Stochastic quantum entanglements give a basic problem in the dynamics of the coherence effect and the randomness of the dynamics. The paper examines the entanglement evolution in stochastic partial differential equations (SPDEs) in terms of dynamical chaos theory. We solve a family of SPDEs modeling quantum systems under Gaussian and Lévy noise, both in the weak and strong coupling limits. Spectral spatial discretization and stochastic Runge-Kutta time integration allow the calculation of numerically accurate solutions and the reconstruction of density matrices and calculation of entanglement measures, such as von Neumann entropy and concurrence. Measurement of chaotic properties is done by maximal Lyapunov exponents, Poincaré-type diagnostics, and sensitivity to initial conditions measures of high-dimensional stochastic dynamics. Findings indicate that stochastic forcing is much more efficient to promote entanglement and weak noise spin-offs non-zero concurrence in more than 65 percent of realizations, and high-noise regimes increase von Neumann entropy means more than 200 percent compared to deterministic baselines. Entanglement amplification is strongly related to positive Lyapunov exponents (Γ≈0.58), and high-chaos flows have oscillatory entanglement with peak-to-peak variations of 45%. Such results define a quantitative correlation between stochastic chaos and quantum correlations, and it is possible to note that the noise-induced instabilities can serve as a constructive mechanism creation of entanglements. The work gives a solid framework for studying quantum systems with stochastic perturbations and has insight that is applicable in quantum information processing, complex network dynamics, and semiclassical quantum-chaotic systems.
Stochastic quantum entanglements give a basic problem in the dynamics of the coherence effect and the randomness of the dynamics. The paper examines the entanglement evolution in stochastic partial differential equations (SPDEs) in terms of dynamical chaos theory. We solve a family of SPDEs modeling quantum systems under Gaussian and Lévy noise, both in the weak and strong coupling limits. Spectral spatial discretization and stochastic Runge-Kutta time integration allow the calculation of numerically accurate solutions and the reconstruction of density matrices and calculation of entanglement measures, such as von Neumann entropy and concurrence. Measurement of chaotic properties is done by maximal Lyapunov exponents, Poincaré-type diagnostics, and sensitivity to initial conditions measures of high-dimensional stochastic dynamics. Findings indicate that stochastic forcing is much more efficient to promote entanglement and weak noise spin-offs non-zero concurrence in more than 65 percent of realizations, and high-noise regimes increase von Neumann entropy means more than 200 percent compared to deterministic baselines. Entanglement amplification is strongly related to positive Lyapunov exponents (Γ≈0.58), and high-chaos flows have oscillatory entanglement with peak-to-peak variations of 45%. Such results define a quantitative correlation between stochastic chaos and quantum correlations, and it is possible to note that the noise-induced instabilities can serve as a constructive mechanism creation of entanglements. The work gives a solid framework for studying quantum systems with stochastic perturbations and has insight that is applicable in quantum information processing, complex network dynamics, and semiclassical quantum-chaotic systems.
It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
The dynamical properties of quantum entanglement in the Dicke model without rotating-wave approximation are investigated in terms of the reduced-density linear entropy. The characteristic time of decoherence process in the early-time evolution is numerically obtained and it is shown that the characteristic time decreases as the coupling parameter increases. The mean entanglement, which is defined to be averaged over time, is employed to describe the influences of both quantum phase transition and corresponding classical chaos on the behavior of entanglement. For a given energy, initial conditions are taken to be minimum uncertainty wave packets centered at regular and chaotic regions of the classical phase space. It is shown that the entanglement has a distinct change at the quantum phase transition, and that the entanglement for regular initial conditions is smaller than that for chaotic ones in the case of weak coupling, while it fluctuates with small amplitude in strong coupling and for chaotic initial conditions.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
This thesis studies randomness in quantum dissipative and chaotic dynamics. We first focus on a detailed study of the effect of noise in the Hamiltonian of a quantum system, going beyond the noise-average and characterizing higher moments. The main quantity we introduce is the Stochastic Operator Variance (SOV), which is an observable characterizing the spread of trajectories around the average evolution. Interestingly, this quantity fulfills different types of uncertainty relations and is related to quantum information scrambling through out-of-time-order correlators (OTOC). We illustrate the SOV-OTOC connection in a stochastic version of the Lipkin-Meshkov-Glick (LMG) model, which shows a positive Lyapunov exponent from an unstable saddle point. We find that under the action of the noise, this Lyapunov exponent can change sign, thus stabilizing the unstable phase of the model while destabilizing the stable one. We then study the interplay between noise and decay in a non-Hermitian Hamiltonian. The noise-average evolution follows an antidephasing master equation beyond Lindblad form. We characterize the purity dynamics and the steady states of this master equation, and study this new evolution in a stochastically driven version of the Dissipative Qubit. By characterizing its spectral and steady state properties, we find that there are three phases: the PT unbroken, PT broken, and a novel Noise Induced (NI) phase where the qubit converges to the lossy state. We further investigate the validity of our model to explain experimental data, such as the residual damping rate of the PT unbroken phase. In the last Chapter, we study randomness as a model for chaotic dynamics. In particular, leveraging a Wigner-like surmise for the k-th neighbor level spacing distribution, we compute analytically the k-th neighbor Spectral Form Factor (knSFF), which characterizes the contribution of the k-th neighbor spacings to the Spectral Form Factor (SFF). We study the properties of the individual knSFF and characterize their role in building the universal ramp of the SFF. Interestingly, we find that the very short-range and very long-range spacings are the ones that contribute the most to the extent of the ramp. We finish by discussing our results and possible pathways for further research.
