Quench from Mott insulator to superfluid

We study a linear ramp of the nearest-neighbor tunneling rate in the Bose-Hubbard model driving the system from the Mott insulator state into the superfluid phase. We employ the truncated Wigner approximation to simulate linear quenches of a uniform system in one, two, and three dimensions, and in a harmonic trap, in three dimensions. In all these setups, the excitation energy decays like one over third root of the quench time. The $-1/3$ scaling arises from an impulse-adiabatic approximation—a variant of the Kibble-Zurek mechanism—describing a crossover from nonadiabatic to adiabatic evolution when the system begins to keep pace with the increasing tunneling rate.

Figure 1

The transition rate $(dJ/dt)/J=J^{-1}{\tau }_Q^{-1}$ is huge at early time and small at late time. In contrast, the relaxation rate ${\tau }^{-1}=J^{1/2}$ is negligible at early time and large at late time. The two rates are comparable near $\widehat{J}\simeq {\tau }_Q^{-2/3}$. Before $\widehat{J}$ the evolution is impulse, i.e., the state does not change and remains the initial Mott state with random phases. After $\widehat{J}$, the evolution becomes adiabatic.

Figure 2

Thermalization in 1D ($L=512$). We show correlation functions $C_R=\frac{1}{n}\langle a_s^{†}{a}_{s+R}\rangle =\overline{{\phi }_s^{☆}{\phi }_{s+R}}$ after an instantaneous quench from the initial Mott state (3),(8) at $J=0$ to a final $J\ll 1$ in the Josephson regime obtained with the truncated Wigner method. As predicted in Eqs. (12) and (13), the three plots for the widely different final $J$ collapse in the rescaled time $J^{1/2}t$, proving that $\tau \simeq J^{-1/2}$ is indeed the characteristic time-scale in the Josephson regime. For large $J^{1/2}t$, the collapsed plots tend to their equilibrium values predicted in Eq. (A6) in Appendix, demonstrating that $\tau$ is indeed the thermalization time.

Figure 3

Thermalization in 3D ($128\times 128\times 128$ lattice). We show short-range correlators $C_1,\cdots ,C_4$ after an instantaneous quench from the initial Mott state (3),(8) at $J=0$ to the final $J=0.1$ in the Josephson regime obtained with the truncated Wigner method. These short-range correlators $C_R$ do thermalize but, unlike in 1D, their relaxation time clearly increases with $R$. There is quick local thermalization, but it takes longer time to thermalize the system over larger scales.

Figure 4

Thermalization in 3D. We show the correlation function $C_R\left(u\right)=\frac{1}{n}\langle a_s^{†}{a}_{s+R}\rangle =\overline{{\phi }_s^{☆}{\phi }_{s+R}}$ for several values of the rescaled time $u=J^{1/2}t$ at a fixed $J=0.01$. The initial state at $u=0$ has phase correlations of finite range. It is prepared by a linear quench from the initial Mott state (3, 8) at $J=0$ to $J=0.01$ with ${\tau }_Q=512$. After this ramp, this low-energy initial state thermalizes at the fixed $J=0.01$ towards a low-temperature thermal state with long-range order. While the short range correlations $C_R$ with small $R$ are quick to thermalize, see Fig. 3, the long-range correlations are slow to develop the long-range order expected at low temperature. As we can see in this figure, the range of $C_R$ roughly doubles when the time $u$ increases by a factor of $4,$ i.e., the correlation range grows like the square root of time.

Figure 5

Dependence of the kinetic (left column) and potential (right column) energy densities for 1D (the upper row), 2D (the middle row), and 3D lattice (the bottom row). The lattice size $L=4096,256,128$ in $1D,2D,3D$, respectively. For large ${\tau }_Q\gg 1$, we observe power-law behavior consistent with the predicted $\langle E\rangle \sim {\tau }_Q^{-1/3}$. The best fits to the tails of the energy plots (the solid lines) give the exponents listed in Table . The plots also demonstrate the equipartition $\langle E_{\mathrm{kin}}\rangle \approx \langle E_{\mathrm{pot}}\rangle$ at $J=0.1$ as predicted in the Josephson regime where the energy is approximately quadratic.

Figure 6

Correlation functions $C_R$ in 1D at $J=0.1$ for different quench times ${\tau }_Q$ and the lattice length $L=4096$. The functions are exponential, as expected in a thermal state in the adiabatic stage of the evolution. Their correlation length scales like $\xi \sim {\tau }_Q^{\alpha }$ with the best fit $\alpha =0.329$ close to the predicted $1/3$.

