Limits of sympathetic cooling of fermions by zero-temperature bosons due to particle losses

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 $k_{\mathrm{B}}T∕\mu \propto {({\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}})}^{0.44}$, where ${\gamma }_{\text{loss}}$ is a constant loss rate, ${\gamma }_{\text{coll}}$ is the bare fermion-boson collision rate not including the reduction due to Fermi statistics, and $\mu \sim k_{\mathrm{B}}{T}_{\mathrm{F}}$ 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.

Figure 1

Shown is the minimum value of $k_{\mathrm{B}}T∕\mu$ as a function of the time step $t_e{E}_{\mathrm{F}}∕\hslash$ for a ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ mixture and $\hslash {\gamma }_{\text{coll}}∕E_{\mathrm{F}}\left(0\right)=\frac{3}{8}\times {10}^{-5}$. The ratio ${\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)$ is $\frac{8}{3}\times {10}^{-2}$, $\frac{8}{3}\times {10}^{-3}$, and $\frac{8}{3}\times {10}^{-4}$ in panels (a), (b), and (c), respectively. The case of a finite system, Eq. (15), is shown for ${10}^2$ (dashed line), ${10}^3$ (long-dashed line), and ${10}^4$ (dot-dashed line) fermions, and the thermodynamic limit, Eq. (23), is shown as a solid line. For the finite system, to the right side of the plot, in the discrete regime, the cooling is blocked; to the left, in the continuous regime, the minimum temperature is dominated by the evaporation rate, according to Eq. (25), and rises as $t_e{E}_{\mathrm{F}}∕\hslash$ decreases. The data obtained in the thermodynamic limit are independent of $t_e$ except where the evaporation-limited temperature is larger than the loss-limited temperature. Note that the actual data points are represented by open circles [25].

Figure 2

Shown is the mean occupation number for $N_{\mathrm{F}}=1000$ ${}^6\mathrm{Li}$ atoms in contact with a ${}^7\mathrm{Li}$ reservoir in a finite system. The initial state (plus symbols) is a thermal distribution of $k_{\mathrm{B}}T∕\mu =0.324$; $\hslash {\gamma }_{\text{coll}}∕E_{\mathrm{F}}\left(0\right)=\frac{3}{8}\times {10}^{-5}$. After an evolution time (solid circles) such that $k_{\mathrm{B}}T∕\mu$ passes through a minimum according to Eq. (15) with the parameters ${\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)=\frac{8}{3}\times {10}^{-3}$ and (a) $t_e{E}_{\mathrm{F}}∕\hslash ={10}^2$ (continuous regime), (b) $t_e{E}_{\mathrm{F}}∕\hslash ={10}^3$, and (c) $t_e{E}_{\mathrm{F}}∕\hslash ={10}^4$ (discrete regime), it is clear that in the continuous regime the distribution, though nonthermal, depends only on $\mid \mathbf{k}\mid$. In contrast, in the discrete regime the distribution is fully $\mathbf{k}$ dependent.

Figure 3

Time evolution of the number distribution of a Fermi gas sympathetically cooled by a perfect Bose reservoir for two experimentally important cases in the thermodynamic limit:(a) ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ and (b) ${}^{23}\mathrm{Na}\text{−}{}^6\mathrm{Li}$. Shown are the initial state of $(k_{\mathrm{B}}T∕\mu )\left(0\right)=0.7$ (thin line), the state at which the temperature reaches a minimum (thicker line), and a later stage where the gas has heated (thickest line). The hole in the distribution near $k=0$ is particularly evident in panel (b), where the loss processes balance evaporative cooling, as described by Eq. (27). Here $\left[{\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right),{(k_{\mathrm{B}}T∕\mu )}_{\min }\right]=[8.27\times {10}^{-3},0.081]$, and $[1.25\times {10}^{-2},0.089]$ for panels (a) and (b), respectively, and $\hslash {\gamma }_{\text{coll}}∕E_{\mathrm{F}}\left(0\right)=\frac{3}{8}\times {10}^{-5}$.

Figure 4

The numerically determined maximum Fermi degeneracy as a function of loss rate for ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ (open circles) and ${}^{23}\mathrm{Na}\text{−}{}^6\mathrm{Li}$ (black diamonds) mixtures in the thermodynamic limit, according to the quantum Boltzmann equation. One finds ${\left(k_{\mathrm{B}}T∕\mu \right)}_{\min }\propto {\left[{\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)\right]}^{0.44}$. The dashed line shows the analytic prediction of ${(k_{\mathrm{B}}T∕\mu )}_{\min }\propto {\left[{\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)\right]}^{1∕3}$ based on the assumption of a Fermi distribution. The difference is due to the nonthermal nature of the quasistatic mean occupation number distribution, an example of which is illustrated in Fig. 3. Here $\hslash {\gamma }_{\text{coll}}∕E_{\mathrm{F}}\left(0\right)=\frac{3}{8}\times {10}^{-5}.$

Figure 5

Time evolution of the degeneracy in the thermodynamic limit as determined by numerical evolution of Eq. (23) (solid line), and the analytical prediction of Eq. (37) (dashed line) based on a Fermi ansatz. The parameters are ${\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)=8.27\times {10}^{-3}$, $(k_{\mathrm{B}}T∕\mu )\left(0\right)=0.7$, $t_e{E}_{\mathrm{F}}\left(0\right)\hslash ={10}^3$, and $\hslash {\gamma }_{\text{coll}}∕E_{\mathrm{F}}\left(0\right)=\frac{3}{8}\times {10}^{-5}$.

Figure 6

Evolution of the total energy of the fermions for a finite system for one Monte Carlo realization of the exact occupation number distribution under the secular approximation (see text) according to Eq. (57). (a) ${\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)=8∕3\times {10}^{-4}$: the loss events (in this case 6) are marked by arrows. Clearly cooling dominates the decrease in the energy at early times while loss plays a stronger role at later times. (b) ${\gamma }_{\text{loss}}∕{\gamma }_{\text{coll}}\left(0\right)=8∕3\times {10}^{-2}$: the upper curve shows the evolution of the energy and the lower curve shows the evolution of the number of atoms. At early times, the slope of the energy is compounded of the effect of cooling and a trivial decrease of the Fermi energy $\propto \exp (-2{\gamma }_{\text{loss}}t∕3)$ due to atom loss. In both (a) and (b), $(k_{\mathrm{B}}T∕\mu )\left(0\right)=0.324$, $N_{\mathrm{F}}\left(0\right)=100$, and $t_e{E}_{\mathrm{F}}∕\hslash ={10}^3$.

Figure 7

The numerically determined maximum Fermi degeneracy as a function of loss rate for $N=100$ (dashed line) and $N=1000$ (solid line) ${}^6\mathrm{Li}$ fermions in contact with ${}^7\mathrm{Li}$ (open circles) or ${}^{23}\mathrm{Na}$ (black diamonds), according to the quantum master Boltzmann equation. The value of $t_e{E}_{\mathrm{F}}∕\hslash$ which gives the minimal temperature has been chosen. Arrows indicate the long term cooling limits for ${\gamma }_{\text{loss}}=0$ for ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ and ${}^{23}\mathrm{Na}\text{−}{}^6\mathrm{Li}$ mixtures, respectively. The minimal temperature is much higher in comparison to that predicted by the quantum Boltzmann equation in the thermodynamic limit, as was shown in Fig. 4 (note scale change).