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

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.

Figure 1

Shown is a diagram of the evaporative cooling process in our model. In the upper two panels, a gas of $N_f$ fermions at temperature $T$ is in thermal equilibrium with a coolant of $N_b$ bosons or Boltzmann particles. In the middle two panels, the tail of the number distribution of the coolant is cut and $\delta N_b$ particles are removed. In the lower two panels, the systems are permitted to rethermalize, resulting in a reduced common temperature $T-\delta T$. The values of $\delta T$ and $\delta N_b$ have been exaggerated for the purposes of illustration; the units are arbitrary.

Figure 2

Shown is the temperature of a degenerate Fermi gas sympathetically cooled by a Bose gas with a condensate present, as a function of the number of bosons evaporated in a harmonic trap, that is, of the difference between the initial number of bosons $N_{b,0}$ [assumed to fulfill Eqs. (13, 14)] and the final number $N_b$. Two cases are depicted: ${\omega }_f∕{\omega }_b=1$ (solid line) and ${\omega }_f∕{\omega }_b=5$ (dashed line) where ${\omega }_f∕{\omega }_b$ is the ratio of the mean trapping frequencies of the fermions and bosons. It is apparent that ${\omega }_f∕{\omega }_b\lesssim 1$ requires a greatly reduced number of bosons in order to cool to $T=0$, and may therefore be considered as a more efficient experimental configuration. Here the choice of parameters was $\eta =6$ for the energy cutoff and $T_0∕T_F=0.2$, where $T_0$ is the initial temperature.

Figure 3

Shown is the temperature of a degenerate Fermi gas subject to sympathetic cooling by a Boltzmann gas as a function of the number of Boltzmann particles. Three cases are depicted: $C_{f,0}∕C_{b,0}=\left(\lambda -1\right)∕2$ (solid curve), $C_{f,0}∕C_{b,0}=\lambda -1$ (dashed curve), and $C_{f,0}∕C_{b,0}=2\left(\lambda -1\right)$ (dot-dashed curve). In the latter case the number of Boltzmann particles is insufficient to cool to $T=0$. Here the choice of parameters was $\eta =6$ for the energy cutoff parameter, $T_0∕T_F=0.2$ for the initial temperature, and a harmonic trapping potential.