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CRC Press eBooks

Entangled Polarizations: Probability Amplitudes and Experimental Configurations

F.J. Duarte

Published Oct 29, 2013• 0 citations

Abstract

This chapter provides the ubiquitous equation for the probability amplitude of polarization entanglement of two photons moving away in different directions, from a common source. The first derivation is performed from a Hamiltonian approach using the two-state infrastructure taught to by R. P. Feynman. The second derivation utilizes an interferometric approach, while the third approach is based on the original analysis used by Ward. The initial link between quantum mechanical concepts and the polarization correlation of photons propagating in opposite directions was given by Wheeler: “According to the pair theory, if one of these photons is polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.” The theory of entangled quantum polarizations, central to A. Einstein, B. Podolsky, and N. Rosen-type optical experiments, was established in the 1947–1949 period by M. H. Pryce and J. C. Ward and independently by H. S. Snyder et al.

DOI: 10.1201/b16055-17

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This chapter provides the ubiquitous equation for the probability amplitude of polarization entanglement of two photons moving away in different directions, from a common source. The first derivation is performed from a Hamiltonian approach using the two-state infrastructure taught to by R. P. Feynman. The second derivation utilizes an interferometric approach, while the third approach is based on the original analysis used by Ward. The initial link between quantum mechanical concepts and the polarization correlation of photons propagating in opposite directions was given by Wheeler: “According to the pair theory, if one of these photons is polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.” The theory of entangled quantum polarizations, central to A. Einstein, B. Podolsky, and N. Rosen-type optical experiments, was established in the 1947–1949 period by M. H. Pryce and J. C. Ward and independently by H. S. Snyder et al.

The experimental origin of quantum mechanics is highlighted, and essential features of the photon are described. The philosophical path to quantum entanglement, as defined by the works of EPR–Bohm and Aharonov–Bell, and the physics path to quantum entanglement, as defined by the works of Dirac–Wheeler–Pryce and Ward, are introduced and discussed. The probability amplitude for quantum Entanglement, discovered in 1947, is then introduced prior to a high-level overview of the field. This introduction then concludes with a chapter-by- chapter description of the book and its purpose.

The experimental origin of quantum mechanics is highlighted and essential features of the photon are described. The philosophical path to quantum entanglement, as defined by the works of EPR to Bohm and Aharonov to Bell, and the physics path to quantum entanglement, as defined by the works of Dirac to Wheeler to Pryce and Ward, are introduced and discussed. The probability amplitude for quantum entanglement, discovered in 1947, is then introduced prior to a high-level overview of the field. The introduction then concludes with a chapter-by-chapter description of this book and its aims.

An explanation of polarization entanglement is presented using Maxwells classical electromagnetic theory.Two key features are required to understand these classical origins.The first is that all waves diffract and weakly diffracting waves,with a principal direction of propagation in the laboratory frame, travel along that direction at speeds ever so slightly less than c.This allows nontrivial Lorentz transformations that can act on selected forward F waves or selected waves R traveling in the opposite direction to show that both can arise from a single zero momentum frame where all the waves are transverse to the original principal direction.Such F and R waves then both belong to a single relativistic entity where correlations between the two are unremarkable.The second feature requires the avoidance of using the Coulomb gauge.Waves, tending to plane waves in the limit of zero diffraction,can then be shown to be composed of two coupled sets of E and B fields that demonstrate the classical entanglement of F and R waves.Being derived from Maxwells equations,the theory is compatible with special relativity.This is used to account for entanglement between waves traveling at arbitrary angles from a source.In spite of a classical explanation,selection of appropriate F and R waves means entanglement is likely to remain a quantum phenomenon.

One of the most intriguing concepts of quantum mechanics is quantum entanglement. Two physical systems are said to be entangled with respect to a certain variable, if their individual outcomes of the variable are undetermined before measurement, but strictly correlated. Measurement of the variable for only one system immediately determines the outcome for the other system, irrespective of the distance between the two systems. It is this strong correlation that provides a basis for perspective, powerful applications like quantum cryptography and quantum computation. The most studied entangled systems, often called EPR systems (after the pioneers Einstein, Podolsky and Rosen), are pairs of light particles or photon pairs. These photon pairs, usually generated via the optical process of a spontaneous parametric down-conversion, are, in principal, simultaneously entangled in three variables, being polarization (polarization entanglement), energy or time (time entanglement), and momentum or transverse space (spatial entanglement). The contents of this thesis cover new insights in the polarization and spatial entanglement of photons. We have investigated both the entanglement quality and the yield of entangled photons for different experimental geometries that are associated with the production and detection of the photons.