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Abstract
The ability to discriminate the number of photons in a radiation field has a critical role in the implementation of quantum optical technologies. True photon-number-resolving detectors are rare and complex devices, while a quasi-photon-number-resolving detector (qPNRD) is a practical alternative for real-world applications. Our qPNRD is composed of a fiber demultiplexer and individual non-photon-number-resolving detectors. We perform quantum tomography on our qPNRD based on the positive operator-valued measure and extend the analysis using the Bayesian formalism to uncover how the measurement influences knowledge of the measured photon probability distribution.
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1. Introduction
Resolving the number of excitations in a radiation field is a key ingredient for further developments in quantum mechanics research [1], quantum metrology [2–7], and quantum technologies where for example, state-of-the-art quantum key distribution protocols require precise characterization of signal statistics [8]. The implementation of a true photon-number-resolving detector is technologically demanding [3,5,9–14], hence a quasi-photon-number-resolving detector (qPNRD) is a practical alternative for real-world applications [15]. A qPNRD [15] can be implemented by spatial [16,17] and/or temporal [18–23] demultiplexing of the beam onto multiple non-photon-number-resolving detectors, where each detector outputs a single ‘click’ when one or more photons are detected.
For practical use, the qPNRD must be characterized and a mathematical description of its response to an input field must be achieved. To this end, quantum tomography is performed through the computation of the positive operator-valued measure (POVM), which relates the detector’s response to the number of photons in a pulse [16,17,21,24–27].
The Bayesian formalism has been successfully applied in many fields such as artificial intelligence [28–30], statistics [30–32], and classical statistical mechanics [33]. In the quantum technology context, the relation between Bayesian probability and quantum probability has been discussed [34], and applications to quantum state tomography [35,36], parameter estimation [37,38], and learning quantum systems [39] have been demonstrated. In contrast to standard quantum tomography of a qPNRD, the Bayesian approach facilitates time-dependent POVM reconstruction [40], discrimination of different states of light [41,42], and assessment of the importance of the photon probability distribution before a photon is detected [43].
In this work, we perform the POVM computational analysis previously reported [16,21] and extend it by Bayesian interpretation of the measurement. We have investigated how the ‘clicks’ from the qPNRD affect the knowledge of the photon distribution of the state by exploiting the learning concept of the Bayesian approach.
2. Experimental setup
Light originated from a gain-switched laser (PicoQuant LDH P-F-N 1550) emitting $\sim 95$ ps duration pulses with a repetition rate $\Gamma =1$ MHz at $1550$ nm. A Hewlett-Packard 8158B variable optical attenuator (VOA) controls the optical power sent to a $50:50$ beamsplitter. One beamsplitter output is measured by an optical power meter and the other output is sent to a balanced splitter system with four outputs (Fig. 1). The optical power meter (Keysight N7747A) has a sensitivity of $10$ fW with $<0.1$ pW peak-to-peak noise and drift. Finally, four Photon Spot superconducting nanowire single-photon detectors (SNSPDs) [44–48] detect the photons in the four beams. A Swabian Instruments Time Tagger Ultra with a resolution of $1$ ps, root-mean-square jitter of $8$ ps, and a deadtime of $2$ ns records the detection events.
Fig. 1. Schematic of the experimental setup. The system is entirely fiber-coupled, thus the beamsplitters and mirrors are symbolic representations. The set of elements enclosed by the dashed red line is the qPNRD. The histogram curve (red bars) is an example of the power distribution obtained from the power meter for a fixed attenuation value in the VOA.
