All-order momentum correlations of three ultracold bosonic atoms confined in triple-well traps. I. Signatures of emergent many-body phase transitions and analogies with three-photon quantum-optics interference

The higher-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 a three-site Hubbard model. The methodology introduced here contrasts with and goes beyond that based on the standard Wick's factorization scheme; it enables determination of third-, second-, and first-order momentum correlations for entangled states beyond the product states. 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 interferometric interrogations of the interaction-strength-induced quantum phase transition from the Mott insulating phase to the superfluid one. Furthermore, we discuss that our interferometric interrogations establish strong analogies with the quantum-optics interference of three photons, including the aspects of genuine three-photon interference.