Isotherms, Order Parameter and Density Profiles for Weakly Interacting Bose Gases within Three Mean-Field Theories

In this work we consider three mean field approximations of standard use in the literature to describe the weakly interacting Bose gas confined in a box of volume V. For these approximations we calculate the corresponding isotherms ??=??(??,T), where ?? is the chemical potential, ?? the particle density and T the absolute temperature. In particular we address this calculation at the nanokelvin regime for a gas parameter $\gamma= \rho a_{s}^{3}\leq 1$ where the Bose-Einstein Condensation in alkali atoms is observed. The non singled value observed for the equation of state ??=??(??,T) suggests strongly that the approximations considered here do not capture properly the thermodynamic behavior in the vicinity of the BEC transition. In order to support this statement we calculate the so-called order parameter ??(T)=N ₀(T)/N from 0??T??T _c where N ₀ represents the number of particles within the condensate, N the total number of particles and T _c the critical temperature. Both results suggest that the three mean-field theories considered here, Hartree-Fock (HF), Popov (P) and Yukalov-Yukalova (Ykv) do not predict a second order phase transition as the BEC transition in weakly interacting gases is expected to show. Using the Local Density Approximation (LDA) we extend these calculation to obtain the density profiles $\rho(\vec{r})$ for an inhomogeneous Bose gas trapped in a harmonic external potential $V_{ext}(\vec{r})$ . As expected the density profiles show that the confinement is not enough to override the anomalies observed in the thermodynamic quantities for the gas confined in a box of volume V.     

Bogoliubov Theory for a Superfluid Bose Gas Flowing in a Random Potential: Stability and Critical Velocity

We investigate the stability and critical velocity of a weakly interacting Bose gas flowing in a random potential. By applying the Bogoliubov theory to a disordered Bose system with a steady flow, the condensate density and the superfluid density are determined as functions of the disorder strength, flow velocity, and temperature. The critical velocity, at which the steady flow becomes unstable, is calculated from the spectrum of hydrodynamic excitation. We also show that in two dimensions the critical velocity strongly depends on the system size.     

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.     

Bose-Einstein Condensation Measurements and Superflow in Condensed Helium

We review the formulation and measurement of Bose-Einstein condensation (BEC) in liquid and solid helium. BEC is defined for a Bose gas and subsequently for interacting systems via the one-body density matrix (OBDM) valid for both uniform and non-uniform systems. The connection between the phase coherence created by BEC and superflow is made. Recent measurements show that the condensate fraction in liquid ⁴He drops from 7.25±0.75 % at saturated vapor pressure (p≈0) to 2.8±0.2 % at pressure p=24 bars near the solidification pressure (p=25.3 bar). Extrapolation to solid densities suggests a condensate fraction in the solid of 1 % or less, assuming a frozen liquid structure such as an amorphous solid. Measurements in the crystalline solid have not been able to detect a condensate with an upper limit set at n ₀≤0.3 %. Opportunities to observe BEC directly in liquid ⁴He confined in porous media, where BEC is localized to patches by disorder, and in amorphous solid helium is discussed.     

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 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.     

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.     

Pair Excitations and Vertex Corrections in Fermi Fluids

Based on an equations–of–motion approach for time–dependent pair correlations in strongly interacting Fermi liquids, we have developed a theory for describing the excitation spectrum of these systems. Compared to the known “correlated” random–phase approximation (CRPA), our approach has the following properties: (i) The CRPA is reproduced when pair fluctuations are neglected. (ii) The first two energy–weighted sumrules are fulfilled implying a correct static structure. (iii) No ad–hoc assumptions for the effective mass are needed to reproduce the experimental dispersion of the zero sound mode in ³He. (iv) The density response function displays a novel form, arising from vertex corrections in the proper polarisation. Our theory is presented here with special emphasis on this latter point. We have also extended the approach to the single particle self-energy and included pair fluctuations in the same way. The theory provides a diagrammatic superset of the familiar GW approximation. It aims at a consistent calculation of single particle excitations with an accuracy that has previously only been achieved for impurities in Bose liquids.     

Magnetic Field Effect on the Evolution of the Magnon Population Properties in the [Co/Ag] Superlattice

In this work, the effect of applied magnetic field on the magnetic properties of [Co/Ag] superlattice is studied in the framework of the theory of spin waves for ferromagnetic Heisenberg systems, where the exchange and the dipolar interactions and the magneto-crystalline and surface anisotropies are all taking into account. The obtained corresponding Hamiltonian is treated using Green’s functions method. The analytical expressions of the excitation spectrum and the magnetization per spin are obtained when the exchange is present alone. While in the case where the above-cited other interactions are also present, a numerical study is done. The adjustment of the results of calculated and measured magnetization per spin obtained for various magnetic layer thickness (t _Co) is good. The deduced values of the exchange integral J are in line of the values usually given for this type of superlattices based on 3d-transition metals. The combined effect of dipolar interactions and surface anisotropy is studied numerically.     

