Quantifying quantum coherence of multiple-charge states in tunable Josephson junctions

Coherence and tunneling play central roles in quantum phenomena. In a tunneling event, the time that a particle spends inside the barrier has been fiercely debated. This problem becomes more complex when tunneling repeatedly occurs back and forth, and when involving many particles. Here we report the measurement of the coherence time of various charge states tunneling in a nanowire-based tunable Josephson junction; including single charges, multiple charges, and Cooper pairs. We studied all the charge tunneling processes using Landau-Zener-St\"uckelberg-Majorana (LZSM) interferometry, and observed high-quality interference patterns. In particular, the coherence time of the charge states was extracted from the interference fringes in Fourier space. In addition, our measurements show the break-up of Cooper pairs, from a macroscopic quantum coherent state to individual particle states. Besides the fundamental research interest, our results also establish LZSM interferometry as a powerful technique to explore the coherence time of charges in hybrid devices.


Introduction
In a quantum tunneling event, a fundamental open question is the time spent by the particle(s) inside the barrier [1][2][3][4][5][6] .Another unsolved issue is the coherence time 7 of the back-and-forth tunneling process through the barrier, especially for multi-particles.Josephson junctions (JJs) provide a platform for studying tunneling processes of various charges; including single charges, coherent multiple charges, and Cooper pairs [8][9][10] .When a voltage is applied to a JJ, the well-known multiple Andreev reflections (MARs) provide a mechanism for the coherent transfer of multiple charges, up to infinity 11 .At the gap edge, the tunneling of single charges (Giaever tunneling) contributes to a coherence peak 12 .In addition, a JJ usually inherits the macroscopic quantum coherent nature of the superconductor.
However, for an ultra-small JJ with a low capacitance and a weak Josephson coupling, the phase fluctuates significantly and is no longer a good quantum number, leading to a loss of macroscopic quantum coherence of Cooper pairs, regressing to individual particle states 9,12,13 .
Accordingly, the Shapiro-step picture breaks down under microwave driving [14][15][16] .Therefore, a tunable JJ can serve as a unique platform for studying the coherence of tunneling processes of many charges, and for monitoring the disappearance and establishment of macroscopic quantum coherence of the Cooper pairs.In recent years, scanning-tunneling-microscope (STM) experiments utilizing a superconducting tip have been used to distinguish the tunneling processes of the charges under microwave drive.The results were interpreted based on the Tien-Gordon theory and/or the microwaveassisted MAR model 13,[17][18][19][20] .But the coherence time of various charges that tunnel back and forth through the barrier has not yet been examined.A systematic investigation of the onset and disappearance of macroscopic quantum coherence of the Cooper pairs in an ultra-small JJ is missing.
Interferometry can probe the coherence times of quantum states.Recently, Landau-Zener-Stückelberg-Majorana (LZSM) interferometry [21][22][23] has gained great success as a tool to diagnose the physical parameters and also to realize fast coherent manipulation of isolated qubits [24][25][26][27][28][29][30][31] .In this work, we implemented LZSM interferometry in a gate-tunable nanowire-based JJ, which is an open system (i.e., directly connected to the measurement circuit), in contrast to the generally isolated qubits.We constructed the JJ down to the single ballistic-channel limit, allowing us to resolve LZSM interference for single charges, multiple charges, and Cooper pairs −− in the weak coupling regime and under the drive of microwaves.Through analyzing the interference fringes in the two-dimensional (2D) Fourier space, for the first time we extracted the coherence times of all these charge states that tunnel back and forth through the junction.We further uncovered the loss of macroscopic quantum coherence of the Cooper pairs in the JJs.

