ψ over the part B s, n the canoncal base ρ A = ( ) 1/2 0 = 1 0 1/2 2 1 (1.27) and the entanglement entropy becomes E( ψ AB ) = ln 2, as expected. The entanglement entropy s a good measure when the system s n a pure state, but t becomes useless when the state s mxed. Indeed, ths measure s unable to make the dstncton between quantum and classcal correlatons. For example, f the two prevous spns are n the classcal separable mxed state ρ AB = 1 41, the reduced densty matrx ρ A yelds to an entanglement entropy of S = ln 2, although the spns are n a separable state Entanglement of formaton and concurrence The entanglement entropy falng to detect entanglement of a mxed state, we need to ntroduce a proper measure able to dstngush quantum from classcal correlatons. The entanglement of formaton [HW97, Woo98, Woo01, PV07] of a mxed state ρ s defned lke E f (ρ) = mn p E( ψ ), (1.28) { ψ } where the mnmzaton s taken over all the possble pure-state decompostons {p, ψ } of ρ, and E(ρ) s the entanglement entropy defned n the prevous secton. In other words, t s the mnmal average entanglement over the pure state decomposton of the state ρ. The decomposton of a mxed state nto pure states beng not unque, the mnmzaton procedure appearng n the defnton of E f (ρ) s n practce a dffcult task. However, an exact expresson of the entanglement of formaton has been establshed for systems composed of two levels systems, lke 1/2 spns. For ths knd of systems, the entanglement of formaton has been found to be [HW97, Woo98] where E(C) s the functon E(C) = h E f (ρ) = E(C(ρ)), (1.29) ( ) 1 C 2, (1.30) 2 and h(x) = x ln x (1 x) ln(1 x) s the bnary entropy functon. The number C appearng n the equaton (1.30), rangng from 0 to 1, s called the concurrence and s defned by C(ρ) = max{0, λ 1 λ 2 λ 3 λ 4 }, (1.31)

36 1.1. Quantum entanglement 11 where the λ s are the egenvalues n decreasng order of the Hermtan matrx R = ρ ρ ρ where ρ s the flpped densty matrx defned by ρ = (σ y σ y )ρ (σ y σ y ), (1.32) and the conjugate s taken n the standard base. Alternatvely, the λ s can be seen as the square roots of the generally non Hermtan matrx 1 R = ρ ρ. The entanglement of formaton E f beng a monotoncally ncreasng functon of C, the concurrence can be taken as an entanglement measure by tself. More the value of C s mportant, more the state s entangled. In partcular, f C = 0, the two spns are n a separable state, whereas f C = 1, they are maxmally entangled. In the rest of ths work, we wll use the concurrence as soon as we wll have to determne the entanglement between two spns 1/ Entanglement and quantum phase transtons Whle a classcal phase transton occurs at non zero temperature, quantum phase transtons can only be observed at zero temperature [Sac00] when a physcal parameter s vared n the Hamltonan. In the frst case, the transton s drven by the thermal fluctuatons, whereas the quantum fluctuatons are responsble of the transton n the quantum case. One example of such a transton wll be gven n secton 1.3 where we wll present the quantum XY model. The Isng pont κ = h = 1 s the crtcal pont separatng the ferromagnetc phase from the paramagnetc one. At ths pont, the ground and frst excted states cross and the energy gap closes accordng to h h c zν, (1.33) where z s the dynamcal exponent and ν s the exponent of the correlaton length ξ. Ths correlaton length dverges at the crtcal pont ξ h h c ν, (1.34) whereas t s fnte far from the crtcal regon, and leads to exponentally damped correlatons. The quantum phase transton s characterzed by a sgnfcant change n the ground state propertes of a quantum many-body system. Then, f ths ground state s an entangled state, t must exst a sgnature of ths crtcalty n the entanglement propertes. A bg number of works has been devoted to the study of ground state entanglement n the vcnty or at the crtcal pont (see for example the revew [AFOV08]). One of the semnal work has been done by Osterloh et al. n [OAFF02], where they study the concurrence between pars of spns n the one dmensonal Isng chan n a transverse magnetc feld. Entanglement s found only between nearest and next nearest neghbour spns. The nearest neghbour concurrence vanshes for h = 0 and n the lmt h snce the many-body ground state s a tensor product of sngle states. Between these two lmts the concurrence s non zero and behaves smoothly. 1. Note that even f ths matrx s non Hermtan, ts egenvalues are guaranteed to be non negatve, the matrx R beng the product of two postve defnte matrces [Woo01].

37 12 Chapter 1. Quantum Entanglement and Models It presents a maxmum occurrng close but, surprsngly, not at the crtcal pont. One argument justfyng ths feature has been proposed n [ON02]. The fact that the concurrence s not maxmum exactly at the crtcal pont comes from the monogamy property of entanglement [CKW00], lmtng the amount of entanglement that one system can share wth two or more others. At the crtcal pont, as the correlatons ncrease, the global entanglement s maxmum over the lattce. As a consequence of the aforementoned monogamy property, the parwse entanglement decreases. The sgnature of the second order phase transton s not encoded n the concurrence drectly, but n ts frst dervatve wth respect to the magnetc feld. Ths dervatve shows a logarthmc dvergence n the vcnty of the crtcal pont n the thermodynamc lmt. For fnte systems, one observes rather a peak whose poston s shfted wth respect to the crtcal value. The poston of the peak h max and ts maxmum h C hmax obey to the fnte sze scalng h c h max N µ, h C hmax c ln(n)+cst, (1.35) where N s the number of spns n the chan. Note that the same knd of scalng of the entanglement entropy has been observed n [SSG06, Che07] where the local entanglement between two spns and the rest of the chan s analyzed n the quantum Isng model. The entanglement propertes of other models exhbtng quantum phase transton have been studed. For example the XXZ model [GLL03, GTL06] or the Lpkn- Meshkov-Glck model [VMD04, VPM04], whch s a generalzaton of the Isng model where all spns are nteractng together. The correspondng Hamltonan s H LMG = J N <j ( ) σ x σx j + γσ y σy j h σ z. (1.36) The nature of the phase transton of ths model depends of the sgn of the mcroscopc couplng J. Indeed, n the antferromagnetc case (J < 0), the phase transton s of the frst order whereas t s of the second order n the ferromagnetc case (J > 0). Obvously, any par of spns wll be entangled n the same way due to the symmetry of the Hamltonan. It s then necessary to rescale the par concurrence by multplyng t by the coordnaton number, C R = (N 1)C, n order to have a fnte value when N. In the second order phase transton, t s shown [VPM04] that the concurrence s maxmal at the crtcal pont and ts frst dervatve presents a dvergence close to h c = 1 lke n the Isng case. Nevertheless, the dvergence s not logarthmc but obeys to a power law, reflectng the fact that the two models belong to two dfferent unversalty classes. In the antferromagnetc case where the transton s of the frst order at the transton pont h = 0, the dscontnuty occurs drectly n the par concurrence, and not n ts dervatve [VMD04]. From the prevous observatons, t seems that there s a deep relaton between the nature of the quantum phase transton and the dscontnuty observed (n the entanglement measure for a frst order phase transton or n ts dervatve for a second order phase transton). Wu et. al have establshed n [WSL04] a theorem statng that, under certan condtons, a dvergence of the concurrence (resp. ts frst dervatve)

38 1.1. Quantum entanglement 13 s a necessary and suffcent sgnal of the presence of a frst order (resp. second order) phase transton. The basc dea behnd ths theorem s that a quantum phase transton s assocated to a dvergence of at least one of the elements of the reduced densty matrx, or of one of ts dervatve. The condtons aforementoned guaranty that no accdental or artfcal dvergence occurs n the entanglement measure (or ts dervatve) used, or n the elements of the reduced densty matrx (or ts dervatve). If one of these condtons s not fulflled, one can for example fnd dscontnuty n the dervatve of the par concurrence assocated wth no phase transton, or to a frst order phase transton [Yan05]. Fnally, we pont out some works about thermal entanglement at low but non zero temperature [ABV01, Wan01, GKVB01, ON02, AP07], where quantum effects are competng wth thermal fluctuatons Quantum teleportaton As mentoned n the ntroducton, entanglement can be use as a resource to buld quantum technologes, openng the world of quantum communcaton [GT07]. One of the most famous example of such a technology s the quantum teleportaton. The teleportaton protocol as been frst descrbed by Bennett et al. n [BBC + 93]. Here, we brefly sketch the prncpal steps of ths protocol. Suppose that Alce has a qubt n her hands, and she wants to transfer t to Bob, spatally separated from her. The communcaton between them can only be performed usng classcal channel, lke phone or nternet. The qubt of Alce s labeled by 1, and s n the state φ 1 = a + b, wth the usual normalzaton a 2 + b 2 = 1. The frst step s to construct a par of qubts n the entangled Bell state ψ 23 = 1 2 ( ), and to gve the qubt 2 to Alce, and the qubt 3 to Bob. Alce has then the qubt she wants to send and her consttuent of the Bell par, whereas Bob has only the second consttuent. At ths pont, the full state of the three qubts systems s gven by Ψ 123 = a 2 ( )+ b 2 ( ). We can rewrte ths state usng the Bell base for the spn 1 and 2 (1.37) ψ ± 12 = 1 2 ( 1 2 ± 1 2 ), (1.38) φ ± 12 = 1 2 ( 1 2 ± 1 2 ), (1.39) and we see easly that Ψ 123 becomes Ψ 123 = 1 [ φ (a 3 b 3 )+ φ 12 (a 3 +b 3 ) ] + ψ + 12 ( a 3 +b 3 )+ ψ 12 ( a 3 b 3 ). (1.40)

39 14 Chapter 1. Quantum Entanglement and Models Now, Alce performs a jont measurement on the spns 1 and 2 she has on her sde. Accordng to the decomposton (1.40), the outcome of the measurement wll be one of the four states composng the Bell base ψ ± 23 or φ± 23 wth the same probablty 1/4. Because the quantum channel was ntally made of two entangled qubts, the qubt 3 of Bob wll nstantaneously be projected nto one of the pure states appearng n eq.(1.40) parameterzed by a and b. Once Alce knows the result of her measurement, she sends t to Bob by the classcal channel, and he wll perform a untary transformaton on hs qubt φ 3 accordng to ths result n order to reconstruct the ntal state of the qubt of Alce. For example, f he receves that the result s ψ 12, then hs qubt s already n the desred state, up to a trval phase factor, and no transformaton s done. For the three other outcomes, he has to use the transformatons ( ) φ : φ 3 φ = σ y φ 3, (1.41) ( ) φ : φ 3 φ = σ x φ 3, (1.42) ( ) ψ : φ 3 φ = σ z φ 3, (1.43) to reconstruct the ntal state of the qubt of Alce φ 3 = a 3 +b 3. Note that the state of the qubt of Alce has been destroyed durng the measurement, the fnal state of the qubt of Bob does not result from a copy, and then, the teleportaton protocol s not n contradcton wth the non clonng theorem [WZ82]. Quantum teleportaton has been acheved expermentally, for example n [BPM + 97, MDRT + 03, UJA + 04, MHS + 12]. In these experments, the quantum channel s made by two photons prepared n an entangled state by parametrc down converson, and the polarzaton state of a photon s teleported from a place to another. 1.2 Quantum decoherence The lnearty of the Hlbert space used to descrbe a quantum system leads to the prncple of quantum superposton. More precsely, f Ψ 1 and Ψ 2 are both accessble states of a quantum system, then a lnear superposton a Ψ 1 + b Ψ 2 s an accessble state as well. Ths prncple s the orgn of purely quantum effects lke quantum nterferences or entanglement, ntroduced just before. But ths prncple seems to be n contradcton wth the vson we have of our macroscopc world, where all objects are n a defnte state. Relevant questons are then: How the classcal world emerges from the quantum world? Is there a clear boundary between classc and quantum? These questons les n the heart of the measurement problem [Sch05, BR13]. It exsts several theores amng to answer these questons, but the most accepted one by the physcst communty s the theory of decoherence 2 [Zur82, Zur02, Zur03, PZ01, Sch07]. In ths theory, the classcalty of a quantum state comes from the unavodable nteracton of the system under consderaton wth an envronment. Indeed, a quantum system s never really solated, and always nteracts wth the surroundng envronment made of a bg number of degrees of freedom. 2. For an alternatve theory, see for example the many worlds nterpretaton [Eve57].

40 1.2. Quantum decoherence 15 Ths nteracton wll have for consequence to kll the coherence (.e the ablty to do nterferences) and then to destroy the quantum features. The off dagonal elements of the reduced densty matrx of the system decay (untl zero f the decoherence s total), and the state, whch was ntally a quantum superposton of pure states, s brought nto a classcal statstcal mxture. Let us llustrate the decoherence by the example of a smple system, taken from [Zur03]. The system under consderaton s a sngle spn S, coupled to a spn envronment E. The two states of the system are {, }, and t s moreover assumed that local Hamltonans vanshes. The nteracton between the system and the envronment s H SE =( ) g k ( ) k k =σs z g k σk z. (1.44) k The whole system spn plus envronment s ntally prepared nto the state Ψ(0) = (a +b ) (α k k + β k k ), (1.45) k whch wll be at a latter tme t, under the dynamcs set by the nteracton Hamltonan, n the state Ψ(t) = a E (t) +b E (t), (1.46) where E (t) = N k=1 ) (α k e gkt k + β k e gkt k = E ( t). (1.47) If one s nterested only n the propertes of the spn system, the partal trace has to be taken from the whole densty matrx to have access to the reduced densty matrx. After ths operaton, we get ρ S (t) = a 2 + b 2 +ab f(t) +ba f (t) (1.48) where the functon f(t) encodes all the propertes of the envronment, f(t) = N k [ cos 2gk t+ ( α k 2 + β k 2) sn 2g k t ], (1.49) whch behaves at large tmes and for bg number of envronmental spns lke f(t) 2 2 N. One clearly see here that the off dagonal elements of the reduced densty matrx of the system are dramatcally affected by the couplng wth the envronment. In the large tmes lmt, the coherence s totally lost, and the system s n a classcal mxture ρ S (t) = a 2 + b 2 where the quantum effects have been suppressed. A model of decoherence wll be studed n chapter 4.

41 16 Chapter 1. Quantum Entanglement and Models 1.3 The XY model Presentaton and canoncal dagonalzaton The quantum XY model s a model descrbng the behavor of nteractng spns 1/2 on a one dmensonal lattce. It has been frst ntroduced wthout external magnetc feld and solved by Leb, Schultz and Matts n 1961 [LSM61]. In 1970, Pfeuty solved the ansotropc case correspondng to the Isng model wth a transverse magnetc feld [Pfe70], and, one year latter, Barouch and McCoy solved the general XY model n a transverse feld, establshed the crtcal behavor of the correlatons functons and derved the phase dagram of the model [BMD70, BM71a, BM71b, MBA71]. In ths secton, we wll sketch n detals the canoncal dagonalzaton procedure of ths model [Kar06, Pla08, Col12]. The Hamltonan of a the generc quantum XY model s gven by H XY = 1 ( 2 1+κ n 2 σx n σn+1 x + 1 κ ) 2 σy nσ y n+1 h 2 σ z, (1.50) n where σ α n,(α = x, y, z) are the Paul matrces n the α drecton assocated to the ste n, and we consder frst neghbor nteractons wth couplng constant J. Here, κ s a parameter fxng the ansotropy. The value κ = 0 corresponds to the sotropc case, called the XX model. On the other hand, the value κ = 1 descrbes the most ansotropc case, the Isng model. The man propertes of theses models wll be dscussed n the next secton, where the phase dagram s presented. The Hamltonan can be mapped nto a chan of non nteractng fermons by means of the Jordan-Wgner transformaton [JW28]. Let us frst ntroduce the spn ladder operators σ ± n = (σ x n ± σ y n)/2 satsfyng the antcommutatve algebra {σ + n, σ n } = 1, whereas they commute on dfferent stes. The dea behnd ths transformaton s to assocate to these ladder operators creaton and annhlaton fermonc operators. Ths s done by the ntroducton of the Clfford operators Γ 1 n 1 n = =1 Γ 2 n 1 n = =1 ( σ z ) σx n, ( σ z ) σy n, (1.51) satsfyng Clfford algebra {Γ µ n, Γ ν m} = 2δ nm δ µν. Note that Γ µ n = ( Γ µ n), so they can be seen as not properly normalzed Majorana fermons. Usng these new operators, the dfferent terms of the Hamltonan (1.50) become Introducng the 2N dmensonal row vector Γ = σ z n =Γ 1 nγ 2 n, (1.52) σ x n σ x n+1 = Γ2 nγ 1 n+1, (1.53) σ y nσ y n+1 =Γ1 nγ 2 n+1. (1.54) ( Γ 1, Γ 2 ), wth Γ ν = ( ) Γ1 ν,..., Γν N, (1.55)

42 1.3. The XY model 17 the Hamltonan can be rewrtten lke where T s a hermtan matrx H = 1 4 Γ TΓ, (1.56) T = wth the N N matrx C h J y. J.. x... C = J y h and where we have ntroduced the notaton ( ) 0 C C, (1.57) 0 J x, (1.58) J x = 1+κ 2, J y = 1 κ 2. (1.59) Now, we defne the untary matrx V contanng the egenvectors V q of the matrx T where the egenvectors are parameterzed lke V = (V 1, V 2,... V N ), (1.60) V q = 1 ( ) φq. (1.61) 2 ψ q The egenvalue equaton TV q = ε q V q leads to two coupled equatons Cψ q =ε q φ q, (1.62) C φ q = ε q ψ q. (1.63) We remark that the two equatons (1.62) and (1.63) are nvarant under the smultaneous changes ε q ε q and ψ q ψ q. It follows that to each egenvalue ε q 0 s assocated an other one ε q 0 wth a correspondng egenvector gven by V q = 1 ( ) φq. (1.64) 2 ψ q Thanks to the symmetry of the T matrx, the egenvalues are obtaned by the sngular value decomposton Λ = V TV wth Λ = ( 1 E, E = ε.. E). ε N. (1.65)

43 18 Chapter 1. Quantum Entanglement and Models Now, takng advantage that the V matrx s untary (VV = 1), we get H = 1 4 Γ VV TVV Γ = 1 2 η Λη, (1.66) where we have ntroduced the dagonal creaton and annhlaton operators η η 1. η = η N η 1 = 1 V Γ, (1.67) 2.. η N satsfyng the canoncal antcommutatve algebra{η q, η q } = δ qq. More explctly, they are gven by the Bogolubov decomposton η q = 1 2 η q = 1 2 N ( ) φ q (n)γ 1 n + ψ q (n)γ 2 n n=1 N n=1 (1.68) ( ) φ q (n)γ 1 n ψ q (n)γ 2 n. (1.69) These relatons that can be nverted to express the Clfford operators as a functon of the dagonal ones Γ 1 n = Γ 2 n = N q=1 N q=1 ( ) φ q (n) η q + η q (1.70) ( ) ψ q (n) η q η q. (1.71) Usng the explct expresson of the matrx Λ, the Hamltonan s recast n the form η 1. H = 1 (. E 2 (η 1,..., η N, η 1..., η N ) η N E) η 1.. η N = 1 2 = N ε q η q η q ε q η q η q q=1 N q=1 ε q ( η q η q 1 2 (1.72) ), (1.73)

44 1.3. The XY model 19 where we have used the antcommutaton relaton {η q, η q } = 1. The ground state s thus gven by the product of the η s vacuum states 0 : GS = 0 N, (1.74) wth the property η q 0 = 0, 0 η q = 0 q = 1,, N. The assocated ground state energy s gven by E 0 = 1 2 N q=1 ε q Phase dagram The dsperson relaton of the XY model s [LSM61, Kar06, Pla08] ε q = (h+cos q) 2 κ 2 sn 2 q, (1.75) where the quas momentum q s restrcted to the range [0 : π], and the ε q exhbts the symmetry ε q+π = ε q. Ths dsperson relaton allows us to buld the phase dagram, as shown n fgure 1.1. Fgure 1.1 Phase dagram of the XY model n the (κ, h) plan. Note that ths phase dagram s symmetrc wth respect to the change h h. Thanks to the aforementoned symmetry, the XY model s gapless f t exsts a value of the quas momentum q such that ε q = 0. The C lne of equaton h = 1, represented n green, s a crtcal lne separatng a ferromagnetc phase from an paramagnetc one. On ths lne, one can see from the dsperson relaton that the gap closes, and consequently, the correlaton length dverges. Moreover, the correlaton functons present an algebrac decay wth the dstance. Ths lne s carachterzed by the crtcal exponent ν = 1 (correlaton length) and a dynamcal exponent z = 1. For h < 1, the system s n a ferromagnetc phase where the spns prefer to algn n the drecton of the mcroscopc couplng. On the contrary, for h > 1, the system s n the paramagnetc phase where t s more favorable for the spns

45 20 Chapter 1. Quantum Entanglement and Models to algn along the drecton of the magnetc feld. In order to dstngush these two phases, one can ntroduce an order parameter, whch s dfferent from zero only n the ordered phase. In ths case, ths role wll be played by the average lm n σ σ +n. The two dashed lnes represent the Isng lnes correspondng to a ansotropy parameter κ = ±1. The value +1 corresponds to a couplng to the x drecton, whereas the value 1 to a couplng of the y components. The C XX lne, plotted n red n fgure 1.1, represents a contnuous crtcal transton. On ths lne and for any value of h 1, the system s gapless and presents a crtcal behavor, wth dvergng correlaton length and a slow decay of correlaton functons. Fnally, the regon nsde the crcle of equaton h 2 + κ 2 = 1, represents an oscllatory ferromagnetc phase. Ths phase s characterzed by a wave vector modulatng the exponental decay of the correlaton functons, makng them oscllate [Hen99] Dynamcs Two dfferent, but equvalent, representatons are possble to descrbe the dynamcs of a quantum system. The frst one s the Schrödnger representaton where the tme dependence s carred by the state ψ (t) through the dfferental equaton d ψ(t) dt = H ψ (t). (1.76) If the Hamltonan s tme ndependent, the quantum state ψ s gven at a tme t by the applcaton of a untary operator, the tme evoluton operator U(t), on an ntal state ψ(0) ψ(t) = U(t) ψ(0), (1.77) where U(t) = exp( Ht). If the system s ntally n a statstcal mxture and ts state s descrbed by the densty matrx ρ, the state evolves followng the von Neumann equaton [BP02] dρ(t) = [H, ρ(t)], (1.78) dt and the densty matrx at a tme t s ρ(t) = U(t)ρ(0)U (t). (1.79) In the Hesenberg representaton, the tme dependence s no longer carred by the state, but by the operators. An operator O(t) wll then evolve through the Hesenberg equaton of moton do(t) = [H,O(t)]. (1.80) dt The average of the operator O s gven by O (t) = Tr{ρ(t)O} = Tr{ρO(t)}, (1.81) whch defnes the Hesenberg evoluton of the operator O O(t) = U (t)o(0)u(t). (1.82)

46 1.3. The XY model 21 Usng ths last representaton, the operators η q dagonalzng the XY Hamltonan evolve through η q (t) = U (t)η q (0)U(t), (1.83) and wth U q (t) = exp( tε q η q η q ), one has leadng to or, n a matrx form η q (t) = U q(t)η q (0)U q (t), (1.84) η q (t) = e tε q η q (0), η q(t) = e tε q η q(0), (1.85) η(t) = e tλ η(0). (1.86) Wth ths, on can wrte the tme evoluton of the Clfford operators usng the lnear transformaton 1 2 Γ(t) = Vη(t) (1.87) and equaton (1.86) to arrve at Γ(t) = R(t)Γ(0), (1.88) wth R(t) = Ve tλ V = e tvλv = e tt, where T s the matrx (1.57) Entanglement entropy of the XX chan Here we establsh a general expresson for the entanglement entropy of the XX model, and we show that t can be related to the egenvalues of the reduced correlaton matrx [Col12]. Instead of usng the Clfford operators as we dd n secton 1.3, we can use the lattce fermonc creaton and annhlaton operators c and c defned by c = 1 j=1 ( σ z j )σ, c = 1 j=1 leadng to the followng expresson of the XX Hamltonan ( σ z j )σ+, (1.89) H XX = c Tc. (1.90) The egenfunctons of the Hamltonan beng Slater determnant, and defnng the two-pont correlaton matrx as C j = c c j, t follows that, accordng Wck theorem, all the correlators of hgher order are expressed n terms of ths two-pont correlaton functon [Pes03]. By defnton, any average of local operator actng on the subsystem A has to be expressed n terms of the reduced densty matrx ρ A. As a consequence, we must have C j = Tr{ρ A c c j},, j A, (1.91) and all the correlatons of hghest order must factorze. It can be shown [Pes03, CP01] that the prevous property s vald f the reduced densty matrx ρ A s wrtten as the exponental of a quadratc form n terms of fermonc operators,.e ) ρ A = ( 1 Z exp c B jc j, (1.92) j

47 22 Chapter 1. Quantum Entanglement and Models where Z s a normalzaton constant ensurng the unty of the trace. The expresson (1.92) can be rewrtten ) ρ A = ( 1 Z exp ε k η k η k (1.93) k by the ntroducton of the dagonal operators related to the orgnal ones by the relaton c = φ k ()η k, c = φk ()η k. (1.94) k k Usng the fact that η k η k ρ = δ kk 1+e ε k, (1.95) the elements of the reduced correlaton matrx C j of the subsystem A become C j = φ k ()φk (j) 1 1+e k ε. (1.96) k Ths last expresson gves a relaton between the egenvalues ξ k assocated to the C matrx and those assocated to the B matrx ξ k = 1 1+e ε k. (1.97) Let us come back to the entanglement entropy, whch takes the expresson, usng n k = η k η k S(ρ A ) = Tr{ρ A ln ρ A } { ( ( 1 = Tr ρ A ln Z exp The normalzaton constant Z s gven by ))} ε k n k k = ε k Tr{ρ A n k }+ln Z. (1.98) k Z Tr{e k ε k n k } = (1+e ε k ). (1.99) k Usng ths last expresson together wth equatons (1.95) and (1.97), we obtan the fnal expresson S(ρ A ) = [ ξ k ln ξ k (1 ξ k ) ln(1 ξ k )]. (1.100) k The entanglement entropy of a block A s then completely determned by the egenvalues of the two-pont correlaton matrx of the block. Ths form of the entanglement entropy of a XX chan wll be used n the chapter 2.

