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arXiv:cond-mat/0310671 · Physical Review A

Limits of sympathetic cooling of fermions: The role of heat capacity of the coolant

Lincoln D. Carr, Yvan Castin

Published Apr 20, 2004• 9 citations
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Abstract

The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below ${T}_{c}$ or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below ${T}_{c}$ is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve $T=0$ and all of the particles must be evaporated.

DOI
10.1103/physreva.69.043611

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The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below Tc{T}_{c}Tc​ or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below Tc{T}_{c}Tc​ is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve T=0T=0T=0 and all of the particles must be evaporated.
It has been suggested by Timmermans [Phys. Rev. Lett. 87, 240403 (2001)] that loss of fermions in a degenerate system causes strong heating. We address the fundamental limit imposed by this loss on the temperature that may be obtained by sympathetic cooling of fermions by bosons. Both a quantum Boltzmann equation and a quantum Boltzmann master equation are used to study the evolution of the occupation number distribution. It is shown that, in the thermodynamic limit, the Fermi gas cools to a minimal temperature ${k}_{\mathrm{B}}T∕\ensuremath{\mu}\ensuremath{\propto}{({\ensuremath{\gamma}}_{\text{loss}}∕{\ensuremath{\gamma}}_{\text{coll}})}^{0.44}$, where ${\ensuremath{\gamma}}_{\text{loss}}$ is a constant loss rate, ${\ensuremath{\gamma}}_{\text{coll}}$ is the bare fermion-boson collision rate not including the reduction due to Fermi statistics, and $\ensuremath{\mu}\ensuremath{\sim}{k}_{\mathrm{B}}{T}_{\mathrm{F}}$ is the chemical potential. It is demonstrated that, beyond the thermodynamic limit, the discrete nature of the momentum spectrum of the system can block cooling. The unusual nonthermal nature of the number distribution is illustrated from several points of view: the Fermi surface is distorted, and in the region of zero momentum the number distribution can descend to values significantly less than unity. Our model explicitly depends on a constant evaporation rate, the value of which can strongly affect the minimum temperature.
Sympathetic cooling of an atomic Fermi gas by a Bose gas is studied by solution of the coupled quantum Boltzmann equations for the confined gas mixture. Results for equilibrium temperatures and relaxation dynamics are presented, and some simple models are developed. Our study illustrates that a combination of sympathetic and forced evaporative cooling enables the Fermi gas to be cooled to the degenerate regime where quantum statistics and mean-field effects are important. The influence of mean-field effects on the equilibrium spatial distributions is discussed qualitatively.
We study the sympathetic cooling of a trapped Fermi gas interacting with an ideal Bose gas below the critical temperature of the Bose-Einstein condensation. We derive the quantum master equation, which describes the dynamics of the fermionic component, and postulating the thermal distribution for both gases we calculate analytically the rate at which fermions are cooled by the bosonic atoms. The particle losses constitute an important source of heating of the degenerate Fermi gas. We evaluate the rate of loss-induced heating and derive analytical results for the final temperature of fermions, which is limited in the presence of particle losses.
We compare strategies for evaporative and sympathetic cooling of two-species Fermi-Bose mixtures in single-color and two-color optical dipole traps. We show that in the latter case a large heat capacity of the bosonic species can be maintained during the entire cooling process. This could allow one to efficiently achieve a deep Fermi degeneracy regime having at the same time a significant thermal fraction for the Bose gas, crucial for a precise thermometry of the mixture. Two possible signatures of a superfluid phase transition for the Fermi species are discussed.
We discuss the cooling efficiency of ultracold Fermi-Bose mixtures in species-selective traps using a thermodynamical approach. The dynamics of evaporative cooling trajectories is analyzed in the specific case of bichromatic optical dipole traps also taking into account the effect of partial spatial overlap between the Fermi gas and the thermal component of the Bose gas. We show that large trapping frequency ratios between the Fermi and the Bose species allow for the achievement of a deeper Fermi degeneracy, consolidating in a thermodynamic setting earlier arguments based on more restrictive assumptions. In particular, we confirm that the minimum temperature of the mixture is obtained at the crossover between boson and fermion heat capacities, and that below such a temperature sympathetic cooling vanishes. When the effect of partial overlap is taken into account, optimal sympathetic cooling of the Fermi species may be achieved by properly tuning the relative trapping strength of the two species in a time-dependent fashion. Alternatively, the dimensionality of the trap in the final stage of cooling can be changed by increasing the confinement strength, which also results in a crossover of the heat capacities at deeper Fermi degeneracies. This technique may be extended to Fermi-Bose degenerate mixtures in optical lattices.
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.