From ultracold atoms to condensed. matter physics

Transcription

1 From ultracold atoms to condensed matter physics Charles Jean-Marc Mathy A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Professor David A. Huse September 2010

2 c Copyright by Charles Jean-Marc Mathy, All rights reserved.

3 Abstract We study the possibility of realizing strong coupled many-body quantum phases in ultracold atomic systems. Motivated by recent experiments, we first analyze the phase diagram of a Bose-Fermi mixture across a Feshbach resonance, and offer an explanation for the collapse of the system observed close to the Feshbach resonance: we find that phase separation leads to a high density phase which causes the collapse. We then focus on the recent attempts to realize the three-dimensional fermionic Hubbard model in an optical lattice. One milestone on the experimentalists agenda is to access the antiferromagnetic ordered Neél phase, which has so far been hindered by the low ordering temperature. We ask which experimental parameters maximize the antiferromagnetic interactions, which set the scale for the ordering temperature. We find that the maximum is obtained in a regime where the effective Hamiltonian describing the system no longer corresponds to a simple one-band Hubbard model, and we characterize the physics of the system in this regime. The final system we consider is mass imbalanced polarized two-component Fermi gases interacting via a Feshbach resonance. By going to the strongly polarized limit, we use a recently developed method to obtain results which have been shown to be accurate in the mass balanced case, and we find an intriguing set of competing phases in this limit. We discuss what these results imply for the full phase diagram. iii

4 Acknowledgements First off, I would like to thank my advisor, David Huse. David is one of the sharpest and most creative physicists I have ever met. He is a fantastic advisor : always available, and constantly coming up with interesting problems to work on. He also put me in touch with collaborators, and got me involved in the DARPA program, which was tremendously beneficial to my career, and partly funded my Ph.D. Thank you, David, for everything. I would also like to thank Shivaji Sondhi and Duncan Haldane for the physics discussions and for helping me with securing the next step. I owe a tremendous debt of gratitude to Sander Bais for getting me on the condensed matter track, and working with me in my first year at Princeton. In a Ph.D. program, friends come and go, and it would be impossible to thank everyone. But there was a friendship bedrock I would like to acknowledge. I ll miss the road trips with Abhi, listening to Will Smith, Dave Grusin or whatnot, riding into the sunset. I ll miss Fabio s stories on the Peloponnesian war, pigeons used as missile guides, and of course his arroz con tomato. Princeton would have been a lot less exciting without Chris around : no ski trips, no camping, no themed parties. Thanks to Said (sorry, Chris was taken) for all the adventures. I would also like to thank everyone who made jadwin hall a second home: Meera, Aakash, Katerina, Sid, Arijeet, Fiona, Xinxin, Tibi, Pablo, Diego, Arvind, Richard, Anand, soccer capitan John and the rest of the team, I m sure I ve missed out a lot of people. There is also life outside of jadwin : Civo, Alex, Masha, Vanya, Catherine, Ana, thank you all. There is also life outside of Princeton : Samantha, grazie per tutto. Auntie Rebecca, thanks for providing an oasis where I could leave my woes behind. Tio Ruben, gracias por mantener contacto todos estos años. My family deserves an acknowledgment longer than this thesis, for their unwavering love and support throughout my life and my career. My mother and brother iv

5 were instrumental in putting the US in my field of vision. It s hard to live away from family for so long, so I really appreciate that my mom, dad and brother came to visit. Special thanks to mom, for taking the time to arrange that we saw each other on a regular basis: from New York to Montreal, Milan, Buenos Aires, it was very important for me to see you and know that you were with me the whole way. Thanks to the all of my family, in the Netherlands, England, Belgium, Geneva, Argentina, Australia, I dedicate this thesis to you all. v

6 Relation to previously published work Parts of this dissertation can be found in publications in APS journals [49, 51]. APS permits the reproduction of material in its publications for the purpose of a Ph.D. dissertation, provided that one includes the appropriate copyright notices in the bibliography. The results of chapter 2 were published in [49]. Chapter 3 was based on [51] and [50]. Most of the results of chapter 4 can be found in [52]. The work in chapters 3 and 4 was supported under ARO Award W911NF with funds from the DARPA OLE Program. vi

7 vii To my family.

8 Contents Abstract Acknowledgements List of Tables List of Figures iii iv x xi 1 Introduction Tunability and universality Strong coupling and phase transitions Thesis outline Bose-Fermi mixtures across a Feshbach resonance Introduction The model The phase diagram at T= Experimental consequences Comparison to previous work Conclusion Accessing the Néel phase of ultracold fermions in a simple-cubic optical lattice Introduction The incarnation of the Hubbard model in cold atoms viii

9 3.3 Strong lattice expansion and Néel temperature Hartree approximation Experimental consequences Conclusion Polarons, molecules and trimers in strongly polarized Fermi gases Mean field theory of the imbalanced fermi gas FF, LO and FFLO Bare polaron, molecule and trimer Dressed polaron and molecule, and bare trimer Unbinding transitions vs phase separation Conclusion A Single channel model of Feshbach resonances 94 A.1 Single channel model of the contact interaction B Two-channel model of Feshbach resonances 104 Bibliography 107 ix

