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Suppressing relaxation in superconducting qubits by quasiparticle pumping

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Science  23 Dec 2016:
Vol. 354, Issue 6319, pp. 1573-1577
DOI: 10.1126/science.aah5844

Extending qubit lifetime through a shaped environment

Qubits are the quantum two-level systems that encode and process information in quantum computing. Kept in isolation, qubits can be stable. In a practical setting, however, qubits must be addressed and interact with each other. Such an environment is typically viewed as a source of decoherence and has a detrimental effect on a qubit's ability to retain encoded information. Gustavsson et al. used a sequence of pulses as a source of “environment shaping” that could substantially increase the coherence time of a superconducting qubit.

Science, this issue p. 1573

Abstract

Dynamical error suppression techniques are commonly used to improve coherence in quantum systems. They reduce dephasing errors by applying control pulses designed to reverse erroneous coherent evolution driven by environmental noise. However, such methods cannot correct for irreversible processes such as energy relaxation. We investigate a complementary, stochastic approach to reducing errors: Instead of deterministically reversing the unwanted qubit evolution, we use control pulses to shape the noise environment dynamically. In the context of superconducting qubits, we implement a pumping sequence to reduce the number of unpaired electrons (quasiparticles) in close proximity to the device. A 70% reduction in the quasiparticle density results in a threefold enhancement in qubit relaxation times and a comparable reduction in coherence variability.

Since Hahn’s invention of the spin echo in 1950 (1), coherent control techniques have been crucial tools for reducing errors, improving control fidelity, performing noise spectroscopy, and generally extending coherence in both natural and artificial spin systems. All of these methods are similar: They correct for dephasing errors by reversing unintended phase accumulations due to a noisy environment through the application of a sequence of control pulses, thereby improving the dephasing time T2. However, such coherent control techniques cannot correct for irreversible processes that reduce the relaxation time T1, where energy is lost to the environment. Improving T1 requires reducing the coupling between the spin system and its noisy environment, reducing the noise in the environment itself (2), or implementing full quantum error correction. We demonstrate a pumping sequence that dynamically reduces the noise in the environment and improves T1 of a superconducting qubit through an irreversible pumping process. The sequence contains the same type of control pulses common to all dynamical-decoupling sequences, but rather than coherently and deterministically controlling the qubit time evolution, the sequence is designed to shape the noise stochastically via inelastic energy exchange with the environment. Similar methods have been used to extend T2 of spin qubits by dynamic nuclear polarization (3), and irreversible control techniques are commonly used to prepare systems into well-defined quantum states through optical pumping (4, 5) and sideband cooling (6). However, outside of quantum error correction, to our knowledge no dynamic enhancement of T1 has been previously reported.

We implement the pumping sequence in a superconducting flux qubit, with the aim of reducing the population of unpaired electrons or quasiparticles in close vicinity to the device. When a superconducting circuit is cooled well below its critical temperature, the quasiparticle density via Bardeen-Cooper-Schrieffer theory is expected to be exponentially suppressed, but a number of experimental groups have reported higher-than-expected values in a wide variety of systems (710). Although the reasons for the enhanced quasiparticle population and the mechanism behind quasiparticle generation are not fully understood, their presence has a number of adverse effects on the qubit performance, causing relaxation (1116), dephasing (1719), excess excited-state population (20), and temporal variations in qubit parameters (2124). Moreover, quasiparticles are predicted to be a major obstacle for realizing Majorana qubits in semiconductor nanowires (25). Our results provide an in situ technique for removing quasiparticles, especially in conjunction with recent experiments showing that vortices in superconducting electrodes can act as quasiparticles traps, thus keeping the quasiparticles away from the Josephson junctions where they may contribute to qubit relaxation (22, 2628).

We characterize and quantify the quasiparticle population by measuring qubit relaxation. Generally, the relaxation rate is given by a sum of contributions from many different decay channels. Quasiparticles contribute to the relaxation in a process whereby the qubit releases its energy to a quasiparticle tunneling across one of the Josephson junctions (Fig. 1A). Because of the small number of quasiparticles typically present in the device, fluctuations in the quasiparticle population lead to large temporal variations in the qubit decay rate. As a consequence, if the number of quasiparticles changes between trials while one repeats an experiment to determine the average qubit polarization, the time-domain decay no longer behaves as a single exponential but rather takes the following form (21) [see also (29), section S2] Embedded Image(1)Here, Embedded Image is the average quasiparticle population in the qubit region during the experiment; t is the time after qubit excitation; Embedded Image is the relaxation time induced by one quasiparticle; and T1R is the residual relaxation time from other decay channels such as flux noise, Purcell decay, or dielectric losses. Because only quasiparticles are responsible for the nonexponential decay, Eq. 1 provides a direct method for separating out quasiparticle contributions from other relaxation channels.

