Quantum enhanced measurement of an optical frequency comb

Optical frequency combs underpin many precision measurements, including broadband spectroscopy, optical clocks, and time-distance synchronization. Accurately measuring the spectral properties of ultrafast optical pulses—specifically mean energy, central frequency, and spectral bandwidth—is thus essential for precision metrology. The quantum-limited sensitivity in such measurements is set by noise fluctuations within well-defined spectral modes, with precision generally limited by shot noise scaling as √N. Squeezed states can surpass this standard quantum limit (SQL). However, characterizing systems often requires sequential measurements of multiple parameters with setup modifications, limiting flexibility. This work introduces a multimode approach enabling parallel estimation of multiple pulse parameters via a multi-pixel spectrally resolved (MPSR) detector, and demonstrates quantum-enhanced metrology by incorporating a multimode squeezed-light resource. We realize shot-noise-limited single-shot estimation of central frequency, mean energy, and spectral bandwidth via post-processing of MPSR data, reconstruct the full covariance matrix of a quantum frequency comb, and use squeezed light to surpass the SQL for multiple parameters, including configurations where two parameters are simultaneously enhanced.

Prior work established the role of optical frequency combs in precision metrology, including spectroscopy, absolute frequency determination with optical clocks, and distance/time synchronization. Quantum-limited measurement sensitivities are constrained by photon shot noise, with central frequency estimation linked to the derivative of the spectral line shape. Squeezed light has enabled sensitivities beyond the SQL in numerous platforms such as interferometry (including gravitational waves), Raman spectroscopy, magnetometry, beam pointing, biological sensing, and distributed phase sensing. Theoretical frameworks relate parameter estimation sensitivity to the quantum Cramér-Rao bound (QCRB) for Gaussian states, and prior studies have explored time–frequency mode separation and quantum-limited distance measurements with multimode detectors. Multimode squeezed vacuum generated by synchronously pumped OPOs yields Hermite-Gaussian-like spectral modes well-suited for encoding physical parameters (e.g., mean-field, frequency derivative, and bandwidth modes), providing a resource for multiparameter quantum-enhanced metrology and measurement-based quantum information processing.

Theory and quantum limits: The single-pulse field is modeled as E(t)=A u(t) e^{iω0 t}, or in frequency ε(ω)=E0√N u(ω), with normalized Gaussian spectral mode u(ω) of center ω0 and bandwidth Δω. Small variations in mean energy δN, central frequency δω, and bandwidth δ(Δω) induce field deviations carried by specific (generally orthogonal) spectral modes: u (mean-field), u_CF (central-frequency mode ∝ ∂u/∂ω), and u_BD (bandwidth mode ∝ ∂u/∂Δω). For coherent states, the SQL Cramér-Rao bounds are δN_SQL=√N, δω_SQL=Δω/√N, and δ(Δω)_SQL=√2/√N. For Gaussian noise with quadrature variances ⟨Δx^2⟩ in the corresponding modes, sensitivities scale as δN=√N⟨Δx^2⟩, δω=Δω/(√N⟨Δx_CF^2⟩), and δ(Δω)=√2/(√N⟨Δx_BD^2⟩). MPSR detection model: The optical spectrum is dispersed by a grating and detected on a one-dimensional multi-pixel photodiode array; each pixel defines a normalized spectral slice (pixel mode). A target mode v is reconstructed from pixel modes by a real linear combination of demodulated photocurrents with coefficients m_i proportional to overlaps ∫u_i(ω) v(ω) dω. With perfect detection efficiency η=1 (set by mode overlap, optical loss, and pixel fill factor), the reconstructed mode reaches the Cramér-Rao bound; η<1 reduces sensitivity accordingly. Because multiple orthogonal modes can be reconstructed simultaneously, multiparameter estimation proceeds in parallel via post-processing without altering the optics. Experimental setup and shot-noise-limited operation: A ~100 fs pulse train at 795 nm and 76 MHz repetition rate is weakly tapped and spectrally dispersed onto an 8-pixel photodiode array. A micro-lens array fills the pixel active areas; PD amplifiers have ~10 MHz bandwidth. The laser’s center frequency is modulated via intra-cavity mirror tilt at fm=1.5 MHz to avoid technical noise; individual pixel photocurrents are demodulated at fm and low-pass filtered at 50 kHz (effective 20 μs measurement time). Mode-reconstruction coefficients m_i are computed from the measured mean spectrum and the predicted parameter modes (mean field, frequency derivative, bandwidth). Calibration: the detected spectrum has 8.8 nm FWHM; photon flux N0≈4×10^16 s^-1 (~10 mW). From theory, δω_shot≈55.7 kHz Hz; accounting for quantum efficiency, optical losses, and finite pixelization (global intensity efficiency η^2≈70%), the practical sensitivity is δω_meas≈66.5 kHz Hz. For 20 μs integration, the per-shot central-frequency resolution is 14.9 MHz. Quantum-enhanced measurements with a quantum frequency comb: A synchronously pumped OPO (SPOPO) driven by the second harmonic of the 76 MHz laser generates multimode squeezed vacuum with Hermite–Gaussian-like spectral eigenmodes. Two MPSR detectors configured as a spectrally resolved balanced homodyne measure quadratures across eight spectral bands, enabling simultaneous acquisition of amplitude and phase quadrature correlations and reconstruction of the full 8×8 covariance matrix. Bloch–Messiah decomposition yields orthogonal squeezed eigenmodes; the leading four exhibit squeezing of −2.9, −2.2, −1.7, and −1.4 dB (electrical dark-noise corrected). Odd (even) order modes are amplitude- (phase-) squeezed; the mean-field, frequency-derivative, and bandwidth modes align closely to the most squeezed modes. To implement quantum enhancement, the squeezed comb is mixed with the signal on a ~10/90 (T/R) beam splitter, producing a synthetic beam with the original mean field and modified quantum noise; the OPO is seeded and locked such that relevant modes are amplitude-squeezed. A fast shutter blocks the seed during 100 ms acquisition windows while electronic locks are maintained. The same MPSR post-processing extracts the three parameter modes. By adjusting lock conditions, which mode(s) are squeezed can be selected, enabling enhancement of different parameters (e.g., central frequency, or simultaneously mean energy and bandwidth). Multi-pixel homodyne technicals: Dispersive optics are gratings (≈93% efficiency); photodiode arrays have ≈80% QE; overall MPSR detection efficiency ≈80%, 10 MHz bandwidth, and 94% homodyne visibility. The covariance matrix is obtained from simultaneous difference currents of pixel pairs with the LO phase locked at 0 or π/2.

