Quantum Aberrations: Entangling Photons with Zernike Polynomials

Abstract

We introduce Zernike polynomials as a novel degree of freedom for encoding quantum information in the spatial structure of photons. Building on their orthogonality and completeness over the unit disc, we develop a framework for generating, manipulating, and detecting photons in Zernike modes, and propose methods for realizing single-photon and two-photon Zernike wave packets. We demonstrate analytically that two-photon states generated via spontaneous parametric down- conversion exhibit mode entanglement in the Zernike basis, with correlations arising from selection rules enforced by Clebsch-Gordan coefficients. Our results open a new pathway for structured spatial entanglement, complementary to schemes based on Laguerre-Gaussian or Hermite-Gaussian modes, and suggest practical experimental implementations based on holographic modulation and optical Fourier techniques.

Figures (2)

• Figure 1: The basic optical configuration. A wavefront in the exit pupil coming from a point in the object plane (not shown) converges towards the image plane. The position on the exit pupil is defined by the polar coordinates $(\rho, \theta)$; the position in the image plane region is defined by the polar coordinate system $(q, \phi)$.
• Figure 2: Schematic of Two-Photon Zernike mode entanglement via SPDC. A pump beam prepared in a Zernike mode $Z_n^m$ illuminates a thin nonlinear crystal, producing signal and idler photons whose spatial modes are correlated. The dashed lines in front of the pupils represent the interferograms pictured near them. The picture near the non-linear crystal (NLC) represents the propagated pump profile according to Eq. \ref{['ZFraunh']}. The conservation of azimuthal quantum number $(m = m_1 + m_2)$ and triangle conditions for $n-$ indices, arising from Clebsch–Gordan coefficients, govern the resulting two-photon entangled state in the Zernike basis.