48 1.4. The Bose-Hubbard model 23 Fgure 1.2 (left) Phase dagram of the Bose-Hubbard model. Insde the blue lobes, the system s n a Mott phase where the partcle densty s constant, whereas outsde, the system s n a superflud phase wth delocalzed partcles over the lattce. (rght) Pctural representaton of the transton between the Mott nsulator and the superflud states of the Bose-Hubbard model (pcture taken from http: // 1.4 The Bose-Hubbard model We fnsh ths chapter by the presentaton of the Bose Hubbard model. The Hubbard model has been ntroduced n 1963 for the descrpton of the behavor of electrons n a low temperature sold [Hub63]. The extenson of ths model to bosonc felds s called the Bose-Hubbard model. Ths s a smple model descrbng nteractng bosons on a lattce. The correspondng Hamltonan n one dmenson s H BH = t ( b b +1+ b b +1 ) µ n + U 2 n (n 1), (1.101) where the b and b are creaton and destructon bosonc operators on ste satsfyng the bosonc algebra[b, b j] = δ j, and n = b b s the occupaton number operator. The bosons are allowed to jump from ste to ste +1 wth an ampltude t. The second term s proportonal to the chemcal potental µ controllng the number of bosons n the system. Fnally, the last term s local bosonc par nteracton term repulsve for U > 0 and attractve for U < 0. Ths Hamltonan s analytcally solvable n the two lmts t/u 0 and t/u [FWGF89]. In the frst case, the on-ste nteracton term s domnant and the Hamltonan reduces to H = µ n + U 2 n (n 1). The system s characterzed by a constant densty of partcles n 0 on each ste fxed by the mnmzaton constrant ε n = 0, (1.102) n0 where ε(n) = µn Un 0(n 1) s the on-ste energy. The ground state s gven by the product of states wth n 0 densty Ψ = (b )n 0 0, (1.103)

49 24 Chapter 1. Quantum Entanglement and Models where 0 s the vacuum state wth no boson. The state s sad to be ncompressble n the sense that d ˆN /dµ = 0 where ˆN s the total occupaton number ˆN = n. Ths ground state corresponds n consequence to a Mott nsulator state. In the other lmt, the knetc part of the Hamltonan domnates and the system s n a superflud phase. The ground state s gven by Ψ = ( N b ) 0, (1.104) and all the partcles are completely delocalzed over the whole lattce. The pctural phase dagram n the (t/u) (µ/u) plane of the Bose-Hubbard model s presented on fgure 1.2. The Mott nsulator phases are confned n lobes nsde whch the number of bosons on each ste s constant. In ths phase, t exsts an energy gap between the ground and the frst excted states. Consderng non exactly nteger values of µ/u to avod degeneracy 3, the ground state wll move adabatcally wthout level crossng when the hoppng ampltude t s tuned to a small value. Addtonally, the Mott ground state (1.103) s an egenstate of the total partcle number operator ˆN wth egenvalue Nn 0. The small perturbaton brought by a small t commutng wth the Hamltonan, t results that the ground state remans an egenstate of ˆN wth the same egenvalue, explanng the survval of the Mott phase wth fxed densty even for non zero hoppng [Sac00]. The Mott to superflud transton s of second order, and the phase boundary can be determned usng standard Landau theory of phase transtons. One can defne a Landau free energy L = r Ψ B 2 + q Ψ B 4 +O( Ψ B 6 ), (1.105) where Ψ B 2 acts as order parameter. The parameter r can be found usng second order perturbaton theory where z s the coordnaton number, and r = ξ(µ/u)(1 ztξ(µ/u)), (1.106) ξ(µ/u) = n 0(µ/U)+1 Un 0 (µ/u) µ + n 0 (µ/u) µ Un 0 (µ/u) 1. (1.107) The transton occurs at r = 0 gvng the phase boundary µ U = µ U ( t ) U. 3. If µ/u s nteger, there s two degenerate ground states on each lattce ste wth occupaton number dfferng by one [Sac00].

50 Self-trappng of Hard Core bosons on optcal lattce 2 Expermental progresses have relaunched the nterest of physcsts for one dmensonal many-body systems. Among these systems, one can cte for example ultra cold quantum gases on optcal lattces, whch s the subject of many experments [Blo05, BDZ08, PSSV11]. Among all the nterestng phenomenon observed n such a systems, one s partcularly relevant for our study, namely the Bloch oscllatons [Blo28, Zen34, DPR + 96, KK04]. These oscllatons occur when a constant force F s appled to a quas-partcle on a lattce, leadng to a perodc moton wth perod 2π/ F. Some works have been devoted to the studes of the out of equlbrum dynamcs of bosonc systems n a lattce after the release of the confnng trap [HMMR + 09, MG05, RM05], and recently, studes have consdered the dynamcs of Hard core bosons on tlted optcal lattces [CCW11, CARK12]. In ths work, we propose to extend the work done n [CARK12] by consderng an ntal nhomogeneous dstrbuton of partcles set by an harmonc trappng potental. Dependng on the value of the harmonc potental, two dfferent knds of ntal states are possble. The system can be completely n a superflud state, or t can be composed by a Mott phase surrounded by two superflud phases. In the followng, we propose two dfferent knds of quench, the sudden release of the trappng potental, and the release of the trap and the loadng of a lnear ramp wth constant force F on the negatve part of the real axe. We develop an hydrodynamcal theory that allows us to explan the man features of the behavor of the partcles. The chapter s organzed as follow: In a frst part, we present the Bose Hubbard Hamltonan, descrbng bosons on a lattce, and we ntroduce the Hard Core lmt. After that, the man features of the hydrodynamcal theory are sketched, and the two knd of ntal states used n ths chapter are presented. Afterward, n secton 2.4, we turn to the descrpton of the out of equlbrum dynamcs after the quench of the trappng potental, startng wth the free expanson case and fnshng wth the tlted potental case. Fnally, the dfferent results are summarzed n secton

51 26 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce 2.1 Bose-Hubbard model and Hard-Core lmt Our model consst of bosons loaded on a one dmensonal lattce. It s descrbed by the Bose-Hubbard Hamltonan H BH = J N 1( ) b b +1+H.c + U 2 =1 N =1 n (n 1)+ N =1 V n, (2.1) where the b and b are the creaton and destructon bosonc operators, and n = b b s the densty operator countng the partcle on ste. The frst term s the knetc term modelng the hoppng of one partcle from the ste to the next one +1 wth ampltude J. The second one, wth U > 0, ntroduces a local par repulsve nteracton between bosons located at the same ste. Fnally, the last term s an nhomogeneous potental over the lattce. The phase dagram of ths model has been brefly dscussed n secton 1.4. It presents two dfferent phases, a Mott nsulator phase, where the partcles are well localzed on each ste, leadng to a constant nteger value of the partcle densty n 0 = 1, 2, 3..., and a superflud phase, where all the partcles are completely delocalzed over the lattce. The Hamltonan s not ntegrable and then not analytcally solvable. Nevertheless, we can consder the lmt of an nfnte repulson between bosons on one ste U, namely the Hard Core boson lmt. In ths case, the partcle densty s then restrcted to the two values n 0 = 0 or n 0 = 1, and the system s sent to the orgn of the phase dagram (see fgure 1.2). The second term of the Hamltonan dsappears and we end up wth the Hard Core boson Hamltonan ( ) H hc = J b b +1+H.c + V n. (2.2) The Hard Core boson model becomes analytcally solvable and can be mapped nto free fermons model. The frst step s done by the ntroducton of the transformaton between bosonc creaton and destructon operators and Paul operators σ x =b + b = σ + + σ σ y =(b b ) = σ+ σ (2.3) σ z =2b b 1 = σ + σ 1. Pluggng ths transformaton nto equaton (2.2), one gets the Hamltonan of nteractng spns 1/2: H = J 2 N 1( σ x σ x N σy +1) σy + 2 V σ z. (2.4) =1 =1 The usual procedure to treat ths Hamltonan s to use the Jordan Wgner transformaton between spn operators and lattce fermonc ones c = ( σj z)σ+, j< c = ( σj z)σ, (2.5) <j

52 2.2. Contnuum lmt and local equlbrum hypothess 27 n order to map the Hamltonan (2.4) nto free fermonc model H = J N 1 =1 (c c +1+ c +1 c )+ N =1 V c c =c T c. (2.6) The c and c are fermonc creaton and destructon operators satsfyng antcommutatve algebra {c c j} = δ j. We have ntroduced the N components row vector c = ( c 1, c 2,..., N) c and the trdagonal matrx (T )j = V δ j J(δ j,+1 + δ j,+1 ). Ths last Hamltonan can be dagonalzed through Bogolubov transformaton, by the ntroducton of the dagonal operators η q related to the lattce ones by η q = φq()c, η q = φ q ()c, (2.7) leadng to the dagonal form H = ε q η q η q, (2.8) q where the ε q are the exctaton energes assocated to the free Ferm partcles created by η q and destroyed by η q. For general nhomogeneous potental V, the spectrum ε q has to be computed numercally. But f the potental becomes homogeneous,.e V = V, the spectrum can be obtaned analytcally, and one gets assocated to the egenvector ε q = V 2J cos(q), (2.9) φ q () = 2 sn(q), (2.10) N+ 1 wth the quas momentum q n = nπ/(n+ 1) [0 : π]. In the rest of the chapter, the value of the hoppng strength J wll be set to J = 1/2, leadng to a dsperson relaton equal to ε k = V cos(q) and to a band wdth = 2. From the dsperson relaton, all the quas energes ε k are postves for V > 1 leadng to the vacuum state 0 wth a partcle number n 0 = 0. On the contrary, f V < 1, all the exctatons are negatves, and the system s n a Mott phase wth a partcle number equal to n 0 = 1 on each ste. For ntermedate values of the potental 1 < V < 1, the system s n a superflud phase wth a densty gven by ρ 0 = q F π, (2.11) where q F s the Ferm level assocated to the Ferm energy fxed by the constrant q η q η q = N on the total number of partcles. 2.2 Contnuum lmt and local equlbrum hypothess The behavor of the Hard Core bosonc system can be well understood by consderng an hydrodynamcal descrpton, as already ntroduced n [CARK12, Col12],

53 28 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce based on a local equlbrum hypothess. Here we brefly sketch the man features of ths descrpton [Col12]. Suppose an one dmensonal bosonc system wth lattce spacng a 1. Usng contnuous varable x = na, one can splt the real lne nto ntervals [x, x + x] wth x = am, where M s the number of stes. Suppose moreover that we keep the nterval x small enough such that the potental keep an almost constant value over ths nterval, that s V(ja) V(x) ja [x, x+ x]. Takng the contnuums lmt a 0 and x 0 keepng the rato x/a = M 1, the Hamltonan can be rewrtten, wth H(x) = 1 x dy a x 0 H = dxh(x), (2.12) [ ] 1 2 Ψ (x+y)ψ(x+y a)+ H.c+Ψ (x+y)v(x)ψ(x+y), (2.13) where we have ntroduced the creaton and annhlaton Ferm feld operators Ψ (x) and Ψ(x). Ths local Hamltonan densty can be dagonalzed usng the mappng π Ψ(x+y) = Ψ (x+y) = 0 π 0 dqφ q (x+y)η(x, q), (2.14) dqφ q (x+y)η (x, q), (2.15) where the η (x, q) and η(x, q), satsfyng the ant commutaton relaton {η (x, q), η(x, q )} = δ(q q )δ(x x ), respectvely create and annhlate a partcle wth momentum q n the regon [x, x+ x]. The φ q (u) enterng nto the mappng are the Bogolubov functons, whose exact expresson depends essentally on the boundary condtons. Usng these feld operators, the Hamltonan s recast nto the dagonal form H = π dx dq(v(x) cos q) η (x, q)η(x, q). (2.16) 0 The correspondng ground state s then gven by addng to the vacuum state 0 all the quas partcles, startng from the lowest energy up to the Ferm level ε F. In the grand canoncal ensemble, where the number of partcles can fluctuate, the Ferm level has to be set to ε F = 0. Hence, the ground state s gven by addng to the vacuum 0 the quaspartcles wth negatve energy GS = η (x, q) 0, (2.17) V(x) cos q<0 and the densty profle s gven by ρ(x) = 0 f V(x) > 1 1 π arccos(v(x)) f V(x) < 1 1 f V(x) < 1. (2.18)

54 2.3. Intal states 29 Now, ntroducng a regon x q around a pont (x, q) of the phase space, we can defne the coarse graned densty ρ(x, q) = 1 x q x+ x/2 q+ q/2 x x/2 q q/2 Φ p (y) 2 dpdy, (2.19) and state that ths coarse graned densty s almost constant leadng to an almost homogeneous dstrbuton n the phase space ρ(x, q) = 1 π Θ(q)Θ(q F q)θ( x)θ(x+ A), (2.20) where Θ(x) s the Heavsde functon, and A < 0 fx the spacal extenson of the condensate. The spacal densty profle s gven by ntegratng the prevous expresson over the momentum ρ(x) = q F Θ( x)θ(x+ A), π (2.21) wth q F = arccos(v(x)). 2.3 Intal states In the followng, we fx the ntal state of the system wth an harmonc trap. Ths harmonc potental V(x) s parameterzed lke [ V(x) = α x A ] 2 µ 0, α = 4 2 A 2(1+µ 0), (2.22) wth A < 0 and µ 0 > 1. Ths last form ensures that the bosons are loaded between postons x = A and x = 0 wth a condensate centered n x = A/2. Dependng on the value of the parameters, two dstnct stuatons are possble, controlled by µ 0 (see fgure 2.1 for a pctural representaton of the dfferent cases). 1 V(x) ρ(x) 1 V(x) S f 2 Mott S f ρ(x) A 0 x A 0 x 1 Fgure 2.1 Two dfferent possble ntal states dependng on the value of the parameter µ 0. On the left, µ 0 s smaller than 1, leadng to a condensate ntally completely n a superflud phase. On the rght, µ 0 s greater than 1, leadng to a Mott phase of extenson Mott surroundng by two superflud phases of extenson S f. If µ 0 <1, the potental V(x) s never smaller than 1 and the full condensate s then n a superflud phase. Integratng the phase space densty, we get the followng partcle densty θ(x)θ(x+ A) ρ 0 (x) = arccos(v(x)). (2.23) π

55 30 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce On the other hand, when µ 0 > 1, t exsts a spacal regon where the potental s smaller than 1 leadng to a Mott phase wth ρ 0 = 1 surrounded by two superflud phases. The Mott phase extends over the spacal regon x [ A 2 Mott, A 2 + Mott ], where Mott s gven by Mott = A µ µ 0 + 1, (2.24) and s surrounded by a superflud phase of extenson S f = A ( ) 1 µ 0 1, (2.25) 2 1+µ 0 wth a densty profle gven by the equaton We plot n fgure 2.2 the densty profles correspondng to a superflud ntal state (left) and a mxture Mott/superflud ntal state (rght) for dfferent values of µ 0. We see that the hydrodynamcal predctons ft perfectly the numercal results obtaned by exact dagonalzaton. 1 0,8 µ 0 =sqrt(2)/2 µ 0 =0 µ 0 =-sqrt(2)/2 1 ρ 0,6 0,4 0,2 ρ 0,5 µ 0 =3 µ 0 =2 µ 0 = x x Fgure 2.2 Intal densty profles for an ntal state n the superflud phase (left) and n a mxture Mott/superflud (rght) for dfferent values of the parameter µ 0. The spacal extenson of the condensate s A = 200. The hydrodynamcal predcatons are gven by the full lnes whereas the dots represent the numercal data obtaned by exact dagonalzaton. 2.4 Dynamcs after the sudden quench The quantum quench s realzed by suddenly changng the value of the trappng potental at t = 0 +, resultng n an out of equlbrum dynamcs. Usng the fact that the dynamcs s untary, and that all the quaspartcles are non nteractng, they are emtted to the rght and to the left on trajectores wth constant energy Free expanson of the condensate The frst knd of quench we have consdered s the sudden release of the trap at tme t = 0 +, the potental passng from a non zero value to a vanshng one,.e [ V(x) =α x A ] 2 µ 0 f t < 0 2 V(x) =0 f t > 0. (2.26)

56 2.4. Dynamcs after the sudden quench 31 The Evoluton of the densty profle obtaned numercally by exact dagonalzaton s shown n fgure 2.3 for the two ntal states presented above. Fgure 2.3 Evoluton of the densty profle obtaned numercally by exact dagonalzaton. The up snapshot represents an ntal condensate n a superflud phase wth µ 0 = 0, whereas the down snapshot s for a ntal state n a mxture Mott/superflud wth µ 0 = 3. In both cases, the spacal extenson of the condensate s A = 100. After the quench, the potental beng zero, the condensate s allowed to explore spacal regons that were forbdden n the presence of the trappng potental. One can see that the condensate s spreadng to the rght and to the left over the whole lattce. Notce that snce the numercal results are obtaned wth fnte system sze, the partcles are reflected when they reach the two boundares. After the quench, the energy of the quaspartcles s smply gven by the knetc part ε = cos(q) [ 1, 1]. (2.27) The movement of the quaspartcles beng ballstc, ther velocty s gven by the dervatve of the energy wth respect to the momentum v ± = ± ε q, (2.28) where the + sgn (resp. ) counts for the rght (resp. left) movers. Usng the expresson 2.27 for the energy, we fnd that the velocty s gven by v ± q = ± 1 ε 2 = ± 1 cos 2 q, (2.29)

57 32 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce and depends only on the ntal energy of the quaspartcles. Snce the potental vanshes, the dynamcs of the quaspartcles s free. As a consequence, a quaspartcle ntally located at the pont (x 0, q 0 ) at t = 0 wll be at a latter tme t > 0 moved at the pont x ± (x 0, q 0, t) = x 0 ± t 1 ε 2 = x 0 ± t 1 cos 2 q 0. (2.30) The densty at a pont (x, q) of the phase space assocated to the rght and left movers s gven by [CARK12, Col12] ρ ± (x, q, t) = 1 0 qf (x 0 ) dx 0 dq 0 δ(x x ± (x 0, q 0, t))δ(q q ± (x 0, q 0, t)). (2.31) 2π A 0 In other words, we are lookng n the ntal dstrbuton all the ponts(x 0, q 0 ) that wll be at tme t at the pont (x, t) under the effect of the quench dynamcs. The spacal densty s smply gven by ntegratng (2.31) over the momentum q, and we get ρ ± (x, t) = 1 0 qf (x 0 ) dx 0 dq 0 δ(x x ± (x 0, q 0, t)). (2.32) 2π A 0 The total densty at a pont x and tme t s obtaned by summng the rght and left movers contrbutons ρ(x, t) = ρ + (x, t)+ρ (x, t). (2.33) We present on fgure 2.4 the densty profle for dfferent tmes for the two types of ntal states. The numercal evaluaton of the ntegral (2.32) reproduce perfectly the exact dagonalzaton results. ρ 0,5 0,4 0,3 0,2 ρ 1 0,5 0, x x Fgure 2.4 Densty profle for dfferent tmes. In the left plot, the condensate s ntally n a superflud phase wth µ 0 = 0 and the tmes are, from top to bottom t = 10, t = 40, t = 60, t = 100 and t = 150. In the rght plot, the ntal state s a mxture Mott/superflud wth µ 0 = 3 and the tmes are, from top to bottom, t = 10, t = 40, t = 50, t = 80 and t = 120. In both case, we have A = 100. The numercal calculaton of the expresson (2.32) s gven by the full lnes, whereas the symbol are the exact dagonalzaton results. It s nterestng to look at the number of partcles N t present n the ntal trappng regon of the condensate as a functon of tme. Ths number s obtaned by ntegratng the densty ρ(x, t) between postons A and 0 N t (t) = N + t (t)+ N t (t), (2.34)

58 2.4. Dynamcs after the sudden quench 33 wth N t ± = qf (x 0 ) dx dx 0 dq 0 δ(x x ± (x 0, q 0, t)). (2.35) 2π A A 0 The system possessng the mrror symmetry wth respect to the mddle of the condensate n the pont x = A 2, the contrbutons assocated to the rght and left movers are equals, and N t + = Nt. Usng ths symmetry and the propertes of the delta functon, one arrves at N t (t) = 1 0 qf (x 0 ) dx 0 dq 0 Π π [A:0] (x (x 0, q 0, t)), (2.36) where Π [a,b] (x) s the door functon defned by A 0 Π [a,b] (x) = { 1 f x [a, b] 0 f x / [a, b]. (2.37) The dots n fgure 2.5 represent, for the two ntal states, the number of partcles present n the ntal trapped regon as a functon of tme obtaned by exact dagonalzaton, whereas the full lnes gve the hydrodynamcal predctons usng numercal evaluaton of the ntegral (2.36). Interestngly, one can see, after a transent tme where N t s constant, two dfferent regmes for an ntal Mott/superflud state, where the behavor s lnear at the begnnng of the evoluton and turns to a power law after a tme t A, whereas the ntal superflud case presents only the second regme N t N t t t Fgure 2.5 Number of partcles N t, present n the ntal trappng regon as a functon of the tme for an ntal Mott/superflud state (up curves) and an ntal fully superflud condensate (down curves). The dots are obtaned wth exact dagonalzaton, whereas the full lnes gve the hydrodynamcal predctons usng equaton (2.56). The full blue lne s the expresson (2.41), correspondng to an ntal state completely n a Mott phase (case µ ). On the nsert s shown the same plot n log scale. The analytcal evaluaton of the ntegral (2.36) s dffcult for the two ntal states, but we can nevertheless understand ths change of behavor by consderng the case µ 0. Ths case corresponds to an ntal state completely n a Mott phase wth q 0 (x 0 ) = π x 0 and ρ 0 = 1 x 0. In ths case, the condton mposed by the door functon n equaton (2.36) s fulflled for all q 0 [0 : π] consderng ntal postons

59 34 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce x 0 > A+t. For values of x 0 < A+t, ths condton s fulflled n the nterval q 0 [0 : q + (x 0, t)] and q 0 [q (x 0, t) : π] where q ± 0 (x 0, t) are solutons of the equaton x (x 0, q 0, t) = A and gven by q ± 0 (x 0, t) = arccos The ntegral (2.36) can then be rewrtten ( ± 1 (x 0 A) 2 t 2 ). (2.38) N t (t) = 1 ( x(t) ( q + (x 0,t) π 0 π ) dx 0 dq 0 + dq 0 )+Θ( A t) dx 0 dq 0, π A 0 q (x 0,t) x(t) 0 (2.39) where x(t) = (A+t)Θ( A t). Usng the property arccos(x)+arccos( x) = π of the arccosnus functon, one arrves to N t (t) = 1 π x(t) A dx 0 2 arccos ( Performng the ntegral, we fnally fnd N t (t) = A 2 π t Θ( A t) ( ( + A+ 2A π arctan ( t 2 1 (x 0 A) 2 t 2 A 2 1 ) Θ( A t)(a+t). (2.40) ) t+ t 2 A 2 )) Θ(t A ), (2.41) whch behaves, for t, lke N t (t) 1/t. We recover here the lnear behavor for t < A and the power law decay for t > A. The expresson (2.41) s plotted n dashed lne n fgure 2.5, and we can see that the slope n the lnear regme and the exponent n the power law decay are the same n the pure Mott and mxture Mott/superflud cases Self Trappng wth a lnear potental In ths secton, we turn to the second quench consdered n ths study. Startng from the two ntal states mentoned prevously, the trappng potental s suddenly released at tme t = 0 + and a lnear ramp of equaton V(x, t > 0) = Fx Θ( x) (2.42) s loaded n the negatve part of the x axe. Just after the quench, the energy of a partcle n poston x s shfted by the quantty Fx, as shown n the fgure 2.6 where we present a pctural representaton of the energy band of a Mott/superflud ntal state just after the loadng of the ramp. Lke n the prevous case treated, the sudden quench leads to an out-of-equlbrum untary dynamcs, whose man features are perfectly caught by the hydrodynamcal theory developed above. We present n fgure 2.7 the dynamcs of the densty profle just after the quench for the two ntal states and for dfferent forces of the lnear potental. One can see from ths pcture that the ntal condensate s splttng nto

60 2.4. Dynamcs after the sudden quench 35 ε=1 A x m 0 Fgure 2.6 The partcles n the blue regon leave the trap. The poston x m corresponds to the most rght poston of the ntal condensate wth energy ε = 1. Fgure 2.7 Snapshot of the tme evoluton of the densty profle obtaned numercally by exact dagonalzaton startng from the two ntal states and for dfferent forces. Up/left: Mott/superflud state wth µ 0 = 3 and F = 0.1, top/rght: Mott/superflud state wth µ 0 = 3 and F = 0.04, bottom/left: superflud state wth µ 0 = 0 and F = 0.1, and bottom/rght: superflud state wth µ 0 = 2/2 and F = For all cases, we set A = 200. two parts: a frst part of the condensate leaves the ntal regon and spreads n tme toward the rght of the x axs, whereas a second part stays trapped nto the regon x [A, 0] and performs Bloch oscllatons. Note that lke n the prevous case, the escapng partcles are reflected nto the rght boundary snce the system s numercally fnte. The reason why a part of the condensate escapes the ntal regon can be understood

61 36 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce n the followng way: the partcles wth an energy n the range ε [ 1, 1] are connected wth the propagatve band startng at poston x = 0 wth zero potental, and then can escape toward the rght sde of the system. On the contrary, partcles wth an energy hgher than ε = 1 stay trapped nsde the ntal regon due to the energy conservaton. The number of escapng partcles N esc s then gven by summng all the partcles wth an energy ε between 1 and 1 n the ntal densty dstrbuton. Usng energy varable, t yelds N esc = 1 1 dε dq 0, (2.43) Fπ 1 Q(ε) where Q(ε) s the ntegraton contour shown n blue n fgure 2.6. In partcular, f the most energetc locus of the ntal dstrbuton has an energy smaller than ε = 1, the regon connected to the propagatve band covers the complete condensate, and, as a consequence, N esc = N, meanng that all the partcles are lvng the trap and are escapng to the rght. Of course, f the system s ntally n a Mott/superflud state, the hghest energy s always ε = 1 and then the only value of the force leadng to N esc = N s F = 0. On the contrary, f the system s n a pure superflud state, the most energetc pont just after the quench s gven by the soluton of the equaton ε/ x = 0, that s x max = { A 2 FA2 8(1+µ 0 ) f F < F 2, A f F > F 2, (2.44) where F 2 = 4 A (1+µ 0) s the force makng the most energetc pont located n x = A. The force below whch all the partcles escape has then to fulfll the equaton Fx max cos q F (x max ) = 1. We fnally fnd the values F M/s f esc = 0 (2.45) { ) 4 Fesc S f = A 0)( 1 (1+µ 2 µ 0 +1 f F < F 2 (2.46) 2 A f F > F 2. We plot n fgure 2.8 N esc as a functon of the force F obtaned numercally for the two ntal states and for several values of µ 0. As predcted, we see that N esc s constant to N for forces smaller than F esc. Once ths force crossed, the number of escapng partcles decreases when the force s ncreased. We fnd a power law N esc F γ where the exponent γ ncreases wth the value of µ 0 untl a saturaton value γ = 1 for µ 0. Indeed, n ths case, the spacal extenson of the superflud phase tends to zero, and the system s ntally n a fully Mott phase, leadng to the behavor N esc 1 F found n [Col12, CARK12] Dynamcs of the trapped partcles We focus n a frst tme our attenton on the dynamcs of the trapped partcles. These partcles wth energy ε > 1 perform Bloch oscllatons due to the constant force F actng on them. For an ntal Mott/superflud state, one observes two oscllatory regons, located at

62 2.4. Dynamcs after the sudden quench N esc 10 1 µ 0 =-sqrt(2)/2 µ 0 =0 µ 0 =sqrt(2)/2 0,0001 0,001 0,01 0,1 1 F N esc 10 µ 0 =1 µ 0 =3 µ 0 =5 µ 0 =10 µ 0 = 1 0,0001 0,001 0,01 0,1 1 F Fgure 2.8 Number of escapng partcles N esc as a functon of the force F for dfferent values of µ 0. The system s ntally n a pure superflud state n the left plot, whereas t s n a Mott/Superflud mxture on the rght one. The full pnk lne represents the case of µ 0 =, correspondng to a condensate ntally n a fully Mott phase. the two edges, surroundng a plateau regon where the densty stays constant and equal to the ntal value of the Mott phase ρ 0 = 1. We can also note dfferent regmes of oscllatons, dependng of the value of the appled force. Indeed, by lookng fgure 2.7, we can see that at hgh forces, the two regons surroundng the plateau oscllate n phase, n the sense that they start to move both to the rght. On the contrary, at lower forces, these two regons oscllate n opposte drecton, the left sde of the plateau startng to move to the left, and the rght sde to the rght, leadng to a breathng regme. When the ntal state s fully superflud, the oscllatons concern now the complete condensate. The dstncton between hgh and low force dscussed above stll holds for ths ntal state. Indeed, for hgh forces, all the condensate oscllates as whole whereas at low forces, t splts nto two parts, one part startng to move to the left and the other one to the rght. One can fnally note that, for both ntal states and at low forces, the leftmost locus of the condensate can explore spacal regon located at the left of the poston x = A delmtng the ntal poston of the condensate. All these last features can be explaned by to the hydrodynamcal theory, as we wll now see. The dynamcs beng untary, the partcles evolve along trajectores of constant energy. We then obtan dε dt = d (V(x) cos q) = 0, (2.47) dt gvng F dx dt + dq sn q = 0. (2.48) dt Usng the expresson (2.28) of the velocty v ± = ± ε q = ± sn q, (2.49)

