## Abstract

Photon detectors are an elementary tool to measure electromagnetic waves at the quantum limit^{1,2} and are heavily demanded in the emerging quantum technologies such as communication^{3}, sensing^{4} and computing^{5}. Of particular interest is a quantum non-demolition (QND)-type detector, which projects an electromagnetic wave onto the photon-number basis^{6,7,8,9,10}. This is in stark contrast to conventional photon detectors^{2} that absorb a photon to trigger a ‘click’. The long-sought QND detection of a flying photon was recently demonstrated in the optical domain using a single atom in a cavity^{11,12}. However, the counterpart for microwaves has been elusive despite the recent progress in microwave quantum optics using superconducting circuits^{13,14,15,16,17,18,19}. Here, we implement a deterministic entangling gate between a superconducting qubit and an itinerant microwave photon reflected by a cavity containing the qubit. Using the entanglement and the high-fidelity qubit readout, we demonstrate a QND detection of a single photon with the quantum efficiency of 0.84 and the photon survival probability of 0.87. Our scheme can serve as a building block for quantum networks connecting distant qubit modules as well as a microwave-photon-counting device for multiple-photon signals.

## Main

Microwave quantum optics in superconducting circuits enables us to investigate unprecedented regimes of quantum optics. The strong nonlinearity brought by Josephson junctions together with the strong coupling of the qubits with resonators/waveguides reveals rich physics not seen in the optical domain before. It has also been applied in demonstrations of the generation and characterization of non-classical states in cavity modes^{13,14,15} and propagating modes^{16,17} as well as the remote entanglement of localized superconducting qubits^{18,19}. However, single-photon detection in the microwave domain is still a challenging task because of the photon energy being four to five orders of magnitude smaller than in optics. The sensitivities of conventional incoherent detectors such as avalanche photodiodes, bolometers and superconducting nanowires are not sufficient for single microwave photons^{2}. Therefore, resonant absorption of a microwave photon with a superconducting qubit was exploited for single-photon detection^{20,21}. Note also that QND measurements of cavity-confined microwave photons have been realized by using a Rydberg atom or a superconducting qubit as a probe^{22,23}.

For a QND detection of an itinerant microwave photon, we use a circuit quantum electrodynamics architecture with a transmon qubit in a far-detuned 3D cavity^{24}. An input pulse mode through a one-dimensional (1D) transmission line to the cavity is entangled with the qubit upon the reflection and is projected to a number state by the subsequent qubit readout without destroying the photon (Fig. 1).

In our set-up, the qubit–cavity interaction is described with the Hamiltonian

where *a*^{†}(*a*) is the creation (annihilation) operator of the cavity mode, *σ*_{
z
} is the Pauli operator of the transmon qubit, *ω*_{c} is the cavity resonance frequency, *ω*_{q} is the qubit resonance frequency and *χ* is the dispersive shift due to the interaction. We control the qubit state with a Rabi oscillation driven by a resonant pulse and read out the qubit non-destructively via the dispersive shift of the cavity frequency. A readout pulse reflected by the cavity is led to a flux-driven Josephson parametric amplifier (JPA)^{25} and is measured in the quadrature by a heterodyne detector. The nearly quantum-limited amplifier enables us to read out the qubit state in a single shot, with the assignment fidelity^{26} of 0.988 ± 0.001.

The interaction between an itinerant microwave field and the superconducting qubit through the cavity is first characterized by the cavity reflection of weak continuous microwaves. Figure 1b shows the spectra, with the qubit being in the ground state $\u2223\mathrm{g}\u27e9$ (blue) or the excited state $\u2223\mathrm{e}\u27e9$ (red). The dispersive shift of the cavity frequency is observed in accordance with equation (1). With the optimal configuration where the external coupling rate of the cavity *κ*_{ex} is adjusted to twice the dispersive shift, 2*χ*, the qubit-dependent phase shift (phase difference in Fig. 1b) of the reflected field is close to π within the bandwidth centred at the cavity frequency *ω*_{c} (green region in Fig. 1b).

