Hierarchy of Relaxation Timescales in Local Random Liouvillians

Abstract

To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size with up to n-body Lindblad operators, which are n-local in the complexity-theory sense. Without locality (n = ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403; arXiv:1905.02155]. However, for local Liouvillians (n < ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit.Introduction. For unitary quantum many-body dynamics, the characterization of generic features common to the vast majority of systems is well developed, in the form of an effective random matrix theory [1-6]. It is for example a crucial ingredient to our understanding of thermalization in unitary quantum systems, manifest in the eigenstate thermalization hypothesis (ETH) [7][8][9][10][11][12][13].For open quantum many-body systems analogous organizing principles are yet missing. Only very recently the first developments in this direction appeared with the investigation of spectral features of a purely random Liouvillian [14-18], describing generic properties of trace preserving positive quantum maps. The purely random Liouvillians considered in [14,15,17] constitute the least structured models aiming at describing generic features of open quantum many-body systems. A main result of these recent studies is that the spectrum of purely random Liouvillians is densely covering a lemon-shaped support whose form is universal, differentiating random Liouvillians from completely random Ginibre matrices with circular spectrum [1,[19][20][21].How close is a completely random Liouvillian to a physical system? In this Letter, we make a step towards answering this question by adding the minimal amount of structure to a purely random Liouvillian, namely a notion of locality. We consider a dissipative spin system of size with Lindblad operators given by Pauli strings classified by their length n ≤ , the latter being a measure of their n-locality in a complexity-theory sense [22].In this language, purely random Liouvillians are completely nonlocal and structureless, since they include the full operator basis including Pauli strings of all lengths 1 ≤ n ≤ .Here, we restrict the maximal number of non-identity operators in the Lindblad operators to n max and examine

Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm

“…Although the speedup at the second level is not ideal, parallelization solves the main problem, allowing us to fit into the limitations of memory size. The proposed parallel algorithm opens up new perspectives to numerical studies of large open quantum models and allows us to advance further into the territory of “Dissipative Quantum Chaos” [ 17 , 27 , 28 , 29 ].…”

“…A generalization of the random Lindbladian model to many-body setup was proposed very recently [ 27 ]. It allows us to take into account the topological range of interaction in a spin chain model, by controlling the number of neighbors n involved in an n -body Lindblad operator acting on the i -th spin.…”

Full Spectrum of the Liouvillian of Open Dissipative Quantum Systems in the Zeno Limit

“…The eigenvalues of a Liouvillian with a locally acting dissipator at large dissipation strength Γ are arranged in a set of stripes, see Fig. 1, indicating the existence of a hierarchy of relaxation timescales in the system [30]. The number of stripes coincides with the number of different eigenvalues of the Lindblad dissipator D in (1).…”

Spectral transitions and universal steady states in random Kraus maps and circuits

“…Following pioneering works on non-Hermitian Hamiltonians [8][9][10][11], some studies focused on structureless discrete-time quantum maps [12,13], and decoherence effects induced by a random environment [14][15][16]. Recently, the focus has shifted to continuous-time random Lindblad dynamics of Markovian open quantum systems [17][18][19][20][21]. These efforts led to a thorough numerical characterization of the spectral support, the spectral density, spectral gaps, and steady states, complemented by analytical calculations based on perturbative arguments and non-Hermitian RMT.…”

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