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At least 631 records · Page 35Linked to original sources
arXiv1999

Can the trace formula describe weak localisation?

We attempt to systematically derive perturbative quantum corrections to the Berry diagonal approximation of the two-level correlation function (TLCF) for chaotic systems. To this end, we develop a ``weak diagonal approximation'' based on a recent description of the first weak localisation correction to conductance in terms of the Gutzwiller trace formula. This semiclassical method is tested by using it to derive the weak localisation corrections to the TLCF for a semiclassically disordered system. Unfortunately the method is unable to correctly reproduce the ``Hikami boxes'' (the relatively small regions where classical paths are glued together by quantum processes). This results in the method failing to reproduce the well known weak localisation expansion. It so happens that for the first order correction it merely produces the wrong prefactor. However for the second order correction, it is unable to reproduce certain contributions, and leads to a result which is of a different form to the standard one.

cond-mat.mes-hall↗
arXiv1999

Quantization Ambiguity, Ergodicity, and Semiclassics

A simple argument shows that eigenstates of a classically ergodic system are individually ergodic on coarse-grained scales. This has implications for the quantization ambiguity in ergodic systems: the difference between alternative quantizations is suppressed compared with the $O(\hbar^2)$ ambiguity in the integrable case. For two-dimensional ergodic systems in the high-energy regime, individual eigenstates are independent of the choice of quantization procedure, in contrast with the regular case, where even the ordering of eigenlevels is ambiguous. Surprisingly, semiclassical methods are shown to be much more precise for chaotic than for integrable systems.

quant-ph↗
arXiv1993

Is Schrödinger's Conjecture for the Hydrogen Atom Coherent States Attainable

We construct the most general SO(4,2) hydrogen atom coherent states which are the counterpart of Schrödinger's harmonic oscillator coherent states. We show that these states cannot be localized and cannot follow the classical orbits. Thus, Schrödinger's conjecture for the hydrogen atom coherent states is unattainable.

nucl-th↗
arXiv1995

Localized States in the Chaotic Ce Atom

We show that the energy averaged entropy localization length for the eigenstates of the Ce atom is well approximated by the prediction of the Wigner ensemble.

chao-dyn↗
arXiv1996

From Heisenberg matrix mechanics to EBK quantization: theory and first applications

Despite the seminal connection between classical multiply-periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several new or modified applications of EBK quantization.

chem-ph↗
arXiv1996

Quantum-Classical Correspondence via Liouville Dynamics: I. Integrable Systems and the Chaotic Spectral Decomposition

A general program to show quantum-classical correspondence for bound conservative integrable and chaotic systems is described. The method is applied to integrable systems and the nature of the approach to the classical limit, the cancellation of essential singularities, is demonstrated. The application to chaotic systems requires an understanding of classical Liouville eigenfunctions and a Liouville spectral decomposition, developed herein. General approaches to the construction of these Liouville eigenfunctions and classical spectral projectors in quantum and classical mechanics are discussed and are employed to construct Liouville eigenfunctions for classically chaotic systems. Correspondence for systems whose classical analogs are chaotic is discussed, based on this decomposition, in the following paper.

chao-dyn↗
arXiv1996

Quantum-Classical Correspondence via Liouville Dynamics: II. Correspondence for Chaotic Hamiltonian Systems

We prove quantum-classical correspondence for bound conservative classically chaotic Hamiltonian systems. In particular, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the $h\rightarrow 0$ limit. Correspondence is shown to occur via the elimination of essential singularities. In addition, applications to matrix elements of observables in chaotic systems are discussed.

chao-dyn↗
arXiv1997

Quantum anti-Zeno effect

Prevention of a quantum system's time evolution by repetitive, frequent measurements of the system's state has been called the quantum Zeno effect (or paradox). Here we investigate theoretically and numerically the effect of repeated measurements on the quantum dynamics of the multilevel systems that exhibit the quantum localization of the classical chaos. The analysis is based on the wave function and Schroedinger equation, without introduction of the density matrix. We show how the quantum Zeno effect in simple few-level systems can be recovered and understood by formal modeling the measurement effect on the dynamics by randomizing the phases of the measured states. Further the similar analysis is extended to investigate of the dynamics of multilevel systems driven by an intense external force and affected by frequent measurement. We show that frequent measurements of such quantum systems results in the delocalization of the quantum suppression of the classical chaos. This result is the opposite of the quantum Zeno effect. The phenomenon of delocalization of the quantum suppression and restoration of the classical-like time evolution of these quasiclassical systems, owing to repetitive frequent measurements, can therefore be called the 'quantum anti-Zeno effect'. From this analysis we furthermore conclude that frequently or continuously observable quasiclassical systems evolve basically in a classical manner.

quant-ph↗
arXiv1997

Periodic Orbit Theory of the circular billiard in homogeneous magnetic fields

We present a semiclassical description of the level density of a two-dimensional circular quantum dot in a homogeneous magnetic field. We model the total potential (including electron-electron interaction) of the dot containing many electrons by a circular billiard, i.e., a hard-wall potential. Using the extended approach of the Gutzwiller theory developed by Creagh and Littlejohn, we derive an analytic semiclassical trace formula. For its numerical evaluation we use a generalization of the common Gaussian smoothing technique. In strong fields orbit bifurcations, boundary effects (grazing orbits) and diffractive effects (creeping orbits) come into play, and the comparison with the exact quantum mechanical result shows major deviations. We show that the dominant corrections stem from grazing orbits, the other effects being much less important. We implement the boundary effects, replacing the Maslov index by a quantum-mechanical reflection phase, and obtain a good agreement between the semiclassical and the quantum result for all field strengths. With this description, we are able to explain the main features of the gross-shell structure in terms of just one or two classical periodic orbits.

