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Physical Review A

Characterization of opening angle correlations of a biphoton state decomposed in Bessel modes

Baghdasar Baghdasaryan, Fabian Steinlechner, S. Fritzsche

Published Apr 29, 2020• 4 citations
Publisher

Abstract

The spontaneous parametric down-conversion of photons has been widely applied for generating entangled photon pairs. We theoretically explore the entangled down-converted state of the photon pair, also known as biphoton state, for both degenerate and nondegenerate photon pairs. In particular, the spatial structure of the biphoton state has been expressed in Bessel modes to better understand the correlation with regard to the opening angle ${\ensuremath{\vartheta}}_{k}$ of Bessel modes. In fact, the opening angles of the down-converted photon pair are not independent of each other, but rather are correlated. Furthermore, we confirm the experimentally observed conditions concerning the optimization of generating high-degree spatial entanglement by controlling the beam waist of the pump beam. We also introduce a new experimental setup for efficient measurement of the spatial entanglement of the biphoton state using geometrical optics arguments.

DOI
10.1103/physreva.101.043844

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The spontaneous parametric down-conversion of photons has been widely applied for generating entangled photon pairs. We theoretically explore the entangled down-converted state of the photon pair, also known as biphoton state, for both degenerate and nondegenerate photon pairs. In particular, the spatial structure of the biphoton state has been expressed in Bessel modes to better understand the correlation with regard to the opening angle \ensuremathϑk{\ensuremath{\vartheta}}_{k}\ensuremathϑk​ of Bessel modes. In fact, the opening angles of the down-converted photon pair are not independent of each other, but rather are correlated. Furthermore, we confirm the experimentally observed conditions concerning the optimization of generating high-degree spatial entanglement by controlling the beam waist of the pump beam. We also introduce a new experimental setup for efficient measurement of the spatial entanglement of the biphoton state using geometrical optics arguments.
In this paper we consider geometrical two-photon optics of Bessel-Gaussian modes generated in spontaneous parametric down-conversion of a Gaussian pump beam. We provide a general theoretical expression for the orbital angular momentum (OAM) spectrum and Schmidt number in this basis and show how this may be varied by control over the radial degree of freedom, a continuous parameter in Bessel-Gaussian modes. As a test we first implement a back-projection technique to classically predict, by experiment, the quantum correlations for Bessel-Gaussian modes produced by three holographic masks: a blazed axicon, a binary axicon, and a binary Bessel function. We then proceed to test the theory on the down-converted photons using the binary Bessel mask. We experimentally quantify the number of usable OAM modes and confirm the theoretical prediction of a flattening in the OAM spectrum and a concomitant increase in the OAM bandwidth. The results have implications for the control of dimensionality in quantum states.
Abstract We experimentally demonstrate the quantum entanglement in various spatial modes of the correlated photons generated via spontaneous parametric down-conversion in a type-II phase-matched periodically-poled KTP crystal. The two-fold coincidence measurement reveals the spatial entanglement between the photon pairs by projecting one of the correlated photons onto a Gaussian mode while scanning the spatial mode of its partner photon through selective projection (Gaussian, Laguerre–Gaussian (LG), Bessel-Gaussian (BG), and Airy–Gaussian (AG)) using a spatial light modulator (SLM). We also show the spatially resolved measurement of the non-conservation of orbital angular momentum via modal decomposition of the correlated photon in LG and BG bases.
Partially coherent optical beams present phenomena that may not appear in perfectly coherent beams. In this regard, spontaneous parametric downconversion is an intriguing physical process, since the optical properties of the pump laser beam are transferred to the quantum properties of the down-converted photon pairs. Here, the authors study the transfer of twist phase, a novel property of the so-called twisted Gaussian Schell model beams, from a pump beam to down-converted photons. The authors show that a non-zero twist phase can lead to highly entangled yet highly mixed states. The correlations that appear in near/far-field intensity measurements, in the two-photon covariance matrix as well as in orbital angular momentum modes, are explored. The authors derive expressions for the purity of the two-photon state and relate it to the transverse spatial coherence of the pump beam. Their main results utilize a Wigner function formalism that can be applied to a number of scenarios involving partially coherent downconversion sources and contribute to the recent interest in spatially correlated photon pairs produced in mixed states, with possible applications in quantum communication and imaging.
