Driven-dissipative quantum Monte Carlo method for open quantum systems

We develop a real-time full configuration-interaction quantum Monte Carlo approach to model driven-dissipative open quantum systems with Markovian system-bath coupling. The method enables stochastic sampling of the Liouville–von Neumann time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the nonequilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce the statistical error. Finally, we demonstrate the efficiency of our approach by applying it to the driven-dissipative two-dimensional $XYZ$ spin-$\frac{1}{2}$ model on a lattice.

Figure 1

The initiator approach used to extrapolate the $M_z$ magnetization (in units of $\hslash$) in the case of the $4\times 4$ dissipative $XYZ$ Heisenberg model. Parameters of the model are $J_x/\gamma =0.225$, $J_y/\gamma =0.335$, $J_z/\gamma =0.25$, $h=0.1$, $\theta =0$. Parameters of the simulation are $p=2.5$, with a population of ${10}^6$ walkers. The straight line is a linear extrapolation of the lowest initiator values.

Figure 2

The amount of simultaneously occupied density-matrix elements, with and without using the initiator approach and importance sampling in the case of the $4\times 4$ $XYZ$ Heisenberg lattice. Parameters of the model are $J_x/\gamma =0.225$, $J_y/\gamma =0.225$, $J_z/\gamma =0.25$, $h=0.1$, $\theta =0$. Parameters of the simulation are $p=1.5$, $I_{\mathrm{limit}}=25;75$ with a population of ${10}^8$ walkers.

Figure 3

The exact and the DDQMC magnetization values (in units of $\hslash$) for the $3\times 3$ dissipative $XYZ$ Heisenberg lattice with periodic boundary conditions. The coupling parameters are $J_x/\gamma =0.225$, $J_y/\gamma =0.335$, and $J_z/\gamma =0.25$. The diagonal population was limited to (a) ${50}^4$ and (b) ${20}^6$ walkers.

Figure 4

The magnetization $M_z$ per site (in units of $\hslash$) as a function of the normalized coupling parameter $J_y/\gamma$ for different lattice sizes. The other coupling parameters are $J_x/\gamma =0.225$ and $J_z/\gamma =0.25$. The exact ($2\times 2$ and $3\times 3$) and MCWF ($4\times 4$) results are plotted for comparison. Error bars, when not shown, are smaller than the symbol size.

Figure 5

The angle-averaged susceptibility ${\chi }_{av}$ per site as a function of the normalized coupling parameter $J_y/\gamma$ for different lattice sizes. The other coupling parameters are $J_x/\gamma =0.225$ and $J_z/\gamma =0.25$. Each point on the plot was determined by 21 simulations, which corresponds to 525 calculations per lattice size. For each point, we considered $3+3+1$ values of the applied field (three for each in-plane direction and one with no external field), and for each setting an extrapolation over three different initiator values was carried out.