M. P. Pato

The nuclear two-neutron removal cross-section of 11Li: A dynamic polarization potential approach

Abstract We determine the dynamic polarization potential associated to the two-neutron removal process from 11 Li. This potential is then employed to calculate the nuclear break-up cross section of 11 Li when it collides with different targets, as a function of its bombarding energy. An analytic expression for this cross section is obtained and shown to be in good agreement with the calculation based on the exact expression. We also discuss the relevance of this potential in the study of the fusion of 11 Li with a heavy target.

Family of generalized random matrix ensembles.

Using the generalized maximum entropy principle based on the nonextensive q entropy, a family of random matrix ensembles is generated. This family unifies previous extensions of random matrix theory (RMT) and gives rise to an orthogonal invariant stable Lévy ensemble with new statistical properties. Some of them are analytically derived.

Attenuation of the intensity within a superdeformed band.

The attenuation of the intra-band intensity of a superdeformed band which results from mixing with normally deformed configurations is calculated using reaction theory. It is found that the sharp increase of the attenuation is mostly due to the tunnelling through a spin dependent barrier and not to the chaotic nature of the normally deformed states. It is now well established that the intensities of E2 gamma transitions within a superdeformed (SD) rotational band show cascades down to low angular momentum [1–7]. These cascades exhibit the distinct feature that the intensity remains constant until a certain spin is reached where-after the intensity drops to zero within a few transitions. The sharp drop in intensity is commonly referred to as the decay out of a superdeformed band and is believed to arise from mixing of the SD states with normally deformed (ND) states of identical spin. The earliest theoretical work to implement such an interpretation [8–11] used a statistical model of the coupling between the SD and ND states. More recently, Refs. [12,13] used a framework originally developed for the study of compound nuclear reactions to derive formulae for the intensity in a more rigorous fashion (the expressions for the intensity in Refs. [8–11] are deduced from probability arguments). Ref. [13] concluded that Refs. [8–11] are valid in the non-overlapping resonance region. Refs. [8–11] further calculate the spin dependence of the relevant parameters (the electromagnetic widths of the SD and ND states, the level density of the ND states and spin dependence of the barrier separating the SD and ND wells) which Refs. [12,13] do not. Two features common to Refs. [8–13] are (i) the use of the Gaussian Orthogonal Ensemble (GOE) to simulate the ND states (ii) the use of the “golden rule” to extract a width for the the SD states due to mixing with the ND states. Here, as in Refs. [12,13] we exploit the similarity between the decay out of superdeformed bands and compound nuclear reactions to write the intensity as the sum of average and fluctuation contributions. However we use an energy average in place of the ensemble average used in Refs. [12,13]. The energy average approach allows the inclusion of the following features which are more difficult to incorporate into an ensemble average. (i) A hierarchy of complexity in the ND spectrum may be introduced. (ii) A statistical model different from the GOE may be used to simulate the ND states, as was proposed in Refs. [14,15]. (iii) A width for the SD states due to mixing with the ND states arises naturally without appealing to the “golden rule” whose range of validity has been found to be restricted [16,17].

Level density for deformations of the Gaussian orthogonal ensemble.

Formulas are derived for the average level density of deformed, or transition, Gaussian orthogonal random matrix ensembles. After some general considerations about Gaussian ensembles, we derive formulas for the average level density for (i) the transition from the Gaussian orthogonal ensemble (GOE) to the Poisson ensemble and (ii) the transition from the GOE to m GOEs.

Transition between Hermitian and non-Hermitian Gaussian ensembles

The transition between Hermitian and non-Hermitian matrices of the Gaussian unitary ensemble is revisited. An expression for the kernel of the rescaled Hermite polynomials is derived which expresses the sum in terms of the highest order polynomials. From this Christoffel–Darboux-like formula some results are derived including an extension to the complex plane of the Airy kernel.

Perturbation treatment of symmetry breaking within random matrix theory

We discuss the applicability, within the random matrix theory, of perturbative treatment of symmetry breaking to the experimental data on the flip symmetry breaking in quartz crystal. We found that the values of the parameter that measures this breaking are different for the spacing distribution as compared to those for the spectral rigidity. We consider both two-fold and three-fold symmetries. The latter was found to account better for the spectral rigidity than the former. Both cases, however, underestimate the experimental spectral rigidity at large L. This discrepancy can be resolved if an appropriate number of eigenfrequencies is considered to be missing in the sample. Our findings are relevant for symmetry violation studies in general.

Energy averages and fluctuations in the decay out of superdeformed bands.

We derive analytic formulas for the energy average ~including the energy average of the fluctuation contribution! and variance of the intraband decay intensity of a superdeformed band. Our results may be expressed in terms of three dimensionless variables: G # /G S , G N /d, and G N /(G S1G # ). Here G # is the spreading width for the mixing of a superdeformed ~SD! state u0& with the normally deformed ~ND! states uQ& whose spin is the same as u0&’s. The uQ& have mean lever spacing d and mean electromagnetic decay width G N whilst u0& has electromagnetic decay width G S . The average decay intensity may be expressed solely in terms of the variables G # /G S and G N /d or, analogously to statistical nuclear reaction theory, in terms of the transmission coefficients T0(E) and TN describing transmission from the uQ& to the SD band via u0& and to lower ND states. The variance of the decay intensity, in analogy with Ericson’s theory of cross section fluctuations, depends on an additional variable, the correlation length G N /(G S1G # )5(d/2p)TN /(G S1G # ). This suggests that analysis of an experimentally determined variance could yield the mean level spacing d as does analysis of the cross section autocorrelation function in compound nucleus reactions. We compare our results with those of Gu and

Random pure states: Quantifying bipartite entanglement beyond the linear statistics.

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

Structure of trajectories of complex-matrix eigenvalues in the Hermitian-non-Hermitian transition.

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

Level repulsion and the dynamical diagonalization of matrices

A set of first-order coupled equations of motion for eigenvalues and eigenvectors of a generic matrix is derived in terms of the equation of motion for the matrix itself. An efficient method of diagonalization is then constructed by defining an appropriate dynamics for the matrix. A comparison with the standard diagonalization method based on Jacobi transformations is made.