Quantum Variance and Ergodicity for the Baker's Map

We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order |ln | −1 on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch. We then extend the results to some classes of quantised maps on the torus and obtain a logarithmic rate for perturbed cat-maps and a sharp algebraic rate for parabolic maps.

“…The same result has been recently proved for the baker's map too, see [DENW04]. For cat maps much stronger results are known, due to their arithmetic nature, see [KR00,KR05].…”

Section: Quantum Mapssupporting

confidence: 57%

Upper Bounds on the Rate of Quantum Ergodicity

Schubert

2006

“…The same result has been recently proved for the baker's map too, see [DENW06]. For cat maps much stronger results are known for a special choice of a basis of eigenfunctions, due to their arithmetic nature, see [KR00,KR05] and Section 4.3 below.…”

Section: Perturbed Cat Mapssupporting

confidence: 53%

On the Rate of Quantum Ergodicity for Quantised Maps

Schubert

2008

We study the distribution of expectation values and transition amplitudes for quantised maps on the torus. If the classical map is ergodic then the variance of the distribution of expectation values will tend to zero in the semiclassical limit by the quantum ergodicity theorem. Similarly the variance of transition amplitude goes to zero if the map is weak mixing. In this paper we derive estimates on the rate by which these variances tend to zero. For a class of hyperbolic maps we derive a rate which is logarithmic in the semiclassical parameter, and then show that this bound is sharp for cat maps. For a parabolic map we get an algebraic rate which again is sharp.

“…They suffer from diffraction effects due to the classical discontinuities of B (the Egorov property is slightly problematic, but still allows one to prove Quantum Ergodicity [9]). It was recently observed [29] that some eigenstates of the 2-baker in the case N = 2 k , (k ∈ N) have an interesting multifractal structure in phase space.…”

Section: Weyl Quantization Of the 2-torusmentioning

confidence: 99%

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

Anantharaman

Nonnenmacher

2006

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".