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 resonator2,3 (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 T1 in which the qubit relaxes from |1〉 to |0〉, and with a power low enough to avoid spurious qubit transitions8.

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 readouts9,10. In our experiment, the bistable system is a Josephson bifurcation amplifier6,7 (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 T1; the selected resonator state is then held by reducing the measuring power during a time tH long enough to determine this state with certainty.

Figure 1: Principle of a single-shot readout for a transmon qubit.
figure 1

A transmon qubit (magenta) is capacitively coupled to a coplanar resonator (green-bordered grey strips) made anharmonic by inserting a Josephson junction (green cross) at its centre. This qubit is coherently driven by a source VQ and measured by operating the resonator as a cavity JBA: a microwave pulse with properly adjusted frequency f and time-dependent amplitude (rise, sampling and holding times tR, tS and tH, respectively—see inset and the Methods section) is applied by a second source VR; this pulse is reflected by the system and routed to a cryogenic amplifier and to a homodyne detection circuit yielding the two quadratures I and Q. During the ‘sampling’ time tS, the electromagnetic field in the resonator has a probability pB to bifurcate from a low-amplitude state to a high-amplitude one B, both states corresponding to different amplitudes of I and Q. The ‘holding’ time tH is then used to average I(t) and to determine with certainty if the resonator has bifurcated or not. a, Oscillogram showing filtered I(t) traces of both types (obtained here with tR=30 ns and tS=tH=250 ns). b, The probability pB depends on f and on the sampling power PS. The two qubit states |0〉 and |1〉 shift the resonator frequency, resulting in two displaced S-curves S0 and S1. When their separation is large enough, PS can be chosen (vertical dashed line) so that and B map |0〉 and |1〉 with a high fidelity.

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JBAs were used previously to readout quantronium11,12,13 and flux qubits, obtaining for the latter fidelities up to 87% (ref. 14) with quantum non-demolition character15. 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 resonator16, 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 transmon4,5 of frequency f01 tunable with a magnetic flux φ is coupled with a coupling constant g=44±3 MHz to the nonlinear CPWR of fundamental frequency fC=6.4535 GHz, quality factor Q0=685±15 and Josephson-junction critical current IC=0.72±0.04 μA. In this work, the qubit is operated at positive detunings Δ=fC−f01 larger than g. In this dispersive regime, the resonator frequency fCi depends on the qubit state |i〉, and the difference 2χ=fC0−fC1 (so-called cavity pull) is a decreasing function of Δ. Readout pulses (Fig. 1, inset) of frequency f and maximum power PS 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 resonator17,18, and occurs during the sampling time tS with a probability pB that increases with PS. The position of the so-called S-curve pB(PS) depends on the detuning fCi−f (ref. 6) and thus on the qubit state. When the two S-curves Sf0 and Sf1 corresponding to |0〉 and |1〉 are sufficiently separated, one can choose a value of PS 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 Sf0 and Sf1 (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 f1 that makes Sf+Δ f10 coincide with Sf1 at low bifurcation probability. This indirect determination of the cavity pull gives Δ f1=4.1 MHz, in good agreement with the value 2χ=4.35 MHz calculated from the experimental parameters. At high pB, 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 Sf2 and a 92% contrast. This technique, already used with other Josephson qubits10, 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.

Figure 2: Best single-shot visibility obtained at Δ=0.38 GHz and fC−f=17 MHz.
figure 2

a, S-curves pB(PS) obtained with the qubit prepared in state |0〉, |1〉 or |2〉 (solid lines Sf0, Sf1 and Sf2, respectively) with the proper resonant π-pulses (top diagram). The maximum differences between Sf0 and Sf1 (red vertical dashed line) and between the Sf0 and Sf2 (green vertical dashed line) define two readout contrasts of 86 and 92%. The readout fidelity is thus increased by using a composite readout where the measurement pulse is preceded by a π-pulse at frequency f12 that transfers |1〉 to |2〉. The dotted blue curve obtained after a single π-pulse at frequency f12, starting from |0〉, shows that this technique has almost no effect on |0〉. Also plotted are the curves obtained for |0〉 when shifting the readout frequency f by Δ f1=4.1±0.1 MHz (red dashed line) and Δ f2=5.1±0.1 MHz (green dashed line) to match at low pB the curves obtained for |1〉 and |2〉. The difference between the corresponding solid and dashed curves is a loss of visibility mostly due to qubit relaxation before bifurcation. b, Rabi oscillations at 29 MHz measured with the composite readout, as sketched on top. The circles are experimental values of pB(Δt), whereas the solid line is a fit by an exponentially damped sine curve with a 0.5 μs decay time and an amplitude of 94% (best visibility). The total errors in the preparation and readout of the states are 2% and 6.5% for |0〉 and |1〉, respectively.

