Correlation function

Abstract: In a previous paper we developed a method, Filter‐Diagonalization, for extracting eigenvalues and eigenstates of a given operator at any desired energy range. In essence, the method eliminates correlation between distant eigenstates through a short‐time filter while correlations between closely lying states are eliminated by diagonalization. Here we extend Filter‐Diagonalization. When used to extract eigenvalues for a given operator H, we show that all eigenvalue information is directly extracted from a short segment of the correlation functionC(t)=(ψ(0)‖e −iHt ‖ψ(0)), or alternately from a small number of residues (ψ(0)‖R n (H)‖ψ(0)), where ψ(0) is a random initial function and R n (H) is any desired polynomial expansion in H. The implications of this feature are twofold. First, in contrast to the previous version the wave packet needs only to be propagated once (to prepare C(t)), and eigenstates at all desired energy windows can then be extracted with negligible extra computation time (and negligible storage requirements). In a simulation presented here, accurate eigenvalues are extracted using propagation times which are only a 0.0041 fraction of the ‘‘natural’’ time, i.e., the time by which the relative phase of the two closest eigenstates reaches 2π. The second and more important feature is that the method is automatically suitable for extracting eigenvalues (or normal modes) using a short‐time segment of any signal C(t) which is a sum of (unknown) Fourier components (C(t)=∑ nd ne −ie nt ) regardless of its origin. In addition to its use for determining eigenvalues of known operators, the method may also be utilized to extract normal modes from classical‐dynamics simulations, eigenstates from real‐time Quantum Monte‐Carlo studies, frequencies from experimental optical or electrical signals, or be utilized in any other circumstance where a correlation function or general signal is only known for short times (or expensive to generate at long times).

Abstract: Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via Mopen channels; a=1,2,…,M. A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the χ2 distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer–Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases θa of the S-matrix (taken modulo 2π). We use it to find the distribution of derivatives τa=∂θa/∂E of these eigenphases with respect to the energy (“partial delay times”) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner–Smith time delay τw(E)=(1/M)∑a ∂θa/∂E at two different energies E−Ω/2 and E+Ω/2 as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.

Abstract: The problem of the interaction between a two-level atom and a quantum electromagnetic field is treated without the use of perturbation theory, without introduction of classical fields or factorization conditions for the states, and without assumptions about loss of memory. The calculation is carried out in the Heisenberg picture, without mode decomposition, and the conclusions all refer to physically measurable quantities, such as the fluorescence detected in the far field of the atom. It is shown that in a coherent field of constant amplitude the system always settles down to a quasistationary state, and that the stationarity is a manifestation of the quantum fluctuations. A solution for the growth of the fluorescent light intensity is presented that holds for any coherent exciting field. The two-time correlation function and the spectral density of the fluorescence are calculated, and are found to agree in the long-time limit with earlier results of Mollow. The two-time intensity correlation function of the field is derived, which corresponds to measurable photoelectric pair correlations, and it is found that this reflects several quantum features of the field. It is shown that quantum fluctuations are manifest more explicitly in two-time correlations in the steady state than in transient effects, like spontaneous emission in the vacuum. The measurement of such correlations therefore presents an opportunity for further experimental tests of quantum electrodynamics.