Non-local spin transport with two coupled channels: Manifestation of the inter-channel tunneling in the shapes of the Hanle curves

Dynamics of charge-density fluctuations in a system of two tunnel-coupled channels contains two diffusion modes with dispersion $i\omega=Dq^2$ and $i\omega =Dq^2+\frac{2}{\tau_t}$, where $D$ is the diffusion coefficient and $\tau_t$ is the tunneling time between the channels. The dispersion of corresponding spin-density modes depends on magnetic field as a result of spin precession with Larmour frequency, $\omega_L$. The presence of two modes affects the shape of the Hanle curve, describing the non-local resistance between the injector and the detector. We calculate the shapes, $R_{11}(\omega_L)$ and $R_{12}(\omega_L)$, of the Hanle curves, for geometries in which detector is located, respectively, in the same and in the different channel than the detector. We demonstrate that the relative shapes of $R_{11}(\omega_L)$ and $R_{12}(\omega_L)$ depend on the ratio $\tau_t/\tau_s$, where $\tau_s$ is the spin-diffusion time. If the coupling between the channels is local, i.e. only at the point $x=0$, then the difference of the shapes of $R_{11}(\omega_L)$ and $R_{12}(\omega_L)$ curves reflects the difference in statistics of diffusive trajectories which ``switch" or do not switch near $x=0$.