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\title{M\"unchhausen effect: tunneling in an asymmetric SQUID}
\classification{85.25.Dq, 74.50.+r}
\keywords{SQUID, Macroscopic Quantum Tunneling}
\author{A.U. Thomann}{address=
{Institute for Theoretical Physics, ETH Zurich, 8093 Zurich,
Switzerland}}
\author{V.B. Geshkenbein}{address=
{Institute for Theoretical Physics, ETH Zurich, 8093 Zurich,
Switzerland \\
L.D. Landau Institute for Theoretical Physics, 117940 Moscow,
Russia}}
\author{G. Blatter}{address=
{Institute for Theoretical Physics, ETH Zurich, 8093 Zurich,
Switzerland}}





%\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{abstract} 
%
A classical system cannot escape out of a metastable state at zero
temperature.  However, a composite system made from both classical and
quantum degrees of freedom may drag itself out of the metastable state by a
sequential process. The sequence starts with the tunneling of the
quantum component which then triggers a distortion of the trapping
potential holding the classical part. Provided this distortion is
large enough to turn the metastable state into an unstable one, the 
classical component can escape. This process reminds
of the famous baron M\"unchhausen who told the story of rescuing himself
from sinking in a swamp by pulling himself up by his own hair---we thus term
this decay the `M\"unchhausen effect'.  We show that such a composite system
can be conveniently studied and implemented in a dc-SQUID featuring asymmetric
dynamical parameters.  We determine the dynamical phase diagram of this
system for various choices of junction parameters and system preparations.
%
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Introduction}
\label{sec:introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Consider a very heavy particle ($M \to \infty$) trapped in a metastable
potential minimum at temperature $T=0$. Can it escape? The answer seems to be
obvious: thermal activation is forbidden, since we are at zero
temperature and tunneling is not possible either, because the particle is
heavy. Matters are not that simple, however, if this heavy object is a
composite one, with a quantum (light) degree of freedom coupled to the
classical (heavy)
one. Then the light part may tunnel out of the metastable minimum and
exert a pulling force on the classical part. Once the latter is large
enough to suppress the trapping barrier, the classical
object is able to leave the potential well---hence a classical object
may escape from a metastable state  
if helped by a coupled quantum degree of freedom. Two interacting
particles with coordinates $x_1, x_2$ in 1D can be viewed as one
particle in a 2D potential $V(x_1,x_2)$. Here  $V(x_1,x_2)$ takes into
account both the external potential and the interaction between the
particles. Then the heavy and light
particles map to an anisotropic mass in this problem. The M\"unchhausen
decay corresponds to tunneling ``along the light axis'' and sliding
thereafter along the heavy one. It is clear, that there is no problem
to imagine such a 2D potential.




The above situation can be implemented experimentally in a dynamically 
asymmetric dc-SQUID; it is this specific realization which we will study in
detail in this paper, see Fig.\ \ref{fig:setup}. The dynamical degrees
of freedom in the dc-SQUID are the gauge-invariant phase differences 
$\varphi_{i},\  i=1,2$, across the two Josephson junctions. 
The potential energy (of a single Josephson junction) is given by 
$\mathcal{V}_{i}=E_{J}(1-\cos\varphi_{i}),\ i=1,2$, 
involving the Josephson energy $E_{J}=\Phi_{0}I_{c}/2 \pi c$ (with the 
flux unit $\Phi_{0}=hc/2e$ and the critical current $I_{c}$ of the
junction). 
The kinetic energy reads
$\mathcal{T}_{i}=(\hbar/2e)^2C_{i}\dot\varphi_{i}^2/2$,
 where the capacitances $C_{i}$ assume the role of effective masses. 
Hence, a SQUID featuring two Josephson junctions with equal critical 
current $I_{c}$ but adequately chosen and strongly asymmetric
capacitances, one large and one small ($C_{1}\gg C_{2}$),
 effectively provides us with a classical and a quantum degree of freedom. 
%
\begin{figure}[htb]
\centering
\includegraphics{SQUID}
\centering
\caption{Schematics of the dynamically asymmetric dc-SQUID. 
Two Josephson junctions with equal critical current $I_{c}$ but
strongly asymmetric capacitances ($C_{1}\gg C_{2}$) and resistances
$R_{i}$ are integrated in a biased (current $I$) superconducting loop 
with symmetric inductance $L$.} 
\label{fig:setup}
\end{figure}
%

