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. {\bf 119}, 257201 (2017); M. D. Petrovi\'{c} {\em et al.}, Phys. Rev. X {\bf 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 comprises a spin-polarizer (FM$_p$) and a spin-analyzer (FM$_a$) FM layers separated by 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 non-equilibrium 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 {\em collinear but antiparallel} arrangements of the FM$_p$ and FM$_a$ 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 multi-electron 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.