Josephson junction with magnetic-field tunable current-phase relation
A. Lipman
We consider a 0-$\pi$ Josephson junction consisting of asymmetric $0$ and $\pi$ regions of different lengths $L_0$ and $L_\pi$ having different critical current densities $j_{c,0}$ and $j_{c,\pi}$. If both segments are rather short, the whole junction can be described by an effective current-phase relation for the spatially averaged phase $\psi$, which includes the usual term $\propto\sin(\psi)$, a negative second harmonic term $\propto\sin(2\psi)$ as well as the unusual term $\propto H \cos\psi$ tunable by magnetic field $H$. Thus one obtains an electronically tunable current-phase relation. At $H=0$ this corresponds to the $\varphi$ Josephson junction. \iffalse%arxiv Further, by means of numerical simulaton we extend our results to a practically relevant 0-$\pi$ junction of moderate length. Finally, we present experimental data obtained on superconductor-insulator-ferromagnet-superconductor 0-$\pi$ Josephson junction. \EG{My feeling is that the paper will be long enough without experiment. Also, we will discuss a lot how mederate length changes CPR, but in experiment CPR will ne be measured.} \fi
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varphiEx03d.tex
% 17.07.2012 v2 by EG
% 04.08.2012 v3 by EG after progress in analytics at L->infty with RM
% 20.08.2012 v3 for arxiv (only analytics for short L)

%\documentclass[aps,prb,twocolumn,showpacs,showkeys,superscriptaddress]{revtex4}
\documentclass[aps,prb,preprint,showpacs,showkeys,superscriptaddress]{revtex4-1}

\usepackage{amsmath}
\usepackage{graphics}

\include{MyMc}
\include{we}



%%% Abbreviations for this paper only:

\newcommand{\gxi}{\ensuremath{\av{g\xi}}}


\begin{document}

\title{Josephson junction with magnetic-field tunable current-phase relation}

\author{A. Lipman}
\affiliation{%
  The Raymond and Beverly Sackler School of Physics and Astronomy,
  Tel Aviv University, Tel Aviv 69978, Israel
}

\iffalse%arxiv{
\author{H. Sickinger}
\affiliation{%
  Physikalisches Institut and Center for Collective Quantum Phenomena in LISA$^+$,
  Universit\"at T\"ubingen, Auf der Morgenstelle 14, D-72076 T\"ubingen, Germany
}

\author{M. Weides}
\affiliation{%
  Peter Gr\"unberg Institute and JARA-Fundamentals of Future Information Technology,
  Forschungszentrum J\"ulich GmbH, 52425 J\"ulich, Germany
}
\altaffiliation{%
  Current address:
  Physikalisches Institut, Karlsruher Institut f\"ur Technologie, 76131 Karlsruhe, Germany
}
\fi%arxiv}

\author{R. G. Mints}
\affiliation{%
  The Raymond and Beverly Sackler School of Physics and Astronomy,
  Tel Aviv University, Tel Aviv 69978, Israel
}

\iffalse%arxiv{
\author{H. Kohlstedt}
\affiliation{%
  Nanoelektronik, Technische Fakult\"at,
  Christian-Albrechts-Universit\"at zu Kiel,
  D-24143 Kiel, Germany
}
\fi%arxiv}

\author{R. Kleiner}
\author{D. Koelle}
\author{E. Goldobin}
\affiliation{%
  Physikalisches Institut and Center for Collective Quantum Phenomena in LISA$^+$,
  Universit\"at T\"ubingen, Auf der Morgenstelle 14, D-72076 T\"ubingen, Germany
}

\date{%
  \today\ %File: \textbf{\jobname.\TeX}
}

\begin{abstract}
  We consider a 0-$\pi$ Josephson junction consisting of asymmetric $0$ and $\pi$ regions of different lengths $L_0$ and $L_\pi$ having different critical current densities $j_{c,0}$ and $j_{c,\pi}$. If both segments are rather short, the whole junction can be described by an \emph{effective} current-phase relation for the spatially averaged phase $\psi$, which includes the usual term $\propto\sin(\psi)$, a \emph{negative} second harmonic term $\propto\sin(2\psi)$ as well as the unusual term $\propto H \cos\psi$ tunable by magnetic field $H$. Thus one obtains an electronically tunable current-phase relation. At $H=0$ this corresponds to the $\varphi$ Josephson junction. \iffalse%arxiv
  Further, by means of numerical simulaton we extend our results to a practically relevant 0-$\pi$ junction of \emph{moderate} length. Finally, we present experimental data obtained on superconductor-insulator-ferromagnet-superconductor 0-$\pi$ Josephson junction.
  \EG{My feeling is that the paper will be long enough without experiment. Also, we will discuss a lot how mederate length changes CPR, but in experiment CPR will ne be measured.}
  \fi

\end{abstract}

\pacs{
  74.50.+r,   %Proximity effects, weak links, tunneling phenomena,
              %and Josephson effect
  85.25.Cp    %Josephson devices
%  05.45.-a    %Nonlinear dynamics and chaos
%  74.20.Rp    %Pairing symmetries (other than s-wave)
}

\keywords{$\varphi$ Josephson junction}

\maketitle



\section{Introduction}
\label{Sec:Intro}

Recently we proposed\cite{Goldobin:2011:0-pi:H-tunable-CPR} to implement a $\varphi$ Josephson junction (JJ) \cite{Buzdin:2003:phi-LJJ} with magnetic-field tunable current-phase relation (CPR) based on an 0-$\pi$ JJ with the 0 and $\pi$ segments of different length $L_0\neq L_\pi$. This proposal was made keeping in mind \YBCO-Nb ramp zigzag JJ technology\cite{Smilde:ZigzagPRL,Hilgenkamp:zigzag:SF} (or a similar one\cite{Ariando:Zigzag:NCCO} with \NCCO-Nb) established recently in our group also\cite{Scharinger:ramp-zigzag:Ic(H)}. However, in experiment we were more successful\cite{Sickinger:varphiExp} in employing superconductor-insulator-ferromagnet-superconductor (SIFS) 0-$\pi$ JJs\cite{Weides:2006:SIFS-0-pi,Weides:2010:SIFS-jc1jc2:Ic(H),Kemmler:2010:SIFS-0-pi:Ic(H)-asymm}, where the lengths of 0 and $\pi$ segments are equal, but critical current densities $j_{c,0}$ and $j_{c,\pi}$ in the 0 and $\pi$ parts are different. \iffalse%arxiv{
Also, the region in parameter space where the theory\cite{Goldobin:2011:0-pi:H-tunable-CPR} works and predicts $\varphi$ JJ is so slim, that it is almost impossible experimentally to get a 0-$\pi$ JJ that lays in this right region of the parameter space.
\fi%arxiv}

