Statistics of impedance, local density of states, and reflection in quantum chaotic systems with absorption

Abstract

We are interested in finding the joint distribution function of the real and imaginary parts of the local Green function for a system with chaotic internal wave scattering and a uniform energy loss (absorption). For a microwave cavity attached to a single-mode antenna the same quantity has a meaning of the complex cavity impedance. Using the random matrix approach, we relate its statistics to that of the reflection coefficient and scattering phase and provide exact distributions for systems with β=2 and β=4 symmetry class. In the case of β=1 we provide an interpolation formula which incorporates all known limiting cases and fits excellently available experimental data as well as diverse numeric tests. PACS numbers: 05.45.Mt, 42.25.Bs, Characterising statistical fluctuations of physical observables in quantum systems with underlying chaotic classical dynamics remains a very active field of research in theoretical and experimental physics. A considerable progress in understanding the phenomenon was underpinned by revealing the apparent universality of the fluctuations in systems of very diverse microscopic nature, ranging from atomic nuclei and Rydberg atoms in strong external fields, to complex molecules, quantum dots, and mesoscopic samples, see e.g. [1]. From theoretical side, the universality allows one to exploit the random matrix theory (RMT) as a powerful tool for analysis of generic features of the energy spectra of such systems [2,3].In many atomic, molecular, and mesoscopic systems the quantity which is readily obtained experimentally is the absorption spectrum for transitions from a given initial state |g to highly excited chaotic states at the energy E. For high-resolution experiments chaotic spectra consists of wellresolved narrow resonance peaks and one can, in principle, study statistics of the peak heights and widths as well as that of spacings between consecutive peaks. Most frequently, however, the absorption spectra look practically continuous due to both inevitable level broadening and finite experimental resolution. Then the relevant statistics is the distribution and correlation functions of the absorption probability σ(E) (also known as the strength function of the dipole operatorμ) which in the simplest situation of uniform level broadening Γ is given by σ(E) ∝ Im g|μĜ(E)μ|g , see e.g. Ref.[4] and discussion therein. Assuming the validity of the RMT, the problem then amounts to studying statistical properties of the resolvent (Green function) operatorĜ(E) ≡ (E +iΓ/2−Ĥ) −1 associated with the random matrixĤ which replaces the actual chaotic Hamiltonian. In particular, the imaginary part of the diagonal entries ofĜ(E) is well known in the solid state physics as the local density of states (LDoS) and in this capacity its statistics enjoyed many studies [4,5,6,7,8].From the experimental point of view the same universality, which makes the use of the RMT legitimate, provides one with an attractive possibility to employ simple model systems for analyzing generic statistics of the fluctuating quant...

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