Abstract
The tunnelling of quasiparticles across Josephson junctions in superconducting quantum circuits is an intrinsic decoherence mechanism for qubit degrees of freedom. Understanding the limits imposed by quasiparticle tunnelling on qubit relaxation and dephasing is of theoretical and experimental interest, particularly as improved understanding of extrinsic mechanisms has allowed crossing the 100 microsecond mark in transmontype charge qubits. Here, by integrating recent developments in highfidelity qubit readout and feedback control in circuit quantum electrodynamics, we transform a stateoftheart transmon into its own realtime chargeparity detector. We directly measure the tunnelling of quasiparticles across the single junction and isolate the contribution of this tunnelling to qubit relaxation and dephasing, without reliance on theory. The millisecond timescales measured demonstrate that quasiparticle tunnelling does not presently bottleneck transmon qubit coherence, leaving room for yet another order of magnitude increase.
Introduction
Quasiparticle (QP) excitations adversely affect the performance of superconducting devices in a wide range of applications. They limit the sensitivity of photon detectors in astronomy^{1,2}, the accuracy of current sources in metrology^{3}, the cooling power of microrefrigerators^{4} and could break the topological protection of Majorana qubits^{5}. In superconducting quantum information processing (QIP), the preservation of charge parity (even or odd number of electrons) has historically been a primary concern. In the first superconducting qubit, termed the Cooperpair box (CPB)^{6}, maintaining the parity in a small island connected to a reservoir via Josephson junctions is essential to qubit operation. The qubit states 0〉 and 1〉 consist of symmetric superpositions of charge states of equal parity, brought into resonance by a controlled charge bias n_{g} and split by the Josephson tunnelling energy E_{J} (≲E_{C}, the island Cooperpair charging energy). QP tunnelling across the junction changes the island parity, ‘poisoning’ the box until parity switches back or n_{g} is offset by ±e (ref. 7). QP poisoning has been extensively studied in CPBs and similar devices, such as singleCooperpair transistors and charge pumps, with most experiments^{8,9,10,11,12,13} finding parity switching times of 10 μs–1 ms, and some >1 s (refs 14, 15, 16). While these times are long compared with qubit gate operations (∼10 ns), the sensitivity of the CPB qubit transition frequency ω_{01} to background charge fluctuations limits the dephasing time to <1 μs, severely restricting the use of traditional CPBs in QIP.
Engineering the CPB into the transmon regime E_{J}≫E_{C} (refs 17, 18) exponentially suppresses the sensitivity of ω_{01} to chargeparity and background charge fluctuations. However, recent theory^{19,20,21} predicts that QP tunnelling remains a relevant source of relaxation and pure dephasing of the qubit degree of freedom. The contribution of QP tunnelling on qubit decoherence has become particularly interesting as control of the Purcell effect^{22} in circuit quantum electrodynamics (cQED)^{23} and the reduced contribution of dielectric losses in threedimensional geometries^{24} have allowed reaching the 100 μs scale. To guide further improvements, it is imperative to precisely pinpoint the timescale for QP tunnelling and its contribution to qubit decoherence. To date, only upper and lower bounds on QP tunnelling rates have been placed^{18,25} in transmon qubits, while the effect of QP tunnelling on transmon decoherence remains unexplored.
Here, we transform a stateoftheart singlejunction transmon qubit into a realtime chargeparity detector. We measure both the characteristic time for QP tunnelling across the junction and the effect of such tunnelling on qubit decoherence at the millisecond timescale. Our qubit is controlled and measured in a threedimensional cQED architecture^{24}, an emerging platform for QIP, without need for any electrometer or other circuitry. At the heart of our detection scheme is a very small but detectable parity dependence of the qubit transition frequency (up to 0.04% of the average ω_{01}/2π=4.387 GHz), obtained by choosing E_{J}/E_{C}=25.
Results
Evidence of QP tunnelling
Standard Ramsey fringe experiments provide the first evidence of QP tunnelling across the qubit junction, as shown in Fig. 1 for a refrigerator temperature T_{r}=20 mK. Instead of the usual single decaying sinusoid, we observe two. Repeated Ramsey experiments always reveal two frequencies, fluctuating symmetrically about the average ω_{01} (Fig. 1c). The double frequency pattern results from QP tunnelling events causing n_{g} to shift by ±e. The fluctuation in the difference Δf between the two frequencies is owing to background charge motion slow compared with QP tunnelling. The observation of two frequencies in every experiment shows that QP tunnelling is fast compared with the averaging time (∼15 s), but slow compared with the maximum time 1/2Δf∼5 μs (ref. 26). From the similar amplitude of the sinusoids, we can already deduce that the two parities are equally likely. Clearly, these timeaveraged measurements only loosely bound the timescale for QP tunnelling, similarly to refs 18, 25.