The decay process of the biased dynamically asymmetric SQUID
proceeds in the following manner: The bias $I$, leading to a term 
$\propto -I (\varphi_{1}+\varphi_{2})$
in the potential, turns a stable state of the washboard potential into
a metastable one. 
As junction 1 features a large capacitance, we assume its dynamics to 
be strictly 
classical. If the bias $I$ is large enough, a (imaginary time) 
decay process involving only the quantum junction (at constant
$\varphi_{1}$) is enabled.
This phase slip leads to the entry of magnetic flux into the ring.
Given the inductive coupling $\propto1/L$  (inductance $L$),  
the current through the classical junction is enhanced and it may
eventually become 
overcritical, thus decaying via a classical real time trajectory. 


\section{Setup}
\label{sec:setup}

We start from the capacitively shunted junction model (CSJ), where the 
dc-SQUID, biased with a current $I$, is described by the Lagrangian
%
\begin{multline}
   \mathcal{L}=\sum_{i=1}^2 \bigg[\left(\frac{\Phi_{0}}{2\pi
       c}\right)^2\frac{C_{i}}{2}\dot\varphi_{i}^2 
   -E_{J}(1-\cos\varphi_{i})  \bigg]\\
   +\frac{\Phi_{0} I}{2 \pi c} \frac{\varphi_{1}+\varphi_{2}}{2}-
   \left(\frac{\Phi_{0}}{2\pi
       c}\right)^2\frac{(\varphi_{1}-\varphi_{2})^2}{2 L};
\end{multline}
%
here, we have assumed that the inductance $L$ of the SQUID is
symmetrically distributed.
The Lagrangian $\mathcal{L}$ generates the equations of motion
%
\begin{equation}
   m_{i}\ddot\varphi_{i}+\eta_{i}\dot\varphi_{i}=-\partial_{\varphi_{i}}
   v(\varphi_{1},\varphi_{2}),
\label{eq:motion}
\end{equation}
%
with the `masses' $m_{i}=\Phi_{0}C_{i}/2\pi c I_{c}$ and where we have 
added the dissipative terms $\eta_{i}\dot\varphi_{i}$ with the damping 
parameters $\eta_{i}=\Phi_{0}/2 \pi c I_{c}R_{i}
\propto 1/R_{i}$, $R_{i}$ the normal resistances of the junctions;
the potential (illustrated in Fig.\ \ref{fig:v}) is given by
%
\begin{multline}
   v(\varphi_{1},\varphi_{2})=2-\cos\varphi_{1}-\cos\varphi_{2}\\
   -j (\varphi_{1}+\varphi_{2})+\frac{k}{2}(\varphi_{1}-\varphi_{2})^2,
\label{eq:v}
\end{multline}
%
with the dimensionless current $j=I/2 I_{c}$, the coupling constant 
$k=\Phi_{0}/2 \pi c I_{c}L$ 
and where energies are measured in units of $E_{J}$.

%
\begin{figure}[htb]
\centering
\includegraphics[width=5.5cm]{contour4}
\centering
\caption{Illustration of the decay sequence of the dynamically
asymmetric SQUID ($j=0.5,\ k=0.04$). The initial metastable well is 
unstable w.r.t. the macroscopic quantum tunneling 
of the small junction 2. A continuous sequence of phase slips takes
the system to a state which is classically unstable. In the following,
the (classical) relaxation of the large junction 
1 and the quantum decay of junction 2 alternate and lead to a finite
voltage state of the SQUID.} 
\label{fig:v}
\end{figure}
%



Residing in a symmetric ($\varphi_{1}=\varphi_{2}=\arcsin i$)
metastable state of the potential $v(\varphi_{1},\varphi_{2})$ at
finite bias current $i$, the classical version of 
the system described through Eq.\ \eqref{eq:motion} cannot decay at
zero temperature. 
Here, we are investigating the case where junction one, featuring a
large capacitance $C_{1}$, is assumed to behave strictly classical, 
whereas the dynamics of junction 2 is characterized by large quantum 
fluctuations. This can be achieved through a suitable choice 
of parameters, i.e., a small capacitance $C_{2}$, while 
keeping $E_{J}\gtrsim E_{C2}=e^2/2 C_{2}$ such that we remain in a 
quasi-classical regime.

