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Semiklassisk model av spontan parametrisk nedkonvertering i fri-elektronlaser

Caspar Albinsson

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

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.

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Semi-classical modelling of second-order spontaneous down-conversion measured over 10 decades of pump intensity in a type II phase-matched KTP crystal

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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.

arXiv:2502.04815•Feb 7, 2025•0 citations

Theory of Spontaneous Parametric Down-Conversion in Thin Etalons

Elkin A. Santos, Frank Setzpfandt, Sina Saravi

The process of spontaneous parametric down-conversion (SPDC) in$X$(2)-materials, in which a pump (P) photon of wavelength λpcan split into a pair of entangled signal (S) and idler (I) photons of lower energy, is an instrumental tool in optical quantum technologies. Recently, there has been a growing interest in pair generation in sub-wavelength-thin nonlinear crystals [1], [2], especially because they are not restricted by the phase-matching condition and exhibit SPDC in a very wide spectral and angular range. This also motivates their use in quantum imaging applications, for pushing the resolution of such schemes to the limit [3], [4]. Despite its seemingly simple appearance, there has not yet been a comprehensive theoretical analysis of photon-pair generation in a thin unstructured nonlinear slab. Specifically, a proper theory should embed the Fabry-Pérot type of interferences that the signal and idler photons experience in such a nonlinear etalon during SPDC. In our work, we develop the theory for description of SPDC in thin nonlinear etalons and calculate far-field properties of the generated pairs. We use the Green's function (GF) quantization approach for description of pairgeneration [5], combined with analytical expressions for the GF of multilayer systems [6], to develop our formalism. Because of the generality of the GF method, this formalism can describe any thickness of the slab, ultra-thin or thick, and far-field radiation in the non-paraxial regime. Fig. 1 (a) schematic of spdc in a nonlinear etalon.$\epsilon_{1},\epsilon_{2}$and$\epsilon_{3}$are linear permittivities of the mediums. Medium 2 with thickness$h$has a second-order$\chi^{(2)}$nonlinearity. (b-e) Joint detection probabilities of different detection configurations and etalon thicknesses.

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Spontaneous parametric downconversion (SPDC) using periodically poled nonlinear optical crystals under the quasiphase- matching condition has found wide use in quantum optics. High efficiencies and good coupling to single-mode fibers resulted from using channel waveguides in crystals. It is often desirable to have a very narrow bandwidth for the signal and idler photons, but under the typical operating conditions, phase matching dictates the bandwidth of the SPDC to be of the order of <1 nm. This occurs because the co-propagating signal and idler photons are entangled, and an increase of the signal wave-vector is compensated by a decrease of the idler wave-vector. One way to reduce the bandwidth is by forming either external or internal cavities. Additionally, bandwidth reduction is possible without cavities when the signal and idler are counter-propagating, and the changes in the wave-vector with frequency are additive. To accomplish this a domain inversion on the wavelength scale is required. In this work, we experimentally demonstrate SPDC in one-dimensional KTP-based waveguides with sub-micron poling for forward and backward interactions. Some of the spectral features of the generated light are accounted for by mode coupling theory in periodically poled waveguides but other features are as yet not explained.

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Evolution of Spatially Entangled Photons Beyond Standard Approximations

Christian Howard

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.

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Quantum states of light, such as single photons, pairs of entangled photons, and squeezed light are a key resource for photonic quantum technologies. All these states can be obtained via the nonlinear optical effect of spontaneous parametric down-conversion (SPDC). In SPDC, a higher-energy photon of a coherent beam (pump) splits in two lower-energy daughter photons, called signal and idler. The photons of such a pair can be entangled in polarization, time/frequency, and position/momentum. Efficient generation of photon pairs requires a high second-order susceptibility$\chi^{(2)}$of the material where it occurs, and the phase matching between the pump and the down-converted photons.

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Entangled X-ray Photon Pair Generation by Free Electron Lasers

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Abstract Einstein, Podolsky and Rosen's prediction on incompleteness of quantum mechanics was overturned by experimental tests on Bell's inequality that confirmed the existence of quantum entanglement. In X-ray optics, entangled photon pairs can be generated by X-ray parametric down conversion (XPDC), which is limited by relatively low efficiency. Meanwhile, free electron laser (FEL) has successfully lased at X-ray frequencies recently. However, FEL is usually seen as a classical light source, and its quantum effects are considered minor corrections to the classical theory. Here we investigate entangled X-ray photon pair emissions in FEL. We establish a theory for coherently amplified entangled photon pair emission from microbunched electron pulses in the undulator. We also propose an experimental scheme for the observation of the entangled photon pairs via energy and spatial correlation measurements. Such an entangled X-ray photon pair source is of great importance in quantum optics and other X-ray applications.

