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Physical Review A

Quantum reflection times and space shifts for Casimir–van der Waals potential tails

Alexander Jurisch, Harald Friedrich

Published Sep 23, 2004• 53 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.70.032711

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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 \ensuremathρ=\ensuremathρ(C3,C4)\ensuremath{\rho}=\ensuremath{\rho}({C}_{3},{C}_{4})\ensuremathρ=\ensuremathρ(C3​,C4​), which describes the relative importance of the \ensuremath−C3∕r3\ensuremath{-}{C}_{3}∕{r}^{3}\ensuremath−C3​∕r3 and \ensuremath−C4∕r4\ensuremath{-}{C}_{4}∕{r}^{4}\ensuremath−C4​∕r4 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 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.
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 photoassociative process between cold atoms and a surface. A mechanism to confine free cold atoms in a planar waveguide, characterized by the long-range atom-surface van der Waals potential and based on radiative transitions, is presented. We propose that the vibrational states of this potential can be probed by reflection spectroscopy, giving information about the C3{C}_{3}C3​ van der Waals coefficients.
Collisions between ultracold atoms and material surfaces are characterized by the reflection of the atomic matter wave from the attractive Casimir-Polder potential. This quantum reflection is particularly relevant to experiments such as GBAR, which will determine the gravitational acceleration of a cold antihydrogen atom by timing its fall onto a detection plate. In this thesis, the Casimir-Polder potential is computed from the electromagnetic scattering properties of the atom and surface and it is found to depend notably on the dielectric response, thickness and density of the medium. We show that reflection on this potential is associated with a breakdown of the semiclassical approximation and that it is enhanced for slow atoms and weak potentials. Liouville transformations relate Schrödinger equations with different potential landscapes but identical scattering properties. We gain new insights on the problem of quantum reflection on a potential well by mapping it onto an equivalent problem of tunneling through a wall. We also discuss the effect of gravity on the atomic wavepacket and its implications for free fall experiments with atoms. When combined with quantum reflection from a horizontal mirror, gravity can be used to trap particles in long lived states with promising applications for metrology. In particular, we suggest a scheme to improve the precision of the GBAR experiment by reducing the velocity dispersion of the falling atoms.
The modification of quantum fluctuations by a surface of finite reflectivity shifts the energy levels of atoms and changes their radiative properties (Casimir-Polder effect). At nanometric distances away from a surface (near field), the Casimir-Polder energy shift is given by −C3/z3 (where C3 is the van der Waals coefficient and z is the atom-surface distance) corresponding to the interaction of the fluctuating atomic dipole with its surface induced image. Near field Rydberg-surface interactions are of fundamental interest because the atomic size, scaling as n∗2 (where n∗ is the principle quantum number corrected by the quantum defect), is no longer negligible compared to the atom-surface separation, z. In this case, the dipole approximation, (−C3/z3 law of interaction), is no longer valid and higher-order terms, corresponding to the interaction of the fluctuation quadrupole with its own image, need to be taken into account. Quadrupole interaction terms in the atom-surface interaction manifest themselves with a novel distance dependence of z5 that has not been experimentally investigated yet1. Casimir-Polder experiments were first performed with low-lying Rydberg states (n= 10-14), demonstrating the −C3/z3 law of interaction2. Nevertheless, more recent experiments with high-lying rubidium Rydbergs (n=32-43)3 were not in agreement with Casimir-Polder theory thus highlighting the need for further experimental studies.