  We present a new scheme for controlling the quantum state of a harmonic
oscillator by coupling it to an anharmonic multilevel system (MLS) with first
to second excited state transition frequency on-resonance with the oscillator.
In this scheme that we call "ef-resonant", the spurious oscillator Kerr
non-linearity inherited from the MLS is very small, while its Fock states can
still be selectively addressed via an MLS transition at a frequency that
depends on the number of photons. We implement this concept in a circuit-QED
setup with a microwave 3D cavity (the oscillator, with frequency 6.4 GHz and
quality factor QO=2E-6) embedding a frequency tunable transmon qubit (the MLS).
We characterize the system spectroscopically and demonstrate selective
addressing of Fock states and a Kerr non-linearity below 350 Hz. At times much
longer than the transmon coherence times, a non-linear cavity response with
driving power is also observed and explained.