•Sep 12, 2022•0 citations
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.
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.
The process of spontaneous parametric down-conversion (SPDC) inXXX(2)-materials, in which a pump (P) photon of wavelength λpcan split into a pair of entangled signal (S) and idler (I) photons of lower energy, is an instrumental tool in optical quantum technologies. Recently, there has been a growing interest in pair generation in sub-wavelength-thin nonlinear crystals [1], [2], especially because they are not restricted by the phase-matching condition and exhibit SPDC in a very wide spectral and angular range. This also motivates their use in quantum imaging applications, for pushing the resolution of such schemes to the limit [3], [4]. Despite its seemingly simple appearance, there has not yet been a comprehensive theoretical analysis of photon-pair generation in a thin unstructured nonlinear slab. Specifically, a proper theory should embed the Fabry-Pérot type of interferences that the signal and idler photons experience in such a nonlinear etalon during SPDC. In our work, we develop the theory for description of SPDC in thin nonlinear etalons and calculate far-field properties of the generated pairs. We use the Green's function (GF) quantization approach for description of pairgeneration [5], combined with analytical expressions for the GF of multilayer systems [6], to develop our formalism. Because of the generality of the GF method, this formalism can describe any thickness of the slab, ultra-thin or thick, and far-field radiation in the non-paraxial regime. Fig. 1 (a) schematic of spdc in a nonlinear etalon.ϵ1,ϵ2\epsilon_{1},\epsilon_{2}ϵ1​,ϵ2​andϵ3\epsilon_{3}ϵ3​are linear permittivities of the mediums. Medium 2 with thicknesshhhhas a second-orderχ(2)\chi^{(2)}χ(2)nonlinearity. (b-e) Joint detection probabilities of different detection configurations and etalon thicknesses.
Spontaneous parametric downconversion (SPDC) using periodically poled nonlinear optical crystals under the quasiphase- matching condition has found wide use in quantum optics. High efficiencies and good coupling to single-mode fibers resulted from using channel waveguides in crystals. It is often desirable to have a very narrow bandwidth for the signal and idler photons, but under the typical operating conditions, phase matching dictates the bandwidth of the SPDC to be of the order of <1 nm. This occurs because the co-propagating signal and idler photons are entangled, and an increase of the signal wave-vector is compensated by a decrease of the idler wave-vector. One way to reduce the bandwidth is by forming either external or internal cavities. Additionally, bandwidth reduction is possible without cavities when the signal and idler are counter-propagating, and the changes in the wave-vector with frequency are additive. To accomplish this a domain inversion on the wavelength scale is required. In this work, we experimentally demonstrate SPDC in one-dimensional KTP-based waveguides with sub-micron poling for forward and backward interactions. Some of the spectral features of the generated light are accounted for by mode coupling theory in periodically poled waveguides but other features are as yet not explained.
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.
Quantum states of light, such as single photons, pairs of entangled photons, and squeezed light are a key resource for photonic quantum technologies. All these states can be obtained via the nonlinear optical effect of spontaneous parametric down-conversion (SPDC). In SPDC, a higher-energy photon of a coherent beam (pump) splits in two lower-energy daughter photons, called signal and idler. The photons of such a pair can be entangled in polarization, time/frequency, and position/momentum. Efficient generation of photon pairs requires a high second-order susceptibilityχ(2)\chi^{(2)}χ(2)of the material where it occurs, and the phase matching between the pump and the down-converted photons.
Abstract Einstein, Podolsky and Rosen's prediction on incompleteness of quantum mechanics was overturned by experimental tests on Bell's inequality that confirmed the existence of quantum entanglement. In X-ray optics, entangled photon pairs can be generated by X-ray parametric down conversion (XPDC), which is limited by relatively low efficiency. Meanwhile, free electron laser (FEL) has successfully lased at X-ray frequencies recently. However, FEL is usually seen as a classical light source, and its quantum effects are considered minor corrections to the classical theory. Here we investigate entangled X-ray photon pair emissions in FEL. We establish a theory for coherently amplified entangled photon pair emission from microbunched electron pulses in the undulator. We also propose an experimental scheme for the observation of the entangled photon pairs via energy and spatial correlation measurements. Such an entangled X-ray photon pair source is of great importance in quantum optics and other X-ray applications.