Spectroscopy of Ultracold Atoms by Periodic Lattice Modulations

We present a nonperturbative analysis of a new experimental technique for probing ultracold bosons in an optical lattice by periodic lattice depth modulations. This is done using the time-dependent density-matrix renormalization group method. We find that sharp energy absorption peaks are not unique to the Mott insulating phase at commensurate filling but also exist for superfluids at incommensurate filling. For strong interactions, the peak structure provides an experimental measure of the interaction strength. Moreover, the peak height of the peaks at $\hslash \omega \gtrsim 2U$ can be employed as a measure of the incommensurability of the system.

Figure 1

Absorption rate vs the modulation frequency $\omega$ at a lattice depth $V_0=15E_r$; the corresponding values of the Bose-Hubbard model are $U\approx 0.71E_r$ and $J\approx 0.0074E_r$, i.e., $U/J\approx 95$. The lines are analytical results obtained by linear response treating the hopping term as a perturbation [10]. The symbols are the results obtained using the adaptive t-DMRG. Inset: Energy absorbed versus time. $\omega =0.71E_r/\hslash$ corresponds to a modulation frequency at resonance and $\omega =0.5E_r/\hslash$ away from resonance.

Figure 2

Energy absorption rate vs modulation frequency $\omega$ at incommensurate filling $N=38$ particles on $L=32$ sites for a lattice height of (a) $V_0=15E_r$ and (b) $V_0=6E_r$. Compared to the system at commensurate filling, additional peaks at $\hslash \omega \approx 2U$ and in (b) at $\hslash \omega \approx 2.6U$ arise. The solid lines are a guide to the eyes.

Figure 3

Saturation effects for strong modulations. The absorption rate for a weak modulation (a) $\delta V_1=0.01$ (circles) and a strong modulation (b) $\delta V_2=0.2$ (upward triangles) are compared. The rate for (a) is scaled by the ratio $(\delta V_2/\delta V_1{)}^2$ to remove trivial scaling. Clear saturation effects in the peak height and width can be seen. Saturation of the integrated energy absorption in time: (c) is the integrated energy absorbed up to time $t_m=600\hslash /E_r$ per unit time (downward triangles), i.e., $[E(t_m)-E(0)]/t_m$ for $\delta V_2=0.2$. Because of saturation effects in time, it is smaller than the energy absorption rate shown in (a).

Figure 4

The integrated energy absorbed up to time $t_m=50\hslash /E_r$, i.e., [$E(t_m)-E(0)$] for $\delta V_2=0.2$ in a system with confining potential and $N=65$ (squares) and in a homogeneous system with $N=46$ and $L=32$ (circles). The inset shows the initial density distribution in the trapped system.