Abstract
Superconducting circuits offer a scalable platform for the construction of largescale quantum networks, where information can be encoded in multiple temporal modes of propagating microwaves. Characterization of such microwave signals with a method extendable to an arbitrary number of temporal modes with a single detector and demonstration of their phaserobust nature are of great interest. Here, we show the ondemand generation and Wigner tomography of a microwave timebin qubit with superconducting circuit quantum electrodynamics architecture. We perform the tomography with a single heterodyne detector by dynamically switching the measurement quadrature independently for two temporal modes through the pump phase of a phasesensitive amplifier. We demonstrate that the timebin encoding scheme relies on the relative phase between the two modes and does not need a shared phase reference between sender and receiver.
Introduction
In the past few decades, quantum bits implemented as superconducting circuits have become promising candidates for building blocks of largescale quantum computers^{1,2,3,4}. To increase the scalability of these architectures, robust methods of generating single photons for quantum computation in the propagating modes and for transferring information between multiple superconducting qubits over relatively long distances are of recent interest. In the optical domain, different photonic qubit encodings have been demonstrated before for such purposes^{5}. However, optical singlephoton generation protocols are often probabilistic rather than deterministic, limiting success probability^{6}. Moreover, conversion of quantum information stored in superconducting qubits operated in the microwave regime to optical photons suffers from low efficiency and limited bandwidth^{7,8}. Schemes focused on generating photons at microwave frequencies and their characterization are therefore of great interest.
Photonic qubit encoding can be realized by constructing a set of computational basis states with one or more orthogonal modes of light. In the microwave regime, singlerail (singlemode) encoding has been demonstrated by using the photon number states of a propagating microwave qubit to transfer information between two superconducting qubits over a transmission line with fidelity close to 0.8^{9,10,11,12}. However, photon loss reduces the transfer fidelity greatly, as decayed photon states cannot be distinguished readily. In addition, the phase information in a singlerail photonic qubit state is stored as the relative phase between the propagating qubit mode and a separate phase reference. Thus, the reference must be shared between any hardware operating the nodes of a quantum network that the photonic qubit will interact with, reducing the practicality of singlerail encoding in large networks.
As an alternative to the singlerail encoding, dualrail (dualmode) encoding has been demonstrated in the optical regime in the form of polarization^{13,14,15} and timebin qubits^{16,17}. Occupation of a single photon in one of two orthogonal temporal modes functions as the basis of the timebin qubit. Timebin encoding allows one to readily determine loss of information during transfer with a photon number parity measurement^{5,18}, and the qubit state is more robust against dephasing, as the phase information is stored in the relative phase between the two temporal modes. Thus, timebin qubits do not require sharing of a phase reference^{19}. Owing to these favorable properties, a linear optical scheme for quantum computation with timebin qubits has been proposed^{5}. However, only the lossrobustness of the microwave timebin qubit has been demonstrated. The demonstration was based on discretevariable measurements of superconducting qubits as a part of a transfer protocol, thus being limited to a single qubit of information^{20}. A different approach is necessary for full state tomography of a general two temporal mode state or cluster states with multiple modes and qubits of information. Ideally, for a scalable characterization method, only a single detector should be necessary regardless of the number of modes.
In this work, we experimentally demonstrate ondemand generation of microwave timebin qubits with a superconducting transmon qubit^{21} and show how the timebin qubit retains phase information and can be losscorrected. Our scheme allows us to generate and shape the singlephoton wave packet as well as to generate any superposition state of the timebin qubit with variable spacing between the temporal modes. We perform Wigner tomography of microwave signal in two temporal modes by measuring the quadrature distributions with a fluxdriven Josephson parametric amplifier^{22} and a single heterodyne detector^{23}. With the Josephson parametric amplifier (JPA), we can rapidly change the measurement quadrature for each temporal mode independently in a single shot. We reconstruct the quantum state of the signal with a maximumlikelihood method^{23,24}. We compare the state preparation fidelity of the dualrail timebin qubit with a singlerail numberbasis qubit and a transmon qubit. We demonstrate that correcting photon loss of the timebin qubit state improves the fidelity significantly. By removing the phaselocking between the singlephoton source and the detector, we observe that the singlerail photonic qubit state dephases completely owing to the lack of a stable phase reference, whereas the timebin qubit state is unaffected. This demonstrates that the phase information of the dualrail qubit is contained in the relative phase between the two modes and that using the timebin qubit in a quantum network does not require a shared phase reference.
