Quantum Gases

The following sections are included:

• Foreword

• Preface

• Participants of FINESS 2009 (Durham)

• Contents

• Common Symbols/Expressions and their Meanings

With Introductory Material, we provide important general background material, particularly in the field of ultracold atomic gases which forms the backbone of much of this book. Relative newcomers to the field will likely find this a necessary prelude before moving on to the generally more advanced material that follows, and even relative sophisticates should find much of the material contained within this part to be a useful resource…

We give a brief historical overview of the physical realisations of quantum degeneracy and Bose–Einstein condensation observed to date, with the aim of showing why quantum gases is a key rapidly evolving interdisciplinary field of physics. We motivate the need for developing more advanced theories to understand all the features observed in ultracold gases, and present some of the unresolved issues where the theories discussed in this book can play an important role.

Experiments with ultracold gases have assumed a significant role in atomic physics with direct and major relevance in condensed matter and many other areas of physics. We give a few examples of studies that have been carried out with these systems since the first production of a Bose–Einstein condensate in a dilute atomic vapour in 1995. We discuss in particular early studies of the formation and growth of condensates, excitations of Bose–Einstein condensates, and more recent investigations of strongly correlated systems, arising in optical lattices, near Feshbach resonances, and in restricted geometries of effiective low dimensionality. The understanding of such experiments requires advanced finitetemperature non-equilibrium theories, the discussion of which forms the basis for this book.

We review the basic experimental techniques that have led to the development and study of large coherent ensembles of atoms which opened up the way to probe experimentally the physics of ultracold quantum gases. In particular, we discuss techniques for achieving quantum degeneracy (laser cooling and trapping, magnetic and optical traps, evaporative cooling), for manipulating (Feshbach resonances and stimulated rapid adiabatic passage) and visualising (e.g. absorption imaging) quantum gases. We discuss some application areas including matter-wave interferometry, optical lattices, rotational experiments, microtraps, ‘atom lasers,’ and cold molecules, that are representative of the breadth of experiments that can be performed with quantum gases, and give a brief overview of the areas where significant experimental activity is likely to occur in the coming years.

We briefly overview commonly encountered theoretical notions arising in the modelling of quantum gases, intended to provide a unified background to the ‘language’ and diverse theoretical models presented elsewhere in this book, and aimed particularly at researchers from outside the quantum gases community.

This part reviews the development, and key applications, of some of the most common theoretical models used in the context of ultracold bosonic atomic gases. These theories stem from the early developments in the modelling of weakly interacting Bose–Einstein condensates (initially aimed at liquid helium), and developments in the fields of condensed-matter physics, quantum optics, and high-energy physics. All the presented theories are based on the binary interaction Hamiltonian described in Chapter 4; their selection is based mainly on their suitability for describing finite-temperature and non-equilibrium settings (selected equilibrium approaches, such as quantum Monte Carlo, are thus left out)…

The kinetic theory of gases, remarkably successful in understanding classical gases, is based on a separation-of-timescales argument valid for sufficiently dilute samples. One would therefore expect an appropriate generalisation to dilute quantum gases to be equally promising. The exact form and applicability of such a generalisation naturally depends on the approximations made and the exact physical scenario under consideration. With the exception of Chapter 9, the approaches presented in this part are of a perturbative nature, assuming a large Bose–Einstein condensate fraction. The condensate is typically defined in terms of symmetry breaking, which sacrifices the concept of an exactly defined total particle number, although this can be avoided, as is addressed in Chapter 8…

We review a self-consistent scheme for modelling trapped weakly interacting quantum gases at temperatures where the condensate coexists with a significant thermal cloud. This method has been applied to atomic gases by Zaremba, Nikuni, and Griffin, and is often referred to as ZNG. It describes both meanfield-dominated and hydrodynamic regimes, except at very low temperatures or in the regime of large fluctuations. Condensate dynamics are described by a dissipative Gross–Pitaevskii equation (or the corresponding quantum hydrodynamic equation with a source term), while the non-condensate evolution is represented by a quantum Boltzmann equation, which additionally includes collisional processes which transfer atoms between these two subsystems. In the mean-field-dominated regime, collisions are treated perturbatively, and the full distribution function is needed to describe the thermal cloud, while in the hydrodynamic regime the system is parametrised in terms of a set of local variables. Applications to the finite-temperature-induced damping of collective modes and vortices in the mean-field-dominated regime are presented.

