Hierarchical relaxation dynamics in a tilted two-band Bose-Hubbard model

We numerically examine slow and hierarchical relaxation dynamics of interacting bosons described by a tilted two-band Bose-Hubbard model. The system is found to exhibit signatures of quantum chaos within the spectrum and the validity of the eigenstate thermalization hypothesis for relevant physical observables is demonstrated for certain parameter regimes. Using the truncated Wigner representation in the semiclassical limit of the system, dynamics of relevant observables reveal hierarchical relaxation and the appearance of prethermalized states is studied from the perspective of statistics of the underlying mean-field trajectories. The observed prethermalization scenario can be attributed to different stages of glassy dynamics in the mode-time configuration space due to dynamical phase transition between ergodic and nonergodic trajectories.

Figure 1

Distribution of the ratio of adjacent level spacings $P\left(r\right)$. Top: Different system size $L$ and fixed interaction $Ng/E_R=4$. Bottom: For $N=40,L=2$, and varying interaction strengths. The GOE (dark-dashed) and the Poissonian (bright-dashed) averages are shown for comparison. Inset: Mean value $\langle r\rangle$ as a function of $L$ (top) and $Ng/E_R$ (bottom).

Figure 2

Coarse-grained distribution of the EEV for $Ng/E_R=8$. Top: Upper-band manifold $M$. Middle: Upper band of the second site $\{\ell =2,r=2\}$. Bottom: Lower band of the second site $\{\ell =1,r=2\}$. Left: $L=2$. Right: $L=3$.

Figure 3

Distributions of consecutive EEV gaps of the upper-band manifold $M$. Top left: Comparison between different interaction couplings for $L=2$ (top left) and $L=3$ (top right). Bottom: Comparison between $L=2$ and 3 with fixed $Ng/E_R=8$.

Figure 4

Distributions of consecutive EEV gaps for $Ng/E_R=8$ and $L=2$ with varying number of bosons $N$. Top: Upper-band manifold $M$. Middle: Upper band of the second site $\{\ell =2,r=2\}$. Bottom: Lower band of the second site $\{\ell =1,r=2\}$.

Figure 5

Variances of the distributions of consecutive EEV gaps of different observables for varying $N$ and fixed $Ng/E_R=8$. Left: $L=2$. Right: $L=3$.

Figure 6

Examples of dynamics of the mode occupation number with initial Fock states. Top: Comparison between exact quantum dynamics (ED) and TWA simulation for $L=2,Ng/E_R=8$, and initial state of $\{n_1^1,n_2^1,n_1^2,n_2^2\}/N=\{0.55,0.25,0.05,0.15\}$. Bottom: TWA results for the dynamics of mode occupation number for $L=3,Ng/E_R=8$, and initial state of $\{n_1^1,n_2^1,n_3^1,n_1^2,n_2^2,n_3^2\}=\{4,2,0,1,1,6\}\times {10}^4$.

Figure 7

Time evolution of the distance $d_r^{\ell }\left(t\right)$ between two different initial Fock states with the same energies for $L=2$. Top: $Ng/E_R=4$ with initial states $\left\{{\psi }_1\right\}=\{n_1^1,n_2^1,n_1^2,n_2^2\}=\{0,14,13,13\}\times {10}^4$ and $\left\{{\psi }_2\right\}=\{6,10,4,20\}\times {10}^4$. Bottom: $Ng/E_R=8$ with initial states $\left\{{\psi }_1\right\}=\{23,10,2,5\}\times {10}^4$ and $\left\{{\psi }_2\right\}=\{18,13,0,9\}\times {10}^4$. Inset (top): Semilogarithmic scale in $d_r^{\ell }\left(t\right)$. Top: Log-log scale in $d_r^{\ell }\left(t\right)$ and $t$.

