Introduction

In a quantum tunneling event, a fundamental open question is the time spent by the particle(s) inside the barrier1,2,3,4,5,6. Another unsolved issue is the coherence time7 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 pairs8,9,10. 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 infinity11. At the gap edge, the tunneling of single charges (Giaever tunneling) contributes to a coherence peak12. 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 states9,12,13. Accordingly, the Shapiro-step picture breaks down under microwave driving14,15,16. 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 microwave-assisted MAR model13,17,18,19,20,21. 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) interferometry22,23,24 has gained great success as a tool to diagnose the physical parameters and also to realize fast coherent manipulation of isolated qubits25,26,27,28,29,30,31,32. 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, 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.

The LZSM model assumes a system with a discrete energy spectrum whose temporal modulation produces interlevel transitions. In this sense, it allows the description of the tunneling between states as soon as the relevant states are identified. This approach allows us to consider various-charge states in one context, namely, we will consider within this model the tunneling of Cooper-pairs, 2- and 3-charges through MARs, and single charges. These three different phenomena were previously well studied18,33,34,35,36,37,38 using more microscopic models, providing various derivations and results. The LZSM model allows considering the tunneling of various charges from a different, but complementary, perspective. Moreover, the involvement of coherence and decoherence is enabled in the quantum interference process.

Results and discussion

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 particles9,12,13 (Fig. 1a). MARs enable a coherent transfer of 2-charges through a 1st-order reflection (Fig. 1b) and 3-charges through a 2nd-order reflection (Fig. 1c). Single charge tunneling is activated when the voltage reaches Vb = 2Δ/e (Δ 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 required39. 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 Fig. 1a–d. The charges can tunnel between these two states under a microwave drive.

Fig. 1: Illustration of LZSM interference in a Josephson junction.
figure 1

a–d Charge tunneling at typical bias voltages Vb. 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 adiabatic 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 t1 where LZSM transitions occur, and back to the anti-crossing at time t2, where the two trajectories interfere. f Linear dependence of ln(WFT) on |kε|, whose slope is ±Γ2. 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.

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Figure 1e illustrates LZSM interference. When a harmonic drive \({V}_{{\rm{RF}}}\cos \left(\omega t\right)\) (ω = 2πf, with f being the frequency of the microwave) is applied, the Hamiltonian of the system can be expressed as \(H=-\frac{\hslash }{2}\left(\begin{array}{cc}\beta \left(t\right) & \alpha \\ \alpha & -\beta \left(t\right)\end{array}\right)\), where \(\beta \left(t\right)=\varepsilon -A\cos \left(\omega t\right)\), \(\varepsilon ={me}{V}_{0}\) is the detuning energy, V0 is the voltage relative to the anti-crossing point (see Fig. 1e), \(A={me}{V}_{{\rm{RF}}}\), m = 1, 2, 3… represents the number of charges in an elementary tunneling event, and α/2 is the coupling strength of the JJ. The two-level system as shown in Fig. 1e can be directly obtained after a rotation of the Hamiltonian. The two states |L> (black dashed line) and |R> (green dashed line) anti-cross, with a minimal distance α, which is twice the coupling strength. The junction is detuned by V0 relative to the anti-crossing point, and a harmonic microwave (brown curve) with an amplitude VRF drives the system adiabatically to, and nonadiabatically through, the anti-crossing at time t = t1, with a transition probability of \({P}_{{\rm{LZSM}}}=\exp \left(-\pi {\alpha }^{2}/2\nu \hslash \right)\), where \(\nu \) is the sweeping velocity. The two trajectories accumulate a phase difference (the shaded area) until t = t2, 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>40,41

$$W(\varepsilon ,A)=\frac{{\alpha }^{2}}{2}\mathop{\sum }\limits_{n=-\infty }^{+\infty }\frac{{\varGamma }_{2}\,{J}_{n}^{2}(\frac{A}{\hslash \omega })}{{(\varepsilon -n\hslash \omega )}^{2}+{\hslash }^{2}{\varGamma }_{2}^{2}},\,$$
(1)

where \({\varGamma }_{2}=1/{T}_{2}\) is the decoherence rate, \(n=\pm 1,\pm 2\ldots \) denotes the satellite replicas of the n = 0 Lorentzian-shaped peaks, Jn is the Bessel function of the first kind. Evidently, the observability (broadening) of the interference fringes depends on the competition between ω and \({\varGamma }_{2}\).

