Brother Allah has made the night for rest & day for work. If a person gets up during the night for an hour or so for tahajjud its natural that during the daytime he 'll feel sleepy and not be as productive. He wont be able as alert in the daytime and his day-night schedule will be unbalanced. Please explain how to reconcile these two.
People during the time of the Prophet 
used to sleep for a number of hours in the afternoon, so in this way they got a healthy amount of sleep during the 24 hours. But you are right, if you stay up to pray then it can affect you negatively the next day. Personally I’m very sensitive to bad sleep and I can barely get anything done the next day if I don’t get my 8 hours of sleep.
I don’t think there is any virtue in ruining one’s sleep and productivity for the sake of worship. What matters is to make the effort to do extra worship in whatever way that fits your schedule.