Shot-noise-driven escape in hysteretic Josephson junctions
J.P. Pekola
We have measured the influence of shot noise on hysteretic Josephson %%@ junctions initially in macroscopic quantum tunnelling (MQT) regime. Escape %%@ threshold current into the resistive state decreases monotonically with %%@ increasing average current through the scattering conductor, which is another %%@ tunnel junction. Escape is predominantly determined by excitation due to the %%@ wide-band shot noise. This process is equivalent to thermal activation (TA) %%@ over the barrier at temperatures up to about four times above the critical %%@ temperature of the superconductor. The presented TA model is in excellent %%@ agreement with the experimental results.
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\begin{document}
\preprint{ }


\title{Shot-noise-driven escape in hysteretic Josephson junctions}

\author{J.P. Pekola}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}
\author{T.E. Nieminen}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}
\author{M. Meschke}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}
\author{J.M. Kivioja}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}
\author{A.O. Niskanen}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}
\affiliation{VTT Information Technology, Microsensing, P.O. Box 1207, 02044 %%@
VTT, Finland}
\author{J.J. Vartiainen}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O. Box 3500, 02015 TKK, Finland}

\pacs{72.70.+m,73.23.-b,05.40.-a}

\begin{abstract}
We have measured the influence of shot noise on hysteretic Josephson %%@
junctions initially in macroscopic quantum tunnelling (MQT) regime. Escape %%@
threshold current into the resistive state decreases monotonically with %%@
increasing average current through the scattering conductor, which is another %%@
tunnel junction. Escape is predominantly determined by excitation due to the %%@
wide-band shot noise. This process is equivalent to thermal activation (TA) %%@
over the barrier at temperatures up to about four times above the critical %%@
temperature of the superconductor. The presented TA model is in excellent %%@
agreement with the experimental results.
\end{abstract}
\maketitle
Shot noise and full counting statistics (FCS) in mesoscopic conductors are %%@
currently intensively studied, because such measurements yield fingerprints %%@
of conduction mechanisms of these scatterers %%@
\cite{blanter,nazarov,levitov,reulet}. Yet direct measurements of the second %%@
and higher moments of current or voltage are typically much more difficult %%@
and prone to errors as compared to the measurement of DC transport %%@
properties. This is because, in order to extract information about %%@
fluctuations, one generally needs to perform high frequency measurements on %%@
remotely connected samples at cryogenic temperatures, which is in practise a %%@
non-trivial task. Therefore, detectors of noise and FCS that operate directly %%@
on-chip near the noise source are of great importance.  A recently proposed %%@
read-out device of these fluctuations is a Josephson junction (JJ) threshold %%@
detector \cite{tobiska,jp04}. A mesoscopic non-hysteretic Josephson junction %%@
in the Coulomb blockade regime is another choice %%@
\cite{schoelkopf,lindell,heikkila}.

The dynamics of the JJ can be described as that of a phase ($\varphi$) %%@
particle in a tilted cosine potential, see Fig.~\ref{fig1}. Under the %%@
influence of equilibrium environment fluctuations at low temperatures the %%@
particle resides in the quantum mechanical ground state, from where the %%@
escape mechanism is tunnelling (MQT) through the barrier. At higher %%@
temperatures it assumes a nearly thermal population of the states, and the %%@
escape mechanism is predominantly thermal activation (TA) over the barrier %%@
top. Below we discuss the regime where excited states are not accessed by the %%@
influence of the thermal noise of the dissipative environment, which is an %%@
appropriate assumption based on our present measurements in the absence of %%@
shot noise and those of Ref.~\cite{kivioja} in the same set-up at low %%@
temperatures.

\begin{figure}
\begin{center}
\includegraphics[width=0.4
\textwidth]{fig1.eps}
\caption{(a) Measurement configuration. A split JJ in MQT regime measures the %%@
current fluctuations due to the shot noise of the scattering junction in the %%@
right arm. (b) Processes leading to escape from zero-voltage (supercurrent) %%@
state into resistive state. (c) Sketch of current pulses $I_{\rm P}$, with %%@
superimposed noise of $I_{\rm N}$, and (d) the associated escape %%@
characteristics. The central quantities relating to the pulses and escape %%@
histograms have been indicated in the figure.}\label{fig1}
\end{center}
\end{figure}