Results
System
To generate a single photon, we consider a coherently driven circuit quantum electrodynamical setup where a superconducting transmon qubit is dispersively coupled to a 3D microwave cavity with a resonance frequency ω_{c}∕2π = 10.619 GHz. The dynamics of the system are described in the rotating frame of the drive by the Hamiltonian
The qubit is coupled to the cavity with coupling strength g∕2π = 156.1 MHz and it is driven by coherent microwaves at frequency ω_{d} with timedependent complex amplitude Ω(t) through the cavity. In Eq. (1), a and b are defined as the cavity and transmon annihilation operators, and ω_{ge}∕2π = 7.813 GHz is the qubit \(\left{\rm{g}}\right\rangle\)−\(\left{\rm{e}}\right\rangle\) transition frequency separated from the \(\left{\rm{e}}\right\rangle\)−\(\left{\rm{f}}\right\rangle\) transition frequency ω_{ef} = ω_{ge} + α by the transmon anharmonicity α∕2π = −340 MHz. The cavity and qubit are dispersively coupled, i.e., \(\left{\omega }_{{\rm{ge}}}{\omega }_{{\rm{c}}}\right\gg g\), which allows us to readout the qubit state based on the qubit statedependent dispersive shift of the cavity resonance frequency. The cavity is coupled to an external transmission line with an external coupling rate κ_{ex}∕2π = 2.91 MHz. The relaxation and coherence times between the \(\left{\rm{g}}\right\rangle\)–\(\left{\rm{e}}\right\rangle\) and \(\left{\rm{e}}\right\rangle\)–\(\left{\rm{f}}\right\rangle\) states are \({T}_{1}^{{\rm{ge}}}=\) 26 μs, \({T}_{1}^{{\rm{ef}}}=\) 15 μs, and \({T}_{2}^{{\rm{ge}}}=\) 15 μs, \({T}_{2}^{{\rm{ef}}}=\) 16 μs, respectively.
Dynamics of timebin qubit generation
The state of two timebin modes can be represented in the photon number basis in two orthogonal temporal modes
where \(\leftnm\right\rangle := {\leftn\right\rangle }_{{\rm{E}}}\otimes {\leftm\right\rangle }_{{\rm{L}}}\) represents the photon number states of the earlier (E) and later (L) modes, respectively, with \(\mathop{\sum }\nolimits_{n,m = 0}^{\infty } {C}_{nm}{ }^{2}=1\).
The protocol for quantum state transfer from a superconducting qubit to a timebin qubit is shown in Fig. 1a. We prepare the superconducting qubit in a superposition state \({\alpha }_{{\rm{q}}}\left{\rm{g}}\right\rangle +{\beta }_{{\rm{q}}}\left{\rm{e}}\right\rangle\) and transfer the state to \({\alpha }_{{\rm{q}}}\left{\rm{e}}\right\rangle +{\beta }_{{\rm{q}}}\left{\rm{f}}\right\rangle\) with a sequence of π_{ef} and π_{ge} pulses at frequencies ω_{ef} and ω_{ge}, respectively.
We induce the transition between the \(\left{\rm{f}}0\right\rangle\) and \(\left{\rm{g}}1\right\rangle\) states of the combined qubit–cavity system with a drive pulse to generate a shaped single photon inside a transmission line^{25}. The \(\left{\rm{f}}0\right\rangle\)–\(\left{\rm{g}}1\right\rangle\) transition frequency is defined as ω_{f0g1} = 2ω_{ge} + α − ω_{c}. When the drive frequency matches this transition, the microwaveinduced effective coupling between \(\left{\rm{f}}0\right\rangle\) and \(\left{\rm{g}}1\right\rangle\) can be derived from the system Hamiltonian in Eq. (1)
Here, the complex amplitude \(\Omega (t)=\exp [i\phi (t)] \Omega (t)\) has a phase degree of freedom ϕ(t). By applying this coupling pulse to the sample we can generate a photon inside the cavity. The photon in the cavity will decay to the waveguide at the external coupling rate κ_{ex}. Thus, the coefficient β_{q} is transferred to the photon in the E mode of the timebin qubit. The second coefficient, α_{q}, is transferred to the propagating microwave mode by driving the qubit with a π_{ef} pulse and the coupling pulse once afterwards.