Therefore, in this paper we present a more general theory, which describes an effective $\varphi$ JJ made of asymmetric $0$ and $\pi$ regions of different lengths $L_0$ and $L_\pi$ having different critical current densities $j_{c,0}$ and $j_{c,\pi}$.
\iffalse%arxiv
and further extend our results to the regions of parameter space where the simple analytical approach (based on the smallness of the deviation of the phase from its average value) does not work. The paper is organized as follows. In Sec.~\ref{Sec:Model} we present a model and derive the effective equation for the general case $j_{c,0}\neq j_{c,\pi}$ and $L_0\neq L_\pi$. In Sec.~\ref{Sec:NumSim} we, by means of numerical simulations, consider practically relevant regions in parameter space, where the theory presented in Sec.~\ref{Sec:Model} does not work. Experimental results obtained on SIFS 0-$\pi$ JJs are presented and discussed in Sec.~\ref{Sec:ExpRes}. Finally, Sec.~\ref{Sec:Conclusions} concludes the work.
\fi



\section{Model}
\label{Sec:Model}



The static sine-Gordon equation that describes the behavior of the Josephson phase $\phi$ in a 0-$\pi$ JJ is
%
\begin{equation}
  \frac{\Phi_0}{2\pi\mu_0 d_J}\phi'' - j_c(x)\sin\phi = -j
  .\label{Eq:sine-Gordon:Phys}
\end{equation}
%
Here $\mu_0$ is the magnetic flux quantum, $\mu_0 d_J$ is the specific inductance (per square) of the superconducting electrodes forming the JJ and $j$ is the bias current density. The prime denotes the partial derivatives with respect to coordinate $x$. We assume that the critical current density $j_c(x)$ has the form of a step-function
%
\begin{eqnarray}
  &j_c=j_{c,0}>0,\quad\ &0\le x \le L_0
  ,\label{Eq:eq01}\\
  \label{eq02}
  &j_c=j_{c,\pi}<0,\quad\ &-L_\pi\le x < 0.
\end{eqnarray}
%
We write the critical current density $j_c(x)$ as
%
\begin{equation}
  j_c(x)=\av{j_c(x)} [1+g(x)]
  , \label{eq04}
\end{equation}
where
%
\begin{equation}
  \av{j_c} = \frac{1}{L}\int_{-L_\pi}^{L_0} j_c(x)\,dx
  = \frac{1}{L}\left( j_{c,0}L_0 + j_{c,\pi} L_\pi \right)
  \label{Eq:j_av}
\end{equation}
%
is the average critical current density, $L=L_0+L_\pi$ is the total length of the junction, and
$\av{g(x)}=0$. The function $g(x)$ is defined as
%
\begin{equation}\label{eq07}
  g(x) = \frac{j_c(x)}{\av{j_c(x)}} - 1
\end{equation}
%
that results in
%
\begin{equation}
  g(x) =
  \begin{cases}
    g_0,   &      0<x<L_0,\\
    g_\pi, & -L_\pi<x<0.
  \end{cases}
  \label{Eq:g(x)}
\end{equation}
%
where
\begin{equation}
  g_0   = \frac{(j_{c,0}-j_{c,\pi})L_\pi}{j_{c,0}L_0+j_{c,\pi} L_\pi}
  ;\quad
  g_\pi =-\frac{(j_{c,0}-j_{c,\pi})L_0  }{j_{c,0}L_0+j_{c,\pi} L_\pi}
  . \label{Eq:g}
\end{equation}


Then we divide Eq.~\eqref{Eq:sine-Gordon:Phys} by $|\av{j_c}|$ and normalize the coordinate $x$ to the Josephson length calculated using $|\av{j_c}|$, \ie,
%
\begin{equation}
  \lambda_J = \sqrt{\frac{\Phi_0}{2\pi\mu_0 d_J |\av{j_c}|}}
  . \label{Eq:lambda_J_av}
\end{equation}
%
Thus, we obtain a normalized sine-Gordon equation for the phase difference $\phi(x)$
% eqn 3
\begin{equation}
  \phi'' -\sgn(\av{j_c})\left[1+g(x)\right]\sin\phi =-\gamma
  , \label{Eq:sG(g):sgn}
\end{equation}
%
where $\gamma =j/|\av{j_c}|$ is the normalized bias current density. It is worth mentioning that $\av{j_c}$ can be positive as well as negative. Below, for the same of simplicity, we assume $\av{j_c}>0$. Thus, Eq.~\eqref{Eq:sG(g):sgn} becomes
% eqn 3
\begin{equation}
  \phi'' -\left[1+g(x)\right]\sin\phi =-\gamma
  . \label{Eq:sG(g)}
\end{equation}
%
In the case $\av{j_c}<0$ the substitution $\phi\to\pi-\phi$ converts Eq.~\eqref{Eq:sG(g):sgn} to the same Eq.~\eqref{Eq:sG(g)}.

We look for a solution of Eq.~\eqref{Eq:sG(g)} in the form
% eqn 10
\begin{equation} \label{Eq:PhaseSplit}
  \phi(x) = \psi + \xi(x)\sin\psi,
\end{equation}
%
where
%
\begin{equation}
  \psi = \av{\phi(x)}
  \label{Eq:Def:psi}
\end{equation}
%
is a constant \emph{average phase}, while $\xi(x)\sin\psi$ describes the deviation of the phase from the average value, \ie, $\av{\xi(x)}=0$. Further we assume that the deviation is small, \ie,  $| \xi(x)\sin\psi| \ll 1$. Then we plug the relation \eqref{Eq:PhaseSplit} into Eq.~\eqref{Eq:sG(g)}, expand it in series in $\xi(x)\sin\psi$, and keep the terms of zero and first order. We get
%
\begin{equation}
  \xi''\sin\psi - [1+g(x)][1+\xi(x)\cos\psi]\sin\psi = -\gamma.
  \label{Eq:sG-mix}
\end{equation}
%
The constant terms (zero order of $\xi$ in Eq.~\eqref{Eq:sG-mix}) are
%
\begin{equation}
  \gamma = \sin\psi + \av{\xi(x)g(x)}\cos\psi\sin\psi.
  \label{Eq:const-terms}
\end{equation}
%
The terms of first order of $\xi(x)$ in Eq.~\eqref{Eq:sG-mix} are
%
\begin{equation}
  \xi'' - g(x) = \{\xi + \xi(x)g(x)-\av{\xi(x)g(x)}\}\cos\psi.
  \label{Eq:dev-terms}
\end{equation}

Numerical calculations show that the two terms $\propto\cos\psi$ have an extremely weak effect on solutions of Eq.~\eqref{Eq:dev-terms}. We neglect these terms and obtain for $\xi(x)$
%
\begin{equation}
  \xi'' - g(x) = 0
  .\label{Eq:for-xi}
\end{equation}
%
We treat solutions of Eq.~\eqref{Eq:for-xi} by using the matching continuity (at $x=0$) and boundary (at $x=-l_\pi\equiv -L_\pi/\lambda_J$, $x=l_0\equiv L_0/\lambda_J$) conditions
% eqn 16
\begin{equation}\label{eq16}
  \xi_{\pi}(0) = \xi_0(0),\qquad \xi_{\pi}'(0) = \xi_0'(0),
\end{equation}
% eqn 17
\begin{equation}\label{eq17}
  \xi_\pi'(-l_\pi)\sin\psi = h,\qquad \xi_0'(l_0)\sin\psi = h.
\end{equation}

