Hamiltonian engineering of collective XYZ spin models in an optical cavity

Quantum simulations offer opportunities both for studying many-body physics and for generating useful entangled states. However, existing platforms are usually restricted to specific types of interaction, fundamentally limiting the models they can mimic. Here we realize an all-to-all interacting model with an arbitrary quadratic Hamiltonian, thus demonstrating an infinite-range tunable Heisenberg XYZ model. This was accomplished by engineering cavity-mediated four-photon interactions between an ensemble of 700 rubidium atoms with a pair of momentum states serving as the effective qubit degree of freedom. As one example of the versatility of this approach, we implemented the so-called two-axis counter-twisting model, a collective spin model that can generate spin-squeezed states that saturate the Heisenberg limit on quantum phase estimation. Furthermore, our platform allows for including more than two relevant momentum states by simply adding additional dressing laser tones. This approach opens opportunities for quantum simulation and quantum sensing with matter–wave interferometers and other quantum sensors, such as optical clocks and magnetometers.