## Abstract

Complete and accurate quantum state characterization is a key requirement of quantum information science and technology. The Wigner quasi-probability distribution function provides such a characterization. We reconstructed the Wigner function of a narrowband single-photon state from photon-number-resolving measurements with transition-edge sensors (TESs) at system efficiency 58(2)%. This method makes no assumption on the nature of the measured state, although a limitation on continuous-wave photon flux was imposed by the TES. The negativity of the Wigner function was observed in the raw data without any inference or correction for decoherence.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Single and multiphoton sources prepared in Fock states are of fundamental importance: not only do they enable experiments that epitomize the wave-particle “duality” of quantum mechanics, they also can only be described by quantum theory due to the nonpositivity of their Wigner quasi-probability distribution [1,2].

Eugene Wigner originally defined the continuous phase-space quasi-probability distribution function to study quantum corrections to classical statistical systems [3]. For a quantum state of density operator $\widehat{\rho}$, the Wigner function is given by

An issue with BHD-based tomography is that the reconstruction process is computationally intensive, using the inverse Radon transform, or maximum likelihood algorithms [17]. A more direct approach to reconstruct the Wigner function was proposed by Wallentowitz and Vogel [18] and by Banaszek and Wodkiewicz [19]. It is based on the following expression of the Wigner function [20]:

This technique is commonplace in quantum optics and was used, for example, to implement Bob’s CV unitary in the first unconditional quantum teleportation experiment [22]. In all rigor, the resulting Wigner function is a more general one, the $s$-ordered Wigner function, $W(s\alpha ;s)$, which tends toward $W(\alpha )$when $s=-t/r\to 0$ [23]. This method was implemented for quantum state tomography of phonon Fock states of a vibrating ion [24], as well as microwave photon states in cavity QED [25,26]. For quantum states of light, it has been experimentally realized for the positive Wigner functions of vacuum and coherent states, as well as phase-diffused coherent-state mixtures, initially detecting no more than one photon [27] and subsequently detecting several photons [28,29]. The nonpositive Wigner function of a single-photon state was confirmed using PNR measurements by time-multiplexing non-PNR, low efficiency avalanche photodiodes, albeit with the use of *a priori* knowledge of the input state in order to deconvolve the effect of losses [30]. Our work is the first demonstration of state-independent photon-counting quantum state tomography of a nonpositive Wigner function. The only assumption made here is that the initial quantum state consist of low photon numbers to avoid the saturation limit of the detector, which is less than five photons per microsecond for the superconducting transition-ddge sensor (TES) used in our experiment. Since no other prior knowledge is assumed about the state to be measured, this technique is equally applicable to any arbitrary quantum state with low photon flux. We directly observe negativity of the Wigner function with no correction for detector inefficiency.

## 2. EXPERIMENTAL SETUP AND METHODS

#### A. Setup Description

The experiment, depicted in Fig. 2, built upon our previous demonstration of coherent-state tomography [29] with the addition of the heralded single-photon source: a type II (YZY) quasi-phase-matched periodically poled ${\mathrm{KTiOPO}}_{4}$ (PPKTP) crystal, of period 450 μm, was used in a doubly resonant optical parametric oscillator (OPO), as detailed in Supplement 1 [31]. The OPO was pumped by a stable frequency-doubled 532 nm Nd:YAG nonplanar ring oscillator laser (1 kHz FWHM). The two-mirror OPO cavity was one-ended, with a finesse of 300, a free spectral range of 1.5 GHz, and an FWHM of 5 MHz. One mirror’s inside facet was 99.995% reflective for the signal and idler fields near 1064 nm and 98% transmissive for the pump field at 532 nm (the outside mirror facet was uncoated); the other mirror’s inside facet was 98% reflective at 1064 nm and 99.95% reflective at 532 nm (its outside facet was antireflection-coated at 1064 nm). The cavity was near-concentric with a super-Invar structure, the mirrors’ radius of curvature being 5 cm and the mirrors $\simeq 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$ apart. The filter cavity (FC) was made of two 5 cm-curvature, 99% reflective mirrors placed $\simeq 0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ apart.

The OPO mode was aligned and mode-matched to all parts of the experiment (FC, TES fibers) by using a seed beam that was injected into the OPO through its highly (99.995%) reflecting mirror and exited through its output coupler. The seed beam was carefully mode-matched to the OPO so as to be a pure ${\mathrm{TEM}}_{00}$ mode before being sent to the rest of the setup. It was also used for interference visibility optimization with the displacement field. A significant contribution to photon loss is any mode mismatch between the OPO and the FC, which must also be locked on resonance simultaneously, as detailed in the next section. By careful mode-matching of a seed OPO beam to the FC, we achieved 83% transmission of the OPO mode through the FC.

#### B. Stabilization Procedure

Both the OPO and the FC cavities were Pound–Drever–Hall (PDH)-locked [32] to a reference laser beam provided by the undoubled output of the pump laser. This was achieved by way of an “on/off” locking system, effected by a system of computer-controlled diaphragm shutters. In the “on” locking phase, the input to the single-photon sensitive PNR detectors was closed, and the reference laser was unblocked and sent into both the OPO and the FC (dotted lines in Fig. 2), whose PDH lock loops were closed for a few seconds. Because of its super-Invar structure, the OPO drift was low and the PDH loops could then be open, in the “off” phase, with their correction signals held constant. The shutter of the reference laser was closed and the paths between the OPO and the PNR detectors were open for data acquisition, for as long as 3 s; see Fig. 3. This procedure allowed us to lock the OPO to its doubly resonant, frequency-degenerate mode at ${\omega}_{s}={\omega}_{i}={\omega}_{p}/2$. This was essential, as the displacement field, also provided by the undoubled output of the pump laser, had to be at the same frequency as the OPO’s quantum signal beam and phase-coherent with it. Note that finding this frequency-degenerate, doubly resonant OPO mode is nontrivial since the double resonance condition

features two different indices of refraction ${n}_{s}\ne {n}_{i}$ ($L$ is the cavity length in air only, $\ell $ the crystal length, and ${m}_{s,i}\in \mathbb{N}$ the mode numbers). It is, however, possible to temperature-tune the OPO crystal to achieve stable frequency degeneracy [33–35]. This required temperature control of the PPKTP crystal to the level a few millidegrees, around 27.810°C, using a commercial temperature controller.#### C. PNR Detection