Figure 8

Time evolution of the distance $d_r^{\ell }\left(t\right)$ between two different initial Fock states with the same energies for $L=3$. Top: $Ng/E_R=4$ with initial states $\left\{{\psi }_1\right\}=\{n_1^1,n_2^1,n_3^1,n_1^2,n_2^2,n_3^2\}=\{1,1,1,2,5,4\}\times {10}^4$ and $\left\{{\psi }_2\right\}=\{0,3,1,2,5,3\}\times {10}^4$. Bottom: $Ng/E_R=8$ with initial states $\left\{{\psi }_1\right\}=\{4,2,0,1,1,6\}\times {10}^4$ and $\left\{{\psi }_2\right\}=\{2,1,5,1,3,2\}\times {10}^4$.

Figure 9

Time evolution of the distance $d_r^{\ell }\left(t\right)$ between an initial Fock and coherent state with the same initial energies for $Ng/E_R=8$. Top: $L=2$ with Fock initial state of $\left\{{\psi }_1\right\}=\{n_1^1,n_2^1,n_1^2,n_2^2\}=\{23,10,2,5\}\times {10}^4$ and coherent initial state of $\left\{{\psi }_2\right\}=\{23,13,2,2\}\times {10}^4$. Bottom: $L=3$ with Fock initial state of $\left\{{\psi }_1\right\}=\{n_1^1,n_2^1,n_3^1,n_1^2,n_2^2,n_3^2\}=\{4,2,0,1,1,6\}\times {10}^4$ and coherent initial state of $\left\{{\psi }_2\right\}=\{4,2,3,1,1,3\}\times {10}^4$. The black dash-dotted line represents an exponential decay as a guide to the eye. Inset (top): Short-time oscillations of $d_r^{\ell }\left(t\right)$. Bottom: Comparison between the short-time decay of $d_r^{\ell }\left(t\right)$ in $\langle {\widehat{n}}_1^1\rangle$ and $\langle {\widehat{n}}_3^2\rangle$.

Figure 10

Time evolution of the distance, $d_r^{\ell }\left(t\right)$, in semilogarithmic scale between two different initial Fock states with the same initial energies for $L=3$ as in Fig. 8. Top: $Ng/E_R=4$. Bottom: $Ng/E_R=8$. The black broken lines correspond to exponential fits of the form $\sim \exp (-t/\tau )$. The dotted lines emphasize the prethermalization plateaus.

Figure 11

Mean-field trajectories of two slightly different initial Fock states up to $t/T=20$ for $L=2$. Left):$Ng/E_R=4$. Right: $Ng/E_R=8$. Note that we rescaled $b_r^{\ell }\to b_r^{\ell }/\sqrt{N}$.

Figure 12

Mean-field solution of Eq. (11). Top: $L=2$. Bottom: $L=3$. Left column: $Ng/E_R=4$. Right column: $Ng/E_R=8$. The individual curve in each plot represents the mean-field dynamics of a particular mode occupation number. Legend labels are the same as Fig. 9.

Figure 13

Density plot of the local mobility in mode-time for representative mean-field trajectories with $L=3$. Top: $Ng/E_R=4$ with Fock initial state $\{n_1^1,n_2^1,n_3^1,n_1^2,n_2^2,n_3^2\}=\{1,1,1,2,5,4\}\times {10}^4$. Bottom: $Ng/E_R=8$ with Fock initial state $\{n_1^1,n_2^1,n_3^1,n_1^2,n_2^2,n_3^2\}=\{2,1,5,1,3,2\}\times {10}^4$.

Figure 14

Distribution of activity $K\left[\psi \right(t\left)\right]$ of the mean-field trajectories at $t_{\mathrm{obs}}/T=150$ for $L=3$ with the same initial states as in Fig. 13 and $\mathrm{\Delta }t/T=0.4$. Inset: Semilogarithmic scale of the distribution for $Ng/E_R=8$.

Figure 15

Mean activity $K_s$ (top) and susceptibility ${\chi }_s$ (bottom) as a function of $s$ for different observation time $t_{\mathrm{obs}}$ for $L=3,Ng/E_R=8$, and the same initial Fock state as in Figs. 13 and 14.