Light scattering from ultracold gases in disordered optical lattices

We consider a gas of bosons in a bichromatic optical lattice at finite temperatures. As the amplitude of the secondary lattice grows, the single-particle eigenstates become localized. We calculate the canonical partition function using exact methods for the noninteracting and strongly interacting limits and analyze the statistical properties of the superfluid phase, localized phase, and the strongly interacting gas. We show that those phases may be distinguished in experiment using off-resonant light scattering.

Figure 1

Energy spectrum of the Aubry-Andre Hamiltonian (4) for $M=610$ lattice sites and for different values of the disorder amplitude $\Delta$.

Figure 2

Probability distribution $|c_i^0{|}^2$ of finding atoms in different lattice sites for atoms in the ground state and for various disorder strengths $\Delta$. We use logarithmic scale for the $y$ axis to show exponential localization.

Figure 3

Similar to Fig. 2, but for the first excited state. For $\Delta =4.5J$, the dominating left peak is two orders of magnitude stronger than the other.

Figure 4

Top: number of ground-state atoms obtained in canonical ensemble for different values of $\Delta$, compared with the analytical model predictions. Bottom: fluctuations of the number of ground-state atoms. Dashed lines show the results obtained within the Maxwell-Demon approach. $N=300$.

Figure 5

Mean number of particles in each lattice obtained in the canonical ensemble for $M=144$ sites, $N=300$ particles, fixed value of the disorder strength $\Delta =2.5J$, calculated for various temperatures.

Figure 6

Number fluctuations of particles in each lattice site obtained in the canonical ensemble for $M=144$ sites, $N=300$ particles, fixed value of the disorder strength $\Delta =2.5J$, calculated for various temperatures. The blue (dotted) lines show the thermal approximation.

Figure 7

Correlations between sites $\langle n_i{n}_j\rangle -\langle n_i\rangle \langle n_j\rangle$ for the Aubry-Andre model with $k_{B}T=0.05J$ and $\Delta =0.5J$ (left); $k_{B}T=0.05J$ and $\Delta =2.2J$ (middle); and for strongly interacting gas with $k_{B}T=0.5U$, $M=89$, $N=2M$, and $\Delta =5U$ (right).

Figure 8

Sample graph for mean occupation number (top) and its fluctuations (bottom) for $M=89$ sites, $N=2M$, $k_{B}T=0.5U$, and various $\Delta$.

Figure 9

Experimental setup for light scattering. The optical lattice is illuminated by the laser set at angle $\alpha$. The detector is set at angle $\beta$.

Figure 10

The averaged spectrum of scattered photons for the Aubry-Andre model for different values of $\Delta$ and $T$. The probing beam is set at the angle $\alpha =\pi /2$. For $\Delta <2J$, the system is in the superfluid state, while for larger $\Delta$ it is in the localized regime.

Figure 11

The angular distribution of scattered photons for the strongly interacting gas for different values of $\Delta$ and $T$. The probing beam is set at the angle $\alpha =\pi /2$.