Electron-mediated entanglement of two distant macroscopic ferromagnets within a nonequilibrium spintronic device

Using the nascent concept of quantum spin -transfer torque [A. Zholud et al., Phys. Rev. Lett. 119, 257201 (2017); M. D. Petrovic ' et al., Phys. Rev. X 11, 021062 (2021)], we demonstrate that a current pulse can be harnessed to entangle quantum localized spins of two spatially separated ferromagnets (FMs) which are initially unentangled. The envisaged setup is composed of a spin-polarizer (FMp) and a spin -analyzer (FMa) FM layers separated by a normal metal (NM) spacer. The injection of a current pulse into the device leads to a time -dependent superposition of many -body states characterized by a high degree of entanglement between the spin degrees of freedom of the two distant FM layers. The nonequilibrium dynamics are due to the transfer of spin angular momentum from itinerant electrons to the localized spins via a quantum spin -torque mechanism that remains active even for collinear but antiparallel arrangements of the FMp and FMa magnetizations (a situation in which the conventional spin torque is absent). We quantify the mixed -state entanglement generated between the FM layers by tracking the time evolution of the full density matrix and analyzing the build-up of the mutual logarithmic negativity over time. The effect of decoherence and dissipation in the FM layers due to coupling to bosonic baths at finite temperature, the use of multielectron current pulses, and the dependence on the number of spins are also considered in an effort to ascertain the robustness of our predictions under realistic conditions. Finally, we propose a "current-pump-x-ray-probe" scheme, utilizing ultrafast x-ray spectroscopy, that can witness nonequilibrium and transient entanglement of the FM layers by extracting its time -dependent quantum Fisher information.