In this licenciate thesis, based on two articles, we discuss the properties of open quantum systems and investigate how their dynamics can be used to design a synthetic lattice framework and how their spectrabehave based on two. The first article presents an extension of the Fock state lattice (FSL) framework to open quantum systems. Using the Liouville space representation, density matrices can be vectorized and the dynamics are generated by a linear operator, the Liouvillian, allowing for the introduction of Liouville-Fock state lattices (LFSLs). This framework provides an intuitive and visual understanding of the dynamics of open quantum systems and shows how environmental effects can give rise to phenomena such as directional transport, sources and sinks, typical phenomena occurring in classical processes. As an example, we show how frustration in LFSL can be induced by the environment. The second article in the work focuses on non-normality of Liouvillians and spectral diagnostics of chaos. Non-normal operators are characterized by the fact that their eigenvectors are not orthogonal, which leads to the spectrum exhibiting strong sensitivity to perturbations. We show that strong non-normality can give rise to random matrix-like spectral statistics without the system dynamics exhibiting chaotic signatures. Therefore, we conclude that this violates established assumptions and points to the need for alternative methods to characterize chaotic dynamics in open quantum systems.
In this licenciate thesis, based on two articles, we discuss the properties of open quantum systems and investigate how their dynamics can be used to design a synthetic lattice framework and how their spectrabehave based on two. The first article presents an extension of the Fock state lattice (FSL) framework to open quantum systems. Using the Liouville space representation, density matrices can be vectorized and the dynamics are generated by a linear operator, the Liouvillian, allowing for the introduction of Liouville-Fock state lattices (LFSLs). This framework provides an intuitive and visual understanding of the dynamics of open quantum systems and shows how environmental effects can give rise to phenomena such as directional transport, sources and sinks, typical phenomena occurring in classical processes. As an example, we show how frustration in LFSL can be induced by the environment. The second article in the work focuses on non-normality of Liouvillians and spectral diagnostics of chaos. Non-normal operators are characterized by the fact that their eigenvectors are not orthogonal, which leads to the spectrum exhibiting strong sensitivity to perturbations. We show that strong non-normality can give rise to random matrix-like spectral statistics without the system dynamics exhibiting chaotic signatures. Therefore, we conclude that this violates established assumptions and points to the need for alternative methods to characterize chaotic dynamics in open quantum systems.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
For Hamiltonian systems, level statistics provide a faithful diagnostic of quantum chaos. By analogy, the statistics of the Lindbladian spectrum are often used in open quantum systems, and the Grobe-Haake-Sommers conjecture proposes that systems with chaotic classical counterparts should exhibit level repulsion in the Lindbladian spectrum. Here we point out an important flaw in this analogy: Hamiltonian and Lindbladian spectra behave differently and have distinct physical interpretations, and one should therefore not expect the latter to provide a reliable diagnostic. For Lindbladians, the late-time dynamics are not determined by the bulk of the eigenvalues but only by those eigenvalues -- and their corresponding eigenvectors -- with small real parts. Combined with the strong non-normality typical of Lindbladians, this allows situations in which the level statistics can be tuned almost arbitrarily without affecting the dynamics on either short or long time scales. We explicitly demonstrate this phenomenon and provide examples in which Ginibre level repulsion arises while the system dynamics at no time show signatures of chaos. We further relate this mechanism to the emergence of a non-Hermitian skin effect in Liouville space, linking boundary-induced eigenvector localization to the observed spectral instability. Our results show that level statistics cannot universally serve as a reliable diagnostic of quantum chaos in open quantum systems and highlight the need for alternative diagnostics that remain robust in strongly non-normal regimes.
