Super-Selberg trace formula from the chaotic model

The arbitrary trajectory quantization method (ATQM) is a time dependent approach to quasiclassical quantization based on the approximate dual relationship that exists between the quantum energy spectra and classical periodic orbits. It has recently been shown however, that, for polygonal billiards, the periodicity criterion must be relaxed to include closed almost-periodic (CAP) orbit families in this relationship. In light of this result, we reinvestigate the ATQM and show that at finite energies, a smoothened quasiclassical kernel corresponds to the modified formula that includes CAP families while the delta function kernel corresponding to the periodic orbit formula is recovered as E-->infinity. Several clarifications are also provided.

Path integration over closed loops and Gutzwiller's trace formula

Classical and quantum mechanics of a strongly chaotic billiard system

It is possible to measure in the laboratory the periods Tk or the classical actions Sk of classical periodic orbits of an atomic electron. (We discuss highly excited states of the atom, in which the electron behaves much like a classical particle, and classical orbits become observable.) We will examine situations in which classical dynamics of the electron makes a transition from orderly to chaotic. In this circumstance, periodic orbits proliferate—existing orbits bifurcate, sending out new ones, and new periodic orbits appear ‘‘out of nowhere.’’ In this paper we briefly review experimental measurements carried out by various groups that display these recurrences, and we review the theory that gives a quantitative description of these recurrences and their bifurcations.

The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.

The scattering of a point particle from three hard discs in a plane is studied in the semiclassical approximation, using the Gutzwiller trace formula. Using a previously introduced coding of the classical dynamics, the needed summation over the classical periodic orbits is performed. The trace function is then given in terms of Ruelle zeta functions. A semiclassical limit upper bound is obtained on the lifetimes of the scattering resonances. This bound is larger than the classical lifetime when the classical repellor is chaotic but coincides with it when the repellor is periodic. We conclude that classical chaos dramatically influences the lifetimes of the scattering resonances. Our upper bound for the resonance lifetime is compared with the results of numerical calculation of the full quantum dynamics. The distribution of the imaginary parts of the complex wave numbers of the resonances is also calculated.

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

15 - Explicit Formulae as Trace Formulae

We first give a short outline of the semiclassical periodic orbit theory that allows one to relate quantum shell fluctuations in a mean-field system to the periodic orbits of the corresponding classical system. Earlier applications to gross-shell effects in nuclei (ground-state deformations), metal clusters (ground-state deformations and supershells), and quantum dots (conductance oscillations in weak magnetic fields) are briefly summarized. We then report on a recent simple and transparent semiclassical interpretation of the origin of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we explain the onset of left-right asymmetric shapes at the fission isomer minimum by the principle of constant action of the shortest orbits.

In this paper we obtain a simple analytic formula for the photodetachment cross section of H 2 in parallel electric and magnetic fields. The three-dimensional semiclassical approximation predicts oscillations in the spectrum and correlates these oscillations with closed classical orbits. The cylindrical symmetry of the Hamiltonian produces some interesting effects. In particular, at boundary energies the semiclassical approximation fails as a focused cusp approaches the origin. @S1050-2947~97!03306-4# PACS number~s!: 32.80.Gc I. INTRODUCTION Experimental measurements of the photodetachment cross section of H 2 in strong static electric fields were made by Bryant et al. @1#. For energies above the threshold energy the resulting cross section was found to be a smooth background upon which was superposed sinusoidal oscillations. Theoretical discussions of the measurements have been given by a number of authors @2,3#. The oscillations arise as an interference effect because the outgoing electron can move against the electric force and then return to the atom. It was shown in Ref. @3# that the cross section could be expressed in the following way: s~ E!5s 0~ E!1C~ E!sinF~ E!, where s 0 is the cross section in the absence of any external fields andC(E) and F(E) are called the recurrence strength and recurrence phase associated with the returning orbit. Theoretical calculations @4,5# show that analogous oscillations occur for photodetachment in crossed electric and magnetic fields, and that the oscillations are again associated with closed orbits. Quantum calculations have shown that very strong oscillations are present in parallel fields @6#, but the relationship to closed orbits was not made clear. Our original motivation in the present paper was to complete this subject by calculating the closed orbits and the resulting spectral oscillations for photodetachment of an electron from H 2 in parallel electric and magnetic fields. This is a system in which the recurrences are simple, strong, relatively easy to observe, and easy to understand. We found results which go far beyond this topic, and which are connected with many other problems of current interest. We know from study of nonlinear dynamics that even when long-time motion is chaotic, short-time motion remains relatively simple and predictable. Periodic orbits play a central role: in Poincare ´ ’s words, they offer ‘‘the only opening through which we might try to penetrate the fortress ~chaos! which has the reputation of being impregnable.’’ This classical statement holds also in quantum mechanics, wherein the periodic-orbit theory of Gutzwiller, Balian and Bloch, and Berry and Tabor @7# provides a general theoretical framework for studying quantum manifestations of classical chaos. Besides producing oscillations in absorption spectra @8#, periodic orbits produce scars in wave functions @9#, oscillations in the density of states @10#, and real-time recurrences that have been observed in many atoms and molecules @11#. @Similar phenomena are observed in microwaves in cavities @12~a!#, in microjunctions @12~b!#, and they are calculated to be consequences of certain models in nuclear physics @12~c!#.# Bifurcations of periodic orbits are of particular interest. A

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

The time variable is considered in the quantum gravity theory and calculated explicitly in the framework of the chaotic inflationary scenario where the scalar matter field has a contribution to the time variable in addition to the gravity field. The time formulated under the semiclassical approximation is a natural extension of that in the classical orbital motion. A perturbation method is introduced in terms of the scalar mass to obtain analytically solvable expressions for the time. The Wheeler-DeWitt equation is solved numerically to ensure that the semiclassical approximation is well justified. We examine the obtained time in detail and find that it is reasonable to consider it as time in the region where the semiclassical approximation is well justified.

An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creping waves. A new family of resonances in the two disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used.

The relationship between the hyperbolic automorphisms of the unit 2-torus and their quantum counterparts is studied. An exact and explicit representation of the quantum density of states in terms of the periodic orbits of the classical motion is derived. This is then used to obtain exact formulae for certain statistics of the distribution of the eigenvalues of the quantum propagator. It is shown that these statistics are strongly dependent upon the arithmetical nature of the dimensionless Planck's constant. The authors investigate their semiclassical form and, using results concerning the distribution of the long period orbits, find that they do not correspond to those of any of the universality classes previously studied. Finally, formulae are derived which relate properties of the quantum eigenfunctions to the periodic orbits.

The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.