  All-order momentum correlation functions associated with the time-of-flight
spectroscopy of three spinless ultracold bosonic interacting neutral atoms
confined in a linear three-well optical trap are presented. The underlying
Hamiltonian employed for the interacting atoms is an augmented three-site
Hubbard model. Our investigations target matter-wave interference of massive
particles, aiming at the establishment of experimental protocols for
characterizing the quantum states of trapped attractively or repulsively
interacting ultracold particles, with variable interaction strength. The
manifested advantages and deep physical insights that can be gained through the
employment of the results of our study for a comprehensive understanding of the
nature of the quantum states of interacting many-particle systems, via analysis
of the all-order (that is 1st, 2nd and 3rd) momentum correlation functions for
three bosonic atoms in a three well confinement, are illustrated and discussed
in the context of time-of-flight inteferometric interrogations of the
interaction-strength-induced emergent quantum phase transition from the Mott
insulating phase to the superfluid one. Furthermore, we discuss that our
inteferometric interrogations establish strong analogies with the
quantum-optics interference of three photons, including the aspects of genuine
three-photon interference, which are focal to explorations targeting the
development and implementation of quantum information applications and quantum
computing.
