Ultracold gases far from equilibrium

Ultracold gases far from equilibrium

Abstract

Ultracold atomic quantum gases belong to the most exciting challenges of modern physics. Their theoretical description has drawn much from (semi-) classical field equations. These mean-field approximations are in general reliable for dilute gases in which the atoms collide only rarely with each other, and for situations where the gas is not too far from thermal equilibrium. With present-day technology it is, however, possible to drive and observe a system far away from equilibrium. Functional quantum field theory provides powerful tools to achieve both, analytical understanding and numerical computability, also in higher dimensions, of far-from-equilibrium quantum many-body dynamics. In the article, an outline of these approaches is given, including methods based on the two-particle irreducible effective action as well as on renormalisation-group theory. Their relation to near-equilibrium kinetic theory is discussed, and the distinction between quantum and classical statistical fluctuations is shown to naturally emerge from the functional-integral description. Example applications to the evolution of an ultracold atomic Bose gas in one spatial dimension underline the power of the methods.

1 Introduction

Five millimeters of mere nothing separate the micrometer-scale ultracold cloud of sodium atoms from the glass walls of its surrounding vacuum cell in a typical experiment in the basement of maybe the reader’s research institution. Five millimeters between the glass at 293 K and the Bose-Einstein condensed gas at a few nanokelvin. These eleven orders of magnitude in temperature are to be compared to the eight orders the temperature in a supernova is higher than that in our office. And the vacuum pressure of Pa which is quantifying ”mere nothing” is similar to the atmospheric pressure at the surface of the moon.

Bose-Einstein condensation, a phenomenon predicted over 70 years ago by Albert Einstein (1) on the basis of Bose’s new statistical formulation of a photon gas (2), has revolutionised atomic physics since its ground breaking experimental achievement in dilute alkali gases at JILA (Boulder) (3), MIT (4), and Rice University (5). By today, more than 80 groups worldwide dispose of techniques to produce ultracold and Bose-Einstein condensed gases, mostly of alkalis like Rb, Na, Li but also including metastable He, as well as K, Cr, Rb, Cs, Yb. Another 100 groups need to be included when counting the wider range of experiments studying cold atomic gases, atom optics, trapping, cooling, and many more subjects (6). The past decade has seen an exploding range of experiments studying many different properties of such systems, thereby varying densities, atom numbers, dimensionality, interaction strengths, internal (electronic) state multiplicities, as well as character, geometry, shape, size and temporal behaviour of the external trapping potentials, both, between different runs and during the experiment’s timeline. A range of review articles and monographs on theory and experiment can be consulted (7); (8); (9); (10); (11). During the past five years, a growing number of experiments with ultracold Fermi gases has been added to the spectrum of activities, see, e.g., Refs. (12); (13); (14); (15); (16). For recent reviews see Ref. (17). Today, ultracold fermions are regarded as a promising tool to design many-body quantum systems exhibiting many of the phenomena relevant in solid state systems, in particular in the context of superconductivity, and to explore these beyond the range of parameters realistic for degenerate electron gases. Moreover, new efforts aim at a cross-fertilisation between high-energy physics, in particular concerning the phase structure of quantumchromodynamics, and the physics of strongly correlated atomic gases.

The quantum degeneracy of ultracold gases results from the (anti-)symmetrisation principle for the wave function describing indistinguishable particles, together with the statistical behaviour of the many-body system. A non-interacting gas is characterised, for given mean energy and particle number, by the occupation numbers of the single-particle eigenmodes. Bose-Einstein condensation emerges as the macroscopic occupation of a single mode, at zero as well as finite temperatures (18). Although interactions are not required for the existence of this quantum degeneracy, they are, in a realistic physical system, indispensable for reaching the respective degenerate equilibrated state. In fact, in experiment, the key last step to degeneracy is induced by evaporative cooling (7), where collisional relaxation following a removal of the high-frequency tail of atoms restores an equilibrium distribution.

Trapped atomic gases provide the unique possibility to tune both, the interaction strength between the particles and the external boundary conditions fixed by the trapping potential: External electromagnetic fields can be used to considerably vary, in particular near (Feshbach) zero-energy scattering resonances, the scattering length which quantifies the collisional interactions (19). Thereby, laser light, combined with elaborate lensing technology, allows for almost arbitrary trapping geometries. Most of these tuning knobs can be turned so quickly as to excite many-body dynamics far away from a thermal or metastable equilibrium state.

What do we mean by “far-from-equilibrium” dynamics? Consider a non-integrable many-body system. Far from equilibrium, in contrast to close to it, there is no longer the notion of a precisely defined spectrum of quasiparticle modes whose damping can be described on the grounds of a linear-response analysis. The latter generally relies on a perturbative expansion in some small parameter, and analytic relations between the fluctuation and response functions reflect the principle of detailed balance in Boltzmann’s kinetic theory (20); (21). In far-from-equilibrium or, as it is also often called, nonequilibrium time evolution there is no such fluctuation-dissipation relation, while microcausality and microreversibility are still conserved. When studying the transition from far-from- to near-equilibrium dynamics one of the key questions is how the known near-equilibrium features are recovered during the time evolution of the system, given specific interaction properties. An interesting observation gives rise to the distinction between short- and long-time evolution after a sudden quench of some boundary conditions which produces an initial state far from equilibrium. In particular long-time many-body evolution and equilibration are demanding and still largely unresolved problems.

