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arXiv:2411.09457 · Journal of Statistical Mechanics Theory and Experiment

Scaling theory for the collapse of a trapped Bose gas in a synthetic magnetic field: a critical study at the condensation point

Bikram Keshari Behera, Shyamal Biswas

Published Apr 1, 2025• 0 citations
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Abstract

Abstract We have analytically explored both the zero temperature and the finite temperature scaling theory for the collapse of an attractively interacting 3D harmonically trapped Bose gas in a synthetic magnetic field. We have considered short ranged (contact) attractive inter-particle interactions and Hartree–Fock approximation for the same. We have separately studied the collapse of both the condensate and the thermal cloud below and above the condensation point, respectively. We have obtained an anisotropy, artificial magnetic field, and temperature dependent critical number of particles for the collapse of the condensate. We have found a dramatic change in the critical exponent (from α = 1 to 0) of the specific heat ( C v ∝ | T − T c | α ) when the thermal cloud is about to collapse with the critical number of particles ( N = N c ) just below and above the condensation point. All the results obtained by us below and around the condensation point are experimentally testable within the present-day experimental set-up for the ultracold systems in the magneto-optical traps.

DOI
10.1088/1742-5468/adc711

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Abstract We have analytically explored both the zero temperature and the finite temperature scaling theory for the collapse of an attractively interacting 3D harmonically trapped Bose gas in a synthetic magnetic field. We have considered short ranged (contact) attractive inter-particle interactions and Hartree–Fock approximation for the same. We have separately studied the collapse of both the condensate and the thermal cloud below and above the condensation point, respectively. We have obtained an anisotropy, artificial magnetic field, and temperature dependent critical number of particles for the collapse of the condensate. We have found a dramatic change in the critical exponent (from α = 1 to 0) of the specific heat ( C v ∝ | T − T c | α ) when the thermal cloud is about to collapse with the critical number of particles ( N = N c ) just below and above the condensation point. All the results obtained by us below and around the condensation point are experimentally testable within the present-day experimental set-up for the ultracold systems in the magneto-optical traps.
We investigate condensation and finite-temperature magnetic phase transitions in a coherently (Rabi) coupled Bose gas confined in a three-dimensional harmonic trap. For the noninteracting system, we derive analytical expressions for the critical temperature and condensate fraction using a semiclassical description of the single-particle spectrum and density of states. At fixed particle number, coherent coupling enhances the critical temperature relative to the uncoupled system, and this enhancement decreases with increasing particle number NNN. The finite-size correction, in contrast, lowers the transition temperature, with its effect diminishing for larger NNN. We then incorporate repulsive interactions within the Hartree--Fock--Bogoliubov--Popov framework to investigate the finite-temperature phase diagram in the temperature--Rabi coupling plane. The system undergoes successive transitions from a ferromagnetic to a paramagnetic condensate and, at higher temperature, to a thermal gas. The transition boundaries obtained from the vanishing of the condensate fraction show excellent agreement with the analytical Hartree-Fock predictions. We find that the interactions substantially suppress the variation in the condensation temperature with an increase in the coherent coupling.
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.
A simple picture describes the results of recent treatments of partially--condensed, dilute, trapped Bose gases at temperature T > 0. The condensate wave function is nearly identical to that of a T = 0 condensate with the same number of condensate atoms, Ν0.The cloud of non-condensed atoms is described by the statistical mechanics of an ideal Bose gas in the com- bined potentials of the magnetic trap and the cloud-condensate interaction.We provide a physical motivation for this result, show how it emerges in the Hartree-Fock-Bogo liubov-Popov approximation, and explore some of its implications for future experiments.
We study the condensed fraction of a harmonically trapped atomic Bose gas at the critical point predicted by mean-field theory. The nonzero condensed fraction f(0) is induced by critical correlations which increase the transition temperature T(c) above T(c) (MF). Unlike the T(c) shift in a trapped gas, f(0) is sensitive only to the critical behavior in the quasiuniform part of the cloud near the trap center. To leading order in the interaction parameter a/λ(0), where a is the s-wave scattering length and λ(0) the thermal wavelength, we expect a universal scaling f(0) proportionally (a/λ(0))(4). We experimentally verify this scaling using a Feshbach resonance to tune a/λ(0). Further, using the local density approximation, we compare our measurements with the universal result obtained from Monte Carlo simulations for a uniform system, and find excellent quantitative agreement.
In this paper, the temperature dependence for synthetic magnetization \bar M of a rotating Bose gas in harmonic trap and the magnetic field dependence of the critical temperature are investigated. A simple semiclassical approximation for the density of state approach is used in this study. The suggested approach simultaneously gives corrections to quantities which arise from the finite size and interatomic interaction effects. The calculated results show that the critical temperature decreases due to the increase of the synthetic magnetic field. The synthetic magnetization also exhibits a characteristic temperature dependence. It is proportional to T^3 if \bar T < \bar T_c, while it is proportional to T if \bar T > \bar T_c. The calculated results provide a solid theoretical foundation of the current experiments.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.