Optomechanical Pair-Coherent State Generation

In the pursuit of fault-tolerant quantum computing, platforms like superconducting circuits, trapped ions, and photonic qubits face significant scaling challenges. A promising alternative involves cat states in mechanical resonators, offering noise-biased error correction, long lifetimes, compact footprints, and minimal crosstalk. Furthermore, entangled coherent states in two modes, known as pair-coherent states (PCS), enable pair-cat quantum computing codes that can offer robustness to single-quanta loss. This work introduces a novel method to dissipatively engineer mechanical PCS, leveraging optomechanical interactions. Here we derive an analytical reduced master equation using a Schrieffer-Wolff transformation of the optomechanical Hamiltonian, followed by adiabatic elimination of the cavity mode. The optimal parameter space for PCS generation is verified through numerical solutions of the full optomechanical master equation using QuTiP. We demonstrate simulated PCS achieved fidelity exceeding 99% and 94%, for the adiabatic and full models, respectively, confirming the protocol's suitability for generating optomechanical cat codes. These results highlight the platform's potential for fault-tolerant qubits with extended coherence times, providing a pathway for scalable quantum computing through mechanical pair-cat codes.