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Physical Review A

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

Satadal Bhattacharyya, M. L. Lekala, Barnali Chakrabarti, Gaotsiwe Joel Rampho

Published Mar 14, 2016• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.93.033624

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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 (△E2)(△{E}^{2})(△E2) of the interacting trapped bosons and plot △E2kB2T2\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}kB2​T2△E2​ as a function of TTc\frac{T}{{T}_{c}}Tc​T​. In the classical limit △E2△{E}^{2}△E2 is related to the specific heat per particle cv{c}_{v}cv​ through the relation △E2=kBT2cv△{E}^{2}={k}_{B}{T}^{2}{c}_{v}△E2=kB​T2cv​. We have obtained a distinct hump in △E2kB2T2\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}kB2​T2△E2​ around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number N\ensuremath≃5000N\ensuremath{\simeq}5000N\ensuremath≃5000 and above which corresponds to the second-order phase transition. However for finite-size interacting bosons (N\ensuremath≃N\ensuremath{\simeq}N\ensuremath≃ a few hundred) the hump is not sharp, and the maximum in △E2kB2T2\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}kB2​T2△E2​ can be interpreted as a smooth increase in the scaled fluctuation below Tc{T}_{c}Tc​ and then a decrease above Tc{T}_{c}Tc​. To illustrate the justification we also calculate cv{c}_{v}cv​, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.
We present a correlated many-body approach to calculate the distribution function and fluctuations for a Bose-Einstein condensate with NNN interacting atoms in the harmonic confinement. The present formulation uses the recursion relation for the canonical ensemble partition function (Z)(Z)(Z). ZZZ is calculated from the energy spectrum of the many-body effective potential, which keeps all possible two-body correlations and uses the realistic interatomic interaction. The condensate statistics are in very good agreement with earlier results of an ideal gas for which exact statistical moments for all temperature are known. We also present the numerical results of condensate statistics for real experimental situations. The calculated moments nicely exhibit the mesoscopic effect for a few hundred atoms, whereas the sharp fall in the variance for the large condensate near the critical temperature shows the possibility of phase transition. We also calculate the critical temperature for the mesoscopic regime. Our present calculation mimics the JILA experiment with 87Rb^{87}\mathrm{Rb}87Rb atoms.
A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates---dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density.
We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.
It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s)P(s)P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s)P(s)P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s=0s=0s=0, which signifies the presence of a large number of quasidegenerate states.
The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.