Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems

We present a theoretical method to study driven-dissipative correlated quantum systems on lattices with two spatial dimensions (2D). The steady-state density matrix of the lattice is obtained by solving the master equation in a corner of the Hilbert space. The states spanning the corner space are determined through an iterative procedure, using eigenvectors of the density matrix of smaller lattice systems, merging in real space two lattices at each iteration and selecting $M$ pairs of states by maximizing their joint probability. The accuracy of the results is then improved by increasing the dimension $M$ of the corner space until convergence is reached. We demonstrate the efficiency of such an approach by applying it to the driven-dissipative 2D Bose-Hubbard model, describing lattices of coupled cavities with quantum optical nonlinearities.

Figure 1

Sketch of the corner-space renormalization method.

Figure 2

Evolution of $n$ and $g_2$ versus time (units of $1/\gamma$) for lattices of various sizes. Parameters: $U/\gamma =20$, $J/\gamma =3$, $F/\gamma =2$, $\mathrm{\Delta }\omega /\gamma =5$. The black-dotted lines represent the mean-field values.

Figure 3

Probabilities $p_r$ (top panels, logarithmic scale) and expectation value of the total boson population $⟨n_{\mathrm{tot}}⟩=\sum _j⟨n_j⟩$ for the orthonormal eigenvectors $|{\mathrm{\Psi }}_r⟩$ of the steady-state density matrix. Lattice size: $6\times 3$. Driving parameters: $F/\gamma =2$, $\mathrm{\Delta }\omega /\gamma =5$. Left panels: $U/\gamma =20$ and $J/\gamma =3$. Right panels: hardcore bosons with $J/\gamma =1$.