Dynamics of disordered states in the Bose-Hubbard model with confinement

Observations of center-of-mass dynamics offer a straightforward method to identify strongly interacting quantum phases of atoms placed in optical lattices. We theoretically study the dynamics of states derived from the disordered Bose-Hubbard model in a trapping potential. We find that the edge states in the trap allow center-of-mass motion even with insulating states in the center. We identify short- and long-time-scale mechanisms for edge-state transport in insulating phases. We also argue that the center-of-mass velocity can aid in identifying a Bose-glass phase. Our zero-temperature results offer important insights into mechanisms of transport of atoms in trapped optical lattices while putting bounds on center-of-mass dynamics expected at nonzero temperature.

Figure 1

Schematic showing two distinct mechanisms for center-of-mass dynamics after an initial shift of the trapping potential to the left. The center of the system is in an insulating phase (either the Mott insulator or the Bose glass), but the edges are superfluid. Left: We find that initially (on times scales shorter than the intersite tunneling), the edge superfluid quickly moves around the insulator to the left to redistribute the total center of mass. Right: We find that (on time scales much larger than the intersite tunneling) a two-stage process shifts the center of mass. First, particles slowly tunnel out of the central insulator into the edges (vertical arrows). Then the particles quickly move to the left along the edge to redistribute the center of mass.

Figure 2

Schematic mean-field phase diagram of the disordered Bose-Hubbard model [Eq. (1)] in the absence of a trap, $\mathrm{\Omega }=0$. The black regions indicate the incompressible Mott insulators found at integer densities. The gray regions indicate the compressible Bose glass that separates the Mott from the superfluid. The three vertical arrows depict how the trapping potential lowers the local chemical potential as we move from the center of the system (top of the arrow) to the edge (bottom of the arrow). The three arrows represent regimes studied here, where the center of the system contains a disordered Mott insulator (left), Bose glass (center), and disordered superfluid (right).

Figure 3

(a)–(c) plot the disorder-averaged local density in the initial state as a function of distance from the center of the trap. (a) and (c) have a superfluid and Mott insulator in the trap center, respectively. (b) is in an intermediate regime. Model parameters are chosen to be disorder strength $\mathrm{\Delta }=0.3U$, trap strength $\mathrm{\Omega }=0.02U$, and central chemical potential $\mu =0.5U$. (d)–(f) plot the same but for the disorder-averaged local-density fluctuations in the initial state. The density fluctuations in (e) and (f) show the edge superfluid. (g)–(i) plot the disorder-averaged local superfluid order parameter in the initial state. Error bars result from disorder averaging.

Figure 4

The disorder-averaged center-of-mass position along the $x$ direction plotted as a function of time for various interaction strengths $U$. The trap shift is chosen to be $\mathrm{\Delta }{R}_0=a$, and other model parameters are the same as in Fig. 3. (a)–(c) are initially in the disordered superfluid state. In (d) the center of the system is initially in the Bose-glass state, whereas (f) is initially in the Mott state. The inset in (d) shows the short-time behavior. (e) is initially in an intermediate state. In (d)–(f), the center of mass has a nearly constant velocity for $\tau \gg \hslash /t$.

Figure 5

The disorder-averaged local-density deviation from the initial state, $\mathrm{\Delta }{n}_i\equiv {\langle \langle n_i\rangle \rangle }_{\tau }-{\langle \langle n_i\rangle \rangle }_{\tau =0}$, plotted as a function of position in the $x-y$ plane for three different times. All model parameters are the same as in Fig. 4, where a superfluid occupies nearly the entire trap. These parameters correspond to the right arrow in the phase diagram in Fig. 2. Here we see how the superfluid as a whole oscillates in the trap even in the presence of disorder.

Figure 6

Top: the same as Fig. 5 but for an interaction strength of $U=26t$ [parameters in Fig. 4], which corresponds to a Bose glass in the center of the trap (central arrow in Fig. 2). Bottom: the disorder-averaged local current flow with the same parameters as in the top panel. At $\tau =0.8\hslash /t$ we see the first mechanism of transport where at short times the edge state quickly slips around the central insulator, depicted in the left panel of Fig. 1. At $\tau =8\hslash /t$ and $80\hslash /t$ we see a different two-step mechanism of transport, depicted in the right panel of Fig. 1. Here the particles first slowly tunnel out of the central insulator to get pushed, in the second step, quickly along the edges. In the two right snapshots, lateral tunneling events and a shrinking of the central insulator can be seen.

Figure 7

The disordered-averaged center-of-mass velocity at short times [Eq. (9)] and long times [Eq. (10)] plotted as a function of the trap shift for an interaction strength of $U=26t$. For short times, the center-of-mass velocity responds linearly to the length of the trap shift. For long times, the center-of-mass velocity barely changes with the trap shift.

Figure 8

The disorder-averaged center-of-mass position plotted as a function of time for (a) several different trap shifts for $U=26$ and (b) several different $U$ for $\mathrm{\Delta }{R}_0=a$. The model parameters are otherwise the same as in Fig. 3. In (a) we see that different trap shifts do not change the long-time center-of-mass velocity (the slope is nearly constant). But in (b) we see that the slope depends on $U$.

Figure 9

The disorder-averaged center-of-mass velocity at long times [Eq. (10)] plotted as a function of interaction strength. For weak interactions the entire system is in the superfluid (SF) phase, and frequent center-of-mass oscillations result in a zero net velocity. For intermediate interaction strengths the Bose glass (BG) lies in the trap center. Here the two-step edge flow mechanism depicted in the right panel of Fig. 1 allows a small but finite center-of-mass velocity. At large interaction strengths the center of the system is in the Mott regime (MI), where tunneling into the edge (and therefore the center-of-mass velocity) is strongly suppressed.

Figure 10

Schematic showing sites of a lattice (dashes) along one direction filled up to site 1 with particles. Site 2 is an empty edge site. Tunneling from site 1 to site 2 is controlled with a potential $V$ at site 2. The dotted line depicts the parabolic trapping potential which zeros the edge density.

Figure 11

Comparison of the center-of-mass velocity at long times between the estimate of the two-site model and numerical simulations for various interaction strengths.