Quantization of the kicked rotator with dissipation

The effect of dissipation on a quantum system exhibiting chaos in its classical limit is studied by coupling the kicked quantum rotator to a reservoir with angular momentum exchange. A master equation is derived which maps the density matrix from one kick to the subsequent one. Several limiting cases are investigated. The limits of 0 and of vanishing dissipation serve as tests of consistency, in reproducing the maps of the classical kicked damped rotator and of the kicked quantum rotator, respectively. In the limit of strong dissipation the classical map reduces to a circle map. A quantum map corresponding to the circle map is therefore obtained in this limit. In the limit of infinite dissipation the density matrix becomes independent of the initial condition after a single application of the map, allowing for a simple analytical solution for the density matrix. In the semi-classical limit the quantum map reduces to a classical map with quantum mechanically determined classical noise terms, which are evaluated. For sufficiently small dissipation the physical character of the leading quantum corrections changes. Quantum mechanical interference effects then render the Wigner distribution negative in some parts of phase space and prevent its interpretation in classical terms. Numerical results will be presented in a subsequent paper.     

Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator

The Hamiltonian H = 2πp + V(θ) Σ^∞_−∞ δ(t − n)(0 θ < 2π) is solved exactly, classically and quantally; the so lutions depend strongly on . There is no classical chaos and the phase cylinder p, θ is filled with invariant curves, which are finite loops around the cylinder if is sufficiently irrational and are translates of the infinitely long p axis if is rational. Quantal quasi-energy states correspond exactly to these invariant curves: localized in p and extended in θ if is sufficiently irrational, and extended in p and localized in θ if is rational. For a classical or quantal initial pure-momentum state, the energy at time t = n grows as n² if is rational (resonance) and remains bounded if is sufficiently irrational (non-resonance). If is very nearly rational (marginal resonance), the energy may grow as n^λ where λ is expressed in terms of exponents describing the irrationality of and the continuity class of V(θ). If the value of is uncertain, ensemble-averaging over shows that the energy grows ultimately as n, i.e. diffusively, as though under random impulses.     

Wave packets and Bohmian mechanics in the kicked rotator

Quantum-mechanical kicked-rotator wave packets are studied using Bohmian mechanics and a novel numerical approach. A method for extending the results into the classical regime is developed. A clear physical picture of packet behavior, including a new expression for packet spread times, emerges.     

Statistical properties of a dissipative kicked system: Critical exponents and scaling invariance

A new universal empirical function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal empirical function describes remarkably well a phase transition from limited to unlimited growth of the average action.     

Dynamics of rotator chain with dissipative boundary

We study the deternfinistic dynanfics of rotator chain with purely mechanical driving on the boundary by stability analysis and numerical sinmlation. Globally synchronous rotation, clustered synchronous rotation, and split synchronous rotation states are identified. In particular, we find that the single-peaked wariance distribution of angular momenta is the consequence of the deterministic dynamics. As a result, the operational definition of temperature used in the previous studies on rotator chain should be revisited.     

The periodically kicked rotator: Recurrence and/or energy growth

We explore the properties of the quantum kicked rotator, its classical equivalent being the standard map. Its behavior, as found by computer studies, depends very much on the strength of the external forcing. At low strength it is seemingly recurrent in the sense of Hogg and Huberman. However, its energy increases with time at large forcings. For quantum systems, a unitary map defines the evolution over one period of time. The spectrum of this map in an infinite space does not seem to change continuously when one approaches the ratio of the frequencies of the external and of the unperturbed system by rational approximations of the golden mean.     

量子Harper模型的量子计算鲁棒性与耗散退相干

运用量子轨迹和量子Monte Carlo仿真的方法，研究耗散退相干对周期驱动的量子Harper (quantum kicked Harper, QKH)模型量子计算的影响.数值仿真结果表明，一定强度的耗散干扰将破坏QKH特征状态的动态局域化以及相空间的随机网结构.以相位阻尼信道噪声模型为例分析了保真度的衰减规律以及可信计算时间尺度.与静态干扰相比，在干扰强度小于某一阈值时，耗散干扰下的可信计算时间尺度随量子比特的增加而快速下降；而在干扰强度大于该阈值时，静态干扰下的可信计算时间尺度下降更快.     