The phase-shift condition also holds for a pulse mode as long as its spectral bandwidth fits inside the cavity bandwidth. A single photon in the reflected pulse mode acquires the π-phase shift conditioned on the excited state of the qubit (Fig. 1c), while maintaining the temporal and spatial mode shapes. It corresponds to the controlled-*Z* gate between the superconducting qubit and the pulse mode. As a result of the symmetry between the control and the target qubits in a controlled-*Z* gate, the interaction can also be interpreted as a phase-flip gate of the qubit induced by the reflection of the single photon (Fig. 1d). There is a trade-off between the quantum efficiency of the single-photon detection and the detection bandwidth (inverse pulse length). A longer single-photon pulse acquires a more ideal phase flip at the cost of increased qubit decoherence. We optimize the input pulse length in terms of the quantum efficiency (see Supplementary Section 4).

The protocol for the QND detection of an itinerant photon is shown in Fig. 2a,b. (i) The qubit is initialized to the ground state $\u2223\mathrm{g}\u27e9$ via the non-destructive readout and post-selection. The input state of the microwave pulse mode is a coherent state in the weak power limit with the single-photon occupancy *p*_{1}, which well approximates a superposition of the vacuum and single-photon states, $\sqrt{{p}_{0}}\u22230\u27e9+\sqrt{{p}_{1}}\u22231\u27e9$. (ii) The qubit state is rotated by π/2 about the *y* axis and the composite system becomes $\u2223+\u27e9\left(\sqrt{{p}_{0}}\u22230\u27e9+\sqrt{{p}_{1}}\u22231\u27e9\right)$. (iii) The pulse mode is reflected by the cavity, and the state after the controlled-*Z* gate becomes entangled, $\sqrt{{p}_{0}}\u2223+\u27e9\u22230\u27e9+\sqrt{{p}_{1}}\u2223-\u27e9\u22231\u27e9$. (iv) Finally, the qubit is rotated by −π/2 about the *y* axis to obtain $\sqrt{{p}_{0}}\u2223\mathrm{g}\u27e9\u22230\u27e9+\sqrt{{p}_{1}}\u2223\mathrm{e}\u27e9\u22231\u27e9$ and is then measured in the *Z* basis. The presence of a single photon in the pulse mode is correlated with the excited state of the qubit and is detectable.

The phase-flip probability of the qubit as a function of the average photon number ${\u2223{\alpha}_{\mathrm{in}}\u2223}^{2}$ in the input pulse is shown in Fig. 2c. The slight deviation from the linear relationship is due to the two-photon occupation in the pulse mode. By fitting the slope in the weak power limit, we evaluate the quantum efficiency of the detection scheme to be 0.84 ± 0.02. The reduction of the efficiency from unity is attributed to a few mechanisms. First, the external coupling rate is not perfectly adjusted to twice the dispersive shift (*κ*_{ex}/2*χ* = 1.1), which causes an incomplete phase flip of the qubit. Second, an input photon is probabilistically absorbed in the cavity due to the finite internal loss rate *κ*_{in}/*κ*_{ex} = 0.07, which also gives rise to the incomplete phase flip. Finally, the qubit dephasing during the gate interval results in an erroneous phase flip of the qubit. The qubit dephasing also contributes dominantly to the dark-count probability of 0.0147 ± 0.0005.

To verify the QND property of the photon detector, we analyse the reflected pulse mode by using Wigner tomography via the quadrature measurements. We measure the quadrature *α*_{
θ
} of the reflected pulse mode, which is amplified by the phase-sensitive amplifier (JPA) with the various pump phases *θ* to obtain the necessary information. Then, we characterize the quantum state with the iterative maximum-likelihood tomography^{27}, correcting for the measurement inefficiency of the quadratures 1 − *η*_{meas}, where *η*_{meas} = 0.43 ± 0.01.

As the input signal, we use a coherent pulse with the average photon number of ${\u2223{\alpha}_{\mathrm{in}}\u2223}^{2}=0.165\pm 0.003$. First, in Fig. 3a,b, we plot the Wigner functions of the reflected pulse mode when the qubit is prepared in the ground state $\u2223\mathrm{g}\u27e9$ and in the excited state $\u2223\mathrm{e}\u27e9$, respectively. The outcomes are the coherent states with π-phase difference depending on the qubit states. Next, in Fig. 3c,d, we show the Wigner function and the photon-number distribution when the qubit is prepared in the superposition state $\u2223+\u27e9$. Without being conditioned on the outcome of the qubit readout, the obtained state is an incoherent mixture of the states in Fig. 3a,b. Importantly, the interaction for the photon detection retains the photon-number distribution with the survival probability of 0.87 ± 0.03, which is calculated from the ratio of the average photon number of the reflected pulse mode to that of the input.

