Ultra-cold atomic gases are an ideal platform for
precision measurement devices and analogue quantum simulators,
which will prove instrumental in unlocking the secrets of quantum
gravity and high-temperature superconductivity. However, current
experimental techniques cannot cool atomic gases to
simulatenously both the low entropies and high particle numbers
necessary for these applications. A promising alternative is
feedback cooling: using continuous-measurement feedback control
to damp out energy fluctuations and cool a gas. So far, feedback
cooling has been primarily studied at zero temperature, with the
only finite temperature simulation achieved via a computationally
expensive numerical method restricted to bosons. This thesis
develops a perturbative model for the feedback cooling of a
finite temperature condensed Bose gas using Bogoliubov theory,
with the aim of deriving dynamic equations for the system that
are both analytically tractable, and allow for fast numerical
prototyping of new feedback control schemes. Using the
measurement-feedback model of Szigeti \textit{et al.}
\cite{szigeti_continuous_2009,szigeti_feedback_2010}, we derive a
low temperature perturbative model for feedback cooling of a Bose
gas in an arbitrary trapping and control potential. Using this
general model, we then derive a model for the dynamics of a Bose
gas in a hard box trap being cooled with an energy damping
control. We complete preliminary simulations of this model in the
no-backaction conditional measurement limit, damping out density
fluctuations in a gas of 90% condensate fraction and cooling
it to 93.5±1%. We find that, in this limit, the dynamics
of the gas are largely independent of number but significantly
depend on the inter-particle interaction. We also find an optimal
energy damping control strength in this limit. However, our model
is not very efficient for simulation, particularly for a large
number of particles and measurement strength. As an alternative,
we propose an approximation scheme in which steady-state analytic
solutions could be obtained from the model. Finally, we propose
two methods to develop a Bogoliubov model for the feedback
cooling of fermions, which would be the first finite temperature
model for the Fermi gas case
