Nondemolition measurements of a single quantum spin using Josephson oscillations

We consider a Josephson junction containing a single localized spin 1/2 between conventional singlet superconducting electrodes. We study the spin dynamics and measurements when a dc-magnetic field $\mathbf{B}\parallel z$ acts on the spin and the junction is embedded into a dissipative circuit. We show that when tunneling or a voltage are turned on at time $t=0$ the Josephson current starts to oscillate with an amplitude depending on the initial ($t=0$) value of the spin $z$-component, $S_z=\pm 1\mathrm{/}2$. At low temperatures, when effects of quasiparticles may be neglected, this procedure realizes a quantum-non-demolition (QND) measurement of $S_z$.