# Quantum reflection

Quantum reflection

Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical physics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, you do not expect one of the magnets to suddenly (i.e. before the magnets `touch') turn around and retreat in the opposite direction.

Definition

Quantum reflection becomes important branch of physics in XXI century; special workshop Quantum Reflection, workshop;October 22-24, 2007, Cambridge, Massachusetts, USA; http://cfa-www.harvard.edu/itamp/QuantumReflection.html ] is dedicated to this topic. The preamble suggests the following definition of quantum reflection:

Quantum reflection is a classically counterintuitive phenomenon whereby the motion of particles is reverted "against the force" acting on them. This effect manifests the wave nature of particles and influences collisions of ultracold atoms and interaction of atoms with solid surfaces.

Observation of quantum reflection has become possible thanks to recent advances in trapping and cooling atoms. Utilization of this effect has only begun and holds many exciting promises.

Reflection of slow atoms

Although the principles of quantum mechanics apply to any particles, usually the termquantum reflection means reflection of atoms from a surface of condensed matter (liquid or solid).It should be noted that the full potential experienced by the incident atom "does" become repulsive at a very small distance from the surface (of order of size of atoms). This is when the atom becomes aware of the discrete character of material. This repulsion is responsible for the classical scattering one would expect for particles incident on a surface. Such scattering is diffuse rather than specular, and so this component of the reflection is easy to distinguish. Indeed to reduce this part of the physical process, 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.…

ingle-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 $y$ and $z$), such that only a single coordinate (say $x$) is important. In this case one can examine the specular reflection of a slow neutral atom from a solid state surfacecite journal
url=http://prola.aps.org/abstract/PRL/v86/i6/p987_1
author= F.Shimizu
comment=6
title=Specular Reflection of Very Slow Metastable Neon Atoms from a Solid Surface
journal=PRL
volume=86
pages=987–990
year=2001
doi=10.1103/PhysRevLett.86.987
] cite journal
url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLRAAN000071000005052901000001&idtype=cvips&gifs=yes
author=H.Oberst
coauthors=Y.Tashiro, K.Shimizu, F.Shimizu
title=Quantum reflection of He* on silicon
journal=PRA
volume=71
pages=052901
year=2005
doi=10.1103/PhysRevA.71.052901
] .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 $x$ axis,

::$lambdaleft\left(x ight\right)=frac\left\{h\right\}\left\{sqrt\left\{2mleft\left(E-Vleft\left(x ight\right) ight\right)$

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

::$left|frac\left\{dlambdaleft\left(x ight\right)\right\}\left\{dx\right\} ight|sim 1$

That is, in regions of space where the variation of the atomic wavelength is significant over its own length (i.e. the gradient of $V\left(x\right)$ is steep), there is no meaning in the approximation of a local wavelength. This breakdown occurs "irrespective of the sign of the potential", $~V\left(x\right)~$. 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.

Experiments with grazing incidence

Practically, in many experiments with quantum reflection from Si, the grazing incidence angle is used (figure 0). 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). Movable edges are used to stop atoms which do not go toward the sample (for example a Si plate), providing the collimated atomic beam. The He-Ne laser was used to control the orientation of the sample and measure the grazing angle $~ heta~$. 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), and 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 was considered as efficiency of quantum reflection, i.e., reflectivity. This reflectivity strongly depends on the grazing angle and speed of atoms.

In the experiments with Ne atoms, usually just fall down, when the MOP is suddenly switched-off. Then, the speed of atoms is determined as $~v=sqrt\left\{2gh\right\}~$, 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 $~k=sin\left( heta\right)frac\left\{mv\right\}\left\{hbar\right\}~$, where $~m~$ is mass of the atom, and $hbar$ is the Planck constant.

In the case 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, $~v=frac\left\{1\right\}\left\{t!~h\right\}~$, 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 cite journal
url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000094000001013203000001&idtype=cvips&gifs=yes
author= H.Oberst
coauthors=D.Kouznetsov, K.Shimizu, J.Fujita, and F.Shimizu
title=Fresnel Diffraction Mirror for an Atomic Wave
journal=PRL
volume=94
pages=013203
year=2005
doi=10.1103/PhysRevLett.94.013203
] cite journal
url=http://annex.jsap.or.jp/OSJ/opticalreview/TOC-Lists/vol12/12e0363tx.htm
author= D.Kouznetsov
coauthors= H.Oberst
title=Reflection of Waves from a Ridged Surface and the Zeno Effect
journal=Optical Review
volume=12
pages=1605–1623
year=2005
doi=10.1007/s10043-005-0363-9
] cite journal
url=http://stacks.iop.org/0953-4075/39/1605
author= D.Kouznetsov
coauthors=H.Oberst, K.Shimizu, A.Neumann, Y.Kuznetsova, J.-F.Bisson, K.Ueda, S.R.J.Brueck
title=Ridged atomic mirrors and atomic nanoscope
journal=JOPB
volume=39
pages=1605–1623
year=2006
doi=10.1088/0953-4075/39/7/005
] .

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 job in the experiments with quantum reflection of cold atoms. The possibility of an experiment with the quantum reflection with just a pinhole instead of MOT are discussed in the literaturecite journal
url=http://stacks.iop.org/0953-4075/39/1605
author= D.Kouznetsov
coauthors=H.Oberst, K.Shimizu, A.Neumann, Y.Kuznetsova, J.-F.Bisson, K.Ueda, S.R.J.Brueck
title=Ridged atomic mirrors and atomic nanoscope
journal=JOPB
volume=39
pages=1605–1623
year=2006
doi=10.1088/0953-4075/39/7/005
] .

