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arXiv:2502.11631 · New Journal of Physics

Advancing the heralded photon-number-state characterization by understanding the interplay of experimental settings

Daniel Fernando Borrero Landazabal, Kaisa Laiho

Published May 22, 2025• 1 citation
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Abstract

Abstract We theoretically explore the properties of heralded number states including up to three photons that are generated from single-mode twin beams. We investigate the effects of different parameters involved in the state preparation by using the fidelity, normalized second-order factorial moment of photon number for the heralded state ( g h ( 2 ) ) , and photon-number parity as figures of merit. Especially, the photon-number parity offers a practical and robust tool for inferring the target state quality by capturing the contamination of all undesired photon-number contributions. We focus on expressing our results in terms of experimentally easily accessible parameters such as the coincidences-to-accidentals ratio and the detection efficiencies. Our results identify the optimal parameter regions for generating high quality photon-number states by heralding and provide useful insights for advancing their use in quantum technologies.

DOI
10.1088/1367-2630/addc11

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Abstract We theoretically explore the properties of heralded number states including up to three photons that are generated from single-mode twin beams. We investigate the effects of different parameters involved in the state preparation by using the fidelity, normalized second-order factorial moment of photon number for the heralded state ( g h ( 2 ) ) , and photon-number parity as figures of merit. Especially, the photon-number parity offers a practical and robust tool for inferring the target state quality by capturing the contamination of all undesired photon-number contributions. We focus on expressing our results in terms of experimentally easily accessible parameters such as the coincidences-to-accidentals ratio and the detection efficiencies. Our results identify the optimal parameter regions for generating high quality photon-number states by heralding and provide useful insights for advancing their use in quantum technologies.
Abstract Due to their strict photon-number correlation, the twin beams produced in parametric down-conversion (PDC) work well for heralded state generation. Often, however, this state manipulation is distorted by the optical losses in the herald and by the higher photon-number contributions inevitable in the PDC process. In order to find feasible figures of merit for characterizing the heralded states, we investigate their normalized factorial moments of the photon number that can be accessed regardless of the optical losses in the detection. We then perform a measurement of the joint photon statistics of twin beams from a semiconductor Bragg-reflection waveguide with transition-edge sensors acting as photon-number-resolving detectors. We extract the photon-number parity of heralded single photons in a loss-tolerant fashion by utilizing the moment generating function. The photon-number parity is highly practicable in quantum state characterization, since it takes into account the complete photon-number content of the target state.
The manifold coincidences and singles counting provides a resource-saving quantum-optics analysis tool, since it is appropriate even in the presence of heavy experimental imperfections. Here, we prepare cross-polarized twin beams in the telecommunication wavelength range via parametric down-conversion in a periodically-poled KTiOPO4_44​ waveguide and herald photon-number states up to three photons. First, we show that the single-click probability is a versatile tool not only for extracting the state's mean photon number but also for sampling values of the moment generating function being the core behind any quantum optical state. Second, we measure values of the normalized factorial moments of photon number, gh(m)g^{(m)}_{\text{h}}gh(m)​, up to the order m=n+1m = n+1m=n+1 for the heralded nnn-photon state. These normalized photon correlations provide an expedient method for examining the higher-order non-classicality of light by violating the condition gh(m+1)≥gh(m)≥1g^{(m+1)}_{\text{h}} \ge g^{(m)}_{\text{h}} \ge 1gh(m+1)​≥gh(m)​≥1.
Generation of highly non-classical quantum states of light is essential for optical quantum information processing and quantum metrology. Given the lack of sufficiently strong nonlinear interactions between optical fields, the commonly employed optical quantum-state preparation schemes are conditional, based on nonlinearity induced by heralding photon number measurement on a part of a multimode squeezed Gaussian state. Development and optimization of such probabilistic quantum-state engineering schemes represents one of the central challenges in current quantum optics. As technology advances and experiments progress to detection of higher numbers of photons, the maximization of the heralding probability becomes essential to ensure sufficiently high state-preparation rates. Here, we show that for the conditional quantum state preparation schemes based on Gaussian states and photon number measurements the maximization of the heralding probability can be formulated as finding solution to a system of polynomial equations, which offers an efficient way to find the optimal configuration and allows us to apply techniques dedicated specifically to solving such systems of equations. Our approach can seamlessly incorporate bounds on the available single-mode quadrature squeezing, which is highly experimentally relevant. We mainly consider generation of finite superpositions of Fock states but show that the approach can be straightforwardly extended to generation of squeezed superpositions of Fock states. We focus on Gaussian states with vanishing coherent displacements, hence the conditionally generated states have well defined photon number parity. We illustrate our general methodology through examples of generation of single-mode states with two and three heralding modes, and generation of two-mode states with two heralding modes.
Coincidence-count discrimination is a practical and straightforward tool for the characterization of photon-pair processes and heralded single photons. Here, we implement a heralded single photon source based on parametric down-conversion in a periodically poled KTiOPO4_44​ waveguide in the telecom wavelength range involving a low number of optical modes. We extend the toolbox for the loss-tolerant state characterization by combining conventional figures-of-merit in order to access the heralded state's mean photon number and its photon-number parity. Our experiment demonstrates that an accurate determination of these characteristics is possible just through simple photon-correlation measurements. We believe that our results can find usage in the calibrated creation of heralded single photons and in determining the expectation values of observables that are crucial for attributing a single quantum.
We consider photon heralding with quasi-number-resolving detection schemes and account for detection efficiencies and dark count probabilities. With a straightforward formalism, we develop closed-form expressions for the heralding probability, photon number distribution of the resulting heralded state, and fidelity of this heralded state to a single-photon state. We calculate that this fidelity has a maximum as a function of the number of multiplexed detection modes and that, for experimental configurations in which the maximum is reached for quite a large number of detection modes, it can still be highly beneficial to multiplex fewer modes.
We investigate different strategies suitable to generate pure heralded single photons through spontaneous parametric down-conversion, comparing the counterpropagating geometry with more conventional copropagating configurations which enhance the purity of the heralded photon state through the technique of group-velocity matching. We derive general results for the Schmidt number associated to the temporal modes, which provide a quantitative estimate of the purity. An analysis of the correlation of twin photons in the temporal domain provides a more physical view of the mechanisms that permit to eliminate the temporal entanglement of the state and to generate high-purity heralded photons. The efficiency of the various strategies and the individual properties of the heralded photons thereby generated are then compared.