The applied field $H$ is normalized by $H_{c1}/2$, \ie,
% eqn 18
\begin{equation} \label{eq18}
  h=\frac{2H}{H_{c1}},\qquad H_{c1}=\frac{\Phi_0}{\pi\Lambda\lambda_J},
\end{equation}
where $\Lambda$ is the effective magnetic thickness of the JJ. We integrate Eq. (\ref{Eq:for-xi}) once and obtain
% eqn 19
\begin{eqnarray}\label{eq19}
  \xi_0'(x) = g_0(x-l_0) + \frac{h}{\sin\psi},\quad 0<x<l_0,
\end{eqnarray}
% eqn 20
\begin{eqnarray}\label{eq20}
  \xi_{\pi}'(x) = g_{\pi} (x+l_\pi) + \frac{h}{\sin\psi},\quad -l_\pi<x<0.
\end{eqnarray}
The second integration results in
% eqn 21
\begin{eqnarray}\label{eq21}
  \xi_0(x) = g_0 \left(\frac{x^2}{2}-l_0 x\right) + \frac{hx}{\sin\psi}+C,\\
  \text{for }0<x<l_0,\nonumber
\end{eqnarray}
% eqn 22
\begin{eqnarray}\label{eq22}
  \xi_{\pi}(x) = g_{\pi}\left(\frac{x^2}{2}+l_\pi x\right) + \frac{hx}{\sin\psi}+C,\\
  \text{for }-l_\pi<x<0.\nonumber
\end{eqnarray}
The integration constant $C$ can be obtained using the condition $\langle\xi(x)\rangle = 0$
% eqn 23
\begin{equation}\label{eq23}
  C=\frac{l_0 -l_\pi}{2}\left(\frac{g_0l_0+g_\pi l_\pi}{3}-\frac{h}{\sin\psi}\right).
\end{equation}

We use  Eqs.~\eqref{Eq:g(x)}, \eqref{eq21}, and \eqref{eq22} and obtain the average $\av{\xi(x)g(x)}$ in the form
%
\begin{equation}\label{Eq:xig}
  \av{\xi(x)g(x)} = \Gamma_0 + \Gamma_h\frac{h}{\sin\psi},
\end{equation}
%
where the coefficients $\Gamma_0$ and $\Gamma_h$ are given by
% eqn 25
\begin{eqnarray}%{
  %\begin{subequations}
    \Gamma_0 &=& -\frac{l_0^2l_\pi^2}{3} \frac{(j_{c,0}-j_{c,\pi})^2}{(j_{c,0}l_0+j_{c,\pi} l_\pi)^2}
    ,\label{Eq:Gamma0}\\
    \Gamma_h &=& \frac{l_0l_\pi}{2} \frac{j_{c,0}-j_{c,\pi}}{j_{c,0}l_0+j_{c,\pi} l_\pi}
    .\label{Eq:GammaH}
  %\end{subequations}
  \label{Eq:Gammas}
\end{eqnarray}%}

Using Eqs.~\eqref{Eq:const-terms} and \eqref{Eq:xig} we find the current-phase relation in the form
%
\begin{equation}
  j=\av{j_c}\left(\sin\psi +\Gamma_0\sin\psi\cos\psi +h\Gamma_h\cos\psi\right)
  .\label{Eq:CPR}
\end{equation}

It is worth noting that there is a simple relation between the coefficients $\Gamma_0$ and $\Gamma_h$. Indeed, it follows from Eqs. (\ref{Eq:Gamma0}) and (\ref{Eq:GammaH}) that
% eqn 28
\begin{equation}
  \Gamma_0 = -\frac{4}{3}\Gamma_h^2
  .\label{Eq:GammaH(Gamma0)}
\end{equation}

In the case of equal lengths of 0 amd $\pi$ parts ($l_0=l_\pi =l/2$) we find
% eqn 29
\begin{equation}
  \Gamma_0=-\frac{l^2}{12} \left(\frac{j_{c,0}-j_{c,\pi}}{j_{c,0}+j_{c,\pi}}\right)^2 , \qquad
  \Gamma_h= \frac{l}{4} \frac{j_{c,0}-j_{c,\pi}}{j_{c,0}+j_{c,\pi}}\,.
  \label{Eq:Gammas@EqualL}
\end{equation}

The energy $U(\psi)$ corresponding to the current-phase relation (\ref{Eq:CPR}) is given by
% eqn 30
\begin{equation}
  U(\psi)=\av{j_c}\left(1-\cos\psi +h\Gamma_h\sin\psi+\frac{\Gamma_0}{2}\sin^2\psi\right)
  . \label{Eq:U(psi)}
\end{equation}

\iffalse%arxiv
In summary, the dependence of the CPR on the phase and applied field is the same as in our previous study\cite{Goldobin:2011:0-pi:H-tunable-CPR}. The difference is in the formulas for $\Gamma_0$ and $\Gamma_h$. In addition, we have done a detailed comparison of the numerically and analytically calculated data.
\fi%arxiv

\iffalse% the energy part is commented out
\subsection{Energy}
\par
The JJ energy is given by
% eqn 28
\begin{equation}\label{Eq:Energy}
{\cal{E}} ={\cal E}_0\int dx \left[\frac{1}{2}\left(\frac{d\phi}{dx}\right)^2\!\!\! + [1+g(x)](1-\cos\phi)\right],
\end{equation}
where
% eqn 29
\begin{equation}\label{Eq:E0}
{\cal E}_0=\frac{\hbar\lambda_J\av{j_c}}{2e}.
\end{equation}
We calculate the energy $\cal{E}$ in two ``steps'' writing
% eqn 30
\begin{equation}\label{Eq:U(psi)}
{\cal E}={\cal E}_1+{\cal E}_2,
\end{equation}
where ${\cal E}_1$ is for the boundary contribution and ${\cal E}_2$ is for the bulk contribution
% eqn 31
\begin{eqnarray}
\label{eq31}
{\cal E}_1&=&\frac{{\cal E}_0}{2}\int_{-l_\pi}^{l_0}dx\left(\frac{d\phi}{dx}\right)^2\approx
\nonumber\\
&\ &\frac{{\cal E}_0}{2}\,\sin^2\psi\int_{-l_\pi}^{l_0}dx \left(\frac{d\xi}{dx}\right)^2\approx
\nonumber\\
&-&\frac{{\cal E}_0}{2}\sin^2\psi\,\int_{-l_\pi}^{l_0}\!\!\! dx\,\xi''\xi =
\nonumber\\
&-&\frac{{\cal E}_0}{2}\sin^2\psi\,l\langle g(x)\xi(x)\rangle\,.
\end{eqnarray}
% eqn 32
\begin{eqnarray}\label{eq32}
{\cal E}_2={\cal E}_0\,\int_{-l_\pi}^{l_0} dx\,[1+g(x)](1-\cos\phi)\approx
\nonumber\\
{\cal E}_0\,l\left[1-\cos\psi +\sin^2\psi\,\langle g(x)\xi(x)\rangle\right].
\end{eqnarray}
\par
Summing Eqs. (\ref{Eq:U(psi)}) and (\ref{eq31}) we find
% eqn 33
\begin{equation}\label{eq33}
{\cal E}= {\cal E}_0l\left[1 -\cos\psi +\frac{1}{2}\sin^2\psi\langle g(x)\xi(x)\rangle \right]\,.
\end{equation}
%
\fi%Energy part

