Low-Loss Superconducting Nanowire Circuits Using a Neon Focused Ion Beam

We present low-temperature measurements of low-loss superconducting nanowire-embedded resonators in the low-power limit relevant for quantum circuits. The superconducting resonators are embedded with superconducting nanowires with widths down to 20nm using a neon focused ion beam. In the low-power limit, we demonstrate an internal quality factor up to 3.9x10^5 at 300mK [implying a two-level-system-limited quality factor up to 2x10^5 at 10 mK], not only significantly higher than in similar devices but also matching the state of the art of conventional Josephson-junction-embedded resonators. We also show a high sensitivity of the nanowire to stray infrared photons, which is controllable by suitable precautions to minimize stray photons in the sample environment. Our results suggest that there are excellent prospects for superconducting-nanowire-based quantum circuits.

We present low-temperature measurements of low-loss superconducting nanowire-embedded resonators in the low-power limit relevant for quantum circuits. The superconducting resonators are embedded with superconducting nanowires with widths down to 20 nm using a neon focused ion beam. In the low-power limit, we demonstrate an internal quality factor up to 3.9×10 5 at 300 mK (implying a TLS-limited quality factor up to 2×10 5 at 10 mK), not only significantly higher than in similar devices, but also matching the state of the art of conventional Josephson-junction-embedded resonators. We also show a high sensitivity of the nanowire to stray infrared photons, which is controllable by suitable precautions to minimise stray photons in the sample environment. Our results suggest that there are excellent prospects for superconducting-nanowire-based quantum circuits.

I. INTRODUCTION
Quantum circuits based on conventional Josephsonjunctions have begun to tackle real-world problems 1 . This has been despite high decoherence produced by the loss 2,3 and noise 4,5 caused by parasitic two-level systems (TLS) 6,7 . In principle, superconducting nanowires can provide a route to low-decoherence quantum circuits due to their monolithic structure and lack of a TLS-hosting oxide layer. To date, superconducting nanowires with cross-sectional areas approaching the coherence length have demonstrated a variety of Josephson 8,9 and phaseslip [10][11][12][13] effects, but features such as their unconventional current-phase relationships 14 remain unexploited in quantum circuits. Previous demonstrations of superconducting nanowire-embedded resonators exhibit unusually high dissipation, with internal quality factors (Q i ) below 5×10 3 , 10-12,15 , far lower than in similar conventional Josephson-junction-based circuits 16,17 . In general, the performance of nanowire-embedded resonators can be limited by material quality, interface imperfections, resist residues and the measurement environment.
We demonstrate superconducting nanowire-embedded circuits with single photon Q i up to 3.9×10 5 , comparable to or even better than conventional Josephsonjunction resonators. Superconducting nanowires with widths down to 20 nm were fabricated with a neon focused ion beam (FIB). We study the loss in our devices within the well-established framework of loss mechanisms in superconducting resonators 2,3,7,18,19 to determine which factors are significant in limiting their performance. The vastly improved Q i demonstrates that the detrimental effects can be sufficiently reduced and shows that competitive quantum circuits could be based on monolithic nanowire technology.

II. METHODS
Superconducting 20-nm-thick NbN films were deposited on sapphire by dc magnetron sputtering from a 99.99%-pure Nb target in a 1:1 Ar:N 2 atmosphere. The vacuum chamber was pumped to 6×10 −7 mbar before sputtering at a pressure of 3.5×10 −3 mbar and power of 200 W. The superconducting critical temperature, T c , was 10 K with a sheet resistance of 450 Ω/sq. Electronbeam lithography (EBL) was used to pattern λ/4 and λ/2 coplanar microwave resonators capacitively coupled to a common microwave feed line (shown in Fig. 1d). The width of the central conductor was 10 µm and the gap was 5 µm. This pattern was transferred from a 300nm-thick-layer of polymethyl methacrylate (PMMA) into the film by a reactive ion etch (RIE) using a 2:1 ratio of SF 6 :Ar, at 30 W and 30 mbar.
A neon FIB was used to directly pattern 20 nanowires in the central conductor of the microwave resonators at the current antinode -see Fig. 1b. With an acceleration voltage of 15 kV, the clearance dose for the NbN film is ≈ 0.3 nC/µm 2 . 15 kV was chosen as a compromise between minimising the spot size and minimising lateral milling of the nanowire 21 , leading to a fewminute mill time per µm 2 for a ∼1 pA beam current. By prior patterning of a sub-200-nm-wide precursor wire in the same EBL step as the resonator (shown in Fig. 1c), we minimise the mill time and the total neon flux that the nanowire is subject to. Several devices were measured, and Table I shows important parameters including nanowire dimensions. The nanowire devices all feature two nanowires, configured either in parallel so that the nanowires complete a superconducting loop 12 , or in series with a wider segment in between 22 . Here, there is no external flux-or gate-bias, so the nanowires are treated as simple constrictions within the superconductor.
Samples were enclosed within a brass box and cooled using a 3 He refrigerator containing a heavily attenuated microwave in-line and an out-line with a cryogenic highelectron-mobility transistor (HEMT) amplifier. Figure 1a shows the forward transmission (S 21 ) magnitude response of a nanowire-embedded resonator, at 307 mK and for an applied microwave drive of arXiv:1708.02502v1 [cond-mat.supr-con] 8 Aug 2017 −105 dBm, demonstrating Q i = 5.2×10 5 . This Q i is significantly higher than in comparable nanowire-based devices [10][11][12]15 . This highlights the promise of the neon FIB and demonstrates that superconducting nanowires are not intrinsically lossy. The complex S 21 notch response of the superconducting resonators is fitted by 23

