The two Josephson relations are and

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Problem

The two Josephson relations are

I = I_c\sin\varphi

and

V = \dfrac{\hbar}{2e}\dfrac{d\varphi}{dt}

, with

\varphi

the gauge-invariant phase difference across the junction and

I_c

its critical current. ((a)) Using

V = L_J\,dI/dt

for an inductor, derive the phase-dependent Josephson inductance

L_J(\varphi)

. ((b)) Using

dE/dt = IV

, derive the junction's potential energy

-E_J\cos\varphi

(up to an additive constant) and identify

E_J

in terms of

I_c

. ((c)) A symmetric split junction (two identical junctions of critical current

I_{c0}

, joined into a loop threaded by an external flux

\Phi

, so that the pair behaves as one effective junction of critical current

I_c(\Phi) = I_c|\cos(\pi\Phi/\Phi_0)|

, with

I_c = 2I_{c0} = 30.0\,\mathrm{nA}

its value at zero flux and

\Phi_0 = h/2e

the flux quantum) is biased at

\Phi = \Phi_0/3

. Using

\hbar = 1.0546\times10^{-34}\,\mathrm{J\,s}

h = 6.6261\times10^{-34}\,\mathrm{J\,s}

and

e = 1.6022\times10^{-19}\,\mathrm{C}

, find the effective critical current at

\Phi = \Phi_0/3

, and hence

E_J/h

and

L_J(0)

there (still

\hbar/2eI_c(\Phi)

, the linear inductance of the effective junction at zero phase). Then, for a transmon built on this split junction with

E_C/h = 0.250\,\mathrm{GHz}

fixed and

f_{01} \approx \sqrt{8E_JE_C}/h - E_C/h

, find

f_{01}

at zero flux and at

\Phi = \Phi_0/3

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