Realtime detection of chargeparity fluctuations
In order to accurately pinpoint the timescale for QP tunnelling, we have devised a scheme to monitor the charge parity in real time using the qubit itself (Fig. 2a). The scheme takes advantage of recent developments in highfidelity nondemolition readout^{27,28} and feedback control^{29}. Starting from 0〉, the qubit is prepared in the superposition state ( with a π/2 ypulse at ω_{01}. The Rabi frequency of 16 MHz is sufficient to drive both odd and evenparity qubit transitions, which differ by 2Δf ≤1.76 MHz. The qubit then acquires a phase ±π/2 during a chosen idle time Δt=1/4Δf, where the + (−) sign corresponds to even (odd) parity. A second π/2 xpulse completes the mapping of parity into a qubit basis state, even →0〉, odd →1〉. A following projective qubit measurement ideally matches the result P=1 (−1) to even (odd) parity. Feedbackbased reset^{29} reinitializes the qubit to 0〉 and allows repeating this sequence every Δt_{exp}=6 μs.
The time evolution of charge parity is encoded in the series of results P (Fig. 2b). The time series has zero average, confirming that the two charge parities are equally probable. Both the QP dynamics and the detection infidelity determine the distribution of dwell times t_{+1} and t_{−1} (Fig. 2d). The measured identical histograms match a numerical simulation of a symmetric random telegraph signal (RTS) with transition rate Γ_{rts}, masked by uncorrelated detection errors occurring with probability (1−F)/2. These two noise processes contribute distinct signatures to the spectral density of P (Fig. 2c). The best fit of the form
shows excellent agreement, giving 1/Γ_{rts}=0.79 ms and F=0.92.
Measurement of QPtunnellinginduced qubit decoherence
While the above scheme detects a characteristic time for QP tunnelling, our goal is to determine the effect of such QP tunnelling on the performance of the qubit degree of freedom. Specifically, we aim to determine the rates connecting level kl〉 to level k′l′〉 (k (k′) and l (l′) denote the initial (final) qubit and parity state, respectively, as illustrated in Fig. 3b). For example, denotes the QPtunnellinginduced qubit relaxation rate. Based on the identical distribution of dwell times, we safely approximate symmetric rates .
To extract the above rates, we measure the autocorrelation function of charge parity, conditioned on specific initial and final qubit states (Fig. 3). We first execute the chargeparity sequence illustrated in Fig. 2. Conditioning on the result of the projective measurement P_{1}=+1 postselects the qubit in 0〉 and even parity. After a waiting time τ, another measurement M determines the qubit state. Conditioning also on M=+1 ensures that the qubit ends in 0〉. A second instance of the chargeparity sequence, ending with P_{2}, completes the scheme. The average result, once corrected for detector infidelity (see Methods), is the parity autocorrelation R_{00}(τ)=〈P(0)P(τ)〉_{00}, with first (second) subscript indicating initial (final) qubit state. Neglecting qubit excitation, that is, setting , R_{00}(τ) simply decays as . The exact solution shows that this remains a valid approximation when including the measured Γ_{01}=1/6 ms^{−1}, as the probability of multiple qubit transitions in τ is negligible. Similarly, we measure the parity autocorrelation with qubit initially and finally in 1〉, . To do this, we use the same conditioning, but apply a π pulse after P_{1} and before M. Exponential decay fits give and .
To quantify the contribution of QP tunnelling to the measured net qubit relaxation time T_{1}=1/Γ_{10}=0.14±0.01 ms (see Methods), we apply the same method, but condition on initial state 1〉 and final state 0〉. The ratio of QPinduced to total relaxation rates can be extracted from R_{10}(τ→0)=1−2α. The best fit of the model R_{10}(τ) to the data, with α as only free parameter, gives and . This result clearly demonstrates that QP tunnelling does not dominate qubit relaxation at T_{r}=20 mK, contributing only 5% of qubit relaxation events.
To facilitate comparison of the measured rates to theory, we perform the above experiments at elevated T_{r} (Fig. 4). We observe that , and have similar magnitude and jointly increase with T_{r} in the range 20–170 mK. However, T_{1} remains insensitive to T_{r} until 150 mK. The observed sign reversal in R_{10}(τ→0) near this temperature (Fig. 4b) indicates that QP tunnelling becomes the dominant qubit relaxation process.