Fig. 1 Nonexponential decay in a superconducting flux qubit.

(A) Schematic drawing of device A, consisting of a flux qubit (lower loop) coupled to a dc superconducting quantum interference device (SQUID) for qubit readout (outer loop). Red crosses mark the position of the Josephson junctions. Qubit relaxation is induced by quasiparticles tunneling across the qubit junctions, as illustrated by the blue circle. (B) Qubit decay, as measured by applying a π pulse and delaying the qubit readout. The decay is clearly nonexponential, with the solid line showing a fit to the decay function expected from quasiparticle tunneling [Eq. 1 in the main text]. (C) Pulse sequence for pumping quasiparticles away from the qubit junctions, consisting of multiple qubit π pulses separated by a fixed period ΔT. (D) Average qubit population during the pumping sequence, measured by repeating the experiment over 40,000 trials. The remaining population after each pulse interval steadily increases, demonstrating that the qubit decay becomes progressively slower. The solid lines are fits to Eq. 1, with Embedded Image decreasing as {2.4,1.9,1.7,1.6} from the first to the last decay.

The experiments are performed using two different flux qubits. Device A is a traditional flux qubit with switching-current readout using a dc superconducting quantum interference device (SQUID), whereas device B is a capacitively shunted (C-shunt) flux qubit (24). Qubit A was operated at a frequency of 5.4 GHz, whereas qubit B was operated at 4.7 GHz [for more information on qubit parameters, see sections S1 and S6 in (29)]. Figure 1B shows the measured relaxation of qubit A, together with a fit to Eq. 1. The decay is clearly nonexponential: The fast initial decay due to quasiparticle fluctuations is followed by a slower, constant decay attributable to residual relaxation channels. From the fit, we find an average quasiparticle population Embedded Image, with Embedded Image and T1R = 55 μs. We have also measured the qubit decay as a function of flux and temperature to further validate its sensitivity to quasiparticles [sections S3 and S4 in (29)].

The same mechanism that leads to qubit relaxation also provides an opportunity for reducing the quasiparticle population. When the qubit relaxes through a quasiparticle tunneling event, the quasiparticle both tunnels to a different island and acquires an energy ħω0 from the qubit (ω0/2π is the qubit frequency and ħ is Planck’s constant h divided by 2π). The increase in energy leads to a higher quasiparticle velocity (at a constant mean free path) so that a quasiparticle can move more quickly away from the regions close to the qubit junctions where it may cause qubit relaxation. The situation is depicted in Fig. 1A, where the quasiparticle tunneling out from the section of the qubit loop containing the junctions may diffuse away toward the normal-metal ground electrode. We make use of this mechanism by applying a pulse sequence (Fig. 1C) consisting of several qubit π pulses separated by a fixed period (in this case, ΔT = 30 μs). The first π pulse excites the qubit into state Embedded Image and, during the subsequent waiting time, it has some probability of relaxing to the ground state. Because Embedded Image, this most likely occurs through a quasiparticle tunneling event, which transfers a quasiparticle across a junction, increases the quasiparticle energy, and thereby enhances its diffusion rate. The process is stochastic and may transfer quasiparticle in any direction, but by repeating the sequence we expect to pump quasiparticles away from the qubit junctions. The measured average qubit polarization during the pumping sequence (Fig. 1D) starts with the qubit in the ground state, the first π pulse brings the qubit to Embedded Image, and during the following waiting period the qubit relaxes back to an average polarization of 9%. The second π pulse inverts the polarization to 91%, and the qubit starts decaying again. However, at the end of the second waiting period the remaining polarization is 11%, demonstrating that the decay is slower during the second interval. The third and fourth π pulses further retard the decay, yielding a remaining polarization of 12 and 13%, respectively. Note that the excess population can be removed by using single-shot readout techniques to reset the qubit state after the pumping sequence ends (30).

