Mean-field dynamics of a Bose-Hubbard chain coupled to a non-Markovian environment

We study the dynamics of an interacting Bose-Hubbard chain coupled to a non-Markovian environment. Our basic tool is the reduced generating functional expressed as a path integral over spin-coherent states. We calculate the leading contribution to the corresponding effective action, and by minimizing it, we derive mean-field equations that can be numerically solved. With this tool at hand, we examine the influence of the system's initial conditions and interparticle interactions on the dissipative dynamics. Moreover, we investigate the presence of memory effects due to the non-Markovian environment.

Figure 1

Dependence of the real part of the averaged dissipation kernel $\overline{\mu }\left(t\right)$. (a) Comparison between sub-Ohmic, $n=0.5$ (solid blue line), Ohmic, $n=1$ (red dashed line), and super-Ohmic, $n=2$ (green dash-dotted line) non-Markovian environments. (b)–(d) Comparison between Markovian (red dashed line) and non-Markovian (blue solid line) dynamics for sub-Ohmic ($\mathrm{\Gamma }=0.0855J$) (b), Ohmic ($\mathrm{\Gamma }=0.0161J$) (c), and super-Ohmic ($\mathrm{\Gamma }=5.6\times {10}^{-4}J$) (d) environments. In all cases $U{N}_0=6J$.

Figure 2

Evolution of the normalized total particle number, $n_{\mathrm{tot}}^{\mathrm{norm}}=n_{\mathrm{tot}}/N_0$. (a) Comparison between sub-Ohmic, $n=0.5$ (solid blue line), Ohmic, $n=1$ (red dashed line), and super-Ohmic, $n=2$ (green dash-dotted line) non-Markovian environments. (b)–(d) Comparison between Markovian (red dashed line) and non-Markovian (blue solid line) dynamics for sub-Ohmic (b), Ohmic (c), and super-Ohmic (d) environments. In all cases $U{N}_0=6J$ and the initial conditions are $(p,q)=(0.5,0.58\pi )$.

Figure 3

The normalized total particle number, $n_{\mathrm{tot}}^{\mathrm{norm}}=n_{\mathrm{tot}}/N_0$, after a fixed propagation time $J{t}_{\mathrm{f}}=2$, for different initial conditions and interaction strengths. (a), (b) We have fixed $p$ and $q$ and we depict $n_{\mathrm{tot}}^{\mathrm{norm}}\left(t_f\right)$ as a function of $q$ and $p$, respectively, for three different interaction strengths: $U{N}_0=0$ (blue solid line), $U{N}_0=3J$ (red dashed line), and $U{N}_0=6J$ (green dash-dotted line). In (c) and (d) the color map shows the value of $n_{\mathrm{tot}}^{\mathrm{norm}}\left(t_f\right)$ for all $(p,q)$ initial conditions, for $U{N}_0=0$ and $U{N}_0=6J$, respectively. In (c), (d) the black lines depict the classical phase space of the nondissipative DNLS.

Figure 4

Evolution of (a), (b) the phase of the fields, (c), (d) the normalized fields, and (e), (f) decay rate, for two different interaction strengths: $U{N}_0=3J$ (left column) and $U{N}_0=6J$ (right column). The vertical black dot-dashed line depicts the time when the phase difference is zero, while the vertical black solid line the time when the population difference is zero. The initial conditions in both cases are $(p,q)=(0.5,0.64\pi )$.

Figure 5

Time evolution of the normalized total particle number (second column) and of the normalized particle density in each lattice site (third column) for four different initial states (first column): (a)–(c) $\mathit{a}\left(0\right)=\sqrt{N_0}(1,-1,1,-1)/2$, (d)–(f) $\mathit{a}\left(0\right)=\sqrt{N_0}(0,1,-1,0)/\sqrt{2}$, (g)–(i) $\mathit{a}\left(0\right)=\sqrt{N_0}(1,0,0,-1)/\sqrt{2}$, and (j)–(l) $\mathit{a}\left(0\right)=\sqrt{N_0}(1,-1,-1,1)/2$. In all cases $U{N}_0=6J$ and we have hard-wall boundary conditions.

Figure 6

Time evolution of the normalized total particle number. In all cases the initial condition is $\mathit{a}\left(0\right)=\sqrt{N_0}(1,0,0,0)$ and $U{N}_0=0$. For the other parameters vertically we have the sub-Ohmic case ($n=0.5$, first column), the Ohmic case ($n=1$, second column), and the super-Ohmic one ($n=2$, third column). Horizontally we have $\lambda =0.001$ (a)–(c), $\lambda =0.005$ (d)–(f), and $\lambda =0.01$ (g)–(i).