Hard-sphere Bose Gas Research Articles

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {a}$$\\end{document} in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\\pi \\rho \\mathfrak {a}$$\\end{document} and shows that corrections are smaller than Cρa(ρa3)1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\rho \\mathfrak {a} (\\rho {{\\mathfrak {a}}}^3)^{1/2}$$\\end{document}, for a sufficiently large constant C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C > 0$$\\end{document}. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order ρa(ρa3)1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\mathfrak {a}(\\rho {{\\mathfrak {a}}}^3)^{1/2}$$\\end{document}, in agreement with the Lee–Huang–Yang prediction.

The explicit expression of the fugacity for weakly interacting Bose and Fermi gases

In this paper, we calculate the explicit expression for the fugacity for two- and three-dimensional weakly interacting Bose and Fermi gases from their equations of state in isochoric and isobaric processes, respectively, based on the mathematical result of the boundary problem of analytic functions --- the homogeneous Riemann-Hilbert problem. We also discuss the Bose-Einstein condensation phase transition of three-dimensional hard-sphere Bose gases.

Variational Monte Carlo study of soliton excitations in hard-sphere Bose gases

By using a full many-body approach, we calculate the excitation energy, the effective mass and the density profile of soliton states in a three dimensional Bose gas of hard spheres at zero temperature. The many-body wave function used to describe the soliton contains a one-body term, derived from the solution of the Gross-Pitaevskii equation, and a two-body Jastrow term which accounts for the repulsive correlations between atoms. We optimize the parameters in the many-body wave function via a Variational Monte Carlo procedure, calculating the grand-canonical energy and the canonical momentum of the system in a moving reference frame where the soliton is stationary. As the density of the gas is increased, significant deviations from the mean-field predictions are found for the excitation energy and the density profile of both dark and grey solitons. In particular, the soliton effective mass $m^\ast$ and the mass $m\Delta N$ of missing particles in the region of the density depression are smaller than the result from the Gross-Pitaevskii equation, their ratio being however well reproduced by this theory up to large values of the gas parameter. We also calculate the profile of the condensate density around the soliton notch finding good agreement with the prediction of the local density approximation.

Ground state of a homogeneous Bose gas of hard spheres

The ground state of a homogeneous Bose gas of hard spheres is treated in self-consistent mean-field theory. It is shown that this approach provides an accurate description of the ground state of a Bose-Einstein condensed gas for arbitrarily strong interactions. The results are in good agreement with Monte Carlo numerical calculations. Since all other mean-field approximations are valid only for very small gas parameters, the present self-consistent theory is a unique mean-field approach allowing for an accurate description of Bose systems at arbitrary values of the gas parameter.

DRESSED BOSONS AND EFFECTIVE TEMPERATURE — A FORMALISM FOR HELIUM II

The thermodynamics of a free Bose gas with effective temperature scale [Formula: see text] and hard-sphere Bose gas with the [Formula: see text] scale are studied. [Formula: see text] arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction V osc , which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential V He between 4 He atoms. The repulsive part of V He is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the [Formula: see text] temperature scale (HSET), and particle mass and density equal to those of 4 He , is investigated, first, by the Bogoliubov–Huang method and next by an improved version of this method, which describes He II in terms of dressed bosons and takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators a0, [Formula: see text] (originally rejected by Bogoliubov). Theoretical heat capacity CV(T) exhibits good agreement, below 1.9 K, with the experimental heat capacity graph observed in 4 He at saturated vapour pressure. The phase transition to the He II phase, occurs in the HSET at Tλ = 2.17 K, and is accompanied, in the modified HSET version, by a singularity of CV(T). The fraction of atoms in the momentum condensate at 0 K equals 8.86% and agrees with other theoretical estimates for He II. The fraction of normal fluid falls to 8.37% at 0 K which exceeds the value 0% found in He II.

Application of the static fluctuation approximation to the computation of the thermodynamic properties of an interacting trapped two-dimensional hard-sphere Bose gas

The static fluctuation approximation (SFA) is applied to compute the thermodynamic properties of a trapped two-dimensional (2D) interacting hard-sphere (HS) Bose gas in the weakly and strongly interacting regime. A mean-field approach involving a variational wave function is used to compute the mean-field energy as a function of temperature for each harmonic oscillator (HO) state plugged into the SFA technique. In the variational approach, a parameter $\ensuremath{\alpha}$ is introduced into the harmonic oscillator wave function in order to take into account the changes in the width when the repulsive interactions between the bosons are increased. In the weakly interacting regime, below the critical temperature, the total energy of all HO states (evaluated by our model) matches the noninteracting result very well. However, beyond the critical temperature, we ``fit'' our energies to the classical limit for 2D bosons in a trap by using a suitably proposed weighting function. We compare our results to earlier results of mean-field theory. Further, we evaluate the density matrix arising from correlations between the HO orbitals.

