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.
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.
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.
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.
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 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.
Spontaneous parametric down-conversion (SPDC) is a widely used method of creating correlated and entangled photon pairs, but its extension to the X-ray regime is very limited by the weak nonlinear response of conventional birefringent crystals, and by absorption and scattering of the high energy photons. This work investigates an alternative approach based on free-electron lasers (FELs), where a relativistic electron beam in an undulator acts as an effective nonlinear medium. A semiclassical model for FEL sub-harmonic generation is developed by introducing pump, signal, and idler radiation fields and deriving their coupled propagation equations directly from the electron equations of motion and Maxwell’s equations. The nonlinear interaction is shown to arise from collective phase matching of the ponderomotive phases, leading to three effective susceptibilities, one for each modal interaction to the electron beam, that replace the second-order nonlinear susceptibility of ordinary SPDC. Analytical expressions for the nonlinear beam currents and coupling coefficients are derived using the slowly varying envelope approximation (SVEA). The resulting coupled wave equations show the same modal relations as optical SPDC and predict exponential growth of vacuum-seeded signal and idler fields under near phase-matched conditions. Simplified scaling laws of the effective susceptibilities show that the interaction strength increases with undulator strength and decreases strongly with electron-beam energy, while efficient generation requires near resonant conditions for the signal and idler. By introducing stochastic vacuum fluctuations as initial conditions, the model can be derived semi-classically, connecting the FEL gain dynamics and spontaneous photon-pair generation. The results indicate that x-ray free-electron lasers may support spontaneous parametric down-conversion and could become an alternate more efficient source of correlated photons in the x-ray regime, with an exponential dependence on the pump amplitude, and without the limitations of the ineffective conventional nonlinear optical materials. The semi-classical model forms an understanding of the SPDC process and its dependence on the FEL dynamics, and a foundation for future fully quantum optical derivations in order to investigate photon statistics, coherence, and entanglement generation rigorously in FEL-based systems.
Pair creation by spontaneous parametric down-conversion (SPDC) has become a reliable source for single-photon states, used in many kinds of quantum information experiments and applications. In order to be spectrally pure, the two photons within a generated pair should be as frequency-uncorrelated as possible. For this purpose most experiments use narrow bandpass filters, having to put up with a drastic decrease in count rates. This article elaborates (theoretically and by numerical evaluation) the alternative method to engineer a setup such that the SPDC-generated quantum states are intrinsically pure. Using pulsed pump lasers and periodically poled crystals this approach makes bandpass filtering obsolete and allows for significantly higher output intensities and therefore count rates in the detectors. After numerically scanning all common wavelength regimes, polarisation configurations and three different non-linear crystals, we present a broad variety of setups which allow for an implementation of this method.