## Abstract

We study the problem of determining the photon number statistics of an unknown quantum state using conjugate optical homodyne detection. We quantify the information gain in a single-shot measurement and show that the photon number statistics can be recovered in repeated measurements on an ensemble of identical input states without scanning the phase of the input state or randomizing the phase of the local oscillator used in homodyne detection. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction and illustrate our approach by conducting experiments to study the photon number distributions of a weak coherent state and a thermal state source.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A single photon detector (SPD) is a workhorse of modern quantum optics experiments. It is commonly used to determine the number of photons in a given light pulse in a single-shot measurement scenario [1,2]. While photon-number-resolving detection schemes have been demonstrated using either a single SPD or SPD arrays [3–8], most commercial SPDs are threshold detectors which can only distinguish between vacuum and non-vacuum states. Alternatively, given a large ensemble of identical input states, optical homodyne detection (OHD) can be used to completely reconstruct the quantum state of the ensemble, including the photon number statistics. This method is known as the optical homodyne tomography [9–11].

In OHD, a strong local oscillator (LO) is mixed with a weak input signal at a beam splitter. The interference signals can be strong enough to be detected by low cost, highly efficient photo-diodes working at room temperature. This makes OHD an appealing solution in practice, for example, in chip-size implementation [12]. Note that the LO in OHD also functions as a mode selector: only the signals in the same spectral-temporal mode as the LO will be detected. This can be advantageous in certain applications such as quantum key distribution (QKD) [13–16], where the broadband background noise originating from the communication channel should be suppressed [17–20].

Given the LO is sufficiently strong, a DC-balanced homodyne detector measures quadrature $X_\theta$ of the input signal, where $\theta$ is the phase of the LO [21,22]. To reconstruct the full quantum state, repeated measurements are required for all values of $\theta \in (0,2\pi ]$. Obviously, a crucial requirement for such a reconstruction scheme is that the phase of the input state is well controlled. However, in many applications, this requirement cannot be easily satisfied. For example, in QKD, the quantum signals detected by the receiver come from a channel controlled by an adversary. In this case, we cannot make any assumptions about the phase of incoming signals. Fortunately, it has been shown that the photon number statistics of an ensemble of states with unknown phases can still be fully recovered by using either one or two optical homodyne detectors, given the phase of the LO is uniformly randomized [23–27]. In the case of QKD, phase randomization can be implemented by using a phase modulator driven by a random pattern, as demonstrated in [28]. Nevertheless, the requirements of truly random numbers and high-speed phase modulation introduce additional complexities.

In this paper, we show that the conjugate optical homodyne detection scheme [26,29,30] can be used to determine the photon number statistics without controlling the phase of the input quantum state or randomizing the phase of the LO. In this scheme, two homodyne detectors are employed to measure conjugate quadratures of the input state simultaneously. We define a measurement observable $Z$ as the sum of the squared outputs of the two homodyne detectors. In classical electrodynamics, the outputs of two conjugate homodyne detectors correspond to the in-phase and out-of-phase components of an electromagnetic wave, so the observable $Z$ defined above is proportional to the intensity (or the photon number) of the input signal. In quantum mechanics, canonically conjugate quadrature components of quantum optical fields do not commute with each other and thus cannot be determined simultaneously and noiselessly due to Heisenberg’s uncertainty principle. So, a single-shot measurement of $Z$ is intrinsically noisy. Nevertheless, it can still provide partial information about the photon number of the input state. By repeating the $Z$ measurement on a large ensemble of identical states, the photon number statistics can be determined.

This paper is organized as follows: in Sec. 2, we present the theory of conjugate homodyne detection. In Sec. 3, we study the case of single-shot measurement and quantify the information gain on the input photon number. In Sec. 4, we study the case of repeated measurements on an ensemble of identical input states. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction of photon number statistics and illustrate our approach by conducting experiments with a weak coherent state and a thermal state source. Finally, we conclude this paper with a discussion in Sec. 5.

