Abstract
We investigate the physics of quantum imaging with N > 2 entangled photons in position space. It is shown that, in paraxial approximation, the space-time propagation of the quantum state can be described by a generalized Huygens-Fresnel principle for the N-photon wave function. The formalism allows the initial conditions to be set on multiple reference planes, which is very convenient to describe the generation of multiple photon pairs in separate thin crystals. Applications involving state shaping and spatial entanglement swapping are developed.
Highlights
In the recent years, there has been a growing interest in producing quantum states of light in which more than two photons are entangled
We mean processing the photons through arbitrary optical systems that modify the wave fronts of the single-photon wave functions and possibly projects the N-photon state on a M-photon state (M < N)) by detecting some of the particles
We present a general and self-consistent formalism that describes the propagation of multi-photon states in position representation and uses methods closely related the Huygens-Fresnel principle and Fourier techniques in coherent optics
Summary
There has been a growing interest in producing quantum states of light in which more than two photons are entangled. In the case of an arbitrary optical system (made of lenses, mirror, masks, beam splitters, ...) between the crystal plane and the detection region, the use of the momentum representation for the field in the crystal plane will involve mixed propagators g(k, r j), where r j are the positions of the photon-counters and k wave vectors [22, 23] Such propagators arise because the photons generated through nonlinear interactions are first regarded as populating an ensemble of plane-wave modes (momentum representation). Application of standard Fourier optics techniques to the position representation wave function shows that the propagation of N photons through an optical system can be described by a generalized Huygens-Fresnel principle This has been first noticed for biphoton states produced by parametric down-conversion in nonlinear crystals [26,27,28].
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