THERMODYNAMIC PROPERTIES OF AN INTERACTING HARD-SPHERE BOSE GAS IN A TRAP USING THE STATIC FLUCTUATION APPROXIMATION

A hard-sphere (HS) Bose gas in a trap is investigated at finite temperatures in the weakly interacting regime and its thermodynamic properties are evaluated using the static fluctuation approximation. The energies are calculated with a second-quantized many-body Hamiltonian and a harmonic oscillator wave function. The specific heat capacity, internal energy, pressure, entropy, and the Bose–Einstein occupation number of the system are determined as functions of temperature and for various values of interaction strength and number of particles. It is found that the number of particles plays a more profound role in the determination of the thermodynamic properties of the system than the HS diameter characterizing the interaction, that the critical temperature drops with the increase of the repulsion between the bosons, and that the fluctuations in the energy are much smaller than the energy itself in the weakly interacting regime.

Two super-Tonks-Girardeau states of a trapped one-dimensional spinor Fermi gas

A harmonically trapped ultracold 1D spinor Fermi gas with a strongly attractive 1D even-wave interaction induced by a 3D Feshbach resonance is studied. It is shown that it has two different super Tonks-Girardeau (sTG) energy eigenstates which are metastable against collapse in spite of the strong attraction, due to their close connection with 1D hard sphere Bose gases which are highly excited gaslike states. One of these sTG states is a hybrid between an sTG gas with strong $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ attractions and an ideal Fermi gas with no $(\ensuremath{\uparrow}\ensuremath{\uparrow})$ or $(\ensuremath{\downarrow}\ensuremath{\downarrow})$ interactions, the sTG component being an exact analog of the recently observed sTG state of a 1D ultracold Bose gas. It should be possible to create it experimentally by a sudden switch of the $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ interaction from strongly repulsive to strongly attractive, as in the recent Innsbruck experiment on the bosonic sTG gas. The other is a trapped analog of a recently predicted sTG state which is an ultracold gas of strongly bound $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ fermion dimers which behave as bosons with a strongly attractive boson-boson interaction leading to sTG behavior. It is proved that the probability of a transition from the ground state for strongly repulsive interaction to this dimer state under a sudden switch from strongly repulsive to strongly attractive interaction is $\ensuremath{\ll}$$1$, contrary to a previous suggestion.

Wave functions of the super-Tonks-Girardeau gas and the trapped one-dimensional hard-sphere Bose gas

Recent theoretical and experimental results demonstrate a close connection between the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gas with hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ of the sTG gas, a highly excited gas-like state with nodes only at interparticle separations $|x_{j\ell}|=x_{node}\approx a_{1D}$. It is shown herein that when the coupling constant $g_B$ in the Lieb-Liniger interaction $g_B\delta(x_{j\ell})$ is negative and $|x_{12}|\ge x_{node}$, the sTG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the sTG and HSB wave functions is that the sTG wave function allows a small penetration into the region $|x_{12}|<x_{node}$, whereas for a HSB gas with hard sphere diameter $a_{h.s.}=x_{node}$, the HSB wave function vanishes when all $|x_{12}|<a_{h.s.}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The sTG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ge 2$, $x_{node}$ is given accurately by a simple parabola. The metastability of the sTG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.

WAVE FUNCTION AND PAIR DISTRIBUTION FUNCTION OF A DILUTE BOSE GAS

The wave function of a dilute hard sphere Bose gas at low temperatures is revisited. Errors in an early 1957 paper are corrected. The pair distribution function is calculated for two values of [Formula: see text].

Diffusion Monte Carlo analysis of the crossover from three to two dimensions for trapped Bose gases

We investigate the crossover from three to two dimensions for harmonically trapped hard-sphere Bose gases by varying the aspect ratio of the trapping potential. The diffusion Monte Carlo method is used to calculate both the ground-state energy and structural properties. The effect of trap anisotropy, interparticle interaction, and number of particles on the ground-state properties is discussed. Our results show that the minimum value of the aspect ratio at which the system reaches an asymptotic equilibrium distribution in the weakly confined direction decreases with increasing scattering length, while the minimum value of the aspect ratio at which the system enters the quasi-two-dimensional (2D) regime increases as both the scattering length and the number of particles increase. Additionally, the role played by particle correlations is proved to be more pronounced in the quasi-2D system than in the three-dimensional (3D) system by directly comparing the ground-state properties for the two cases.