63 38 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce we fnd ± F+ dq dt = 0, (2.50) leadng, after ntegraton, to the evoluton equaton of the momentum q ± (t) q ± (t) = ±q 0 + Ft. (2.51) Ths last equaton descrbes the converson of the potental energy V(x) nto knetc one cos q(t). We remnd that the densty correspondng to the rght and left movers s gven at a tme t by ρ ± (x, q, t) = 1 0 qf (x 0 ) dx 0 dq 0 δ(x x ± (x 0, q 0, t))δ(q q ± (x 0, q 0, t)), (2.52) 2π A 0 and the spacal densty by the ntegraton over the momentum q: Now, we use the fact that ρ ± (x, t) = 1 0 qf (x 0 ) dx 0 dq 0 δ(x x ± (x 0, q 0, t)). (2.53) 2π A 0 dx 0 δ(x x ± (x 0, q 0, t)) = 1 F dεδ(x x± (ε, q 0, t)) (2.54) wth x ± (ε, q 0, t) = 1 F (ε+cos(±q 0+Ft)) (2.55) together wth the energy at the ntal tme ε = Fx 0 cos(q 0 ), and we fnd that the rght and left denstes can be rewrtten ρ ± (x, t) = 1 π dq 0 2π 0 where the functon g ± (x, q 0, t) s gven by dx 0 δ(x 0 g ± (x, q 0, t)) (2.56) g ± (x, q 0, t) = x 1 F [cos(q 0) cos(±q 0 + Ft)]. (2.57) Fnally, performng the ntegraton, we get the expresson ρ ± (x, t) = 1 π dq 0 Π 2π [ f2 (q 0 ), f 1 (q 0 )](g ± (x, q 0, t)), (2.58) q n f where we have used to propertes of the delta dstrbuton, and the functons f 1 (q 0 ) and f 2 (q 0 ) are respectvely gven by f 1 (q) = 1+cos q 0 F A cos 2 + q0 + µ 0 α q 0 [q n f, q 0 ] q 0 [ q 0, π] (2.59) f 2 (q) = A 2 cos q0 + µ 0 α (2.60)

64 2.4. Dynamcs after the sudden quench 39 wth q 0 satsfyng the equaton Fx m + cos q 0 = 1 where x m s the rghtmost locus of the condensate wth energy ε = 1 (see fgure 2.6), and the lower bound q n f of the ntegral s equal to q n f = { 0 f F > 2 A arccos(1 Fx ) f F < 2 A. (2.61) The reason why the expresson on the functon f 1 (q 0 ) depends of the range of q 0 s to take nto account the escapng partcles. Indeed, these partcles have left the trap by the propagatng band, and then they do not contrbute to the oscllatng behavor. It s then necessary to remove them from the ntal densty profle when the ntegraton s performed. In fgures 2.9 and 2.10, we plot the tme evoluton of the densty profle of the self trapped partcles over a half perod of oscllatons for the two ntal states Mott/superflud and superflud, both numercal (full lnes) and theoretcal (dashed lnes), together wth a pctural representaton of the energy band just after the quench. The frst observaton that can be made s that the hydrodynamcal predctons perfectly reproduce the oscllatory behavor of the densty profle. Secondly, the dfferent oscllatory regmes correspondng to low and hgh forces are clearly vsble for the two knds of ntal states. We can also see that the condensate extents beyond the poston x = A n the low forces case. The key pont to understand these features s to look at the poston of the most energetc locus of the condensate x max. We fnd x M/s f max = A 2 ( ) µ µ A 2 FA2 8(1+µ 0 ) A f F > F 2 for the case of a ntal Mott/superflud state, and x s f max = f F < F 1 = 4 A f F 1 < F < F 2 A 2 FA2 8(1+µ 0 ) f F < F 2 µ (2.62) A f F > F 2 (2.63) for a fully superflud ntal state. Here, F 1 s the force makng the most energetc poston n the separatng poston x = x [M S f] between Mott and superflud phases. If the most energetc poston s located at x max > A,.e for forces F < F 2, the condensate can explore spacal regon on the left of the pont x = A thanks to the untary evoluton. The leftmost locaton of the condensate may extend untl the poston x L (see down plot of fgure 2.9) defned by the equaton Fx L 1 = Fx max cos(q F (x max )), (2.64)

65 40 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce 1 ρ 0,5 0 1 t=62 t=70 t=76 t=85 t=94 t=94 (hydro) x A=x max 0 ρ 0,5 0 1 t=92 t=106 t=121 t=140 t=92 (hydro) x A x max 0 ρ 0,5 0 t=156 t=196 t=200 t=215 t=237 t=156 (hydro) x x L A x max 0 Fgure 2.9 Top: Self-trapped densty profle for an ntal SF-Mott state wth µ 0 = 3 at dfferent tmes for F = 0.1. Mddle: Same as top for F = Down: Same as top for F = In each cases, we set A = 200, and the hydrodynamcal predctons are represented n each case by the orange dashed lnes. gvng x M/s f L = A 2 ( ) µ µ F A 2 FA 2 16(1+µ 0 ) 1+µ 0 F f 0 < F < F 1 f F 1 < F < F 2 (2.65)

66 2.4. Dynamcs after the sudden quench 41 ρ 0,5 0,4 0,3 0,2 0,1 0 0 t=130 t=155 t=168 t=186 t=130 (hydro) x 0,6 t=418 t=458 0,5 t=471 0,4 t=538 t=418 0,3 (hydro) 0,2 0, A=x max A x max Fgure 2.10 Top: Self-trapped densty profle for an ntal superflud state wth µ 0 = 0 at dfferent tmes for F = Down: Same as top for µ 0 = 2/2 and F = In each cases, we set A = 200, and the hydrodynamcal predctons are represented n each case by the orange dashed lnes. and x s f L = A 2 FA 2 16(1+µ 0 ) 1+µ 0 F f F S f esc < F < F 2. (2.66) In ths range of forces, the energy of the pont x max s greater than the energy of the pont x = A. As a consequence, there s accessble regon on the left and on the rght of the pont x = x max (see for nstance down plot of fgure 2.10), and the trapped densty of the ntal superflud condensate splts nto two parts, one starts to move to the left, whereas the other one starts to move to the rght. Ths dynamcs wll be called the splttng regme. For an ntal Mott/superflud state, one has to make the dstncton between the two force regmes F < F 1 and F 1 < F < F 2. Indeed, n the frst case, the most energetc pont s the separatng pont x [M s f] between the Mott and superflud phase, and s then located on the rght tlted edge (see down plot n fgure 2.9). Therefore, the part of the condensate located on the left of the plateau starts to move to the left whereas the rght sde of the plateau starts to move to the rght, leadng to a breathng regme. In the other case F 1 < F < F 2, one recovers the splttng

67 42 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce regme but only n the regon on the left of the plateau. Indeed, ths part of the ntal condensate splts nto two parts movng n opposte drecton, whereas the movement of the rght sde s unchanged. We can note that n ths case, the oscllatons n the left part of the condensate are weak. Ths feature s due to the fact that the energy n the three ponts x = A, x = x max and x = x [M s f] are close to each other, and then the part of the condensate partcpatng to the oscllaton s small, leadng to almost no oscllatons. In the strong forces regme F > F 2, the most energetc pont s x max = A and s then located on the left tlde band. In ths case, and whatever the ntal state, the condensate oscllates n phase. Indeed, n the superflud case, the condensate oscllates as a whole, and n the Mott/superflud case, the two sdes of the plateau move n the same drecton at a gven tme. Note fnally that the two boundares x p1 and x p2 of ths plateau are also force dependent, and gven by { A+ 2 F f F < F 2 x p1 = ( ) A 1+ µ0 1 (2.67) f F > F 2 x p2 = A 2 2 ( 1 µ 0 +1 µ 0 1 µ ) 2 F. (2.68) The oscllatory behavor of the densty profle s assocated to a flow of partcles gvng rse to a current densty j(x, t). Ths current s defned through the contnuty equaton dn k = [H, n dt k ] = j k = j k 1 j k, (2.69) where n k s the occupaton number at ste k. The explct calculaton of the commutator gves j k = 1 2 ( ) c k c k+1 c k+1 c k, (2.70) and ts average reads ) j k = Im ( c k c k+1. (2.71) We present n fgure 2.11 snapshots of the tme evoluton of the current profle, obtaned numercally, for a superflud ntal state, and for a Mott/superflud ntal state at dfferent forces. In all the cases, we clearly see the oscllatory behavor of the partcles n the trapped regon. For an ntal superflud state at hgh forces, we observe a strp structure of the oscllatons showng that all the condensate s oscllatng as a whole, as already observed n the densty profle. For low forces, there are two regons n the trapped profle of opposte sgn current at the begnnng of the evoluton, ndcatng movement n opposte drecton (see second plot of fgure 2.11). In the Mott/superflud case, the current vanshes n the plateau regon x [x p1, x p2 ] where the densty stays constant at the ntal value ρ 0 = 1 and where no oscllatons of the densty s observed. At hgh forces F > F 2, the two sdes of the condensate oscllate n phase, and the sgn of the current s the same on both sdes of the plateau at a gven tme, whereas at low forces F < F 1, the two sdes surroundng the plateau

68 2.4. Dynamcs after the sudden quench 43 Fgure 2.11 Snapshot of the tme evoluton of the current densty profle. In the two frst plots startng from the top the ntal states are superflud wth µ 0 = 0 and F = 0.1 and µ 0 = 2/2 and F = respectvely. In the two followng, the ntal states are a Mott/superflud mxture wth µ 0 = 3 and F = 0.1 and F = 0.04 respectvely. The last plot s a zoom n the left part of the plateau for a Mott/superflud ntal state wth µ 0 = 3 and F = In the three cases, A = 200.

69 44 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce oscllate n opposte drecton leadng to current of opposte sgn at a gven tme. For ntermedate forces F 1 < F < F 2, we see the perfect alternaton of currents wth dfferent sgn on both sdes of the mddle of the superflud phase reflectng the splttng regme takng place on the left sde of the plateau (see last plot of fgure 2.11). In the framework of the hydrodynamcal theory, the current densty s obtaned by summng the current contrbutons of all the partcles j(q) = ρ(q)v(q) wth veloctes v ± (q) = ± sn(±q 0 + Ft). Then the current s gven by addng a factor± sn(±q 0 + Ft) to the ntegral (2.56), leadng to π dq 0 j(x, t) = q n f 2π sn(q 0+Ft)Π [ f2 (q 0 ), f 1 (q 0 )](g + (x, q 0, t)) π q n f dq 0 2π sn(q 0 Ft)Π [ f2 (q 0 ), f 1 (q 0 )](g (x, q 0, t)). (2.72) We plot n fgure 2.12 the behavor of the current for dfferent stuatons. The hydrodynamcal predctons ft perfectly the numercal data for all the plots except the case of a ntal superflud state at low forces where the curves are dfferent at the begnnng of the evoluton. The reason for that comes from the escapng partcles. Indeed, n the theoretcal development, these partcles are removed from the ntal densty profle n order to not take them nto account durng the Bloch oscllatons of the trapped partcles. Because the force s small, some of the escapng partcles contrbute to the current untl they leave the trap. Once these partcles far n the rght regon (at t 200), the predctons match the numercal data. We can remark a perfect snusodal tme evoluton n the mddle of the condensate x = A/2 n the case of a ntal pure superflud state. In ths case, the condton gven by the door functon n equaton (2.72) s fulflled for q 0 [0 : π/2] leadng to j SF (A/2, t) = 1 π/2 dq 0 (sn(q 0 + Ft) sn(q 0 Ft)) 2π 0 = 1 sn(ft) (2.73) π On the contrary, for a Mott/superflud, the mddle of the condensate s located n the plateau regon and the condton s always fulflled, leadng to a vanshng current. Fnally, on fgure 2.13 we resume all the dfferent features we have observed prevously n a phase dagram n the (F A, µ 0 ) plan Dynamcs of the escapng partcles Here we descrbe the dynamcs of the escapng partcles. We remnd that these partcles are those wth an energy ε [ 1, 1], and that the evoluton equaton leads to a drft of the momentum accordng to q ± (t) = q 0 ± Ft. The rght movers movng to the rght, they wll leave drectly the trap, whereas the left movers wll be frst reflected on the left tlde band edge before escapng. When a partcle reaches the poston x = 0, all the potental energy has been converted nto knetc one, and the dynamcs becomes free, lke n the case treated n secton If we call t ± the tme

70 2.4. Dynamcs after the sudden quench 45 j(a/2,t) j(x=-185,t) 0,4 0,2 0-0,2-0, t ,4 0,2 0-0,2 j(x,t) j(x,t) -0, t 0,4 0,3 0,2 0,1 0 0,2 0,1 0-0,1-0,2 F=0.015, t=t(n+0.25) F=0.1, t=t(n+0.25) x F=0.04, t=t(n+0.25) F=0.068,t=T(n+0.25) F=0.1, t=t(n+0.25) x Fgure 2.12 (Top) Current n a ntal superflud state as a functon of the tme (left) n the mddle of the condensate n two dfferent cases (µ 0 = 0 and F = 0.1 n black dots and µ 0 = 2/2 and F = n red ones) and as a functon of the poston (rght) at quarter of the perod T for two dfferent forces correspondng to the dfferent oscllatons regme. (Bottom) Current n a ntal mxture Mott/superflud as a functon of the tme (left) at poston x = 185 (correspondng to the mddle of the left superflud phase) for two forces (F = 0.04 n black dots and F = 0.1 n red ones) and as a functon of the poston (rght) at quarter of the perod T for three dfferent forces correspondng to the dfferent oscllatons regmes. The dots are the numercal data obtaned by exact dagonalzaton, and the full lnes are the predctons gven by the ntegral (2.72). needed for a partcle to reach the poston x ± = 0, we fnd that t ± s equal, thanks to equaton (2.55), to t ± = 1 F ( q 0+ arccos( ǫ)). (2.74) The potental beng zero n the regon x > 0, we then fnd x ± = 1 ε 2 (t t ± ), (2.75) that can be rewrtten as a functon of x 0 and q 0 lke ( x ± (x 0, q 0, t) = 1 (Fx 0 + cos q 0 ) 2 t 1 ) F ( q 0+ arccos(fx 0 + cos q 0 )). (2.76) The densty s then gven by ρ esc (x, t) = ρ + esc(x, t)+ρ esc(x, t), where the rght and left contrbutons are gven by, usng energy varable 1 ρ ± dε dq 0 esc(x, t) = F 2π δ(x x± (ε, q 0, t)), (2.77) 1 Q(ε)

71 46 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce Fgure 2.13 Phase dagram showng the dfferent behavors of the trapped part of the condensate n the (F A, µ 0 ) plan. In the green regon, all the partcles are leavng the trap because the most energetc pont has an energy smaller than ε = 1. In the red regon, the condensate oscllates n phase, the most energetc pont s the left-most locus x max = A. In the blue regon, the poston of the most energetc pont s force dependent, and the splttng dynamcs takes place. Fnally, n the yellow regon, concernng only the Mott/superflud case, the condensate oscllates wth a breathng regme where the left and rght sdes of the plateau start to move n opposte drecton. and the ntegraton doman Q(ε) s represented n dashed blue n fgure 2.6. We show n fgure 2.14 the comparson of the escapng densty at dfferent tmes between numercal exact dagonalzaton and hydrodynamcal predctons gven by equaton (2.77). We see a good agreement between the hydrodynamcal predctons and numercal results, up to nterferences whch dsappear n the contnuum lmt. It s nterestng to note that the rght propagatng front of the escapng densty profle has a star lke structure (the phenomenon s well observable for tme t = 156 n fgure 2.14). The ntegraton of the densty over each star beng equal to 1, they represent an ejected partcle movng ballstcally wth a velocty v(ε) = 1 ε 2. Note that ths star lke structure of propagatng front has already been observed n dfferent contexts [HRS04, PK05] Entanglement dynamcs As we saw prevously, the quench wll have for consequence the splttng of the ntal condensate nto an escapng part and a trapped one. It results from the correlatons presents n the ntal state an entanglement between these two parts. Ths entanglement between escapng partcles and trapped ones s quantfed by the entanglement entropy ntroduced n chapter 1, whch s completely determned by the egenvalues of the reduced correlaton matrx (see equaton (1.100)). We present the

72 2.5. Concluson 47 0,4 0,3 t=74 t=110 t=156 ρ esc 0,2 0, x Fgure 2.14 Densty profle of the escapng partcles for an ntal Mott/superflud state. The dots correspond to the numercal data, whereas the full lne are the results of the ntegral (2.77). tme evoluton of the entanglement profle n fgure 2.15 for the two dfferent ntal states wth A = 50. In both cases, we clearly see the entanglement growth between the trappng regon and the propagatve band by the ballstc movement of the escapng partcles. Insde the trapped regon, one has to dstngush between the two ntal states. In the case of a pure superflud state, the entanglement s non zero n the complete zone and t exhbts the same knd of oscllaton already observed n the dynamcs of the densty profle. In the Mott/superflud ntal state, the entanglement vanshes n the plateau regon. Indeed, n ths regon, the whole state of the system can be wrtten as a drect product of sngle-occupancy local states and s, as a consequence, separable. Interestngly, the entanglement present ntally wthn the self trapped condensate s conserved n tme, ndcatng that no entanglement s created or lost n ths regon. Indeed, we can see n fgure 2.16, where the entanglement entropy s plotted at nteger multple of the perod and of the half perod, the perfect superposton of the profles, ndcatng that the entanglement shows a trval tme evoluton smply related to the Bloch oscllatons of the trapped partcles. 2.5 Concluson In ths chapter, we have studed the behavor of hard core bosons after the sudden quench of the trappng potental. The use of the hydrodynamcal descrpton, based on a local equlbrum hypothess, gave us access to observable lke densty or current, and perfectly explan the man features of the dynamcs. Two dfferent knds of ntal states, dependng on the value of µ 0 have been consdered : an ntal state completely n a superflud phase, and a ntal state showng a mxture between superflud and Mott phases. We frst studed the case of a quench consstng n the release of the trappng potental. We observed a spreadng of the condensate to the left and the rght, and we derved the number of partcles wthn the ntal trappng regon n the case µ 0.

73 48 Chapter 2. Self-trappng of Hard Core bosons on optcal lattce Fgure 2.15 Tme evoluton of the entanglement entropy profle obtaned numercally by usng (1.100) for an ntal superflud state wth F = 0.3 and µ 0 = 0 (top) and an ntal Mott/superflud state (bottom) wth F = 0.5 and µ 0 = 2. In both cases, we set the spacal extenson of the condensate to A = S 0,5 S 0, x x Fgure 2.16 Entanglement entropy at nteger multples of the Bloch perod and of the half perod for an ntal superflud state wth µ 0 = 0 and F = 0.1 (left) and for an ntal mxture Mott/superflud wth µ 0 = 2 and F = 0.5. In both case, the tmes multple of the perod are plotted wth symbols, and those multple of the half perod wth full lnes, and A s set to 50. Secondly, we consdered the case of the release of the ntal potental and the loadng of a lnear ramp n the negatve sde of the real axe. As already observed n the case of an ntal homogeneous dstrbuton of partcles, the condensate splts nto a escapng

74 2.5. Concluson 49 part leavng the regon [A : 0] and a part whch stays trapped due to the untary evoluton. The trapped partcles perform Bloch oscllatons due to the constant force actng on them, We dentfed dfferent scenaro for these oscllatons, dependng of the ntal state and the appled force. Fnally, we have analyzed the entanglement entropy between the escapng part and the trapped one.

75

76 Entanglement creaton between two spns embedded n an Isng chan 3 Snce the development of quantum nformaton processng [NC00], quantum entanglement s not only used for a better understandng of the foundaton of quantum mechancs, but t becomes also the key ngredent of concrete applcatons, lke quantum communcaton [GT07], quantum cryptography [GRTZ02, BB84] or teleportaton [BBC + 93]. Creatng and manpulatng entangled quantum states s then a major ssue of modern physcs. Snce entanglement s related to strong correlatons, the smplest way to entangle two parts of a quantum system s the drect nteracton between them. However, from a quantum nformaton pont of vew, t s better to create entanglement between parts located at fnte dstance. Whereas t s commonly beleved that nteracton of a quantum system wth an envronment leads to a death of quantum features lke entanglement, t has recently been shown that the couplng to a bath can endorse the creaton of entanglement between two dstant systems nteractng wth t [Bra02, CVDEBR06, CVGIZ07, CVDEBR07, PR08, PR09, WDCK + 11, KWLM12, FKT + 13, TFK + 14]. The envronment can be modeled by dfferent physcal systems, such as an assembly of harmonc oscllators, spn or on chans. For nstance, n [Bra02], D. Braun studed the creaton of entanglement between two spns coupled at the same pont to a chan of harmonc oscllators. He dentfed a decoherence free subspace whch protects a part of the ntal coherence and created entanglement. In [WDCK + 11, KWLM12], the authors nvestgated the entanglement between two harmonc oscllators coupled to an harmonc crystal n the thermodynamc lmt. Dependng on the parameters, dfferent scenaros have been found for the steady state entanglement: sudden death, no sudden death or sudden death and revvals. In ths chapter, we propose to extend the model studed n [Bra02] by consderng the case of two non nteractng defect spns 1/2 coupled at two dfferent locatons n a quantum Isng chan. The chapter s organzed as follows: n a frst secton, we gve the model and the theoretcal treatment of the problem. We show how the spn bath can be mapped, n the hgh magnetzaton and low temperature lmt, nto a collecton of nteractng harmonc oscllators. Then we ntroduce a new set of coordnates leadng to a natural decouplng of the systems, and we derve the tme evoluton of the reduced densty matrx of the two defect spns, allowng us to have access to 51

77 52 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan ther entanglement evoluton. In the second secton, we present the results concernng the entanglement dynamcs, for both vanshng and non vanshng dstance between the defects. The behavor of the concurrence as a functon of the parameters of the systems s n partcular analyzed. 3.1 Model and theoretcal treatment Hamltonans and ntal states We consder n ths study a chan of 2N spns wth Isng nteractons and transverse magnetc feld descrbed by the Hamltonan H b = J 2 <j> σ x σ x j 1 2 σ z, (3.1) where the ferromagnetc couplng (J > 0) runs over the frst neghbors only. The prme n the sum symbol ndcates that the label = 0 has been removed for further convenence. Moreover, we assume perodc boundary condtons σn+1 α = σα N, wth α = x, y, z. The two spns l and l of ths chan are locally coupled to "defect" spns, labeled A and B, thought the Hamltonan H = γ ( σa x σx l + ) σx B σx l, (3.2) wth couplng strength γ. Moreover, the two defect spns are subject to a Zeeman term H d = h(σ z A + σz B). (3.3) The whole system beng close, the dynamcs s untary and s set by the total Hamltonan H = H b + H d + H. The spn chan wll act as an envronment for the two defect spns by medatng an effectve nteracton between them. In the followng, the chan s prepared nto a thermal state at temperature T. Its densty matrx s the Gbbs state ρ b = 1 Z e H b/t, (3.4) where Z Tr{e H b/t } s the normalzaton constant. The two defect spns are ntally supposed to be n a pure separable state Ψ AB = ψ A ψ B, (3.5) where ψ A,B = α A,B A,B + β A,B A,B wth the usual normalzaton α A,B 2 + β A,B 2 = 1 and A,B and A,B the egenstates of the operator σa,b z wth egenvalues 1 and 1 respectvely.

78 3.1. Model and theoretcal treatment Hgh magnetzaton lmt As llustrated n [CVDEBR06, CVGIZ07, GI10, SG09], bath-medated entanglement between the boundary spns of a fnte spn chan s found n the ground state of hghly symmetrc Hamltonans where the reservor presents long-range correlatons. A major lmtaton faced n these systems s the vanshng energy gap between the ground state and the excted ones [GI10] as the chan sze s ncreased. End-toend entanglement s therefore very senstve to attenuaton of long-range correlatons due to thermal fluctuatons. In the case we consder here, decreasng the correlaton length n the chan wll decrease the correlatons between spns l and l. Ths wll naturally lead to a drastc attenuaton of the entanglement between the defects. In order to avod ths decrease of correlatons, we opt for the low temperature and the paramagnetc regme J 1 where the spn of the envronment are hghly polarzed along the drecton of the transverse feld. Ths lmt allows us to perform a bosonzaton of the spns of the envronment by means of the Holsten-Prmakoff transformaton [HP40]. The dea of the transformaton s to ntroduce a mappng between the spn operators and bosonc operators a n and a n : σ n + = 1 a na n a n, (3.6) 1 a na n, (3.7) σ n =a n σ z n =1 2a na n. (3.8) Here, σ + n and σ n are the spn ladder operators, and a n and a n are creaton and annhlaton bosonc operator respectvely, satsfyng the algebra [a n, a m] = δ nm. Note that the average n the operator a na n countng the number of bosons n the mode n represents the devaton of the projecton of the spn polarzaton wth respect to ts maxmal value. From our prevous requrement of a hgh polarzaton of the spns and a low temperature, t follows then that the average a na n s small. The expressons (3.6)-(3.8) can n consequence be truncated up the to zeroth order, leadng to σ n a n, (3.9) σ + n a n. (3.10) Ther s then a drect correspondence between lowerng (resp. rsng) spn operators and creaton (resp. destructon) bosonc operators. Pluggng ths transformaton nto the total Hamltonan, we get a system of 2N nteractng bosons H = J 2 n ( ) a n a n+1 + a na n+1 + a n a n+1 + a na n+1 (( ) ( ) ) γ a l + a l σa x + a l + a l σb x (σa z + σz B), + a n a n (3.11) n where we have dropped an rrelevant constant. For further convenence, we ntroduce

79 54 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan the poston and momentum operators x n and p n defned by a n = 1 2 (x n + p n ), (3.12) a n = 1 2 (x n p n ), (3.13) and satsfyng the canoncal algebra[x n, p m ] = δ nm. Usng these operators, the Hamltonan s recast nto H = J xn x n n 2 (x 2 n + p 2 n) 2γ(x l σa x + x lσb) h(σ x A z + σz B). (3.14) n One can see from the prevous expresson that the bath has been mapped nto a set of lnearly coupled harmonc oscllators. The two defects spns are now locally coupled to the poston varables of the l and l oscllators. One can note that ths system s close to the model studed n [KWLM12] where, nstead of spns, the two defects are also of harmonc oscllator nature Characterzaton of the bath In ths secton, we focus our attenton to the descrpton of the bath Hamltonan H b. To determne the tme evoluton of the spn defects, we frst need to wrte the bath Hamltonan s terms of ts normal modes. Let us frst ntroduce the poston and momentum vectors x = (x N,..., x 1, x 1,..., x N ), p = (p N,..., p 1, p 1,..., p N ), (3.15) such that the bath Hamltonan s wrtten nto a matrx form H b = J x n x n n 2 (xn 2 + p 2 n) n = 1 2 p p+ 1 2 x V b x = 1 2 p x V b x, (3.16) where we have ntroduced the potental matrx 1 J J. J V b = J J J 1 R 2N 2N. (3.17) Ths potental matrx can be rewrtten nto a block matrx form ( ) A B V b =, (3.18) B A

