Nonadiabatic quantum chaos in atom optics

Coherent dynamics of atomic matter waves in a standing-wave laser field is studied. In the dressed-state picture, wave packets of ballistic two-level atoms propagate simultaneously in two optical potentials. The probability to make a transition from one potential to another one is maximal when centroids of wave packets cross the field nodes and is given by a simple formula with the single exponent, the Landau-Zener parameter κ. If κ ? 1, the motion is essentially adiabatic. If κ ? 1, it is (almost) resonant and periodic. If κ ? 1, atom makes nonadiabatic transitions with a splitting of its wave packet at each node and strong complexification of the wave function as compared to the two other cases. This effect is referred as nonadiabatic quantum chaos. Proliferation of wave packets at κ ? 1 is shown to be connected closely with chaotic center-of-mass motion in the semiclassical theory of point-like atoms with positive values of the maximal Lyapunov exponent. The quantum-classical correspondence established is justified by the fact that the Landau-Zener parameter κ specifies the regime of the semiclassical dynamical chaos in the map simulating chaotic center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found in the behavior of the momentum and position probabilities.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

On the classical limit of quantum mechanics,fundamental graininess and chaos: Compatibility of chaos with the correspondence principle

The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess) to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become “amoeboid-like”. This deformations and their consequences can be considered as a threat to the correspondence principle unless we take into account the characteristic timescales of quantum chaos. Essentially we present an analysis of the problem similar to the one of Omnès (1994,1999), but with a simpler mathematical structure.     

Experimental study of quantum chaos with cold atoms

We investigate experimentally the quantum behavior of laser-cooled atoms in a pulsed standing wave, a system that is an atomic analog of the quantum kicked rotor. In particular, it may display the well-known phenomenon of “dynamical localization”, when the standing wave is driven periodically. Furthermore, we study some interesting properties of a quasi-periodically driven kicked rotor, which presents resonances that are shown to be sharper than the inverse of the driven–excitation duration, thus presenting a sub-Fourier character.     

Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

We study nonlinear dynamics of the fundamental cavity quantum-electrodynamical system consisting of a point-like collection of identical two-level atoms moving through a lossless single-mode cavity. Taking into account the interatomic and the atom-field quantum correlations of the first order, we go beyond the semiclassical model and derive a dynamical system that is able to describe the vacuum Rabi oscillations with atoms moving in a spatially inhomogeneous cavity field. A simple expression for the equilibrium points of this system provides a class of initial conditions for atoms and a cavity mode under which the atomic population and radiation may be trapped. In the strong-coupling limit and the rotating-wave approximation, the model is shown to be integrable with atoms moving through a resonant cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact solution is derived in an explicit form in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, that are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, out of resonance, a mechanism responsible for Hamiltonian chaos in the interaction of two-level atoms with cavity vacuum. These chaotic vacuum Rabi oscillations may be considered as a new kind of reversible spontaneous emission.     

Bifurcation continuation,chaos and chaos control in nonlinear Bloch system

A detailed analysis is undertaken to explore the stability and bifurcation pattern of the nonlinear Bloch equation known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance. After the initial analysis of the parameter space and stability region identification, we utilize the MATCONT package to analyze the detailed bifurcation scenario as the two important physical parameters γ (the normalized gain) and c (the phase of the feedback field) are varied. A variety of patterns are revealed not studied ever before. Next we explore the structure of the chaotic attractor and how the identification of unstable periodic orbit (UPO) can be utilized to control the onset of chaos.     

Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems

In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p-Rényi entropy, for all p>1. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten S₁→Sp norms.     