Spectral statistics probe integrability versus chaos and have recently been extended to Markovian open quantum systems described by Lindbladians, whose quantum-trajectory unraveling decomposes the evolution into no-jump dynamics generated by an effective non-Hermitian Hamiltonian and recycling jumps. In this work, we perform spectrum-statistics diagnostics for Lindbladians and their effective non-Hermitian Hamiltonians. We show that recycling processes, symmetry constraints, and the Liouville-space structure crucially shape the spectral correlations. In particular, we identify a family of spectrally separable Lindbladians whose spectra exhibit robust Poisson statistics, despite the effective non-Hermitian Hamiltonian varying from Poisson to strongly correlated complex-spectrum statistics. Our work establishes a unified spectral-statistics characterization for Lindbladians and their associated effective non-Hermitian Hamiltonians, deepening our understanding of spectral properties in open many-body systems.
We point out that the quantum dynamical map of an open quantum system can be generated by an effective Liouville operator. The effective Liouville shows the dynamical breaking of time reversibility. This breaking of reversibility is expressed by the effective Liouville's discrete spectrum having negative imaginary parts. This generated dynamics of open quantum systems is capable of memory effects described by a frequency dependence of the spectrum. When memory effects can be neglected or smoothed out, the effective Liouville generates the well known semi-group dynamics of open systems in the Markov approximation. The spectral analysis of the effective Liouville -with or without memory- allows to represent the quantum dynamical map and expectation values of physical quantities by metric expressions using complex eigenvalues and bi-orthonormal eigenmodes. The long time dynamics is dominated by the zero-mode, which forms the stationary state of the open system. The relaxation time scale τ is set by the inverse of the smallest negative imaginary part of the non-vanishing effective eigenvalues of the effective Liouville. In generic cases of system-environment-coupling this time scale τ is the relevant scale for both, the relaxation of diagonal elements (populations) and the decay of off-diagonal elements (coherences) of an initial density matrix in the stationary state's diagonal representation. It also sets the time scale after which the entropy tends to its stationary value. The effective Liouville and its spectral content thus give a framework to study the dynamic breaking of time reversibility, memory effects, decoherence, relaxation to a stationary state and the role of entropy in open quantum systems.
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 perform a thorough numerical study, focusing on global spectral features, the spectral gap, and the steady-state purity and statistics. We establish that all properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular observable. Within each regime, we determine the scaling exponents with the dissipation strength and system size. On the analytical side, we compute the average spectral gap at arbitrary dissipation and provide simple closed-form expressions for the asymptotic values at strong and weak dissipation strength. We also consider spectral correlations in generic complex spectra, such as that of the Liouvillian. We introduce and study, both analytically and numerically, the complex ratio of the nearest-neighbor spacing by next-to-nearest-neighbor spacing. Besides the usual level repulsion, we find that complex spacing ratios have a nontrivial angular dependence which offers a distinct signature of quantum chaos. It distinguishes integrable systems, which support the flat ratio distribution characteristic of Poisson statistics, and chaotic dissipative quantum systems, which conform to Random Matrix Theory statistics. We apply this new signature of quantum chaos to random Liouvillians and boundary-driven dissipative spin-chains. The results of this thesis can help understand the long-time dynamics, steady-state properties, and spectral correlations of generic dissipative systems.
We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the physical eigenvalue spectrum can be compared. We explain the ideas of the symmetry classification of symmetric matrix spaces and how that yields Dyson's threefold and Altland-Zirnbauer's tenfold way. We also outline how the joint probability density function of the eigenvalues can be calculated from a given probability density function on the matrix space. Furthermore, we dive into the subtleties of the unfolding procedure. For this purpose, we explain the ideas of the local mean level spacing, the local level spacing distribution and the k-point correlation functions. We outline the techniques of orthogonal polynomials, determinantal and Pfaffian point processes and their related Fredholm determinants and Pfaffians as well as the supersymmetry method. Moreover, we relate the local spectral statistics to effective Lagrangians that give the relation to non-linear σ-models. In all these discussions, we also make brief excursions to non-Hermitian random matrix theory which are useful when studying open quantum systems, for instance.