**Aberrations of parametric down-conversion probabilities in generation of entangled photon pairs.**The quantum states of light have long been playing an important role in the physics of lasers and parametric processes in quantum optics, e.g. in noise reduction in measurements. Concerning quantum imaging [1] or quantum information science [2], the application of entangled photon pairs has become a part of everyday practice. The subject of the talk belongs to this quite broad area of research, and the results to be presented may be useful in several sub-areas.

As an introduction, a brief summary of some basic photon correlation experiments will be given. Then, we shall discuss the photon statistics of the entangled photon pairs in a close connection with the quantum phase formalism already introduced earlier [3]. Recently we have derived the probability amplitudes of general squeezing transitions [4] and the photon statistics of a degenerate parametric amplifier. In the meantime, we have used an analogous mathematical procedure and [3] for describing the non-degenerate down-conversion process, resulting in ‘Einstein-Podolsky-Rosen (EPR)-type entangled photon pairs’. We will show that in a special case, the derived probability amplitudes reduce to the Zernike functions, well-known in the classical theory of aberration in optical imaging [5]. In the case under discussion, the ‘aberration’ stems from induced emission, so it quantifies the deviation from the spontaneous process. The derived ‘quantum Zernike polynomials’ form a complete set of orthogonal aberration functions of the phase-space variables of the photons. Finally, we point out that, though these functions do not depend on the coordinates in real space where the electromagnetic wave propagates, they likely leave ‘fingerprints’ of noise on the wavefront. Thus, the ‘quantum Zernike moments’ may be used for quality-assessment of parametric sources.

References.

[1] Lugiato L A, Gatti A and Brambilla E, Quantum imaging. J. Opt. B: Quantum Semiclass. Opt. 4 (2002) S176–S183.

[2] > The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics 2022 to Alain Aspect, John F. Clauser and Anton Zeilinger; “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”. < https://www.nobelprize.org/uploads/2022/10/press-physicsprize2022-2.pdf

[3] Varró S : Regular phase operator and SU(1,1) coherent states of the harmonic oscillator. Physica Scripta 90, 074053 (2015).

[4] Varró S, Coherent and incoherent superposition of transition matrix elements of the squeezing operator. New Journal of Physics 24, 053035 (2022).

[5] Born M and Wolf E, Principles of Optics (Cambridge University Press, Cambridge, 2002).