Benatti-Narnhofer-Sewell quantum Arnol'd cat map: Physical interpretation

A Tomita (standard) Hilbert space representation of a physical system with a toral phase space is presented, both for the classical and for the quantal case. The dynamics of such a system, subject to a regular series of linear kicks, is derived in a discrete-time form, and is shown to be identical to the Benatti-Narnhofer-Sewell quantum Arnol'd cat map in the quantum case. It is shown that this quantum system is chaotic in the sense of having exponentially decaying autocorrelation functions.     

Quantum accelerator modes from the Farey tree

We show that mode locking finds a purely quantum nondissipative counterpart in atom-optical quantum accelerator modes. These modes are formed by exposing cold atoms to periodic kicks in the direction of the gravitational field. They are anchored to generalized Arnol'd tongues, parameter regions where driven nonlinear classical systems exhibit mode locking. A hierarchy for the rational numbers known as the Farey tree provides an ordering of the Arnol'd tongues and hence of experimentally observed accelerator modes.     

Arnol’d diffusion in a system with 2.5 degrees of freedom: Classical and quantum mechanical approaches

Arnol’d diffusion, a universal phenomenon in nonlinear dynamics, is analyzed for a model system with 2.5 degrees of freedom. Only the three primary order resonances are taken into account, and the results obtained by using classical and quantum mechanical approaches are compared. It is shown that the parameter dependence of the rate of quantum Arnol’d diffusion is similar to the classical one, but the quantum diffusion coefficient is smaller by approximately an order of magnitude. It is found that the existence of a threshold with respect to perturbation parameters, pointed out earlier, is not an indispensable feature of quantum Arnol’d diffusion. It is shown that a quantum system with weakly overlapping resonances can exhibit mixed dynamics that has no classical counterpart (diffusion along a resonance superimposed by oscillations across the overlapped resonances).     

Directional emission from whispering-gallery modes in deformed fused-silica microspheres

Highly directional emissions from whispering-gallery modes are demonstrated in deformed nonaxisymmetric fused-silica microspheres. Ray trajectory analysis indicates that ray dynamics in a three-dimensional reso-nator differs qualitatively from that of a two-dimensional system, with Arnol'd diffusion playing an essential role in the three-dimensional system.     

Influence of experimental resolution on the quantum-to-classical transition in the quartic oscillator

The quantum-classical transition problem is investigated for the quartic oscillator coupled to a thermal reservoir. We show for this model that the combination of relevant diffusion, classical action (represented by the amplitude of an initial coherent state) and the experimental uncertainties is necessary to achieve the classical regime. In order to study the role of limited resolutions of measurement apparatuses on the correspondence between the quantum and classical dynamics, we consider experimental errors due the preparation of the initial state of the quartic oscillator and the inaccuracies in the time measurements. A quantum break time depending on the diffusion constant, the amplitude of the initial coherent state and the inaccuracy of measurements is defined. We found, for this model, a regime where the increasing of diffusion does not anticipate classicality. In such regime, there is a minimum value for the classical action associated to classical behavior of the system.     

Dynamics of an atom coupled through a parametric frequency converter with quantum and classical fields

In this communication we introduce a model of a two-level system which represents the interaction between an atom and external quantum and classical electric fields through a parametric frequency converter. We show that the system can be reduced to an effective Jaynes-Cummings model by adequate adjustment of the field coupling in the frequency converter. The atom dynamics and the atom-quantum field entanglement are then shown to be controlled through the classical field.     

Dynamics of quantum discord in two Tavis-Cummings models with classical driving fields

We investigate the dynamics of quantum discord in a system consisting of two Tavis-Cummings models, each of which contains two atoms driven by a classical field. We compare the dynamics of quantum discord for the system with that of entanglement and show that quantum discord vanishes only asymptotically although entanglement disappears suddenly during the time evolution. Furthermore, we examine the influence of the initial states and the classical field on the discord dynamics and find that the value of quantum discord can be improved by adjusting the classical driving field. Finally, the quantum discord of two atoms in dissipative cavity is also discussed.     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.     

Exact c-number representation of non-Markovian quantum dissipation

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr?dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.     

Classical and quantum chaos in atom optics

The interaction of an atom with an electro-magnetic field is discussed in the presence of a time periodic external modulating force. It is explained that a control on atom by electro-magnetic fields helps to design the quantum analog of classical optical systems. In these atom optical systems chaos may appear at the onset of external fields. The classical and quantum chaotic dynamics is discussed, in particular in an atom optics Fermi accelerator. It is found that the quantum dynamics exhibits dynamical localization and quantum recurrences.     

Anomalous diffusion and dynamical localization in polygonal billiards

We study numerically classical and quantum dynamics of a piecewise parabolic area preserving map on a cylinder which emerges from the bounce map of elongated triangular billiards. The classical map exhibits anomalous diffusion. Quantization of the same map results in a system with dynamical localization and pure point spectrum.     