80 3.1. Model and theoretcal treatment 55 where the A and B matrces assume the expresson 1 J. A = J J J 1, B = J J R N N. (3.19) The bath Hamltonan H b s nvarant under the exchange of the two bosons n and n. Wth ths n mnd, we descrbe the bath usng symmetrc (center-of-mass) and antsymmetrc (relatve) varables by the ntroducton of x S,A n = x n ± x n, pn S,A = p n ± p n, (3.20) 2 2 where the upperscrpt S (A) refers to center-of-mass (relatve) coordnates. Ths transformaton can be wrtten nto a vectoral form ξ = Rx, π = Rp, (3.21) where we have ntroduced the vectors ( ξ = x S, x A ) ( ) = x1 S,..., xs N, xa 1,..., xa N, (3.22) ( π = p S, p A ) ( ) = p1 S,..., ps N, pa 1,..., pa N, (3.23) together wth the orthogonal (R 1 = R T ) transformaton matrx R R = 1 ( ) 1 1, wth 1 = R N N. (3.24) Pluggng the relaton R T R = 1 on both sdes of the potental matrx, the nteracton part of equaton (3.16) becomes 1 2 x V b x = 1 2 ξ Λ b ξ (3.25) where Λ b = RV b R T s the transformed potental matrx. Its explct expresson can be determned usng the block representaton (3.18) Λ b = RV B R T = 1 ( )( ) 1 1 A B )( B A 1 1 = 1 ( ) 1A 1+B 1+ 1B+ A 1A 1 B 1+ 1B+ A. (3.26) 2 1A 1+B 1 1B+ A 1A 1 B 1 1B+ A

81 56 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan One can easly show that 1A 1 = A and B 1 = 1B, leadng to wth the two matrces Λ b = ( ) V S b 0 0 Vb A, (3.27) V S b = A+K, V A b = A K, (3.28) and K j = ( 1B) j = J(δ 1 δ j1 + δ N δ jn ). The potental matrx n the new representaton s block dagonal. The ntroducton of symmetrc and antsymmetrc coordnates leads then to a natural splttng of the ntal bath nto two ndependent baths composed of N partcles and such that the Hamltonan s H b = Hb S + HA b wth H S(A) b = 1 2 (ps(a) ) xs(a) V S(A) b x S(A). (3.29) For latter convenence, we wrte the bath Hamltonan n term of ts normal modes. Introducng the normal center-of-mass and relatves coordnates x S(A) and p S(A) defned by x S(A) = (O S(A) ) x S(A), p S(A) = (O S(A) ) p S(A), (3.30) where O S and O A dagonalze the potental matrces Vb S that and V A b respectvely such D S(A) = (O S(A) ) V S(A) b O S(A). (3.31) Wth theses coordnates, the two bath Hamltonans are gven by a set of ndependent oscllators H S(A) b = 1 2 ( ps(a) ) ( xs(a) ) S(A) D S(A) x ( 1 = n 2 ( ps(a) n ) ) 2 ( ωs(a) n ) 2 ( x S(A) n ) 2, (3.32) where the normalfrequences are gven by ( nπ ) ( ω n) S 2 =1 2J cos, n = 0,, N 1 (3.33) ( N nπ ) ( ω n) A 2 =1 2J cos, n = 1,, N. (3.34) N assocated wth the egenvectors Q S(A) n (k) = [O S(A) ] kn Q S 1 n(k) = for n = 0 N 2 ( nπ ) Q S n(k) = N cos 2N (2k 1) for n = 1... N 1 Q A n(k) = 2 N sn ( nπ 2N (2k 1) ) Q A n(k) = ( 1)k N for n = N. for n = 1... N 1 (3.35) (3.36)

82 3.1. Model and theoretcal treatment Full Hamltonan n normal coordnates In the same sprt than the ntroducton of symmetrc and antsymmetrc coordnates for the bath varables, we also ntroduce a base composed by non local symmetrc and antsymmetrc states for the defect spns: φ S(A) +( ) =, (3.37) 2 ψ S(A) +( ) =. (3.38) 2 Note that these four states are the four maxmally entangled Bell states. Defnng the two operators S S x =( ψ S φ S + φ S ψ S ), (3.39) S A x =( ψ A φ A + φ A ψ A ), (3.40) actng as spn flp operators n the symmetrc and antsymmetrc Hlbert space respectvely, the nteracton Hamltonan H takes the form ( H = 2γ S S xxl S + Sx A xl A ). (3.41) As for the bath, the nteracton Hamltonan presents a clear decouplng, the operator n the symmetrc (resp. antsymmetrc) Hlbert space beng coupled only to symmetrc (resp antsymmetrc) bath operators. By the ntroducton of the couplng vector ( γ T = 0,, 2 ) 2γ,, 0, (3.42) where the only non zero entry occupes the poston l, the couplng Hamltonan s recast n a more compact form H = (S S xγ T x S + S A x γ T x A ). (3.43) Introducng the matrx O S(A), the Hamltonan s wrtten n terms of the normal modes of the baths by H = H S + H A where H S(A) = S S(A) x ( γ S(A) ) T x S(A) (3.44) wth the new couplng vectors to the normal modes γ S(A) = (O S(A) ) T γ explctly gven by γ S = 2γ N 1 ( π ) 2 cos 2N (2l 1) (. (N 1)π 2 cos (2l 1) 2N ), γ A = 2γ N ( π ) 2 sn 2N (2l 1). ( ) (N 1)π. 2 sn (2l 1) 2N ( 1) N+1 (3.45)

83 58 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan Note that 2l 1 appearng n the expresson of the couplng vectors s the dstance d between the two defect spns. Ths stuaton s the startng pont of the model studed n [Bra02] where the two defect spns are coupled to the same locaton n the bath. The last step s to wrte down the local Zeeman term. It reads n the symmetrc and antsymmetrc base ( ) H d = 2h φ A φ S + φ S φ A = 2hS z, (3.46) where we have defned the operator S z = φ A φ S + φ S φ A 1. Now that all our Hamltonans are wrtten n the proper bases, we can wrte the total Hamltonan as H = H S + H A + H d wth H S = 1 2 ( ps ) ( xs ) T D S x S S S x( γ S ) T x S, (3.47) H A = 1 2 ( pa ) ( xa ) T D A x A S A x ( γ A ) T x A. (3.48) Because they are actng n two dfferent Hlbert spaces, the two Hamltonans H S and H A represents two decoupled dynamcs. On the contrary, the Zeeman term couples the two subspaces and breaks the ndependent dynamcs snce [ H d, H S,A ] = Tme evoluton of the spn defects The fact that H d and H S,A do not commute renders the calculaton of the tme evoluton of the spn defects dffcult. However, at short tme scales, the dynamcs s easly derved. Indeed, for tmes smaller than the typcal tme scale of the defects gven by the nverse of the energy gap h/2, the contrbuton of the local Hamltonan H d can be gnored n the total Hamltonan and a decoupled soluton of the form Ũ(t) = Ũ S (t)ũ A (t) (3.49) can be obtaned n the lowest order n h. In the followng, we set the transverse magnetc feld h of the defects to zero, leadng to perfectly degenerate spns, allowng us to assess the dynamcs for an arbtrary long tme. Moreover, n all our numercal smulatons of the dynamcs, the number of harmonc oscllators n the bath wll be fxed to N = 10 4, suffcently bg to avod fnte sze effects. Fnally, the temperature wll be set to a low value to fulfll the hgh polarzaton lmt, and we choose T = We remnd here that the two defect spns are ntally prepared nto a separable state Ψ AB = ψ A ψ B and the bath nto a thermal state ρ b = exp( H b /T) /Z. The defects and the bath are moreover supposed to be ntally uncorrelated such that ρ tot (0) = ρ b (0) ρ d (0). (3.50) 1. Note that f nstead of the symmetrc and antsymmetrc base we would have used the snglet trplet base j, m j, the Zeeman Hamltonan would have been H d = 2h( 1, 1 1, 1 1, 1 1, 1 ) and then ntroduces an energy gap between the two states 1, 1 and 1, 1 of the trplet sector.

84 3.2. Tme evoluton of the spn defects 59 The tme evoluton of the elements of the reduced densty matrx are wrtten n the 4-dmensonal common egenbase { s } of the couplng operators S S(A) x defned by s = s S(A) s wth egenvalues s S(A) S S(A) x reduced densty matrx = 0,±1. In ths base, the elements the ρ d (t) = Tr b {Ũ(t)ρ d (0)Ũ(t) } (3.51) evolve through (see appendx A for detals of the calculatons) s ρ d (t) s j =exp { [ f S (t)(s S ss j )2 + f A (t)(s A sj A)2] + [ ϕ S (t)((s S )2 (s S j )2 )+ϕ A (t)(s A ) 2 (s A j )2 ) ]} s ρ d (0) s j, (3.52) where the four tme dependent functons are gven by f S(A) (t) = ϕ S(A) (t) = ( ) 2(2ñ γ S(A) S(A) + 1) ( ) 2 ω S(A) 3 (1 cos ( ( 2 ) γ S(A) 2 ) 2 ω S(A) ( t sn ω S(A) t ω S(A) ( ) )) ω S(A) t, (3.53). (3.54) ( ( exp ω S(A) ) 1) 1 s the In the expresson of the f S(A) (t) functons, n S(A) = /T thermal occupaton of the bosonc mode of the symmetrc (antsymmetrc) bath. Note that the dynamcs s then completely set by the bath through these functons encodng all the propertes of the envronment lke frequency or temperature. The elements of ρ d have an oscllatory term dependng on the ϕ S(A) functon together wth an exponental decay set by f S(A) (t). One can remark that the dagonal elements (s S(A) = s S(A) j ) do not evolve n tme. In the base { s }, the populaton elements are then constant n tme and only the coherence elements are affected by the couplng to the bath. The spn chans consttutes then n ths base a purely dephasng envronment. The tme evoluton of the four functons f S(A)(t) and ϕ S(A) (t) s shown n fgure 3.1 for two dfferent dstances. One can remark that the lnear part of both functons ϕ S and ϕ A are the same n the case l = 5, as well as the constant part of the f S and f A functons. We plot the slope of the ϕ S(A) and the constant part of the f S(A) functons as a functon of the parameter l n fgure 3.2. As we already guessed, the lnear part of the ϕ and the constant part of the f functons become equal for the symmetrc and antsymmetrc baths when l 5. Moreover, they become ndependent of the value of l for large dstances. Before turnng to the entanglement dynamcs of the defects, let us frst study ther dynamcs for short tmes. The dynamcs of one ndvdual spn s obtaned by tracng the densty matrx ρ d (t) over the second one, for example ρ A (t) = Tr B {ρ d (t)}. If the two defects are, for nstance, prepared n the state, one can show that the

85 60 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan 0,04 0,03 4 l=1 l=1 3 f S(A) 0,02 ϕ S(A) 2 0,01 1 f S(A) ,03 t 4 t l=5 l=5 3 0,02 0,01 ϕ S(A) t t Fgure 3.1 Tme evoluton of the four functons f S(A) (left panels) and ϕ S(A) (rght panels) for l = 1 (up panels) and l = 5 (down panels). The black curves correspond to the symmetrc bath whereas the red curves correspond to the antsymmetrc bath. The others parameters are γ = 0.05, J = 0.4, T = 10 5 and N = ,025 0,02 lnear part of ϕ S lnear part of ϕ A constant part of f S constant part of f A 0,015 0,01 0, l Fgure 3.2 Lnear part of the φ S,A functons and constant part of the f S,A functons a functon of the parameter l. The couplngs are fxed to γ = 0.05, J = 0.4 and the temperature s T = 10 5.

86 3.3. Entanglement dynamcs 61 populaton elements evolve as ( ( )) ρ A (t) =1 1+e f S (t) f A (t) cos ϕ S (t) ϕ A (t), 2 (3.55) ρ A (t) =1 ρa (t). (3.56) The exponental appearng n the prevous expresson depends on the sum f S (t)+ f A (t), whch n the case of the couplng vector gven by equaton (3.45), s proportonal to cos 2 ((2l 1) k 2 )+sn2 ((2l 1) k 2 ) and s as a consequence ndependent of the value of l. For short tmes and for suffcently low temperature such that ñ S,A = 0, on can expend the functons f S(A) (t) and ϕ S(A) (t) around zero, and we fnd, takng the thermodynamc lmt N ( ) f S (t)+ f A (t) 2γ2 K 4J 2J+1 t 2, (3.57) π 2J+ 1 ϕ S (t) =ϕ A (t) γ2 3 t3, (3.58) where K(x) s the complete ellptc ntegral of the frst knd. The sum f S (t)+ f A (t) s, as expected, ndependent of the value of l, but we also fnd that the ϕ S(A) functons are dstance ndependent as well. For short tmes, the defect s dynamcs does not dependent on the dstance between them. The evoluton of one spn does not nfluence the one of the second spn, and the dynamcs comes only from the couplng wth the bath through the decoherence process. Ths ndependent dynamcs holds untl the frst exctaton emtted n poston l (resp. l) reaches the poston l (resp. l), that s untl a tme tnd l gven by t l nd = 2l 1 v g, (3.59) where v g = J s the group velocty along the chan n the lmt J < 1. We compare n fgure 3.3 the evoluton of the populatons ρ of the spn A n the case of two spns coupled to the same bath, obtaned by tracng the reduced densty matrx ρ d (t), and the evoluton of the populaton ρ nd of a sngle spn coupled to the bath. As expected, the curves collapse untl a tme gven approxmatvely by tnd l, as shown n the nset, ndcatng that the spn evoluton s only the consequence of the couplng wth the bath. We also add the curve correspondng to the short tme dynamcs, obtaned wth expansons (3.57) and (3.58), whch reproduces correctly the dynamcs untl a tme t Entanglement dynamcs In ths secton, we analyze the entanglement dynamcs, measured by the concurrence, of the two defect spns as a functon of the parameters of the system. We frst start by the smplest case of two defects coupled to the same spn of the chan, and we turn to the general case of a non vanshng dstance d = 2l 1 afterward.

87 62 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan Fgure 3.3 Evoluton of the populaton ρ of the spn A for dfferent dstances (full lnes) compared to the case of an ndvdual spn (dots). In the nset s shown ρ ρ nd for several dstances. All the spns are ntally prepared n the state Spns coupled to the same pont Here, the defect spns are coupled to the same spn l of the chan. Note that ths s the case treated n [Bra02], where the author used a phenomenologcal bath. In the case of a vanshng dstance, the nteracton Hamltonan (3.41) s transformed nto H = S S x( γ S x S + γ A x A ). (3.60) Note that only the couplng operator n the symmetrc space S S x appears now n the nteracton Hamltonan. The two antsymmetrc states ψ A and φ A are no longer nvolved n the dynamcs. The couplng of the two defects n the same pont has then for effect the creaton of a two dmensonal decoherence free subspace [LW03], composed by the two antsymmetrc states, protected from the non untary dynamcs set by the couplng wth the bath. Usng the same method developed n appendx A, the elements of the reduced densty matrx of the defects n the egenbase { s } of the couplng operators are gven by s ρ d (t) s j =exp { [ ( f S (t)+ f A (t))(s S ss j )2] + [ (ϕ S (t)+ϕ A (t))((s S)2 (s S j )2 ) ]} s ρ d (0) s j. (3.61) We show on fgure 3.4 the tme evoluton of the concurrence between the two defects coupled to the same pont for several values of γ (left plot) and J (rght plot). One observes that the concurrence starts to grow at t = 0, ndcatng an nstantaneous creaton of entanglement between the two defects. The concurrence has an oscllatory

88 3.3. Entanglement dynamcs 63 Fgure 3.4 Entanglement dynamcs for the two defect spns coupled to the same pont. Parameters are l = 10, and (left) J = 0.4, γ = 0.04 (red), γ = 0.02 (green), γ = 0.01 (blue) and (rght) γ = 0.04 (red), J = 0.32 (red), J = 0.16 (green), J = 0.08 (blue). In both plots the ntal states s ϕ A ϕ B = A B. behavor whch hghly depends on the system s parameters. Ths dependence of the oscllaton perod wll be analyzed n the next secton when the defect are separated by a non vanshng dstance d. We can fnally note that the entanglement dynamcs s ndependent on the couplng poston l snce the boundary condtons are perodc whch s reflected by the fact that the matrx elements of ρ d (t) depend only on the sum ( f S + f A ) and (ϕ S + ϕ A ), whch, as already seen n the prevous secton, are ndependent on the value of l Spns coupled at two dfferent ponts Now we ncrease the dstance between the defects, settng the dstance d = 2l 1 between the defects. The decoherence free subspace created by the couplng to the same pont n the chan dsappears n ths case, and the two states φ A and ψ A are now partcpatng n the dynamcs. The reduced densty matrx evolves n ths case followng the evoluton equaton (3.52). We present n fgure 3.5 the concurrence dynamcs of the two defects when the parameters l s vared for couplng constant J = 0.4 and γ = As n the case of a vanshng dstance, we observe oscllatons n the tme evoluton of the concurrence, wth maxma close to unty. One of the man dfference wth the prevous case s that now, entanglement does not start to grow a t = 0, but the defects need an establshment tme t ent to become entangled due to the fnte dstance between them (see rght plot of fgure 3.5). Ths tme grows exponentally wth the dstance t ent exp(c 1 l), as shown n fgure 3.6. Ths exponental ncrease ndcates that the entanglement s not medated by the fastest exctaton travellng from one defect to the other. As the sound velocty does not change wth the dstance, ths would rather lead to a lnear growth. As n the case treated prevously, the perod P of the oscllatons of the concurrence depends on the parameters of the system, as we can see n fgure 3.7. The decrease of the couplng strength γ leads to a reducton of the nformaton transfer between the defects medated by the chan, leadng to an ncrease of the

89 64 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan Fgure 3.5 (left) Tme evoluton of the concurrence for J = 0.4, γ = 0.04 and l = 1 (red), l = 2 (green), l = 3 (blue), l = 4 (magenta), l = 5 (lght blue) and l = 6 (orange). (rght) Zoom at the begnnng of the growth of C(t). Intal state s ϕ A ϕ B = A B t nt l Fgure 3.6 Establshment tme t ent of the entanglement as a functon of the parameter l. The other parameters are γ = 0.04, and J = 0.4. perod of the oscllaton of C(t). We fnd numercally a power law scalng followng P(γ) γ 2. (3.62) On the other hand, the ncrease of the defect-chan couplng buld more correlatons (and then more entanglement) between the defects and the chan. Due to the monogamy property of the entanglement [CKW00], ths has for effect to lnearly decay the maxmum of the concurrence C max (γ), as we can see on the down left panel of fgure 3.7. Ths last feature does not occur when the ntra-chan couplng J s vared snce t s only responsble of the transport of the nformaton n the chan, and not on the of the buld-up of correlatons between the defects and the chan. The J parameters nfluences then only the perod P by makng the transport of the nformaton easer. We found an exponental decay of ths perod when the couplng

90 3.3. Entanglement dynamcs 65 Fgure 3.7 Varaton of the oscllaton perod P of the concurrence as a functon of γ (up left), J (up rght), l (down left), and varatons of the maxmum of the concurrence C max as a functon of the couplng γ. J s ncreased. As for the tme needed for the growth of entanglement, we fnd that the perod P of the oscllatons ncreases exponentally wth the dstance P(l) exp(c 2 l). Interestngly, for fxed couplng parameters, we fnd that the numercally ftted values of c 1 and c 1 are very close (1.36 and 1.38 respectvely wth the parameters of the fgures 3.6 and 3.7). Ths would ndcate a close relaton between these two quanttes. The ntal state of the defects s nfluencng the oscllatory behavor of the concurrence too. Changng ths state has an effect n the maxmum reached by the concurrence, but not on the perod of the oscllatons. We parameterze then the ntal state of the defects as Ψ = ψ A ψ B wth ψ = cos α + sn α, = A, B, (3.63) and we show the maxmum of the concurrence as a functon of tan α A and tan α B n the left panel of fgure 3.8 for J = 0.2 and γ = One can see that entanglement s found to be non zero for every ntal state, except when one of the defects s prepared n the equal superposton ψ = 1 2 ( + ), = A, B. (3.64) Ths state beng an egenstate of the couplng Hamltonan, t wll not evolve and

91 66 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan Fgure 3.8 (left) Maxmum of the concurrence as a functon of tan α A and tan α B for J = 0.2 and γ = (rght) Maxmum of the concurrence as a functon of the state of the spn B when the state of the spn A s fxed to ψ A = wth the same parameters as the left plot. s then unable to develop correlatons wth the other defect. On the other hand, the concurrence reaches ts bggest value when the two spns A and B are n one of the egenstates of the σ z operator ψ = ± or ψ = ±. The maxmum of the concurrence s untouched by the exchange of the state of the two defects ψ A ψ B, as we can see from the symmetry axe of equaton tan α A = tan α B. The other axes correspond to the transformaton cos α + sn α cos α + sn α or cos α + sn α cos α sn α wth = A, B. 3.4 Spectral densty theory The theory developed above s a feature of the degeneracy of the two states 1, 1 and 1, 1 of the trplet sector, whch come from the choce to work wth a vanshng defect free Hamltonan (h = 0). Ths approxmaton s usually not vald, and the free evoluton of the defect spns has to be taken nto account n the total dynamcs. Recently, t has been shown [WDCK + 11, KWLM12, FKT + 13] that envronments may support decoherence free subspaces whch can be used to generate correlatons and entanglement between defects coupled to t. For the creaton of the mentoned decoherence free subspace, t s necessary to tune the defect s frequences n a value for whch the modes of the envronment nterfere destructvely, leadng to an effectve decouplng of the defects wth the bath. These specal frequences are found by analyzng the spectral densty of the bath [We99], whch measures how strong the defects are coupled to the dfferent modes of the envronment. Its expresson s gven by [We99] I S(A) ( ω) = π 2 n ( γ S(A) n ) 2 ω S(A) n ( δ ) ω ω S(A) n. (3.65) Usng the bath spectra (3.33) and (3.34) and takng the thermodynamc lmt N,

92 3.4. Spectral densty theory 67 the contnuous lmt of I S(A) s found to be [ ( 1 1 γ 2 cos 2 I S 2 (2l 1) arccos ( 1 ω 2 ))] 2J ( ω) = (, (3.66) 1 ( J 1 1 ω 2 )) 2 2J [ ( 1 1 γ 2 sn 2 I A 2 (2l 1) arccos ( 1 ω 2 ))] 2J ( ω) = (. (3.67) 1 ( J 1 1 ω 2 )) 2 2J The fgure 3.9 shows the spectral denstes for dfferent defect separatons l for the Fgure 3.9 Spectral densty assocated to the-center-of mass (black) and relatve coordnates (red) baths for several values of the dstance. The parameters are γ = 0.04, and J = 0.2. center-of-mass and relatve baths. The zeros n I S(A) ndcate the effectve decouplng to the symmetrc (antsymmetrc) bath and the presence of the decoherence-free subspaces. For example for the relatve bath, the zeros are found by vanshng the numerator of expresson (3.67), leadng to ω A 0 (p) = 1 2J cos ( ) 2pπ, (3.68) 2l 1 wth p = 0,..., l 1. We can note that the spectral densty has l zeros for a dstance d = 2l 1.

93 68 Chapter 3. Entanglement creaton between two spns embedded n an Isng chan As mentoned prevously, one has to tune the transton frequency of the defects to match one zero of the spectral denstes I S(A) ( ω) n order to recover the presence of a decoherence free subspace. For example, f the transton frequences 2h s engneered to match one ω A 0 (p), the two global states of the antsymmetrc subspace φa and ψ A wll be effectvely decoupled from the antsymmetrc bath, and then they wll form the decoherence free subspace. The defnton (3.65) s vald for frequences n the range ω [ 1 2J, 1+2J]. On the other hand, the bosonzaton of the bath made n secton mposes J 1 rendng ths nterval confned around 1. The theory used n the prevous secton s vald for tmes shorter than the nverse of the energy gap 1/2h, whch turns to be close to 1 f the magnetc feld s tuned to match a zero of the spectral densty. As a consequence, the tme evoluton of the reduced densty matrx (3.52) s not vald to determne the entanglement dynamcs for tmes t 1. Nevertheless, t can be used n order to determned the dervatve of the concurrence at t = 0 t C(t) t=0 n order to nvestgate the effect of the decoherence free subspace on the creaton of entanglement between the two defect spns. The effectve decouplng of the global states of the antsymmetrc subspace φ A and ψ A s equvalent to set the two correspondng functons f A (t) and ϕ A (t) to zero n the tme evoluton of the matrx elements of ρ d. We show n fgure 3.10 the dervatve at t = 0 of the concurrence for dfferent values of the two couplngs J and γ. Fgure 3.10 Dervatve of the concurrence at t = 0 as a functon of the dstance. On the left plot, we set J = 0.2, and γ = 0.08 (red crcles), 0.07 (green squares) and 0.06 (blue damonds). On the rght plot, we set γ = 0.08 and J = 0.16 (red crcles), 0.2 (green squares) and 0.4 (blue damonds). Intal state s ϕ A ϕ B = A B. We can see here a sgnfcant dfference wth the case of a vanshng magnetc feld on the defect spns. Indeed, and as a drect consequence of the creaton of the decoherence free subspace, the dervatve of the concurrence s non zero even for fnte dstances, ndcatng an nstantaneous creaton of entanglement. At a gven value of the parameters, the entanglement creaton seems to be not very senstve to the dstance between the defects, the dervatve of the concurrence reachng an almost constant value for l > 2. Once agan, because we are lookng at the dynamcs over a small tme scale, the two defect spns evolve ndependently for suffcently large dstance, leadng the an ndependent value of the dervatve of the concurrence

94 3.5. Concluson 69 wth respect to l. We can fnally note that, as we already observed n the prevous secton, an ncrease of γ or a decrease of J leads to a speed up of the establshment of the entanglement between the defects. 3.5 Concluson The entanglement dynamcs between two defect spns locally coupled to an Isng chan has been studed n ths chapter. After an Holsten-Prmakoff transformaton, the spn chan has been mapped nto an assembly of nteractng harmonc oscllators, and the ntroducton of new coordnates leads to a natural decouplng of the full dynamcs. For a certan tme regme, the dynamcs of the two defect spns has been derved, and the entanglement evoluton deduced from t. We analyzed the dependence of the concurrence on the parameters lke the couplng constants, the dstances between the defects, or ther ntal states. Fnally, we used the spectral densty and ts zeros to create artfcally a decoherence free subspace, whch has for effect the nstantaneous creaton of entanglement between the defects, even at fnte dstances.

