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Journal of Physics A Mathematical and Theoretical

Probability amplitudes for broadbandN-photon processes in linear optics

Jesús Urı́as, Nehemías Leija

Published Jul 18, 2012• 0 citations

Abstract

Advances in the production of multiple-photon states with a broadband spectrum and its subsequent processing through linear optical systems demand simple rules to automate the calculation of probability amplitudes. A methodology to write down directly the probability amplitude of any broadband multi-photon process in linear optics is proved. A physically intuitive notion of multi-photon paths is derived from Wick's theorem and amplitudes are formulated as sums over paths. Every permutation of the out-state photon frequencies over the photon occupation indices that the in-state wavepacket assigns to the inports contributes a path. A simple general rule is provided to compute them. Broadband two-photon processes in the interferometers by Hong, Ou and Mandel (HOM) and by Mach and Zehnder (MZ) are considered. We provide a rigorous description of the HOM interferometer as a probe of the amount of frequency entanglement contained in some two-photon wavepackets, establishing an integral transform between the interference profile and the spectral density of frequency-entangled photon pairs. For the MZ interferometer, our method provides the exact rate of coincidence counts for a broadband two-photon process that is relevant for pairs produced with a wide two-dimensional spectral bandwidth by spontaneous parametric downconversion.

DOI: 10.1088/1751-8113/45/30/305303

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Advances in the production of multiple-photon states with a broadband spectrum and its subsequent processing through linear optical systems demand simple rules to automate the calculation of probability amplitudes. A methodology to write down directly the probability amplitude of any broadband multi-photon process in linear optics is proved. A physically intuitive notion of multi-photon paths is derived from Wick's theorem and amplitudes are formulated as sums over paths. Every permutation of the out-state photon frequencies over the photon occupation indices that the in-state wavepacket assigns to the inports contributes a path. A simple general rule is provided to compute them. Broadband two-photon processes in the interferometers by Hong, Ou and Mandel (HOM) and by Mach and Zehnder (MZ) are considered. We provide a rigorous description of the HOM interferometer as a probe of the amount of frequency entanglement contained in some two-photon wavepackets, establishing an integral transform between the interference profile and the spectral density of frequency-entangled photon pairs. For the MZ interferometer, our method provides the exact rate of coincidence counts for a broadband two-photon process that is relevant for pairs produced with a wide two-dimensional spectral bandwidth by spontaneous parametric downconversion.

The prominent role that linear optical networks have acquired in the engineering of photon states calls for physically intuitive and automatic methods to compute the probability amplitudes for the multiphoton quantum processes occurring in linear optics. A version of Wick's theorem for the expectation value, on any vector state, of products of linear operators, in general, is proved. We use it to extract the combinatorics of any multiphoton quantum processes in linear optics. The result is presented as a concise rule to write down directly explicit formulae for the probability amplitude of any multiphoton process in linear optics. The rule achieves a considerable simplification and provides an intuitive physical insight about quantum multiphoton processes. The methodology is applied to the generation of high-photon-number entangled states by interferometrically mixing coherent light with spontaneously down-converted light.

Richard Feynman’s method of path integrals is based on the fundamental assumption that a system starting at a point 𝐴𝐴 and arriving at a point 𝐵𝐵 takes all possible paths from 𝐴𝐴 to 𝐵𝐵, with each path contributing its own (complex) probability amplitude. The sum of the amplitudes over all these paths then yields the overall probability amplitude that the system starting at 𝐴𝐴 would end up at 𝐵𝐵. We apply Feynman’s method to several optical systems of practical interest and discuss the nuances of the method as well as instances where the predicted outcomes agree or disagree with those of classical optical theory. Examples include the properties of beam-splitters, passage of single photons through Mach-Zehnder and Sagnac interferometers, electric and magnetic dipole scattering, reciprocity, time-reversal symmetry, the optical theorem, the Ewald-Oseen extinction theorem, far field diffraction, and the two-photon interference phenomenon known as the Hong-Ou-Mandel effect.

We present experimental observations of a nonclassic and nonlocal interference effect with frequency-entangled photon pairs. We observed two-photon interference in a Hong–Ou–Mandel interferometer by monitoring the coincidence-counting rate as we varied the path-length difference or the relative time delay between two photons. The quantum-mechanical probabilities of joint detection by two detectors that respond to different frequency components exhibit differences that depend not only on the arrangement of the detector pairs but on the spectral filtering. The experimental results provide more understanding of two-photon entanglement and of the interference effects that arise from superposition of indistinguishable probability amplitudes.

We propose and demonstrate a method for measuring the joint spectrum of photon pairs via Fourier spectroscopy. The biphoton spectral intensity is computed from a two-dimensional interferogram of coincidence counts. The method has been implemented for a type-I downconversion source using a pair of common-path Mach-Zehnder interferometers based on Soleil compensators. The experimental results agree well with calculated frequency correlations that take into account the effects of coupling into single-mode fibers. The Fourier method is advantageous over scanning spectrometry when detectors exhibit high dark count rates leading to dominant additive noise.

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Abstract Frequency entangled biphoton sources have played an important role in quantum information. There are two key characteristics which determine the extent of their applications: the spectral indistinguishability and the degree of frequency entanglement. Previously, the spectral indistinguishability of the biphoton state is measured by the Hong–Ou–Mandel (HOM) interference visibility, whereas the degree of frequency entanglement is normally characterized with independent spectral measurements. However, these two quantum features have not been successfully measured with a single experimental setup. In this paper, we deduce the temporal distribution of a short-pulse pumped type-II spontaneous parametric down converted (SPDC) biphoton source and its relevant temporal entanglement parameter R t for operationally quantifying the degree of frequency entanglement. Furthermore, we theoretically and experimentally demonstrate, due to the inherent group delay, the difference between the signal and idler photons at the exit of the type-II nonlinear crystal, resulting in a HOM-shape dip at both sides of the center fringed envelope of the MZ interferometric coincidence diagram. By measuring the HOM-shape dip depth, the separation between the two sideband dips as well as the MZ fringed envelope width, both the spectral indistinguishability and the temporal entanglement parameter can thus be simultaneously quantified. This implementation provides us a unified and convenient method to characterize the frequency entanglement of the short-pulse pumped biphotons.