Quantum Reflection - Single-dimensional Approximation

Single-dimensional Approximation

So far, one usually considers the single-dimensional case of this phenomenon, that is when the potential has translational symmetry in two directions (say and ), such that only a single coordinate (say ) is important. In this case one can examine the specular reflection of a slow neutral atom from a solid state surface . Where one has an atom in a region of free space close to a material capable of being polarized, a combination of the pure van der Waals interaction, and the related Casimir-Polder interaction attracts the atom to the surface of the material. The latter force dominates when the atom is comparatively far from the surface, and the former when the atom comes closer to the surface. The intermediate region is controversial as it is dependent upon the specific nature and quantum state of the incident atom.

The condition for a reflection to occur as the atom experiences the attractive potential can be given by the presence of regions of space where the WKB approximation to the atomic wave-function breaks down. If, in accordance with this approximation we write the wavelength of the gross motion of the atom system toward the surface as a quantity local to every region along the axis,

where is the atomic mass, is its energy, and is the potential it experiences, then it is clear that we cannot give meaning to this quantity where,

That is, in regions of space where the variation of the atomic wavelength is significant over its own length (i.e. the gradient of is steep), there is no meaning in the approximation of a local wavelength. This breakdown occurs irrespective of the sign of the potential, . In such regions part of the incident atom wave-function may become reflected. Such a reflection may occur for slow atoms experiencing the comparatively rapid variation of the van der Waals potential near the material surface. This is just the same kind of phenomenon as occurs when light passes from a material of one refractive index to another of a significantly different index over a small region of space. Irrespective of the sign of the difference in index, there will be a reflected component of the light from the interface. Indeed, quantum reflection from the surface of solid-state wafer allows one to make the quantum optical analogue of a mirror - the atomic mirror - to a high precision.