## Abstract

The future development of quantum information using superconducting circuits requires Josephson qubits^{1} with long coherence times combined with a high-fidelity readout. Significant progress in the control of coherence has recently been achieved using circuit quantum electrodynamics architectures^{2,3}, where the qubit is embedded in a coplanar waveguide resonator, which both provides a well-controlled electromagnetic environment and serves as qubit readout. In particular, a new qubit design, the so-called transmon, yields reproducibly long coherence times^{4,5}. However, a high-fidelity single-shot readout of the transmon, desirable for running simple quantum algorithms or measuring quantum correlations in multi-qubit experiments, is still lacking. Here, we demonstrate a new transmon circuit where the waveguide resonator is turned into a sample-and-hold detector—more specifically, a Josephson bifurcation amplifier^{6,7}—which allows both fast measurement and single-shot discrimination of the qubit states. We report Rabi oscillations with a high visibility of 94%, together with dephasing and relaxation times longer than 0.5 μs. By carrying out two measurements in series, we also demonstrate that this new readout does not induce extra qubit relaxation.

## Main

A common strategy to readout a qubit consists of coupling it dispersively to a resonator, so that the qubit states |0〉 and |1〉 shift the resonance frequency differently. This frequency change can be detected by measuring the phase of a microwave pulse reflected on (or transmitted through) the resonator. Such a method, successfully demonstrated with a Cooper pair box capacitively coupled to a coplanar waveguide resonator^{2,3} (CPWR), faces two related difficulties that have so far prevented measurement of the qubit state in a single readout pulse (so-called single-shot regime): the readout has to be completed in a time much shorter than the time *T*_{1} in which the qubit relaxes from |1〉 to |0〉, and with a power low enough to avoid spurious qubit transitions^{8}.

This issue can be solved by using a sample-and-hold detector consisting of a bistable hysteretic system in which the two states of the system are brought in correspondence with the two qubit states. Such a scheme has been implemented in various qubit readouts^{9,10}. In our experiment, the bistable system is a Josephson bifurcation amplifier^{6,7} (JBA) obtained by inserting a Josephson junction in the middle of the CPWR (see Fig. 1). When driven by a microwave signal of properly chosen frequency and power, this nonlinear resonator can bifurcate between two dynamical states and *B* with different intra-cavity field amplitudes and reflected phases. To exploit the hysteretic character of this process, we carry out the readout in two steps (see inset in Fig. 1): the qubit state |0〉 or |1〉 is first mapped onto or *B* in a time much shorter than *T*_{1}; the selected resonator state is then held by reducing the measuring power during a time *t*_{H} long enough to determine this state with certainty.

JBAs were used previously to readout quantronium^{11,12,13} and flux qubits, obtaining for the latter fidelities up to 87% (ref. 14) with quantum non-demolition character^{15}. Here, we couple capacitively a transmon to a JBA, combining all of the advantages of the circuit quantum electrodynamics architecture (long coherence times, scalability) with the single-shot capability of a sample-and-hold detector. A crucial characteristic of this new design is its very low back-action during readout. Indeed, the qubit frequency depends only on the slowly varying photon number inside the resonator^{16}, yielding less relaxation than in previous experiments where the qubit was coupled to a rapidly varying variable of the JBA (the intra-resonator current). Furthermore, we designed the resonator to make it bifurcate at a low photon number, thus avoiding unwanted qubit-state transitions during readout.

The complete set-up is shown in Fig. 1: the transmon^{4,5} of frequency *f*_{01} tunable with a magnetic flux *φ* is coupled with a coupling constant *g*=44±3 MHz to the nonlinear CPWR of fundamental frequency *f*_{C}=6.4535 GHz, quality factor *Q*_{0}=685±15 and Josephson-junction critical current *I*_{C}=0.72±0.04 μA. In this work, the qubit is operated at positive detunings *Δ*=*f*_{C}−*f*_{01} larger than *g*. In this dispersive regime, the resonator frequency *f*_{Ci} depends on the qubit state |*i*〉, and the difference 2*χ*=*f*_{C0}−*f*_{C1} (so-called cavity pull) is a decreasing function of *Δ*. Readout pulses (Fig. 1, inset) of frequency *f* and maximum power *P*_{S} are sent to the circuit; after reflection on the resonator, their two quadratures *I* and *Q* are measured by homodyne detection. They belong to two clearly resolved families of trajectories (Fig. 1a) corresponding to both oscillator states and *B*. The escape from to *B* is a stochastic process activated by thermal and quantum noise in the resonator^{17,18}, and occurs during the sampling time *t*_{S} with a probability *p*_{B} that increases with *P*_{S}. The position of the so-called *S*-curve *p*_{B}(*P*_{S}) depends on the detuning *f*_{Ci}−*f* (ref. 6) and thus on the qubit state. When the two *S*-curves *S*_{f}^{0} and *S*_{f}^{1} corresponding to |0〉 and |1〉 are sufficiently separated, one can choose a value of *P*_{S} at which these states are well mapped onto and *B* (Fig. 1b).