Results
LZSM interferometry.Before presenting the experimental results, we first interpret the physical picture of LZSM interference in an ultra-small JJ.Phase fluctuations lead to the loss of macroscopic quantum coherence and Cooper pair tunneling as individual microscopic particles 9,12,13 (Fig. 1a).MARs enable a coherent transfer of 2-charges through a 1 st -order reflection (Fig. 1b) and 3-charges through a 2 nd -order reflection (Fig. 1c).
Single charge tunneling is activated when the voltage reaches V b = 2/e ( is the superconducting gap and e is the elementary charge) (Fig. 1d).
However, to enable LZSM interferometry, two coupled energy levels plus a fast ac drive are required 32 .For a JJ, an open system, we argue that thanks to the singularities of the BCS single particle density of states and the Cooper pair condensate, the effective two levels can be mapped to the charges being at the left and the right sides of the junction, respectively.These two states, |L> and |R>, are shown in the bottom row of Figs.1a-1d.
The charges can tunnel between these two states under a microwave drive.
Figure 1e illustrates LZSM interference.The two states |L> (black dashed line) and |R> (green dashed line) anti-cross, with a minimal distance , which is twice the coupling strength.The junction is detuned by V 0 relative to the anti-crossing point, and a harmonic microwave (brown curve) with an amplitude of V RF drives the system adiabatically to and nonadiabatically through the anti-crossing at time t = t 1 , with a transition probability of ) , where  is the sweeping velocity.The two trajectories accumulate a phase difference (the shaded area) until t = t 2 when the system is brought back to the anti-crossing, where LZSM interference occurs.
The continuous microwave drive sustains the interference of the alternating tunneling until the charges lose coherence.To incorporate the environmental decoherence, classical noise can be introduced to the microwave drive, and a white noise model is employed under perturbation, to obtain the rate of transitions between |L> and |R> 33,34 (, ) =  2 where ε = meV 0 is the detuning energy, A = meV RF ,  = 2f (with f being the frequency of the microwave),  = 1 ∕  is the decoherence rate,  = ± 1, ± 2 … denotes the satellite replicas of the n = 0 Lorentzianshaped peaks, J n is the Bessel function of the first kind, and  = 1, 2, 3 … represents the number of charges in an elementary tunneling event.
Evidently, the observability (broadening) of the interference fringes depends on the competition between  and  .
The InAs 0.92 Sb 0.08 nanowires were grown by molecular beam epitaxy, followed by an in situ epitaxy of ~15 nm-thick Al at a low temperature, which guarantees a hard superconducting gap 35,36 .Narrow Al gaps (JJs) were formed naturally during growth due to shadowing between the dense nanowires, eliminating the wet etching process and possible degradations.
Figure 1g shows the nanowire possessing two JJs studied in this work, with Al gaps of ~20 nm (JJ1) and ~40 nm (JJ2), respectively.The yellow rectangles show post-fabricated Ti/Au (10/80 nm) contacts.Standard lockin techniques and a microwave antenna were applied to carry out the transport measurements at ~10 mK.We will focus on JJ1 in the main text and present typical results for JJ2 in the Supplementary Information.
One of the advantages of the nanowire-based JJ is its full gate tunability.
By applying a gate voltage V G (Figs. 2a and 2b), the JJ could be tuned to the strong coupling regime, the single ballistic-channel limit, and the weakcoupling regime.Figure 2a displays the correlation between the conductance dI/dV measured at the normal state and the critical supercurrent I C .A quantized conductance plateau at 2e 2 /h corresponds to a single ballistic channel, and the associated I C stays around 15 nA, which is smaller than the theoretical expectation 37 of ∆/ℏ = 52 nA ( ≈ 215 eV).
