# Suppression of thermal smearing in a feedback-driven atomic size Josephson
junction

Samuel D. Escribano  Departamento de Física Teórica de la Materia Condensada,
Instituto Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC),
Universidad Autónoma de Madrid, E-28049 Madrid, Spain Department of Condensed
Matter Physics, Weizmann Institute of Science, Rehovot 7610, Israel  Víctor
Barrena  Laboratorio de Bajas Temperaturas, Departamento de Física de la
Materia Condensada, Instituto Nicolás Cabrera and Condensed Matter Physics
Center (IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma de Madrid,
E-28049 Madrid, Spain  David Perconte  Laboratorio de Bajas Temperaturas,
Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera and
Condensed Matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Institut Néel, Univ.
Grenoble Alpes, CNRS, Grenoble INP, 25 avenue des martyrs, Grenoble, 38000
France  Jose Antonio Moreno  Laboratorio de Bajas Temperaturas, Departamento
de Física de la Materia Condensada, Instituto Nicolás Cabrera and Condensed
Matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma
de Madrid, E-28049 Madrid, Spain  Marta Fernández Lomana  Laboratorio de Bajas
Temperaturas, Departamento de Física de la Materia Condensada, Instituto
Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Unidad Asociada
UAM-CSIC, Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Miguel Águeda
Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia
Condensada, Instituto Nicolás Cabrera and Condensed Matter Physics Center
(IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma de Madrid, E-28049
Madrid, Spain  Edwin Herrera  Laboratorio de Bajas Temperaturas, Departamento
de Física de la Materia Condensada, Instituto Nicolás Cabrera and Condensed
Matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma
de Madrid, E-28049 Madrid, Spain  Beilun Wu  Laboratorio de Bajas
Temperaturas, Departamento de Física de la Materia Condensada, Instituto
Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Unidad Asociada
UAM-CSIC, Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Jose Gabriel
Rodrigo  Laboratorio de Bajas Temperaturas, Departamento de Física de la
Materia Condensada, Instituto Nicolás Cabrera and Condensed Matter Physics
Center (IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma de Madrid,
E-28049 Madrid, Spain  Elsa Prada  Instituto de Ciencia de Materiales de
Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC), E-28049
Madrid, Spain  Isabel Guillamón  Laboratorio de Bajas Temperaturas,
Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera and
Condensed Matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Alfredo Levy Yeyati
Departamento de Física Teórica de la Materia Condensada, Instituto Nicolás
Cabrera and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de
Madrid, E-28049 Madrid, Spain Hermann Suderow  Laboratorio de Bajas
Temperaturas, Departamento de Física de la Materia Condensada, Instituto
Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Unidad Asociada
UAM-CSIC, Universidad Autónoma de Madrid, E-28049 Madrid, Spain

The ultimate spatial limit to establish a Josephson coupling between two
superconducting electrodes is an atomic-scale vacuum tunneling junction. The
Josephson effect in such ultrasmall junctions has been widely used to study
superconducting materials, providing a powerful probe of the spatial
variations of the superfluid density. Furthermore, studies of the supercurrent
and charging effects unveiled new, frequency dependent, switching dynamics and
thermal processes emerging due to the small size of the junctions. However,
the Cooper pair current magnitude was considerably reduced by thermal
fluctuations. Here we show that a feedback element induces a time-dependent
bistable regime which periodically restores the zero voltage state. We find
that the time-averaged current within the bistable regime is independent of
thermal smearing, opening new avenues to study superconducting phenomena at
the nanoscale.

## Main

Atomic-scale superconducting Josephson junctions are highly susceptible to
phase fluctuations due to their small size. For small typical junctions in Al
or Pb (with critical temperature Tc of 1.2 K and 7.2 K respectively), the
critical current $I_{\rm c}$ is often small, in the nA range, and the
Josephson coupling energy $E_{\rm J}=\frac{\Phi_{0}}{2\pi}I_{\rm c}$
($\Phi_{0}$ being the flux quantum) can fall down to the mK range. Thus, even
at temperatures well below Tc, there are significant thermal fluctuations of
the superconducting phase, which suppress the Josephson current.

To see how to address the problem of thermal fluctuations in Josephson
junctions, we start by discussing the standard resistively and capacitively
shunted junction (RCSJ) model, shown in Fig. 1a (black lines) [1]. The bias
current $I_{\rm b}$ can be separated through Kirchoff’s law into the sum of
the supercurrent through the junction $I_{\rm c}\sin(\varphi)$ (from the first
Josephson relation, $\varphi$ is the phase difference across the junction),
the current through a resistance $\frac{\Phi_{0}}{2\pi}I_{\rm
c}\frac{d\varphi}{dt}$ (following the second Josephson relation, a resistive
voltage drop $V$ at the junction corresponds to a current
$\frac{\Phi_{0}}{2\pi}I_{\rm c}R_{\rm S}\frac{d\varphi}{dt}$, with $R_{\rm S}$
the resistance), and the current through a capacitance $C_{\rm
J}\frac{dV}{dt}$ (given by $\frac{\Phi_{0}}{2\pi}I_{\rm c}R_{\rm S}C_{\rm
J}\frac{d^{2}\varphi}{dt^{2}}$. Thermal phase fluctuations induce a time
dependence of the phase difference $\varphi(t)$, whose influence in the
circuit can be reduced by designing the electromagnetic environment to obtain
large damping, for instance through a large $RC$ impedance [2, 3, 4, 5].
However, it is not always feasible to modify damping, for instance in
Josephson junctions made with a Scanning Tunneling Microscope (STM).

Scanning Josephson spectroscopy (SJS) probes superconducting samples using a
superconducting STM tip while recording the near-zero voltage current with
atomic precision. This requires a vacuum atomic size junction, which
essentially fixes $R_{\rm S}$ to the inverse of the vacuum impedance shuntimg
the circuit at high frequency, and $C_{\rm J}$ to the capacitance between tip
and sample. Even with reduced Josephson currents, SJS has contributed to the
understanding of pair density waves in cuprates and chalcogenides [6, 7],
inhomogeneous superconductivity in pnictides [8], photon assisted Cooper pair
tunneling [9], individual bound states tunneling [10], or a diode effect [11].
However, in these and other small scale Josephson junctions the avoidable
reduced Josephson current poses a considerable limitation for further
studies[12, 13].