We quantify the reduction in qubit decay by extending the pump sequence to contain more π pulses and fitting the decay to Eq. 1. The measured qubit decay, using up to 40 pumping pulses (Fig. 2), demonstrates a more than threefold enhancement in qubit decay time compared with the bare decay (the decay time is defined as the time T1/e it takes for the signal to decay by a factor of 1/e). The solid lines in Fig. 2B are fits to Eq. 1; Fig. 2, C and D, show the resulting fitting parameters Embedded Image and Embedded Image as a function of the number of pumping pulses. The average quasiparticle population drops from Embedded Image to Embedded Image after 40 pulses and then saturates at this level. Simultaneously, the decay time associated with one quasiparticle drops from Embedded Image to Embedded Image. The reduction of Embedded Image is somewhat surprising, as one might generally expect the decay time per quasiparticle to remain constant as the quasiparticles are pumped away. However, as the number of π pulses increases, the quasiparticles remaining near the junctions generally have higher energy and, hence, cause qubit excitation as well as qubit relaxation. Because Embedded Image is the sum of decay and excitation rates, this conceptually explains, at least in part, the suppression of Embedded Image. Note that despite the introduction of an excitation rate, the qubit will still eventually decay to Embedded Image due to nonquasiparticle relaxation channels (T1R), preventing us from determining the excitation and decay rate separately from the steady-state qubit population. Additionally, the range of values of Embedded Image reported here is consistent with previous measurements in flux qubits (31).

Fig. 2 Dynamic improvement in qubit decay time.

(A) Pulse sequence for pumping quasiparticles. The last π pulse acts as a probe pulse to measure the qubit polarization. (B) Normalized population versus readout delay, showing qubit decay after the pump sequence, measured with ΔT = 10 μs for an increasing number of pulses N. The decay time steadily increases from T1/e = 8 to 26 μs after 40 pump pulses. The decay traces have been normalized to the population at τ = 0 to allow direct comparison. Solid lines are fits to Eq. 1. Each data point is averaged over 105 trials. (C) Average quasiparticle number Embedded Image and (D) decay rate per quasiparticle Embedded Image, extracted from the fits shown in (B). T1R is held constant at 55 μs for all fits.

To further validate the quasiparticle pumping model and rule out alternate explanations of the data, we have also implemented the same pumping scheme, using pulses corresponding to 2π instead of π rotations. If the qubit’s environment were directly influenced by the microwave pulses through a mechanism other than quasiparticle pumping (e.g., heating, or saturation of two-level systems), we would expect both the π and 2π pulses to affect the qubit decay time. However, in the experiment we only observe an improvement in the qubit decay when driving the system with π pulses, consistent with the quasiparticle pumping model [(29), section S7].

Having demonstrated that the pumping sequence can substantially reduce the quasiparticle population, we introduced a variable delay before the final probe pulse to investigate how long the reduction in Embedded Image persists before it returns to the equilibrium value [see (29), section S5]. With the exception of an initial, faster rate for tdelay < 50 μs, the return to its steady state is well described by an exponential function with a time constant of 300 μs. The time scale for quasiparticle recovery is much longer than the qubit lifetime, thus justifying the use of the steady-state solution in Eq. 1 for estimating the quasiparticle population.

The measured quasiparticle population range of Embedded Image corresponds to an upper bound on the normalized quasiparticle density of χqp ~ 10–4 to 10–5, where χqp is the number of quasiparticles divided by the number of Cooper pairs, and we assume that all decay-inducing quasiparticles are confined to the qubit islands. This is higher than the typical values of χqp ~ 10–6 to 10–7 reported in the literature (710). The difference may possibly be attributed to the switching current readout of device A, where the qubit state is inferred by applying a short current pulse to the SQUID and determining its probability to switch to the normal state. Whenever a switching event occurs, quasiparticles are created in close vicinity to the SQUID junctions, leading to an increase in the overall quasiparticle density.

We next investigate quasiparticle pumping in a dispersively read-out C-shunt flux qubit (device B), consisting of a flux qubit loop shunted by a large capacitance (Fig. 3A). Although the capacitor improves the qubit coherence by reducing its sensitivity to charge noise, the C-shunt flux qubit is still affected by quasiparticle fluctuations. As reported in (24), the qubit was observed to switch between a stable configuration, with a purely exponential decay with T1 > 50 μs, and an unstable configuration, with nonexponential decay and temporal fluctuations. The switching between the various configurations was found to occur on a slow time scale, ranging from hours to several days. Similar switching events between stable and unstable configurations have also been observed in a fluxonium qubit and were attributed to fluctuations in the quasiparticle density (22).

Fig. 3 Improvement in qubit decay time for a C-shunt flux qubit.

(A) Scanning electron microscopy image of device B, showing the large square capacitor plates (bottom) and a magnification of the qubit loop containing the three Josephson junctions (top). The qubit is coupled to a half-wavelength (λ/2) resonator. (B) Qubit relaxation, measured with and without quasiparticle pumping pulses. The trace with N = 5 pumping pulses was taken with a pulse period of ΔT = 30 μs. The data are averaged from 15 individual traces, acquired over a 1-hour period. The fit was performed assuming the same value of T1R for both traces. The uncertainties in fitting parameters are Embedded Image, Embedded Image, and Embedded Image.