High-dimensional spatial entanglement generated via spontaneous parametric down-conversion (SPDC) provides a powerful resource for quantum information processing, yet its mode structure becomes increasingly complex when the pump occupies a higher-order Laguerre--Gaussian (LG) mode. Here, we develop an algebraic framework for characterizing biphoton spatial-mode entanglement by factorizing LG-pumped SPDC into a spatial-mode beam-splitter transformation followed by spatial two-mode squeezing. By choosing the biphoton LG basis radius as the geometric mean of the pump-beam radius and the crystal-induced correlation width, we obtain a natural modal basis in which the two operations can be treated separately. When the pump-beam radius matches the correlation width, spatial squeezing vanishes, and the two circular-mode components of the pump are independently conserved across the signal and idler photons, leading to conservation of both orbital angular momentum and the total spatial mode number. We further derive an analytical expression for the Schmidt number for arbitrary LG pump modes and squeezing strengths, revealing how the radial and azimuthal pump indices govern the dimensionality of the spatial entanglement. This algebraic characterization provides a unified description of the mode structure, conservation laws, and entanglement dimensionality of higher-order-LG-pumped SPDC.
Spontaneous parametric downconversion is the primary source to generate entangled photon pairs in quantum photonics laboratories. Depending on the experimental design, the generated photon pairs can be correlated in the frequency spectrum, polarization, position-momentum, and spatial modes. Exploring the spatial modes' correlation has hitherto been limited to the polar coordinates' azimuthal angle, and a few attempts to study Walsh mode's radial states. Here, we study the full-mode correlation, on a Laguerre-Gauss basis, between photon pairs generated in a type-I crystal. Furthermore, we explore the effect of a structured pump beam possessing different spatial modes onto bi-photon spatial correlation. Finally, we use the capability to project over arbitrary spatial mode superpositions to perform the bi-photon state's full quantum tomography in a 16-dimensional subspace.
Spontaneous parametric down-conversion (SPDC) is one of the most important sources of entangled photons in quantum optics and underpins a wide range of applications in quantum communication, quantum imaging, quantum sensing, and photonic quantum information processing. In this nonlinear optical process, a pump photon is converted into a pair of lower-energy photons that exhibit strong correlations in multiple degrees of freedom. While the generation of photon pairs is governed by energy and momentum conservation, the resulting biphoton state possesses a rich spatial structure whose properties are determined by the interplay among the pump field, the nonlinear medium, and subsequent propagation. This thesis investigates the spatial properties of entangled photon pairs generated through SPDC, focusing on the evolution of their correlations in the near field and on the behavior of orbital-angular-momentum (OAM) entangled states under tight focusing conditions. A common theme throughout this work is the examination of physical effects that emerge when the full biphoton wavefunction is retained rather than reduced through the simplifying approximations often employed in theoretical treatments. The principal contribution of this thesis concerns the near-field propagation of SPDC photon pairs. The spatial correlations of SPDC-generated photons are commonly analyzed within the thin-crystal approximation, where the biphoton wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. We demonstrate that birefringence-induced transverse walk-off breaks this factorization and introduces an intrinsic coupling between these coordinates. More importantly, this coupling survives free-space propagation and remains observable even for crystals that nominally satisfy the thin-crystal condition. As a result, the joint spatial correlations develop a characteristic tapering near the crystal image plane that cannot be captured by conventional factorized models. Numerical simulations and experimental measurements are found to be in excellent agreement, revealing a previously overlooked feature of SPDC and providing a more complete description of photon-pair generation and propagation in birefringent nonlinear media. The second part of the thesis explores the propagation of OAM-entangled photon pairs beyond the paraxial regime. While OAM correlations are typically studied under weak-focusing conditions, many emerging quantum technologies require structured quantum states to be manipulated in highly confined optical systems. We show that tight focusing modifies the OAM correlation spectrum, leading to a broadening of the accessible mode distribution. This broadening exhibits distinct transverse and longitudinal characteristics and arises from the emergence of nonparaxial field components absent in conventional scalar descriptions. These results provide new insight into the behavior of structured quantum light in strongly focused optical fields and establish a framework for understanding OAM entanglement beyond the paraxial approximation. Together, the results presented in this thesis reveal previously unexplored aspects of the structure and propagation of entangled photon pairs. By moving beyond widely used approximations, this work uncovers physical effects that influence the generation, evolution, and measurement of spatially entangled states, with implications for quantum imaging, high-dimensional quantum communication, and the broader development of structured-light-based quantum technologies.