Laser probing of the single-particle energy gap of a Bose gas in an optical lattice in the Mott-insulator phase

We study single-particle excitations of a Bose gas in an optical lattice in the Mott-insulator phase. The characteristic feature of the single-particle spectrum in the Mott-insulator phase is the existence of an energy gap between the particle and hole excitations. We show that the single-particle excitation energies and associated energy gap in the Mott-insulator phase can be directly probed by an output-coupling experiment. We apply the general expression for the output current derived by Luxat and Griffin, which is given in terms of the single-particle Green’s functions of a trapped Bose gas, to the Mott-insulator phase using the Bose-Hubbard model. The energy spectrum of the momentum-resolved output current exhibits two characteristic peaks corresponding to the particle and hole excitations, and thus it can be used to detect the transition point from the Mott insulator to superfluid phase where the energy gap disappears.

Figure 1

The phase diagram of the Bose-Hubbard Hamiltonian. The solid line is the phase boundary for $\beta zt=10.0$ and the dashed line is for $\beta zt=2.0$. The Mott phase has the filling factor (average particle number per site) $n=1$.

Figure 2

The dispersion relations of the particle excitation ${ϵ}_{\mathrm{p}}\left(\mathbf{k}\right)$ (upper branch) and the hole excitation ${ϵ}_{\mathrm{h}}\left(\mathbf{k}\right)$ (lower branch) for $\mathbf{k}=(k,0,0)$. (a) shows the $U$ dependence at the fixed temperature $\beta zt=10.0$, while (b) shows the temperature dependence at $U=U_{\mathrm{c}}=5.83zt$.

Figure 3

Momentum-resolved output-coupling current for $\mathbf{p}=(0,0.01,0.01)\pi ∕a$ and $\mathbf{q}=\mathbf{0}$. The detuned energy $\delta$ is defined as $\delta \equiv \hslash \omega -\Delta ϵ$. (a)–(c) are plots with the fixed values of $U$ and various temperatures $\beta zt=10.0,4.0,3.0,2.0$. (d) is a plot with the fixed temperature with $U∕zt=5.83,6.5,8.0$. Here we take $\lambda =852\mathrm{nm}$ and $m$ and $a_s$ for the ${}^{87}\mathrm{Rb}$ atom, and vary $V_0$ to adjust the ratio $U∕zt$.