10 List of Tables 3.1 The values of the various energies at the two T N maxima x

11 List of Figures 2.1 Phase diagrams for different values of ν/γ 2 at λm 3/2 b γ = Bose Fermi hase diagram in the parameter space {ν (r), µ (r) b, λ (r) } {(ν µ b )/ µ f, µ b /γ µ f 1/2, m 3/2 b λ µ f 3/2 /γ 2 } Density profiles in a harmonic trap at ν = Sketch of the generic phase diagram of High Temperature Superconductors, as a function of doping x and temperature T The three lowest maximally localized Wannier functions in a one dimensional sinusoidal potential Wannier states in a 2D square lattice Contour plots of Wannier states in a 3D simple cubic lattice Hopping in the lowest band Interaction terms in a three-dimensional optical lattice with atoms scattering in the s-wave channel Approximate phase diagram for filling one fermion per lattice site Our estimates of the optimal Néel temperature, T N, as a function of a s /d The strongest higher-order process contributing to the energy of the antiferromagnetic Mott insulator at the maxima of T N Ground-state phase diagram for filling one fermion per lattice site Plot of the Hartree estimate of the antiferromagnetic exchange coupling 53 xi

12 4.1 Mean field zero temperature phase diagram of mass imbalanced spin polarized two component fermions, with mass ratio m /m = M 2 P 1 phase diagram Momentum Q of the bare FFLO molecule in units of k F, as a function of r = m /m, along the M 2 P 1 boundary T 3 M 2 P 1 phase diagram P 3 M 4 F F LO T 3 phase diagram Momentum Q of the dressed FFLO molecule in units of k F, as a function of r = m /m, along the M 4 P 3 boundary Schematics of two different scenarios for a molecule unbinding into a polaron + particle The molecular residue Z M4, for 1/(k f a s ) = 1.5, as a function of r The different approximations to the lines that mark the onset of the phase separating region to their right xii

13 Chapter 1 Introduction In the last two decades, the field of ultracold atomic physics has become interwoven with condensed matter physics[6, 30]. What made this mariage possible was a series of breakthroughs in cooling methods in the nineties, which allowed experimentalists to bring a system of many atoms into the quantum regime. Some of the early milestones were the observation of Bose-Einstein Condensation of 87 Rb [2] and 23 Na [19], and a degenerate Fermi gas of 40 K [21]. The next important step was the realization that by using a clever combination of electromagnetic fields, one could tune the background potential that the atoms felt, and vary the interactions between them. This opened up a seemingly endless world of possibilities in the use of ultracold atoms as quantum simulators, with the prospect of resolving long standing issues in condensed matter physics. As cold atom systems are in fact different from conventional condensed matter systems in several respects, there are also new issues and questions which can be addressed. 1.1 Tunability and universality Cold atom systems consist of neutral atoms at densities of about cm 3, cooled down to nanokelvin, or sometimes picokelvin temperatures, using a combination of 1

14 magnetic, laser and evaporative cooling. The atoms are kept in place thanks to a harmonic confinement. On top of this harmonic trap, in analogy to the periodic potential felt by electrons in a solid, one can introduce a periodic potential using a combination of lasers. The lasers that create the lattice are detuned from an optical absorption line and generate an electric dipole in the atoms that become trapped in either the maxima (if it is red detuned) or the minima (if it is blue detuned) of the laser light intensity [4]. The atoms feel an optical potential by way of the Stark effect: a dipole moment d is formed due to the electric field E, given by d = αe, where α is the atom s polarizability. An electric dipole has Hamiltonian d E in an electric field, thus the atom sees a potential αe 2 proportional to the square of the electric field. By superimposing a laser with its own retroreflection, one generates a standing wave. If there is only one retroreflected laser, the atoms feel a one dimensional potential. If the potential is made to be very deep, the system effectively becomes a set of uncoupled pancakes. Thus one can reduce the effective dimensionality of the system. Two orthogonal retroreflected lasers lead to a set of one-dimensional tubes, thus allowing one to study (quasi) one-dimensional physics. In general, it is possible to generate a lattice of one s choice, thus one can for example simulate atoms feeling the potential of the Hubbard model, of frustrated systems, etc. The atoms in cold atom experiments are typically alkali atoms, because they have one electron in their outer shell, which makes their behavior relatively simple to describe and predict. Such atoms have a set of hyperfine states, due to the nuclear spin I and the electron spin S. The total spin is called F, and for a given value of F there will be a manifold of 2F + 1 states. These states would be degenerate if there were no magnetic field, and can be populated using RF spectroscopy. Thus one can simulate systems with internal degrees of freedom, such as fermions with two internal spin states, where in the cold atom context the internal degree of freedom is the hyperfine states that one has chosen to occupy. 2

15 To vary the interaction between two distinguishable atoms, a static magnetic field is tuned to a Feshbach resonance, which is defined as a value of the magnetic field where the s-wave scattering length between the two states diverges[24]. Across a Feshbach resonance, the scattering length a s as a function of magnetic field behaves as a s (B) = a bg (1 B B B 0 ), (1.1) where a bg is a background scattering length. The physics of Feshbach resonances is briefly described in Appendix A. On one side of the resonance, where a s < 0, the low-energy scattering is attractive, while on the other side it is repulsive. Note that the bare interaction is always attractive. Thus, moving around a Feshbach resonance, one can realize interactions which are repulsive or attractive, and one can vary the interaction strength by orders of magnitude, depending on how precisely one can set the magnetic field close to the Feshbach resonance. One of the beautiful properties of cold atom systems is the universality of the results: indeed one typically works in the dilute limit, at low temperatures, such that the s-wave scattering length a s is the only parameter needed to describe the interactions. This means that the results obtained with one set of atoms will not depend on the details of the short-range atomic physics, and can be described by a restricted set of parameters, such as a s, the mass of the species, and their densities. The calculations are carried out using a s-wave pseudopotential, which is chosen to be as simple as possible while capturing the essential features of the interactions at low energies. The first applications of this pseudopotential was in the context of nuclear physics [27, 8], as the interactions between neutrons at relatively low densities (such as in the outer layers of a neutron star) can be modelled by a s-wave pseudopotential [73]. 3

