# Breakdown of the escape dynamics in Josephson junctions

D. Massarotti<EMAIL_ADDRESS>Universitá degli Studi di Napoli
”Federico II”, Dipartimento di Fisica, via Cinthia, 80126 Napoli, Italy  CNR-
SPIN UOS Napoli, Complesso Universitario di Monte Sant’Angelo, via Cinthia,
80126 Napoli, Italy  D. Stornaiuolo  Universitá degli Studi di Napoli
”Federico II”, Dipartimento di Fisica, via Cinthia, 80126 Napoli, Italy  CNR-
SPIN UOS Napoli, Complesso Universitario di Monte Sant’Angelo, via Cinthia,
80126 Napoli, Italy  P. Lucignano  CNR-SPIN UOS Napoli, Complesso
Universitario di Monte Sant’Angelo, via Cinthia, 80126 Napoli, Italy  L.
Galletti  Universitá degli Studi di Napoli ”Federico II”, Dipartimento di
Fisica, via Cinthia, 80126 Napoli, Italy  CNR-SPIN UOS Napoli, Complesso
Universitario di Monte Sant’Angelo, via Cinthia, 80126 Napoli, Italy  D. Born
Leibniz Institute of Photonic Technology e.V., P.O. Box 100239, D-07702 Jena,
Germany  G. Rotoli  Seconda Universitá di Napoli, Dipartimento di Ingegneria
Industriale e dell’Informazione, via Roma 29, 81031 Aversa (Ce), Italy  F.
Lombardi  Department of Microtechnology and Nanoscience, MC2, Chalmers
University of Technology, S-41296 Göteborg, Sweden  L. Longobardi  Seconda
Universitá di Napoli, Dipartimento di Ingegneria Industriale e
dell’Informazione, via Roma 29, 81031 Aversa (Ce), Italy  American Physical
Society, 1 Research Road, Ridge, New York 11961, USA  A. Tagliacozzo
Universitá degli Studi di Napoli ”Federico II”, Dipartimento di Fisica, via
Cinthia, 80126 Napoli, Italy  CNR-SPIN UOS Napoli, Complesso Universitario di
Monte Sant’Angelo, via Cinthia, 80126 Napoli, Italy  F. Tafuri  CNR-SPIN UOS
Napoli, Complesso Universitario di Monte Sant’Angelo, via Cinthia, 80126
Napoli, Italy  Seconda Universitá di Napoli, Dipartimento di Ingegneria
Industriale e dell’Informazione, via Roma 29, 81031 Aversa (Ce), Italy

###### Abstract

We have identified anomalous behavior of the escape rate out of the zero-
voltage state in Josephson junctions with a high critical current density
$J_{c}$. For this study we have employed YBa2Cu3O7-x grain boundary junctions,
which span a wide range of $J_{c}$ and have appropriate electro-dynamical
parameters. Such high $J_{c}$ junctions, when hysteretic, do not switch from
the superconducting to the normal state following the expected stochastic
Josephson distribution, despite having standard Josephson properties such as a
Fraunhofer magnetic field pattern. The switching current distributions (SCDs)
are consistent with non-equilibrium dynamics taking place on a local rather
than a global scale. This means that macroscopic quantum phenomena seem to be
practically unattainable for high $J_{c}$ junctions. We argue that SCDs are an
accurate means to measure non-equilibrium effects. This transition from global
to local dynamics is of relevance for all kinds of weak links, including the
emergent family of nano-hybrid Josephson junctions. Therefore caution should
be applied in the use of such junctions in, for instance, the search for
Majorana fermions.

###### pacs:

74.50.+r, 85.25.Cp, 74.40.Gh, 74.78.Na

##  I Introduction

Rich physics, such as the Josephson effectjosephson , quantum
coherencelikharev ; leggettrmp  and quantum interferencejaklevic , arise when
two coherent quantum systems are weakly coupled. Some examples of systems that
exhibit such physics are superconducting Josephson junctionsbarone , the flow
of superfluid 4He through an array of nano-apertureshokinson , and the
observation of the Josephson effect in Bose-Einstein condensateslevy .
Superconductors have traditionally held a special place as test-bench systems
for these kinds of quantum phenomena due to the ease of scaling and
integrating them into real quantum devices, offering a high degree of
measurability and control of a macroscopic wave function.

