When a type-II superconductor is exposed to magnetic fields, vortices emerge as line defects where the order parameter vanishes, quantized magnetic flux penetrates the superconductor, and localized low-energy bound states form in the vortex cores. The nature of the vortex bound state is mysterious in many unconventional superconductors. Recently, much focus is on iron-based superconductors, where topologically nontrivial superconductivity and elusive Majorana bound states have been predicted to exist in vortex cores1.

So far, the most often reported signature for Majorana bound states is a peak in tunneling differential conductance at zero bias voltage. This signature is readily accessible by experiments, but it is not conclusive proof of the Majorana character of a state2,3,4,5. Other topologically trivial bound states, including Yu–Shiba–Rusinov (YSR) states, can also show the same zero-bias conductance peak, as demonstrated in proximitized superconducting nanowires6,7. Further, Caroli–de Gennes–Matricon (CdGM) states in the vortex cores are difficult to differentiate from Majorana bound states, because the former could also appear at zero energy4,8,9,10,11,12.

Zero-bias conductance peaks in full-flux-quantum vortex cores are also the main evidence for Majorana bound states in the iron-based superconductor FeTe0.55Se0.4513, which is the focus of this study. However, controversy remains as the absence of zero-energy bound states has been reported14,15. It is still being debated whether the additional observation16—a saturating conductance at roughly two-thirds of the expected quantized value 2e2/h—is a strong argument for the Majorana character (h is the Planck constant and e is the elementary charge). The issue is that such saturating behavior at an arbitrary conductance near 2e2/h has been observed for YSR states17 as well; these states are present in FeTe0.55Se0.45 and may also appear as a conductance peak at zero bias18. Furthermore, it was pointed out that the simple approximations of the Fu–Kane model are not likely applicable to the system of vortices in iron-based superconductors19, which brings the exact nature of the zero-energy vortex bound states in FeTe0.55Se0.45 into question.

New local probes are thus desired to investigate the electronic properties of the vortex bound states in iron-based superconductors. It has been widely investigated theoretically how shot noise could act as a tell-tale probe to distinguish between trivial and Majorana bound states in vortex matter and nanowires20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36, but experiments have not been possible. The principle behind most theoretical proposals is that Majorana bound states induce resonant Andreev reflection: an incident electron from the coupling lead, when tunneling into a Majorana bound state, is reflected as a hole with unity probability22. Such a resonant Andreev process is predicted to generate unique Majorana signatures that are absent for trivial fermionic states.

These theoretical studies form the motivation for the shot noise measurements on individual vortex cores that we present here. We measure both the vortex bound states in a conventional superconductor NbSe2 and the putative Majorana bound states in vortices of FeTe0.55Se0.45 at temperature T = 2.3 K. While we argue that our results do not represent a smoking gun experiment for the existence of Majorana bound states, they allow us to exclude YSR states as the origin of the zero-bias conductance peak, and they give an experimental insight into these bound states.

Shot noise is, at its core, a consequence of the discreteness of charge. Because of this, tunneling is a Poissonian process, and the noise spectral density S is proportional to the time-averaged current I,


Shot noise thus allows probing two quantities that are not visible in the time-averaged current: the effective charge q* of the charge carriers and possible correlations between them in electronic matters37. The former has been used to measure fractional charges in mesoscopic quantum hall systems38, and the latter has been used to measure the vanishing noise at the quantum conductance of break junctions39.

Despite a large number of theoretical studies on shot noise tunneling into vortex cores, there has been no experiment so far. The challenge is that one needs high enough sensitivity to measure the change in q* from noise, with nanometer resolution to locate individual vortices. This nanoscale resolution is not feasible in mesoscopic setups where noise measurements have been widely applied.

