  A broad and a narrow level of a quantum dot connected to two external leads
may swap their respective occupancies as a function of an external gate
voltage. By mapping this problem onto a multi-flavored Coulomb-gas we show that
such population switching is not abrupt. However, trying to measure it by
adding a third electrostatically coupled lead may render this switching an
abrupt first order quantum phase transition. This is related to the interplay
of the Mahan mechanism versus the Anderson orthogonality catastrophe, in
similitude to the Fermi edge singularity. A concrete setup for experimental
observation of this effect is also suggested.