A route to reduce thermal fluctuations is to use a delayed and time dependent
feedback [14, 15, 16]. To understand the effect of a feedback on a Josephson
junction, we recall that $\varphi$ behaves as the position of a particle in a
one dimensional potential $U$. We can write $U(\varphi)=\int
d\varphi\left(I_{\rm b}-I_{\rm c}\sin(\varphi)\right)$. $U$ has the form of a
tilted washboard as shown in Fig. 1b [17]. With a small $I_{\rm b}$ (orange
curve in Fig. 1b), the phase difference $\varphi$ remains around a potential
minimum, resulting in zero voltage across the junction. As the bias is
increased (blue and magenta curves), the phase rolls down the potential,
leading to a non-zero voltage. A feedback adds a time delayed current to the
actual Josephson current and periodically inhibits the evolution of the phase
with time by reducing the tilt, as schematically shown in Fig. 1a,b. As we
show below, thermal phase fluctuations are suppressed through a time dependent
oscillatory action on the circuit.

## Feedback driven Josephson junction

In Fig. 1c, we compare a time-averaged current-voltage characteristic for a
Josephson junction formed between a Pb STM tip and a Pb sample (blue line)
when a feedback element is integrated into the circuit, to the result obtained
without a feedback element (dashed blue line, the experiment is described
below and in Methods). We observe a regime with a sustained current in the
circuit with a feedback over a bias range that is considerably larger than in
the circuit without a bias. As we show in the upper right inset the voltage
fluctuates with time in this regime. Importantly, whereas the behavior close
to zero bias is highly temperature dependent (following approximately the same
behavior as in the circuit without a feedback), the current in the regime
induced by the feedback is almost independent of temperature.

To gain a deeper understanding, let us start by establishing the feedback RCSJ
model at zero temperature. We can write

$\frac{dx_{2}}{d\tau}=\frac{-\sin(x_{1})-x_{2}+i_{\rm
b}}{\beta}+F(\tau,\tau_{\rm D}),\\\ $ (1)

where $F(\tau,\tau_{\rm D})$ is given by

$F(\tau,\tau_{\rm
D})=-\frac{1}{2\beta}\int_{\tau}^{\tau+\Delta\tau}\frac{1}{\Delta\tau}\left[x_{2}(\tau^{\prime}-\tau_{\rm
D})-x_{2}(\tau^{\prime})\right]d\tau^{\prime}.\\\ $ (2)

Here, $x_{1}\equiv\varphi$ and $x_{2}\equiv\frac{d\varphi}{d\tau}$ are the
dimensionless variables, the reduced time is $\tau=\frac{t}{\tau_{\rm J}}$,
with $\tau_{\rm J}=\frac{\Phi_{0}}{2\pi I_{\rm c}R_{\rm S}}$ the
characteristic Josephson time (and $\omega_{\rm J}=\frac{2\pi}{\tau_{\rm J}}$
the Josephson frequency), the reduced bias current is $i_{\rm b}\equiv I_{\rm
b}/I_{\rm c}$, and $\beta=\frac{2\pi I_{\rm c}R_{\rm S}^{2}C_{\rm
J}}{\Phi_{0}}$ is the McCumber parameter. Eq. 1 corresponds to the
conventional RCSJ model plus the feedback term $F(\tau,\tau_{\rm D})$.
$\Delta\tau$ is the bandwidth of the measurement circuit. The behavior of the
solutions to the new coupled equations $\frac{dx_{1}}{d\tau}=x_{2}$ and Eq.
(1) is shown in Fig. 2.

In Fig. 2 we show the evolution of voltage $x_{2}$ as function of the bias
current. We identify three distinct regimes. For $i_{\rm b}\leq 1$ (orange in
Fig. 2a and Fig. 2b) we find zero voltage, $x_{2}=0$, and a finite Josephson
current $i=i_{\rm b}$. When $i_{\rm b}\simeq 1$, $x_{2}(\tau)\neq 0$ and
$x_{2}(\tau-\tau_{\rm D})=0$. The feedback reacts and there is an increase of
$x_{2}(\tau)$ up to $\left<x_{2}\right>=2i_{\rm b}$. The current through the
junction first increases and then oscillates at a time scale which is of order
of $\tau_{\rm J}$. After a time $\tau_{\rm D}$, $x_{2}(\tau-\tau_{\rm D})$
becomes $2i_{\rm b}$ too, which drives $x_{2}(\tau)$ down to $0$ through the
feedback $F(\tau,\tau_{\rm D})$ (blue in Fig. 2a and Fig. 2c). The reaction of
the feedback is immediate for $i_{\rm b}\simeq 1$ but requires more time when
$i_{\rm b}$ increases. As an example we show the evolution of the voltage
$x_{2}$ and the current $i$ with time in Fig. 2c over one period of the
feedback approximately in the middle of the bistable regime. We see an initial
increase in $i$ with $x_{2}\approx 0$ and then an increase of $x_{2}$ to
$x_{2}=2i_{\rm b}$. We show the time dependence of $x_{2}$ and of
$F(\tau,\tau_{\rm D})$ in Fig. 2e. The establishment of the feedback (periods
with a finite slope in Fig. 2e) gradually takes over more time when $i_{\rm
b}$ increases. Ultimately, when the required time reaches $\tau_{\rm D}$ we
lose the bistable regime and find the conventional AC periodic Josephson
regime, characterized by a finite time-averaged voltage $x_{2}=i_{\rm b}$ and
a zero time-averaged current (blue in Fig. 2a and Fig. 2d). When decreasing
$i_{\rm b}$ again, $x_{2}(\tau-\tau_{\rm D})\neq 0$, for which the feedback
$F(\tau,\tau_{\rm D})$ (Eq. 2) remains low over a certain range of $i_{\rm
b}$, so that a smaller $i_{\rm b}$ is needed to reach the bistable regime,
obtaining a hysteretic behavior.

We note that the time-averaged current just at the point where the bistable
regime is lost, is propotional to $\frac{\Delta\tau}{\tau_{\rm D}}$. At this
point, the time dependence of $x_{2}$ (Fig. 2e) is close to a triangular wave,
such that there is indeed just a single unit in time
$\frac{\Delta\tau}{\tau_{\rm D}}$ for which $i$ is not oscillating and does
not average to zero. As we show below, this point is asymptotically approached
when increasing $i_{\rm b}$ within the bistable regime, giving a time-averaged
current which is independent of thermal smearing.

We now compare theory and experiment at a finite temperature. In Fig. 3a, we
show a time-averaged current-voltage characteristic taken on an atomic size
tunnel junction between a tip and a sample of Pb ($T_{\rm c}^{\rm(Pb)}=7.2$ K
and $\Delta_{\rm Pb}=1.37$ meV, $\tau_{\rm J}\approx 10^{-9}$ s. Starting from
zero bias, we observe that the average current increases, reaches a peak, and
then decreases and saturates at a plateau, eventually dropping to zero at
higher bias voltages. The point at which the current drops to zero is
hysteretic and depends on the direction of the bias voltage ramp (light and
dark blue lines show the different ramping directions). The time dependence of
the voltage is shown in Fig. 3b and the minimum and maximum values in the
inset of Fig. 3a (we use $\Delta\tau\sim 10^{-5}$ s for the temporal range or
bandwidth of the experimental set-up, see also Methods). We observe a bistable
regime (orange to blue curves) where the junction switches periodically,
approximately every $0.15$ ms, between zero and a finite voltage. In Fig. 3c
we show the time-averaged current vs voltage resulting from our simulations.
The voltage vs time is shown in Fig. 3d and the minimum and maximum values in
the inset of Fig. 3c. We see that experiment and calculations are mostly
identical. The shape of the time-averaged current to voltage curve is similar,
the hysteresis has a similar shape and the oscillatory behavior in the
bistable regime occurs in experiment as well as in simulations.