The discrepancy could be explained by intrinsic reasons though 38 , to the best of our knowledge, the value I C / in our device is the largest for a single channel [39][40][41][42] .Unlike the STM or the break-junction experiments where usually a mix of several channels contributes to the conductance 19,43 , we demonstrate the single channel limit unambiguously, beneficial for the investigation of the coherence.Figure 2b displays the conductance map at the weak-coupling regime, showing the peaks contributed by Cooper pairs, 2-charges (the 1 st -order MARs), and single charges, as marked by the green, red, and blue triangles, respectively.
LZSM interference of single charges.We proceed to turn on the microwave and examine the LZSM interferometry.To verify our implementation of the interferometry, the JJ was first set to the tunneling limit at V G = −31 V. Since the conductance scales roughly to  , where  is the transmission 44 , the conductance peaks of multiple charges were heavily suppressed for small , leaving the single-charge coherence peaks as the prominent characteristics (Fig. 2b). Figure 2c depicts the dI/dV versus microwave power P and V b at a frequency f = 11.755GHz, exhibiting clear interference fringes consistent with Eq. ( 1).After considering an attenuation of the microwave and converting P to V RF , the corresponding 2D FT is plotted in Fig. 2d, and the lemon-shaped ovals nicely follow the predictions of Eq. ( 2).LZSM interference of various charges.The JJ was then configured to a slightly stronger coupled regime (V G = −28.9V) to release the tunneling of multiple charges and Cooper pairs.Figure 3a presents the interference fringes of the Cooper pairs, 2-charges through the 1 st -order MARs and single charges, as marked by the square, circle, and triangle, respectively.
The fringes of 3-charges through the 2 nd -order MARs can also be recognized by the dashed lines in Fig. 3b, a zoom-in of Fig. 3a.The 2D FT maps were plotted in Figs.3d-f, marked in correspondence with Fig. 3a, and T 2 were extracted to be 0.1 ns, 0.049 ns, and 0.051 ns, for Cooper pairs, 2-charges, and single charges, respectively.
Taking the dI/dV versus V b line-cut at P = −45 dBm in Fig. 3a as an input and using the T 2 values, the conductance map can be calculated.For this we assume that dI/dV is proportional to W and its behavior follows Eq. ( 1).This is shown in Fig. 3c, which manifests a good agreement with the measured data in Fig. 3a.So far, we realized the measurement of the coherence times for the tunneling of all these charges.
We next discuss about three more characteristics of the interference fringes.Second, the dI/dV peaks behave as J n 2 , versus the microwave amplitude V RF .In Fig. 3h, the left and right columns show the extracted curves (black) from Fig. 3a for the main peak (n = 0) and the first satellite peak (n = 1), respectively.A good agreement was achieved by a J n 2 fitting, from Eq. ( 1), as shown by the red curves.Note that a dI/dV constant was included in each fitting to account for the background conductance, and the power P has been converted to V RF .Moreover, the J n 2 dependence serves as evidence for Cooper pair tunneling as microscopic particles near V b = 0, instead of the macroscopic Shapiro-step picture whose characteristic width would rather follow J n to the 1 st -order 13,19 , as shown below.
Onset and loss of the macroscopic coherence.Third, there is an opposite trend of the coherence for the Cooper pairs and the other charges To conclude, we realized the measurement of the coherence time of various charge states that tunnel back and forth in a JJ by implementing LZSM interferometry.Then, we revealed the change of the Cooper pairs in the JJ, from a macroscopic quantum coherent state to individual particle states; or conversely, the onset of macroscopic quantum coherence.We expect that the LZSM interferometry is a powerful technique for exploring the coherence time of charges tunneling in various hybrid devices, such as through trivial Andreev and Yu-Shiba-Rusinov bound sates 20 , Floquet states 45 , as well as topological Majorana bound states [46][47][48][49] .Note that these states might also be subjected to the crossover of the Cooper pairs from a macroscopic quantum coherence state to individual particle states.

