Many-body physics in the radio-frequency spectrum of lattice bosons

We calculate the radio-frequency spectrum of a trapped cloud of cold bosonic atoms in an optical lattice. By using random phase and local-density approximations we produce both trap-averaged and spatially resolved spectra, identifying simple features in the spectra that reveal information about both superfluidity and correlations. Our approach is exact in the deep Mott limit and in the dilute superfluid when the hopping rates for the two internal spin states are equal. It contains final state interactions, obeys the Ward identities (and the associated conservation laws), and satisfies the $f$-sum rule. Motivated by earlier work by Sun, Lannert, and Vishveshwara [Phys. Rev. A 79, 043422 (2009)], we also discuss the features that arise in a spin-dependent optical lattice.

Figure 1

Illustration of two types of rf-active excitations of the lattice superfluid near the Mott transition. Open (blue) circles are atoms in the $|a\rangle$ state, filled (red) circles are atoms in the $|b\rangle$ state, and the arrows indicate a delocalized particle whereas other particles are localized. Panel (a) illustrates the initial superfluid state, which consists of a dilute gas of atoms moving in a Mott background. Final states in panels (b) and (c) show the excitation of a core or delocalized atom.

Figure 2

Spectral density of a homogeneous system as a function of $\omega /U_{\mathit{aa}}$ and $t_a/U_{\mathit{aa}}$ (whiter regions indicate larger spectral density) compared with sum-rule prediction (red line). Delta functions are broadened to Lorentzians for visualization purposes. (a) Ground state phase diagram within the Gutzwiller approximation, for reference. In panels (b) and (c), we take $U_{\mathit{ba}}=1.2U_{\mathit{aa}}$ and $t_b=t_a$, with (b) $\mu =1.98U_{\mathit{aa}}$ and (c) $\mu =2.02U_{\mathit{aa}}$. In panels (d) and (e), we take parameters corresponding to typical ${}^{87}\mathrm{Rb}$ experiments, $U_{\mathit{ba}}=0.976U_{\mathit{aa}}$ and $t_b=t_a$, and take (d) $\mu =2.000U_{\mathit{aa}}$ and (e) $\mu =2.004U_{\mathit{aa}}$. In both cases, a double-peak structure is visible, but the region of the phase diagram in which it is important is much smaller for ${}^{87}\mathrm{Rb}$ parameters than for the parameters for panels (b) and (c).

Figure 3

(a) Density $n$ as a function of distance to trap center rescaled by the lattice spacing $r/d$ in a local-density approximation. For all subfigures, we take $t_a/U_{\mathit{aa}}=0.004$, which is moderately smaller than the tip of the first Mott lobe. The left side of panels (b) to (e) show the spectrum of a homogeneous gas with density $n(r)$, representing the spatially resolved spectrum observed in an experiment on a trapped gas. The horizontal axis is position, the vertical axis is frequency, and the color from dark to light represents increasing spectral density. The continuous (red) curve denotes the sum-rule result for $\langle \omega \rangle$. We round the $\delta$-functions to Lorentzians for visualization. The right side of panels (b) to (e) shows the trap-averaged spectrum for a three-dimensional trap within our RPA (black, solid line) compared with the sum rule (red, dashed line). (b) $U_{\mathit{ab}}=1.2U_{\mathit{aa}}$, $t_b=t_a$; (c) $U_{\mathit{ab}}=U_{\mathit{aa}}$, $t_b=t_a+0.1U_{\mathit{aa}}$; (d) $U_{\mathit{ab}}=1.2U_{\mathit{aa}}$, $t_b=t_a+0.1U_{\mathit{aa}}$; (e) ${}^{87}\mathrm{Rb}$ parameters: $U_{\mathit{ab}}=0.976U_{\mathit{aa}}$, $t_b=t_a$.