Signatures of quantum stability in a classically chaotic system

We experimentally and numerically investigate the quantum accelerator mode dynamics of an atom optical realization of the quantum delta-kicked accelerator, whose classical dynamics are chaotic. Using a Ramsey-type experiment, we observe interference, demonstrating that quantum accelerator modes are formed coherently. We construct a link between the behavior of the evolution's fidelity and the phase space structure of a recently proposed pseudoclassical map, and thus account for the observed interference visibilities.     

How chaotic is a state of a quantum system?

A concept related to the entropy is studied. Let A and B be two density matrices, with eigenvalues a₁, a₂,… and b₁, b₂,…, arranged in decreasing order and repeated according to multiplicity. Then A is said to be “more mixed”, or “more chaotic”, than B, if a₁?b₁, a₁+a₂?b₁+b₂,…,a₁+…+a_m?b₁+…+b_m,…; It turns out that if A is more mixed than B, then the entropy of A is larger than the entropy of B. However, more generally, let v be an arbitrary concave function, ?0, and vanishing at 0. Then, if A is more mixed than B, trv(A)?trv(B). It is shown that also the converse is true. Furthermore, a variety of other characterizations of the relation “A is more mixed than B” is obtained, and several applications to quantum statistical mechanics are given.     

A super-Ohmic energy absorption in driven quantum chaotic systems

We consider energy absorption by driven chaotic systems of the symplectic symmetry class. According to our analytical perturbative calculation, at the initial stage of evolution the energy growth with time can be faster than linear. This appears to be an analog of weak anti-localization in disordered systems with spin-orbit interaction. Our analytical result is also confirmed by numerical calculations for the symplectic quantum kicked rotor.     

Influence of phase-space localization on the energy diffusion in a quantum chaotic billiard

The quantum dynamics of a chaotic billiard with moving boundary is considered in this paper. We found a shape parameter Hamiltonian expansion, which enables us to obtain the spectrum of the deformed billiard for deformations so large as the characteristic wavelength. Then, for a specified time-dependent shape variation, the quantum dynamics of a particle inside the billiard is integrated directly. In particular, the dispersion of the energy is studied in the Bunimovich stadium billiard with oscillating boundary. The results showed that the distribution of energy spreads diffusively for the first oscillations of the boundary (=2Dt). We studied the diffusion constant D as a function of the boundary velocity and found differences with theoretical predictions based on random matrix theory. By extracting highly phase-space localized structures from the spectrum, previous differences were reduced significantly. This fact provides numerical evidence of the influence of phase-space localization on the quantum diffusion of a chaotic system.     

On the dynamics of a quantum system which is classically chaotic

We investigate the dynamics of one anisotropic spin in an external time-dependent magnetic field. The classical dynamics of the system is nonintegrable (and very similar to the standard map). We present results on this model for a quantum spin (i.e. for finite values of the spin lengthS). In particular we discuss the semiclassical regime,S1, using the concept of Wigner functions to define a suitable probability distribution. In regular regions of phase space the time evolution of the probability distribution shows an algebraic decay of correlations as in quantum mechanics. In chaotic regions of phase space it is characterised by a positive Lyapunov exponent which depends onS. In these regions semiclassical trajectories coincide with classical ones fort <₀ where ₀InS.     

Survival probability and saturation energy in periodically driven quantum chaotic systems

We study characteristics of the steady state of a random-matrix model with periodical pumping, where the energy increase saturates by quantum localization. We study the dynamics by making use of the survival probability. We found that Floquet eigenstates are separated into the localized and extended states, and the former governs the dynamics.     

Sensitivity of energy eigenstates to perturbation in quantum integrable and chaotic systems

We study the sensitivity of energy eigenstates to small perturbation in quantum integrable and chaotic systems.It is shown that the distribution of rescaled components of perturbed states in unperturbed basis exhibits qualitative difference in these two types of systems:being close to the Gaussian form in quantum chaotic systems,while,far from the Gaussian form in integrable systems.     

The effect of measurement on the quantum features of a chaotic system

We investigate the quantum dynamics of a periodically kicked nonlinear spin system which exhibits regular and chaotic dynamics in the classical regime. The quantum behaviour is characterised by the evolving eigenvalue distributions for the angular momentum components and the features, including recurrences in the quantum means and the presence of quantum tunneling, are discussed. We employ the evolution operator eigenvalue distribution to prove that coherent quantum tunneling occurs between the fixed points in the regular regions of phase space. Continual quantum measurement is included in the model: the classical dynamics are unchanged but a destruction of coherences occurs in the quantum system. Recurrences in the means are destroyed and quantum tunneling is suppressed by measurement, a manifestation of the quantum Zeno effect.     

A quantum transition from localized to extended states in a classically chaotic system

We study the quantum dynamics of a particle in a one dimensional triangular well under a monochromatic perturbation. Though the classical dynamics is described by the Standard Map, the quantum motion is not always localized. At a certain threshold field intensity, a transition takes place, from a, regime of power-localized quasi-energy eigenstates to one of extended states.     

Steady-state relation of a two-level system strongly coupled to a many-body quantum chaotic environment

We study the long-time average of the reduced density matrix(RDM) of a two-level system as the central system, which is locally coupled to a many-body quantum chaotic system as the environment,under an overall Schr?dinger evolution. A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain, as well as a mixed-field Ising chain.     

Decoherence induced by magnetic impurities in a quantum hall system

Scattering by magnetic impurities is known to destroy coherence of electron motion in metals and semiconductors. We investigate the decoherence introduced in a single act of electron scattering by a magnetic impurity in a quantum Hall system. For this, we introduce a fictitious nonunitary scattering matrix for electrons that reproduces the exactly calculated scattering probabilities. The strength of decoherence is identified by the deviation of eigenvalues of the product from unity. Using the fictitious scattering matrix, we estimate the width of the metallic region at the quantum Hall effect inter-plateau transition and its dependence on the exchange coupling strength and the degree of polarization of magnetic impurities.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

Disappearance of quantum chaos in coupled chaotic quantum dots

Statistical properties of the single electron levels confined in the semiconductor (InAs/GaAs, Si/SiO₂) double quantum dots (DQDs) are considered. We demonstrate that in the electronically coupled chaotic quantum dots the chaos with its level repulsion disappears and the nearest neighbor level statistics becomes Poissonian. This result is discussed in the light of the recently predicted “huge conductance peak” by R.S. Whitney et al. [Phys. Rev. Lett. 102 (2009) 186802] in the mirror symmetric DQDs.     

Diffusive energy growth in classical and quantum driven oscillators

We study the long-time stability of oscillators driven by time-dependent forces originating from dynamical systems with varying degrees of randomness. The asymptotic energy growth is related to ergodic properties of the dynamical system: when the autocorrelation of the force decays sufficiently fast one typically obtains linear diffusive growth of the energy. For a system with good mixing properties we obtain a stronger result in the form of a central limit theorem. If the autocorrelation decays slowly or does not decay, the behavior can depend on subtle properties of the particular model. We study this dependence in detail for a family of quasiperiodic forces. The solution involves the analysis of a small-denominator problem that can be treated by fairly elementary methods. In the special case of a periodic force the quantum stability problem can be expressed in terms of spectral properties of the Floquet operator. In the presence of resonances the spectrum is absolutely continuous. We find explicitly the eigenvalues and eigenfunctions for the nonresonant case.