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Physical Review A

Quantum reflection by coupled-channel potentials

Carlo G. Meister, Harald Friedrich

Published Oct 25, 2002• 4 citations
Publisher

Abstract

We study quantum reflection in a system of two attractive step potentials with a steplike coupling term. The amplitude ${R}_{11}$ for elastic reflection generally has the near-threshold behavior $|{R}_{11}|\stackrel{\stackrel{\ensuremath{\rightarrow}}{k}0}{\ensuremath{\sim}}1\ensuremath{-}2bk$ familiar from the single-channel case. Reflectivity in the elastic channel is quenched in the presence of a second adiabatic channel open for transmission, while it is enhanced when the second transmission channel is closed; there is a pronounced minimum in the reflectivity when the potential in the upper adiabatic channel is zero or close to zero. When the upper adiabatic channel is repulsive, the elastic reflectivity can have an exact zero, at which quantum reflection leads to complete transfer into the excited state. Any degree of entanglement of translational and internal motion can be achieved at an appropriate energy between the threshold for inelastic reflection and this zero of the elastic reflectivity. For degenerate reflection channels, the unusual case of incomplete reflection at threshold occurs when the potential in the upper adiabatic channel vanishes.

DOI
10.1103/physreva.66.042718

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We study quantum reflection in a system of two attractive step potentials with a steplike coupling term. The amplitude R11{R}_{11}R11​ for elastic reflection generally has the near-threshold behavior ∣R11∣\ensuremath∼k\ensuremath→01\ensuremath−2bk|{R}_{11}|\stackrel{\stackrel{\ensuremath{\rightarrow}}{k}0}{\ensuremath{\sim}}1\ensuremath{-}2bk∣R11​∣\ensuremath∼k\ensuremath→​0​1\ensuremath−2bk familiar from the single-channel case. Reflectivity in the elastic channel is quenched in the presence of a second adiabatic channel open for transmission, while it is enhanced when the second transmission channel is closed; there is a pronounced minimum in the reflectivity when the potential in the upper adiabatic channel is zero or close to zero. When the upper adiabatic channel is repulsive, the elastic reflectivity can have an exact zero, at which quantum reflection leads to complete transfer into the excited state. Any degree of entanglement of translational and internal motion can be achieved at an appropriate energy between the threshold for inelastic reflection and this zero of the elastic reflectivity. For degenerate reflection channels, the unusual case of incomplete reflection at threshold occurs when the potential in the upper adiabatic channel vanishes.
We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.
We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for sss waves, m=0m=0m=0, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the sss-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than 1∕r21∕{r}^{2}1∕r2) and an effective range (for potentials falling off faster than 1∕r41∕{r}^{4}1∕r4). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to 1∕r41∕{r}^{4}1∕r4. For $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when mmm is neither integer nor half-integer.
We study the phases of the amplitudes for transmission through and near-side quantum reflection by the nonclassical regions of singular attractive potential tails, as occur in the interaction of atoms and molecules with each other and with surfaces. The derivatives of these phases with respect to energy or momentum are interpreted as time and space shifts according to the prescription of Eisenbud and Wigner. The results for transmission through the nonclassical part of the potential follow a straightforward pattern: Near threshold, the time and space shifts are close to those of a free particle and only marginally resolve the influence of the attractive potential; with increasing energy they rapidly approach the expectations for a classical particle accelerated in the potential. The phase of the amplitude for near-side quantum reflection leads to a negative time shift. In a regularized potential obtained by smoothly continuing the attractive tail to a deep constant at negative distances, this can be understood as a small boost for a wave packet approaching the origin from the left---relative to a free wave totally reflected at the origin.
Quantum threshold reflection is a well-known quantum phenomenon which prescribes that at threshold, except for special circumstances, a quantum particle scattering from any potential, even if attractive at long range, will be reflected with unit probability. In the past, this property had been associated with the so-called badlands region of the potential, where the semiclassical description of the scattering fails due to a rapid spatial variation of the de Broglie wavelength. This badlands region occurs far from the strong interaction region of the potential and has therefore been used to ``explain'' the quantum reflection phenomenon. In this paper we show that the badlands region of the interaction potential is immaterial. The extremely long wavelength of the scattered particle at threshold is much longer than the spatial extension of the badlands region, which therefore does not affect the scattering. For this purpose, we review and generalize the proof for the existence of quantum threshold reflection to stress that it is only a consequence of continuity and boundary conditions. The nonlocal character of the scattering implies that the whole interaction potential is involved in the phenomenon. We then provide a detailed numerical study of the threshold scattering of a particle by a Morse potential and an Eckart potential, especially in the time domain. We compare exact quantum computations with incoherent results obtained from a classical Wigner approximation. This study shows that close to threshold the time-dependent amplitude of the scattered particle is negligible in the badlands region and is the same whether the potential has a reflecting wall as in the Morse potential or a steplike structure as in the Eckart smooth step potential. The mean flight time of the particle is not shortened due to a local reflection from the badlands region or due to the lower density of the wave function at short distances. This study should serve to definitely rule out the badlands region as a qualitative guide to the properties of quantum threshold reflection.
Starting from a system of NNN radial Schr\"odinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N\ifmmode\times\else\texttimes\fi{}N$ exactly solvable potential model is obtained with the help of a single nonconservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as does its Jost matrix. It depends on N(N+1)/2N(N+1)/2N(N+1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2\ifmmode\times\else\texttimes\fi{}2$ case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has zero, one, or two bound states, and zero or one resonance; a compact formula for the open-channel scattering length is obtained. As a first physical application, exactly solvable $2\ifmmode\times\else\texttimes\fi{}2$ atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interacts with a bound or virtual state close to threshold, which results in a large background scattering length.
Classically forbidden reflection by a potential step is analyzed by matching WKB waves to solutions of a Schr\"odinger equation involving the tail of the potential on the up side of the step. Analytic expressions for the leading deviation of the reflection probability from unity and the phase of the reflection amplitude at the top of the step are derived for potentials decaying as an inverse power of the coordinate $V(x)\ensuremath{\propto}\ensuremath{-}{1/x}^{\ensuremath{\alpha}}, \ensuremath{\alpha}>2$.