95

96 Dsentanglement of Bell state by nteracton wth a non equlbrum envronment 4 0ne of the man dffcultes faced n the development of quantum nformaton processng [NC00] comes from the unavodable nteracton of a quantum system wth ts surroundng envronment. Ths phenomenon, the so-called decoherence process [Zur82, Zur02, Zur03, PZ01, Sch07], s detrmental for example for quantum computers, snce t s responsble of the loss of quantum features such as entanglement or coherence, necessary for ther achevement. It s n consequence of prmary mportance to understand the decoherence of a quantum system, and to try to lmt ts undesred effects. For example, dynamcal control consstng n pulses appled to the systems have been proposed to lmt the decoherence process [VL98, RFF + 08]. Wth ths n mnd, a bg number of studes have been realzed about dynamcs of open coherent quantum system nteractng wth an envronment. A good canddate for the descrpton of such an envronment s quantum spn chan, snce t descrbes many physcal stuatons [dsds03, KS06]. Models wth one or two spns coupled globally or locally to spn baths have then be extensvely studed n the lterature [CPZ05, QSL + 06, CFVP07, YZL07, RCG + 07b, RCG + 07a, CP08b, DQZ11, MSD12, SMD12, NDD12, FS13]. For example, a specal focus has been set to the effects of the crtcalty of the envronment on the decoherence dynamcs [QSL + 06], the latter beng enhanced close to the crtcal pont. Unversal effects have also been ponted out n [CFVP07, CP08a], where t s shown that the decay of quantum correlatons has an envelope ndependent of the strength of the system-envronment couplng. In Ref [CP08b], Cormck and Paz have studed the decoherence dynamcs of two ntally entangled spns coupled locally to a spn chan envronment. In ths work, and n most of the prevously cted ones, the envronment s ntally "at equlbrum", snce t s prepared n ts ground state. In ths chapter, we propose to look at what happens to the systems studed n [CP08b] when the envronment s set out of equlbrum by a sudden change of one of ts control parameters, realzng the so-called quantum quench [CC05, PSSV11]. The man objectve of ths work s to study the nfluence of the quench on the decoherence process wth respect to the equlbrum stuaton treated n [CP08b]. The chapter s organzed as follows: In a frst secton, we ntroduce the setup of our model, and we show that the decoherence dynamcs of the system can be 71

97 72 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment completely determned by the Loschmdt echo, for whch we gve an expresson n terms of the fermonc covarance matrx. Then, we turn to the descrpton of non equlbrum dsentanglement, startng wth the rough descrpton of the effect of the quench, wth a focus of the case of an ntal crtcal envronment. It wll be followed by the short tme dynamcs of the systems, and the comparson to the ndependent case. Fnally, the chapter s ended by summary and concluson. 4.1 Hamltonan and dynamcs In the study, we consder then the same knd of system already ntroduced n the prevous chapter, namely two non nteractng defect spns locally coupled to a spn chan envronment, see fgure 4.1. Fgure 4.1 Two defect spns labeled A and B are locally coupled to two locatons 0 and d of a spn chan envronment. The total Hamltonan governng the dynamcs s H tot = H E + H I where H E s the Hamltonan of the Isng chan playng the role of the envronment H E (h) = J N 1 j=0 σj x N 1 σx j+1 h σj z. (4.1) j=0 We choose here a postve nearest neghbor couplng J, and h s a transverse magnetc feld. We assume perodc boundary condtons σn = σ 0 wth = x, y, z. The nteracton between the defect spns and the envronment s modeled by the nteracton Hamltonan H I H I = ε ( A σ0 z + B σd) z, (4.2) where the state A,B s an egenstate of the σa,b z operator satsfyng σz A,B =, and ε s the postve couplng constant whch sets the strength of the nteracton. Note that we work wthout local Hamltonan for the defect spns. Indeed, due to the form of the couplng Hamltonan, a Zeeman term turns to be uneffectve on the dsentanglement process. The two spns formng the system are assumed to be ntally n the maxmally entangled Bell state φ AB = 1 2 ( + ) and decoupled from the envronment

98 4.1. Hamltonan and dynamcs 73 such that the total state of the whole system can be wrtten ψ(0) = φ AB G(h ) E, (4.3) where G(h ) E s the ground state of the spn chan envronment H E (h ) wth ntal magnetc feld h. At tme t = 0 +, n addton to the swtch-on of the nteracton, the transverse feld of the envronment s suddenly quenched from ts ntal value h to the fnal one h f, forcng the chan to evolve n a non equlbrum regme. In order to derve the dynamcs of the defects, lets rewrte the total Hamltonan lke [YZL07] 4 H tot = k k H k (h f ), (4.4) k=1 where k s one of the four states {,,, }. One can see here that H tot splts nto four dfferent channels governed by the effectve Hamltonans H (h f ) = H E (h f ) ε(σ0 z + σd z ) f the defects are n the state, (4.5) H (h f ) = H E (h f ) εσ0 z f the defects are n the state, (4.6) H (h f ) = H E (h f ) εσ z d f the defects are n the state, (4.7) H (h f ) = H E (h f ) f the defects are n the state. (4.8) The Hamltonan H (h f ) s then equal to the envronment Hamltonan, whereas the three remanng are the envronment Hamltonan wth a magnetc feld actng at poston 0 (H (h f )), d (H (h f )) or at both postons 0 and d (H (h f )) and takng the value h f + ε nstead of h f. Snce the total system s close and the Hamltonans tme ndependent, the ntal state ψ (0) evolves accordng to ψ(t) = U(t) ψ(0), (4.9) wth the evoluton operator U(t) = exp( H tot t). Usng the expresson (4.4) of the total Hamltonan, one can easly show that the evoluton operator takes the form wth U(t) = 4 k k U k (t), (4.10) k=1 U k (t) = e H k(h f )t. (4.11) At a tme t, gvng the ntal state of the defects, the global state becomes wth the evolved states ψ (t) = 1 2 [ ϕ (t) E + ϕ (t) E ], (4.12) ϕ k (t) E = e H k(h f )t G(h ) E, (4.13) where k =,. Snce the ntal state s a superposton of the two pure states and, only the two correspondng channels appear n the evolved total state ψ (t). The dynamcs of the defect spns s encoded n ther reduced densty matrx obtaned

99 74 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment by tracng out the envronmental degrees of freedom from the total densty matrx ρ s (t) = Tr E { ψ(t) ψ(t) }. We obtan ρ s (t) = 1 2 k,k G(h ) U k (t)u k(t) G(h ) k k, (4.14) or, wrtten n the computatonal base {,,, } D, (t) ρ s (t) = (4.15) where D, (t) D, (t) = ϕ (t) ϕ (t) = G(h ) e H (h f )t e H (h f )t G(h ) = D, (t) (4.16) s the decoherence factor. One can remark that the populatons do not evolve n tme, whereas the coherence elements are affected by a factor between 0 and 1. In ths base, the model descrbes a completely dephasng process. Ths decoherence factor, descrbng completely the dynamcs of the defect spns, can be related to the so-called Loschmdt echo [GJPW12] va L, (t) = D, (t) 2 = G(h ) e H (h f )t e H (h f )t G(h ) 2. (4.17) Note that f we set h f = h, meanng that we do not quench the transverse magnetc feld of the chan, the ntal state G(h ) s an egenstate of the Hamltonan H, and the echo reduces to L(t) = G(h ) e H (h )t G(h ) 2, whch s the case treated by Cormck and Paz n [CP08b]. The decoherence process comng from the nteracton wth the bath leads to a loss of the entanglement ntally present n the Bell state of the defects. In order to measure ths dsentanglement, we use the concurrence ntroduced n chapter 1. We remnd that t s defned through the egenvalues of the matrx R = ρ ρ wth ρ = (σ z σ z )ρ (σ z σ z ). In the case of the densty matrx (4.15), one has ρ = ρ, and then R = ρ 2 = D D D D 2, (4.18) leadng to the egenvalues ε 1 = 1 4 (1+ D )2, ε 2 = 1 4 (1 D )2, ε 3 = ε 4 = 0. Fnally, the concurrence s smply related to the Loschmdt echo va 1 C AB (t) = L(t) = D(t). (4.19) 1. Note that f we would have worked wth a Zeeman term for the defects H d = h d 2 (σ z A + σz B ), ths would have changed the densty matrx element ρ s to ρ s = e 2h dt D, = ρ s. But ths phase factor dsappears n the calculaton of the concurrence, and the relaton (4.19) stll holds n that case.

100 4.2. Loschmdt Echo n the fermonc representaton 75 The complete decoherence and dsentanglement dynamcs s then encoded n the Loschmdt echo, and our goal s to determned t. 4.2 Loschmdt Echo n the fermonc representaton In ths secton, we gve the explct expresson of the Loschmdt echo n terms of fermonc operators. For later convenence, we use the fermonc representaton for the descrpton and the dagonalzaton of the envronment. Indeed, as we mentoned prevously, the two channels Hamltonans are dentcal, except the two shfted magnetc felds n poston 0 and d for H, and they are then dagonalzed n the same way, that s the Jordan-Wgner mappng followed by a Bogolubov transformaton. After the Jordan-Wgner transformaton, the Hamltonan rewrtten n the relevant party sector s H = (c A jc j (c B jc j + h.c.)), (4.20),j where the Ferm operators satsfy the algebra {c, c j } = δ,j, {c, c j } = {c, c j } = 0, and the symmetrc A and antsymmetrc B matrces are gven by A j = 2h δ j J[δ,j 1 + δ,j+1 ] and B j = J[δ,j+1 δ,j 1 ] where the ndce N s dentfed wth ndces 0 n order to take nto account the perodc boundary condtons. One can ntroduce the feld operator Ψ = (C, C ) = (c 0,..., c N 1, c 0,..., c N 1 ), (4.21) to wrte the Hamltonan n a more compact form H = 1 2 Ψ HΨ, (4.22) where the sngle partcle Hamltonan s gven by ( ) A B H = B A The ntroducton of the untary matrx ( ) g h V = h g. (4.23) (4.24) dagonalzng the sngle partcle Hamltonan matrx H: Λ = V HV leads to the dagonalzaton of H through the normal modes η = V Ψ H = 1 2 η Λη. (4.25) These normal modes operators can be related to the orgnal Ferm operators va the Bogolubov coeffcents g j and h j η k = (g k c + h k c ), (4.26)

101 76 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment wth smlar expressons for the adjons η k. The nverson of these relatons gves the orgnal Ferm operators as a functon of the normal modes c = (g k η k + h k η k ). (4.27) k The Loschmdt echo (4.17) s nothng else but the square of the fdelty of the two evolved states Ths fdelty can be rewrtten ϕ (t) =e H (h f )t G(h ), ϕ (t) =e H (h f )t G(h ). ϕ (t) ϕ (t) 2 = ϕ (t) ρ ϕ ϕ (t) (4.28) wth ρ ϕ = ϕ (t) ϕ (t). Snce the two Hamltonans H and H are free fermonc, they are pure fermonc Gaussan states, and t has been shown n [KS10] that expresson (4.28) can be evaluated by Gaussan Grassmann ntegrals nvolvng the covarance matrces and t reads [KS10, Cor09] C k (t) = ϕ k (t) ΨΨ ϕ k (t), k =,, (4.29) L, (t) = det ( 1 C (t) C (t) ), (4.30) where 1 s the 2N 2N dentty matrx. We need then the tme evoluton of the covarances matrces (4.29) n order to determne the Loschmdt echo. To derve the tme dependence, t s better to swtch nto the Hesenberg representaton. In ths representaton, the fermonc creaton and annhlaton operators obey to the evoluton equaton dc j dt = [H, c j], leadng to the dfferental equaton dc j dt = [H, c j ], (4.31) d dt Ψ k = H k Ψ k (4.32) for the fermonc felds Ψ k. In the last expresson, the k ndce referes to the two possble channels and. Gvng the ntal condtons Ψ k (0) = Ψ k, these equatons are easly ntegrated and one has Ψ k (t) = e th k Ψ. (4.33) Wth ths, on can wrte the tme evoluton of the covarance matrces lke C k (t) = e th k C(0)e th k, (4.34)

102 4.3. Quench dynamcs 77 wth C k (0) = G(h ) Ψ k Ψ k G(h ) the matrces at tme t = 0. Usng the feld operators C and C, these matrces are wrtten ( C C(0) = C C C ) CC CC, (4.35) where the brackets ndcates the expectaton value taken n the chan ground state G(h ), and where we have dropped the ndces k. The evoluton of the Loschmdt echo s then derved usng equaton (4.30) together wth (4.34) gven the ntal condton C k (0). 4.3 Quench dynamcs Weak and strong couplng regmes We study frst the nfluence of the couplng strength ε on the decoherence dynamcs of the two defect spns. We show n fgure 4.2 the tme evoluton of the Loschmdt echo for a quench from an ntal transverse feld h = 1.5 to a fnal one h f = 0.5 and a dstance d = 1 for dfferent values of ε. Fgure 4.2 Tme evoluton of the Loschmdt echo after a quench from h = 1.5 to h f = 0.5 for dfferent values of the couplng strength ε. The dstance s set to d = 1 and the sze of the chan s N = 100. One can observe that the decoherence s faster as the couplng strength s ncreased. However the behavor of the echo s dfferent dependng on the strength of the couplng. Indeed, whereas the echo decreases slowly n the regme ε 1, oscllatons start to appear when the couplng s close to unty. When the couplng s ncreased further, ε 1, the echo exhbts faster oscllatons embedded nsde an envelope whch s ndependent of the couplng strength for suffcently strong ε (see for nstance the cases ε = 20 and ε = 50 n fgure 4.2). Note that ths change of behavor s not a consequence of the quench performed n the transverse feld of the chan, snce

103 78 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment t has been observed n the equlbrum stuaton h = h f as well [CP08b]. The fast oscllatons are the consequence of the ntroducton of two hgh frequences by the couplng of the two defect spns, whereas the remanng frequences, ndependent of ε, are responsble of the slow decay of the envelope Effect of the quench on the dsentanglement dynamcs Let us now have a more precse look on the effect of the quench on the dsentanglement propertes of the two spns, through the tme evoluton of the Loschmdt echo obtaned numercally by exact dagonalzaton usng equatons (4.30) and (4.34). The evoluton of the echo n the weak and strong regmes (ε = 0.1 and ε = 20), N = 100 and d = 1 for dfferent magnetc felds s shown n fgures 4.3 and 4.4. Fgure 4.3 Tme evoluton of the echo for dfferent quench protocols. For all plots, we choose N = 100, ε = 0.1 and keep fxed the dstance to d = 1. The two up plots are a varaton of the fnal magnetc feld whereas the two down plots are a varaton of the ntal magnetc feld. For all plots, the vared feld s plotted wth symbols for h > h f, wth dashed lne for h < h f and n full lne n the equlbrum case h = h f. One can observe that the quench n the bath s always detrmental for the two system s spns, n the sense that t ncreases the decoherence, as we can see by comparason to the equlbrum stuaton represented n red lnes n fgures 4.3 and 4.4. More the quench ampltude h h f s mportant, more the decoherence s strong. Note that we observe such a behavor whatever the dstance between the defect spns s. The behavor of the echo wth the defect spns separaton s opposte n weak and

104 4.3. Quench dynamcs 79 Fgure 4.4 Tme evoluton of the echo n the strong couplng regme for dfferent quench protocols. For all plots, we choose N = 100, ε = 20 and keep fxed the dstance to d = 1. The two up plots are a varaton of the fnal magnetc feld whereas the two down plots are a varaton of the ntal one. For a) and c), the vared felds are h, f = 0.5 (black), h, f = 0.7 (red), h, f = 1 (green) and h, f = 1.5 (orange). For b) and d), the vared felds are h, f = 0.7 (black), h, f = 1 (magenta), h, f = 1.5 (red), h, f = 1.7 (green) and h, f = 1.9 (orange). strong couplng regmes. Indeed, n the weak couplng regme, the echo ncreases wth the dstance whereas t decrease wth the dstance n the strong couplng regme [CP08b], as t s shown n fgure 4.5 where, for a gven quench protocol, we have plotted the echo for dfferent dstances and for the two couplng regmes. In the strong couplng regme, one can observe beatng of the envelope contanng the fast oscllatons, see for nstance the red curves n plots a) and c) of fgure 4.4. Ths feature, already observed n the equlbrum stuaton [CP08b] can be explaned n terms of the decomposton of the spectrum of the Hamltonan. As we have already mentoned, the couplng of the defect spns to the chan brngs two hgh frequences, of the order of ε, n the spectrum. The remanng ones can be splt nto two dfferent regons, namely the regon lyng between the two nteracton ponts (at poston 0 and d), and the one outsde these ponts. When the transverse feld s smaller than the crtcal value, t appears that the lowest energy level, belongng to the outsde regon, s the most populated [CP08b]. Ths frequency s assocated to the beatng of the envelope. When the magnetc feld s ncreased, more and more levels, wth frequences of the same order as the lowest energy level, start to be populated,

105 80 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment Fgure 4.5 Tme evoluton of the Loschmdt echo n the weak (left) and strong (rght) couplng regmes as a functon of the dstance for a quench from h = 1.5 to h f = 0.5. The sze of the envronment s set to N = 100. resultng to the dsappearance of the envelope beatng. In the weak couplng regme, the Loschmdt echo decrease monotonously durng the tme evoluton, wth some supermposed oscllatons, whereas t tends to reach a constant value n the equlbrum case [CP08b, RCG + 07b]. In order to check f our observatons lnkng the ampltude of the quench wth the strengh of the decoherence s correct, we study the dependence of the Loschmdt echo wth the quench ampltude n the weak couplng case, where the decrease s monotonous. For ths, we plot on fgure 4.6 the echo at large tme enough (we choose t = 10) as a functon of the ntal feld when the fnal one s fxed (left panel), and as a functon of the fnal feld when the ntal one s fxed (rght panel). Fgure 4.6 Loschmdt echo at tme t = 10 as a functon of the ntal (fnal fxed) (left) and fnal (ntal fxed) (rght) magnetc feld. The vared magnetc felds are 0.5 (red) and 1.5 (green). The dashed lnes represent the lmtng case of a completely polarzed ntal state (J = 0). Other parameters are d = 1 and ε = 0.1. The plots confrm our predctons, we clearly see that the echo ncreases untl t

106 4.3. Quench dynamcs 81 reaches ts maxmal value at the equlbrum pont 2 (h = h f ), and decreases once the equlbrum pont, reflectng the fact that the quench stuaton s always unfavorable for the coherence dynamcs. Ths last pont can be explaned by expendng the ntal ground state G(h ) over the egenstates { φ n } of the Hamltonan H E (h f ) wth egenenerges E n (h f ) G(h ) = a n φ n, (4.36) n such that e H(h f)t G(h ) = a n e E n(h f )t φ n. (4.37) n Then, more the quench ampltude s mportant, more the number of oscllatory terms n the expanson (4.37) wll be mportant, leadng to a decrease of the Loschmdt echo [MSD12]. One can see n the curves of fgure 4.6 that the echo saturates at constant value for very large ntal or fnal felds. The saturaton for ntal strong feld can be easly understood. Indeed, f h s very hgh, the ntal state s close to the completely polarzed state where all the spns of the envronment are pontng n the drecton of the magnetc feld. The ntal covarance matrx assocated to ths state s then C(0) = ( ) 1 0, (4.38) 0 0 whch obvously does not depend on the value of the feld, and as a consequence, the echo nether. The value of the echo of a completely polarzed state, obtaned by settng J = 0 n the ntal Hamltonan H(h ), s shown n dashed lnes n the left panel of fgure 4.6. One can check that, asymptotcally, the echo converges to ths lmtng value. The Loschmdt echo (and then the dsentanglement) exhbts a clear sgnature of the quantum phase transton experenced by the Isng envronment at the crtcal feld h = 1. By lookng at the left panel of fgure 4.6, where the ntal magnetc feld s vared, one can see a jump n the curves when the feld approaches the crtcal value. Ths behavor s better seen by analyzng the dervatve wth respect to the feld h of the curves of fgure 4.6. These dervatves are presented on fgure 4.7 for two dfferent fnal felds n the ordered and dsordered phases. In the two cases, we clearly see a sngularty n d h L at the crtcal value h = 1. One can note that the sgn of the peak s dfferent n the two cases. Indeed, for h f = 0.5, the crtcal pont s located after the equlbrum pont h = h f = 0.5, that s when the echo s decreasng wth the feld leadng to a negatve dervatve. The opposte stuaton occurs for h f = 1.5, the crtcal pont beng located before the equlbrum pont h = h f = 1.5, the dervatve s postve snce the echo s ncreasng wth the feld. On the other hand, the dervatve wth respect to the fnal magnetc feld h f at fxed ntal one are more smooth and does not show such sngularty as we can see n the nset of the fgure 4.7. The crtcal behavor of the Loschmdt echo s then completely set by the ntal feld h, whereas the fnal one s only responsble of dynamcal effects 2. Note that at the equlbrum pont, we recover the value of the echo obtaned n [CP08b].

107 82 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment Fgure 4.7 Frst dervatve of the Loschmdt echo wth respect to the ntal feld h for two fnal felds h f = 0.5 (crcles) and h f = 1.5 (squares). In the nset s shown the frst dervatve of the Loschmdt echo wth respect to the fnal feld h f for two ntal felds h = 0.5 (crcles) and h = 1.5 (squares). The other parameters are N = 100, ε = 0.1, d = 1 and t = 10. through the nformaton transfer medated by the chan. Due to the fnte sze of the envronment, the frst dervatve of the Loschmdt echo reaches a fnte value and does not dverge. Indeed, the dvergence of the correlaton length n the thermodynamc lmt s suppressed by fnte sze effects. In fgure 4.8, we plot the dervatve of the Loschmdt echo wth respect to the ntal feld for several szes of the chan. On can see that the maxmum n the sngularty of the dervatve s rounded and appears at a value of the feld h max shfted from the nfnte crtcal value h c = 1. Numercal analyses show that the maxmum value of the dervatve of the Loschmdt echo dverges logarthmcally wth the envronment sze lke d h L hmax ln N, (4.39) whereas the poston of the maxmum h max approaches the crtcal value as a power law of the envronmental sze : h c h max N γ, (4.40) wth an exponent γ found numercally to be γ 1.1, as shown n fgure 4.9, where we plot the maxmum reached by the dervatve and the shft to the crtcal pont both wth respect to the envronment sze.

108 4.3. Quench dynamcs 83 Fgure 4.8 Frst dervatve of the Loschmdt echo wth respect to the ntal feld h for several szes of the chan, from N = 40 to N = 160. The other parameters are ε = 0.1, d = 1, h f = 1.5 and t = 10. Fgure 4.9 Left: Scalng of the poston of the maxmum h max of the dervatve as a functon of the envronment sze. Rght: Scalng of the maxmum reached by the dervatve d h L hmax as a functon of the envronment sze. Usng arguments of crtcal scalng theory [Hen99], the γ exponent s expected to be related to the crtcal exponent ν of the correlaton length lke γ = 1 ν = 1. (4.41) The numercal departure of γ from the scalng theory predcton can be attrbuted to strong correctons to fnte sze scalng. These correctons are numercally compatble wth a 1/N scalng correcton : N(h c h max ) 1+const. 1 N. (4.42) Note that these scalng relatons are coherent wth those found n [OAFF02, ZZL08].

109 84 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment Short tmes dynamcs In ths secton, we focus our attenton to the short tme dynamcs of the Loschmdt echo. For tmes shorter than the typcal tme of the system t t typ gven by t typ = 1 f ε 1 t typ = 1 ε f ε 1, one can show that the decay s Gaussan [Per84, RCG + 07b] L(t) exp( αt 2 ) where α s the Gaussan rate. The coeffcent α can be determned by expendng the two exponentals appearng n the echo up to the second order n t: e H t e H t = 1+t(H H ) t2 2 (H2 + H2 )+t2 H H +O(t 3 ) = 1+ H I t t2 2 (H 2 + H2 2H H ) +O(t 3 ), (4.43) where H I = H H = ε(σ0 z + σz d ). Remarkng that we can wrte ( ) ( ) 2 H H = H 2 2H H + H 2 + [ ] H, H, (4.44) equaton (4.43) becomes, takng the average over the ground state G(h ), G(h ) e H t e H t G(h ) = 1 t2 2 ( H 2 I [ H, H ] ) + H I +O(t 3 ). (4.45) The two Hamltonans H and H beng Hermtan, ther commutator s anthermtan, [ H, H ] = C wth C = C. It follows G(h ) e H t e H t G(h ) = 1 t2 2 H I + ( H 2 I t+ t22 ) C +O(t 3 ). (4.46) Fnally, takng the square modulus, we obtan L(t) = 1 t 2[ H 2 I H I 2] +O(t 3 ). (4.47) The Gaussan rate α depends only on the varance of the nteracton Hamltonan taken n the ntal state of the bath G(h ), and s as a consequence ndependent of the quench protocol, as we can see on fgure 4.10 where the short tme behavor of the echo s shown for several fnal magnetc felds for both weak and strong couplng regmes. The varance α can be determned usng the expresson of the nteracton Hamltonan H I n terms of the normal modes of the Hamltonan H E (h ). Usng the relaton σ z = 2c c 1 and equaton (4.27), one fnds ] H I = 2ε [(g 0k η k + h 0kη k )(g 0l η l + h 0l η l )+(g dkη k + h dkη k )(g dl η l + h dl η l ) + 2ε. kl (4.48)

110 4.3. Quench dynamcs 85 Fgure 4.10 Short tme evoluton of the Loschmdt echo for dfferent quench protocols n the weak (left) and strong couplng (rght) regmes. Other parameters are N = 100, h = 1.5 and d = 1. Usng the expectaton values of the normal modes nto the ground state η k η l = η k η l = 0 and η kη l = δ kl, on obtans ] 2 H I 2 = 4ε [ 2 ( h0k 2 + h dk 2) 1, (4.49) k and {[ 2 [ H I 2 =4ε 2 ( h 0k 2 + h dk ) 1] 2 + (g 0k h 0l ) 2 +(g dk h dl ) 2 k k =l ]} +2h dl h 0l g dk g 0k 2h dk h 0l g dl g 0k h 0k h 0l g 0k g 0k h dk h dl g dk g dk (4.50) leadng to the fnal expresson for the varance α 3 [ α = 4ε 2 (g 0k h 0l ) 2 +(g dk h dl ) 2 + 2h dl h 0l g dk g 0k k =l 2h dk h 0l g dl g 0k h 0k h 0l g 0k g 0k h dk h dl g dk g dk ]. (4.51) Notce that n terms of spn correlaton functons, the varance α s nothng but α = 2ε 2( 1+ σ z 0 σz d c σ z 0 2), (4.52) where AB AB A B s the connected correlator, and where we have used the fact that σ0 z = σz d due to the translatonal nvarance. In fgure 4.11, α s plotted as a functon of the ntal feld h for several dstances and for ε = 0.1. On can see that the full lnes gven by equaton (4.51) match perfectly the numercal ft of the echo represented by the symbols. For ntal magnetc felds 3. We can note that f we set d = 0, meanng that the two spns are coupled to the same spn n the chan, we recover the formula obtaned n [RCG + 07b], but wth a couplng constant ε two tmes stronger. Indeed, H I = ε(σ z 0 + σz d=0 ) = (2ε)σz 0.

