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IEEE Transactions on Quantum Engineering

Observing the Poisson Distribution of a Coherent Microwave Field With a Parametric Photon Detector

Jiaming Wang, Kirill Viktorovich Petrovnin, Pertti Juhani Hakonen, Gheorghe Sorin Paraoanu

Published Jan 1, 2025• 4 citations
Publisher

Abstract

Single-photon detectors are essential for implementing optical quantum technologies, such as quantum key distribution, and for enhancing optical imaging systems such as lidar, while also playing a crucial role in studying the statistical properties of light. In this work, we show how the underlying photon statistics can be revealed by using a threshold detector, implemented as a Josephson parametric amplifier operating near a first-order phase transition. We describe the detection protocol, which utilizes a series of pumping pulses followed by the observation of activated switching events. The acquired data are analyzed using two binomial tests, and the results are compared to a theoretical model that takes into account the photon statistics of the microwave field, with additional validation provided by computer simulations. We show that these tests provide conclusive evidence for the Poissonian statistics in the case of a coherent state, in agreement with the experimental data. In addition, this method enables us to distinguish between different statistics of the incoming probe field. Our approach is broadly applicable to standard non-photon-number-resolving detectors, offering a practical pathway to characterize photon statistics in quantum microwave and optical systems.

DOI
10.1109/tqe.2025.3549485

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Single-photon detectors are essential for implementing optical quantum technologies, such as quantum key distribution, and for enhancing optical imaging systems such as lidar, while also playing a crucial role in studying the statistical properties of light. In this work, we show how the underlying photon statistics can be revealed by using a threshold detector, implemented as a Josephson parametric amplifier operating near a first-order phase transition. We describe the detection protocol, which utilizes a series of pumping pulses followed by the observation of activated switching events. The acquired data are analyzed using two binomial tests, and the results are compared to a theoretical model that takes into account the photon statistics of the microwave field, with additional validation provided by computer simulations. We show that these tests provide conclusive evidence for the Poissonian statistics in the case of a coherent state, in agreement with the experimental data. In addition, this method enables us to distinguish between different statistics of the incoming probe field. Our approach is broadly applicable to standard non-photon-number-resolving detectors, offering a practical pathway to characterize photon statistics in quantum microwave and optical systems.
The detection of microwave fields at single-photon power levels is a much-sought-after technology, with practical applications in nanoelectronics and quantum information science. Here we demonstrate a simple yet powerful criticality-enhanced method of microwave photon detection by operating a magnetic-field-tunable Kerr Josephson parametric amplifier at the border of a first-order phase transition and close to the critical point. We obtain an efficiency of 73% and a dark-count rate of 167 kHz, corresponding to a responsivity of 1.3 × 10 17 W − 1 and noise-equivalent power of 3.28 zW / Hz . We verify the single-photon operation by extracting the Poissonian statistics of a coherent probe signal. Published by the American Physical Society 2024
Abstract Measurement of photon statistics is an important tool for the verification of quantum properties of light. Due to the various imperfections of real single photon detectors, the observed statistics of photon counts deviates from the underlying statistics of photons. Here we analyze statistical properties of coherent states, and investigate a connection between Poissonian distribution of photons and sub-Poissonian distribution of photon counts due to the detector dead-time corrections. We derive a functional dependence between the mean number of photons and the mean number of photon counts, as well as connection between higher-order statistical moments, for the pulsed or continuous wave coherent light sources, and confirm the results by numerical simulations.