In the system under consideration, different scenarios can arise
depending on the strength of the dissipation as quantified by the 
dimensionless damping parameter 
%
\begin{equation}
\alpha_{i}=(2 R_{i}C_{i}\omega_{pi})^{-1},\ i=1,2,
\end{equation}
%
where, at $j=0$, the plasma frequency 
$\hbar\omega_{pj}=(8 E_{J}E_{Ci})^{1/2}$. 
The simplest case is the overdamped situation $\alpha_{1},\alpha_{2}>1$,
where the dynamics of the classical junction is viscous and both
relaxation and tunneling of the quantum junction are incoherent   
\cite{Leggett:1987, Caldeira:1983}. We will analyze
this situation in detail in the next section; the obtained results are
also relevant for other choices of parameters, c.f. below.

\section{Phase diagram - strong damping}
\label{sec:pd}

In the following, we determine 
for which currents $i$ and coupling constants $k$ a zero temperature 
decay of a symmetric metastable state 
($\varphi_{1}=\varphi_{2}=\arcsin i$, up to an arbitrary multiple of $2\pi$) 
is allowed in the interferometer potential $v(\varphi_{1},\varphi_{2})$. 
The result is displayed in a dynamical phase diagram in the $j$-$k$-plane,
see Fig.\ \ref{fig:pd_vv}, 
where the critical line $j_{c}(k)$ separates regions where this decay
is prohibited (localized state) 
from regions where it is allowed (delocalized).

Assuming junction 1 to behave strictly classical, 
(i.e. considering the limit of very large $C_{1}$), 
a quantum decay of the SQUID in a metastable state can only occur at
fixed $\varphi_{1}$, i.e. 
through an imaginary-time trajectory of $\varphi_{2}$ in the effective
potential $v_\mathrm{eff}
(\varphi_{2})=v(\varphi_{1}=\mathrm{const.},\varphi_{2}$), see Fig.\
\ref{fig:effpot}.
We will adopt this approximation for all tunneling processes
throughout the discussion.
%
\begin{figure}[htb]
\includegraphics{effpot}
%
\caption{Effective potential $v_\mathrm{eff}(\varphi_{2})=v(\varphi_1
=\mathrm{const.}, \varphi_2)$ (solid line) and the parabola remaining
after
dropping $\cos\varphi_2$ (dashed line) for $j=0.5$, $k=0.02$ and
$\varphi_1=\arcsin j$. The bullets and arrows illustrate the sequential
decay
of the quantum phase $\varphi_2$ to the ground state for the case of
strong
damping.}
%
\label{fig:effpot} 
\end{figure}
%
For sufficiently large $j$, the quantum degree of freedom $\varphi_2$
undergoes tunneling to a new local minimum nearby $2\pi n$, while the
classical degree of freedom $\varphi_1 =\arcsin j$ remains localized, thus
allowing a flux $\simeq n\Phi_0,\ n \in \mathbb{N}$ to enter the SQUID
loop,
cf.  Fig.\ \ref{fig:effpot}. If the resulting force on the classical
phase $\varphi_1$ is sufficiently large, the M\"unchhausen decay is
enabled
with a classical decay of $\varphi_1$ and successive iteration of quantum
decay (directed along $\varphi_2$, flux entry) and classical relaxation
(directed mainly along $\varphi_1$, flux exit), cf. Fig.\
\ref{fig:v}.





