⇦Chinese Physics Letters, 2023, Vol. 40, No. 4, Article code 047302Express Letter Gatemon Qubit Based on a Thin InAs-Al Hybrid Nanowire Jierong Huo (霍杰荣)1†, Zezhou Xia (夏泽洲)1†, Zonglin Li (李宗霖)1†, Shan Zhang (张珊)1†, Yuqing Wang (王宇清)2, Dong Pan (潘东)3, Qichun Liu (刘其春)2, Yulong Liu (刘玉龙)2, Zhichuan Wang (王志川)4, Yichun Gao (高益淳)1, Jianhua Zhao (赵建华)3, Tiefu Li (李铁夫)2,5, Jianghua Ying (应江华)6*, Runan Shang (尚汝南)2, and Hao Zhang (张浩)1,2,7* Affiliations 1State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 2Beijing Academy of Quantum Information Sciences, Beijing 100193, China 3State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China 4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5School of Integrated Circuits and Frontier Science Center for Quantum Information, Tsinghua University, Beijing 100084, China 6Yangtze Delta Region Industrial Innovation Center of Quantum and Information, Suzhou 215133, China 7Frontier Science Center for Quantum Information, Beijing 100084, China Received 9 February 2023; accepted manuscript online 7 March 2023; published online 16 March 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: yingjianghua@tgqs.net; hzquantum@mail.tsinghua.edu.cn Citation Text: Huo J R, Xia Z Z, Li Z L et al. 2023 Chin. Phys. Lett. 40 047302    Abstract We study a gate-tunable superconducting qubit (gatemon) based on a thin InAs-Al hybrid nanowire. Using a gate voltage to control its Josephson energy, the gatemon can reach the strong coupling regime to a microwave cavity. In the dispersive regime, we extract the energy relaxation time $T_1\sim0.56$ µs and the dephasing time $T_2^* \sim0.38$ µs. Since thin InAs-Al nanowires can have fewer or single sub-band occupation and recent transport experiment shows the existence of nearly quantized zero-bias conductance peaks, our result holds relevancy for detecting Majorana zero modes in thin InAs-Al nanowires using circuit quantum electrodynamics.
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 DOI:10.1088/0256-307X/40/4/047302 © 2023 Chinese Physics Society Article Text Topological quantum computation[1,2] aims to solve the decoherence problem at the device level by encoding information into Majorana zero modes.[3,4] A promising material candidate is semiconductor-superconductor hybrid nanowires.[5,6] Tremendous efforts have been put into searching for possible Majorana signatures in InAs and InSb nanowires.[7-14] Meanwhile, proposals on topological qubits have been theoretically explored with great enthusiasm.[15-21] A major technique in those proposals is the circuit quantum electrodynamics (cQED), similar to that in the superconducting transmon qubit.[22,23] Moreover, cQED could also be used to probe Majorana signatures if incorporating the nanowire into a transmon-like device.[24-28] Motivated by this, transmon qubits based on InAs-Al nanowires have been realized and studied in recent years.[29-37] The InAs wire diameters in those gate-tunable transmons (gatemons) are typically large, $\sim $ 75–160 nm. Though the junction region can be easily gate-tuned, the proximitized InAs region is heavily screened by the covered Al film and is still in the multi-subband regime. Thick wire and multi-band bring challenges into the Majorana detection.[38-43] To overcome this issue, thin InAs-Al nanowires have been explored and nearly quantized zero bias conductance peaks have been reported.[11,12,44] In this Letter, we report the realization of gatemon qubit based on these thin wires. The InAs diameter is $\sim$ 35 nm, significantly smaller than those in previous gatemons. Our result paves the way for future Majorana cQED experiments. Qubit Device and Measurement Circuit. Figure 1(a) shows the optical image (false colored) of the device chip. One gatemon qubit (device A, the dashed box) was measured with results shown in Figs. 2–4. The other five qubits on this chip were not working. We have characterized three working qubits, see the Supplementary Material (SM) for the other two. For the device fabrication, a 100-nm-thick NbTiN superconducting film (the orange region) was first sputtered onto a sapphire substrate. Reactive ion etching was then performed to etch away part of the film (the dark regions). This lithography step defines the co-planar wave-guide feed line (blue), the resonator/cavity (green) and the shunt capacitor of the gatemon (red). Figure 1(b) is a zoomed-in image of the qubit. The feed line capacitively couples to a $\lambda/4$ cavity for the qubit readout. The cavity internal quality factor $Q_{\rm i}$ is $\sim$ 14000 and the bare resonance frequency $f_{\rm C}$ is $\sim$ 4.6 GHz. The cavity further couples to the T-shape capacitor whose capacitance is estimated to be $\sim$ 100 fF. This capacitor connects to the InAs-Al nanowire Josephson junction [Figs. 1(c) and 1(d)] and together, they form the gatemon qubit. The large capacitance determines the charging energy to $E_{\rm C} \sim e^2/2\,C \sim190$ MHz. The other side of the Josephson junction connects to the ground. A side gate (pink) tunes the junction transparency and therefore controls the Josephson energy $E_{\rm J}$. The junction was defined by removing (etching) a small Al segment on the InAs wire (diameter $\sim$ 35 nm). Transports on these wires show a gate-tunable supercurrent ($I_{\rm c}$) on the order of 100 nA.[45] The corresponding $E_{\rm J}=\hbar I_{\rm c}/2e \sim50$ GHz is much larger than $E_{\rm C}$. This ensures that the qubit can be operated in the transmon regime. For details of the device fabrication, circuit setup and cavity calibration, see the SM (the method session, Figs. S1 and S2).