16 Thus the physics of two-component fermions interacting via a Feshbach resonance has direct consequences for other systems with the same effective description. This universality has been shown time and time again in theory and experiments. One must however work with dilute systems: indeed there is a set of bound states that the atoms typically can fall into. If two atoms come together, they cannot fall into the bound state because of kinematic rescritions. However if three atoms collide, two can form a bound state and the third atom can carry off the excess energy[64]. Thus three-body losses should be carefully avoided. Note that in some cases the three-body losses can be used as a probe: if they suddenly increase, it suggests that the density of the system is increasing, signalling an instability (see Chapter 2 for an example). Thus in cold atoms one can vary parameters that are inaccessible in other experimental contexts, such as the number of species, their internal spin structure, the dimensionality of the system, the masses of the particles, the interaction strength, the potential seen by each species, etc. One can also introduce a rotating lattice, which leads to Quantum Hall physics in the rotating frame [17]. Recently a combination of Raman beams has allowed for the generation of artificial gauge fields [47]. An active area of investigation is the trapping of dipolar molecules, which would allow one to study systems with long range interactions [5]. In short, cold atoms are rapidly coming into contact with many subfields of condensed matter physics. 1.2 Strong coupling and phase transitions One of the main unresolved issues of modern condensed matter physics is the description and analysis of strongly coupled theories. A system is defined as strongly coupled if the average kinetic and interaction energies of the atoms are of the same order. If the kinetic term dominated, one could do perturbation theory in the interactions, such as is done in Fermi liquids, for example. If the interaction term is dominant, 4

17 then the kinetic energy may be treated perturbatively, as in done in lattice models around the atomic limit. Between these two regimes, there is no small parameter, and it is not clear how to proceed. It is precisely in this regime that the most interesting physics, from High temperature superconductivity to Fractional Quantum Hall physics to spin liquids, occurs. Several approaches have been explored to realize strongly coupled physics in cold atoms [6, 30, 36]. If one tunes a system to lie exactly at a Feshbach resonance, the system is called a unitary gas. In such a gas, there is no small parameter to expand around, and in that sense the system is strongly coupled. Perturbation theory will fail around unitarity, and more accurate methods such as Quantum Monte Carlo (QMC) must be employed [3]. However, the ubiquitous minus sign problem for fermions makes the regime of applicability of QMC limited. The canonical example of atoms interacting via a Feshbach resonance is twocomponent fermions (without an optical lattice). When there is an equal number of fermions of both species, as one crosses from the repulsive to the attractive side of the resonance, one realizes a BEC-BCS crossover [74]. The system s ground state is superfluid all the way, and crosses over from being a system of strongly bound fermion pairs behaving like bosons and forming a BEC, to a system of weakly bound Cooper pairs. At unitarity, we have what is called a crossover superfluid, where the size of the Cooper pairs is of the order of the atomic spacing. Instead of increasing the interactions, one can also reach strong coupling by reducing the kinetic energy. For example if one introduces an optical lattice in which the atoms are confined into deep potential wells, with weak tunneling between the wells, the kinetic energy is lowered, while simultaneously increasing the interactions within one well, since the atoms in a well are closer together. Thus one can reach strong coupling this way. If one starts with a BEC and introduces a three-dimensional simple cubic optical lattice, as one deepens the well, the system goes through a quan- 5

18 tum phase transition from Superfluid to Mott Insulator [32]. This has been observed experimentally. In the deep lattice limit, the bosons behave according to a Bose- Hubbard model. The same approach can be applied to two-component fermions, leading to fermions interacting via a Fermi Hubbard model[30, 36]. This model is one of the holy grails of condensed matter physics, and may hold the key to high temperature superconductivity. 1.3 Thesis outline In this thesis, we study the realization of strongly coupled many-body quantum phases in cold atoms, in three specific contexts. In chapter 2 we look at Bose-Fermi mixtures across a Feshbach resonance. In analogy to the BEC-BCS phase diagram of two-component fermions, we look at the phase diagram around unitarity. We were motivated by experiments on Bose-Fermi mixtures showing a collapse as one approached the resonance. We offer an explanation for the collapse, namely that the system is phase separating to a phase with high density, where three-body losses kick in. We propose ways to test our predictions. In chapter 3 we look at the attempts to realize the antiferromagnetically insulating Néel phase of the three-dimensional Fermi Hubbard model in cold atom systems. We find that to increase the robustness of the Néel phase, one must leave the region of the phase diagram where the Hubbard model is a good approximation. We use an expansion valid at relatively strong lattice potential, and a Hartree calculation at weak to intermediate lattice poential to completely map out the phase diagram and find the sweet spot to measure the Néel phase. Our two calculations agree well in the intermediate regime. 6

19 Finally, chapter 4 deals with the strongly polarized limit of two-component fermions interacting via a Feshbach resonance. Introducing mass imbalance, we find an intriguing competition between polaron, molecule and trimer phases. The trimer phase is competing directly with an FFLO phase, a phase which has so far eluded experimental observation. We discuss the experimental consequences of these results. 7