Choice,hull, continuity and fidelity

Continuity and fidelity (i.e., ‘path-independence’) conditions are studied for choices (picking subsets), hulls (picking supersets) and compositions of these. Examples of hulls are topological closure and convex hull, both of which are faithful. Using a continuity theorem of Sertel, a sufficient condition is given for closed convex hull, d, to be both continuous and faithful on the space of compact subsets of a locally convex topological vector space. A sufficient condition is also given for the joint continuity and fidelity of the composition, sd, of a choice, s, and d. In contrast with the Kalai and Megiddo theorem that singleton-valued maps of this form cannot be faithful and at the same time continuous on the space of finite subsets of Eⁿ, the conjunction of (upper semi-)continuity and fidelity is shown to be commonplace for choices or maps of the above form (not constrained to be singleton-valued).     

Permutations,periodicity, and chaos

Let f: I → I be a continuous function on a closed interval I. If there exists x?I which has period 3 with respect to f, then Li and Yorke [1] proved that f is chaotic in the sense that there are not only points x?I of arbitrarily large period, but also uncountably many points x?I which are not even asymptotically periodic with respect to f. By using only elementary combinatorial facts about permutations, it is shown that if there is a point x?I of period p with respect to f, where p is divisible by 3, 5, or 7, then f is chaotic. The proof is followed by a study of some related combinatorial problems in symmetric groups.     

Randomness,chaos, and structure

We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. © 2009 Wiley Periodicals, Inc. Complexity 2009     

A Quantum Version of Spectral Decomposition Theorem of dynamical systems,quantum chaos hierarchy: Ergodic,mixing and exact

In this paper we study Spectral Decomposition Theorem (Lasota and Mackey, 1985) and translate it to quantum language by means of the Wigner transform. We obtain a Quantum Version of Spectral Decomposition Theorem (QSDT) which enables us to achieve three distinct goals: First, to rank Quantum Ergodic Hierarchy levels (Castagnino and Lombardi, 2009, Gomez and Castagnino, 2014). Second, to analyze the classical limit in quantum ergodic systems and quantum mixing systems. And third, and maybe most important feature, to find a relevant and simple connection between the first three levels of Quantum Ergodic Hierarchy (ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the physical relevance of QSDT applying it to two examples: Microwave billiards (Stockmann, 1999, Stoffregen et al. 1995) and a phenomenological Gamow model type (Laura and Castagnino, 1998, Omnès, 1994).     

On chaos, transient chaos and ghosts in single population models with Allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)^∞ occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects.     

Entanglement in Percolation

We study finite and infinite entangled graphs in the bond percolationprocess in three dimensions with density p. After a discussionof the relevant definitions, the entanglement critical probabilitiesare defined. The size of the maximal entangled graph at theorigin is studied for small p, and it is shown that this graphhas radius whose tail decays at least as fast as exp(–n/logn); indeed, the logarithm may be replaced by any iterate oflogarithm for an appropriate positive constant . We explorethe question of almost sure uniqueness of the infinite maximalopen entangled graph when p is large, and we establish two relevanttheorems. We make several conjectures concerning the propertiesof entangled graphs in percolation. http://www.statslab.cam.ac.uk/\simgrg/1991 Mathematics Subject Classification: primary 60K35; secondary05C10, 57M25, 82B41, 82B43, 82D60.     

Correlation and Entanglement

In quantum mechanics, it is long recognized that there exist correlations between observables which are much stronger than the classical ones. These correlations are usually called entanglement, and cannot be accounted for by classical theory. In this paper, we will study correlations between observables in terms of covariance and the Wigner-Yanase correlation, and compare their merits in characterizing entanglement. We will show that the Wigner-Yanase correlation has some advantages over the conventional covariance.     

Multi-separation, centrifugality and centripetality imply chaos

Let be an interval. need not be compact or bounded. Let be a continuous map, and be a trajectory of with or . Then there is a point such that . A point is called a centripetal point of relative to if or , and is centrifugal if or . In this paper we prove that if there exist centripetal points of in , then has periodic points of some odd () period . In addition, we also prove that if ) is multi-separated by Fix(), or there exists a centrifugal point of in , then is turbulent and hence has periodic points of all periods.