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arXiv:quant-ph/0605149 · Physical Review A

Two-photon wave mechanics

Brian J. Smith, Michael G. Raymer

Published Dec 7, 2006• 97 citations
arXivHTMLPDF

Abstract

The position-representation wave function for multiphoton states and its equation of motion are introduced. A major strength of the theory is that it describes the complete evolution (including polarization and entanglement) of multiphoton states propagating through inhomogeneous media. As a demonstration of the two-photon wave function's use, we show how two photons in an orbital-angular-momentum entangled state decohere upon propagation through a turbulent atmosphere.

DOI
10.1103/physreva.74.062104

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The position-representation wave function for multiphoton states and its equation of motion are introduced. A major strength of the theory is that it describes the complete evolution (including polarization and entanglement) of multiphoton states propagating through inhomogeneous media. As a demonstration of the two-photon wave function's use, we show how two photons in an orbital-angular-momentum entangled state decohere upon propagation through a turbulent atmosphere.
The evolution of an entangled photon state propagating through a turbulent atmosphere is formulated in terms of a set of coupled first order differential equations, by using an infinitesimal propagation approach. The orbital angular momentum (OAM) basis is used to described the density matrix of the state. Although the analysis is done in the paraxial limit for a monochromatic optical field, the formalism is comprehensive in the sense that it does not require any assumptions about the strength of the turbulence and it can incorporate any spectral model for the turbulence. As a comparative example the case of entangled qubit OAM biphoton states is considered.
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of entanglement duality and develop schemes to do entanglement sorting based on either OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.
Previously a set of coupled first order differential equations were derived for the decoherence of a pair of spatial mode entangled photons, propagating along different paths through turbulence. Here we extend this analysis to the situation where both photons travel along the same path, which introduces the effect of multiple photon correlations. The resulting equation now contains additional terms that take these multiphoton correlations into account. At the same time, we provide a more thorough formulation of the quantized field, starting from a Lorentz invariant formulation, which is then explicitly broken by the choice of a particular propagation direction. The effect of the latter improvement in the quantization on the form of the final equation is minimal.
Abstract The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.