Kinetic Theory of a Spin-1 Bose Condensed Gas at?Finite Temperatures

We derive a kinetic theory of a spin-1 Bose condensed gas at finite temperatures. The theory consists of coupled equations for the condensate and noncondensate. The condensate dynamics is described by a generalized Gross?CPitaevskii (GP) equation including the spin degree of freedom. The condensate and noncondensate atoms interact with each other through the mean field and collisions. From the coupled equations, we determine the condition for static equilibrium of the spin-1 system. We also use the static thermal cloud approximation to derive a generalized GP equation including a dissipative term due to the coupling to a thermal cloud.     

Vortex Dynamics in Trapped Bose-Einstein Condensate

We perform numerical simulations of vortex motion in a trapped Bose-Einstein condensate by solving the two-dimensional Gross-Pitaevskii Equation in the presence of a simple phenomenological model of interaction between the condensate and the finite temperature thermal cloud. At zero temperature, the trajectories of a single, off-centred vortex precessing in the condensate, and of a vortex-antivortex pair orbiting within the trap, excite acoustic emission. At finite temperatures the vortices move to the edge of the condensate and vanish. By fitting the finite-temperature trajectories, we relate the phenomenological damping parameter to the friction coefficients α and α′, which are used to describe the interaction between quantised vortices and the normal fluid in superfluid helium.     

Dynamical Simulation of Sound Propagation in a Highly Elongated Trapped Bose Gas at Finite Temperatures

We study sound propagation in a Bose-condensed gas confined in a highly elongated harmonic trap at finite temperatures. Our analysis is based on Zaremba-Nikuni-Griffin (ZNG) formalism, which consists of Gross-Pitaevskii equation for the condensate and the kinetic equation for a thermal cloud. We extend ZNG formalism to deal with a highly-anisotropic trap potential, and use it to simulate sound propagation in the two fluid hydrodynamic regime. We use the trap parameters for the experiment that has reported second sound propagation. Our simulation results show that propagation of two sound pulses corresponding to first and second sound can be observed in an intermediate temperature.     

Soliton Dynamics of F=1 Spinor Bose–Einstein Condenstate with Nonvanishing Boundaries

We review the recently studied soliton dynamics of the hyperfine spin F=1 spinor Bose–Einstein condensate in a one-dimensional uniform system. The characters of one-soliton states and the behavior of two-soliton collision are briefly described.     

Thermodynamics of a Trapped Bose Condensate with Negative Scattering Length

We study the Bose–Einstein condensation (BEC) for a system of ⁷Li atoms, which have negative scattering length (attractive interaction), confined in a harmonic potential. Within the Bogoliubov and Popov approximations, we numerically calculate the density profile for both condensate and non-condensate fractions and the spectrum of elementary excitations. In particular, we analyze the temperature and number-of-boson dependence of these quantities and evaluate the BEC transition temperature T _BEC. We calculate the loss rate for inelastic two- and three-body collisions. We find that the total loss rate is strongly dependent on the density profile of the condensate, but this density profile does not appreciably change by increasing the thermal fraction. Moreover, we study, using the quasi-classical Popov approximation, the temperature dependence of the critical number N _c of condensed bosons, for which there is the collapse of the condensate. There are different regimes as a function of the total number N of atoms. For N<N _c the condensate is always metastable but for N>N _c the condensate is metastable only for temperatures that exceed a critical value T _c.     

Simulation of a Stationary Dark Soliton in a Trapped Zero-Temperature Bose-Einstein Condensate

We discuss a computational mechanism for the generation of a stationary dark soliton, or black soliton, in a trapped Bose–Einstein condensate (BEC) using the Gross–Pitaevskii (GP) equation for both attractive and repulsive interaction. It is demonstrated that the black soliton with a “notch" in the probability density with a zero at the minimum is a stationary eigenstate of the GP equation and can be efficiently generated numerically as a nonlinear continuation of the first vibrational excitation of the GP equation in both attractive and repulsive cases in one and three dimensions for pure harmonic as well as harmonic plus optical-lattice traps. We also demonstrate the stability of this scheme under different perturbing forces.     