We now present our best visibility, obtained at *Δ*=0.38 GHz in this work and confirmed on another sample. We measure *S*_{f}^{0} and *S*_{f}^{1} (Fig. 2) after preparing the transmon in state |0〉 or |1〉 using a resonant microwave pulse. The contrast, defined as the maximum difference between both curves, reaches 86%. To interpret the power separation between the *S*-curves, we search the readout frequency *f*+*Δ**f*_{1} that makes *S*_{f+Δf1}^{0} coincide with *S*_{f}^{1} at low bifurcation probability. This indirect determination of the cavity pull gives *Δ**f*_{1}=4.1 MHz, in good agreement with the value 2*χ*=4.35 MHz calculated from the experimental parameters. At high *p*_{B}, however, the two *S*-curves do not coincide, which shows that the limiting factor of our readout fidelity is relaxation of the qubit before the time needed for the resonator to reach its final state. To reduce this effect and improve the readout contrast, we transfer state |1〉 into the next excited state |2〉 with a resonant π-pulse just before the readout pulse, yielding the *S*-curve *S*_{f}^{2} and a 92% contrast. This technique, already used with other Josephson qubits^{10}, is analogous to electron shelving in atomic physics and relies here on the very low decay rate from |2〉 to |0〉 in the transmon. Figure 2b shows Rabi oscillations between |0〉 and |1〉 obtained with such a composite readout pulse. The visibility, defined as the fitted amplitude of the oscillations, is 94%, and the Rabi decay time is 0.5 μs. Of the remaining 6% loss of visibility, we estimate that about 4% is due to relaxation before bifurcation and 2% to residual out-of-equilibrium population of |1〉 and to control pulse imperfections. Such a visibility higher than 90% is in agreement with the width of the *S*-curves estimated from numerical simulations, with their theoretical displacement and with the measured qubit-relaxation time.

As the visibility is limited by relaxation, it is important to determine whether the readout process itself increases the qubit relaxation rate. For that purpose, we compare (at *Δ*=0.25 GHz) Rabi oscillations obtained with two different protocols: the control pulse is followed either by two successive readout pulses yielding curves *R*_{1} and *R*_{2}, or by only the second readout pulse yielding curve *R*_{3} (see Fig. 3a). *R*_{2} and *R*_{3} show almost the same loss of visibility compared to *R*_{1}, indicating that relaxation in the presence of the first readout pulse is the same as (and even slightly lower than) in its absence.

To further investigate this remarkable effect, we measure *T*_{1} in the presence of a microwave field at the same frequency *f* as during readout, and for different input powers *P* (see Fig. 3b). We first roughly estimate the intra-cavity mean photon number by measuring the a.c.-Stark-shifted qubit frequency *f*_{01}(*P*) (ref. 16; the correspondence *f*_{01}(*n*) is obtained by a numerical diagonalization of the Hamiltonian of the transmon coupled to a field mode with *n* photons). Bifurcation is clearly revealed by a sudden jump of from about 5–10 to 50–100 photons, whereas *T*_{1} does not show any decrease up to about 5 dB above bifurcation. It even slightly increases because the qubit frequency is pushed away from the cavity, slowing down spontaneous emission as explained in the next paragraph. This is in strong contrast with all previous experiments using a JBA readout^{18,19}. These results prove that our design achieves very low back-action on the qubit. A similar behaviour was observed for most qubit frequencies, except at certain values of *P* and *f*_{01} where dips in *T*_{1}(*P*) were occasionally observed above bifurcation.

