Ideal Bose Gas Research Articles

A Simple Model of Bose–Einstein Condensation of Interacting Particles

A simple model of Bose-Einstein condensation of interacting particles is proposed. It is shown that in the condensate state the dependence of thermodynamic quantities on the interaction constant does not allow an expansion in powers of the coupling constant. Therefore it is impossible to pass to the Einstein model of condensation in an ideal Bose gas by means of a limiting passage, setting the interaction constant to zero. The account for the interaction between particles eliminates difficulties in the description of condensation available in the model of an ideal gas, which are connected with fulfillment of thermodynamic relations and an infinite value of the particle number fluctuation in the condensate phase.

Momentum Distribution in the Unitary Bose Gas from First Principles

We consider a realistic bosonic N-particle model with unitary interactions relevant for Efimov physics. Using quantum MonteCarlo methods, we find that the critical temperature for Bose-Einstein condensation is decreased with respect to the ideal Bose gas. We also determine the full momentum distribution of the gas, including its universal asymptotic behavior, and compare this crucial observable to recent experimental data. Similar to the experiments with different atomic species, differentiated solely by a three-body length scale, our model only depends on a single parameter. We establish a weak influence of this parameter on physical observables. In current experiments, the thermodynamic instability of our model from the atomic gas towards an Efimov liquid could be masked by the dynamical instability due to three-body losses.

Finite-temperature hydrodynamics for one-dimensional Bose gases: Breathing-mode oscillations as a case study

We develop a finite-temperature hydrodynamic approach for a harmonically trapped one-dimensional quasicondensate and apply it to describe the phenomenon of frequency doubling in the breathing-mode oscillations of its momentum distribution. The doubling here refers to the oscillation frequency relative to the oscillations of the real-space density distribution, invoked by a sudden confinement quench. We find that the frequency doubling is governed by the quench strength and the initial temperature, rather than by the crossover from the ideal Bose gas to the quasicondensate regime. The hydrodynamic predictions are supported by the results of numerical simulations based on a finite-temperature c-field approach, and extend the utility of the hydrodynamic theory for low-dimensional quantum gases to the description of finite-temperature systems and their dynamics in momentum space.

Symmetry breaking and the geometry of reduced density matrices

The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of these convex bodies exhibit non-analyticities, which signal the emergence of symmetry breaking and of an associated order parameter and also show different characteristics for different types of phase transitions. We illustrate this with three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model, the classical q-state Potts model in two-dimensions at finite temperature and the ideal Bose gas in three-dimensions at finite temperature. This state based viewpoint on phase transitions provides a unique novel tool for studying exotic many body phenomena in quantum and classical systems.

Unitary boson-boson and boson-fermion mixtures: third virial coefficient and three-body parameter on a narrow Feshbach resonance

We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters R t appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov-effect threshold. We give a new, exact integral expression of R t for the microscopic narrow Feshbach resonance model. A divergence of R t in the preefimovian regime at a scaling exponent s = 1 / 2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.

Exactly solvable tight-binding model on two scale-free networks with identical degree distribution

We study ideal Bose gas upon two scale-free structures and with identical degree distribution. Energy spectra belonging to tight-binding Hamiltonian are exactly solved and the related spectral dimensions of and are obtained as and . We show that Bose-Einstein condensation only takes place upon instead of . The topology and thermodynamical property of the two structures are proven to be totally different.

Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hat\psi({\bf r})\rangle$ of the field operator $\hat\psi({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hat\psi({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.

Thermodynamic properties of a rotating ideal Bose gas in an anisotropic harmonic trap

In this paper, the thermodynamic properties of a rotating Bose gas in an anisotropic harmonic trap are investigated. Using the density of states method, the thermodynamic potential, transition temperature, condensate fraction and heat capacity are analytically calculated. Consequently, the change of these thermodynamic quantities with reduced temperature, particle number and the rotation frequency are studied. It is found that, unlike the previous studied cases where the trapping potential is planar isotropic, due to the fact that the system has two stable regions as the rotation frequency increases, the thermodynamic quantities behave quite differently in these two stable regions. The validity of our results are also discussed.

