量子混沌系统中的自旋压缩性质

量子信息是21世纪的一门新兴交叉学科,现已经成为世界关注的热门研究领域.近年来,量子计算机的研究正成为大家十分感兴趣的课题.在寻找量子计算的实现方案过程中,量子混沌引起了研究人员的极大关注,因为在量子计算机执行一些量子运算法则的过程中可能产生量子混沌,并可能破坏量子计算机的运算操作条件.近期有关量子纠缠与量子混沌之间的关系已经有所报道,而自旋压缩作为另外一种典型的纯量子效应,是否也与量子混沌之间存在一定关系呢?讨论了量子混沌研究中一个非常典型的QKT模型,研究了量子混沌系统中自旋压缩的性质.通过数值模拟计算,给出了两种不同定义的自旋压缩系数与混沌系数κ之间的变化关系,结果发现在经典相空间中,如果在规则区域占优势的情况下,当初始自旋相干态波包位于椭圆形中心时,随着时间的演化,系统压缩行为表现得非常强;而对于经典相空间中混沌区域占优势的情况下,初始自旋相干态波包同样位于椭圆形中心,则系统的压缩行为表现得非常弱,说明自旋压缩对相应的经典混沌非常敏感.通过比较还发现,采用Wineland等定义的自旋压缩系数比采用Kitagawa和Ueda等定义的自旋压缩系数对经典混沌更敏感一些,从而得出用自旋压缩可以刻画量子混沌的结论.     

Random quantum states: recent developments and applications

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.     

Chaotic versus nonchaotic stochastic dynamics in Monte Carlo simulations: a route for accurate energy differences in N-body systems

We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Quantization of Hénon's map with dissipation II — numerical results

The master equation for a quantized version of Hénon's map with dissipation derived in a preceding paper is here solved numerically for the Wigner quasi-probability density, under conditions of period doubling and classical chaos both in the transient regime and in the dissipative steady state. Approximations of the quantum map by a classical stochastic process are also considered and compared with solutions incorporating non-classical quantum fluctuations.     

Time-reversal characteristics of quantum normal diffusion: time-continuous models

In quantum map systems exhibiting normal diffusion, time-reversal characteristics converge to a universal scaling behavior which implies a prototype of irreversible quantum process [H.S. Yamada, K.S. Ikeda, Eur. Phys. J. B 85, 41 (2012)]. In the present paper, we extend the investigation of time-reversal characteristic to time-continuous quantum systems which show normal diffusion. Typical four representative models are examined, which is either deterministic or stochastic, and either has or not has the classical counterpart. Extensive numerical examinations demonstrate that three of the four models have the time-reversal characteristics obeying the same universal limit as the quantum map systems. The only nontrivial counterexample is the critical Harper model, whose time-reversal characteristics significantly deviates from the universal curve. In the critical Harper model modulated by a weak noise that does not change the original diffusion constant, time-reversal characteristic recovers the universal behavior.     

Synchronisation of chaos and its applications

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronisation, a classical paradigm of order. We survey the main theory behind complete, generalised and phase synchronisation phenomena in simple as well as complex networks and discuss applications to secure communications, parameter estimation and the anticipation of chaos.     

Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Stabilization of periodic and quasiperiodic motion in chaotic systems through changes in the system variables

This work deals with the suppression of chaos in dissipative systems that exhibit a transition from the coexistence of several periodic oscillations to deterministic chaos. The application of changes in the system variables is able to yield the prechaotic behaviour, that can be either quasiperiodic (two inconmensurate frequencies) or periodic (frequency locking), in the same way as for the original system. The performance of the method is shown by application to the two-dimensional Burgers map.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations

The deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of stochastic evolution equations after taking an average over realizations using a theorem. Examples are given that show that deterministic quantum mechanical evolution equations, obtained initially by R.P. Feynman and subsequently studied by Boghosian and Taylor IV [B.M. Boghosian, W. Taylor IV, Phys. Rev. E 57 (1998) 54. See also arXiv:quant-ph/9904035] and Meyer [D.A. Meyer, Phys. Rev. E 55 (1997) 5261], among others, are derived from a set of stochastic evolution equations. In addition, a deterministic classical evolution equation for the diffusion of monomers, similar to the second Fick law, is also obtained.     