Figure 3e–h shows the conditioned results. In the case without a qubit phase flip (Fig. 3e,f), the reflected pulse mode is in the vacuum state with the fidelity of 0.9844 ± 0.0002 (theory: 0.9894). The weak squeezing seen in the Wigner function is due to the finite probability of two-photon occupation (~0.007) in the pulse mode and the coherence between the vacuum and the two-photon state. On the other hand, for the case with a qubit phase flip (Fig. 3g,h), the reflected pulse mode is in the single-photon state with the fidelity of 0.84 ± 0.02 (theory: 0.82). The infidelity is mainly due to the internal loss of the cavity and dark counts. The small anisotropy in the observed Wigner function is attributed to the incomplete phase flip of the qubit, which does not erase the coherence completely. Those results prove that the outcome of the qubit readout is strongly correlated to the photon-number state of the reflected pulse mode and demonstrate a QND single-photon detection. The system also works as a heralded single-photon generator. Since the QND detection maintains the pulse mode as long as the pulse bandwidth is within the cavity bandwidth, we can control the temporal mode shape of the heralded single photon by tuning the envelope of the input coherent pulse.

Finally, we analyse the composite state of the qubit and the reflected pulse mode to verify the entanglement. After the reflection of the pulse, the qubit is measured in three orthogonal bases *X*,*Y* and *Z*, and the reflected pulse mode is measured in the quadrature *α*_{
θ
} with various phases. We characterize the density matrix *ρ* of the composite quantum system by using the iterative maximum-likelihood reconstruction with the composite measurement operators, correcting for the inefficiency in the quadrature measurement of the pulse mode (Fig. 4). The correlation in the diagonal elements enables the QND detection of an itinerant photon. Moreover, the off-diagonal elements indicate the presence of entanglement. We calculate the negativity $\mathcal{N}\left(\rho \right)$ of the composite system from the density matrix and obtain $\mathcal{N}\left(\rho \right)=0.296\pm 0.005>0$, quantifying the entanglement^{28}. Note that for the given value of the average photon number ${\u2223{\alpha}_{\mathrm{in}}\u2223}^{2}$, the maximum possible value of the negativity in the composite system is 0.346. The fidelity of the experimentally obtained density matrix to the one with the ideal controls and measurements is found to be 0.957 ± 0.003.

Here, we focused on a superposition of the vacuum and single-photon states in a pulse mode. However, the QND detection scheme can be readily applied to many-photon states, where the qubit detects the even/odd parity of the photon number in the pulse mode. This can be applied to Wigner tomography of multi-photon states as well as heralded generation of a Schrödinger cat state in an itinerant mode. Moreover, by cascading the QND detectors with different conditional phases, we can realize a number-resolved photon counter for a microwave pulse mode.

*Note added in proof:* After the submission of this work, we became aware of a related manuscript^{29} taking a different approach for the same purpose.

## Methods

### System parameters

An aluminium-made transmon qubit on a sapphire substrate is mounted at the centre of a 3D aluminium cavity that is over-coupled to a 1D transmission line (Fig. 1a). The parameters determined from independent measurements are as follows: the cavity resonance frequency *ω*_{c}/2π = 10.62524 GHz, the qubit resonance frequency *ω*_{q}/2π = 7.8693 GHz, the dispersive shift *χ*/2π = 1.50 MHz, the cavity external coupling rate *κ*_{ex}/2π = 3.32 MHz, the cavity internal loss rate *κ*_{in}/2π = 0.25 MHz, the qubit relaxation time *T*_{1} = 32 μs, the qubit dephasing time ${T}_{2}^{*}=26$ μs and the echo decay time *T*_{2E} = 33 μs. The qubit readout fidelity of the ground state (excited state) is better than 0.998 (0.978). The assignment fidelity is calculated as the average of the two^{26}. The population of the qubit excited state before the first qubit readout is 0.067 in thermal equilibrium. The qubit initialization fidelity via the qubit readout and post-selection is better than 0.998. The coupled system fulfils the optimal conditions of *κ*_{ex} ≈ 2*χ*, and the interaction bandwidth between a microwave pulse mode and the qubit is much larger than the qubit dephasing rate $\left({\kappa}_{\mathrm{ex}}\gg 1\mathrm{\u2215}{T}_{2}^{*}\right)$.

