Perpetual motion and driven dynamics of a mobile impurity in a quantum fluid

Transcription

1 and driven dynamics of a mobile impurity in a quantum fluid Oleg Lychkovskiy Russian Quantum Center Seminaire du LPTMS, Seminaire du LPTMS, /

2 Plan of the talk 1 Perpetual motion 2 Driven dynamics Seminaire du LPTMS, /

3 Outline 1 Perpetual motion 2 Driven dynamics Seminaire du LPTMS, /

4 Question Consider an impurity particle injected with some initial velocity v 0 into a quantum fluid. What velocity v will it have in the asymptotic (t = ) steady state? Seminaire du LPTMS, /

5 What quantum fluids we are interested in We consider quantum fluids with nontrivial dispersion: Superfluids (3D and 2D): Any 1D fluid: Seminaire du LPTMS, /

6 Suggested answers Question: v =? Possible answers: Landau s argument (conservation of momentum and kinetic energy): v = v 0 for v 0 < v c. Seminaire du LPTMS, /

7 Suggested answers Question: v =? Possible answers: Landau s argument (conservation of momentum and kinetic energy): v = v 0 for v 0 < v c. Roberts, Pomeau ( ): v = 0 when impurity-fluid coupling is taken into account. Seminaire du LPTMS, /

8 Suggested answers Question: v =? Possible answers: Landau s argument (conservation of momentum and kinetic energy): v = v 0 for v 0 < v c. Roberts, Pomeau ( ): v = 0 when impurity-fluid coupling is taken into account. Recent studies of dynamics of impurity in 1D (Zvonarev et al., , Burovski et al., 2014, Gamayun et al., 2014): v is a complicated function of v 0. Seminaire du LPTMS, /

9 Scope and notations Hamiltonian of the impurity-fluid system: Ĥ = Ĥh + Ĥi + Û. E - eigenstates of H. Host fluid consists of N particles in volume V with number density ρ = N/V. Hamiltonian of the host fluid Ĥ h is arbitrary. Dispersion of the fluid is the lower edge of spectrum of Ĥ h : where q q. ε(q) inf Φ Ĥ h Φ, Φ: ˆP h Φ=qΦ Seminaire du LPTMS, /

10 Some more notations Hamiltonian of the impurity: Ĥ i = ˆP 2 imp 2m. Impurity-host interaction term: Û = N U( r n r imp ). n=1 Interaction is called everywhere repulsive if U(r) 0 r 0. Periodic boundary conditions are assumed to ensure translation invariance. Seminaire du LPTMS, /

11 Initial condition Initially the impurity is injected with some velocity v 0 (with v 0 v 0 ) into the host fluid at zero temperature: in = GS v 0 = GS, v 0, where GS is the ground state of the fluid. Seminaire du LPTMS, /

12 Landau s argument applied to a motion of impurity of finite mass Consider a single scattering event between impurity and (a (quasi-)particle of) fluid. Seminaire du LPTMS, /

13 Landau s argument applied to a motion of impurity of finite mass Consider a single scattering event between impurity and (a (quasi-)particle of) fluid. Assume that the final state of the impurity-fluid system is a product eigenstate of noninteracting Hamiltonian Ĥ h + Ĥ i. Seminaire du LPTMS, /

14 Landau s argument applied to a motion of impurity of finite mass Consider a single scattering event between impurity and (a (quasi-)particle of) fluid. Assume that the final state of the impurity-fluid system is a product eigenstate of noninteracting Hamiltonian Ĥ h + Ĥ i. Let q and E h ε(q) be final momentum and energy of the fluid. Seminaire du LPTMS, /

15 Landau s argument applied to a motion of impurity of finite mass Consider a single scattering event between impurity and (a (quasi-)particle of) fluid. Assume that the final state of the impurity-fluid system is a product eigenstate of noninteracting Hamiltonian Ĥ h + Ĥ i. Let q and E h ε(q) be final momentum and energy of the fluid. Then conservation of momentum and kinetic energy lead to mv = E h + (mv 0 q) 2 2m v 0 ε(q) + q2 2m q Seminaire du LPTMS, /

16 Generalized critical velocity Generalized critical velocity depends on mass of the impurity [Rayfield, 1966]: v c inf q ε(q) + q2 2m. q Seminaire du LPTMS, /

17 Generalized critical velocity Generalized critical velocity depends on mass of the impurity [Rayfield, 1966]: v c inf q ε(q) + q2 2m. q Physically, v c is the minimal velocity which allows the impurity to create real excitations of the fluid (remind however that impurity-fluid interaction was ignored). Seminaire du LPTMS, /

18 Geometrical sense of critical velocity The line v c q is tangent to the curve ε(q) + q2 2m : Seminaire du LPTMS, /

19 Relation to the conventional Landau criterion of superfluidity What Landau originally defined in 1941 as the critical velocity is v c in the limit m : v cl inf q (ε(q)/q) = v c(m = ). Seminaire du LPTMS, /