We summarise a method of extended mean fields to derive kinetic equations for mean fields and the fluctuations around them. For bosonic gases, this gives a consistent interpolation between the coherent Gross—Pitaevskii mean-field equation and the time-dependent Hartree–Fock–Bogoliubov equations, which are valid at low temperatures, as well as the classical Boltzmann equation, which is valid in the high-temperature regime. This extension to multiple mean fields exhibits long-range order, finite-temperature equilibria, as well as irreversible, non-equilibrium relaxation. Numerical examples from the reversible, coherent dynamics and the irreversible thermalisation of trapped gases illustrate the concepts. The successful prediction of fermionic resonant superfluidity using two channel Feshbach resonances is also based on this method.

We discuss the cumulant approach to the non-equilibrium dynamics of strongly interacting ultracold atomic gases. After a general introduction, we derive a non-Markovian, nonlinear Schrödinger equation for a Bose–Einstein condensate, and a non-Markovian Boltzmann equation for the one-body density matrix of a thermal Bose gas. We apply these equations to the dynamics of Feshbach molecule production.

Assuming the existence of a Bose–Einstein condensate composed of the majority of a sample of ultracold, trapped atoms, perturbative treatments to incorporate the non-condensate fraction are common. Here we describe how this may be carried out in an explicitly number-conserving fashion, providing a common framework for the work of various authors; we also briefly consider issues of implementation, validity, and application of such methods.

We review the multiconfigurational time-dependent Hartree method for bosons, which is a formally exact many-body theory for the propagation of the time dependent Schrödinger equation of N interacting identical bosons. In this approach, the time-dependent many-boson wavefunction is written as a sum of all permanents assembled from M orthogonal orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and determined according to the Dirac–Frenkel time-dependent variational principle. In this way, a much larger effective subspace of the many-boson Hilbert space can be spanned in practice, in contrast to multiconfigurational expansions with timeindependent configurations. We also briefly discuss the extension of this method to bosonic mixtures and resonantly coupled bosonic atoms and molecules. Two applications in one dimension are presented: (i) the numerically exact solution of the time-dependent many-boson Schrödinger equation for the population dynamics in a repulsive bosonic Josephson junction is shown to deviate significantly from the predictions of the commonly used Gross–Pitaevskii equation and Bose–Hubbard model; and (ii) the many-body dynamics of a soliton train in an attractive Bose–Einstein condensate is shown to deviate substantially from the widely accepted predictions of the Gross—Pitaevskii mean-field theory.

The theoretical descriptions presented in Chapters 10–14 have a stark distinction from most approaches presented in Part II.A, in that they abandon the idea of a well-defined single-mode condensate coupled to its excitations, in favour of a cumulative description of the low-lying modes of the system; these modes are intended to encompass both the condensate and the modes that become significantly affected by its presence. While such theories have a common scope and philosophy, there are subtle differences between the models used, and even in their respective implementations…

We review c-field methods for simulating the non-equilibrium dynamics of degenerate Bose gases beyond the mean-field Gross–Pitaevskii approximation. We describe three separate approaches that utilise similar numerical methods, but have distinct regimes of validity. Systems at finite temperature can be treated with either the closed-system projected Gross–Pitaevskii equation (PGPE), or the open-system stochastic projected Gross–Pitaevskii equation (SPGPE). These are both applicable in quantum degenerate regimes in which thermal fluctuations are significant. At low or zero temperature, the truncated Wigner projected Gross Pitaevskii equation (TWPGPE) allows for the simulation of systems in which spontaneous collision processes seeded by quantum fluctuations are important. We describe the regimes of validity of each of these methods, and discuss their relationships to one another, and to other simulation techniques for the dynamics of Bose gases. The utility of the SPGPE formalism in modelling non-equilibrium Bose gases is illustrated by its application to the dynamics of spontaneous vortex formation in the growth of a Bose–Einstein condensate.