Increasing the temperature reduces both the bistable range as well as the size
of the hysteretic behavior, but leads, as shown in Extended Data Fig. 1, to a
temperature independent value of the time-averaged current in the bistable
regime.

A detailed analysis of the different parameters governing the feedback RCSJ
model (Extended Data Fig. 2 and Methods) shows that the appearance of the
bistable regime for different time constants $\tau_{\rm D}$ goes all the way
down to $\tau_{\rm D}$ being one or two orders of magnitude larger than
$\tau_{\rm J}$. Furthermore, there is a large range of $\beta>1$ where the
bistable regime can be observed. On the other hand, $\Delta\tau$ should be
smaller than $\tau_{\rm D}$, but below $\Delta\tau\approx 0.1\tau_{\rm D}$,
the integration does no longer provide a sufficiently large feedback term. In
the experiment, we have observed the bistable regime in Pb-NbSe2 and in Al-Al
atomic size junctions (Extended Data Figs. 3,4 and Supplementary Information).
We discuss the dependence on the circuit parameters in Methods (circuit is
shown in Extended Data Fig. 5). The switching behavior between different
regimes is thermally driven, as shown in the Extended Data Fig. 6 and
discussed in the Supplementary Information.

Note that there is a discrepancy between the symmetric peaks obtained in the
voltage as a function of time in the bistable regime of our simulations, and
the asymmetric peaks observed in the experiment (yellow and green curves in
Fig. 3b and Fig. 3d). We do not have a clear explanation for this discrepancy,
but it might be related to asymmetries in the circuit. For instance, different
resistance and capacitance values in the two branches connecting the junction
(Fig. 1c), which are not included in our feedback RCSJ model. The other
essential features, such as the existence of a bistable regime and its
accompanying hysteretic behavior, are accurately captured by the feedback RCSJ
model.

## Critical current from the bistable regime in feedback driven ultra small
Josephson junctions

In summary, we have unveiled a novel behavior in ultrasmall Josephson
junctions when measured with a circuit that incorporates a delayed feedback
element. This component can induce bistable oscillations between two distinct
states: the conventional one characterized by zero voltage and a finite
Josephson current, and a non-zero voltage state. Notice that the observed
delay time $\tau_{\rm D}$ in the experiments is dictated by the time constant
of the feedback component while $\Delta\tau$ is provided by the bandwidth of
the experimental set-up.

From the feedback RCSJ model at a finite temperature, we find that
$\left<I\right>=I_{\rm c}\frac{\Delta\tau}{4\tau_{\rm D}}$ in the bistable
regime, just before the transition to the periodic regime. This is in sharp
contrast to the value of the current close to zero voltage, for $i_{\rm
b}\approx 1$, which decreases with increasing temperature as $\sim T^{-1}$, or
with the derivative of the current near zero voltage, which decreases with
temperature as $\sim T^{-2}$ (see Supplementary Information). For the
experiment shown in Fig. 3a, we find $I_{\rm c}\sim 50$ nA, which is very
similar to the estimation provided by the Ambegaokar-Baratoff equation $I_{\rm
c}=\frac{G}{G_{0}}\frac{e}{2\hbar}\Delta$ for symmetric junctions at zero
temperature [18] ($G$ being the conductance and $G_{0}$ the quantum of
conductance, we use $G\approx 0.4G_{0}$ and $\Delta_{\rm Pb}=1.37$ meV). By
taking $\Delta\tau=0.06\tau_{\rm D}$ and $I_{\rm c}\sim 50$ nA we can follow
closely the dependence of $I_{\rm c}$ on $G$, as shown in Fig. 4, finding that
the experiment (dots) follows the calculations (line). Thus, the measurement
of the time-averaged current in the bistable regime provides a quantitative
measurement of the critical current $I_{\rm c}$ of the junction.

The bistable regime is found in junctions with different superconductors, as
shown in the Supplementary Information. It is worth pointing out that
performing SJS in the bistable regime can be particularly interesting when
measuring a topological superconductor [19, 20, 21, 22, 23, 24, 25, 26, 27] or
a chain of magnetic adatoms [28, 29]. In such systems, the phase coherent
tunneling between the s-wave state of the tip and p-wave state of the sample
occurs through higher-order processes [26], which considerably reduces the
zero-bias Josephson current. Our method enables the sampling of the (near-
zero) Josephson effect over a larger range of voltages, and allows to better
probe the spatial distribution of the topologically protected states. The
implications of our findings extend further, offering new possibilities for
the measurement of small-size Josephson junctions through the detection of a
signal which is mostly free of thermal broadening.

