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arXiv:2103.02674 · Physical Review A

Sorkin parameter for type-I spontaneous parametric down-conversion biphotons and matter waves

F. C. V. de Brito, Carlos H. S. Vieira, I. G. da Paz, Jonas B. Araujo, Marcos Sampaio

Published Mar 17, 2021• 1 citation
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Abstract

Type-I spontaneous parametric down-converted biphotons can be described approximately by a double-Gaussian wave function in configuration space. Using an effective propagator in the Fresnel approximation, the time evolution and transversal spreading of the two-particle biphoton wave function allow us to evaluate the Sorkin parameter $\ensuremath{\kappa}$, which results from nonclassical path contributions of the kink type and loops to double- and triple-slit interferometry. This simple unidimensional model for the evaluation of $\ensuremath{\kappa}$ predicts that kinked nonclassical paths may lead to $\ensuremath{\kappa}\ensuremath{\approx}{10}^{\ensuremath{-}5}$ for degenerate biphotons. We show that such a model reproduces well the Sorkin parameter for matter waves found in more involved approaches in the literature. Moreover, we establish a hierarchy of approximations based on the shape of the nonclassical paths for matter waves and compare their size with leading relativistic corrections to the propagator.

DOI
10.1103/physreva.103.033707

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Type-I spontaneous parametric down-converted biphotons can be described approximately by a double-Gaussian wave function in configuration space. Using an effective propagator in the Fresnel approximation, the time evolution and transversal spreading of the two-particle biphoton wave function allow us to evaluate the Sorkin parameter \ensuremathκ\ensuremath{\kappa}\ensuremathκ, which results from nonclassical path contributions of the kink type and loops to double- and triple-slit interferometry. This simple unidimensional model for the evaluation of \ensuremathκ\ensuremath{\kappa}\ensuremathκ predicts that kinked nonclassical paths may lead to \ensuremathκ\ensuremath≈10\ensuremath−5\ensuremath{\kappa}\ensuremath{\approx}{10}^{\ensuremath{-}5}\ensuremathκ\ensuremath≈10\ensuremath−5 for degenerate biphotons. We show that such a model reproduces well the Sorkin parameter for matter waves found in more involved approaches in the literature. Moreover, we establish a hierarchy of approximations based on the shape of the nonclassical paths for matter waves and compare their size with leading relativistic corrections to the propagator.
Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where LLL is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the sinc\mathrm{sinc}sinc behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of sinc\mathrm{sinc}sinc and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of $\ensuremath{\alpha}$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.
Quantum correlations encoded in photonic Laguerre-Gaussian modes were shown to be related to the Gouy-phase shifts [Kawase et al., Phys. Rev. Lett. 101, 050501 (2008)], allowing for a nondestructive manipulation of photonic quantum states. In this work, we exploit the relation between phase-space correlations of biphotons produced by spontaneously parametric down-conversion (SPDC) as encoded in the logarithmic negativity (LN) and the Gouy phase as they are diffracted through an asymmetrical double-slit setup. Using an analytical approach based on a double-Gaussian approximation for type-I SPDC biphotons, we show that measurements of Gouy-phase differences provide information on their phase-space entanglement variation, governed by the physical parameters of the experiment and expressed by the LN via covariance matrix elements.
We investigate theoretically the strong anisotropy in the wave packet distribution of biphoton pairs generated via spontaneous parametric down-conversion (SPDC) process with biaxial crystals. From the angle relationship in biaxial crystals, we deduce the wave function of type I SPDC by considering the longitudinal detuning under paraxial approximation. Taking BiB3O6 (BIBO) as an example, we numerically simulate the single-particle and coincidence distributions in two observation planes under the optimum phase-matching scheme. In contrast to uniaxial crystals, a stronger anisotropy effect and a higher entanglement of the photon pair states are obtained, which are attributed to the complex structure of biaxial crystals. These results are important for the generation of highly entangled biphoton pairs, especially for multi-photon pairs generation.
Abstract As a tutorial to the spatial aspects of spontaneous parametric downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.
As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.