We study the Gouy phase that arises from the time evolution of
confined matter waves in a harmonic potential. We consider the quantum
dynamics of a particle represented by a Gaussian wavepacket
position-momentum correlated. By tuning the parameters that govern its
evolution, we uncover interesting effects, with a particular focus on squeezing.
During the squeezing process, a Gouy phase contribution of π/4 accumulates,
establishing a direct connection between the Gouy phase and a purely quantum
phenomenon. The interplay between the wavepacket squeezing and spreading
propagation in one dimension contributes to the total Gouy phase accumulation
of π/2. Both, squeezing and Gouy phase have been shown independently
valuable for quantum metrology and state engeenering. By establishing a direct
and controlable link between these two fundamental phenomena, we broaden
the scope for quantum-enhanced technologies, offering potential applications in
precision measurements and quantum communication..