In this Letter, we experimentally demonstrated that Josephson threshold detectors (JTDs) could be developed to implement the sensitive detection of broadband microwave photons. Beginning with the numerical simulations of the damped dynamics for the macroscopic phase particle generated by a Josephson junction (JJ), we showed that the time-domain threshold voltage state switched measurements of the JJ driven by a low-frequency triangular wave current, which could be utilized to implement the sensitive microwave photons detection. The detectable signal power can be determined by using the Kumar–Caroll index to quantify the distinguishability between two statistical distributions on the measured threshold switched durations of the JJ with and without the signal input. Experimentally, we fabricated the Al/AlOx/Al JJ and measured its threshold responses, at 50 mK temperature, to the input microwave photons at the frequency of ∼5 GHz. The experimental results indicated that the weak microwave signals, which are phenomenologically threatened by the additional thermal noise that modifies the damped phase dynamics of the JJ, can be detected by the fabricated JTD. The detectable power of the microwave signals demonstrated here was calibrated as about −92 dBm (corresponding to the photons rate of ∼2×108 photons/ms). Based on the numerical simulations for optimizing the relevant performance indicators of the detectors, we further argue that the JTDs could be utilized to achieve the sensitive detection of broadband microwave photons at the single-photon level.
The strong interaction of superconducting integrated circuits with microwave photons forms the basis of circuit quantum electrodynamics (cQED), an attractive paradigm for scalable quantum computing and a test bed for quantum optics in the strong coupling regime [1–6]. While singlephoton detectors are an important component of the standard quantum optics toolbox [7], they have not played a role in cQED, as there are no microwave photon counters currently available. In the search for a microwave component that will emulate the nonlinearity of traditional optical photon counters, it is natural to consider the Josephson junction—a nonlinear, nondissipative superconducting circuit element with tunable microwave-frequency transitions. There has been recent interest in the development of on-chip detectors to probe the microwave radiation emitted by mesoscopic quantum circuits [8–14], including proposals to utilize Josephson circuits as photon counters in the context of cQED [8,9]. This study represents the first experimental measurement of the coincidence counting statistics of microwave photons. We exploit microwaveinduced transitions between discrete energy levels in a Josephson junction to realize an efficient, versatile microwave detector. Moreover, we use a parallelized detector circuit to probe the counting statistics of both coherent and thermal microwave sources. Acircuitdiagramofthesingle-channeldetectorisshown in Fig. 1(a). The Josephson junction is current biased near thecriticalcurrent I0.Thejunctionpotentialenergyisgiven by Uð� Þ¼� I
Low-photon-number sources can exhibit non-classical, counterintuitive behavior that can be exploited in the developing field of quantum technology. Single photons play a special role in this arena since they represent the ultimate lowphoton- number source. They are considered an important element in various applications such as quantum key distribution, optical quantum information processing, quantum computing, intensity measurement standards, and others yet to be discovered in this developing field. True deterministic sources of single photons on demand are currently an area of intensive research, but have not been demonstrated in a practical setting. As a result, researchers commonly default to the well-established workhorse: spontaneous parametric down-conversion generating entangled signal-idler pairs. Since this source is thermal-statistical in nature, it is common to use a detected idler photon to herald the production of a signal photon. The need exists to determine the quality of the single photons generated in the heralded signal beam. Quite often, the literature reports a "heralded second-order coherence function" of the signal photons conditioned on the idler photons using readily available single-photon detectors. In this work, we examine the applicability of this technique to single-photon characterization and the consequences of the fact that the most commonly used single-photon detectors are not photon-number resolving. Our results show that this method using non-photonresolving detectors can only be used to characterize the signal-idler correlations rather than the nature of the signalphoton state alone.
Abstract Microwave photons are essential carriers of quantum information in several promising platforms for quantum computing. However, measurement of the quantum statistical properties of microwave photons is demanding owing to their low energy relative to thermal fluctuations of any room-temperature detector, and phase-insensitive voltage amplification necessarily adds noise. Here, we overcome this trade-off with a nanobolometer that directly measures the photon statistics at millikelvin. Using a cryogenic temperature-controlled blackbody radiator, we demonstrate the detection of the mean photon number ⟨n^⟩\langle \hat{n}\rangle⟨n^⟩ ⟨ n ̂ ⟩ and reveal the expected photon number variance (Δn)2=⟨n^⟩(⟨n^⟩+1){(\Delta n)}^{2}=\langle \hat{n}\rangle \left(\langle \hat{n}\rangle+1\right)(Δn)2=⟨n^⟩(⟨n^⟩+1) ( Δ n ) 2 = ⟨ n ̂ ⟩ ⟨ n ̂ ⟩ + 1 , following the Bose–Einstein distribution. By engineering the coherent and incoherent proportions of the input field, we observe a transition between super-Poissonian and Poissonian statistics from the bolometric second-order correlation measurements. This technique is poised to serve in fundamental tests of quantum mechanics and function as a scalable readout solution for a quantum information processor.