Hyperfine spectra of trapped bosons in optical lattices

We calculate the interaction induced inhomogeneous broadening of spectral lines in a trapped Bose gas as a function of the depth of a three-dimensional cubic optical lattice. As observed in recent experiments, we find that the terraced “wedding-cake” structure of Mott plateaus splits the spectrum into a series of discrete peaks. The spectra are extremely sensitive to density corrugations and trap anharmonicities. For example, even when the majority of the cloud is superfluid the spectrum displays discrete peaks.

Figure 1

Theoretical spectra showing the number of ${}^{87}\text{R}\text{b}$ atoms transferred from hyperfine state $|a⟩=|F=1,m_F=-1⟩$ to state $|b⟩=|F=2,m_F=1⟩$ when excited by light detuned from resonance by the frequency $\delta \nu$. The $N=9\times {10}^4$ atoms are confined by a harmonic potential with $\overline{\omega }={\left({\omega }_x{\omega }_y{\omega }_z\right)}^{1∕3}=2\pi \times 100\text{Hz}$ and a three-dimensional periodic potential with lattice depth $V_0=5,10,25,35E_{\text{rec}}$ (from top to bottom).

Figure 2

(a) Gutzwiller mean-field phase diagram with constant density contours. The vertical dashed lines show the spatial dependence of the chemical potential for a trapped gas in the LDA: from left-to-right these correspond to the deep Mott limit, the “corrugated superfluid” situation appropriate to Fig. 4b, and the dilute superfluid. (b) Phase diagram with contours of constant $⟨n⟩g_2=⟨n\left(n-1\right)⟩∕⟨n⟩$. Contours are spaced by 0.1, with additional lines at $m\pm 0.01$, for integer $m$, to emphasize the Mott regions.

Figure 3

Analytically calculated spectra for the harmonically trapped system in the deep Mott limit (solid line), plotted as a function of $\delta \omega ∕\left(U_{ab}-U_a\right)$. Superimposed is the spectrum of the superfluid (dashed line) with the same parameters, but horizontally shifted to the left by $-\left(\delta \omega ∕2\right)\left(U_{ab}-U_a\right)$. The central density is $n_s\left(0\right)=\left[{\mu }_0-ϵ\left(0\right)\right]∕U_a=5.8$. This illustrates that the envelope of the spectral line in the Mott insulating state has the same shape as the superfluid spectrum, shifted horizontally.

Figure 4

(a) The density as a function of distance from the trap center for a harmonic trap in units of the Thomas-Fermi radius ${\ell }_{\text{TF}}$, defined as ${\ell }_{\text{TF}}\equiv \sqrt{\left({\mu }_0-{ϵ}_0\right)∕\left(m{\omega }^2∕2\right)}$. The density profile corresponds to the LDA contour at $t∕U_a=0.018$ of Fig. 2a. The corrugation of the density is observable, but not dramatic; it would be particularly difficult to image in experiments looking at column integrated densities. (b) The hyperfine spectra arising from the density plotted in (a).

Figure 5

(a) A slice of the anharmonic potential $V_{\text{trap}}\left(\mathit{r}\right)$ similar to the one found in experiments. (b) Contour lines of constant density in the $x\text{−}y$ plane for $n=1,2,\dots ,5$ in the anharmonic trap at $V_0=35E_{\text{rec}}$. (c) Corresponding spectra (using the “spherical trap model” discussed in the text) for $V_0=5,10,25,35E_{\text{rec}}$ with $N=1.4\times {10}^7$ particles.