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Quantum Optics Journal of the European Optical Society Part B

Photon localisation in parametric down-conversion

Edward R. Pike, Sarben Sarkar

Published Sep 1, 1989• 3 citations
Publisher

Abstract

A full multimode theory for parametric down-conversion is used to calculate the two-photon detection amplitude. From this amplitude a heuristic argument is given for the configuration space extent of the single photons that are involved. These single phonons have a configuration space localisation consistent with an asymptotic fall off with distance r or r -5/4 in the radial direction.

DOI
10.1088/0954-8998/1/1/007

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A full multimode theory for parametric down-conversion is used to calculate the two-photon detection amplitude. From this amplitude a heuristic argument is given for the configuration space extent of the single photons that are involved. These single phonons have a configuration space localisation consistent with an asymptotic fall off with distance r or r -5/4 in the radial direction.
A new theory for multimode parametric downconversion is given that yields differen, t predictions from existing treatments for photodetection tails. The theory is rigorous and is derived from first principles. The cases of both a single-atom and a nonlinear-crystal source are treated. The detection is taken to be far from the source. For a single atom the photodetection amplitude has a tail in the reduced time that falls off asymptotically as the inverse second power. The corresponding tail for the case of a crystal has an inverse quartic falloff.
With regard to the system of parametric downconversion, we analyze a recently proposed experiment for demonstrating the nonlocal property of a single photon.1 In the original proposal by Tan et al., a light field in a single photon-number state is split into two beams, which are then homodyned with two spatially separated weak local oscillators. The two-photon detection probability with the two homodyned beams provides evidence for nonlocality. It is shown that this phenomenon is in essence a phase-sensitive fourth-order interference effect. A single photonnumber state can be achieved from the pair of photons produced in parametric downconversion processes.2 A multimode model for the downconversion process is used to carry out the calculation of three-photon detection probability that is associated with nonlocality of a single photon.
Single-photon sources are necessary components for many prospective quantum technologies. One candidate for a single-photon source is spontaneous parametric down-conversion combined with a heralding photon detection. The heralded light pulse from such a source is typically treated as single mode; this treatment, however, is incomplete. We develop a full multimode description based on the exact Bogoliubov treatment of the down-conversion process. We then provide a perturbative and effective treatment, which illustrates the most important physical mechanisms and permits analytical estimates of the success probability and purity of single-photon states under practical heralding conditions, both without relying on the precise detection time of the heralding photon and when accepting photons only in a narrow window around the time of the detection. This permits us to characterize the emitted light under three different assumptions for the pump pulse. For spontaneous parametric down-conversion with a very short pump pulse, we find the single-mode description to be accurate, while for longer pump pulses and continuous pumping, a multimode description is necessary. Our findings can be used to guide the design of quantum information protocols based on heralded single-photon sources, as their performance may depend on the multimode nature of the sources.
Photon-based quantum information schemes have increased the need for light sources that produce individual photons, with many such schemes relying on optical parametric down-conversion (PDC). Practical realizations of this technology require that the PDC light be collected into a single spatial mode defined by an optical fibre. We present two possible models to describe single-mode fibres coupling with PDC light fields in a non-collinear configuration, that lead to two different results. These approaches include factors such as crystal length and walk-off, non-collinear phase-matching and transverse pump field distribution. We propose an experimental test to distinguish between the two. The goal is to help clarify open issues, such as how to extend the theory beyond the simplest experimental arrangements and, more importantly, to suggest ways to improve the collection efficiency.