# Tunneling junction in electromagnetic field [closed]

Why in presence of electromagnetic field hamiltonian,describing 2 systems with tunneling(for example josephson junction ) $$H=H_L+H_R+T(I+I^{\dagger})$$ where $I=c_Lc^{\dagger}_R$ is tunneling operator

should be changed as follows $$H=H_L+H_R+T(I e^{-i a}+I^{\dagger}e^{ia})$$ where $$a=\int_{L}^{R}\vec{A} d\vec{x}$$ $\vec{A}$-vector potential. For example when we put voltage $V$ to our system,$\vec{A}=-\vec{E}t$ and $a(t)=V t$.

• Could you please provide some more context for your notation? What is I, what is A? Aug 28, 2018 at 16:13
• Also, it looks like this just accounts for probability of tunneling decreasing exponentially with distance. See Griffiths sec 2.6 and 2.7 (finite square well and scattering). Aug 28, 2018 at 16:14
• you can see,that this coefficients became time dependent,unlike in griffiths. Now i think this coefficients should be modified to ensure gauge invariance,but why in this way? Aug 29, 2018 at 17:15