Random Behaviour in Quantum Chaos

Matrix Element Randomness, Entanglement, and Quantum Chaos

We demonstrate the connection between an operator's matrix element distribution and entangling power via numerical simulations of random, pseudo-random, and quantum chaotic operators. Creating operators with a random distribution of matrix elements is more difficult than creating operators that reproduce other statistical properties of random matrices. Thus, operators that fulfill many random matrix statistical properties may not generate states of high multi-partite entanglement. To quantify the randomness of various statistical distributions and, by extension, entangling power, we use proper…

Chaos in the Quantum Double Well Oscillator: The Ehrenfest View Revisited

We treat the double well quantum oscillator from the standpoint of the Ehrenfest equation but in a manner different from Pattanayak and Schieve. We show that for short times there can be chaotic motion due to quantum fluctuations, but over sufficiently long times the behaviour is normal.

Entanglement, randomness and chaos

Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the d…

Quantum non-Markovian behavior at the chaos border

In this work we study the non-Markovian behaviour of a qubit coupled to an environment in which the corresponding classical dynamics change from integrable to chaotic. We show that in the transition region, where the dynamics has both regular islands and chaotic areas, the average non-Markovian behaviour is enhanced to values even larger than in the regular regime. This effect can be related to the non-Markovian behaviour as a function of the the initial state of the environment, where maxima are attained at the regions dividing separate areas in classical phase space, particularly at the bord…

玻色子相互作用下混沌相中密度关联的动力学行为

Dynamical Behaviour of Density Correlations Across the Chaotic Phase for Interacting Bosons

研究一维玻色-哈伯德模型中密度关联的传播特性，揭示可积极限下关联传输距离呈球面增长，混沌相出现亚球面行为，验证了长时距离依赖关联尾和关联前沿衰减增强导致的传输减慢。

The Large Time Behaviour in Quantum Field Theory and Quantum Chaos

An exact general formula for the matrix elements of the evolution operator in quantum theory is established. The formul ("ABC-formula") has the form <U(t)>=exp(At+B+C(t)). The constants A and B and the decreasing function C(t) are computed in perturbation theory. The ABC-formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory. The formula includes the higher order corrections to the well known Weisskopf-Wigner approximation and to the stochastic (van Hove) limit which are widely used in considerations of problems of radiation, decay, quantum de…

Exploring chaos in Dicke Model using ground state fidelity and Loschmidt echo

We study the quantum critical behaviour of the Dicke Hamiltonian, with finite number of atoms and explore the signature of quantum chaos using measures like the ground state fidelity and the Loschmidt echo. We show that both these quantities clearly point to the chaotic nature of the system in the super-radiant phase.

Quantum Chaotic Systems and Random Matrix Theory

This article is an introductory review of random matrix theory (RMT) and its applications, with special focus on quantum chaos. Random matrices were first used by Wigner to understand the spectra of complex nuclei from a statistical perspective. Subsequently there have been novel applications to diverse areas, e.g., atomic and molecular physics, mesoscopic and nanoscopic systems, microwave cavities, econophysics, biological sciences, communication theory. This article is designed to be accessible at the graduate and post-doctoral level.

Generalized random matrix conjecture for chaotic systems

The eigenvalues of quantum chaotic systems have been conjectured to follow, in the large energy limit, the statistical distribution of eigenvalues of random ensembles of matrices of size $N\rightarrow\infty$. Here we provide semiclassical arguments that extend the validity of this correspondence to finite energies. We conjecture that the spectrum of a generic fully chaotic system without time-reversal symmetry has, around some large but finite energy $E$, the same statistical properties as the Circular Unitary Ensemble of random matrices of dimension $N_{\rm eff} = \tH / \sqrt{24 d_1}$, where …

A hypothesis concerning "quantal" Hilbert space criterion of chaos in nonlinear dynamical systems

Based on the Hilbert space approach to the theory of nonlinear dynamical systems developed by the author a hypothesis is formulated concerning the "quantal" criterion for classical ordinary differential systems to exhibit chaotic behaviour.

Fidelity and level correlations in the transition from regularity to chaos

Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.

Chaos and random matrices in supersymmetric SYK

We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss …

Semiclassical behaviour of quantum eigenstates

Given a quantum Hamiltonian, we explain how the dynamical properties of the underlying classical system affect the behaviour of quantum eigenstates in the semi-classical limit. We study this problem via the notion of semiclassical measures. We mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. We recall standard tools from microlocal analysis and from dynamical systems. We show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.

量子混沌与数字量子模拟中的通用Trotter化行为

Quantum Chaos and Universal Trotterisation Behaviours in Digital Quantum Simulations

本研究通过实验可实现的Trotter化模型，揭示了量子混沌动态导致Trotter化性能崩溃，并展示了预阈值区域中不同准周期性性能行为的统一图景。

基于量子噪声随机相位反馈的具有高斯不变分布的混沌

Chaos with Gaussian invariant distribution by quantum-noise random phase feedback

本研究提出基于量子噪声的随机相位反馈方法，生成具有高斯不变分布的高质量混沌激光，可用于超快随机数生成和安全通信。

混沌、局域与可积开放量子多体系统中的谱形状因子

Spectral form factor in chaotic, localized, and integrable open quantum many-body systems

本文利用耗散谱形状因子研究开放量子多体系统的谱统计，发现混沌系统普遍呈现 Ginibre 系综的二次斜坡-平台行为，并识别出非厄米多体 Thouless 时间，通过十二个模型验证了随机矩阵理论普适性，同时分析了可积与多体局域系统的不同行为。

Semiclassical properties and chaos degree for the quantum baker's map

We study the chaotic behaviour and the quantum-classical correspondence for the baker's map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.