Resonances in open quantum maps

Abstract

We review recent studies about the resonance spectrum of quantum scattering systems, in the semiclassical limit and assuming chaotic classical dynamics. Stationary quantum properties are related to fractal structures in the classical phase space. We focus attention on a particular class of problems that are chaotic maps in the torus with holes. Among the topics considered are the fractal Weyl law, the formation of a spectral gap and the morphology of eigenstates. We also discuss the situation when the holes are only partially transparent and the use of random matrices for a statistical description.Resonances in open quantum maps 5 10686 S Nonnenmacher and M Zworski

Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

“…In particular, width fluctuations govern the decay law [8,9] and give rise to nonorthogonal modes [10,11] leading to enhanced sensitivity to perturbations [12]. Various aspects of lifetime and width statistics are a subject of intensive research, both theoretically and experimentally, with recent studies motivated by practical applications including optical microresonators [13], superconductor superlattices [14], many-body fermionic systems [15], microwave billiards [16,17], and dissipative quantum maps [18,19], as well as by long-standing interest in superradiance-like "resonance trapping" phenomena [4,15,[20][21][22]. Recent advances in experimental techniques have made it possible to test many of theoretical predictions with unprecedented accuracy [23][24][25][26][27][28][29][30], giving further impetus for the theory development.…”

“…Finding resonances inside that gap is a rare event, see [19] for a recent review and further discussion. Inside the cloud the resonance distribution mainly follows the so-called fractal Weyl law [31,32], which is determined by the fractal dimension of the chaotic repeller.…”

“…The above representation is especially convenient for our purposes, since it factorizes into a product of the p-4 1 √ y e −gy/2 and the factor Ψ γ (y), which therefore accumulates all corrections beyond the PTD. Unfortunately, performing exactly the GOE average in (19) at arbitrary coupling γ ∼ 1 is a rather challenging task and remains an interesting outstanding problem of RMT. However, in the weak-coupling limit γ ≪ 1 (i.e.…”

Semiclassical theory of chaotic quantum resonances

“…In spite of the ubiquity of the general setup -open quantum chaos is realized in many of the devices currently explored in mesoscopic physics, quantum optics, and cold atom physics -salient features of these resonances are not fully understood. While the deep quantum regime (the Ehrenfest time, t E , marking the diffractive disintegration of minimal wave packages shorter than classical escape times, t d ) appears to be under reasonable control [1], it is the opposite, semiclassical limit which poses unsettled issues [2].…”

“…Although the quantitative profile of that quantity is not fully understood, the density appears to be gapped against the real axis, Γ = 0 (the existence of rare midgap states notwithstanding [2,3].) The integrated number of resonances at a given value of E has been found to obey the so-called fractal Weyl law, ρ ∝ −d f , where d f is a non-universal fractal exponent.…”