Continuum of classical-field ensembles in Bose gases from canonical to grand canonical and the onset of their equivalence

The canonical and grand-canonical ensembles are two usual marginal cases for ultracold Bose gases, but real collections of experimental runs commonly have intermediate properties. Here we study the continuum of intermediate cases and look into the appearance of ensemble equivalence as interaction rises for mesoscopic $1d$ systems. We demonstrate how at sufficient interaction strength the distributions of condensate and excited atoms become practically identical, regardless of the ensemble used. Importantly, we find that features that are fragile in the ideal gas and appear only in a strict canonical ensemble can become robust in all ensembles when interactions become strong. As evidence, the steep cliff in the distribution of the number of excited atoms is preserved. To make this study, a straightforward approach for generating canonical and intermediate classical field ensembles using a modified stochastic Gross-Pitaevskii equation is developed.

Figure 1

The probability distribution of having $N_{\mathrm{ex}}$ excited atoms in a uniform $1d$ ideal gas at a relatively low temperature $T=0.341T_{*}$. Shown are the GCE (lower, yellow, $\sigma =\infty$) and CE (upper, red, $\sigma =1$) cases. Dotted and dashed lines correspond to exact classical field predictions (B18) and (B14), respectively. They are indiscernible from the full quantum predictions (B9) and (B17) for these parameters. The target total atom number was fixed at $\overline{N}=500$ in both cases. A visible and distinguishable difference between the GCE and CE is confirmed by the histograms.

Figure 2

Probability distribution of the total number of atoms as the $\mathrm{CE}\to \mathrm{GCE}$ parameter $\sigma$ is varied. The inset shows the range of $\sigma$ values for which the properties of the ensemble are very close to an ideal canonical Bose gas. Low-temperature case, $T=0.341T_{*}, \overline{N}=500$.

Figure 3

The progression of probability distributions of the number of excited atoms $N_{\mathrm{ex}}$ in the high-temperature case of $T=1.365T_{*}, \overline{N}=500$. From bottom to top, we have the GCE (yellow), going through intermediate ensembles to the CE (upper, red). Dotted and dashed lines correspond to the exact classical field results (B18) and (B14), respectively.

Figure 4

Relative variance of condensate atom number $N_0$ in the low-temperature ($T=0.341T_{*},\overline{N}=500$) regime as a function of the $\sigma$ parameter for a wide spectrum of interaction values $g$ (in units of $1/L$). Gray lines are exact asymptotic results for the ideal gas system ($g=0$) in the CE and GCE.

Figure 5

Relative variance of total atom number $N$ in the low-temperature ($T=0.341T_{*},\overline{N}=500$) regime. All notation the same as in Fig. 4. The dashed line $\delta N/\langle N\rangle =\sigma /\overline{N}$ is a naive estimation of the variance due to just the effect of the external parameter $\sigma$.

Figure 6

Mean number of atoms as the fluctuation control parameter $\sigma$ changes. Various interaction strengths, and the ideal gas case, are shown. Low-temperature case, $T=0.341T_{*}, \overline{N}=500$. The red line shows the simple estimate (40) for the ideal gas.

Figure 7

Relative variance of condensate atom number $N_0$ in the high-temperature ($T=1.365T_{*},\overline{N}=500$) regime. Notation is the same as in Fig. 4.

Figure 8

A demonstration of the equivalence of ensembles as interaction rises. Low-temperature ($T=0.341T_{*},\overline{N}=500$) probability distributions of $N_{\mathrm{ex}}$ (left) and $N_0$ (right) in the case of large interaction, $g=2$ (upper panel), in comparison to the probability distributions in the noninteracting atom case (lower panel).

Figure 9

Approach to equivalence of ensembles in the high-temperature ($T=1.365T_{*},\overline{N}=500$) case. Notation as in Fig. 8. Probability distributions of $N_{\mathrm{ex}}$ (left) and $N_0$ (right) in the case of large interaction, $g=8$ (upper panel), in comparison to the probability distributions in the noninteracting atom case (lower panel). Note the robust reproduction of the cliff in $P\left(N_{\mathrm{ex}}\right)$ near $N_{\mathrm{ex}}=\overline{N}$.