Recently, we have developed scanning tunneling noise microscopy (STNM), which combines scanning tunneling microscopy (STM) and noise spectroscopy, allowing us to measure the effective tunneling charge with atomic resolution40. To do so, we build a cryogenic megahertz amplifier that works in parallel with the usual dc measurements, as illustrated in Fig. 1a. STNM has revealed paired electrons in superconductors41,42. STNM also allows to measure shot noise exactly at the core of an individual vortex, which provides a direct and local extraction of the effective charge of the tunneling process into vortex bound states.

Fig. 1: Local tunneling shot noise measurements of vortex bound states.
figure 1

a Schematic illustration of the scanning tunneling noise microscope setup. A bias voltage (Vbias) is applied between the superconducting (SC) tip and sample, while the cylinder represents a SC vortex. If the tunneling process is a single-electron (gray) into vortex bound states (red arrow), an effective charge q* = 1e is transferred from the tip to the vortex. When Andreev reflection takes place (blue arrows), a hole (white) is reflected, and the effective charge doubles q* = 2e. HF and LF stand for the high- and low-frequency amplifier, respectively. STM, scanning tunneling microscope. Full flux quantum (h/2e) vortex lattice in NbSe2 (b) and FeSe0.55Te0.45 (c) revealed by spatially resolved differential conductance at a magnetic field of 0.1 T. Setup conditions: b, Vset = −5 mV, Iset = 200 pA; c, Vset = 10 mV, Iset = 250 pA.

Here, we measure two different materials: the iron-based superconductor FeTe0.55Se0.45, which is conjectured to host putative Majorana bound states, and the conventional superconductor 2H-NbSe2 as a comparison. We use a tip with an apex made out of Pb, which is a type-I, s-wave superconductor with a relatively large gap Δt ~ 1.3 meV. We choose to use a superconducting tip in this study for two reasons: first, a superconducting tip provides a superior energy resolution without the limitation from thermal broadening as in the case of a normal-metal tip, i.e. ~0.25 meV (See Supplementary Note 1) instead of ~3.5kBT = 0.70 meV, where kB is the Boltzman constant; second, as a consequence of a convolution with the density of states of the superconducting tip, the tunneling signal into a zero-energy vortex bound state is effectively shifted from the Fermi level to ±Δt (illustrated in Supplementary Fig. 2)6. This shift circumvents the challenge of measuring shot noise at zero bias voltage.


Vortex bound states in NbSe2 and FeTe0.55Se0.45

We first image the subgap electronic structure of the vortices in NbSe2. We introduce vortices by applying an external magnetic field B = 0.1 T perpendicular to the sample surface (the critical field of the tip Bc ~ 0.7 T, see Supplementary Fig. 1). Because vortices have the strongest enhancement in density of states at the Fermi level of the sample, they are visible as enhanced differential conductance at the energy |E | = Δt when using a superconducting tip (Supplementary Fig. 2a). Figure 1b shows a spatially resolved image of the differential conductance taken with a sample bias Vbias = −Δt/e, revealing the full-flux-quantum (h/2e) vortex lattice, with each vortex in the characteristic sixfold star shape for NbSe243,44. We then take differential conductance maps g(E,r) on a fine spatial grid around an individual vortex as shown in Fig. 2. Away from the vortex core, the spectrum in Fig. 2b shows an energy gap with a size of 2(Δt +Δs), where Δs = 1.0 meV is the superconducting gap of the sample. On the other hand, the spectrum measured at the core center develops two peaks at ±Δt, which translates to a zero-bias conductance peak for a spectrum taken with a normal-metal tip6. This translation is confirmed by a deconvolution procedure18,45 that extracts the local density of states of the sample (see Supplementary Note 2); as expected, the resulting density of states has a peak at zero energy (Fig. 2e). Deconvolution of spectra along a linecut through the vortex reveals that the zero-bias peak splits away from the core into two dispersing peaks, which eventually merge to the gap edges. These dispersing states are consistent with previous studies43 of NbSe2 and the expectations of many closely-spaced (on the order of 40 μeV) CdGM bound states from solving the Bogoliubov–de Gennes equations46, where the peak at a longer distance from the core center corresponds to a CdGM bound state with a larger angular momentum.

