Local Versus Global Equilibration near the Bosonic Mott-Insulator–Superfluid Transition

We study the time scales for adiabaticity of trapped cold bosons subject to a time-varying lattice potential using a dynamic Gutzwiller mean-field theory. We explain apparently contradictory experimental observations by demonstrating a clear separation of time scales for local dynamics ($\sim \mathrm{ms}$) and global mass redistribution ($\sim 1\mathrm{s}$). We provide a simple explanation for the short and fast time scales, finding that while density or energy transport is dominated by low energy phonons, particle-hole excitations set the adiabaticity time for fast ramps. We show how mass transport shuts off within Mott-insulator domains, leading to a chemical potential gradient that fails to equilibrate on experimental time scales.

Figure 1

Energy scales in 2D Bose-Hubbard model as a function of lattice depth [13]: Solid: $4J$, Dashed: $U$, Dotted: Two lowest $k=0$ excitations from linearizing at unity filling. Parameters: ${}^{87}\mathrm{Rb}$ in a $d=680\mathrm{nm}$ lattice. In the superfluid state, the Goldstone mode has zero energy. In the Mott state, these modes represent the particle or hole excitations.

Figure 2

Population dynamics at unity density $n=1$ (Top): Probability of having one particle per site at the end of a lattice ramp from $V_i/E_R=11$ lattice to (top to bottom) $V_f/E_R=13$ (yellow), 15 (green), 17 (blue), 19 (purple) and 25 (red) after different lattice ramp times ${\tau }_r=0.1/U_i\sim 0.3\mathrm{ms}$ to $10/U_i$. Inset: Characteristic time scale ${\tau }_a$ extracted from exponential fits to main figure. Fig. (3) of 6.

Figure 3

Slow transport across Mott-insulator region (Top-Left) Evolution of an initial (solid) superfluid state at $V_i=11E_R$ and $N=500$ in a 25 Hz radial trapping potential. Final density profile (dashed) after a ramp ${\tau }_r=120\times 2\pi /U_i\sim 400\mathrm{ms}$, is very different from the equilibrium state (dotted) at $V_f=16E_R$. (Right) Density plot of the time evolution of the coherences (${\mathcal{C}}_i\equiv -⟨a_i⟩\sum _j⟨a_j^{*}⟩$) features a growing Mott-insulator region which cuts off transport between the superfluid regions in the center and the edge, leading to a nonequilibrium final state at late times. Brighter colors correspond to larger $\mathcal{C}$. (Bottom-Left): A chemical potential gradient is established between the superfluid regions after dynamics ($t=400\mathrm{ms}$, dots). Within the Mott-insulator domain, $\mu$ is not a unique function of $n$, and vertical lines illustrate the range of values of the ordinate. In the initial state $\mu +V(r)={\mu }_0$ is a constant. At the wings there are no particles and $\mu +V(r)\sim V(r)$. The fact that $\mu +V(r)$ is roughly constant in the superfluid regions confirms that they are in local equilibrium.

Figure 4

Fast equilibration in absence of transport (Left): Evolution of an initial superfluid state for $V_i=11E_R$ and $N=800$ (solid) in a 25 Hz radial trapping potential in a linear ramp with ${\tau }_r=25\times 2\pi /U_i=80\mathrm{ms}$. The dotted profile is the $T=0$ equilibrium Gutzwiller profile at $V_0=16E_R$ for the same parameters. The final density profile (dashed) agrees with the $T=0$ equilibrium Gutzwiller profile. (Right): Time evolution of the spatial coherence distribution, showing the formation of an $n=1$ and $n=2$ Mott plateaus. Lighter colors imply larger coherences, cf. Fig. (2) in 7.