If the generation protocol has ideal efficiency, the coefficients α_{q} and β_{q} are transferred to the modes \(\left01\right\rangle\) and \(\left10\right\rangle\) as C_{01} = α_{q} and C_{10} = β_{q}. As the original qubit state is normalized, \({\left{C}_{{\rm{01}}}\right}^{2}+{\left{C}_{{\rm{10}}}\right}^{2}=1\), and all of the other coefficients in Eq. (2) become zero. Thus, the transfer process of the qubit state to propagating microwave mode in the temporal mode basis represents the mapping \({\alpha }_{{\rm{q}}}\left{\rm{g}}\right\rangle +{\beta }_{{\rm{q}}}\left{\rm{e}}\right\rangle \mapsto {\alpha }_{{\rm{q}}}\left01\right\rangle +{\beta }_{{\rm{q}}}\left10\right\rangle\). We can therefore define the temporal modes \(\left01\right\rangle \equiv \left{\rm{L}}\right\rangle\) and \(\left10\right\rangle \equiv \left{\rm{E}}\right\rangle\) as the basis states of a dualrail timebin qubit. One should note that the timebin qubit basis states have a single photon, meaning that a valid qubit state can be confirmed with a parity measurement of the total photon number in the two temporal modes.
Characterization of the experimental setup
A schematic of the experimental configuration for generating and measuring the propagating timebin qubit state is shown in Fig. 1b. We input the qubit control pulses, qubit state readout pulse, and coupling pulse, to the cavity cooled down to 30 mK inside a dilution refrigerator. We amplify the generated timebin qubit signal with a fluxdriven JPA operated in the degenerate mode by driving the JPA with two successive microwave pulses at frequency \({\omega }_{{\rm{p}}}=2{\omega }_{{\rm{c}}}^{{\rm{g}}}\) where \({\omega }_{{\rm{c}}}^{{\rm{g}}}/2\pi =\) 10.628 GHz is the dressed cavity frequency when the qubit is in the ground state. The measured signal is demodulated with a local oscillator at frequency \({\omega }_{{\rm{c}}}^{{\rm{LO}}}\) shifted from \({\omega }_{{\rm{c}}}^{{\rm{g}}}\) by the sideband frequency −2π × 50 MHz.
We estimate the measurement efficiency for our generation and characterization system by measuring the marginal distribution along a given quadrature in phase space and reconstructing the Wigner function of a singlerail singlephoton state \(\left1\right\rangle\) in Fig. 1c, d. We only consider measurements where the qubit is in the ground state both before and after the measurement. In the marginal distribution of the measured signal, we extract from a theoretical fit^{26} a singlephoton probability of \({P}_{\left1\right\rangle }=\mathrm{0.591 \pm 0.038}\) with 95% confidence intervals. We obtain a fidelity of 0.556 ± 0.009 for the reconstructed Wigner function and observe a negative region in the quasiprobability distribution near the origin of the phase space (Fig. 1d), demonstrating negativity of the measured state without loss correction for detection inefficiency. We define the error interval of the fidelity as three times the standard deviation obtained from bootstrapping^{27} of the tomography data. We obtain from an analytical calculation (see Section 2 of Supplementary Methods) the possible maximum generation efficiency of η_{gen} = 0.83 ± 0.02 with the parameters in our system, resulting in the minimum measurement efficiency of η_{meas} = 0.67 ± 0.01, comparable to recent experiments in similar systems^{28,29,30} and mostly explained by the insertion loss of the circulators and isolators.
Quadrature distribution of microwave timebin qubit signal
The pulse sequence used in the experiment for timebin qubit generation is shown as a quantum circuit in Fig. 2a and as temporal waveforms with different angular frequencies in Fig. 2b. We perform a zbasis dispersive readout on the qubit state^{28,31} with an assignment fidelity of 0.99 to initialize the qubit, and at the end of the generation sequence to measure whether the transfer sequence results in the qubit being in the ground state or not.
