Abstract
The future development of quantum information using superconducting circuits requires Josephson qubits^{1} with long coherence times combined with a highfidelity 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 wellcontrolled electromagnetic environment and serves as qubit readout. In particular, a new qubit design, the socalled transmon, yields reproducibly long coherence times^{4,5}. However, a highfidelity singleshot readout of the transmon, desirable for running simple quantum algorithms or measuring quantum correlations in multiqubit experiments, is still lacking. Here, we demonstrate a new transmon circuit where the waveguide resonator is turned into a sampleandhold detector—more specifically, a Josephson bifurcation amplifier^{6,7}—which allows both fast measurement and singleshot 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 (socalled singleshot 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 sampleandhold 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 intracavity 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 nondemolition 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 singleshot capability of a sampleandhold detector. A crucial characteristic of this new design is its very low backaction 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 intraresonator current). Furthermore, we designed the resonator to make it bifurcate at a low photon number, thus avoiding unwanted qubitstate transitions during readout.
The complete setup 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 Josephsonjunction 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 2χ=f_{C0}−f_{C1} (socalled 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 socalled Scurve p_{B}(P_{S}) depends on the detuning f_{Ci}−f (ref. 6) and thus on the qubit state. When the two Scurves 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 Scurves, we search the readout frequency f+Δf_{1} that makes S_{f+Δf1}^{0} coincide with S_{f}^{1} at low bifurcation probability. This indirect determination of the cavity pull gives Δf_{1}=4.1 MHz, in good agreement with the value 2χ=4.35 MHz calculated from the experimental parameters. At high p_{B}, however, the two Scurves 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 πpulse just before the readout pulse, yielding the Scurve 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 outofequilibrium population of 1〉 and to control pulse imperfections. Such a visibility higher than 90% is in agreement with the width of the Scurves estimated from numerical simulations, with their theoretical displacement and with the measured qubitrelaxation 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 intracavity mean photon number by measuring the a.c.Starkshifted 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 backaction 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 Scurves 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 Δf_{1} determined from the Scurves shifts (see Figure 2) is in quantitative agreement with the value of 2χ 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 highvisibility readout frequency window.
We have demonstrated the highfidelity singleshot 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 singleshot 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 120nmthick niobium film is sputtered on an oxidized highresistivity 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 electronbeam lithography and doubleangle evaporation of two aluminium thin films, the first one being oxidized to form the junction tunnel barrier. The chip is glued on and wirebonded to a microwave printedcircuit 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 bandpass 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 bandpass 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 singleshot traces of Fig. 1a were obtained with an extra 10 MHz lowpass 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 squareshaped pulses of length t_{π}∼20 ns are applied before the readout pulse. The dotted blue Scurve of Fig. 1 was recorded with a single resonant πpulse 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 Gaussianshaped 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 relaxationinduced 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 signaltonoise 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 πpulse 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/2pulses are applied at a frequency slightly offresonance 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}−(2T_{1})^{−1} (ref. 22).
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Acknowledgements
We acknowledge financial support from European projects EuroSQIP and Midas, from ANR08BLAN007401 and from Region IledeFrance 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, CEASaclay, 91191 GifsurYvette cedex, France
 François Mallet
 , Florian R. Ong
 , Agustin PalaciosLaloy
 , 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.
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Correspondence to Denis Vion.
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