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arXiv:cond-mat/0304010 · Physical Review Letters

Linear versus Nonlinear Coupling Effects in Single- and Multiphonon Atom-Surface Scattering

Antonio Šiber, B. Gumhalter

Published Mar 25, 2003• 30 citations
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Abstract

We present a comparative assessment of the features of inelastic atom-surface scattering spectra that are produced by different forms of linear and nonlinear phonon coupling to the projectile atom. Starting from a simple theoretical model of atom-surface scattering and employing recently developed exact numerical and approximate analytical methods we calculate and compare the scattering probabilities ensuing from each form of interaction. This enables us to demonstrate that in the regime of thermal energy atom scattering from surfaces the dominant contributions to the zero-, one-, and multiphonon excitation probabilities arise from linear coupling treated to all orders in the interaction.

DOI
10.1103/physrevlett.90.126103

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We present a comparative assessment of the features of inelastic atom-surface scattering spectra that are produced by different forms of linear and nonlinear phonon coupling to the projectile atom. Starting from a simple theoretical model of atom-surface scattering and employing recently developed exact numerical and approximate analytical methods we calculate and compare the scattering probabilities ensuing from each form of interaction. This enables us to demonstrate that in the regime of thermal energy atom scattering from surfaces the dominant contributions to the zero-, one-, and multiphonon excitation probabilities arise from linear coupling treated to all orders in the interaction.
The zero energy scattering limit of inelastic molecule–surface scattering is studied within the context of a multiphonon expansion of the molecule–bath wave function. By assuming that at low scattering energies the expansion may be truncated at first order in the phonon operators, we derived a closed form solution to the Lippmann Schwinger equation for the scattering wave function which includes a nonlocal and energy dependent self-energy term which correctly incorporates virtual phonon transitions in the elastic channel. The closure relation results from the use of a discrete spectral (ℒ2) form of the inelastic channel Greens functions. We compute the zero energy limit of these wave functions and discuss the trapping and reflection of cold atoms from ultracold surfaces. Our results indicate that for realistic atom surface interactions the low energy limit of the sticking coefficient, s, can deviate markedly from the expected s∝E1/2 behavior and is shown to approach a constant nonzero limiting value. This trend is consistent with recent experimental work involving the sticking of spin polarized hydrogen atoms on liquid He films.
We develop a multidimensional coupled-channel method suitable for studying the interplay of bound-state resonance and phonon-assisted scattering of inert gas atoms from solid surfaces in one, two, and three dimensions. This enables us to get insight into the features that depend on the dimensionality of inelastic resonant processes typically encountered in low-energy He atom scattering from surfaces, in general, and to elaborate on the observability of recently conjectured near threshold resonances in scattering from Einstein phonons, in particular.
The gas-surface scattering kernel for Na atoms interacting with a dielectric surface is obtained by measuring and theoretically reproducing two-photon evanescent-volume-wave spectra. From a detailed analysis of the spectral line shapes the contribution of inelastically scattered atoms is identified and a mean energy transfer of 36 meV is deduced for atoms having initial mean thermal energies of 79 meV. The mean probability of direct scattering mediated by surface phonons increases from 0.03 to 0.05 by increasing the surface temperature from 386 K to 398 K.
The effect of phonons on a nonlinear optical response of a quantum dot-cavity system in quantum strong coupling regime can be accounted for by a fully analytical treatment, provided that the exciton-phonon dynamics is much faster than the exciton-cavity dynamics. Modern experiments involving semiconductor quantum dots embedded in optical microcavities typically meet this criterion. We find that, for a relatively small exciton-cavity coupling, the effect of phonons is concentrated mainly in the polaron shift of the exciton frequency and reduction of exciton-cavity coupling by the Huang-Rhys factor. We have generalized this result to an arbitrary optical nonlinearity and demonstrated a good agreement with the exact solution in a wide range of temperatures. This generalization provides access to higher rungs of the Jaynes-Cummings ladder, where exact numerical approaches are impractical or even impossible. At larger coupling strengths and low temperatures, our approximation is also in good agreement with the exact solution, which makes it a very useful tool for addressing the phonon contribution to the coherent dynamics of a nonlinear optical system. We demonstrate our results for third-order optical polarization with varying observation time and delay time between excitation pulses in the form of two-dimensional spectra. These spectra provide useful information about the coherent coupling between the exciton and the cavity modes. The presented analytical approach is also compared with another useful approximation, having the form of a matrix product, which is a special case of the asymptotically exact solution in the limit of a short phonon memory time.
Quantum scattering calculations for strongly interacting molecular systems are computationally demanding due to the large number of molecular states coupled by the anisotropy of atom - molecule interactions. We demonstrate that thermal rate coefficients for total (elastic + inelastic) atom - molecule scattering are insensitive to the interaction anisotropy of the underlying potential energy surface. In particular, we show that the rate coefficients for Rb-H2_22​ and Rb-N2_22​ scattering at room temperature can be computed to 1% accuracy with anisotropy set to zero, reducing the complexity of coupled channel quantum scattering calculations to numerical solutions of a single differential equation. Our numerical calculations and statistical analysis based on Gaussian process regression elucidate the origin and limitations of the invariance of the total scattering rate coefficients to changes in atom - molecule interaction anisotropy.
The conventional notion of elastic, coherent atom-surface scattering originates from the scattering particles acting as a quantum-mechanical matter wave, which coherently interfere to produce distinct Bragg peaks which persist at finite temperature. If we introduce inelastic scattering to this scenario, the result is that the surface particles become displaced by the scattering atoms, resulting in emission or absorption of phonons that shift the final energy and momentum of the scatterer. As the lowest-lying phonons are gapless excitations, the ability to measure these phonons is very difficult and this difficulty is exacerbated by the roughly 1-eV resolution found in high-energy helium scattering experiments. Even though the surface has in effect measured the presence of the scatterer which decoheres the particle, we retain the diffraction spots which are referred to as coherent scattering. How do we reconcile these disparate viewpoints? We propose an experiment to more precisely examine the question of coherence in atom-surface scattering. We begin with an initially coherent superposition of helium particles with equal probabilities of interacting with the surface or not interacting with the surface. The beams are directed so that after the scattering event, the atoms are recombined so that we can observe the resulting interference pattern. The degree to which phonons are excited in the lattice by the scattering process dictates the fringe contrast of the interference pattern of the resulting beams. We use semiclassical techniques to simulate and test the viability of this experiment and show that for a wide range of conditions, despite the massive change in the momentum perpendicular to the surface, we can still expect to have coherent (in the superposition sense) scattering.