Quantization of a class of piecewise affine transformations on the torus

Bièvre, Stephan De; Esposti, Mirko Degli; Giachetti, R.

doi:10.1007/bf02099363

Cited by 28 publications

(26 citation statements)

References 15 publications

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“…To obtain the Hilbert space we take the subspace of the wave functions on a line whose probability densities, |Ψ(x)| 2 , |Ψ(p)| 2 are periodic in both position and momentum representations, respectively: Ψ(x+1) = exp(i2πϕ q )Ψ(x), Ψ(p + 1) = exp(i2πϕ p )Ψ(p), where ϕ q , ϕ p ∈ [0, 1) are phases parameterizing quantization. The quantization of the baker map requires the phase space volume to be an integer multiple of the quantum of action [8,9,18,19,20,21]. Therefore the effective Planck constant is h = 1/N, for a baker map on a unit torus, where N, an integer, is the dimension of the Hilbert space.…”

Section: The Quantum Multi-baker Mapmentioning

confidence: 99%

Crossover from diffusive to ballistic transport in periodic quantum maps

Wójcik¹,

Dorfman²

2004

Physica D: Nonlinear Phenomena

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We derive an expression for the mean square displacement of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, t, and Planck's constant, , and allows a study of both the long time, t → ∞, and semi-classical, → 0, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results.

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Section: The Quantum Multi-baker Mapmentioning

confidence: 99%

Crossover from diffusive to ballistic transport in periodic quantum maps

Wójcik¹,

Dorfman²

2004

Physica D: Nonlinear Phenomena

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“…There are other quantization schemes of skew translations in the literature[3,2]. However, they do not satisfy Theorem 3.1 and so their relevance to the classical dynamics is unclear.…”

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confidence: 99%

Quantum unique ergodicity for parabolic maps

Marklof

Rudnick²

2000

GAFA, Geom. funct. anal.

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We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space average with respect to Liouville measure of the corresponding classical observable.The possible existence of any exceptional subsequences of eigenstates is an important issue, which until now was unresolved in any example. The absence of exceptional subsequences is referred to as quantum unique ergodicity (QUE). We present the first examples of maps which satisfy QUE: Irrational skew translations of the two-torus, the parabolic analogues of Arnold's cat maps. These maps are classically uniquely ergodic and not mixing. A crucial step is to find a quantization recipe which respects the quantum-classical correspondence principle.In addition to proving QUE for these maps, we also give results on the rate of convergence to the phase-space average. We give upper bounds which we show are optimal. We construct special examples of these maps for which the rate of convergence is arbitrarily slow. QUANTUM UNIQUE ERGODICITY FOR PARABOLIC MAPS 1555Consider for instance the geodesic flow on a compact Riemannian manifold M (or rather, on its co-tangent bundle), whose quantum Hamiltonian is, in suitable units, represented by −∆, the positive Laplacian on M . Let ψ j be a sequence of normalized eigenfunctions: ∆ψ j + λ j ψ j = 0,

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“…[8]). The basic problem comes from the fact that the projection operators L and E p (for example) do not commute as Ä 0 (even weakly).…”

Section: The Classical Limitmentioning

confidence: 93%

A Canonical Quantization of the Baker's Map

Rubin

Salwen²

1998

Annals of Physics

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We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref.1 ) and Saraceno (ref.2 ). We first construct a natural "baker covering map" on the plane R 2 . We then use as the quantum algebra of observables the subalgebra of operators on L 2 (R) generated by {exp (2πi x) , exp (2πi p)} . We construct a unitary propagator such that as → 0 the classical dynamics is returned. For Planck's constant h = 1/N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N × N matrix propagator is the same as given in ref.

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Quantization of a class of piecewise affine transformations on the torus

Cited by 28 publications

References 15 publications

Crossover from diffusive to ballistic transport in periodic quantum maps

Crossover from diffusive to ballistic transport in periodic quantum maps

Quantum unique ergodicity for parabolic maps

A Canonical Quantization of the Baker's Map

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