Particle-hole symmetric localization in optical lattices using time modulated random on-site potentials

The random hopping models exhibit many fascinating features, such as diverging localization length and density of states as energy approaches the band center due to its particle-hole symmetry. Nevertheless, such models are yet to be realized experimentally because the particle-hole symmetry is easily destroyed by diagonal disorder. Here we propose that a pure random hopping model can be effectively realized in ultracold atoms by modulating a disordered onsite potential in particular frequency ranges. This idea is motivated by the recent development of the phenomena called “dynamical localization” or “coherent destruction of tunneling.” Investigating the application of this idea in one dimension, we find that if the oscillation frequency of the disorder potential is gradually increased from zero to infinity, one can tune a noninteracting system from an Anderson insulator to a random hopping model with diverging localization length at the band center, and eventually to a uniform-hopping tight-binding model.

Figure 1

Phase diagram of model (4) studied in this work. At zero frequency, the system is an Anderson insulator; when the frequency $\omega$ is comparable to the disorder width $\gamma$, the system behaves as a random hopping model; when $\omega$ is much larger than $\gamma$, the system crosses over to the uniform-hopping tight-binding regime, which is fully achieved when $\omega =\infty$. We also present some interesting and puzzling results for the regime $0<\omega ⪡\gamma$.

Figure 2

[(a), (c), and (e)] Inverse of the localization length $1/\lambda$ and [(b), (d), and (f)] inverse of the density of states $1/\rho$ vs quasienergy $E$ (in units of the hopping amplitude $J$). Disorder width $\gamma =10J$, hopping amplitude $J=1$, oscillation frequency $\omega =3J,7J,35J$. Solid lines are from the Floquet calculation of the original model (4); solid triangles are from the effective Hamiltonian $H_{\text{eff}}^{\left(0\right)}$ (27). Averaged over 1000 realizations of disorder.

Figure 3

(Color online) Inverse of the localization length $1/\lambda$ vs quasienergy $E$ (in units of the hopping $J$). Averaged over 1000 realizations of disorder. Hopping amplitude $J=1$, disorder width $\gamma =10J$, oscillation frequency $\omega =3J,5J,7J,9J,15J,35J$.

Figure 4

Fitting the inverse of the localization length ${\lambda }^{-1}$ and the inverse of the density of states ${\rho }^{-1}$ near the band center to their analytical form $\rho \left(E\right)=N2{\sigma }^2/\left|E{\left[\text{ln}{\left(E/E_0\right)}^2\right]}^3\right|$ and $\lambda \left(E\right)=2\left|\text{ln}{\left(E/E_0^{\prime }\right)}^2\right|/{\sigma }^2$. Here, $E_0$ and $E_0^{\prime }$ are energy scales (Ref. 5), and the theoretical value of $\sigma$ is the standard deviation of $\text{ln}{J}_{\text{eff}}^2$, which is 1.535 for our choice of parameters here. The fitted value of $\sigma$ is 1.496 for $1/\lambda$ and 1.677 for $1/\rho$. Hopping amplitude $J=1$, oscillation frequency $\omega =7$, and disorder width $\gamma =10$.

Figure 5

(Color online) Inverse of the localization length $1/\lambda$ vs the label (e.g., 1st, 2nd, 3rd, …, 100th) of every Floquet eigenstate for $\omega =0.05,0.1,0.5,1,3,7$. System size $N=100$, hopping amplitude $J=1$, oscillation frequency $\omega =7$, and disorder width $\gamma =10$.