Every experiment or application using a superconducting weak link is based on
how the phase difference $\varphi$ between the electrodes evolves in time and
spacejosephson ; barone ; likharev ; libroBez . The large variety of barriers
now available between the superconducting electrodes offer novel
functionalities and efficient tuning of physical processes occurring at the
nanoscale and at different interfacesdefranceschi2 ; doh ; topological ;
morpurgo ; bouchiat . A recent example is the proposal to use the Josephson
effect for the detection of the Majorana fermionskane .

Figure 1: a) Three-dimensional representation of the SCDs measured for various
temperatures on three GB YBCO JJs. b) At low $J_{c}$ values (5$\cdot$102
A/cm2), the SCDs of JCT A are confined to a small range of currents, with
$\sigma$ of the order of 10 nA. With increasing $J_{c}$ the histograms
progressively cover a larger interval of currents, both absolutely and
relatively (JCTs B and C in panels c and d respectively). In panel e) the
escape rate curves $\Gamma(I)$ computed from the SCDs of JCT C have been shown
for the same temperature range as in panel d.

Progress in material science in producing a larger variety of interfaces and
in nanotechnologies applied to superconductivity, is promoting a rethinking of
the phase dynamics of Josephson junctions (JJs). Here we give evidence of a
breakdown of a fundamental tenet of the Josephson effect: the transition from
the superconducting to the normal state does not follow the expected
stochastic Josephson phase dynamics, but has a more intriguing balance between
local and global energy processes. The onset of non-equilibrium effects is the
key to describe the local processes, which may radically change perspectives
on how to interpret the physics and make predictions on the performances of a
large variety of ”smart” Josephson devices. We measure switching current
distributions (SCDs) which codify the very general process of the escape of a
particle (phase) from a potential well in a JJbarone ; likharev , keeping
track in our case of non-equilibrium effects. Roughly speaking, SCDs are
obtained in JJs with hysteretic current-voltage (I-V) characteristics by
counting the number of times the system switches from the superconducting to
the resistive state within a small window of bias current, when ramping forth
and back the bias current. Thermally activated processes are well understood
in JJs both in the underdampedkurk ; fulton ; jackel  and in the moderately
dampedkautz_secondo  regime. The transition to the macroscopic quantum
tunneling regime has been theoreticallyLeggett ; Leggett_b ; Leggett_c  and
experimentallywebb2 ; devoret1985 ; bauch_2005  widely investigated. SCD
measurements focus on ”the very moment” at which resistance originates in
superconducting weak links. Thus we use the power of encoding information of
non-equilibrium local processes in fluctuations to characterize these
dynamical processes.

In the resistively and capacitively shunted junction (RCSJ) modelbarone , the
damping parameter $Q=\omega_{p}RC$ is proportional to the square root of
$I_{c}$ via the plasma frequency $\omega_{p}=(2eI_{c}/\hbar C)^{1/2}$ at zero
bias current, where $R$ and $C$ are the resistance and capacitance,
respectively. In a more general approach, $Q$ has a frequency
dependencekautz_secondo ; dani_PRB , which includes the effects of the
external shunting impedance. A junction cannot sustain an unlimited increase
in the critical current $I_{c}$ and thus in the quality factor $Q$ through
larger critical current density $J_{c}$ while still preserving all the
properties of the Josephson effect and all the features of the underdamped
regime in the I-V curves.

The breakdown of the Josephson dynamics is found at higher values of the
critical current density. The classical Josephson phase dynamics, which takes
place in junctions characterized by lower critical current densities $J_{c}$,
are replaced at high $J_{c}$ values by a regime driven by non-equilibrium
dynamics where phase information is lost. Non-equilibrium effects produce
hysteretic I-V characteristics and modify the influence of dissipation, thus
becoming measurable through modeling of the SCD in terms of heating modes. The
heating modes follow a heat diffusion-type equation in analogy to phase slip
eventsphase_slips ; langer . In the present work, this transition from
classical to non-equilibrium phase dynamics is found for high critical
temperature superconductors (HTS) grain boundary (GB) junctions, but should be
expected for any kind of JJskleinsasser1 ; kleinsasser2 . Specific thresholds
may depend on the type of junctions and materials, but the features of the
transition are universal. HTS GB junctions are the ideal system to identify
this transition because of the possibility of varying the critical
supercurrent density $J_{c}$ over a wide range of valueshans ; rop .

