Quantum reflection

Quantum reflection is physical phenomenon of specular reflection of a matter wave from the material surfaces. In the simplest case, such a surface can be represented with some attractive potential. This phenomenon can be described in terms of quantum mechanics and has no classical analogy.

For observation of quantum reflection, the atoms of speed of few meters per second are used. Usually, the magneto-optical atomic traps [MOP] are used to cool the atoms.

Reflection of slow atoms
Although the principles of quantum mechanics apply to any particles, usually the term quantum reflection means reflection of atoms from a surface of condensed matter (liquid or solid).

The effective potential for the incident atom becomes repulsive at a very small distance from the surface (of order of size of atoms). At such a scale, the surface does not look as a uniform plane; the atom would scatters at other atoms, thus would cause the diffuse reflection.

At small values of the normal component of velocity, wavelength of the atom is large compared to the thickness of the layer, where the potential varies from almost zero to values huge compared to the kinetic energy of the atom. In these conditions, the specular reflection is possible.

In order to reduce the normal component of velocity, a grazing angle of incidence is used; this enhances the quantum reflection. This requirement of small incident velocities for the particles means that the non-relativistic approximation to quantum mechanics is all that is required.

Single-dimensional approximation
Many features of the quantum reflection can be described in the single–dimensional approximation. This implies that the potential has translational symmetry in two directions (say $$y$$ and $$z$$), such that only a single coordinate (say $$x$$) is important .

In the first approximation, the interaction between an atom and the surface can be described in terms of the van der Waals attraction, and the related Casimir-Polder interaction attracts the atom to the surface of the material. This force dominates when the atom is comparatively far from the surface, while the repulsion takes place only at the distance or orderof atomic size. In the intermediate region, the reflection may show properties specific for the material of the surface and the kind (and state) or atom, and do not allow the simple description.

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 $$x$$ axis,
 * \( \displaystyle \lambda\left(x\right)=\frac{h}{\sqrt{2m\big(E-V(x)\big)}} \)

where \(~m~\) is the atomic mass, $$~E~$$ is its energy, and $$~V(x)~$$ is the potential it experiences, then it is clear that we cannot give meaning to this quantity where,

\left|\frac{\mathrm{d}\lambda(x)}{\mathrm{d}x}\right|\sim 1 $$

In region, where the variation of the atomic wavelength is significant over its own length (i.e. the gradient of $$V(x)$$ is steep), there is no meaning in the approximation of a local wavelength. This breakdown occurs irrespective of the sign of the potential, $$~V(x)~$$. 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. At any 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.

Experiments with grazing incidence


Practically, in many experiments with quantum reflection from Si, the grazing incidence angle is used (figure 1).

The set-up is mounted in a vacuum chamber to provide several meter free path of atoms; the good vacuum (at the level of 10−7 mm Hg ) is required. The magneto-optical trap (MOT) is used to collect cold atoms, usually excited He or Ne, approaching the point-like source of atoms.

The excitation of atoms is not essential for the quantum reflection but it allows the efficient trapping and cooling using optical frequencies.

In addition, the excitation of atoms allows the registration at the micro-channel plate (MCP) detector (bottom of the figure).

The movable edges are used to stop atoms which do not go toward the sample (for example a Si plate), providing the collimated atomic beam.

At the MCP, there was observed relatively intensive strip of atoms which come straightly (without reflection) from the MOT, by-passing the sample, strong shadow of the sample. The thickness of this shadow could be used for rough control of the grazing angle. In order to improve the control of the orientation of the sample, the He-Ne laser was used to measure the grazing angle $$~\theta~$$.

The relatively weak strip produced by the reflected atoms. The ratio $$~r~$$ of density of atoms registered at the center of this strip to the density of atoms at the directly illuminated region is considered as efficiency of quantum reflection, i.e., reflectivity. The reflectivity strongly depends on the grazing angle and speed of atoms.

In the experiments with Ne atoms, they usually just fall down, when the MOT is suddenly switched-off. Then, the speed of atoms is determined as $$~v=\sqrt{2gh}~$$, where $$~g~$$ is acceleration of free fall, and $$~h~$$ is distance from the MOT to the sample. In experiments described, this distance was of order of 0.5 meter, providing the speed of order of 3 m/s. Then, the transversal wavenumber can be calculated as $$~k=\sin(\theta)\frac{mv}{\hbar}~$$, where $$~m~$$ is mass of the atom, and  $$\hbar$$ is the Planck constant.