We review the stochastic Gross–Pitaevskii approach for non-equilibrium finitetemperature Bose gases, focusing on the formulation of Stoof; this method provides a unified description of condensed and thermal atoms, and can thus describe the physics of the critical fluctuation regime. We discuss simplifications of the full theory, which facilitate straightforward numerical implementation, and how the results of such stochastic simulations can be interpreted, including the procedure for extracting phase-coherent (‘condensate’) and density-coherent (‘quasi-condensate’) fractions. The power of this methodology is demonstrated by successful ab initio modelling of several recent atom chip experiments, and by analysing dark soliton decay within a phase-fluctuating condensate.

We discuss an implementation of the ‘classical-field’ method for Bose gases. This approach treats all modes up to a certain energy cutoff classically (i.e. the atoms do not relax to the full Bose–Einstein distribution, but instead to the equipartition distribution). We present our approach to choosing the cutoff, and show that this can be used to reproduce the probability distribution for the number of non-condensate atoms for an ideal gas. We describe how to generate an initial state for a given non-zero temperature for classical dynamics, and we mention some recent applications of the method.

We discuss stochastic phase-space methods within the truncated Wigner approximation and show explicitly that they can be used to solve for the non-equilibrium dynamics of bosonic atoms in one-dimensional (1d) traps. We consider systems both with and without an optical lattice, and address different approximations in the stochastic synthesisation of quantum statistical correlations of the initial atomic field. We also present a numerically efficient projection method for analysing correlation functions of the simulation results, and demonstrate physical examples of the non-equilibrium quantum dynamics of solitons and atom number squeezing in optical lattices.

We review several number-conserving stochastic field methods for equilibrium and time-dependent Bose gases; these range from classical-field to exact methods, and include truncated Wigner (with an explicitly number-conserving implementation), and, only for equilibrium, a semiclassical field method. Stochastic elements in the initial state mimic thermal fluctuations in the classical field, or thermal and quantum fluctuations in the Wigner, semiclassical and exact methods. Time evolution is deterministic with the nonlinear Schrödinger equation for the classical field and Wigner methods, while it is stochastic for the exact method. We illustrate each method by relevant applications to quantum gases.

The positive-P representation approach to quantum dynamics in Bose gases is presented. This first-principles stochastic method allows real-time quantumdynamical calculations to be carried out. Examples are given for experimentally tested predictions of soliton quantum squeezing and quantum correlations in Bose–Einstein condensate collisions.

We discuss the functional-integral approach to far-from-equilibrium quantum many-body dynamics. Specific techniques considered include the two-particle-irreducible effective action and the real-time flow-equation approach. Different applications, including equilibration after a sudden parameter change and non-equilibrium critical phenomena, illustrate the potential of these methods.

This part is intended to provide an unbiased view (insofar as that is possible) of the relative merits of the various theoretical approaches presented thus far, in order to provide the reader with as much assistance as possible in forming their own views on this complicated and not uncontroversial matter…

One of the aims of the organisers of the FINESS conferences, and of the editors of this book, has been to encourage dialogue between researchers working in the general area of non-equilibrium superfluids. The researchers come from widely different backgrounds, and come with a broad array of favourite theoretical techniques. In this chapter we present an incomplete survey of figures from previously published papers that make a comparison between selected subsets of different theoretical methods. The goal is that this compilation of figures, when placed in a broader context, will provide some background for the reader to understand the physical conditions that determine when various theories are useful.

The standard theoretical basis for understanding superfluidity in Bose systems was formulated by Beliaev in 1957, based on splitting the quantum field operator into a macroscopically occupied condensate component and a non-condensate component. This leads to a description of the condensate in terms of a ‘single-particle state,’ the so-called macroscopic wavefunction. Since the discovery of Bose-condensed gases, an alternative theoretical picture has been developed which is based on a ‘coherent band’ of classically occupied states. This is often called the classical or c-field approach. The goal of this chapter is to review the differences between the Beliaev broken-symmetry and c-field approaches, and to argue that the c-field concept of a coherent condensate band of states has problems as a description of Bose superfluidity. However, the c-field idea of treating the lowest-energy excitations classically can be used to advantage to simplify calculations within the Beliaev broken-symmetry formalism.