The coherence time \({T}_{2}=1/{\varGamma }_{2}\) can be conveniently characterized in Fourier space by inverting the energy variable to the time variable. A 2D Fourier transform (2D FT) of \(W\left(\varepsilon ,A\right)\) yields40

$${W}_{{\rm{FT}}}\left({k}_{A},{k}_{\varepsilon }\right)=\frac{{\alpha }^{2}\omega {e}^{-{\varGamma }_{2}\left|{k}_{\varepsilon }\right|}}{2\sqrt{\frac{4}{{\omega }^{2}}{\sin }^{2}\left(\frac{1}{2}\omega {k}_{\varepsilon }\right)-{k}_{A}^{2}}}$$
(2)

for \(\left|{k}_{A}\right| < \frac{2}{\omega }\left|\sin \left(\frac{1}{2}\omega {k}_{\varepsilon }\right)\right|\) and zero otherwise. The reciprocal-space variables kA and kε correspond to the energy variables A and ε, respectively. Therefore, lemon-shaped ovals following the singular boundary \({\frac{\omega }{2}k}_{A}=\pm \sin \left(\frac{1}{2}\omega {k}_{\varepsilon }\right)\), and an exponential decay on kε as \({e}^{-{\varGamma }_{2}\left|{k}_{\varepsilon }\right|}\), are expected (see Supplementary Note 3). Figure 1f illustrates a linear dependence of ln(WFT) on |kε|, whose slope generates the coherence time \({T}_{2}=1/{\varGamma }_{2}\).

Epitaxial nanowire JJs

Next, we present the experimental results. InAs0.92Sb0.08 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 gap42,43. 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 lock-in 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 Supplementary Notes 7 and 8.

One of the advantages of the nanowire-based JJ is its full gate tunability. By applying a gate voltage VG (Fig. 2a, b), the JJ could be tuned to the strong coupling regime, the single ballistic-channel limit, and the weak-coupling regime. Figure 2a displays the correlation between the conductance dI/dV measured at the normal state and the critical supercurrent IC. A quantized conductance plateau at 2e2/h corresponds to a single ballistic channel, and the associated IC stays around 15 nA, which is smaller than the theoretical expectation44 of \(e\triangle /{{\hslash}}\) = 52 nA (Δ ≈ 215 μeV). The discrepancy could be explained by intrinsic reasons though45, to the best of our knowledge, the value IC/Δ in our device is the largest for a single channel46,47,48,49. Unlike the STM or the break-junction experiments, where usually a mix of several channels contributes to the conductance19,50, 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 1st-order MARs), and single charges, as marked by the green, red, and blue triangles, respectively.

Fig. 2: LZSM interference of single charges in the tunneling limit.
figure 2

a Conductance dI/dV, measured in the normal state, and critical supercurrent Ic, both versus the gate voltage VG. b dI/dV versus VG and Vb at the weak coupling regime. The green, red and blue triangles mark the peaks of Cooper pairs, 2-charges of the 1st-order MARs, and single charges, respectively. c Interference fringes at VG = −31 V and f = 11.755 GHz versus microwave power P and Vb. d The 2D FT of (c). For clarity, kA and kε are in units of 1/f. e Averaged line-cut (black) near kA = 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 ΔVb = Vb − 2Δ/e at P = −55 dBm of (c). The data for ΔVb < 0 are extracted from (c), and the ΔVb > 0 branch is simply mirrored. g–j Interference fringes and the corresponding 2D FT at f = 7.665 GHz (4.0 GHz).