For any fixed attenuation value, a time series measurement of the power with an integration time of $1$ ms produces an empirical probability distribution function with a standard deviation of $147(1)$ fW, which corresponds to about $1.147(8)\times 10^{6}$ photons/s (Fig. 1). The correlation coefficient ($\mathcal {C}$) between each power time series and their corresponding total count rate array, given by
where $K_{ij}$ is an element of the covariance matrix, are represented by blue dots in Fig. 2(a). The power series, represented by the index $i$, and the count rate (index $j$) were measured simultaneously, and for every mean photon number $\mu$, the small values of $\mathcal {C}$ indicate that the power fluctuations are primarily caused by the uncertainty on the power meter readings, not by fluctuations from the pulsed laser.Fig. 2. (a) The blue dots are the correlation coefficients between the power time trace and the qPNRD count rate as a function of the mean number of photons per pulse, $\mu$. The red dots are values of $1-g^{(2)}(0)$ providing information about the photon statistics of the light source. (b) The red dots are the measured count rate as a function of $\mu$. The solid black line is the theoretical curve with $\eta =0.373$, and the shaded region is the hypothetical shot-noise limit of a true photon-number-resolving detector. (c) The response of the qPNRD, $\xi _{n}(\mu )$. The dashed black lines are theoretical curves with $\eta =0.373$.
The second-order correlation function at zero delay time, $g^{(2)}(0)$, quantifies the deviation of the laser photon distribution from a Poisson distribution [49]. The red dots in Fig. 2(a) show the results for a single pair of detectors. Different detector pairs yield similar graphs. The average values of $1-g^{(2)}(0)$ and their standard deviations are obtained by dividing the area of the correlation peak located at $\tau =0$ by the area of its ten nearest neighbouring peaks. Each measurement has the same duration, corresponding to $60$ million pulses. Hence, the more accentuated deviations at small powers are attributed to the small number of coincidences measured. Nevertheless, the deviation from the ideal case is negligible, so the combination of the laser with the VOA approaches an ideal Poisson light source.
For the tomography, all the optical elements enclosed by the red dashed line in Fig. 1 are considered to be a single device, the qPNRD. The count rate as a function of the mean photon number, $\mu$, is presented in Fig. 2(b). The shaded region corresponds to the hypothetical shot-noise limit of a true photon-number-resolving detector. For high values of $\mu$, the standard deviation of the count rate decreases because the probability of having a click from any of the four single-photon detectors converges to one. The bias current of each SNSPD was manipulated to balance the count rates of the individual detectors for practical modelling. Before starting the measurement, we intentionally decreased the bias current - and consequently the efficiency - of the SNSPDs to avoid discontinuities in the detection time trace due to latching in high-excitation-power regimes [50,51].
Given an incident mean photon number, the probability of obtaining $n$ clicks from a single laser pulse, $\xi _{n}(\mu )$, is computed from the time series of the number of clicks with a time bin equal to the inverse of the light pulse’ s repetition rate, $1$ microsecond. The dots in Fig. 2(c) represent $\xi _{n}(\mu )$ for $n$ up to four. The probability of obtaining $n>4$ clicks is negligible because of the low dark count rate of $30$ Hz per single-photon detector. For $D$ single-photon detectors with the same average count rate,
Fitting the measured counts per pulse, $\langle n\rangle$, using
leads to estimates of $\eta =0.373$ and $\sigma =0$ with standard deviation for the estimates of $0.001$ and $9\times 10^{-4}$, respectively (black solid line in Fig. 2(b)). The low value of $\sigma$ allows us to neglect any power fluctuation and consider Eq. (2) suitable for modelling the detector’s response (black dashed lines in Fig. 2(c)).3. Results
3.1 POVM computation
The tomographic reconstruction of the positive operator-valued measure (POVM) is crucial to fully characterize the qPNRD since it relates the detector’s response to the mean number of photons per pulse [16,17,21,25]. If the qPNRD is not phase-sensitive, its POVM can be written as
with where $\hat {\mathcal {I}}$ is the identity operator.The coefficient $\Theta _{m,n}=\langle {m}|\hat \Theta _n|{m}\rangle$ is the probability of detecting $n$ photons if the input pulse has $m$ photons. The response of the qPNRD to the field described by the density operator, $\hat \rho$, is