## 2. Conjugate homodyne detection

The basic setup of a conjugate homodyne detection system is shown in Fig. 1. The input quantum state $\vert \psi \rangle$ is split into two by a symmetric beam splitter ($\textrm {BS}_1$ in Fig. 1). One output of the beam splitter (mode 3 in Fig. 1) is measured by an optical homodyne detector with an LO phase $\theta$; the other output state (mode 4 in Fig. 1) is measured by another optical homodyne detector with an LO phase $\theta +\pi /2$. The two LOs can be generated from a common laser using a beam splitter and a $\pi /2$ phase shifter. The common phase $\theta$ is defined using the phase of the input state as a reference. Provided that the input state has an unknown phase which may change from pulse to pulse, we have no control of $\theta$ (i.e. $\theta$ is a random variable with an unknown distribution). We remark that the setup shown in Fig. 1 (plus the beam splitter and the $\pi /2$ phase shifter for generating two LOs from a single laser) can be conveniently implemented with a compact commercial $90^{0}$ optical hybrid [31]. In this Section, we assume noiseless homodyne detectors with unity efficiency. We will discuss the case of non-unity detection efficiency in Sec. 4.

Given the LOs are sufficiently strong, the outputs of the two homodyne detectors are quadrature components of mode 3 ($X_{3,\theta }$) and mode 4 ($X_{4,\theta +\pi /2}$). For simplicity, we use $X_3$ and $P_4$ to represent $X_{3,\theta }$ and $X_{4,\theta +\pi /2}$ in the rest of the paper.

We define a new parameter

Intuitively, $Z$ is proportional to the intensity (thus the photon number) of the input light.In quantum optics, the two homodyne outputs are represented by operators $\hat {X}_{3}$ and $\hat {P}_{4}$, which are defined in terms of photon annihilation operator $\hat {a}$ and photon creation operator $\hat {a}^{\dagger }$ as

#### 2.1 Expectation value of $\hat {Z}$

As shown in Fig. 1, the joint input state of mode 1 and 2 is given by $\vert \psi _1 0_2\rangle$, where $\vert 0\rangle$ represents vacuum state. From Eq. (8), the expectation value of $\hat {Z}$ can be determined to be

where $\langle n_1\rangle$ is the average photon number of the input state. The constant $1$ on the RHS of Eq. (9) can be interpreted as vacuum noise contribution [33]. Eq. (9) shows that the average photon number of the input state can be estimated by subtracting $1$ from the expectation value of $\hat {Z}$.#### 2.2 Variance of $\hat {Z}$

The variance of $\hat {Z}$ is given by

#### 2.3 Second-order correlation function $g^{(2)}(0)$

The single-time second-order correlation function $g^{(2)}(0)$ is an important parameter for characterizing a photon source [34]. In the case of a single homodyne detector with a phase randomized LO, it has been shown that $g^{(2)}(0)$ can be determined from the measurement statistics [35,36]. Here, we show that $g^{(2)}(0)$ can also be determined from the statistics of $Z$ measurement.

As defined in [34]

#### 2.4 Probability density function of $\hat {Z}$

Unlike photon number $n$, the parameter $Z$ measured in our scheme is a continuous variable. Given an arbitrary input state described by the density matrix $\rho$, we would like to determine the probability density function (PDF) of $Z$.

The joint PDF of quadrature components $X_3$ and $P_4$ of conjugate homodyne detection has been derived in [26] as

## 3. Single-shot measurement

In most of previous studies on OHD, one assumption is that an ensemble of identical input states are available. This allows precise quantum state characterization based on repeated measurements. However in some applications, we may only have one copy of the state. In this section, we study the case of single-shot measurement and quantify the information gain on the input photon number.

Given the input state is a Fock state $\rho =\vert n \rangle \langle n \vert$, the likelihood of a measurement output of $z$ can be determined from Eq. (19) as

Using Bayes’ rule the likelihood of $n$ photons coming in given the measurement output $z$ is In the lase step of Eq. (21), we have assumed that the prior $P_N(n)$ is a uniform distribution. This leads to a uniform PDF of $P_Z(z)$ from Eq. (19).The distribution in Eq. (21) is the Poisson distribution, which quantifies the uncertainty of the photon number $n$ given a single measurement of $Z$. This shows that our measurement scheme is intrinsically noisy. The uncertainty of the photon number can be determined from Eq. (21) as

Next, we evaluate the performance of the proposed scheme as a threshold SPD which can discriminate vacuum state from non-vacuum states but cannot resolve photon number. This kind of detector is commonly using in discrete-variable (DV) QKD. The basic strategy is to choose an optimal threshold value $T\in \left [ 0,\infty \right )$ for the $Z$ measurement: if the measurement result $z$ is smaller (larger) than $T$, the input state is assigned as vacuum (non-vacuum) state.Two important parameters of a threshold SPD are detection efficiency and dark count probability. The detection efficiency $\eta$ is defined as the conditional probability that the detector reports a non-vacuum state given the input is a single photon Fock state. The dark count probability $D$ is defined as the conditional probability that the detector reports a non-vacuum state given the input is vacuum. From Eq. (20), these two parameters can be determined by