An important issue when studying nonequilibrium dynamics is the level of approximation on which interaction effects are taken into account. As pointed out above, interactions are inextricably linked with nonequilibrium dynamics. If they are weak, i.e., occur rarely which is the case in a dilute gas, low-order perturbative approximations in the diluteness parameter can provide a reliable description over a certain time. At large times, however, such perturbative descriptions are expected to break down. Moreover, if the interactions are sufficiently strong, the time at which this breakdown occurs can be shorter than the time at which near-equilibrium kinetic theory, which in general involves perturbative approximations, sets in to be valid.

For systems with large, i.e., classical occupations of the kinematically relevant modes, quantum fluctuations typically play a minor role and, if classical fluctuations are relevant, Monte Carlo simulations are often the method of choice. However, if quantum fluctuations become important, no such methods are at hand, and this is generically the case for long-time evolutions and dense, strongly interacting systems. (Note that recently, for a certain range of applications, stochastic simulation techniques have been studied (22); (23); (24); (25).) Functional quantum field theoretical techniques represent a powerful approach to such dynamics. Moreover, they provide analytical insight and make numerical computations feasible, in particular in more than one spatial dimension.

This article provides an introduction to functional quantum field theoretical methods to describe far-from-equilibrium many-body quantum dynamics. It is beyond its scope to represent a full review and to provide a satisfactory account of the relevant literature. In Section 2, we will define the relevant observables and give a brief summary of mean-field theory for the short-time and near-equilibrium evolution of a Bose gas. In Section 3, we shall then discuss the nonequilibrium two-particle irreducible (2PI) effective action (26); (27); (28) in the nonperturbative approximation introduced in Refs. (29); (30). This nonequilibrium approach has been developed and extensively applied, in the context of relativistic quantum field theories, as reported in Refs. (29); (30); (31); (32); (33); (34); (35); (36); (37). Their extension to abelian and non-abelian gauge theories (38); (39); (40); (41) is the subject of recent and ongoing research. We will, furthermore, summarise a new functional renormalisation-group approach to far-from-equilibrium dynamics introduced in (42). For applications, the focus will be set on interacting ultracold atomic Bose gases (43); (44); (45); (46) and an overview be given of the results first presented in Refs. (46); (47); (48); (49). Example applications to the long-time evolution of an interacting Bose gas will be discussed. The relation to near-equilibrium evolution is described in Section 4. Quantum statistical fluctuations and their distinction from their classical counterparts are the subject of Section 5. A summary will be given in Section 6.

2 Mean-field dynamics of Bose-Einstein condensates

In this section we give an introduction to mean-field theory of time-evolving ultracold Bose gases. Mean-field theory is generically valid for the description of near-equilibrium dynamics and has been applied successfully to understand and predict an enormous variety of experimental observations, see, e.g. (10); (11). Discussing it allows us to lay the foundations for the later development of far-from-equilibrium dynamics and dynamics of strongly interacting systems. We will first define the observables of interest in the context of most experiments and then give a concise introduction to the Gross-Pitaevskii and Hartree-Fock-Bogoliubov theories of Bose-Einstein condensates near their ground-state configuration. We close the section with an outlook beyond mean-field theory, focusing on atomic gases near a Feshbach resonance as well as on an exact method to calculate the nonequilibrium dynamics of an interacting Bose gas in one spatial dimension.

2.1 Observables

The application of statistical many-body theory to quantum degenerate states, e.g. of non-interacting Bose-Einstein condensates (BEC), aims at the occupation number distribution over the available energy eigenstates. Quantum states, however, contain additional information which manifests itself in the phase of the wave function. This phase gives rise to coherence and interference phenomena which, in a number of experiments with degenerate atomic gases, has been made visible in a macroscopic manner, e.g., in the experiments reported in Refs. (50); (51); (52); (53); (54). Let us discuss in more detail the observables required to describe such properties of the many-body system.

Large occupation numbers, together with the phase of the single-particle wave function, lead to an approximate description of coherent matter in terms of a scalar complex field . Here and in the following time and space variables are included in the four-vector . In view of the statistical properties of BEC, forms an order parameter. With respect to coherence properties as well as large occupation numbers, the matter-wave field exists in full analogy with, e.g., Maxwell’s classical electromagnetic field . Recall that an ideal, single-mode laser can be described by a coherent state which characterises it as a coherent superposition of photon number states,

(1)

The expansion coefficients are chosen such that is an eigenstate of the Fock annihilation operator with eigenvalue . For photons propagating in vacuum, the time evolution of the coherent state reads , being the frequency of the mode. This corresponds, in spatial representation, to a classical oscillation of the wave packet in the oscillator potential . Hence, the expectation value of the electric field operator performs the harmonic oscillations of the macroscopic classical field.