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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 R1 and R2, or by only the second readout pulse yielding curve R3 (see Fig. 3a). R2 and R3 show almost the same loss of visibility compared to R1, indicating that relaxation in the presence of the first readout pulse is the same as (and even slightly lower than) in its absence.

Figure 3: Effect of the readout process on the qubit at Δ=0.25 GHz and fC−f=25 MHz.
figure 3

a, Rabi oscillations pB(Δ t) obtained at PS=−30.5 dB with the protocols sketched on top, that is, with two successive readout pulses placed immediately after the control Rabi pulse (red and blue circles), or with the second pulse only (green circles). The loss of Rabi visibility between the red curve (83%) and the blue (44%) and green (37%) curves is due to qubit relaxation during the first readout or the delay. b, Top panel: spectroscopic determination of the qubit frequency f01 when it is a.c.-Stark-shifted by an auxiliary microwave with frequency f and power P (protocol on top). The shift provides an in situ estimate of the average photon number in the resonator (right scale) with a precision of ±30%. The bifurcation is seen as a sudden jump. Bottom panel: qubit relaxation time T1 (measurement protocol not shown) in the presence of the same auxiliary field. T1 does not show any strong decrease even at power well above bifurcation.

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To further investigate this remarkable effect, we measure T1 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 f01(P) (ref. 16; the correspondence f01(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 T1 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 readout18,19. 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 f01 where dips in T1(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 circuit20,21. The smaller Δ, the stronger the coupling between the qubit and the resonator, implying a larger separation between the Sf0 and Sf1 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, T1 is in quantitative agreement with calculations of the spontaneous emission through the resonator. However, it shows a saturation, as observed in previous experiments20, but at a smaller value of around 0.7 μs. The effective cavity pull Δ f1 determined from the S-curves 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 T1=0.5 μs. Larger T1 can be obtained at the expense of a lower contrast and reciprocally. Another important figure of merit is the pure dephasing time Tφ (ref. 22), which also controls the lifetime of a superposition of qubit states. Tφ 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 elsewhere23. 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%, T1 is between 0.7 and 0.3 μs and Tφ is between 0.7 and 1.5 μs. Further optimization of the JBA parameters IC and Q0 could increase this high-visibility readout frequency window.

Figure 4: Trade-off between qubit coherence and readout fidelity.
figure 4

a, Experimental relaxation time T1 (red circles) and dephasing time Tφ (blue circles) of the qubit as a function of f01 (or equivalently Δ/g). Note that Tφ≈2.5±0.5 μs at the flux optimal point24 (Δ≈−0.75 GHz, data not shown). The error bars on Tφ are absolute minima and maxima resulting from the maximum experimental uncertainties on the coherence times T1 and T2 (see the Methods section). The solid red line is the value of T1 obtained by adding to the expected spontaneous emission through the resonator (dashed red line) a relaxation channel of unknown origin with T1=0.7 μs (horizontal dotted line). The blue line is the pure dephasing time Tφ corresponding to a 1/f flux noise with an amplitude set to at 1 Hz. b, Left scale: readout contrast with (green filled circles) and without (green open circles) transfer from state |1〉 to |2〉 (see Fig. 2). Right scale: effective cavity pull Δ f1 (blue squares) determined as shown in Fig. 2. For the sake of comparison, the predicted cavity pull 2χ in the dispersive approximation is also shown as a cyan region, taking into account the maximal experimental uncertainty on g. The pink area denotes the region where the readout contrast is higher than 85%.

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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 TN=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 EJ=21 GHz and charging energy Ec=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 fC 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 tπ∼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 f12 (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 pulses9 (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 fC−f=17 MHz, and we used tR=15 ns, tS=250 ns and tH=700 ns. We stress that although tS is of the same order of magnitude as T1, 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 tS). We nevertheless keep tS=250 ns because the bifurcation process itself needs such a duration to develop properly. The effective measurement time tM is thus shorter than tS. We verified that weighted sums of Sf0 and Sf+Δ fi0 fit properly the Sfi curves (i=1,2) of Fig. 2, allowing us to quantify the population of each level at readout. Using the experimentally determined relaxation times T12→1∼0.3 μs and T11→0∼0.45 μs, we thus estimate tM∼40 ns.

In Fig. 3, readout is carried out at fC−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 tH to 50 ns. We also used for these data tR=10 ns and tS=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 sequences24. For the relaxation time T1, 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 T1. The coherence time T2 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 T2. We then extract the pure dephasing contribution Tφ to decoherence (as well as the corresponding maximum uncertainty) using the relation Tφ−1=T2−1−(2T1)−1 (ref. 22).