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arXiv:2509.01792

Can We Conjugate the Orbital Angular Momentum of a Single Photon with Nonlinear Optics? A Theoretical Analysis

Tobey J. B. Luck, Aarón A. Aguilar-Cardoso, Cheng Li, Manuel F. Ferrer-Garcia, Jeremy Upham, Jeff S. Lundeen, Robert W. Boyd

Published Sep 1, 2025• 0 citations
arXivHTMLPDF

Abstract

Stimulated parametric down-conversion (StimPDC) produces an idler beam that exhibits a phase conjugate of the spatial structure of the classical seed beam. Extending this effect to the single-photon regime can, in principle, enable the control of the orbital angular momentum of single photons. However, its experimental implementation remains challenging due to the lower photon flux of realistic single-photon sources compared to a classical pump beam, which causes the StimPDC process to occur less frequently compared to its spontaneous counterpart. To overcome this, we propose a cascaded parametric down-conversion scheme, in which the signal photon from a spontaneously down-converted pair seeds the StimPDC process, while the idler photon heralds the seed's presence. This configuration uses coincidence detection to effectively post-select for stimulated events among the background of spontaneous contributions. We show that phase conjugation at the single-photon level manifests as a bias towards equal orbital angular momentum between the heralding idler and the stimulated idler. Our results offer a practical route to manipulate single-photon correlations in the orbital angular momentum degree of freedom and could have implications for applications such as free-space quantum communication and quantum-enhanced sensing.