For an ideal-gas BEC, the field can be written as , where has the functional form of the quantum mechanical single-particle wave function whose macroscopic mean occupation number is . Hence, while describes the particle density at , its phase gives rise to the same interference phenomena as the wave function does for single particles. The complex matter-wave field describes the spatial and temporal density and phase distributions of the coherent cloud of atoms in a particular internal state. This analogy with photons gives rise to the notion of an atom laser in experiments where, e.g., a coherent beam of particles is coupled out from a trapped BEC (51); (52); (53).

A remark is in order: Only compact systems are experimentally relevant. According to the above picture, is the expectation value of the complex non-relativistic quantum field operator obeying the bosonic equal-time commutation relations . In finite, closed, non-relativistic gatherings of atoms, the total number of atoms is a conserved quantity, i.e., the expectation value of this field operator with respect to the reduced density matrix describing any subsystem necessarily vanishes, . Therefore, a coherent state can not describe such a system. This is closely related to the fact that an isolated system does by definition not interfere with any other system such that its total phase can not be measured. Phase can only be measured by means of interference effects, and hence only the relative phase between different (sub)systems is a meaningful quantity.

This leads to the concept of the phase coherence of a BEC. This coherence manifests itself in the off-diagonal elements of the reduced single-particle density matrix, i.e., the single-time two-point correlation function

(2)

The local particle number density is given by the diagonal elements, . For an infinite uniform ideal-gas BEC, one finds that the first-order coherence function derived from the off-diagonal elements, for . Since the momentum distribution of particles is given by the spatial Fourier transform of with respect to the relative coordinate , this asymptotic off-diagonal finiteness implies a macroscopic occupation of the zero mode, i.e., Bose-Einstein condensation. It is termed Off-Diagonal Long-Range Order (ODLRO) and allows the asymptotic factorisation of the single-particle density matrix,

(3)

ODLRO was introduced by Penrose and Onsager as a general criterion for interacting BEC (55). It is applicable also in non-uniform systems alternative to the field expectation value and remains meaningful for number-conserving states. The asymptotic factorisation in Eq. (3) defines as an in general complex order parameter for BEC. In finite systems, once “long-range” is quantified, it can be taken as a measure for local order. It clearly expresses the fixed phase relation between distant points in the BEC. In the thermodynamic limit can be identified with the field expectation value .

In summary, the field expectation value is a useful concept for most cases where a system involves macroscopic occupations . When comparing a formulation in terms of with that based on a density matrix with fixed total particle number differences in the observables are suppressed at least with a factor .

In the following, the classical field

(4)

will be regarded as the non-vanishing expectation value of the field operator evaluated at some time with respect to the density operator at some initial time , where the time dependence of is implied to include the time evolution from to . Besides this one-point function we will use, in the following, the time-ordered connected Green function

(5)

where denotes time ordering of the subsequent operators and is the operator of fluctuations around the classical field . For the first, the field indices are chosen to number and as independent components, . The time ordering allows us to write as

(6)

where and involve the anticommutator and commutator of the fields, respectively,

(7)

In the --basis, one has , , etc., and is related to the single-particle density matrix , , and to the pair function as follows

(8)

As will be discussed in more detail in Sect. 4, is called the statistical correlation function, containing, near equilibrium, information about the occupation of the available states, while the spectral correlation function provides the frequencies and widths of these states. Near equilibrium, and are linked by a fluctuation-dissipation relation. quantifies the amount of pair correlations in the system, as, e.g., the number of atoms bound in pairs or, for fermions, the number of long-range correlated (Cooper) pairs, see Sect. 2.4.1.

Connected -point functions with contain information about higher-order correlations and are defined analogously. In this section we will concentrate on .

2.2 The Gross-Pitaevskii equation

This subsection intends to give a concise summary of the non-relativistic classical equation of motion for the field expectation value which represents the leading-order approximation to the full quantum dynamics in the case that . It was first studied by Gross (56) and Pitaevskii (57) in the context of vortices in superfluid helium. The equation for a field describing an ideal-gas BEC according to our above discussion has the same form as the Schrödinger equation for the single-particle wave function and is a special case of the Gross-Pitaevskii equation (GPE) which, in addition, includes the leading-order effects of interactions between the particles.

We define the system to be studied by the many-body Hamiltonian for a single-species of non-relativistic particles,

(9)

with the Hamiltonian for a single particle exposed to an external, e.g., trapping potential ,

(10)

and the two-body potential describing the interactions between the particles. The interactions are assumed to depend only on the relative coordinate between the collision partners as, e.g., in the Born-Oppenheimer approximation of atom-atom collisions. The GPE is then obtained from the Liouville equation , where denotes the expectation value with respect to the density operator at time , by approximating the 3-point function as a product of field expectation values, :

(11)

Here, the contact-potential approximation has been chosen for , with , where is the -wave scattering length. In Sect. 2.4.1 below, we will discuss in more detail the scattering theory behind this parametrisation, and note here only, that this approximation renders the GPE to be a low-energy effective theory: The typical length scale characterising the motion of atoms in a BEC, the thermal de Broglie wave length , is generally much larger than the scale determining the short-distance behaviour of the interatomic potential. The atoms therefore only see an averaged interaction potential which at large internuclear distance manifests itself in a scattering phase shift. At low energies, given that the potential is sufficiently short ranged, only the -wave scattering amplitude survives which tends to a constant related to the phase shift and equal to the minus the -wave scattering length at vanishing scattering momentum. The scattering length also quantifies the spatial extent of the highest excited dimer bound state close to the dissociation threshold and therefore the pair correlation length. Setting , Eq. (11) has the form of the Schrödinger equation.

