Steven Tomsovic

Manifestations of classical phase space structures in quantum mechanics

Abstract Using two coupled quartic oscillators for illustration, the quantum mechanics of simple systems whose classical analogs have varying degrees of non-integrability is investigated. By taking advantage of discrete symmetries and dynamical quasidegeneracies it is shown that Percivals semiclassical classification scheme, i.e., eigenstates may be separated into a regular and an irregular group, basically works. This allows us to probe deeply into the workings of semiclassical quantization in mixed phase space systems. Some observations of intermediate status states are made. The standard modeling of quantum fluctuation properties exhibited by the irregular states and levels by random matrix ensembles is then put on a physical footing. Generalized ensembles are constructed incorporating such classical information as fluxes crossing partial barriers and relative fractions of phase space volume occupied by interesting subregions. The ensembles apply equally well to both spectral and eigenstate properties. They typically show non-universal, but nevertheless characteristic level fluctuations. In addition, they predict “semiclassical localization” of eigenfunctions and “quantum suppression of chaos” which are quantitatively borne out in the quantum systems.

Postmodern Quantum Mechanics

Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

Ray dynamics in a long-range acoustic propagation experiment.

A ray-based wave-field description is employed in the interpretation of broadband basin-scale acoustic propagation measurements obtained during the Acoustic Thermometry of Ocean Climate programs 1994 Acoustic Engineering Test. Acoustic observables of interest are wavefront time spread, probability density function (PDF) of intensity, vertical extension of acoustic energy in the reception finale, and the transition region between temporally resolved and unresolved wavefronts. Ray-based numerical simulation results that include both mesoscale and internal-wave-induced sound-speed perturbations are shown to be consistent with measurements of all the aforementioned observables, even though the underlying ray trajectories are predominantly chaotic, that is, exponentially sensitive to initial and environmental conditions. Much of the analysis exploits results that relate to the subject of ray chaos; these results follow from the Hamiltonian structure of the ray equations. Further, it is shown that the collection of the many eigenrays that form one of the resolved arrivals is nonlocal, both spatially and as a function of launch angle, which places severe restrictions on theories that are based on a perturbation expansion about a background ray.

Ray dynamics in long-range deep ocean sound propagation.

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focused on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (timelike) variable. Topics discussed include integrable and nonintegrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.

Statistical properties of many-particle spectra V. Fluctuations and symmetries

Abstract The manner in which energy-level and strength fluctuations change as a good symmetry is gradually broken is studied for a wide variety of chaotic systems describable in terms of random-matrix models. The local parameter Λ which governs the transition is identified as the mean-square symmetry-breaking matrix element in units of the local average spacing; its basic significance is clarified by exhibiting it as the “time” variable in a hierarchic set of diffusion equations. While many transitions, including Poisson → GOE, are considered, the transition GOE → GUE, from Gaussian orthogonal to Gaussian unitary ensembles, is studied in particular detail because of its connection with the breaking of time-reversal invariance (TRI), especially in complex nuclei for which the relevance of the GOE has been experimentally confirmed (GOE → GUE enters also into the study of chaotic magnetic systems). It is shown that the ensemble theory is directly applicable to real systems so that the transition equations derived for various spectral and strength measures can be applied to fluctuation data to give multiparticle bounds on the TRI-breaking part of the Hamiltonian. Analyses of spectral and strength fluctuations in the nuclear neutron-resonance and proton-resonance regions give bounds on the symmetry-breaking matrix elements as approximately one-tenth of the local average spacing. The analysis however must not stop with determining a bound on Λ; although the mere existence of a nonzero Λ could be of interest, indicating the occurrence of a symmetry breaking, the significance, if any, of an upper bound will usually not be obvious. It is therefore essential to determine, from the local multibody parameter Λ, the global (two-body) quantity α, the relative norm of the time-reversal noninvariant (TRNI) part of the nucleon-nucleon interaction. We do that in the following paper.

Statistical properties of many-particle spectra VI. Fluctuation bounds on N-NT-noninvariance

Abstract The preceding paper (V) shows that fluctuation analysis applied to experimental data in a chaotic region of the nuclear spectrum, determines values or bounds for a parameter Λ, proportional to the mean-squared near-diagonal matrix element of a symmetry-breaking interaction. The immediate interest is with time-reversal invariance (TRI) for which bounds, not values, are found. We show here how this information can be reduced to a bound on α, the relative magnitude of the time-reversal noninvariant (TRNI) part of the nucleon-nucleon interaction. This is done in terms of the level and transition-strength densities, both given as explicit functions of the Hamiltonian parameters. It is shown that, just as univariate Gaussian densities are essential ingredients of the state density, so also bivariate Gaussians enter naturally into a theory for the transition strengths. Calculations made for a subset, namely 167,169Er, 233Th, and 239U, of the nuclei considered in (V), give α≲2×10−3, a value which by some arguments is at the boundary of fundamental interest. It is found that ΛD α 2 is roughly constant over the periodic table and varies only slowly with energy. This implies that, for α-bounds better by perhaps a factor 4 (or α-values), we need more small strengths in nuclei with small spacings such as 236U (which presently yields no Λ-bound). The methods introduced here, which admit a number of simplifying approximations, are applicable to TRI studies via detailed-balance experiments proceeding through a compound nucleus, to other symmetries, and to a wide class of other problems.

Tunneling in complex systems

Some general aspects of quantum tunnelling and coherence - application to the BEC atomic gases, A.J. Leggett tunnelling in two-dimensions, S.C. Creagh quantum environments - spin baths, oscillator baths and applications to quantum magnetism, P. Stamp coherent magnetic moment reversal in small particles with nuclear spins, A. Garg tunnelling from super- to normal-deformed minima in nuclei, T.L. Khoo solitons in the bose condensate, W.P. Reinhardt.

Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier

We investigate the behavior of the quantum bakers transformation, a system whose classical analogue is completely chaotic, for time scales where the classical mechanics generates phase space structures on a scale smaller than Plancks constant (i.e., past the log time t∗ ≈ ln ħ-1). Surprisingly, we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime.

A uniform approximation for the fidelity in chaotic systems

In quantum/wave systems with chaotic classical analogues, wavefunctions evolve in highly complex, yet deterministic ways. A slight perturbation of the system, though, will cause the evolution to diverge from its original behaviour increasingly with time. This divergence can be measured by the fidelity, which is defined as the squared overlap of the two time evolved states. For chaotic systems, two main decay regimes of either Gaussian or exponential behaviour have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively. The accuracy of the approximations is illustrated with the standard map.

The unusual nature of the quantum Baker's transformation

Abstract We investigate the quantum Bakers transformation which has been introduced as a paradigm for studies of the quantum mechanics of classically chaotic systems. A semiclassical theory of the dynamics, good to times logarithmic in ħ, is given in the form of a linear wave packet dynamics. The consequences for the implied heavy eigenfunction scarring and for the trace of the propagator are compared to the quantum results. Unexpectedly, the semiclassical trace remains accurate beyond this logarithmic time. Some exotic quantum properties are found that cannot be explained by the short time semiclassics. Further study reveals level clustering, large recurrences in the trace of the propagator, and excesses of small eigenvector components. Many of the unusual behaviors have a very strong and peculiar ħ dependence.