🔬 optics

Pump quantization changes optimal measurements for quantum sensing

This paper demonstrates that accounting for pump quantization in quantum phase sensing reveals that Heisenberg scaling ( $N^2$ ) is achieved not at maximum signal squeezing, but at maximal pump depletion, thereby necessitating a shift in both the optimal operating point and the measurement strategy to a joint nonlinear detection of the entire pump-signal system.

Original author: S. A. Wadood, L. Bello, T. Ilias, J. W. Fleischer

Published 2026-11-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original author: S. A. Wadood, L. Bello, T. Ilias, J. W. Fleischer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to measure the universe with extreme precision, scientists have long relied on a fundamental truth: light is not a smooth, continuous stream, but a collection of discrete packets called photons. When these photons are counted, they arrive with a natural, random jitter known as shot noise, which sets a hard limit on how precisely we can measure things like the distance to a star or the subtle ripples of a gravitational wave. To push past this limit, researchers use a special trick called "squeezing." Imagine a balloon filled with air; if you squeeze it on one side, it bulges out on the other. In the quantum world, squeezing light reduces the uncertainty in one property of the light wave while increasing the uncertainty in another. By carefully managing this trade-off, scientists can create a beam of light that is quieter than nature usually allows, offering a path to measurements far more sensitive than anything possible with ordinary light.

For decades, the standard way of analyzing these experiments treated the powerful laser beam used to create this squeezed light as a perfect, unchanging resource. It was assumed that the laser was so strong that it would never run out of energy, acting merely as a silent, classical driver that produced the squeezed light without being affected by the process itself. However, as technology has advanced, particularly in tiny, highly efficient optical devices, this assumption has begun to break down. In these modern, high-intensity regimes, the laser beam actually loses a significant amount of energy to create the squeezed light, and in doing so, it becomes entangled with the light it produces. This raises a critical question for the future of ultra-precise sensing: does the quantum advantage disappear when we stop treating the laser as a perfect, infinite resource and acknowledge that it is a quantum object subject to the same fluctuations as the light it creates?

A team of researchers at Princeton University and the Weizmann Institute of Science has tackled this question by simulating the full quantum behavior of the system, treating the pump laser not as a fixed background but as a dynamic participant. Their work reveals that the quantum advantage does indeed survive, but the rules for achieving it change completely. They found that the most intuitive approach—trying to generate the maximum amount of squeezing—actually fails to provide the highest possible sensitivity when the laser is depleted. Instead, the path to the ultimate precision lies in a different operating regime where the laser gives up most of its energy to the signal, creating a state that is no longer a simple squeezed wave but a complex, entangled mixture of the pump and the signal.

The researchers discovered that when the laser is allowed to deplete significantly, the system reaches a point where the signal flux is maximized. In this specific regime, the information available about the phase of the light scales in a way that was previously thought impossible for this type of setup. While standard methods of measuring the light, such as simply counting the photons at the output, fall short of the theoretical limit, the team showed that a more sophisticated measurement strategy can unlock the full potential. By using a nonlinear measurement that jointly analyzes the remaining pump light and the generated signal, the system achieves a level of precision that scales with the square of the total number of photons. This is the famous Heisenberg limit, the gold standard of quantum metrology, which represents a quadratic improvement over the standard limits of classical physics.

Crucially, the study demonstrates that the optimal way to run the experiment depends entirely on how the laser is depleted. If the goal is simply to get the most squeezing, the system operates at a point where the sensitivity improves, but only moderately, scaling with the number of photons to the power of one and a half. This is better than the classical limit but falls short of the ultimate quantum potential. However, by tuning the system to the point of maximum energy transfer, where the laser is most depleted, the sensitivity jumps to the optimal scaling. The catch is that this high-performance state is non-Gaussian and highly complex, meaning that the simple, linear detectors used in most current experiments are no longer sufficient. To harvest this extra sensitivity, one must employ a measurement technique that reverses the generation process, effectively recombining the pump and signal in a way that extracts information hidden in their entanglement.

This finding reshapes the understanding of how to design quantum sensors. It suggests that the best performance does not come from trying to keep the generating laser as strong and unchanged as possible, nor from chasing the maximum squeezing alone. Instead, the most powerful sensors will be those designed to operate deep in the depletion regime, where the laser and the signal are inextricably linked. The researchers emphasize that this requires a shift in strategy: the sensor must be designed to account for the entire lifecycle of the quantum resource, from its creation by a quantized pump to its final readout through a correlated, nonlinear measurement. By treating the pump as a quantum object that evolves and interacts, rather than a static tool, the team has shown that the path to the highest precision lies in embracing the complexity of the quantum system itself.

