Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U_3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the CKM matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the MNS matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.Comment: 33 pages, 6 figures version 2 - misprints corrected, explicit formulae for phases provide

On the Courant-Fischer theory for Krein spaces

http://www.sciencedirect.com/science/article/B6V0R-4V462G8-2/2/25c16be9e99d2fbaa89b7c1a6a47e95

Curvature effect on nuclear pasta: Is it helpful for gyroid appearance?

In supernova cores and neutron star crusts, nuclei are thought to deform to rodlike and slablike shapes, which are often called nuclear pasta. We study the equilibrium properties of the nuclear pasta by using a liquid drop model with curvature corrections. It is confirmed that the curvature effect acts to lower the transition densities between different shapes. We also examine the gyroid structure, which was recently suggested as a different type of nuclear pasta by analogy with the polymer systems. The gyroid structure investigated in this paper is approximately formulated as an extension of the periodic minimal surface whose mean curvature vanishes. In contrast to our expectations, we find from the present approximate formulation that the curvature corrections act to slightly disfavor the appearance of the gyroid structure. By comparing the energy corrections in the gyroid phase and the hypothetical phases composed of d-dimensional spheres, where d is a general dimensionality, we show that the gyroid is unlikely to belong to a family of the generalized dimensional spheres.Comment: 14 pages, 8 figure

Dzyaloshinskii-Moriya and dipole-dipole interactions affect coupling-based Landau-Majorana-Stückelberg-Zener transitions

It has been theoretically demonstrated that two spins (qubits or qutrits), coupled by exchange interaction only, undergo a coupling-based joint Landau-Majorana-Stuckelberg-Zener (LMSZ) transition when a linear ramp acts on one of the two spins. Such a transition, under appropriate conditions on the parameters, drives the two-spin system toward a maximally entangled state. In this paper, effects on the quantum dynamics of the two qudits, stemming from the Dzyaloshinskii-Moriya (DM) and dipole-dipole (d-d) interactions, are investigated qualitatively and quantitatively. The enriched Hamiltonian model of the two spins shares with the previous microscopic one the same C2 symmetry which once more brings about an exact treatment of the new quantum dynamical problem. This paper transparently reveals that the DM and d-d interactions generate independent, enhancing or hindering, modifications in the dynamical behavior predicted for the two spins coupled exclusively by the exchange interaction. It is worthwhile to notice that, on the basis of the theory here developed, the measurement of the time evolution of the magnetization in a controlled LMSZ scenario can furnish information on the relative weights of the three kinds of couplings describing the spin system. This possibility is very important since it allows us in principle to legitimate the choice of the microscopic model to be adopted in a given physical scenario

Exponential behavior of a quantum system in a macroscopic medium

An exponential behavior at all times is derived for a solvable dynamical model in the weak-coupling, macroscopic limit. Some implications for the quantum measurement problem are discussed, in particular in connection with dissipation.Comment: 8 pages, report BA-TH/94-17

An quantum approach of measurement based on the Zurek's triple model

In a close form without referring the time-dependent Hamiltonian to the total system, a consistent approach for quantum measurement is proposed based on Zurek's triple model of quantum decoherence [W.Zurek, Phys. Rev. D 24, 1516 (1981)]. An exactly-solvable model based on the intracavity system is dealt with in details to demonstrate the central idea in our approach: by peeling off one collective variable of the measuring apparatus from its many degrees of freedom, as the pointer of the apparatus, the collective variable de-couples with the internal environment formed by the effective internal variables, but still interacts with the measured system to form a triple entanglement among the measured system, the pointer and the internal environment. As another mechanism to cause decoherence, the uncertainty of relative phase and its many-particle amplification can be summed up to an ideal entanglement or an Shmidt decomposition with respect to the preferred basis.Comment: 22pages,3figure

Hidden evidence of non-exponential nuclear decay

The framework to describe natural phenomena at their basics being quantum mechanics, there exist a large number of common global phenomena occurring in different branches of natural sciences. One such global phenomenon is spontaneous quantum decay. However, its long time behaviour is experimentally poorly known. Here we show, that by combining two genuine quantum mechanical results, it is possible to infer on this large time behaviour, directly from data. Specifically, we find evidence for non-exponential behaviour of alpha decay of 8Be at large times from experiments.Comment: 12 pages LaTex, 3 figure

Timelapse

We discuss the existence in an arbitrary frame of a finite time for the transformation of an initial quantum state into another e.g. in a decay. This leads to the introduction of a timelapse $\tilde{\tau}$ in analogy with the lifetime of a particle. An argument based upon the Heisenberg uncertainty principle suggests the value of $\tilde{\tau}=1 / M_0$. Consequences for the exponential decay formula and the modifications that $\tilde{\tau}$ introduces into the Breit-Wigner mass formula are described.Comment: 5 pages [2 figs], ReV-Te