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arXiv:2503.18828 · Physical Review A

Isoenergetic model for optical down-conversion and error-specific limits of the parametric approximation

D. B. Horoshko, V. S. Shchesnovich

Published Aug 18, 2025• 1 citation
arXivHTMLPDF

Abstract

Optical down-conversion is widely used for generating photon pairs, squeezed and entangled states of light, making it an indispensable tool in quantum optics and quantum information. In the regime where the pump is much stronger than the generated field, the standard parametric approximation treats the pump amplitude as a fixed parameter of the model. This approximation has a limited domain of validity since it assumes a nondepleted and nonentangled pump. By finding an approximate solution to the Schrödinger equation of the down-conversion process, we obtain an improved analytical model beyond the parametric one, which accounts for pump depletion and pump-signal entanglement. Our model is advantageous, first, because it allows one to compute averages of field operators far beyond the domain of validity of the parametric approximation, and second, because it allows one to establish error-specific limits of the latter domain. For a given pump amplitude, we find a maximum squeezing parameter, up to which the approximation remains valid within a specified acceptable error. Our results confirm that recent experiments on Gaussian boson sampling, with a squeezing parameter of r ≈ 1.8 and a coherent pump amplitude of α ≈ 2 × 10 6 , can still be accurately described by the standard parametric approximation. However, we observe a sharp decline in validity as the squeezing parameter increases. For pump amplitudes of α ≈ 2 × 10 6 , the parametric approximation breaks down when the squeezing parameter exceeds r ≈ 4.5 , whereas the new approximation remains valid up to r ≈ 6 with an acceptable error of 1%.

