Bohr-Sommerfeld, WKB, and current-biased Josephson junction

[EightBells]

Homework Statement

Using the Bohr-Sommerfeld approximation, evaluate numerically values of the lowest three energy levels as a function of J in the interval J < Ic. Compare this result with the perturbation value η from problem 4. Also, using the WKB
approximation, calculate the tunneling coefficient from the ground state energy to the continuous states. When setting up the WKB integral for the tunneling rate, assume that the left turning point is located at the minimum of the potential energy δmin and the right turning point δr is defined by the condition UJ (δr) = UJ (δmin)

Relevant Equations

Hamiltonian for the current-biased junction: $H=\frac {Q^2} {2C} + U_J(\delta)$
One dimensional Josephson potential: $U_J(\delta)=-\frac {\Phi} {2 \pi} \left( J \delta + I_C \cos \delta \right)$, where Q and C are charge and capacitance of the junction, $\Phi$ the magnetic flux quantum, J the current flowing through the junction, and $I_C$ the critical current of the junction.
Considering the case where $|J|\lt I_C$.

I've already found the turning points, in the case of the left turning point, the local minimum of the potential, $\delta_{min}=1.11977$ when evaluating for an arbitrary value of current $J=0.9I_C$. The left turning point is therefore $\delta_r=2.48243$.

I know the Bohr-Sommerfeld approximation states $\int_{x_1}^{x_2} p(x,E_n) \,dx = \hbar \pi (n+ \frac 1 2)$. I'm unsure what equation for momentum I should use in the case of the Josephson Junction. I know in the case of the harmonic oscillator, $p(x)=\sqrt{2m(E-U)}$ from the structure of the Hamiltonian $H=\frac {p^2} {2m} + U$. I can see the Josephson junction has a similarly structured Hamiltonian, where the charge Q is analogous to the momentum. However when I try to evaluate $\int_{\delta_{min}}^{\delta_r} \sqrt{2C(E-U_J(x))} \,dx=\hbar\pi(n+ \frac 1 2)$, I can't seem to find a way (i.e. Mathematica, Wolfram, other online integral calculators) to solve the integral and find an expression for the energies $E_n$. This makes me think I'm using the wrong expression for the momentum?

I'm also confused as to what the "WKB integral for the tunneling rate" is. I understand the WKB approximation can be used to approximate the wavefunction, and in the tunneling case when $E \lt V$ the momentum is of the form I attempted to use in the Bohr-Sommerfeld approximation, but how do I find the tunneling rate or tunneling coefficient?

*note: using values of $I_C=30.572 \mu A$ and $C=47 pF$.

 

[mark726]

Thank you for sharing your findings and questions regarding the Josephson Junction. It seems like you have a good understanding of the Bohr-Sommerfeld approximation and how it can be applied to the Josephson Junction. However, you are correct in questioning the expression for the momentum in this system.

In the case of the Josephson Junction, the momentum is not simply the charge Q, but rather the phase difference δ between the superconducting leads. This is because the Hamiltonian for the Josephson Junction is not a simple harmonic oscillator, but rather a sine-Gordon potential of the form H=−EJcos(δ). Therefore, the correct expression for the momentum in this case is p=√2m(EJ−E).

I would suggest using this expression for the momentum in your Bohr-Sommerfeld approximation and see if it helps in solving the integral. Additionally, for the WKB integral for the tunneling rate, you can use a similar approach and incorporate the expression for the momentum in the integral.

I hope this helps in your calculations and understanding of the Josephson Junction. Keep up the good work and don't hesitate to reach out if you have any further questions.