Our PNR detection system comprises two TESs, consisting of tungsten chips in a cryostat, coupled through standard telecom fiber. A detailed description of the TES system can be found in Refs. [29,36]. The TESs are cooled using an adiabatic demagnetization fridge at a stable temperature of 100 mK, at the bottom edge of the steep superconducting transition slope (resistance versus temperature). When one photon is detected, its energy is absorbed by the tungsten chip, yielding a sharp increase in its resistance that is detected by a superconducting quantum interference device over a rise time on the order of 100 ns. The heat is then dissipated through a weak thermal link, over a time on the order of 1 μs. During this time, the TES is still active (as opposed to, say, nanowire detectors or avalanche photodiodes). Due to the finiteness of its superconducting transition slope, the TES can resolve up to five photons. The absolute maximum photon flux sustainable by the TES without the tungsten driven into the normal conductive regime is therefore 5 photons/μs in the continuous-wave regime, i.e., a power of 1 pW. The OPO’s average power was kept at 100 fW by setting the pump power to 200 μW (the OPO threshold was 200 mW). We observed that the background counts were negligible when the TES signal was suppressed by rotating the pump’s linear polarization by 90°, thereby completely phase-mismatching the nonlinear interaction in PPKTP.

## 3. EXPERIMENTAL RESULTS

#### A. Heralding Ratio

The heralding ratio determines the quality of the single-photon source. It is the probability of seeing one photon in the OPO signal (heralded) beam with no displacement field, provided one photon was detected in the filtered idler (heralding) beam. The pump power was kept low enough so as to suppress two-photon events in the OPO signal in the absence of a displacement field. Results are displayed in Table 1. We can see that the heralded channel has many more counts than the heralding channel, as expected, since the latter is filtered by the FC and the interference filter (IF). The heralding efficiency was

and can also be considered the overall detection efficiency of the heralded channel, i.e., of the quantum signal.#### B. Quantum Tomography of a Single-Photon State

We now turn to the state tomography results. Figure 4 shows the reconstructed Wigner function along with a fit with the following Wigner function of the single-photon state mixed with vacuum [31],

where we took the overall detection efficiency of the signal, $\eta $, as a fit parameter [31]. The Wigner function is plotted for experimentally measured values of $|\alpha |$, where phase-space coordinates are $(q,p)=(\sqrt{2}|\alpha |\mathrm{cos}\text{\hspace{0.17em}}\varphi ,\sqrt{2}|\alpha |\mathrm{sin}\text{\hspace{0.17em}}\varphi )$, where $\varphi $ is the tomographic angle. We can clearly see the negativity around the origin of the phase space, Errors in the displacement amplitudes were considered to be negligible due to the long-term amplitude stability of the laser producing the displacement field and the high accuracy of the calibration as mentioned in the “displacement calibration” section of Supplement 1 [31]. The Wigner function error bars ($1\sigma $) at zero displacement were obtained from the statistics of multiple data sets with the displacement field blocked. At nonzero displacement, in order to speed up the measurement process and minimize experimental drifts, we decided to use the statistics of the measurement results at 10 different phases for the same displacement amplitude. This procedure yields a conservative estimate of the Wigner function error bars ($1\sigma $), in the particular case of a single-photon Fock state, because it assumes that the measured Wigner function has the required cylindrical symmetry about the origin of phase space. The results are plotted in Fig. 5. Note that the fact that Wigner function is not significantly altered by this averaging—in fact, both the two-dimensional (2D) fit in Fig. 4 and the one-dimensional (1D) fit in Fig. 5 yield $\eta =0.57(3)$—speak to the high quality of the phase-space rotational symmetry of our data. One can notice that the fit residuals are reasonably small around the origin of phase space but grow larger in the outskirts of the function, near our maximum displacement values. These correspond to larger detected photon numbers on the TES, for which photon pileups during the TES’ cooling time make data analysis more arduous [29].## 4. CONCLUSION

We have demonstrated state-independent photon-counting quantum state tomography with PNR measurements using a superconducting TES system and evidenced clear negativity in the single-photon Fock Wigner function with no correction for photon loss. This work has been limited by two factors: when working with continuous-wave detection, photon fluxes become overwhelming to the TES when $|\alpha |\to 1$. Moreover, photon pileups, in particular during the TES cooling time, greatly complicate data analysis [29]. In the future, we will multiplex several TES channels in order to access larger displacement amplitudes, i.e., larger regions of phase space. This will also reduce the photon pileup effect. Finally, owing to the intrinsic simplicity of photon-counting quantum tomography, we believe it is possible to herald and visualize Fock state Wigner functions in real time for quantum information applications.

## Funding

National Science Foundation (PHY-1708023, PHY-1521083).

## Acknowledgment

The authors thank Rafael Alexander, Carlos Andreas González Arciniegas, Xu Yi, Avi Pe’er, Chun-Hung Chang, Jacob Higgins, Chaitali Joshi, and Xuan Zhu for helpful discussions. We would also like to thank Scott Glancy and Arik Avagyan for valuable comments and feedback during the revision process.

See Supplement 1 for supporting content.

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