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} et al., Phys. Rev. X {\bf 11}, 021062 (2021)], we predict that a charge current pulse can be harnessed to entangle localized quantum spins of two spatially separated ferromagnets (FMs) within a FM$_p$/NM/FM$_a$ spin-valve spintronic device, where a normal metal (NM) spacer separates a spin-polarizer (FM$_p$) from a spin-analyzer (FM$_a$) FM layer. The injection of a current pulse endows the spin-valve system with rich nonequilibrium dynamics, where a quantum superposition of many-body states places the spatially separated FM layers into a mixed entangled state, after tracing out the degrees of freedom of the conduction electrons. This is due to a transfer of spin angular momentum from conduction electrons to the localized spins via a quantum spin-transfer 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. The dynamical build-up of mixed-state entanglement between the FM layers is quantified by calculating the mutual logarithmic negativity, entanglement entropy and mutual information over time via fully quantum many-body approaches. The effect of decoherence on our scheme, the use of multi-electron pulses and the scaling with system size are also analyzed in an effort to ascertain the robustness of our predictions under realistic experimental conditions. Finally, we propose a ``current-pump/X-ray-probe'' scheme, utilizing ultrafast X-ray spectroscopy, which can witness nonequilibrium entanglement of the FM layers by extracting their time-dependent quantum Fisher information.