Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains

Abstract

The unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (nonunitary) dynamics, which can be radically different from closed-system scenarios. Such open quantum system dynamics is generally described by Lindblad master equations, whose dynamical and steady-state properties are challenging to obtain, especially in the many-particle regime. Here, we introduce a method to deal with these systems based on the calculation of a (dissipative) lattice Green's function with a real-space decimation technique. Compared to other methods, such a technique enables us to obtain compact analytical expressions for the dynamics and steady-state properties, such as asymptotic decays or correlation lengths. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity, including the Hatano-Nelson model. The latter is especially illustrative because its surface and bulk dissipative behavior are linked due to its nontrivial topology, which manifests in directional amplification.

Non-Hermitian topological phases in traveling-wave parametric amplifiers

“…In the Hatano-Nelson model, these two regions have been shown to display different dynamical properties, related with the steady state of the system [36,56]. However, in contrast with the Hatano-Nelson model, we find that stability in this model is size-dependent.…”

“…1 by green arrows. The local term naturally appears in the adiabatic elimination due to the possibility to hop back and forth between a site and the auxiliary reservoir [36,48]. As we will show below, the incoherent pump term is not required to engineer the dissipative BdG topological phase, but it is crucial to produce a topological phase with larger winding number.…”

“…The calculation requires to first determine the surface Green's function, Ĝ0,0 (ω), from the non-linear matrix equation [36]: Ĝ0,0 (ω) = ĝ0,0 (ω) + ĝ0,0 (ω) V + Ĝ0,0 (ω) V − Ĝ0,0 (ω) .…”

“…Importantly, topological phases in photonic systems typically include gain and loss, and this makes them fundamentally different to the ones typically considered in materials science [26]. In fact, the dissipative nature of the systems can enrich the physics of the topological phases [27][28][29][30][31][32], and this can be detected in the steady state [33,34] or in the transient properties [35][36][37]. An interesting case of a dissipative topological phase, present in photonic systems, is that of topological amplification [33,34,[38][39][40].…”

“…In Section IV, we characterize the gain, noise-to-signal ratio, and squeezing of the topological phases. In addition, we describe how combining a Green's function approach with decimation techniques allows us to find analytical expressions in the semi-infinite case, which perfectly capture the numerical results for finite systems [36]. In Section V we discuss some possible experimental implementations with state-of-the-art techniques.…”