The method used to compute the POVM elements is to invert Eq. (7) in a matrix equation. The least-squares method with the addition of normalization constraints and a smoothness condition leads to a solution for $\Theta _{m,n}$ [17]. The normalization constraints guarantee that the sum of the probabilities is one, and the smoothness condition, given by
prevents abrupt fluctuations between neighbouring data points. In the least-squares method, the weight - a multiplying factor used to emphasize some equations relative to others - given to the equations from the POVM problem and to the smoothness conditions was $1$, while to the normalization constraints it was $40$ to achieve a precision of $10^{-6}$ on the sum of the probabilities.Following previous works [17,24,25], the influence of the choice of $\epsilon$ on the POVM elements is shown in Fig. 3(a)-(c) for values of $n$ up to $2$. We observe that the probability curves are spiky at $\epsilon =0$, then there is a relative stability in the range $0<\epsilon \leq 1$, and a broadening in the POVM curves followed by a decrease in their heights for $\epsilon >1$. One of the shortcomings of this approach is that the value of $\epsilon$ is arbitrarily determined. The parameter we use to find an optimum $\epsilon$ value is the fidelity
where $\xi _{n}^{(L)}(\mu )$ is the qPNRD response reconstructed from the computed $\Theta _{m,n}$ and Eq. (7). Small $\epsilon$ values favour high fidelity (Fig. 3(d)), while the changes on the POVM curves degrades the fidelity when $\epsilon >1$. A remarkable observation is that the fidelity is nearly one for $\epsilon =0$, where the POVM curves are spiky. This happens because the reconstructed detector’s response does not inherit the irregular shape of the POVM after the multiplication with the matrix representing the state of the excitation field. The bars in Fig. 3(e)-(i) correspond to the obtained POVM curves with $\epsilon ^2=0.2$, and the bump (dip) for $n=3$ ($n=4$) at $m\approx 38$ is an experimental artifact. However, they are reproducible, only occur at high values of μ, and are accounted for in the POVMs. The black dots are computed from the theoretical model [15,19,22,42]Fig. 3. The POVM elements $\Theta _{m,n}$ computed via the least-squares method for the number of clicks, $n$, equal to $0$ (a), $1$ (b), and $2$ (c) as a function of the smoothness parameter, $\epsilon$, and the exact number of photons per pulse, $m$. (d) Fidelity as a function of the smoothness parameter and the mean photon number, $\mu$. (e) POVM element $\Theta _{m,n}$ as a function of $m$ for $n=0$; (f) $n=1$; (g) $n=2$; (h) $n=3$; (i) $n=4$. The black dots are theoretical predictions for a qPNRD composed of four identical single-photon detectors with efficiency $\eta =0.373$. The dashed black lines are guides-to-the-eye.
3.2 Learning from qPNRD clicks
The POVM curves obtained in the previous section are important because they map the exact number of photons to the number of clicks on the qPNRD. However, for the qPNRD to serve its purpose, we should be capable of estimating the number of photons in the pulse that is detected by the single-photon detectors. In the Bayesian formalism, we can write the probability function as $\Theta _{m,n}=P({n|m,\mu,I})=P({n|m,I})$, where $I$ encloses all the background knowledge about the experiment [52] and the last equality means that the information about the mean photon number, $\mu$, is not relevant if one knows the exact number of photons in the pulse, $m$. The independence of $\Theta _{m,n}$ on $\mu$ is asserted by its theoretical description in Eq. (10).
For the qPNRD response, $\xi _n(\mu )=P({n|\mu,I})$, the mean photon number measured from the optical power meter and the second-order correlation function shown in Fig. 2(a) guide to a Poisson distribution, implicit in the density matrix elements $\rho _m(\mu )$ in Eq. (7). Its corresponding probability function is $P({m|\mu,I})$. To fully characterize the system, we need to compute the posterior probability $P({m|n,\mu,I})$. Different from the POVM curves, the information about the value of $\mu$ is important in this case because of the use of imperfect single-photon detectors and the geometry of the qPNRD, which has influence on the overall efficiency of the system due to the randomized photon path. To compute $P({m|n,\mu,I})$, we use Bayes’ theorem
Notice that solving the POVM problem is equivalent to obtaining the posterior probability $P({m|n,\mu,I})$ since the multiplying factor is known. If the prior probability $P({m|\mu,I})$ is unknown, a possible approach is to use a guessed distribution and apply corrections iteratively as the number of clicks is measured.