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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 VG = − 31 V. Since the conductance scales roughly to \(m{\tau }^{m}\), where τ is the transmission51, the conductance peaks of multiple charges were heavily suppressed for small τ, leaving the single-charge coherence peaks as the prominent characteristics (Fig. 2b). Figure 2c depicts the dI/dV versus microwave power P and Vb at a frequency f = 11.755 GHz, exhibiting clear interference fringes consistent with Eq. (1). After considering an attenuation of the microwave and converting P to VRF, the corresponding 2D FT is plotted in Fig. 2d, and the lemon-shaped ovals nicely follow the predictions of Eq. (2).

Figure 2e shows ln(WFT) as a function of kε, averaged near kA = 0 to smooth out the singularities. Since \({W}_{{FT}}\propto {e}^{-{\varGamma }_{2}\left|{k}_{\varepsilon }\right|}\), a linear fit (red line) of the peak values (blue squares) generates a coherence time T2 = 0.075 ns. In addition, the Lorentzian shape of Eq. (1) versus ε = meV0 depends on T2 as well. Figure 2f illustrates a Lorentzian fit (red curve) of the dI/dV versus ΔVb = Vb − 2Δ/e curve (black) at P = −55 dBm (VRF ≈ 0), which yields a similar T2 = 0.083 ns.

Although T2 is short, we demonstrate LZSM interferometry as a powerful approach to extract the coherence time of the charges in an open JJ. Two lower frequencies were further used, as shown in Fig. 2g, h for 7.665 GHz and Fig. 2i, j for 4.0 GHz. When the frequency decreases, the interference fringes become blurred because the time interval between two subsequent LZSM transitions increases and thus the interference loses coherence gradually40. The number of ovals in the Fourier space reveals a fingerprint of how many LZSM transitions (up to a factor of 2) can develop before decohering completely. Therefore, the observable ovals reduce from Fig. 2d to Fig. 2h, and to Fig. 2j. Please refer to Supplementary Note 5 for a detailed data analysis.

LZSM interference of various charges

The JJ was then configured to a slightly stronger coupled regime (VG = −28.9 V) to release the tunneling of multiple charges and Cooper pairs. Figure 3a presents the interference fringes of the Cooper pairs, 2-charges through the 1st-order MARs and single charges, as marked by the square, circle, and triangle, respectively. The fringes of 3-charges through the 2nd-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 Fig. 3d–f, marked in correspondence with Fig. 3a, and their T2 values were extracted as 0.1 ns, 0.049 ns, and 0.051 ns, for Cooper pairs, 2-charges, and single charges, respectively.

Fig. 3: LZSM interference of various charges.
figure 3

a Interference fringes at VG = −28.9 V and f = 11.755 GHz. The square, circle and triangle depict the contributions of Cooper pairs, 2-charges through the 1st-order MARs, and single charges, respectively. b Zoom-in of (a). The dashed lines highlight the features of 3-charges through the 2nd-order MARs. c, Calculated dI/dV based on both the dI/dV versus Vb curve at P = −45 dBm in (a) and the T2 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 Jn2 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 VG = −27.5 V and f = 11.755 GHz.

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Taking the dI/dV versus Vb line-cut at P = −45 dBm in Fig. 3a as an input and using the T2 values, the conductance map can be calculated. For this we assume that dI/dV is proportional to W and its behavior follows Eq. (1). For example, if we consider the transport of single charges at Vb = 2Δ/e, the number of electrons transferred from the left side to the right side of the JJ in unit time is W; i.e., the conductance is proportional to W. 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 various charges.

