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arXiv:2403.15047 · Journal of Physics B Atomic Molecular and Optical Physics

Atom number fluctuations in Bose gases—statistical analysis of parameter estimation

Toke Vibel, Mikkel Berg Christensen, Rasmus Malthe Fiil Andersen, Laurits Nikolaj Stokholm, Krzysztof Pawłowski, Kazimierz Rza̧żewski, Mick Kristensen, J. Arlt

Published Aug 28, 2024• 5 citations
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Abstract

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.

DOI
10.1088/1361-6455/ad7458

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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.
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.
Fluctuations are a key property of both classical and quantum systems. While the fluctuations are well understood for many quantum systems at zero temperature, the case of an interacting quantum system at finite temperature still poses numerous challenges. Despite intense theoretical investigations of atom number fluctuations in Bose-Einstein condensates, their amplitude in experimentally relevant interacting systems is still not fully understood. Moreover, technical limitations have prevented their experimental investigation to date. Here we report the observation of these fluctuations. Our experiments are based on a stabilization technique, which allows for the preparation of ultracold thermal clouds at the shot noise level, thereby eliminating numerous technical noise sources. Furthermore, we make use of the correlations established by the evaporative cooling process to precisely determine the fluctuations and the sample temperature. This allows us to observe a telltale signature: the sudden increase in fluctuations of the condensate atom number close to the critical temperature.
Bose-Einstein condensation (BEC) in dilute gases stands as a fascinating phenomenon of quantum mechanics. In this state, hundreds of thousands of atoms occupy the lowest energy level, establishing a collective macroscopic quantum state, with numerous intriguing properties. BEC has become a fundamental tool for the exploration of quantum physics and it may facilitate further advancements in quantum technology, for instance, in quantum simulation and sensing applications. Due to the potential of BECs in both basic research and technology, a fundamental understanding of this phenomenon is essential to guarantee the precision and reliability of its applications. Atom number fluctuations in BECs stand as an unresolved fundamental problem that has posed significant theoretical challenges for nearly eight decades. The experimental observation of this effect has recently become feasible. However, a direct comparison between theoretical predictions and experimental results remains outstanding. In particular, technical challenges persist in the accurate detection of the BEC, and a thorough understanding of estimation errors is lacking. This PhD thesis addresses some of these challenges in the field. Firstly, it introduces an enhanced calibration method for time-of-flight absorption images of ultracold atomic clouds, improving the accuracy of such measurements. Additionally, a thorough statistical analysis of uncertainties in the preparation and parameter estimation of partially condensed Bose gases is presented, leading to an improved understanding of atom number fluctuations in BEC. Finally, this thesis presents the experimental advances toward a new project within the research group: Electromagnetically induced transparency (EIT). EIT is a concept where an otherwise opaque cloud can be made transparent through manipulation of its quantum states. Notably, these advancements include the construction of setups for an optical dipole trap and a hybrid trap, along with the development of systems enabling precise control of magnetic fields. This laid the groundwork for recent demonstrations of EIT within the group.
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.
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.