The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. Contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with nonsmall eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literature between the inverse of the Liouvillian gap and the relaxation time in dissipative many-body quantum systems.
The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. Contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with nonsmall eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literature between the inverse of the Liouvillian gap and the relaxation time in dissipative many-body quantum systems.
Markovian open many-body quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate towards the stationary state, but it has recently been pointed out that the spectral gap does not necessarily determine the overall relaxation time. Our understanding on the relaxation process before the asymptotically long-time regime is still limited. We here present a collective relaxation dynamics of autocorrelation functions in the stationary state. As a key quantity in the analysis, we introduce the instantaneous decay rate, which characterizes the transient relaxation and converges to the conventional asymptotic decay rate in the long-time limit. Our theory predicts that a bulk-dissipated system generically shows an accelerated decay before the asymptotic regime due to the scrambling of quantum information associated with the operator spreading.
It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with L being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.
The out-of-time-order correlator (OTOC) quantifies information scrambling in quantum systems and serves as a key diagnostic of quantum chaos. In one-body systems with a classical counterpart, the relaxation of the OTOC is governed by Ruelle-Pollicott resonances. For many-body systems lacking a semiclassical limit, recent studies have identified an analogous role played by the Liouvillian spectrum of weakly open extensions of the dynamics, where the slowest decay rate-the Liouvillian gap-encodes relaxation. Here we study the kicked Ising spin chain and show that the long-time exponential decay of the OTOC in the isolated system occurs at a rate equal to twice this intrinsic gap. This correspondence is demonstrated across parameter regions exhibiting distinct level spacing statistics, indicating that the Liouvillian spectrum provides a robust framework for characterizing relaxation and irreversibility in closed many-body quantum systems.
We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than ∼1/L. In integrable systems with boundary dissipation one typically observes scaling of ∼1/L(3), while in chaotic ones one can have faster relaxation with the gap scaling as ∼1/L and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.
Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.
Abstract Understanding how dissipation affects quantum many-body dynamics is crucial for quantum information processing, yet explicit quantitative predictions remain scarce. While closed quantum systems with spectral gaps exhibit robust dynamical properties, the transition to open systems governed by Lindblad dynamics has lacked constructive bounds relating Hamiltonian gaps to Liouvillian relaxation timescales. Scientific novelty. We derive the first explicit, parameter-dependent bound connecting a constructive Hamiltonian gap to the Liouvillian spectral gap in weakly dissipative quantum lattice systems. Starting from a frustration-free Hamiltonian with Nemirovsky gate gap Δgate = m0 − 2zgVmax > 0 and adding local Lindblad dissipation with strength γ ≪ Δgate, we prove that the Liouvillian gap satisfies ΛL ≥ Δgate − 2π·z·d·γ·Vdiss with explicit constant C = 2π, where z is the lattice coordination number, d is the number of dissipators per site, and Vdiss bounds the dissipator norms. This inequality provides quantitative control over open-system dynamics directly from microscopic parameters, enabling predictions for finite systems without asymptotic limits. We further establish a dissipative spectral weight bound WAdiss(E) ≤ K(A)·min(1, E/Λeff), where Λeff = Δgate − O(γ) characterizes the effective gap reduction. This directly yields the relaxation time lower bound τrelax ≥ ℏ/Λeff, with immediate applications to Rydberg atom arrays, trapped ions, and neutral atom quantum processors. Our results bridge constructive Hamiltonian spectral theory and open quantum dynamics, providing experimentalists with concrete design criteria for quantum gate timing, many-body scar lifetimes, and error correction protocols. Unlike previous existence theorems, our bounds are fully constructive, perturbative, and applicable to experimentally relevant parameter regimes. Keywords: Open quantum systems, Liouvillian gap, spectral bounds, Lindblad dynamics, dissipative quantum many-body systems, relaxation times, quantum simulators, Rydberg atoms, constructive quantum mechanics