Discussion
The effect of QP dynamics on the qubit degree of freedom in superconducting circuits has been extensively studied theoretically^{7,19,20,21}. For transmon qubits, the predicted QPinduced relaxation rate is^{19,20}
where x_{qp}=n_{qp}/2ν_{0}Δ is the QP density n_{qp} normalized to the Cooperpair density, with ν_{0}=1.2 × 10^{4} μm^{−3}μeV^{−1} the singlespin density of states at the Fermi energy^{11} and Δ the Al superconducting gap. This relation holds for any energy distribution of QPs. For T_{r}≥150 mK, the data closely match equation (2) using the thermal equilibrium and Δ=170 μeV, the value estimated from the normalstate resistance of the junction (see Methods). The suppression of at lower T_{r} is much weaker than expected from a thermal QP distribution. Using equation (2), we estimate n_{qp}=0.04±0.01 μm^{−3} at T_{r}=20 mK, matching the lowest value reported for Al in a Cooperpair transistor for use in metrology^{30}. Improved shielding against infrared radiation^{31} could further decrease n_{qp} at low T_{r}, consequently suppressing the contribution of QP tunnelling to qubit relaxation, and will be pursued in future work.
QP tunnelling events that do not induce qubit transitions still contribute to pure qubit dephasing. Calculations based on refs 19, 21 predict , in good agreement with the data (Fig. 4c). It is presently not understood whether such QP tunnelling events completely destroy qubit superposition states (case A) or simply change the qubit precession frequency (case B). In either case, in the regime of strongly coupled RTS valid for our experiment ( (ref. 26)) the QPinduced dephasing time is . For case B, this time would further increase in the weakcoupling regime (attained at E_{J}/E_{C}≳60) owing to motional averaging^{26}.
In conclusion, we have converted a stateoftheart transmon qubit into its own chargeparity detector to answer whether QP tunnelling already limits qubit coherence. We measure the contribution of QP tunnelling to relaxation and dephasing to be in the millisecond range. We stress that these times are directly measured, without relying on any theory. Thus, transmon qubit coherence can increase by at least another order of magnitude before QP tunnelling begins to limit coherence. Such an increase would facilitate the realization of faulttolerant quantum computing in the solid state. The implemented scheme also provides an essential ingredient in the envisioned toptransmon architecture for manipulation and readout of Majorana qubits^{32}.
Methods
Device parameters
The transmon has Josephson energy E_{J}=8.442 GHz and charging energy E_{C}=0.334 GHz. Using the Ambegaokar–Baratoff relation E_{J}R_{n}=Δ/8e^{2} and the measured roomtemperature resistance R_{n,300K}=15.2 kΩ of the single Josephson junction, we estimate Δ=170 μeV. The qubit couples to the cavity fundamental mode ω_{r}/2π=6.551 GHz (decay rate κ/2π=720 kHz) with strength g/2π=66 MHz, inducing a dispersive shift χ/π=−1.0 MHz. The qubit relaxation time T_{1} may be limited by the multimode Purcell effect^{22}. A simple estimate including only the fundamental mode gives 240 μs. The dephasing time, μs, is limited by background charge fluctuations (see Supplementary Fig. S1).
Experimental setup
The device and the experimental setup are similar to those described in refs 28, 29. Here, we detail the changes we made since these earlier reports. In an effort to lower the transmon residual excitation, we replaced the Al cavity with a Cu cavity^{33}, improved thermal anchoring to the mixing chamber plate and added lowpass filters (K&L Microwave 6L2508000/T18000O/O) on the input and output ports of the cavity. As a result, the transmon effective temperature decreased from 127 to 55 mK, corresponding to a reduction of total steadystate excitation from ∼16 to 2%, respectively. As these changes were made simultaneously, we cannot pinpoint the individual contributions to the improved thermalization.Projective readout with 99% fidelity is achieved by homodyne detection with a 400 ns pulse at ω_{r}−χ, aided by a Josephson parametric amplifier^{28}. To perform qubit reset faster, we replaced the ADwin processor with a homebuilt feedback controller based on a complex programmable logic device (CPLD, Altera MAX V). The CPLD integrates the last 200 ns of the readout signal and conditionally triggers a π pulse (all resonant pulses are Gaussian, with σ=8 ns, and total duration 32 ns). The CPLD allows a response time, from the end of signal integration to the πpulse trigger, of 0.11 μs. The total loop time, from the start of the measurement pulse to the end of the triggered π pulse at the cavity input, is 0.98 μs. However, a delay is added to reach 2 μs (∼10/κ) between the end of measurement and the start of the conditioned π pulse, ensuring that the cavity is devoid of readout photons.