We now discuss the dependence of the readout contrast and qubit coherence on the detuning *Δ*. Besides acting as a qubit state detector, the resonator also serves as a filter protecting the qubit against spontaneous emission into the 50 Ω impedance of the external circuit^{20,21}. The smaller *Δ*, the stronger the coupling between the qubit and the resonator, implying a larger separation between the *S*_{f}^{0} and *S*_{f}^{1} curves but also a faster relaxation. We thus expect the contrast to be limited by relaxation at small *Δ*, by the poor separation between the *S*-curves at large *Δ*, and to show a maximum in between. Figure 4 shows a summary of our measurements of contrast and coherence times. At small *Δ*, *T*_{1} is in quantitative agreement with calculations of the spontaneous emission through the resonator. However, it shows a saturation, as observed in previous experiments^{20}, but at a smaller value of around 0.7 μs. The effective cavity pull *Δ**f*_{1} determined from the *S*-curves shifts (see Figure 2) is in quantitative agreement with the value of 2*χ* calculated from the sample parameters. The contrast varies with *Δ* as anticipated and shows a maximum of 92% at *Δ*=0.38 GHz, where *T*_{1}=0.5 μs. Larger *T*_{1} can be obtained at the expense of a lower contrast and reciprocally. Another important figure of merit is the pure dephasing time *T*_{φ} (ref. 22), which also controls the lifetime of a superposition of qubit states. *T*_{φ} is extracted from Ramsey fringes experiments (see the Methods section), and shows a smooth dependence on the qubit frequency, in qualitative agreement with the dephasing time deduced from a 1/*f* flux noise of spectral density set to at 1 Hz, a value similar to those reported elsewhere^{23}. To summarize our circuit performances, we obtained a 400 MHz frequency range (pink area in Fig. 4) where the readout contrast is higher than 85%, *T*_{1} is between 0.7 and 0.3 μs and *T*_{φ} is between 0.7 and 1.5 μs. Further optimization of the JBA parameters *I*_{C} and *Q*_{0} could increase this high-visibility readout frequency window.

We have demonstrated the high-fidelity single-shot readout of a transmon qubit in a circuit quantum electrodynamics architecture using a bifurcation amplifier. This readout does not induce extra qubit relaxation and preserves the good coherence properties of the transmon. The high fidelity achieved should allow a test of Bell’s inequalities using two coupled transmons, each one with its own JBA single-shot readout. Moreover, our method could be used in a scalable quantum processor architecture, in which several transmon–JBAs with staggered frequencies are read by frequency multiplexing.

## Methods

### Sample fabrication.

The sample was fabricated using standard lithography techniques. In a first step, a 120-nm-thick niobium film is sputtered on an oxidized high-resistivity silicon chip. It is patterned by optical lithography and reactive ion etching of the niobium to form the CPWR. The transmon and the Josephson junction of the JBA are then patterned by electron-beam lithography and double-angle evaporation of two aluminium thin films, the first one being oxidized to form the junction tunnel barrier. The chip is glued on and wire-bonded to a microwave printed-circuit board enclosed in a copper box, which is thermally anchored to the mixing chamber of a dilution refrigerator at typically 20 mK.

### Electrical lines and signals.

Qubit control and readout microwave pulses are generated by mixing the output of a microwave source with ‘d.c.’ pulses generated by arbitrary waveform generators, using d.c. coupled mixers. They are then sent to the input microwave line that includes band-pass filters and attenuators at various temperatures. The powers given in decibels in this letter are arbitrarily referred to 1 mW (on 50 Ω) at the input of the dilution refrigerator; the total attenuation down to the sample is about −77 dB. The pulses are routed to the resonator through a circulator to separate the input and output waves.

The readout output line includes a band-pass filter (4–8 GHz), two isolators and a cryogenic amplifier (CITCRYO 1–12 from California Institute of Technology) with 38 dB gain and noise temperature *T*_{N}=3 K. The output signal is further amplified at room temperature with a total gain of 56 dB, and finally mixed down using an *I/Q* mixer with a synchronized local oscillator at the same frequency. The *I* and *Q* quadratures are further amplified by 20 dB, and sampled by a fast digitizer. The data are then transferred to a computer and processed. The single-shot traces of Fig. 1a were obtained with an extra 10 MHz low-pass filter.

### Sample characterization.

The characteristic energies of the system, namely the transmon Josephson energy *E*_{J}=21 GHz and charging energy *E*_{c}=1.2 GHz (for a Cooper pair), as well as the qubit–resonator coupling constant *g*, were determined by spectroscopic measurements. The bare resonator frequency *f*_{C} was determined at a magnetic field such that the qubit was far detuned from the resonator.