DOI
10.48550/arxiv.2509.01792

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Stimulated parametric down-conversion (StimPDC) produces an idler beam that exhibits a phase conjugate of the spatial structure of the classical seed beam. Extending this effect to the single-photon regime can, in principle, enable the control of the orbital angular momentum of single photons. However, its experimental implementation remains challenging due to the lower photon flux of realistic single-photon sources compared to a classical pump beam, which causes the StimPDC process to occur less frequently compared to its spontaneous counterpart. To overcome this, we propose a cascaded parametric down-conversion scheme, in which the signal photon from a spontaneously down-converted pair seeds the StimPDC process, while the idler photon heralds the seed's presence. This configuration uses coincidence detection to effectively post-select for stimulated events among the background of spontaneous contributions. We show that phase conjugation at the single-photon level manifests as a bias towards equal orbital angular momentum between the heralding idler and the stimulated idler. Our results offer a practical route to manipulate single-photon correlations in the orbital angular momentum degree of freedom and could have implications for applications such as free-space quantum communication and quantum-enhanced sensing.
The ability to manipulate the orbital angular momentum (OAM) of a single photon is of vital importance to photonic quantum technologies. A particularly relevant operation is the phase conjugation of OAM. One can in principle achieve this using stimulated parametric down-conversion (StimPDC), which produces an idler beam that exhibits a phase conjugate of the seed beam’s spatial structure. However, implementing StimPDC for single-photon OAM phase conjugation remains challenging. This is because the photon flux of realistic single-photon sources is low compared to that of a classical pump beam, which causes the StimPDC process to occur less frequently compared to its spontaneous counterpart. To overcome this, we propose a cascaded parametric down-conversion scheme, in which the signal photon from a spontaneously down-converted pair seeds the StimPDC process, while the idler photon heralds the seed’s presence. This configuration uses coincidence detection to effectively post-select for stimulated events among the background of spontaneous contributions. We show that single-photon OAM phase conjugation manifests as a bias toward equal OAM between the heralding idler and the stimulated idler. Our results offer a practical route to manipulate single-photon correlations in the OAM degree of freedom and could have implications for applications such as free-space quantum communication and quantum-enhanced sensing.
Stimulated parametric down-conversion (StimPDC) is the seeded version of the process that underlies much of photonic quantum information. Replacing the vacuum at one input port with a coherent seed leaves the interaction Hamiltonian, the phase-matching function and the two-photon amplitude unchanged, while raising the output to levels accessible with ordinary classical detectors. This review treats the transverse spatial physics of that process. We show that a single relation, in which the idler amplitude is the pointwise product of the pump field with the complex conjugate of the seed, organizes results usually presented separately: the image and coherence transfer experiments of the 1990s; phase conjugation and its use for wavefront correction; the conservation and conjugation of orbital angular momentum, together with the generation of radial modes and geometric intensity patterns; and the processing of vector vortex beams carrying spatially varying polarization. We then consider StimPDC as an instrument rather than as an object of study, covering stimulated emission tomography, the experimental realization of Klyshko's advanced-wave picture, and the transfer of partial coherence, including the conservation of the twist phase of twisted Gaussian Schell-model beams. We close with the quantum regime, reached when the seed approaches the single-photon level or the idler is used as a herald, and with an assessment of open problems and prospective applications.
Stimulated parametric down-conversion (StimPDC) can act as a phase-conjugating nonlinear process when one of the down-converted modes is driven by a bright classical field. In this regime, the stimulated contribution to the nonlinear interaction dominates over the spontaneous background and the generated idler reproduces the conjugated phase of the seed with high fidelity. This behaviour has enabled classical turbulence-mitigation schemes in free-space optical channels, where a probe beam acquires a distorted wavefront that is deterministically conjugated in a nonlinear crystal and corrected upon reverse propagation. This thesis examines whether the same mechanism can be extended to the single-photon regime. The analysis begins by reviewing the spatial-mode structure of classical StimPDC using a Laguerre–Gaussian description of the nonlinear interaction. The turbulence-mitigation protocol is then summarized to clarify why high-fidelity phase conjugation is central to its operation. The main theoretical development introduces the single-photon–seeded version of the process, referred to as SPS-PDC. Using the full biphoton amplitude of SPS-PDC, the idler mean photon number is derived and shown to contain coherent spontaneous and stimulated contributions of comparable magnitude when only one photon occupies the seeded mode. Because the stimulated term no longer overwhelms the spontaneous background, the generated idler does not reproduce a unique conjugated wavefront. This result identifies a fundamental limit: perfect phase conjugation cannot be achieved with a single-photon seed in a second-order nonlinear interaction of this type. The implications of this scaling behaviour are examined in the context of turbulence mitigation and quantum key distribution. Although a single-photon version of the classical protocol would offer intrinsic security, the absence of deterministic conjugation prevents reliable wavefront correction. The thesis concludes by outlining an experimental approach for resolving the joint spatial structure of the two idler photons produced in a cascaded parametric down-conversion process. Such measurements would directly test the predicted division between spontaneous and stimulated contributions and further clarify the boundary between classical and quantum phase-conjugation regimes.
Identifying conservation laws is central to every subfield of physics, as they illuminate the underlying symmetries and fundamental principles. A prime example can be found in quantum optics: the conservation of orbital angular momentum (OAM) during spontaneous parametric down-conversion (SPDC) enables the generation of a photon pair with entangled OAM. In this Letter, we report on the observation of OAM conservation in SPDC pumped on the single-photon level by a preceding SPDC process. We implement this cascaded down-conversion scheme in free space, without waveguide confinement, and thereby set the stage for experiments on the direct generation of multiphoton high-dimensional entanglement using all degrees of freedom of light.
We show that an optical vortex beam, implemented classically, can be transferred to the transverse amplitude of a heralded single photon. For this purpose we have relied on the process of spontaneous parametric downconversion (SPDC) for the generation of signal and idler photon pairs, using a pump in the form of a Bessel-Gauss (BG) beam with orbital angular momentum (specifically, with topological charge l = 1 and l = 2). We have designed our source so that it operates within the short SPDC crystal regime for which, the amplitude and phase of the pump may be transferred to a heralded single photon. In order to verify the vortex nature of our heralded single photon, we have shown that the conditional angular spectrum and the transverse intensity at the single-photon level match similar measurements carried out for the pump. In addition, we have shown that when our heralded single photon is diffracted through a triangular aperture, the far-field single-photon transverse intensity exhibits the expected triangular arrangement of intensity lobes associated with the presence of orbital angular momentum.
Several wave-function approximations describing spontaneous parametric down-conversion can be found in the literature. Basically all cases are derived from the standard Hamiltonian for parametric down-conversion. Most frequently, particular cases describing collinear or paraxial approximations are described. This work presents a wave function in compact form, valid for all cases of single photon-pair conversion (Type I or Type II), for all angles allowed by the phase-matching conditions and for all orbital angular momentum values lll. Examples are given of coincidence structures to be expected for signal and idler photons. Partial transfer of orbital angular momentum from the pump laser to the photon pair is discussed. Some hypothesis for the decay channels of the nontransferred part of the orbital angular momentum is made.