Quantum Phases of Time-Reversal Invariant Bose-Einstein Condensates

Recent experimental realization of spin-orbit coupling (SOC) for ultracold atomic gases with the use of synthetic gauge fields provides a powerful platform for the study of novel quantum phenomena and the simulation of exotic condensed matter phases. However, in conventional schemes of SOC in ultracold bosonic gases, time-reversal symmetry, which plays a critical role in topologically nontrivial states, is broken by an effective transverse Zeeman field. We study the quantum phases of SOC Bose-Einstein condensates (BECs) with the use of a Hermite-Gaussian (HG) beam to induce Raman transitions. This treatment allows for SOC in bilayer BECs with inter-layer tunneling where time-reversal symmetry is preserved. New ground-state phases are introduced that are not seen in conventional SOC BECs. We propose a experimentally feasible setup and discuss the physical parameters under which time-reversal symmetry can be preserved. The Hamiltonians for SOC BECs are often nonlinear and the methods used for calculating the ground-state wavefunctions are computationally expensive. The wavefunctions need to be calculated on an individual basis to study the ground-state quantum phases on a granular level. We propose the use of convolutional-neural-networks (CNN) to train SOC BEC systems and reduce the computational cost of these ground-state calculations. We show the overall network setup and discuss the ranges over which the model is realizable. The proposed CNN uses a reverse-flow algorithm that allows for complex phases of the wavefunction and thus permits for a broader study of SOC BEC systems. In summary, this dissertation details how time-reversal symmetry can be preserved in SOC BECs and how predictive analytics can be used to further understand the ground-state properties of these systems