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arXiv:2601.10415 · Journal of Physics B Atomic Molecular and Optical Physics

Cloud parameter estimation for interacting BEC after time-of-flight

Rasmus Malthe Fiil Andersen, Stine Frederiksen, Laurits Stokholm, Ilja Zebergs, Mick Kristensen, Carrie Weidner, Jan Joachim Arlt

cond-mat.quant-gasphysics.atom-phquant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026• 0 citations

Abstract

Experiments on Bose-Einstein condensates at finite temperature typically extract the system parameters, such as temperature, atom number, and condensed fraction from time-of-flight images taken after a free expansion time. This paper systematically examines the effect of repulsive interactions between the condensed and thermal atoms in partially condensed clouds on the expansion profile of the thermal cloud. An analytical expression for the expansion can be obtained only if the interactions between the Bose-Einstein condensate and thermal atoms are neglected, resulting in a Bose-enhanced distribution for the thermal component. Here, the deformation of the cloud due to interactions and the effects on estimated parameters are investigated by simulating the expansion using a ballistic approximation. By fitting the simulated expansion profiles with a Bose-enhanced distribution, the errors of using such a fit are estimated, and the results are explained phenomenologically. The simulation was also used as a fitting function for experimental data, showing better agreement of the extracted condensed fraction with the semi-ideal model than results from a Bose-enhanced fit.

Journal reference: J. Phys. B: At. Mol. Opt. Phys. 59 095502 (2026)
DOI: 10.1088/1361-6455/ae637b

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Experiments on Bose-Einstein condensates at finite temperature typically extract the system parameters, such as temperature, atom number, and condensed fraction from time-of-flight images taken after a free expansion time. This paper systematically examines the effect of repulsive interactions between the condensed and thermal atoms in partially condensed clouds on the expansion profile of the thermal cloud. An analytical expression for the expansion can be obtained only if the interactions between the Bose-Einstein condensate and thermal atoms are neglected, resulting in a Bose-enhanced distribution for the thermal component. Here, the deformation of the cloud due to interactions and the effects on estimated parameters are investigated by simulating the expansion using a ballistic approximation. By fitting the simulated expansion profiles with a Bose-enhanced distribution, the errors of using such a fit are estimated, and the results are explained phenomenologically. The simulation was also used as a fitting function for experimental data, showing better agreement of the extracted condensed fraction with the semi-ideal model than results from a Bose-enhanced fit.

Present experiments with Bose condensed gases can be largely described by a semi-ideal two-gas model. In this model, the condensate is influenced only by the mean-field repulsion among condensed atoms, while the thermal cloud is considered an ideal gas confined by an effective potential that consists of the external trap and the mean-field repulsion by the condensate. This simple, intuitive model provides explicit analytical expressions for the density distributions of the condensate and the thermal component. It describes the reduction of the condensate fraction relative to that of an ideal gas as a consequence of the positive chemical potential due to interactions in the condensate.

We have investigated experimentally the finite-temperature properties of a Bose-Einstein condensed cloud of

{}^{87}\mathrm{Rb}

atoms in a harmonic trap. Focusing primarily on condensed fraction and expansion energy, we measure unambiguous deviations from ideal-gas thermodynamics and obtain good agreement with a Hartree-Fock description of the mixed cloud. Our results offer clear evidence of the mutual interaction between the condensed and thermal components. To probe the low-temperature region unaccessible to the usual time-of-flight technique, we use coherent Bragg scattering as a filtering technique for the condensate. This allows us to separate spatially the condensed and normal components in time of flight and to measure reliably temperatures as low as

0.2{T}_{c}^{0}

and thermal fractions as low as 10%. Finally, we observe evidence for the limitations of the usual image analysis procedure, pointing out to the need for a more elaborate model of the expansion of the mixed cloud.

Ultracold quantum gases provide a unique setting for studying and understanding the properties of interacting quantum systems. Here, we investigate a multi-component system of 87 Rb– 39 K Bose–Einstein condensates (BECs) with tunable interactions both theoretically and experimentally. Such multi-component systems can be characterized by their miscibility, where miscible components lead to a mixed ground state and immiscible components form a phase-separated state. Here we perform the first full simulation of the dynamical expansion of this system including both BECs and thermal clouds, which allows for a detailed comparison with experimental results. In particular we show that striking features emerge in time-of-flight (TOF) for BECs with strong interspecies repulsion, even for systems which were separated in situ by a large gravitational sag. An analysis of the centre of mass positions of the BECs after expansion yields qualitative agreement with the homogeneous criterion for phase-separation, but reveals no clear transition point between the mixed and the separated phases. Instead one can identify a transition region, for which the presence of a gravitational sag is found to be advantageous. Moreover, we analyse the situation where only one component is condensed and show that the density distribution of the thermal component also shows some distinct features. Our work sheds new light on the analysis of multi-component systems after TOF and will guide future experiments on the detection of miscibility in these systems.

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.

We study the free expansion of a dilute two-component Fermi gas with attractive interspecies interaction in the Bardeen-Cooper-Schrieffer--Bose-Einstein condensate (BCS-BEC) crossover. We apply a time-dependent parameter-free density-functional theory by using two choices of the equation of state: an analytic formula based on Monte Carlo data and the mean-field equation of state resulting from the extended BCS equations. The calculated axial and transverse radii and the aspect ratio of the expanding cloud are compared to experimental data on vapors of

^{6}\mathrm{Li}

atoms. Remarkably, the mean-field theory shows a better agreement with the experiments than the theory based on the Monte Carlo equation of state. Both theories predict a measurable dependence of the aspect ratio on expansion time and on scattering length.