Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma

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.

Figure 1

(Color online) (Left) The average boson number in the condensate is plotted versus temperature. The master equation approach (solid line) is compared with numerical simulation of canonical results (dots), as well as the thermodynamic limit (dashed line). (Right) The variance of the distribution is plotted versus temperature. The grand canonical result (dashed line) deviates near and below the critical temperature, leading to the grand canonical catastrophe. The results of ter Haar [10] (dash dotted line) and Politzer [11] (small dots) (derived in the thermodynamic limit) are also shown. The results are shown for $N=200$ particles in a harmonic trap.

Figure 2

(Color online) The average condensate particle number is plotted versus temperature for $N=200$ particles in an isotropic harmonic trap. The approximate analytic result given by Eqs. (7) [or (13)] and (14) (solid line) is compared with the “exact canonical dots” computed numerically, with good agreement.

Figure 3

(Color online) The variance of the condensate distribution is plotted versus temperature for $N=200$. The approximate analytic result within the generalized grand canonical approach (solid line) is compared with the exact dots computed numerically in the canonical ensemble. The analytic result is in good agreement for low temperature, but fails at temperatures comparable to the critical temperature and above. The reason for the high-temperature discrepancy is that the modified grand canonical theory neglects correlations between excited levels.

Figure 4

(Color online) The present hybrid theory, which we call CNB5 (solid line), is compared with the exact numerical results obtained in the canonical ensemble (dots). (left) The average condensate particle number $⟨n_0⟩$ and (right) the variance $\mathrm{\Delta }{n}_0$ are plotted for $N=200$ versus temperature. The predictions of CNB3 [17] (dashed line) are also shown.

Figure 5

(Color online) As in Fig. 4, for the third, fourth, fifth, and sixth centered moments, denoted by $\mathbf{⟨}(n_0-⟨n_0⟩{)}^n\mathbf{⟩}$, as well as cumulants ${\kappa }_n$ [21].

Figure 6

(Color online) The average condensate particle number is plotted versus temperature for $N=200$. The approximate analytic result (A6) within the grand canonical approach (solid line) is compared with the exact canonical dots. The triple sum together with the simple expansion gives a poor fit.