We review recent progress in analysing wave scattering in systems with both intrinsic chaos and/or disorder and internal losses, when the scattering matrix is no longer unitary. By mapping the problem onto a nonlinear supersymmetric σmodel, we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system. One of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel. The developed theory has far-reaching consequences for many quantities of interest, including local Green functions and time delays. In particular, we point out the role played by absorption as a sensitive indicator of mechanisms behind the Anderson localisation transition. We also provide a random-matrix based analysis of S-matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken timereversal invariance, completing previous works on the subject. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas.

We study the correlations of time delays in a model of chaotic resonance scattering based on the random matrix approach. Analytical formulae which are valid for arbitrary number of open channels and arbitrary coupling strength between resonances and channels are obtained by the supersymmetry method. We demonstrate that the time delay correlation function, though being not a Lorentzian, is characterized, similar to that of the scattering matrix, by the gap between the cloud of complex poles of the S-matrix and the real energy axis.

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...

We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the nonorthogonality of resonance wavefunctions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems, and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum. PACS numbers: 05.45.Mt, 03.65.Nk, 05.60.Gg The classical question of how energy levels of a quantum system get shifted under the action of a perturbation kept attracting renewed attention during the last two decades, mostly due to the established universality of such a parametric motion for systems with chaotic dynamics or intrinsic disorder [1,2]. In particular, the distributions and correlation functions of parametric derivatives of energy levels ("level velocities") [1][2][3] and their second derivatives ("level curvatures") [4] were found explicitly using the methods of random matrix theory (RMT) [5], and also verified, e.g., in microwave billiard experiments [6]. The other reason for such an interest is the recent development of the fidelity concept as the measure of the susceptibility of internal dynamics to perturbations [7].Experimentally, the energy levels are mostly accessible by means of a scattering setup [8]. From such a viewpoint, parametric dependencies of scattering characteristics, like phase shifts and time delays [9], conductances [10] and S matrix elements [11] were under intensive study. As to the parental discrete energy levels, they are converted into the resonances with finite lifetimes, since the original closed system becomes open (unstable). Such resonances manifest themselves in the energy-dependent S matrix as its poles in the complex energy plane, and can be analytically described as the complex eigenvalues of an effective non-Hermitian Hamiltonian [12][13][14]. Notably, the corresponding eigenfunctions are not orthogonal in the conventional sense but rather form a biorthogonal system. Their nonorthogonality plays an important role in many applications, e.g., describing interference in neutral kaon systems [15], influencing branching ratios of nuclear cross sections [16], and yielding excess noise in open laser resonators [17,18]. It also features in decay laws of quantum chaotic systems [19] and in dissipative quantum chaotic maps [20].In such a context the question of parametric motion of resonances and associated resonance states in open systems arises very naturally, but to the best of our knowledge has never been properly addressed. Our goal here is to begin filling in that gap by considering universal statistics of the shifts in the resonance widths under a generic perturbation in chaotic systems. In particular, we will demonstrate that such shifts are a clear manifestation of eigenstate nonorthogonality, thus providing a promising way to probe this ...

We report on an analytical study of the statistics of conductance, g, and shot-noise power, p, for a chaotic cavity with arbitrary numbers N1,2 of channels in two leads and symmetry parameter β = 1, 2, 4. With the theory of Selberg's integral the first four cumulants of g and first two cumulants of p are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For 0 < g < 1 a power law for the conductance distribution is exact. All results are also consistent with numerical simulations.