The influence of noisy current on escape characteristics is typically %%@
considered in two different limits: (i) adiabatic (quasistationary) regime, %%@
where fluctuations of bias current that change the tilt of the cosine %%@
potential are so slow and weak that the phase particle remains in the ground %%@
state of the metastable well, and the escape rate is dictated by MQT from %%@
this varying ground state \cite{martinis88,jp04}, or (ii) (nearly) resonance %%@
excitation limit, where most of the escape events are consequences of %%@
shot-noise-driven excitations of the phase particle into higher levels in the %%@
well. Due to the high frequency resonance character of our on-chip circuit, %%@
it turns out that the latter approach is more appropriate in describing the %%@
dynamics of the detector in this case.

In this Letter we show that a hysteretic Josephson junction in MQT regime can %%@
measure high frequency current fluctuations generated by a mesoscopic %%@
scatterer in non-equilibrium. We present a model, in which the noisy current %%@
excites the phase particle in a metastable, nearly parabolic well out from %%@
the ground state, and subsequently the particle escapes from the well. This %%@
occurs via thermal activation at an equivalent temperature of the phase %%@
particle which is determined by the competition between shot-noise %%@
excitation, and relaxation due to the dissipative environment. This model is %%@
in excellent agreement with our experimental results, and we are able to %%@
investigate Josephson phase dynamics up to equivalent temperatures which are %%@
about four times higher than the critical temperature of the superconductor. %%@
This is the temperature responsible for phase dynamics: the electrons and %%@
lattice of the superconductor are only weakly coupled to this subsystem, and %%@
the former two remain essentially at their base temperature.

Our measurement configuration with an electron micrograph of a split JJ %%@
detector is shown in Fig.~\ref{fig1}. In a typical experiment two currents, %%@
$I_{\rm N}$ and $I_{\rm P}$, are injected into the circuit through large %%@
resistances ($> 1 $ M$\Omega$) at room temperature. Current $I_{\rm N}$ is %%@
applied constantly and it runs through another tunnel junction at a distance %%@
of 120 $\mu$m from the detector. This junction plays the role of the %%@
shot-noise source in the circuit. The DC-component of this current, %%@
$\bar{I}_{\rm N}$, returns through a long (3 mm) and narrow (width 2 $\mu$m) %%@
superconducting line such that no DC current due to $I_{\rm N}$ passes %%@
through the detector. We verify this balance frequently during the %%@
measurements. Thus, only the fluctuations of $I_{\rm N}$ are admitted through %%@
the detector. We probe these fluctuations by applying repeatedly trapezoidal %%@
current pulses of height $I_{\rm P}$ through the detector as shown in %%@
Fig.~\ref{fig1}~(c). These pulses are slow, typically 10 $\mu$s - 10 ms long %%@
and with long transient times at the beginning and at the end of the pulse to %%@
secure adiabatic response to them. Typically 1000 pulses at each value of %%@
$I_{\rm P}$ are repeated, and the escape probability $P(I_{\rm P})$ is %%@
obtained as the fraction of those pulses that lead to escape from the %%@
supercurrent state.

Let us discuss the dynamics of a hysteretic JJ in more detail. In the common %%@
RCSJ-model (Resistively and Capacitively Shunted Junction model), the phase %%@
particle in the tilted cosine potential is either trapped and oscillating in %%@
one of the wells, or alternatively it runs down the potential. The latter %%@
regime is called resistive state. The measurable quantities, {\it i.e.}, %%@
current $I$ and voltage $V$, are related to $\varphi$ via the Josephson %%@
equations: $I = I_{\rm C}\sin \varphi$ and $2eV = \hbar \frac{d\varphi}{dt}$, %%@
where $I_{\rm C} = \frac{2e}{\hbar}E_{\rm J}$ is the critical current of the %%@
junction and $E_{\rm J}$ is its Josephson coupling energy. The quality factor %%@
$Q \equiv \omega _{\rm p}RC$ is assumed to be $Q\gg 1$ to assure hysteretic %%@
dynamics. This means that once the particle leaves the well, it runs freely %%@
such that the voltage across the junction is close to twice the energy gap of %%@
the superconductor, and it can be retrapped to the zero-voltage supercurrent %%@
state only when current is lowered virtually to zero. Here $R$ is the %%@
resistive shunt of the junction, $C$ is the parallel capacitance, and $\omega %%@
_{\rm p} \equiv \sqrt{8E_{\rm J}E_{\rm C}q_0}/\hbar$ is the plasma frequency, %%@
where $E_{\rm C}=e^2/2C$ and $q_0=\sqrt{2(1-I/I_{\rm C})}$. Quantum %%@
mechanically the phase particle has energy states in the nearly parabolic %%@
wells, with energies close to those of a harmonic oscillator: $E_n \simeq %%@
(n+1/2)\hbar \omega _{\rm p}$.

