Quantum dynamics of solitons in strongly interacting systems on optical lattices

Mean-field dynamics of strongly interacting bosons described by hard-core bosons with nearest-neighbor attraction has been shown to support two species of solitons: one of Gross-Pitaevskii type (GP type) where the condensate fraction remains dark, and a non-Gross-Pitaevskii type (non-GP type) characterized by brightening of the condensate fraction. Here we study the effects of quantum fluctuations on these solitons using the adaptive time-dependent density matrix renormalization group method, which takes into account the effect of strong correlations. We use local observables as the density, condensate density, and correlation functions as well as the entanglement entropy to characterize the stability of the initial states. We find both species of solitons to be stable under quantum evolution for a finite duration, their tolerance to quantum fluctuations being enhanced as the width of the soliton increases. We describe possible experimental realizations in atomic Bose-Einstein condensates, polarized degenerate Fermi gases, and in systems of polar molecules on optical lattices.

Figure 1

Bright (top) and dark (bottom) soliton solution in the continuum Eq. (7) for ${\rho }_0=0.25$ and $V/J=0.4$. Left panels show the density, right panels show the condensate density as a function of position and speed. Note that the condensate density of the bright soliton shows a “brightening,” around the notch (i.e., it grows above the background value), whereas the dark soliton does not show this effect.

Figure 2

Comparison of the soliton profiles for a background density ${\rho }_0=0.25$ at times $t=20/J$ obtained in the continuum [black dashed line, Eq. (6)] and using the equations of motion approach Eq. (5) on a lattice of $L=40$ sites. The left panels show the results for $V/J=0.75$ (narrow soliton), the right panels show the case $V/J=0.95$ (broad soliton). The top panels show the bright soliton, the bottom panels show the dark soliton solution of Eq. (7). Both the particle density $\rho$ $(*)$ and the condensate density ${\rho }^s$ $(+)$ are shown.

Figure 3

Differences resulting from Eq. (10) of (a) the local density $\rho$ and (b) the condensate density ${\rho }^s$ between the mean-field continuum evolution and the mean-field lattice evolution on a lattice of $L=40$ sites at times $t=20/J$ as a function of $V/J$. Panels (c) and (d) show the lattice mean-field evolution of the local density for a broad soliton (c) with $V/J=0.65$ and a narrow soliton (d) with $V/J=0.45$.

Figure 4

Local particle density $\rho$ (blue $*$) and condensate density ${\rho }^s$ (red $+$) obtained by DMRG (symbols) and by the mean-field ansatz (solid line) for a stationary bright soliton ($\overline{v}=0$) at times $t=0$ (left panels) and at times $t=20/J$ (right panels). The plots show results for lattice sizes of $L=100$ sites (top) and $L=40$ sites (bottom). The parameters are $V/J=0.95$ and ${\rho }_0=0.45$.

Figure 5

Local particle density $\rho$ (top) and condensate density ${\rho }^s$ (bottom) obtained by the DMRG (symbols) and by the mean-field (solid line) propagation of the stationary ($\overline{v}=0$) bright (red $+$) and dark (blue $*$) solitons for times $t=0$ and $20/J$ for a system with $L=40$ sites. The parameters are $V/J=0.95$ and ${\rho }_0=0.25$.

Figure 6

Differences using Eq. (13) between the DMRG results and the lattice mean-field results for the total density (left) and for the condensate density (right) for a stationary (top) and for moving (bottom) bright solitons (${\rho }_0=0.25$) on a system with $L=40$ sites.

Figure 7

Entanglement entropy in ground states of the spin system Eq. (1) for $V/J=0.95$, $L=40$ sites for values of $S_{\mathrm{total}}^z$ corresponding to the background density ${\rho }_0=0.1$, 0.25, and $0.45$, respectively.

Figure 8

Entanglement entropies as a function of the subsystem size of the stationary bright soliton (top) and dark solitons (bottom) at different times for $V/J=0.95$.

Figure 9

Top: Contour plot of the time evolution of the nearest-neighbor spin correlations $\langle S_i·S_{i+1}\rangle -\langle S_i\rangle ·\langle S_{i+1}\rangle$ on the whole lattice for the stationary bright soliton (left) and the dark soliton (right) for ${\rho }_0=0.25$ and $V/J=0.95$. Bottom: Time evolution of the entanglement entropy for the same parameters.

Figure 10

Comparison of the $t$-DMRG evolution of a bright (top) and dark (bottom) discrete HGPE soliton ($\#$) to the evolution of an initial Gaussian state with $(+)$ and without $(*)$ phase imprinting for ${\rho }_0=0.25$ and $V/J=0.95$.