Quantum Geometric Phases as a New Window on Gravitational Waves

We investigate how low-frequency gravitational waves (LFGWs), originating from distant astrophysical or cosmological sources, can induce purely quantum geometric phases in mesoscopic optomechanical systems. These phases represent subtle imprints with no classical counterpart, going beyond standard dynamical or Berry-type contributions that admit Hannay-angle analogues. Such ultra-weak waves couple to the motion of a mechanical mirror and generate distinctive phase shifts in the system's quantum state that cannot arise in any classical description. To access this effect, we propose a Ramsey-type interferometric protocol in which the photon-number states of a quantized optical mode become entangled with the mirror's center-of-mass motion, enabling a direct readout of the LFGW-induced geometric phase. This framework establishes a distinctly quantum approach for probing low-frequency gravitational wave modes, offering an alternative to conventional detection strategies based on spacetime strain.