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The Journal of Chemical Physics

Complex quasiprobability for atoms trapped on surfaces: A novel application of the complex coordinate method

Erik Engdahl, T. Maniv, Nimrod Moiseyev

Published May 1, 1988• 9 citations
Publisher

Abstract

A novel concept of complex quasiprobability Q(x) is proposed for atoms selectively trapped on a static, corrugated surface. The real part of Q(x) is the probability that the position vector of a trapped atom will have a projection x on the nominal surface plane, while Im[Q(x)]/‖Q(x)−1‖ is a local measure of the strength of the coupling between different diffraction channels, which leads to desorption of the trapped atoms from the surface. A method of calculating Q(x), based on the complex coordinate method, is presented and discussed. Applying this method to a model atom–surface potential which includes one-dimensional steps, it is shown that while the probability to find a trapped atom is enhanced around the top of a step, desorption is most likely to occur from the middle of the terraces.

DOI
10.1063/1.454549

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•Jan 1, 2009•1 citation
A novel concept of complex quasiprobability Q(x) is proposed for atoms selectively trapped on a static, corrugated surface. The real part of Q(x) is the probability that the position vector of a trapped atom will have a projection x on the nominal surface plane, while Im[Q(x)]/‖Q(x)−1‖ is a local measure of the strength of the coupling between different diffraction channels, which leads to desorption of the trapped atoms from the surface. A method of calculating Q(x), based on the complex coordinate method, is presented and discussed. Applying this method to a model atom–surface potential which includes one-dimensional steps, it is shown that while the probability to find a trapped atom is enhanced around the top of a step, desorption is most likely to occur from the middle of the terraces.
The complex rotation method is applied to the calculation of complex poles of the scattering matrix for atoms selectively trapped at a corrugated, static surface. The method is found to be extremely efficient and accurate even for highly corrugated surfaces, for which the use of more conventional methods is known to be in trouble. The method also provides insight into the trapping processes, revealing that for a simple harmonic corrugation there is a critical value of the corrugation amplitude, below which the trapping process is dominated by the coupling between adjacent diffraction channels, while above this critical value the coupling between more distant channels dominates.
The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu is considered. In this way, the final angular distribution of trajectories is obtained by box counting, which is in excellent agreement with the results calculated by standard SSS matrix methods of scattering theory. More interestingly, the accumulation of quantum trajectories at the different diffraction peaks is explained in terms of the corresponding quantum potential. This nonlocal potential ``guides'' the trajectories causing a transition from a distribution near the surface, which reproduces its shape, to the final diffraction pattern observed in the asymptotic region, far from the diffracting object. These two regimes are homologous to the Fresnel and Fraunhofer regions described in undulatory optics. Finally, the turning points of the quantum trajectories provide a better description of the surface electronic density than the corresponding classical ones, usually employed for this task.
A short time propagation algorithm for the reduced density matrix is derived to model the interaction of a quantum particle with a moving corrugated surface. The algorithm includes dissipative terms, which can be derived directly from the full Hamiltonian. The scattering of He from a corrugated Cu surface is examined as a function of incident energy and angle and the temperature of the substrate, with a focus on the nature of trapping. It is found that corrugation can make a significant contribution to trapping, even on a metal surface. Energy exchange with the phonons is shown to significantly modify the nature of diffraction mediated selective adsorption.
The elastic resonant scattering of He atoms off the Cu(117) surface is fully described with the formalism of quantum trajectories provided by Bohmian mechanics. Within this theory of quantum motion, the concept of trapping is widely studied and discussed. Classically, atoms undergo impulsive collisions with the surface, and then the trapped motion takes place covering at least two consecutive unit cells. However, from a Bohmian viewpoint, atom trajectories can smoothly adjust to the equipotential energy surface profile in a sort of sliding motion; thus the trapping process could eventually occur within one single unit cell. In particular, both threshold and selective adsorption resonances are explained by means of this quantum trapping considering different space and time scales. Furthermore, a mapping between each region of the (initial) incoming plane wave and the different parts of the diffraction and resonance patterns can be easily established, an important issue only provided by a quantum trajectory formalism.
We calculate the dispersive force between a ground-state atom and a nonplanar surface. We present explicit results for a corrugated surface, derived from the scattering approach at first order in the corrugation amplitude. A variety of analytical results in different limiting cases, including the van der Waals and Casimir-Polder regimes, is derived. We compute numerically the exact first-order dispersive potential for arbitrary separation distances and corrugation wavelengths for a rubidium atom on top of a silicon or gold corrugated surface. We discuss in detail the inadequacy of the proximity force approximation, and present a simple but adequate approximation for computing the potential.
We develop the scattering approach for the dispersive force on a ground state atom on top of a corrugated surface. We present explicit results to first order in the corrugation amplitude. A variety of analytical results are derived in different limiting cases, including the van der Waals and Casimir-Polder regimes. We compute numerically the exact first-order dispersive potential for arbitrary separation distances and corrugation wavelengths, for a Rubidium atom on top of a silicon or gold corrugated surface. We consider in detail the correction to the proximity force approximation, and present a very simple approximation algorithm for computing the potential.