  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 Green's 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.