We present our views on the issues raised in the previous chapter by Griffin and Zaremba. We review some of the strengths and limitations of the Bose symmetry-breaking assumption, and explain how such an approach precludes the description of many important phenomena in degenerate Bose gases. We discuss the theoretical justification for the classical-field (c-field) methods, their relation to other non-perturbative methods for similar systems, and their utility in the description of beyond-mean-field physics. Although it is true that present implementations of c-field methods cannot accurately describe certain collective oscillations of the partially condensed Bose gas, there is no fundamental reason why these methods cannot be extended to treat such scenarios. By contrast, many regimes of non-equilibrium dynamics that can be described with c-field methods are beyond the reach of generalised mean-field kinetic approaches based on symmetry breaking, such as the ZNG formalism.

Much of this book considers, as a common system, a trapped, dilute gas of bosonic atoms. Since the observation of Bose–Einstein condensation in ⁸⁷Rb and ²³Na vapours in 1995, such systems have received the most experimental and theoretical attention within the ultracold atoms community. However there has also been an explosion of activity in the study of atoms in optical lattices, as well as important progress in the study of atomic Fermi gases, and low-dimensional and near-integrable systems. Formally similar systems, in terms of their theoretical study, also include exciton–polariton condensates in the solid state, and of course superfluid liquid helium…

We review exact approaches and recent results related to the relaxation dynamics and description after relaxation of various one-dimensional lattice systems of hard-core bosons after a sudden quench. We first analyse the integrable case, where the combination of analytical insights and computational techniques enable the study of large system sizes. Thermalisation does not occur in this regime. However, after relaxation, observables can be described by a generalisation of the Gibbs ensemble. We then utilise full exact diagonalisation to study what happens as integrability is broken. We show that thermalisation does occur in finite nonintegrable systems provided that they are sufficiently far away from the integrable point. We argue that the onset of thermalisation can be understood in terms of the eigenstate-thermalisation hypothesis.

We give an introduction to the time-evolving block decimation algorithm, which can be used to compute ground states and time-dependent many-body dynamics for one-dimensional (1d) systems. The method is particularly well suited to lattice and spin models, and has been applied to the computation of the dynamics of cold atoms in optical lattices for realistic experimental parameters and system sizes.

We review the basic theory of matrix product states (MPS) as a numerical variational ansatz for time evolution, and present two methods to simulate finite temperature systems with MPS: the ancilla method and the minimally entangled typical thermal state method. A sample calculation with the Bose–Hubbard model is provided.

We derive the bosonic dynamical mean-field equations for bosonic atoms in optical lattices with arbitrary lattice geometry. The equations are presented as a systematic expansion in 1/z, z being the number of lattice neighbours. Hence the theory is applicable in sufficiently high-dimensional lattices. We apply the method to a two-component mixture, for which a rich phase diagram with spin order is revealed.

The first successful macroscopic theory for the motion of superfluid helium was that of Lev Landau (1941) in which the fluid is modelled phenomenologically as an interpenetrating mixture of a superfluid and a normal fluid. It was later shown that Landau's two-fluid model can be derived from a one-fluid model within the classical-field approximation. Assuming a separation of scales exists between the slowly varying, large-scale, background (condensate) field, and the short rapidly evolving excitations, a full description of the kinetics between the condensate and the thermal cloud can be obtained. The kinetics describes three-wave and fourwave interactions that resemble the C₁₂ and C₂₂ terms, respectively, in the collision integrals of the ZNG theory (Chapter 5). The scale-separation assumption precludes analysis of the healing layer and thus does not include the dynamics of quantised vortices. Whilst the analysis required the use of small parameters arising from the scale-separation assumption and the assumption of a weakly depleted condensate, we expect the results to hold over a wider range of parameters. This is motivated by the validity of Landau's two-fluid model which can be derived from a one-fluid model using nothing more than the principle of Galilean invariance. Indeed, we argue that similar arguments can be used to recover a two-fluid model directly from a classical field simply by invoking a local gauge transformation. This derivation does not require any small parameters to be introduced, suggesting that the results that lead to the kinetic equations may turn out to be more general.

