Deterministic and scalable generation of large Fock states

TL;DR

This work tackles the challenge of deterministically generating large Fock states in bosonic modes, a key resource for quantum metrology and quantum information. It introduces a multi-pulse Jaynes-Cummings–displacement protocol that engineers photon-number–dependent phases to create targeted Fock states through interference in Fock space. The design is enabled by a hybrid Genetic-Adam optimization (GAdam) that seeds from a single-layer solution, evolves across generations with local refinement, and enforces a fixed total evolution time. The resulting fidelities exceed 0.9 up to N about 100 with high post-selection success, and the approach shows robustness to detuning, control noise, and dissipation, offering a scalable route across cavity QED, circuit QED, and trapped ions.

Abstract

The scalable and deterministic preparation of large Fock-number states represents a long-standing frontier in quantum science, with direct implications for quantum metrology, communication, and simulation. Despite significant progress in small-scale implementations, extending such state generation to large excitation numbers while maintaining high fidelity remains a formidable challenge. Here, we present a scalable protocol for generating large Fock states with fidelities exceeding 0.9 up to photon numbers on the order of 100, achieved using only native control operations and, when desired, further enhanced by an optional post-selection step. Our method employs a hybrid Genetic-Adam optimization framework that combines the global search efficiency of genetic algorithms with the adaptive convergence of Adam to optimize multi-pulse control sequences comprising Jaynes-Cummings interactions and displacement operations, both of which are native to leading experimental platforms. The resulting control protocols achieve high fidelities with shallow circuit depths and strong robustness against parameter variations. These results establish an efficient and scalable pathway toward high-fidelity non-classical state generation for precision metrology and fault-tolerant quantum technologies.

Figures (3)

• Figure 1: Multi-pulse Jaynes--Cummings--displacement protocol for Fock-state generation.The control sequence consists of $p$ layers, each comprising a resonant Jaynes--Cummings evolution $U(\tau_k)$ acting on the joint qubit--oscillator system, followed by a phase-space displacement ${D}(\beta_k)$ applied to the bosonic mode. The qubit is initialized in the excited state and the oscillator in a coherent state. Repeated accumulation of photon-number--dependent phases arising from the nonlinear Jaynes--Cummings spectrum, combined with displacement-induced interference, progressively reshapes the oscillator state in phase space. The second row shows representative Wigner-function snapshots at different layers of the protocol, illustrating the emergence of nonclassical interference fringes characteristic of high-number Fock states. A final projective measurement on the qubit can be used to disentangle the qubit from the oscillator.
• Figure 2: Performance and internal dynamics of the multi-pulse Jaynes--Cummings--displacement protocol.(a) Fidelity $\mathcal{F}_N$ as a function of the target photon number $N$. Black circles and gray squares show results obtained with fixed circuit depths $p=1$ and $p=2$, respectively. Colored markers connected by the dashed green line indicate the best fidelity achieved for each $N$ among circuits with depths up to $p\leq 10$, with the corresponding optimal depth $p(N)$ encoded by color. The data demonstrate that shallow multi-pulse circuits substantially outperform single-pulse protocols and sustain high fidelities up to $N\gtrsim 100$. (b) Post-selection success probability $P_{\rm succ}$ corresponding to the optimal protocols shown in panel (a). Across the full range of target photon numbers, the success probability remains high, $P_{\rm succ}\gtrsim 0.9$, indicating near-deterministic operation. (c) Photon-number distributions $P_n=\rho_{nn}$ in the Fock basis for the representative case $N=100$ using a nine-pulse sequence ($p=9$). Distributions are shown after the JC evolution of layers $k=1,4,6,$ and $9$ (blue), and after the subsequent displacement operation (red). The progressive reshaping from a broad distribution into a sharply localized peak at the target photon number illustrates how displacement converts the engineered Fock-space interference into population localization. (d) Magnitude $|\rho_{nm}|$ of the reduced cavity density matrix after each JC evolution for the same four layers. Coherence is progressively concentrated into mirror-symmetric components, forming ridges along $n+m\simeq 2N$ that signal the emergence of a symmetric Fock-space wavepacket. (e) Corresponding phase maps $\arg(\rho_{nm})$. The phase structure evolves from an initial near-linear diagonal shear (layer 1) to strongly curved and ultimately anti-diagonal phase bands (layers 4 and 6), reflecting the nonlinear $\sqrt{n}$ dispersion of the JC spectrum. By the final layer, well-defined phase alignment along $n+m=\mathrm{const}$ enables the last displacement to induce constructive interference at $n=N$ and destructive interference elsewhere, yielding the target Fock state. For clarity, only matrix elements with sufficiently large magnitude are shown in the phase map of layer 9.
• Figure 3: Robustness of the multi-pulse protocol against control noise and dissipation.(a) Post-selected fidelity as a function of qubit--cavity detuning $\Delta/\Omega$ for representative target photon numbers. The inset highlights the fidelity response for a fixed circuit depth $p=10$. (b) Average post-selected fidelity under pulse-timing and displacement errors for $N=100$, $p=9$, and $l=5$. The control parameters are perturbed by independent Gaussian noise with standard deviations $\sigma_\tau$ and $\sigma_\beta$, and the fidelity is averaged over 200 noise realizations.