We next discuss about three more characteristics of the interference fringes. First, the number, m, of charges can be conveniently determined from the spacing δVb of the satellite peaks, since δVb ∝ 1/m. Figure 3g plots dI/dV versus the Vb curves taken from Fig. 3a, b, as indicated by the black and red bars. The spacings δVb of 49 μV, 25 μV, 15 μV and 24 μV are consistent with charge numbers m of 1, 2, 3, and 2, at Vb = −2Δ/e, −2Δ/(2e), −2Δ/(3e) and 0, respectively.

Second, the dI/dV peaks behave as Jn2, versus the microwave amplitude VRF. 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 Jn2 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 VRF. Moreover, the Jn2 dependence serves as evidence for Cooper-pair tunneling as microscopic particles near Vb = 0, instead of the macroscopic Shapiro-step picture whose characteristic width would rather follow Jn to the 1st-order13,19, 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 (quasiparticles) when the coupling strength of the JJ increases. A clue can be found by examining the T2 extracted above. When increasing the coupling strength of the JJ from VG = −31 V to −28.9 V, T2 for single charges drops from 0.075 ns to 0.051 ns. At VG = −28.9 V, T2 = 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. 3i at VG = −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 VG = 6 V, and a good agreement was achieved by fitting the step width to the 1st-order of the Bessel function, |Jn|, as shown in Fig. 4.

Fig. 4: Shapiro steps in the strong-coupling regime.
figure 4

a Differential resistance dV/dI versus P and current I, measured at VG = 6 V and f = 11.56 GHz. The numbering indicates the Shapiro steps. b Step width ΔI versus VRF and the corresponding Bessel function fitting to the 1st-order. A − 45 dB attenuation was assumed and subtracted.

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We now comment on the advantages and drawbacks of the application of the LZSM model to Josephson junctions. (1) Advantages: as explained above, the LZSM model allows considering the tunneling of various charges from a different, but complementary, perspective; compared to the well-established microscopic theories. The involvement of coherence and decoherence is enabled in the quantum interference process. We can extract the coherence time of these various charges by the Fourier transform of the interference fringes and the Lorentzian fitting. It effectively includes the influence of various decoherence sources, such as tunnel coupling, thermal broadening, noise, etc. (2) Drawbacks: as we pointed out earlier, we treated the BCS singularity of the density of states as a δ-function; that is, the superconducting gap edge was modeled as a single level, and the continuous density of states outside the superconducting gap was omitted. The dependence of the coupling strength on bias voltage was also not taken into account. This leads to the missing of the background conductance curve which was studied in analytical microscopic theories. The LZSM model only focused on the conductance peaks and their evolution under microwaves to study the interference effect and the coherence/decoherence. As such, we restricted the analysis of the measured data inside the gap.

Conclusions

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 sates20, Floquet states52, as well as topological Majorana bound states53,54,55,56. 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

InAs0.92Sb0.08-Al nanowire growth

InAs0.92Sb0.08 nanowires were grown in a solid source molecular beam epitaxy system (VG V80H) on p-Si (111) substrates using Ag as catalysts57. 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 InAs0.92Sb0.08 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 cooling58. 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 ~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/SiO2 (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 cryo-free 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 ≡ Iac/Vac, with a small excitation ac voltage Vac and a dc bias voltage Vb applied and the ac current Iac 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.

Data analysis

The measured LZSM interference for the various charges in our small JJ manifests as sets of dI/dV fringes as a function of microwave power P and bias voltage Vb. However, P is the output of the signal generator, and the effective power Peff on the JJ needs to be determined. To do so, we calculated the interference fringes for a given measured data set using Eq. (1), and compare their power difference of the n = 0 peaks to extract the attenuation of the microwave, i.e., P–Peff. Afterwards, the measured interference fringes were selected and converted from power (Peff) dependence to amplitude (VRF) dependence. And the data set was further symmetrized to the four quadrants and scaled to a maximum of 1 to carry out the 2D FT. A detailed interpretation can be found in the Supplementary Note 5.