Doubly uniform semiclassical quantization formula for resonances

The radial Schrödinger equation with an effective potential containing a single well and a single barrier is treated with an improved uniform semiclassical method. The improved quantization formula for complex energies (or resonances) contains a correction term that originates from a uniform treatment of the classically forbidden region near the origin in addition to the more familiar uniform treatment of the barrier region. In the present case the origin has a second-order pole, due to the centrifugal barrier potential term, and/or a Coulomb-type singularity, and these terms dominate the region inside the innermost classical turning point.Numerical results for first-order and third-order approximate complex resonance energies are compared with those of a standard (first- and third-order) barrier-uniform semiclassical method and also with those of ‘exact’ numerical computations.The improved quantization formula provides results in significantly better agreement with the exact results as the angular momentum quantum number l approaches zero.     

Semiclassical analysis of edge state energies in the integer quantum Hall effect

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x_c and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=π[n+ γ(E,x_c)] where S is the WKB action and 0 < γ < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of γ[E_n(x_c),x_c] ≡γ_n(x_c) on x_c is analyzed between its two extreme values as x_c ↦-∞ far inside the sample and as x_c ↦∞ far outside the sample. The edge-state energiesE_n(x_c) obey an almost exact scaling law of the form and the scaling function f(y) is explicitly elucidated.     

Quantum—classical correspondence of the time-dependent linear potential

Using the trial-function method, the general solution of the Schrödinger equation for the time-dependent linear potential is obtained. Based on the Heisenberg correspondence principle, the solution of the classical equation of motion is derived from the quantum matrix elements.     

Electron flux distributions in photodetachment of H<Superscript>-</Superscript> in parallel electric and magnetic fields

Photo-detached electrons of negative hydrogen ion in parallel electric and magnetic fields show quite complicated classical dynamical behavior, and a sequence of bifurcation and anti-bifurcation occurs. We investigate the effects of bifurcations on the flux distribution of photo-detached electrons by using position and momentum diagrams. Detached-electron flux distributions are calculated based on a uniform semi-classical theory. The flux distributions exhibit patterns with multiple rings. The bright rings correspond to special points in the diagrams. The flux distributions can be controlled by adjusting the magnetic field strength while fixing the electric field.     

Quantum thermodynamic processes: a control theory for machine cycles

The minimal set of thermodynamic control parameters consists of a statistical (thermal) and a mechanical one. These suffice to introduce all the pertinent thermodynamic variables; thermodynamic processes can then be defined as paths on this 2-dimensional control plane. Putting aside coherence we show that for a large class of quantum objects with discrete spectra and for the cycles considered the Carnot efficiency applies as a universal upper bound. In the dynamic (finite time) regime renormalized thermodynamic variables allow to include non-equilibrium phenomena in a systematic way. The machine function ceases to exist in the large speed limit; the way, in which this limit is reached, depends on the type of cycle considered.     

Universal diffusive decay of correlations in gapped one-dimensional systems

We apply a semiclassical approach to express finite temperature dynamical correlation functions of gapped spin models analytically. We show that the approach of [á. Rapp, G. Zaránd, Phys. Rev. B 74, 014433 (2006)] can also be used for the S = 1 antiferromagnetic Heisenberg chain, whose lineshape can be measured experimentally. We generalize our calculations to O(N) quantum spin models and the sine-Gordon model in one dimension, and show that in all these models, the finite temperature decay of certain correlation functions is characterized by the same universal semiclassical relaxation function.     

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.     

Simulations of Sisyphus cooling including multiple excited states

We extend the theory for laser cooling in a near-resonant optical lattice to include multiple excited hyperfine states. Simulations are performed treating the external degrees of freedom of the atom, i.e., position and momentum, classically, while the internal atomic states are treated quantum mechanically, allowing for arbitrary superpositions. Whereas theoretical treatments including only a single excited hyperfine state predict that the temperature should be a function of lattice depth only, except close to resonance, experiments have shown that the minimum temperature achieved depends also on the detuning from resonance of the lattice light. Our results resolve this discrepancy.     