Quantum coherence,evolution of the Wigner function,and transition from quantum to classical dynamics for a chaotic system

The paper deals with dynamics of a quantum chaotic system under influence of an environment. The effect of an environment is known to destroy the quantum coherence and can convert the quantum dynamics of a system to classical. We use a semiclassical technique for studying the process of decoherence. The condition for transition from quantum to classical dynamics is obtained in general form and checked numerically for a particular chaotic system, known as quantum the standard map on a torus. The relevance of the obtained results to the problem of correspondence between quantum and classical mechanics is briefly discussed. (c) 1996 American Institute of Physics.     

Quantum to Classical Transition by a Classically Small Interaction

We investigate the quantum-classical transition of a kicked rotor (KR) under perturbation by a second one. The influence of such a chaotic KR makes decoherence of the first one, resulting in the emergence of classical diffusion from its quantum dynamics. Such quantum-classical transition persists by decreasing the effective Planck's constant \hbar, and at the same time, decreasing the mass of the second KR and the interaction strength proportionally. In the limit of \hbar → 0, due to vanishing small mass and interaction, the second KR has almost no effect on the classical dynamics of the first one. We demonstrate this via two different coupling potentials.     

Classical harmonic vibrations with micro amplitudes and low frequencies monitored by quantum entanglement

We study the entanglement dynamics of the two two-level atoms coupled with a single-mode polarized cavity field after incorporating the decoupled atomic centers of mass classical harmonic vibrations with micro amplitudes and low frequencies. We discover a new quantum mechanical measurement effect for the entanglement dynamics. We propose a quantitative vibrant factor to modify the concurrence of the two atomic states. When the vibrant frequencies are very low, we obtain that: (1) the factor depends on the relative vibrant displacements and the initial phases rather than the absolute amplitudes, and reduces the concurrence to three orders of magnitude; (2) the concurrence increases with the increase of the initial phases; (3) the frequency of the harmonic vibration can be obtained by measuring the maximal value of the concurrence during a small measurement time. These results indicate that the extremely weak classical harmonic vibrations can be monitored by the entanglement of quantum states. The effect reported in the paper always works well as long as the internal degrees of freedom of the system (regardless of unitary evolution or non-unitary evolution with time) are decoupled with the external classical harmonic vibrations of atomic centers of mass.     

Generalized quantum measurement

We overcome one of Bell's objections to ‘quantum measurement’ by generalizing the definition to include systems outside the laboratory. According to this definition, a generalized quantum measurement takes place when the value of a classical variable is influenced significantly by an earlier state of a quantum system. A generalized quantum measurement can then take place in equilibrium systems, provided the classical motion is chaotic. This Letter deals with this classical aspect of quantum measurement, assuming that the Heisenberg cut between the quantum dynamics and the classical dynamics is made at a very small scale. For simplicity, a gas with collisions is modelled by an “Arnold gas”.     

Quantum recurrences in driven power-law potentials

The recurrence phenomena of an initially well-localized wave packet are studied in periodically driven power-law potentials. For our general study we divide the potentials in two kinds, namely tightly binding and loosely binding potentials. In the presence of an external periodically modulating force, these potentials may exhibit classical and quantum chaos. We show that in the dynamics of a quantum wave packet in the modulated power law potentials quantum recurrences occur at various time scales. We develop general analytical relations for these times and discuss their parametric dependence.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Dynamics of quantum discord in a two-qubit system under classical noise

We study the quantum discord dynamics of two noninteracting qubits that are, respectively, subject to classical noise. The results show that the dynamics of quantum discord are dependent on both the coupling between the qubits and classical noise, and the average switching rate of the classical noise. In the weak-coupling Markovian region, quantum discord exhibits exponent decay without revival, and can be well protected by increasing the average classical noise switching rate. While in the strong-coupling non-Markovian region, quantum discord reveals slowly decayed oscillations with quick revival by decreasing the average switching rate of the classical noise. Thus, our results provide a new method of protecting quantum discord in a two-qubit system by controlling the coupling between the qubits and classical noise, and the average switching rate of the classical noise.     

Geometric phase of a qubit driven by a phase noise laser under non-Markovian dynamics

Robustness of the geometric phase (GP) with respect to the environmental effects is a basic condition for an effective quantum computation. Here, we study quantitatively the GP of a two-level atom system driven by a phase noise laser under non-Markovian dynamics in terms of different parameters involved in the whole system. We find that with the change of the damping coupling, the GP is very sensitive to its properties exhibiting long collapse and revival phenomena, which play a significant role in enhancing the stabilization and control of the system dynamics. Moreover, we show that the GP can be considered as a tool for testing and characterizing the nature of the qubit–environment coupling. Due to the significance of how a system is quantum correlated with its environment in the construction of a scalable quantum computer, the entanglement dynamics between the qubit with its environment under external classical noise is evaluated and investigated during the time evolution.