We note that, in order to have Bose condensation in an ideal gas, needs to exceed the interatomic distance. In turn, for the GPE to be a good approximation, the interatomic distance must be much larger than the scattering length . This means, the gas needs to be dilute, such that multiple scattering effects play no role. One says, the gas is weakly interacting.

The quartic interaction term in the Hamiltonian shows, that the local energy density rises with increasing particle density if is positive. On the other hand, if is negative, the interaction part is lowered by increasing the local density of particles. Therefore, a positive scattering length is said to describe repulsive interactions while negative corresponds to attractive ones. Note that despite this, also at positive scattering lengths the interatomic potential can support bound states and therefore exert attractive forces. Also the trap potential plays a crucial role as it, e.g., can stabilise a BEC of atoms with , since the kinetic part of the energy rises with increasing curvature of the field at the density peak, see Eq. (9).

We briefly discuss stationary solutions of the GPE in the presence of a trapping potential, with the time dependence , i.e., solutions of the time-independent GPE

(12)

For positive , the dilute-gas BEC is often termed strongly interacting1 if the kinetic part can be neglected within the total energy. This is called the Thomas-Fermi (TF) regime, in which the density distribution obtained from (12) reflects directly the shape of the potential:

(13)

Only at the edge of the cloud, where the density vanishes, the GP approximation breaks down. A BEC in a harmonic-oscillator trap with frequency is in the TF regime if the total number of particles times the scattering length is much larger than the oscillator length , i.e., . The TF radius characterising the extent of the cloud results from the requirement that particles fit into the profile (13). Clearly, in the TF regime, the shape and size of the atom cloud is distinctly different from that of a nearly ideal gas, which is, to a good approximation, given by the modulus squared of the single-particle wave function.

Consider, now, the GP dynamics disclosed by the non-linear field equation (11). Writing the field in terms of density and phase, , one derives, by inserting this into the GPE, the hydrodynamic equations

(14)
(15)

where the velocity field is proportional to the gradient of the phase, . Eq. (14) is the continuity equation expressing local number conservation while Eq. (15) represents a quantum version of the Euler equation describing a frictionless fluid. The quantum contribution adding to the classical Euler equation reflects the zero-point fluctuations encoded in the term proportional to in Eq. (15): A curved mean-field profile is subject to a quantum pressure which aims at flattening the density distribution. We note that, since the velocity is a conservative or gradient field, the Euler equation describes, on a singly connected region of space, irrotational flow. As an example which exhibits in a nice way the consequences of this irrotationality consider the motion of a Bose-Einstein condensate which resembles that of a scissors mode first discussed in the context of nuclear physics. A condensed cloud trapped within an anisotropic, ellipsoidal potential which is excited by suddenly rotating the trap away from its prior position, starts oscillating around the new equilibrium orientation (58); (59); (60). Although the oscillation of the density distribution resembles that of a rotation of the cloud, the velocity field shows that the flow pattern of particles is rather irrotational.

Irrotationality of the flow is one signature of the superfluidity present in a system with Bose-Einstein condensation. A closely related and experimentally demonstrated property of such a system is the possibility of vortex formation. The Gross-Pitaevskii equation possesses non-linear solutions describing a circular flow around a singularity at which the density vanishes, i.e., the phase accumulates an integer multiple of along one turn around the singular point (in 2 dimensions) or line (in 3D). Vortices can be excited in trapped BECs (61) using circular polarised laser beams (62) and have been observed to form Abrikosov lattice structures (63) as known from liquid Helium (64). For a recent report on simulations see Ref. (65).

The GPE describes a colourful range of other nonlinear classical phenomena like solitons and nonlinear atom optics, phenomena which have been studied in many experiments and provide the frame of a research field in its own. See, e.g., Refs. (8); (11). With the advent of the formation of bosonic pairs in ultracold Fermi gases classical nonlinear dynamics can be studied in even more systems.

2.3 Beyond the GPE: Hartree-Fock-Bogoliubov mean-field theory

As discussed in the previous section, condensates exhibit superfluidity. The Gross-Pitaevskii equation for the order parameter field includes an Euler-like equation for a perfect fluid. Superfluidity, in turn, does not require a non-vanishing condensate order parameter as is known from the low-temperature physics of helium which is a non-dilute and therefore strongly interacting system. To describe BEC away from zero temperature and vanishing interactions, as well as away from thermal equilibrium, fluctuations need to be considered beyond the Gross-Pitaevskii approximation. In Section 2.1 we identified as a practically suitable measure for the asymptotic off-diagonal long-range order contained in the full two-point correlation function. Describing a Bose gas beyond the GP approximation requires additional information about the more local properties and dynamics of the two-point function, i.e., about the connected correlation function as well as about the back-reaction of this onto the evolution of . In the GPE the terms accounting for this back-reaction were neglected when approximating as a product of field expectation values.

