周期驱动的Harper模型的量子计算鲁棒性与量子混沌

以周期驱动的量子Harper(quantum kicked Harper, QKH)模型为例，研究复杂量子动力系统的量子计算在各种干扰下的稳定性.通过对Floquet算子本征态的统计遍历性及其Husimi函数的分析，比较随机噪声干扰和静态干扰对量子计算不同程度的影响.进一步的保真度摄动分析表明，在随机噪声干扰下保真度随系统演化呈指数衰减，而静态干扰下的保真度为高斯衰减，并通过数值计算得到了干扰下的可信计算时间尺度.与经典混沌仿真中误差使状态产生指数分离不同，量子计算对状态干扰的稳定性和仿真模型的动力学特性无关. 关键词： 量子Harper模型 量子计算 量子混沌 保真度     

Combined Effect of Classical Chaos and Quantum Resonance on Entanglement Dynamics

We use linear entropy of an exact quantum state to study the entanglement between internal electronic states and external motional states for a two-level atom held in an amplitude-modulated and tilted optical lattice.Starting from an unentangled initial state associated with the regular 'island' of classical phase space,it is demonstrated that the quantum resonance leads to entanglement generation,the chaotic parameter region results in the increase of the generation speed,and the symmetries of the initial probability distribution determine the final degree of entanglement.The entangled initial states are associated with the classical 'chaotic sea',which do not affect the final entanglement degree for the same initial symmetry.The results may be useful in engineering quantum dynamics for quantum information processing.     

Evaluation of an algebraic model for the vibrations of water,effects of a discrete symmetry

We evaluate the dynamics of an algebraic model Hamiltonian for the vibrational motion of the water molecule. We pay special attention to the effects of the discrete symmetry of order 2 of the model. For a comparison between the quantum dynamics and the classical dynamics it is necessary to desymmetrize such quantum states which are based on types of motion which come in symmetry related pairs. For the other states based on motion invariant under the symmetry operation a desymmetrization would be meaningless. The desymmetrized quantum states show a simple connection to the guiding motions of the classical dynamics which can be used for a complete assignment of the states even though the system is not integrable in the sense of Liouville and shows chaotic behaviour in large parts of the classical phase space.     

Relaxation and localization in interacting quantum maps

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems, for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but-quantally shows resonant transport, no saturation, and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.     

Chaotic Eigenfunctions in Phase Space

We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on chaotic eigenconstellations, we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g., the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.     

Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos

We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate the Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase-space-averaged multifractal dimensions, which are well described by random matrix theory in a strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.     

Generation of the ballistic particle transport in a periodic Hamiltonian system subjected to small time-dependent perturbation

The problem of the motion of an ensemble of classical particles in a periodic potential field has been considered. A method is proposed for generating directed ballistic transport by means of a perturbation oscillating in time and space. This method makes it possible to significantly reduce the perturbation intensity required to generate a particle flux. In particular, it has been shown that, even if the ensemble of particles is initially near the stable-equilibrium states, a directed flux appears at a perturbation amplitude of about 10^?2 of the potential barrier height. The flux generation mechanism is associated with the creation of global chaotic diffusion due to resonances between spatial and time oscillations of perturbation. A nonlinear pendulum is considered as an example.     

Review: Characterizing and quantifying quantum chaos with quantum tomography

We explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. The tomographic record consists of a time series of expectation values of a Hermitian operator evolving under the application of the Floquet operator of a quantum map that possesses (or lacks) time-reversal symmetry. We find that the rate of information gain, and hence the fidelity of quantum state reconstruction, depends on the symmetry class of the quantum map involved. Moreover, we find an increase in information gain and hence higher reconstruction fidelities when the Floquet maps employed increase in chaoticity. We make predictions for the information gain and show that these results are well described by random matrix theory in the fully chaotic regime. We derive analytical expressions for bounds on information gain using random matrix theory for different classes of maps and show that these bounds are realized by fully chaotic quantum systems.     

Semiclassical structure of chaotic resonance eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as variant Planck's over 2pi-->0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates [psi(variant Planck's over)] 2pi-->0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker's map, for which the probability density in position space is observed to have self-similarity properties.     

Quantum Signatures of Chaos in Adiabatic Interaction Between a Trapped Ion and a Laser Standing Wave

We study quantum motion around a classical heteroclinic point of a single trapped ion interacting with a strong laser standing wave. We construct a set of exact coherent states of the quantum system and from the exact solutions reveal that quantum signatures of chaos can be induced by the adiabatic interaction between the trapped ion and the laser standing wave, where the quantum expectation values of position and momentum correspond to the classically chaotic orbit. The chaotic region on the phase space is illustrated. The energy crossing and quantum resonance in time evolution and the exponentially increased Heisenberg uncertainty are found. The results suggest a theoretical scheme for controlling the unstable regular and chaotic motions.     

Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

热场动力学理论中的Husimi分布函数及Wehrl熵的研究

利用量子相空间技术和信息熵理论,研究了热场动力学理论中量子纯态与相应混合态的Husimi分布函数及Wehrl熵的一致性问题.结果表明,热相干态与相应混合态的Husimi分布函数及Wehrl熵完全相同,支持了热场动力学理论.且热相干态的Wehrl熵与平移因子无关,故在热相干态中,量子系统的可观测量的量子涨落及不确定关系也与平移因子无关.     

Quantum to classical correspondence for the Fermi-acceleration model

A quantum particle which is confined to the interior of a box with infinitely high but periodically oscillating walls can have an unusual semiclassical limit: For the special case of a one-dimensional linear wall motion we show that the semiclassical domain corresponds to a classical motion in phase space where the initial momentum depends on the particle's position in the box. Another result is that quantum states which correspond to classical cycle-1 fixed points have maximum stability against the boundary induced perturbation (caused by the moving wall). Higher cycle-n fixed points are calculated by numerical bookkeeping up to n = 20. The classical motion is marginally stable. We show how a slight change in the boundary condition will lead to chaotic motion.     

Classical chaos with Bose-Einstein condensates in tilted optical lattices

A widely accepted definition of "quantum chaos" is "the behavior of a quantum system whose classical limit is chaotic." The dynamics of quantum-chaotic systems is nevertheless very different from that of their classical counterparts. A fundamental reason for that is the linearity of Schr?dinger equation. In this paper, we study the quantum dynamics of an ultracold quantum degenerate gas in a tilted optical lattice and show that it displays features very close to classical chaos. We show that its phase space is organized according to the Kolmogorov-Arnold-Moser theorem.     

Floquet spectrum for two-level systems in quasiperiodic time-dependent fields

We study the time evolution ofN-level quantum systems under quasiperiodic time-dependent perturbations. The problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged Hilbert space, or equivalently of a generalized Floquet operator. We discuss criteria for the appearance of pure point as well as continuous spectrum, corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior. We discuss two types of mechanisms that lead to instability. The first one is due to near resonances, while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation. We treat explicitly an example of this type. The stability of the pure point spectrum under small perturbations is proven using KAM techniques.     

Quantization of a free particle interacting linearly with a harmonic oscillator

We investigate the quantization of a free particle coupled linearly to a harmonic oscillator. This system, whose classical counterpart has clearly separated regular and chaotic regions, provides an ideal framework for studying the quantization of mixed systems. We identify key signatures of the classically chaotic and regular portions in the quantum system by constructing Husimi distributions and investigating avoided level crossings of eigenvalues as functions of the strength and range of the interaction between the system's two components. We show, in particular, that the Husimi structure becomes mixed and delocalized as the classical dynamics becomes more chaotic.     

玻色-爱因斯坦凝聚系统的量子Fisher信息与混沌

量子Fisher信息作为经典Fisher信息的自然推广,与量子信息中的纠缠判断具有密切联系.在表现为典型量子混沌特征的受击两分量玻色-爱因斯坦凝聚系统中,研究了与经典相空间对应的纠缠和量子Fisher信息动力学性质. 结果表明,初次撞击后的系统量子态是纠缠的,与初态所处相空间中的混乱程度无关.而量子Fisher信息的动力学演化对系统初态非常敏感,当初态处于混沌区域时,量子Fisher信息值比初态处于规则区域时大.利用这种较好的量子-经典对应关系,得到量子Fisher信息可以刻画量子混沌的结论. 关键词： 量子Fisher信息 玻色-爱因斯坦凝聚 量子混沌 量子-经典对应     

Dicke模型的量子经典对应关系

非旋波近似条件下Dicke模型表现为量子混沌动力学特征.在详细考察Dicke模型经典相空间结构特点的基础上,采用经典-量子"一对多"的思想,即经典相空间中的一点对应于量子体系两个初始相干态的演化,利用对两个初态量子纠缠动力学演化取统计平均的方法,得到了与经典相空间对应非常好的量子相空间结构.数值计算结果表明:经典混沌有利地促进系统两体纠缠的产生,平均纠缠可以作为量子混沌的标识,利用平均纠缠可以得到一种较好的量子动力学与经典相空间的对应关系. 关键词： Dicke模型 非旋波近似 量子混沌 经典量子对应     

Floquet theory for short laser pulses

We develop adiabatic perturbation theory for quantum systems responding to short laser pulses, with or without a frequency chirp. Our approach rests on lifting the time-dependent Schr?dinger equation to an extended Hilbert space, then applying standard perturbational techniques to Floquet states in this extended space, and finally projecting back to the physical Hilbert space. The same strategy also allows us to construct superadiabatic bases for monitoring the quantum evolution in the course of a pulse. These bases provide a diagnostic tool for improving the efficiency of pulse-induced population transfer. The formalism is applied to the selective excitation of molecular vibrational states by chirped laser pulses, which exploit either successive single-photon resonances or a multiphoton resonance, and by a STIRAP-like process. Received: 23 June 1998 / Revised: 18 August 1998 / Accepted: 25 August 1998