111 86 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment α 0,02 0,015 0,01 0, ,5 1 1,5 2 h d h α 0-0,01-0,02-0,03 0,01 0,1 1 1-h Fgure 4.11 Gaussan rate α as a functon of the ntal magnetc feld h (left). The symbols represent a numercal ft of the echo for d = 1 (dots) and d = 5 (squares) whereas the full lne represents the numercal computaton of equaton (4.51). On the rght panel are plotted the dervatve of α wth respect to h c h for the two dstances. The other parameters are N = 100, h f = 0.5 and ε = 0.1. close enough to the crtcal value h = 1, the dervatve of α wth respect to the feld exhbts a logarthmc dvergence characterstc of the 2d-Isng unversalty class. For large dstances compared to the correlaton length of the ground state envronment d ξ, one expects a saturaton of the Gaussan rate. Indeed, for large dstances and because we look at short tme scales, one defect spn evolves wthout nfluencng the other one, and the dynamcs s ndependent, leadng to the saturaton. Ths can be seen by analyzng equaton (4.52). For d ξ, the connected correlator σ z 0 σz d c vanshes and the saturaton values of α becomes α(d 1) = 2ε 2( 1 σ z 0 2). (4.53) On the other hand, when the envronment approaches crtcalty h 1, the stuaton s dfferent snce long-range correlatons are present n the ntal state, makng the evoluton not ndependent even for large dstances. Close to the crtcal pont, the decay of the connected correlator s algebrac wth [Hen99] σ z 0 σz d c d 2, (4.54) leadng to an algebrac decay of α toward the nfnte value α(d ). The correlaton functons and the local magnetzaton of the crtcal Isng model are known analytcally [Pfe70] σ z 0 σz d c = 4 π 2 1 4d 2 1, σz 0 = 2 π, (4.55) leadng to the crtcal value of α ( α crt = 2ε )) 1 (1+ π 2 1 4d 2. (4.56) The correlators out of crtcalty are more complcated to determned analytcally, and requre the computaton of Toepltz determnants [LSM61, Pfe70]. We plot on the left

112 4.3. Quench dynamcs 87 panel of fgure 4.12 the Gaussan rate α as a functon of the dstance for several ntal felds. The predctons gven by equaton (4.51) match the numercal ft of the echo, wtch saturate n the value gven by equaton (4.53). On the rght panel, we show the crtcal algebrac decay of α toward the nfnte value wth a power law wth exponent 2, as expected from crtcal phenomena. 0,04 0,03 α 0,02 0,01 0,01 0,001 α α nf 0,0001 h = d e-05 1 d 10 Fgure 4.12 (left) Gaussan rate α as a functon of the dstance. The ntal felds are, from top to bottom h = 0.7, h = 1 (crtcal envronment) and h = 1.5. The dots are obtaned by a numercal ft, the full lnes are the calculaton of the equaton (4.51), and the dashed lnes are the saturaton values (4.53). On the rght s shown on logarthmc scales the algebrac decay toward the nfnte value of α n the crtcal case h = 1. The dashed lne has a slope equal to 2. Other parameters are N = 100, h f = 0.7 and ε = Revval tme The dynamcs descrbed n the prevous secton referred to tmes shorter than a revval tme. Indeed, dependng on the separaton d between the spns and the sze N of the envronment, we observe a sgnfcant change n the behavor of the Loschmdt echo at long tmes, as we can see on fgure 4.13, where we plot the echo for dfferent separaton dstances and two szes of the envronment (N = 100 and N = 200). Note that the followng consderatons wll be exampled n the weak couplng case (ε = 0.1), but the phenomenology of the revval s the same n the strong couplng regme. One can see n fgure 4.13 that for t < N/4, the decay of the echo s bascally lnear for ntal state far from crtcalty and ts seems to be weakly dependent on the dstance. At tme t N/4, one clearly see that, for separaton dstances far from the symmetrc poston (.e d = N/2), the echo turns to a lnear ncrease ndcatng a recoverng of the coherence between the spns. Note that ths phenomenon appears every t n n N/4. On the other hand, when the dstance between the spns gets close to the symmetrc stuaton d = N/2, we observe a supplementary sngularty followng by a speed up of the lnear decrease of the echo appearng at half the prevous revval tme τ t /2 N/8. The maxmum of the slope s reached when the two spns are facng each other,.e when d = N/2. We plot on the left panel of fgure 4.14 the

113 88 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment Fgure 4.13 Loschmdt Echo for dstances N/2 (red crcles), N/2 1 (green squares), N/2 2 (blue damonds), N/2 5 (magenta up trangles) and N/2 15 (orange left trangles) for N = 100 (left) and N = 200 (rght). Note that due to ther almost perfect matchng, the two curves for d = N/2 5 and d = N/2 15 are nor dstngushable. The other parameters are set to ε = 0.1, h = 1.5 and h f =0.99. dervatve of the Loschmdt echo wth respect to tme at fxed quench protocol for dfferent dstances N/2, N/2 1, N/2 2 and N/2 15. Fgure 4.14 Frst dervatve of the Loschmdt Echo wth respect to tme. In the left plot, we keep fxed h = 1.5 and h f = 0.99 and we vary the dstance. The dfferent plots are d = N/2 (red crcles), d = N/2 1 (green squares), d = N/2 2 (blue damonds) and N/2 15 (magenta trangles). In the rght plot, the dstance s d = 1, h f = 0.99 and h = 0.99 (red crcles), h = 0.9 (green squares), and h = 0.7 (blue damonds). The others parameters are ε = 0.1 and N = 100. One can clearly see the sngularty n the dervatve at t = τ for dstances close to the symmetrc stuaton. Ths sngularty has already dsappeared for the dstance d = N/2 15, see the magenta trangles n fgure Note that whatever the dstance between the spn s, we observe a sngularty n the dervatve at tme t = t 25, reflectng the dstance ndependent recovery of the coherence. In the rght panel of the fgure 4.14 s plotted the tme dervatve of the echo at dstance fxed to d = 1 for dfferent quench protocols, ncludng the equlbrum stuaton h = h f. As expected from the prevous observaton, we do not see any

114 4.3. Quench dynamcs 89 effect at tme t = τ because we are far from the symmetrc stuaton. For the two out of equlbrum cases (h = 0.7 and h = 0.9 to h f = 0.99), we only see the effect at tme t = t. On the other hand, n the equlbrum stuaton, the revval occurs at a tme t eq that s twce the value of the non equlbrum stuaton, t eq = 2t. Ths dfference between equlbrum and non equlbrum n the revval tme can be understood n terms of quaspartcles emsson [CC05, SLRD13]. Indeed, the non equlbrum case corresponds to a global quench, where the transverse magnetc feld s suddenly changed everywhere n the chan. As a consequence of ths global change n the energy, a par of quaspartcles wth momentum ±k s emtted at every poston n the chan. The fastest group velocty of these quaspartcles s gven by v g = max k ( εk k ) k = { 2h f f h f < 1 2 f h f 1, (4.57) and because every ste n the chan acts as a local emtter, the quaspartcles need to travel only the half of the chan to start to reconstruct the ntal state, leadng to a revval tme t = N/2v g. On the contrary, the equlbrum case corresponds to a local quench occurrng only at the postons where the two spns are coupled. As a consequence, the exctatons are emtted only at these two postons, and they need to travel the complete chan to restore the ntal state, leadng to a revval tme twce bgger than n the quenched case. When the ntal state s long range, that s for ntal feld close to the crtcal value h = 1, the stuaton s very close to what we observed for short range ntal states, namely a revval at t = t, a sngular behavor at t = τ = t /2 for dstances close to the symmetrc stuaton and a doublng of t n the equlbrum case. The man dfference between crtcal and non crtcal ntal envronment les n the shape of the decay of the echo. Indeed, n the crtcal case, t s no longer lnear as t were for the non crtcal envronment, but t s rather a power law decay, as we can see on fgure Fgure 4.15 Left: Loschmdt echo for a crtcal ntal envronment for dstances d = N/2 (red crcles), d = N/2 1 (green squares), d = N/2 5 (blue damond) and d = N/2 15 (magenta trangles). Rght: Tme dervatve of the Loschmdt echo for the prevous dstances. Other parameters are N = 100, h f = 1.5 and ε = 0.1. The almost lnear decay of the tme dervatve of the echo n the rght panel of the fgure 4.15 suggests a parabolc decay of L n the crtcal ntal state case.

115 90 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment Independent dynamcs A part of the decoherence dynamcs between the two defect spns s a consequence of ther drect couplng to the envronment, whereas the remanng part comes from ther mutual couplng medated by the spn chan. In order to quantfy the part of the decoherence whch comes from the drect nteracton wth the chan, we compute the dfference between the echos n the stuaton where the spns are coupled to a common envronment and the case where the spns are coupled to two non nteractng envronments L = L L nd. The results are presented n fgure 4.16 where L s plotted for dfferent ntal and fnal felds and for several dstances d. Fgure 4.16 Dfference between the Loschmdt echo L n the stuaton where the two spns are coupled to the same bath and to two ndependent baths as a functon of tme for dfferent quench protocols and dstances. Up left: h f = 1.5 and d = 10, up rght: h f = 1.5 and d = 20, down left: h f = 0.5 and d = 10 and down rght: h f = 0.5 and d = 20. The ntal magnetc felds are: h = 0.5 (red crcles for UR and UL), h = 0.4 (red crcles curves for DR and DL), h = 0.7 (green squares), h = 0.8 (blue damonds), h = 0.9 (magenta up trangles), h = 0.95 (orange left trangles) and h = 1 (ndgo down trangles). In all plots, we also add n dashed lne the theoretcal value of t nd = d/(2v g ) One can see n fgure 4.16 that for ntal felds far from the crtcal value h = 1, L s equal to zero at the begnnng of the evoluton untl a tme t nd when t starts to grow (see for example red curves n fgure 4.16). Ths means that, untl ths tme t nd, the two defect spns are evolvng ndependently n the same way as f they were coupled to two non nteractng envronments. For t > t nd, the dynamcs of one spn s nfluenced by the second one through the chan, leadng to a non ndependent dynamcs. The physcal meanng of t nd can be understood followng more or less the same reasonng than for the revval tme, namely n term of the travel of the

116 4.3. Quench dynamcs 91 quas partcle exctatons. The two spns evolve n an ndependent way untl a par of entangled quas partcles created by the quenched envronment n the mddle of them creates correlatons between them and, as a consequence, breaks the ndependent dynamcs. The tme t nd needed for the exctatons to reach the two postons where the spns are coupled s then gven by half the dstance between them dvded by the sound velocty t nd = d 2v g, (4.58) where v g s gven by equaton (4.57). Note that ths tme depends only of the fnal value on the feld and the dstance, and s ndependent of the ntal state. The theoretcal predcton (4.58) of t nd s shown n dashed lnes n fgure 4.16, where we see that t s n good agreement wth the numercal data. Notce that n the equlbrum stuaton, due to the localty of the exctatons emsson, the quas partcles need to travel along the complete dstance d to correlate the spns, leadng to a t nd twce bgger. When the ntal state s prepared wth a magnetc feld close to the crtcal value, the stuaton s dfferent. Indeed, one can see n fgure 4.16 that the departure from zero of L starts already at t = 0 +, ndcatng that the spns are never evolvng ndependently. Ths s a consequence of the long-range correlatons present n the crtcal ntal state of the chan. When the correlaton length ξ of the ntal state, whose typcal value s gven n term of the ntal feld by ξ = ln(h ) 1 [Pfe70], s bgger than the spn separaton d, the defect are already correlated ntally through ths correlaton length, leadng the ndependent dynamcs mpossble even at short tmes. A pctural representaton of the dfference between short and long range stuatons s shown n fgure 4.17 Fgure 4.17 In the left stuaton, the dstance between the two spns connected to the defect spns s larger than the ntal correlaton length (schematzed by the blue and red arrow). Then, the two defect spns wll evolve ndependently untl the exctaton created by the whte spn correlated them. On the contrary, n the rght plot, the two spns already feel each other at t = 0 because the dstance s shorter than the correlaton length. In ths last stuaton, they wll never evolve ndependently. More large s d, more we need to have a large ntal correlaton length (and then a magnetc feld h close to crtcalty) n order to be n the regme where the ndependent dynamcs s broken. For example, n the case of h f = 0.5 and d = 10, ths regme t already reached for h = 0.8, whereas t s not n the case h f = 0.7, d = 20, because the ntal correlatons are not long range enough. Fnally, we can note that, even n ths long range ntal correlatons regme, the sgnature of the correlaton between the defect spns through the emsson of exctaton by the spn n the mddle of them

117 92 Chapter 4. Dsentanglement by nteracton wth a non equlbrum envronment dscussed for the ntal feld far from crtcalty s stll present. Indeed, one clearly see an ncrease of L for t > t nd (see for example magenta curve n the down left panel). 4.4 Concluson In ths chapter, we have studed the decoherence dynamcs of two spns 1/2 locally coupled to an envronment set out of equlbrum after the sudden change of ts transverse magnetc feld. Ths dynamcs has been nvestgated through the tme evoluton of the Loschmdt echo, whch gave us all the nformatons about the entanglement between the two spns. We dentfed n a frst tme two regmes of the couplng strength, a weak one, where the echo decrease monotoncally, and a strong one when the echo performes fast oscllatons dependng on ε, embedded n an envelope ndependent of the couplng. We also observed that the decoherence at large tme n both regmes s enhanced n the case of a quenched envronment wth respect to the equlbrum stuaton. When the envronment s prepared at crtcalty, we observed clear sgnatures n the Loschmdt echo (at both short and long tmes) of the quantum phase transton experenced by the bath, sgnatures that where nonexstent when the fnal feld was set to the crtcal value. At large enough tmes, the systems exhbts revvals, due to the fnal sze of the envronment, at tme t, whch s twce smaller than n the equlbrum case. Ths can be explaned trough the global emsson of quaspartcles, wth velocty set by the fnal feld, whereas the emsson s local (at the defect postons) n the equlbrum stuaton. The quaspartcles need then to travel n the non equlbrum stuaton half the chan for the revval to appear. One also observe a sngular change of the echo when the spns are coupled n opposte postons n the chan, whch does not seem to be explaned n terms of the quaspartcles propagaton. We fnd then that the two magnetc felds settng the quench protocols have two dstnct roles, the ntal one sets the length of the correlatons, whereas the fnal one s responsble of dynamcal effect trough the exctatons propagaton.

118 Non Equlbrum and Equlbrum Steady State entanglement drven by quantum repeated nteractons 5 In the two prevous chapters, we have analyzed the behavor of a small quantum system nteractng wth an envronment usng a Hamltonan approach [AJP06a]. In ths knd of approach, the global system (small system+envronment, both descrbed by a Hamltonan) s closed and ts dynamcs s untary. The evoluton equaton governng the state of the small system s obtaned by the trace over the envronmental degrees of freedom of the complete state. The effects of the couplng to the envronment are then encoded n the evoluton equaton governng ρ S. An other descrpton of open quantum system s the Markovan approach [AJP06b]. In ths approach, t s unnecessary to gve a descrpton of the envronment, snce ts effects are descrbed by dsspatve terms n the dfferental equaton governng the small system dynamcs, lke Quantum Langevn [FK87] or Lndblad [Ln76] equatons. In the last decade, the quantum repeated nteractons process [AP06, AJ07, AD10, BJM14] has been ntroduced by Attal et. al to descrbe the nteracton of a system wth an envronment. The studes based on ths descrpton are essentally mathematcal [Dha08, BJM08, BJM10], but some results concernng physcal systems exst. For nstance, Karevsk and Platn have studed the dynamcs of an open XX chan coupled at both ends to two reservors at dfferent temperatures va the quantum repeated nteractons [KP09]. They derve the long tme behavor of the system and found that ts state reaches a non-equlbrum-steady-state wth observables fxed by the reservor s propertes. More recently, entanglement propertes of a bpartte quantum system subject to a repeated nteracton wth an envronment has been studed [ADP14]. In a recent work, Zppll et. al studed the steady state propertes of two quantum systems coupled at one edge to a common entangled quantum feld [ZPAI13, ZI14]. Ths couplng has for effect the perfect replcaton of the entanglement along the array by the creaton of two-partcles Bell states. In ths chapter, we propose to study f the repeated nteractons process can lead to the same knd of entanglement duplcaton over two non nteractng arrays of spns. The chapter s organzed as follows: the frst secton s devoted to a detaled descrpton of the repeated nteractons process. We show n partcular that, under cer- 93

119 94 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons tan condtons, the system can be descrbed by means of the two-pont fermonc correlators. After that, we ntroduce a toy model and solve completely ts dynamcs, focusng ourself to entanglement propertes of the system. Fnally, n the last secton, we study the most general case of two arrays of sze N, and show that there s very rch transent entanglement dynamcs close to the boundary before the system reaches ts steady state. 5.1 Quantum repeated nteractons In ths secton and n the followng one, we descrbe n detals the repeated nteractons process [Pla08] Descrpton of the repeated nteractons process The repeated nteractons process has recently been ntroduced to descrbe the nteracton of a system wth an envronment (also called bath). Ths envronment s assumed to be made of an nfnte number of copes, all these copes beng dentcal and ndependent. The dynamcs of the system part s drven by the Hamltonan H S, leavng n the Hlbert space H S, whereas the Hamltonan of the envronment leaves n the Hlbert space E = N H j, where the H j are the Hlbert spaces of the ndvdual copes. Because the consttuents of the envronment are all ndependent, the Hamltonan of the envronment s H B = N H n, (5.1) where the H n are the Hamltonans of the ndvdual copes. We suppose that the system and the envronment are ntally not correlated, such that the total densty matrx ρ(0) can be wrtten where ρ S s the densty matrx of the system part, and ρ(0) = ρ S ρ B, (5.2) ρ B = N ρ j, (5.3) where the ρ j are the densty matrces of each copy. The dea of the repeated nteractons s that the system nteracts wth every copy of the envronment one after the other, over a tme scale τ. Once the nteracton tme wth one copy over, the couplng wth ths copy s suppressed, and an other one takes ts place, and the process s repeated n ths way. A physcal pcture of ths process s gven by a laser beam fallng nto a surface. Each photon nteracts wth the surface one after the other before beng absorbed or reflected and not partcpatng anymore to the dynamcs of the system. A pctural representaton of the repeated nteractons process s shown n fgure 5.1. The whole dynamcs s drven by the tme dependant total Hamltonan H = H S + H B + H I (t), (5.4)

120 5.1. Quantum repeated nteractons 95 Fgure 5.1 Pctural representaton of the repeated nteractons process. A system, descrbed by a Hamltonan H s nteracts wth every copy of the envronment over a tme scale τ. where H I (t) s the nteracton Hamltonan. Ths nteracton Hamltonan s constant over the tme nterval ](n 1)τ, nτ], H I (t) = HI n, descrbng the couplng between the system and the n th copy of the bath. The temporal evoluton s governed by the tme evoluton operator K n (τ) = UI(τ) n U k (τ) (5.5) N \n where U n I(τ) = e (H S+H n +H n I )τ (5.6) s the evoluton operator couplng the system and the envronment, and U k (τ) = e H kτ, k N \n (5.7) s the tme evoluton operator of all the copes whch are not nteractng wth the system. The total dynamcs from tme t = 0 untl tme t = nτ s then governed by the strng U(nτ) = K n (τ)k n 1 (τ) K 1 (τ). (5.8) In the Schrödnger representaton, the densty matrx at tme t = nτ s gven by the Louvlle equaton ρ(nτ) = U(nτ)ρ(0)U (nτ), (5.9) whch can be rewrtten usng the expresson (5.8) lke an teratve equaton ρ(nτ) = K n (τ)ρ((n 1)τ)K n(τ). (5.10) In the Hesenberg representaton, where the tme dependence s carred by the observable through O(t) = Tr{Oρ(t)} = Tr{O(t)ρ(0)}, (5.11) we obtan O(nτ) = U (nτ)ou(nτ) = K n(τ)o((n 1)τ)K n (τ). (5.12)

121 96 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons Tme evoluton of the system Here, we focus our attenton to the dynamcs of the system part only. Its densty matrx s obtaned by tracng ρ(t) over the envronmental degrees of freedom ρ S (nτ) = Tr B {ρ(nτ)}. (5.13) Explctly, the densty matrx of the system durng the nteracton wth the n th copy of the envronment s } ρ S (nτ) = Tr B {U(nτ)ρ S ρ k U (nτ) ρ k k n k>n = Tr k n {U(nτ)ρ S (0) ρ k U (nτ) k n }. (5.14) After some algebra, we arrve to the fundamental equaton of evoluton of the reduced densty matrx { } ρ S (nτ) = Tr n UI(τ)ρ n S ((n 1)τ) ρ n U n (τ), (5.15) where the trace s now performed on the n th copy of the envronment only. To proceed further, lets ntroduce the dagonal base of the n th copy of the envronment ρ n φ n k = p k φ n k, k = 1,, dm(h n) = Ω. (5.16) In the base S j φk n, where the vectors S j form a base of the system s Hlbert space H S, the densty matrx ρ ( (n 1)τ) s wrtten lke p 1 ρ S ((n 1)τ) p 2 ρ S ((n 1)τ) ρ n ρ S ((n 1)τ) =.... pωρs((n 1)τ) I Introducng the decomposton V1 1 V1 2 V1 Ω UI(nτ) n = VΩ 1 V2 Ω VΩ Ω (5.17), (5.18) where the V j are operators lvng n H S, one can show that the densty matrx at tme t = nτ s gven by the applcaton of a super operator on the densty matrx at prevous tme t = (n 1)τ ρ S (nτ) = L(ρ S ((n 1)τ)), (5.19) wth L(X) = Ω p V j XVj. (5.20) j

122 5.2. XY model 97 Fnally, because all the copes are dentcal and prepared n the same state, we obtan by successve teratons ρ S (nτ) = L n (ρ S (0)). (5.21) The adjont of the super-operator L s defned by the scalar product (X, Y) = Tr{XY } through (X, LY) = (L X, Y), (5.22) leadng to the followng average of an observable O n the Hesenberg representaton wth O(nτ) = Tr S {O(nτ)ρ S (0)} = Tr S {L n (O)ρ S (0)}, (5.23) L (O) = Ω j p V j OV j. (5.24) The contnuous lmt of the evoluton equaton of an observable O s taken by lettng the nteracton tme τ gong to zero τ 0. Here, we wll not show the demonstraton, but we just expose the result [AP06, Pla08] assumng that the envronment s prepared n one of ts egenstate α. In ths case, p = δ,α and equaton (5.24) smplfes to L (O) = Ω V α OV α. (5.25) It has been shown n [AP06] that the contnuous lmt of the evoluton equaton s descrbed by the dfferental equaton where the Lndblad generator L(X) s defned lke t O(t) = L(O), (5.26) L(X) = [H, X]+ 1 2 (2L α XL α {L α L α, X}) (5.27) wth the lmts L α Vα α = α 1 lm, L α = lm, = α. (5.28) τ 0 τ τ τ 0 V α 5.2 XY model In the followng, we consder that the system s a chan of N nteractng spns wth XY nteractons. Its dynamcs s governed by the Hamltonan H S = 1 2 N 1( 1+κ 2 σx σx κ ) 2 σy σy +1 h N 2 σ z, (5.29) =1 =1 where κ s the ansotropy parameter. The envronment s modeled by an nfnte set of copes made by ndependent spns wth Hamltonan H n B = h B 2 µz n, (5.30)

123 98 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons where the µ are the Paul matrces n the bath Hlbert space. Moreover, we assume that only the frst spn of the chan s nteractng wth the envronment. Ths nteracton s drven by the Hamltonan Intal states H I = λ I 2 ( 1+κI 2 µ x σ1 x + 1 κ ) I µ y σ y 2 1. (5.31) In the rest of ths secton, we wll assume that the system and every copy of the bath are prepared nto a Gbbs state at nverse temperature β S and β B ρ S = 1 Z S e β SH S, ρ n = 1 Z n B e β BH n B, (5.32) where Z S,B s a normalzaton constant. The ntal state s then ρ(0) = ρ S ρ B wth ρ B = N ρ n. (5.33) Here we can note that the states of the system and the bath are Gaussan, snce they can be wrtten as an exponental of a quadratc form n terms of fermonc operators Dynamcs of the Clfford operators We remnd that n the tme nterval](n 1)τ, τ],.e when the system nteracts wth the n th copy of the envronment, the dynamcs s governed by the total Hamltonan whch can be rewrtten usng Clfford operators Γ H = H S + H n B + H I, (5.34) Γ 1 k 1 k = ( σ j z) σx k, j=1 j=1 k 1 Γ2 k = ( σ j z) σy k, (5.35) wth Γ = Γ, and where we have defned σ =1,,N+1 = {µ, σ =1,,N }. The average of an operator wrtten n terms of these Clfford operators O = f(γ) wll be at tme t = nτ O (nτ) = Tr S,n {OU n I[ρ S ((n 1)τ) ρ n ]U n I } = Tr S,n {O(τ)ρ S ((n 1)τ) ρ n }. (5.36) We see here that we adopt an "hybrd" representaton for the dynamcs of the average of the operator. Indeed, the tme dependence s carred by the densty matrx untl the tme (n 1)τ, whereas the operator are evolvng n the tme nterval ](n 1)τ, τ]. We need then to know what s the dynamcs of the Clfford operators. We remnd that, for tme ndependent Hamltonan, the operators Γ evolve lke Γ = R(t)Γ(0) = e Tτ Γ, (5.37)

124 5.2. XY model 99 where T s the Hamltonan matrx and ( ) Γ 1 Γ =, Γ α = Tme evoluton of the reduced densty matrx Γ 2 Γ α 1 Γ α 2.. Γ α N+1. (5.38) The ntal state of the global system (5.32) can be wrtten, usng the fact that [H S, H B ] = 0, lke ρ(0) = 1 Z S Z B e β SH S β B H B = 1 Z e H 0 (5.39) wth H 0 = β S H S + β B H B. In terms of Clfford operators, t reads H 0 = 1 4 Γ TΓ, (5.40) wth and T = 0 0 β S C S β B C B β S C C β B C B 0 0 h J y J x h J y C S = J x h J y J x h, (5.41), C B = h, (5.42) where we have set J x(y) = (1+( )κ)/2. After the nteracton wth the frst copy of the envronment, the densty matrx s ρ(τ) = 1 U (1) Z I e H 0,1 U (1) I 0,1 N /1 ρ k, (5.43) where H 0,1 = β S H S + β B H (1) B s the Hamltonan of the system and the frst copy. The densty matrx can be rewrtten ρ(τ) = 1 e H 0,1(τ) Z 0,1 ρ k = 1 e Γ ( τ)t 0,1 Γ( τ) Z N /1 0,1 ρ k, (5.44) N /1 where we have used the fact that U I (τ) = U I( τ). Usng the matrx representaton (5.37), together wth ts adjont, the Hamltonan H 0,1 becomes H 0,1 (τ) = 1 4 Γ T 0,1 (τ)γ, (5.45)

125 100 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons wth T 0,1 (τ) = R(τ)T 0,1 R (τ). Then, after the frst nteracton, the densty matrx of the whole system s ρ 0,1 (τ) = 1 ( exp 1 ) Z S,B 4 Γ T 0,1 (τ)γ. (5.46) The state of the global system beng a quadratc form of fermonc operators, t follows that the reduced densty matrx assocated wth the system part ρ S (τ) = Tr 1 {ρ 0,1 (τ)} s quadratc as well [Pes03], and can be wrtten ρ S (τ) = 1 K(τ) exp ( 14 Γ S T S(τ)Γ S ), (5.47) where the Γ S belongs to the system part only, and where T S (τ) s the restrcton of the matrx T 0,1 (τ) obtaned after tracng over the bath degrees of freedom. The teraton of the process leads to ρ S (nτ) = exp( 1 4 Γ S T ) { S(nτ)Γ S, K(nτ) = Tr exp ( 14 )} K(nτ) Γ S T S(nτ)Γ S. (5.48) It s clear from the prevous equaton that the state of the system part s Gaussan. It follows that the Wck theorem apples, and the state can be completely descrbed by the means of the two-pont correlaton functons. Indeed, usng the Wck theorem, any strng of an even number of Clfford operators factores nto a product of two pont correlators, for example Γ Γ j Γ k Γ l = Γ Γ j Γ k Γ l Γ Γ k Γ j Γ l + Γ Γ l Γ j Γ k, (5.49) whereas the average of a strng made by an odd number of operators vanshes, n partcular Γ = 0. Then, we just need to know how the correlaton matrx of the system evolves to have access to the tme evoluton of the observables Tme evoluton of the two-pont correlaton matrx The system beng Gaussan, we can completely descrbe the state by means of the two-pont correlaton functons. We defne the matrx G on the space formed by the system and the nteractng copy by G k,k = Γ k Γ k +δ kk, k, k = 1,..., N+ 1. (5.50) The tme evoluton of ths matrx s gven by the applcaton of the R(τ) matrx wth, by defnton G 0,n (nτ) = R(τ)G (0,n) ((n 1)τ)R (τ), (5.51) G (0,n) kk ((n 1)τ) = Tr S Tr n {Γ k Γ k (ρ S ((n 1)τ) ρ n )}+δ kk. (5.52) To go further, we reorganze the Γ operator lke ( ) ( ) ΓB Γ 1 Γ =, Γ Γ S = S S Γ 2, (5.53) S

126 5.2. XY model 101 and the same thng for Γ B. The system and the n th copy of the envronment beng uncorrelated for tmes t < (n 1)τ, t follows that, n the state ρ S ((n 1)τ) ρ n, all the correlators of type Tr S Tr n {Γ B Γ S (ρ S ((n 1)τ) ρ n )} vansh, and the correlaton matrx G (0,n) ((n 1)τ) assumes as a consequence the block dagonal structure ( ) G (0,n) GB 0 ((n 1)τ) =, (5.54) 0 G S ((n 1)τ) where (G S ) kk ((n 1)τ) = Tr S {Γ k Γ k ρ S ((n 1)τ)}+δ kk. (5.55) Decomposng the rotaton matrx R(τ) n the same way ( ) RB (τ) R R(τ) = BS (τ), (5.56) R SB (τ) R S (τ) we obtan the fundamental evoluton equaton of the two-pont correlaton matrx G S (nτ) = R S (τ)g S ((n 1)τ)R S (τ)+r SB(nτ)G B R SB (nτ). (5.57) The matrx at tme nτ s then gven by the applcaton of a super operator on the matrx at tme (n 1)τ G S (nτ) = L(G S ((n 1)τ)), (5.58) gvng by successve teratons G S (nτ) = L n (G S (0)), (5.59) wth L(X) = R S XR S + R SBG B R SB. (5.60) Contnuous lmt of the evoluton equaton The contnuous lmt s obtaned by lettng the nteracton τ gong to zero. In ths lmt, the matrx G S obeys to the dfferental equaton L(G t G S = S (nτ)) G S (nτ) lm. (5.61) τ 0 τ The contnuous lmt has to be taken carefully. Indeed, one needs to renormalze the couplng constant between the system and the envronment. If one takes navely the lmt τ 0 wthout any rescalng of the couplng, ths wll lead to a decouplng of the system wth the envronment, and then to the trval free evoluton of the system wthout any nfluence of the bath. It has been shown n [AP06, Pla08] that the the only possble rescalng leadng to the correct contnuum lmt s to dvde the couplng constant by the square root of the nteracton tme λ I λ I. (5.62) τ After ths proper renormalzaton, we obtan the followng fundamental evoluton equaton for the correlaton of the system part [Pla08, KP09] t G S (t) = [T S, G S (t)] 1 ({ } ) G S (t), Θ Θ 2Θ G B Θ, (5.63) 2

