Semiclassical approximation for the scattering volume in cold-atom collisions

We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in an different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system “charged particle + electromagnetic radiation”, induced by a (strong) coherent field or by a black-body radiation component (with a Planck-Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it completely fails.

The field-theoretical result for the low-$T$ behavior of the correlation length of the quantum Heisenberg antiferromagnet on the square lattice was recently improved by Hasenfratz [Eur. Phys. J. B 13, 11 (2000)], who corrected for cutoff effects. We show that starting from his expression, and exploiting our knowledge of the classical thermodynamics of the model, it is possible to take into account non-linear effects which are responsible for the main features of the correlation length at intermediate temperature. Moreover, we find that cutoff effects lead to the appearance of an effective exchange integral depending on the very same renormalization coefficients derived in the framework of the semiclassical pure-quantum self-consistent harmonic approximation: The gap between quantum field-theoretical and semiclassical results here is eventually bridged.

Semiclassical quantization of complex Henon–Heiles systems

We have used a semiclassical method developed earlier to compute the spectrum of a field theory in two-dimensional space-time defined by the (sine-Gordon) Lagrangian $\frac{1}{2}{({\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\varphi})}^{2}+(\frac{{m}^{4}}{\ensuremath{\lambda}}){cos[(\frac{\sqrt{\ensuremath{\lambda}}}{m})\ensuremath{\varphi}]\ensuremath{-}1}$. For weak coupling we find a heavy particle, the soliton, corresponding to a peculiar classical field configuration and an antisoliton. Below the soliton-antisoliton threshold there are a large number of further states. They can be viewed either as soliton-antisoliton bound states or as bound states of $n$ of the usual quanta of the theory. The elementary particle $\ensuremath{\varphi}$ is the lowest of these. As the coupling increases, the higher states successively unbind, decaying into soliton-antisoliton pairs. At $\frac{\ensuremath{\lambda}}{{m}^{2}}=4\ensuremath{\pi}$, the elementary particle unbinds leaving only solitons and antisolitons for $\frac{\ensuremath{\lambda}}{{m}^{2}}g4\ensuremath{\pi}$. Comparing our semiclassical results with recent exact results of Coleman and with perturbation theory, we find that the semiclassical calculations are exact. This field theory seems similar to the hydrogen atom for which the Bohr-Sommerfeld quantization rules give the energy levels exactly. We also treat a ${\ensuremath{\varphi}}^{4}$ theory in weak coupling and carry out a number of calculations which provide nontrivial illustrations of the semiclassical method.

We present approximate off-shell wave functions and $T$-matrix elements obtained using a uniform semiclassical propagator. The approximate off-shell wave function is valid near turning points and stationary-phase points for the incoming or outgoing momenta. Our results for the off-shell $T$-matrix elements reduce to the primitive semiclassical results in appropriate limits but extend the range of validity of the available semiclassical approximations.

General expressions for all moments are calculated for both semiclassical and classical approximations to absorption and emission band shapes due to impurities whose vibrational interactions with the surrounding lattice of a solid contain both linear and quadratic terms. The first five moments are presented and compared to the exact quantum-mechanical results. The semiclassical approximation is found to be most accurate for vibrational interactions having either linear or linear plus quadratic terms with strong coupling and/or high temperature. The semiclassical approximation is rarely, if ever, appropriate if only quadratic terms occur. The classical approximation is a valid representation of the semiclassical approximation at high temperatures. These conclusions are shown to be consistent with the correspondence principle.

This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.

The calculation of the anomaly of the axial current of Dirac fermions in external antisymmetric tensor fields of odd rank is considered. A sequence of one-dimensional supersymmetric models is presented and it is shown that their supercharges are, after quantization, Dirac operators in the external tensor fields, while their Witten partition function is the anomaly of the axial current. It is shown that the action in the corresponding functional integral for the partition function is different from the classical action. The semiclassical approximation gives the anomaly only in the case of an external tensor field of third rank with zero exterior derivative, and it is calculated in this case for arbitrary dimension. The interpretation of this field as an external gravitational field with torsion and the connection with the results of Witten and Alvarez-Gaume and the Atiyah-Singer theorem are discussed.

We revisit the vacuum structure of the N = 1 Intriligator-Seiberg-Shenker model on ℝ3× S1. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and Ünsal, we argue that previously overlooked non-perturbative effects generate a Higgs-type potential on the classical Coulomb branch of the low-energy effective 3d N = 2 theory. In particular, on part of the Coulomb branch we encounter the first instance of a dynamically-generated quintic monopole superpotential.

Reduced chiral U( N)×U( N) model

Using the classical (CSC), primitive (PSC), and uniform (USC) semiclassical expressions for transition probabilities given by Miller and co-workers, we have calculated the reactive and nonreactive 0 → 0 and 0 → 1 transition probabilities for the collinear H + H2 exchange reaction. Comparison with previously calculated exact quantum and quasiclassical results for the reactive and nonreactive 0 → 0 transitions reveals that the semiclassical approximations are not very good, especially the CSC and PSC ones. All three semiclassical probabilities for the reactive 0 → 0 transition exceed unity in the collision energy range from 0.0 to 0.2 eV above the quasiclassical reaction threshold. This feature coupled with the failure of any of the semiclassical approximations to produce the marked quantum effects present in this transition causes these results to be less accurate than the corresponding quasiclassical ones. For the reactive and nonreactive 0 → 1 transitions the USC results are in qualitative agreement with the exact quantum ones and are better than the standard quasiclassical results. However, the reverse quasiclassical results are almost as good as the USC ones for these transitions. A probable reason for the inability of the USC expression to produce the strong oscillations observed in the exact quantum results is that the latter are due to interference between direct and resonant (i.e., compound state) processes whereas the present formulation of the semiclassical method does not encompass such phenomena. A comparison of the total reaction probabilities obtained by the USC and quasiclassical methods with the exact quantum one indicates that the USC result is more accurate than the quasiclassical one, except at collision energies less than 0.50 eV. This improved accuracy is due to a partial cancellation of errors in the contributing 0 → 0 and 0 → 1 USC reactive transition probabilities.

The partition function of N=6 supersymmetric Chern-Simons-matter theory (known as ABJM theory) on S^3, as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super-matrix model is closely related to a matrix model describing topological Chern-Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS_4 x CP^3 and gives the correct N^{3/2} scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in CP^3. We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi-Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau, and leads to an expansion around topological Chern-Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

Black hole evaporation by thermal bath removal

The presence of small amounts of strongly electronegative gases such as SF6 and CCl4 can greatly perturb from equilibrium the electron energy distribution function in a variety of electron devices. The effect of the addition of these electron attaching gases on the relaxation behaviour of a nonequilibrium electron distribution is studied with a linear Fokker-Planck equation. The time dependence of the electron mobility as well as other transport properties can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. This eigenvalue problem can be transformed to a Schrodinger equation with a potential function parametrized by the electron-atom cross sections, the electric field strength and the gas temperature. In the present paper, the eigenvalues and eigenfunctions are calculated with a WKB semiclassical approximation and compared with exact results.

Tunneling and energy splitting in Ising models