  A quantum harmonic oscillator (spring subsystem) is stabilized towards a
target Fock state by reservoir engineering. This passive and open-loop
stabilization works by consecutive and identical Hamiltonian interactions with
auxiliary systems, here three-level atoms (the auxiliary ladder subsystem),
followed by a partial trace over these auxiliary atoms. A scalar control input
governs the interaction, defining which atomic transition in the ladder
subsystem is in resonance with the spring subsystem. We use it to build a
time-varying interaction with individual atoms, that combines three
non-commuting steps. We show that the resulting reservoir robustly stabilizes
any initial spring state distributed between 0 and 4n+3 quanta of vibrations
towards a pure target Fock state of vibration number n. The convergence proof
relies on the construction of a strict Lyapunov function for the Kraus map
induced by this reservoir setting on the spring subsystem. Simulations with
realistic parameters corresponding to the quantum electrodynamics setup at
Ecole Normale Superieure further illustrate the robustness of the method.
