Feshbach resonances in an optical lattice

We present a theory for ultracold atomic gases in an optical lattice near a Feshbach resonance. In the single-band approximation the theory describes atoms and molecules that can both tunnel through the lattice. Moreover, an avoided crossing between the two-atom and the molecular states occurs at every site. We determine the microscopic parameters of the generalized Hubbard model that describes this physics, using the experimentally known parameters of the Feshbach resonance in the absence of the optical lattice. As an application we also calculate the zero-temperature phase diagram of an atomic Bose gas in an optical lattice.

Figure 1

The relative energy levels of the atom-molecule system as a function of the detuning $\delta$. This figure was calculated for $g^2∕\sqrt{2}\pi l^3{\left(ħ\omega \right)}^2=0.1$.

Figure 2

Details of the physical content of our theory. We show the avoided crossing between the molecular level and the lowest two-atom trap state. The inset shows the probability $Z_{\sigma }$ as a function of the detuning $\delta$. This figure was calculated for $g^2∕\sqrt{2}\pi l^3{\left(ħ\omega \right)}^2=0.1$ Note that the center-of-mass contribution to the energy has been taken into account here.

Figure 3

Zero-temperature phase diagram as a function of the filling fraction per site and the detuning $\delta$ in units of $ħ\omega$. The different curves that separate the MC and the $\mathrm{AC}+\mathrm{MC}$ phases correspond to values of $g^{\prime }∕ħ\omega =0.10$ (full curve) and $g^{\prime }∕ħ\omega =0.12$ (dashed curve), respectively. In both cases we have taken $\omega$ to be ${10}^4\mathrm{rad}∕\mathrm{s}$.