Fractal Weyl law for quantum fractal eigenstates

Abstract

The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate. [9,10,11]. Thus the understanding of their properties in the semiclassical limit represents an important challenge.According to the fractal Weyl law [4,5] the number of Gamow eigenstates N γ , which have escape rates γ in a finite band width 0 ≤ γ ≤ γ b , scales aswhere d is a fractal dimension of a classical strange repeller formed by classical orbits nonescaping in future (or past) times. By numerical simulations it has been shown that the law (1) In view of the recent results described above I study numerically a simple model of the quantum Chirikov standard map (kicked rotator) with absorption introduced in [18] which allows to vary continuously the fractal dimension of the classical strange repeller. In this way the fractal Weyl law (1) is verified in the whole interval 1 ≤ d ≤ 2. The model also allows to establish the limiting semiclassical distribution over escape rates γ and find its links with the fractal properties of the classical strange repeller. The Chirikov standard map is a generic model of chaotic dynamics and it finds applications in various physical systems including magnetic mirror traps, accelerator beams, Rydberg atoms in a microwave field [19,20,21,22]. The quantum model has been built up in experiments with cold atoms [23]. Thus the results obtained for this model should be generic and should find applications for various systems.The quantum dynamics of the model is described by the evolution matrix:wheren = −i∂/∂θ and the operatorP projects the wave function to the states in the interval [−N/2, N/2]. The semiclassical limit corresponds to k → ∞, T → 0 with the chaos parameter K = kT = const and absorption boundary a = N/k = const. Thus N is inversely proportional to the effective Planck constant T =h ef f , it gives the number of quantum eigenstates and the number of quantum cells inside the classical phase space. The classical dynamics is described by the Chirikov standard map [19,20] in its symmetric form: n = n + k sin θ + T n 2 ,θ = θ + T 2 (n +n).Physically, the map describes a free particle propagation in presence of periodic kicks with period T (e.g. kicks of optical lattice in [23]). In this model all trajectories (and quantum probabilities) escaping the interval [−N/2, N/2] are absorbed and never return back. It is convenient to fix K = 7 so that the phase space have no visible stability

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…The fractal Weyl law has been numerically confirmed for a number of physical model systems: a three-bump scattering potential (Lin, 2002;Lin and Zworski, 2002), a three-disk system , open quantum maps (Nonnenmacher, 2006;Schomerus and Tworzydlo, 2004;Shepelyansky, 2008), a Hénon-Heiles Hamiltonian with Coriolis term (Ramilowski et al, 2009), and a four-sphere system (Eberspächer et al, 2010). The asymptotic form (27) has been rigorously proven only for a simplified variant of the open quantum baker's map (Nonnenmacher and Zworski, 2007).…”

Short periodic orbit approach to resonances and the fractal Weyl law

“…In this way, we isolate the relevant information needed to construct just the long lived resonances, without calculating the others. According to the fractal Weyl law, the number of such resonances grows like N d/2 [3][4][5][6][7][8][9], where d is a fractal dimension of the classical repeller. Our method takes advantage of this fact: the number of scar functions that we need to obtain a reasonable approximation to the long-lived sector of the spectrum, denoted N s , is of the order of N d/2 .…”

“…In chaotic systems the number of states with a prescribed decay rate grows as a power of the energy which is conjectured to be related to a fractal dimension of the classical repeller, the set of initial conditions which remains trapped in the scattering region for all, positive and negative, times. This fractal Weyl law has been investigated in several systems [3][4][5][6][7][8][9]. One the other hand, the (right) eigenfunctions are supported by the unstable manifold of this repeller [10,11], and its scarring properties have also been under investigation [12][13][14].…”

Semiclassical theory of chaotic quantum resonances

“…Previous work on the phenomenon includes the formulation of lower bounds on the resonance gap [4][5][6], semiclassical approaches based on short periodic orbits trapped in the open system [7], a description in terms of non-unitarily evolving Husimi functions [8], phenomenology based on a mixture of phase space dynamics and random matrix theory, resp. [9], and numerical analyses [2,5,[8][9][10][11][12]. However, a unified theory of resonance formation in terms of first principle semiclassical dynamics appears to be missing and the formulation of such a theory is the subject of the present work.…”