Cavity-mediated Interaction in a Bragg Interferometer: Hamiltonian Engineering and Mössbauer-like Suppression of Doppler Dephasing*

Laser-cooled atoms interacting via photon-mediated interactions are versatile platforms for quantum simulation and sensing. By harnessing momentum states as an effective qubit degree of freedom in an optical cavity quantum simulator, we realize an all-to-all interaction with arbitrary quadratic Hamiltonian or effectively a tunable collective Heisenberg XYZ model. With this capability, we realize for the first time the two-axis counter-twisting model, an iconic XYZ collective spin model that can generate spin-squeezed states that saturate the Heisenberg limit. An experimentally observable many-body energy gap also emerges, effectively binding interferometer matter-wave packets together to suppress Doppler dephasing with analogies to Mössbauer spectroscopy. The versatility of our platform to include more momentum states, combined with the flexibility of the simulated Hamiltonians by adding cavity tones opens rich opportunities for quantum information processing and quantum sensing using photon-mediated interactions with synthetic momentum states.