Variational Principle for Steady States of Dissipative Quantum Many-Body Systems

We present a novel generic framework to approximate the nonequilibrium steady states of dissipative quantum many-body systems. It is based on the variational minimization of a suitable norm of the quantum master equation describing the dynamics. We show how to apply this approach to different classes of variational quantum states and demonstrate its successful application to a dissipative extension of the Ising model, which is of importance to ongoing experiments on ultracold Rydberg atoms, as well as to a driven-dissipative variant of the Bose-Hubbard model. Finally, we identify several advantages of the variational approach over previously employed mean-field-like methods.

Figure 1

Up-spin density $n_{\uparrow }$ in the steady state of the dissipative Ising model on a two-dimensional square lattice depending on the transverse field $g$. Results are shown for the variational methods using product states ${\rho }_p$ and correlated states ${\rho }_c$, as well as solutions to the full quantum master equation (QME) for $3\times 3$ and $4\times 4$ lattices ($h=0$, $V=5\gamma$).

Figure 2

Phase diagram of the dissipative transverse-field Ising model ($h=0$). For large interaction strength $V$, there is a first-order transition between a lattice gas and a lattice liquid of up spins. This first-order transition ends in a critical point, around which the system forms a critical fluid.

Figure 3

Steady state density $n$ of the driven-dissipative Bose-Hubbard model ($F=0.4\mu$, $\gamma =0.2\mu$). The system undergoes a first-order jump in the density when the hopping $J$ is increased over a critical value.

Figure 4

Comparison of the norm of the two-site dynamics $\parallel {\dot{\rho }}_{ij}\parallel$ of the variational solution (left) and the mean-field solution (right) for the two-dimensional dissipative Ising model ($V=5\gamma$).