In Fig. 2c, we show the measured and simulated mean field amplitude squared ∣〈a_{out}(t)〉∣^{2} of the state \((1/\sqrt{2})\left00\right\rangle +(1/2)\left10\right\rangle +(1/2)\left01\right\rangle\) as a function of time. The magnitude is calculated according to the theory in Section 2 of the Supplementary Methods. The measured amplitude is normalized to match the simulated amplitude by defining that the integrals calculated over the time interval for the squared amplitudes must be equal. We only consider here measurement events where the transmon qubit was measured as being in the ground state both before and after the generation sequence. We utilize the shape of the measured temporal mode amplitudes to calculate the quadrature distributions of the timebin qubit. The correlation between the measurements changes based on the selected quadratures, as shown in Fig. 2d.
Characterization of microwave photonic qubit states
We experimentally prepare the transmon, singlerail number basis and timebin qubits in the six cardinal states of the Bloch sphere, as shown in Fig. 3a. We define the numberbasis qubit state basis as \(\left0\right\rangle \equiv {\left0\right\rangle }_{{\rm{S}}}\) and \(\left1\right\rangle \equiv {\left1\right\rangle }_{{\rm{S}}}\), corresponding to no excitation or a single excitation in a single mode. The numberbasis qubit states are generated with a sequence similar to the timebin generation sequence in Fig. 2, but with only the first two qubit control pulses and the first coupling and JPA pump pulses. A series of qubit state readouts along the three Bloch sphere axes are performed to reconstruct the transmon qubit state. All of the measurements are performed in singleshot.
We calculate the fidelity of each prepared state as
where the pure target state is defined as \(\left{\psi }_{{\rm{t}}}\right\rangle\) and ρ is the measured qubit state.
Transmon qubit tomography
For the transmon qubit states, we only consider measurement events where the qubit is initially measured to be in the ground state. On average, 87.5% of our measurement events fulfill this condition. The relatively high initial excited state population may be explained by noise from the qubit control line^{32}. The population can be reduced with cooling techniques^{33,34}. Given the above condition, we measure a state preparation fidelity of \({{\mathcal{F}}}_{{\rm{T}}}^{{\rm{avg}}}=\mathrm{0.987 \pm 0.001}\) averaged over the six cardinal states (Fig. 3b), limited mainly by the qubit control pulse fidelity and readout assignment fidelity.
Singlerail numberbasis qubit tomography
For the singlerail states, we postselect the measurement events where both of the readouts before and after the generation sequence result in the qubit state being assigned to the ground state. On average, we keep 82.6% of all data in the tomography process.
We prepare the singlerail numberbasis qubit states with a fidelity of \({{\mathcal{F}}}_{{\rm{SR}}}^{{\rm{avg}}}=\mathrm{0.781 \pm 0.003}\), noticeably lower than the transmon qubit states. The difference in fidelity is caused by relaxation and dephasing of the transmon qubit state during singlephoton generation and photon loss during photon transfer from the qubit to the JPA and heterodyne detector. The effect of photon loss can be observed in the Bloch sphere as a bias towards the \(\left0\right\rangle\) state for all of the six cardinal states.
Timebin qubit tomography
We postselect the timebin measurement events where both of the readouts result in the transmon qubit being in the ground state corresponding to 80.4% of all measurements. We discuss the other measurement events in more detail in Section 7 of the Supplementary Methods.
Without loss correction, we measure an average state preparation fidelity of \({{\mathcal{F}}}_{{\rm{TB}}}^{{\rm{avg}}}=\mathrm{0.434 \pm 0.001}\). As the generation sequence is longer than that of the singlerail qubit, the effect of qubit control pulse infidelity and qubit dephasing and relaxation on the state preparation fidelity also becomes stronger. Furthermore, we emulate an effective photon number parity measurement on the timebin qubit density matrices by projecting the full twomode density matrix to the timebin qubit subspace spanned by \(\left{\rm{E}}\right\rangle\) and \(\left{\rm{L}}\right\rangle\), as detailed in Section 6 of the Supplementary Methods. After the effective parity measurement, we obtain a losscorrected timebin qubit average state fidelity of \({{\mathcal{F}}}_{{\rm{TB,LC}}}^{{\rm{avg}}}=\mathrm{0.910 \pm 0.002}\).
Phase robustness of the timebin qubit
We measure and reconstruct the density matrices of the singlerail qubit and timebin qubits for the coherent superposition states \((1/\sqrt{2})({\left0\right\rangle }_{{\rm{S}}}+{\left1\right\rangle }_{{\rm{S}}})\) and \((1/\sqrt{2})(\left{\rm{L}}\right\rangle +\left{\rm{E}}\right\rangle )\) when the photon source does and does not share the same relative phase reference with the detector, as shown in Fig. 4a, b, respectively. To experimentally realize this condition, we use a separate reference clock for the microwave source, which generates the coupling pulse carrier signal than for the other two microwave sources used for qubit control, JPA operation, and demodulation of singlephoton signal. We also perform photon loss correction on the timebin qubit density matrices.
