Editing Josephson effect (section)

==Josephson energy==
Based on the similarity of the Josephson junction to a non-linear inductor, the energy stored in a Josephson junction when a supercurrent flows through it can be calculated.<ref>[[Michael Tinkham]], Introduction to superconductivity, Courier Corporation, 1986.</ref>

The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):

:<math>I = I_c \sin\varphi.</math>

The superconducting phase evolution equation is analogous to [[Faraday's law of induction|Faraday's law]]:

:<math>V=\operatorname{d}\!\Phi/\operatorname{d}\!t\,.</math>

Assume that at time <math>t_1</math>, the Josephson phase is <math>\varphi_1</math>; At a later time <math>t_2</math>, the Josephson phase evolved to <math>\varphi_2</math>. The energy increase in the junction is equal to the work done on the junction:

:<math>
\Delta E = \int_1^2 I V\operatorname{d}\!{t}
= \int_{1}^{2} I\operatorname{d}\!\Phi
= \int_{\varphi_1}^{\varphi_2} I_c\sin \varphi \operatorname{d}\!\left(\Phi_0\frac{\varphi}{2\pi}\right)
= -\frac{\Phi_0 I_c}{2\pi} \Delta\cos\varphi\,.
</math>

This shows that the change of energy in the Josephson junction depends only on the initial and final state of the junction and not the [[Thermodynamic process path|path]]. Therefore, the energy stored in a Josephson junction is a [[state function]], which can be defined as:

:<math>E(\varphi)=-\frac{\Phi_0 I_c}{2\pi}\cos\varphi=-E_J\cos\varphi
\,.</math>

Here <math>E_J = |E(0)|=\frac{\Phi_0 I_c}{2\pi}</math> is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by <math>E_J = L_JI^2_c</math>. An alternative but equivalent definition <math>E(\varphi)=E_J(1-\cos\varphi)</math> is also often used.

Again, note that a non-linear [[Inductor|magnetic coil inductor]] accumulates [[potential energy]] in its magnetic field when a current passes through it; However, in the case of Josephson junction, no magnetic field is created by a supercurrent — the stored energy comes from the kinetic energy of the charge carriers instead.