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     

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.     

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.     

Thermodynamic Measurements in a Strongly Interacting Fermi Gas

Strongly interacting Fermi gases provide a clean and controllable laboratory system for modeling strong interparticle interactions between fermions in nature, from high temperature superconductors to neutron matter and quark-gluon plasmas. Model-independent thermodynamic measurements, which do not require theoretical models for calibrations, are very important for exploring this important system experimentally, as they enable direct tests of predictions based on the best current non-perturbative many-body theories. At Duke University, we use all-optical methods to produce a strongly interacting Fermi gas of spin-1/2-up and spin-1/2-down ⁶Li atoms that is magnetically tuned near a collisional (Feshbach) resonance. We conduct a series of measurements on the thermodynamic properties of this unique quantum gas, including the energy E, entropy S, and sound velocity  c. Our model-independent measurements of E and S enable a precision study of the finite temperature thermodynamics. The E(S) data are directly compared to several recent predictions. The temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation T=? E/? S by parameterizing the E(S) data using two different power laws that are joined with continuous E and T at a certain entropy S _c, where the fit is optimized. We observe a significant change in the scaling of E with S above and below S _c. Taking the fitted value of S _c as an estimate of the critical entropy for a superfluid-normal fluid phase transition in the strongly interacting Fermi gas, we estimate the critical parameters. Our E(S) data are also used to experimentally calibrate the endpoint temperatures obtained for adiabatic sweeps of the magnetic field between the ideal and strongly interacting regimes. This enables the first experimental calibration of the temperature scale used in experiments on fermionic pair condensation, where the ideal Fermi gas temperature is measured before sweeping the magnetic field to the strongly interacting regime. Our calibration shows that the ideal gas temperature measured for the onset of pair condensation corresponds closely to the critical temperature T _c estimated in the strongly interacting regime from the fits to our E(S) data. We also calibrate the empirical temperature employed in studies of the heat capacity and obtain nearly the same T _c. We determine the ground state energy by three different methods, using sound velocity measurements, by extrapolating E(S) to S=0 and by measuring the ratio of the cloud sizes in the strongly and weakly interacting regimes. The results are in very good agreement with recent predictions. Finally, using universal thermodynamic relations, we estimate the chemical potential and heat capacity of the trapped gas from the E(S) data.     

Critical behavior of a charged Bose gas

A fully self-consistent Hartre-Fock theory, using the Coulomb interaction screened by the polarization insertions calculated in the self-consistent random-phase approximation, is applied to thed-dimensional, dense, charged Bose gas at temperatures close to the transition temperatureT _c . The quasiparticle energy spectrum is calculated and shown to behave atT _c like ε(k)=Ak ^σ for smallk, and σ is calculated as a function of the dimensionalityd. The change in transition temperature from that of an ideal gas at the same density, and of the chemical potential are shown to be given by (T _c ?T _c0 )/T _c0 ≈Xr _s ^(d?2)/3 and μ _c ≈Yr _s ^2/3 , wherer _s is the ratio of the interparticle spacing to the Bohr radius. Approximate expressions are given for the coefficientsX andY. The critical exponents are calculated, and the system is shown to obey exact scaling.     

Soft excitation modes of quantum Bose fluids in the liquid-solid phase transition

We search for signatures of the liquid-solid phase transition in a group of strongly correlated quantum Bose fluids. We identify the non-spherically symmetric excitation mode which becomes soft at the critical density as the driving mechanism of the phase transition. The symmetry of the soft mode indicates the point-group symmetry of the arising solid phase. We place special emphasis on examining the effect of the short- and long-range features of the interaction on the crystal structure. Where possible, comparison with experiments and Monte Carlo results is made and a good agreement in the critical densities is found. The analysis reveals new features in the phase diagram of the charged Bose fluid.     

Quantum Phases and Collective Excitations of a Spin-Orbit-Coupled Bose–Einstein Condensate in a One-Dimensional Optical Lattice

The ground state of a spin-orbit-coupled Bose gas in a one-dimensional optical lattice is known to exhibit a mixed regime, where the condensate wave function is given by a superposition of multiple Bloch-wave components, and an unmixed one, in which the atoms occupy a single Bloch state. The unmixed regime features two unpolarized Bloch-wave phases, having quasimomentum at the center or at the edge of the first Brillouin zone, and a polarized Bloch-wave phase at intermediate quasimomenta. By calculating the critical values of the Raman coupling and of the lattice strength at the transitions among the various phases, we show the existence of a tricritical point where the mixed, the polarized and the edge-quasimomentum phases meet, and whose appearance is a consequence of the spin-dependent interaction. Furthermore, we evaluate the excitation spectrum in the unmixed regime and we characterize the behavior of the phonon and the roton modes, pointing out the instabilities occurring when a phase transition is approached.     

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.