The Sixth Amendment guarantees "an impartial jury of the State and district whereof the crime shall have been committed."  It doesn't say anything about them being awake.  Nicole Hong reports for the WSJ:

The right to a jury trial is a pillar of America's justice system, enshrined in the Constitution from a tradition dating back more than 1,000 years.

The problem these days is making sure jurors stay awake.

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"From time immemorial, jurors have been falling asleep because from time immemorial, lawyers have been boring," says John Gleeson, who was a federal judge in Brooklyn for 22 years. "We're the dullest people in the world, for Christ's sake."