In Fig. 4, the bars are the Poisson probability mass functions, the prior probabilities. The dots are the posterior probabilities $P({m|n,\mu,I})$ obtained from the POVM curves shown in Fig. 3(e)-(i) and the detector response in Fig. 2(c). The dashed lines are theoretical predictions obtained by substituting Eq. (2) and Eq. (10) into Eq. (11). In all the cases presented, the shift of the probability function with the number of clicks $n$ is evident. However, the discrimination between different values of $m$ is more pronounced for lower values of $\mu$ due to the smaller standard deviation associated with the distribution. The most significant deviations from the model happen when the specific detector responses are rare, such as four clicks when $\mu =0.5$ and no clicks when $\mu =10.6$.
Fig. 4. Posterior probabilities $P({m|n, \mu,I})$ for $n<5$ and mean photon number (a) $\mu =0.5$; (b) $\mu =2.4$; (c) $\mu =5.4$; (d) $\mu =10.6$. The bars are the prior probabilities, which are Poisson distributions with the corresponding mean $\mu$.
The expectation value of the posterior distribution is
The variable $\langle m\rangle$ is interpreted as the corrected value of $\mu$ after measuring $n$. The correction term vanishes when $\langle n\rangle \rightarrow n$, where the Shannon entropy of the detector response $\xi _{n}(\mu )$ is maximum. For small values of $\eta \mu /D$, the corrected mean photon number is $\langle m\rangle \approx \mu (1-\eta )+n$, which is the average number of photons not causing the detectors to click plus the number of clicks observed. Increasing the number of detectors enlarges the $\mu$ range where this approximation is valid, whilst using high-efficiency single-photon detectors decreases the importance of the power meter in the mean photon number measurement.
The variance of the posterior probability is
As the detector clicks ($n>0$), the variance decreases as a consequence of learning. For negligible dark count rates, the average number of clicks is less than or equal to the number of single-photon detectors, so $P({m|n,\mu,I})$ cannot have a variance higher than the mean in any condition. For small values of $\eta \mu /D$, we have that $\sigma ^2_m\approx \mu (1-\eta )+n\eta \mu /D$. In this case, the first term corresponds to the variance of the undetected photons, whilst the second term is the measurement contribution. As the laser power is increased, $\langle n\rangle$ approaches $D$, and the variance approaches $\langle m\rangle \approx \mu$. This happens because, at this regime, the clicks of the qPNRD are not informative about the number of photons.
4. Conclusions
After confirming the Poisson distribution and power stability of the pulsed light source, we performed quantum tomography on a spatially demultiplexed qPNRD. The fidelity obtained from the reconstructed detector’s response via POVM is comparable to state-of-the-art measurements. In addition, the analysis was extended using the Bayesian formalism to derive posterior probability distributions of the photon number given $n$ click events on the qPNRD. As a result, we observed that the mean photon number obtained from the posterior probability $P({m|n,\mu,I})$ increases with the number of photons detected. On the other hand, the difference between the corrected variance and the corrected mean photon number decreases with the number of clicks, leading to sub-Poissonian posterior probability distributions for relatively low values of $\mu$. Therefore, the accuracy of inferring the number of photons in a pulse increases with the inclusion of data on the number of clicks from the characterized quasi-photon-number-resolving detector.
Funding
European Metrology Programme for Innovation and Research (19NRM06 MeTISQ); Horizon 2020 Framework Programme; Department for Business, Energy and Industrial Strategy, UK Government through the U.K. national quantum technologies programme.
Acknowledgments
This research was funded by EMPIR project 19NRM06 MeTISQ which received funding from the EMPIR program co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program, and the U.K. government department for Science, Innovation and Technology through the U.K. national quantum technologies programme.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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