Methods
InAs 0.92 Sb 0.08 -Al Nanowire Growth: InAs 0.92 Sb 0.08 nanowires were grown in a solid source molecular beam epitaxy system (VG V80H) on p-Si (111) substrates using Ag as catalysts 50 .The nanowires were grown for 40 min at a temperature of 465 °C with the beam fluxes of In, As and Sb sources of 1.1×10 -7 mbar, 4.6×10 -6 mbar and 4.7×10 -7 mbar, respectively.After the growth of InAs 0.92 Sb 0.08 nanowires, the sample was transferred from the growth chamber to the preparation chamber at 300 °C to avoid arsenic condensation on the nanowire surface.The sample was then cooled down to a low temperature (~ -30 °C) by natural cooling and liquid nitrogen cooling 51 .Al was evaporated from a Knudsen cell at an angle of ∼20° from the substrate normal (∼70° from the substrate surface) and at a temperature of ∼1150 °C for 180 s (giving approximately 0.08 nm/s).During the Al growth, the substrate rotation was kept disabled.When the growth of nanowires with Al was completed, the sample was rapidly pulled out of the MBE growth chamber and oxidized naturally.
Fabrication of the devices: The as-grown nanowires were transferred onto Si/SiO 2 (300 nm) substrates simply by a tissue.Standard electron-beam lithography was applied to fabricate the Ti/Au (10 nm/80 nm) electrodes using electron-beam evaporation.Note that an ion source cleaning and a soft plasma cleaning were performed to improve the contact prior to the deposition.
Transport measurements: The measurements were carried out in a cryofree dilution refrigerator at a base temperature of ~10 mK.Low-frequency lock-in techniques were utilized to measure the conductance of the JJ, dI/dV  I ac /V ac , with a small excitation ac voltage V ac and a dc bias voltage  S1 shows a high-angle annular dark-field scanning transmission electron microscope (HAADF-STEM), energy dispersive spectrum (EDS) and high-resolution TEM data of a typical shadow InAs 0.92 Sb 0.08 -Al nanowire.
As shown in Fig. S1a, a half Al shell with a narrow Al gap can be clearly observed on the facet of the InAs 0.92 Sb 0.08 nanowire.The false-color EDS elemental maps of In (Fig. S1b), As (Fig. S1c), Sb (Fig. S1d) and Al (Fig. S1e) of the InAs 0.92 Sb nanowire has a pure zinc-blende crystal structure due to its non-<111> growth direction, although the Sb content is low, which is consistent with our previous work 1 .As shown in Fig. S1g, an abrupt interface between the Al shell and the InAs 0.92 Sb 0.08 nanowire can be observed.In this section, we interpret the theoretical treatment of the LZSM interference in the small JJ studied in our work.For ease of reading, we illustrate the LZSM interference again in Figs.S3a-S3c.As explained in the main text, we map the effective two states as the charges being on the left and the right side of the junction based on the Bardeen-Cooper-Schrieffer singularities of the density of states (see Fig. S3a for the case of Cooper pairs).The coupling strength between the two states, |L> and |R>, is defined as /2, resulting in an anti-crossing of .The red and blue energy levels in Fig. S3b are thus the two effective levels for LZSM interference.
We consider a harmonic microwave driving with an amplitude of V RF and a frequency of f.The detuning can be controlled by applying a bias voltage relative to the anti-crossing point, V 0 .As displayed in Figs.S3b and S3c, a LZSM transition takes place at time t 1 and (after accumulating a phase difference of ) a subsequent LZSM transition and LZSM interference occur at time t 2 , when the system is driven back to the anti-crossing point.The extraction of the coherence time in Fourier space for a driven qubit has been theoretically studied by Rudner et al 2 .Here we follow a similar procedure and theoretically analyze the LZSM interference for various charges in our small JJs.When a harmonic drive  RF cos(ω) ( = 2f) is applied, the Hamiltonian of the system can be expressed as: where ε =  0 ,  =  RF , and m is the number of charges, and e is the elementary charge.When  RF >  0 , the system will be driven to pass through the anti-crossing twice at  1 and  2 in one period, respectively.
The times  1,2 satisfy The phase difference (gray shaded area in Fig. S3c) accumulated between the two LZ transitions can be expressed as:  , where   is a Bessel function of the first kind, can be used to obtain It is this equation, Eq. (S9), that relates to the conductance of the JJ, a measurable parameter in the experiment.It describes the interference fringes as a function of the microwave drive  =  RF and the detuning (voltage bias) ε =  0 relative to the anti-crossing point.Figure S3e presents one example calculated directly from Eq. (S9), using a frequency f = 10 GHz and a decoherence rate Γ 2 = 1/(4).Note that the data has been scaled to a maximum of 1, and the x-axis is the microwave amplitude V RF (A = meV RF ), while in our measured data it is the power P. can help to directly extract the coherence time T 2 , which demonstrates that this is a powerful technique.
Figure S3f shows the 2DFT results of Fig. S3e.Note that the integral to obtain  FT is from −∞ to +∞; therefore, the fringes shown in Fig. S3e need to be symmetrized to the four quadrants before performing the 2DFT.
Lemon-shaped ovals are consistent with Eq. (S10) and also with our experimental results.In addition, for a given , there is a ray ε =  in energy space (, ε) and a set of periodic points in the temporal space.As shown in Figs.S3e and S3f, the colored rays of the interference fringes contribute to the colored dots in the Fourier space correspondingly.For example, the red ray stands for ε =   Step 2: select the proper data set and convert to the four quadrants.

