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Journal of the European Optical Society Rapid Publications

Twin-photon generation measured and modeled over 10 orders of magnitude of pump intensity

Julien Bertrand, Véronique Boutou, Benoı̂t Boulanger

Published Jan 1, 2025• 2 citations
Publisher

Abstract

We proposed a semi-classical model of spontaneous parametric down-conversion (SPDC) based on both the quantum fluctuations of vacuum for the seeding of the process and a propagation step ensured by classical fields. We obtained an agreement over a broad pump intensity range, i.e., 2.4 W cm −2 to 3.7 GW cm −2 , which led to the generation of twin-photons with a flux from 1.1 × 10 4 Hz to 1.2 × 10 21 Hz, respectively. The predictivity of our model is much better than the best published quantum models at high intensity, while it is similar at low intensity.

DOI
10.1051/jeos/2025045

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We proposed a semi-classical model of spontaneous parametric down-conversion (SPDC) based on both the quantum fluctuations of vacuum for the seeding of the process and a propagation step ensured by classical fields. We obtained an agreement over a broad pump intensity range, i.e., 2.4 W cm −2 to 3.7 GW cm −2 , which led to the generation of twin-photons with a flux from 1.1 × 10 4 Hz to 1.2 × 10 21 Hz, respectively. The predictivity of our model is much better than the best published quantum models at high intensity, while it is similar at low intensity.
We performed a second-order spontaneous parametric down-conversion (SPDC) experiment in a Type II phase-matched KTiOPO4 crystal pumped at 532 nm giving birth to a signal beam at 1037 nm and an idler one at 1092 nm along the x-axis of the crystal. Three pump sources have been used: in the continuous-wave, nanosecond (200 ns, 1 kHz) and picosecond (15 ps, 10 Hz) regimes. That allowed us to cover a pump intensity range from 2.4 W.cm-2 to 3.7 GW.cm-2 which led to the generation of twin photons with a flux from 1.1x104 Hz to 1.2x1021 Hz, respectively, the corresponding quantum efficiencies being of 1.8x10-11 and 9.2x10-4. We identified two SPDC regimes regarding the pump intensity: one below 30 MW.cm-2 for which the flux increases linearly, and the other one above 30 MW.cm-2 for which the flux behaves exponentially. We proposed a semi-classical model under the undepleted pump approximation based on both the quantum fluctuations of vacuum for the seeding of the process and a propagation step ensured by classical fields, which allowed a satisfying description of these two regimes, with a much better agreement at high intensity than the quantum models based on the nonlinear momentum or the nonlinear Hamiltonian operators.
This article is a theoretical and quantitative exploration of the limit regarding the pump intensity between the two regimes of spontaneous-parametric down-conversion (SPDC) as well as of four-wave-mixing (FWM) in the framework of a semi-classical model and analytical calculations. A dimensionless parameter has been defined at this limit, corresponding to the photon-pairs flux per frequency unit: it has been found equal to 0.369. The ratio between the electric field of the generated photons and the quantum fluctuations of vacuum calculated at the limit is equal to 1.718. These quantitative results confirm that below the limit, the pump photon splitting leading to photon-pairs can be considered as spontaneous, i.e. mainly seeded by the quantum fluctuations of vacuum, while it is stimulated by the generated signal and idler photons above the limit, which corresponds to an optical parametric amplification regime. Our calculations also show that this limit can be easily reached in the case of SPDC according to the typical values of non-linearities and available crystal dimensions. In the case of FWM, it would be only possible in kilometric optical fibers. This corpus is a useful tool box for designing further quantum experiments performed from either side of the limit, as well as at the limit it-self where the influence of the quantum fluctuations of vacuum and of the generated photons should have the same weight. Furthermore, the quantum significance of the numerical values of the two criteria defined here remains to be established, which should motivate future theoretical quantum studies.
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.
This article is a theoretical and quantitative exploration of the limit regarding the pump intensity between the two regimes of spontaneous-parametric down-conversion (SPDC) as well as of four-wave-mixing (FWM) in the framework of a semi-classical model and analytical calculations. A dimensionless parameter has been defined at this limit, corresponding to the photon-pairs flux per frequency unit: the corresponding number of signal and idler photons was found close to unity. The ratio between the electric field of the generated photons and the quantum fluctuations of vacuum calculated at the limit is also close to unity. These quantitative results confirm that below the limit, the pump photon splitting leading to photon-pairs can be considered as spontaneous, i.e. mainly seeded by the quantum fluctuations of vacuum, while it is stimulated by the generated signal and idler photons above the limit, which corresponds to an optical parametric amplification regime. Our calculations also show that this limit can be easily reached in the case of SPDC according to the typical values of non-linearities and available crystal dimensions. In the case of FWM, it would be only possible in kilometric optical fibers. This corpus is a useful tool box for designing further quantum experiments performed from either side of the limit, as well as at the limit it-self where the influence of the quantum fluctuations of vacuum and of the generated photons should have the same weight. Furthermore, the quantum significance of the numerical values of the two criteria defined here remains to be established, which should motivate future theoretical quantum studies.
Summary form only given. Spontaneous parametric down-conversion (SPDC), one of the most important nonlinear processes, is of paramount interest especially in the field of quantum optics due to its intrinsic capability in generating entangled photon pairs in different degree of freedoms (DoFs). The spatial distribution of the paired photons generated through SPDC process are highly influenced by different crystal parameters including birefringence and length, and the spatial structure of the pump beam [1]. Recent studies have shown the transfer of pump properties such as phase profile and doughnut intensity distribution [2] into the transverse amplitude of heralded single photon. Here we report, first time to best of our knowledge, the direct transfer of the non-separable state in OAM and polarization DoFs of classical pump beam to generate two photons state with hybrid entanglement in both OAM and polarization. Using same scheme one can produce macroscopic entanglement with very high quanta of OAM.