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Figure 1:  | Feedback-driven resistively and capacitively shunted junction (RCSJ) model. a Schematic representation of the RCSJ model with feedback. The resistance $R_{\rm S}$, capacitance $C_{\rm J}$ and the Josephson junction (J) are enclosed within the grey dashed rectangle. The feedback subtracts the voltage after a delay from the actual voltage and the result is added to the circuit. b The RCSJ model is often compared to the motion of a particle (black dots) in a periodic potential $U$ (lines) along a coordinate which represents the phase difference across the Josephson junction $\varphi$. At low bias (orange), thermal excitation induces phase fluctuations (grey dashed arrow, the black dots schematically represents the value of $\varphi$ at a certain time). By applying a bias to the junction, the periodic potential $U$ is tilted downwards, as illustrated by the blue and magenta lines, causing the particle to slip along the potential. Incorporating a feedback mechanism with a time constant $\tau_{\rm D}$ periodically reduces the tilt in $U$ (violet dashed line). c Time-averaged current $\left<I\right>$ vs voltage $\left<V\right>$ measured in a junction between a Pb tip and a Pb sample at different temperatures: $T=0.15$ K (solid blue), $T=1$ K (cyan), $T=2$ K (orange) and $T=3$ K (red). Tunneling conductance is $G\approx 0.4G_{0}$, being $G_{0}$ the conductance quantum. Further details of the experiment are given in Methods. These curves are a zoom around zero voltage of the $I-V$ curve shown in the upper central inset. Dashed blue line are data taken without a feedback (see also Methods), at $T\approx 0.15$ K and $G\approx 0.4G_{0}$. The upper right inset shows the time-dependent trace of the voltage at the point in the current-voltage curve marked by a blue disk in the main panel. We show, for clarity, only curves obtained by ramping up the bias. Figure 2:  | Feedback driven RCSJ regimes. a The minimum and maximum values of $x_{2}$ (proportional to the junction voltage) as a function of the reduced bias current $i_{\rm b}=I_{\rm b}/I_{\rm c}$. We identify three different regimes, shadowed with orange, blue, and purple backgrounds. b-d Evolution of the current [$i=\sin(x_{1})$] and the voltage ($x_{2}$) through the junction at different times. The color in each panel represents schematically the regime, following the color of the dots in a. In b the junction quickly reaches a stable state with a time-independent finite current and zero voltage (stable regime). In c the junction leaves the zero voltage state and switches between a zero and a finite voltage state (bistable regime). In d both the current and the voltage oscillate at frequency $\frac{2\pi}{\tau_{\rm J}}$ (periodic regime), but the current averages to zero while the voltage averages to a finite value. e $x_{2}$ vs time (blue line) together with the feedback term multiplied by $2\beta$ (black line), see Eq. (1). Parameters in these simulations are $T=0$, $\beta=2$, $\tau_{\rm D}=10^{5}$ and $\Delta\tau=6\cdot 10^{3}$ (both $\tau_{\rm D}$ and $\Delta\tau$ in units of $\tau_{\rm J}$). Figure 3:  | Feedback driven Josephson effect. a Time-averaged tunneling current vs bias voltage near zero bias (solid blue lines) for a STM Josephson junction made between two Pb electrodes. The blue arrows indicate the direction of the ramp. Colored dots represent the locations at which we stop the ramp and make time-dependent measurements, shown in b. The junction conductance $G$ is $\sim$0.4$G_{0}$, being $G_{0}=\frac{2e^{2}}{h}$ the conductance quantum and $T=$150 mK. The bottom right inset shows the maximum and minimum of the voltage vs $I_{\rm b}$. b Voltage vs time at the colored dots marked in the main panel of a. The curves are vertically shifted by 0.25 mV for clarity. c Time-averaged current vs voltage obtained from the model described in the text. The arrows provide the direction of the ramp. The bottom right inset shows the maximum and minimum of the voltage vs $I_{\rm b}$. d Voltage as a function of time for the colored dots marked in c. Curves are shifted for clarity by 0.5 mV. Note that $\tau$ is in units of $\tau_{\rm J}\approx 10^{-9}$ s, see text and Methods for further details. Parameters in the simulations are $T=1.2$ K, $\beta=2$, $\tau_{\rm D}=10^{5}$ and $\Delta\tau=6\cdot 10^{3}$ (both $\tau_{\rm D}$ and $\Delta\tau$ in units of $\tau_{\rm J}$). Figure 4:  | Current in the bistable regime vs conductance. Time-averaged tunneling current $\left<I\right>_{bi-stable}$ just before the loss of the bistable regime as a function of the conductance $G$ normalized to the conductance quantum $G_{0}$, for STM Josephson junctions made with tip and sample of Pb. The colored circles correspond to the experimental results whereas the line corresponds to the expression $I_{\rm c}\frac{\Delta\tau}{4\tau_{\rm D}}$. $I_{\rm c}$ is determined as a function of $G$ following the Ambegaokar-Baratoff theory [18]. In the inset we show the time-averaged tunneling current $\left<I\right>$ vs the bias voltage $\left<V\right>$. Each line corresponds to a different tunneling conductance, following the color code of the circles in the main panel [$G$: $G=0.61G_{0}$ (violet), 0.50$G_{0}$ (blue), 0.43$G_{0}$ (light blue), 0.37$G_{0}$ (cyan), 0.31$G_{0}$ (light cyan), 0.27$G_{0}$ (green), 0.22$G_{0}$ (light green), and 0.18$G_{0}$ (yellow)]. Dashed lines in the inset are for decreasing bias.

## Methods

### Temperature dependence in the feedback-driven RCSJ model

We model the experiments through a feedback-driven RCSJ model following Refs.
[1, 2]. By using the Kirchhoff equations on the circuit of Fig. 1b and
introducing the feedback term and thermal noise, we obtain the set of
differential equations

$\displaystyle V$ $\displaystyle=\frac{\Phi_{0}}{2\pi}\frac{d\varphi}{dt},$
(3) $\displaystyle I_{\rm b}-k\frac{1}{\Delta t}\int_{t}^{t+\Delta
t}\left[V(t^{\prime}-t_{\rm D})-V(t^{\prime})\right]dt^{\prime}$
$\displaystyle=$ $\displaystyle I_{\rm c}\sin{\varphi}+\frac{V}{R_{\rm
S}}+C_{\rm J}\frac{dV}{dt}-\sqrt{\frac{2k_{\rm B}T}{R_{\rm S}h_{t}}}$
$\displaystyle f_{\rm RND}(t).$ (4)

The first equation corresponds to the voltage drop through the circuit while
the second one accounts for the total current. The first term in the second
equation is the current bias $I_{\rm b}$ and the second one is the feedback
term. This term compares the voltage at two different times, $t$ and $t-t_{\rm
D}$, and adds to the circuit, multiplied by a coupling constant $k$ that has
units of inverse resistance. Since the feedback element may not allow for a
full resolution of $\tau_{\rm J}$, we introduce an average for time-scales
$\Delta t>\tau_{\rm J}$. The third term gives the current through the
Josephson junction, with $I_{\rm c}$ the critical current and $\varphi$ the
phase difference. The fourth and fifth terms provide the currents through the
resistance $R_{\rm S}$ and the capacitor $C_{\rm J}$, respectively. The last
term simulates the current fluctuations in the junction as a result of thermal
noise, with $T$ the temperature and $f_{\rm RND}(t)$ a time-dependent function
that provides normally distributed random values that average to zero with
standard deviation of one. This term is modeled following Ref. [2], which
offers an expression especially suitable for Runge-Kutta solvers ($h_{t}$ is
the time discretization).

