Quantum-enhanced metrology with large Fock states

Precision measurement underpins advances across physics and technology. Quantum metrology can surpass the standard quantum limit (SQL) and approach the Heisenberg limit (HL) by exploiting quantum resources. Multimode entangled states (GHZ, NOON, spin-squeezed states) can enable such enhancements but are difficult to scale. An alternative, hardware-efficient route uses a single bosonic mode prepared in nonclassical states (cat, squeezed, maximum-variance, or Fock states). Prior demonstrations of bosonic-mode metrology with Fock states were limited to modest excitation numbers (N ~ 10), leaving the intrinsic scaling advantages unrealized. This work addresses the challenge of preparing and using large-photon-number Fock states in a superconducting cavity to realize quantum-enhanced displacement and phase sensing approaching HL scaling.

The paper reviews approaches to quantum-enhanced metrology: entanglement-based probes (GHZ-type states, NOON states, spin-squeezed states, and other multimode entangled resources) and single-mode bosonic probes exploiting nonclassical states (Schrödinger cat states, squeezed states, maximum-variance states, and energy eigenstates/Fock states). Previous experiments used phononic and microwave-photonic Fock states for sensing but at small N (≈10), limiting achievable scaling gains. Single-mode platforms in circuit QED offer hardware efficiency (simple controls, parity measurements) but lacked practical preparation of high-N Fock states beyond N>10 due to control and decoherence challenges.

Platform: A high-Q 3D superconducting microwave cavity (probe mode) dispersively coupled to an ancillary transmon qubit and with a fast readout resonator. The effective Hamiltonian in the rotating frame with respect to drives is H/ħ = Δ|e⟩⟨e| − χ a†a |e⟩⟨e| + εp(a† + a) + εq σx, enabling cavity displacements, photon-number-dependent qubit shifts, and qubit control. Device parameters include cavity ω/2π = 6.597 GHz, T1 = 1.2 ms, Tφ = 4.0 ms; qubit ω/2π = 4.878 GHz, T1 = 93 μs, Tφ = 445 μs; dispersive coupling χ0/2π = 0.626 MHz; readout κr/2π = 1.92 MHz. Programmable photon number filters (PNFs): The core state-preparation tool is a sinusoidal PNF implemented via the dispersive interaction by driving the qubit at detuning Δ = Nχ and postselecting the qubit ground outcome. This realizes a projection Pg(θ,N) diagonal in the Fock basis with a photon-number-dependent phase, which acts as a periodic grating filtering photon numbers modulo a chosen parity. Sequential PNFs with θ = π/2^j (j = 1..m) project the cavity state onto a subspace with parity 2^m, effectively isolating target photon numbers N + k·2^m. This permits efficient resolution of photon number with circuit depth scaling d ≈ log2√N for N ≫ 1 when starting from a coherent state with α ≈ √N. Gaussian PNF: To confine the state to a narrow photon-number window around the target N and reduce the number of required operations, a Gaussian PNF is introduced using a qubit pulse with a Gaussian envelope (standard deviation σ) and detuning Nχqc. It realizes P(σ,N) = Σn exp(−(n−N)^2/2σ^2) |n⟩⟨n|, selecting a finite photon-number band that, in combination with the sinusoidal PNF, robustly yields |N⟩ with fewer steps and resilience to decoherence. State preparation and characterization: Starting from a coherent state |α⟩ with α ≈ √N, two PNFs [angles (π, N) and (π/2, N)] combined with the Gaussian PNF probabilistically prepare |N⟩. Photon-number distributions Pn are obtained by qubit spectroscopy with selective Gaussian qubit pulses; spectra are fitted to sums of Gaussians S(f) = Σn An exp(−(f − fn)^2/2σ^2) + B, with normalization Pn = An/ΣAn. Additional Ramsey experiments on the qubit validate photon-number-dependent phase evolutions (Ramsey frequency scales with N), confirming large-N Fock state generation (N = 30, 50, 70, 100). Displacement sensing: The probe is initialized in |N⟩, subjected to a small displacement D(β), and measured via parity mapped onto the qubit using Pg(π). The measured ground-state probability Pg(β) is fitted by A exp(−2|β|^2) LN(4|β|^2) + B, where LN is the Nth Laguerre polynomial, and A,B absorb imperfections. Fisher information Fβ is computed from Pg and used to estimate the precision δβ = 1/√Fm (Fm is the maximal Fisher information over β). Phase sensing: Because |N⟩ is phase-rotation symmetric around the origin, an initial displacement prepares D(γ)|N⟩ with γ = √N (so mean photon number ñ = 2N). A small phase rotation φ is applied, and parity is measured. The measured Pg(φ) is fitted by A exp(−2N(φ+φ0)^2) LN(4N(φ+φ0)^2) + B, with φ0 accounting for self-Kerr-induced phase shifts. Fisher information Fφ yields δφ = 1/√Fm. Photon-number-resolved metrology (quantum jump tracking): For practical sensing without discarding outcomes, multiple PNFs are used to resolve photon number sectors from an initial coherent probe (e.g., α = √3). Recording the sequence of outcomes identifies n; sensing and analysis are conditioned on n. The total Fisher information is upper bounded by the probability-weighted sum Σn Pn F(|n⟩), enabling deterministic operation and scaling enhancements without postselection. Precision and SQL/HL analysis: Using quantum Fisher information (QFI), the SQLs for displacement and phase are δβSQL = 1/2 and δφSQL = 1/(2√Ncoh) for a coherent probe with mean Ncoh. For Fock probes, ideal QFIs give δβFock = 1/(2√(2N+1)) and δφFock = 1/[2√(N(2N+1))], indicating √N enhancements and approach to HL. With probabilistic preparation (success probability ≈ 1/√(2πN)), postselected metrology still yields an N1/4 scaling advantage, and adaptive photon-number-resolved schemes can approach N1/2 scaling.