FIFTY YEARS OF HARD-SPHERE BOSE GAS: 1957–2007

Fifty years ago, Yang and I worked on the dilute hard-sphere Bose gas, which has been experimentally realized only relatively recently. I recount the background of that work, subsequent developments, and fresh understanding. In the original work, we had to rearrange the perturbation series, which was equivalent to the Bogoliubov transformation. A deeper reason for the rearrangement has been a puzzle. I can now explain it as a crossover from ideal gas to interacting gas behavior, a phenonmenon arising from Bose statistics. The crossover region is infinitesimally small for a macroscopic system, and thus unobservable. However, it is experimentally relevant in mesoscopic systems, such as a Bose gas trapped in an external potential, or on an optical lattice.

Interacting quantum gases in confined space: Two- and three-dimensional equations of state

In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.

VARIATIONAL DESCRIPTION OF WEAKLY INTERACTING BOSE GASES IN 3 DIMENSIONS

Static and dynamic properties of a weakly interacting Bose gas of Hard Spheres in three dimensions are studied in the framework of the Correlated Basis Functions (CBF) approximation. Results are compared with explicit expressions for the same quantities derived within the Bogoliubov model. Despite the good agreement in the energy of the groundstate and the excited states, other quantities such as the dynamic structure function present important differences that become more significant when the density is raised.

Particle densities for dilute hard-sphere Bose or Fermi gases in an external potential

We prove that if the diameter of a hard-sphere is much smaller than the size of an external potential, the s-wave pseudopotential reduces to the Huang-Yang s-wave pseudopotential. We obtain the first-order virial expansions of particle densities for dilute hard-sphere Bose or Fermi gases in an arbitrary external potential. In the absence of an external potential, the results reduce to the Huang-Yang-Luttinger and Lee-Yang virial expansions. In the quasi-classical limit, the results reduce to the results of the local density approximation.

Energy and Structure of Hard-Sphere Bose Gases in three and two dimensions

The energy and structure of dilute gases of hard spheres in three dimensions is discussed, together with some aspects of the corresponding 2D systems. A variational approach in the framework of the Hypernetted Chain Equations (HNC) is used starting from a Jastrow wavefunction that is optimized to produce the best two--body correlation factor with the appropriate long range. Relevant quantities describing static properties of the system are studied as a function of the gas parameter $x=\rho a^d$ where $\rho$, $a$ and $d$ are the density, $s$--wave scattering length of the potential and dimensionality of the space, respectively. The occurrence of a maximum in the radial distribution function and in the momentum distribution is a natural effect of the correlations when $x$ increases. Some aspects of the asymptotic behavior of the functions characterizing the structure of the systems are also investigated.

Bogoliubov inequality and Bose-Einstein condensates with repulsive and attractive interactions

The Hohenberg theorem on the absence of Bose-Einstein condensation (BEC) in homogeneous systems of space dimensions $Dl~2$ is based on a well-known Bogoliubov inequality. Applied to an assembly trapped in a harmonic potential we show that the Bogoliubov inequality rules out BEC in all dimensions at finite temperatures. However, this conflicting result with both theory and experiment disappears when the effect of the order parameter is properly taken into account in the boson-field commutation relations. For a hard-sphere Bose gas the theory is consistent with the expansion of the condensate when a positive scattering length is increased, as well as the collapse of the condensate when the sign of the scattering length is reversed and it reaches a minimum critical value.

Dilute Bose gas: short-range particle correlations and ultraviolet divergence

The modified Bogoliubov model where the primordial interaction is replaced by the t matrix is reinvestigated. It is shown to provide a negative value of the kinetic energy for a strongly interacting dilute Bose gas, contrary to the original Bogoliubov model. To clear up the origin of this failure, the correct values of the kinetic and interaction energies of a dilute Bose gas are calculated. It is demonstrated that both the problem of the negative kinetic energy and the ultraviolet divergence, dating back to the well-known paper of Lee, Yang and Huang, is connected with an inadequate picture of the short-range boson correlations. These correlations are reconsidered within the thermodynamically consistent model proposed earlier by the present authors. Found results are in absolute agreement with the data of the Monte-Carlo calculations for the hard-sphere Bose gas.

Quantum hard spheres as a model for a homogeneous Bose gas

Results for the energy per particle and the condensate fraction for a hard-sphere Bose gas are presented for a wide range of densities using the diffusion Monte Carlo method. Numerical fits to these results show the relevance of the expected series continuation beyond the well-known universal terms.

Diagrammatic derivation of Bose condensate fractions.

We describe a diagrammatic calculation of the condensate fraction of dense and strongly interacting Bose systems at zero temperature using the framework of parquet theory. Starting from the proper self-energy of parquet theory we perform a major rearrangement of diagrams to ensure proper counting and to express the final result in terms of well-behaved quantities. A number of interesting limiting cases are considered, in particular the hard sphere Bose gas at low densities where analytical results can be obtained. The present formalism is well-suited for extension to finite temperatures, and we consider possible strategies in that direction. {copyright} {ital 1996 The American Physical Society.}