When all of the microwave sources share the same external rubidium clock (Fig. 4a), phase coherence is maintained between the generated photons, and the tomography results in a singlerail qubit state fidelity of \({{\mathcal{F}}}_{{\rm{SR}},X}^{{\rm{shared}}}=\mathrm{0.811 \pm 0.007}\) and timebin qubit state fidelity of \({{\mathcal{F}}}_{{\rm{TB}},X}^{{\rm{shared}}}=\mathrm{0.901 \pm 0.006}\). The timebin qubit is slightly more coherent than the singlerail qubit, because of a slow phase reference drift, which occurs even with a shared external clock and perhaps also inpart owing to the photon loss correction. In Fig. 4b, we disconnect the microwave source for the coupling pulse from the shared clock. Owing to the phase drift between the two clocks, the singlephoton signal generated by the coupling pulse has a different phase reference each time. Thus, as we observe in the measured offdiagonal matrix elements, the measured singlerail qubit is dephased completely, resulting in a singlerail preparation fidelity of \({{\mathcal{F}}}_{{\rm{SR}},X}^{{\rm{sep}}}=\mathrm{0.500 \pm 0.008}\). In contrast, for the timebin qubit, the phase information is not lost since the relative phase between the two temporal modes determines the phase information of the qubit, resulting in a timebin qubit state fidelity of \({{\mathcal{F}}}_{{\rm{TB}},X}^{{\rm{sep}}}=\mathrm{0.899 \pm 0.006}\).
Discussion
We successfully performed ondemand generation of microwave timebin qubits by driving a 3D circuitQED system in dispersive regime and characterized the resulting quantum states with maximumlikelihood estimation of twomode signal amplified by a JPA in heterodyne measurement. Our tomography method allowed us to perform Wigner tomography of a general two temporal mode microwave state with a single detector by switching the measurement quadrature in time between the temporal modes. We measured an average timebin qubit state preparation fidelity of 0.910 after loss correction. We also demonstrated that the phase information of the timebin qubit is stored in the relative phase of the temporal modes and that the lack of a shared phase reference does not cause the timebin qubit to dephase. By performing a quantum nondemolition measurement of the photon number parity in the timebin qubit with the method in refs ^{23,29,35}, it is possible to perform lossdetection on the timebin qubits to increase the fidelity of information transfer and distributed computation in a superconducting qubit network by an amount corresponding to the photon loss. Our tomography method can also be extended to microwave cluster states with an arbitrary number of temporal modes without any additional detector hardware by adding a new JPA pump pulse for each additional temporal mode. The quadrature detection can be used to realize remote state preparation schemes^{36,37}. It is also possible to combine our method with an entanglement witness^{38} or select the measurement quadratures adaptively to characterize the entanglement and state with minimal number of measurements.
Methods
Pulse calibration
We define the qubit control pulses as Gaussianshaped pulses while the shape of the coupling pulse is defined as a cosine pulse \([1\cos (2\pi t/w)]/2\) with width w for t ∈ [0, w]. We optimize the width, separation, amplitude, phase, and frequency of all pulses in parameter sweep experiments by maximizing the assignment and state preparation fidelities for each parameter separately. The experimental setup used for pulse generation is detailed in Section 1 of the Supplementary Methods. In addition, we apply DRAG^{39} to the qubit control pulses and chirp the coupling pulses to limit the effect of the \(\left{\rm{f}}\right\rangle\)state Stark shift affecting the phase of the generated photon wave packet^{9}. We also add a constant phase shift to the second coupling pulse relative to the first to reduce the effect of \(\left{\rm{e}}\right\rangle\)state Stark shift. The optimization of the coefficients for DRAG and chirping is detailed in Sections 3, 4, and 6 of the Supplementary Methods.
Wigner tomography of two temporal modes
We use JPA phasesensitive amplification together with heterodyne measurement to reconstruct the full quantum state of the singlerail number basis and timebin qubits with iterative maximumlikelihood estimation performed on measured quadrature distributions of the temporal modes based on refs ^{24,40}.
