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Physical Review A

Theory of two-photon down-conversion in the presence of mirrors

Peter W. Milonni, H. Fearn, Anton Zeilinger

Published Jun 1, 1996• 27 citations
Publisher

Abstract

A theory of two-photon down-conversion in the presence of mirrors is developed and applied to recent observations of Herzog et al. [Phys. Rev. Lett. 72, 629 (1994)]. The experimentally observed results for counting and coincidence rates as functions of mirror-crystal separations are obtained, and it is shown how the same results may be derived in a simplified formulation that presumes phase-matched signal and idler modes. We account for the observed effects of the finite coherence length of the pump field as well as the signal and idler coherence lengths, which are much shorter than the pump coherence length and have a different physical origin. Our analysis also supports the interpretation of the phenomena observed as being analogous to the modification of single photon spontaneous emission of atoms in cavity QED. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.4556

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A theory of two-photon down-conversion in the presence of mirrors is developed and applied to recent observations of Herzog et al. [Phys. Rev. Lett. 72, 629 (1994)]. The experimentally observed results for counting and coincidence rates as functions of mirror-crystal separations are obtained, and it is shown how the same results may be derived in a simplified formulation that presumes phase-matched signal and idler modes. We account for the observed effects of the finite coherence length of the pump field as well as the signal and idler coherence lengths, which are much shorter than the pump coherence length and have a different physical origin. Our analysis also supports the interpretation of the phenomena observed as being analogous to the modification of single photon spontaneous emission of atoms in cavity QED. \textcopyright{} 1996 The American Physical Society.
A proposed experiment is analyzed theoretically. In the proposed experiment two coherent pump waves fall on two identical nonlinear crystals, down-converted signal and idler beams from the two crystals are mixed by two beam splitters, and the coincidence counting rate for photons leaving the beam splitters is measured. We show that this counting rate depends on the phase difference between the two coherent pump waves, and results from the interference of the vacuum with the down-converted photons. The experiment could be used to look for locality violations along the lines recently proposed by Grangier, Potasek, and Yurke [Phys. Rev. A 38, 3132 (1988)], but without the need for a coherent reference beam for homodyning.
We report two-photon correlation experiments using spontaneous parametric down-conversion under a severe manipulation of the input pump field. Considering the case of passing the laser beam through a focusing lens before the down-conversion crystal, theoretical calculations and a series of imaging experiments demonstrate two-photon geometric optics effects. In particular, the imaging in coincidence counts of an aperture placed in one of the down-conversion beams is found to be the analog of a simple spherical mirror system, which displays a ``vacuum dispersion'' effect in that the object and image distances are wavelength weighted. \textcopyright{} 1996 The American Physical Society.
Using a multidimensional Gaussian approximation of the wave function for the signal and idler light generated by spontaneous parametric down-conversion, we derive analytical expressions for the second-order coherence function and the fourth-order coherence function (which is proportional to the signal-idler photon coincidence rate). The magnitudes of these functions are expressed as products of Gaussian functions of the azimuthal angles, the polar angles, and the time delay. Their widths determine six parameters: the coherence angles and coherence time, and the entanglement angles and entanglement time. We show how these parameters are governed by the pump-beam waist, the pump spectral width, and the crystal length. We thereby derive relations analogous to the van Cittert--Zernike theorem and the Siegert relation for thermal light. We show that the normalized photon coincidence rate decreases sharply as the signal and idler apertures become mismatched or misaligned. We experimentally confirm this latter prediction by using parametrically down-converted light obtained from a LiIO3{\mathrm{LiIO}}_{3}LiIO3​ crystal pumped by Kr+{\mathrm{Kr}}^{+}Kr+-ion laser radiation at 413 nm. \textcopyright{}1996 The American Physical Society.
A theoretical study is made of the process in which incident pump photons that interact with a nonlinear medium (such as a crystal lacking inversion symmetry) are spontaneously split into lower-frequency signal and idler photons. The down-converted fields are quantized and described by a continuum of modes, a subset of which interacts with each photodetector. It is shown that when two ideal photodetectors are appropriately located so that they receive the conjugate signal and idler photons, then the joint probability of two-photon detection by the two detectors can equal the single-photon detection probability. The time correlation between the two detected photons is shown to be limited either by the resolving time of the detectors, or by the bandwidth of the down-converted light, and to be independent of the coherence time of the pump field or of the length of the nonlinear medium. These conclusions are compared with the results of recent experiments.
Although the signal and idler light beams produced in the process of spontaneous parametric down-conversion are mutually incoherent and exhibit no one-photon interference, interference effects should be observable after either the signal or the idler is reflected by a phase-conjugate mirror. The degree of coherence can, in principle, be as large as unity. We refer to the relationship between the signal and the idler produced by down-conversion as anticoherent and introduce a natural measure for the degree of anticoherence. \textcopyright{} 1996 The American Physical Society.
In this paper we study photon pairs generated in a nonlinear cavity composed of a nonlinear crystal surrounded by mirrors, by the process of spotaneous parametric downconversion. We analyze two different regimes: singly-resonant cavities where the signal and idler modes are resonant and doubly-resonant cavities where the pump mode is also resonant. We present analytic expressions for the joint spectral amplitude in these two cases and study the reduction in emission bandwidth as the cavity coefficient finesse is increased. We also study the enhancement of the source brightness from the presence of the cavity.