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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 $a(t)=V t$.

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    $\begingroup$ Could you please provide some more context for your notation? What is I, what is A? $\endgroup$ Aug 28, 2018 at 16:13
  • $\begingroup$ 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). $\endgroup$ Aug 28, 2018 at 16:14
  • $\begingroup$ 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? $\endgroup$ Aug 29, 2018 at 17:15

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