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arXiv:2106.13575 · Physical Review A

Stimulated parametric down-conversion for spatiotemporal metrology

Filippus Stefanus Roux

Published Oct 14, 2021• 10 citations
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Abstract

A detailed analysis of the stimulated parametric down-conversion (PDC) process is performed to investigate the effects of the spatiotemporal degrees of freedom. The analysis provides information that would be useful for PDC-based metrology applications. Using a Wigner functional approach, we obtain the parametric down-converted state as the Bogoliubov transformed input state in terms of Bogoliubov kernel functions. The result is used to consider the case for a coherent-state seeding stimulated PDC. We also compute the background which is obtained from spontaneous PDC.

DOI
10.1103/physreva.104.043514

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A detailed analysis of the stimulated parametric down-conversion (PDC) process is performed to investigate the effects of the spatiotemporal degrees of freedom. The analysis provides information that would be useful for PDC-based metrology applications. Using a Wigner functional approach, we obtain the parametric down-converted state as the Bogoliubov transformed input state in terms of Bogoliubov kernel functions. The result is used to consider the case for a coherent-state seeding stimulated PDC. We also compute the background which is obtained from spontaneous PDC.
Stimulated parametric down-conversion (StimPDC) is the seeded version of the process that underlies much of photonic quantum information. Replacing the vacuum at one input port with a coherent seed leaves the interaction Hamiltonian, the phase-matching function and the two-photon amplitude unchanged, while raising the output to levels accessible with ordinary classical detectors. This review treats the transverse spatial physics of that process. We show that a single relation, in which the idler amplitude is the pointwise product of the pump field with the complex conjugate of the seed, organizes results usually presented separately: the image and coherence transfer experiments of the 1990s; phase conjugation and its use for wavefront correction; the conservation and conjugation of orbital angular momentum, together with the generation of radial modes and geometric intensity patterns; and the processing of vector vortex beams carrying spatially varying polarization. We then consider StimPDC as an instrument rather than as an object of study, covering stimulated emission tomography, the experimental realization of Klyshko's advanced-wave picture, and the transfer of partial coherence, including the conservation of the twist phase of twisted Gaussian Schell-model beams. We close with the quantum regime, reached when the seed approaches the single-photon level or the idler is used as a herald, and with an assessment of open problems and prospective applications.
Pulsed parametric down-conversion (PDC) processes generate photon pairs with a rich spectral-temporal structure, which offer an attractive potential for quantum information and communication applications. We investigate the four-dimensional chronocyclic Wigner function ${W}_{\mathrm{PDC}}({\ensuremath{\omega}}_{\mathrm{s}},{\ensuremath{\omega}}_{\mathrm{i}},{\ensuremath{\tau}}_{\mathrm{s}},{\ensuremath{\tau}}_{\mathrm{i}})$ of the PDC state, which naturally lends itself to the pulsed characteristics of these states. From this function we derive the conditioned time-bandwidth product of one of the pair photons, a quantity which not only is a valid measure of entanglement between the PDC photons but also allows us to highlight a remarkable link between the discrete- and continuous-variable descriptions of PDC. We numerically analyze PDC processes with different conditions to demonstrate the versatility of our approach, which is applicable to a large number of current PDC sources.
Parametric down-conversion is a widely used source of nonclassical light in quantum optics and photonic quantum technologies. While stimulated parametric down-conversion with strong classical seeds is well studied, the elementary physical mechanism of stimulation by individual photons has remained largely unexplored experimentally. Here, we study continuous-wave, low-gain down-conversion seeded by an ultra-weak coherent field with an average photon number well below one per coherence time. By measuring third-order temporal correlations, we access the regime in which the single-photon component of the seed can stimulate the emission of a photon pair, giving rise to a temporally correlated three-photon event. We observe a clear enhancement that cannot be explained by spontaneous processes or accidental coincidences alone, providing time-domain evidence of the fundamental stimulation mechanism at the single-photon level. Beyond its fundamental significance, these results suggest new ways to characterize seed–PDC mode overlap, phase matching, and weak nonlinear coupling in regimes where bright stimulated-emission measurements are not appropriate. Multimode and interferometric extensions could enable correlation-based imaging and phase-sensitive spectroscopy.
The role of the spatiotemporal degrees of freedom in the preparation and observation of squeezed photonic states, produced by parametric down-conversion, is investigated. The analysis is done with the aid of a functional approach under the semiclassical approximation and the thin-crystal approximation. It is found that the squeezed state loses its minimum uncertainty property as the efficiency of down-conversion is increased, depending on the conditions of the homodyne measurements with which the amount of squeezing is determined.
Using a perturbative approach, we investigate the parametric down-conversion process without the semi-classical approximation. A Wigner functional formalism, which incorporates both the spatiotemproal degrees of freedom and the particle-number degrees of freedom is used to perform the analysis. First, we derive an evolution equation for the down-conversion process in the nonlinear medium. Then we use the perturbative approach to solve the equation. The leading order contribution is equivalent to the semi-classical solution. The next-to-leading order contribution provides a solution that includes the evolution of the pump field as a quantum field.
We investigate the spatiotemporal structure of the biphoton correlation in type-II parametric down-conversion (PDC). As in type-I PDC [Phys. Rev. Lett. 102, 223601 (2009)], we find that the correlation is nonfactorizable in space and time. Differently from type I, the type-II correlation in the spontaneous regime displays an asymmetric V-shaped structure in any cross section including time and one transverse dimension. This asymmetry along the temporal coordinate originates from the signal-idler group velocity mismatch and tends to disappear as the parametric gain is raised. We observe a progressive transition toward a symmetric X-shaped geometry similar to that found in type I when stimulated PDC becomes dominant. We also give quantitative evaluations of the localization in space and in time of the correlation, analyze its behavior for different crystal tuning angles, and underline qualitative differences with respect to type-I PDC.