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arXiv:2112.09610 · Physical Review A

Interferometric signature of different spectral symmetries of biphoton states

Nicolas Fabre

Published May 20, 2022• 5 citations
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Abstract

In this paper, we investigate the influence of the symmetry of the biphoton wave function on the coincidence measurement of the generalized Mach-Zehnder (MZ) interferometer, where there are a temporal and frequency shift operations between the two beam splitters. We show that the generalized MZ interferometer allows the measurement of the short-time Fourier transform of the function modeling the energy conservation of a spontaneous parametric down-conversion process if the full biphoton state is symmetric, and of the symmetric characteristic distribution of the phase-matching function if the state is antisymmetric. Thus, this technique is phase sensitive to the spectral distribution of the photon pairs. Finally, we investigate in detail the signature of a pair of anyons whose peculiar statistics can be simulated by engineering the spectrum of photon pairs.

DOI
10.1103/physreva.105.053716

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In this paper, we investigate the influence of the symmetry of the biphoton wave function on the coincidence measurement of the generalized Mach-Zehnder (MZ) interferometer, where there are a temporal and frequency shift operations between the two beam splitters. We show that the generalized MZ interferometer allows the measurement of the short-time Fourier transform of the function modeling the energy conservation of a spontaneous parametric down-conversion process if the full biphoton state is symmetric, and of the symmetric characteristic distribution of the phase-matching function if the state is antisymmetric. Thus, this technique is phase sensitive to the spectral distribution of the photon pairs. Finally, we investigate in detail the signature of a pair of anyons whose peculiar statistics can be simulated by engineering the spectrum of photon pairs.
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.
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.
Spectral and temporal coincidence and single-particle photon wave packets are described and their widths and durations are found. The degree of entanglement is characterized by the experimentally measurable parameter R defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration. This function is shown to have a minimum and even in the minimum, at rather natural conditions, the parameter R is found to be very high (R=73>>1). The Schmidt number K is found for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of R and K are found to be identical and numerical difference between them is shown to be not exceeding 20%. Two-time temporal wave function of a biphoton state is investigated in details, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter R is defined as the ratio of durations of the single-particle and coincidence signals, and the defined in such a way parameter R is shown to be very close to the Schmidt number K.
We propose a simple scheme to swap the non-local correlations, characteristic of a Franson interferometric setup, between pairs of frequency entangled photons emitted by distinct nonlinear crystals in a parametric downconversion process. Our scheme consists of two distinct sources of frequency entangled photons. One photon of each pair is sent to a separate Mach--Zehnder interferometer while the other photons of the pairs are mixed by a beam splitter and then detected in an Ou--Mandel interferometer. For suitably postselected joint measurements, the photons sent at the Mach--Zehnder show a coincidence photocount statistics which depends non-locally on the settings of the two interferometers.
Fourth-order interference patterns of biphotons generated by type-I spontaneous parametric downconversion are examined for beams with partial angular and spectral entanglement, both for degenerate and nondegenerate constituent photons. Two-beam configurations using a single beam-splitter interferometer and a Mach-Zehnder interferometer are explicitly studied, as are four-beam configurations using pairs of beam splitters and Mach-Zehnder interferometers. The interference pattern generally comprises a number of harmonic functions of the path-length difference with frequencies and visibilities that usually depend on direction, and may have a finite or infinite duration (fourth-order coherence length). The relation between the visibility and the spectral indistinguishability of the two beams is established. Certain components of the interference pattern are independent of direction so that they are not washed out by the use of apertures of finite size.
We propose theoretically a quantum interferometer in which the NOON state interferometer is combined with the Hong-Ou-Mandel interferometer. With this interferometer, the temporal interference patterns associated with biphoton frequency sum and difference can be shown in different parts of a single interferogram. It is thus possible to simultaneously obtain the spectral correlation information of biphotons in both frequency sum and difference by taking the Fourier transform from such an interferogram. This provides a method for complete spectral characterization of an arbitrary two-photon state with exchange symmetry and might be useful in quantum Fourier-transform spectroscopy where direct spectral measurement is difficult. Furthermore, as it can realize the measurement of time intervals on three scales at the same time, we expect that it can provide a method in quantum metrology. Finally, we discuss another potential application of such an interferometer in the generation and characterization of high-dimensional and phase-controlled frequency entanglement.