Phase-space localization: Topological aspects of quantum chaos

Lebœuf, P.; Kurchan, Jorge; Feingold, Mario; Arovas, Daniel P.

doi:10.1103/physrevlett.65.3076

Cited by 149 publications

(215 citation statements)

References 16 publications

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“…It will be of great interest to clarify whether a similar universal scaling law can be found for other timedependent systems, such as the close-to-resonant dynamics of the kicked harmonic oscillator [26], or the driven Harper model [27,28]. As with the atom-optics kicked rotor, both of the latter systems may be readily realized in laboratory experiments [29,30].…”

Section: Discussionmentioning

confidence: 99%

Experimental verification of a one-parameter scaling law for the quantum and “classical” resonances of the atom-optics kicked rotor

et al. 2005

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We present experimental measurements of the mean energy in the vicinity of the first and second quantum resonances of the atom optics kicked rotor for a number of different experimental parameters. Our data is rescaled and compared with the one parameter ǫ-classical scaling function developed to describe the quantum resonance peaks. Additionally, experimental data is presented for the "classical" resonance which occurs in the limit as the kicking period goes to zero. This resonance is found to be analogous to the quantum resonances, and a similar one-parameter classical scaling function is derived, and found to match our experimental results. The width of the quantum and classical resonance peaks is compared, and their Sub-Fourier nature examined.

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Section: Discussionmentioning

confidence: 99%

Experimental verification of a one-parameter scaling law for the quantum and “classical” resonances of the atom-optics kicked rotor

et al. 2005

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“…This map has been studied extensively and develops full fledged chaos for large τ [23]. For completeness we illustrate this transition to classical chaos in Fig.…”

Section: B Nonintegrable Hamiltoniansmentioning

confidence: 99%

Entanglement sharing in one-particle states

Lakshminarayan

Subrahmanyam

2003

Phys. Rev. A

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Entanglement sharing among sites of one-particle states is considered using the measure of concurrence. These are the simplest in an hierarchy of number-specific states of many qubits and corresponds to "one-magnon" states of spins. We study the effects of onsite potentials that are both integrable and nonintegrable. In the integrable case we point to a metal-insulator transition that reflects on the way entanglement is shared. In the nonintegrable case the average entanglement content increases and saturates along with a transition to classical chaos. Such quantum chaotic states are shown to have universal concurrence distributions that are modified Bessel functions derivable within random matrix theory. Time-reversal breaking and time evolving states are shown to possess significantly higher entanglement sharing capacity that eigenstates of time-reversal symmetric systems. We use the ordinary Harper and kicked Harper Hamiltonians as model systems.

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“…Specifically, we consider the kicked rotor given by T (p) = 1 2 p 2 and V (q) = K cos q with τ ≡ 1, and the kicked Harper given by T (p) = cos p and V (q) = cos q [20]. Using Bloch's theorem, we restrict our study to eigenfunctions of U that are periodic in position.…”

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confidence: 99%

Resonance- and Chaos-Assisted Tunneling in Mixed Regular-Chaotic Systems

2005

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We present evidence that nonlinear resonances govern the tunneling process between symmetryrelated islands of regular motion in mixed regular-chaotic systems. In a similar way as for nearintegrable tunneling, such resonances induce couplings between regular states within the islands and states that are supported by the chaotic sea. On the basis of this mechanism, we derive a semiclassical expression for the average tunneling rate, which yields good agreement in comparison with the exact quantum tunneling rates calculated for the kicked rotor and the kicked Harper.PACS numbers: 05.45. Mt, 03.65.Sq, 03.65.Xp Despite its genuine quantal character, dynamical tunneling [1] is strongly sensitive to details of the underlying classical phase space [2]. A particularly prominent scenario in this context is "chaos-assisted" tunneling [3,4,5,6] which takes place between quantum states that are localized on two symmetry-related regular islands in a mixed regular-chaotic phase space. The presence of an appreciable chaotic layer between the islands dramatically enhances the associated tunneling rate as compared to the integrable case, and induces strong fluctuations of the rate at variations of external parameters [3,4]. This phenomenon is attributed to the influence of "chaotic states" that are distributed over the stochastic sea. Since such chaotic states typically exhibit an appreciable overlap with the boundary regions of both islands, they may provide efficient "shortcuts" between the two regular quasimodes in the islands [4,5,6]. Indeed, chaosassisted tunneling processes arise in a number of physical systems, e.g. in the ionization of resonantly driven hydrogen [7], in microwave or optical cavities [8,9], as well as in the effective pendulum dynamics describing tunneling experiments of cold atoms in optical lattices [10,11].While the statistical properties of the chaos-assisted tunneling rates are well reproduced by a random matrix description of the chaotic part of the Hamiltonian [12], the formulation of a tractable and reliable semiclassical theory for the average tunneling rate is still an open problem. Promising progress in this direction was reported by Shudo and coworkers [13] who obtain a good quantitative reproduction of classically forbidden propagation processes in mixed systems by incorporating complex trajectories into the semiclassical propagator Their approach requires, however, the study of highly nontrivial structures in complex phase space, and cannot be straightforwardly connected to single coupling matrix elements between regular and chaotic states. A complementary ansatz, based on a Bardeen-type expression for the coupling to the chaos, was presented by Podolsky and Narimanov [14]. In comparison with tunneling rates from the driven pendulum, good agreement was obtained for large and moderate , whereas significant deviations seem to occur deep in the semiclassical regime [14].In the present Letter, we shall point out that nonlinear resonances between different classical degrees of freedom play a cr...

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Phase-space localization: Topological aspects of quantum chaos

Cited by 149 publications

References 16 publications

Experimental verification of a one-parameter scaling law for the quantum and “classical” resonances of the atom-optics kicked rotor

Experimental verification of a one-parameter scaling law for the quantum and “classical” resonances of the atom-optics kicked rotor

Entanglement sharing in one-particle states

Resonance- and Chaos-Assisted Tunneling in Mixed Regular-Chaotic Systems

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