### Calibration of the average photon number

To evaluate the quantum efficiency of the QND detection precisely, the calibration of the average photon number ${\u2223{\alpha}_{\mathrm{in}}\u2223}^{2}$ in the input pulse is crucial. We calibrate the photon flux of a continuous cavity drive by measuring the microwave-induced dephasing rate of the qubit^{30}, from which we calculate the average photon number ${\u2223{\alpha}_{\mathrm{in}}\u2223}^{2}$ by integrating the photon flux within the input temporal mode.

### Pulse sequence

The pulse lengths for the qubit control and readout are 25 ns and 500 ns, respectively. The length of the JPA pump pulse accompanying the qubit readout pulse is 650 ns. The amplitude envelope of the input pulse mode is defined to be a Gaussian with the full-width at half-maximum of 500 ns. The gate intervals of the Ramsey sequence are set to 800 ns for the evaluation of the quantum efficiency, and to 1,100 ns for the quantum state tomography of the reflected pulse mode in order to avoid the overlap with the readout pulse. To obtain the quantum efficiency of the QND detection, the sequence is repeated 10^{5} times and the qubit phase-flip probability is determined. For the quantum state tomography of the reflected pulse mode, the phase of the quadrature measurement is swept from 0 to π with a step of π/100. The sequence is repeated 10^{4} times for each phase. To characterize the density matrix of the composite system, the sequence is repeated 10^{4} times for each phase of the quadrature measurement and each of three orthogonal measurement bases, *X*, *Y* and *Z*, of the qubit. *X*- and *Y*-basis readouts of the qubit are implemented by the Z-basis readout with a qubit rotation.

### Quadrature measurement efficiency

As shown in Fig. 1a, the reflected pulse is led to the JPA operated in the degenerate mode, and the quadrature *α*_{
θ
} is measured with the heterodyne detector in the same temporal mode shape as the input. The large gain and small added noise by the JPA in the quadrature measurement suppress the effect of the imperfections in the measurement chain following the JPA^{31}. The remaining propagation loss and Gaussian noise can be modelled with an insertion of a beamsplitter with a transmittance *η*_{meas} in front of an ideal quadrature detector^{32}. From the calibration with a weak coherent pulse (see Supplementary Section 5), the measurement efficiency is found to be *η*_{meas} = 0.43 ± 0.01. The dominant factor for the inefficiency is the propagation loss through the series of circulators between the cavity and the JPA.

### Iterative maximum-likelihood reconstruction^{27}

For the quantum state tomography of the reflected pulse mode, we use the quadrature-measurement operators represented in the photon-number basis. The measurement operators are corrected for the measurement inefficiency 1 − *η*_{meas}. For the quantum state tomography of the composite system, we use the composite operators of the quadrature measurements and the qubit measurements. In both cases, the iterative algorithm is repeated 10^{5} times.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We acknowledge the fruitful discussions with T. Serikawa, T. Sugiyama, Y. Shikano, R. Yamazaki and K. Usami. This work was supported in part by the Advanced Leading Graduate Course for Photon Science (ALPS), the University of Tokyo, the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) (no. 16K05497 and no. 26220601), and the Japan Science and Technology Agency (JST) Exploratory Research for Advanced Technology (ERATO) (grant no. JPMJER1601).

## Author information

### Affiliations

#### Research Center for Advanced Science and Technology (RCAST), University of Tokyo, Tokyo, Japan

- S. Kono
- , Y. Tabuchi
- , A. Noguchi
- & Y. Nakamura

#### College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Ichikawa, Japan

- K. Koshino

#### Center for Emergent Matter Science (CEMS), RIKEN, Wako, Japan

- Y. Nakamura

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### Contributions

S.K. performed the experiments. K.K. provided the theoretical support. S.K., Y.T. and A.N. participated in discussions on the analysis. S.K., K.K. and Y.N. wrote the manuscript with feedback from all authors. Y.N. supervised the project.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to S. Kono or Y. Nakamura.

## Supplementary information

### Supplementary Information

Supplementary notes, figures and references

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