LC-Time Behavior of Weak Superconducting Loops

We use Anderson's operator representation to obtain the radiative properties of a superconducting loop with a single Josephson junction. We treat n, the number of displaced pairs, and φ, the relative phase across the junction, as canonically conjugate operators. Since n is related to the voltage across the junction by V=2en and φ is related to the current in the loop via the fluxoid quantization, the Hamiltonian for the system, H=1/2CV²+1/2LI²-EJ cosφ, can be reduced to operator form. By applying the Hamiltonian formalism in the usual manner, the equations of motion are obtained. In the limit of small oscillation (〈φ²〉<1), the Hamiltonian can be expanded and yields a solution which corresponds to a resonant frequency ̃ \~ωLC=[(1/LC)+(2e)²EJC]1/2. This mode corresponds to an "LC" oscillation of the loop modified by the Josephson junction. For physical situations, the frequency of the oscillation can be adjusted to be as high as 10¹²{sec}^{{-}1}$, with a purity of 1 part in ${10}⁵$ and at a power output of ${10}^{{-}13}$ W. Under appropriate conditions, in addition to the modified $LC$ resonance, we obtain solutions corresponding to the familiar ac Josephson radiation as well as the metastable flux states of a loop with a weak link.