1. Quantum Reflection off 2D Corrugated Surfaces
Emanuele Galiffi2,3, Christoph Sünderhauf3, Benedikt Herwerth4, Sandro Wimberger1,3, Maarten DeKieviet2 1University of Parma, 2Imperial College London, 3University of Heidelberg, 4University of Munich
ABSTRACT
The two-dimensional (2D) numerical propagation of wavepackets finds a wide range of applications in surface scattering problems. In this project, a highly optimised, norm-preserving algorithm is used to solve numerically the Time-Dependent Schrödinger Equation (TDSE) in order to propagate a wavepacket across a two-dimensional spatial domain. The aim of our study is to investigate the quantum reflection (QR) of slow atoms as they approach an attractive surface. This will finally enable experimental tests of potentials which describe Casimir interactions of neutral atoms with ‘’0periodically corrugated surfaces. These are carried out via the Atomic Beam Spin Echo method in Heidelberg.
1D OSCILLATING SURFACES
The study of QR off an oscillating surface revealed sidebands in the momentum distribution of the wavepacket, showing that energy is transferred when the frequency of oscillation corresponds to integer multiples of the energy of the incoming particle, as shown in Fig. 4.
This problem could be remapped to a 2D corrugated surface by means of a change of coordinates is the particle approaches the surface at grazing incidence.
FIGURE 4: Probability distribution functions in coordinate and momentum space of a Gaussian wavepacket which is quantum reflected off an oscillating surface. Sidebands form at oscillation frequencies which correspond to integer multiples of the energy of the particle. The wave was suppressed for negative values of x in order to eliminate spurious reflections off the boundaries of the numerical grid.
QUANTUM REFLECTION
Quantum reflection (QR) is a well-known, though still poorly understood effect, which lies at the heart of the wave interpretation of quantum mechanics (QM). The most striking feature of QR is the absence of a classical analogue in the particle picture of QM.
In fact the occurrence of QR is determined by the conditions in which the wave nature of the particle starts playing a significant role, namely when its De Broglie wavelength changes too abruptly as a result of a steep potential.
FIGURE 1: Typical reflection curves from a Casimir-Van der Waals potential for different numbers of gridpoints. Initially the numerical background results in a small measurement of R, until the wavepacket is progressively reflected by the attractive potential. The breakdown of the method due to insufficient sampling of the potential results in the reflectivity to tend to 1 as can be seen for 216 and 217 gridpoints.
NUMERICAL METHODS FOR 2D PROPAGATION
Typical time-indepdent methods such as S-matrix fail in 2D since stationary boundary conditions are not usually known in the second dimension.
We use a norm-preserving Crank-Nicolson approach to solve the TDSE, which consists of solving the finite difference equation:
1+𝑖 𝐻 2ℏ ∆𝑡 𝜓 𝒓,𝑡+Δ𝑡 = 1−𝑖 𝐻 2ℏ ∆𝑡 𝜓 𝒓,𝑡
Second derivatives are evaluated by three-point approximation,so that the Hamiltonian matrix H has the peculiar structure in Fig.5.
FIGURE 5: Shape of the Hamiltonian matrix with PBC in both dimensions. The main diagonal elements contain the potential, while the non-zero off-diagonal elements arise from the three point approximation to the second derivatives. White areas only contain zero elements. PBCs in x increase the band of the matrix, which does not occur for the y case.
ATOM-SURFACE INTERACTIONS
The interaction of an atom with a metallic surface is described by Casimir forces. These are caused by vacuum fluctuations in the energy of a quantised electric field. Boundary conditions imposed by the presence of the atom and the surface result in a reduced number of photon modes between the two, so that, a net attractive force acts between them. Depending on the distance to the surface, retardation effects reduce the strength of the interaction.
Computational costs are strongly asymmetric in the two dimensions:
However, the cost of periodic boundary conditions (PBC) in along y is negligible, hence this method lends itself to the study of QR off periodicallly corrugated surfaces.
FIGURE 3:.Potential responsible for atom-surface interactions(blue). The black curve includes the repulsive regime, while the dotted red curve includes the parabolic continuation.
CORRUGATED SURFACE POTENTIALS
At the present time, periodic geometries have not been solved analytically. Nevertheless, the form of the potential is inferred to be as shown in Fig.6.
The form of the Casimir-Polder potential for a flat surface geometry is given by:
The singularity of the potential at the surface is avoided by means of a parabolic continuation at a distance x0 (Fig.3). Such continuation induces oscillations in the behaviour of R, which die out as 𝑥0→0 .
FIGURE 2: 1D results for quantum reflection off a Casimir-Van der Waals potential. The dots are the values obtained via our 1D Crank Nicolson scheme, while the TI results are given in the blue curve. The correct value of R can be computed by averaging over the oscillations at different continuation points.
FIGURE 6: Expected form of the 2D Casimir-Van der Waals potential for a corrugated surface.
ACKNOWLEDGEMENTS
We thank the Atomic and Nuclear Physics group of the Phsyikalisches Institut in Heidelberg for the financial support to this project.
References
[1] B. Herwerth, M. DeKieviet, J. Madroñero, S. Wimberger, Quantum reflection from an oscillating surface, Journal of Physics B: Atomic, Molecular and Optical Physics 46, (2013)
[2] B. Herwerth, Dynamics of Quantum Reflection in Atom-Surface Interactions, Bachelor
Thesis (Institute for Theoretical Physics, Heidelberg, 2011).
[3] C. Sünderhauf, Quantum Reflection in Time and Space, Bachelor Thesis (University of Heidelberg (2014).