Spin-Spin Correlators in the Majorana Representation

In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit’s state.

Figure 1

The lowest-order nonvanishing contributions to the self-energies ${\Sigma }_{\Psi }$ and ${\Sigma }_{\eta }$. The solid lines denote the propagators of the bath ($X$), the dashed line—of the $f$ fermion, and the dotted lines—of the Majorana fermion $\eta$. (See text.)