Due to the nearly harmonic potential, transitions between the neighbouring %%@
levels are dominating. The rates of these transitions, see Fig.~\ref{fig1} %%@
(b), are determined by the spectral density $S_{I}(\omega)=\int %%@
_{-\infty}^{\infty}\langle I(t)I(0)\rangle \exp(i\omega t)dt$ of the current %%@
noise at the corresponding level separation $\omega = \omega _{j,j-1}$ as %%@
$\gamma _{j,j-1} \simeq (j/2\hbar\omega _{j,j-1}C)S_{I}(-\omega _{j,j-1})$ %%@
and $\gamma _{j-1,j} \simeq (j/2\hbar\omega _{j,j-1}C)S_{I}(+\omega %%@
_{j,j-1})$ for excitation and relaxation, respectively %%@
\cite{schoelkopf,martinis03}. The noise spectrum has two contributions, one %%@
due to the equilibrium environment, $S_I^{\rm env}$, and the other due to %%@
shot noise, $S_I^{\rm shot}$, which add incoherently: $S_I = S_I^{\rm %%@
env}+S_I^{\rm shot}$. At low temperature, $k_{\rm B}T \ll \hbar \omega _{\rm %%@
p}$, the thermal excitation is strongly suppressed, $S_I^{\rm env} (-\omega %%@
_{j,j-1})\rightarrow 0$, and the relaxation is due to zero point %%@
fluctuations, $S_I^{\rm env}(+\omega _{j,j-1})\rightarrow 2 \hbar \omega %%@
_{j,j-1}{\rm Re} Y(\omega _{j,j-1})$, where $Y(\omega)$ is the admittance of %%@
the circuit surrounding the Josephson junction. The shot noise responsible %%@
for transitions in the Josephson junction can be described as $S_I^{\rm shot} %%@
(\pm \omega _{j,j-1}) = Fe\bar{I}_{\rm N}$. Here $\bar{I}_{\rm N}$ is the %%@
average current through the scatterer, and $F$ is the "Fano factor" of the %%@
noise source and the circuit surrounding the JJ, at the frequency %%@
corresponding to level separation. Factor $F$ can be calculated for a known %%@
experimental circuit. Here we determine it experimentally as a fit parameter. %%@
It would be unity in the case of Poissonian tunnel junction source %%@
\cite{blanter} and if all the noise current would run through the detector %%@
junction. Combining the results above, we have
\begin{equation} \label{rateup}
\gamma _{j,j-1} \simeq \frac{jFe\bar{I}_{\rm N}}{2\hbar\omega _{j,j-1}C}
\end{equation}
and
\begin{equation} \label{ratedown}
\gamma _{j-1,j} \simeq \frac{jFe\bar{I}_{\rm N}}{2\hbar\omega _{j,j-1}C} + %%@
\frac{j\omega _{j,j-1}}{Q}.
\end{equation}
The level dynamics described by Eqs.~(\ref{rateup}) and (\ref{ratedown}) can %%@
be described by the equivalent temperature $T^*$ and effective quality factor %%@
$Q^*$ by requesting $\gamma _{j,j-1}\equiv j\frac{\omega %%@
_{j,j-1}}{2Q^*}[\coth(\hbar \omega _{j,j-1}/2k_{\rm B}T^*)-1]$ and $\gamma %%@
_{j-1,j}\equiv j\frac{\omega _{j,j-1}}{2Q^*}[\coth(\hbar \omega %%@
_{j,j-1}/2k_{\rm B}T^*)+1]$. This yields $Q^*=Q$ and, with $\omega %%@
_{j,j-1}\simeq \omega _{\rm p}$
\begin{equation} \label{tstar}
k_{\rm B}T^* \simeq \frac{\hbar \omega _{\rm p}}{2 {\rm arcoth}(1+ %%@
QFe\bar{I}_{\rm N}/\hbar\omega _{\rm p}^2C)}.
\end{equation}
Following the standard results of the decay from a cubic metastable well, we %%@
can infer that thermal activation is the dominant escape mechanism provided %%@
$T^* > T_0 \equiv \hbar \omega _{\rm p}/2\pi k_{\rm B}$. This yields a %%@
condition for the validity of the TA model: $ {\rm arcoth}(1+ QFe\bar{I}_{\rm %%@
N}/\hbar\omega _{\rm p}^2C)/\pi < 1$. This is fulfilled at all currents %%@
$\bar{I}_{\rm N}$ that were employed in the experiments, as we will show %%@
below. With this procedure it is then straightforward to obtain the switching %%@
probability of the threshold detector, in the limit of many levels in the %%@
well, as $P(\bar{I}_{\rm P}) = 1- \exp(-\Gamma \Delta t)$. Here $\Gamma = %%@
\frac{\omega _{\rm p}}{2\pi}\exp(-\Delta U/k_{\rm B}T^*)$ is the standard TA %%@
escape rate, $\Delta t$ is the length of the current pulse, and $\Delta U = %%@
4\sqrt{2}E_{\rm J}(1-I_{\rm P}/I_{\rm C})^{3/2}/3$ is the height of the %%@
potential barrier at the particular bias point $I_{\rm P}$. The expressions %%@
above allow us to evaluate the position $I_{0.5}$ and the width $\Delta I = %%@
I_{0.9} - I_{0.1}$ of the escape threshold, where $I_x$ is defined by %%@
$P(I_x)\equiv x$, see Fig. \ref{fig1}~(d). These approximate results were %%@
compared to those obtained by full level dynamics calculation \cite{larkin} %%@
numerically. They were identical, although there the results depended %%@
sensitively on the exact values of the level positions and widths near the %%@
barrier top.

