Scalable Generation of Macroscopic Fock States Exceeding 10,000 Photons

Abstract

The scalable preparation of bosonic quantum states with macroscopic excitations poses a fundamental challenge in quantum technologies, limited by control complexity and photon-loss rates that severely constrain prior theoretical and experimental efforts to merely dozens of excitations per mode. Here, based on the duality of the quantum state evolution in Fock state space and the optical wave-function propagation in a waveguide array, we introduce a Kerr-engineered multi-lens protocol in a single bosonic mode to deterministically generate Fock states exceeding $10,000$ photons. By optimizing phase and displacement operations across lens groups, our approach compensates for non-paraxial aberrations, achieving fidelities above $73\%$ in numerical simulations for photon numbers up to $N=100,000$. Counterintuitively, the protocol's execution time scales as $N^{-1/2}$ with the target photon number $N$, exhibiting robustness against the photon loss. Our framework enables exploration of quantum-to-classical transitions of giant Fock states, paving the way for advanced quantum metrology with significant quantum gains, and error-corrected quantum information processing in high-dimensional Hilbert spaces.

Figures (3)

• Figure 1: Fock-space lens. a, Illustration of Fock-space lens in the Fock-state lattice of a bosonic mode. Photon number population is squeezed by a quadratic phase accumulation followed by coherent coupling between neighborhood sites. b, Evolution of the photon number distribution $P(n,t)$. The coherent state $|\alpha\rangle$ with $\alpha=100$ accumulates a quadratic phase during $0\leq t\leq0.5$ and single-photon coherent drive is applied after $t=0.5$. Probability higher than $0.1$ is plotted as the same color for better illustration. Lower panel: the $P(n)$ at $t=0.5$, $t=1.0$, $t=1.4$ are multiplied by 50, 25, and 6, respectively. c, The accumulative distribution function $\text{CDF}(n)$ of $P(n)$ at different times. d, Maximum of $|c_{n}|^{2}$ for different phase $\phi_0$ and single-photon drive time $t$. The phase is normalized to $\phi_{m}=2.45\times 10^{-3}$. The strength of coherent drive is $\varepsilon_{p}=1$. The condition for maximal $|c_{n}|^{2}$ coincides well with the analytical expression between focal time and phase shown in Eq. (\ref{['eq:focal_lattice']}).
• Figure 2: Fock state generation using a group of Fock-space lenses. a, A series of lens operations are applied to the initial coherent state (upper panel), resembling a lens group for tight focusing of a light beam (lower panel). By optimizing the phase and displacement of each lens, the fidelity of the target Fock state can be greatly improved. b, Optimized fidelity to obtain a Fock state with photon numbers $N=10,000$ and $N=100,000$ for different number of lenses. Using three Fock-space lenses, the state fidelity reaches $73.3\,\%$ and $61.5\,\%$ for $10,000$- and $100,000$-photon Fock states, respectively. c, Relationship between the optimized fidelity and the target Fock state for different lens-group configurations.
• Figure 3: Experimental feasibility. a, Microwave realization: a superconducting circuit comprising a capacitively-coupled microwave cavity and a SNAIL. The SNAIL provides the Kerr nonlinearity required for quadratic phase accumulation. b, Optical realization: a strong laser beam passes through a beam splitter with transmittance $\eta \ll 1$ and obtain the parabolic phase from a cavity-QED system. The reflected field is then displaced by the reflected laser from the downside mirror. c, The normalized time duration of the Fock lens, defined as the ratio of the optimal second-order coefficient $\phi_{0}$ to the Kerr coefficient $\chi$. Dots: simulation. Line: power-law fit for photon numbers $> 2500$, yielding a decay exponent of $-0.5525$. b, Relationship between the fidelity and the Kerr-to-loss ratio $\chi/\kappa$ for Fock states of $2500$, $10000$, and $40000$ photons.