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

Abstract: 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 $n_t=\mathcal{O}(10^4)$ 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.

• The paper presents a deterministic adaptive GPM protocol that isolates macroscopic Fock states (>10,000 excitations) with near unit fidelity.
• The methodology leverages repeated projective measurements and adaptive timing to filter Fock sectors, achieving convergence in fewer than 10 measurement rounds.
• The protocol demonstrates robustness against thermal noise and logarithmic scaling with photon number, offering significant advantages for quantum metrology and simulation.

Adaptive Generalized-Parity Measurement for High-Fidelity Macroscopic Fock State Generation

Introduction and Motivation

The generation of macroscopic Fock states in bosonic systems is a fundamental capability for quantum information processing, quantum metrology, and quantum simulation. While preparing Fock states with moderate photon numbers has been achieved in cavity and circuit QED platforms, scaling to Fock states with photon numbers exceeding $10^4$ has posed severe challenges, particularly when maintaining high fidelity and deterministic operation. Previous approaches have relied on postselection-based protocols, carefully engineered Kerr interactions, or resonant subspace engineering, all of which suffer from practical limitations such as exponential decay in success probability, requirement of pure-state initialization, and complex external control.

The paper "Generating Fock state exceeding 10000 excitations with near unit fidelity by adaptive generalized-parity measurement" (2606.01341) introduces an efficient protocol based on adaptive generalized-parity measurement (GPM), enabling deterministic preparation of macroscopic Fock states with photon numbers exceeding $10^4$ using only repeated projective measurements on a coupled ancillary qubit. The protocol does not require postselection, adaptive driving, gate operation, or measurement-basis variation, and is robust against moderate thermal admixture.

Theoretical Framework

The adaptive GPM protocol operates on a composite system comprising a bosonic mode (e.g., cavity photon field) coupled to an ancillary qubit via an exchange-type interaction. The key ingredients are:

• Exchange Coupling Hamiltonian: The system evolves under $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$, where $Q$ is the operator acting on the bosonic mode.
• Repeated Measurement Procedure: The ancillary qubit, initialized in a fixed state, is subject to a sequence of projective measurements following intervals of joint evolution with the bosonic mode.
• Adaptive Timing Rule: The intervals between measurements are recursively adjusted based on the prior measurement outcome, ensuring that each measurement trajectory is retained and the randomness of outcomes is absorbed into the update of the GPM label.

For systems where the coupling operator $Q^\dagger$ possesses a ladder structure (as in the Jaynes-Cummings model), the measurement-induced Kraus operators are diagonal or displaced in the Fock basis, allowing deterministic filtering of Fock sectors. The adaptive rule enables tracking of the survived Fock components with a GPM label, which specifies the modulo-parity subspace centered on a dynamically updated photon number.

Macroscopic Fock State Generation in Jaynes-Cummings Systems

Application to the JC model demonstrates the protocol's constructive power:

• Initialization: The target bosonic mode begins in a coherent state $|\alpha\rangle$ with $|\alpha|^2 = n_t(0) \gg 1$; the ancillary qubit is prepared in $|e\rangle$.
• Adaptive Measurement Sequence: For each measurement round $k$, the evolution interval $\tau_k$ is set according to the GPM label, and the label is updated recursively following the adaptive rule $10^4$0, where outcomes $10^4$1 are $10^4$2 or $10^4$3.
• Population Filtering: After $10^4$4 rounds, the effective measurement operator retains only Fock components separated by $10^4$5 indices, with the center determined by the final GPM label.

Numerical results confirm rapid convergence: with $10^4$6 up to $10^4$7, a single dominant Fock component $10^4$8, $10^4$9, is isolated in fewer than $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$0 measurement rounds.

Numerical Performance and Robustness

Key performance metrics reported:

• Average Fidelity: For $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$1, average fidelity of the generated Fock state approaches $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$2 within $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$3 rounds.
• High-Fidelity Probability: Probability for achieving fidelity above $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$4 saturates at $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$5, substantially higher than postselection-based protocols where probability is limited by initial population distribution.
• Logarithmic Scaling: Number of measurement rounds required grows logarithmically with $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$6, confirming substantial efficiency for large photon numbers.
• Thermal Robustness: The protocol maintains high-fidelity Fock state generation even when starting from displaced thermal states; the average fidelity remains above $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$7 for moderate temperatures, and the probability of near-unit fidelity exceeds $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$8 for $$H = \frac{\Delta}{2}\sigma_z + g(Q\sigma_+ + Q^\dagger\sigma_-)$$9 at inverse temperature $Q$0.

The protocol leverages deterministic adaptive filtering, retaining all measurement trajectories, and achieves scalable, high-efficiency Fock state preparation for both pure and mixed initial states.

Implications and Future Directions

Practically, this protocol provides an accessible pathway for macroscopic Fock state engineering in platforms such as circuit and cavity QED, trapped ions, and optomechanical systems, without the necessity of finely tuned nonlinear interactions or stringent state initialization. The ability to generate large Fock states deterministically and with high fidelity expands the experimental toolkit for quantum-enhanced metrology, robust quantum simulation, and bosonic code-based quantum error correction.

Theoretically, the adaptive GPM mechanism establishes a general recipe for deterministic quantum state engineering via measurement-induced control, applicable to a broad class of discrete-spectrum systems. The conversion of measurement outcome randomness into adaptive filtering may inspire further developments in measurement-based quantum control and preparation of nonclassical states.

Future studies may extend the adaptive protocol to multi-mode systems, optimize measurement strategies for further fidelity enhancement, and incorporate decoherence effects and realistic noise models. Applications to quantum metrology, e.g., employing large Fock states for Heisenberg-limited sensing, will benefit from this protocol's scalability.

Conclusion

The adaptive generalized-parity measurement protocol detailed in "Generating Fock state exceeding 10000 excitations with near unit fidelity by adaptive generalized-parity measurement" (2606.01341) achieves deterministic generation of macroscopic Fock states exceeding $Q$1 excitations with fidelities near unity in the Jaynes-Cummings paradigm. The approach relies solely on adaptive projective measurements, demanding no nonlinear driving or postselection, and demonstrates robustness to thermal admixture. The protocol's scalability and practical performance mark a significant advancement in measurement-based quantum state engineering.