In experiments with He, the additional resonant laser could be used to release the atoms and provide them an additional velocity; the delay since the release of the atoms till the registration allowed to estimate this additional velocity; roughly, $$~v=\frac{1}{t\!~h}~$$, where $$~t~$$ is time delay since the release of atoms till the click at the detector. Practically, $$v$$ could vary from 20 m/s to 130 m/s.

Although the scheme at the figure looks simple, the extend facility is necessary to slow atoms, trap them and cool to millikelvin temperature, providing a micrometre size source of cold atoms. Practically, the mounting and maintaining of this facility (not shown in the figure) is the heaviest part of the job. The possibility of an experiment with the quantum reflection with just a pinhole instead of MOT are discussed in the literature.

Casimir and van der Waals attraction
Despite this, there is some doubt as to the physical origin of quantum reflection from solid surfaces. As was briefly mentioned above, the potential in the intermediate region between the regions dominated by the Casimir-Polder and Van der Waals interactions requires an explicit Quantum Electrodynamical calculation for the particular state and type of atom incident on the surface. Such a calculation is very difficult. Indeed, there is no reason to suppose that this potential is solely attractive within the intermediate region.

Furthermore, a similar dependence for reflectivity on the incident velocity can be simulated with a non-Hermitian potential (i.e. one where probability is not conserved). Such an approximation could take into account both, the absorption of atoms at the surface and the relaxation of the metastable state due to the break of the symmetry (that prohibits the relaxation) at the surface.

Until 2006, the published papers interpreted the reflection in terms of a Hermitian potential this assumption allows to build a quantitative theory ..

The analysis of the reflection due to the relaxation of excited atoms may be good subject for the theoretical and experimental research in this field.

Efficient quantum reflection


A qualitative estimate for the efficiency of quantum reflection follows from the dimensional analysis. Let $$m$$ be mass of the atom and $$k=2\pi/\lambda$$ be the normal component of its wave-vector, then the energy of the normal motion of the particle,


 * $$E=\frac{(\hbar k)^2}{2m}$$

should be compared to the potential, $$V(x)$$ of interaction. The distance, $$w=|x|$$ at which $$E=V(x)$$ can be considered as the distance where the atom will come across a strong variation of the potential. At this point the WKB approximation begins to fail. The condition for efficient quantum reflection can be written as $$kw<1$$. In other words, the wavelength is small compared to the distance at which the atom may become reflected from the surface. This indicates that roughly, the reflectivity should be determined y the product \(kw\). The simple fit
 * $$r=\frac{1}{(1+kw)^4}$$

shows very good agreement with experimental data for excited neon and helium atoms, reflected from a flat silicon surface (fig.2), see and references therein. Such a fit is also in good agreement with a single-dimensional analysis of the scattering of atoms from an attractive potential.

The success of the simple fit indicates, that, at least in the case of noble gases and Si surface, the quantum reflection can be described with single-dimensional Hermitian potential, as the result of attraction of atoms to the surface.

Ridged mirror
The effect of quantum reflection can be enhanced using ridged mirrors .

If one produces a surface consisting of a set of narrow ridges, then the resulting non-uniformity of the material allows the reduction of the effective van der Waals constant; this extends the working ranges of the grazing angle.

For this reduction to be valid, we must have small distances, $$L$$ between the ridges. Where $$L$$ becomes large, the non-uniformity is such that the ridged mirror must be interpreted in terms of multiple Fresnel diffraction or the Zeno effect; these interpretations give similar estimates for the reflectivity . See ridged mirror for the details.

Similar enhancement of quantum reflection takes place where one has particles incident on an array of pillars . This was observed with very slow atoms (Bose-Einstein condensate) at almost normal incidence.

Application of quantum reflection
Quantum reflection makes the idea of solid-state atomic mirrors and atomic-beam imaging systems (atomic nanoscope) possible. The use of quantum reflection in the production of atomic traps has also been suggested. Up to year 2011, no commercial application of quantum reflection is reported.