Figure 2 shows the simulation results of $\eta$ and $D$ as a function of the threshold value $T$. By choosing an appropriate threshold value, we can either achieve a high detection efficiency or a low dark count. Unfortunately, we cannot have both at the same time. Similar conclusions had been drawn in previous studies based on the single homodyne detection scheme [37]. In Fig. 2, we also present the ratio $R=\eta /D$, which is an important figure of merit in applications like QKD. A state-of-the-art SPD can provide a R-value as high as $10^8$ [1]. In comparison, the R-value of the proposed scheme is less than 10 in the region where the detection efficiency is not too low.In next section, we discuss the case of determining photon number distribution from repeated measurements.

## 4. Repeated measurements

Given a sequence of M independent measurements of $Z$, denoted as $\{z_{i}\} = z_{1},\dots ,z_{M}$, we would like to infer the underlying photon number statistics given by the distribution $p(n) = \rho _{nn}$ for $n = 0,1,\dots ,N_{max}$. The observed data statistics can be described in terms of a mixture model of $N_{max}+1$ gamma distributions $\textrm {Gamma}(n+1,1)$ with mixing coefficients given by the photon number distribution $p(n)$. In this model the conditional PDF of observing an outcome $Z_{k}=z_{k}$ reads,

The complete data likelihood function $\mathcal {L}_{c}$ for the entire measurement sequence can be written as, Here, $p(n), n = 0,\ldots ,N_{max}$ are the unknown parameters (Fock state probabilities) we wish to infer and $n_{1},\dots ,n_{M}$ are random variables, each in range $[0,N_{max}]$, that determine which mixture component $n_{i}\in [0,N_{max}]$ has generated an outcome $z_{i}$. Have we known the values of $n_{1},\dots ,n_{M}$ then we could use the maximum-likelihood estimation (MLE) based on $\mathcal {L}_{c}$ to infer the most likely values of the parameters $p(n)$. Unfortunately the variables $n_{i},i={1,M}$ are unobserved (latent). Therefore, in order to use MLE we first need to marginalize the complete data likelihood $\mathcal {L}_{c}$ over the latent variables. The resulting marginal likelihood of the observed data reads, where we used the following convention, In principle one now can try and maximize the function $\mathcal {L}$ with respect to parameters $p(n)$, but in practice the complexity of this task will grow exponentially with the number of measurements $M$. Fortunately, there is a way to determine the maximum of $\log \mathcal {L}$ that avoids explicit maximization of $\mathcal {L}$. This method is widely used for inference in mixture models and is called expectation-maximization (EM) algorithm [38]. Applicability of EM in the context of homodyne measurement of photon statistics has been advocated previously [39]. EM is an iterative procedure that uses the expected value of the complete data likelihood $\mathcal {L}_{c}$ with respect to the latent random variables $n_{1},\dots ,n_{M}$ as a maximization objective function for determining $p^{t}(n)$ – the $t$-th iteration estimate of the parameters $p(n)$. Here is how it works: First, set initial estimates of the parameters $p^{0}(n)$ to some values. We chose a uniform prior as it is uninformative and easy to implement. Due to the multimodality of Eq. (27) the choice of prior will bias the EM reconstruction given below. It is an open problem to identify the best prior for this application. Next, repeat the following steps until convergence criteria are satisfied.- • At the $t$-th iteration, update probabilities $p(n_{k}|z_{k},\{p^{t}(n)\})$ for all latent variables $n_{1},\dots ,n_{M}$ using Bayes rule with $p^{t}(n)$ as a prior,
$p(n_{k}|z_{k},\{p^{t}(n)\}) = \frac {p(Z_{k}=z_{k}|\{p^{t}(n)\},n_{k})}{\sum \limits_{n_{k}=0}^{N_{max}}p(Z_{k}=z_{k}|\{p^{t}(n)\},n_{k})}$

- • Calculate the expected value of the complete data $\log$ likelihood with respect to the updated distribution $p(n_{k}|z_{k},\{p^{t}(n)\})$,
$Q(\{p(n)\}|\{p^{t}(n)\}) = \textrm {E}_{\{n_{k}\}|\{z_{k}\},\{p^{t}(n)\}}[\log \mathcal {L}_{c}]=$

$\sum \limits_{k=1}^{M}\sum \limits_{j=0}^{N_{max}}p(j|z_{k},\{p^{t}(n)\})\log [p(Z_{k}=z_{k}|\{p(n)\},j)]$

- • Find parameter values $\{p_{max}(n)\}$ that maximize $Q(\{p(n)\}|\{p^{t}(n)\})$ and set the next iteration estimates of photon number probabilities $\{p^{t+1}(n)\}=\{p_{max}(n)\}$. Note that the values of $\{p_{max}(n)\}$ can be calculated analytically,
$p^{t+1}(n) = \frac {1}{M}\sum \limits_{j=1}^{M}p(n|z_{j},\{p^{t}(n)\})$.