### Qubit state preparation.

We prepare the qubit in its ground state with a high fidelity at the beginning of each experimental sequence by letting it relax during about 20 μs. We estimate at about 1% the equilibrium population in state |1〉 due to residual noise coming from measurement lines.

To prepare the qubit in its excited state |1〉 or |2〉, one or two successive resonant square-shaped pulses of length *t*_{π}∼20 ns are applied before the readout pulse. The dotted blue *S*-curve of Fig. 1 was recorded with a single resonant π-pulse at *f*_{12} (see text): it reveals that this pulse induces a spurious population of the |1〉 state of order 1%. We checked that this effect is corrected by using Gaussian-shaped pulses^{9} (data not shown).

### Readout pulses.

We give here more information on the timing of the readout pulses used is this work. In Fig. 2, readout is carried out at *f*_{C}−*f*=17 MHz, and we used *t*_{R}=15 ns, *t*_{S}=250 ns and *t*_{H}=700 ns. We stress that although *t*_{S} is of the same order of magnitude as *T*_{1}, the observed relaxation-induced loss of contrast is rather low, which may seem surprising. This is due to an interesting property of our readout: when the qubit is in state |1〉, the JBA bifurcates with a high probability, implying that all bifurcation events occur at the very beginning of the readout pulse (instead of being distributed exponentially during *t*_{S}). We nevertheless keep *t*_{S}=250 ns because the bifurcation process itself needs such a duration to develop properly. The effective measurement time *t*_{M} is thus shorter than *t*_{S}. We verified that weighted sums of *S*_{f}^{0} and *S*_{f+Δfi}^{0} fit properly the *S*_{f}^{i} curves (*i*=1,2) of Fig. 2, allowing us to quantify the population of each level at readout. Using the experimentally determined relaxation times *T*_{1}^{2→1}∼0.3 μs and *T*_{1}^{1→0}∼0.45 μs, we thus estimate *t*_{M}∼40 ns.

In Fig. 3, readout is carried out at *f*_{C}−*f*=25 MHz, to reduce the total measurement duration. Indeed, as a larger readout detuning implies a higher driving power and thus a higher reflected power, the signal-to-noise ratio is increased, which allows us to shorten *t*_{H} to 50 ns. We also used for these data *t*_{R}=10 ns and *t*_{S}=40 ns to shorten the overall measurement time, which also decreases the maximal contrast to approximately 83%. Finally, a delay time of 120 ns between the two readout pulses has been optimized experimentally to empty the resonator of all photons due to the first measurement, and thus avoid any spurious correlations between the two outcomes of the sequence.

### Coherence time measurement.

The qubit coherence times are measured using standard experimental sequences^{24}. For the relaxation time *T*_{1}, we apply a π-pulse and measure the qubit state after a variable delay, yielding an exponentially decaying curve for which the time constant is *T*_{1}. The coherence time *T*_{2} is obtained by a Ramsey experiment: two π/2-pulses are applied at a frequency slightly off-resonance with the qubit and with a variable delay; this yields an exponentially damped oscillation for which the time constant is *T*_{2}. We then extract the pure dephasing contribution *T*_{φ} to decoherence (as well as the corresponding maximum uncertainty) using the relation *T*_{φ}^{−1}=*T*_{2}^{−1}−(2*T*_{1})^{−1} (ref. 22).

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## Acknowledgements

We acknowledge financial support from European projects EuroSQIP and Midas, from ANR-08-BLAN-0074-01 and from Region Ile-de-France for the nanofabrication facility at SPEC. We gratefully thank P. Senat and P. Orfila for technical support, and acknowledge useful discussions within the Quantronics group and with A. Lupascu, I. Siddiqi, M. Devoret, A. Wallraff and A. Blais.

## Author information

## Affiliations

### Quantronics group, Service de Physique de l’État Condensé (CNRS URA 2464), DSM/IRAMIS/SPEC, CEA-Saclay, 91191 Gif-sur-Yvette cedex, France

- François Mallet
- , Florian R. Ong
- , Agustin Palacios-Laloy
- , François Nguyen
- , Patrice Bertet
- , Denis Vion
- & Daniel Esteve

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### Contributions

F.M., P.B., D.V. and D.E. designed the experiment, F.R.O. fabricated the sample, F.M., F.N., A.P.-L., F.R.O. and P.B. carried out the measurements, and all of the authors contributed to the writing of the manuscript.

## Corresponding author

Correspondence to Denis Vion.

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