  In this work, we provide an answer to the question: how sudden or adiabatic
is a change in the frequency of a quantum harmonic oscillator (HO)? To do this,
we investigate the behavior of a HO, initially in its fundamental state, by
making a frequency transition that we can control how fast it occurs. The
resulting state of the system is shown to be a vacuum squeezed state in two
bases related by Bogoliubov transformations. We characterize the time evolution
of the squeezing parameter in both bases and discuss its relation with
adiabaticity by changing the rate of the frequency transition from sudden to
adiabatic. Finally, we obtain an analytical approximate expression that relates
squeezing to the transition rate as well as the initial and final frequencies.
Our results shed some light on subtleties and common inaccuracies in the
literature related to the interpretation of the adiabatic theorem for this
system.