#### 4.1 Simulation results

To illustrate our EM-based photon number statistics inference scheme we applied it to a sequence of $32768$ simulated homodyne measurement outcomes for a coherent state $\rho = |\alpha \rangle \langle \alpha |$ with the mean photon number $|\alpha |^{2}=5$. To reconstruct the photon number statistics from the simulated measurement data, we have selected a mixture model with $N_{max}=20$ ($21$-component model). The results of EM reconstruction (after 9 iterations) are plotted on Fig. 3 and demonstrate a good quantitative agreement with the true state.

#### 4.2 Experimental results

We apply the theory described above in experiments to reconstruct photon number distributions of a weak coherent state and a thermal state.

In Sec. 2, we have assumed perfect homodyne detectors with unity efficiency. Here we consider practical detector with non-unity detection efficiency. It is well known that a realistic photo-detector with efficiency $\eta$ can be modeled by placing a virtual beam splitter (with a transmittance of $\eta$) in front of an ideal detector [40]. By assuming the four photo-detectors have identical efficiency, we can model the conjugate homodyne detection using the setup shown in Fig. 4(a). In Appendix A we will show that given the LOs are strong enough, the setup in Fig. 4(a) is equivalent to that in Fig. 4(b), where the four virtual beam splitters in front of the photo-detectors are replaced by a common virtual beam splitter (with the same transmittance) at the input of the first beam splitter. This is convenient in practice since we can apply the theory in Sec. 2 directly to the experimental results by assuming the photo-detectors are ideal. The photon number distribution reconstructed this way is related to that of the input state by the Bernoulli transformation [41]. By further applying the inverse Bernoulli transformation, the photon number distribution of the input state can be determined.

In our experiments, either a weak coherent source or a thermal source is employed as the input. The state after the virtual beam splitter in Fig. 4(b) is still a coherent state (or a thermal state) with a reduced average photon number. This is convenient since we can simply redefine the state after the virtual beam splitter as the input state and assume the detection efficiency is one.

The experimental setup is shown in Fig. 5. A 1550 nm continuous wave (CW) laser is employed as the LO. The conjugate optical homodyne detection system is constructed by a commercial $90^o$ optical hybrid (Optoplex) and two $350$ MHz balanced amplified photodetectors (Thorlabs). Variable optical attenuators are used to balance the detection efficiency of different channels and control the average photon number of the input state. The outputs of the two balanced photodetectors are sampled by a two-channel data acquisition board (Texas Instruments). The overall efficiency of the detection system is 0.5, with electrical noises variances (in shot-noise unit) of $\sigma ^{2}_{X_3} = 0.21, \sigma ^{2}_{P_4} = 0.16$ for the quadratures $X_{3}$ and $P_{4}$ respectively.

In the first experiment, a heavily attenuated laser source is used to provide a weak coherent state input. By adjusting the variable optical attenuator, the average photon number within one sampling window has been set to 5 (after correction of detection efficiency). In the second experiment, an amplified spontaneous emission (ASE) source is used to provide a thermal state input (with an average photon number of 15.3). Note, while the output of the ASE source contains multiple spectral-temporal modes, the homodyne detector selectively measures the one matched with the mode of LO. Limited by the memory size of the data acquisition board, 32728 quadrature pairs are sampled in each measurement.

Using Eq. (15), the $g^{(2)}(0)$ factors of the two sources have been determined to be $1.11\pm 0.02$ (weak coherent source) and $1.94\pm 0.02$ (thermal source) correspondingly, where the uncertainty quantifies statistic fluctuation. These results match with the theoretical values of 1 (ideal coherent state) and 2 (ideal thermal state) reasonably well.

