Open quantum systems with nearly degenerate energy levels have been shown to exhibit long-lived metastable states in the approach to equilibrium, even when modelled with certain Lindblad-form quantum master equations. This is a result of dramatic separation of timescales due to differences between Liouvillian eigenvalues. These metastable states often have nonzero coherences that die off only in the long-time limit once the system reaches thermal equilibrium. We examine two distinct situations that give rise to this effect: one in which dissipative dynamics couple together states only within a nearly degenerate subspace, and one in which they give rise to jumps over finite-energy splittings, between nearly degenerate subspaces. We find, in each case, that a change of basis can often lead to a representation that more naturally captures the impact of the system-bath interaction than does the energy eigenbasis, revealing that separate timescales are associated with separate processes (e.g., decoherence into a nonenergy eigenbasis, decay of population correlations to the initial state). This approach is paired with the inspection of quantum trajectories, which further provide intuition as to how open system evolution is characterized when coherent oscillations, thermal relaxation, and decoherence all occur simultaneously. Published by the American Physical Society 2024
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with
k-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the
k locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
We analyze the multiple timescales that appear in the dissipative dynamics of an open Kitaev chain. Starting from the unified quantum master equation (UQME), we reduce the problem to a singularly perturbed three‐dimensional system that describes the nearly degenerate subspace spanned by a pair of Majorana zero modes (MZMs). Application of Chiba’s renormalization‐group (RG) method yields an explicit β ‐function for the slow amplitude and the asymptotic expansions of the slow and fast eigenvalues, together with rigorous error bounds on the timescale t = O (1/ ɛ 2 ). A weak longitudinal field, which renders the underlying closed system nonintegrable, produces a quadratic correction of structural type to the same β ‐function. The sign of the correction implies a shortening of the slow timescale; the prefactor k is not evaluated numerically. The analytic predictions are confirmed by numerical integration of both the reduced master equation and a short chain. The resulting framework provides a mathematically controlled description of how dissipation and the loss of integrability jointly determine the long‐time fate of topological edge modes. MSC2020 Classification: 81T17, 34E13, 82C10, 81Q80, 37A60