Statistical properties of resonances in quantum irregular scattering

Abstract

The close relations between statistical properties of quantum dissipative systems and scattering systems is discussed. It is conjectured that for quantum chaotic scattering the distribution of the resonance poles of the S matrix is generic and follows the predictions of the Ginibre ensemble of random nonHermitian matrices. This phenomenon has been demonstrated on a simple example of a single particle scattered by eight randomly distributed point obstacles in three dimensions.PACS numbers: 05.45.+b, 03.65.Nk Irregular scattering is nowadays one of the most interesting questions in the field of quantum chaos. It has been demonstrated, for instance [1], that the existence of classical irregular scattering leads in the quantum case to an S matrix whose structure can be described by the Dyson ensemble of random matrices [2]. It is also known that the fluctuations of the quantum cross section resemble the so-called Ericson fluctuations (discovered for nuclear systems [3]). This indicates that many overlapping resonances contribute. Also, the universal fluctuations of the conductivity in mesoscopic samples has been recently interpreted in terms of chaotic scattering [4]. It is therefore natural to ask whether one can predict something about the universal features of the distribution of the corresponding resonance poles in the complex energy plane.It is intuitively clear that the resonance poles have much in common with the eigenvalues of a quantum dissipative system. From the physical point of view, the resonance can be considered as a two-step process. During the first step an unstable intermediate (virtual) state is excited which then decays (second step) and leads for an isolated resonance to a peak in the cross section. The correspondence with the theory of the dissipative systems becomes apparent as soon as one starts with the description of the virtual state.In order to describe the intermediate state in the scattering process, Livsic [5] introduced a dissipative operator (the so-called Livsic matrix [6]), the eigenvalues of which coincide with the position of the resonance poles.The resonances which can be obtained as the poles of the analytically continued Green's function are defined according to Livsic with the help of a "restricted" quantum dynamics. Let P be a projection on a finite-dimensional subspace of the state Hilbert space J4 and let H be the relevant quantum Hamiltonian on ft. The restricted Green's function is then defined as P(H-z)~]P.(1)The Livsic matrix Biz) represents the effective dissipative and energy-dependent Hamiltonian describing the quantum dynamics on the restricted state space: lB(z)-z)~]=P(H-z)-] P(the rest of the Hilbert space is understood as the "heat bath"). The resonances are then obtained by solving the effective eigenvalue equationThis approach is equivalent to the complex scaling method introduced later [7]. The Livsic method shows a close similarity with the treatment of the dissipative system [8]. It is therefore natural to conjecture that the statistical properties of the e...

Random matrices close to Hermitian or unitary: overview of methods and results

“…(43) derived for a finite-rank perturbation and its Gaussian counterpart, Eq. (17). For this we notice, that the kernel Eq.…”

“…Naturally, such a picture stimulated various groups to develop a statistical description of resonance parameters [13,14,15,16,17].…”

“…Each of these poles, or resonances, E k = E k − i 2 Γ k , is characterized not only by energy E k but also by a finite width Γ k defined as (twice) the imaginary part of the corresponding complex energy and reflecting the finite lifetime of the states in the open system. Naturally, such a picture stimulated various groups to develop a statistical description of resonance parameters [13,14,15,16,17].…”

Critical statistics for non-Hermitian matrices

“…We mention several examples. The statistical properties of the poles of S-matrices have been analyzed in great detail in [3,4,5]. In QCD, the Euclidean Dirac operator in QCD at nonzero chemical potential (which can be interpreted as an imaginary vector potential), is nonhermitian resulting in the failure of the quenched approximation [6].…”

Embedding quantum systems with a non-conserved probability in classical environments

“…Nowadays, this field of research is constantly growing. A first general book on the topic has appeared [6]; applications of nonHermitian quantum mechanics involve the study of scattering by complex potentials and quantum transport [7][8][9][10][11][12][13][14][15][16][17], description of metastable states [18][19][20][21][22][23], optical waveguides [24][25][26], multi-photon ionization [27][28][29], and nano-photonic and plasmonic waveguides [30]. The theoretical investigations are also undergoing rapid developments: non-Hermitian quantum mechanics has been investigated within a relativistic framework [31] and it has been adopted by various researchers as a means to describe open quantum systems [32][33][34][35][36][37][38][39][40][41][42].…”