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Physical Review A

Phases of the amplitudes for transmission and near-side quantum reflection in attractive potential tails

Eskender Mesfin, Harald Friedrich

Published Sep 14, 2006• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.74.032103

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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.
The time delay of an almost monochromatic wave packet can be defined unambiguously via the energy derivative of the phase of the amplitude defining the superposition of stationary states making up the wave packet. We illustrate how this concept, originally formulated by Eisenbud and Wigner in the context of scattering theory, can be applied to quantum reflection by the nonclassical region of a long-ranged attractive potential tail. We pay special attention to the case of a wave packet incident from the near side of the nonclassical region and interpret the results with the help of regularized step potentials.
When cold atoms approach a surface, they can be quantum reflected by quantal regions in the tail of the atom-surface potential. We study the phase of the reflection amplitude for Casimir--van der Waals potential tails, depending on the critical parameter $\ensuremath{\rho}=\ensuremath{\rho}({C}_{3},{C}_{4})$, which describes the relative importance of the $\ensuremath{-}{C}_{3}∕{r}^{3}$ and $\ensuremath{-}{C}_{4}∕{r}^{4}$ parts of the potential. The phase is related to observable kinematic quantities, the space and time shifts, the reflected atom experiences. We study three different models for the shape of the potential between the asymptotic limits and observe that the phases are more sensitive to the potential shape than the quantum reflection probabilities. At threshold, there are always time delays in comparison to the free movement. This is in contrast to the classical movement, which shows time gains. Further above threshold, the quantum reflected atom experiences a time gain relative to free motion, but this time gain is generally smaller than that of the classical particle.
When a wave packet with a narrow momentum distribution is quantum reflected in a purely attractive potential proportional to -1/r(alpha), alpha>2, it generally experiences a time gain compared to a free particle reflected at r=0; for alpha=3 and very low energies there are large time delays. In quantum reflection of an atomic beam by a surface, such a time gain (delay) represents an apparent plane of reflection which is shifted in front of (behind) the surface. The quantum reflected wave is always delayed with respect to the classical particle accelerated in the attractive potential.
In a recent work, we derived a formula for the elastic sss-wave phase shifts for deep potentials with attractive tails vanishing faster than 1/r21/{r}^{2}1/r2 asymptotically [Phys. Rev. A 84, 032701 (2011)]. As an addendum to that work, we reformulate its key result, i.e., the formula for the scattering phase shift ${\ensuremath{\delta}}_{0}$, in such a way that the contributions independent of the short-range influence of the full potential are explicitly expressed in terms of the amplitudes for transmission through and quantum reflection by the nonclassical region of the potential tail.
For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the sss-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length aaa, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number nth{n}_{\mathrm{th}}nth​. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to 1/r61/{r}^{6}1/r6 or to 1/r41/{r}^{4}1/r4, and also for a mixed potential tail consisting of a 1/r41/{r}^{4}1/r4 term together with a non-negligible 1/r61/{r}^{6}1/r6 contribution.
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 $\ensuremath{-}1∕{r}^{5}$.