We apply the EM algorithm described above to the experimental data in order to reconstruct the photon number statistics of the two light sources. In Fig. 6 we plot reconstruction results for a weak coherent state. The blue bars represent the photon number statistic of a coherent state $|\alpha \rangle$ with the mean number of photons $|\alpha |^2 = 5$, assuming noiseless detectors. The red bars correspond to the numerically synthesized data from the coherent state $|\alpha \rangle$ with added Gaussian noise which mimics the effect of noisy photo detectors. The green bars depict the photon number statistics reconstructed from raw experimental data by using the EM algorithm. We note that the red and green histograms look remarkably similar which implies that experimental data come from a coherent state affected by the detector noise. In Fig. 7 we depict EM reconstruction results for a thermal state with a mean photon number of 15.3. The black line represents the theoretical distribution of an ideal thermal state. The red bar are EM reconstruction results using numerically simulated quadrature measurement data with added detector noise. Finally, the blue bars correspond to photon number statistics reconstructed from actual experimental data. We notice that for the thermal state the three distributions are very close visually. This is because the detector noise is much smaller than the mean number of photons in this case. Therefore, the noise effects are not as pronounced as in the case on a weak coherent state.

## 5. Discussion

Classically, the intensity of a single-mode light pulse can be determined by measuring two conjugate quadratures simultaneously. Quantum mechanically, the above measurement process is intrinsically noisy. In this paper, we develop theoretical tools to reconstruct photon number statistics of a single-mode quantum state by performing conjugate homodyne detection. Comparing with previous studies based on single homodyne detection, no LO phase randomization is required in our scheme. Furthermore, partial information about input photon number can be acquired in a single-shot measurement.

While the main goal of this paper is to develop the relevant theoretical tools and illustrate how to apply them by conducting experiments with two different classical light sources, the technology described here may have practical applications in optical sensing and communication. For example, in continuous-variable (CV) QKD [42–44] based on conjugate homodyne detection (also called “heterodyne detection” in literature) [45,46], Bob measures both X and P quadratures simultaneously for key generation. So the $Z$ information is automatically available to Bob without any changes to the QKD system or the measurement procedures. The acquired photon number information could be used to detect eavesdropper’s attacks [47,48], or reduce the dimension of the relevant quantum space, as required in some recent security proofs of CV-QKD with discrete modulations [49–51]. The application of the proposed technology in QKD could be a future research topic.

Finally, we would like to emphasis that the conjugate homodyne technique is essentially a Gaussian measurement, which provides Gaussian-distributed outputs when applied to Gaussian states [52]. It cannot entirely replace a non-Gaussian measurement like photon counting. In fact, recent studies have shown that some important tasks in quantum information, such as CV entanglement distillation [53–55] and quantum repeater [56], cannot be achieved with Gaussian operations alone. Similarly, the discovery that CV quantum computing with only Gaussian elements can be efficiently simulated with a classical computer [57] implies that non-Gaussian elements (such as photon counting) are required to demonstrate quantum advantage [58]. In brief both homodyne detection and photon counting will play important roles in classical and quantum worlds.

## Appendix A: Detector efficiency in the conjugate homodyne scheme

We show that given the LOs are strong enough, the setup shown in Fig. 4(a) is equivalent to that in Fig. 4(b), where the four virtual beam splitters in front of the photo-detectors are replaced by a common virtual beam splitter (with the same transmittance) at the input path of the first beam splitter.

Given the LO is strong enough, a single DC-balanced homodyne detector using two lossy photo-detectors can be modeled by the one with ideal photo-detectors by placing a virtual beam splitter at the signal input [59]. This allows us to replace the four virtual beam splitters in Fig. 4(a) by two virtual beam splitters with the same transmittance (one at each of the output port of beam splitter one). We will show that these two virtual beam splitters can be further replaced by one as shown in Fig. 4(b). More specifically, we will show the two models in Fig. 8 are equivalent to each other.

For simplicity, we define the transmittance of the virtual beam splitter as $\eta =\cos \gamma$. From Fig. 8(a) and using the transmission relations of lossless beam splitter, we have

Similarly, the P-quadrature of mode 4 in Fig. 8(a) is given by

We can apply the same process in the model shown in Fig. 8(b) and have the following relations:

It is easy to show $X_{V1}$, $P_{V2}$, $X_{V3}$, $P_{V4}$ are independent and identically distributed random variables. From Eqs. (29)–(32), the joint probability of $X_3$ and $P_4$ is the same as that of $X_{3'}$ and $P_{4'}$. So the two models given in Fig. 8 are equivalent.

## Funding

Office of Electricity Delivery and Energy Reliability.

## Acknowledgments

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy (DOE) will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). We acknowledge helpful comments from Ryan Bennink, Warren Grice, Charles Lim and Nicholas Peters.

## Disclosures

The authors declare no conflicts of interest.

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