  We study electro-mechanical entanglement in a system where a massive membrane
is capacitively coupled to a {\it low frequency} LC resonator. In opto- and
electro-mechanics, the entanglement between a megahertz (MHz) mechanical
resonator and a gigahertz (GHz) microwave LC resonator has been widely and well
explored, and recently experimentally demonstrated. Typically, coupling is
realized through a radiation pressure-like interaction, and entanglement is
generated by adopting an appropriate microwave drive. Through this approach it
is however not evident how to create entanglement in the case where both the
mechanical and LC oscillators are of low frequency, e.g., around 1 MHz. Here we
provide an effective approach to entangling two low-frequency resonators by
further coupling the membrane to an optical cavity. The cavity is strongly
driven by a red-detuned laser, sequentially cooling the mechanical and
electrical modes, which results in stationary electro-mechanical entanglement
at experimentally achievable temperatures. The entanglement directly originates
from the electro-mechanical coupling itself and due to its quantum nature will
allow testing quantum theories at a more macroscopic scale than currently
possible.