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Fig. 1. (a) Optical image of a gatemon chip and measurement setup. The grounding plane is in orange. The common feed line is in blue (false colored). The $\lambda/4$ cavity (for device A) is in green. The T-shape qubit capacitor (for device A) is in red. The gate line is in pink. All these elements are NbTiN (thickness 100 nm) and were fabricated in one lithography step using reactive ion etching. (b) An enlargement of (a) (the dashed box). (c) SEM (false colored) of the InAs-Al region of the gatemon [the dashed box in (b)]. The top contact connects to the T-shape capacitor. The bottom contact is grounded. A side gate (pink) connects to the gate line. The gate and contacts were fabricated in another lithography step by sputtering Ti/NbTiN (1/100 nm). (d) SEM of the Josephson junction region of the qubit [the blue box in (c)]. (e) Schematic of the measurement circuit. (f) Feed line transmission as a function of the readout power. $V_{\rm G} = -4.554$ V. The cavity shift indicates the coupling of the cavity to a nonlinear circuit (the qubit).
Figure 1(e) draws the equivalent circuit diagram. For the cavity and qubit readout, a microwave tone of frequency $f_{\rm r}$ (near the cavity resonant frequency $f_{\rm C}$) is applied to the feed line [see Fig. 1(a)]. The transmission of this microwave tone, $S_{21}$, is measured by a vector network analyzer (VNA). Figure 1(f) is such a “single-tone” measurement by sweeping $f_{\rm r}$ and its power while monitoring the transmission amplitude $|S_{21}|$. At high power, the qubit is “overwhelmed” and the dip in $|S_{21}|$ corresponds to the bare resonant frequency of the cavity $f_{\rm C}$.[46] At low power, the resonant frequency is qubit-state dependent due to the cavity-qubit dispersive interaction. The repulsion of the qubit and cavity causes the shift of the resonance frequency as shown in Fig. 1(f). To excite and control the qubit, a second microwave tone of frequency $f_{\rm d}$ can be applied in the standard “two-tone” spectroscopy. Vacuum Rabi Splitting and Qubit Spectroscopy. In Fig. 2(a), we keep the readout power low and scan the gate voltage $V_{\rm G}$. The resonant frequency of the cavity is gate tunable, indicating the presence of the gatemon. For better visibility, a signal background, contributed by standing waves in the circuit, was subtracted from $|S_{21}|$ (see Fig. S3 in the SM for details). The jumps in the spectrum are due to charge instabilities in the InAs-Al devices, which are also commonly observed in the transport characterizations. The qubit frequency ($f_{\rm Q}$) is given by the energy difference between the ground state and the first excited state, $hf_{\rm Q}=E_{01}\sim \sqrt{8E_{\rm C}E_{\rm J}}-E_{\rm C}$. $V_{\rm G}$ tunes $E_{\rm J}$, therefore controls $f_{\rm Q}$. When $f_{\rm Q}$ is tuned close to the cavity frequency $f_{\rm C}$, the strong qubit-cavity hybridization leads to the anti-crossings. This anti-crossing is observed in the single-tone spectrum shown in Fig. 2(a). See Fig. 2(b) for a fine scan (an enlargement) of an anti-crossing. Figure 2(c) is the line cut at $V_{\rm G} = -3.310$ V, where the peak spacing is the smallest. In this strong coupling regime, the frequencies of the two peaks read $f_{\pm}=[f_{\rm Q}+f_{\rm C}\pm \sqrt{(f_{\rm Q}-f_{\rm C})^2+4(g/2\pi)^2}]/2$. The peak spacing $\delta=f_{+}-f_{-}= \sqrt{(f_{\rm Q}-f_{\rm C})^2+4(g/2\pi)^2}$ is a function of $f_{\rm Q}$. The qubit frequency can be obtained by $f_{\rm Q}=f_{+}+f_{-}-f_{\rm C}$. Figure 2(d) plots the extracted $\delta$ and $f_{\rm Q}$. The red line is the theoretical fit based on the above formula. We extract the qubit-cavity coupling strength $g/2\pi \sim74$ MHz. Next we tune $V_{\rm G}$ to bias $f_{\rm Q}$ away from $f_{\rm C}$ and reach the dispersive regime. The large detuning, $|\varDelta/2\pi| = |f_{\rm Q}-f_{\rm C} | \gg g$, could effectively suppress the energy relaxation due to the Purcell effect.[47] Figure 2(e) shows the two-tone spectroscopy, $|S_{21}|$, as a function of the qubit drive $f_{\rm d}$ (the second tone) in different three $V_{\rm G}$ ranges. The readout frequency $f_{\rm r}$ (the first tone) was fixed near the cavity resonance ($f_{\rm C}$). When $f_{\rm d}$ is scanned on resonance with $f_{\rm Q}$, the qubit can be excited and the resonator frequency is shifted. A signal in the readout tone can be observed due to the cavity shift. The spectroscopy in Fig. 2(e) reveals the gate tunable nature of $f_{\rm Q}$. The non-monotonic fluctuations are associated with the non-ballistic property of the InAs-Al junction, indicating the presence of disorder.
