Context for Parametric conversion in semi-classical optics

Expansion and references to the 2009 ABB conference abstract

While quantum theory is an incomplete description of reality (Einstein), its validity as a description of light/photons in non-linear crystals is questionable.  In the parametric down-conversion (PDC) phenomenon, a single-frequency laser beam
generates conjugate pairs of lower frequencies at particular angles, which in the photon description satisfy momentum conservation.  The beam is thought to contain single photon states, in which knowledge of the phase of the wave function is lost, since phase and number are conjugate quantities in quantum optics.  The
electromagnetic (Maxwell) theory of PDC depicts the crystal as an oscillator array represented by a complex refractive tensor whose non-linear properties result in secondary emissions at ‘down-converted’ frequencies.

Sulcs (2003) has reviewed differences that should distinguish between the two theories, and we focus here on a new and stronger difference. In the semi-classical (Maxwell) theory, two input frequencies (the second being background zero-point) combine to give a pair of output frequencies, satisfying a phase-matching relation.  Perturbation analysis predicts both the sum and difference frequencies, ie. PDC
and PUC (parametric up-conversion). PUC does not satisfy a ‘momentum’ condition.  We summarise the differences:

● only classical optics predicts PUC with frequencies above the pump-laser frequency

● quantum optics gives PDC angles from “conservation” of photon momentum, but has no similar photon language for PUC

● only classical optics explains changes with crystal temperature etc. (changing refractive parameters)

● quantum optics specifies equal ‘numbers’ at conjugate frequencies, classical optics describes emissions as correlated waves, in phase and amplitude, the intensities dependent on emission angle

● Semi-classical optics with zeropoint field (~ quantum vacuum fluctuations) explains spontaneous PDC and PUC, taking into account a detector threshold.

There is plenty of scope for experimental verification, which has been so far (ie. 2009) rather inconclusive, and further experimentation should differentiate between the theories. 

T W Marshall, E Santos, Semi-classical optics as an alternative to nonlocality,  Recent
Res. Devel. Optics
2, 683-717, 2002

T W Marshall, Parametric up conversion of the vacuum,1998.;

S Sulcs, The Nature Of Light And Twentieth Century Experimental Physics Foundations
of Science
8: 365–391, 2003.

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