Figure 2: a) Temperature dependence of $\sigma$ measured on GB JCTs A (blue
squares), B (red triangles) and C (black points) respectively. The blue and
red solid lines are Monte Carlo simulations of the phase dynamics, according
to multiple escape and retrapping processes in the washboard potential (shown
in panel b), with $Q$ = 1.30 and 1.17 for JCTs A and B respectively. Some data
from sample A have been presented in Ref. luigi2,  previously. c) Simulations
of the thermal dependence of $\sigma$ as a function of $J_{c}$ for different
values of the $Q$ damping parameter (full color lines), confined to the
moderately damped regime, are compared with experimental data of JCT C (black
points). Keeping all the other junction parameters fixed, an enhancement of
$J_{c}$ leads to an increase of $I_{c}$ and $Q$. An increase in $Q$ produces
steeper $\sigma$ tails above $T^{*}$ and cannot reproduce the broadened
experimental data of JCT C. This is even more evident in the two dimensional
($\sigma-T$) projection. For completeness, in the ($\sigma-T$) projection the
color points referred to the results of Monte Carlo simulations have been
reported for $Q$ values ranging from 1 to 10, and compared with the data of
JCT C (black points). The colored lines are guides for the eye. At high
$J_{c}$ the Langevin approach does not hold anymore and non-equilibrium
concepts apply. Simulations fitting the SCDs of JCT C (see Fig. 4) are
consistent with multiple heating events as reported pictorially in panel d. As
explained in the text, $T_{b}$ is the bath temperature and $T_{th}$ is the
threshold temperature above which the transition to the resistive state
occurs. In panel d) the arrows qualitatively sketch the temperature jump due
to a single heating event and the number of events necessary to induce the
switching to the resistive state.

##  II Josephson phase dynamics

YBa2Cu3O7-x (YBCO) off axis biepitaxial GB junctions provide a large variety
of transport regimes because of d-wave order parameter effects as well as the
modulation of the barrier transparency through different relative orientation
of the electrodeshans ; nuovo ; rop . Details on the preparation and
properties of the GB junctions used in this work are given in Refs. rop ;
supplemental ; LSAT . We present here data from junctions (JCT) with low (JCT
A), intermediate (JCT B) and high (JCT C) values of $J_{c}$. For each junction
$J_{c}$ has been determined as the ratio between the measured $I_{c}$ in the
I-V curves and the geometrical area of the junction (width $\times$ thickness
of the YBCO film)supplemental . For JCT C we have employed nanofabrication
techniques to be in the appropriate range of $I_{c}$dani_PRB ; gustafsson2 .
To study their escape rates we have thermally anchored the samples to the
mixing chamber of a He3/He4 Oxford dilution refrigerator and performed
measurements of SCDs. Measurement procedures and filtering used in this
experiment are described elsewhereluigi1 .

Switching histograms of JCTs A, B and C are shown in Fig. 1. The 3-D view
gives an intuitive picture: SCDs cover distinct current ranges, and when
$J_{c}$ increases, they become broader. In the right part of the figure, each
set of SCDs is displayed with appropriate scales for a better view of the
details of their temperature dependence. In Fig. 1e the escape rate $\Gamma$
out of the zero-voltage state as a function of the bias current $I$ has been
plotted for JCT C in the same temperature range. $\Gamma(I)$ curves have been
computed from the SCDs following Fulton and Dunkleberger fulton . The standard
deviation $\sigma$ of the histograms is reported for all junctions in Fig. 2a.
The combined analysis of $\sigma$, of the skewness $\gamma=m_{3}/\sigma^{3}$
($m_{3}$ being the third central moment of the distribution), and of their
temperature evolution characterize the phase dynamics.

