Second order Josephson effect in excitonic insulators

We show that in electron-hole bilayers with excitonic order arising from conduction and valence bands formed by atomic orbitals that have different parities, nonzero interlayer tunneling leads to a second order Josephson effect. This means the interlayer electrical current is related to the phase of the excitonic order parameter as $J = J_c \sin2\theta$ instead of $J = J_c \sin \theta$, and that the system has two degenerate ground states at $\theta=0, \pi$ that can be switched by an interlayer voltage pulse. When generalized to a three dimensional stack of alternating electron-hole planes or a two dimensional stack of chains, AC Josephson effect implies that electric field pulses perpendicular to the layers and chains can steer the order parameter phase between the two degenerate ground states, making these devices ultrafast memories. The order parameter steering also applies to the excitonic insulator candidate Ta$_2$NiSe$_5$.