Periodic orbit quantization of chaotic maps by harmonic inversion

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker’s map as a prototype example of a chaotic map. The harmonic inversion method for signal processing [1,2] has proven to be a powerful tool for the semiclassical quantization of chaotic as well as integrable dynamical systems [3–5]. Starting from Gutzwiller’s trace formula for chaotic systems, or the Berry-Tabor formula for integrable systems [6], the harmonic inversion method is able to circumvent the convergence problems of the periodic orbit sums and to directly extract the semiclassical eigenvalues from a relatively small number of periodic orbits. The technique has successfully been applied to a large variety of Hamiltonian systems [4,5]. It has been shown that the method is universal in the sense that it does not depend on any special properties of the dynamical system. In this Letter we demonstrate that the range of application of the harmonic inversion method extends beyond Hamiltonian systems also to quantum maps. Starting from the analogue of Gutzwiller’s trace formula for chaotic maps, we show that the semiclassical eigenvalues of chaotic maps can be determined by a procedure very similar to the one for flows. As an example system we consider the well known baker’s map. For this map we can take advantage of the fact that the periodic orbit parameters can be determined analytically. We briefly review the basics of quantum maps that are relevant to what follows (for a detailed account of quantum maps see, e.g. Ref. [7]). We consider quantum maps, acting on a finite dimensional Hilbert space of dimension N , which possess a well-defined classical limit for N → ∞. The quantum dynamics is Preprint submitted to Physics Letters A 8 February 2008 determined by the equation ψn+1 = Uψn , (1) where U is a unitary matrix of dimension N , and ψn is the N -dimensional discretized wave vector. The eigenvalues uk of U lie on the unit circle, uk = exp(−iφk). The density of eigenphases φk on the unit circle is given by ρqm(φ) = N 2π + 1 π Re ∞ ∑