Atomic coherent states studied by virtue of the EPR entangled state and their Wigner functions

Based on the newly constructed Einstein, Podolsky and Rosen (EPR) entangled state representation we introduce macroscopic classical functions associated with atomic coherent state τ with angular momentum value j. These functions are proportional to the ordinary one-variable Hermite polynomials of order 2j. The corresponding Wigner quasiprobability function for τ in phase space is also derived which turns out to be a two-variable Hermite polynomial H _{2j, 2j}. In so doing, a new classical-quantum correspondence scheme for angular momentum system is established. Received 7 August 2002 / Received in final form 14 December 2002 Published online 24 April 2003 RID="a" ID="a"Work supported by the National Natural Science Foundation of China under grant 10175057. RID="b" ID="b"e-mail: fhym@sjtu.edu.en     

Atomic density of a harmonically trapped ideal gas near Bose-Einstein transition temperature

We have studied the atomic density of a cloud confined in an isotropic harmonic trap at the vicinity of the Bose-Einstein transition temperature. We show that, for a non-interacting gas and near this temperature, the ground-state density has the same order of magnitude as the excited states density at the centre of the trap. This holds in a range of temperatures where the ground-state population is negligible compared to the total atom number. We compare the exact calculations, available in a harmonic trap, to semi-classical approximations. We show that these latter should include the ground-state contribution to be accurate.     

Wave equations with energy-dependent potentials

We study wave equations with energy-dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy-dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory. This work was supported by the agreement between IN2P3 and ASCR (collaboration no. 97-13) and by the Grant Agency of ASCR (J.M., grant No.A1048305).     

Projection operator approach to transport in complex single-particle quantum systems

We discuss the time-convolutionless (TCL) projection operator approach to transport in closed quantum systems. The projection onto local densities of quantities such as energy, magnetization, particle number, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. In particular, the lowest order contribution of this expansion is used as a strategy for the analysis of transport in “modular” quantum systems corresponding to quasi one-dimensional structures which consist of identical or similar many-level subunits. Such modular quantum systems are demonstrated to represent many physical situations and several examples of complex single-particle models are analyzed in detail. For these quantum systems lowest order TCL is shown to represent an efficient tool which also allows to investigate the dependence of transport on the considered length scale. To estimate the range of validity of the obtained equations of motion we extend the standard projection to include additional degrees of freedom which model non-Markovian effects of higher orders.     

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002     

Semiclassical calculation of ionization rate for Rydberg hydrogen atoms near a metal surface

The ionization of Rydberg hydrogen atoms near a metal surface at different scaled energies above the classical saddle point energy has been discussed by using the semiclassical method. The results show that the atoms ionize by emitting a train of electron pulses. In order to reveal the chaotic and escape dynamical properties of this system in detail, the sensitive dependence of the ionization rate upon the scaled energy is discussed. As the scaled energy is close to the saddle point energy, the ionization process of the hydrogen atom is nearly the same as the case of hydrogen atom in an electric field. There is only a single pulse of electrons, with an exponentially decaying tail. With the increase of the scaled energy, the ionization rates are similar to the case of the hydrogen atom in parallel electric and magnetic field, a series of electron pulses appear in the ionization process. This is caused by classical chaos, which occurs for the metal surface. Our studies also suggest that the metal surface can play the role of both the electric and the magnetic fields. Our theoretical analysis will be useful for guiding experimental studies of the ionization of atoms near the metal surface.     

Recurrence spectra and dynamical simulation of Rydberg hydrogen atom near a metal surface

By using the closed-orbit theory including the effect of Coulomb scattering together with an electrical image potential approach, the recurrence spectra and the dynamical behaviours of the Rydberg hydrogen atom near a metal surface are presented. Theoretical analysis and numerical simulation reveal that the impacts of the image potential contributing to the recurrence spectrum are qualitatively analogous to that of the parallel electrical and magnetic fields on the Rydberg atom. The recurrence spectra are computed for a few selected scaled energies and the results demonstrate that the scaled energy dominates the dynamical properties of system. With the increase of the scaled energy e from small to large, the whole trend of spectral structure is from simple to complex,and then simple.     

The thermal Green functions in nonextensive quantum statistical mechanics

The thermal Green functions of the quantum-mechanical harmonic oscillator are constructed within the framework of nonextensive statistical mechanics with normalized q -expectation values. For the Tsallis index q greater than unity, these functions are found to be expressed analytically in terms of the Hurwitz zeta function. It is found that influence of the nonextensivity on the time-ordered thermal propagator is relevant only at the “on-shell” states. In particular, the finite-temperature contribution to the thermal propagator becomes enhanced for the strong nonextensivity. Received 30 September 1998     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Short orbit distribution in the semiclassical limit of the SU(3) nuclear model

Three different families of short periodic orbits in the semiclassical SU(3) nuclear model were studied and their stability calculated. Then, knowing the shortest periodT _min of the closed trajectories, the long-range behaviour of the ₃ statistic was determined.The authors are greatly indebted to Prof. G. Benettin for many enlightening discussions and to Mr. G. Salmaso for his valuable computational assistance. This work has been partially supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (MURST).