Generating Fock state exceeding 10000 excitations with near unit fidelity by adaptive generalized-parity measurement

Macroscopic Fock states provide valuable resources for quantum information processing and quantum metrology. We here propose an adaptive generalized-parity-measurement protocol to create macroscopic Fock states with more than 10000 excitations. For a general system with a discrete spectrum, e.g., a bosonic mode, that is coupled to an ancillary qubit, we derive a construction rule of either a diagonal generalized parity measurement (GPM) or a displaced GPM with intervals adaptive to the last outcome of repeated measurements on the qubit. Different from the probabilistic protocols based on postselection, in which only a single prescribed sequence of free-evolution-measurement is survived, our protocol retains every measurement trajectory by converting the outcome randomness of the ancillary-qubit measurement to the adaptive update of GPM. Using the resonant Jaynes-Cummings (JC) model, our protocol can transform a large coherent state to a large Fock state of photon numbers up to n_t=mathcal{O}\(10⁴\) within 10 rounds of measurements, where the averaged fidelity reaches about 80\%. The probability for obtaining such a large Fock state with a fidelity above 99\% remains about 35\% with respect to the ensemble sampling. Our protocol also applies to displaced thermal states, indicating its robustness against a moderate thermal mixture.