The histograms from JCTs A and B match well the predictions of the RCSJ model
for moderately damped JJs. Low values of $I_{c}$kautz_secondo ; kivioja2005 ;
mannik ; luigi1 ; vion , especially in HTS d-wave junctions, where intrinsic
sources contribute to dissipationdani_PRB ; luigi2 , lead to the moderately
damped (MD) regime ($Q\simeq 1$). In the MD regime the SCDs become narrower
with increasing temperature for temperatures above $T^{*}$, which is defined
as the temperature at which $\sigma(T)$ is largest (in the case of JCT B
$T^{*}$ is about 1.0 K). This is often reported in the literature as the phase
diffusion (PD) regimekautz_secondo ; kivioja2005 ; mannik ; luigi1 ; luigi2 ,
indicating a phase diffusive dynamics due to multiple escape and retrapping
events in the washboard potential (sketched in Fig. 2b), or equivalently
described in terms of thermal activation above a dissipation barriervion .
Below $T^{*}$, SCDs obey the thermal activation (TA) regime. Upon further
lowering the temperature the junction can eventually enter the macroscopic
quantum tunneling regimedevoret1985 ; luigi2 ; Leggett ; Leggett_b ; Leggett_c
(see Fig. 1b, JCT A). The MD regime gives a very important reference for our
discussion of sample C. In both cases the width of the SCD decreases upon
increasing the temperature, as shown in Figs. 1 and 2. Only an exact fitting
of the SCDs at different temperatures can reveal the very different physics
occurring in the various junctions.

JCTs A and B of the present work have been chosen since they are fabricated on
different substrates (see Ref. supplemental, ) and both show the transition to
the PD regime, in which the derivative $d\sigma/dT$ is negative, but with a
different detailed $T$ dependenceluigi1 . The choice of the substrate affects
the effective capacitance of the circuit the junction is embedded
intopekolaLTP . As a consequence of this choice, for JCT B we observed the
transition from TA to the PD regime, while for JCT A the TA regime is
completely suppressed and we observed a direct transition from MQT to the PD
regimeluigi2 . We show below that the “anomalous” thermal behavior of high
$J_{c}$ JCT C cannot be explained in terms of frequency dependent damping or
as a consequence of the shell circuit, and the numerical simulations of Fig.
2c fully support this.

In JCTs A and B the rate of decrease in $\sigma$ above $T^{*}$ with increasing
$T$ is well described by the Monte Carlo fit of the phase dynamicssupplemental
(blue and red solid lines in Fig. 2a, respectively). The temperature
dependence of the skewness $\gamma$ gives additional distinctive criteria to
confirm the intermediate dissipation regimeluigi1 ; Bezryadin_ultimo . This is
signaled by a transition for JCT B for instance from $\gamma$ $\simeq$ -0.9 at
low temperatures ($T\simeq$ 0.1 K) in the TA to $\gamma$ $\simeq$ -0.1 above
the transition temperature $T^{*}\simeq$ 1.0 K in the PD, indicating a
progressive symmetrization of the SCDsluigi1  (see Fig. 3). Therefore, the
moderately damped regime gives the opportunity of comparing $\sigma(T)$ and
$\gamma(T)$ dependencies of SCD spectra at higher temperatures, which turn out
as unambiguous distinctive criteria for the different switching modes.

JCT C is characterized by high values of $J_{c}$ close to those observed in
nanowiressahu ; chang . This device exhibits radically different phase
dynamics above 3 K, which represents a transition temperature we indicate as
$T^{**}$. Also in the case of JCT C, $T^{**}$ is defined as the maximum of
$\sigma(T)$, but we prefer to use a different symbol to stress the distinct
switching dynamics, as pointed out below. The rate of decrease of $\sigma$
above $T^{**}$ turns to be a distinctive marker of the phase dynamics. In
device C the slope of $\sigma(T)$ above $T^{**}$ is much smaller when compared
to those of JCTs Aluigi2  and B and of all moderately damped JJs. The smooth
decrease of $\sigma$ for $T>T^{**}$ cannot be described in terms of the
intermediate dissipation regime, as evident from the numerical simulations
reported in Fig. 2c. According to the RCSJ model, keeping all the other
junctions parameters fixed, an increase of $J_{c}$ leads to an enhancement of
$I_{c}$ and $Q$. The increase of $Q$ moves $T^{*}$ to higher values, and the
negative slope of $\sigma(T)$ above $T^{*}$ becomes steeper and steeper (as
shown by the numerical simulations for $Q$ ranging between 1 and 10 reported
in Fig. 2c).