Bosonic Clouds with Attractive Interaction Beyond the Local Interaction Approximation

We study the properties of a Bose–Einstein condensed cloud of atoms with negative scattering length confined in a harmonic trap. When a realistic non local (finite range) effective interaction is taken into account, we find that, besides the known low density metastable solution, a new branch of Bose condensate appears at higher density. This state is self–bound but its density can be quite low if the number N of atoms is not too big. The transition between the two classes of solutions as a function of N can be either sharp or smooth according to the ratio between the range of the attractive interaction and the length of the trap. A tight trap leads to a smooth transition. In addition to the energy and the shape of the cloud we study also the dynamics of the system. In particular, we study the frequencies of collective oscillation of the Bose condensate as a function of the number of atoms both in the local and in the non local case. Moreover, we consider the dynamics of the cloud when the external trap is switched off.     

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.     

Bose-Condensed Gases in a 1D Optical Lattice at Finite Temperatures

We study equilibrium properties of Bose-Condensed gases in a one-dimensional (1D) optical lattice at finite temperatures. We assume that an additional harmonic confinement is highly anisotropic, in which the confinement in the radial directions is much tighter than in the axial direction. We derive a quasi-1D model of the Gross-Pitaeavskii equation and the Bogoliubov equations, and numerically solve these coupled equations to obtain the condensate fraction as a function of the temperature. We also discuss the importance of the radial excitations in the thermodynamic properties at finite temperatures.     

High- and low-frequency behavior of response functions in a Bose condensed liquid

Recent studies of superfluid ⁴He have shown that it is useful to separate S(Q, ) into condensate and regular parts. We discuss the f-sum rule and the compressibility sum rule which the condensate part satisfies at T = 0K and at finite temperatures. Our work is based on the microscopic approach of Gavoret and Nozières, as amended by Nepomnyashchii and Nepomnyashchii. We also discuss the implications of the Wong-Gould sum rule for the condensate part of S(Q, ).This research was supported by a grant from NSERC, Canada.     

Remnants of Bose condensation and off-diagonal long range order in finite systems

We briefly review some general aspects of Bose condensation and off-diagonal long range order in infinite bulk systems and then discuss to what extent these features might be manifested in finite systems with free surfaces such as droplets of liquid ⁴ He at low temperatures. We focus on the structure of the one-body density matrix and the macroscopic wave function in the surface region, and note the possibility of a surface condensate.     

Dephasing of Bose-Einstein condensates

Abstract

We discuss some recent work on the dephasing of Bose-Einstein condensates of interacting atoms due to fluctuations in the chemical potential μ. At vanishing temperature such fluctuations are due to the uncertainty of the number of particles in the condensate caused by the depletion effect. The dephasing is in this case a non-diffusive (and in principle reversible) effect called ‘collapse’ (and ‘revival’). Above a certain cross-over temperature the dephasing proceeds mainly via a diffusive process which follows in time a short and ineffective phase collapse. The phase-diffusion constant is dominated by the temperature dependent condensate-number fluctuations caused by the scattering of low-energy excitations off the condensate. We also discuss the (in general subdominant) contribution of the occupation-number fluctuations in quasi-particle states in the thermal cloud. For simplicity we discuss here results for a box-like trapping potential.     

Stability and Excitations of Solitons in 2D Bose-Einstein Condensates

The small oscillations of solitons in 2D Bose-Einstein condensates are investigated by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. We show that the soliton is stable and that the lowest excited states obey the same dispersion law as the one of the stable branch of excitations of a 1D gray soliton in a 2D condensate. The role of these states in thermodynamics is discussed.     

Thermal conductivity measurements of pitch-bonded graphites at millikelvin temperatures: Finding a replacement for AGOT graphite

Pitch-bonded graphites are among the best known thermal insulators at sub-kelvin temperatures, but are very good conductors at higher temperatures. This makes them ideal for mechanical supports which must provide good thermal isolation at an operating temperature below 1 K, but must have good conductance at higher temperatures to aid in initially cooling down an instrument (a “passive heat switch”). One type of graphite, AGOT, has been known as having the lowest thermal conductivity below 1 K not only among graphites, but also compared with any other material. It is, however, no longer available. We have carried out thermal conductivity measurements at temperatures between 60 mK and 4 K on a proposed replacement, POCO AXM-5Q graphite, as well as a sample of AGOT graphite. Our measurements show that both graphites have a difference of about six orders of magnitude in conductivity between room temperature and 100 mK, but that AGOT graphite is not as good an insulator as previously believed. We conclude that AXM-5Q graphite is not only a suitable replacement for AGOT, but in fact is somewhat superior.     