To change the quadrature of amplification independently for two temporal modes, we select pairs of JPA pump pulse phases (φ_{E}, φ_{L}) ∈ [0, 2π] × [0, 2π] corresponding to amplification of each temporal mode. For a single phase pair, we obtain measured quadrature values (q_{E}, q_{L}). A measured twomode state ρ_{EL} matches a given set of the four values above with a probability amplitude \({\rm{Tr}}[{\rho }_{{\rm{EL}}}\Pi ({\varphi }_{{\rm{E}}},{q}_{{\rm{E}}},{\varphi }_{{\rm{L}}},{q}_{{\rm{L}}})]\). Here, we have defined the projection operator
In order to find the optimal density matrix ρ_{opt} that matches the measured probabilities of all outcomes \(\left({\varphi }_{{\rm{E}},{\rm{j}}},{q}_{{\rm{E}},{\rm{j}}},{\varphi }_{{\rm{L}},{\rm{j}}},{q}_{{\rm{L}},{\rm{j}}}\right)\) in our experiment, we iteratively search for the density matrix that maximizes the logarithm of the likelihood function
where we have assumed that the set \(\left({\varphi }_{{\rm{E}},{\rm{j}}},{q}_{{\rm{E}},{\rm{j}}},{\varphi }_{{\rm{L}},{\rm{j}}},{q}_{{\rm{L}},{\rm{j}}}\right)\) forms a dense parameter space. We perform the maximization by defining an iterative operator
which has a property R(ρ_{opt})ρ_{opt}R(ρ_{opt}) ∝ ρ_{opt}. This property allows us to iteratively calculate ρ_{opt} by \({\rho }_{k+1}=R({\rho }_{k}){\rho }_{k}R({\rho }_{k})/{\rm{Tr}}[R({\rho }_{k}){\rho }_{k}R({\rho }_{k})]\). We start the iteration from an initial state \({\rho }_{0}=I/{\rm{Tr}}(I)\) and calculate the logarithm of the likelihood for each density matrix in the iteration until convergence.
We perform the above iteration numerically and solve the reconstructed density matrix in the Fock basis of two temporal modes by calculating the twomode matrix elements \(\left\langle {n}_{{\rm{E}}},{n}_{{\rm{L}}}\rightR(\rho )\left{m}_{{\rm{E}}},{m}_{{\rm{L}}}\right\rangle\) of the iterative operator for photon numbers n_{E}, n_{L}, m_{E}, and m_{L} in the two modes.
We measured a JPA gain of 26.8 dB for the singlephoton signal (See Section 5 of the Supplementary Methods). To reduce the amount of measurements, we perform the tomography for 12 × 12 different quadratures in the two modes. For each quadrature pair, we measure 10^{4} samples. We truncate the twomode density matrix in the tomography by limiting the maximum amount of photons in a single temporal mode to two.
Bootstrapping of the reconstructed density matrices
We estimate the error of the reconstructed density matrices and state preparation fidelity of the qubit states by bootstrapping the tomography measurement events^{27}. We resample the data measured for each qubit state and perform maximumlikelihood estimation on the resampled data set to obtain a bootstrapped density matrix. By performing this procedure for a number of bootstrapping samples, we can calculate the distribution and standard deviation of the reconstructed density matrix elements. We use 250 bootstrapping samples for each tomography measurement.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge fruitful discussions with Y. Tabuchi, M. Fuwa, D. LachanceQuirion, and N. Gheeraert. This work was supported in part by UTokyo ALPS, JSPS KAKENHI (no. 19K03684 and no. 26220601), JST ERATO (no. JPMJER1601), and MEXT Quantum Leap Flagship Program (MEXT QLEAP no. JPMXS0118068682). J.I. was supported by the Oskar Huttunen Foundation.
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J.I., S.K., S.Y., and M.K. designed and performed the experiments. J.I., S.K., and Y.S. analyzed the data. S.K. and S.Y. fabricated the device. K.K. performed the analytical calculations. J.I. wrote the manuscript with feedback from the other authors. Y.N. supervised the project.
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Ilves, J., Kono, S., Sunada, Y. et al. Ondemand generation and characterization of a microwave timebin qubit. npj Quantum Inf 6, 34 (2020). https://doi.org/10.1038/s4153402002664
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DOI: https://doi.org/10.1038/s4153402002664
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