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\heading{E166: Josephson junction and current} 
\auname{Yair Vered , Arnon Mizrahy}

{\bf The problem:}

\Dn
A capacitor assembled from two superconducting plates.
In a superconductor the charge carriers are cooper pairs, which can be refered as bosons with the
charge e*=2e.\\
At a neutral state $|n=0\rangle$ some of the pairs are at the left plate and some at the right one. \\
The charged state $|n\rangle$ obtained by moving n pairs from the left plate to the right plate. \\
In order to describe the system we use the basis$|n\rangle$, $n=$\;integer\\
WE define the translation operator D as: $D|n\rangle=|n+1\rangle$\\
Assume that the shift Amplitude(per time unit)between the plates(by tunnling) is $\Omega$, and the capacitance of the\\      capacitor is C.\\

1. Define the eigenvalues $exp(i\varphi)$ and the eigenstates $|\varphi\rangle$ of D.\\

2. Write the Hamiltonian of the system using the c.c operatores $\varphi$,N.\\

3. Define the current operator J as $\frac{d\langle n \rangle}{dt}.$\\

4.How does the capacitance effect the current as function of time?


 


\Dn\Dn


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{\bf The solution:}

\Dn

(1) The problem is similar to a sites system, where D is the translation operator 
\[
D|\varphi \rangle = e^{(i\varphi)}|\varphi\rangle
\]

\Dn

(2)The operator N can be identified as the position operator, and operator $\varphi$ as the momentum operator.

\[
H=E_0 +\Omega D + \Omega^* D^\dag
\]
assuming $\Omega$ is real,and the average energy of the capacitor is $\frac{Q^2}{2C}$ we'll get:
\[
H=\frac{\left (e^*N \right)^2}{2C}+\Omega(e^{i\varphi}+e^{-i\varphi})= \frac{\left (e^*N \right)^2}{2C}
+2\Omega\cos\varphi\]
(3)  
\[
J=\frac{\partial \langle N \rangle}{\partial t} = i\left[2\Omega \cos\varphi,N\right]=
i(-i2\Omega\sin\varphi)=2\Omega\sin\varphi\]
(4)
\[ 
\frac{\partial J }{\partial t}=i\left[H,J\right]=\frac{(e^*)^2}{2C}\Omega\left[N^2,D+D^\dag\right]=
\]
	
\[
-\frac{(e^*)^2}{2C}\Omega\left[\left(DN-ND\right)N+N\left(DN-ND\right)-\left(D^\dag N-ND^\dag\right)N-N\left(D^\dag N-ND^\dag\right)\right]=
\]

\[
\frac{(e^*)^2 \Omega}{C}\left(N cos\varphi +cos\varphi N\right)
\]
The current decreased with the increase of the capacitance.

\end{document} 