Figure 3: Temperature dependence of the skewness $\gamma$ of the SCDs measured
on JCTs B (red triangles) and C (black circles) respectively. Retrapping
processes in the PD regime cause a progressive symmetrization of the SCDs of
JCT B, signaled by the thermal dependence of $\gamma$, which is almost 0 above
the transition temperature $T^{*}$. Asymmetric SCDs, as the histogram measured
at $T=0.1K<T^{*}$ (shown in the top left corner of the figure), become
Gaussian-like distributions above $T^{*}$ (see the histogram measured at
$T=4K>T^{*}$ in the top right corner). The histograms of JCT C are asymmetric
over the whole temperature range, including for temperatures above $T^{**}$
(as shown by the SCD measured at $T=7$ K reported in the lower right corner of
the figure). The black dashed line indicates the mean value of the skewness
($\gamma\simeq-0.8$).

In addition, while moderately damped JJs show a progressive symmetrization of
the switching histograms near to and above $T^{*}$ (see JCT B in Fig. 3), SCDs
of JCT C are asymmetric over the entire temperature range. In JCT C, the
$\gamma$ factor is temperature independent, consistent with what is observed
in pure phase slip systemsBezryadin_ultimo  (see Fig. 3, the black dashed line
indicates the mean value $\gamma\simeq$-0.8 for JCT C). These behaviors are
quite distinct and do not fall in the framework of any regime of the RCSJ
model.

##  III Switching dynamics of high critical current density Josephson
junctions

We find that the numerical simulation of a transition driven by local heating
events accounts well for devices in the $J_{c}$ interval (104 A/cm2-105
A/cm2), as shown for JCT C in Fig. 4a. Hysteresis in I-V curvesdoh ; pekola2 ;
jung  does not necessarily indicate canonical Josephson phase dynamics, even
in the presence of a Fraunhofer magnetic field patternkleinsasser2 . It may
rather arise as a result of local heating processes, possibly induced by
intrinsic inhomogeneous composition unavoidable for high $J_{c}$ junctions.
Large values of Joule power density deposited in the weak link can induce a
self-heating process during the switch to the resistive branchpekola2 . The
absence of a set of self-consistent electrodynamics parameters to describe JCT
C is a strong indication of the failure of the standard Josephson dynamics.
This failure is of general relevance, applying both to conventional low
$T_{c}$ JJskleinsasser1 ; kleinsasser2  and to the emergent class of hybrid
nanoscale junctionsdoh ; morpurgo ; bouchiat . For larger values of $J_{c}$,
heating driven mechanisms become dominant with a transition to the normal
state locally in the junction area. These events are the mirror of non-
equilibrium processes and can be modeled as ”phase slips entities” (PSEs), as
confirmed by the details of the simulations which are reported in Refs.
supplemental, ; goldbart, ; lau, ; Leggett_RMP, , in the sense that they are
local processes, break the coherence of the phase information and are
described by a heat diffusion-like equation. In particular, the probability
for a single heating event can be still described in terms of the Langer-
Ambegaokar-McCumber-Halperin (LAMH) theorylanger  and further extensionszaikin
, and PSEs can still be approximately assumed to be far apart in time.

The typical time scale of a PSE is of the order of the Ginzburg Landau
relaxation time $\sim\tau(u)=(1-u)^{-1}\>\tau_{0}$ where
$\tau_{0}=\pi\hbar/8k_{B}T_{c}$ and $u=T/T_{c}<1$, with $T_{c}$ the
superconducting critical temperature. The energy dissipated by a PSE
$E_{PSE}(I)=\phi_{o}I$ is rather high, as the modulus of the order parameter
has to vanish over a length $\xi(T)=\xi_{0}(1-u^{4})^{1/2}/(1-u^{2})$, where
$\xi_{0}$ is the zero temperature coherence length in the superconductor and
$\phi_{o}$ is the flux quantum. Following an approach proposed in Ref.
goldbart,  for low critical temperature superconducting (LTS) wires, our
numerical simulation of the temperature jump induced by a PSE obeys the
phenomenological diffusive equation for the relaxation of the temperature
gradient:

$\frac{d\delta T}{dt}+\alpha\left(T,T_{b}\right)~{}\delta
T=r(T_{b},t)+\eta(T,I)\sum_{i}\delta(t-t_{i})\>.$ (1)

Here $\delta T=T-T_{b}$ is the deviation from the temperature of the bath
$T_{b}$. The relaxation coefficient $\alpha\left(T,T_{b}\right)$ depends on
the thermal conductivity $K(T)$, on the thermal capacity ${\cal{C}}_{v}(T)$ of
the phase slip volume, and on $T_{b}$supplemental . $r(T_{b},t)$ is the noise
source due to the environment with an admittance $Y(\omega)$, while
$\eta(T,I)$ is the temperature jump due to the PSEs which occur at the
stocasticaly distributed times $t_{i}$supplemental . $\eta(T,I)$ is implicitly
defined by the following equation:

$\phi_{o}\>I=\int^{T+{\eta}(T,I)}_{T}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dT^{\prime}\>\>{\cal{C}}_{v}(T^{\prime}).$
(2)

In Fig. 4a we show the SCDs derived from the experiment on JCT C over a wide
temperature range, $T\in[0.25K,10K]$. The continuous red curves correspond to
the fit obtained integrating eq. 1. To quantify how many heating events are
needed to escape to the finite voltage state, we define the threshold
temperature $T_{th}$, i.e. the temperature above which the system is
definitely out of the zero voltage state. At low temperatures, $T<T^{**}$, our
fitting procedure does not depend on $T_{th}$. This is the regime when a
single heating event is enough to drive the transition. Following Ref.
goldbart, , we assume that both the specific heat and the thermal conductivity
$K(T)$ are weighted averages of their BCS and Fermi liquid limits. Two main
effects discriminate between the low temperature and the high temperature
behavior.

Figure 4: a) SCDs measured on JCT C (black circles) along with the fit (red
solid lines) resulting from numerical simulations of heat diffusion-like
dynamics, according to eq. 1. b) Fitting procedure above $T^{**}$: single
heating event dynamics does not reproduce the experimental histogram,
signaling a multiple PSE process. Dissipation due to heat relaxation plays a
relevant role. c) Temperature dependence of $T_{eff}$ estimated from numerical
simulations of eq. 1. Below $T^{**}$, a single heating event induces the
switching and $T_{eff}>T_{b}$. Above $T^{**}$, $T_{eff}$ and $T_{b}$ coincide
since the system is able to thermalize in the time interval between well
separated heating events, and multiple PSEs are needed to overcome $T_{th}$.

The temperature jump $\eta(T)$ depends on temperature because the specific
heat is strongly temperature dependent. At low temperatures the specific heat
is quite low, thus with each PSE there is a considerable increase in the
temperature. In addition, the thermal conductivity and the thermal capacity
are both quite low as the system is deeply into the superconducting phase. In
a rough sense, the system is rather isolated from the environment and the
thermal shock due to a heating event is destructive for the superconducting
state. For this reason a single heating event can induce a direct jump to the
resistive state at low temperatures: it induces a large local heating that is
difficult to dissipate. The system is not at equilibrium with its environment,
and we can define an effective temperature $T_{eff}$ for the system, which is
higher than $T_{b}$ as shown in Fig. 4c.

At high temperatures we are in the opposite regime of small $\eta$(T) per
heating eventsupplemental . In addition both the thermal conductivity $K(T)$
and the thermal capacity ${\cal{C}}_{v}(T)$ increase with increasing
temperature as well. Thermal diffusion and contact with the environment is
more effective and multiple PSEs are required for switching. This occurs above
$T^{**}$, where the derivative $d\sigma/dT$ is negative. $T_{eff}$ and $T_{b}$
coincide above $T^{**}$ (see Fig. 4c), which we interpret as the temperature
at which the system is able to thermalize during the time interval between
well separated heating events. In this temperature range, the value chosen for
the fitting parameter $T_{th}$ becomes important to quantify the number of
successive PSEs, which are responsible for the transition. In Fig. 4b the
difference between a single and multiple heating events is shown. At 7 K for
instance about 4-6 PSEs are needed to reproduce the experimental SCD.
Experimental SCDs complemented by the numerical simulations follow the passage
from single to multiple heating events. This is similar to what has been
observed in LTS nanowiressahu ; chang ; goldbart . A consistent set of the
junction parameters (temperature jump $\eta$, number of heating events) can be
extracted from these simulations, as discussed in Ref. supplemental, .