For undercritical currents $j<1$ the described initial state, 
$\varphi_{1}=\varphi_{2}=\arcsin j$, 
is stable against a decay involving the classical junction; however it
is, for $k<j/(\pi-\arcsin j)$, 
unstable with respect to a quantum decay of $\varphi_{2}$ since the minimum of 
$v_\mathrm{eff}(\varphi_{2})$ near 
$\varphi_{2}\approx 2\pi$ is lowered below the initial one
and one or more phase slips of $\varphi_{2}$ are possible.
In order to determine whether the SQUID will remain in a stable or
enter a finite voltage state at given $j,k$
we proceed in two steps: First, we have to determine where the successive 
quantum tunneling of $\varphi_{2}$ comes to a halt, 
i.e., which side minimum is quantum-stable. This is equivalent to
finding the global minimum of the effective potential 
$v_\mathrm{eff}(\varphi_{2})$. Fixing the phase across the classical
junction $\varphi_{1}=\arcsin j$, one immediately sees that 
the quantum-stable minimum is the one near $\varphi_{2}\approx 
2 \pi n$ if $n$ is the largest integer, such that
%
\begin{equation}
   k<k^+_{c,n}(j)\approx\frac{j}{(2 n-1)\pi-\arcsin j}.
   \label{eq:approx-globmin}
\end{equation}
%

The sequence of phase slips of the quantum junction leads to an
accumulation of magnetic flux in the SQUID loop, inducing a screening 
current. Consequently, the current through junction 
2 is reduced whilst that through junction 1 is increased. The
magnitude of the induced current 
depends strongly on the coupling constant $k$; for given $n$, 
it is only large enough to drive the classical junction overcritical if 
%
\begin{equation}
  k > k^-_{c,n}(j)\approx\frac{1-j}{(2n-1/2)\pi+\arcsin(2j-1)}.
   \label{eq:approx-hesse}
\end{equation}
%
If this is the case, the SQUID enters a finite voltage state where
quantum tunneling of junction 2 
(an approximate flux unit enters the loop)
and classical relaxation of junction 1 (flux leaves the loop)
interchange sequentially (see Fig.\ \ref{fig:v}). 
If $k<k^-_{c,n}(j)$, the SQUID resides in a
localized state and a further increase in $j$ is necessary to drive
the system unstable.
The two conditions Eqs. \eqref{eq:approx-globmin} and
\eqref{eq:approx-hesse} generate a web of 
crossing lines in the $j,k$-plane ($j^+_{c,n}(k)$ and $j^-_{c,n}(k)$ are
inversed expression to Eqs. \eqref{eq:approx-globmin} and
\eqref{eq:approx-hesse} correspondingly), determining the critical line
$j_{c}(k)$ marking the dynamic transition 
from a localized to a delocalized state (Fig.\ \ref{fig:pd_vv}).
In the limit $k\rightarrow 0$, where the cosine in the potential 
Eq. \eqref{eq:v} becomes a small 
correction to the parabola, the critical line $j_{c}(k)$ 
approaches 1/2; this indicates that all current is redirected through 
junction 1 and delocalization takes place at $I=I_{c}$, the critical
current of a single junction,
%
\begin{figure}[htb]
\includegraphics{vv2}%
\centering
%
\caption{Phase diagram of the dynamically asymmetric dc-SQUID as a
function of
bias current $j=I/2 I_c$ and inductive coupling $k=\Phi_0 c/2\pi  LI_c$.
Here,
we assume strong damping $\alpha_{p,1}\gg1$ and $\alpha_{p,2}>1$. The
effective critical current $j_c(k)$ (solid line) marks the boundary
between a
localized classical junction (lower bias $j < j_c$) and a delocalized
classical junction ($j > j_c$), corresponding to a finite voltage state of
the
SQUID. Branches with negative slope are determined by a classical
instability
(mainly along $ \varphi_1$), while those with positive slope are
determined by
a quantum instability of the light junction.  For $j<j_c(k)$, the dotted
lines
$j_{c,n}^+(k)$ mark the entry of flux through the quantum junction (the
integer $n$ approximately quantifies the flux through the ring in
the stable state); these lines can be measured via monitoring of the flux
threading the loop. For $j>j_c(k)$, the dashed lines $j_{c,m}^-(k)$ mark
the
minimum number $m$ of flux units necessary to delocalize the classical
junction. The inset shows a comparison between the approximate result 
(solid line) and the exact numerical result (dashed).}
\label{fig:pd_vv}
\end{figure}
%

