Nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects

The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.     

Persistent entanglement in two coupled squid rings in the quantum to classical transition: a quantum jumps approach

We explore the quantum–classical crossover of two coupled, identical, superconducting quantum interference device (SQUID) rings. The motivation for this work is based on a series of recent papers. In [1] we showed that the entanglement characteristics of chaotic and periodic (entrained) solutions of the Duffing oscillator differed significantly and that in the classical limit entanglement was preserved only in the chaotic-like solutions. However, Duffing oscillators are a highly idealized toy system. Motivated by a wish to explore more experimentally realizable systems, we extended our work in [2, 3] to an analysis of SQUID rings. In [3] we showed that the two systems share a common feature. That is, when the SQUID ring’s trajectories appear to follow (semi)classical orbits, entanglement persists. Our analysis in [3] was restricted to the quantum-state diffusion unraveling of the master equation – representing unit efficiency heterodyne detection (or ambi-quadrature homodyne detection). Here we show that very similar behavior occurs using the quantum jumps unraveling of the master equation. Quantum jumps represents a discontinuous photon counting measurement process. Hence, the results presented here imply that such persistent entanglement is independent of measurement process and that our results may well be quite general in nature.     

Signatures for a classical to quantum transition of a driven nonlinear nanomechanical resonator

We seek the first indications that a nanoelectromechanical system (NEMS) is entering the quantum domain as its mass and temperature are decreased. We find them by studying the transition from classical to quantum behavior of a driven nonlinear Duffing resonator. Numerical solutions of the equations of motion, operating in the bistable regime of the resonator, demonstrate that the quantum Wigner function gradually deviates from the corresponding classical phase-space probability density. These clear differences that develop due to nonlinearity can serve as experimental signatures, in the near future, that NEMS resonators are entering the quantum domain.     

Parameter scaling in the decoherent quantum-classical transition for chaotic systems

The quantum to classical transition for a system depends on many parameters, including a scale length for its action, variant Planck's over 2 pi, a measure of its coupling to the environment, D, and, for chaotic systems, the classical Lyapunov exponent, lambda. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of (Planck's over 2 pi,lambda,D) and searching for transformations that collapse this hypersurface into a function of a composite parameter zeta= Planck's over 2 pi alpha)lambda beta D gamma. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition.     

基于量子粒子群算法的自适应随机共振方法研究

为提升随机共振理论在微弱信号检测领域中的实用性,以随机共振系统参数为研究对象,提出了基于量子粒子群算法的自适应随机共振方法.首先将自适应随机共振问题转化为多参数并行寻优问题,然后分别在Langevin系统和Duffing振子系统下进行仿真实验.在Langevin系统中,将量子粒子群算法和描点法进行了寻优结果对比;在Duffing振子系统中,Duffing振子系统的寻优结果则直接与Langevin系统的寻优结果进行了对比.实验结果表明:在寻优结果和寻优效率上,基于量子粒子群算法的自适应随机共振方法要明显高于描点法;在相同条件下,Duffing振子系统的寻优结果要优于Langevin系统的寻优结果;在两种系统下,输入信号信噪比越低就越能体现出量子粒子群算法的优越性.最后还对随机共振系统参数的寻优结果进行了规律性总结.     

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits, respectively. The energy variation of the classical periodicity (τ) is also dramatic, having the special limiting case of τ→∞ at the ‘top’ of the classical motion (i.e., the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to fourth order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the ‘top’ in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.     

Transmission of classical and quantum information through a quantum memory channel with damping

We consider the transfer of classical and quantum information through a memory amplitude damping channel. Such a quantum channel is modeled as a damped harmonic oscillator, the interaction between the information carriers — a train of qubits — and the oscillator being of the Jaynes-Cummings kind. We prove that this memory channel is forgetful, so that quantum coding theorems hold for its capacities. We analyze entropic quantities relative to two uses of this channel. We show that memory effects improve the channel aptitude to transmit both classical and quantum information, and we investigate the mechanism by which memory acts in changing the channel transmission properties.     

Environmental recording of the angular momentum in quantum mechanics

We discuss the coupling of a quantum system through the angular momentum to the reservoir of quantum harmonic oscillators. In classical mechanics an observation of the oscillator trajectories allows one to determine the system's angular momentum. We discuss the quantum dynamics of the model. We show that the model of an observation of environmental coordinates can be related to some models of angular momentum measurement based on a stochastic Schrödinger equation.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

The semiclassical limit of a quantum Fermi accelerator

A classical Fermi accelerator model (FAM) is known to show chaotic behavior. The FAM is defined by a free particle bouncing elastically from two rigid walls, one fixed and the other oscillating periodically in time. The central aim of this paper is to connect the quantum and the classical solutions to the FAM in the semiclassical limit. This goal is accomplished using a finite inverted parametric oscillator (FIPO), confined to a box withfixed walls, as an alternative representation of the FAM. In the FIPO representation, an explicit correspondence between classical and quantum limits is accomplished using a Husimi representation of the quasienergy eigenfunctions.     

