Flux-enhanced localization and reentrant delocalization in the quench dynamics of two interacting bosons on a Bose-Hubbard ladder

We study the quench dynamics of two bosons possessing on-site repulsive interaction on a two-leg ladder and show that the presence of uniform flux piercing through the plaquettes of the ladder favors the localization of the bound states in the dynamics. We find that, when the two bosons are symmetrically initialized on the edge rung of the ladder, they tend to edge-localize in their quantum walk, a phenomenon which is not possible in the absence of flux. However, when the bosons are initialized on the bulk rung they never localize and exhibit linear spreading in their quantum walk. Interestingly, however, we find that, in the later case, a finite flux favors localization of the bulk bound states in the presence of sufficiently weak quasiperiodic disorder, which is otherwise insufficient to localize the particles in the absence of flux. In both cases, we obtain that the localization in the dynamics strongly depends on the combined effect of the flux and interaction strengths, as a result of which we obtain a signature of reentrant delocalization as a function of flux (interaction) for fixed interaction (flux) strengths.

Figure 1

Figure depicts a two-leg Bose-Hubbard ladder in the presence of uniform flux. $J$ and $K$ are the leg and rung hopping strengths, respectively. $U$ is the on-site interaction between the bosons and $\mathrm{\Phi }$ is the flux piercing through each plaquette and $\varphi$ is the phase acquired by the particles.

Figure 2

(a)–(c) Show the density evolution of two particles in a two-leg ladder for $U=0,3,$ and 30, respectively, with the initial states $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (3). (d)–(f) Show the correlation matrix ${\mathrm{\Gamma }}_{i,j}$ at time $t=20\left(J^{-1}\right)$ for the same values of $U$ as in the upper panel. Here we consider $\varphi =\pi$.

Figure 3

$P$ as a function of $t\left(J^{-1}\right)$ for different values of $U$ and for $\varphi =\pi$.

Figure 4

$P_{\text{avg}}$ is plotted as a function of $U$ and $\varphi /\pi$ for the initial state $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (3). The red central patch corresponds to the range of $U$ for which the localization is maximum for each values of $\varphi$. Here the $P_{\text{avg}}$ is calculated by averaging time between 0 to $T=1000\left(J^{-1}\right)$ with a step size of $\mathrm{\Delta }t=2\left(J^{-1}\right)$.

Figure 5

IPR of the states plotted as a function of energy eigenvalues and the eigenstate index for $U=3$ and $\varphi =\pi$.

Figure 6

(a) and (b) show the particle density at each rung for the eigenstate with higher IPR values.

Figure 7

(a)–(c) Show the density evolution of the two particles with the initial state $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (9) for $U=0, U=3$, and $U=30$, respectively, and for $\varphi =\pi$.

Figure 8

(a) IPR of the states is plotted as a function of energy eigenvalues $E$ and $U$ for fixed values of rung hopping strength $K=5$ and $\varphi =\pi$. The isolated localized states are shown in the insets for clarity. (b,c) show the behavior of $\left|\mathrm{\Delta }E\right|$ as a function of $U$ corresponding to the localization of particles at one edge and both edges of the ladder, respectively.

Figure 9

The figure depicts $P$ as a function of $t\left(J^{-1}\right)$ for different values of $K$ keeping $U=3$ and $\varphi =\pi$.

Figure 10

IPR of the states is plotted as a function of energy eigenvalues $E$ and the rung-hopping $K$ for $U=3$ and $\varphi =\pi$.

Figure 11

(I)–(III) Figure shows the density evolution of the initial state $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (10) for $U=0,3,$ and 30, respectively, and $\varphi =\pi$. (a)–(c) correspond to the case when $\lambda =0$ and and (d)–(f) correspond to the case when $\lambda =0.25$ and $\chi =0$.

Figure 12

(a) and (b) depict the survival probability $S{P}_3$ as a function of time $t\left(J^{-1}\right)$ corresponding to disorder strengths $\lambda =0.25$ and $\lambda =0$, respectively, and $\varphi =\pi$ with the initial state $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (10). In (a), the survival probability is computed by averaging over 500 random $\chi$ values.

Figure 13

Shows $S{P}_{3-\text{avg}}$ as a function of $\varphi /\pi$ and $U$ with the initial state $\left|\mathrm{\Psi }\right(0)\rangle$ given in Eq. (10). Here the values of $\lambda =0.25$ and $\chi =0$ are considered and the time average is obtained for time between $t=0\left(J^{-1}\right)$ to $1000\left(J^{-1}\right)$ with interval $\mathrm{\Delta }t=2\left(J^{-1}\right)$.

Figure 14

(a)–(d) show the survival probabilities $S{P}_3$ as a function of $t(J^{-1}$) for $U=3,10,15,$ and 20, respectively, with different values of $\lambda$ for each case. Here we start the dynamics from the initial state $|\mathrm{\Psi }\left(0\right)\rangle ={\widehat{b}}_{12,A}^{†}{\widehat{b}}_{12,B}^{†}|\text{vac}\rangle$ and fix the parameter values $\varphi =\pi$ and $\chi =0$.

Figure 15

(a) and (b) show the band structures in the absence of disorder for $U=3$ and $U=30,$ respectively. In both cases, we consider $\varphi =\pi$. The zoomed in data are shown in the insets for clarity.