D. Huard, H. Kroger, G. G. Melkonyan, K. J. M. Moriarty, L. P. Nadeau
We discuss how the concept of the quantum action can be used to characterize quantum chaos. As an example we study quantum mechanics of the inverse square potential in order to test some questions related to quantum action. Quantum chaos is discussed for the 2-D hamiltonian system of harmonic oscillators with anharmonic coupling.
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We discuss the questions: How to compare quantitatively classical chaos with quantum chaos? Which one is stronger? What are the underlying physical reasons?
Dynamical formation of entanglement is studied for quantum chaotic bi-particle systems. We find that statistical properties of the Schmidt eigenvalues for strong chaos are well described by the random matrix theory of the Laguerre ensemble. This implies that entanglement formation for quantum chaos has universal properties, and does not depend on specific aspects of the systems.
We shortly review the progress in the domain of deterministic chaos for quantum dynamical systems. With the appropriately extended definition of quantum Lyapunov exponent we analyze various quantum dynamical maps. It is argued that, within Quantum Mechanics, irregular evolution for properly chosen observables can coexist with regular and predictable evolution of states.
What We Talk About When We Talk About Dissipative Quantum Chaos
本文探讨了耗散量子混沌理论的核心问题，介绍了其基于开放量子动力学的谱性质方法，并总结了该领域近年来的发展和基础理论。
Achieving full control of the time-evolution of a many-body quantum system is currently a major goal in physics. In this work we investigate the different ways in which the controllability of a quantum system can be influenced by its complexity, or even its chaotic properties. By using optimal control theory, we are able to derive the control fields necessary to drive various physical processes in a spin chain. Then, we study the spectral properties of such fields and how they relate to different aspects of the system complexity. We find that the spectral bandwidth of the fields is, quite gene…
Chaos and Quantum Tunneling
本文探讨了非完全可积也不完全混沌的哈密顿系统中混沌与量子隧穿的关系，分析了混沌辅助和共振辅助隧穿现象，并讨论了混沌对隧穿概率的增强机制。
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A statistical analysis of the prime numbers indicates possible traces of quantum chaos. We have computed the nearest neighbor spacing distribution, number variance, skewness, and excess for sequences of the first N primes for various values of N. All four statistical measures clearly show a transition from random matrix statistics at small N toward Poisson statistics at large N. In addition, the number variance saturates at large lengths as is common for eigenvalue sequences. This data can be given a physical interpretation if the primes are thought of as eigenvalues of a quantum system whose …
Long-time average behavior of quantum correlations in a multi-qubit system, collectively modeled as a kicked top, is addressed here. The behavior of dynamical generation of quantum correlations such as entanglement, discord, concurrence, as previously studied, and Bell correlation function and tangle, as identified in this study, is a function of initially localized coherent states. Their long-time average reproduces coarse-grained classical phase space structures of the kicked top which contrast, often starkly, chaotic and regular regions. Apart from providing numerical evidence of such corre…
A new regularization - renormalization method with no explicit divergence, no counterterm, no bare parameter and no arbitrary running mass scale is discussed >. There is no difficulty of triviality and the Higgs mass in the standard model is calculated to be 138 Gev.
Interferometric Quantum Polynomial Chaos Expansion as a Generative Model for Calorimeter Shower Simulation
本研究提出干涉式量子多项式混沌展开生成模型，用于量热计簇射模拟，其表达能力随电路深度增长，训练后模型违反经典生成模型的贝尔界限，可消除光滑生成器的尾部相关性限制。
We study the correction of errors that have accumulated in an entangled state of spins as a result of unknown local variations in the Zeeman energy (B) and spin-spin interaction energy (J). A non-degenerate code with error rate kappa can recover the original state with high fidelity within a time kappa^1/2 / max(B,J) -- independent of the number of encoded qubits. Whether the Hamiltonian is chaotic or not does not affect this time scale, but it does affect the complexity of the error-correcting code.
We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breakin…
The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more r…
We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-1/2 lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but take place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures, namely the reduced Von Neumann entropy and the concurrence, and discuss quantum entanglement sharing. Our results suggest that the behavior of th…
Coherent control of chaotic molecular systems, using laser-assisted alignment of sulphur dioxide (SO$_2$) molecules in the presence of a static electric field as an example, is considered. Conditions for which the classical version of this system is chaotic are established, and the quantum and classical analogs are shown to be in very good correspondence. It is found that the chaos present in the classical system does not impede the alignment, neither in the classical nor the quantum system. Using the results of numerical calculations, we suggest that laser-assisted alignment is stable against…
Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient cha…
Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with displaceable wall ("piston"). The motion is completely chaotic but with small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity, respectively. For $g$ we use a resistor network ca…
We test the applicability of the Bohigas conjecture to systems whose Hamiltonian is not written as a closed form analytic expression. A class of such Hamiltonians is created and appear to violate the conjecture. Numerical methods are employed to find the spectra of a two-dimensional, classically chaotic "billiard" system whose Hamiltonian is in this class. We find that the spectral fluctuations are not in agreement with the conjecture.
In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born's rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite of that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Ou…
We study nonlinear phenomena of bistability and chaos at a level of few quanta. For this purpose we consider a single-mode dissipative oscillator with strong Kerr nonlinearity with respect to dissipation rate driven by a monochromatic force as well as by a train of Gaussian pulses. The quantum effects and decoherence in oscillatory mode are investigated on the framework of the purity of states and the Wigner functions calculated from the master equation. We demonstrate the quantum chaotic regime by means of a comparison between the contour plots of the Wigner functions and the strange attracto…
A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal c…
In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit, N->infinity, the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.
Recent development of the resonance scattering theory with a transient from the regular to chaotic internal dynamics inspires renewed interest to the problem of the elastic enhancement phenomenon. We reexamine the question what the experimentally observed value of the elastic enhancement factor can tell us about the character of dynamics of the intermediate system. Noting first a remarkable connection of this factor with the time delays variance in the case of the standard Gaussian ensembles we then prove the universal nature of such a relation. This reduces our problem to that of calculation …
Simulating quantum chaos without chaos
本文提出一类计算上难以与GUE区分、但不具传统混沌特征的“伪混沌”哈密顿量，并给出高效量子模拟算法，揭示量子混沌中计算与信息论视角的分离。
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