- Single-shot, parallel estimation of three spectral parameters (mean energy, central frequency, spectral bandwidth) from a single measurement using an 8-pixel MPSR detector and post-processing, without changing the optical setup. - Shot-noise-limited sensitivity for central frequency with coherent input: theoretical δω_shot≈55.7 kHz Hz; practical measured δω_meas≈66.5 kHz Hz (accounting for η^2≈70%). With 20 μs measurement time, per-event resolution ≈14.9 MHz. - Quantum-enhanced central-frequency sensitivity using a multimode squeezed resource: δω_quan≈57.8 kHz Hz, an ≈15% SNR improvement; with 20 μs integration, ≈12.9 MHz per event. - Mean energy sensitivity: practical shot-noise-limited δN_meas≈1.7×10^8 photons Hz^{-1/2}; quantum-enhanced δN_quan≈1.4×10^8 photons Hz^{-1/2} (≈19% improvement). For 20 μs, per-event resolutions ≈7.5×10^5 (shot) and ≈6.1×10^5 (quantum) photons. - Spectral bandwidth sensitivity: δ(Δω)_SQL≈39.4 kHz Hz; practical shot-noise δ(Δω)_shot≈47.1 kHz Hz; quantum-enhanced δ(Δω)_quan≈33.4 kHz Hz (≈29% improvement). For 20 μs, per-event resolutions ≈10.5 MHz (shot) and ≈7.46 MHz (quantum). - Full eight-partite covariance matrix of the quantum frequency comb reconstructed via spectrally resolved homodyne; leading eigenmodes’ squeezing levels: −2.9, −2.2, −1.7, −1.4 dB. - By choosing lock conditions, two parameters (mean energy and spectral bandwidth) can be enhanced simultaneously, demonstrating simultaneous multiparameter quantum advantage.

The study addresses the challenge of efficiently and flexibly estimating multiple, distinct spectral parameters of ultrafast pulses by introducing a wavelength-multiplexed detection and post-processing framework. By mapping parameter variations onto orthogonal or near-orthogonal spectral modes and reconstructing those modes from multi-pixel data, the method achieves the corresponding Cramér-Rao bounds for Gaussian noise. Incorporating a multimode squeezed resource tailors the quadrature noise in the specific parameter modes, thereby surpassing the shot-noise limit. Experimentally, central-frequency, mean-energy, and bandwidth estimates reach or exceed SQL performance; notably, with appropriate locking, simultaneous enhancement of two parameters (energy and bandwidth) is achieved. This confirms that multimode quantum resources can be harnessed for parallel, quantum-enhanced metrology without reconfiguring the photonic architecture. The results are significant for ultrafast precision metrology and suggest that similar multiplexed strategies could be extended to additional parameters (e.g., temporal jitter, phase) and integrated into broader multimode quantum information protocols that require simultaneous interrogation of many modes.

This work demonstrates a general and practical framework for single-shot, parallel estimation of multiple spectral parameters of an optical frequency comb using multi-pixel spectrally resolved detection and post-processing. The approach reaches the shot-noise limit with coherent light and achieves quantum-enhanced sensitivities—improving central frequency, mean energy, and bandwidth estimates by roughly 15%, 19%, and 29%, respectively—by injecting a multimode squeezed quantum frequency comb. The team reconstructs the full covariance matrix of the quantum resource and shows that appropriate lock conditions enable simultaneous enhancement of two parameters. These advances highlight the utility of multimode detection for ultrafast quantum metrology and open pathways toward multimode quantum information processing leveraging wavelength multiplexing. Future work could focus on increasing detection efficiency and pixel number, improving squeezing levels and mode matching, extending to additional parameters and non-Gaussian resources (e.g., photon subtraction), and integrating MPSR homodyne schemes into scalable, real-time quantum-enhanced sensing and computation platforms.

- Overall detection efficiency below unity: global intensity efficiency η^2≈70% (due to optical losses, finite pixel number/fill factor, and mode mismatch) limits achievable sensitivity relative to ideal CRBs. - Photodiode array quantum efficiency (~80%) and 10% loss of the squeezed state at the 10/90 beam splitter reduce observed squeezing and quantum advantage. - Finite pixelization (8 pixels) constrains mode-reconstruction fidelity for complex spectral modes, contributing to non-perfect mode matching. - Sensitivity calibrations and enhancements are demonstrated at a specific demodulation frequency (1.5 MHz) and integration time (20 μs); performance may vary at other operating points. - The theoretical sensitivity bounds adopted assume Gaussian noise and no amplitude–phase cross-correlations; non-Gaussian noise or technical drifts could degrade performance. - Multiple phase locks (OPO, seed/signal relative phase, homodyne phases) are required; residual lock errors may impact stability and reproducibility.