  We consider a model of a Bloch band subjected to an oscillating electric
field and coupled to a featureless fermionic heat bath, which can be solved
exactly. We demonstrate rigorously that in the limit of vanishing coupling to
this bath (so that it acts as an ideal thermodynamic bath) the occupation of
the Floquet band is not a simple Fermi-Dirac distribution function of the
Floquet energy, but instead it becomes a ``staircase'' version of this
distribution. We show that this distribution generically leads to a finite
rectified electric current within the optical gap of a metal even in the limit
of vanishing carrier relaxation rates, providing a rigorous demonstration that
such rectification is generically possible and clarifying previous statements
in the optoelectronics literature. We show that this current remains non-zero
even up to the leading perturbative second order in the amplitude of electric
fields, and that it approaches the standard perturbative expression of the Jerk
current obtained from a simpler Boltzmann description within a relaxation time
approximation when the frequencies are small compared to the bandwidth.