量子Harper模型的量子计算鲁棒性与耗散退相干

运用量子轨迹和量子Monte Carlo仿真的方法,研究耗散退相干对周期驱动的量子Harper (quantum kicked Harper, QKH)模型量子计算的影响.数值仿真结果表明,一定强度的耗散干扰将破坏QKH特征状态的动态局域化以及相空间的随机网结构.以相位阻尼信道噪声模型为例分析了保真度的衰减规律以及可信计算时间尺度.与静态干扰相比,在干扰强度小于某一阈值时,耗散干扰下的可信计算时间尺度随量子比特的增加而快速下降;而在干扰强度大于该阈值时,静态干扰下的可信计算时间尺度下降更快. 关键词： 量子计算 量子Harper模型 主方程 量子Monte Carlo方法     

Growth of ordered domains beyond a dynamic instability in dissipative systems

We furnish evidence supporting our conjecture that the growth of structure which results at the onset of a center manifold in driven systems operating far from equilibrium can be described by a power law with a single exponent. More specifically, if the dissipative structure is contained in a center manifold and thus its stability is warranted, the linear dimensions for domains of organized spatial cells increase ast ^1/2.     

周期驱动玻色-爱因斯坦凝聚系统的棘齿效应

研究了周期脉冲驱动下的玻色-爱因斯坦凝聚体系（BEC）的动力学演化.其中着重考虑了BEC原子间的非线性相互作用对量子棘齿效应的影响.数值计算结果表明,较弱的非线性相互作用可以减弱定向动量流的强度.而较强的非线性相互作用则会使量子棘齿效应消失甚至发生反转,即系统会出现反向的定向动量流,而且随着时间的演化,动量流会表现出微弱的饱和趋势.计算还发现,高阶量子共振下系统的棘齿效应变得很不明显,而且外部驱动势的周期噪声很容易破坏体系的棘齿效应. 关键词： 玻色-爱因斯坦凝聚 量子混沌 量子共振 棘齿效应     

Transition to instability in a kicked bose-einstein condensate

A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.     

Ergodic properties of a kicked damped particle

We investigate a class of nonlinear dynamical systems describing the movement of a particle in a viscous medium under the influence of a kick force. These systems can be regarded as a generalization of the Langevin approach to Brownian motion in the sense that the fluctuating force on the particle is not Gaussian white noise but an arbitrary non-gaussian process generated by a nonlinear dynamical system. We investigate how certain properties of the force (periodicity, ergodicity, mixing property) transfer to the velocity of the particle. Moreover, the relaxation properties of the system are analysed.Address after October 1, 1989: Institut für Theoretische Physik, RWTH, D-5100, Aachen, FRG     

Leggett-Garg inequality with a kicked quantum pump

A kicked quantum nondemolition measurement is introduced, where a qubit is weakly measured by pumping current. Measurement statistics are derived for weak measurements combined with single-qubit unitary operations. These results are applied to violate a generalization of the Leggett-Garg inequality. The violation is related to the failure of the noninvasive detector assumption, and may be interpreted as either intrinsic detector backaction, or the qubit entangling the microscopic detector excitations. The results are discussed in terms of a quantum point contact kicked by a pulse generator, measuring a double quantum dot.     

Floquet states of a periodically kicked particle

We present explicit expressions for the Floquet states of a periodically kicked particle in coordinate and momentum representations. These states have been used to evaluate the energy of the particle after an arbitrary number of kicks.     

Periodically kicked hard oscillators

A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval; the topological structure of this object is generated by the finite size of the repelling set, and is therefore also intrinsic to hard oscillators.     

Localization-delocalization transition in a system of quantum kicked rotors

The quantum dynamics of atoms subjected to pairs of closely spaced delta kicks from optical potentials are shown to be quite different from the well-known paradigm of quantum chaos, the single delta-kick system. We find the unitary matrix has a new oscillating band structure corresponding to a cellular structure of phase space and observe a spectral signature of a localization-delocalization transition from one cell to several. We find that the eigenstates have localization lengths which scale with a fractional power L approximately h(-0.75) and obtain a regime of near-linear spectral variances which approximate the "critical statistics" relation summation2(L) approximately or equal to chi(L) approximately 1/2 (1-nu)L, where nu approximately 0.75 is related to the fractal classical phase-space structure. The origin of the nu approximately 0.75 exponent is analyzed.