Kapitsa–Dirac effect
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The Kapitza–Dirac effect is a
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
effect consisting of the diffraction of matter by a standing wave of light. The effect was first predicted as the diffraction of electrons from a standing wave of light by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
and
Pyotr Kapitsa Pyotr Leonidovich Kapitsa or Peter Kapitza ( Russian: Пётр Леонидович Капица, Romanian: Petre Capița ( – 8 April 1984) was a leading Soviet physicist and Nobel laureate, best known for his work in low-temperature physics ...
(or Peter Kapitza) in 1933. The effect relies on the wave–particle duality of matter as stated by the de Broglie hypothesis in 1924.


Explanation

In 1924, French physicist
Louis de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to Old quantum theory, quantum theory. In his 1924 PhD thesis, he pos ...
postulated that matter exhibits a wave-like nature given by: :\lambda = \frac h p, where ''λ'' is the particle wavelength, ''h'' is the Planck constant, and ''p'' is the particle momentum. From this, it follows that interference effects between particles of matter will occur. This forms the basis of the Kapitza–Dirac effect. Specifically, Kapitza–Dirac scattering operates in the Raman–Nath regime. This is to say that the interaction time of the particle with the light field is sufficiently short in duration such that the motion of the particles with respect to the light field can be neglected. Mathematically, this means the kinetic energy term of the interaction Hamiltonian can be neglected. This approximation holds if the interaction time is less than the inverse of the recoil frequency of the particle, \tau\ll 1/\omega_\text. This is analogous to the thin lens approximation in optics. A coherent beam of particles incident on a standing wave of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
(typically light) will be diffracted according to the equation: : n\lambda = 2d\sin\Theta, where ''n'' is an integer, ''λ'' is the de Broglie wavelength of the incident particles, ''d'' is the spacing of the grating and ''θ'' is the angle of incidence. This matter wave diffraction is analogous to optical diffraction of light through a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structur ...
. Another incidence of this effect is the diffraction of ultra-cold (and therefore almost stationary) atoms by an
optical lattice An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congrega ...
that is pulsed on for a very short duration. The application of an optical lattice transfers momentum from the photons creating the optical lattice onto the atoms. This momentum transfer is a two-photon process meaning that the atoms acquire momentum in multiples of 2ħk, where ''k'' is the wavevector of the electromagnetic. The recoil frequency of the atom as can be expressed by: :\omega_\text = \frac where ''m'' is the mass of the particle. The recoil energy is given by :E_\text=\hbar \omega_\text.


Mathematics

The following is based on the mathematical description by Gupta ''et al.'' The AC Stark shift of the standing wave potential can be expressed as :U(z,t)=\fracf^2(t)\sin^2(kz), where \omega_\text is the single-photon Rabi frequency and the detuning of the light field \delta \gg \Gamma^2/4 (\Gamma is particle resonance). The particle
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
immediately after interaction with the light field is given by :\left, \psi\right\rangle = \left, \psi_0\right\rangle e^ = \left, \psi_0\right\rangle e^ e^, where \tau=\int dt'f^2(t') and the integral is over the duration of the interaction. Using the identity for Bessel functions of the first kind, e^ = \sum^\infty_i^n J_n(\alpha)e^, the above
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
becomes : \begin \left, \psi\right\rangle & = \left, \psi_0\right\rangle e^ \sum^\infty_i^nJ_n \left( \frac\tau\right) e^ \\ & = e^ \sum^\infty_ i^n J_n\left( \frac\tau\right) \left, g,2n\hbar k\right\rangle \end It can now be seen that 2n\hbar k momentum states are populated with a probability of P_n = J^2_n(\theta) where n = 0,\pm 1, \pm 2, \ldots and the pulse area (duration and amplitude of the interaction) \theta = \frac\tau = \omega^_\text\tau. The transverse RMS momentum of the diffracted particles is therefore linearly proportional to the pulse area: p_\text = \sum^\infty_ (n\hbar k)^2 P_n = \sqrt\theta\hbar k.


Realisation

The invention of the
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
in 1960 allowed the production of coherent light and therefore the ability to construct the standing waves of light that are required to observe the effect experimentally. Kapitsa–Dirac scattering of sodium atoms by a near resonant standing wave laser field was experimentally demonstrated in 1985 by the group of D. E. Pritchard at the Massachusetts Institute of Technology. A supersonic atomic beam with sub-recoil transverse momentum was passed through a near resonant standing wave and diffraction up to 10ħk was observed. The scattering of electrons by an intense optical standing wave was experimentally realised by the group of M. Bashkansky at AT&T Bell Laboratories, New Jersey, in 1988.


References

Diffraction Quantum mechanics Paul Dirac {{quantum-stub