Novel Spin Dynamics in a Josephson Junction

We address the dynamics of a single spin embedded in the tunneling barrier between two superconductors. As a consequence of pair correlations in the superconducting state, the spin displays a rich and unusual dynamics. To properly describe the time evolution of the spin we find the generalized Wess-Zumino-Witten-Novikov term in the effective action for the spin on the Keldysh contour. The superconducting correlations lead to an effective spin action which is nonlocal in time leading to unconventional precessions. Our predictions might be directly tested for macroscopic spin clusters.

Figure 1

Magnetic spin coupled to two superconducting leads. In the presence of a magnetic field $\mathbf{B}$, the spin precesses around the field direction. As we illustrate here, in a Josephson junction the standard precession is accompanied by a polar motion leading to a full blown rigid body like dynamics.

Figure 2

The unit sphere for the vectors ${\mathbf{n}}^{u,l}(t)$ is shown. The contour $C$ is the Keldysh contour for the forward ($u$) and backward ($l$) evolution. To properly describe the spin on this closed contour we analyze the WZWN term, see Eq. (13). For clarity, we draw a small piece of the closed trajectories.

Figure 3

The spin dynamics. Top panels: the azimuthal ($\varphi$) and polar ($\theta$) angles as a function of time for an exaggerated $\alpha =0.4$ (red/light gray) versus the standard precessions in the absence of the coupling ($\alpha =0$ in blue/dark gray). Note that unlike the standard case, the semiclassical spin trajectory is not confined to planar motions. The polar angle oscillates between its two extreme values ${\theta }_1$ and ${\theta }_2$. In the bottom panels we display $⟨S_x(t)⟩$ and $⟨S_z(t)⟩$. Here we take ${\omega }_L={\omega }_J$.

Figure 4

The resulting spin motion on the unit sphere in the general case. As in the motion of classical spinning top, the spin exhibits undulations along the polar direction.