  Periodically-driven oscillators are commonly described in a frame co-rotating
with the drive and using the rotating-wave approximation (RWA). This
description, however, is known to induce errors for off-resonant driving. Here
we show that the standard quantum description, using creation and annihilation
of particles with the oscillator's natural frequency, necessarily leads to
incorrect results when combined with the RWA. We demonstrate this on the simple
harmonic oscillator and present an alternative operator basis which reconciles
the RWA with off-resonant driving. The approach is also applicable to more
complex models, where it accounts for known discrepancies. As an example, we
demonstrate the advantage of our scheme on the driven quantum Duffing
oscillator.