We can now rewrite the set of first-order differential equations of Eqs. (3)
and (4) as a (dimensionless) second-order differential equation

$\frac{d^{2}\varphi}{d\tau^{2}}=\frac{i_{\rm
b}-\sin(\varphi)-\frac{d\varphi}{d\tau}}{\beta}\\\ -\frac{kR_{\rm
S}}{\beta}\frac{1}{\Delta\tau}\int_{\tau}^{\tau+\Delta\tau}\left(\frac{d\varphi}{d\tau}(\tau^{\prime}-\tau_{\rm
D})-\frac{d\varphi}{d\tau}(\tau^{\prime})\right)d\tau^{\prime}\\\
+\frac{1}{\beta}\sqrt{\frac{2\pi}{I_{\rm c}\Phi_{0}}}\sqrt{\frac{2k_{\rm
B}T}{h_{\tau}}}f_{\rm RND}(\tau),$ (5)

where $\tau=t/\tau_{\rm J}$ is the reduced time (with $\tau_{\rm
J}=\frac{\Phi_{0}}{2\pi I_{\rm c}R_{\rm S}}$ the inverse of the characteristic
Josephson frequency), $\beta=\frac{2\pi I_{\rm c}R_{\rm S}^{2}C_{\rm
J}}{\Phi_{0}}$ the McCumber parameter, and $i_{\rm b}\equiv I_{\rm b}/I_{\rm
c}$ the reduced bias current. Notice that $\tau_{\rm D}=t_{\rm D}/\tau_{\rm
J}$ and $\Delta\tau=\Delta t/\tau_{\rm J}$ are also written in units of
$\tau_{\rm J}$.

Defining $x_{1}=\varphi$ and $x_{2}=\frac{d\varphi}{dt}$, we can write

$\frac{dx_{2}}{d\tau}=\frac{i_{\rm b}-\sin(x_{1})-x_{2}}{\beta}\\\
-\frac{kR_{\rm
S}}{\beta}\frac{1}{\Delta\tau}\int_{\tau}^{\tau+\Delta\tau}\left[x_{2}(t^{\prime}-\tau_{\rm
D})-x_{2}(t^{\prime})\right]dt^{\prime}\\\
+\frac{1}{\beta}\sqrt{\frac{2\pi}{I_{\rm c}\Phi_{0}}}\sqrt{\frac{2k_{\rm
B}T}{h_{\tau}}}f_{\rm RND}(\tau).$ (6)

This equation is equivalent to Eq. (1) of the main text but including the
effect of a finite temperature through a white noise term. Moreover, we assume
in the main text perfect coupling between the feedback element and the
circuit, provided by $kR_{\rm S}=0.5$, in which the feedback perfectly
compensates the voltage drop through the circuit (at $\Delta\tau\rightarrow
0$).

We solve this non-linear delayed differential equation using a fourth-order
Runge-Kutta method as a function of time and sweeping for different bias
current $i_{\rm b}$. For each current bias $i_{\rm b}$, we impose as initial
condition the values of $(x_{1},x_{2})$ found for the previous value of
$i_{\rm b}$. In this way we obtain a hysteretic behavior as also found in the
experiments. We employ a small time discretization step $h_{\tau}$ so that the
time scale $\tau_{\rm J}$ is well-resolved ($h_{\tau}=0.1\tau_{\rm J}$) and we
compute the solution for very large time scales so that one can account for
the phenomena happening at scales of $\tau_{\rm D}$. Notice that there are
five orders of magnitude between $\tau_{\rm J}$ and $\tau_{\rm D}$, which
makes these simulations computationally challenging.

Once $x_{1}(\tau)$ and $x_{2}(\tau)$ are found we compute the current and
voltage through the Josephson junction as

$I(\tau)=I_{\rm c}\sin\left(x_{1}(\tau)\right),\;\;\;V(\tau)=R_{\rm S}I_{\rm
c}x_{2}(\tau),$ (7)

and find their time average $\left<I\right>$ and $\left<V\right>$. We remove
the initial solutions to the differential equation in the time average (a few
$\tau_{\rm D}$ intervals) as the simulations may take some time to reach a
possible different regime when changing $i_{\rm b}$. We also run several times
(10 times) the same simulation using the same $i_{\rm b}$ and initial
condition, and average among the solutions to remove the dependence on a
particular realization of thermal noise.

### Parameters of the feedback-driven RCSJ model

To explore the impact of different parameters of the feedback on the results
of the model we calculate the time-averaged current $\left<I\right>$ for
different bias currents $i_{\rm b}$ and parameters: $kR_{\rm S}$ (shown in
Extended Data Figure 2a, $\left<I\right>$ is given by the color scale),
$\Delta\tau$ (Extended Data Figure 2b), $\beta$ (Extended Data Figure 2c) and
$\tau_{\rm D}$ (Extended Data Figure 2d) at zero temperature. In the stable
regime, $i_{\rm b}\leq 1$ (Extended Data Figure 2a), the coupling constant has
no significant influence since the feedback does not play a significant role.
However, for $i_{\rm b}>1$ we observe a substantial bistable regime that
appears within the range of approximately $kR_{\rm S}\simeq 0.4$ to $kR_{\rm
S}=0.5$, as indicated by the red area in Extended Data Figure 2a (red
indicates a finite $\left<I\right>$, see color scale on the right). The saddle
points of the differential equation determine the bistablility regime, which
follows in this case $i_{\rm b}^{\rm(max)}\propto\left(\frac{1-kR_{\rm
S}}{1-2kR_{\rm S}}\right)$. There is a divergence at $kR_{\rm S}=0.5$, which
serves as the upper boundary for the bistablility range. Conversely, the lower
boundary for $kR_{\rm S}$ is when $i_{\rm b}^{\rm(max)}=1$. Below this value
the range for the bistable regime is infinitesimally close to the stable
regime. In the present simulation this occurs at approximately $kR_{\rm
S}\simeq 0.4$.

The effect of the integration interval $\Delta\tau$ is very similar to that of
$kR_{\rm S}$ as it ultimately modifies the coupling between $x_{2}(\tau)$ and
an average near $x_{2}(\tau-\tau_{\rm D})$. Thus, even at perfect coupling
$kR_{\rm S}=0.5$, see Extended Data Figure 2b, the bistability is possible
when $\Delta\tau$ is smaller than the characteristic delay time $\tau_{\rm
D}$, as $x_{2}(\tau-\tau_{\rm D})$ is averaged to a very different quantity.
Notice, nevertheless, that the bistable regime disappears when
$\Delta\tau\sim\tau_{\rm D}$, as the feedback element averages to a time-
independent constant.

The relationship between the bistable regime and the McCumber constant $\beta$
(Extended Data Figure 2c) is more intricate. In principle, one might argue
that the saddle points of the differential equation are independent of
$\beta$, suggesting that the bistablility range should remain unaffected by
it. However, the value of $\beta$ does influence the rate at which the
differential equation converges to the saddle points, which in turn influences
the extension in parameter space of the bistable regime.

