Dynamics of localization phenomena for hard-core bosons in optical lattices

We investigate the behavior of ultracold bosons in optical lattices with a disorder potential generated via a secondary species frozen in random configurations. The statistics of disorder is associated with the physical state in which the secondary species is prepared. The resulting random potential, albeit displaying algebraic correlations, is found to lead to localization of all single-particle states. We then investigate the real-time dynamics of localization for a hardcore gas of mobile bosons which are brought into sudden interaction with the random potential. Regardless of their initial state and for any disorder strength, the mobile particles are found to reach a steady state characterized by exponentially decaying off-diagonal correlations and by the absence of quasicondensation; when the mobile particles are initially confined in a tight trap and then released in the disorder potential, their expansion is stopped and the steady state is exponentially localized in real space, clearly revealing Anderson localization.

Figure 1

(Color online) Average participation ratio (a) and density of states (b) of the single-particle energy eigenstates in the correlated random potential created by frozen bosons with disorder strength $W=0.5\mathit{J}$. The density of states is calculated for a system size $L=2002$.

Figure 2

(Color online) Scaling of the disorder-averaged largest eigenvalue ${\lambda }_0$ of the OPDM (a) and the occupation of the zero momentum state $n_{k=0}$ (b) for the ground state of a half-filled system of mobile bosons, interacting with the disorder potential. The cases of a correlated potential resulting from the superfluid state, Eq. (19), and that of a fully uncorrelated potential, Eq. (20), are compared. Here the strength of disorder is $W=0.5\mathit{J}$. The black lines are fits to $n_{k=0},⟨{\lambda }_0⟩\propto \sqrt{N}$ for the first four data points.

Figure 3

(Color online) Scaling of the steady-state zero-momentum occupation $n_{k=0}$ with the particle number $N$ in a homogeneous system of mobile and frozen bosons $(V=V^{\mathrm{f}}=0)$ at half-filling $(N=N^{\mathrm{f}}=L∕2)$, with interaction strength $W=0.5\mathit{J}$. The line corresponds to a fit $n_{k=0}\sim \sqrt{N}$ to the first four data points.

Figure 4

(Color online) Snapshot of the evolution of the real-space density $n_i$ for a single particle, initially in the ground state of a harmonic trap with $V=0.01\mathit{J}$, and interacting with a correlated disorder potential. Here the interaction strength is $W=0.5\mathit{J}$, and the system size is $L=502$.

Figure 5

(Color online) Time evolution of the participation ratio of a single particle for various disorder strengths. Other parameters as in Fig. 4.

Figure 6

(Color online) Steady state participation ratio (recorded at $t=500{\mathit{J}}^{-1}$) for $N$ particles, time-evolved in a system of size $L=702$ starting from the ground state of a harmonic trap $(V=0.01\mathit{J})$ and interacting with a correlated random potential or with a fully uncorrelated random potential. The lines show a fit to the data points with an algebraically decaying function.

Figure 7

(Color online) (a) Steady-state participation ratio (recorded at $t=400{\mathit{J}}^{-1}$) of $N$ particles as a function of the particle number for various disorder strengths. Other parameters as in Fig. 6. (b) Total, kinetic, and potential energy of the $M\text{th}$ single-particle eigenstate of the initial system in the trap.

Figure 8

(Color online) Snapshots of the time evolution of the momentum distribution $n_k$ for a system of $N=100$ hardcore bosons, initially in a trap with density $\tilde{\rho }=0.5$, interacting with a correlated disorder potential. Here the system size is $L=1002$ and the interaction strength is $W=0.5\mathit{J}$. The momentum distribution for the corresponding spinless fermions is shown for comparison.

Figure 9

(Color online) Decay of the modulus of the OPDM $\mid {\rho }_{i_0,i_0+r}\mid$ from the system center $i_0$ at various times during evolution. Parameters as in Fig. 8.

Figure 10

(Color online) Scaling of the steady-state ${\lambda }_0$ (recorded at time $t=500{\mathit{J}}^{-1}$) for varying number of particles in a system of size $L=1502$; other parameters as in Fig. 8. The data for correlated frozen bosons are compared with those for fully uncorrelated ones. The line corresponds to a fit to ${\lambda }_0\sim \sqrt{N}$.

Figure 11

(Color online) Time evolution of the density profile (a) and the momentum distribution function (b) of $N=35$ hardcore bosons, initially in a perfect Mott insulator, in a system of size $L=502$ without disorder.

Figure 12

(Color online) Time evolution of the density profile (a) and the momentum distribution function (b) of $N=35$ hardcore bosons, initially in a Mott insulator, interacting with the disorder potential created by frozen particles at half-filling. Here the interaction strength is $W=0.5\mathit{J}$.

Figure 13

(Color online) Time evolution of ${\lambda }_0$ for various disorder strengths. Other parameters as in Fig. 12.

Figure 14

(Color online) Time evolution of ${\lambda }_0$ for various numbers of particles, and for disorder strength $W=0.1\mathit{J}$. Other parameters as in Fig. 12.

Figure 15

(Color online) Scaling of the maximal value of ${\lambda }_0$ during time evolution of $N$ hardcore bosons, initially in a Mott insulator. All parameters as in Fig. 14. The line corresponds to a fit to ${\lambda }_0\sim \sqrt{N}$ of the first four data points.

Figure 16

(Color online) Sketch of the preparation of the disorder potential and of the tuning of its strength using state-dependent optical lattices for a mixture of atoms in two internal states. (a) The two species are first prepared in two shifted optical lattices, (b) then one of the two optical lattice is increased in strength so as to freeze one of the two bosonic species, and (c) finally the spatial phase of the two lattices is changed to bring the two species into interaction.