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arXiv:cond-mat/0610238 · Physical Review Letters

Condensation of N Interacting Bosons: A Hybrid Approach to Condensate Fluctuations

Anatoly A. Svidzinsky, Marlan O. Scully

Published Nov 7, 2006• 33 citations
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Abstract

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.

DOI
10.1103/physrevlett.97.190402

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The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.

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Physical Review A•Mar 14, 2011•8 citations
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 present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.
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.
The atom fluctuation statistics of an ideal, mesoscopic, Bose-Einstein condensate are investigated from several different perspectives. By generalizing the grand canonical analysis (applied to the canonical ensemble problem), we obtain a self-consistent equation for the mean condensate particle number that coincides with the microscopic result calculated from the laser master equation approach. For the case of a harmonic trap, we obtain an analytic expression for the condensate particle number that is very accurate at all temperatures, when compared with numerical canonical ensemble results. Applying a similar generalized grand canonical treatment to the variance, we obtain an accurate result only below the critical temperature. Analytic results are found for all higher moments of the fluctuation distribution by employing the stochastic path integral formalism, with excellent accuracy. We further discuss a hybrid treatment, which combines the master equation and stochastic path integral analysis with results obtained based on the canonical ensemble quasiparticle formalism [Kocharovsky et al., Phys. Rev. A 61, 053606 (2000)], producing essentially perfect agreement with numerical simulation at all temperatures.
The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.
We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number NNN of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N=200N=200N=200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.