\iffalse%arxiv
\subsection{Bulaevskii normalization}

The Mints\cite{Mints:1998:SelfGenFlux@AltJc} normalization of length to $\lambda_J(\av{j_c})$ given by Eq.~\eqref{Eq:lambda_J_av} is very useful for theoretical analysis. However for comparing with experiment it may be more convenient to use Bulaevskii normalization\cite{Bulaevskii:0-pi-LJJ}, namely to the local $\lambda_{J,0}=\lambda_J(j_{c,0})$ and $\lambda_{J,\pi}=\lambda_J(j_{c,\pi})$:
%
\begin{equation}
  \ell_0   = \frac{l_0}{\lambda_{J,0}}    =l_0  \sqrt{\frac{j_{c,0}}{\av{j_c}}}
  ,\quad
  \ell_\pi = \frac{l_\pi}{\lambda_{J,\pi}}=l_\pi\sqrt{\frac{j_{c,\pi}}{\av{j_c}}}
  . \label{Eq:def:ell}
\end{equation}
%
In these terms
%
\begin{equation}
  \av{j_c} = \frac{l_0 j_0 + l_\pi j_\pi}{l_0 + l_\pi}
  = \frac{l_0 j_0\sqrt{|j_\pi|}+l_\pi j_\pi\sqrt{j_0}}{l_0\sqrt{|j_\pi|}+l_\pi\sqrt{j_0}}
  , \label{Eq:j_av:Bulaevskii}
\end{equation}
%
while our key result \eqref{Eq:Gamma0} can be rewritten as
%
\begin{equation}
  \Gamma_0 = -\frac{\ell_0^2\ell_\pi^2}{3}
  \frac{(j_0-j_\pi)^2}{
    \left| \ell_0 j_0 \sqrt{|j_\pi|}+\ell_\pi j_\pi \sqrt{j_0} \right|
    \left( \ell_0\sqrt{|j_\pi|}+\ell_\pi\sqrt{j_0} \right)
  }
  , \label{Eq:Gamma0:Bulaevskii}
\end{equation}
%
while Eq.~\eqref{Eq:GammaH(Gamma0)} stays the same.
\fi%arxiv

\iffalse%arxiv
\section{Beyond analytics}
\label{Sec:NumSim}

\EG{This is a big section where Roman and Adi should insert a lot of simulation data}

Let us now consider the regions in the parameter space where the condition $|\xi(x)\sin\psi|$ is not satisfied. This can be the case of 0-$\pi$ JJ of moderate length, or of a rather high applied magnetic field $H$. In the latter case magnetic field sets a high value of the phase derivative at the edges of the JJ, which results in a high amplitude of $|\xi(x)\sin\psi|$.

To treat these cases numerically we perform a numerical simulation of the complete static sine-Gordon Eq.~\eqref{Eq:sine-Gordon:Phys} for given $L_0$, $L_\pi$, $j_{c,0}$, $j_{c,\pi}$ and applied bias current $j$. As a result we obtain one or more solutions $\phi(x)$ depending on initial/seed value of $\phi(x)$. For each of these solutions we then calculate $\psi=\av{\phi(x)}$ and plot these values of $\psi$ on a $\psi(\gamma)$ plot. By repeating this procedure for different values of $\gamma$ one obtaines a $\psi(\gamma)$ plot, \ie, the \emph{effective current-phase relation} (e-CPR).

%
\begin{figure}[!htb]
  %\includegraphics{NumCPR}
  \caption{%
    Numerically calculated e-CPR for 0-$\pi$ with $l_0=..$, $l_\pi=...$, $j_{c,0}=...$, $j_{c,\pi}=...$ for different values of applied mahnetic field $h$. The e-CPR predicted by analytical formula \eqref{Eq:CPR} is shown by dashed lines for comparison.
  }
  \label{Fig:NumCPR}
\end{figure}
%

An example of the e-CPR is shown in Fig.~\ref{Fig:NumCPR}. One can see that qualitatively the behavior is similar, but quantitatively there are deviations. In particular, the ground state phase and harmonic content of e-CPR are different.


\subsection{Ground state}

%
\begin{figure*}[!tb]
  \includegraphics{GroundState@j_c,pi=1.00.eps}
  \caption{%
    Phase diagram with the contour lines of constant ground state phase $\varphi$ calculated numerically (solid lines) and analytically in the ``short'' limit (dashed lines) and long limit (dotted lines). The numbers next to each contour line indicate the value of $\varphi$.
    $j_{c,0}=-j_{c,\pi}$.
  }
  \label{Fig:PhaseDiag10}
\end{figure*}
%

%
\begin{figure*}[!tb]
  \includegraphics{GroundState@j_c,pi=0.50.eps}
  \caption{(Color online)%
    (a) A phase diagram with the contour lines of constant ground state phase $\varphi$ calculated numerically (solid lines). The filled region is a domain of interest where the phase smoothly chages between 0 and $\pi$. The numbers next to each line indicate the value of $\varphi$.
    (b)--(h) a comparison the numerically calculated lines of constant phase with those calculated analytically in the short limit (dotted lines) and long limit (dashed lines).
    $j_{c,0}=-2j_{c,\pi}$.
  }
  \label{Fig:PhaseDiag05}
\end{figure*}
%

\EG{Do we need both symmetric and asymmetric cases?}
In Figs.~\ref{Fig:PhaseDiag10} and \ref{Fig:PhaseDiag05} one can see the phase diagrams with the lines of the constant ground state phase $\varphi$. Then for several selected values of $\varphi$ one can see a comparison of numerically calculated constat phase line with the lines calculated analytically in the limits $\ell_0\lesssim 1$, $\ell_\pi\lesssim 1$ and $\ell_0\gg 1$, $\ell_\pi\gg 1$.