We next investigated how the quasiparticle pumping sequence affects the coherence of device B, both in stable and unstable configurations. Because the switching between different configurations is random but slow, we were careful to average only over intervals when no switching event occurred. Figure 3B shows the decay of device B, measured with and without N = 5 quasiparticle pumping pulses. The data were taken when the device was in a configuration where the qubit decay was clearly nonexponential, which is well captured by fits to Eq. 1 (solid lines in Fig. 3B). We observed a drop in the quasiparticle population from 0.87 to 0.35, leading to a twofold enhancement in the qubit decay time. Note that the long-time decay rate is identical for both traces, as expected because the pumping scheme does not affect nonquasiparticle relaxation channels. The results demonstrate that the pumping scheme works even though device B does not have a ground electrode for trapping quasiparticles, but it has been shown that vortices in the capacitor pads can also act as quasiparticle traps (26). The pumping scheme should also be applicable to other qubit modalities in which quasiparticle tunneling contributes to qubit relaxation.

The data in Fig. 3B were acquired by continuously measuring qubit decay traces over a 1-hour period and averaging them together. Figure 4 shows similar repeated measurements of the qubit decay with and without pumping pulses, but these traces were acquired about a week after the data in Fig. 3. In the more recent data set, the qubit is in a configuration where the averaged decay function is relatively well described by a single exponential, both with and without pumping pulses (Fig. 4A), and the five pumping pulses improve the decay time by only ~6%. However, when investigating the individual decay traces (Fig. 4B), we found substantial amounts of noise and temporal fluctuations in the readout signal for the data without pumping pulses. These random variations vanish when implementing the pumping sequence (right panel of Fig. 4B).

Fig. 4 Reduction of qubit coherence variations with quasiparticle pumping.

(A) Averaged qubit decay, measured with and without pumping pulses. The decay function is close to exponential in both cases. The decay time increases by ~6% with the pumping pulses. The traces have not been normalized to account for the decay during the pulse sequence, causing reduced contrast for the data with N = 5. The data were measured with ΔT = 30 μs. (B) Individual traces of the averaged decay data shown in (A), measured without (left) and with five pumping pulses (right). The pumping sequence substantially reduces the temporal fluctuations observed in the decay without pumping pulses. (C) Standard deviation of the data in (B), demonstrating the strong reduction in temporal shot-to-shot fluctuations in the presence of the pumping pulses.

To quantify the improvements in variability, we calculated the standard deviation of the readout signal over 9 hours of data (Fig. 4C). With pumping pulses, the standard deviation is independent of the readout delay τ and can be ascribed to the noise of the high–electron-mobility (HEMT) amplifier used for amplification. Without pumping pulses, the standard deviation is substantially larger for τ < 50 μs but approaches the same level as for N = 5 for long delay times. The increased noise is caused by variations in the qubit T1 time, which lead to strong fluctuations in the qubit population directly after the initial π pulse. The fluctuations are reduced as the qubit decays to the ground states for long τ, leaving only the contributions from the HEMT noise.

Our implementation of a stochastic scheme to dynamically shape the environment by pumping quasiparticles in a superconducting flux qubit lead to substantial improvements in both qubit coherence times and coherence variability. In addition to applications in superconducting qubits, we anticipate our results to be of practical importance for implementing Majorana fermions in hybrid semiconducto–superconductor systems, where the presence of a single quasiparticle is detrimental to the device performance (25).

Supplementary Materials

www.sciencemag.org/content/354/6319/1573/suppl/DC1

Supplementary Text

Figs. S1 to S4

References (3237)

References and Notes

  1. Supplementary materials are available on Science Online.
Acknowledgments: We thank M. Blencowe, D. Campbell, M. Devoret, J. Grover, P. Krantz, and I. Pop for useful discussions and P. Baldo, V. Bolkhovsky, G. Fitch, J. Miloshi, P. Murphy, B. Osadchy, K. Parrillo, R. Slattery, and T. Weir at MIT Lincoln Laboratory for technical assistance. This research was funded in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), and the Assistant Secretary of Defense for Research and Engineering via MIT Lincoln Laboratory under Air Force contract no. FA8721-05-C-0002; by the U.S. Army Research Office grant no. W911NF-14-1-0682; and by NSF grant no. PHY-1415514. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. government. G.C. acknowledges partial support by the European Union (EU) under Research Executive Agency (REA) grant agreement no. CIG-618258. J.By. acknowledges partial support by the EU under REA grant agreement no. CIG-618353.
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