Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory

Abstract

The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period T hyp is order of | ln h|. And, for large time, a new period Trev for the quantum dynamics appears: the initial wave packets form again at t = Trev. Moreover, for the time t = p q Trev a fractional revivals phenomenon of the initial wave packets appears: there is a formation of a finite number of clones of the original wave packet. Keywords: Schrödinger's dynamics, revivals of wave packets, semi-classical analysis, hyperbolic trajectory, Schrödinger operator with double wells potential 0921-7134/11/$27.50 © 2011 -IOS Press and the authors. All rights reserved O. Lablée / Semi-classical behaviour of Schrödinger's dynamics 61 2. Quantum dynamics and autocorrelation function 2.1. The quantum dynamics For a quantum Hamiltonian P h : D(P h ) ⊂ H → H, H is a Hilbert space, the Schrödinger dynamics is governed by the Schrödinger equation ih ∂ψ(t) ∂t = P h ψ(t). With the functional calculus, we can reformulate this equation with the unitary group UIndeed, for a initial state ψ 0 ∈ H, the evolution given by Return and autocorrelation functionWe now introduce a simple tool to understand the behaviour of the vector ψ(t): a quantum analog of the Poincaré return function.Definition. The quantum return function of the operator P h and for an initial state ψ 0 is defined by r(t) := ψ(t), ψ 0 H , and the autocorrelation function is defined byThe previous function measures the return on initial state. This function is the overlap of the time dependent quantum state ψ(t) with the initial state ψ 0 . Since the initial state ψ 0 is normalized, the autocorrelation function takes values in the compact set [0, 1]. Then, if we have an orthonormal basis of eigenvectors (e n ) n∈N :we get, for all integer nSo for a initial vector ψ 0 ∈ D(P h ) ⊂ H, let us denote by (c n ) n∈N = (c n (h)) n∈N the sequence of 2 (N) given (c n ) n = π(ψ 0 ), where π is the projector (unitary operator):π : H → 2 (N), ψ → ψ, e n H .

The dynamics of the Schrödinger flow from the point of view of semiclassical measures

“…In the most general framework, it is known [CR97, BGP99, HJ99, HJ00, BR02] that (1.9) holds uniformly for (1.10) |t| ≤ T hn E := (1 − δ) λ −1 max log (1/h n ) , where δ ∈ (0, 1) and λ max stands for the maximal expansion rate of the geodesic flow on the spheres { ξ x = ξ 0 x0 }. This upper bound T hn E , known as the Ehrenfest time, has been shown to be optimal for some one-dimensional systems, see [dBR03,Lab11].…”

“…, where δ ∈ (0, 1) and λ max stands for the maximal expansion rate of the geodesic flow on the spheres { ξ x = ξ 0 x0 }. This upper bound T hn E , known as the Ehrenfest time, has been shown to be optimal for some one-dimensional systems, see [dBR03,Lab11].…”

Quantum revivals in two degrees of freedom integrable systems: The torus case

“…The article [Lab4] deals with the quantum dynamics for the hyperbolic case. So, in dimension 1, we get the full and fractionnals revivals phenomenon (see [Av-Pe], [LAS], [Robi1], [Robi2], [BKP], [Bl-Ko], [Co-Ro], [Rob], [Pau1] for the elliptic case and see [Lab4] or [Pau2] for the the hyperbolic case). For an initial wave packets localized in energy, the dynamics follows the classical motion during short time, and, for large time, a new period T rev for the quantum dynamics appears : the initial wave packets form again at t = T rev .…”