We consider a ground-state wide-gap band insulator turning into a nonequilibrium excitonic insulator (NEQ-EI) upon visiting properly selected and physically relevant highly excited states. The NEQ-EI phase, characterized by self-sustained oscillations of the complex order parameter, neatly follows from a Nonequilibrium Greens Function treatment on the Konstantinov-Perel contour. We present the first {em ab initio} band structure of LiF, a ground-state bulk insulator, in different NEQ-EI states and show that these states can be generated by currently available pump pulses. We highlight two general features of time-resolved ARPES spectra: (1) during the pump-driving the excitonic spectral structure undergoes a convex-to-concave shape transition and {em concomitantly} the state of the system goes through a BEC-BCS crossover; (2) attosecond pulses shone after the pump-driving at different times $t_{rm delay}$ generate a photocurrent which {em oscillates} in $t_{rm delay}$ with a pump-tunable frequency -- we show that this phenomenon is similar to the AC response of an exotic Josephson junction.
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