The Two-Dimensional Bose–Einstein Condensate

We study the Hartree–Fock–Bogoliubov mean-field theory as applied to a two-dimensional finite trapped Bose gas at low temperatures and find that, in the Hartree–Fock approximation, the system can be described either with or without the presence of a condensate; this is true in the thermodynamic limit as well. Of the two solutions, the one that includes a condensate has a lower free energy at all temperatures. However, the Hartree–Fock scheme neglects the presence of phonons within the system, and when we allow for the possibility of phonons we are unable to find condensed solutions; the uncondensed solutions, on the other hand, are valid also in the latter, more general scheme. Our results confirm that low-energy phonons destabilize the two-dimensional condensate.     

Fermi–Bose Correspondence and Bose–Einstein Condensation in the Two-Dimensional Ideal Gas

The ideal uniform two-dimensional (2D) Fermi and Bose gases are considered both in the thermodynamic limit and the finite case. We derive May's Theorem, viz. the correspondence between the internal energies of the Fermi and Bose gases in the thermodynamic limit. This results in both gases having the same heat capacity. However, as we shall show, the thermodynamic limit is never truly reached in two dimensions and so it is essential to consider finite-size effects. We show in an elementary manner that for the finite 2D Bose gas, a pseudo-Bose–Einstein condensate forms at low temperatures, incompatible with May's Theorem. The two gases now have different heat capacities, dependent on the system size and tending to the same expression in the thermodynamic limit.     

Two-Fluid Hydrodynamics in Trapped Bose Gases and in Superfluid Helium

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.     

Quasi 1 and 2d Dilute Bose Gas in Magnetic Traps: Existence of Off-Diagonal Order and Anomalous Quantum Fluctuations

Current magnetic traps can be made so anisotropic that dilute Bose gases confined in these traps will occupy the lowest quantum state in the tightly confining direction, while still in the Thomas-Fermi limit in the loosely confining direction. As a result, the trapped Bose gas behaves like a quasi one or two dimensional systems. Unlike the homogeneous case, quantum phase fluctuations do not destroy macroscopic off-diagonal order of trapped Bose gases in d2 because they are suppressed by the the trapping potential. In the dilute limit, quantum fluctuations increase, remain constant, and decrease with size for 3, 2, 1 d respectively. These behaviors are due to the combination of a finite gap and the universal spectrum of the collective mode.     

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.     

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.     

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.     

The Schrödinger Cat Family in Attractive Bose Gases

We show that the ground state of an attractive Bose gas in a double well evolves from a coherent state to a Schrödinger Cat like state as the tunneling barrier is decreased. The latter exhibits super-fragmentation similar to the ground state of a spin-1 Bose gas with antiferromagnetic interactions. We also show that the fragmented condensates of attractive and repulsive Bose gases in double wells lead to very different interference patterns.     

Dissipationless Flow and Superfluidity in Gaseous Bose-Einstein Condensates

We study dissipation in a dilute Bose gas induced by the motion of a macroscopic object. A blue-detuned laser beam focused on the center of a trapped gas of sodium atoms was scanned both above and below the BEC transition temperature. The measurements allow for a comparison between the heating rates for the superfluid and normal gas.     

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.     

Superfluid-Insulator Transition of a Bose–Einstein Condensation in a Periodic Potential and Its Interference Pattern

Recent experiments have succeeded in observing the superfluid-Mott insulator quantum phase transition of an alkali atomic Bose–Einstein condensate in an optical lattice potential. Motivated by this work, we studied the two-dimensional Bose gas in a periodic potential by analyzing the Gross–Pitaevskii equation. We found evidence of a superfluid-insulator transition that occurs as the potential depth of the lattice is increased. For the periodic potential, the phase of the macroscopic wave function in the ground state is localized in each potential minimum. Also, according to the resugts using the Hartree–Fock–Bogoliubov equation, an energy gap appears in the lowest excitation state. We then added a parabolic trapping potential to the periodic potential and studied how the dynamics of the wave function and its interference pattern depend on the initial ground state. For the initial ground state localized by the deep periodic potential, the wave function oscillates in the central potential minimum after removing only the trapping potential. After turning off both the trapping and periodic potentials, the wave packets with periodicity escape from the condensate.     

Anisotropic Spin Diffusion in Trapped Boltzmann Gases

No Heading Recent experiments in a mixture of two hyperfine states of trapped Bose gases show behavior analogous to a spin-1/2 system, including transverse spin waves and other familiar Leggett-Rice-type effects. We have derived the kinetic equations applicable to these systems, including the spin dependence of interparticle interactions in the collision integral, and have solved for spin-wave frequencies and longitudinal and transverse diffusion constants in the Boltzmann limit. We find that, while the transverse and longitudinal collision times for trapped Fermi gases are identical, the Bose gas shows diffusion anisotropy. Moreover, the lack of spin isotropy in the interactions leads to the non-conservation of transverse spin, which in turn has novel effects on the hydrodynamic modes.PACS numbers: 03.75.Mn,05.30Jp,05.60.Gg,51.10.+y.67.20.+k.     

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.     

A Study of Bose-Einstein Condensation in a Two-Dimensional Trapped Gas

We examine the possibility of Bose-Einstein condensation (BEC) in two-dimensional (2D) system of interacting particles in a trap. We use a self-consistent mean-field theory of Bose particles interacting by a contact interaction in the Popov and WKB approximations. The equations show that the normal state has a phase transition at some critical temperature T _c but below T _c the Bose-Einstein condensed state is not a consistent solution of the equations in the thermodynamic limit. This result agrees with a theorem recently discussed by the author that shows that a BEC state is impossible for an interacting gas in a 2D trap in the thermodynamic limit.     

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.     

Polarized Dipolar Bose Gas with Strong Interactions

We present calculations of properties of a polarized dipolar Bose gas, trapped in the polarization direction by a harmonic potential while translationally invariant in the perpendicular direction. This system is of particular interest because the dipolar interaction is not only anisotropic, but also long-ranged and it already showed an interesting behaviour in the weakly interacting limit (Santos et al. in Phys. Rev. Lett. 90:250403, 2003), where a roton-maxon like excitation spectrum was found. Here we stabilize the dipolar Bose gas by a repulsive core of the form (σ/r)¹² to avoid a collapse of the system. For our calculation we use the hypernetted-chain Euler-Lagrange method which is not limited to weakly interacting or dilute systems, but is also valid for strongly interacting systems. We find strong evidence that under certain conditions a quantum phase transition occurs to a state where pairs of dipoles bind to form dimers. Close to this phase transition a roton-maxon like excitation spectrum is observed.     

Dynamics of Fluctuating Bose–Einstein Condensates

We present a generalized Gross–Pitaevskii equation that describes the dissipative dynamics of a trapped partially Bose-condensed gas. It takes the form of a complex nonlinear Schrödinger equation with noise. We consider an approximation to this Langevin field equation that preserves the correct equilibrium for both the condensed and the noncondensed parts of the gas. We then use this formalism to describe the reversible formation of a one-dimensional Bose condensate, and compare with recent experiments. In addition, we determine the frequencies and the damping of collective modes in this case.