In the limit of large noise currents, $\bar{I}_{\rm N} \gg \frac{\hbar \omega %%@
_{\rm p}^2C}{QFe}$, we obtain $T^* \simeq \frac{QFe\bar{I}_{\rm N}}{2k_{\rm %%@
B}\omega _{\rm p}C}$, which can be written in a more familiar looking form %%@
$T^* \simeq \frac{Fe\bar{I}_{\rm N}}{2k_{\rm B}/R}$. This is a valid %%@
approximation in a wide range of experimental parameters. The expressions of %%@
TA escape and the current dependence of the barrier height combined with this %%@
approximate expression of $T^*$ finally allow us to approximate $I_{0.5}$ and %%@
$\Delta I$ as
\begin{equation} \label{I50}
I_{0.5}/I_{\rm C} \simeq 1 - (\frac{3}{4\sqrt{2}})^{2/3} %%@
[\ln(\frac{\omega_{\rm p}\Delta t}{2\pi\kappa _{0.5}})]^{2/3} \big{(} %%@
QF\frac{E_{\rm C}}{E_{\rm J}}\frac{\bar{I}_{\rm N}}{e\omega _{\rm %%@
p}}\big{)}^{2/3}
\end{equation}
and, further assuming $\omega _{\rm p}\Delta t \gg 1$ yields
\begin{equation} \label{deltaI}
\Delta I/I_{\rm C} \simeq  \frac{\ln(\kappa _{0.9})-\ln(\kappa %%@
_{0.1})}{[12\ln(\frac{\omega_{\rm p}\Delta t}{2\pi})]^{1/3}} %%@
\big{(}QF\frac{E_{\rm C}}{E_{\rm J}}\frac{\bar{I}_{\rm N}}{e\omega _{\rm %%@
p}}\big{)}^{2/3}.
\end{equation}
Here $\kappa _x \equiv -\ln(1-x)$.