The simple arguments above have to be refined in order to obtain the
precise location of the critical line $j_{c}(k)$. 
First, the condition of classical stability is 
  but the standard determination of the critical current of a dc-SQUID's 
asymmetric minimum \cite{Tsang:1975}. 
In our case, where classical stability along the $\varphi_{2}$-direction 
is guaranteed, the relevant set of equations is given by
%
\begin{eqnarray}
   \sin(\bar\varphi_{1}^{n})&=&j-k_{{c,n}}(\bar\varphi_{1}^{n}-
   \bar\varphi_{2}^{n}),\label{eq:hesse1}\\
   \sin(\bar\varphi_{2}^{n})&=&j+k_{{c,n}}(\bar\varphi_{1}^{n}-
   \bar\varphi_{2}^{n}),\label{eq:hesse2}\\
   \cos \bar\varphi_{1}^{n}\cos \bar\varphi_{2}^{n}
   &=& -k_{c,n}( \cos \bar\varphi_{1}^{n}+\cos \bar\varphi_{2}^{n}),
  \label{eq:hesse3}
\end{eqnarray}
%
and has to be solved (numerically) for $k_{c,n}(j)$ and  
$\bar\varphi_{1,2}^n$, the coordinates of the true minima 
near $\varphi_{1}=\arcsin j$,  $\varphi_{2}=2 \pi n$.
Eq.\ \eqref{eq:approx-hesse} is
an approximate solution to Eqs.\ \eqref{eq:hesse1}-\eqref{eq:hesse3}
in the limit of $k\ll 1$.
Second, as tunneling of the quantum junction might be enabled only after the 
relaxation of the classical junction to a minimum, condition
Eq.\ \eqref{eq:approx-globmin} has to be corrected to 
%
\begin{equation}
   k^+_{c,n}(j)=\frac{j}{(2 n-1)\pi-\bar\varphi_{1}^{n-1}},
   \label{eq:globmin}
\end{equation}
%
taking into account the change of the effective potential 
$v_\mathrm{eff}(\varphi_{2})$ upon 
a change in $\varphi_{1}$.
The exact numerical solutions 
of Eqs.\ \eqref{eq:hesse1}-\eqref{eq:hesse3} and \eqref{eq:globmin}
are shown in the inset 
of Fig.\ \ref{fig:pd_vv}, where the approximate solution is seen to be
rather precise.


The critical current line $j_c(k)$ is constructed from interchanging
segments
of $j_{c,n}^+(k)$ and $j_{c,n}^-(k)$, resulting in the dynamical phase
diagram, Fig.\ \ref{fig:pd_vv}. Note that the different nature
of the decay, classical or quantum, associated with the two types of critical
lines may allow for an experimental distinction: Ramping the current past
a $^+$-type segment of $j_c(k)$ triggers a \emph{quantum} decay with a broad
histogram describing multiple measurements. A $^-$-type segment of
$j_c(k)$ triggers a \emph{classical} decay with a sharp histogram (the quantum
decay of $\varphi_2$ needs to have occurred already, which limits the 
ramping speed before reaching the critical line). Note that the lines 
$j_{c,n}^+(k)$ are detectable throughout all the stable portion of the
phase diagram, e.g., via a measurement of the flux threading the loop 
(the flux increases by approximately one flux unit upon crossing the 
dotted lines in Fig.\ \ref{fig:pd_vv}).