Time-dependent HFB equations

The dynamic equation for can be derived, as before, from the Liouville equation , where is now to be replaced by the respective products of field operators . Rewriting all correlation functions appearing in the equations in terms of their cumulants, i.e., in terms of connected Green functions and neglecting all cumulants of order three and higher one obtains the equations

(16)
(17)
(18)

where denotes the one-body Hamiltonian. The generalised GPE (16) and the equations (2.3.1) and (2.3.1) for the connected propagator contain, through the normal and anomalous density matrices and , respectively, only the statistical two-point function , cf. Eqs. (6), (2.1). The above equations are commonly termed the (time-dependent) Hartree-Fock-Bogoliubov (HFB) equations (66); (67); (68) which describe the mean-field dynamics beyond the GPE in leading order in the coupling . They form a closed system of partial differential equations which describe the coupled dynamics of the condensate and noncondensate components of an ultracold Bose gas. The set of equations preserves important conservation laws such as the total number of particles and energy. The exchange between the condensate and the noncondensed fractions is caused by the elastic direct and exchange collision processes between a condensate atom and an excited atom, as well as pair excitations out of the condensate, see Fig. 1.

Figure 1: Schematic representation of the different scattering processes contributing to the dynamics of a homogeneous gas within the Hartree-Fock-Bogoliubov (HFB) mean-field approximation. The left panel illustrates the Gross-Pitaevskii (GP) approximation, which includes scattering between atoms in the condensate () mode. The right panel distinguishes between the three possible channels for elastic scattering between a condensate and an excited atom, the direct (Hartree), exchange (Fock), and pair production (Bogoliubov) processes. In an operator language, these processes are described by the respective vertex operator contributions to the interaction Hamiltonian quoted above the diagrams. All vertex operators are at most quadratic in excited-mode operators (), such that the resulting Hamiltonian can be diagonalised, and it describes an effectively free system.

Linearised HFB equations

Let us finally consider the case that Eqs. (16)–(2.3.1) describe small-amplitude oscillations around their stationary solutions. This approximation is, at first sight, irrelevant for the later discussion of far-from-equilibrium dynamics. We will discuss it here since it has widely been used for ultracold gases, and in order to point to the difference between the so called Landau and Beliaev damping processes and the collisional damping we will discuss in the later sections. See, e.g., Refs. (69); (11) for a concise summary of the procedure outlined in the following.

On the right-hand side of Eq. (16), only diagonal elements of the normal and anomalous density matrices appear. We can therefore focus on the closed set of equations for and the diagonal elements and . To study small-amplitude deviations one linearises the generalised GPE (16) as well as Eqs. (2.3.1) and (2.3.1) in small displacements from the equilibrium values of , , and ,

(19)

The time-independent part of the field expectation value is determined by the generalised stationary GPE

(20)

with . We work in the grand canonical ensemble, with the Hamiltonian replaced by which is equivalent to factor out a phase from in order to make it time-independent.

In order to diagonalise the equations for the stationary densities and one transforms the fluctuation operators to a quasiparticle basis by means of the Bogoliubov transformation

(21)

Here, and are quasiparticle operators which satisfy the Bose commutation relations , provided the mode functions , are subject to the normalisation conditions .

Defining the normal and anomalous quasiparticle density matrices and , respectively, the Bogoliubov-deGennes eigenvalue problem

(22)

with , , fixes the quasiparticle amplitudes and , and yields a diagonal stationary part of the quasiparticle density matrix, and a vanishing stationary anomalous quasiparticle density matrix .

The resulting linearised coupled equations for the time dependent variations , and read

(23)
(24)
(25)

The quantities are the equilibrium quasiparticle occupations, in terms of which the equilibrium non-condensate density is obtained as

(26)

The variations of the normal and anomalous particle densities expressed in terms of the quasiparticle density variations read

(27)

Note that Eqs. (20), (22)–(27) form a closed set of equations of motion.

Neglecting the cross terms coupling the condensate and noncondensate oscillations and , , the equations for and can be disentangled by a Bogoliubov-like transformation. The Bogoliubov frequencies are the resulting eigenvalues and therefore the frequencies of the BEC’s elementary excitations. For a translationally invariant gas, they read

(28)

where the Popov approximation has been chosen (70); (71). This approximation ensures that the spectrum (28) is gapless as required by the Hugenholtz-Pines (72) and Goldstone (73); (74) theorems: At low momenta, the energy (28) is linear in the momentum . In this limit the elementary oscillations are collective sound modes with dispersion , where is the sound velocity. At high momenta, the Bogoliubov dispersion assumes the quadratic form of free particles, . The approximations made in deriving the above equations are valid if the gas is weakly interacting, i.e., for a small diluteness parameter .