Fig. 2: Identifying vortex bound states in NbSe2 with a superconducting tip.
figure 2

a Spatially resolved differential conductance around an individual vortex at the energy E = +Δt, where Δt and Δs stand for the superconducting gaps of the tip and sample, respectively. b High-resolution differential conductance spectra acquired at the two locations marked by the crosses in a: the center of the vortex core (red) and off vortex (blue). The off-vortex location is 60 nm away from the core center, in the midpoint between two neighboring vortices of Fig. 1b. The red and blue dashed lines indicate the coherence peaks. c Differential conductance spectra along the dashed line (purple) in a showing the spatial dispersion of the vortex bound states. The gray dashed lines in b and c indicate the peaks at ±Δt where tunneling into the vortex bound states occurs at the core center. df Local density of states (DOS) plots corresponding to ac, after deconvolution using the tip DOS. The vortex bound states are indicated by the peak in the local DOS at around zero energy (gray dashed line). b, e and c, f share the same horizontal axes. Setup conditions: Vset = 5 mV, Iset = 200 pA.

In contrast to the dispersing CdGM bound states in vortex cores of NbSe2, a Majorana bound state is topologically protected such that its energy is locked at the Fermi level1. This is exactly what we observe, in agreement with the literature13,16, in tunneling differential conductance measurements on vortices of FeTe0.55Se0.45 (Fig. 3). The zero-bias conductance peak does not split (Fig. 3c, f) as in the case of NbSe2 (See Supplementary Note 7); instead, the non-split bound state extends ~8 nm spatially across the vortex core (Supplementary Fig. 2), identical to the states observed and interpreted as Majorana bound states in refs. 13,15,16. In principle, one expects a pair of peaks at ±Δt with an equal amplitude in the differential conductance spectrum when tunneling into a Majorana state47. We find, that the pairs of peaks ±Δt we observe at every spectrum on vortex are asymmetric (e.g., the red spectrum in Fig. 3b), which may indicate, the presence of accompanying states such as CdGM or YSR states, assuming a superconducting tip with a particle-hole symmetric density of states. We also note that the hybridization between Majorana bound states in a vortex lattice could also split the conductance peaks owing to the spatial overlap of Majorana wavefunctions. However, since the average distance between vortices in Fig. 1c is about 120 nm, the energy splitting for the putative Majorana bound states is on the order of 1 μeV48.

Fig. 3: Identifying the putative Majorana bound state in FeTe0.55Se0.45.
figure 3

a Spatially resolved differential conductance around an individual vortex at the energy E = +Δt, where Δt and Δs stand for the superconducting gaps of the tip and sample, respectively. b High-resolution differential conductance spectra acquired at the two locations marked by the crosses in a: the center of the vortex core (red) and off vortex (blue). The red and blue dashed lines indicate the coherence peaks. c Differential conductance spectra along the dashed line (purple) in a showing the spatial extent of the zero-energy bound state. The gray dashed lines in b and c indicate the peaks at ±Δt, where tunneling into the putative Majorana bound state occurs. df Local density of states (DOS) plots corresponding to ac, after deconvolution using the tip DOS. The putative Majorana bound state is indicated by the peak in the local DOS at zero energy (gray dashed line). b, e and c, f share the same horizontal axes. Setup conditions: a, c, Vset = 10 mV, Iset = 250 pA; b, Vset = 5 mV, Iset = 250 pA.