We report on data of two samples, A and B, which were fabricated by standard %%@
electron beam lithography and shadow evaporation with aluminium as the %%@
superconductor. The samples were measured via adequately filtered signal %%@
lines in dilution refrigerators at bath temperatures of 30 mK - 1 K. Sample A %%@
had a split JJ detector and one scattering junction as in the scheme of Fig. %%@
\ref{fig1}. Sample B had a single JJ as a detector, and it had two scattering %%@
junctions located symmetrically with respect to the detector (at a distance %%@
of 120 $\mu$m), and with respect to {\sl two} long injection lines. In this %%@
sample the two scattering junctions were made intentionally very different to %%@
check the invariance of the results with respect to junction properties. The %%@
parameters of the two samples are listed in Table \ref{table}.

\begin{table}
\caption{Parameters of the two samples.}
\label{table}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
 Sample & $I_{\rm C}$ & $C$ & $I_{\rm C}$ & $C$ \\
    & detector & detector & scatterer & scatterer \\
\noalign{\smallskip}\hline\noalign{\smallskip}
A & 3.9 $\mu$A & 230 fF & 600 nA & 40 fF\\
\noalign{\smallskip}\hline
B & 1.5 $\mu$A & 230 fF & 15 nA (1) & 10 fF (1)\\
 & & & 90 nA (2)& 40 fF (2)\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
Data in Fig.~\ref{fig2}~(a) show results of a control experiment on Sample B %%@
at the base temperature $T \simeq 30$ mK, where the trapezoidal pulse current %%@
$I$ was injected through (i) one of the long injection lines, (ii) through %%@
scatterer 1, and (iii) through scatterer 2. The histogram of case (i) lies at %%@
higher currents than those of (ii) and (iii), which in turn overlap %%@
practically with each other. This demonstrates that shot noise tends to push %%@
the threshold towards lower values of current, as predicted, {\it e.g.}, by %%@
Eq.~(\ref{I50}). Furthermore these data demonstrate that the noise is %%@
predominantly, and equally in (ii) and (iii), generated by the scattering %%@
tunnel junctions. The calculated lines run through the corresponding %%@
experimental histograms assuming MQT without any noise in (i) and
TA with $T^*$ evaluated from Eq. (\ref{tstar}) with $QF=5$ in (ii) and (iii). %%@
The latter value is realistic in terms of the expected $Q\simeq 10$ %%@
\cite{kivioja} and $F\le 1$.

Figure~\ref{fig2}~(b) shows data on Sample B at the temperature $T= 40$ mK. %%@
The threshold current $I_{0.5}$, {\it i.e.}, the pulse current at which the %%@
switching probability is 50\%, has been plotted as a function of %%@
$\bar{I}_{\rm N}$ through the scatterer (2). The data, shown by solid %%@
symbols, follow the prediction of the model presented above, again with %%@
$QF=5$. The initial plateau in the experimental data below $I_{\rm N} \simeq %%@
100$ nA arises because the scattering junction is not in the linear %%@
quasiparticle tunnelling regime here.
\begin{figure}
\begin{center}
\includegraphics[width=0.5
\textwidth]{fig2.eps}
\caption{Histogram positions under shot noise injection. (a) Escape %%@
histograms of Sample B when 800 $\mu$s long current pulses have been injected %%@
through the long injection line (open circles), and through the two noise %%@
sources (1 - squares, 2 - filled circles), respectively. The lines are the %%@
corresponding theoretical results. (b) Experimental results (circles) on the %%@
switching threshold current $I_{0.5}$ against the average current %%@
$\bar{I}_{\rm N}$ through the scatterer 2 in Sample B. Pulse length was %%@
$\Delta t = 800$ $\mu$s. The solid line is again the result of the used %%@
theoretical model. The inset shows the $IV$ curve of the noise source. (c) %%@
Similar data as in (b) but for Sample A. The rising curve shows the %%@
corresponding equivalent temperature $T^*$, which ranges from 2 to 5 K in %%@
this measurement.}\label{fig2}
\end{center}
\end{figure}
The inset of Fig.~\ref{fig2}~(b) shows the $IV$ curve of this junction.