One can show that, with a linear dispersion, energy and momentum conservation restrict the possibilities for the excitation of particle modes in a flowing BEC when encountering obstacles, e.g., atoms at a cavity wall (see, e.g., Ref. (11)). A linear dispersion implies a maximum critical velocity for frictionless flow, i.e., for superfluidity. A weakly interacting BEC therefore obeys Landau’s criterion for superfluidity in the same way as superfluid He which, in addition, shows a pronounced roton minimum at finite wave vectors. Note that the Bogoliubov dispersion (28) is already obtained in the Bogoliubov approximation where the time evolution of the excited modes as well as the back action of the static excitation numbers on the condensate oscillation frequencies are neglected.

We emphasise that the HFB equations (24), (25) are local in time and only involve single-time correlation functions. The full HFB equations (2.3.1), (2.3.1), however, also determine the off-diagonal time dependence of . The fact that this does not feed back into the equations (2.3.1), (2.3.1) for the density matrices reflects that the HFB approximation does not account for direct scattering required for collisional dissipation and thermalisation. Hence, the HFB approximation is expected to be valid in the collisionless regime, where the mean free path is much larger than the scattering length. Note, however, that the linearised equations (23)–(25), if the coupling of excitations of the condensate and noncondensed fractions account is taken into account, also describe one-to-two and two-to-one collision processes between the excitations, provided a BEC phase is present, i.e., . These give rise to the so-called Landau and Beliaev damping caused by the mixing of superfluid and normal fluid phases, see, e.g., (69); (11) and Refs. cited therein. This damping is different in nature from the collisional dissipation obtained beyond the HFB approximation of the dynamic equations and discussed further in Sect. 3.6.3.

2.4 Beyond mean field

The collisionless regime discussed so far is easily left behind in present-day experiments. The preparation of ultracold atomic Bose and Fermi gases in various trapping environments allows to precisely study quantum many-body dynamics of strongly correlated systems, see, e.g., Refs. (75); (76); (77); (78). In particular, techniques exploiting zero-energy (magnetic and photoassociative, i.e., optical Feshbach) scattering resonances have helped to provide ultracold atomic gases with the importance they nowadays bear and the attraction they exert on physicists in most different areas of physics. Such techniques allow, by means of external electromagnetic fields, to tune the scattering length freely between large negative and large positive numbers. Special trapping configurations such as quasi one- and two-dimensional traps as well as optical lattices add to these possibilities and ask for descriptions beyond the mean-field level.

Figure 2: (Color online) Radial zero-energy scattering wave function (black solid line) at small internuclear distances , for the square-well potential (drawn in blue). Shown are four different potential depths , resulting in different -wave scattering lengths . The scattering length is given by the intersection radius of the extrapolated wave function with the zero-energy axis, cf. Eq. (29). (Red) horizontal lines in the potential wells indicate bound-state energy levels.

Feshbach resonances

Feshbach resonant scattering is most easily understood by realising that the value of the -wave scattering length is directly related to the energy of the highest bound state the Born-Oppenheimer scattering potential supports below the zero-energy threshold.2 To illustrate this consider the simple case of a square-well potential which is non-zero only for and tends to infinity at , see Fig. 2. The radial zero-energy scattering wave function oscillates within the square well, with a frequency determined by the depth of the well, while its wave length outside the well is much larger, . Hence, outside but close to the well, it is approximately linear in . As can be seen in Fig. 2, the continuity conditions for the scattering wave function at the edge of the well imply that crosses the -axis at . This can also be expressed in terms of the scattering amplitude approaching, for , a constant, the -wave scattering length, :

(29)

The square-well example shows clearly that, if the potential depth is changed, the energy of the uppermost bound state shifts, and the scattering length goes through infinity when crosses zero. We remark that the number of nodes within the well corresponds to the number of bound states left in the potential well. Very close to the resonance, the energy of the uppermost bound state with respect to threshold is proportional to the inverse of ,

(30)

The wave function of the uppermost bound state, as can be imagined from Fig. 2, is very similar to the wave function of the zero-energy scattering state for radii considerably smaller than . Close to , however, since , the bound-state wave function starts to differ from and approaches zero for larger . As a consequence, the scattering length, if positive and larger than the extent of , measures the spatial extent of the bound state, i.e., the size of the respective dimer molecules. Feshbach resonances have been exploited at large to produce degenerate molecular gases consisting of dimers of bosons, fermions, and of diatomic molecules of different species. See, e.g., Refs. (81); (83); (84); (85); (86); (87) for reviews and further references on cold molecules. We remark that in these experiments, dimers could be identified at values of the scattering length on the order of one thousand Bohr radii (88). These states are the largest and presumably most fragile molecules ever produced and measured in physics, see Fig. 4.

Figure 3: Coupled-channel bound states of Rb at magnetic field strengths of mT and mT. The dotted (dashed) curves indicate the closed ( open) channel components. Note the extreme size at . See text, Figs. 6, 6, and Refs. (89); (88) for more details.
Figure 4: (Color online) Coupled-channels picture of a magnetic Feshbach resonance. The ultracold atoms collide, with an almost vanishing relative momentum, near the threshold of the background channel. Intramolecular forces couple them to bound states of the closed-channel potential in which they are lacking the energy to separate to asymptotical distances. By tuning the Zeeman shift between the asymptotic channel energies, a closed-channel bound state can be brought into resonance with the colliding atoms, causing a Feshbach-resonant increase of the -wave scattering length .