Before discussing shot-noise, we comment on what conventional conductance spectroscopy can contribute to distinguishing Majorana, CdGM, or YSR states. CdGM states are expected to be at finite energy instead of zero energy, but the energy difference can be small, and additional effects might shift the energy8. CdGM states have been observed in FeTe0.55Se0.45—surprisingly only in a subset of vortices13,14. In these vortices, the lowest energy levels have been reported as small as ~0.1 meV. From Lorentzian fits (Supplementary Fig. 2), our results show that the energy of the zero-bias peak is 0 ± 50 μeV, much smaller than the energy of the lowest-lying CdGM bound states reported. Furthermore, a previous high-energy-resolution study15 has shown that while some vortices show non-zero-bias peaks, associated with CdGM states, the majority (~80% at B = 1.0 T) show zero-bias peaks having an energy of 0 ± 20 μeV, which is evidence to exclude CdGM states. Based on these studies and the statistics therein, combined with our measured electronic structure, we deduce that the probability that all the vortices measured here only have CdGM states is less than 0.8%. Therefore, the zero-energy state we observe here is in agreement with the putative Majorana bound state previously reported—with the caveat that it has recently been shown that a CdGM state can imitate a zero-energy state10,12. We end this discussion by noting that the possibility of trivial YSR states, which can exist at zero bias18, has been investigated much less.

Effective charge inside and outside of vortex cores

The key advance of this study is high-sensitivity, atomic-scale noise spectroscopy that allows to extract the effective charge q* transferred when tunneling into vortex bound states. In the tunneling regime where the transparency is small, we parametrize any changes in noise, including the so-called Fano factor, via the effective charge q* in Eq. (1) (see Methods for different definitions of the Fano factor). For example, in the simplest case of electron tunneling, the transferred charge of each tunneling event is a single electron charge (q* = 1e), as expected from Poissonian statistics. In contrast, when Andreev reflection takes place such that an incident electron is reflected as a hole, two electron charges (q* = 2e) are effectively transferred per event. Since our measurements are performed at a finite temperature T, the current noise in the tunnel junction with resistance RJ consists of shot noise and thermal current noise 4kBT/RJ, and takes the form of37

$$S=2{q}^{*}({V}_{{{{{{\rm{bias}}}}}}}/{R}_{{{{{{\rm{J}}}}}}})\,{{\coth }}\,({q}^{*}{V}_{{{{{{\rm{bias}}}}}}}/2{k}_{{{{{{\rm{B}}}}}}}T).$$

This equation, which reduces to Eq. (1) at zero temperature, allows us to extract the effective charge q* as a function of bias voltage. Note that we keep RJ constant during noise spectroscopy by the changing tip-sample distance in a slow feedback loop (See Methods and Supplementary Note 6 for details).

We start by measuring current noise at B = 0.1 T at locations far away from the vortex cores. There, one expects the noise to correspond to an effective charge of q* = 1e at bias energies larger than the gap, |eVbias | > (Δt +Δs). At these energies, the tunneling of Bogoliubov quasiparticles dominates the noise. Around the gap energy ±(Δt +Δs), one then expects a step in noise from q* = 1e outside the gap, to q* = 2e inside. This is because, inside the gap, single-electron processes are not allowed anymore, and only Andreev processes contribute to the noise. As shown in Fig. 4, our measurements are in qualitative agreement with this picture, both in NbSe2 and FeTe0.55Se0.45. Outside the gap, our data follows the q* = 1e line, in agreement with Eq. (2). At the gap energy ±(Δt +Δs), a broadened step is visible in the extracted effective charge spectrum (Supplementary Fig. 5).

Fig. 4: Local noise spectroscopy on and off vortices in NbSe2 and FeTe0.55Se0.45.
figure 4

a, c Current noise spectra in the tunnel junction (with a resistance RJ = 2.5 MOhm) taken on (red) and off (blue) the vortex shown in Fig. 2a for NbSe2 and Fig. 3a for FeTe0.55Se0.45, respectively. The locations of these spectra are marked by the crosses in Figs. 2a and 3a with the same colors. Gray curves are the expected noise from Eq. 2 with an effective charge q* of 1e and 2e at T = 2.3 K. The dashed lines in a and c are replicated from Figs. 2b and 3b, respectively, serving as guides for the coherence peaks (red and blue) and the bound states (gray). The error bars are determined by the fluctuation of the current noise in time before each experiment, yielding a standard deviation of 9.25 and 6.77 fA2/Hz for a and c, respectively. b, d Effective charge q* derived by numerically solving Eq. 2 at the energy E = ±Δt on (red) and off (blue) vortex for three different vortices in NbSe2 and FeTe0.55Se0.45, respectively. The error bars are determined by the standard deviation of the extracted q* (Supplementary Figs. 35) in the energy ranges (Δt ± 0.1 meV) and -(Δt ± 0.1 meV).