20 Relation to the conventional Landau criterion of superfluidity What Landau originally defined in 1941 as the critical velocity is v c in the limit m : v cl inf q (ε(q)/q) = v c(m = ). Observe that v cl is an attribute of the fluid alone while v c is an attribute of the impurity-fluid system. Seminaire du LPTMS, /

21 Landau s argument: limitations The argument follows from conservation of kinetic energy, while it is total energy which is in fact conserved. Impurity-fluid interaction is disregarded. Seminaire du LPTMS, /

22 Landau s argument: limitations The argument follows from conservation of kinetic energy, while it is total energy which is in fact conserved. Impurity-fluid interaction is disregarded. Can the effect of interaction, however small, build up with time to stop the impurity completely? D. C. Roberts, Y. Pomeau: yes, v = 0 due to scattering of quantum fluctuations D. C. Roberts, Y. Pomeau, Casimir-Like Force Arising from Quantum Fluctuations in a Slowly Moving Dilute Bose-Einstein Condensate, Phys.Rev.Lett.95, (2005). D.C.Roberts, Force on a moving point impurity due to quantum fluctuations in a Bose-Einstein condensate, Phys. Rev. A 74, (2006). D.C.Roberts, When superfluids are a drag, Contemp. Phys. 50, 453 (2009). Seminaire du LPTMS, /

23 Evidence for 0 < v < v 0 in recent studies of 1D models Integrable model [M. Zvonarev et. al, 2012; O. Gamayun, OL et al., in preparation] Non-integrable case: perturbation theory in impurity-fluid coupling E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy, Phys. Rev. A89, (R) (2014); O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, (2014). Seminaire du LPTMS, /

24 Rigorous bound on v 0 v Main result of Part I For an everywhere repulsive impurity-fluid interaction U(x) 0 and for v 0 v 0 < v c where U ρ dr U( r ). v 0 v U m(v c v 0 ), (1) Seminaire du LPTMS, /

25 Rigorous bound on v 0 v v 0 v U m(v c v 0 ), Corollary 1: The bound implies that although the velocity of the impurity can decrease, it does not vanish, despite finite impurity-fluid interactions. Seminaire du LPTMS, /

26 Rigorous bound on v 0 v v 0 v U m(v c v 0 ), Corollary 2: In the limit of vanishing impurity-fluid coupling the bound amounts to v = v 0. Thus it can be viewed as a rigorous proof of the Landau criterion of superfluidity. Seminaire du LPTMS, /

27 Proof of the bound First rigorously define v : v 1 m lim 1 t dt GS, v 0 e iĥt ˆP imp e iĥt GS, v 0. t t 0 Expanding the initial state in eigenstates E of the total Hamiltonian, Ĥ, and integrating out oscillating exponents, one obtains v = 1 GS, v0 E 2 E ˆP imp E. m E If Ĥ has degenerate eigenvalues, one should adjust the eigenbasis to diagonalize the matrix E GS, v 0 GS, v 0 E in every degenerate subspace. Seminaire du LPTMS, /

28 Proof of the bound v 0 v = = (v 0 v) E E h, v 2 GS, v 0 E 2 E E h,v v 0 v E E h, v 2 GS, v 0 E 2. (2) E E h,v The sums are performed over the eigenstates E of Ĥ and over the eigenstates E h, v of Ĥ h + Ĥ i with the total momentum mv 0. Seminaire du LPTMS, /

29 Proof of the bound The key step: the definition of v c implies v 0 v 1 m(v c v 0 ) (E h + mv2 2 mv2 0 2 ) for any E h, v with the total momentum mv 0. This inequality is of pure kinematical origin. It leads to v 0 v E E h, v 2 E h,v = 1 m(v c v 0 ) 1 m(v c v 0 ) E Ĥ h + Ĥ i mv2 0 2 E h, v E h, v E E h,v ( ) E mv2 0 2 E Û E. (3) Seminaire du LPTMS, /

30 Proof of the bound Substituting eq. (3) into eq. (2) one obtains v 0 v 1 ( GS, v 0 m(v c v 0 ) Û GS, v 0 GS, v0 E ). 2 E Û E E (4) If the impurity-fluid coupling is everywhere repulsive, one obtains the bound (1) from the bound (4) by omitting the second term in the brackets in the r.h.s. of (4) and rewriting the first term as. GS, v 0 Û GS, v 0 = U Seminaire du LPTMS, /

31 (Nice) features of the bound The bound is equally valid in 1D, 2D and 3D; The bound is valid for long-range interactions provided U(r) decreases with distance faster than 1/r D. The bound is valid both in thermodynamic limit and for finite systems. Seminaire du LPTMS, /

32 Limitations of the bound An explicit formula is available only for everywhere repulsive interactions (see however eq. (4)). Useless for hardcore interaction potential. Not very natural initial conditions. These limitations call for further work! Seminaire du LPTMS, /

33 A related result Define a total dispersion of the impurity-fluid system: E(p) inf Ψ Ĥ Ψ Ψ: ˆPΨ=pΨ Then [Zvonarev et al., Nat. Phys. 8, 881 (2012)] v GS (p) GS(p) ˆV i GS(p) = E(p) p. (5) (obtained by virtue of the Hellmann-Feynman theorem). Since E(p) p is generically nonzero whenever ε(p) is nonzero, these proves the very possibility of perpetual motion. The exact relation between this result and bound on v 0 v remains to be elucidated. Seminaire du LPTMS, /

