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We consider a ground-state wide-gap band insulator turning into anonequilibrium excitonic insulator (NEQ-EI) upon visiting properly selected andphysically relevant highly excited states. The NEQ-EI phase, characterized byself-sustained oscillations of the complex order parameter, neatly follows froma Nonequilibrium Green's Function treatment on the Konstantinov-Perel' contour.We present the first {\em ab initio} band structure of LiF, a ground-state bulkinsulator, in different NEQ-EI states and show that these states can begenerated by currently available pump pulses. We highlight two general featuresof time-resolved ARPES spectra: (1) during the pump-driving the excitonicspectral structure undergoes a convex-to-concave shape transition and {\emconcomitantly} the state of the system goes through a BEC-BCS crossover; (2)attosecond pulses shone after the pump-driving at different times $t_{\rmdelay}$ generate a photocurrent which {\em oscillates} in $t_{\rm delay}$ witha pump-tunable frequency -- we show that this phenomenon is similar to the ACresponse of an exotic Josephson junction.

Pump driven normal-to-excitonic insulator transition: Josephson oscillations and signatures of BEC-BCS crossover in time-resolved ARPES