Sewing sound quantum flesh onto classical bones

Semiclassical transformation theory implies an integral representation for stationary-state wave functions $\psi_m(q)$ in terms of angle-action variables ($\theta,J$). It is a particular solution of Schr\"{o}dinger's time-independent equation when terms of order $\hbar^2$ and higher are omitted, but the pre-exponential factor $A(q,\theta)$ in the integrand of this integral representation does not possess the correct dependence on $q$. The origin of the problem is identified: the standard unitarity condition invoked in semiclassical transformation theory does not fix adequately in $A(q,\theta)$ a factor which is a function of the action $J$ written in terms of $q$ and $\theta$. A prescription for an improved choice of this factor, based on succesfully reproducing the leading behaviour of wave functions in the vicinity of potential minima, is outlined. Exact evaluation of the modified integral representation via the Residue Theorem is possible. It yields wave functions which are not, in general, orthogonal. However, closed-form results obtained after Gram-Schmidt orthogonalization bear a striking resemblance to the exact analytical expressions for the stationary-state wave functions of the various potential models considered (namely, a P\"{o}schl-Teller oscillator and the Morse oscillator).Comment: RevTeX4, 6 page

Wave Mechanics of a Two Wire Atomic Beamsplitter

We consider the problem of an atomic beam propagating quantum mechanically through an atom beam splitter. Casting the problem in an adiabatic representation (in the spirit of the Born-Oppenheimer approximation in molecular physics) sheds light on explicit effects due to non-adiabatic passage of the atoms through the splitter region. We are thus able to probe the fully three dimensional structure of the beam splitter, gathering quantitative information about mode-mixing, splitting ratios,and reflection and transmission probabilities

Strong-field dipole resonance. I. Limiting analytical cases

We investigate population dynamics in N-level systems driven beyond the linear regime by a strong external field, which couples to the system through an operator with nonzero diagonal elements. As concrete example we consider the case of dipolar molecular systems. We identify limiting cases of the Hamiltonian leading to wavefunctions that can be written in terms of ordinary exponentials, and focus on the limits of slowly and rapidly varying fields of arbitrary strength. For rapidly varying fields we prove for arbitrary $N$ that the population dynamics is independent of the sign of the projection of the field onto the dipole coupling. In the opposite limit of slowly varying fields the population of the target level is optimized by a dipole resonance condition. As a result population transfer is maximized for one sign of the field and suppressed for the other one, so that a switch based on flopping the field polarization can be devised. For significant sign dependence the resonance linewidth with respect to the field strength is small. In the intermediate regime of moderate field variation, the integral of lowest order in the coupling can be rewritten as a sum of terms resembling the two limiting cases, plus correction terms for N>2, so that a less pronounced sign-dependence still exists.Comment: 34 pages, 1 figur

Space Charge Limited 2-d Electron Flow between Two Flat Electrodes in a Strong Magnetic Field

An approximate analytic solution is constructed for the 2-d space charge limited emission by a cathode surrounded by non emitting conducting ledges of width Lambda. An essentially exact solution (via conformal mapping) of the electrostatic problem in vacuum is matched to the solution of a linearized problem in the space charge region whose boundaries are sharp due to the presence of a strong magnetic field. The current density growth in a narrow interval near the edges of the cathode depends strongly on Lambda. We obtain an empirical formula for the total current as a function of Lambda which extends to more general cathode geometries.Comment: 4 pages, LaTex, e-mail addresses: [email protected], [email protected]

Implications of invariance of the Hamiltonian under canonical transformations in phase space

We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of the classical analogue of a quantum system manifest themselves in an integral equation for its stationary state eigenfunctions. We restrict ourselves to that subclass of these dynamical symmetries for which the corresponding effective generating functions are necessaarily free of quantum corrections. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the geometry of the canonical transformations and the symmetry of the effective generating function can be exploited to pin down the precise form of the integral equations for stationary state eigenfunctions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We end with a brief discussion (relevant to string theory) of the generalization to scalar field theories in 1+1 dimensions.Comment: REVTeX v3.1, 13 page

Adiabatically coupled systems and fractional monodromy

We present a 1-parameter family of systems with fractional monodromy and adiabatic separation of motion. We relate the presence of monodromy to a redistribution of states both in the quantum and semi-quantum spectrum. We show how the fractional monodromy arises from the non diagonal action of the dynamical symmetry of the system and manifests itself as a generic property of an important subclass of adiabatically coupled systems

Counterintuitive transitions in multistate curve crossing involving linear potentials

Two problems incorporating a set of horizontal linear potentials crossed by a sloped linear potential are analytically solved and compared with numerical results: (a) the case where boundary conditions are specified at the ends of a finite interval, and (b) the case where the sloped linear potential is replaced by a piecewise-linear sloped potential and the boundary conditions are specified at infinity. In the approximation of small gaps between the horizontal potentials, an approach similar to the one used for the degenerate problem (Yurovsky V A and Ben-Reuven A 1998 J. Phys. B 31,1) is applicable for both problems. The resulting scattering matrix has a form different from the semiclassical result obtained by taking the product of Landau-Zener amplitudes. Counterintuitive transitions involving a pair of successive crossings, in which the second crossing precedes the first one along the direction of motion, are allowed in both models considered here.Comment: LaTeX 2.09 using ioplppt.sty and psfig.sty, 16 pages with 5 figures. Submitted to J. Phys.

Two-dimensional atom trapping in field-induced adiabatic potentials

We show how to create a novel two-dimensional trap for ultracold atoms from a conventional magnetic trap. We achieve this by utilizing rf-induced adiabatic potentials to enhance the trapping potential in one direction. We demonstrate the loading process and discuss the experimental conditions under which it might be possible to prepare a 2D Bose condensate. A scheme for the preparation of coherent matterwave bubbles is also discussed

Curve crossing in linear potential grids: the quasidegeneracy approximation

The quasidegeneracy approximation [V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and Y. B. Band, J. Phys. B {\bf 32}, 1845 (1999)] is used here to evaluate transition amplitudes for the problem of curve crossing in linear potential grids involving two sets of parallel potentials. The approximation describes phenomena, such as counterintuitive transitions and saturation (incomplete population transfer), not predictable by the assumption of independent crossings. Also, a new kind of oscillations due to quantum interference (different from the well-known St\"uckelberg oscillations) is disclosed, and its nature discussed. The approximation can find applications in many fields of physics, where multistate curve crossing problems occur.Comment: LaTeX, 8 pages, 8 PostScript figures, uses REVTeX and psfig, submitted to Physical Review

Nonadiabatic losses from radio-frequency-dressed cold-atom traps: beyond the Landau-Zener model

Nonadiabatic decay rates for a radio-frequency-dressed magnetic trap are calculated using Fermi’s golden rule: that is, we examine the probability for a single atom to make transitions out of the dressed trap and into a continuum in the adiabatic limit, where perturbation theory can be applied. This approach can be compared to the semiclassical Landau-Zener theory of a resonant dressed atom trap, and it is found that, when carefully implemented, the Landau-Zener theory overestimates the rate of nonadiabatic spin-flip transitions in the adiabatic limit. This indicates that care is needed when determining requirements on trap Rabi frequency and magnetic-field gradient in practical atom traps