Quantum dynamics of the phase of a Bose–Einstein condensate

Abstract

In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.     

Probing first-order spatial coherence of a Bose-Einstein condensate

Abstract

We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.     

Kinetics of phase transition in ideal and weakly interacting bose gas

The time evolution of a Bose system passing through the critical point is considered. The solution of the nonlinear integrodifferential equation that governs the kinetics demonstrates that the new phase formation proceeds by the set of essentially nonequilibrium states. The phase transition in an ideal Bose gas is of first order and can be completed att= only if there are no nuclei of the new phase at the beginning of the cooling process. With nuclei the Bose condensate formation takes a finite time. A Bose gas with interaction between Bose particles exhibits a second-order phase transition with a finite time of new phase formation even without nuclei. The time evolution of an energy spectrum of a Bose system following the variation of its distribution function is considered and it is shown that the appearance of superfluidity coincides with the instant of Bose condensate formation.     

Instability of a Bose condensate of neutral atoms in an external light field of nonuniform intensity

The existence of a new of type of instability of a Bose condensate in a rarefied atomic gas due to momentum exchange with an external resonant spatially nonuniform radiation field is demonstrated theoretically. Pis’ma Zh. Tekh. Fiz. 25, 77–81 (May 12, 1999)     

The dynamics of three-mode coupling in a trapped Bose gas

ABSTRACT

Multiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.     

Recent Progress and New Challenges in Quantum Fluids and Solids

The exact expression for the average kinetic energy of an inhomogeneous Bose gas in the ground state is obtained as a functional of the inhomogeneous density of the Bose–Einstein condensate. The result is based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. On this basis, the explicit expressions for the ground-state energy and the local pressure of an inhomogeneous Bose gas are derived within the self-consistent Hartree–Fock approximation.     

On Transport Properties of Exciton - Phonon Condensate in Cu2O

A moving condensate of para-excitons in a 3D Cu ₂ O crystal turns out to be spatially inhomogeneous in the direction of motion, and the registered velocities of coherent exciton packets are approximately equal to the longitudinal sound speed of the crystal. We explore a simple theoretical model to describe such properties of the Bose condensed excitons. Taking into account the exciton - phonon interaction and introducing the coherent phonon part of the moving condensate, we derive the dynamic equations for the exciton - phonon condensate. Within the Bose approximation for excitons, we discuss the conditions for the moving condensate to appear in the crystal. We calculate the condensate wave function and energy and a collective excitation spectrum in the semiclassical approximation. The stability conditions of the moving condensate are analyzed as well.     

Classification of Phase Transitions for an Ideal Bose Gas in a d-Dimensional Quartic Potential

Based on the classification scheme of phase transitions, we study the phase transitions for an ideal Bose gas with a finite number N of particles trapped in a d-dimensional quartic potential. We find that the presence and nature of phase transition depend on the dimensionality of the quartic potential. Proposing three different definitions of transition temperature, we discuss either N or d dependence of transition temperature for the ideal Bose condensate in the d-dimensional quartic potential.     

Subhomogeneous spectroscopy of a Bose condensate of neutral atoms

A method is proposed for resolving experimentally the details of the resonant fluorescence spectrum of a Bose condensate within a homogeneous line whose width is greatly increased because of collective effects. A discussion is presented concerning the application of this method for determining the shifts of the resonant frequencies of the transitions in the condensate as compared to the usual case of nondegenerate atom ensembles. Pis’ma Zh. Tekh. Fiz. 23, 33–38 (January 26, 1997)     

Coherence of bose condensates

Abstract

We study the coherence of interacting Bose condensates in recent magnetic trap experiments. The coherent evolution manifests itself in the macroscopic interference of two independent Bose condensates. The theoretical predictions from the time-dependent Gross–Pitaevskii equation are in excellent agreement with the measured interference patterns. A coherent coupling of two condensates represents the atomic analogon of a Josephson junction. The dependence of the magnetic confinement on the nuclear spin orientation allows one to build a controllable beam splitter by magnetic resonance. The application of this beam splitter to realize an atom laser is studied theoretically. The coherence of the output beam is limited only by phase diffusion of the condensate.     

Interacting Bose Gas in an Optical Lattice

A grand canonical system of hard-core bosons in an optical lattice is considered. The bosons can occupy randomly N equivalent states at each lattice site. The limit N is solved exactly in terms of a saddle-point integration, representing a weakly-interacting Bose gas. In the limit N there is only a condensate if the fugacity of the Bose gas is larger than 1. Corrections in 1/N increase the total density of bosons but suppress the condensate. This indicates a depletion of the condensate due to increasing interaction at finite values of N.     

Critical Velocities in Two-Component Superfluid Bose Gases

On the ground of the Landau criterion we study the behavior of critical velocities in a superfluid two-component Bose gas. It is found that under motion of the components with different velocities the velocity of each component should not be lower than a minimum phase velocity of elementary excitations (s ₋). The Landau criterion yields a relation between the critical velocities of the components (v _c1, v _c2). The velocity of one or even both components may exceed s ₋. The maximum value of the critical velocity of a given component can be reached when the other component does not move. The approach is generalized for a two-component condensate confined in a cylindrical harmonic potential.      

Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap

Path-Integral-Monte-Carlo simulation has been used to calculate the properties of a two-dimensional (2D) interacting Bose system. The bosons interact with hard-core potentials and are confined to a harmonic trap. Results for the density profiles, the condensate fraction, and the superfluid density are presented. By comparing with the ideal gas we easily observe the effects of finite size and the depletion of the condensate because of interactions. The system is known to have no phase transition to a Bose-Einstein condensation in 2D, but the finite system shows that a significant fraction of the particles are in the lowest state at low temperatures.     

Kinetic theory and dynamic structure factor of a condensate in the random phase approximation

No Heading We present the microscopic kinetic theory of a homogeneous dilute Bose condensed gas in the generalized random phase approximation (GRPA), which satisfies the following requirements: 1) the mass, momentum and energy conservation laws; 2) the H-theorem; 3) the superfluidity property and 4) the recovery of the Bogoliubov theory at zero temperature ¹. In this approach, the condensate influences the binary collisional process between two normal atoms, in the sense that their interaction force results from the mediation of a Bogoliubov collective excitation traveling throughout the condensate. Furthermore, as long as the Bose gas is stable, no collision happens between condensed and normal atoms. In this paper, we show how the kinetic theory in the GRPA allows to calculate the dynamic structure factor at finite temperature and when the normal and superfluid are in a relative motion. The obtained spectrum for this factor provides a prediction which, compared to the experimental results, allows to validate the GRPA.PACS numbers:03.75.Hh, 03.75.Kk, 05.30.–d     

Thermodynamics of a Trapped Bose-Condensed Gas

We investigate the thermodynamic behaviour of a Bose gas interacting with repulsive forces and confined in a harmonic anisotropic trap. We develop the formalism of mean field theory for non uniform systems at finite temperature, based on the generalization of Bogoliubov theory for uniform gases. By employing the WKB semiclassical approximation for the excited states we derive systematic results for the temperature dependence of various thermodynamic quantities: condensate fraction, density profiles, thermal energy, specific heat and moment of inertia. Our analysis points out important differences with respect to the thermodynamic behaviour of uniform Bose gases. This is mainly the consequence of a major role played by single particle states at the boundary of the condensate. We find that the thermal depletion of the condensate is strongly enhanced by the presence of repulsive interactions and that the critical temperature is decreased with respect to the predictions of the non-interacting model. Our work points out an important scaling behaviour exhibited by the system in large N limit. Scaling permits to express all the relevant thermodynamic quantities in terms of only two parameters: the reduced temperature t = T/T _c ⁰ and the ratio between the T = 0 value of the chemical potential and the critical temperature T _c ⁰ for Bose-Einstein condensation. Comparisons with first experimental results and ab-initio calculations are presented.     

Existence of True Bose-Einstein Condensate in a 2D Interacting Gas in an Anti-Trap Geometry

The possibility of Bose-Einstein condensation of a nonideal 2D Bose gas in an anti-trap external field is studied. The problem is solved exactly for the external potential of a special form u(x)=–u cosh ²(x/l), where is the chemical potential. The spectrum and eigenfunctions of elementary excitations are found. It is shown that at T=0 there exist a true long range order and a Bose condensate in the system.     

Coarse-Grained Finite-Temperature Theory for the Bose Condensate in Optical Lattices

In this paper, we derive a coarse-grained finite-temperature theory for a Bose condensate in a one-dimensional optical lattice, in addition to a confining harmonic trap potential. We start with a two-particle irreducible (2PI) effective action on the Schwinger-Keldysh closed-time contour path. In principle, this action involves all information of equilibrium and non-equilibrium properties of the condensate and noncondensate atoms. In constructing a theory for the condensate and noncondensate in a periodic lattice potential, a difficulty arises from the rapid spatial variation due to a lattice potential, compared to the length scale of the harmonic potential. We employ a coarse-graining procedure to eliminate this rapid variation. By introducing a variational ansatz for the condensate order parameter in an effective action, we derive a coarse-grained effective action, which describes the dynamics on the length scale much longer than a lattice constant. Using the variational principle, coarse-grained equations of motion for condensate variables are obtained. These equations include a dissipative term due to collisions between condensate and noncondensate atoms, as well as noncondensate mean-field. As a result of a coarse-graining procedure, the effects of a lattice potential are incorporated into equations of motion for the condensate by an effective mass, a renormalized coupling constant, and an umklapp scattering process. To illustrate the usefulness of our formalism, we discuss a Landau instability of the condensate moving in optical lattices by using the coarse-grained generalized Gross-Pitaevskii hydrodynamics. We find that the collisional damping rate due to collisions between the condensate and noncondensate atoms changes its sign when the condensate velocity exceeds a renormalized sound velocity, leading to a Landau instability consistent with the Landau criterion. Our results in this work give an insight into the microscopic origin of the Landau instability.      

Particle number fluctuations in an ideal Bose gas

Abstract

We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.     

The role of elastic interparticle interactions in the dynamics of raman atom-molecule conversion in a Bose condensate

The influence of elastic interparticle interactions on the process of atom-molecule conversion in a Bose condensate has been studied. It is shown that allowance for these collisions can lead to both suppression and enhancement of the amplitude of oscillations in the density of atoms and molecules in the condensate in periodic regimes of conversion.