III. RESULTS & DISCUSSION
where ν is the applied frequency, ν 0 the resonance frequency, Q L the loaded quality factor and |Q c | the absolute value of the coupling quality factor; φ accounts for impedance mismatches, a describes a change in amplitude, θ describes a change in phase and τ a change in the electronic delay. The internal quality factor, Q i , is defined by 1/Q L = 1/Q i + Re(1/Q c ) and the energy within the resonator is W sto = 2P app S min Q L /ω 0 , where P app is the applied microwave power (in W) and S min the normalized minimum of the resonator magnitude response. We describe the microwave power in terms of the average number of photons in the resonator, n , given To examine the effect of the neon FIB on the NbN film, we measured the resonator response as a function of temperature (shown in Fig. 2). As temperature decreases from 2 K to 1 K, the resonant frequency increases due to changes in the complex conductivity which are described by ∆ν ν0 = α 2 ∆σ2 σ2 , where ∆ν ν0 is the normalised resonance frequency, α is the kinetic inductance fraction and σ 2 is the imaginary part of the complex conductivity as given by Mattis-Bardeen (MB) theory 24 . The inset of Fig. 2 shows the temperature dependence of the resonant frequency for all resonators on chip 1. The bunching of data points indicates a very similar T c whether the resonator contains nanowires or not, implying that the neon FIB has not significantly suppressed the superconductivity.
Further decreasing temperature from 1 K, the resonant frequency decreases due to a thermal desaturation of TLS, which can be described by where F is the filling factor which typically relates to device geometry and electric field density, T 0 is a reference temperature, Ψ is the complex digamma function and F δ i TLS is the intrinsic loss tangent. Fig. 2 shows a fit to both the MB and TLS frequency shifts, and the extracted F δ i TLS is shown in Table I. Barends et al. 25 have previously showed that, to determine F δ i TLS using both MB and TLS models, it is not necessary to obtain data in the temperature range covering the frequency upturn below 100mK seen in the TLS fit curve in Fig. 2.
The thermal desaturation of TLS below 1 K results in absorption of microwave photons, leading to a power-and temperature-dependent resonator loss rate 2,3 . At low microwave drive, the unsaturated TLS dominate the loss, but as the microwave drive increases these TLS become saturated and therefore their loss rate decreases. At high microwave drives, where the TLS are saturated, the loss becomes dominated by residual quasiparticles, with a loss rate δ qp which is temperature-dependent but assumed to be independent of microwave power 19 . The TLS and quasiparticle loss behaviour can be described by where n c is the number of photons equivalent to the saturation field of the TLS, β describes how quickly the TLS saturate with power and F δ 0 TLS is the TLS loss tangent (F δ 0 TLS is power-and temperature-independent). TLS models were originally based on the anomalous properties of glasses at low temperatures 6 and assumed noninteracting TLS, which leads to a prediction of β = 0.5. However, as superconducting circuits have improved, this model has failed to accurately describe the power dependence of dielectric losses: a weaker power dependence with β < 0.5 is frequently found 3,26-28 . This showed the need to consider TLS interactions 4,5,7,26,29 , changing the loss model to 7, 29 1 where χ is the dimensionless TLS parameter, P γ is the TLS switching rate ratio, C is a large constant and δ qp is the log-scaled quasiparticle loss rate. This loss is examined in more detail by fitting the resonator S 21 response as a function of microwave drive and temperature (shown in Figs. 3a-c). Fig. 3a (Fig. 3c) show measurements of δ i tot where δ i tot = 1/Q i as a function of n on bare (nanowire-embedded) resonators. Each resonator has its own symbol, with solid (hollow) symbols corresponding to measurements in a normal (Eccosorb-lined) sample box. Eccosorb CR-117 (see supplemental 30 ) is a microwave absorber which has been shown to reduce quasiparticle excitation from stray infrared (IR) photons 31 , the Eccosorb lining is shown in Fig. 1e We first consider bare resonators measured in a standard sample box (solid symbols in Fig. 3b). Resonators on the same chip show a fabrication-based variability, also found in the literature 2,19,32 : high-n Q i = 1.2-3.1×10 6 and low-n Q i = 3.6-5.7×10 5 at 307 mK. Increasing the temperature leads to an increase in low-n Q i because, as thermal occupation of the TLS increases, their ability to absorb microwave photons decreases 3,32 , as described by the tanh temperature term. Increasing temperature also leads to a decrease in high-n Q i . This is due to a higher quasiparticle density, meaning that more energy is lost to the quasiparticle system 19 . In the centre is a chip, which is wirebonded to a microwave printed circuit board, the dark material is Eccosorb.  We next consider nanowire-embedded resonators in the standard sample box (solid symbols in Fig. 3c). At 307 mK, at low n , we find Q i = 2.7-3.9×10 5 , in good agreement with the results from the bare resonators, so the FIB-based fabrication of the nanowire has produced very little additional TLS loss. At high n , we find Q i = 4.1-7.2×10 5 , 3-5 times lower than the bare resonators, indicating a higher residual quasiparticle density for the nanowire-embedded resonators.
