Quantum Confocal Microscopy in Fock Space with a 19 dB Metrological Gain

Abstract

Quantum metrology promises measurement precision beyond classical limits by exploiting large-scale quantum states, yet realizing this advantage faces two fundamental challenges: the deterministic preparation of non-trivial quantum probes and the efficient extraction of metrological information in high-dimensional Hilbert spaces. Here, we introduce quantum confocal microscopy in Fock space that simultaneously resolves both challenges. Drawing a direct analogy between classical wave optics and quantum state evolution in a bosonic mode, we construct a confocal system with two Fock-space lenses. The first lens deterministically focuses a coherent state into a quantum probe with a tightly concentrated photon-number distribution, while the second lens maps the metrological information back to the vacuum state for efficient readout. Using a superconducting circuit QED platform, we prepare focused probe states with mean photon numbers up to ${N} = 500$, achieving a 21.5$\pm$1.1 dB compression of the photon-number uncertainty relative to a coherent state, with a scalable quantum circuit of $\mathcal{O}(1)$ operational depth. We demonstrate a displacement sensitivity scaling as $N^{-0.416}$, approaching the Heisenberg scaling ($N^{-0.5}$), and achieve a record metrological gain of 19.06$\pm$0.13 dB beyond the standard quantum limit. This work establishes quantum confocal microscopy as a scalable and practical framework for quantum-enhanced precision measurement, readily extendable to other bosonic platforms and high-dimensional quantum many-body systems.

Figures (4)

• Figure 1: Fock-space confocal system for state preparation and detection. a, Changes of photon-number distribution during the Fock-space confocal circuit. The circuit maps a coherent state back to an approximate coherent state using two Fock-space convex lenses with the same focal point, analogous to the confocal convex lens system in optics. b-f, Close-up of photon-number distribution at various moments. b, At the starting point, the system is in the vacuum state, with all population being in the Fock state $\left|0\right\rangle$. c, A displacement transformation maps the vacuum state to a coherent state with 500 average photons and a broad distribution. The inset shows the Wigner distribution of the coherent state. d, After passing through a Fock-space convex lens, the coherent state is focused into an N-focused state, with photon-number distribution tightly concentrated around several Fock states near $N=500$. The inset shows the Wigner distribution of the N-focused state, which has fine fringes for detecting small signals. e, After passing through another Fock-space convex lens, the N-focused state diverges into a coherent-like state, with a broad photon-number distribution. The inset shows its Wigner distribution, which is generally similar to that of a coherent state except for slight distortions. f, A final backward displacement transformation maps the coherent-like state to a near vacuum state. Theoretically, over 90% of the population returns to the vacuum state, facilitating efficient state measurement.
• Figure 2: Scaling of N-focused state.a, Schematic of the superconducting circuit QED setup: a 3D cavity coupled to a transmon qubit and a readout resonator. b, The circuit of preparing and calibrating the N-focused states. The N-focused state is prepared by compressing a coherent state through a convex Fock-space lens, then calibrated with a photon number selection pulse and measurement of the transmon qubit. c, Ancilla-qubit-assisted detection results for an N-focused state with an average photon number of $\bar{N}=300$. d, Measured $P(N)$ (dots) and Gaussian fits (solid lines) for the N-focused state at various $\bar{N}$ from 50 to 500, demonstrating deep sub-Poissonian statistics compared to coherent states (dashed lines). The peak probability and the width of the distribution remain nearly the same at various target photon numbers, confirming the scalability of the state preparation process.
• Figure 3: Fock-space scanning tomography (N-tomo) of various states.a, The circuit of calibrating an unknown target state with N-tomo. The target state passes through a Fock-space convex lens and then a displacement operation, followed by measuring the vacuum probability of the final state. The N-tomo circuit maps the N-focused state component in the target state back to the vacuum state, similar to the optical conjugation of the N-focused state preparation circuit. By adjusting the circuit parameters, the N-tomo circuit can detect the proportions of different N-focused components in the target state, thereby reconstructing the photon-number distribution. b-e, The theoretical Wigner distribution of the N-focused state, the coherent state, the parallel-displaced cat (PDC) state, and the orthogonal-displaced cat (ODC) state. f-i, The ideal and reconstructed photon-number distribution of each state. The distribution of the N-focused state is used for reconstructing other unknown states. The PDC state has two components with different photon numbers, resulting in a two-peak photon-number distribution. The ODC state has two components with the same photon numbers, resulting in a three-peak photon-number distribution. The N-tomo results clearly capture the difference between PDC and ODC states.
• Figure 4: Quantum sensing of displacement.a, Measured vacuum population $P(0)$ vs. displacement $\beta$. N-focused states ($\bar{N}$ from 50 to 500) show a steeper slope (higher sensitivity) than the coherent state (SQL benchmark). b, Fisher Information $I_c$ (derived from a) shows a massive quantum enhancement for N-focused states over the coherent state at optimal bias points $\beta_0$. c, Focused $\bar{N}=350$ state detection result. The same signal $\delta\beta$ (bottom) induces a dramatic, measurable change in the photon-number distribution of the final state, illustrating the mechanism of quantum gain. d, Coherent state (SQL) detection result. A small signal $\delta\beta$ (bottom) causes a negligible change to the final state. e, Displacement sensitivity $\delta\beta$ versus $\bar{N}$. The N-focused states (blue dots) demonstrate a clear quantum advantage, surpassing the SQL (black dashed line). A maximum quantum gain of $19.06\pm0.13$ dB is achieved at $\bar{N} = 350$. Linear fitting (blue dashed line) reveals that the sensitivity scales as $\bar{N}^{-0.416}$, approaching the Heisenberg-limit scaling ($N^{-0.5}$). This scaling is well-described by lossless simulation (yellow circles and dashed line, $\bar{N}^{-0.417}$).