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 $\varphi$, the relative phase across the junction, as canonically conjugate operators. Since $n$ is related to the voltage across the junction by $V=\frac{2en}{\hslash }$ and $\varphi$ is related to the current in the loop via the fluxoid quantization, the Hamiltonian for the system, $H=\frac{1}{2}C{V}^2+\frac{1}{2}L{I}^2-E_J cos\varphi$, 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 ($〈{\varphi }^2〉<1\mathrm{}$), the Hamiltonian can be expanded and yields a solution which corresponds to a resonant frequency ${\tilde{\omega }}_{\mathrm{LC}}={\left[\left(\frac{1}{\mathrm{LC}}\right)+\frac{{\mathrm{}\left(2e\right)\mathrm{}}^2{E}_J}{\hslash C}\right]}^{\frac{1}{2}}.$ This mode corresponds to an "$\mathrm{LC}$" 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 ${10}^{12}$ ${\sec }^{-1\mathrm{}}$, with a purity of 1 part in ${10}^5$ and at a power output of ${10}^{-13\mathrm{}}$ W. Under appropriate conditions, in addition to the modified $\mathrm{LC}$ 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.