Quantum dynamics of lattice states with compact support in an extended Bose-Hubbard model

We study the dynamical properties, with special emphasis on mobility, of quantum lattice compactons (QLCs) in a one-dimensional Bose-Hubbard model extended with pair-correlated hopping. These are quantum counterparts of classical lattice compactons (localized solutions with exact zero amplitude outside a given region) of an extended discrete nonlinear Schrödinger equation, which can be derived in the classical limit from the extended Bose-Hubbard model. While an exact one-site QLC eigenstate corresponds to a classical one-site compacton, the compact support of classical several-site compactons is destroyed by quantum fluctuations. We show that it is possible to reproduce the stability exchange regions of the one-site and two-site localized solutions in the classical model with properly chosen quantum states. Quantum dynamical simulations are performed for two different types of initial conditions: “localized ground states” which are localized wave packets built from superpositions of compactonlike eigenstates, and SU(4) coherent states corresponding to classical two-site compactons. Clear signatures of the mobility of classical lattice compactons are seen, but this crucially depends on the magnitude of the applied phase gradient. For small phase gradients, which classically correspond to a slow coherent motion, the quantum time scale is of the same order as the time scale of the translational motion, and the classical mobility is therefore destroyed by quantum fluctuations. For a large phase instead, corresponding to fast classical motion, the time scales separate so that a mobile, localized, coherent quantum state can be translated many sites for particle numbers already of the order of 10.

Figure 1

Time evolution of $|{\Psi }_i{|}^2$ in a four-site lattice for $Q_2^{\text{MF}}=-Q_5^{\text{MF}}=0.3$ and $\theta =0.1$. One can distinguish the IS region (left figure, $Q_4^{\text{MF}}=-0.13$) where the compacton gets trapped on two sites, the OS region (right figure, $Q_4^{\text{MF}}=-0.07$) where it is trapped on one site, and the mobile region (middle figure, $Q_4^{\text{MF}}=-0.10$) where it travels through the lattice without much energy loss. Note that in the mobile region, it periodically returns to something very close to the compacton structure with a population of $\approx$$0.5$ on two adjacent sites.

Figure 2

Time evolution of $|{\Psi }_i{|}^2$ for $Q_2^{\text{MF}}=-Q_5^{\text{MF}}=0.3$ and $\theta =1$. The upper (lower) row shows a four-site (100-site) lattice with, from left to right, $Q_4^{\text{MF}}=-0.2,-0.1,-0.06$. A mobile localized solution is found in both the four-site and 100-site lattice (middle figures). For these figures, the $i$th peak corresponds to the $(i+1)$th site (modulo 4 and 100, respectively). Signs of the IS- and OS-stable regions can be distinguished in the 100-site lattice since the solution gets trapped on two and one sites, respectively. This cannot be done in the four-site lattice.

Figure 3

Quantities related to the localized ground state (see text for details) for 20 particles and $Q_2=0.3$. The figure shows (a) $\langle {\widehat{n}}_1\rangle$, (b) $\langle {\widehat{n}}_2\rangle$, (c) $\langle {\widehat{n}}_3\rangle$, (d) $\langle {\widehat{n}}_4\rangle$, (e) $\langle 4{\widehat{n}}_1{\widehat{n}}_2\rangle$, (f) $\langle 4{\widehat{n}}_2{\widehat{n}}_3\rangle$, (g) $\langle 4{\widehat{n}}_3{\widehat{n}}_4\rangle$, (h) $\langle 4{\widehat{n}}_4{\widehat{n}}_1\rangle$, and (i) two-site compactness. Expectation values are taken with localized ground states. Lines in (i) indicate $Q_2=-Q_{5}N$ (solid) and $Q_2=-Q_5(N-1)$ (dashed), two candidates of a two-site QLC condition. The solid line in (a) indicates $Q_2=-2Q_5(N-1)$, one of the two conditions (the other being $Q_4=0$) for having an exact one-site QLC.

Figure 4

Time evolution of $\langle {\widehat{n}}_i\rangle$ for the localized ground state (lines) with 20 particles, $Q_2=-Q_{5}N=0.3$ and $Q_{4}N=$ (a) $-0.16$, (b) $-0.123$, (c) $-0.122$. Circles indicate $0.25\{1+\cos [(E_0-E_2)t/2]\cos [(E_0-2E_1+E_2)t/2]\}$ and crosses $0.25\{1-\cos [(E_0-E_2)t/2]\cos [(E_0-2E_1+E_2)t/2]\}$, which overlap very well with the curves in (a), reasonably well in (b), and not at all for (c). In (c) we have crossed over into the OS region, where the solution initially is located mainly on one site. The figures are given for the same parameter values as Table .

Figure 5

(a) $1/(E_2-E_0)$ and $1/(E_2-2E_1+E_0)$ for 20 particles and $Q_2=-Q_{5}N=0.3$. $E_i$ is the lowest energy in $k$-space $i$. (b) shows $1/(E_2-E_0)$ for the same values of $Q_2$ and $Q_{5}N$, $Q_{4}N=-0.16$, $-0.13$, and $-0.123$ as a function of the number of particles. The increase of $1/(E_2-E_0)$ with the number of particles slows down drastically as one approaches the exact crossing.

Figure 6

Time evolution of $\langle {\widehat{n}}_i\rangle$ for a localized ground state with 20 particles and $Q_2=-Q_{5}N=0.3$. The top row is for phase 0.1 and the bottom row phase 1; columns show $Q_{4}N=-0.16$ (left), $-0.123$ (middle, IS region), and $-0.122$ (right, OS region).

Figure 7

The figure shows how many sites a localized ground state with $\theta =1$, for 20 particles and $Q_2=-Q_{5}N$, travels before the population is less than 0.4 at an intersection point, i.e., a point where $\langle {\widehat{n}}_i\rangle$ is equal on the two highest populated sites. The black and dark red region in the upper part corresponds to the OS regime, where the LGS initially has a large fraction of the particles on one site [cf. Fig. 6f].

Figure 8

Fidelity, ${\left|\langle \text{LGS}|\text{CS}\left(0\right)\rangle \right|}^2$, for $Q_2=0.3$ and 20 particles.

Figure 9

Dynamics of a coherent QLC state for 20 particles and $Q_2=-Q_{5}N=0.3$. The top row is for phase $\varphi =0.1$ and the bottom row $\varphi =1$. The columns show $Q_{4}N=-0.16$ (left), $-0.123$ (middle), and $-0.11$ (right).

Figure 10

Same as Fig. 7 but for a coherent QLC state.