To better understand this, let us explain in more detail the evolution of the
system with time (for $\Delta\tau\rightarrow 0$). As discussed in the main
text, the system undergoes a periodic evolution (with period $2\tau_{\rm D}$)
in the bistable regime, oscillating between two distinct saddle points.
Initially, the system reaches a periodic fixed point that persists for a time
$\tau_{\rm D}$. This fixed point can be linked to an AC Josephson regime but
with an average voltage that is renormalized to $\left<x_{2}\right>=i_{\rm
b}/(1-kR_{\rm S})$ instead of the conventional $i_{\rm b}$. After this time
$\tau_{\rm D}$, the action of the feedback element changes, leading the system
to transition towards a different fixed point for an additional period
$\tau_{\rm D}$. This latter saddle point is stable and exhibits similarities
to a DC Josephson regime albeit with a renormalized current given by
$x_{1}=i_{\rm b}\left(\frac{1-kR_{\rm S}}{1-2kR_{\rm S}}\right)$. Notably, the
transition between these two saddle points occurs continuously in time. The
system passes through an intermediary saddle point during this evolution. This
intermediate point corresponds to the conventional AC Josephson regime (i.e.,
with $\left<x_{2}\right>=i_{\rm b}$), but it only serves as a fixed point for
a (brief) period of time when the condition $x_{2}(t-\tau_{\rm D})\simeq
x_{2}(t)$ is (approximately) satisfied. As the bias current $i_{\rm b}$ is
increased the transition between the two primary fixed points is slower until
reaching a critical value $i_{\rm b}^{\rm(max)}$ beyond which the evolution
becomes so sluggish that the system remains solely in the conventional AC
regime.

The value $i_{\rm b}^{\rm(max)}$ is also influenced by $k$, $\Delta\tau$ and
$\beta$ since the rate of the transition is dependent on these parameters as
well. In fact, one can analytically show that the eigenvalues of the Jacobian
of the linearized differential equations near the saddle points exhibit a
dependence on $\beta$, and specifically the real part scales as $\sim
1/\beta$. This implies an exponential decay of the eigenstates towards the
saddle points with a rate of $\sim 1/\beta$, and so does $i_{\rm
b}^{\rm(max)}$. Our numerical simulations, see Extended Data Figure 2c,
confirm this intuition by showing that the extent of the bistable regime (red
area) decreases approximately like $\sim 1/\beta$. Note however that $\beta=0$
is a especial case in which there is no bistable regime.

We finally explore the dependence of the bistable regime with the delay time
$\tau_{\rm D}$ in Extended Data Figure 2d. We observe that the value of
$\tau_{\rm D}$ does not influence the bistable regime except when $\tau_{\rm
D}$ is comparable to $\tau_{\rm J}$, i.e., $\tau_{\rm D}\lesssim 10^{2}$,
where the behavior is more complex. This is because the evolution between the
stable points becomes slower than $\tau_{\rm D}$ itself, thereby disrupting
the bistablility regime.

### Feedback driven RCSJ model and experiment as a function of temperature

We show the dependence of the current-voltage $\left<I\right>-\left<V\right>$
characteristic for different temperatures in Extended Data Figure 1. We
observe that temperature fluctuations suppress the value of the current at the
peak (the peak reaches $\sim 50$ nA at $T\rightarrow 0$). To a lesser degree,
the extension of the bistable regime is also decreased with temperature and
may actually disappear for too high temperatures. On the contrary, the time-
averaged current $\left<I\right>$ in the bistable regime is barely affected by
temperature. To understand this behavior, we note that at $T\rightarrow 0$ the
entire stable regime happens just at $\left<V\right>=0$, while the bistable
regime happens for a wide range of $\left<V\right>$. Temperature mixes the
regimes close to their transitions, thus suppressing the peak at
$\left<V\right>\rightarrow 0$ (and moving it to larger $\left<V\right>$
values) as it hybridizes with the bistable regime. On the other hand,
$\left<I\right>$ in the central part of the bistable regime is barely affected
by temperature, as it hybridizes with other $\left<V\right>$ values inside the
same bistable regime. This also implies that whenever the thermal energy is
comparable to or larger than the voltage range of the bistable regime, no
bistability is possible. Analytically, we find that the time-averaged current
within the bistable regime, just prior to transitioning to the AC Josephson
regime, is roughly given by

$\left<I\right>_{\rm bistable}\simeq I_{\rm c}\left(\frac{1-2kR_{\rm
S}}{1-kR_{\rm S}}+\frac{\Delta\tau}{2\tau_{\rm D}}\frac{\left(kR_{\rm
S}\right)^{2}}{1-kR_{\rm S}}\right).$ (8)

We have checked that this analytical expression agrees with our numerical data
either when $\Delta\tau$ is negligible, and thus the first term dominates, or
when $\Delta\tau$ is comparable to $\tau_{\rm D}$ so that the second one does.
For intermediate values, the behaviour of the system is more complex than the
one this analytical equation can predict. Notably, this equation aligns well
with the experimental data, as in our experimental setup $\Delta\tau$ is
comparable to $\tau_{\rm D}$. Particularly, assuming perfect coupling $kR_{\rm
S}=0.5$ and $\Delta\tau\simeq 0.05\tau_{\rm D}$, this equation allows us to
estimate a critical current $I_{\rm c}$ of 50 nA for the junction discussed in
the Fig. 3a of the main text, in agreement with Ambegaokar-Baratoff theory.
Further support for this expression is demonstrated in Fig. 4 of the main
text, where the current within the bistable regime obtained through
experimental measurements closely follows the theoretical prediction across a
wide range of tunneling conductances.

### Measurement of the Josephson current with STM

We use a STM set-up in a dilution refrigerator with superconducting tip and
sample. The tips are always prepared and cleaned through repeated indentation
inside a pad of the same material as the tip. As described in Refs. [3, 4], we
indent the tip in a controlled way and follow the conductance steps
characteristic of atomic-size junctions. The same preparation method is used
for tip and sample when measuring Pb-Pb, Pb-2H-NbSe2, and Al-Al junctions. The
STM set-up, the software used and aspects of the cryogenics are described in
Refs. [4, 5, 6, 7, 8, 9]. The 2H-NbSe2 sample has been cleaved in-situ at low
temperatures.

In our setup (Extended Data Figure 5), we employ a circuit featuring an
operational amplifier for the feedback element, which is commonly utilized for
SJS measurements [6, 7, 8, 9, 10, 11, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27]. There is nevertheless an important difference
between our circuit and similar ones. Typically, the resistor $R$ is chosen
within the range of $1$ to $10$ M$\Omega$ in order to prevent excessive
bandwidth reduction, and an additional amplification stage is employed to
achieve current-to-voltage conversions in the $10^{8}$, $10^{9}$ or higher
ranges. This amplification stage often incorporates a filter, which can
eliminate any oscillatory behavior present in the junction like the bistable
regime discussed in this work.

In absence of such an additional filtering stage, the current flowing through
the atomic-size tunnel junction is directly measured through the operational
amplifier in a current-to-voltage configuration. In our setup we use a voltage
source with a 10 k$\Omega$ resistor connected in series. The voltage drop on
this resistor is subtracted from the bias voltage to obtain the
$\left<I\right>-\left<V\right>$ curves. We determine the voltage as a function
of time by multiplying the output of the operational amplifier by the tunnel
resistance of the junction $R_{\rm J}$ and subtracting an offset. The feedback
of the operational amplifier ensures that the current flows through the
resistor $R$ and reaches ground, thereby simultaneously nullifying the voltage
difference at the amplifier’s two inputs.