In order to meet the conditions for a magnetic Feshbach resonance, the effective Born-Oppenheimer potential of the gas atoms is modified by means of external magnetic fields coupling to the magnetic moment. The atoms are usually trapped in a well-defined hyperfine state, such that the field causes a Zeeman shift relative to the energy of atoms in different polarisation states. By applying an external magnetic field, different scattering channels, corresponding to different asymptotic hyperfine states, can be shifted in energy relative to each other, see Fig. 4. Intramolecular electromagnetic forces couple these potentials, with a strength depending on the internuclear distance as well as on the energies of bound states supported by the system. In this way, the effective scattering potential can be changed, and Feshbach resonances occur whenever a bound state of the coupled system crosses the energy of the asymptotically separated atom pair, see, e.g., Refs. (82); (85). We finally note, that photoassociation scattering is analogous to the magnetic Feshbach scattering described here. There, the coupling between the channels is provided by polarised laser light. In the most simple case of induced dipole transitions, which require asymptotically -wave closed channels, spontaneous decay of the closed-channel bound state results in a complex scattering length (90); (91); (92).

Figure 5: Scheme of a typical magnetic field pulse shape in the low density () experiments in Ref. (93). The minimum magnetic field strength of the first and second pulse is mT. In the evolution period the field strength is chosen as mT. In the course of the experiments the evolution time as well as were varied. The dashed line indicates the position of the resonance at mT.
Figure 6: The remaining fraction of condensate atoms, , (solid line) together with the noncondensate fraction (dotted line), and the total density of unbound atoms (dashed line), as a function of the final time , at the end of the magnetic-field pulse in Fig. 6. All densities are given relative to the initial density. The figure shows the results of simulations of the HFB dynamic equations in the form described in Ref. (89), for the experiment reported in Ref. (93). See the main text for an outline of the experiment, and (89) for more details.

Ultracold gases near a Feshbach resonance

In a many-body system, the description in terms of binary scattering becomes unreliable close to the Feshbach resonance. As follows from the above discussion, the scattering length can reach and exceed the mean atomic separation, and bound states are no longer binary but should be regarded as extended clusters involving a macroscopic number of particles. For short evolution times, theoretical comparisons with experimental results for molecule formation in condensates, see, e.g., Refs. (93); (94); (95); (89); (96), indicate that the HFB dynamic equations discussed above can be applied even if the scattering length exceeds the mean interatomic spacing. In this experiment, performed in the group of Carl Wieman at JILA, Boulder, Bose-Einstein condensed Rb atoms were, for the first time, observed to coherently bind to dimer molecules when the scattering length was tuned close to a Feshbach resonance (93). The setup worked as a Ramsey interferometer: In the first step, the magnetic field was tuned, for a few microseconds, close to such that was on the order of Bohr radii . This constituted the first (coupling) Ramsey pulse. A longer evolution time followed, during which was ramped back to a few hundred , and in the end, a second Ramsey pulse, identical to the first one, was applied, see Fig. 6.

The length of the pulses were sufficiently short such that the HFB mean-field equations yield results quantitatively close to measurements (94); (95); (89); (96). Before the effects of multiple collisions and higher correlations become important a certain time after the quench a description within mean-field approximation remains valid. The calculations showed that the two Ramsey pulses coupled the colliding BEC atoms into the uppermost bound state of the two-channel system and coherently transferred atom pairs into molecules and vice versa. Due to the short pulse times, the coupling evolution resembled that of the fraction of a Rabi oscillation between two energy eigenstates. During the intermediate quasi free evolution, atoms and molecules could evolve such that a relative phase built up which was given by the binding energy relative to the free atoms multiplied by . Depending on the value of this phase, the second Ramsey pulse lead to a further production of molecules or dissociated the previously formed dimers. As a result, sinusoidal oscillations of the remaining fraction of atoms at the end of the pulse sequence were observed as a function of , see Fig. 6. The frequency of these oscillations precisely reproduced the expected binding energy of the Rubidium dimers (96).

However, discrepancies between theory and experiment remained in other cases, in particular for longer evolution times under strong interactions (97), which may indicate that descriptions beyond mean field are required to interpret experimental data. We will discuss such methods in the following sections.

Feshbach resonances have become, for experimenters, a versatile and convenient tool to control the collisional interactions. The have, in particular, opened the door to the exploration of rich physics in parameter regimes never explored before. For example, ultracold atomic Fermi gases can be manipulated such that they not only show the superconductor-like properties known from electron gases in solids but can cross from a BCS-like state containing Cooper pairs, over into a BEC of tightly bound molecules (For recent reviews cf. Refs. (98); (99); (17)). As experiments are usually conducted in more than one spatial dimension and since the dynamics of Fermion gases can not be simulated by means of classical equations of motion, the functional field theoretical methods to be described in the following sections are expected to represent the most promising theoretical approach to the dynamics of strongly correlated Fermi gases beyond mean-field theory.

