On the fluctuations of the number of atoms in the condensate

TL;DR

This work surveys how condensate number fluctuations in Bose gases depend on the statistical ensemble, with a focus on the ideal gas where GC predictions can be unphysical below Tc and CN/MC frameworks are required to obtain meaningful fluctuations, as expressed through quantities like $\delta^2 N_0$ and $\langle N_0\rangle$.It catalogues historical and modern methods for describing fluctuations, including analytic approaches tied to partition functions, recurrence relations, stochastic ensemble sampling, master equations, and Bogoliubov-type theories for weakly interacting gases, and it emphasizes spectral classification and the Maxwell Demon rescue ensemble as keys to understanding ensemble inequivalence.The review also highlights experimental progress, notably the ability to stabilize and measure BEC fluctuations in ultracold atoms and the use of photonic condensates as a platform for exploring canonical versus grand-canonical statistics, while outlining future directions toward higher moments, nonequilibrium ensembles, and interaction-driven restoration of ensemble equivalence.Overall, the paper anchors a nuanced view of condensate statistics that connects exact and approximate theories with cutting-edge experiments, underscoring the importance of ensemble choice and advanced numerical methods for accurately capturing fluctuation phenomena in finite, interacting, and open quantum gases.

Abstract

Bose-Einstein condensation represents a remarkable phase transition, characterized by the formation of a single quantum subsystem. As a result, the statistical properties of the condensate are highly unique. In the case of a Bose gas, while the mean number of condensed atoms is independent of the choice of statistical ensemble, the microcanonical, canonical, or grand canonical variances differ significantly among these ensembles. In this paper, we review the progress made over the past 30 years in studying the statistical fluctuations of Bose-Einstein condensates. Focusing primarily on the ideal Bose gas, we emphasize the inequivalence of the Gibbs statistical ensembles and examine various approaches to this problem. These approaches include explicit analytic results for primarily one-dimensional systems, methods based on recurrence relations, asymptotic results for large numbers of particles, techniques derived from laser theory, and methods involving the construction of statistical ensembles via stochastic processes, such as the Metropolis algorithm. We also discuss the less thoroughly resolved problem of the statistical behavior of weakly interacting Bose gases. In particular, we elaborate on our stochastic approach, known as the hybrid sampling method. The experimental aspect of this field has gained renewed interest, especially following groundbreaking recent measurements of condensate fluctuations. These advancements were enabled by unprecedented control over the total number of atoms in each experimental realization. Additionally, we discuss the fluctuations in photonic condensates as an illustrative example of grand canonical fluctuations. Finally, we briefly consider the future directions for research in the field of condensate statistics.

Figures (17)

• Figure 1: Illustration of the average number of condensed atoms (left) and its variance $\delta^2 N_0$ (right) as a function of temperature $T$ in the three archetypical statistical ensembles. The choice of ensemble is irrelevant for $\langle N_0\rangle$. On the other hand, when the temperature of an ultracold Bose gas is lowered towards the critical temperature $T_c$, a grand canonical ensemble (green) predicts unphysically large fluctuations. A canonical ensemble (orange) does not suffer from this problem and has fluctuations peaking just below $T_c$. A microcanonical ensemble shows the same temperature-dependent trend but with quantitatively lower fluctuations. Data for $N=1000$ non-interacting atoms in an isotropic harmonic trap. The right panel is adopted from Kruk23 ( https://creativecommons.org/licenses/by/4.0/).
• Figure 2: The microcanonical probability distribution of having $N_{\rm ex}$ atoms in a 1d harmonically trapped Bose gas: exact result (blue solid) obtained via Eq. (\ref{['eq:1d-recurrence']}) and asymptotic formulas for the number of restricted Szekeres1953Jan partitions (dashed red). The inset shows comparison of the entropy $S(E, N):=k_{\rm B} \ln \Gamma\left( E,\,N \right)$, where $\Gamma\left( E,\,N \right)$ is computed via exact method (blue line) or approximated with the asymptotic formula $A(E)$Hardy1918 (red dashed line). Parameters $E=200$ and $N>E$.
• Figure 3: The probability distribution of microstates with energy $E$ and $N=100$ particles, $\mathcal{P}(E)=\Gamma e^{-\beta E}/\mathcal{Z}$, in a CN ensemble in a 1D harmonic trap given by Eq. (\ref{['eq:cn']}). The figure illustrates the shrinking relative width of the energy distribution, shown using the normalised probability distribution $\tilde{p}(x)=\langle E\rangle\, \mathcal{P}(\,E(x)\,)$ in the relative variable $x=(E-\langle E\rangle)/\langle E\rangle$.
• Figure 4: The narrowing of the probability distribution of microstates with $N$ particles, $\mathcal{P}(N)=z^N\mathcal{Z}/\Xi$, in a 3D harmonic trap given by Eq. (\ref{['eq:gk']}). We illustrate the shrinking particle number distribution $\tilde{p}(x)$ width (relative to the mean particle number) in the variable $x=(N-\langle N\rangle)/\langle N\rangle$, ${\tilde{p}(x)=\langle N\rangle \mathcal{P}(N(x))}$ for constant, uncondensed, $T/T_c>1$. The chemical potential $\mu$ is fixed by the choice of $\langle N\rangle$.
• Figure 5: The change of microcanonical fluctuations $\delta_{MC}N_0$ with $N=200,500,1000$ (left to right) for an ideal gas in a 3d harmonic trap. Note the perfect overlap at low $T$. Reproduced from Grossmann97prl © American Physical Society, https://dx.doi.org/10.1103/PhysRevLett.79.3557.