Figure
Figure 2e shows ln(W FT ) as a function of k, averaged near k A = 0 to smooth out the singularities.Since  ∝  | | , a linear fit (red line) of the peak-values (blue squares) generates a coherence time T 2 = 0.075 ns.In addition, the Lorentzian shape of Eq. (1) versus  = meV 0 depends on T 2 as well.Figure 2f illustrates a Lorentzian fit (red curve) of the dI/dV versus V b = V b −2/e curve (black) at P = −55 dBm (V RF ≈ 0), which yields a similar T 2 = 0.083 ns.

First, the
number, m, of charges can be conveniently determined from the spacing (V b ) of the satellite peaks, since V b  1/m. Figure 3g plots dI/dV versus the V b curves taken from Figs. 3a and 3b, as indicated by the black and red bars.The spacings V b of 49 V, 25 V, 15 V and 24 V are consistent with charge numbers m of 1, 2, 3 and 2 at V b = −2/e, −2/(2e), −2/(3e) and 0, respectively.

(
quasiparticles) when the coupling strength of the JJ increases.A clue can be found by examining the T 2 extracted above.When increasing the coupling strength of the JJ from V G = −31 V to −28.9 V, T 2 for single charges drops from 0.075 ns to 0.051 ns.At V G = −28.9V, T 2 = 0.049 ns and 0.1 ns for 2-charges through MARs, and Cooper pairs, respectively.When the coupling strength increases further, as shown in Fig.3iat V G = −27.5 V, the fringes for the Cooper pairs become more evident while the others fade away.This is consistent with the scenario that when the Josephson coupling and the related capacitance increase, the Cooper pairs tend to recover the macroscopic quantum coherence, but the quasiparticles decohere faster due to the energy broadening and the background conductance, etc.Eventually, at the strong-coupling regime the macroscopic quantum coherence was fully restored and regular Shapiro steps were obtained at V G = 6 V, and a good agreement was achieved by fitting the step width to the 1 st -order of the Bessel function, |J n |, as shown in Fig.4.

Figure 1 .
Figure 1.Illustration of LZSM interference in a Josephson junction.a-d, Charge tunneling at typical bias voltages V b .The arrows in b and c indicate successive Andreev reflections.|L> and |R> represent the two effective states where charges tunnel back and forth under a microwave drive.e, The dashed lines show the diabatic energy levels for the states |L> and |R>; the solid blue and red curves show the energy levels when taking tunneling into account.A microwave (brown) drives the system through the anti-crossing at the time t 1 where LZSM transitions occur, and back to the anti-crossing at time t 2 , where the two trajectories interfere.f, Linear dependence of ln(W FT ) on |k|, whose slope is  2 .g, The nanowire (white) contains JJ1 and JJ2.Aluminum is shown in violet, and the yellow rectangles denote post-fabricated Ti/Au contacts.Scale bar: 200 nm.

Figure 2 .
Figure 2. LZSM interference in the tunneling limit.a, Conductance dI/dV, measured in the normal state, and critical supercurrent I c , both versus the gate voltage V G .b, dI/dV versus V G and V b at the weak coupling regime.The green, red and blue triangles mark the peaks of Cooper pairs, 2-charges of the 1 st -order MARs, and single charges, respectively.c, Interference fringes at V G = −31 V and f = 11.755GHz versus microwave power P and V b .d, The 2D FT of c.For clarity, k A and k are in units of 1/f.e, Averaged line-cut (black) near k A = 0 of d.The red line is a linear fit of the peak-values as marked by the blue squares.f, Lorentzian fit (red curve) of the dI/dV data (black curve) versus V b = V b −2/e at P = −55 dBm of c.The data for V b < 0 are extracted from c, and the V b > 0 branch is simply mirrored.g-h (i-j), Interference fringes and the corresponding 2D FT at f = 7.665 GHz (4.0 GHz).

