  We show the existence of a majorization ladder in bosonic Gaussian channels,
that is, we prove that the channel output resulting from the $n\text{th}$
energy eigenstate (Fock state) majorizes the channel output resulting from the
$(n\!+\!1)\text{th}$ energy eigenstate (Fock state). This reflects a remarkable
link between the energy at the input of the channel and a disorder relation at
its output as captured by majorization theory. This result was previously known
in the special cases of a pure-loss channel and quantum-limited amplifier, and
we achieve here its nontrivial generalization to any single-mode
phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our
proof is the explicit construction of a column-stochastic matrix that relates
the outputs of the channel for any two subsequent Fock states at its input.
This is made possible by exploiting a recently found recurrence relation on
multiphoton transition probabilities for Gaussian unitaries [M. G. Jabbour and
N. J. Cerf, Phys. Rev. Research 3, 043065 (2021)]. Possible generalizations and
implications of these results are then discussed.