Interestingly, q* does not reach 2e inside the gap, but saturates at a value of 1.3e ~ 1.6e (Fig. 4b, d), despite a vanishing conductance within ±(Δt +Δs) in Figs. 2b and 3b. This is in contrast to the measurement at B = 0 T on FeTe0.55Se0.45, where the extracted q* reaches 1.97e at ±Δt (see Supplementary Note 4). We hypnotize that the presence of the magnetic field leaves behind a small fraction of delocalized quasiparticles49, which allows charges of 1e to tunnel. Even a very small fraction of quasiparticles will decrease the noise substantially, because for a given tunneling transparency τ, the single-particle processes occur with probability τ, while the Andreev processes occur with probability τ2 (see Supplementary Note 5 for an estimation of fractions). Future experiments at different magnetic fields, and mapping the exact spatial dependence of current noise around vortex cores are necessary to test this hypothesis relating to the effective charge away from the vortex core. In this study, we focus on the noise spectra in the centers of individual vortex cores.

To investigate the tunneling process into vortex bound states, we then measure the current noise at the vortex cores, first for NbSe2. The experimental data in Fig. 4a show that noise in the core center follows the 1e-noise behavior until reaching well within ±Δt. We observe a transition from q* = 1e to q* > 1e around ±1.0 meV, within which Andreev reflection at the tip side starts to dominate (Supplementary Fig. 5a). Nevertheless, the q* remains at 1.05e at E = ±Δt, where tunneling into the CdGM bound states occurs (Fig. 4b). As a comparison, we then measured the shot noise of tunneling into the vortex bound states in FeTe0.55Se0.45. To our surprise, the behaviors of noise and effective charge (Fig. 4c, d) at the vortex cores of FeTe0.55Se0.45 are very similar to those of NbSe2, i.e., without any Andreev-reflection enhanced noise at E = ±Δt. We extract an effective charge q* = 0.99e into the zero-energy vortex bound states in FeTe0.55Se0.45, even closer to a single electron charge than that into CdGM bound states in NbSe2.


We proceed to our discussion of which states are compatible with our results from noise measurements for FeTe0.55Se0.45. We start with YSR states, which have been observed to cause the zero-bias peak and the saturating conductance, and are present in FeTe0.55Se0.4518. YSR states originate from the resonant coupling between a superconductor and a magnetic impurity. One can tune the coupling by a local gate, such as a voltage-biased STM tip, and the energy levels of the YSR states shift correspondingly to the local field felt by the impurity18. Thus, we expect to see spatially dispersing in-gap conductance peaks when moving the tip away from the impurity. The spatial extent of YSR states on FeTe0.55Se0.45 is about 8 nm (Supplementary Fig. 6), comparable to the size of a vortex. One way is to examine the tunneling process into YSR states, which is expected to be dominated by Andreev reflection in the strong tunneling limit17. Naively, the difference compared to the tunneling process into CdGM states can be explained by the different natures of the two states: CdGM states live in the vortex core that extends throughout the superconductor, so that the tunneled electron can leave the superconductor via the one-dimensional vortex core; YSR states localize on the surface of a superconductor around a magnetic impurity, so the tunneled electron cannot go through the superconductor but Andreev-reflected as a hole. Therefore, we expect shot noise with an effective charge of q* = 2e when tunneling into YSR states50.

