Dynamical and steady-state properties of a Bose-Hubbard chain with bond dissipation: A study based on matrix product operators

We study a dissipative Bose-Hubbard chain subject to an engineered bath using a superoperator approach based on matrix product operators. The dissipation is engineered to stabilize a Bose-Einstein condensate wave function in its steady state. We then characterize the steady state emerging from the interplay between incompatible Hamiltonian and dissipative dynamics. While it is expected that interactions lead to this competition, even the kinetic energy in an open boundary condition setup competes with the dissipation, leading to a nontrivial steady state. We also present results for the transient dynamics and probe the relaxation time revealing the closing of the dissipative gap in the thermodynamic limit.

Figure 1

Green's Function $G(L/2-j,L/2+j)$ for the steady-state at filling $n=0.1$ and $L=20$ and 40. Upper panel: Simulation results for $J=0$, i.e., the OBC kinetic energy does not contribute to the time evolution. Lower panel: Finite nearest-neighbor hopping (here $\kappa /J=2$) suppresses qODLRO even for vanishing interaction. The orange symbols represent the density-normalized Green's function $\tilde{G}(j,l)$ normalized such that $\tilde{G}(L/2,L/2)=G(L/2,L/2)$. The horizontal dashed line denotes $G(j,L/2)=0.1$.

Figure 2

Real space density distribution $n_j$ for the steady state of a chain with $L=40$ sites at filling $n=0.1$ for $J=0$ (upper panel) and $\kappa /J=2$ (lower panel) and different values of the interaction $U$.

Figure 3

Imaginary part of the Green's function $\langle b_j^{†}{b}_l-b_j{b}_l^{†}\rangle$ at filling $n=0.1$. Upper: Left (right) panel shows steady state results for $U=0 \left(U/J=4\right)$ and $\kappa /J=2$. Lower: Left (right) panel shows the lattice divergence of the unitary current $\langle {\mathcal{J}}_u\rangle$.

Figure 4

Comparison of the real steady-state Green's function $G(j,l)$ for filling $n=1$, different particle number cutoffs $N=\sqrt{D}-1,L=16$, and $U=0$. The circles denote data obtained with finite $J$, whereas the data for $J=0$ are represented by squares. The finite particle cutoff acts as an effective interaction and suppresses the Green's function similarly to a finite $U$.

Figure 5

Real space density distribution $n_j$ for the steady state of a chain with $L=16$ sites at filling $n=1$ for $\kappa /J=2$ and 10. The inset shows the local density for $U/J=4$ and $\kappa /J=2$ for different system sizes.

Figure 6

Distances of correlation functions ${\Delta }_P$(left) and ${\Delta }_G$ (right) for $n=1,L=14$, and $U/J=6$ as a function of the effective interaction $U_{\mathrm{eff}}/J$ and inverse temperature $\beta J$.

Figure 7

Left: Particle number projector $P_n$ for the steady state at $L=14,n=1$, and $U/J=6$ with thermal results at an effective $U_{\mathrm{eff}}/J$. Right: Comparison of the steady-state Green's function for the same parameters compared to the thermal result at $U_{\mathrm{eff}}/J=3$ at various temperatures.

Figure 8

Steady-state operator space entropy (left) and bipartite fluctuations $\mathcal{F}$ for a block size of $L/2$. The dashed lines in the left panel denote fits to $alnL+b$ where the solid lines on the right-hand side are linear fits to $AL+B$.

Figure 9

Increase of operator space entanglement entropy $S_{L/2}$ with respect to the initial state for $L=40,n=0.1 \kappa /J=2$, and different values of the interaction strength. Inset (a): The figure shows $1-[S_{L/2}\left(t\right)-S_{L/2}(t=0)]/[S_{L/2}^{\mathrm{ss}}-S_{L/2}(t=0)]$ in order to illustrate the exponential convergence of the entropy towards the steady-state values for the data shown in the main panel. Inset (b): Relaxation rate $\alpha$ of the long-time behavior of $S$. The $n=0.1$ data corresponds to $\kappa /J=2$ (solid symbols) with interaction parameters $U/J=2$ (red circles), 4 (green squares), and 6 (blue diamonds) and $J=0$ (hollow symbols) with $U/\kappa =1$ (red circles), 2 (green squares), and 3 (blue diamonds). For $n=1$, we show data for $J=0$ and $U/\kappa =2.5$ (blue triangles) and 3 (red triangles). They obey a power law scaling $\alpha \sim L^{-2}$, indicated by a dashed line, compatible with the low-momentum damping spectrum of $\mathcal{L}$.

Figure 10

Time evolution of the operator space entanglement (top) and energy (bottom) for $U/J=4$ and $\kappa /J=2$ for different initial thermal density matrices. The upper dashed line corresponds to the $T=0$ and the lower dash-dotted line to the $T=\infty$ initial state, whereas the soliod lines correspond to temperatures $\beta J=0.2$, 0.4, 0.6, 0.8, 1, 2, 4, 6, 8, and 10. The arrows indicate increasing temperatures.

Figure 11

$F=\mathrm{tr}{\rho }^2$ for the single-particle problem for $J=0$ (upper panel) $J=\kappa$ for different system sizes.

Figure 12

Finite-size extrapolation of $F=\mathrm{tr}{\rho }^2$ for the single-particle problem at $J=\kappa$.

Figure 13

Convergence of $F=\mathrm{tr}{\rho }^2$ for the single-particle problem at $J=\kappa$. Inset: Power-law behavior of the exponent $\alpha ={\tau }^{-1}\propto L^{-2}$.