  We study the thermally fluctuating state of a bi-layer cuprate superconductor
under the periodic action of a staggered field oscillating at optical
frequencies. This analysis distills essential elements of the recently
discovered phenomenon of light enhanced coherence in YBa$_2$Cu$_3$O$_{6+x}$,
which was achieved by periodically driving infrared active apical oxygen
distortions. The effect of a staggered periodic perturbation is studied using a
Langevin and Fokker-Planck description of driven, coupled Josephson junctions,
which represent two neighboring pairs of layers and their two plasmons. In a
toy model including only two junctions, we demonstrate that the external
driving leads to a suppression of phase fluctuations of the low-energy plasmon,
an effect which is amplified via the resonance of the high energy plasmon. When
extending the modeling to the full layers, we find that this reduction becomes
far more pronounced, with a striking suppression of the low-energy
fluctuations, as visible in the power spectrum. We also find that this effect
acts onto the in-plane fluctuations, which are reduced on long length scales.
All these findings provide a physical framework to describe light control in
cuprates.