\paragraph{Analytics for $L_0\lesssim\lambda_{J,0}$ and $L_\pi\lesssim\lambda_{J,\pi}$.} To calculate a curve of constant ground state phase $\varphi$, we use \eqref{Eq:CPR} with $h=0$ (ground state) and $\psi=\varphi$ to get
%
\begin{equation}
  1+\Gamma_0\cos(\varphi)=0
  . \label{Eq:GrStt}
\end{equation}
%
Since $\Gamma_0<0$ Eq.~\eqref{Eq:GrStt} results in solutions $|\varphi|\leq\pi/2$. We remind that this is the case when $\av{j_c}>0$. For $\av{j_c}<0$ one should use $\pi-\varphi$ instead of $\varphi$ in Eq.~\eqref{Eq:GrStt}.\EG{Guys, shall we introduce a new phase $\delta=\psi$ for $\av{j_c}>0$ and $\delta=\pi-\psi$ for $\av{j_c}<0$?} Thus, for arbitrary $\av{j_c}$ one can write
%
\begin{equation}
  1+\Gamma_0|\cos(\varphi)|=0
  . \label{Eq:GrStt:gen}
\end{equation}
%
Using the expression \eqref{Eq:Gamma0:Bulaevskii} for $\Gamma_0$ we solve the above equation to find
%Bulaevskii units
\begin{subequations}
  \begin{eqnarray}
    \ell_\pi(\ell_0) = \ell_0 \frac{\sqrt{3j_0|j_\pi|}}{2}
    \frac{j_\Sigma\sqrt{3}+|j_\Delta|\sqrt{3+4\ell_0^2\cos\varphi}}
    {\ell_0^2 j_\Delta^2 \cos\varphi -3j_0j_\pi}
    ;\label{Eq:ell<1:ell_pi(ell_0)}\\
    \ell_0(\ell_\pi) = \ell_\pi \frac{\sqrt{3j_0|j_\pi|}}{2}
    \frac{\sqrt{3}j_\Sigma-|j_\Delta|\sqrt{3-4\ell_\pi^2\cos\varphi}}
    {\ell_\pi^2 j_\Delta^2 \cos\varphi +3j_0j_\pi}
    .\label{Eq:ell<1:ell_0(ell_pi)}
  \end{eqnarray}
  \label{Eq:ell<1:ell_0-ell_pi}
\end{subequations}
%
where
%
\begin{equation}
  j_\Sigma = j_0+j_\pi, \quad j_\Delta = j_0-j_\pi
  . \label{Eq:Def:j_Sigma&Delta}
\end{equation}
%
Note that Eqs.~\eqref{Eq:ell<1:ell_0-ell_pi} are valid for any $0\leq\varphi\leq\pi$. Still, for $\varphi\leq\pi/2$ it is more convenient to use Eq.~\eqref{Eq:ell<1:ell_pi(ell_0)}, while Eq.~\eqref{Eq:ell<1:ell_0(ell_pi)} has a singularity at $\ell_\pi=\sqrt{-3j_0 j_\pi/(j_\Delta^2\cos\varphi)}$. For $\varphi>\pi/2$ the situation is opposite: Eq.~\eqref{Eq:ell<1:ell_0(ell_pi)} always give $\ell_0$ for given $\ell_\pi$, while Eq.~\eqref{Eq:ell<1:ell_pi(ell_0)} diverges at $\ell_0=\sqrt{2j_0 j_\pi/(j_\Delta^2\cos\varphi)}$.

The same formulas in Mints units read
%this is in Mints's units:
\begin{eqnarray}
  l_\pi(l_0) = j_0 l_0
  \frac%
    {3j_\pi-l_0 j_\Delta \sgn(\cos\varphi)\sqrt{3|\cos\varphi|}}
    {l_0^2 j_\Delta^2 |\cos(\varphi)| - 3j_0^2}
  ; \label{Eq:L<1:Lpi(L0)}\\
  l_0(l_\pi) = j_\pi l_\pi
  \frac%
    {3j_0+l_\pi j_\Delta \sgn(\cos\varphi)\sqrt{3|\cos\varphi|}}
    {l_\pi^2 j_\Delta^2 |\cos(\varphi)| - 3j_0^2}
  . \label{Eq:L<1:Lpi(L0)}
\end{eqnarray}
%

\paragraph{Analytics for $L_0\gg\lambda_{J,0}$ and $L_\pi\gg\lambda_{J,\pi}$.}
In this limit the phase solution in the ground state $\phi(x)$ is given by the two fluxon tails properly joined at $x=0$ as described in appendix \ref{Sec:Asym-0-pi-LJJ.Semifluxon}. The average phase in the limit $L_0,L_\pi\to\infty$ is calculated in appendix \ref{Sec:Asym-0-pi-LJJ.AvPhase} and is given by Eq.~\eqref{Eq:Res:varphi@L=inf} (in physical units). To obtain the lines of constant phase $\varphi$, we express $L_\pi$ as a function of $L_0$, \ie,
%
\begin{equation}
  L_\pi = \frac{\varphi}{\pi-\varphi}L_0 + \Delta L_\pi
  , \label{Eq:L_pi(L_0).units.RM}
\end{equation}
%
where
%
\begin{eqnarray}
  \Delta L_\pi
  &=& \frac{-\lambda_{J,0  }}{\pi-\varphi}\left[ 4\ln(z_0)\arctan(z_0) +M(z_0)\right]
  \label{Eq:DeltaL_pi.units.phys}\\
  &+& \frac{-\lambda_{J,\pi}}{\pi-\varphi}\left[ \ln(z_\pi)(2\pi-4\arctan(z_0)) -M(1/z_\pi)\right]
  , \nonumber
\end{eqnarray}
%
In the units of Bulaevskii we get
%
\begin{equation}
  l_\pi = \frac{\varphi}{\pi-\varphi}l_0\sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}} + \Delta l_\pi
  , \label{Eq:L_pi(L_0).units.LB}
\end{equation}
%
where
%
\begin{eqnarray}
  \Delta l_\pi
  &=& \frac{-1}{\pi-\varphi}\left[ 4\ln(z_0)\arctan(z_0) +M(z_0)\right] \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}}
  \label{Eq:DeltaL_pi.units.LB}\\
  &+& \frac{-1}{\pi-\varphi}\left[ \ln(z_\pi)(2\pi-4\arctan(z_0)) -M(1/z_\pi)\right]
  . \nonumber
\end{eqnarray}
%
Note that for $j_{c,0}=|j_{c,\pi}|$ the values $z_0=\sqrt2-1$, $z_\pi=\sqrt2+1$, so that $z_\pi=1/z_0$ and, after simple algebra, $\Delta l_\pi=0$.


\EG{One can make also the simpler approximation assuming that the (semi)fluxon tails are centered at $x=0$.} Then the average phase is

%
\begin{equation}
  \varphi = \frac{4\lambda_{J,0} + \pi L_\pi - 4\lambda_{J,\pi}}{L_0+L_\pi}
  . %\label{Eq:}
\end{equation}
%
From here
%
\begin{equation}
  L_\pi = \frac{\varphi}{\pi-\varphi}L_0 - 4\frac{\lambda_{J,0}-\lambda_{J,\pi}}{\pi-\varphi}
  . %\label{Eq:}
\end{equation}
%
In Bulaevskii notations
%
\begin{equation}
  l_\pi
  = \frac{\varphi}{\pi-\varphi}l_0\sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}}
  - \frac{4}{\pi-\varphi} \left( \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}} - 1\right)
  . \label{Eq:l_pi(l_0).simple}
\end{equation}
%

\EG{End of simple approximation. In fact, Eq.~\eqref{Eq:l_pi(l_0).simple} describes the numerical results very well at least for not very disbalanced $j_{c,0}$ and $j_{c,\pi}$.}



\subsection{Harmonics analysis of e-CPR}

%
\begin{figure}[!tb]
  %\includegraphics{FFT}
  \caption{%
    The amplitude of (a) $\sin(n\psi)$ and (b) $\cos(n\psi)$ in the e-CPR obtained numerically.
  }
  \label{Fig:CPR:FFT}
\end{figure}
%

To see harmonic content of CPR we have performed FFT of the numerically obtained CPR. The results are presented in Eq.~\eqref{Fig:CPR:FFT}. Analytical model, discussed in Sec.~\ref{Sec:Model} predics only $\sin(\psi)$ and $\sin(2\psi)$ harmonics present. One can see that in the longer JJ one also obtains higher harmonics $\sin(n\psi)$, first of all with even $n$. In the same time the amplitude of $\sin(\psi)$ and $\sin(2\psi)$ decreases. \EG{Do we get $\cos$-terms without magnetic field?}

When a magnetic field is applied, in addition to $\cos(\psi)$ term predicted by the model, one gets also $\cos(n\psi)$ terms. Also $\sin$-terms are modified....

