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.     

Unified theory of bound and scattering molecular Rydberg states as quantum maps

Using a representation of multichannel quantum defect theory in terms of a quantum Poincaré map for bound Rydberg molecules, we apply Jung's scattering map to derive a generalized quantum map, that includes the continuum. We show that this representation not only simplifies the understanding of the method, but moreover produces considerable numerical advantages. Finally we show under what circumstances the usual semi-classical approximations yield satisfactory results. In particular we see that singularities that cause problems in semi-classics are irrelevant to the quantum map.     

Detecting nonclassical system-environment correlations by local operations

We develop a general strategy for the detection of nonclassical system-environment correlations in the initial states of an open quantum system. The method employs a dephasing map which operates locally on the open system and leads to an experimentally accessible witness for genuine quantum correlations, measuring the Hilbert-Schmidt distance between pairs of open system states. We further derive the expectation value of the witness for various random matrix ensembles modeling generic features of complex quantum systems. This expectation value is shown to be proportional to a measure for the quantum discord which reduces to the concurrence for pure initial states.     

Statistical properties of eigenvalues for an operating quantum computer with static imperfections

We investigate the transition to quantum chaos, induced by static imperfections, for an operating quantum computer that simulates efficiently a dynamical quantum system, the sawtooth map. For the different dynamical regimes of the map, we discuss the quantum chaos border induced by static imperfections by analyzing the statistical properties of the quantum computer eigenvalues. For small imperfection strengths the level spacing statistics is close to the case of quasi-integrable systems while above the border it is described by the random matrix theory. We have found that the border drops exponentially with the number of qubits, both in the ergodic and quasi-integrable dynamical regimes of the map characterized by a complex phase space structure. On the contrary, the regime with integrable map dynamics remains more stable against static imperfections since in this case the border drops only algebraically with the number of qubits. Received 19 June 2002 / Received in final form 30 September 2002 Published online 17 Decembre 2002 RID="a" ID="a"e-mail: dima@irsamc.ups-tlse.fr RID="b" ID="b"UMR 5626 du CNRS     

Translation map in quantum principal bundles

The notion of a translation map in a quantum principal bundle is introduced. A translation map is then used to prove that the cross sections of a quantum fibre bundle E((B, V, A) associated to a quantum principal bundle P (B, A) are in bijective correspondence with equivariant maps V → P, and that a quantum principal bundle is trivial if it admits a cross section which is an algebra map. The vertical automorphisms and gauge transformations of a quantum principal bundle are discussed. In particular it is shown that vertical automorphisms are in bijective correspondence with Ad_R-covariant maps A → P.     

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.     

Chirality Quantum Phase Transition in Noncommutative Dirac Oscillator

The charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic field is studied in this paper. We map the noncommutative plane to a commutative one by means of Bopp shift and study this problem on the commutative plane. We find that this model can be mapped onto a quantum optics model which contains Anti-Jaynes-Cummings (AJC) or Jaynes-Cummings (JC) interactions when a dimensionless parameter ζ (which is the function of the intensity of the magnetic field) takes values in different regimes. Furthermore, this model behaves as experiencing a chirality quantum phase transition when the dimensionless parameter ζ approaches the critical point. Several evidences of the chirality quantum phase transition are presented. We also study the non-relativistic limit of this model and find that a similar chirality quantum phase transition takes place in its non-relativistic limit.     

Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map

We report on a new approach to the calculation of Chern-Simons theory expectation values, using the mathematical underpinnings of loop quantum gravity, as well as the Duflo map, a quantization map for functions on Lie algebras. These new developments can be used in the quantum theory for certain types of black hole horizons, and they may offer new insights for loop quantum gravity, Chern-Simons theory and the theory of quantum groups.     

The classical skeleton of open quantum chaotic maps

We have studied two complementary decoherence measures, purity and fidelity, for a generic diffusive noise in two different chaotic systems (the baker map and the cat map). For both quantities, we have found classical structures in quantum mechanics-the scar functions-that are specially stable when subjected to environmental perturbations. We show that these quantum states constructed on classical invariants are the most robust significant quantum distributions in generic dissipative maps.     

Border between regular and chaotic quantum dynamics

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power-law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with [1+(q-1)(t/tau)2](1/(1-q)) (with the nonextensive entropic index q and tau depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.     

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.     

Quantum Variance and Ergodicity for the Baker's Map

We prove an Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.     

Random quantum operations

We define a natural ensemble of trace preserving, completely positive quantum maps and present algorithms to generate them at random. Spectral properties of the superoperator Φ associated with a given quantum map are investigated and a quantum analogue of the Frobenius-Perron theorem is proved. We derive a general formula for the density of eigenvalues of Φ and show the connection with the Ginibre ensemble of real non-symmetric random matrices. Numerical investigations of the spectral gap imply that a generic state of the system iterated several times by a fixed generic map converges exponentially to an invariant state.     

Exponential speedup with a single bit of quantum information: measuring the average fidelity decay

We present an efficient quantum algorithm to measure the average fidelity decay of a quantum map under perturbation using a single bit of quantum information. Our algorithm scales only as the complexity of the map under investigation. Thus for those maps admitting an efficient gate decomposition, it provides an exponential speedup over known classical procedures. Fidelity decay is important in the study of complex dynamical systems, where it is conjectured to be a signature of eigenvector statistics. Our result also illustrates the role of chaos in the process of decoherence.     

The Scattering Matrix and Associated Formulas in Hamiltonian Mechanics

We prove two new identities in scattering theory in Hamiltonian mechanics and discuss the analogy between these identities and their counterparts in quantum scattering theory. These identities involve the Poincaré scattering map, which is analogous to the scattering matrix. The first of our identities states that the Calabi invariant of the Poincaré scattering map can be expressed as the regularised phase space volume. This is analogous to the Birman-Krein formula. The second identity relates the Poincaré scattering map to the total time delay and is analogous to the Eisenbud-Wigner formula.     

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L ² functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.