Exact Solutions of Interacting Dissipative Systems via Weak Symmetries

We demonstrate how the presence of continuous weak symmetry can be used to analytically diagonalize the Liouvillian of a class of Markovian dissipative systems with strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method can be viewed as implementing an exact, sector-dependent mean-field decoupling, or alternatively, as a kind of quantum-to-classical mapping. We focus on two canonical examples: a nonlinear bosonic mode subject to incoherent loss and pumping, and an inhomogeneous quantum Ising model with arbitrary connectivity and local dissipation. In both cases, we calculate and analyze the full dissipation spectrum. Our method is applicable to a variety of other systems, and could provide a powerful new tool for the study of complex driven-dissipative quantum systems.

Figure 1

(a) Schematic of the second model analyzed in this work: $N$ two-level systems interact via arbitrary Ising interactions $J_{ij}$, and are also subject to local dissipation, cf. Eq. (10). (b) Using weak symmetry, one can make an exact mean-field decoupling for each symmetry-constrained dynamical sector, leaving one with an easily-solved but unusual independent, dissipative spin problem. (c) A similar solution method can be used for an incoherently-driven nonlinear bosonic mode [cf. Eq. (1)], enabling an exact calculation of the Liouvillian eigenvalues ${\lambda }_{m,\mu }$. We plot these here for $|m|\le 10$ and $\mu \le 15$. Each color corresponds to a different value of $|m|$. By fixing $m$, the level spacing ${\lambda }_{m,\mu +1}-{\lambda }_{m,\mu }=-{\tilde{\kappa }}_m-i{\tilde{U}}_m$ is constant, reflecting the noninteracting nature of the problem in each symmetry-constrained block. We work in a rotating frame where ${\omega }_0$ is shifted to 0, and set $U=\kappa$, ${\overline{n}}_{\mathrm{th}}=0.1$.