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.
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.
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.
Abstract We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation channels, the spectral gap increases with the system size. The spectral distribution of the steady state is Poissonian at low dissipation strength and conforms to that of a random matrix once the dissipation is sufficiently strong. Our results can help to understand the long-time dynamics and steady-state properties of generic dissipative systems.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
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
We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian L(t). We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed-system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of L(t) will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time T. In the case of a finite number of level crossings, the scaling becomes ${T}^{\ensuremath{-}\ensuremath{\eta}}$ with an exponent $\ensuremath{\eta}$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time T should be long compared to the corresponding minimum inverse gap squared of L(t). Our results are illustrated with several examples.
Abstract Relaxation processes in quantum systems coupled to external environments represent one of the most fundamental nonequilibrium phenomena in condensed matter physics. The Lindblad master equation provides a powerful framework for characterizing such open quantum dynamics. In this work, we systematically investigate how different types of quantum jump operators and system geometries influence the Liouvillian gap and the properties of the nonequilibrium steady state (NESS) in finite-size systems. We demonstrate that, due to the intricate structure of the Liouvillian superoperator, multiple NESSs with unphysical characteristics can emerge. The physically meaningful steady state must instead be understood as a superposition of these NESSs that collectively satisfy the required physical constraints. Furthermore, we find that the Liouvillian gap does not necessarily increase monotonically with the system–environment coupling strength. Instead, it can exhibit a nontrivial peak structure, corresponding to a minimum in the relaxation time. The magnitude of this peak is closely related to the symmetry properties of the system. Our results provide a deeper understanding of nonequilibrium behavior in finite quantum systems and offer new insights into the design and control of open quantum dynamics.