20 Chapter 2 Bose-Fermi mixtures across a Feshbach resonance In this chapter, we analyze the zero-temperature phase diagram of a gas of bosonic and fermionic atoms interacting through a Feshbach resonance, in a two-channel model which explicitly includes the closed channel molecule as a separate species. We find a rich phase diagram, comprising a mixture of Bose-condensed and non Bose-condensed phases separated by both second order and first order phase transitions, and Fermi Surface changing phase transitions. We show that close to unitarity there is a regime in which the system phase separates. Finally we study the density profile in a trap using LDA, and discuss in which experimentally available systems one is most likely to see the predicted behaviour. 2.1 Introduction The discovery of Feshbach resonances between bosonic and fermionic species has led to a flurry of activity, both theoretical and experimental, in the study of Bose-Fermi mixtures. Theoretical investigations have led to a prediction of a rich variety of phases: phase separation of bosons and fermions [55, 78], BCS type Cooper pairing 8

21 mediated by the bosons [34], density waves in optical lattices [46], and polar molecules with long range dipolar interactions. We will be considering a single species of boson and a single species of fermion interacting through a Feshbach resonance, in the low temperature limit where the only interaction is in the s channel. In this limit, the fermions do not interact because of Pauli exclusion. The bosons interact repulsively amongst themselves, with a background scattering length a bb. For details on the two-channel model of a Feshbach resonance, see Appendix. The physical picture is that if there exist bound states between the boson and the fermion, and a static magnetic field is applied to the system, the energy of the bound state will change, as it carries a certain angular momentum. If the energy of the bound state is made to cross the bottom of the continuum (i.e. the energy that the boson and fermion have when they are far apart and at rest), then the s-wave scattering length of the boson and fermion will diverge. We will define a parameter ν, called the detuning of the bound state, which when varied will take us across the Feshbach resonance. We include the bound state explicitly in the Hamiltonian, considering a so-called two-channel model. 2.2 The model Around the Feshbach resonance a bound state of a fermion and boson appears around zero energy. Thus the fermions and bosons in the system can interact by forming a molecule. The two-channel Hamiltonian is[67] Ĥ = +λ d 3 k ( ξ f (2π) 3 k f k f k + ξb k b k b k + ξψ k ψ k ψ ) k + g d 3 k d 3 k d 3 q (2π) 3 (2π) 3 (2π) 3 b k b k b k +q b k q d 3 k d 3 k ) (ψ (2π) 3 (2π) 3 k+k f k b k + h.c. (2.1) 9

22 b, f, and ψ are respectively the destruction operators for the bosonic atom, the fermionic atom, and the closed channel fermionic molecule. The molecule has a binding energy which is called the detuning ν. ν will vary when a magnetic field is applied, as the magnetic field couples to the total spin of the molecule, which we assume to be nonzero. The dispersion relations are given by ξ f k = h2 k 2 /2m f µ f ξ b k = h2 k 2 /2m b µ b ξ ψ k = h2 k 2 /2(m b + m f ) µ ψ. m b and µ b are respectively the masses and chemical potentials for the bosons, and similarly for the fermions and molecules. The chemical potential for the molecules is given by µ ψ = µ b + µ f ν, where ν is the detuning. It is given to lowest order in g by [24] ν = µ(b B 0 ) (2.2) where µ is the difference in magnetic moments between open and closed channels, and B 0 is the value of B at which the Feshbach resonance occurs (see Eq.(1.1)). For alkali atoms, to a good approximation one can think of the scattering problem as being between the triplet and singlet state, thus µ = µ B, the Bohr magneton. Positive ν corresponds to negative a s, and negative ν to negative a s vice versa. ν = 0 thus corresponds to unitarity 1. We define a mass ratio r = m f m b, and a mass parameter m = 2 m f m b m f +m b. We will work in units where h = m = 1. The relationship between the microscopic parameters and the s-wave scattering length is derived in Appendix B. 1 ν = 0 only corresponds to unitarity to lowest order in g, in fact unitarity occurs at ν = 0, where ν is defined in Appendix B. 10

23 To study the mean field theory of this model, we equate the boson operator to a scalar: b k δ k,0 φ. Defining ρ = φ 2, the mean field Hamiltonian becomes Ĥ MF = d 3 k ( ξ f (2π) 3 k f k f k + ξψ k ψ k ψ ) k + gφ d 3 k ( ψ (2π) 3 k f k + f k ψ ) k µb ρ + λρ 2. This Hamiltonian is quadratic, and can be diagonalized by defining mixed fermionic operators: Ĥ MF = d 3 k ( ξ F (2π) 3 k F k F k + ξψ k Ψ k Ψ ) k µb ρ + λρ 2. (2.3) The new operators are defined by F k = cos θ k f k + sin θ k ψ k (2.4) Ψ k = sin θ k f k + cos θ k ψ k, (2.5) where θ k is the mixing angle between the bands: cos 2 θ k = ξ k f ξψ k 2 (ξ f k ξ k ψ )2 + 4g 2 ρ. (2.6) The dispersion relations for the F and Ψ bands are ξ k F,Ψ = 1 2 (ξf k + ξψ k ) ± 1 (ξ f k 2 ξψ k )2 + 4g 2 ρ (2.7) At zero temperatures, the F and Ψ bands are occupied up to their respective Fermi momenta, k F and k Ψ. The (free) energy becomes E = kf 0 dk 2π 2 k2 ξ k F + kψ 0 dk 2π 2 k2 ξ Ψ k µ bρ + λρ 2, (2.8) 11

