Statistical mechanics of a Feshbach-coupled Bose-Fermi gas 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. The phase diagram of the resulting atom-molecule mixture in chemical and thermal equilibria is determined numerically in the absence of interactions under the constraint of particle conservation. In the limiting cases of vanishing or large lattice depth we derive simple analytical results for important thermodynamic quantities. One such quantity is the dissociation energy, defined as the detuning of the molecular energy spectrum with respect to the atomic one for which half of the atoms have been converted into dimers. Importantly we find that the dissociation energy has a nonmonotonic dependence on lattice depth.

Figure 1

(Color online) The density of states of a 3D optical lattice calculated by binning the energy levels into intervals of length $\Delta E=0.13E_{R,a}$. The blue (dark gray) and red (light gray) curves show the density of the atomic and molecular states, respectively.

Figure 2

(Color online) Molecule fraction in a $\left(E_{\text{res}},T\right)$ phase diagram for $V_0=5E_{R,a}$ and $\eta =\frac{1}{2}$.

Figure 3

(Color online) Schematic illustration of how $\mu$ changes when the resonance energy is varied across $E_{\text{F}}$. For clarity we consider a deep lattice where the density of states of the lowest atomic and molecular bands are narrow peaks as illustrated by the solid lines. The distributions $f_a\left(E\right)$ and $f_m\left(E\right)$ are indicated by the dashed lines, and the shading under the peaks indicates the population of the atomic and molecular levels. Note that the $E_{\text{res}}$ dependence is included in the position of the peak in the molecular density of states and not in the Bose-Einstein distribution.

Figure 4

Chemical potential as a function of $E_{\text{res}}$ for $\eta =\frac{1}{2}$ and different potential depths in the limit $T\approx 0$. This corresponds to following a horizontal line in the bottom of phase diagrams like in Fig. 2. The linear behavior in the left side is characteristic for the pure Bose gas regime, and the difference in the slopes arises because the Fermi energy, which scales the abscissa, varies with the potential depth. Inset: chemical potential curves for $V_0=5E_{R,a}$, $10E_{R,a}$, $15E_{R,a}$, and $20E_{R,a}$ (from bottom to top) for a finite temperature of $5.6E_{R,a}/k_{\text{B}}$.

Figure 5

(Color online) Zero-temperature molecule fractions as a function of $E_{\text{res}}$ and $V_0$. The color coding indicates the molecule fraction, and the dissociation energy is shown by the solid white curve. For deep lattice potentials $E_{\text{dis}}$ can be approximated by Eq. (27) (dashed white curve), while the limiting behavior for a vanishing lattice depth is given by Eq. (25). Note that since the Fermi energy increases with increasing lattice depth, the scaling on the vertical axis depends on $V_0$.