- Efficient, programmable generation of large Fock states up to N = 100 photons in a superconducting 3D cavity using sinusoidal and Gaussian photon number filters (PNFs). State fidelity and number distributions validated by qubit spectroscopy and Ramsey measurements. - Displacement sensing with Fock probes |N⟩ surpasses the SQL. Extracted precision δβ shows a maximum metrological gain of 14.8 ± 0.2 dB at N = 40 relative to the SQL, with precision scaling approximately N^-0.35 for N ≤ 40, approaching Heisenberg-like scaling. - Phase sensing with displaced Fock probes D(√N)|N⟩ (mean ñ = 2N) surpasses the SQL. Achieved a maximum metrological gain of 12.3 ± 0.5 dB at ñ = 60, with precision scaling ñ^-0.87 for ñ ≤ 60, close to the HL scaling. - Photon-number-resolved metrology (quantum jump tracking) using an initial coherent state (α = √3) and three PNFs to resolve 0–7 photons yields a Fisher information gain of 5.04 ± 0.06 dB over the SQL (FSQL = 4), without discarding outcomes. - Success probability for preparing |N⟩ from |α≈√N⟩ scales as ≈ 1/√(2πN). Even with postselection, displacement and phase sensing retain an N^1/4 scaling advantage; adaptive schemes can approach N^1/2 scaling. - Demonstrated sensitivities and scaling behaviors are consistent with theoretical QFI predictions showing √N enhancement of Fock-state metrology over coherent-state SQLs.

The work demonstrates that single-mode bosonic probes prepared in large-photon-number Fock states enable quantum-enhanced metrology approaching Heisenberg scaling for both displacement and phase estimation. The programmable PNF method overcomes the long-standing challenge of preparing high-N Fock states in circuit QED with low circuit depth and robustness. Experimentally observed precision gains (14.8 dB for displacement at N=40; 12.3 dB for phase at ñ=60) validate theory and highlight that fine sub-Planck structures in Fock-state Wigner functions translate into higher phase-space resolution. Limitations in scaling (e.g., deviation from ideal N^-1) are primarily due to decoherence (cavity and qubit decay/dephasing) and measurement imperfections; improving coherence should extend gains towards HL scaling for larger N. Importantly, the authors show that practical metrology need not rely on discarding outcomes: photon-number-resolved/adaptive schemes can deterministically harness number-specific sensitivities, preserving scaling advantages. The results position high-N Fock states as a hardware-efficient resource competitive with, and more experimentally accessible than, other nonclassical states for precision sensing, with broad applicability from fundamental physics (e.g., dark photon searches modeled as weak cavity displacements) to quantum technologies.

The paper introduces a programmable photon number filtering technique that efficiently prepares large Fock states (up to 100 photons) in a superconducting cavity and leverages them for quantum-enhanced displacement and phase sensing near Heisenberg scaling. Measured metrological gains reach 14.8 dB (displacement) and 12.3 dB (phase), with favorable precision scaling. The approach is hardware-efficient, compatible with adaptive, photon-number-resolved protocols, and extensible to other bosonic platforms (trapped ions, mechanical and optical systems via transduction). Future work should improve cavity and qubit coherence to push N beyond 100 toward 1,000+ and deploy adaptive control to realize deterministic N^1/2 scaling enhancements and further approach HL limits in practical sensing tasks, including weak-force and dark matter detection.

- State preparation is probabilistic with success probability ~ 1/√(2πN), requiring postselection or adaptive number-resolved protocols for deterministic operation. - Decoherence (cavity photon loss and qubit dephasing/relaxation) limits achievable metrological scaling, with observed precision scaling deviating from ideal N^-1 beyond N≈40; decay yields population leakage to N−1 and N−2. - Measurement imperfections necessitate fitting offsets (A, B) and introduce bias; parity mapping fidelity and self-Kerr-induced phase shifts (φ0) affect phase sensing. - Current gains rely on high-Q cavities and well-calibrated dispersive interactions; scaling to much larger N will require further improvements in device quality and control robustness.