24 where ρ = φ 2 is chosen so as to minimize E. We call φ min the value of φ that minimizes the mean field energy, and define ρ min = φ 2 min. The chemical potentials are fixed by setting the total number of fermionic and bosonic atoms: n f = 1 ( k 3 6π 2 F + kψ) 3 (2.9) n b = ρ + kf 0 dk k2 2π 2 sin2 θ k + kψ 0 dk k2 2π 2 cos2 θ k (2.10) Since the molecular and fermionic bands are mixed, one finds bosons in both bands, and in the BEC determined by ρ. This model has a rather rich mean field phase diagram, as we will now see. There are seven different phases. The phases are firstly characterized by whether the condensate φ min = 0 or not. One then has to state the number of Fermi surfaces in the phase: there can be no Fermi Surface (FS), in which case one has either vacuum, if φ min = 0, or a pure BEC with no fermions, φ min 0. If there is one FS, once again there can be a BEC or no BEC. If there is a BEC, then there is no clear distinction between a FS of fermions or molecules, since the bands are hybridized. However, if there is no BEC, then one has to distinguish between having a FS of fermions or a FS of molecules. Finally, one van have two FS and either a BEC or no BEC. All told, we have seven different phases. The way these phases are connected is rather intricate, and involves a phase diagram with both second order and first phase transitions from a phase without to a phase with a BEC, as well as a series of phase transitions where the number of FS changes. We will elucidate the phase diagram in the next section. 12

25 2.3 The phase diagram at T=0 Our task is to determine the phase diagram, as a function of the parameters r, µ f, µ ψ, µ b, g, λ. We will fix r to be the mass ratio relevant for 87 Rb 40 K, though we have checked that the physics is qualitatively the same for different r. It turns out that for fixed r, we can rescale our problem (by rescaling ρ and the energy) so that we are left with three parameters. We have a certain freedom with regards to which parameters we pick, which we will exploit later on. In fact, when studying the phase diagram relevant to experiments it turns out to be favorable to work with four parameters. If there were only second order phase transitions present, we would find the line of phase transitions by solving de(ρ)/dρ ρ=0 = 0. This would be the whole story if higher order derivatives were always positive, but this is not the case. In fact, one can simultaneously solve de/dρ ρ=0 = 0 and de/dρ 2 ρ=0 = 0 and obtain a line tricritical points. One can finally solve de/dρ ρ=0 = de/dρ 2 ρ=0 = de/dρ 3 ρ=0 = 0 and obtain tetracritical points. The existence of tetracritical points signals the richness of the phase diagram to come. Although it is possible, as we have just discussed, to plot the full phase diagram in terms of three parameters, to relate the results to experiments it is more convenient to work with four parameters: we choose the dimensionless parameters { λ, ν, µ b, µ f } = {λm 3/2 b γ, ν/γ 2, µ b /γ 2, µ f /γ 2 }, (2.11) where we define (remember that h = 1) γ = g2 8π m3/2. (2.12) 13

26 γ is related to the width B of the Feshbach resonance [67]. Namely, within a mean field approximation [24], g is given by 2 4πabg µ B g = h. (2.13) m Thus, large γ corresponds to a wide Feshbach resonance. This choice of parameters is physically sensible, because for a given system and fixed magnetic field, λ and ν are set, and µ b and µ f are fixed by the total number of bosons and fermions one loads into the trap. To image the phase diagram, we fix λ and ν, and look for phase transitions as one varies µ b and µ f. We set the mass ratio to be the one for the 87 RB 40 K system: r = We also set λ = , which is a typical value [67]. Note that for a given system, one can alter λ by choosing a Feshbach resonance with a different width. The resulting phase diagram, for different values of the detuning ν, is shown in Fig.2.1. We show the phase diagram in chemical potential space for four values of ν: ν = 80, 0, 100 and 140, from left to right. Below each of these diagrams we show the corresponding phase diagram in number space. The experiments so far have focused on the ν > 0 (attractive) side, where they see collapse as they approach ν = 0. Let us look at the diagram on the top right, with ν = 140. At any µ f, for µ b negative enough we are in the Normal (N) phase, where there is no BEC. As one increases µ b, at some point the BEC appears, characterized by a nonzero value of ρ, in the gray region. This phase transition is second order for µ f away from the red line, and first order along the red line. The second order lines join the first order line at conventional tricritical points, indicated by the circles. Now for µ b < 0 and µ f < 0, we are in vacuum, because all states in the bands have positive energy. This persists as we increase µ b until µ b > 0, where a BEC appears, due to the µ b ρ term 2 The discrepancy between the equation given here and the one cited in the reference is due to the fact that we are dealing with scattering of distinguishable particles. 14

27 in the Hamiltonian. As µ b increase, ρ increases, which pushes the Ψ band down, until it crosses the zero energy line, which happens at the grey dotted line in the figure. Above this line there is one Fermi Surface (1 FS) for the Ψ band. This is an example of a Fermi Surface changing phase transition. Since an increase in ρ pushes the F band up and the Ψ band down, there are two possible Fermi Surface changing phase transitions, induced by the appearance of a BEC: either the Ψ band starts filling up, or the F band becomes empty. In this particular diagram, we haven t indicated the line where the second possibility takes place, it appears at positive values of µ f beyond the values shown here. We do show this line in the number space diagram. Another way of changing the number of FS is by varying the chemical potentials in the normal region, where there is no BEC: the lines µ f = 0 and µ f + µ b ν = 0 are Fermi Surface phase transition lines. Let us now move closer to the resonance, to ν = 100 (the second diagram from the right). for µ f > 0 we once again have a conventional tricritical point. For µ f < 0, however, the tricritical point gets preempted by a critical point, indicated by the little square. The second order line joins the first order line at what is referred to as a critical endpoint. Around this point, the energy has two minima as a function of ρ. To the left of the critical endpoint, as one increase µ b one first encounters a conventional second order phase transition, as the first minimum (i.e. the minimum at a smaller value of ρ) shifts from ρ = 0 to nonzero ρ. Increasing µ b further, the value of the energy at the second minimum decreases, until it becomes the global minimum of the energy, at which point we have a first order phase transition. To the right of the critical endpoint, the second minimum is always the global minimum, and the conventional second order phase transition is preempted by the first order transition. The first order line is surrounded by spinodal lines, which are the dotted blue lines in the diagram. Along each line, one of the minima discussed above either appears or disappears. The lower spinodal line indicates the appearance of the second minimum, 15

