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Frontiers in Optics

Phase-Space Optics, One Photon at a Time

Harrison H. Barrett, Luca Caucci, Kyle J. Myers

Published Jan 1, 2015• 0 citations
Publisher

Abstract

The relationship between the Wigner Distribution Function and the spectral photon radiance is discussed for both physical optics and quantum optics. Computational methods and applications are surveyed, and photon-by-photon phase-space imagers are introduced.
DOI
10.1364/fio.2015.fw5g.5

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The Wigner distribution function (WDF) is a quasi-probability distribution that provides a phase-space representation of quantum states. It offers insights into the state dynamics and contains statistical information of it, such as the marginal probabilities. This thesis investigates a protocol for estimating the WDF of quantum states using symplectic transformations, which are linear canonical transformations that preserve phase-space structure and can be implemented experimentally using the technique of Gaussian optics. These transformations are studied because the time evolution of the WDF can be described by means of these transformations in cases where the Hamiltonian is a second-order (quadratic) polynomial in the phase-space variables. Furthermore, it was found that the Fourier transform of the marginal probability (measured at the laboratory) is related to the characteristic function (Fourier transform of the Wigner function). This relationship is given by Equation 2.46, which states that, with each intensity data obtained by applying a given symplectic transformation, the value of the characteristic function (Fourier transform of the WDF) at a given point of the phase space can be obtained. Based on this, an optical setup was implemented to perform symplectic transformations, such as free-space propagation and passing through a lens. The study focuses on the WDF of a one-dimensional electric field of two Gaussian beams, separated by a distance 2d, where d = 1.388 ± 0.004mm. The two beams are passed through the transformation system (propagation-lens-propagation) and then the marginal probability (intensity) is measured with a CCD camera. After this, the Fourier transform was computationally applied to the intensity data and compared with the theoretical plots of the characteristic function (Equation 2.2). In doing so, it was found that it is necessary to multiply the data obtained by an additional scaling factor to improve them to better fit the theoretical results. Finally, the protocol to be followed to reconstruct the Wigner function of this state with the obtained data was described.