Extraction of QP tunnelling rates
To convert 〈P_{2}(τ)〉_{kk′} into R_{kk′}(τ), we correct for the overall detection errors, distributed among readout (<1%) and reset (∼1%) infidelities, suboptimal Δt (<2%) and dephasing during Δt (remaining 1−3%). For this correction, we first fit an exponential decay to 〈P_{2}(τ)〉_{00} and 〈P_{2}(τ)〉_{11}. The average of the bestfit value at τ=0 is used to renormalize the data in Figs 3c and 4a. The fitted decay times are and , respectively. To extract and , we fit the solution of equation (2) to R_{10}(τ), using . Γ_{10} is obtained from the equilibration time T_{eq} after inverting the steadystate populations P_{0〉,ss}, P_{1〉,ss} with a π pulse:
The total excitation 1−P_{0〉,ss} is obtained by measurement and postselection^{29}. Equation (3) remains a valid approximation even for the highest temperatures in Fig. 4, at which population of higher excited states becomes relevant. In this case, the populations P_{0〉,ss}, P_{1〉,ss} are estimated from the total excitation, assuming that the populations are thermally distributed^{29}. Error bars for are calculated from the s.d. of repeated T_{1} measurements and the fit uncertainty in α.
Validation of the chargeparity detector
We perform several control experiments to validate the use of the qubit as a chargeparity detector. First, the parity to qubitstate conversion is tested with suboptimal choices of the Ramsey interval Δt (Supplementary Fig. S2). As expected from equation (1), the white noise level in S_{P} increases at the expense of the signal contrast as Δt deviates from the optimal choice 1/4Δf. Remarkably, the extracted rate Γ_{rts} is approximately constant down to F∼0.4. This is consistent with the model of charge parity as a symmetric RTS, with time constant determined solely by QP tunnelling.In a second test, we replace the Ramseylike sequence with a single pulse, with rotation angle θ. Time series of M for θ=0, π and π/2 are shown in Supplementary Fig. S3a. The very high occurrence (∼99%) of 1 (−1) for θ=0 (π) equals the efficiency of reset, following each measurement M. For θ=π/2, the qubit is repeatedly prepared in an equal superposition of 0〉 and 1〉, and the measurement produces uncorrelated projection noise. The spectra of these control experiments are compared with the QP tunnelling measurement in Supplementary Fig. S3b, clearly showing that the observed RTS is owing to the signal acquired during Δt.As a final test of the chargeparity detector, we subject the qubit to an externally generated RTS, similar to ref. 25. Symmetric RTS sequences with switching rate Γ_{π} are generated in LabVIEW and sent to an ADwin controller. The ADwin samples the RTS at 9 μs interval. When the signal is +1, the ADwin triggers an AWG520 (also used for reset^{29}), which then applies a π pulse on the qubit. As a result, the measured qubit state in M is conditioned on the RTS state, mimicking the paritycontrolled π pulse implemented in Fig. 2. In all cases, the fitted rates Γ_{fit} match the programmed Γ_{π} within 3% (Supplementary Fig. S4).
Additional information
How to cite this article: Ristè, D. et al. Millisecond chargeparity fluctuations and induced decoherence in a superconducting transmon qubit. Nat. Commun. 4:1913 doi: 10.1038/ncomms2936 (2013).
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Acknowledgements
We thank G. Catelani, A. Endo, F. Hassler, G. de Lange, J. M. Martinis, O. P. Saira, L. M. K. Vandersypen, P. J. de Visser and the Yale cQED team for discussions. We acknowledge funding from the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO, VIDI scheme), the EU FP7 project SOLID and the DARPA QuEST program.
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D.R. fabricated the device, M.J.T. and R.N.S. realized the feedback controller, K.W.L. designed the Josephson parametric amplifier, D.R. and C.C.B. performed the experiment and data analysis, D.R. and L.D.C. wrote the manuscript, and L.D.C. designed and supervised the experiment.
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Ristè, D., Bultink, C., Tiggelman, M. et al. Millisecond chargeparity fluctuations and induced decoherence in a superconducting transmon qubit. Nat Commun 4, 1913 (2013). https://doi.org/10.1038/ncomms2936
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