Tunneling measurement of quantum spin oscillations

We consider the problem of tunneling between two leads via a localized spin 1/2 or any other microscopic system (e.g., a quantum dot) which can be modeled by a two-level Hamiltonian. We assume that a constant magnetic field ${\mathbf{B}}_0$ acts on the spin, that electrons in the leads are in a voltage driven thermal equilibrium, and that the tunneling electrons are coupled to the spin through exchange and spin-orbit interactions. Using the nonequilibrium Keldysh formalism we find the dependence of the spin-spin and current-current correlation functions on the applied voltage between leads V, temperature T, ${\mathbf{B}}_0,$ and on the degree and orientation ${\mathbf{m}}_{\alpha }$ of spin polarization of the electrons in the right (α=R) and left (α=L) leads. We show the following (a) The spin-spin correlation function exhibits a peak at the Larmor frequency, ${\omega }_L,$ corresponding to the effective magnetic field $\mathbf{B}$ acting upon the spin as determined by ${\mathbf{B}}_0$ and the exchange field induced by tunneling of spin-polarized electrons. (b) If the ${\mathbf{m}}_{\alpha }\mathrm{’}\mathrm{s}$ are not parallel to $\mathbf{B}$ the second-order derivative of the average tunneling current $I\left(V\right)\mathrm{}$ with respect to V is proportional to the spectral density of the spin-spin correlation function, i.e., exhibits a peak at the voltage $V=ħ{\omega }_L/e.\mathrm{}$ (c) In the same situation when $V>B$ the current-current correlation function exhibits a peak at the same frequency. (d) The signal-to-noise (shot-noise) ratio R for this peak reaches a maximum value of order unity, $R<~4,$ at large V when the spin is decoupled from the environment and the electrons in both leads are fully polarized in the direction perpendicular to B. (e) $R\ll 1$ if the electrons are weakly polarized, or if they are polarized in a direction close to ${\mathbf{B}}_0,$ or if the spin interacts with the environment stronger than with the tunneling electrons. Our results of a full quantum-mechanical treatment of the tunneling-via-spin model when $V\gg B$ are in agreement with those previously obtained in the quasiclassical approach. We discuss also the experimental results observed using scanning tunneling microscopy dynamic probes of the localized spin.