In Fig.~\ref{fig2}~(c) we plot, in addition to similar data as in (b), the %%@
equivalent temperature $T^*$ in the measurement of Sample A when current %%@
$\bar{I}_{\rm N}$ is varied between 2 $\mu$A and 8 $\mu$A. There are two %%@
interesting points to note. (i) We are able to study Josephson dynamics of %%@
the JJ up to temperatures $T^* \simeq 5$ K, about four times above the %%@
critical temperature $T_{\rm C}$ of aluminium. (ii) At all currents employed %%@
we are well above the threshold $T_0$ of thermal activation: for this %%@
particular sample $T_0 \simeq 0.3$ K.
\begin{figure}
\begin{center}
\includegraphics[width=0.5
\textwidth]{fig3.eps}
\caption{The width of the histograms when shot noise and thermal noise are %%@
applied.
(a) The width of the grey zone, $\Delta I$, against the average current %%@
$\bar{I}_{\rm N}$ through the scatterer. The dots are the experimental %%@
results and the solid line is the result of the theoretical model. (b) %%@
$\Delta I$ vs. $I_{0.5}$ under two different experimental conditions. The %%@
filled circles are data with elevated shot-noise temperature $T^*$, whereas %%@
the open circles are with variable bath temperature $T$. The solid line is %%@
the result of thermal activation model with constant energy gap, and the %%@
dashed line takes into account the suppression of the BCS gap \cite{tinkham} %%@
in the measurement with increased $T$.}\label{fig3}
\end{center}
\end{figure}

Figure~\ref{fig3}~(a) shows the width of the switching threshold, $\Delta I %%@
$, of sample A. Again the experimental (circles) and theoretical (line) %%@
results are in good agreement between each other using the same fit %%@
parameters as in Fig.~\ref{fig2}.
In Fig.~\ref{fig3}~(b) data on switching of Sample A has been plotted in two %%@
different experimental conditions, in each case as $\Delta I$ vs. $I_{0.5}$. %%@
The open circles are data where the bath (lattice) temperature has been %%@
varied by heating the sample stage to several temperatures $T$ in the range %%@
0.03 - 1 K. The full circles are data taken at the base temperature of the %%@
cryostat ($T\simeq 30$ mK), but by varying the equivalent temperature $T^*$ %%@
injecting different levels of $\bar{I}_{\rm N}$. The line closely following %%@
the latter data is that originating from the thermal activation model. At the %%@
first sight it is surprising that the truly thermal data (open circles) fall %%@
far below this line and the other set of data. This is, however, accounted %%@
for by the fact that when increasing the bath temperature close to $T_{\rm %%@
C}$ of aluminium ($\sim 1.2$ K), the BCS energy gap is progressively %%@
diminishing, thus leading to decrease of the critical current. The line %%@
following closely this data set is obtained by using the very same thermal %%@
activation model, but by taking into account suppression of the BCS gap due %%@
to temperature $T$ \cite{tinkham}. This figure thus demonstrates that our %%@
thermal activation model is in good agreement with the shot noise data, and %%@
that it is indeed possible, by promoting shot noise, to study Josephson %%@
dynamics at super-$T_{\rm C}$ temperatures without suppressing %%@
superconductivity.

Finally there are a few more topics to address. We found out that the %%@
adiabatic models of rocking slowly the current bias of the JJ do not account %%@
for our observations on shot noise: the adiabatic models, involving no %%@
excitations, predict exponential dependence in variance-of-the-current of the %%@
tunnelling rate from the ground state through the barrier %%@
\cite{jp04,martinis88}. Our experimental results demonstrate, however, much %%@
weaker dependence, which we derived in this Letter. The (nearly) resonant %%@
model works in this case because the circuit responding to the shot noise %%@
resonates at frequencies, which are of the same order as the plasma frequency %%@
of the junction. Furthermore, the low frequency noise does not run through %%@
the detector junction but rather leaks through the bias line. Note that at DC %%@
we suppress the detector current due to $I_{\rm N}$ totally. The second issue %%@
is the assessment of an MQT detector as an absolute noise detector. Based on %%@
the invariance of the obtained results, especially in terms of results on %%@
different scatterers on the same sample, see Fig.~2~(a), we suggest that this %%@
kind of a detector could be made into an absolute on-chip detector of %%@
Fano-factors \cite{blanter}, and noise in general, by careful tailoring of %%@
the circuit surrounding the JJ and by measuring $Q$ independently.

We thank T. Heikkil\"a, T. Ojanen, E. Sonin and A. Savin for useful %%@
discussions, and H. Grabert and D. Esteve for their insightful comments at an %%@
early stage of this work. We acknowledge Academy of Finland for financial %%@
support.

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