DOI
10.1103/wtcf-5pkh

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Optical down-conversion is widely used for generating photon pairs, squeezed and entangled states of light, making it an indispensable tool in quantum optics and quantum information. In the regime where the pump is much stronger than the generated field, the standard parametric approximation treats the pump amplitude as a fixed parameter of the model. This approximation has a limited domain of validity since it assumes a nondepleted and nonentangled pump. By finding an approximate solution to the Schrödinger equation of the down-conversion process, we obtain an improved analytical model beyond the parametric one, which accounts for pump depletion and pump-signal entanglement. Our model is advantageous, first, because it allows one to compute averages of field operators far beyond the domain of validity of the parametric approximation, and second, because it allows one to establish error-specific limits of the latter domain. For a given pump amplitude, we find a maximum squeezing parameter, up to which the approximation remains valid within a specified acceptable error. Our results confirm that recent experiments on Gaussian boson sampling, with a squeezing parameter of r ≈ 1.8 and a coherent pump amplitude of α ≈ 2 × 10 6 , can still be accurately described by the standard parametric approximation. However, we observe a sharp decline in validity as the squeezing parameter increases. For pump amplitudes of α ≈ 2 × 10 6 , the parametric approximation breaks down when the squeezing parameter exceeds r ≈ 4.5 , whereas the new approximation remains valid up to r ≈ 6 with an acceptable error of 1%.
Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where LLL is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the sinc\mathrm{sinc}sinc behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of sinc\mathrm{sinc}sinc and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of $\ensuremath{\alpha}$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.
A quantum theory of parametric amplification and frequency conversion of an optical image in coupled nonlinear optical processes that include one parametric amplification process at high-frequency pumping and two up-conversion processes in the same pump field is developed. The field momentum operator that takes into account the diffraction and group velocities of the waves is used to derive the quantum equations related to the spatial dynamics of the images during the interaction. An optical scheme for the amplification and conversion of a close image is considered. The mean photon number density and signal-to-noise ratio are calculated in the fixed-pump-field approximation for images at various frequencies. It has been established that the signal-to-noise ratio decreases with increasing interaction length in the amplified image and increases in the images at the generated frequencies, tending to asymptotic values for all interacting waves. The variance of the difference of the numbers of photons is calculated for various pairs of frequencies. The quantum entanglement of the optical images formed in a high-frequency pump field is shown to be converted to higher frequencies during the generation of sum frequencies. Thus, two pairs of entangled optical images are produced in the process considered.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
Quantum optics has been, from its beginning, a driving force both for the exploration of fundamental limits of the quantum world and for conceiving seminal ideas and applications of the so-called quantum technologies. The last 20 years have seen a rapid development of ideas and proof-of-principle experiments involving the fields of quantum communication, quantum computation, quantum metrology and quantum simulation. The so called continuous variable (CV) approach to quantum optics, which uses many photon in collective states, and continuous observables like the quadratures of the electric field to encode information, has many interesting properties, especially from a communications perspective. It is inherently broadband and compatible with standard telecom infrastructures. Moreover, in CV, entanglement, one of the fundamental resource of quantum optics, can be generated deterministically. One of the main challenges for all quantum information technologies is scalability, being able to generate and manipulate a large number of quantum resources to achieve practical tasks efficiently. One approach to solve the issue of scalability is to use highly multimode quantum states. The quantum description of the electromagnetic field associate each photon (particle of light) with a mode (way for light to propagate). In a mutlimodal approach, we look at the quantum state bases and optical modes bases conjointly and tailor quantum fields not only in given modes, but also optimize the spatio-temporal shapes of the modes in which the state is defined. This opens wide perspectives for treating complex quantum states. In particular, using ultra-fast pulses of light which contain many temporal/spectral modes, we are able to generate large and complex entangled states of light using simple resources. In this thesis, we used an optical parametric oscillator pumped synchronously (SPOPO) with a frequency comb to generate multimode squeezed vacuum states which can be used to form cluster states: a large collection of modes (the nodes) entangled to each other in a certain way. These state are the basic resource for Measurement Based Quantum Computation (MBQC). This set-up has the advantage to generate entanglement in many mode with a single device. It is also highly tunable. Indeed, by tuning the spectrum of the OPO pump with a pulse shaper, one can tailor the properties of the generated quantum state. In this work, we focus on the optimization of the pump spectral shape to generate specific states. Using simulations based on Machine Learning Algorithms (MLA), we find optimal pump profile for typical target states. We then implement those shapes on the experimental set-up and measure the resulting states using multipixel homodyne detection. We also study intra-cavity dispersion effects. Dispersion inside the SPOPO cavity is indeed one of the main factor that limits the number of entangled modes in the generated quantum states. A systematic study of dispersion effects is therefore necessary to model the SPOPO output accurately. This works paves the way toward a fully tunable device that can be optimized in real time to generate specific quantum resources tailored for specific tasks.
We investigate the behavior of squeezing in combined quadratures of biphotons generated in the parametric downconversion process for partially coherent pump beams in spatial, temporal, and spatiotemporal domains compared to fully coherent beams. These beams involve both stationary (continuous-wave) and non-stationary (pulsed) processes. This study also involves the derivation of complex Gaussian representation for a partially spatially coherent pump modeled as a Gaussian Schell model beam, which is more resistant to the effects of turbulence. Our findings illustrate that by employing the partially coherent pump, one can generate a squeezed state possessing partially coherent properties without compromising in squeezing by optimizing pump field amplitude, interaction length, and degree of coherence of the pump. We anticipate these results to be useful in limiting the possibility of eavesdropping in noisy channels and optical high-precision measurements in the case of continuous-variable quantum key distribution based on two-mode squeezed states.
We have analyzed the evolution of quantum operators in a three-wave mixing process by using the nonlinear polarization driven wave equations and linearization of the quantum operators. We have theoretically shown that a nondegenerate optical parametric amplifier can generate amplitude-squeezed light when operated in the backconversion regime. Furthermore, a nondegenerate optical parametric oscillator, where only the signal wave is resonant, is proved to generate amplitude-squeezed light when the pump intensity is above the value at which 100% photon conversion efficiency is achieved. The calculated limit for amplitude-squeezing in this case is 3 d B.