To confirm this picture, we carry out tunneling conductance and noise measurements on the YSR states in FeTe0.55Se0.45 in the strong tunneling limit (see Supplementary Note 4). The YSR states appear as differential conductance peaks with a ring shape around an impurity site18. Our noise measurements (Supplementary Fig. 6) when tunneling into these YSR states show enhanced noise and q* ≈ 2e and are indistinctive from those of tunneling into the bare superconductor. With their stark contrast to the noise and effective charge of q* = 1e measured for vortex bound states. This leads to the first conclusion of our paper: we can exclude YSR states as the origin of the zero-bias conductance peak.

We then turn to the possibility of Majorana bound states as the origin of the zero-bias peak, as put forward by refs. 13,15,16. Theoretical calculations20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36 show that shot noise for tunneling into an isolated Majorana bound state vanishes for Majorana-induced Andreev reflection with unity probability, when the bias energy lies within the width of the Majorana bound state. For typical STM measurements where the bias energy is much larger than the intrinsic width of Majorana bound states16, the tunneling shot noise is Poissonian, i.e., q* = 1e. A second finding is thus that the shot-noise noise we measure is consistent with Majorana bound states.

However, we emphasize that our observation of the identical noise behavior and effective charge for CdGM bound states and the putative Majorana bound states clearly implies consistency with CdGM states as well, at least from a shot-noise point of view. The possible accompanying CdGM states10 leading to the asymmetry in our differential conductance spectra, would not change the noise behavior but still lead to an effective charge q* = 1e. No theoretical work has focused on tunneling processes into CdGM states. Thus, more theoretical and experimental studies are needed to understand the tunneling process into CdGM and Majorana bound states in vortices. Still, our work excludes YSR states, and therefore, taken together with high-resolution conductance measurements15, points towards Majorana modes as likely candidate for the zero-bias peak.

More generally, our work represents a step towards determining the exact nature of a zero-energy state, following theoretical work for Majorana bound states in vortex cores and nanowires. We have measured local shot noise when tunneling into vortex bound states in individual vortices of NbSe2 and FeTe0.55Se0.45. Using a superconducting tip, we demonstrate the feasibility of measuring shot noise even for states close to the Fermi level, which is usually overwhelmed by thermal noise. First, our data exclude YSR states as the origin of the zero-bias conductance peak at the vortex cores of FeTe0.55Se0.45. Second, while our data are in agreement with the theoretical prediction for Majorana bound states, we emphasize that we observe an identical shot noise behavior of topologically trivial CdGM bound states in NbSe2.

More theoretical work, especially including a superconducting tip, might allow to gain more information from shot-noise studies. Such a model for the tunneling process from a superconducting tip into a Majorana bound state has already been developed51, but only in the limit of low temperature. In the future, unambiguous identification of Majorana bound states by shot noise, might be possible in the low-bias limit where temperature and bias energy are both lower than the intrinsic width Γ of the zero-bias state, i.e. eVbias « Γ and kBT « Γ. This limit can be reached if shot noise measurements are enabled at milli-Kelvin temperatures. There, Majorana-induced resonant Andreev reflection leads to a vanishing shot noise because of unity transmission, which distinguishes itself from q* =1e when only CdGM states exist in the vortex core. A further proposal in the low-bias limit21 suggested a two-tip shot noise measurement setup on two different vortices: each tip tunnels into one localized vortex state, and a positive cross-correlated current noise is expected exclusively for Majorana states. One could further investigate the spin-resolved current-current correlation by a more sophisticated approach combining spin-polarized spectroscopy52,53 with shot-noise measurements54,55.


Different definitions of the Fano Factor

While the Fano factor was originally defined as the ratio between the variance and the mean value of a quantity, specific definitions vary in dealing with electrical current and its shot noise. An often applied definition of the Fano factor F is the ratio between the shot noise power S (precisely the Fourier transform of the current-current correlation function) and the Poisson noise SP due to independent single electrons37,