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weak damping}
\label{sec:underdamped}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The behavior of the system for weak damping $\alpha_{LC,2}\ll 1,\
\alpha_{p,i}\ll 1$ strongly depends on preparation. Here we are not
going to discuss all the possible situations. Instead, we
concentrate on the most interesting case.
Depending on parameters, after tunneling the quantum degree of freedom may
become delocalized between  different local wells of the `super-well' 
Fig.\ \ref{fig:effpot} . These side-well states have different lifetimes. 
 The dissipation described by the normal
resistance $R$ in the RCSJ-equation of motion, cf.\ Eq.\
\eqref{eq:motion},
leads to typical finite lifetimes $\tau_2 \sim RC_2= 1/2\alpha_2
\omega_{p,2}$ of the excited states $|l \rangle$ of the quantum
junction.\cite{esteve:1986}  A much longer lifetime $\tilde\tau_2$ shows
up if
the quantum junction is trapped in a local ground state of a side-well in
$v_\mathrm{eff}(\varphi_2)$, cf.\ Fig.\ \ref{fig:decay}. The decay then is
protected through a large barrier $E_b$, enhancing the typical lifetime to
a
value $\tilde \tau_2$, with $\tau_2 \ll \tilde \tau_2 \propto
 \exp(2\gamma\sqrt{E_J/4E_{c,2}})$.\cite{Averin:2000}
Thus, there are two (extreme) ways how a highly exited state in the
`super-well' of $v_\mathrm{eff}(\varphi_2)$ can decay, see Fig.\
\ref{fig:decay}, either via states within the super-well involving the
typical lifetime $\tau_2$, or via states involving a tunneling
process, resulting in an exponentially larger decay time of the order
of $\tilde\tau_2$.
%
\begin{figure}[htb]
\begin{center}
\includegraphics{effpot3}
%
\caption{Illustration of decay sequences of an excited `super-well'-state
of
the quantum junction in the effective potential
$v_\mathrm{eff}(\varphi_2)$. Solid arrows indicate a decay sequence involving
only states within the `super-well' with a typical decay time $\tau_2$. The
dashed arrows illustrate another (extreme) decay sequence, involving localized
states
in the side-minima of $v_ \mathrm{eff}(\varphi_2)$. The decay time then
involves the typical lifetime $\tilde \tau_2 \gg \tau_2$ of local ground
states within side-wells. }
\label{fig:decay}
%
\end{center}
\end{figure}
%

The quantum junction may be  trapped in any side-well for a
significant amount of time. If it gets trapped in a distant side-well
(large values of $\varphi_2$ in Fig.\ \ref{fig:decay})  it exerts a
larger force on the classical junction. The classical junction is
pulled out of its metastable state more efficiently in this case. 
We can define two extreme values for the critical current, a lower limit
$j_{c}^{\scriptscriptstyle (1)}(k)$ arising from the largest attainable
fluxon index $n$, and an upper limit $j_{c}^{\scriptscriptstyle (2)}(k)$
associated with the index $n$ of the global minimum of the quantum junction. 
The resulting phase diagram is displayed in Fig.\
\ref{fig:farthest}.
%
\begin{figure}[htb]
\begin{center}
\includegraphics{farthest_mod}
%
\caption{Effective critical currents of the dynamically asymmetric SQUID
for
$\tilde \tau_2\gg \omega_{p1}^{-1}$ and fast ramping. The system always
turns resistive for $j>j_{c}^{\scriptscriptstyle (2)}(k)$ but never turns
resistive for $j<j_{c}^{\scriptscriptstyle (1)}(k)$. 
For $j_{c}^{\scriptscriptstyle
(1)}<j<j_{c}^{\scriptscriptstyle (2)}$ the delocalization of the system is
determined by the statistical nature of the decay process of the quantum
junction involving the side minima.}
%
\label{fig:farthest}
\end{center}
\end{figure}
%

The two critical lines $j_{c}^{\scriptscriptstyle (1)}(k)$ and
$j_{c}^{\scriptscriptstyle (2)}(k)$ define a broad intermediate regime,
see the grey area in Fig.\ \ref{fig:farthest}, where the decay of the system
is of probabilistic nature.  The system is localized for
$j<j_c^{\scriptscriptstyle (1)}(k)$ and delocalized for 
$j>j_{c}^{\scriptscriptstyle (2)}(k)$.  The decay probability in the
intermediate regime is neither 0 nor 1, but depends on which minimum
the quantum junction $\varphi_2$ decays to. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental Implementation}
\label{sec:remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The running state following a M\"unchhausen decay is associated with a
finite
time-averaged voltage across the device. The above results thus can be
tested
experimentally by measuring the voltage drop as a function of the applied
bias
current $I$ and of the inductance $L$ of the SQUID.