Figure 3 .
Figure 3. LZSM interference of various charges.a, Interference fringes at V G = −28.9V and f = 11.755GHz.The square, circle and triangle depict the contributions of Cooper pairs, 2-charges through the 1 st -order MARs, and single charges, respectively.b, Zoom-in of a.The dashed lines highlight the features of 3-charges through the 2 nd -order MARs.c, Calculated dI/dV based on both the dI/dV versus V b curve at P = −45 dBm in a and the T 2 values shown in d-f, which display the 2D FT maps of the sets of the fringes in a. g, Line-cuts taken from a and b, as indicated by the black and red bars.h, Extracted curves (black) from the fringes of a and the J n 2 fit (red).The left column is for the n = 0 main peaks, and the right for the n = 1 satellite peaks.i, Interference fringes at V G = −27.5 V and f = 11.755GHz.

Figure 4 .
Figure 4. Shapiro steps in the strong-coupling regime.a, Differential resistance dV/dI versus P and current I, measured at V G = 6 V and f = 11.56GHz.The numbering indicates the Shapiro steps.b, Step width I versus V RF and the corresponding Bessel function fitting to the 1 st -order.A −45 dB attenuation was assumed and subtracted.

Figure S1 .Section 2 .
Figure S1.Chemical composition and crystal structure of the InAs 0.92 Sb 0.08 -Al nanowires.a, HAADF-STEM image of an InAs0.92Sb0.08-Alnanowire.b-e, False-color EDS elemental maps of In (yellow), As (green), Sb (purple), and Al (orange) of the InAs0.92Sb0.08-Alnanowire, respectively.f, High-resolution TEM image of the InAs0.92Sb0.08nanowire.The inset of f is its corresponding fast Fourier transform.g, High-resolution TEM image of the InAs0.92Sb0.08nanowire taken from the InAs0.92Sb0.08-Alinterface area.The blue and red rectangles in a highlight the regions where high-resolution TEM images were recorded.

FigureSection 3 .
Figure S2 shows the transport measurement setup.A small ac voltage V-ac and a dc voltage V-dc were applied through a voltage divider, after which the actual V ac and V b were subjected onto the device.The ac current I ac was measured.A serial resistance from the dc wires and the low-pass filters was subtracted during the data analysis.A gate voltage V G was applied through a 300 nm-thick SiO 2 to control the coupling strength of the Josephson junction (JJ).The shaded InAsSb nanowire segments were brought into proximity with the epitaxial Al to possess a hard superconducting gap of size .A microwave antenna with an open end of a coaxial cable was implemented to radiate the junction with microwaves.For the critical current measurement and the Shapiro step measurement, the dc current was converted.

Figure S3 .
Figure S3.LZSM interference and its Fourier transform.a, Illustration of the two effective states in a small JJ.b, c, Schematics of the two-level system and the microwave driven LZSM interference.d, e, Calculated interference fringes.A harmonic driving signal (inset in e) with a frequency f = 10 GHz, and a decoherence rate Γ2 = 1/(4) ( = 2f) were used.f, The two-dimensional Fourier transform (2DFT) of e.The colored dots stand for the corresponding contributions from the lines in e.

Section 4 .
Choice of the microwave frequencies.In order to choose proper frequencies of the microwave, we measured the dI/dV dependence on both the microwave frequency f and power P at a back-gate voltage V G = −30 V and a bias voltage near the gap edge, as shown in Fig.There is a clear frequency-dependent effective attenuation of the microwave, presumably due to the coaxial lines and the details of the coupling between the microwave and the device.Nevertheless, in order to get rid of the heat load on the dilution refrigerator, we selected frequencies corresponding to a local minimum attenuation, as marked by the green arrows for 7.665 GHz and 11.755 GHz, respectively.