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. Thus the reflection could simply be explained by a repulsive force, which would make the phenomenon not quite so surprising. Furthermore, a similar dependence for reflectivity on the incident velocity is observed in the case of the adsorption of particles in vicinity of a surface. In the simplest case, such absorption could be described with a non-Hermitian potential (i.e. one where probability is not conserved). Until 2006, the published papers interpreted the reflection in terms of a Hermitian potentialcite journal
author=H.Friedrich
coauthors=G.Jacoby, C.G.Meister
journal=PRA
title=quantum reflection by Casimir–van der Waals potential tails
year=2002
volume=65
url=http://prola.aps.org/abstract/PRA/v65/i3/e032902
pages=032902
doi=10.1103/PhysRevA.65.032902
] ;this assumption allows to build a quantitative theorycite journal
author=F.Arnecke
title=Effective-range theory for quantum reflection amplitudes
journal=PRA
volume=74
pages=062702
year=2006
doi=10.1103/PhysRevA.74.062702
] .

Efficient quantum reflection

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

::$E=frac\left\{\left(hbar k\right)^2\right\}\left\{2m\right\}$

should be compared to the potential, $V\left(x\right)$ of interaction. The distance, $|x_\left\{t\right\}|$ at which $E=V\left(x\right)$ can be considered as the distance the which the atom will come across a troublesome discontinuity in the potential. This is the point at which the WKB method truly becomes nonsense. The condition for efficient quantum reflection can be written as $k|x_\left\{t\right\}|<1$. In other words the wavelength is small compared to the distance at which the atom may become reflected from the surface. If this condition holds, the aforementioned effect of the discrete character of the surface may be neglected. This argument produces a simple estimate for the reflectivity, $r$,

::$r=frac\left\{1\right\}\left\{\left(1+k|x_\left\{t\right\}|\right)^4\right\}$

which shows good agreement with experimental data for excited neon and helium atoms, reflected from a flat silicon surface (fig.1), seecite journal
url=http://stacks.iop.org/0953-4075/39/1605
author= D.Kouznetsov
coauthors=H. Oberst, K. Shimizu, A. Neumann, Y. Kuznetsova, J.-F. Bisson, K. Ueda, S. R. J. Brueck
title=Ridged atomic mirrors and atomic nanoscope
journal=JOPB
volume=39
pages=1605–1623
year=2006
doi=10.1088/0953-4075/39/7/005
] 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,cite journal
coauthors=H.Friedrich
title=Influence of realistic atom wall potentials in quantum reflection traps
journal=PRA
volume=75
pages=022902
year=2007
doi=10.1103/PhysRevA.75.022902
] . Such agreement indicates, that, at least in the case of noble gases and Si surface, the quantun 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 cite journal
comment=8
author= F.Shimizu
coauthors=J. Fujita
title=Giant Quantum Reflection of Neon Atoms from a Ridged Silicon Surface
journal=Journal of the Physical Society of Japan
volume=71
pages=5–8
year=2002
doi=10.1143/JPSJ.71.5
] .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 cite journal
url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000094000001013203000001&idtype=cvips&gifs=yes
author= H.Oberst
coauthors=D.Kouznetsov, K.Shimizu, J.Fujita, and F. Shimizu
title=Fresnel Diffraction Mirror for an Atomic Wave
journal=PRL
volume=94
pages=013203
year=2005
doi=10.1103/PhysRevLett.94.013203
] orthe Zeno effect cite journal
url=http://annex.jsap.or.jp/OSJ/opticalreview/TOC-Lists/vol12/12e0363tx.htm
author= D.Kouznetsov
coauthors= H.Oberst
title=Reflection of Waves from a Ridged Surface and the Zeno Effect
journal=Optical Review
volume=12
pages=1605–1623
year=2005
doi=10.1007/s10043-005-0363-9
] ; these interpretations give similar estimates for the reflectivitycite journal
comment=8
url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLRAAN000072000001013617000001&idtype=cvips&gifs=yes
free=http://www.ils.uec.ac.jp/~dima/PhysRevA_72_013617.pdf
author= D.Kouznetsov
coauthors=H.Oberst
title=Scattering of waves at ridged mirrors.
journal=PRA
volume=72
pages=013617
year=2005
doi=10.1103/PhysRevA.72.013617
] . See ridged mirror for the details.

Similar enhancement of quantum reflection takes place where one has particles incident on an array of pillarscite journal
author= T.A.Pasquini
coauthors=M.Saba, G.-B.Jo, Y.Shin, W.Ketterle, D.E.Pritchard,T.A.Savas, N. Mulders.
title= Low Velocity Quantum Reflection of Bose-Einstein Condensate
journal=PRL
volume=97
pages=093201
year=2006
doi= 10.1103/PhysRevLett.97.093201
] .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) possiblenanoscope] . The use of quantum reflection in the production of atomic traps has also been suggestedcite journal
url=http://www.ils.uec.ac.jp/~dima
coauthors=H.Friedrich
title=Influence of realistic atom wall potentials in quantum reflection traps
journal=PRA
volume=75
pages=022902
year=2007
doi=10.1103/PhysRevA.75.022902
] . Up to year 2007, no commercial application of quantum reflection is reported.

References

*atom optics
*ridged mirror
*Casimir force
*van der Waals potential
*slow atoms
*cold atoms

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