Quasiparticles generated from pair-breaking events are an important consideration in conventional Josephsonjunction devices 33 , where Eccosorb is typically used to reduce quasiparticle-based losses caused by stray IR photons 31,33 . We examined whether quasiparticles generated from IR photons are important for nanowireembedded resonators by measuring them in an Eccosorblined sample box. As the hollow dotted symbols in Fig. 3c show, losses at high n are much lower than for the standard sample box and Q i ≈ 6-9×10 5 . This value matches that of the bare resonators for the same n , suggesting that the density of residual quasiparticles has been reduced to that of the bare resonators (see Table I). A saturated high-n Q i is not observed, due to nonlinearities in the resonance lineshape of the nanowire-embedded resonators. With the smaller quasiparticle-based loss, the TLS-based low-n trend of loss increasing as n decreases is once again found. The high-n Q i is found to increase with increasing temperature, consistent with losses from thermally generated quasiparticles as found in the bare resonators, indicating that increased quasiparticle losses in nanowire-embedded resonators in the normal sample box arose from quasiparticles excited by IR pho- tons. As Table I shows, δ qp of the nanowire-embedded resonators in the Eccosorb environment match those of the bare resonators (both with and without the Eccosorb environment) and are therefore limited by another mechanism which is not unique to the nanowire. Figure 3b shows the loss for the same bare resonator with and without the Eccosorb enclosure. In contrast to nanowire-embedded resonators, the high-n loss decreases only slightly when the Eccosorb-lined sample box is used. This is actually unsurprising since the energy gap of NbN is ∼ 10× larger than in Al. On the other hand, the reason for the sensitivity to IR photons in the nanowire-embedded resonators is not immediately obvious. Our results demonstrate the importance of IR filtering even when nanowires have a large superconducting energy gap such as those in NbN. This is relevant to all nanowire-based devices. We note that a small suppression of T c in our nanowire (below the precision of our T c determination) could give some enhanced sensitivity to IR photons. Alternative explanations for the sensitivity include the nanowire exhibiting a different quasiparticle lifetime 34 or non-equilibrium superconductivity 35 , but these are beyond the scope of this study, although, since Q i remains high, the number of quasiparticles created from IR photons must still be quite small 31 .
Finally, we compare the consistency of the TLS-loss rates (Table I and  TLS differ by less than 20%, this difference is because F δ 0 TLS is only sensitive to near-resonant TLS, whereas F δ i TLS is also sensitive to a broad spectrum of off-resonant TLS 3,32,36 . Next, we note that δ i TLS = χ, 7 so that the ratio F P γ χ/F δ i TLS gives P γ . We find an average value of P γ = 0.093. This agrees well (see supplemental 30 ) with the charge noise spectra of singleelectron transistors that give P γ ≈ 0.10. 37,38 Therefore, all TLS-loss rates are consistent with each other. The TLS loss rates imply a TLS-limited Q i up to ≈2×10 5 in the quantum limit (at temperatures down to 10 mK and at single-photon energies). This is approximately 100× larger than in equivalent nanowireembedded resonators and compares favourably with Josephson-junction-embedded resonators.

IV. CONCLUSION
To conclude, we have used a neon FIB to create superconducting nanowires with widths down to 20 nm within superconducting resonators. In the low-power limit, these devices demonstrated Q i up to 3.9×10 5 at 300 mK, with δ i TLS and δ 0 TLS corresponding to a TLSlimited Q i up to 2×10 5 at 10 mK. These TLS losses arise from the NbN thin-film technology rather than the neon FIB, meaning a higher Q i should be possible with better resonator technology 32 . By obtaining such a high Q i using nanowires, we have demonstrated a critical step towards realising nanowire-based, superinductance, phaseslip or Dayem-bridge circuits with coherence times comparable to conventional Josephson-junction-type devices.