\subsection{Multivalued e-CPR}

%
\begin{figure}[!htb]
  \caption{%
    (a) e-CPR showing a multivalued region for $\psi=xxx \ldots yyy$. $l_0=..$, $l_\pi=...$, $j_{c,0}=...$, $j_{c,\pi}=...$
    (b) magnetic field profiles $\phi'(x)$ corresponding to two branches of e-CPR, shown by arrows in (a).\EG{We can show profiles at critical current or at $\gamma=0$? what do you think?}
    (c) e-CPR showing several multivalued regions. $l_0=..$, $l_\pi=...$, $j_{c,0}=...$, $j_{c,\pi}=...$
  }
  \label{Fig:MultiCPR}
\end{figure}

At larger $H$ and $l$ the e-CPR may become not single valued, see Fig.~\ref{Fig:MultiCPR}(a). This corresponds to two (3?) different static solutions of Eq.~\eqref{Eq:sine-Gordon:Phys}. As an example, Fig.~\ref{Fig:MultiCPR}(b) shows the magnetic field profile $\phi'(x)$ corresponding to the same values of the phase $\psi$, but different total supercurrent flowing through the structure, see arrows in Fig.~\ref{Fig:MultiCPR}(a). The number of such a double valued regions in e-CPR grows with the length of the JJ and with applied magnetic field. As an example, in Fig.~\ref{Fig:MultiCPR}(a) we show a rather extreem e-CPR for $l=8$ and $H=1.8$ \EG{Adi, try different}.


\section{Experiment}
\label{Sec:ExpRes}

Results on the 2nd 0-$\pi$ JJ.

\subsection{Self-field effect}

It turns out that the so-called \emph{self-field effect}, \ie, parasitic magnetic field created by the bias current circuitry, may skew an $I_c(H)$ dependence in a manner similar to the one prediced by the model (see Sec.~\ref{Sec:Model}) and numerical simulations (see Sec.~\ref{Sec:NumSim}). Here we investigate in detail the difference between self-field effect and the $I_c(H)$ asymmetry caused by $j_{c,0}\neq j_{c,\pi}$ or $L_{0}\neq L_{\pi}$.

In most cases the self-field effect is modelled by an additional \emph{uniform} applied magnetic field $\propto$ bias curent $I$, \ie,
%
\begin{equation}
  H_\mathrm{tot} = H + K_\mathrm{SF}I
  , \label{Eq:H_tot}
\end{equation}
%
where $H_\mathrm{tot}$ is the total magnetic field applied to the JJ, $H$ is magnetic field applied externally \eg by the coil, while $k_\mathrm{SF}$ is a self-field coefficient. What one measures in experiment is $I_c(H)$. To measure $I_c$ at given $H$ one ramps the bias current $I$ up to find $I_c$, bute simultaneously the total applied field changes according to Eq.~\eqref{Eq:H_tot}. As a result the whole $I_c(H)$ will be skewed along $H$ axis. Note several properties of this transformation: (a) the self-field effect preserves point-symmetry if $I_c(H)$ curve relative to origin, if any; (b) the height of all $I_c(H)$ maxima stays the same; (c) The minima with $I_c=0$ do not move.

The first suspicion that one may have is that the the point-symmetric shift of the crossing point (further x-point) of $I_c(H)$ branches (minimum between two main maxima) is caused fully by self-field effect alone. That is one assumes that the junction is symmetric ($l_0=l_\pi$ and $j_{c,0}= j_{c,\pi}$) and should have symmetric $I_c(H)$ with a minimum at $H=0$. From the experimental position $I_\mathrm{c,min}$ and $H_\mathrm{min}$ one can easily get $k_\mathrm{SF}=H_\mathrm{min}/I_\mathrm{c,min}$. However, the difference in maxima cannot be reproduced in this way. Therefore in the following we consider a mixture of self-field effect and asymmetry of the junctions parameters trying to obtain a better fit to experimental $I_c(H)$ dependence.

%
\begin{figure}[!htb]
  \caption{%
    Experimentally measured $+I_c(H)$ and $-I_c(H)$ dependences. In addition $-I_c(H)$ dependence is point-reflected to compare it with $+I_c(H)$ curve.
  }
  \label{Fig:ExpIc(H)}
\end{figure}
%

First, if one assumes uniform remanent magnetization of the F-layer, the experimental $I_c(H)$ is expected to be point-symmetric with respect to the origin $I_c=H=0$. In our experiment it is roughly so, \cf, $+I_c(H)$ and point-reflected $-I_c(H)$ in Fig.~\ref{Fig:ExpIc(H)}. The asymmetry, if any, is caused by different megnetization of $0$ and $\pi$ parts or even more general $M(x)$ profile. At this moment we ignore the small point-asymmetry.

\EG{Here Hanna's fitting results and more pictures for JJ 2.}

%
\begin{figure}[!htb]
  \caption{%
    Different fits with self-field...
  }
  \label{Fig:SelfField}
\end{figure}
%


Finally, we would like to stress that the the self-field effect, if any, only shifts $H_\mathrm{min}$, which may affect the value of $\varphi$ that we find from experimental data not taking a self-field into account. The fact that we still have x-point and, therefore, two critical currents remains. This means that we are still dealing with a bistable system with two energy minima.

\fi%arxiv

\section{Conclusions}
\label{Sec:Conclusions}

\iffalse%arxiv
Actually we've made a lot of stuff: extended theory for $|\xi(x)\sin\psi|\ll1$ to the case of $L_0\neq L_\pi$ and $j_{c,0}\neq|j_{c,\pi}|$; wrote solution for semifluxon in infinite asymmetric LJJ, ...
\else
We have extended our previous results \cite{Goldobin:2011:0-pi:H-tunable-CPR} to the case of arbitrary critical current densities $j_{c,0}\neq j_{c,\pi}$ more relevant for experiment\cite{Sickinger:varphiExp}. The dependence \eqref{Eq:CPR} of the CPR on the phase and applied field is the same as in our previous study\cite{Goldobin:2011:0-pi:H-tunable-CPR}. The difference is in the formulas \eqref{Eq:Gammas} for $\Gamma_0$ and $\Gamma_h$.

