  We investigate the effects of a low frequency (acoustic) phonon bath on the
dynamics of a quantum dot modelled as a cascade three-level system. We show
that the phonon bath appears to the upper transition of the cascade system as a
broadband reservoir of inverted rather than conventional harmonic oscillators.
The action of the annihilation and creation operators of the inverted
oscillator are interchanges relative to those of the usual harmonic oscillator
that it serves as a linear amplifier to the system, and thereby gives rise to
unusual features in the dynamics of the quantum dot. We find that the phonon
bath, although being in a thermal state, affects the quantum dot as a
correlated-type reservoir which results in the decay of the system to a
correlated two-photon state with the population distribution no longer obeying
a Boltzmann distribution. It is particularly interesting that even for a zero
temperature phonon reservoir the steady state is a correlated state which under
appropriate conditions on the Rabi frequencies and the damping rates can reduce
to a strongly correlated pure state. It is shown that the two-photon
correlations result in a significant squeezing and strong two-photon
correlations in the radiation field emitted by the quantum dot. The presence of
the correlations in the system is manifest in the presence of quantum beats in
the time evolution of the populations and the radiation intensity. The effect
of the ordinary spontaneous emission on the features induced by the phonon bath
is also discussed.
