Precise measurement of optical frequency combs is crucial for various applications, including broadband spectroscopy, optical clock frequency determination, and time-distance synchronization. The sensitivity of these measurements is often limited by photon number fluctuations, scaling as the square root of the number of photons (√N). Squeezed states of light offer a pathway to surpass this shot-noise limit, enhancing the precision of numerous measurements. Existing methods often involve measuring multiple parameters sequentially, requiring modifications to the experimental setup. This research introduces a multimode approach for parallel estimation of multiple parameters, using a multi-pixel spectrally resolved (MPSR) detector. This MPSR detector, combined with a multimode squeezed light source, enables simultaneous measurement of central frequency, mean energy, and spectral bandwidth of an optical frequency comb, exceeding the shot-noise limit without altering the optical architecture. This technique is broadly applicable to simultaneous interrogation of multiple parameters and holds potential for other quantum information processing tasks.

The paper extensively reviews the significance of optical frequency combs in precision metrology, highlighting their use in diverse applications such as broadband spectroscopy, optical clock frequency determination, and time-distance synchronization. It cites previous work on quantum-limited sensitivity in spectral measurements and the use of squeezed states of light to overcome the shot-noise limit in various measurement scenarios, including laser interferometry, Raman spectroscopy, gravitational wave interferometry, and optical magnetometry. The limitations of conventional single-parameter measurement schemes are also discussed, motivating the need for a parallel, multi-parameter approach.

The researchers developed a theoretical framework based on the Cramér-Rao bound to define the quantum limit for the simultaneous measurement of multiple parameters (mean photon number N, central frequency ω₀, and spectral bandwidth Δω) within an optical pulse. They derived expressions for the minimum standard deviations (δN, δω, δ(Δω)) for these parameters, both at the standard quantum limit (SQL) and considering Gaussian noise. A multi-pixel spectrally resolved (MPSR) detection system was designed and implemented. This system uses a diffraction grating to disperse the light and a photodiode array to simultaneously detect multiple spectral components. Post-processing of the multichannel data allows for the extraction of the desired parameters. The MPSR system's model is detailed, outlining the linear basis change from pixel modes to the modes carrying the parameters of interest. To achieve quantum-enhanced measurements, a quantum frequency comb generated by a synchronously pumped optical parametric oscillator (OPO) was used. This comb intrinsically contains multiple squeezed states in a family of Hermite-Gaussian modes. The quantum frequency comb was then combined with the signal field on a beam splitter, resulting in a synthetic beam with noise properties determined by the quadrature squeezing. The experimental setup is shown, highlighting the components used (pulsed laser, beam splitter, grating, multi-pixel detector). To measure the sensitivity of the system, the modulation depth of the central frequency in the laser cavity was varied, and the signal-to-noise ratio (SNR) was calculated using 1000 data points. The quantum state of the multimode quantum resource was characterized using spectrally resolved homodyne detection, reconstructing the eight-partite covariance matrix. The measurement of the quantum frequency comb's covariance matrix provided information about the squeezing levels in different modes. Quantum-enhanced measurements were performed by combining the quantum frequency comb with the signal field, allowing for sensitivity beyond the shot-noise limit. The data analysis involved reconstructing modes associated with central frequency fluctuations, mean energy variations, and spectral bandwidth shifts, utilizing the MPSR data and the derived mode projections.

The experimental results demonstrate that the MPSR detector achieves shot-noise limited measurement of central frequency, mean energy, and spectral bandwidth variations simultaneously. Using the quantum frequency comb, the sensitivity of the measurements was significantly enhanced beyond the shot-noise limit. The SNR of the central frequency was improved by approximately 15%. The mean energy and spectral bandwidth measurements showed improvements of approximately 19% and 29%, respectively, compared to the shot-noise limited case. Interestingly, a configuration was found where the mean energy and spectral bandwidth estimations were simultaneously quantum enhanced. The full eight-partite covariance matrix of the multimode quantum resource was successfully reconstructed, showcasing the capability of simultaneous multimode interrogation. The leading squeezed modes of the quantum frequency comb closely resembled Hermite-Gaussian spectral shapes, aligning well with the modes used for parameter estimation. The experimental sensitivities are reported with corresponding error bars less than 2%. The achieved quantum enhancement demonstrates exceeding the standard quantum limit for multiple parameters simultaneously using a single experimental setup.

The study successfully demonstrates the ability to surpass the standard quantum limit in the measurement of multiple parameters associated with an optical frequency comb, achieving simultaneous quantum enhancement in two out of three parameters. The use of the MPSR detector is shown to be versatile and generalizable, enabling parallel estimation of several parameters without changing the optical setup. The results highlight the significant potential of this approach for improving the precision of ultrafast quantum metrology and enhancing various applications leveraging optical frequency combs. The observation that linear combinations of photocurrents reveal variations in field bandwidth, temporal jitter, and overall phase suggests further possibilities for multidimensional light field interrogation. The demonstrated approach also has implications for measurement-based quantum computing and multipartite quantum secure communication.

This research provides a proof-of-principle demonstration of surpassing the standard quantum limit in the simultaneous measurement of multiple parameters (central frequency, mean energy, and spectral bandwidth) within an optical frequency comb. The use of a multi-pixel spectrally resolved detector combined with a multimode squeezed light source allows for parallel estimation beyond the shot-noise limit without altering the experimental configuration. Future research could explore the application of this technique with more complex quantum resources, such as non-Gaussian states, to further enhance measurement precision and explore its potential in quantum computing and quantum communication.

The current study is limited by the quantum efficiency of the photodiode array (approximately 80%), which impacts the overall sensitivity of the measurements. The efficiency of the optical components also contributes to the reduction in sensitivity. While the study demonstrates simultaneous quantum enhancement for two out of three parameters, achieving simultaneous enhancement for all three parameters could be a focus for future work. The number of pixels in the detector array also influences the quality of mode reconstruction. Increased pixel density might lead to improved sensitivity and more accurate parameter retrieval.