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Journal of the Optical Society of America B

Quantum optical formulation of difference-frequency generation and optimal cloning of structured light

Tanita Permaul, Andrew Forbes, Thomas Konrad

Published Feb 6, 2026• 0 citations
Publisher

Abstract

We present a quantum optical formulation of difference-frequency generation (DFG) that incorporates the spatial modes of light. It reproduces the well-established result for classical light beams and establishes the relation of DFG to stimulated and spontaneous parametric down-conversion. As shown here, these relations determine that stimulated parametric down-conversion can realize N → M d -dimensional optimal quantum cloning.

DOI
10.1364/josab.583026

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We present a quantum optical formulation of difference-frequency generation (DFG) that incorporates the spatial modes of light. It reproduces the well-established result for classical light beams and establishes the relation of DFG to stimulated and spontaneous parametric down-conversion. As shown here, these relations determine that stimulated parametric down-conversion can realize N → M d -dimensional optimal quantum cloning.
We present a quantum optical formulation of difference-frequency generation (DFG) that incorporates the spatial modes of light. It reproduces the well established result for classical light beams and establishes the relation of DFG to stimulated and spontaneous parametric downconversion. As shown here, these relations determine that stimulated parametric down-conversion can realise $N\xrightarrow{} M $ ddd-dimensional optimal quantum cloning.
We model spontaneous parametric down-conversion (SPDC) as classical difference frequency generation (DFG) of the pump field and a hypothetical stochastic ``vacuum'' seed field. We analytically show that the second-order spatiotemporal correlations of the field generated from the DFG process replicate those of the signal field from SPDC. Specifically, for low gain, the model is consistent with the quantum calculation of the signal photon's reduced density matrix; and for high gain, the model's predictions are in good agreement with our experimental measurements of the far-field intensity profile, orbital angular momentum spectrum, and wavelength spectrum of the SPDC field for increasing pump strengths. We further theoretically show that the model successfully captures second-order SU(1,1) interference and induced coherence effects in both gain regimes. Intriguingly, the model also correctly predicts the linear scaling of the interference visibility with object transmittance in the low-gain regime---a feature that is often regarded as a quintessential signature of the nonclassicality of induced coherence. Our model may not only lead to fundamental insights into the classical-quantum divide in the context of SPDC and induced coherence, but can also be a useful theoretical tool for numerous experiments and applications based on SPDC.
We analyze the dynamics of sum-frequency generation (SFG) and difference-frequency generation (DFG) for broadband input fields. We demonstrate that amplitude modulation of the input fields significantly affects conversion efficiency of SFG and DFG, but frequency modulation does not affect conversion efficiency. Analytic results for SFG and DFG output-field intensities are thereby available for arbitrary fields. Self- and cross-phase modulation can affect the dynamics in the limit of very high field strengths, but these effects are of higher nonlinearity and therefore usually negligible. The quantum efficiency of DFG is shown to be generally higher than SFG, but the energy efficiency is much lower if the difference frequency is significantly less than the input frequencies. Optimal SFG quantum conversion efficiency for such systems approaches that of second-harmonic generation only if the quantum fluence (total number of photons) in the two input beams are equal. Optimal DFG quantum conversion efficiency occurs when the number of photons of frequency ${\mathrm{\ensuremath{\omega}}}_{2}$ is small yet sufficient for significant stimulation. A recently developed method for obtaining efficient SFG for multimode input fields is theoretically analyzed. The method involves using an arrangement with two or more nonlinear mixing crystals with a time-delay line situated between the crystals that delays one of the fundamental fields relative to the other. The efficiency in the second crystal depends on the cross-correlation function of the two fundamental fields upon leaving the first nonlinear crystal. The time-delay method is not effective for multimode DFG.
Two-dimensional lattices of the optical nonlinearity implemented by periodic poling inχ(2)\chi^{(2)}χ(2)materials afford additional degrees of freedom for enhanced gains in optical parametric generation [1–2] and the generation of multipartite photon entanglement in compact device formats [3]. So far all investigations have considered single-frequency excitation of the crystals in classical or quantum parametric downconversion regimes. However, the effects underpinned by the coherent coupling of multiple nonlinear resonances unique to 2D lattices, may be expected to further enlarge the spectrum of possibilities also in suitably seeded coherent parametric amplification and difference frequency generation (DFG) processes.
Spontaneous parametric down-conversion in quantum optics is an invaluable resource for the realization of high-dimensional qudits with spatial modes of light. One of the main open challenges is how to directly generate a desirable qudit state in the SPDC process. This problem can be addressed through advanced computational learning methods; however, due to difficulties in modeling the SPDC process by a fully differentiable algorithm that takes into account all interaction effects, progress has been limited. Here, we overcome these limitations and introduce a physically-constrained and differentiable model, validated against experimental results for shaped pump beams and structured crystals, capable of learning every interaction parameter in the process. We avoid any restrictions induced by the stochastic nature of our physical model and integrate the dynamic equations governing the evolution under the SPDC Hamiltonian. We solve the inverse problem of designing a nonlinear quantum optical system that achieves the desired quantum state of down-converted photon pairs. The desired states are defined using either the second-order correlations between different spatial modes or by specifying the required density matrix. By learning nonlinear volume holograms as well as different pump shapes, we successfully show how to generate maximally entangled states. Furthermore, we simulate all-optical coherent control over the generated quantum state by actively changing the profile of the pump beam. Our work can be useful for applications such as novel designs of high-dimensional quantum key distribution and quantum information processing protocols. In addition, our method can be readily applied for controlling other degrees of freedom of light in the SPDC process, such as the spectral and temporal properties, and may even be used in condensed-matter systems having a similar interaction Hamiltonian.