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Physical Review Letters

Quantum Reflection Times for Attractive Potential Tails

Harald Friedrich, Alexander Jurisch

Published Mar 12, 2004• 39 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.92.103202

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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.
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.
We explore the analogy between following the motion of a reflected wave packet and a quantum measurement of the spatial delay imposed on the particle by the scattering potential. It is shown that converting such delays into temporal durations can lead to “negative times,” and best be avoided. It is also demonstrated that a “soft” potential can be replaced by a superposition of hard walls. This representation is used for calculating the Goos-Hänchen shift of a Gaussian beam incident on a potential step or a barrier at an oblique angle.
In low energy atom-surface scattering, it is possible for the atom to be reflected in a region of attractive potential with no classical turning point. This phenomenon has come to be known as quantum reflection and it can reduce the sticking probability of atoms to surfaces, as well be used for atom trapping. We simulate the quantum reflection process in a one-dimensional model with a slow-moving atom moving in a Morse potential in the presence of an applied laser field. We show that in the case of laser-assisted quantum reflection, the laser field imparts additional momentum and kinetic energy to the atom. This results in a decreased distance of closest approach between the atom and surface. Our results show that the distance of closest approach and can be controlled through the timing and intensity of the laser pulse, which may result in enhanced sticking probability and/or reduced quantum reflection probability.
We study the reflection of narrow (in space) quantum wavepackets from nonabsorptive mirrors (totally or partially reflecting). If the initial mean value of the momentum component perpendicular to the mirror surface is less than the momentum uncertainty, then the mean value of this component gradually increases with time at the expense of shrinking the packet in the momentum space. As a consequence, very slow particles moving initially in the direction parallel to the mirror surface will be deflected to appreciable angles, even when they pass at macroscopical distances from the mirror. We give analytical expressions describing the asymptotical behaviour of wavefunctions and density matrices in the coordinate and momentum representations for arbitrary narrow initial packets. We show that the asymptotical mean values and variances do not depend on the phases of the complex reflection and transmission coefficients. Moreover, they are insensitive to the concrete form of the reflective potential in the case of totally reflecting mirrors. For partially reflecting mirrors we introduce the concept of conditional wavefunctions and mean values. The dependences of the asymptotical values of different quantities characterizing the packet (e.g. the `momentum transformation coefficient' and the `invariant uncertainty product') on the parameters of the reflecting potential (the height and width or characteristic length of the transition region) are analysed in the examples of the potentials of Epstein's type and their limit cases (the ideal reflecting wall and the delta potential). A possibility for the verification of the effect of quantum deflection in experiments with ultracold atoms is briefly discussed.