Mapping any part of $j_c(k)$ requires changing the coupling $k\propto
L^{-1}$. It is difficult to vary the induction of the loop, which is
usually fixed after fabrication. One (although rather difficult) way
to modify it is to install a diamagnetic shield.
Another way to trace the critical
current line is obtained by applying an external magnetic flux $\Phi_e$ to
the
sample, in which case the potential Eq.\ \eqref{eq:v} has to be replaced
by
%
\begin{multline}
v(\varphi_1,\varphi_2)=1-\cos\varphi_1+1-\cos\varphi_2\\ 
-j(\varphi_1+\varphi_2)+\frac{k}{2}(\varphi_1-\varphi_2-2\pi\Phi_e/\Phi_0)^2.
\label{eq:v_phi_e}
\end{multline}
%
The critical current $j_c$ then can be studied as a function of $\Phi_e$.
The
analysis proceeds in the same way as before: a change in $k$
altering the opening angle of the parabola in $v(\varphi_1,\varphi_2)$ is
replaced by a shift in the parabola's position due to the applied flux
$\Phi_e$.
%
\begin{figure}[htb] 
\begin{center} 
\includegraphics{penew} 
%
\caption{Phase diagram of the dynamically asymmetric SQUID in the strong
damping case as a function of the externally applied flux $\Phi_e$ and
bias
current $j$. Here, the coupling $k \propto L^{-1}$ is fixed at a value 
$k=0.06$; the critical lines in the diagram correspond to $n=2$. The
critical line is periodic in $\Phi_e$ with a period $\Phi_0$, a
consequence of
the invariance under the replacements $\Phi_e/\Phi_0 \rightarrow
\Phi_e/\Phi_0\pm 1$ and $n\rightarrow (n \pm1)$.}
%
\label{fig:pe} 
\end{center} 
\end{figure}
%

The results  are invariant under the
shifts $\Phi_e/\Phi_0 \rightarrow \Phi_e/\Phi_0\pm 1$ and $n \rightarrow
(n\pm1)$. The M\"unchhausen decay thus can be studied within a finite
interval, e.g. $\Phi_e/ \Phi_0\in[-0.5,0.5]$, and the resulting phase
diagram
is displayed in Fig.\ \ref{fig:pe}.

Biasing the SQUID with an external magnetic flux has a convenient side
effect:
For negative flux $\Phi_e$, the crossing point of $j_{c,n}^+(k)$ and
$j_{c,n}^-(k)$ is shifted to larger values of $k$ and the tunneling
barrier
for $\varphi_2$ is lowered, allowing to study the M\"unchhausen effect
using a
SQUID with considerably smaller inductance and hence smaller size.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have shown that a system  consisting of two degrees of freedom 
can escape out of a metastable state, 
even if one of the degrees  behaves \emph{fully} classical, provided the
second one shows quantum behavior. This can be realized in
a dynamically asymmetric dc-SQUID. The ``M\"unchhausen-decay''
involves tunneling of the quantum junction, where magnetic flux  
accumulates in the superconducting ring and eventually redirects 
enough current through the classical junction 
as to drive it overcritical. The resulting phase diagram of the SQUID
appears to be unexpectedly rich.\\
\\ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theacknowledgments}
%
We thank A.\ Larkin, A.\ Ustinov, G.\ Lesovik, A.\ Lebedev, A.\ Wallraff,
and
E.\ Zeldov for interesting discussions and acknowledge support of the
Fonds
National Suisse through MaNEP.
%
\end{theacknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{10}



\bibitem{Caldeira:1983}
A.O.\ Caldeira and A.J.\ Leggett,
 \emph{Ann.\ Phys.\ (NY)} \textbf{149}, 374 (1983).

\bibitem{Leggett:1987}
A.J.\ Leggett, S.\ Chakravarty, A.T.\ Dorsey, M.P.A.\ Fisher, A.\ Garg,
and W.\
  Zwerger,
 \emph{Rev.\ Mod.\ Phys.\} \textbf{59}, 1 (1987).

\bibitem{Tsang:1975}
W.-T.\ Tsang and T.V.\ Duzer,
 \emph{J.\ Appl.\ Phys.\ }\textbf{46}, 4573 (1975).

\bibitem{esteve:1986}
D.\ Esteve, M.H.\ Devoret, and J.M.\ Martinis,
 \emph{Phys.\ Rev.\ B} \textbf{34}, 158 (1986).

\bibitem{Averin:2000}
D.V.\ Averin, J.R.\ Friedman, and J.E.\ Lukens,
 \emph{Phys.\ Rev.\ B} \textbf{62}, 11802 (2000).
  



\end{thebibliography}

\end{document}




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