However, the feedback’s action is not immediate but rather delayed by a
certain time constant $\tau_{\rm D}$, limited by the junction’s resistance and
the capacitance $C$. The capacitance $C$ in our circuit is mainly due to the
capacitance in parallel to the resistor in Extended Data Figure 5, which
mostly consists of the wiring to ground. This can be significant due to the
typical long wiring in a dilution refrigerator setup [9, 5]. We observe the
oscillatory behavior when $C_{\rm wire}\simeq 0.3$ nF. In the Al-Al experiment
(discussed in the Supplementary information), we use a capacitor of $C\approx
2$ nF and get an approximately ten times larger $\tau_{\rm D}$. When the setup
has a capacitance of $\sim 10$ nF, $\tau_{\rm D}$ further increases and the
bistable regime is lost, resulting in the Josephson behavior reported in
previous works, discussed in more detail in the Supplementary Information and
shown in Fig.1b of the main text (blue dashed line).

In our experiment, taking $\beta=\frac{2\pi}{\Phi_{0}}I_{\rm c}R_{\rm
S}^{2}C_{\rm J}=4$, we estimate that $C_{\rm J}$ is on the order of, or below,
a few fF. The capacitance of tunnel junctions made with STM have been measured
and calculated for typical tip shapes and sample-tip distances, yielding
values in the range of tens of fF [28]. These capacitance values indicate that
the tip can be modelled by a sphere with an effective radius somewhat below
$1$ $\mu$m. This suggests that the junction capacitance $C_{\rm J}$ is
primarily determined by the geometry of the tip at a distance slightly below
that value, typically in the range of a few tens to hundreds of nm.

We use the OPA 111 amplifier in our setup, featuring a low bias current [29].
In the operational amplifier, two field effect transistors (FET) are connected
to the input and to the power supply. The inputs ($+$ and $-$ in Extended Data
Figure 5) are connected to the gate of each FET while the power supply is
connected to the source and the output to the drain of each FET. Before
reaching the output, the signal is amplified through a low-noise cascode. As
the gain is very large in the used circuit, we expect the settling time to be
of order of the RC time of the circuit.

#### Acknowledgments.

Authors particularly acknowledge Juan Antonio Higuera from SEGAINVEX at UAM
for the insight into the amplification circuitry used in STM. We also
acknowledge discussions with Sebastián Vieira about SJS. Support by the
Spanish Research State Agency (PID2020-114071RB-I00, PID2020-117671GB-100,
PDC2021-121086-I00, TED2021-130546BI00 and CEX2018-000805-M), the European
Research Council PNICTEYES through grant agreement 679080, grants No.
PID2021-122769NB-I00 and No. PID2021-125343NB-I00 funded by
MCIN/AEI/10.13039/501100011033, and by the Comunidad de Madrid through program
NANOFRONTMAG-CM (S2013/MIT-2850) is acknowledged. We have benefitted from
collaborations through EU program Cost CA21144 (superqumap), and from
SEGAINVEX at UAM in the design and construction of STM and cryogenic
equipment.

#### Author contributions.

Calculations were carried out by S.D.E and D.P., being supervised by E.P. and
A.L.Y. The oscillatory signal was observed by V.B. and I.G. in Pb. Further
experiments were carried out by J.A.M., M.F.L., M.A. and B.W., with the
supervision of H.S. and I.G. The electronics, data and set-up were analyzed by
J.A.H., E.H. and J.G.R. The concept was devised and the manuscript was written
by S.D.E., D.P., A.L.Y. and H.S. All authors have read and approved the final
manuscript.

#### Corresponding authors.

Correspondence to<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
Figure 1:  | Bistable regime at different temperatures. a Time-averaged current $\left<I\right>$ vs voltage $\left<V\right>$ measured in a junction between a Pb tip and a Pb sample taken at different temperatures: $T=1$ K (blue), $T=2$ K (green) and $T=3$ K (red). To obtain the curve at 3 K we have subtracted the temperature-induced quasiparticle current. Solid lines are data taken with increasing bias while dashed lines are with opposite ramping direction. Tunneling resistance is $G\approx 0.1G_{0}$, being $G_{0}$ the conductance quantum. b Results from the model calculations. We plot $\left<I\right>$ vs voltage $\left<V\right>$ at $T=0.5$ K (blue), $T=1.25$ K (green) and $T=2.5$ K (red). Figure 2:  | Evolution of the junction current for different parameters. Evolution of the time-averaged current through the junction $\left<I\right>/I_{\rm c}$ as a function of the bias current $i_{\rm b}$ and: a the feedback constant $k$ (while fixing $\Delta\tau=0$, $\tau_{\rm D}=10^{5}$ and $\beta=4$), b the interval of integration $\Delta\tau$ ($kR_{\rm S}=0.5$, $\tau_{\rm D}=10^{5}$ and $\beta=4$), c the McCumber parameter $\beta$ ($kR_{\rm S}=0.5$, $\Delta\tau/\tau_{\rm D}=10^{-1}$ and $\beta=4$), and d the delay time $\tau_{\rm D}$ ($kR_{\rm S}=0.48$, $\Delta\tau=0$ and $\beta=4$). Figure 3:  | Feedback-driven Josephson effect in Pb-2H-NbSe2. a Josephson current in a STM obtained with tip of Pb and sample of 2H-NbSe2. The colored dots represent the positions where we have measured the time-dependent voltage shown in b. In the inset we show the maximum and minimum voltages vs $I_{b}$. The curves are vertically shifted for clarity by 0.25 mV. Temperature is 280 mK and the tunneling conductance is $0.4G_{0}$. Figure 4:  | Feedback driven Josephson effect in Al-Al. a Josephson current in a STM obtained with tip and sample of Al. The colored dots represent the positions where we have measured the time-dependent voltage shown in b. In the inset we show the maximum and minimum voltages vs $I_{\rm b}$. The curves are vertically shifted for clarity by 0.25 mV. Temperature is 400 mK and the tunneling conductance is $0.6G_{0}$. Figure 5:  | Scheme of a STM circuit. Scheme of a typical measurement circuit used in STM. The junction is shown by a triangle (tip), the tunneling current (dashed line) and the sample (horizontal bar), enclosed in a dashed rectangle. The electromagnetic environment of the junction is modelled using a RCSJ model (upper right scheme in the dashed rectangle). The operational amplifier is shown schematically on the bottom right. The junction is driven by a source (circles) and the current through the junction is measured by the usual current-voltage converter circuit of the operational amplifier. The amplifier uses a feedback circuit whose time constant is given by the resistance and capacitance connected to the operational amplifier. Figure 6:  | Switching behavior of the bistable regime. a Switching probability (left $y$-axis) and switching time (right $y$-axis) as a function of the time-averaged bias voltage at 280 mK. We show the entry into the bistable regime with squares and the exit out of the bistable regime with circles. b Switching probability (circles) vs time-averaged voltage for different temperatures: $T=280$ mK (dark red), $T=700$ mK (orange), and $T=1.2$ K (green). Data fits with Gaussians are shown with solid lines with the same colors and the width (dispersion) of the Gaussians vs temperature is shown in the inset.