28 i.e. at the lower spinodal line there is a nonzero ρ for which de/dρ = 0. Above the lower spinodal line, this point becomes a local minimum, and as one crosses the first order line this local minimum becomes the global minimum. The upper spinodal line corresponds to the disappearance of the first minimum, which is determined by the same criterion as the lower spinodal line. Numerically the first order lines take a long time to calculate, but the calculation of the spinodal lines is much faster. This comes in very handy, since one can first calculate the spinodal lines, after which one can look for the first order line between them. The discussion of the Fermi Surface transitions is the same here as it was for ν = 140, except that in that case the 0F S 1F S transition at µ f < 0 joined up wit the tricritical point, while here this transition line does not join up with the critical point (this is clearer in the graphs for lower value of ν). It joins the first order line at some point between the critical point and the critical endpoint. Thus if we sit very slightly to the right of the critical point, and vary µ b from negative to positive values, we encounter three phase transitions: a second transition from vacuum to a BEC, then a FS changing phase transition where the Ψ band gets occupied, and finally a first order BEC BEC transition in which the value of ρ jumps. Closer but still to the left of the critical endpoint, the FS transition disappears, and the rest is the same. To the right of the critical endpoint, one encounters one first order phase transition. Now for the resonance ν = 0. At this point, we have two critical points, acoompanied by two critical endpoints. In this case, both BEC induced FS transitions connect to the first order line between a critical point and a critical endpoint. Furthermore, here we actually see the 1F S 2F S transition line in the normal phase. Finally, the leftmost diagrams are at ν = 80. Once again we have one critical point, and one tricritical point. 16

29 If we decrease ν far beyond 80 or increase it far beyond 140, the first order line will shrink until it disappears completely, leaving us with second order phase transitions, and FS transitions. By using the equations for number densities given above, we can translate the phase diagram to number space. The first order line becomes a region of phase separation, which one obtains by calculating the numbers densities for the chemical potentials just above and below the first order line. The darkened lines within the Phase Separating (PS) region connect the two phases on the first order line that the system will separate into, if one starts with (ñ b, ñ f ) on that line ( n b,f = n b,f /(m 3/2 γ 3 )). The spinodal lines delineate an unstable region. Inside the unstable region, there is no local minimum of the energy, and it will immediately phase separate. Outside of the unstable region, but still within the first order region, there is a metastable minimum of the energy. In the metastable region, phase separation occurs through nucleation, as there is an activation energy required to roll out of the metastable state. The remaining lines in number space denote the FS transition lines. Let us now address the full phase diagram. To this end, it is convenient to revert to different parameters, this time three instead of four: {ν (r), µ (r) b, λ (r) } {(ν µ b )/ µ f, µ b /γ µ f 1/2, m 3/2 b λ µ f 3/2 /γ 2 }. (2.14) As discussed earlier, the tritrical points will form a line in parameter space, because they are set by fixing two derivatives of the energy at ρ = 0. Similarly the critical endpoints, critical points and points where the FS transition lines join the first order line, which we will call the FS endpoint, form lines in parameter space. However, these points cannot be determined by studying derivatives of the energy at ρ = 0, so instead one has to find the lines numerically. To obtain a planar phase diagram, one can project down to the (ν (r), λ (r) ) plane. We have scaled away µ f, but there are still 17

30 Figure 2.1: Phase diagrams for different values of ν/γ 2 at λm 3/2 b γ = , where the top and bottom rows correspond to chemical potential and density space, respectively, while the columns represent different detunings: ν/γ 2 = 80 (first), ν/γ 2 = 0 (second), ν/γ 2 = 100 (third) and ν/γ 2 = 140 (fourth). In chemical potential space, the phase transition from the normal phase (in white) to the BEC phase (in gray) can either be second order, along the thin (gray) lines, or first order, along the thick (red) line. In (rescaled) number density space, this thick (red) line encircles a region of phase separation (PS). Dark dotted lines within this region connect points on the first order lines, such that a system whose total number densities lie on this line will phase separate into the phases where the line intersects the red curve. The dotted (blue) lines around the first order line in chemical potential space are the spinodal lines, and in number space they encompass an unstable region within the PS region, outside of which the system is metastable. The other dotted lines represent lines where the number of Fermi Surfaces changes. The (blue) circles tricritical points, while the (red) squares are critical points. 18