Virial Coefficients from Unified Statistical Thermodynamics of Quantum Gases Trapped under Generic Power Law Potential in d Dimension and Equivalence of Quantum Gases

From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases, trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in d = 3 and d = 2. But more importantly we found that, the virial coefficients of Bose and Fermi gases become identical (except the second virial coefficient, where the sign is different) when the gases are trapped under harmonic potential in d = 1. This result suggests the equivalence between Bose and Fermi gases established in d = 1 (J. Stat. Phys. DOI 10.1007/s10955-015-1344-4). Also, it is found that the virial coefficients of two-dimensional free Bose (Fermi) gas are equal to the virial coefficients of one-dimensional harmonically trapped Bose (Fermi) gas.

Bose gas in disordered, finite-layered systems

Disorder effects in the thermodynamic properties of an ideal Bose gas confined in a semi-infinite multi-layer structure within a box of thickness [Formula: see text] and infinite lateral extent, are analyzed. The layers are first modeled by a periodic array of [Formula: see text] Dirac-delta functions of equal intensity. Then, we introduce structural and compositional disorder, as well as a random set of layer vacancies in the system to calculate the internal energy, chemical potential and the specific heat for different configurations. Whereas structural and compositional disorder do not reveal a significant change, a dramatic increase in the maximum of the specific heat is observed when the system is depleted a fraction of the order of 0.1–0.2 of random layers compared to the original, fully periodic array. Furthermore, this maximum, which is reminiscent of a Bose–Einstein condensation for an infinite array, occurs at higher temperatures.

Vortex rings in Bose gas

We consider excitations that exist, in addition to phonons, in the ideal Bose gas at zero temperature. These excitations are vortex rings whose energy spectrum is similar to the roton one in liquid helium.

Semiclassical and quantum description of an ideal Bose gas in a uniform gravitational field

We consider an ideal Bose gas contained in a cylinder in three spatial dimensions, subjected to a uniform gravitational field. It has been claimed by some authors that there is discrepancy between the semiclassical and quantum calculations in the thermal properties of such a system. To check this claim, we calculate the heat capacity and isothermal compressibility of this system semiclassically as well as from the quantum spectrum of the density of states. The quantum calculation is done for a finite number of particles. We find good agreement between the two calculations when the number of particles are taken to be large. We also find that this system has the same thermal properties as an ideal five dimensional Bose gas.

Calorimetry of a Bose-Einstein-condensed photon gas.

Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behaviour of thermodynamic quantities. For liquid helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example, for ultracold atomic Bose gases. Here we examine the thermodynamic behaviour of a trapped two-dimensional photon gas, a system that allows us to spectroscopically determine the specific heat and the entropy of a nearly ideal Bose gas from the classical high temperature to the Bose-condensed quantum regime. The critical behaviour at the phase transition is clearly revealed by a cusp singularity of the specific heat. Regarded as a test of quantum statistical mechanics, our results demonstrate a quantitative agreement with its predictions at the microscopic level.

Self-similarity of the statistics in the critical region of Bose condensation of ideal gas in mesoscopic traps: Canonical and grand canonical ensembles

For an ideal Bose gas confined in a trap of arbitrary dimension and shape, the features of probabilistic distributions of the number of particles at the ground and all excited levels are studied within canonical and grand canonical ensembles. It is shown that in the critical range of system parameters the statistics is self-similar, and its characteristic scales (mathematical expectation, variance, etc.) are independent of an ensemble used in the description. At the same time, the selfsimilar distributions themselves differ for different ensembles.

Energy fluctuation of a finite number of interacting bosons: A correlated many-body approach

We calculate the energy fluctuation of a truly finite number of interacting bosons and study the role of interaction. Although the ideal Bose gas in thermodynamic limit is an exactly solvable problem and analytic expression of various fluctuation measures exists, the experimental Bose-Einstein condensation (BEC) is a nontrivial many-body problem. We employ a two-body correlated basis function and utilize the realistic van der Waals interaction. We calculate the energy fluctuation $(△{E}^{2})$ of the interacting trapped bosons and plot $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ as a function of $\frac{T}{{T}_{c}}$. In the classical limit $△{E}^{2}$ is related to the specific heat per particle ${c}_{v}$ through the relation $△{E}^{2}={k}_{B}{T}^{2}{c}_{v}$. We have obtained a distinct hump in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number $N\ensuremath{\simeq}5000$ and above which corresponds to the second-order phase transition. However for finite-size interacting bosons ($N\ensuremath{\simeq}$ a few hundred) the hump is not sharp, and the maximum in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ can be interpreted as a smooth increase in the scaled fluctuation below ${T}_{c}$ and then a decrease above ${T}_{c}$. To illustrate the justification we also calculate ${c}_{v}$, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.