Figure 7

Long-range tail of the correlation functions $C_R$ in 2D (lattice size $L=256$) and 3D (lattice size $L=128$) at $J=0.1$ for different quench times ${\tau }_Q$. They are in the adiabatic stage, in the sense that local observables have equilibrated, but they have not had enough time to develop infinite-range (quasi-)long-range order expected in the low-temperature phase. Instead, the correlations have finite range limited by a finite rate at which they can spread across the system. The range grows with ${\tau }_Q$ faster than ${\tau }_Q^{1/2}$.

Figure 8

Quench on a 2D periodic $256\times 256$ lattice in the Josephson regime at $J=0.1$. (Top) phase ${\theta }_{\mathbf{s}}$ for a quench time ${\tau }_Q=1638.4$. (Middle) phase ${\theta }_{\mathbf{s}}$ for a quench time ${\tau }_Q=52428.8$. (Bottom) atomic density $|{\phi }_{\mathbf{s}}{|}^2$ associated with the phase in the middle panel. The top and middle panels demonstrate that the range of phase correlations increases with increasing ${\tau }_Q$. Average size of domains of constant phase is consistent with the range of 2D correlations in Fig. 7. In the faster quench (top panel), where the range of correlations is comparable to the lattice spacing, there is plenty of random vorticity. In the slower quench (middle panel), where the range of correlations is much longer than the lattice spacing, it is possible to identify smooth topological vortices (marked by the arrows). The associated density fluctuations in the bottom panel are small, within $10\%$ of the average density $|{\phi }_{\mathbf{s}}{|}^2=1$, and very weakly correlated between different sites. This is not really surprising because in the Josephson regime the healing length $\simeq J^{1/2}$ is less than the lattice spacing and, consequently, densities at different sites are only weakly coupled and vortex cores are less than the lattice spacing.

Figure 9

A quench with ${\tau }_Q=3276.8$ at $J=1$ on a part of a 2D periodic $256\times 256$ lattice. In the top panel the phase ${\theta }_{\mathbf{s}}$ and in the bottom panel the density distribution $|{\phi }_{\mathbf{s}}{|}^2$. The three arrows mark an isolated vortex and a vortex-antivortex pair. Unlike in the Josephson regime at $J=0.1$ where density fluctuations are small and vortex cores are less than the lattice spacing, see Fig. 8, here, at $J=1$ the topological vortices in the upper panel are associated with clear dips in atomic density marked in the bottom panel.

Figure 10

Excitation energy above the discrete Gross-Pitaevskii ground state at a given $J$, $\langle E\rangle -E_{\mathrm{GS}}\left(J\right)$, as a function of ${\tau }_Q$ for a 3D lattice in a harmonic trap. The tails $\langle E\rangle -E_{\mathrm{GS}}\sim {\tau }_Q^{-\alpha }$ for large ${\tau }_Q\gg 1$ are best fitted with the exponents: $\alpha =0.31$ for $J=0.1$, $\alpha =0.32$ for $J=1.0$, and $\alpha =0.31$ for $J=10.0$. Here, the lattice size is ${64}^3$ and the initial Thomas-Fermi radius $R_{\mathrm{TF}}=10$.

Figure 11

A cross section of $|{\phi }_{\mathbf{s}}{|}^2$ along the line $s_y=s_z=0$ in a single realization of the statistical ensemble for ${\tau }_Q=10$. The initial Thomas-Fermi profile at $J=0$ spreads into a near-Gaussian wave packet at $J=10$. The simulation performed for a 3D lattice of $64\times 64\times 64$ sites.

Figure 12

Linear quench in the 3D trap with ${\tau }_Q=204.8$ and the initial Thomas-Fermi radius $R_{\mathrm{TF}}=25$ at the tunneling rate $J=1$ for $128\times 128\times 128$ lattice. (Top): phase in the $x-y$ cross-section across the center of the trap. (Middle) density in the $x-y$ cross section across the center of the trap. (Bottom) cumulative column density in the direction perpendicular to the $x-y$ plane. The arrows in the top and middle panels point to a vortex.

Figure 13

The same excitation energy as in Fig. 10 but with constant initial phases ${\theta }_{\mathbf{s}}=0$. For large ${\tau }_Q$ the energy decays approximately like ${\tau }_Q^{-0.5}$ at $J=0.1$ and ${\tau }_Q^{-0.4}$ at $J=10$ i.e. faster than for the random initial phases in Fig. 10. The size of the lattice is $64\times 64\times 64$.