Quasi-integrability and nonlinear resonances in cold atoms under modulation.

Quantum dynamics of a collection of atoms subjected to phase modulation has been carefully revisited. We present an exact analysis of the evolution of a two-level system (represented by a spinor) under the action of a time-dependent matrix Hamiltonian. The dynamics is shown to evolve on two coupled potential energy surfaces (PESs): one of them is binding, while the other one is scattering type. The dynamics is shown to be quasi-integrable with nonlinear resonances. The bounded dynamics with intermittent scattering at random moments presents a scenario reminiscent of Anderson and dynamical localization. We believe that a careful analytical investigation of a multi-component system that is classically non-integrable is relevant to many other fields, including quantum computation with multi-qubit systems.

Phase space trajectories and Lyapunov exponents in the dynamics of an alpha-helical protein lattice with intra- and inter-spine interactions.

The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.

Random reverse-cyclic matrices and screened harmonic oscillator.

We have calculated the joint probability distribution function for random reverse-cyclic matrices and shown that it is related to an N-body exactly solvable model. We refer to this well-known model potential as a screened harmonic oscillator. The connection enables us to obtain all the correlations among the particle positions moving in a screened harmonic potential. The density of nontrivial eigenvalues of this ensemble is found to be of the Wigner form and admits a hole at the origin, in contrast to the semicircle law of the Gaussian orthogonal ensemble of random matrices. The spacing distributions assume different forms ranging from Gaussian-like to Wigner.

Random cyclic matrices.

We present a Gaussian ensemble of random cyclic matrices on the real field and study their spectral fluctuations. These cyclic matrices are shown to be pseudosymmetric with respect to generalized parity. We calculate the joint probability distribution function of eigenvalues and the spacing distributions analytically and numerically. For small spacings, the level spacing distribution exhibits either a Gaussian or a linear form. Furthermore, for the general case of two arbitrary complex eigenvalues, leaving out the spacings among real eigenvalues, and, among complex conjugate pairs, we find that the spacing distribution agrees completely with the Wigner distribution for a Poisson process on a plane. The cyclic matrices occur in a wide variety of physical situations, including disordered linear atomic chains and the Ising model in two dimensions. These exact results are also relevant to two-dimensional statistical mechanics and nu -parametrized quantum chromodynamics.

Pseudounitary symmetry and the Gaussian pseudounitary ensemble of random matrices.

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudounitary group. Further, we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices the pseudounitary ensemble. We obtain exact results for the nearest-neighbor level-spacing distribution for (2 x 2) PT-invariant Hamiltonian matrices that have forms, approximately Sln(1/S) near zero spacing for three independent elements and approximately S for four independent elements. This shows a level repulsion in a marked distinction with an algebraic form S(beta) in the Wigner surmise. We believe that this paves the way for a description of varied phenomena in two-dimensional statistical mechanics, quantum chromodynamics, and so on.

Quantum modes on chaotic motion: Analytically exact results.

We discover a class of chaotic quantum systems for which we obtain some analytically exact eigenfunctions in closed form. These results have been possible due to connections shown between random matrix models, many-body theories, and dynamical systems. We believe that these results and connections will pave the way to a better understanding of quantum chaos.