Figure S4 .
Figure S4.Choice of the microwave frequencies.The dI/dV was measured as a function of the microwave frequency f and power P using a back-gate voltage VG = −30 V and a bias voltage near the gap edge.

Figure S5 .
Figure S5.Determination of the attenuation.a, Calculated interference fringes corresponding to Fig. 2c in the main text, without an attenuation of the microwave.b, Comparison between the experimental data (black) taken from Fig. 2c in the main text and the calculated curve taken from a, as indicated by the green dashed line.An attenuation of −45.5 dB can be extracted by the power difference between the first peaks, as marked by the green and black arrows.

Figure S6 .
Figure S6.Selecting the proper data set and converting to the four quadrants.a, The same data as in Fig. 2c in the main text, but with the effective microwave power Peff = P − 45.5 (dBm).The green dashed rectangle displays the data set chosen to perform the 2DFT.The coordinates of this data set were converted to the microwave amplitude VRF and relative bias voltage Vb-rel by setting the n = 0 interference fringes to Vb-rel = 0 mV, as indicated in b by the green dashed rectangle.The data set was further symmetrized to the four quadrants to carry out the 2DFT.

Figure S7 .Section 7 .
Figure S7.Effect of the magnetic field on the LZSM interference.a, Interference fringes at VG = −30 V, B = 0 T, and f = 11.755GHz.b, 2DFT of a. c, The same as a, but at B = 0.2 T. d, 2DFT of c.

Figure
Figure S8 shows the results for JJ2 (see Fig. of the main text).The conductance dI/dV shown in the upper panel was measured in the normal state at a high-bias voltage: V b =3 mV (black) and V b =4.5 mV (red).Thequantized conductance plateau at 2e 2 /h demonstrates the single ballistic channel in the JJ.However, the plateau for 4e 2 /h can be barely recognized, and the conductance evolves to the plateau of 6e 2 /h when V G increases.Such transition from a single channel to three channels could be attributed

Figure
FigureS9shows the results of another typical device.An Al gap of ~50 nm was formed during the epitaxial growth of the Al shell due to the shadowing of the dense nanowires, as indicated by the red arrow in the scanning electron microscope image of the device shown in Fig.S9a.The side-gate voltage was set to 0 and a back-gate voltage V G was applied to control the coupling strength.FigureS9bdisplays the interference fringes coherence time  = 1/ can be conveniently characterized in In this work, all the nanowires were grown on p-type Si (111) substrates.The <111> and non-<111>-oriented InAs 0.92 Sb 0.08 -Al nanowires coexisted on the substrate surface because the thin natural oxide layer on the Si substrates cannot be removed completely before the nanowire growth.According to detailed transmission electron microscope (TEM) observations, we find that continuous half Al shells can be successfully grown on the facets of InAs 0.92 Sb 0.08 nanowires with non-<111> growth directions, while the Al shells are discontinuous and look like 'pearls on a string' on the side walls of the <111> oriented nanowires.Particularly, narrow Al gaps can form naturally in these non-<111> InAs 0.92 Sb 0.08 -Al nanowires due to shadowing between the dense nanowires.Figure b applied and the ac current I ac measured.The microwave driving was supplied through a semi-rigid coaxial cable with an open end which is several millimeters above the JJ, serving as an antenna.Wang, Z., Huang, W.-C., Liang, Q.-F.&Hu, X. Landau-Zener-Stückelberg Interferometry for Majorana Qubit.Sci Rep 8, 7920, (2018).Abay, S. et al.Quantized Conductance and Its Correlation to the Supercurrent in a Nanowire Connected to Superconductors.Nano Letters 13, 3614-3617, (2013).Section 1.Growth of the epitaxial nanowires.
0.08 -Al nanowire further confirm that a narrow Al gap indeed formed in the continuous Al shell.Figures S1f and S1g are highresolution TEM images of the nanowire taken from the InAs 0.92 Sb 0.08 region and InAs 0.92 Sb 0.08 -Al interface area, respectively.The InAs 0.92 Sb 0.08