Random matrix ensemble

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

The preceding considerations are essentially based on the model of random-matrix ensembles proposed by Dyson and others in the 1960s. Recent works, in particular by Casati and co-workers [89], have focused on band random matrices. Such matrices naturally arise in quantum systems with subspaces coupled only to next-neighboring subspaces such as for electronic states in a chain of atoms or in the kicked rotator. In such systems, localized states are observed that present a level statistics interme-... 

We still lack a proper understanding of how classical chaos induces the universal fluctuations in the energy levels that are so accurately described by the statistics of random matrix ensembles. Berry (1985) was the first to prove semiclassically some important results on the connection between chaos and energy level correlation functions of classically chaotic systems. The fundamental question, however, of how classical chaos generates quantum spectra amenable to a random matrix description is still not completely solved in all its details. 

Using eqn (20.69) for and averaging over the random matrix ensemble gives ... 

J. Stockmann, 1999). The main achievement of this field is the establishment of universal statistics of energy levels the typical distribution of the spacing of neighbouring levels is Poisson or Gaussian ensembles for integrable or chaotic quantum systems. This statistics is well described by random-matrix theory (RMT). It was first introduced by... 

Very accurate results were obtained for the classically chaotic Sinai billiard by Bohigas, Giannoni, and Schmit (see Fig. 2) which led them to the important conclusion (Bohigas, Giannoni and Schmit, 1984) Spectra of time-reversal invariant systems whose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussian orthogonal ensemble (GOE) of random-matrix theory... 

Here the authors consider the possibility of inferring such statistical characteristics from the spectral features of probe photons or particles that are scattered by the density fluctuations of trapped atoms, notably in optical lattices, in two hitherto unexplored scenarios, (a) The probe is weakly (perturbatively) scattered by the local atomic density corresponding to the random occupancy of different lattice sites, (b) The probe is multiply scattered by an arbitrary (possibly unknown a priori) multi-atom distribution in the lattice. The highlight of the analysis, which is based on this random matrix approach, is the prediction of a semicircular spectral lineshape of the probe scattering in the large-fluctuation limit of trapped atomic ensembles. Thus far, the only known case of quasi-semicircular lineshapes in optical scattering has been predicted [Akulin 1993] and experimentally verified [Ngo 1994] in dielectric microspheres with randomly distributed internal scatterers. 

Since the starting structure and the initial atom pair was casually selected, distance matrix generation and random metrization should be performed several times in order to get an ensemble of metric matrices. 

The diagonal element of the density matrix, W(n) — a nan is the probability that a system chosen at random from the ensemble occurs in the state characterized by n, and implies the normalization... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

The necessary summations reflect the fact that no complete information exists with respect to these magnetic quantum numbers statistically significant information can be derived from initial states with all quantum numbers M-, represented with equal probability, l/(2Jj + 1), the detection of final states with quantum numbers Mf being independent of the actual Mf value, and the summations over Mj and Mf taking care of all possible combinations of matrix elements leading from Mj to Mf substates. The appropriate formalism for such statistical information is that of density matrices. In the special representation in which the basic states for defining the density matrix coincide with the actual states of the ensemble, one obtains forms for the density matrices which are easy to interpret the density matrix attached to the initial, randomly oriented state has the following... 

When one studies random objects, the first question is always what is the probability of occurrence of individual objects in the ensemble In our context the question is what is the probability of occurrence of a specific Hamiltonian matrix H So, clearly, we need a probability density p H) which assigns a probability to a given matrix H. Apparently the matrix H is specified by stating the three matrix elements H, H 2 and H22-Accordingly, the density p depends on three arguments. 

If the transition dipoles of the chromophores in a solid polymer matrix are randomly oriented, the main source of depolarization in these experiments will be due to excitation transport. The initially excited ensemble is polarized along the direction of the excitation E field and gives rise to polarized fluorescence. Transport occurs into an ensemble of chromophores with randomly distributed dipole directions and the fluorescence becomes unpolarlzed. The random distribution is assured by the low concentration of the chromophores. To a slight extent, on the time scale of interest, depolarization also occurs as a result of chromophore motion. In this case the fluorescence anisotropy is approximately... 

---