Self-similar Expansion of the Density Profile in a Turbulent Bose-Einstein Condensate

In a recent study we demonstrated the emergence of turbulence in a trapped Bose-Einstein condensate of ⁸⁷Rb atoms. An intriguing observation in such a system is the behavior of the turbulent cloud during free expansion. The aspect ratio of the cloud size does not change in the way one would expect for an ordinary non-rotating (vortex-free) condensate. Here we show that the anomalous expansion can be understood, at least qualitatively, in terms of the presence of vorticity distributed throughout the cloud, effectively counteracting the usual reversal of the aspect ratio seen in free time-of-flight expansion of non-rotating condensates.     

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.     

Hydrodynamics of Superfluid Bose Gases in an Optical Lattice at Finite Temperatures

Starting from an effective action for the order parameter field, we derive a coupled set of generalized hydrodynamic equations for a Bose condensate in an optical lattice at finite temperatures. Using the linearized hydrodynamic equations, we study the microscopic mechanism of the Landau instability due to the collisional damping process between the condensate and noncondensate atoms. It is shown that the Landau criterion for the stability of a superfluid in a uniform system is modified due to the presence of the periodic optical lattice potential.     

Boson Number in Bose–Einstein Condensate of Liquid Helium Near the λ Point

Bose–Einstein condensation has been well investigated in dilute atomic gases. For liquid helium system, the superfluid component is considered to be a background flow in the Landau theory. We study the relation between Bose–Einstein condensate and superfluid component. The concept of dressed bosons is introduced, which are eigenstates of the total Hamiltonian. The total energy is the sum of the kinetic energy of the center of mass and Galilean invariant terms. Therefore, the energy of the dressed boson has a nonlinear form for their number distribution function. If it is required that the excitation energy of a dressed boson be in agreement with the elementary excitation energy of the Landau theory near 0.0 K, then the functional form of the dressed boson energy can be determined. Because of this functional form, the dressed bosons with zero momentum have no friction against a vessel only in existence of a Bose–Einstein condensate. Consequently, the condensate of the dressed bosons with zero momentum is the superfluid component. The number n ₀ of dressed bosons with zero momentum is calculated. It shows temperature dependence (1−(T/T _λ ))^1/3 near the λ point, where T is the temperature and T _λ is the λ transition temperature.     

Quasiparticle Kinetic Equation in a Trapped Bose Gas at Low Temperatures

Recently the authors used the Kadanoff–Baym non-equilibrium Green's function formalism to derive kinetic equation for the non-condensate atoms, in conjunction with a consistent generalization of the Gross–Pitaevskii equation for the Bose condensate wavefunction. This work was limited to high temperatures, where the excited atoms could be described by a Hartree–Fock particle-like spectrum. Following the approach of Kane and Kadanoff in 1965, we present the generalization of our recent work which is valid at low temperatures, where the input single-particle spectrum is now described by the Bogoliubov–Popov approximation. We derive a kinetic equation for the quasiparticle distribution function with collision integrals describing scattering between quasiparticles and the condensate atoms. From the general expression for the collision integral for the scattering between quasiparticle excitations, we find the quasiparticle distribution function corresponding to local equilibrium. This expression includes a quasiparticle chemical potential that controls the non-diffusive equilibrium between condensate atoms and the quasiparticle excitations. We derive a generalized Gross–Pitaevskii equation for the condensate wavefunction that also includes the damping effects due to collisions between atoms in the condensate and the thermally excited quasiparticles. For a uniform Bose gas, our kinetic equation for the thermally excited quasiparticles reduces to that found by Eckern, as well as by Kirkpatrick and Dorfman.     

Relaxation of Bose-Einstein Condensates of Magnons in Magneto-Textural Traps in Superfluid 3He-B

In superfluid ³He-B externally pumped quantized spin-wave excitations or magnons spontaneously form a Bose-Einstein condensate in a 3-dimensional trap created with the order-parameter texture and a shallow minimum in the polarizing field. The condensation is manifested by coherent precession of the magnetization with a common frequency in a large volume. The trap shape is controlled by the profile of the applied magnetic field and by the condensate itself via the spin-orbit interaction. The trapping potential can be experimentally determined with the spectroscopy of the magnon levels in the trap. We have measured the decay of the ground state condensates after switching off the pumping in the temperature range (0.14÷0.2)T _c. Two contributions to the relaxation are identified: (1) spin diffusion with the diffusion coefficient proportional to the density of thermal quasiparticles and (2) the approximately temperature-independent radiation damping caused by the losses in the NMR pick-up circuit. The measured dependence of the relaxation on the shape of the trapping potential is in a good agreement with our calculations based on the magnetic field profile and the magnon-modified texture. Our values for the spin diffusion coefficient at low temperatures agree with the theoretical prediction and earlier measurements at temperatures above 0.5T _c.