Thermostatting the atomic spin dynamics from controlled demons

Deterministic dynamics in extended phase space of a constant temperature interacting spin system is formulated. The spin temperature is recovered through the constrained equation of motion and is in agreement with Rugh’s geometrical approach to temperature for classical Heisenberg spin systems. Detailed comparisons are investigated between state-of-the-art stochastic spin dynamics and deterministic dynamics using a chain of thermostats, for which an accelerated convergence structure is found.     

Quantum Lyapunov Exponents

We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (quantum flow) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.     

Quantum Mechanics and Stochastic Mechanics for Compatible Observables at Different Times

Bohm mechanics and Nelson stochastic mechanics are confronted with quantum mechanics in the presence of noninteracting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with quantum mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of noninteracting systems at different times.     

Irreversibility and randomness in linear response theory

Recall that the fluctuation-dissipation theorem connects the response function of a passive linear system and the spectral density of the stationary stochastic process which describes the thermal fluctuations in the system. It is shown that the classical limit (=0) of the fluctuation-dissipation theorem implies a correspondence between systems which are reversible in the sense that the energy used to drive them away from equilibrium is completely recoverable as work and processes which are deterministic in the sense of Wiener's prediction theory, while irreversible systems correspond to nondeterministic processes. This correspondence is expressed by a simple transformation between the operator kernel which determines the optimal choice of the time-dependent force and the linear predictor for the stochastic process. For quantum systems this correspondence does not hold; the fluctuations are always of the deterministic type for any finite temperature, but the system is not necessarily reversible. For irreversible systems a formula is derived for the instantaneous entropy production which is a generalization of the standard one for Markovian dynamics.     

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.     

Ehrenfest ‘phase-space’ trajectories and quantum chaos in He atom under strong,oscillating magnetic fields: an application of time-dependent quantum fluid density functional theory (TDQFDFT)

Using a dynamical signature proposed earlier from our laboratory, quantum chaos in He atom interacting with strong, oscillating magnetic fields has been studied through a comparison between the nonlinear divergence of two neighbouring Ehrenfest ‘phase-space’ (EPS) trajectories differing slightly in initial conditions and the Loschmidt echo. The dynamical EPS signature can detect quantum chaos independently of the Loschmidt echo and in agreement with the latter, even for low-lying states, in the same spirit as that of classical chaos. This time-dependent signature extends the concept of quantum chaos to systems which have no classical counterparts and brings the concept of quantum chaos closer to that of classical chaos.     

一类位于加周期分岔中的貌似混沌的随机神经放电节律的识别

识别非周期神经放电节律是混沌还是随机一直是一个重要的科学问题. 在神经起步点实验中发现了一类介于周期k和周期k+1(k=1,2)节律之间非周期自发放电节律, 其行为是长串的周期k簇和周期k+1簇的交替. 确定性理论模型Chay模型展示出了周期k和周期k+1节律的共存行为. 噪声在共存区诱发出了与实验结果类似的非周期节律, 说明该类节律是噪声引起的两类簇的跃迁. 非线性预报及其回归映射揭示该节律具有确定性机理; 将两类簇分别转换为0和1得到一个二进制序列, 对该序列进行概率分析获得了两类簇跃迁的随机机理. 这不仅说明该节律是具有确定性结构的随机节律而不是混沌, 还为深入识别现实神经系统的混沌和随机节律提供了典型示例和有效方法.     

Gibbs measure as quantum ground states

We study certain quantum spin systems which are equivalent to stochastic Ising models. We show that any translationally invariant quantum ground state is given by integration of Gibbs measure. The existence of mass gap is shown to be the same as exponentially fast convergence of stochastic models to invariant states.