Lattice modulation spectroscopy of strongly interacting bosons in disordered and quasiperiodic optical lattices

We compute the absorption spectrum of strongly repulsive one-dimensional bosons in a disordered or quasiperiodic optical lattice. At commensurate filling, the particle-hole resonances of the Mott insulator are broadened as the disorder strength is increased. In the noncommensurate case, mapping the problem to the Anderson model allows us to study the Bose-glass phase. Surprisingly, we find that a perturbative treatment in both cases, weak and strong disorders, gives a good description at all frequencies. In particular, we find that the infrared-absorption rate in the thermodynamic limit is quadratic in frequency. This result is unexpected since for other quantities, like the conductivity in one-dimensional systems, perturbation theory is only applicable at high frequencies. We discuss applications to recent experiments on optical lattice systems and, in particular, the effect of the harmonic trap.

Figure 1

Sketch of the full absorption spectrum at incommensurate filling. Here, $W\sim \text{max}\left(4J,2\Delta \right)$ is the effective bandwidth. The peak at $\omega \sim U$ stems from particle-hole excitations, whereas the low-frequency absorption appears as a consequence of the formation of a Bose glass at incommensurate filling.

Figure 2

(Color online) Energy absorption rate of hard-core bosons in a random potential: the response function $F$ [see Eq. (14)] is plotted versus modulation frequency for different filling factors and increasing values of disorder strength: $\Delta =0.1J$ (top panel), $\Delta =J$, and $\Delta =5J$. Calculations are done on a ring of $M=500$ lattice sites yielding negligible finite-size effects. The number of disorder realizations used was $N_r=500$.

Figure 3

(Color online) Comparison between numerics (symbols) and analytics [Eq. (9), solid line] for weak disorder. Here, $\Delta /J=0.01$ and we consider two filling factors $\nu =0.3$ and $\nu =0.5$. The length of the chain is $M=2000$ and the number of disorder realizations is $N_r=2000$. The inset is a zoom of the low-frequency regime, where the absorption rate is quadratic in frequency as given by Eq. (17).

Figure 4

(Color online) Comparison between numerics (symbols) and analytics [solid line, Eq. (19)] for strong disorder. The value of the disorder is $\Delta /J=100$ and the filling factors are $\nu =0.3$ and $\nu =0.5$. The length of the chain is $M=2000$ and the number of disorder realizations is $N_r=2000$. In the inset, we compare our numerics with the quadratic expansion (20) expected at low frequency. In both cases, the agreement is quite good.

Figure 5

(Color online) Energy absorption rate in the Mott insulator phase for hard-core bosons in a random potential: the response function $F$ [see Eq. (14)] is plotted versus modulation frequency for fixed $J=0.01U$ and increasing values of the disorder strength $\Delta /U=0\left(\text{black}\text{dotted}\text{line}\right),0.03,0.06,0.1$. The length of the chain is $M=60$ and the number of disorder realizations is $N_r=200$. A broadening of the absorption peak is observed for increasing disorder.

Figure 6

(Color online) Energy absorption rate of hard-core bosons in a quasiperiodic potential with $\sigma =0.77145245$: the response function $F$ [see Eq. (14)] is plotted versus modulation frequency for different filling factors and increasing values of disorder strength: $\Delta =0.1J$ (top panel), $\Delta =J$, and $\Delta =10J$. Calculations are done on a ring of $M=1200$ lattice sites yielding almost negligible finite-size effects. The number of disorder realization is $N_r=500$.

Figure 7

(Color online) Energy absorption in the Mott insulator phase for bosons in a quasiperiodic potential with $\sigma =0.77145245$: the response function $F$ [see Eq. (14)] is plotted versus modulation frequency for fixed $J=0.01U$ and increasing disorder strength $\Delta /U=0\left(\text{black}\text{dotted}\text{line}\right),0.03,0.06,0.1$. Here, convergence is achieved for system size $L=70$. As disorder increases, the response function broadens and changes convexity, as predicted by Eq. (32).

Figure 8

(Color online) Energy absorption rate of disordered and quasidisordered hard-core bosons in a parabolic trap. The response function $F$ is plotted versus modulation frequency for different fillings. Upper panel: $\Delta =0.1J$, middle panel: $\Delta =J$, and lower panel: $\Delta =10J$. The system size is $M=500$, the number of disorder realizations is $N_r=4000$, and ${\alpha }^{\text{ho}}=2.88\times {10}^{-4}J$. The inset in the upper panel shows the low-frequency absorption in the absence of disorder, which is finite for trapped gases. As $\Delta$ increases, this sharp peak fragments out and is no longer visible on the scale of the bulk contribution.