This chapter provides a brief overview of the theoretical tools used to describe a superfluid Fermi gas. Based on mean-field theory, we present results for the equilibrium state of a Fermi gas with Cooper pairing for both spatially homogeneous and inhomogeneous systems. We discuss how these results provide a qualitative description of the crossover from a weakly coupled BCS superfluid to a molecular condensate in the strong-coupling limit. Finally, we introduce the time-dependence of the mean-field equations for non-equilibrium situations, and make a connection to superfluid hydrodynamics.

The time-dependent superfluid local density approximation (TDSLDA) is an extension of the Hohenberg–Kohn density functional theory (DFT) to time-dependent phenomena in superfluid fermionic systems. Unlike linear-response theory, which is only valid for weak external fields, the TDSLDA approach allows the study of nonlinear excitations in fermionic superfluids, including large-amplitude collective modes, and the response to strong external probes. Even in the case of weak external fields, the TDSLDA approach is technically easier to implement. We illustrate the implementation of the TDSLDA for the unitary Fermi gas, where dimensional arguments and Galilean invariance simplify the form of the functional, and ab initio input from quantum Monte Carlo simulations fix the coefficients to quite high precision.

We review phase-space simulation techniques for fermions, showing how a Gaussian operator basis leads to exact calculations of the evolution of a many-body quantum system in both real and imaginary time. We apply such techniques to the Hubbard model and to the problem of molecular dissociation of bosonic molecules into pairs of fermionic atoms.

In recent years, experiments by several groups have demonstrated spontaneous coherence in polariton systems, which can be viewed as a type of non-equilibrium Bose–Einstein condensation. In these systems, the polariton lifetime is longer than, but not much longer than, the polariton–polariton scattering time which leads to the thermalisation. By contrast, over the past 20 years several groups have pursued experiments in a different system, consisting of indirect excitons in coupled quantum wells, which has very long exciton lifetime, up to 30 μs or more, which is essentially infinite compared with the thermalisation time of the excitons. Thermal equilibrium of this type of exciton in a trap has been demonstrated experimentally. In coupled quantum wells, the interactions between the excitons are not short-range contact interactions, but instead are dipole–dipole interactions, with the force at long range going as 1/r³. Up to now there has not been a universally accepted demonstration of BEC in coupled quantum wells, and the way forward will require better understanding of the many-body effects of the excitons. This chapter reviews what has been learned and accomplished in the past two decades in the search for an equilibrium BEC in this promising system.

In this chapter, we review the recent experimental and theoretical advances in the study of the effects of non-equilibrium Bose–Einstein condensation in gases of exciton–polaritons in semiconductor microcavities. Given the short lifetime of polaritons, some pumping mechanism is in fact required to compensate for losses and keep the system in a non-equilibrium steady state. This makes polaritons a unique testbed in which to study effects of quantum statistical mechanics in a novel non-equilibrium framework. After a short historical account of the route towards polariton condensation, we summarise the most notable dynamical properties of polariton condensates: special focus is given to those features that originate from their driven-dissipative nature. Finally, the many questions that are still open about the superfluidity of polariton condensates are reviewed, together with the first evidence of superfluidity that has been recently observed in experiments.

As discussed in Chapters 28 and 29, solid-state quantum condensates can differ from other condensates, such as helium, ultracold atomic gases, and superconductors, in that the condensing quasiparticles have relatively short lifetimes, and so, as for lasers, external pumping is required to maintain a steady state. In this chapter we present a non-equilibrium path-integral approach to condensation in a dissipative environment and apply it to microcavity polaritons, driven out of equilibrium by coupling to multiple baths, describing pumping and decay. Using this, we discuss the relation between non-equilibrium polariton condensation, lasing, and equilibrium condensation.

The following sections are included:

• References

• Author Index

• Subject Index