Adiabatic cooling of a tunable Bose-Fermi mixture in an optical lattice

We consider an atomic Fermi gas confined in a uniform optical lattice potential, where the atoms can pair into molecules via a magnetic field controlled narrow Feshbach resonance. Thus by adjusting the magnetic field the portion of fermionic and bosonic particles in the system can be continuously varied. We analyze the statistical mechanics of this system and consider the interplay of the lattice physics with the atom-molecule conversion. We study the entropic behavior of the system and characterize the temperature changes that occur during adiabatic ramps across the Feshbach resonance. We show that an appropriate choice of filling fraction can be used to reduce the system temperature during such ramps.

Figure 1

(Color online) Schematic of the system under consideration: fermionic atoms in two different spin states occupy states in an optical lattice and are coupled by a Feshbach resonance into bosonic dimers. Because the bosonic dimers have a larger polarizability they experience a deeper lattice potential with a different spectrum to that for the atoms.

Figure 2

(Color online) The density of states for a $31\times 31\times 31$ optical lattice as a function of the potential depth for the particle type $\sigma$. The density of states is found by binning the energy levels in intervals of width $0.04E_{R,\sigma }$.

Figure 3

(Color online) The system entropy $S_{\text{tot}}$ in the $\left(E_{\text{res}},T\right)$ plane for $V_0=10E_{R,a}$ and $\eta =0.8$.

Figure 4

(Color online) The atomic and molecular contributions to the entropy as well as the total entropy as functions of $E_{\text{res}}$ at constant temperature, $k_{\text{B}}T=0.04E_{\text{F}}$, lattice depth $V_0=10E_{R,a}$, and for two different filling fractions. The horizontal dashed lines are the atomic and molecular entropy plateaus, Eqs. (15a, 15b), respectively, and the vertical dashed lines indicate the resonance energy maximizing the entropy as approximated by Eq. (14).

Figure 5

(Color online) Entropic behavior for $V_0=10E_{R,a}$. Left: the total entropy at $E_{\text{res}}=-4E_{\text{F}}$ (red curve) and $E_{\text{res}}=+4E_{\text{F}}$ (blue curve), corresponding to a pure Bose and Fermi gas, respectively. The curves have plateaus, which are marked by the two dashed horizontal lines. Right: the height of the plateaus is read off manually [(red) ◻ for the molecular and (blue) ○ for the atomic entropy plateau]. These are well approximated by Eqs. (15a, 15b) (dashed curves) and intersect at the filling fraction ${\eta }_c=0.726$.

Figure 6

(Color online) The dependence of the final temperature on the initial temperature in adiabatic $E_{\text{res}}$ sweeps from the atomic to the molecular regime for different filling fractions and for a lattice depth $V_0=20E_{R,a}$. For $\eta <{\eta }_c$ the temperature increases while it decreases (for sufficiently low $T_i$) for $\eta >{\eta }_c$.

Figure 7

(Color online) Schematic illustration of the adiabatic cooling. If the atomic entropy plateau lies above the molecular one $\left(\eta <{\eta }_c\right)$ the final temperature in an adiabatic $E_{\text{res}}$ sweep will be higher than the initial temperature as shown to the left. However, if the molecular entropy plateau lies above the atomic one $\left({\eta }_c<\eta \le 1\right)$ the final temperature will be comparable to the bandwidth of the molecular spectrum, ${\delta }_m$, as shown to the right.