We study two classes of quantum phenomena associated with classical chaos in a variety of quantum models: (i) dynamical localization and its extension and generalization to interacting few- and many-body systems and (ii) quantum exponential divergences in high-order correlators and other diagnostics of quantum chaos. Dynamical localization (DL) is a subtle phenomenon related to Anderson localization. It hinges on quantum interference and is typically destroyed in presence of interactions. DL often manifests as a failure of a driven system to heat up, violating the foundations of statistical physics. Kicked rotor (KR) is a prototypical chaotic classical model that exhibits linear energy growth with time. The quantum kicked rotor (QKR) features DL instead: its energy saturates. Multiple attempts of many-body generalizations faced difficulties in preserving DL. Recently, DL was shown in a special integrable many-body model. We study non-integrable models of few- and many-body QKR-like systems and provide direct evidence that DL can persist there. In addition, we show how a novel related concept of localization landscape can be applied to study transport in rippled channels. Out-of-time-ordered correlator (OTOC) was proposed as an indicator of quantum chaos, since in the semiclassical limit, this correlator's possible exponential growth rate (CGR) resembles the classical Lyapunov exponent (LE). We show that the CGR in QKR is related, but distinct from the LE in KR. We also show a singularity in the OTOC at the Ehrenfest time tᴱ due to a delay in the onset of quantum interference. Next, we study scaling of OTOC beyond tᴱ. We then explore how the OTOC-based approach to quantum chaos relates to the random-matrix-theoretical description by introducing an operator we dub the Lyapunovian. Its level statistics is calculated for quantum stadium billiard, a seminal model of quantum chaos, and aligns perfectly with the Wigner-Dyson surmise. In the semiclassical limit, the Lyapunovian reduces to the matrix of uncorrelated finite-time Lyapunov exponents, connecting the CGR at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference. Finally, we consider quantum polygonal billiards: their classical counterparts are non-chaotic. We show exponential growth of the OTOCs in these systems, sharply contrasted with the classical behavior even before quantum interference develops.
We study the real time quantum dynamics of a matrix model consisting two bosonic fields on the fuzzy sphere using the Gaussian state approximation. Starting from the Hamiltonian formulation and using Wick's theorem, we derive a closed set of coupled nonlinear differential equations governing the time evolution of the one- and two-point correlation functions. Thermal equation of state is found by maximizing the von Neumann entropy over Gaussian states and solving algebraic self-consistency equation(s) leading to a complete determination of the symplectic spectrum of the covariance matrix. We identify near thermal initial conditions and use them to solve the equations of motion and employ our findings to probe chaos by calculating the largest Lyapunov exponent at various temperatures. Our results demonstrate that the latter tends to zero at a finite temperature indicating that the quantum dynamics respect the Maldacena,Shenker,Stanford bound across all temperatures, while approaching toward the classically chaotic regime at high temperatures. Finally, we examine the entanglement dynamics of the model in real-time by considering a sequence of bipartitions of the Hilbert space and computing the entanglement entropy and clearly exhibit the fast scrambling features that emerge in due detail.