Simple models of quantum chaos: Spectrum and eigenfunctions

Saturation of fidelity in the atom-optics kicked rotor

“…Specifically, we derive an analytic expression for the fidelity at the fundamental quantum resonances of the KR. At these resonances, a subclass of the experimental atomic ensemble [7,8,9] experiences an enhanced acceleration by the kicking field, at precisely defined kicking periods [5]. We show that this resonantly driven subclass induces a slow algebraic decay of fidelity, whilst the bulk of the initial atomic ensemble produces the novel phenomenon of a perfect saturation of fidelity at a finite and experimentally easily detectable level.…”

“…(1) induces the characteristic properties of the quantum KR, which sensitively depend on the parameter τ [5,10]. If τ = 4πr/q, with r, q integers, then the kicking period is rationally related to the propagation time of the kicked atoms across the lattice constant (which is 2π in our units), and quantum transport is enhanced.…”

“…If τ = 4πr/q, with r, q integers, then the kicking period is rationally related to the propagation time of the kicked atoms across the lattice constant (which is 2π in our units), and quantum transport is enhanced. Otherwise classically diffusive energy growth is suppressed by dynamical localisation [5]. In the following, we derive analytical results for the fidelity, for the experimentally relevant quantum resonances at τ = 2πℓ (ℓ ∈ N) [8], i.e., for large effective Planck constants τ .…”

“…If ξ 0 = 0, then |W N | = |sin(Nξ 0 /2)/sin(ξ 0 /2)|, else if ξ 0 = 0 then |W N | = N, and the average kinetic energy increases like k 2 N 2 /4 [5]. The latter occurs at the resonant values β = 1/2 + n/ℓ mod(1), n = 0, 1, .., ℓ − 1, which support ballistic motion [10].…”

“…We study the fidelity for cold, periodically kicked atoms, which mimic a paradigmatic model of quantum chaos theory, the δ-Kicked Rotor (KR) [5]. In this context, we show that bona fide fidelity is readily accessible, yet has never been measured in an experiment such as reported in [6].…”