The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3). In order to reveal level repulsion under conditions of global classical chaos special care may be necessary: (i) subspectra referring to different values of the quantum numbers related to symmetries must be dealt with separately and (ii) for systems with quantum localization only levels whose wavefunctions have overlapping support must be admitted. A “level” may either be an energy eigenvalue E in the case of autonomous systems or, for periodically driven systems, a quasi-energy φ, i.e. an eigenphase of the unitary Floquet operator transporting the wavevector from period to period.

Quantum signatures of chaos

Random-matrix theories in quantum physics : common concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

We investigate the quantum-chaotic properties of the Dicke Hamiltonian; a quantum-optical model that describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. This model exhibits a zero-temperature quantum phase transition in the $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite N, and by analyzing the level statistics, we demonstrate that the system undergoes a transition from quasi-integrability to quantum chaotic, and that this transition is caused by the precursors of the quantum phase transition. Our considerations of the wave function indicate that this is connected with a delocalization of the system and the emergence of macroscopic coherence. We also derive a semiclassical Dicke model that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.

http://export.arxiv.org/pdf/cond-mat/0301273

Chaos and the quantum phase transition in the Dicke model.

Nonperturbative quantum gravity

The joint distribution of energy eigenvalues of a Hamiltonian is derived by means of the usual statistical laws of classical many-body systems. It makes a transition from the Poisson type to the Gaussian type depending on the value of a single parameter characteristic of the Hamiltonian.

New approach to the statistical properties of energy levels.

Lax form of the quantum mechanical eigenvalue problem

We propose an evolutional scenario of the universe which starts from quantum states with conformal invariance, passing through the inflationary era, and then makes a transition to the conventional Einstein space-time. The space-time dynamics is derived from the renormalizable higher-derivative quantum gravity on the basis of a conformal gravity in four dimensions. Based on the linear perturbation theory in the inflationary background, we simulate evolutions of gravitational scalar, vector, and tensor modes, and evaluate the spectra at the transition point located at the beginning of the big bang. The obtained spectra cover the range of the primordial spectra for explaining the anisotropies in the homogeneous cosmic microwave background.

Space-time evolution and CMB anisotropies from quantum gravity

Statistics of energy spectra are studied for quantum billiards, i.e., free motion of a particle in two dimensions surrounded by a hard wall. The transition from regular to irregular spectra observed in moments of nearest-neighbor spacing distributions for different shapes of the boundary is analyzed by a superposition of independent level sequences with Poisson and Wigner distributions. The rate of mixture of the two sequences is compared with the rate of phase volumes spanned by quasiperiodic and chaotic orbits of the corresponding classical motion.

Tetsuyuki Yukawa

Papers

New approach to the statistical properties of energy levels.

Lax form of the quantum mechanical eigenvalue problem

Space-time evolution and CMB anisotropies from quantum gravity

Transition from regular to irregular spectra in quantum billiards.

Simulation of 2D dynamical triangulation with higher order curvature terms