Doublon production rate in modulated optical lattices

We theoretically study lattice modulation experiments with ultracold fermions in optical lattices. We focus on the regime relevant to current experiments when interaction strength is larger than the bandwidth and temperature is higher than magnetic superexchange energy. We obtain analytical expressions for the rate of doublon production as a function of modulation frequency, filling factor, and temperature. We use local density approximation to average over inhomogeneous density for atoms in a parabolic trap and find excellent agreement with experimentally measured values. Our results suggest that lattice modulation experiments can be used for thermometry of strongly interacting fermionic ensembles in optical lattices.

Figure 1

The slave-particle densities as a function of temperature for different chemical potentials: (a) the slave boson, (b) holon, and (c) doublon density. The chemical potentials are chosen as follows: $\mu /U=0.5$, $0.2$, $0.1$, and 0 from the top to bottom. The MF results $n_{\sigma }^{\mathrm{MF}}$, $n_{\mathrm{h}}^{\mathrm{MF}}$, and $n_{\mathrm{d}}^{\mathrm{MF}}$ given by the slave-particle technique and the exact results in the atomic limit, $n_{\sigma }$, $n_{\mathrm{h}}$, and $n_{\mathrm{d}}$, are compared. $n_{\sigma }^{\mathrm{MF}}$, $n_{\mathrm{h}}^{\mathrm{MF}}$, and $n_{\mathrm{d}}^{\mathrm{MF}}$ are calculated numerically and analytically. The solid, dashed, and dotted lines, respectively, denote the exact result by the atomic limit calculation, the slave-particle approach (A3).

Figure 2

The NC diagrams giving the doublon self-energy. The solid, double dotted, and dashed lines, respectively, denotes the full propagators of the doublon, holon, and slave bosons. The left diagram (a) describes the scattering between a doublon and slave boson. The right diagram (b) represents the higher energy scattering to a holon than the left (a). Thus, as long as charge excitations of energy $\sim$$U$ are taken for large $U$, the diagram (b) would be irrelevant. The holon self-energy is also given by the same type of diagrams.

Figure 3

The doublon-holon bandwidth $W/\sqrt{z}J$ as a function of $k_{\mathrm{B}}T$ for $\mu /U=0.5$. The chemical potential dependence of the bandwidth for $k_{\mathrm{B}}T/U=0.01$ (solid), $0.05$ (dashed), $0.1$ (dashed-dotted), and $0.5$ (dotted) is also shown in the inset.

Figure 4

The spectral function of the original fermion as a function of $k_{\mathrm{B}}T/U$ for $\mu /U=0.5$ and $J/U=0.02$ (top panel), $\mu /U$ for $k_{\mathrm{B}}T/U=0.1$ and $J/U=0.02$ (middle panel), and $J/U$ for $k_{\mathrm{B}}T/U=0.1$ and $\mu /U=0.5$ (bottom panel).

Figure 5

The diagrams contributing to the considered correlation function (18): The diagrams (a), (b), (c), and (d) correspond to the first, second, third, and fourth terms. The solid, double dotted, and dashed lines denotes the doublon, holon, and slave boson propagators, respectively.

Figure 6

The DPR spectrum as a function of modulation frequency. The solid line and points denote the theoretical and experimental results, respectively. The temperature necessary to draw the theoretical curve is determined in the inset; the temperature dependence of the theoretically given DPR at $\omega =U/\hslash$ and the temperature is determined from the crossing point to the experimental data (dotted line), which is $k_{\mathrm{B}}T/U\approx 0.052$.