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Fig. 2. (a) Cavity transmission in the single-tone measurement as a function of the cavity drive frequency and $V_{\rm G}$. The anti-crossings are the vacuum Rabi splittings. (b) Fine scan of an anti-crossing. The gate voltage has a shift compared to the same feature in (a) due to hysteresis or charge jumps. The two dashed lines are the bare cavity frequency (pink, $f_{\rm C}$) and the extracted qubit frequency (yellow, $f_{\rm Q}$). (c) A line cut from (b) at the crossing point of the two dashed lines. The peak spacing $\delta = f_+ - f_-$ is indicated by the black arrow in (b). (d) Behavior of $\delta$ versus the qubit frequency. The red line is the theoretical fit. Inset: energy schematic of the cavity-qubit hybridization and the parameters. (e) Gatemon qubit spectroscopy (two-tone) as a function of $V_{\rm G}$ in three different ranges. (f) A line cut of (e) (at the black arrow) with a Lorentzian fit (blue dashed line).
The multiple peaks in Fig. 2(e) are likely caused by the photon-number-dependent frequency shift of the qubit.[48] This is obvious when the cavity readout tone was continuously applied (instead of pulsed) on the feed line, see Fig. S4 in the SM for detailed analysis. The extracted peak spacing ($\sim$ 12 MHz) roughly matches our estimation of $2|\chi|/2\pi=g^2/|\varDelta|\pi \sim10$ MHz. Figure 2(f) shows a line cut of the qubit excitation peak from Fig. 2(e). From the Lorentzian fit we extract the full width at half maximum (FWHM $\sim$ 3.8 MHz). This width corresponds to a coherence time $\sim$ 0.26 µs, consistent with the time-domain measurement in Figs. 3 and 4. Rabi Oscillations. We now manipulate the gatemon qubit in time domain. A qubit drive pulse ($f_{\rm d}$) was first applied for a duration time of $t$ and then followed by a readout pulse of the cavity. Figure 3(a) shows the typical Rabi oscillations as functions of $f_{\rm d}$ and $t$. The Rabi oscillation frequency $\omega \propto \sqrt{(f_{\rm d}-f_{\rm Q})^2+{{\rm const}}}$. The constant term is proportional to the square of the driving amplitude. The oscillation pattern shows a Chevron feature. We note that the pattern gets disrupted near 4.40 GHz, probably due to the presence of another cavity on the chip whose resonant frequency is around 4.40 GHz. The split/shift of the oscillation pattern near 4.41 GHz is likely due to the gate-instability-induced shift/jump of the Josephson energy. Another possibility could be an effect of photon-number-dependent shift (see Fig. S4) due to, e.g., un-calibrated IQ-mixer. From Fig. 3(a), we can estimate the qubit frequency $f_{\rm Q} \sim4.426$ GHz. Note that this is not in the dispersive regime yet since the detuning is not large enough. Figure 3(b) shows the line cut near $f_{\rm Q}$. Fitting the decaying oscillation using an empirical formula $y=Ae^{-t/T_{{\rm R}}}{\cos}(\omega t+\phi)+C$ (the dashed line), we extract the Rabi coherence time $T_{{\rm R}}\sim0.23$ µs. In Fig. 3(c), the qubit drive power is varied. Higher power drives the qubit faster, resulting in a shorter oscillation period. We extract the oscillation frequency (the inverse of the period) and plot it as a function of its driving amplitude (converted from the power) in Fig. 3(d). The dashed line is a linear fit, confirming its Rabi oscillation nature. Note that the cavity readout power was kept to be constant in Fig. 3(c) and the pulse-method was used for the measurement. Therefore, the ac-Stark effect should not play a role.
