Collective Oscillations of Strongly Correlated One-Dimensional Bosons on a Lattice

We study the dipole oscillations of strongly correlated 1D bosons, in the hard-core limit, on a lattice, by an exact numerical approach. We show that far from the regime where a Mott insulator appears in the system, damping is always present and increases for larger initial displacements of the trap, causing dramatic changes in the momentum distribution, $n_k$. When a Mott insulator sets in the middle of the trap, the center of mass barely moves after an initial displacement, and $n_k$ remains very similar to the one in the ground state. We also study changes introduced by the damping in the natural orbital occupations, and the revival of the center-of-mass oscillations after long times.

Figure 1

Evolution of the c.m. position (a)–(c), maximum value of $n_k$ (d)–(f), and the occupations of the three lowest NO (g)–(i), vs time ($\tau$, in units of $\hslash /t$). The initial trap displacements are $x_0=10a$ ($x_0/R_0\sim 0.07$) (a), (d), (g), $x_0=29a$ ($x_0/R_0\sim 0.2$) (b), (e), (h), and $x_0=80a$ ($x_0/R_0\sim 0.55$) (c), (f), (i). These systems have $N_b=101$, and $V_2{a}^2={10}^{-4}t$.

Figure 2

$n_k$ of the ground state (GS) compared with the ones at $\tau =4000\hslash /t$ for two different initial displacements of the trap ($N_b=101$, and $V_2{a}^2={10}^{-4}t$). As shown in Figs. 1b, 1c, at $\tau =4000\hslash /t$ the motion of the c.m. is totally damped. The inset shows the OPDM for the same systems, the straight line is ${\rho }_{ij}\sim 1/\sqrt{|x_i-x_j|/a}$.

Figure 3

(a) Ratio between the c.m. position after damping ($x_F$) and the initial trap displacement ($x_0$) vs $\tilde{\rho }$. The insets show $x_{\mathrm{c}.\mathrm{m}.}/R_0$ vs $\tau$ for the signaled $\tilde{\rho }$. (b) Ratio between $n_{k=0}$ after damping ($n_{k=0}^F$) and $n_{k=0}$ at $\tau =0$ ($n_{k=0}^0$) vs $\tilde{\rho }$. The inset shows $n_k^m$ vs $\tau =0$ for the signaled $\tilde{\rho }$.

Figure 4

Density and momentum profiles for a system with $N_b=101$, and $\tilde{\rho }=4$. See also results for $\tilde{\rho }=4$ in the insets of Fig. 3.

Figure 5

Long time evolution of the c.m. position (a), maximum value of $n_k$ (b), and the occupation of the lowest NO (c), vs time ($\tau$, in units of $\hslash /t$). The initial trap displacement is $x_0=4a$ ($x_0/R_0\sim 0.1$), and the system has $N_b=20$, and $\tilde{\rho }=1.0$.