\fi%arxiv

\iffalse%arxiv

\appendix
\section{Semifluxon in asymmetric 0-$\pi$ JJ}
\label{Sec:Asym-0-pi-LJJ.Semifluxon}

Consider an infinite asymmetric 0-$\pi$ LJJ with $j_{c,0}\neq j_{c,\pi}$. We assume that 0-part is situated at $x<0$ and $\pi$ part at $x>0$. Then the phase is given by the fluxon tails
%
\begin{equation}
  \phi(x) =
  \begin{cases}
    4\arctan\exp\left( \ratio{x-x_0  }{\lambda_{J,0  }} \right)      & x<0\\
    \\
    4\arctan\exp\left( \ratio{x-x_\pi}{\lambda_{J,\pi}} \right) -\pi & x>0
  \end{cases}
  . \label{Eq:mu(x):general}
\end{equation}
%
At 0-$\pi$ boundary at $x=0$ the phase and its derivative (magnetic field) must be continuous, \ie,
%
\begin{eqnarray}
  \arctan(z_0) &=& \arctan(z_\pi)-\frac{\pi}{4}
  ; \label{Eq:BC0:phase}\\
  \frac{1}{\lambda_{J,0}}\frac{z_0}{1+z_0^2}&=&\frac{1}{\lambda_{J,\pi}}\frac{z_\pi}{1+z_\pi^2}
  , \label{Eq:BC0:field}
\end{eqnarray}
%
where
%
\begin{equation}
  z_0  = \exp\left( \frac{-x_0  }{\lambda_{J,0  }}     \right)
  ,\quad
  z_\pi= \exp\left( \frac{-x_\pi}{\lambda_{J,\pi}}     \right)
  .\label{Eq:Def:z}
\end{equation}
%
By taking a $\tan(\ldots)$ of the both sides of Eq.~\eqref{Eq:BC0:phase} we get
%
\begin{eqnarray}
  z_0 &=& \frac{z_\pi-1}{z_\pi+1}
  ; \label{Eq:e0(e_pi)}\\
  \frac{1}{\lambda_{J,0}}\frac{z_0}{1+z_0^2}&=&\frac{1}{\lambda_{J,\pi}}\frac{z_\pi}{1+z_\pi^2}
  , \label{Eq:BC0:field:1}
\end{eqnarray}
%
Solving for $z_0$ and $z_\pi$ we obtain two roots (for each of them). The negative root makes no sense since $z_0$ and $z_\pi$ are positive by definition. Thus, the remaining positive roots are
%
\begin{subequations}
  \begin{eqnarray}
    z_\pi = \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}+1} + \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}} \geq 1
    ; \label{Eq:Sol:z_pi}\\
    z_0   = \sqrt{\frac{j_{c,0}}{|j_{c,\pi}|}+1} - \sqrt{\frac{j_{c,0}}{|j_{c,\pi}|}} \leq 1
    . \label{Eq:Sol:z_0}
  \end{eqnarray}
  \label{Eq:Sol:z}
\end{subequations}
%
From here, taking into account definitions \eqref{Eq:Def:z}, we get
%
\begin{subequations}
  \begin{eqnarray}
    x_\pi = -\lambda_{J,\pi} \ln
    \left( \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}+1} + \sqrt{\frac{|j_{c,\pi}|}{j_{c,0}}} \right) \leq0
    ; \label{Eq:Sol:x_pi}\\
    x_0   = -\lambda_{J,0  } \ln
    \left( \sqrt{\frac{j_{c,0}}{|j_{c,\pi}|}+1} - \sqrt{\frac{j_{c,0}}{|j_{c,\pi}|}} \right) \geq0
    . \label{Eq:Sol:x_0}
  \end{eqnarray}
  \label{Eq:Sol:x}
\end{subequations}
%



\section{Approximating the integral of $\ln(z)/(1+z^2)$}
\label{Sec:M(z)}

Here we derive an approximate expression for the following function $(Z\geq0)$
%
\begin{equation}
  M(z) = -\int_0^z \frac{\ln(u)}{1+u^2} \,du
  . \label{Eq:Def:M(z)}
\end{equation}
%
Note that $u=1$ is a special point where $\ln(u)$ changes sign so that for $z<1$ the integral $M(z)>0$ and grows with $z$, while for $z>1$ the function $M(z)$ decreases, see Fig.~\ref{Fig:M(z)}. Moreover, by introducing a new variable $v=1/u$ one can immediately prove that
%
\begin{equation}
  M(z) = M(1/z)
  . \label{Eq:M(z).reciprocity}
\end{equation}
%
In particular, $M(0)=M(+\infty)=0$.

%
\begin{figure}[!tb]
  \centering\includegraphics{M(Z)}
  \caption{%
    The function $M(z)$ (continuous line) calculated directly from definition \eqref{Eq:Def:M(z)} and its approximations $M_0(z)$ (dotted line) and $M_1(z)$ (dashed line) calculated using Eqs.~\eqref{Eq:_M(z)} and \eqref{Eq:M_N(z)}.
  }
  \label{Fig:M(z)}
\end{figure}
%

The integrand of Eq.~\eqref{Eq:Def:M(z)} can be represented as a series
%
\begin{equation}
  \frac{\ln(u)}{1+u^2} = \sum_{n=0}^{N} (-1)^n u^{2n}\ln(u)
  , \label{Eq:Integrand.SerPresent}
\end{equation}
%
with $N=\infty$, which converges only for $u \leq 1$.

If $z\leq1$, then $u \leq 1$ too, and the series \eqref{Eq:Integrand.SerPresent} always converges. By integrating it, one obtains
%
\begin{equation}
  {\cal M}_N(z) = -\sum_{n=0}^N (-1)^n \frac{z^{2n+1}}{(2n+1)^2} \left[ (2n+1)\ln(z)-1 \right]
  . \label{Eq:_M(z)}
\end{equation}
%

If $z>1$ one uses relation \eqref{Eq:M(z).reciprocity} to make argument $z<1$ and again use the sum \eqref{Eq:_M(z)} with the argument $1/z<1$, \ie,
%
\begin{equation}
  M_N(z) = \begin{cases}
    {\cal M}_N(z)   & Z\leq1\\
    {\cal M}_N(1/z) & Z>1
  \end{cases}
  , \label{Eq:M_N(z)}
\end{equation}
%

The sum \eqref{Eq:_M(z)} converges very fast. Therefore, one can approximate it very well by taking only the few first terms, \ie, finite $N$ instead of $N=\infty$. The plots of $M(z)$ and its approximation by finite $N$ are shown in Fig.~\ref{Fig:M(z)}. One can see that the deviation has a maximum at $Z=1$ (boundary of convergence interval) and is less than few percents even for $N=0$. For example, $M(1)=0.916$, while $M_0(1)=1.000$, $M_1(1)=0.889$, $M_2(1)=0.929$, $M_3(1)=0.908$. Thus, for most practical purposes one can simply use $M_0(z)$ or $M_1(z)$.





