Expansion dynamics of strongly correlated lattice bosons: A selfconsistent density-matrix approach

TL;DR

This work tackles the non-equilibrium expansion of strongly interacting lattice bosons near the superfluid-to-Mott insulator transition by developing a self-consistent density-matrix framework. By deriving a time-local quantum master equation for local site densities, the approach incorporates both condensate dynamics and non-condensed fluctuations via a dynamical bath, enabling spatio-temporal simulations of inhomogeneous 2D lattices. Application to the 2D Bose-Hubbard model reveals a ballistic expansion of the condensate halo coexisting with a slow, diffusive expansion of non-condensed atoms and a robust melting of Mott regions, accompanied by growth in the von Neumann number entropy. The method is scalable and benchmarks beyond Gutzwiller mean-field and traditional DMFT, offering a practical tool for exploring strongly correlated lattice bosons in inhomogeneous settings and potentially disordered or more complex models.

Abstract

We study the spatio-temporal dynamics of interacting bosons on a two-dimensional Hubbard lattice in the strongly interacting regime, taking into account the dynamics of condensate amplitude as well as the direct transport of non-condensed fluctuations. To that end we develop a selfconsistent density-matrix approach which goes beyond the standard Gutzwiller mean-field theory. Starting from the Liouville-von-Neumann equation we derive a quantum master equation for the time evolution of the system's local density matrix at each lattice site, with a dynamical bath that represents the rest of the system. We apply this method to the expansion dynamics of an initially prepared cloud of interacting bosons in an optical lattice. We observe a ballistic expansion of the condensate, as expected, followed by slow, diffusive transport of the normal bosons. We discuss, in particular, the robustness of the Mott insulator phase as well as its melting due to incoherent transport. The method should be applicable to various models of lattice bosons in the strongly correlated regime.

Figures (8)

• Figure 1: Schematic of the setup. (a) Sketch of the expansion of an interacting, bosonic cloud in an optical lattice. (b) and (c) show the zoomed-in spatial profiles of the particle density $N_{\bf r}(0)$ and of the superfluid density $|\Phi_{\bf r}(0)|^2$, respectively, for an initial Mott insulating core surrounded by a superfluid ring. The initial profile is computed using the Gutzwiller mean-field theory for $U/J=50$, $V_0/J=0.2$ and chemical potential $\mu/J=15$, see the text.
• Figure 2: Density-matrix vs. Gutzwiller mean-field theory. Spatial profiles of (a) particle-number density $N_{\bf r}(t)$, (b) condensate density $|\Phi_{\bf r}(t)|^2$ and (c) non-condensed density $N_{\bf r}^{\rm fl}(t)$ at different times (see at the top) are shown as a color scale in a square lattice using the density matrix approach for $U/J=50$. The particle-number densities $N_{\bf r}(t)$ obtained using the Gutzwiller mean-field method are shown in (d) for comparison.
• Figure 3: Spatio-temporal evolution of atomic densities. The spatio-temporal dynamics of (a) the particle-number density $N_{\bf r}(t)$, (b) condensate density $|\Phi_{\bf r}(t)|^2$ and (c) non-condensed density $N_{\bf r}^{\rm fl}(t)$ corresponding to \ref{['fig:GW-DM-Profiles']}\ref{['fig:GW-DM-Profiles']}, respectively, are shown along the diagonal cut of the square lattice.
• Figure 4: Dynamics of condensate fraction and number entropy. Time evolution of (a) the condensate fraction $\bar{\nu}(t)$ and (b) average number entropy $\bar{S}_{\rm N}(t)$ corresponding to fig:GW-DM-Profilesfig:DM-Timeevol are shown. The insets show the representative dynamics of (a) the condensate density $|\Phi_{{\bf r}_0}(t)|^2$ and non-condensed density $N_{{\bf r}_0}^{\rm fl}(t)$, and (b) the number entropy $S_{\rm N}^{{\bf r}_0}(t)$ at ${\bf r}_0=(10,10)$. The corresponding Fock state distributions $\rho_{{\bf r}_0}^{n,n}(t)$ are shown in (c) at different time instants as given and marked by filled circles in (a) inset.
• Figure 5: Slow melting of Mott insulator. The spatio-temporal dynamics of the particle-number density $N_{\bf r}(t)$ from an initially prepared Mott insulating core ($N_{\bf r}(0)=1$) of size $11\times 11$ are shown for $U/J=100$ along the diagonal cut of the square lattice in (a) and (b) with time plotted in linear and square-root scale, respectively. The straight, white lines in panel (b) are guides to the eye. The spatial density profiles are shown in panel (c) at different time instants as given and marked by arrowheads in (a).