127 102 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons where T S s the Hamltonan matrx of the system part, and Θ s the nteracton matrx contanng the couplng between the system and the envronment. Note that wthout the proper renormalzaton, the evoluton equaton wll smply reduce to t C S (t) = [T S, G S (t)], and then to the trval evoluton. Defnng the super operator L(.) = [T S,.] 1 2 {., Θ Θ } and the constant C(G B ) = Θ G B Θ, equaton (5.62) can be rewrtten wth formal soluton, gven the ntal condton G S (0) t G S = L(G S )+C(G b ), (5.64) G S (t) = e L t (G S (0) L 1 (C(G B )))+L 1 (C(G B )), (5.65) where the matrx L 1 (C(G B )) contans all the nformatons about the steady state propertes of the system. 5.3 Toy model After the ntroducton of the general formalsm of the quantum repeated nteractons process, we wll n ths secton nvestgate the case of a smple model, for whch the dynamcs can be completely determned analytcally. Note that untl the end of ths chapter, we wll use, except f the contrary s precsed, the contnuous lmt τ Model and shape of the reduced densty matrx The smplest model we can magne conssts of two non nteractng spns, each of them coupled to one consttuent of a par of spns formng one copy of the bath, as shown n fgure 5.2. Fgure 5.2 Pctural representaton of our toy model. The two spns of the system (n red), labeled 1 and 4 are each coupled to one consttuent of a par of spn, labeled 2 and 3 formng the copy n of the envronment. The untary dynamcs of the system+envronment s drven by the tme dependent Hamltonan H = H S + H B + H I (t), where the local Hamltonans of the system and the copes of the bath are smply gven by a Zeeman term H S = h 2 (σz 1 + σz 4 ), (5.66) H B = k=1 H k B, (5.67)

128 5.3. Toy model 103 where HB k = h 2 (σz k,2 + σz k,3 ), (5.68) s the Hamltonan of the par k. The tme dependent nteracton Hamltonan s constant over one tme step [(k 1)τ, kτ[ and s gven by H (k) I = γ 2 (σx 1 σx k,2 + σy 1 σy k,2 + σx k,3 σx 4 + σy k,3 σy 4 ). (5.69) Here, because we work wth XX type nteractons, t s more judcous to use the Jordan-Wgner transformaton n terms of c fermonc operators nstead of Clfford ones, as we dd n the presentaton of the repeated nteractons. After the mappng, we obtan H S =h(c 1 c 1+ c 4 c 4), H B =h(c 2c 2 + c 3c 3 ), H I = γ(c 1 c 2+ c 2c 1 + c 3c 4 + c 4 c 3), (5.70) such that the total Hamltonan H takes the matrx form, after the ntroducton of the row vector Ψ = ( c B, ( S) c = c 2, c 3, c 1, ) c 4 H = ( ) c 2, c 3, c 1, c 4 h 0 γ 0 0 h 0 γ γ 0 h 0 0 γ 0 h c 2 c 3 c 1 c 4 ( = TB Ψ Θ Θ T S ) Ψ. (5.71) In the followng, we are nterested n the dynamcs of the two spns labeled 1 and 4, and more partcularly to ther entanglement propertes. In order to have access to the concurrence between them, we need to know the densty matrx ρ S = Tr 2,3 {ρ tot }. When the system nto consderaton s modeled by a spn chan wth XY nteracton, t has been shown [OW01, AOP + 04] that, thanks to the party symmetry of the XY Hamltonan, the reduced densty matrx of two consttuents and j of the chan assumes the structure a 0 0 c ρ j = 0 x z 0 0 z y 0. (5.72) c 0 0 b Wth ths smple structure, the concurrence reduces to C = 2 max{0, z ab, c xy}. (5.73) We are now left wth the calculaton of the matrx elements a, b, c, x, y, z to get the concurrence. These sx elements can be expressed n terms of the spn-spn correlaton

129 104 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons functons thanks to the defnton σ ασβ j Tr{ρ jσ ασβ j }. We obtan a = 1 4 σz σz j σz σz j + 1 4, x = 1 4 σz 1 4 σz j 1 4 σz σz j + 1 4, y = 1 4 σz j 1 4 σz 1 4 σz σz j + 1 4, b = 1 4 σz 1 4 σz j σz σz j + 1 4, z = 1 4 ( σx σx j + σy σy j +( σx σy j σy σx j )), c = 1 4 ( σx σx j σy σy j ( σx σy j + σy σx j )). (5.74) Note that another argument justfyng the cross structure of the densty matrx s to remark that all the vanshng elements n ρ j can be expressed as a functon of σ x j, σ zσx j or σz σy j, whch, as soon as we are workng wth Gaussan states, vansh snce they are wrtten as a strng of an odd number of fermonc operators Intal state The system and the envronment are ntally uncorrelated such that the ntal state s the product state ρ(0) = ρ S (0) η B. All the copes of the envronment beng ndependent, the ntal densty matrx of the bath s gven by η B = k N η. (5.75) We choose to prepare the system nto a factorzed state ρ S (0) = ρ 1 ρ 4 where ρ j (j = 1, 4) s the reduced densty matrx assocated to the spn j. We shall work wth a thermal state for each system s spn ρ j = 1+m0 j m0 j 2, j = 1, 4, (5.76) where m j s the magnetzaton of the spn j gven by m j Tr{σj z ρ(0)}. The two spns formng the copes of the bath are supposed to be prepared nto the maxmally entangled Bell state ψ B = 1 ( + ). (5.77) 2 As mentoned n the presentaton secton, n order to determne the evoluton of the system n terms of the evoluton of ts two-pont correlaton matrx, the total system has to be prepared nto a Gaussan state. For the two spns 1 and 4, ths s obvous snce ρ j = 1 Z e β jhσ z (5.78)

130 5.3. Toy model 105 leadng to m j = tanh(β j h/2). For the bath part, one has to check that the reduced densty matrx of each par η = ψ B ψ B = (5.79) can be wrtten as the low temperature lmt (β ) of a thermal densty matrx ρ th e η = lm ρ th = β H lm β β Z, (5.80) wth H a quadratc operator n terms of fermons. One can show (see appendx B for detals) that the operator H = (σx 2 σx 3 σ y 2 σy 3 ), (5.81) whch s well quadratc n terms of fermonc operators, leads to the correct densty matrx n the lmt β. by Startng wth the ntal Bell state ψ B, the correlators of the bath s spns are gven σ2 x σx 3 = σ y 2 σy 3 = 1, (5.82) σ2 x σy 3 = σy 2 σx 3 = 0, (5.83) σ z 2 = σ z 3 = 0. (5.84) Usng the Jordan-Wgner transformaton, one can, from these correlators, reconstruct the ntal fermonc two-pont correlaton functon of the bath (G B ) j = c c j (, j = 2, 3): leadng to σ x 2 σx 3 = c 2c 3 c 2 c 3 = 2Re c 2c 3, σ x 2 σy 3 = c 2c 3 + c 2 c 3 = 2Im c 2c 3, σ z 2 = 2 c 2c 2 1, σ z 3 = 2 c 3c 3 1, (5.85) G B (0) = 1 2 ( ) For the ntal two-pont correlaton matrx of the system part (G S ) j = c c j (, j = 1, 4), we obtan 1+m G S (0) = 2 1+m

131 106 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons Tme evoluton of the correlaton matrx Here, we focus on the dervaton of the tme evoluton of the two-pont correlaton matrx of the system. We remnd that t evolves followng the evoluton equaton t G S (t) = [T S, G S (t)] 1 2 ({ } ) G S (t), Θ Θ 2Θ G B Θ. (5.86) In the smple case consdered here, the T S matrx s proportonal to the dentty T S = h1 2 2, then the frst term of the r.h.s of equaton (5.86) obvously vanshes Usng [T S, G S ] = h[1 2 2, G S ] = 0. (5.87) Θ Θ = γ , Θ G B Θ = γ 2 G B, (5.88) the equaton governng the dynamcs of G S s found to be t G S (t) = γ 2 (G B G S (t)), (5.89) leadng to the followng evoluton equatons of the fermonc correlators t c 1 c 1 = γ 2 c 1 c 1 +γ 2 c 2c 2, (5.90) t c 4 c 4 = γ 2 c 4 c 4 +γ 2 c 3c 3, (5.91) t c 1 c 4 = γ 2 c 1 c 4 +γ 2 c 2c 3. (5.92) The resoluton of the two frst equatons gves the evoluton of the local fermonc occupaton c j c j (t) = 1 ( ) 1+m 0 t j 2 e γ2, j = 1, 4, (5.93) whereas the off-dagonal equaton leads to c 1 c 4 (t) = 1 2 ( 1 e γ2 t ). (5.94) Note that equaton (5.93) s related to the local magnetzaton by m j (t) = 2 c j c j (t) 1, gvng m j (t) = m 0 j e γ2t, j = 1, 4. (5.95) We compare on fgure 5.3 the magnetzaton on spn 1 and the fermonc correlators c 1 c 4 obtaned numercally by exact dagonalzaton, and the theoretcal predctons gven by equatons (5.95) and (5.94). We see a very good agreement wth the numercs Spn-spn correlaton functons and evoluton of the concurrence As mentoned n the prevous secton, the reduced densty matrx of spns 1 and 4 s reconstructed wth the spn-spn correlaton functons, see equatons (5.74). To

132 5.3. Toy model ,8 <c 1 + c4 > m 1 0,6 0,4 0, γ 2 t Fgure 5.3 Magnetzaton of the frst spn m 1 and fermonc correlators c 1 c 4 as a functon of the rescaled tme γ 2 t. The dots are the numercal results obtaned by exact dagonalzaton whereas the full lnes are the formula (5.95) and (5.94). The two spns 1 and 4 are ntally prepared n a state wth m 0 = 1. proceed, we have to re-express all these correlators n terms of fermonc correlators usng the Jordan-Wgner transformaton. For example, consderng σ1 xσx 4, we have σ x 1 σx 4 = (c 1 + c 1)σ z 1 σz 2 σz 3(c 4 + c 4) = (c 1 c 1)σ z 2 σz 3(c 4 + c 4) = (c 1 c 1)(2c 2c 2 1)(2c 3c 3 1)(c 4 + c 4). (5.96) Snce the Jordan-Wgner transformaton s non local and the two system s spns are not consecutve, the stes belongng to the bath part are enterng nto the expresson of the system s correlators. Usng the fermonc commutaton relatons, and the fact that the matrces are Hermtan, t comes, after some algebra ( ( σ1 x σx 4 = 2Re c 1 c c 2c 2 + c 3c 3 )+4 c 2c 2 c 3c 3 4 c 2c 3 2) ( )( ) ( )( ) + 4Re c 1 c 2 c 2c c 2c 2 + 4Re c 1 c 3 c 3c c 3c 3 ( ( ) ( + 8 Re c 1 c 2 c 3c 4 c 2c 3 +Re c 1 c 3 c 2c 4 c 2c 3 )). (5.97) Here we recognze three types of fermonc correlators c c j : those who descrbe correlatons wthn the system (, j = 1, 4), wthn the bath (, j = 2, 3) and those who represent correlatons between system and bath. In the repeated nteractons process, the state of each par nteractng wth the system s refreshed after the nteracton tme τ. Then, contrary to the "system-system" correlators, there s no cumulatve effects concernng "bath-bath" and "system-bath" correlators and they stay close to ther ntal value durng the evoluton, for example c 2 c 2 = 1/2+O(τ) and c 1 c 2 = 0+O(τ).

133 108 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons In the contnuous lmt τ 0, one can thus approxmate equaton (5.97) by ( ( ) ( ) σ1 x σx 4 (t) = 2Re c 1 c c 2c 2 + c 3c c 2c 2 c 3c 3 4 c 2c 3 2). (5.98) Usng the ntal values of the bath correlators c 2 c 2 = c 3 c 3 = c 2 c 3 = 1/2, together wth equaton (5.94), we obtan fnally σ x 1 σx 4 (t) = e γ2t 1. (5.99) Usng the same reasonng for σ y 1 σy 4, σx 1 σy 4 and σy 1 σx 4, we fnd σ y 1 σy 4 (t) = σx 1 σx 4 (t), (5.100) σ1 x σy 4 (t) = σy 1 σx 4 (t) = 0. (5.101) The last correlator to determned s σ1 zσz 4 (t). Its expresson n terms of fermonc operators s σ1 z σz 4 = (2c 1 c 1 1)(2c 4 c 4 1). (5.102) Note that here, only operators actng on stes 1 and 4 are present, the Jordan-Wgner transformaton of the σ z operator beng local. It becomes, usng Wck theorem σ1 z σz 4 ( c = 4 1 c 1 c 4 c 4 c 1 c 4 2) ( ) 2 c 1 c 1 + c 4 c = m 0 1 m0 4 e 2γ2t (1 e γ2t ) 2 (5.103) Now that all the spn-spn correlaton functons are known, we are n poston to wrte the matrx elements of ρ S (t) and determne the concurrence between spns 1 and 4. Takng expressons (5.74), the matrx elements of ρ S are leadng to a concurrence of a = 1 ( 1+(m )e γ2t + m 0 1 m0 4 e 2γ2t (1 e γ2t ) 2), (5.104) x = 1 ( 1+(m )e γ2t m 0 1 m0 4 e 2γ2t +(1 e γ2t ) 2), (5.105) y = 1 ( 1+(m )e γ2t m 0 1 m0 4 e 2γ2t +(1 e γ2t ) 2), (5.106) b = 1 ( 1 (m )e γ2t + m 0 1 m0 4 e 2γ2t (1 e γ2t ) 2), (5.107) c =0, (5.108) z = 1 ( ) e γ2t 1, 2 (5.109) C 14 (t) = max { 0, 1 e γ2t 1 2 ( [ 2e γ2t +(m 0 1 m0 4 1)e 2γ2 t ] 2 (m m0 4 )2 e 2γ2 t ) 1/2 }. (5.110) One can see that f the two system s spns are each prepared n the state wth magnetzaton m 0 1,4 = 1 (resp. wth magnetzaton m0 1,4 = 1), the matrx element

134 5.3. Toy model 109 b (resp. a ), correspondng to the probablty to be n the state (resp. ), s zero durng the evoluton. The concurrence wll be, as a consequence, always bgger for states wth m 0 1,4 = 1 than for states wth m0 1,4 = 1. One can also remark that the concurrence s untouched by the smultaneous change m 0 1 m0 1 and m0 4 m0 4. For long tmes, and ndependently of the ntal state of the system, the statonary concurrence reaches s maxmal value C 14 (t ) = 1, (5.111) reflectng the fact that G S (t ) = G B. Ths ndcates a transfer and a replcaton of the entanglement present ntally n the bath to the two non nteractng system s spn. The steady state value of the matrx elements are z = 1/2, x = y = 1/2 and a = b = c = 0, leadng to the densty matrx ρ S = (5.112) The steady state of the spns 1 and 4 s then pure and gven by the Bell state ψ = 1 2 ( ). Notce that the Bell state reached by the system s not the Bell state of the bath s copes ψ B = 1 2 ( + ). Ths dfference comes from the mcroscopc couplng chosen to descrbe the nteracton between the system and the bath. Indeed, by tunng the system-bath couplng properly, one can recover the Bell state ψ B as steady state of the system, for example by settng the twsted nteracton Hamltonan H I = γ ( σ x 2 1 σ y 2 σy 1 σx 2 + σ3 x σy 4 ) σy 3 σx 4. (5.113) In ths case, G S (t ) = (1 σ x )/2, leadng to σ1 xσx 4 = 1 and then to ρ S = = ψ B ψ B. (5.114) We plot on the left panel of fgure 5.4 the concurrence C 14 (t) as a functon of the rescaled tme γ 2 t for two dfferent ntal states. One can see that the entanglement starts to grow drectly at t = 0 n the case m 0 1 = m0 4 = 1, correspondng to the pure state, whereas ther s a delay n the growth of C 14 n the case m 0 1 = m0 4 = 0.5, correspondng to an ntal mxed state. We plot n the rght panel of fgure 5.4 the tme t ent, defned by C 14 (t) = 0, t < t ent, as a functon of the two ntal magnetzatons. Ths tme s maxmum for m 0 j = 0 correspondng to the state where the classcal statstcal mxng s the maxmum, snce t s prepared wth an nfnte temperature β 0.

135 110 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons t nt l Fgure 5.4 (Left) Concurrence C 14 as a functon of the rescaled tme γ 2 t for two dfferent ntal magnetzatons. (Rght) Tme t ent needed for the spns 1 and 4 to become entangled as a functon of ther ntal magnetzaton m 0 1 and m0 4. We can look n partcular at the evoluton of the entanglement for the two cases correspondng to dentcal magnetzatons m 0 1 = m0 4 = m and to opposte magnetzatons m 0 1 = m0 4 = m. In the frst case, the concurrence reduces to { C 14 (t) = max 0, 1 e γ2t 1 e 2 2γ2 t (m 2 1) e 2γ2 t + 4 ( e γ2 t 1 )}. (5.115) At large tmes, we can keep only terms of the order of e γ2t, and we obtan C 14 (t) max {0, ( 1 e γ2 t 1+ )} 1 m 2. (5.116) In the second case, the concurrence smplfes even more, and we get { C 14 (t) = max 0, 1 2e γ2t 1 } 2 e 2γ2t (1 m 2 ), (5.117) whch becomes at large tmes { } C 14 (t) max 0, 1 2e γ2 t. (5.118) In ths last case, and contrary to the frst one, the approach to the steady state value s ndependent of the ntal magnetzaton m. The prefactor of the exponental n equaton (5.116) lyng between 1 and 2, the convergence of the concurrence toward the steady state value s always faster when the spns have the same magnetzaton than when they have an opposte one. We plot n the left panel of fgure C 14 as a functon of exp( γ 2 t) for dfferent magnetzatons. One can check that the relaxaton toward the statonary value s faster when the ntal magnetzaton s m = 1, and becomes slower and slower when m s decreased, the slowest dynamcs beng for m = 0, mmckng the opposte magnetzaton case. In the case of an opposte ntal magnetzaton, thanks to the smple expresson taken by the concurrence (5.117), we can get an analytcal estmaton of the rescaled

136 5.3. Toy model 111 γ 2 t ent 0,5 0,4 0,3 0,2 0, ,5 0 0,5 1 m Fgure 5.5 (Left)1 C as a functon of exp( γ 2 t) for dfferent ntal magnetzatons of the system s spns. (Rght) Rescaled tme γ 2 t ent needed for the establshment of the entanglement n the case of opposte ntal magnetzaton as a functon of m. The dots are the values extracted from the numercal data whereas the full lne s gven by equaton (5.119). tme γ 2 t ent needed for the entanglement to be establshed. By solvng the equaton C 14 (γ 2 t) = 0, one fnds ( ) 1 m γ 2 2 t nt = ln 1+, (5.119) 2 whch vanshes wth a square root sngularty close to the pure state m = 1 lke γ 2 t nt 1 m. (5.120) We compare on the rght panel of fgure 5.22 the estmaton gven by equaton (5.119) wth the numercal data, and we see a very good agreement Loss of entanglement n the envronment As we mentoned prevously, one can consder that n the contnuous lmt τ 0, the state of each spn of the envronment s not affected by the couplng wth the system, such that the fermonc correlators stay very close to ther ntal value. If we work now wth a fnte value of τ, ths consderaton does not hold anymore and the state of one copy can change after the nteracton wth the system. In partcular, the couplng of each copy of the envronment to the system may has for consequence a partal loss of entanglement between the spns of the Bell pars. For small but fnte value of the nteracton tme τ, one can show that the dfference between the fermonc correlaton matrx descrbng the n th copy after and before the nteracton at tme t = nτ s gven by [Pla08] δg B = τ2 2 ( ) {G B, Θ Θ} 2Θ G S (nτ)θ, δg B = G B ((n+1)τ) G B (nτ). (5.121) After calculatons, we obtan δg B = τ 2 γ 2 ( G B + G S (nτ)). (5.122)

137 112 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons In order to compute ths varaton, one need the evoluton of the fermonc correlaton matrx of the system part. By solvng the evoluton equaton of the system for fnte nteracton tme (5.57), we fnd c c (nτ) = 1 ( 1+m 0 2 cos 2n (γτ) ), = 1, 4, (5.123) c 1 c 4 (nτ) = 1 ( 1+cos 2n (γτ) ), 2 (5.124) leadng to the followng values for the bath correlators of the copy n just after the nteracton at tme t = nτ c 2c 2 ((n+1)τ) = 1 ( 1+γ 2 τ 2 m (γτ) ) (5.125) c 3c 3 ((n+1)τ) = 1 ( 1+γ 2 τ 2 m (γτ) ) (5.126) c 2c 3 ((n+1)τ) = 1 ( 1 γ 2 τ 2 cos 2n (γτ) ). 2 (5.127) We show on the left panel of fgure 5.6 the value of the correlators c 2 c 2 and c 2 c 3 of the copy n after the nteracton at tme t = nτ for an nteracton tme τ = We can see that the theoretcal predctons (full lnes) perfectly match the data obtaned by exact dagonalzaton (symbols). After the calculaton of the spn-spn correlaton functons, we fnally fnd that the concurrence of the copy n after the nteracton at tme t = nτ s gven by C (n) B ( = 1 γ 2 τ 2 cos 2n (γτ) (γ 2 τ 2 cos 2n (γτ) ( m 0 1 m0 4 1) + 2 ) ) 2 ( m m 0 ) 2 4. (5.128) Snce cos(γτ) < 1, one can see from the prevous equaton that after many nteractons, the copes do not loose any entanglement after the couplng wth the system. Indeed, for long tme, the two spns of the system have reached a steady state and consequently, they do not evolve anymore, as well as the copes of the envronment. In the rght panel of fgure (5.57), we plot the concurrence after the nteracton at tme t = nτ for τ = 0.25, m 0 1 = m0 4 = 0.5. On can check that the concurrence goes to the ntal value C B = 1 after many nteractons, reflectng the fact that the spns of the system have reached the steady state. 5.4 General case: System of two chans of sze N Model an ntal states We now turn to the study of the most general case of two non nteractng chans made by N spns. The spns of the frst chan are labeled from 1 to N, and those of the second one from N + 3 to 2N + 2. The envronment wll be modeled exactly as t was n the prevous secton, and the nteracton wll be done n the followng way: the last spn of the frst chan nteracts wth the frst spn of the envronmental par, whereas the frst spn of the second chan s coupled to the second one. A pcture of the system s presented n fgure 5.7.

138 5.4. General case: System of two chans of sze N 113 Fgure 5.6 Values of the correlators c 2 c 2 and c 2 c 3 (left) and concurrence (rght) of the copy n just after the nteracton at tme t = nτ. In the left panel, the symbols are the numercal data whereas the full lnes are gven by equatons (5.126) and (5.127). We used γ = 0.5, m 0 1 = m0 4 = 0.5, and the nteracton tme s fxed to τ = Fgure 5.7 The repeated nteractons process n the case of two chans made of N spns. The last spn of the frst chan nteracts wth the frst spn of the envronmental par, whereas the frst spn of the second chan s coupled to the second one. The Hamltonan of the system formed by the two chans s H S = h 2 σn z J 2 ( σ x n σn+1 x + σy nσ y ) n+1, (5.129) n S n S wth S = {1, 2,..., N} {N+ 3, N+ 4,..., 2N+ 2} and S = S\{N, 2N+ 2}, whereas and HB k = h 2 (σz k,n+1 + σz k,n+2 ), (5.130) H (k) I = γ 2 (σx N σx k,n+1 + σy N σy k,n+1 + σx k,n+2 σx N+3+ σ y k,n+2 σy N+3 ). (5.131) are the Hamltonan of the copy k of the envronment and the couplng Hamltonan respectvely. After the ntroducton of fermonc operators by the Jordan-Wgner transformaton, the total Hamltonan takes the form H = Ψ TΨ wth Ψ = (c S, c B ) where c S = (c 1,..., c N, c N+3,..., c 2N+2 ) and c B = (c N+1, c N+2 ). The (2N + 2) (2N + 2) matrx T s gven by ( TS Θ T = Θ T B ), T S = ( ) A 0 0 A (5.132)

139 114 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons wth A j = hδ j J(δ,j+1 + δ,j 1 ) of sze N N, T B = h1 2 2, and the 2N 2 couplng matrx Θ s gven by Θ j = γ(δ,n δ j,1 + δ,n+1 δ j,2 ). The whole system s ntally prepared nto an uncorrelated state such that wth η B = k N η, and ρ(0) = ρ S (0) η B, (5.133) η = ψ B ψ B, ψ B = 1 2 ( + ). (5.134) The two XX chans are prepared nto a fully factorzed thermal mxture wth ρ S (0) = n S ρ n (5.135) ρ n = 1+m0 n + 1 m0 n, (5.136) 2 2 wth m 0 n = Tr{ρ(0)σn} z the ntal magnetzaton of the spn n. The ntal state of the system and the bath beng Gaussan, one can, as we dd for the toy model, descrbe completely the system by means of ts 2N 2N fermonc correlaton matrx (G S ) j (t) = (c S ) (c S ) j (t). We remnd that ths correlaton matrx evolves, n the contnuous lmt τ 0 followng the evoluton equaton t G S (t) = [T S, G S (t)] 1 ({ } ) G S (t), Θ Θ 2Θ G B Θ, (5.137) 2 where s the correlaton matrx of the bath. G B = 1+σx 2 (5.138) In the followng, we are nterested n the cross entanglement between two spns facng each other n the chans, as represented by the dashed double arrows n fgure 5.7. We call C (p) the assocated concurrence between the spns N p (belongng to the left chan) and N + 3+ p (belongng to the rght chan), wth p = 0,..., N 1. The value p = 0 corresponds for example to the par formed by the two spns n contact wth the envronment. Moreover, we ntroduce the longtudnal concurrence C (p) L, measurng the entanglement between to neghborng stes p and p 1 n a gven chan. These two concurrences are, as for the toy model, computed thanks to the reconstructon of the reduced densty matrx usng the spn-spn correlaton functons, themselves obtaned wth the tme evolved fermonc correlators. Concernng the cross concurrence, the fact that the two spns under consderaton are not frst neghbors n the chan renders the analytcal determnaton of the spn-spn correlaton functons very dffcult. For a gven p, t requres the calculaton of a Pfaffan of sze 4p+2 [AOP + 04]. Ths step wll therefor be realzed numercally. We assume n the followng that all spns of the two chans are prepared wth the same ntal magnetzaton,.e m 0 j = m 0 j. Moreover, the value of the ntra-chan couplng J wll be set to J = 1/2 n order to fx the sound velocty n the chans to 1.