## Supplementary Information

## Temperature dependence of the bistable regime and results in Al and in
2H-NbSe2

We show data taken with a tip of Pb ($T_{\rm c}^{\rm(Pb)}=7.2$ K) and a sample
of 2H-NbSe2 ($T_{\rm c}^{\rm(NbSe_{2})}=7.15$ K) in Extended Data Figure 3.
2H-NbSe2 is known to exhibit multiple superconducting gaps
$\Delta_{\mathrm{2H-NbSe_{2}},i}$, which are distributed over the Fermi
surface, ranging from $\Delta_{\mathrm{2H-NbSe_{2}},1}\approx 0.5$ meV to
$\Delta_{\mathrm{2H-NbSe_{2}},2}\approx 1.5$ meV [30, 26, 31, 32, 33, 34, 35,
36]. This leads to the rounded bias-voltage dependence of the current-voltage
characteristics above about $1$ mV shown in the upper inset of Extended Data
Figure 3a. Despite these differences, the behavior regarding the bistable
regime is similar to Pb-Pb junctions.

In Extended Data Figure 4, we show the results for the Al-Al junction. Due to
the reduced superconducting gap compared to Pb, the voltage range is modified.
Furthermore, the peak in the Josephson current is considerably reduced and the
hysteresis is practically absent (Extended Data Figure 4). These curves were
also taken at a relatively large temperature of 0.4 K (note that $T_{\rm
c}^{\rm(Al)}=1.2$ K). Nonetheless, the time-dependent oscillatory behavior in
the bistable regime is still observed (Extended Data Figure 4b). We employed
for these measurements a capacitor with ten times higher capacitance in the
operational amplifier circuit than in the other experiments. This increased
the time constant $\tau_{\rm D}$ by about an order of magnitude.

## Other approaches to measure the Josephson current

The current vs voltage characteristic in SJS is often understood through the
phase diffusive transport approximation derived by Ivanchenko and Zilberman
[37]. According to this model, the current is given by $I=\frac{I_{\rm
c}^{2}}{2}Z\left(\frac{V}{V^{2}+V_{\rm c}^{2}}\right)$, where $Z$ represents a
high-frequency impedance and $V_{\rm c}=\frac{2e}{\hbar}k_{\rm B}TZ$ is a
characteristic voltage. The conductance at zero bias $\frac{dI}{dV}(V=0)$ is
often used to map the Josephson current as a function of the position.
However, $Z$ is typically unknown [11]. Furthermore, the maximal current (for
$V=V_{c}$) decreases as $1/T$ and the conductance at zero bias as
$\frac{dI}{dV}(V=0)$ as $1/T^{2}$.

In the Fig. 1c of the main text we compare curves taken in different STM set-
ups using tip and sample of Pb. We have tried to match as far as possible the
conductance $G$, which is around $0.4G_{0}$. In the set-up having a
capacitance of around $C\approx 0.3$ nF (blue curve) we observe the bistable
regime. In the set-up having a capacitance of about $C\approx 10$ nF (dashed
blue curve) we observe no bistable regime. The current is then time-
independent for all bias voltages and there is no hysteresis. The dashed blue
curve is well captured by the Ivanchenko-Zilberman approximation.

Our model also requires an effective temperature of 1 K to obtain the shape of
the curve at low bias voltages (Fig. 3c in the main text). It is important to
note that the energy resolution of our experiment is far below this value, as
evidenced by the sharp increase in the current at the gap edge (upper inset of
Fig. 1c, and Ref. [5]). The increased temperature arises from the equivalent
temperature of the electromagnetic environment at the Josephson frequency
$\omega_{\rm J}$. However, we can extract the value of $\left<I\right>$ in the
bistable regime from the oscillatory signal, which is given by $I_{\rm c}$ and
the time constants of our circuit, and is thus nearly independent of
temperature for $T<T_{\rm c}$/2, following the dependence of $I_{\rm c}$ with
temperature.

## Switching in and out of the bistable regime

We can analyze in more detail the effect of temperature by studying the entry
and exit into and out of the bistable regime. The escape dynamics of small
Josephson junctions has been extensively studied [4, 38, 39, 40, 41, 5]. The
current jumps from the Josephson regime into the resistive regime with its
dynamics determined by thermal activation at high temperatures and quantum
tunneling at very low temperatures [4, 38].

In the new bistable regime, the junction leaves the zero voltage regime and
enters the bistable regime. In Extended Data Figure 6a, we show that this
occurs through a stochastic process. Pulses of width $\tau_{D}$ are measured
at time intervals that seem to be random. The inverse of the time between
pulses, $\frac{1}{\tau_{\rm Sto}}$, increases exponentially with bias. We can
find the probability of switching $P_{1}(V)$ by taking $P_{1}(V)=\tau_{\rm
Sto}^{-1}\left(\int_{0}^{1}P(t)dt\right)$. $P_{1}(V)$ follows a Gaussian
behavior within a range of time-averaged voltage of about $5$ $\mu V$ (see
Extended Data Figure 6a).

The junction leaves the bistable regime producing the hysteresis discussed
above. To analyze this behavior, we have swept through the current-voltage
characteristics many times and traced at each sweep the voltage where we lose
oscillations at $\tau_{\rm D}$. From this, we define a switching probability
$P_{2}$ and the corresponding time scale (Extended Data Figure 6a). As we
observe in Extended Data Figure 6b, the Gaussian function that describes the
switching probability becomes sharper when decreasing temperature.

## Relation to relaxation oscillators and other dynamic phenomena in Josephson
junctions

The nonlinear dynamics of a Josephson junction in itself, without feedback,
can be extremely rich and complex, with regimes of chaotic behavior [42, 43,
44, 45, 46, 47]. Additionally, unstable regions in the current-voltage
characteristics of Josephson junctions can lock into passive elements leading
to an oscillatory behavior [48, 49, 50, 51, 52, 53, 54]. Often, they are
called relaxation oscillators and imply the passage of the current to the
dissipative regime. This is not possible in our junctions, as the voltage
moves in a narrow interval close to zero, far below the superconducting gap
value. Furthermore, these behaviors occur at frequencies between four and six
orders of magnitude higher than those observed here [48, 49, 50, 51, 52, 53,
54].

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