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arXiv:2603.20446

Semi-classical evaporative cooling: classical and quantum distributions

A. A. Arvizu-Velazquez, Alejandra del Río‐Lima, S. Dondé-Rodríguez, Freddy Jackson Poveda-Cuevas

Published Mar 20, 2026• 0 citations
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Abstract

We develop a semiclassical thermodynamic framework for the evaporative cooling of trapped atomic gases that treats Maxwell--Boltzmann, Bose--Einstein, and Fermi--Dirac statistics on equal footing across box, harmonic, mixed, and linear-quadrupole potentials. Using the global thermodynamic variables of inhomogeneous confinement, we show that all geometries are unified by a single parameter $s$, which fixes the polylogarithm order, the density of states exponent, and the number of degrees of freedom $2s$. Modeling evaporation as a recursive sequence of energy truncation and rethermalization, we derive closed-form recurrence relations for the particle number and internal energy that track the full thermodynamic state, with the classical energy budget set by a virial factor $C_{\mathrm{trap}} = 1 + 3/(2s)$. Quantum degeneracy emerges not as a singularity in the global susceptibilities, but as a smooth, geometry-dependent crossover in which bosons and fermions display opposite thermodynamic signatures. The results provide a versatile theoretical tool for modeling evaporative cooling across experimentally relevant geometries and offer quantitative guidance for optimizing the cooling process in ultracold atomic systems.

DOI
10.48550/arxiv.2603.20446

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The evaporative cooling of a two-component mixture of trapped atomic fermions into quantum degeneracy is investigated using the semiclassical Boltzmann equation for fermions. The influence of Fermi blocking for temperatures well below the Fermi temperature is discussed. Comparison between collisional and relaxation rates for a Fermi degenerate gas are made, and the similarities and differences between the cooling kinetics of the Fermi gas mixture and a mixture of classical gases is investigated, and discussed using a simple model. The latter also serves to clarify the processes which terminate the cooling in the quantum degenerate regime. Finally, a brief discussion of mean-field effects is given.
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We apply kinetic theory to the problem of evaporative cooling of a dilute collisional gas in a trap. Assuming ``sufficient ergodicity'' (phase-space distribution only a function of energy) and s-wave collisions with an energy-independent cross section, an equation for the evolution of the energy distribution of trapped atoms is derived for arbitrary trap shapes. Numerical integration of this kinetic equation demonstrates that during evaporation the gas is accurately characterized by a Boltzmann distribution of atom energies, truncated at the trap depth. Adopting the assumption of a truncated Boltzmann distribution, closed expressions are obtained for the thermodynamic properties of the gas as well as for the particle and energy loss rates due to evaporation. We give analytical expressions both for power-law traps and for a realistic trapping potential (Ioffe quadrupole trap). As an application, we discuss the evaporative cooling of trapped atomic hydrogen gas. \textcopyright{} 1996 The American Physical Society.
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