34 Driven dynamics Outline 1 Perpetual motion 2 Driven dynamics Seminaire du LPTMS, /

35 Driven dynamics Question Consider an impurity particle pulled through a quantum fluid with a small force F. How will it move? In other words, what will be v(t)? Seminaire du LPTMS, /

36 Driven dynamics 1D: Universal Bloch oscillations? A striking effect was predicted by Gangardt, Kamenev and Schecter to occur in one dimension Bloch oscillations of v(t) in the absence of an external periodic potential [Phys. Rev. Lett.102, (2009), Ann. Phys. 327, 639 (2012)]: v(t) = E p (6) p=ft Alleged justification: In 1D joint impurity-fluid dispersion E(Q) is periodic in Q. Adiabatically following the ground state leads to (6). Seminaire du LPTMS, /

37 Driven dynamics Bloch oscillations: controversy Detailed studies of specific impurity-fluid model, point impurity in a TG gas, revealed a controversy: While oscillations are present for a heavy impurity, m > (1 + γ)mh they are absent for a light impurity, m < (1 γ)mh (mh being mass of a gas particle, γ - effective impurity-fluid interaction). Xp\HΤL Η=mi mh =0.5 Xp\HΤL Τ Η=mi mh = Τ O. Gamayun, O. Lychkovskiy, and V. Cheianov, Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas, Phys. Rev. E 90, (2014). Oleg Lychkovskiy (RQC) Seminaire du LPTMS, Perpetual motion and driven dynamics of a mobile impurity in a quantum fluid 29 / 35

38 Driven dynamics Breakdown of adiabaticity Resolution of the controversy In fact, in thermodynamic limit it is not possible to adiabatically follow the ground state of the many-body system, for however small force. In particular, for the impurity-fluid system GS(p) Ψ(p) 2 = exp ( δ2 p 2 ) 2π 2 (log N f (F )), (7) (mv c ) 2 where p = Ft is acquired momentum, δ is a scattering phase, N is number of host particles and f (F ) is some function of force [O. Gamayun, OL et al., in preparation]. In the thermodynamic limit N for any fixed F the overlap vanishes. Seminaire du LPTMS, /

39 Driven dynamics Adiabaticity is broken. What about oscillations? Breakdown of adiabaticity implies that the original justifications for the oscillations is incorrect. Seminaire du LPTMS, /

40 Driven dynamics Adiabaticity is broken. What about oscillations? Breakdown of adiabaticity implies that the original justifications for the oscillations is incorrect. The oscillations, however, do survive in the case of TG gas, but only for sufficiently heavy impurity. Seminaire du LPTMS, /

41 Driven dynamics Adiabaticity is broken. What about oscillations? Breakdown of adiabaticity implies that the original justifications for the oscillations is incorrect. The oscillations, however, do survive in the case of TG gas, but only for sufficiently heavy impurity. Questions: why so? What does determine whether oscillations present or absent? What is a physical mechanism for oscillations? In what follows we assume impurity-fluid coupling to be weak. Seminaire du LPTMS, /

42 Driven dynamics Backscattering oscillations and saturation without oscillations We demonstrate that the results obtained within the specific model are in fact generic. Namely, two dynamical regimes are generically possible: Backscattering oscillations (BO) we advocate a self-explanatory name Backscattering oscillations as opposed to misleading Bloch oscillations Saturation of velocity without oscillations (SwO) Seminaire du LPTMS, /

43 Driven dynamics Simple kinematical explanation HaL HqL+ HbL m>mc q2 HqL+ 2m arctanhvcl HqL qc q2 2m HqL arctanhvcl q vhtl vc m< mc q vhtl qc m vc m< mc vchml vs m>mc Oleg Lychkovskiy (RQC) HcL t vc L 0 mc m HdL t Seminaire du LPTMS, Perpetual motion and driven dynamics of a mobile impurity in a quantum fluid 33 / 35

44 Driven dynamics What determines which regime is realized? Which regime is realized depends (1) on the properties of the fluid (namely ratio v cl /v s, v s being speed of sound) and (2) on the mass of the impurity. v cl = v s v cl < v s m < m c m > m c regime SwO SwO BO A dynamical QPT (or crossover?) occurs at a critical mass m c determined from v c (m c ) = v s. Seminaire du LPTMS, /

45 Summary Summary An impurity injected in a quantum fluid will never stop completely, although its velocity can somewhat decrease due to interactions. This follows from a rigorous bound on v 0 v proven for v 0 below a generalized critical velocity. Two generic regimes of driven impurity dynamics exist (in the limit of small force and weak impurity-fluid coupling): Backscattering oscillations and saturation of velocity without oscillations. What regime is realized depends on the ratio v cl /v s and mass of the impurity. The results are published in O. Lychkovskiy, Phys. Rev. A 91, (R) (2015), O. Lychkovskiy, Phys. Rev. A 89, (2014). Seminaire du LPTMS, /