Semiclassical theory of the Coulomb anomaly

An effective action approach for the problem of Coulomb blocking of tunneling is discussed. The method is applied to the strong-coupling problem arising near zero bias, where perturbation theory diverges. We find an instanton for the electrodynamics in imaginary time, and express the anomaly in terms of the exact conductivity of the system σ(ω,q) and the exact interaction. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 3, 200–205 (10 August 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Semiclassical theory of the Anderson transition

We study analytically the metal-insulator transition in a disordered conductor by combining the self-consistent theory of localization with the one parameter scaling theory. We provide explicit expressions of the critical exponents and the critical disorder as a function of the spatial dimensionality d. The critical exponent nu controlling the divergence of the localization length at the transition is found to be nu=1/2+1/d-2 thus confirming that the upper critical dimension is infinity. Level statistics are investigated in detail. We show that the two level correlation function decays exponentially and the number variance is linear with a slope which is an increasing function of the spatial dimensionality. Our analytical findings are in agreement with previous numerical results.     

On Quasi-static processes in linear response theory

It is shown that the difference between adiabatic and isothermal susceptibilities for homogeneous external forces is accounted for by aδ-singularity at zero frequency in the spectral function of linear response theory.     

Irreversibility and randomness in linear response theory

Recall that the fluctuation-dissipation theorem connects the response function of a passive linear system and the spectral density of the stationary stochastic process which describes the thermal fluctuations in the system. It is shown that the classical limit (=0) of the fluctuation-dissipation theorem implies a correspondence between systems which are reversible in the sense that the energy used to drive them away from equilibrium is completely recoverable as work and processes which are deterministic in the sense of Wiener's prediction theory, while irreversible systems correspond to nondeterministic processes. This correspondence is expressed by a simple transformation between the operator kernel which determines the optimal choice of the time-dependent force and the linear predictor for the stochastic process. For quantum systems this correspondence does not hold; the fluctuations are always of the deterministic type for any finite temperature, but the system is not necessarily reversible. For irreversible systems a formula is derived for the instantaneous entropy production which is a generalization of the standard one for Markovian dynamics.     

About the exactness of the linear response theory

For quantum lattice systems with finite range potentials and integrable space clustering, we prove the linearity of the response theory when dealing with fluctuation observables.     

Semiclassical theory of nuclear rotation

The semiclassical theory of large amplitude collective motion developed previously by the author is applied to uniform nuclear rotation. In this way a sophisticated foundation of the self-consistent cranking model is given and its quantization is accomplished. The obtained quantization condition already contains the leading order correction to the self-consistent cranking model and also reveals the relation between the total angular momentum and the total signature. In the low frequency limit it yields the I(I + 1) law of a quantum rotor.     

Semiclassical theory of chaotic conductors

We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or "clean," agrees with predictions of random-matrix theory.     

Semiclassical theory for two-anyon system

The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affects the semiclassical quantization condition for energy levels. For some simple cases admitting exact answers, we verify that our semiclassical formulas in fact provide highly accurate results over a broad quantum number range.     

Semiclassical theory of transport in antidot lattices

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conductivities semiclassically starting from the Kubo formula. The leading contribution reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations with Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases exponentially with temperature on the scale of the inverse orbit traversal time/T . The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. Observation of both effects requires that the elastic and inelastic scattering lengths be larger than the lengths of the relevant periodic orbits. The amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.This paper is dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Semiclassical laser theory in the stochastic and thermodynamic frameworks

The thermodynamic properties of the laser distribution in the steadily oscillating state are investigated to determine the minimum characteristic of the entropy production. First, the laser Langevin equation for five random variables is treated in the light of the stochastic calculus to deduce the photon-number rate equationn = – C+(n – n_c) + [A/(1 + sn)](n–n_A), where n_n and n₄ are the two constants of the fluctuation attributed to the noise forces subject to the usual fluctuation-dissipation theorem, withn ₄ < 0 for the inverted atomic population. We then combine the dynamics of the lasing mode with a model open system of the Lebowitz type with two reservoirs for which the entropy production(p) is expressed and made subject to a variational principle: The modified variation scheme, the same as Prigogine's local potential method, is shown to give the exact lasing distributionp as the optimum between two distributions of thermal type with temperatures far from each other.     

Semiclassical S-matrix theory for atomic fragmentation

A semiclassical scattering approach is developed which can handle long-range (Coulomb) forces without the knowledge of the asymptotic wave function for multiple charged fragments in the continuum. The classical cross section for potential and inelastic scattering including fragmentation (ionization) is derived from first principles in a form which allows for a simple extension to semiclassical scattering amplitudes as a sum over classical orbits and their associated actions. The object of primary importance is the classical deflection function which can show regular and chaotic behavior. Applications to electron impact ionization of hydrogen and electron–atom scattering in general are discussed in a reduced phase space, motivated by partial fixed points of the respective scattering systems. Special emphasis, also in connection with chaotic scattering, is put on threshold ionization. Finally, motivated by the reflection principle for molecules, a semiclassical hybrid approach is introduced for photoabsorption cross sections of atoms where the time-dependent propagator is approximated semiclassically in a short-time limit with the Baker–Hausdorff formula. Applications to one- and two-electron atoms are followed by a presentation of double photoionization of helium, treated in combination with the semiclassical S-matrix for scattering.     

Semiclassical theory of chaotic quantum transport

We present a refined semiclassical approach to the Landauer conductance and Kubo conductivity of clean chaotic mesoscopic systems. We demonstrate for systems with uniformly hyperbolic dynamics that including off-diagonal contributions to double sums over classical paths gives a weak-localization correction in quantitative agreement with results from random matrix theory. We further discuss the magnetic-field dependence. This semiclassical treatment accounts for current conservation.     

On the linear response theory of infinite quantum systems

The perturbation of a C¹-dynamics α by a time-dependent unbounded 1-derivation of the form μ?(t)δ′ is considered. The existence of the perturbed dynamics and that of the linear responce oftthe C¹-dynamical system ($\text{A}$, $\text{R}$, α) is shown. The existence of the corresponding generalized susceptibility and some convergence problems are discussed.     

Semiclassical theory of three-level gas lasers

We present numerical solutions of the density matrix equations for a three-level gas laser gain medium with a single resonator mode at each of the two transition frequencies. Population pulsations and interference effects between the two transition amplitudes are included. Comparison with solutions obtained neglecting the interference terms shows these effects to be significant in low-pressure lasers at moderate intensities. Population pulsations also change results for resonator modes tuned near line center.