Universal Behavior of the BEC Critical Temperature for a Multi-slab Ideal Bose Gas

For an ideal Bose-gas within a multi-slab periodic structure, we discuss the effect of the spatial distribution of the gas on its Bose-Einstein condensation critical temperature $T_c$, as well as on the origin of its dimensional crossover observed in the specific heat. The multi-slabs structure is generated by applying a Kronig-Penney potential to the gas in the perpendicular direction to the slabs of width $b$ and separated by a distance $a$, and allowing the particles to move freely in the other two directions. We found that $T_c$ decreases continuously as the potential barrier height increases, becoming inversely proportional to the square root of the barrier height when it is large enough. This behavior is {\it universal} as it is independent of the width and spacing of the barriers. The specific heat at constant volume shows a crossover from 3D to 2D when the height of the potential or the barrier width increase, in addition to the well known peak related to the Bose-Einstein condensation. These features are due to the trapping of the bosons by the potential barriers, and can be characterized by the energy difference between the energy bands below the potential height.

Momentum-Space Correlations of a One-Dimensional Bose Gas

Analyzing the noise in the momentum profiles of single realizations of one-dimensional Bose gases, we present the experimental measurement of the full momentum-space density correlations ⟨δn_{p}δn_{p^{'}}⟩, which are related to the two-body momentum correlation function. Our data span the weakly interacting region of the phase diagram, going from the ideal Bose gas regime to the quasicondensate regime. We show experimentally that the bunching phenomenon, which manifests itself as super-Poissonian local fluctuations in momentum space, is present in all regimes. The quasicondensate regime is, however, characterized by the presence of negative correlations between different momenta, in contrast to the Bogolyubov theory for Bose condensates, predicting positive correlations between opposite momenta. Our data are in good agreement with abinitio calculations.

Investigation of Bose Condensation in Ideal Bose Gas Trapped under Generic Power Law Potential in d Dimension

The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential are studied carefully. Detailed calculation of Kim et al. (J. Phys. Condens. Matter 11 (1999) 10269) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume CV are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.

Erratum: “Thermodynamics of noninteracting bosonic gases in cubic optical lattices versus ideal homogeneous Bose gases”

We have studied the thermodynamic properties of noninteracting gases in periodic lattice potential at arbitrary integer fillings and compared them with that of ideal homogeneous gases. By deriving explicit expressions for the thermodynamic quantities and performing exact numerical calculations, we have found that the dependence of e.g., entropy and energy on the temperature in the normal phase is rather weak especially at large filling factors. In the Bose condensed phase, their power dependence on the reduced temperature is nearly linear, which is in contrast to that of ideal homogeneous gases. We evaluated the discontinuity in the slope of the specific heat which turned out to be approximately the same as that of the ideal homogeneous Bose (IHB) gas for filling factor ν = 1. The discontinuity i.e. the jump in the heat capacity per particle linearly decreases with increasing ν. These results may serve as a checkpoint for various experiments on optical lattices as well as theoretical studies of weakly interacting Bose systems in periodic potentials being a starting point for perturbative calculations.

Specific heat and effects of pairing fluctuations in the BCS-BEC-crossover regime of an ultracold Fermi gas

We investigate the specific heat at constant volume $C_V$ in the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover regime of an ultracold Fermi gas above the superfluid phase transition temperature $T_{\rm c}$. Within the framework of the strong-coupling theory developed by Nozi\`eres and Schmitt-Rink, we show that this thermodynamic quantity is sensitive to the stability of preformed Cooper pairs. That is, while $C_V(T\gesim T_{\rm c})$ in the unitary regime is remarkably enhanced by {\it metastable} preformed Cooper pairs or pairing fluctuations, it is well described by that of an ideal Bose gas of long-lived {\it stable} molecules in the strong-coupling BEC regime. Using these results, we identify the region where the system may be viewed as an almost ideal Bose gas of stable pairs, as well as the pseudogap regime where the system is dominated by metastable preformed Cooper pairs, in the phase diagram of an ultracold Fermi gas with respect to the strength of a pairing interaction and the temperature. We also show that the calculated specific heat agrees with the recent experiment on a $^6$Li unitary Fermi gas. Since the formation of preformed Cooper pairs is a crucial key in the BCS-BEC crossover phenomenon, our results would be helpful in considering how fluctuating preformed Cooper pairs appear in a Fermi gas, to eventually become stable, as one passes through the BCS-BEC crossover region.