31 two cases one has to consider: µ f > 0 and µ f < 0. Thus we obtain two planar phase diagrams, shown in Fig The line of tricritical points joins all the other lines, namely the lines of critical endpoints, critical points, and FS endpoint, at tetracritical points, which are found by solving de dρ ρ=0 = de dρ 2 ρ=0 = de dρ 3 ρ=0 = 0. And that is where the fun stops : we only have three parameters to vary (after rescaling), therefore we can only set three derivatives to zero. The lines in the phase diagram demarcate areas where there is a definite series of phase transitions one encounters, as one varies µ b from to +. Let us neglect the FS transitions within the normal phase, and focus on the phase transitions to the BEC phase and within it. The different sequences of phase transitions are as follows: Below the critical endpoint line: 1st order N BEC Between the critical endpoint and FS endpoint lines: 2nd order N BEC, 1st order BEC BEC Between the FS endpoint and critical point lines: 2nd order N BEC, 2nd order 0F S 1F S or 2F S 1F S, 1st order BEC BEC Away from all lines: 2nd order N BEC, 2nd order 0F S 1F S or 2F S 1F S (2.15) Now that we have fully discussed the phase diagram, we want to address phase separation in an actual trap. In an experiment, the atoms are confined by an optical trap, which creates a harmonic potential, with a variable trap frequency. The potential is given by V (r) = 1 2 mω2 r 2. (2.16) 19

32 (a) µ f > 0 Λ r 8 6 Critical endpoints FS endpoints Tricritical points Critical points Tetracritical points Ν r (b) µ f < 0 Λ r Critical endpoints FS endpoints Tricritical points Critical points Tetracritical points 10 5 Printed by Mathematica for Students Ν r Figure 2.2: Phase diagram in the parameter space {ν (r), µ (r) b, λ (r) } {(ν µ b )/ µ f, µ b /γ µ f 1/2, m 3/2 b λ µ f 3/2 /γ 2 } projected onto the (ν (r), λ (r) ) plane, for (a)µ f > 0 and (b)µ f < 0. For λ (r) above a critical λ (r) crit (λ (r) crit = 8.44 for µ f > 0, and 25.5 for µ f < 0), one only has second order phase transitions, which are independent of λ (r). As λ (r) is lowered below λ (r) crit, one first encounters tricritical points. When λ (r) < λ (r) crit, one first encounters tricritical points, until one reaches a tetracritical point. Beyond such a point one has a line of critical points, a line of critical endpoints, and a line of FS endpoints (see text), although the lines may overlap too strongly to be visible. 20 Printed by Mathematica for Students

33 As one goes out from the center of the trap to the outside regions, the potential energy increases, and the densities will decrease. Within the Local Density Approximation (LDA), one assumes that the chemical potential follows the behaviour opposite to the potential. In other words, within LDA we have µ(r) = µ mω2 r 2. Experiments have shown that the bosons and fermions feel the same potential in the trap[26] (in other words, the trapping frequency depends on the mass precisely so as to cancel the mass dependence of V (r)). Thus the LDA approximation leads to µ f,b (r)/γ 2 = µ f,b (0)/γ 2 (r/r 0 ) 2, (2.17) where r 0 is some arbitrary length scale. To obtain the density profile in the trap, one chooses a value µ f,b (0) of the chemical potentials in the center of the trap. As r varies, so do µ f,b (r), and therefore at each r we can calculate the number densities of the fermions and bosons, and the density of condensed bosons. We will consider values of µ f,b (0) such that one crosses the first order line as r, so as to see the behaviour we are interested in. We choose r 0 to sit at the distance from the center of the trap where one crosses the first order line. We choose to sit at the unitary limit ν = 0, because all possible types of sequence of phase transitions are represented there. The result of three different choices of µ f,b (0) are shown in Fig Experimental consequences After having laid out the physics of our model in detail, we address the question of whether the rich behaviour we are predicting is accessible experimentally. To obtain an estimate for g, we use Eq.(2.13). To estimate λ we use the mean field value[24] λ = 2πa bb m 21 (2.18)

34 Figure 2.3: Density profiles in a harmonic trap at ν = 0 for the rescaled boson number ñ b = n b /(m 3/2 γ 3 ), fermion number ñ f = n f /(m 3/2 γ 3 ) and condensed boson number n cond = ρ, for λm 3/2 b γ The profiles (a), (b) and (c) correspond to three different sequences of phase transitions. The top left diagram shows which trajectory is traced out in chemical potential space, with the arrow pointing along increasing distance r from the center of the trap. r 0 is chosen to coincide with the crossing of the first order line. 22

35 where a bb is the boson-boson scattering length. From Fig. 2.1, we see that the phase separated region occurs at densities of order 100m 3/2 γ 3. In experiments, after cooling the atoms the trap has typically about 10 5 atoms, and the linear size of the cloud is about 0.1 mm, giving a density of about cm 3. The density varies strongly across the trap, particularly if there is a BEC, and the densities at the center of the trap are typically on the order of cm 3. We will now estimate the value of 100m 3/2 γ 3 / h 6 for the three pairs of bosonic and fermionic atoms that have received the most experimental attention: 23 Na 6 Li[29], 7 Li 6 Li[72], and 87 Rb 40 K[26]. Using values quoted in the literature for all the parameters that go into estimating g and λ, we can obtain estimates for 100m 3/2 γ 3 / h 6. The value depends on which Feshbach resonance one considers: 6 Li 7 Li : 100m 3/2 γ cm 3 for B = 1G 87 Rb 40 K : 100m 3/2 γ cm 3 for B = 51mG 87 Rb 40 K : 100m 3/2 γ cm 3 for B = 1G 23 Na 6 Li : 100m 3/2 γ cm 3 for B = 0.6G. We see that these are reasonable densities, except for 87 Rb 40 K at a resonance of width 1G. Therefore, to see the interesting physics in this pair, one would need to tune to a rather narrow Feshbach resonance (the resonance with width B = 51mG has only been predicted theoretically [26], it hasn t been measured yet). 2.5 Comparison to previous work The experiments on 87 Rb 40 K see a collapse of the system[53, 54], as the Feshbach resonance is approached from the a < 0 side. A collapse indicates that the bosons all condense into a BEC. The standard theory used in the literature to study collapse is 23