\section{The dependence of the $\varphi(L_0,L_\pi)$ at $L_0\gg\lambda_{J,0}$, $L_\pi\gg\lambda_{J,\pi}$}

\label{Sec:Asym-0-pi-LJJ.AvPhase}

Our aim is to calculate
%
\begin{equation}
  \varphi=\av{\phi(x)} \equiv \frac{1}{L_0+L_\pi}\int_{-L_0}^{+L_\pi} \phi(x)
  , \label{Eq:Def:phi_av}
\end{equation}
%
in the limit of $L_0,\,L_\pi\to\infty$. In this limit the phase $\phi(x)$ is given by Eq.~\eqref{Eq:mu(x):general}.

First, we find the value of integral in 0 domain, \ie, we integrate from $-L_0$ to 0. By introducing a new integration variable
%
\begin{equation}
  y = \frac{x-x_0}{\lambda_{J,0}}
\end{equation}
%
and, correspondingly, new integration limits
%
\begin{equation}
  y_0   =\frac{-x_0  }{\lambda_{J,0  }},\quad y_{L_0  } = \frac{-L_0  -x_0  }{\lambda_{J,0  }},
\end{equation}
%
after integration by parts, we get
%
\begin{equation}
  \lambda_{J,0}\left[\left. 4 y \arctan e^y \right|_{y_{L_0}}^{y_0}
  -\int_{y_{L_0}}^{y_0} 4 \arctan e^y\, dy\right]
  . \label{Eq:int0(y)}
\end{equation}
%
Further, we introduce a new integration variable, \cf, Eq.~\eqref{Eq:Def:z},
%
\begin{equation}
  z = e^y\text{, so that }z_0=e^{y_0}\text{ and }z_{L_0} = e^{y_{L_0}}
  , %\label{Eq:}
\end{equation}
%
and Eq.~\eqref{Eq:int0(y)} becomes
%
\begin{equation}
  \lambda_{J,0}\left[ \left. 4 \ln(z)\arctan(z) \right|_{z_{L_0}}^{z_0}
  - 4\int_{z_{L_0}}^{z_0} \frac{\ln z}{1+z^2} \,dz\right]
  . \label{Eq:int0(z)}
\end{equation}
%
The value of $z_{L_0}$ is exponentially small for large $L_0$. Therefeore, $\ln(z_{L_0})\arctan(z_{L_0})$ as well as the the lower integration limit $z_{L_0}$ can be substituted by 0 with exponential accuracy. Thus, we obtain
%
\begin{equation}
  \lambda_{J,0}\left[ 4 \ln(z_0)\arctan(z_0) + 4 M(z_0)\right]
  , \label{Eq:int0(z_0)}
\end{equation}
%
where the function $M(z)$ is defined by Eq.~\eqref{Eq:Def:M(z)} and can be calculated as discussed in appendix \ref{Sec:M(z)}. Note, that according to Eq.~\eqref{Eq:Sol:z} $z_0<1$, \ie, one can take a series ${\cal M}_N(z_0)$, see Eq.~\eqref{Eq:_M(z)}.

The calculation of the average phase in $\pi$ domain follows the same procedure. Now
%
\begin{equation}
  y = \frac{x-x_\pi}{\lambda_{J,\pi}}
\end{equation}
%
and, correspondingly, new integration limits
%
\begin{equation}
  y_\pi   =\frac{-x_\pi}{\lambda_{J,\pi}},\quad y_{L_\pi} = \frac{L_\pi-x_\pi}{\lambda_{J,\pi}}
  , \label{Eq:Def:y_pi}
\end{equation}
%
after integration by parts, we get
%
\begin{equation}
  -\pi L_\pi
  +\lambda_{J,\pi}\left[\left. 4 y \arctan e^y \right|_{y_\pi}^{y_{L_\pi}}
  -\int_{y_\pi}^{y_{L_\pi}} 4 \arctan e^y\, dy\right]
  . \label{Eq:int_pi(y)}
\end{equation}
%
Further, we introduce a new integration variable, \cf, Eq.~\eqref{Eq:Def:z},
%
\begin{equation}
  z = e^y\text{, so that }z_\pi=e^{y_\pi}\text{ and }z_{L_\pi} = e^{y_{L_\pi}}
  , \label{Eq:Def:z_pi}
\end{equation}
%
and Eq.~\eqref{Eq:int_pi(y)} becomes
%
\begin{equation}
  -\pi L_\pi
  +\lambda_{J,\pi}\left[ \left. 4 \ln(z)\arctan(z) \right|_{z_\pi}^{z_{L_\pi}}
  - 4\int_{z_\pi}^{z_{L_\pi}} \frac{\ln z}{1+z^2} \,dz\right]
  . \label{Eq:int_pi(z)}
\end{equation}
%
The value of $z_{L_\pi}$ is exponentially large so that the limit of integration can be substituted by $+\infty$ and $\arctan(z_{L_\pi})$ can be substituted by $2\pi$ with exponential accuracy. Thus, we obtain
%
\begin{equation}
  -\pi L_\pi
  + \lambda_{J,\pi}\left[ 2\pi \ln(z_{L_\pi}) - 4 \ln(z_\pi)\arctan(z_\pi) - 4 M(z_\pi)\right]
  . \label{Eq:int_pi(z_pi)}
\end{equation}
%
By using a definitions \eqref{Eq:Def:y_pi} and \eqref{Eq:Def:z_pi} the first term in the braces can be expanded to have $L_\pi$ explicitely. Further, according to Eq.~\eqref{Eq:Sol:z} $z_\pi>1$, \ie, one should make use of relation \eqref{Eq:M(z).reciprocity} and obtain
%
\begin{equation}
  +\pi L_\pi
  + \lambda_{J,\pi}\left[ \ln(z_\pi)(2\pi  - 4 \arctan(z_\pi)) - 4 {\cal M}(1/z_\pi)\right]
  . \label{Eq:int_pi(z_pi).final}
\end{equation}
%

Finally, combining Eqs.~\eqref{Eq:int0(z_0)} and \eqref{Eq:int_pi(z_pi).final} and embedding them into Eq.~\eqref{Eq:Def:phi_av} we obtain
\begin{widetext}
%
\begin{equation}
  \varphi = \frac{
    \lambda_{J,0}\left[ 4 \ln(z_0)\arctan(z_0) + 4 {\cal M}(z_0)\right]
    +\pi L_\pi
    + \lambda_{J,\pi}\left[ \ln(z_\pi)(2\pi  - 4 \arctan(z_\pi)) - 4 {\cal M}(1/z_\pi)\right]
    }{L_0+L_\pi}
  . \label{Eq:Res:varphi@L=inf}
\end{equation}
%
\end{widetext}
Note that the values of $z_0$, $z_\pi$ in Eq.~\eqref{Eq:Res:varphi@L=inf} are just constants defined by $j_{c,0}$ and $j_{c,\pi}$ and given by Eq.~\eqref{Eq:Sol:z}.

\EG{Roman, during writing this piece I had another idea. If one introduces the variable $e$ right away, and only then integrates by parts one arrives to different integrals/series. Shall we try it? May be the answer will be more compact...What do you think?}

\fi%arxiv

\bibliography{SF,pi}

\end{document}

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