In some theory proposals21,22,27,29 for shot noise of Majorana bound states, one different definition appears where the Fano factor is expressed as

$$F=P/e| I |,$$

where P is the shot noise power (time-averaged current-current correlation function). Another definition of the Fano factor is expressed as the ratio between the differential noise power (the derivative of the time-averaged current-current correlation function with respect to the bias voltage) and the differential conductance27,28,

$$F={dP}/{dV}/(e\cdot {dI}/{dV}).$$

In the above definitions, however, the transmission of a single electron at a time is assumed. As a consequence, the correlation between them appears as sub- or super-Poissonian shot noise with F < 1 or F > 1, depending on the details of the transmission probabilities of the conducting channels. In this work, on the other hand, the charge transfer is the quantity of interest, and the STM junction is well in the single-channel, low-transmission regime (our highest tunnel conductance is 0.4 μS, yielding τ < 5.2 × 10−3). In this regime, we include the possible correlation between charge carriers in the effective charge, q* = S/2 | I | , or, more precisely, following Eq. (2).

Sample preparation and STM measurements

The FeTe0.55Se0.45 single crystals with a transition temperature TC = 14.5 K were grown using the Bridgman method. The 2H-NbSe2 samples (TC = 7.2 K) are purchased from HQ Graphene. The samples with a thickness of ~ 0.5 mm are cleaved in an ultrahigh vacuum at ~30 K and immediately inserted into a customized STM (USM-1500, Unisoku Co., Ltd). All measurements are performed in a cryogenic vacuum at a base temperature of T = 2.3 K. We perform scanning tunneling spectroscopy using standard lock-in techniques without the feedback loop enabled. A bias voltage modulation at a frequency of 887 Hz with an amplitude of 100 μV (for maps around vortex) or 50 μV (for high-resolution point spectra) is applied. The resulting differential conductance (dI/dV) values are normalized by setup conductance Iset/Vset. Prior to all the measurements, a Pt-Ir tip is made superconducting by indenting it into a clean Pb(111) surface. Our superconducting tip exhibits a critical field of about 0.7 T, deducted from differential conductance measurements in different magnetic fields on an atomically flat Au(111) surface (see Supplementary Note 1 for details).

Noise measurements

We perform noise spectroscopy at a constant junction resistance RJ in a slow feedback loop (see Supplementary Note 6 for details) when varying the bias voltage Vbias (and hence tunneling current I = Vbias/RJ) using our custom-built cryogenic megahertz amplifier developed recently. Because the Josephson tunnel junction with a low RJ may couple to its environment41,56, which affects the measured noise, we keep our RJ > 2.5 MOhm and Vbias > 0.2 mV, where the Andreev-reflection enhanced conductance at ±Δt and the environmental coupling effect is negligible. The amplifier consists of an LC tank circuit and a high-electron-mobility transistor that converts the current fluctuations in the junction into voltage fluctuations across a 50 Ohm line, as described in detail elsewhere40. To extract the effective charge transferred in the junction we follow a similar procedure as described in refs. 41,42.

The measured total voltage noise is


where G is the total gain calibrated by noise spectrum at a high bias (see Supplementary Fig. 9), and SI is the total current noise

$${S}_{I}=2{q}^{*}I\coth \left(\frac{{q}^{*}V}{2{k}_{{{\mbox{B}}}}T}\right)+\frac{4{k}_{{{\mbox{B}}}}T{R}_{{{\mbox{res}}}}}{{\left|{Z}_{{{\mbox{tot}}}}\right|}^{2}}+{S}_{{{\mbox{amp}}}}.$$

The first term is the junction noise from Eq. (2), the second is the thermal noise originating from the resistive part Rres of the LC tank circuit, and Samp is the intrinsic current noise of our amplifier.

As the first step of the procedure, we measure the background noise by retracting the tip out of tunneling (I = 0 so the first term vanishes), which gives 4kBTRres/|Zres | 2 + Samp, where Ztot = Zres because the junction is an open circuit. Then we measure noise in tunneling and subtract it by the background noise (for low RJ we also consider Ztot in the second term as Zres in parallel with RJ). Thus, the current noise data plotted in Fig. 4a, c consists only of the noise from the junction. Finally, we extract the q* value at each bias by numerically solving Eq. (2).