Interplay between classical and quantum signals: partial trace and measurement channels

We continue the study of similarities between quantum information theory and theory of classical Gaussian signals. The possibility of using quantum entropy for classical Gaussian signals was explored a long time ago. Recently we demonstrated that some basic quantum channels can be represented as linear transforms of classical Gaussian signals. Here we consider bipartite quantum systems and show that an important quantum channel given by the partial trace operation has a simple classical representation, namely, a coordinate projection of a classical “prequantum signal.” We also consider the classical signal realization of quantum channels corresponding to state transforms in the process of measurement. The latter induces a difficult interpretational problem — the output signal corresponding to one system depends on a measurement that has been done on the second system. This situation might be interpreted as a sign of quantum nonlocality, action at a distance. Although we do not exclude such a possibility, i.e., that, in complete accordance with Bell, the creation of a realistic prequantum model is impossible without action at a distance, we found another interpretation of this situation that is not related to quantum nonlocality.     

Classical formulation of quantum mechanics

The concept of quantum state is given in terms of classical probability for position in squeezed and rotated classical reference frames in phase space. Stationary states and energy levels of the quantum system are obtained in a classical formulation of quantum mechanics. The positive probability density of the harmonic oscillator position is obtained by solving a new eigenvalue equation of standard quantum mechanics instead of the Schrödinger equation. The orthogonality and completeness relations are found for the eigendistributions.     

Analytic study of a sequence of path integral approximations for simple quantum systems at low temperature

The sequence of Feynman-Trotter approximations to the thermal Feynman path integral for the simple harmonic oscillator is obtained in an easily analyzable closed form. While it converges pointwise at every non-zero temperature to the quantum thermal propagator, the sequence manifests a highly non-uniform behaviour in the zero temperature limit—every one of its elements tends toward theclassical ground state (static equilibrium). For high order elements of the sequence, there is an abrupt “collapse” from the quantum to the classical ground state with falling temperature, a phenomenon which bears a possibly misleading resemblance to a phase transition. It is shown that Feynman-Trotter sequences for many simple systems other than the harmonic oscillator also have all their elements tending to the classical static equilibrium state in the zero temperature limit.     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

The second law of thermodynamics in the quantum Brownian oscillator at an arbitrary temperature

In the classical limit no work is needed to couple a system to a bath with sufficiently weak coupling strength (or with arbitrarily finite coupling strength for a linear system) at the same temperature. In the quantum domain this may be expected to change due to system-bath entanglement. Here we show analytically that the work needed to couple a single linear oscillator with finite strength to a bath cannot be less than the work obtainable from the oscillator when it decouples from the bath. Therefore, the quantum second law holds for an arbitrary temperature. This is a generalization of the previous results for zero temperature [Ford and O'Connell, Phys. Rev. Lett. 96, 020402 (2006); Kim and Mahler, Eur. Phys. J. B 54, 405 (2006)]; in the high temperature limit we recover the classical behavior.     

Enhancement of feasibility of macroscopic quantum superposition state with the quantum Rabi-Stark model

We propose an efficient scheme to generate a macroscopical quantum superposition state with a cavity optomechanical system, which is composed of a quantum Rabi-Stark model coupling to a mechanical oscillator. In a low-energy subspace of the Rabi-Stark model, the dressed states and then the effective Hamiltonian of the system are given. Due to the coupling of the mechanical oscillator and the atom-cavity system, if the initial state of the atom-cavity system is one of the dressed states, the mechanical oscillator will evolve into a corresponding coherent state. Thus, if the initial state of the atom-cavity system is a superposition of two dressed states, a coherent state superposition of the mechanical oscillator can be generated. The quantum coherence and their distinguishable properties of the two coherent states are exhibited by Wigner distribution. We show that the Stark term can enhance significantly the feasibility and quantum coherence of the generated macroscopic quantum superposition state of the oscillator.     

Overdamping of coherently driven quantum systems

The phenomenon of overdamping, when frictional forces overwhelm the restoring force of an harmonic oscillator, is well known in classical mechanics. Analogous phenomena occur in time-dependent quantum mechanics when probability loss rates (e.g. photoionization rates) become larger than the characteristic inverse times for coherent excitation (the Rabi frequencies). These results often seem surprising, even for simple two-state quantum systems. We discuss examples of this overdamping phenomenon—when an increase of the loss rate actually produces a slower rate of probability loss and even freezes population—for a number of quantum systems: steady and pulsed two-state excitation, time evolution with adiabatic passage, simple multistate chains, and branched chains of excitation linkage. In each situation there occurs for very large damping, often unexpectedly, phenomena akin to the overdamping of the classical harmonic oscillator.