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Fig. 3. (a) Rabi oscillations as a function of the qubit drive frequency and the drive pulse duration. (b) A line cut at the qubit resonant frequency $f_{\rm Q} = 4.426$ GHz [see the arrow in (a)]. The dashed line is a theoretical fit. (c) Rabi oscillations as a function of the drive power at a fixed drive frequency $f_{\rm d}=4.446$ GHz. (d) The Rabi oscillation frequency ($\omega/2\pi$) extracted from (c) versus the driving amplitude (over the range where the oscillations are visible). Here, $10^{P/20}$ has a linear relation with the driving amplitude. The dashed line is a linear fit. $V_{\rm G}=-2.890$ V for all panels.
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Fig. 4. Energy relaxation time $T_1$ and dephasing time $T_2^*$. (a) Pulse sequence schematic for the $T_1$ measurement. [(b), (c)] $T_1$ measurements at $V_{\rm G}$ of $-2.890$ V and $-4.442$ V, respectively. The black dashed lines are the exponential decay fits. The increasing vs decreasing trends between (b) and (c) are due to different selected working points, resulting in reversed readout signal strength. (d) Pulse sequence schematic for the $T_2^*$ measurement. [(e), (f)] Ramsey oscillations at different $V_{\rm G}$ values. The dashed lines are fits with an exponential decaying envelope.
Gatemon Quantum Coherence. To extract the gatemon energy relaxation time $T_1$, a $\pi$ pulse was first applied to excite the qubit to the $|1\rangle$ state. The readout was performed after the waiting time $t_1$ [Fig. 4(a)]. Figures 4(b) and 4(c) show the exponential fit at two different gate voltages. A relaxation time $T_1 \sim0.56$ µs can be extracted. The dephasing time $T_2^*$ was determined by the Ramsey experiment: inserting a waiting time $t_2$ between two slightly detuned $\pi$/2 pulses before the readout [Fig. 4(d)]. Figures 4(e) and 4(f) show two Ramsey oscillations. $T_2^*$ can reach $\sim$ 0.38 µs. The fitting assumes an exponential decaying envelope: $A{\cos}(\omega t+\phi){\exp}(-(t_2/T_2^*)^2)+C$.[49] Note that the gate voltage of Figs. 4(c) and 4(f) corresponds to the dispersive regime ($f_{\rm Q}$ here is $\sim$ 3.446 GHz), therefore a longer coherence time is expected. In comparison, the coherence times for Figs. 4(b) and 4(e) are shorter due to the Purcell effect. $T_2^* < 2T_1$ indicates that the coherence of our qubit is not entirely limited by energy relaxation. For the Rabi oscillations at $V_{\rm G} = -4.442$ V [as same as Figs. 4(c) and 4(f)], see Fig. S5. In Fig. S6, we show the measurement of two more gatemon qubits. In summary, we have studied the gatemon qubit based on a thin InAs-Al hybrid nanowire. The gatemon can reach strong coupling to a cavity. Coherent Rabi oscillations can be observed. The qubit relaxation time $T_1$ and dephasing time $T_2^*$ can reach 0.56 µs and 0.38 µs, respectively. We note that the recent experimental progress on Majorana signatures has addressed the potential advantage of single sub-band occupation.[11,12,50] Though more experiments are needed to build the connection between thinner wires and Majoranas, theoretical models favor the single sub-band (i.e., thinner diameter). However, device fabrication is more challenge for thinner wires than for thick wires due to less contacting area. If thin diameter is the necessary step toward Majoranas, this work will overcome the challenge and fulfill the requirement by demonstrating its feasibility for cQED experiments. Our work, in this sense, can be considered as a step forward compared to previous gatemons based on thick wires. Future work on these thin-wire-based gatemons could aim for possible Majorana signatures in a finite magnetic field. Acknowledgment. We thank Chunqing Deng and Luyan Sun for valuable discussions. We also thank the Teaching Center for Experimental Physics of Tsinghua University for using their equipment. This work was supported by the Tsinghua University Initiative Scientific Research Program, the Alibaba Innovative Research Program, and the National Natural Science Foundation of China (Grant Nos. 12204047, 92065106, and 61974138). D.P. also acknowledges the support from Youth Innovation Promotion Association, Chinese Academy of Sciences (Grant Nos. 2017156 and Y2021043). Raw data and processing codes in this study are available at https://doi.org/10.5281/zenodo.7620737. 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