##  IV Discussion and concluding remarks

The passage above a characteristic $J_{c}$ threshold to a state that does not
sustain Josephson phase-coherence should be universally expected in weak link
systems. The threshold probably depends on the type of junction, responding to
structurally different ways of enhancing barrier transparency and generating
sources of local heating. The threshold of $10^{4}-10^{5}$ A/cm2 found for our
junctions is consistent with the general empiric rule on HTS GBshans ; rop .
In bicrystal junctions, an increase in the misorientation angle between the
two electrodes determines a well-known decrease of the critical current
density. For small misorientation angles devices are found in the flux-flow
regime. For $\theta>20^{\circ}$ Josephson phenomenology starts to appear for
$J_{c}$ values of $10^{3}$ A/cm2 hans ; rop . Several effects have also been
observed in conventional trilayer Nb/Al-Oxide/Nb JJs above the same threshold
of about $10^{4}$ A/cm2 that could be attributed to non-equilibrium
dynamicskleinsasser1 ; kleinsasser2 .

Some analogies can be established with what is observed in 4He
superfluidhokinson . Here the passage from weak to strong coupling manifests
itself in a change in the current-phase relation [$I(\varphi)$]. In the strong
coupling regime, where the healing length of the superfluid is lower than the
diameter of the constriction, deviations from the $\sin\varphi$ relation
appear, while sinusoidal Josephson oscillations have been measured in the
opposite limithokinson . An increase in $J_{c}$ and in the coupling between
the electrodes leads to the presence of other harmonics in the $I(\varphi)$,
which might become multivaluedlikharev .

A heat diffusion-like model breaking phase-coherent information is consistent
with our data on high $J_{c}$ JCT C. Here the switch to the normal state is
accompanied by a local release of energy characteristic of a PSE. When
departing from the supercurrent branch, non-equilibrium processes produce an
unexpected heating. This is surprisingly different from what commonly is
accepted for hysteretic Josephson junctions, where heating can only influence
the retrapping phenomena, as a memory of the history of heating in the
resistive stateLeggett_RMP ; castellano .

In the framework of the RCSJ model the resistance arises from non-local
properties well described by the (frequency dependent) quality factor $Q$ of
the circuit. This reminds us of the Landauer picture of quantum transport, in
which quantum interference acts at the interface while the dissipation and
memory loss occurs in the contacts. On the other hand, when the switch is
driven by PSEs, it is the local properties of the order parameter which are
important: phase memory is lost when the modulus of the order parameter
becomes zero. The model based on local heating events can be in principle
further extended to extract the $I(\varphi)$ and to fully define the role of
dissipation in high $J_{c}$ junctions.

To conclude, standard phase dynamics of a hysteretic Josephson junction
collapses and cannot be sustained above some threshold of the critical current
density $J_{c}$. In high $J_{c}$ devices information is lost in non-
equilibrium dynamics, and can be partly codified in the local heating process.
This is of great relevance for all the experiments using low-dimensional
barriers, which should be concerned about possible heating effects, leading to
distorted phase information. Non-equilibrium effects would obviously
invalidate a large number of key predictions for Majorana fermions and
nanoscale superconductivity in Josephson junctions, since for instance they
can give rise to zero bias anomalies in conductance measurementsargaman1 . The
proof of quantifiable non-equilibrium processes already in the thermal regime
poses severe constraints on the possible occurrence of macroscopic quantum
phenomena at lower temperatures in high $J_{c}$ samples through standard SCD
measurements.

###### Acknowledgements.

The authors are indebted to Professor A. J. Leggett and Professor J. R.
Kirtley for inspiring conversations and for a careful reading of the
manuscript. The authors acknowledge financial support by Marie Curie Grant No.
248933, by Progetto FIRB HybridNanoDev RBFR1236VV, and by COST Action MP1201
(NanoSC COST).

## References

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