cond-mat.mes-hall↗
arXiv1997

The Hydrogen Atom in Combined Electric and Magnetic Fields with Arbitrary Mutual Orientations

For the hydrogen atom in combined magnetic and electric fields we investigate the dependence of the quantum spectra, classical dynamics, and statistical distributions of energy levels on the mutual orientation of the two external fields. Resonance energies and oscillator strengths are obtained by exact diagonalization of the Hamiltonian in a complete basis set, even far above the ionization threshold. At high excitation energies around the Stark saddle point the eigenenergies exhibit strong level repulsions when the angle between the fields is varied. The large avoided crossings occur between states with the same approximately conserved principal quantum number, n, and this intramanifold mixing of states cannot be explained, not even qualitatively, by conventional perturbation theory. However, it is well reproduced by an extended perturbation theory which takes into account all couplings between the angular momentum and Runge-Lenz vector. The large avoided crossings are interpreted as a quantum manifestation of classical intramanifold chaos. This interpretation is supported by both classical Poincaré surfaces of section, which reveal a mixed regular-chaotic intramanifold dynamics, and the statistical analysis of nearest-neighbor-spacing

quant-ph↗
arXiv1998

Complete positivity of nonlinear evolution: A case study

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown on a concrete example.

quant-ph↗
arXiv1998

Statistics of electromagnetic transitions as a signature of chaos in many-electron atoms

Using a configuration interaction approach we study statistics of the dipole matrix elements (E1 amplitudes) between the 14 lower odd states with J=4 and 21st to 100th even states with J=4 in the Ce atom (1120 lines). We show that the distribution of the matrix elements is close to Gaussian, although the width of the Gaussian distribution, i.e. the root-mean-square matrix element, changes with the excitation energy. The corresponding line strengths are distributed according to the Porter-Thomas law which describes statistics of transition strengths between chaotic states in compound nuclei. We also show how to use a statistical theory to calculate mean squared values of the matrix elements or transition amplitudes between chaotic many-body states. We draw some support for our conclusions from the analysis of the 228 experimental line strengths in Ce [J. Opt. Soc. Am. v. 8, p. 1545 (1991)], although direct comparison with the calculations is impeded by incompleteness of the experimental data. Nevertheless, the statistics observed evidence that highly excited many-electron states in atoms are indeed chaotic.

physics.atom-ph↗
arXiv1998

Duality in Perturbation Theory and the Quantum Adiabatic Approximation

Duality is considered for the perturbation theory by deriving, given a series solution in a small parameter, its dual series with the development parameter being the inverse of the other. A dual symmetry in perturbation theory is identified. It is then shown that the dual to the Dyson series in quantum mechanics is given by a recent devised series having the adiabatic approximation as leading order. A simple application of this result is given by rederiving a theorem for strongly perturbed quantum systems.

hep-th↗
arXiv1998

Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system

The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states are expressed as a superposition of an infinte number of squeezed states. Their geometric phases are found to be independent of $\hbar$, and equal to $-(n+1/2)$ times the classical nonadiabatic Hannay angle.

quant-ph↗
arXiv1998

Resonant dynamics of the H atom in an elliptically polarized microwave field

The dynamics of Rydberg states of atomic hydrogen driven by elliptically polarized microwaves of frequency fulfilling 2:1 classical resonance condition is investigated both semiclassically and quantum mechanically in a simplified two-dimensional model of an atom. Semiclassical results for quasienergies of the system are shown to be in a good agreement with exact quantum data. The structures of the quantum states are found to reflect the underlying classical dynamics; especially we show the existence of nonspreading wavepackets propagating on elliptical trajectories.

chao-dyn↗
arXiv1999

Semiclassical Calculation of Transition Matrix Elements for Atoms in External Fields

Closed orbit theory is generalized to the semiclassical calculation of cross-correlated recurrence functions for atoms in external fields. The cross-correlation functions are inverted by a high resolution spectral analyzer to obtain the semiclassical eigenenergies and transition matrix elements. The method is demonstrated for dipole transitions of the hydrogen atom in a magnetic field. This is the first semiclassical calculation of individual quantum transition strengths from closed orbit theory.

quant-ph↗
arXiv1998

The Weyl Series and the Trace formula: can we add them?

Periodic orbit expressions for the density of states lead to spurious results when directly used to calculate quantities of thermodynamic interest. This is because the trace formula is usually valid only for large energies while the calculations make use of it at all energies. We present a prescription for circumventing this problem by isolating contributions which arise from the inaccurate low-energy behavior, and are spurious, from other subdominant contributions which are physical and are not contained in the Weyl series. The method is tested by analyzing the fermionic grand partition function for a disk billiard and may readily be extended to other dynamical systems.

chao-dyn↗