36 a single-channel model, which is a model without a distinct molecular state. We can easily see within mean field theory why such a model leads to collapse. The single channel Hamiltonian is given by[55, 78, 12] Ĥ SC = +λ d 3 k ( ξ f (2π) 3 k f k f k + ξb k b k b k ) + U d 3 k d 3 k d 3 q BF (2π) 3 (2π) 3 (2π) f 3 k b k b k +q f k q d 3 k d 3 k d 3 q (2π) 3 (2π) 3 (2π) 3 b k b k b k +q b k q (2.19) where U BF = 4π h 2 a BF /m. By making the replacement b k δ k,0 ρ, the zerotemperature energy becomes E = µ b ρ + kf 0 d 3 k (2π) 3 ( ξ f k + U BF ρ) + λρ 2 (2.20) For large ρ one can show that k f ρ. Therefore E U BF ρ 5/2. For a BF < 0 (on the attractive side of the resonance), this implies a collapse. For comparison, let us analyze the asymptotic behaviour of the energy for large ρ. As ρ, we have here that k F = 0 and k Ψ ρ 1/4, which leads to E Cρ 5/4. The constant C depends only on the mass ratio r, and a careful analysis reveals that C is negative for all mass ratios. Therefore, for λ = 0 the system is unstable with respect to collapse: the energy is unbounded as a function of ρ, and all bosons condense into a BEC. For λ > 0, the λρ 2 term stabilizes the system, i.e. the energy becomes bounded from below. As a final point on the single channel model, one can show that it is obtained from the two-channel model by taking g, and ν, while keeping g 2 /ν constant, because this keeps the effective interaction in the single channel model constant (see appendix B). To see how the single channel model is obtained, consider the mean field energy of the single channel model, by replacing b k with δ k,0 ρ in Eq. (2.19): 24

37 E SC = d 3 k ( k 2 µ (2π) 3 f + U BF ρ)θ(µ f k2 U BF ρ) + λρ 2. (2.21) 2m f 2m f Now take the limit ν, g, with g 2 /ν held constant. What happens is that ξ k F, which means it drops out the problem, since it only contributes to the two-channel mean field energy when it is negative. On the other hand, ξ Ψ k k2 2m f µ f ρ g2 ν, which implies that here k Ψ ρ 1/2. Plugging this into the expression for the free energy of the two channel model, Eq. (2.8), we recover the same expression as the mean field energy for the single channel model, E SC, provided we set g 2 /ν = U BF, which is precisely the expression derived in the appendix, Eq. (B.7) (with Γ replaced by g, and g by U BF ). Thus our model encompasses the single channel model, and allows for a more detailed explanation of the physics close to resonance. Note that in the two-channel model we showed that the energy is unbounded, but in the limit ν, g, with g 2 /ν held constant, the minimum of the energy in the two-channel model goes off to. Therefore in this limit the two-channel model collapses, as does the single-channel model. 2.6 Conclusion To conclude, we have shown that contrary to the predictions in a single channel model, our two channel model does not see a collapse to arbitrarily high density, since the energy is always bounded from below. Far away from unitarity, we get that there are only second order phase transitions, while close to unitarity and at low enough densities, the system will tend to phase separate. So far the experiments see what they interpret as a collapse, i.e. a sudden unbounded increase in the density. They may actually be seeing phase separation into a 25

38 phase with such a high density that three body effects dominate. In the experiments, once the density has increased too strongly, three-body recombination effects become important, which we have neglected in our treatment. These effects play an important role in the trap at positive ν: as the density increases, three-body interaction leads to a significant loss of atoms, where one atom gains the kinetic energy that is released when the other atoms go into a deep bound state. The rate for three-body losses is given by [25] ṅ/n h m (na2 ) 2 (2.22) where n is the number density, and a is the scattering length. It therefore increases strongly as the density increases. On the a < 0 side of the resonance, close to unitarity the experiments see a sudden increase in density, after which the condensate is lost due to three-body recombination losses[82, 59]. To avoid these losses, we want to sit at low densities. If we want the dimensionless parameters n f,b /(m 3/2 γ 3 ) to remain constant, so that we are sitting in the interesting region of phase separation (n f,b /(m 3/2 γ 3 ) 100), we have to reduce γ as we lower the density, which can be achieved by reducing B. Thus, in general one should tune to narrow Feshbach resonances to see this behaviour, as we saw above for 87 Rb 40 K. The physics in the repulsive regime, corresponding to negative ν, is different and subtle, because the physics on that side depends on how one tunes to ν. Namely, if one starts at large negative ν, where a(b) is small, and moves towards unitarity, one is accessing a phase of strongly repulsive bosons and fermions. The single channel theory then predicts that the atomic bosons and fermions phase separate. Namely, within mean field theory they feel each other s presence through an effective chemical potential, and become immiscible if a is large enough. Close to unitarity there is a bound state with small negative energy, but the only way for a boson and a fermion to go into the bound state is through a three-body process, because of energy conversation. To access the phase we are describing, where there is a macroscopic number 26