Ultracold gases in lower dimensional traps and optical lattices

One- and two-dimensional traps (11); (76); (77); (100); (101); (102); (103); (104); (105); (106); (107) as well as optical lattices (108); (109); (110) allow to realise strongly correlated many-body states of atoms. In an optical lattice, strong effective interactions can be induced by suppressing the hopping between adjacent lattice sites and thus increasing the weight of the interaction relative to the kinetic energy (108); (111). This leads, in the limit of near-zero hopping or strong interactions, to a Mott-insulating state (75). It is beyond the scope of this article to discuss in more detail the theory of ultracold gases trapped in such special configurations. Some remarks concerning one-dimensional (1D) gases, though, are in order, as we will focus on such a system when applying the functional field-theory methods in later sections.

In special cases, the models describing 1D gases (112); (113); (114); (115) allow to determine exact time-dependent solutions of the Schrödinger equation (116); (117) providing insight beyond various approximations, which is particularly important in strongly correlated regimes. These 1D systems are experimentally realized with atoms tightly confined in effectively 1D waveguides (76); (77); (101); (102); (103); (104); (105); (106); (107), where nonequilibrium dynamics is considerably affected by the kinematic restrictions of the geometry (77), while quantum effects are enhanced (118); (119); (120). 1D Bose gases are explored for various interaction strengths, from the Lieb-Liniger (LL) gas with finite coupling (101); (102); (103); (77) up to the so-called Tonks-Girardeau (TG) regime of “impenetrable-core” bosons (121); (113); (106); (107); (77). The 1D gas enters the TG regime if the dimensionless interaction parameter is much larger than one. Here, is the coupling parameter of the one-dimensional gas, e.g., for a cylindrical trap with transverse harmonic oscillator length (11). In the Tonks-Girardeau limit the atoms can no longer pass each other and behave in many respects like a one-dimensional ideal Fermi gas (11).

Exact dynamics of an interacting 1D Bose gas

Most theoretical studies of the exact time-dependence address the Tonks-Girardeau (TG) regime (116); (122); (123); (124); (125); (126); (127); (128)). In this limit, the complex many-body problem is considerably simplified due to the Fermi-Bose mapping property where dynamics follows a set of uncoupled single-particle Schrödinger equations (116). A method for calculating the time-evolution of a LL gas with finite interaction strength has recently been discussed in Refs. (129); (130). It has the potential to yield a valuable comparison to the results obtained from solving the dynamical field equations presented in the following sections. We therefore devote this subsection to a brief excursion and outline the method which generalises Lieb and Liniger’s exact diagonalisation method (112) to time evolving -particle quantum mechanical wave functions.

We consider the dynamics of indistinguishable -interacting bosons in a 1D geometry (112). The Schrödinger equation for this system is usually written as

(31)

where is the many-body wave function, and quantifies the strength of the interaction, which is related to the dimensionless 1D interaction parameter introduced above. We do not impose any boundary conditions, i.e. , the -space is infinite which corresponds to a number of interesting experimental situations where the gas is initially localized within a certain region of space and then allowed to freely evolve (123); (124); (125); (126).

The idea is to construct exact solutions by differentiating a fully antisymmetric (fermionic) time-dependent wave function, which obeys the Schrödinger equation for a free Fermi gas (131). The differential operator used for this depends on the interaction strength and the number of particles. When , the scheme reduces to Girardeau’s time-dependent Fermi-Bose mapping (116), valid for ”impenetrable-core” bosons.

Due to the Bose symmetry, it is sufficient to express the wave function in a single permutation sector of the configuration space, . Within , obeys

(32)

while interactions impose boundary conditions at the borders of (112):

(33)

This constraint creates a cusp in the many-body wave function when two particles touch, which should be present at any time during the dynamics. In the TG limit (i.e., when ) the cusp condition is (113); (116), which is trivially satisfied by an antisymmetric fermionic wave function . Hence, within , which is the famous Fermi-Bose mapping (113); (116). In many physically interesting cases, can be constructed as a Slater determinant

(34)

Since within , must obey , which implies that the (orthonormal) single-particle wave functions evolve according to

(35)

. Thus, in the TG limit, the complexity of the many-body dynamics is reduced to solving a simple set of uncoupled single-particle equations, while the interaction constraint (33) is satisfied by the Fermi-Bose construction.

The simplicity and success of this idea motivates us to choose an ansatz which automatically satisfies constraint (33) for any finite (131); (115). For this, define a differential operator

(36)

where stands for

(37)

It can be shown that the wave function

(38)

where is a normalization constant, obeys the cusp condition (33) by construction (131); (115): Consider an auxiliary wave function

(39)

where the primed operator omits the factor as compared to . The auxiliary function can be written as

(40)

It is straightforward to verify that the operator in front of is invariant under the exchange of and (115). On the other hand, the fermionic wave function is antisymmetric with respect to the interchange of and . Thus, is antisymmetric with respect to the interchange of and , which leads to

(41)

This is fully equivalent to the cusp condition (33), . Thus, the wave function (38) obeys constraint (33) by construction.

In order to exactly describe the dynamics of LL gases, the wave function (38) should also obey Eq. (32) inside . From the commutators and