140 5.4. General case: System of two chans of sze N Study for tmes t < 2N: NESS We start our study by focusng our attenton to the tme regme t < 2N. Ths tme corresponds to the tme needed for an exctaton ntroduced n the system at the couplng pont to travel forward along the chan, be reflected on the opposte edge, and come back nto the njecton pont. Then, for suffcently large system sze N, and tmes t N, the stes close to the boundary where the nteracton takes place wll experence a sem-nfnte stuaton. Before turnng to the descrpton of the dynamcs of the system n ths tme regme, let s frst menton an nterestng and mportant feature: the dynamcs of correlators belongng to one chan only s ndependent of the dynamcs of the correlators belongng to the other chan. To prove that, lets decompose the matrx G S lke ( G 1 G S = S GS 12 G 21 S G 2 S ), (5.139) where the matrces GS contan the correlatons wthn the chan, wth = 1, 2, whereas the matrx GS 12 = (G21 S ) contans the correlatons between the two chans. Remarkng that we can also splt the other matrces nto blocks ( ) ΘΘ L1 = ΘG L B Θ = 1 ( ) L1 K 12, (5.140) 2 2 K 21 L 2 wth L 1 = dag(0,, 0, γ 2 ) R N N, L 2 = dag(γ 2, 0,, 0) R N N and (K 12 ) j = γ 2 δ N δ j1 = (K 21 ) j the evoluton equaton of G S becomes ( G 1 S G 12 ) ( S [G 1 t GS 21 GS 2 = S, A] [GS 12, A] ) [GS 21, A] [G2 S, A] + 1 ( L1 K 12 2 K 21 L 2 ( {G 1 S, L 1 } G 12 S L 1+L 2 GS 12 L 2 GS 21+ G21 S L 1 {GS 2, L 2} ). (5.141) where A s the matrx appearng n equaton (5.132). We obtan thus two ndependent equatons governng GS 1 and G2 S. As a consequence, the dynamcs of the fermonc correlators belongng to the left chan and those belongng to the rght one are decoupled. More precsely, because the spns of the two chans are prepared n the same ntal state, we have c c j = c 2N+3 c 2N+3 j,, j = 1,..., N. (5.142) It appears then that local observables (lke local magnetzaton for nstance) evolve exactly n the same way n the two chans. Thus, f we are nterested n such an observable, we can restrct the system to only one chan coupled at one of ts boundary to a sngle spn wth zero magnetzaton, whch greatly mprove the numercal calculaton tme. The evoluton equaton (5.137) s dffcult to solve analytcally. As a consequence, the dynamcs of the systems wll be descrbed numercally usng exact dagonalzaton. We descrbe here the system through the evoluton of observables lke local )

141 116 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons magnetzaton, currents, fermonc correlatons and cross and longtudnal concurrences. We plot n fgure 5.8 the tme evoluton of the local magnetzaton (left) and the current (rght) n the frst 30 spns of a chan of N = 150 coupled at poston N to a sngle spn wth zero magnetzaton. 1 0,9 m 0,8 0,7 0, t Fgure 5.8 Magnetzaton (left) and current (left) n the 30 frst spn of a chan of N = 150 n the tme regme t < 2N. The ntal magnetzaton s m 0 = 1 and the couplng parameters s fxed to γ = 0.5. The magnetzaton at a gven spn at poston x = N p s equal to the ntal value m 0 untl the tme t = x at whch the frst exctaton njected has reached the poston x. After ths tme, the magnetzaton decreases untl a statonary value, whch seems to be mnmum for the spn n contact wth the envronment, and constant n the bulk. Once the fastest exctaton comes back at poston x at tme 2N x after beng reflected on the opposte edge, ths statonary regme s broken. The transport of exctatons s assocated wth a current of quaspartcles. The behavor of the current s smlar to the magnetzaton. Indeed, t starts to be non zero when the frst exctaton reaches the poston x at tme t = x, and ncreases untl a statonary value. Contrary to the magnetzaton, the current s homogeneous n the chan. In the tme regme t N wth N suffcently large, the stes localzed close to the nteracton pont are then n a non equlbrum steady state (ness) characterzed by a non evolvng magnetzaton on each spn and a statonary current travelng along the chan. For both magnetzaton and current, the relaxaton toward the non-equlbrumsteady-state value m and j s algebrac, as shown n fgure 5.9. We fnd m m t 2 (5.143) j j t 3. (5.144) The dfference of exponent can be understood usng hydrodynamcal argument, the current beng smply gven by j = nv, where v t 1 s the velocty and n the local densty proportonal to the magnetzaton. In order to have a good approxmaton of what s the steady state of the sem-nfnte system, we look at magnetzaton and current profles at tme t = N. Indeed, for suffcently large value of N, the state of the spns close to the nteracton pont has

142 5.4. General case: System of two chans of sze N ln(m-m ) ln( j-j ) ln(t) ln(t) Fgure 5.9 Left: relaxaton of the magnetzaton to the asymptotc value of the spns p = 0 (blue), p = 9 (green) and p = 19 (black). Rght: relaxaton of the current of the spns p = 1 (green) and p = 9 (black). The ntal magnetzaton s m 0 = 1 and the couplng s fxed to γ = 0.5. converged to the steady state, whereas the spns close to the opposte edge are stll nto a transent dynamcs wth observables evolvng n tme. We plot n fgure 5.10 the magnetzaton and current profles taken at t = N for szes of the chan N = 100, N = 150 and N = ,9 1 0,9 m(p) 0,8 0,7 0,6 N=100 N=150 N=500 0,7 a) b) 0, p m(p/t) 0,8 t=n/4 t=n/2 t=3n/4 t=n 0 0,5 1 1,5 p/t j(p) 0,08 0,06 0,04 j(p/t) 0,08 0,06 0,04 0,02 0 0,02 c) d) p 0 0,5 1 1,5 p/t Fgure 5.10 Magnetzaton (a) and current (c) profles measures at t = N for dfferent szes of the chans. Graphs (b) and (d) show the profles at dfferent tmes usng the rescaled varable p/t. The sze of the chan s n ths case N = 300. For all plots, the ntal magnetzaton s m 0 = 1 and γ s fxed to 1/2. Snce m 0 = 1, the plots of graph (d) gve drectly access to the scalng functon g(p/t).

143 118 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons We can check that magnetzaton profle s bent at the nterface wth the envronment (p = 0) whereas the devaton from the statonary value n the bulk s algebrac followng ( ( p ) ) 2 1 m(p) m(p ) = a t (bp) 2, (5.145) where a and b are parameters dependng on the couplng constants. On the contrary, no nterface s observed for the current profle, whch tends to a homogeneous value along the complete chan. In the scalng regon, the system s scale nvarant such that we can ntroduce the rescaled varable p/t and wrte the magnetzaton and current profles lke [Pla08] ( p ( ( p ( p ( p m = m 0 1 f, j = m 0 g, (5.146) t) t)) t) t) where f(p/t) and g(p/t) are two scalng functons whch vansh n the non-causal regon p/t > 1. We plot on the rght panels of fgure 5.9 the two profles as a functon of the varable p/t at dfferent tmes for a chan of sze N = 300. All the curves collapse nto a sngle curve n the bulk, ndcatng that the system s well scale nvarant. We can have access to the f(p/t) functon by evaluatng numercally 1 m(p/t)/m 0. The results are shown n fgure 5.11 for dfferent tmes and for two values of the systemenvronment couplng parameter γ. Note that the scalng functon g(p/t) s drectly gven by the rght down plot of fgure 5.9 snce the system s ntally prepared wth m 0 = ,8 f(p/t) 0,6 0,4 t=n/4 t=n/2 t=3n/4 t=n 0,2 γ=1 γ= p/t Fgure 5.11 Scalng functon f(p/t) at dfferent tmes for a system wth N = 300 spns and two values of the system-envronment couplng γ. The statonary current presents a non monotonc behavor wth the couplng strength γ, as shown n fgure It appears that the good varable to use s the rato between the ntra chan couplng J (whch s set to 1/2 n our case) and the square of the couplng to the envronment γ, whch we wll call K = γ 2 /2J. The statonary current s weak n the lmts K 1 and K 1, as already observed n [KP09].

144 5.4. General case: System of two chans of sze N 119 0,15 0,15 j 0,1 0,05 j 0,1 0, ln K K Fgure 5.12 Statonary current evaluated at t = N as a functon of the parameter K = γ 2 /2J (man graph) and as a functon of ln(k) (nsert). The ntal magnetzaton s m 0 = 1. Indeed, n the frst lmt, t s dffcult to nject a partcle, leadng to a small value of the current. In the second lmt, the contrary stuaton happens, t s easy to nject a partcle at the nteracton pont, but t s dffcult to propagate t along the chan, leadng once agan to a small value of the current. The maxmum of the current s reached when the rato s equal to unty, K = 1. We observe moreover that the current s nvarant under the dualty transformaton K 1 K. (5.147) We check, n the nsert of fgure 5.12 where we plot j as a functon of ln(k) that the curve s well symmetrc wth respect to the pont ln(k) = 0, valdatng the dualty (5.147). The travel of the exctatons njected at the nteracton pont has for consequence the correlaton between the dfferent stes of the system. The correlaton between two stes belongng to the same chan (ntra-chan correlators) s a drect consequence of the same quaspartcle flow, whereas the correlaton between two stes belongng to dfferent chans (nter-chan correlator) s the consequence of the passage of two quaspartcles ntally entangled through the entanglement present nto the two spns formng each par of the bath. We show n fgure 5.13 the real and magnary parts of the correlators c c j, wth = 96 (correspondng to p = 4) as a functon of the poston j taken at tme t = N for a system made of two chans of sze N = 100, and for two dfferent ntal magnetzatons m 0 = 1 and m 0 = 0.5. Consderng ntra-chan correlators (j < N), the real part reaches ts maxmum value for j =, correspondng to the local densty, and decreases wth the dstance d = j, provded that +d N. Interestngly, the real part of the correlator vanshes for odd values of d. For even values of d, the correlatons seem to decay algebracally

145 120 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons Fgure 5.13 Real (left) and magnary (rght) part of the fermonc correlator c 96 c j between ste 96 the other stes taken at tme t = N for two ntal magnetzatons of m 0 = 1 and m 0 = 0.5. The other parameters are N = 100 and γ = 0.5. Here we show only the range j [75 : 125] for clarty reason. wth the dstance followng Re c c ±d d 2. (5.148) The magnary part presents two packs correspondng, up to the sgn, to the current of partcles travelng along the chans. Of course, these two peaks are equal n absolute value but wth dfferent sgn, the frst one beng proportonal to the current of partcles movng from the rght to the left, and the second to the current of partcles movng from the left to the rght. As for the real part, some of the correlators have a vanshng magnary part, namely those wth an even value of d. For odd values of d, the decay of the peaks s, as for the real part, algebrac wth the dstance, but wth a dfferent exponent Im c c ±d d 3. (5.149) The real part of the nter-chan correlators presents a peak at poston j = 2N+ 3, ndcatng that the ste sharng the most correlaton wth one ste s the ste facng t, the correlatons decreasng wth the dstance to ths partcular ste. One can note that the ntra-chan correlators are dependent on the ntal magnetzaton m 0 of the system, whereas the nter-chan are not. Ths can be explaned by lookng at the evoluton equaton of the correlaton matrx (5.141). The evoluton equatons of the matrces GS 1 and G2 S contanng the ntra-chan correlators depend on the ntal magnetzaton whereas the equaton governng GS 12 only depends on the ntal correlatons between the two chans (set to zero n our case) and the couplng strength and then not on the value of m 0. The correlatons between the dfferent stes are responsble for two knds of entanglement, the longtudnal one between two neghborng stes p and p + 1, measured by the concurrence C (p) L, and the cross one, measured by the cross concurrence C(p), between two stes facng each others N p and N + 3+ p. We present n the left panel of fgure 5.14 the longtudnal concurrence profle, extracted at tme t = N, for dfferent szes of the chan. As for the magnetzaton and current, the entanglement between stes close to the boundary nteractng wth the envronment has reached ts steady state value. The longtudnal concurrence s equal to zero between the spns

146 5.4. General case: System of two chans of sze N 121 composng the par (N, N 1), and starts to grow for the next pars. Lke for the magnetzaton, and the current, the longtudnal concurrence s scale nvarant n the scalng regon, as we can see on the rght panel of fgure 5.14 where we plot C (p/t) L as a functon of p/t at dfferent tmes, and where we can observe the perfect collapse of the curves. Fgure 5.14 Longtudnal concurrence profle extracted a tme t = N for dfferent szes of the chans (left). Longtudnal concurrence profle as a functon of the rescaled varable p/t at dfferent tmes n the case of a chan of sze N = 300 (rght). For both plots, the ntal magnetzaton s m 0 = 1, and γ = 0.5. Moreover, we defne the total longtudnal concurrence by summng all the concurrences assocated to frst neghborng spns C (p) N 1 tot = p=0 C (p) L, (5.150) and we study ts tme evoluton. Ths wll gves us nformatons about how ths entanglement s establshed and how t behaves after the breakng of the ness regme,.e after tme t = N. The results are shown on fgure 5.15 for dfferent szes of the chan. One can observe that the total longtudnal entanglement grows lnearly n tme wth a slope ndependent of the sze. Ths lnear regme s broken at the tme t = N when the frst exctaton emtted reaches the edge of the chan. After ths tme, the travel n opposte drecton of the exctatons creates destructve nterferences, whch have for consequence the decrease of the correlatons, and the logarthmc decay of the total longtudnal concurrence. We can observe that the longtudnal concurrence s already klled after a tme t 2N, ndcatng that one return of the frst exctaton s enough to kll all the frst neghbor entanglement. We fnsh our study of the non-equlbrum-steady-state propertes of the system by the cross entanglement, measured by the concurrence C (p), between stes facng each other n the two chans. Ths entanglement s the consequence of the nter chan correlatons created through the travel of entangled quaspartcles along the two chans. We plot n fgure 5.16 the cross concurrence between the twenty frst spn pars (labeled, from top to bottom n the graphs by p = 0 to p = 19) of a system made of two chans of N = 60 spns for two values of the couplng parameter, γ = 0.5 (left)

147 122 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons Fgure 5.15 Tme evoluton of the total longtudnal concurrence for dfferent szes of the chan. The couplng parameter s set to γ = 0.5. and γ = 0.1 (rght). These two couplng strengths correspond respectvely to strong and weak current regme. Fgure 5.16 Cross concurrence C (p) for, from top to bottom, p = 0 to p = 19 for a couplng strength γ = 0.5 (left) and γ = 0.1 (rght). The sze of the two chans s N = 60, and the ntal magnetzaton s m 0 = 1. As for other observables lke magnetzaton or current, the cross concurrence converges to a steady state value. Ths non-equlbrum-steady-state entanglement s maxmum n the par of spns p = 0 drectly nteractng wth the envronment, and decreases wth the dstance to the nteracton pont. One can here make two observatons: the concurrence s about one order of magntude bgger when the current s strong than when t s weak. Nevertheless, the decay of the cross concurrence wth respect to the dstance s faster n the strong current than n the weak current regme. Indeed, when the current s strong, t s easy for the entangled quaspartcles to dstrbute the entanglement along the spns, leadng to a strong concurrence on the few frst pars, but also to a rapd decay of t. On the contrary, for weak current, t s

148 5.4. General case: System of two chans of sze N 123 dffcult to transfer the entanglement, leadng to a small value of the concurrence, but ths transfer can be done n a longer dstance than n the strong current regme. Numercal results seem to ndcate an exponental decay of the concurrence wth the dstance to the nteracton pont (see left panel of fgure 5.17) C (p) exp( p/ξ ent ), (5.151) where ξ ent s a typcal entanglement length dependng on the parameters of the system. Interestngly, n the small current regme, we fnd that the entanglement length s proportonal to the nverse of the statonary current n the chan ξ ent 1/ j, whereas devatons can be observed n the strong current regme, as shown n the rght panel of fgure Fgure 5.17 Left: logarthm of the cross concurrence measured at t = N as a functon of the par p for three values of the couplng strength γ. Rght: Inverse of the current and entanglement length ξ ent. The proportonalty coeffcent s n ths case δ The sze of the chans s N = 60 and the ntal magnetzaton s m 0 = 1. The mpossblty to solve the evoluton equaton for t < 2N and the non localty of the Jordan-Wgner transformaton render the analytcal determnaton of the non equlbrum steady state value of the cross concurrence dffcult. Nevertheless, for the par formed by the two spns n contact wth the envronment,.e p = 0, we can determned the value of the concurrence as a functon of the ntal magnetzaton usng some hypothess. Indeed, because the stes close to the edge nteractng wth the envronment are nto a steady state, one can reasonably set the temporal dervatve of these elements to zero. Solvng the equaton evoluton t G S (t) = [T S, G S (t)] 1 2 ({ } ) G S (t), Θ Θ 2Θ G B Θ (5.152) for spns N and N + 3, we fnd that ther magnetzaton n the ness s drectly proportonal to the steady state current establshed n the chans for t < N m = 2J γ 2 j = 1 j, = N, N+ 3. (5.153) K We can check ths relaton by plottng m N / j as a functon of 1/K (see fgure 5.18). We get, as expected, a lnear relaton wth a slope equal to one, valdatng the relaton (5.153).

149 124 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons m N / j /K Fgure 5.18 Steady state magnetzaton on the spn N dvded by the statonary current as a functon of 1/K (dots). The straght lne s a lnear functon wth slope 1. Followng [Pla08], we can wrte the statonary current lke j m 0 m B = m 0. (5.154) It appears then that the magnetzaton of spns N and N + 3 s proportonal to the ntal magnetzaton of the chan, and we wrte m = (1 α)m 0, = N, N+ 3, (5.155) where the coeffcent α depends only on the couplng constants. In the same way, we do the hypothess that the fermonc correlator c N c N+3 s proportonal to the ntal fermonc correlator of the par of spns formng one copy of the envronment c N c N+3 = α c N+1 c N+2 = α 2, (5.156) wth the same real coeffcent α as prevously. Ths coeffcent s dffcult to fnd analytcally, and wll be extracted numercally by lookng the magnetzaton of spn N or N + 3. We can now wrte down the spn-spn correlaton functons between spns N and N+ 3 σ x N σx N+3 = σ y N σy N+3 = α, σ x N σy N+3 = σy N σx N+3 = 0, σ z N σz N+3 =(1 α) 2 m 2 0 α 2, (5.157) where we have used the Wck theorem for the correlator n the z drecton. Ths leads to the followng reduced densty matrx a 0 0 c ρ j = 0 x z 0 0 z y 0, (5.158) c 0 0 b

150 5.4. General case: System of two chans of sze N 125 wth the elements a = (1+(1 α)m 0) 2 α2 4 4, ( 1+α 2 )( 1 m 2 0) x =y = 4 b = (1 (1 α)m 0) 2 4 c =0, α2 4, αm2 0 2, z = α. (5.159) Fnally, wth these matrx elements, we fnd a steady state concurrence n the par p = 0 gven by { C (0) (m 0, α) = max 0, α 1 ( ( 1+m (1 α) 2 2 (1 m 20 2)) } )2 1+α 2 + 2α 0 1 m 2. 0 (5.160) One can remark that for m 0 = 1 (resp. m 0 = 1), b = 0 (resp. a = 0), leadng to a maxmum concurrence of C (0) = α. The statonary entanglement n the frst par s then maxmum for m 0 = 1 and decreases wth m 0. Dependng on the value of the parameter α (and then on the value of the couplng γ), t may exst a threshold value m thre, satsfyng C (0) = 0 m 0 < m thre, below whch the entanglement between spns N and N+ 3 s lost n the ness. The value α above whch the threshold value m thre does not exst s gven by the soluton of the equaton C (0) (m 0 = 0, α ) = 0, leadng to α = 2 1. (5.161) We have checked numercally that the parameter α s a monotoncally ncreasng functon of the couplng γ. There s, as a consequence, only one value γ leadng to α, and we fnd γ = For γ < γ, the threshold value m thre s found by solvng the equaton C (0) (m 0, α) = 0, and we fnd m thre = α 2 2α 2+1, γ < γ. (5.162) 1 α We plot n fgure 5.19 the concurrence for p = 0, 1, 2 computed numercally as a functon of the ntal magnetzaton for two dfferent values of γ, together wth equaton (5.160). Frst, we can see that the behavor of the pars wth p = 1 and p = 2 s smlar to the behavor of the frst par, the concurrence beng maxmal for m 0 = 1 and decreases wth the ntal magnetzaton. Secondly, the theoretcal predcton (5.160) of the concurrence n the frst par as a functon of the ntal magnetzaton fts pretty well the numercal data, valdatng the hypothess made above. In the case γ = 2, correspondng to the stuaton γ > γ, one can check that the threshold value m thre does not exst, and the frst par s entangled whatever the ntal magnetzaton. In the case γ = 0.5, we extract numercally, by lookng at the magnetzaton of the spn = N, the value α = , leadng to a calculated value of the threshold magnetzaton m thre = Ths value s n relatvely good agreement wth the value found numercally m thre =

151 126 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons Fgure 5.19 Cross concurrence for p = 0, 1, 2 as a functon of the ntal magnetzaton obtaned numercally (dots), and equaton (5.160) (dashed lne). The couplng γ s set to γ = 0.5 n the man graph and to γ = 2 n the nsert. In the case γ = 0.5, the extracted value of α s α = Steady state We now consder the long tmes t N, and we derve what s the steady state of the fnte sze system. For t > 2N, the exctatons njected by the nteracton are travelng forward and backward along the two chans. Ths wll have for effect to destroy the correlaton between spns belongng to the same chan, and ncrease entanglement between spns facng each others. Let s ntroduce the matrces N k and J k (k = N, B) such that the correlaton matrces G k becomes G k = N k + J k, k = S, B, (5.163) wth N T k = N k and J T k = J k. The local densty of fermons s gven by the dagonal of the N S matrx, whereas the partcles current s ncluded n the J S one. Pluggng ths decomposton nto the evoluton equaton, one get two coupled equatons governng the dynamcs of the N S and J S matrces t J S +[A, N S ]+ 1 2 {ΘΘ, J S } = ΘJ B Θ = 0, (5.164) t N S [A, J S ]+ 1 2 {ΘΘ, N S } = ΘN B Θ. (5.165) Snce the system does not evolve any more n the steady state, we can set the tme dervatves n the prevous equatons to zero. Moreover, n the steady state, the J S matrx vanshes as well, snce t represents current-lke of the form c nc m c mc n. Pluggng these assumptons nto equatons (5.164) and (5.165), one arrves to [T S, NS ] =0, (5.166) {Θ Θ, N S } =2ΘN B Θ, (5.167)

152 5.4. General case: System of two chans of sze N 127 where the star ndcates steady state value. For latter convenence, we reorganze the row vector Ψ lke Ψ = ( c S, c B) = (c N, c N+3, c N 1, c N+4,..., c 1, c 2N+2, c N+1, c N+2) (5.168) such that the Θ matrx has now the structure (Θ) j = γ 2 δ j (δ 1 + δ 2 ), and N B 0 ΘN B Θ = γ , ΘΘ = γ (5.169) Explctly, the equaton (5.167) s 0 2(N S ) 11 2(N S ) 12 (N S ) (N S ) 1,2N 2(N ) S 12 2(N ) S 12 (N S ) (N S ) 2,2N.. (N S ) 13 (N S ) 23.. (N S ) 1,2N (N S ) 2,2N = 2 N B (5.170) Ths last result s nterestng, we fnd that the reduced fermonc correlaton matrx assocated to the two stes drectly n contact wth the envronment s equal, n the steady state, to the fermonc correlaton matrx N B of each par formng the bath. Moreover, these two stes are completely decorrelated from the rest of the system. Now, let s have a look on equaton (5.166). Wth the new orderng, the matrx T S takes now the form h 0 J. 0 h J T S = = h N 2N J, (5.171) J J 0 h

153 128 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons where s the matrx = = (5.172) =κ wth (κ) j = δ j,+1 + δ j, 1. Ths leads to J[, NS ] = J[κ 1 2 2, NS ] = 0. (5.173) The prevous result concernng the stes n contact wth the bath, together wth the toy model brng us to put the ansatz NS = N B N B N B = 1 N N N B. (5.174) Calculatng the commutator of the matrx wth the prevous ansatz, one fnds [, N S ] = J[κ 1, 1 N B] = 0. (5.175) The ansatz satsfes the equaton (5.167). The steady state of the system beng unque [PPcv08, Pro08, KP09], ths expresson s then the correct soluton. The fermonc representaton of the steady state of the system s then characterzed n the followng way: the reduced fermonc correlaton matrx assocated to each par of stes facng each other s equal to the fermonc correlaton matrx of the copes of the envronment. Moreover, all these pars are completely decorrelated from each other. The fermonc representaton of the steady state s now known. But to know the state of each par p, one need to determned what are the assocated spn-spn correlaton functon. One can show that, n the steady state, one has σ x N p σx N+3+p = σ y N p σy N+3+p = ( 1) p+1. (5.176) The demonstraton of the last equalty for σ x N p σx N+3+p s presented n appendx C. One can note that the value of these correlators depends of the consdered par of spns. In the same way, we have σ x N p σy N+3+p = σ y N p σx N+3+p = 0, σ z N p σz N+3+p = 1 (5.177) whatever the par nto consderaton. We plot on fgure 5.20 the correlaton functons σ x N p σx N+3+p (left) and σ z N p σz N+3+p (rght) as a functon of tme for two chans of N = 5 spns obtaned numercally.

154 5.4. General case: System of two chans of sze N 129 > x x <σ σn+3+p N-p 1 0,5 0-0,5 p=0 p=1 p=2 p=3 p=4 > z σ N+3+p <σ N-p z 1 0,5 0-0, t t Fgure 5.20 Tme evoluton of the σn p x σx N+3+p (left) and σz N p σz N+3+p (rght) correlators for a system composed of two chans of N = 5 spns. The couplng parameters s fxed to γ = 0.5. One can check that the sgn of the steady state value of the correlator n the x drecton depends of the consdered pars, whereas, n the z drecton, the correlator goes to the value 1 whatever the value of p. We are now n poston to wrte the steady state of the system, and n partcular the reduced densty matrx of one par p. We obtan ρ p = ( 1) p ( 1) p+1 1 0, (5.178) whereas the densty matrx of one spn = N p together wth any spn j = N+ 3+ p s gven by the (separable) thermal state ρj = (5.179) In the steady state, the system s then characterzed by an alternaton of pars p of spns n the two states ψ = 1 2 ( ) for p even, ψ + = 1 2 ( + ) for p odd, these pars beng completely decorrelated from each others. Note that the state ψ + s exactly the state of the copes of the envronment. Nevertheless, these two states are two Bell states and are, as a consequence maxmally entangled. The steady state value of the concurrence s equal to C (p) = 1 whatever the value of p. The full statonary state of the system can then be wrtten lke ρ S = (ρ ) q (ρ + ) N q, (5.180)

155 130 Chapter 5. Steady-state entanglement drven by quantum repeated nteractons where ρ ± = ψ ± ψ ± and q = N/2 f N s odd and q = (N + 1)/2 f N s even. Ths steady state s ndependent of the system sze, the ntal magnetzaton of the two chans or the value of the dfferent couplngs, and s only drven by the state of the envronment. Indeed, the entanglement present ntally n the copes s replcated and transferred through the double array. A pctural representaton of the steady state s presented on fgure Fgure 5.21 Pctural representaton of the steady state of the system (here, N s chosen to be even). The red pars are n the state Ψ + whereas the blue ones are n the state Ψ. We show on the left panel of fgure 5.22 the evoluton of the concurrence obtaned numercally as a functon of the tme and the poston for two chans wth N = 10 spns. The rght panels shows the tme evoluton of the concurrence for p = 0 to p = 4 for two chans wth N = 5 spns. Fgure 5.22 Left: Snapshot of the concurrence as a functon of tme and the par consdered for a system wth N = 10. Rght: Tme evoluton of the concurrence for p = 0 to p = 4 for a system wth N = 5. In both plots, we set m 0 = 1 and γ = 0.5. In both plots, we can see that the concurrence correspondng to a par p converges