Geometric Phonon Energy Pumping in a Layer-Hybridized Moir\'e Exciton Manifold
Slow cyclic modulation of an open quantum system can generate a geometric contribution to its energy transfer statistics, separable from the dynamic background. In this work we construct an experimentally anchored four state open system model for a gate-tunable WSe2/WS2 moir\'e exciton manifold within a population level full counting statistics framework. We first drive the system uniformly through a closed gate and pump loop and separate the geometric contribution from the dynamic background by reversing the loop direction. We then examine how this response changes with environmental rates, exciton coupling, control conditions, and static spectral broadening. A shorter cycle period followed by nonuniform traversal of the driving loop substantially improves detectability while preserving the geometric response. Combined with enhanced phonon relaxation and reduced radiative loss, this gives about a fourteenfold fixed time signal to noise gain relative to the uniform loop. Our results show that the geometric phonon response remains robust against realistic variations in the system and its environment, and that we can improve its detectability more effectively by suppressing dynamical noise through frequency modulated driving.