The fluctuations of the atom number between a Bose-Einstein condensate and the surrounding thermal gas have been the subject of a long standing theoretical debate. This discussion is centered around the appropriate thermodynamic ensemble to be used for theoretical predictions and the effect of interactions on the observed fluctuations. Here we introduce the so-called Fock state sampling method to solve this classic problem of current experimental interest for weakly interacting gases. A suppression of the predicted peak fluctuations is observed when using a microcanonical with respect to a canonical ensemble. Moreover, interactions lead to a shift of the temperature of peak fluctuations for harmonically trapped gases. The absolute size of the fluctuations furthermore depends on the total number of atoms and the aspect ratio of the trapping potential. Due to the interplay of these effect, there is no universal suppression or enhancement of fluctuations.
The fluctuations of the atom number between a Bose-Einstein condensate and the surrounding thermal gas have been the subject of a long standing theoretical debate. This discussion is centered around the appropriate thermodynamic ensemble to be used for theoretical predictions and the effect of interactions on the observed fluctuations. Here we introduce the so-called Fock state sampling method to solve this classic problem of current experimental interest for weakly interacting gases. A suppression of the predicted peak fluctuations is observed when using a microcanonical with respect to a canonical ensemble. Moreover, interactions lead to a shift of the temperature of peak fluctuations for harmonically trapped gases. The absolute size of the fluctuations furthermore depends on the total number of atoms and the aspect ratio of the trapping potential. Due to the interplay of these effect, there is no universal suppression or enhancement of fluctuations.
The fluctuations of the atom number between a Bose-Einstein condensate and the surrounding thermal gas have been the subject of a long standing theoretical debate. This discussion is centered around the appropriate thermodynamic ensemble to be used for theoretical predictions and the effect of interactions on the observed fluctuations. Here we introduce the so-called Fock state sampling method to solve this classic problem of current experimental interest for weakly interacting gases. A suppression of the predicted peak fluctuations is observed when using a microcanonical with respect to a canonical ensemble. Moreover, interactions lead to a shift of the temperature of peak fluctuations for harmonically trapped gases. The absolute size of the fluctuations furthermore depends on the total number of atoms and the aspect ratio of the trapping potential. Due to the interplay of these effect, there is no universal suppression or enhancement of fluctuations.
Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.
Quantum systems are typically characterized by the inherent fluctuation of their physical observables. Despite this fundamental importance, the investigation of the fluctuations in interacting quantum systems at finite temperature continues to pose considerable theoretical and experimental challenges. Here we report the characterization of atom number fluctuations in weakly interacting Bose-Einstein condensates. Technical fluctuations are mitigated through a combination of nondestructive detection and active stabilization of the cooling sequence. We observe fluctuations reduced by 27% below the canonical expectation for a noninteracting gas, revealing the microcanonical nature of our system. The peak fluctuations have near linear scaling with atom number ΔN_{0,p}^{2}∝N^{1.134} in an experimentally accessible transition region outside the thermodynamic limit. Our experimental results thus set a benchmark for theoretical calculations under typical experimental conditions.
Bose-Einstein condensation (BEC) 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 (CN), or grand CN (GC) 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 BECs. 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 GC fluctuations. Finally, we briefly consider the future directions for research in the field of condensate statistics.
Abstract The investigation of atom number fluctuations in quantum gases at finite temperatures showcases the ongoing challenges in understanding complex quantum systems. Recently, the microcanonical nature of atom number fluctuations in weakly interacting Bose–Einstein condensates was observed. This motivates an investigation of the thermal component of partially condensed Bose gases, due to the conservation of the total atom number. Here, we present a combined analysis of both components, including a comprehensive analysis of the uncertainties in the preparation and parameter extraction of partially condensed quantum gases. This enables a complementary observation of the thermal atom number fluctuations and yields and improved value of the peak BEC atom number fluctuations Δ N p , 0 2 = ( 3.7 ± 7 ) × 10 5 close to the critical temperature. This corresponds to a reduction by 41% with respect to previous analysis and corroborates the microcanonical nature of the fluctuations. The analysis of noise contributions due to the preparation and evaluation of partially condensed Bose gases is based on Monte Carlo simulations of optical density profiles. Importantly, this allows for an estimation of the technical noise contributions to the atom number and temperature, which is generally applicable in the field of ultracold atoms.
Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.