Stéphane Nonnenmacher

Chaotic Eigenfunctions in Phase Space

We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on “chaotic eigenconstellations,” we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g., the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.

Decay of correlations for normally hyperbolic trapping

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green’s functions (cut-off resolvents) and polynomial bounds on the growth of those functions in the semiclassical parameter.

Distribution of Resonances for Open Quantum Maps

We analyze a simple model of quantum chaotic scattering system, namely the quantized open baker’s map. This model provides a numerical confirmation of the fractal Weyl law for the semiclassical density of quantum resonances. The fractal exponent is related to the dimension of the classical repeller. We also consider a variant of this model, for which the full resonance spectrum can be rigorously computed, and satisfies the fractal Weyl law. For that model, we also compute the shot noise of the conductance through the system, and obtain a value close to the prediction of random matrix theory.

Entropy of Eigenfunctions

We study the high-energy limit for eigenfunctions of the Laplacian, on a compact negatively curved manifold. We review the recent result of Anantharaman–Nonnenmacher (Ann. Inst. Fourier 57(7):2465–2523, 2007) giving a lower bound on the Kolmogorov–Sinai entropy of semiclassical measures. The bound proved here improves that result in the case of variable negative curvature.

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

Abstract.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”.

Eigenstate structures around a hyperbolic point

Using coherent-state representations of quantum mechanics (Bargmann, Husimi, and stellar representations), we describe analytically the phase-space structure of the general eigenstates corresponding to a 1-dimensional bilinear hyperbolic Hamiltonian, H=pq or equivalently H=1/2(P^2-Q^2). Their semi-classical behaviour is discussed for eigenvalues either near or away from the separatrix energy {H=0}, especially in the phase-space vicinity of the saddle-point (q,p)=(0,0).

From Open Quantum Systems to Open Quantum Maps

For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.

Quantum Variance and Ergodicity for the Baker's Map

We prove an Egorov theorem, or quantum-classical correspondence, for the quantised bakers map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.

Dissipation time and decay of correlations

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the other hand, it is fast for a large class of exponentially mixing systems, which include uniformly expanding maps and Anosov diffeomorphisms.

On the resonance eigenstates of an open quantum baker map

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Plancks constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, |zmin| ≤ |z| ≤ |zmax|. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius r. We prove that if the moduli converge to r = |zmax| then the sequence of eigenstates is associated with a fixed phase space measure ρmax. The same holds for sequences with eigenvalue moduli converging to |zmin|, with a different limit measure ρmin. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius |zmin| < r < |zmax| there is no unique limit measure, and we identify some families of eigenstates with precise self-similar properties.