Rate of quantum ergodicity in Euclidean billiards

Bäcker, Arnd; Schubert, Roman; Stifter, P.

doi:10.1103/physreve.57.5425

Cited by 73 publications

(86 citation statements)

References 58 publications

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“…For bounded systems (µ = λ) and for ǫ = 0 we obtain the expression for the variance of matrix elements derived in [13] and tested and verified in many situations [14][15][16]. If the system is sufficiently open so that ΓT H ≫ 1 the contributions from the off-diagonal terms can be neglected and the integration continued to infinity (as explained in the next paragraph).…”

Section: Semiclassical Correlation Functions For Smooth Operatorsmentioning

confidence: 99%

Semiclassical cross section correlations

2000

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We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth classical limit the autocorrelation function of such matrix elements has two contributions with relative weights determined by classical dynamics. We show how the random matrix result can be obtained if the operator approaches a projector onto a single initial state. The expressions are verified in calculations for the kicked rotor.

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Section: Semiclassical Correlation Functions For Smooth Operatorsmentioning

confidence: 99%

Semiclassical cross section correlations

2000

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“…1b which falls one side of the straight line ∂Ω A \Γ. Our choice of the shape of Ω A was informed by the issue of boundary effects raised by Bäcker et al [5], the main point being that within a boundary layer of order a wavelength, there are Gibbs-type phenomena associated with spectral projections, and by choosing a large angle of intersection of the line with Γ their contribution is minimized. Our classical mean is A = vol(Ω A )/ vol(Ω) ≈ 0.55000.…”

Section: Computing Quantum Matrix Elementsmentioning

confidence: 99%

“…1). Numerical tests of Conjecture 1 (in the form of S 1 (E; A), see Remark 1.1) exist for low eigenvalues (n < 6000) in the Anosov cardioid billiard [5]: various powers were found in the range γ = 0.37 to 0.5, and up to 20% deviations from the prefactor in Conjecture 2.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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The Quantum Unique Ergodicity (QUE) conjecture of RudnickSarnak is that every eigenfunction φ n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E n → ∞), that is, 'strong scars' are absent. We study numerically the rate of equidistribution for a uniformly-hyperbolic Sinai-type planar Euclidean billiard with Dirichlet boundary condition (the 'drum problem') at unprecedented high E and statistical accuracy, via the matrix elements φ n ,Âφ m of a piecewise-constant test function A. By collecting 30000 diagonal elements (up to level n ≈ 7 × 10 5 ) we find that their variance decays with eigenvalue as a power 0.48±0.01, close to the estimate 1/2 of FeingoldPeres (FP). This contrasts the results of existing studies, which have been limited to E n a factor 10 2 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance, as a function of distance from the diagonal, against FP at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method used to calculate eigenfunctions.

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“…For integrable systems these are the invariant tori. For ergodic dynamics the eigenstates become equidistributed on the energy shell [2]. Typical systems have a mixed phase space, where regular islands and chaotic regions coexist.…”

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confidence: 99%

Flooding of Chaotic Eigenstates into Regular Phase Space Islands

2005

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We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect is observed for the example of island chains with just 10 islands.PACS numbers: 05.45. Mt, 03.65.Sq One of the cornerstones in the understanding of the structure of eigenstates in quantum systems is the semiclassical eigenfunction hypothesis [1]: in the semiclassical limit the eigenstates concentrate on those regions in phase space which a typical orbit explores in the longtime limit. For integrable systems these are the invariant tori. For ergodic dynamics the eigenstates become equidistributed on the energy shell [2]. Typical systems have a mixed phase space, where regular islands and chaotic regions coexist. In this case the semiclassical eigenfunction hypothesis implies that the eigenstates can be classified as being either regular or chaotic according to the phase-space region on which they concentrate. Note, that this may fail for an infinite phase space [3].In this paper we study mixed systems with a compact phase space, but away from the semiclassical limit. Here the properties of eigenstates depend on the size of phasespace structures compared to Planck's constant h. In the case of 2D maps this can be very simply stated [4]: a regular state with quantum number m = 0, 1, ... will concentrate on a torus enclosing an area (m + 1/2)h, as can be seen in Fig. 1(c).We will show that this WKB-type quantization rule is not a sufficient condition. We find a second criterion for the existence of a regular state on the m-th quantized torus,Here τ H = h/∆ ch is the Heisenberg time of the chaotic sea with mean level spacing ∆ ch and γ m is the decay rate of the regular state m if the chaotic sea were infinite. Quantized tori violating this condition will not support regular states. Instead, chaotic states will flood these regions, see Fig. 1(a). In terms of dynamics we find that wave packets started in the chaotic sea progressively flood the island as time evolves. Partial and even complete flooding is possible, depending on system properties. These findings are relevant for islands surrounded by a large chaotic sea. We numerically demonstrate the flooding and the disappearance of regular states for the important case of island chains. In typical Hamiltonian systems they appear around any regular island. On larger scales they are relevant for Hamiltonian ratchets [5], the kicked rotor with accelerator modes [6], and the experimentally [7,8,9] and theoretically [10] studied kicked atom systems. The flooding of regular islands by chaotic states is a new quantum signature of a classically mixed phase space. This phenomenon shows that not only local phasespace structures, but also global properties ...

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Rate of quantum ergodicity in Euclidean billiards

Cited by 73 publications

References 58 publications

Semiclassical cross section correlations

Semiclassical cross section correlations

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Flooding of Chaotic Eigenstates into Regular Phase Space Islands

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