The implications of this work extend beyond just measuring light. The framework developed here applies to any system where a classical control signal is actually a quantum object, such as in the study of atoms interacting with light, the behavior of sound waves in crystals, or even the dynamics of superconducting circuits. The study provides a unified view that reconciles the idealized models of the past with the messy, realistic constraints of the present. It confirms that while the quantum advantage is robust, it is not a free lunch; it demands a more careful and holistic approach to both the generation of the quantum state and the method used to measure it. For engineers and physicists building the next generation of sensors, the lesson is clear: to reach the limits of what is measurable, one must design the entire system as a single, interacting quantum entity, rather than a collection of separate parts.

Technical Summary: Pump Quantization Changes Optimal Measurements for Quantum Sensing

Problem Statement
Standard analyses of quantum phase sensing often treat the pump field used to generate squeezed light as an undepleted, classical resource with no fluctuations. This approximation neglects the "hidden costs" of nonclassicality generation, specifically the quantum back-action on the pump. In realistic high-flux regimes, particularly in nanophotonics, the pump undergoes significant depletion and becomes entangled with the generated signal, leading to non-Gaussian states. Prior heuristic arguments suggested that pump quantization introduces fluctuations that suppress quantum advantage, potentially limiting sensitivity to classical shot-noise levels. There is a lack of a unified, information-theoretic framework connecting pump quantization, state generation dynamics, and metrological performance to determine if quantum advantage persists when the pump is treated as a quantum object.

Methodology
The authors employ a fully quantized model of a degenerate parametric oscillator (DPO) driven by a $\chi^{(2)}$ nonlinear interaction. Unlike the standard parametric approximation where the pump is replaced by a classical amplitude ( $a_p \to \alpha_p$ ), this work utilizes the full interaction Hamiltonian:
$\hat{H}_I = i\hbar\frac{\kappa}{2} \left( \hat{a}^\dagger_p \hat{a}^2_s - \hat{a}_p \hat{a}^{\dagger 2}_s \right)$
where $\hat{a}_p$ and $\hat{a}_s$ are annihilation operators for the pump and signal modes, respectively. The system is initialized with a coherent pump state $|\alpha\rangle_p$ (mean photon number $N = |\alpha|^2$ ) and a vacuum signal state.

The study analyzes the phase estimation precision using the Quantum Fisher Information (QFI), which sets the ultimate lower bound on estimation variance via the Cramér-Rao bound. The authors evaluate the QFI for two distinct operational regimes of the DPO:

Maximum Squeezing ( $t_{sq}$ ): The point where the signal quadrature variance is minimized.
Maximum Flux/Depletion ( $t_{fl}$ ): The point where the signal photon number is maximized due to near-complete energy transfer from the pump.

They compare these against two measurement strategies:

Linear Interferometry: A Mach-Zehnder interferometer (MZI) mixing the squeezed signal with a coherent probe, followed by standard photon counting (or homodyne detection).
Nonlinear Joint Measurement: An SU(1,1) interferometer scheme involving a time-reversal evolution (re-injection into a DPO) and joint measurement of the pump-signal system.

Key Contributions and Results
The paper establishes that pump quantization fundamentally alters both the optimal operating point of the sensor and the required measurement strategy to achieve quantum advantage.

Limits of Squeezing with Quantized Pumps:
- When the DPO is biased at the point of maximum squeezing ( $t_{sq}$ ), the QFI for phase sensing scales as $N^{1.48 \pm 0.008}$ (effectively $N^{3/2}$ ) when mixed with a coherent probe in an MZI. This surpasses the classical shot-noise limit (SNL) of $N^1$ .
- However, squeezing alone (without a coherent probe) cannot beat the SNL.
- Crucially, at $t_{sq}$ , the QFI does not reach the Heisenberg limit (HL) of $N^2$ . The authors demonstrate that the interference between the squeezed signal and the coherent probe preserves a quantum advantage, but pump quantization restricts the system from attaining the $N^2$ scaling typically associated with ideal squeezed states.
Emergence of Heisenberg Scaling via Depletion:
- The authors identify a different optimal operating point: maximum pump depletion ( $t_{fl}$ ), where the signal flux is maximized.
- In this regime, the QFI scales as $N^{2 \pm 0.00024}$ , achieving Heisenberg scaling.
- This result contradicts the intuition that maximum sensitivity requires maximum squeezing. Instead, the non-Gaussian nature of the state at $t_{fl}$ , characterized by large photon number variances, enables the $N^2$ scaling.
Measurement Strategy Dependence:
- Sub-optimality of Standard Detection: At the maximum depletion point ( $t_{fl}$ ), standard photon counting (or homodyne detection) is shown to be suboptimal, failing to achieve Heisenberg scaling and reverting to shot-noise scaling.
- Necessity of Nonlinear Measurement: To saturate the QFI at $t_{fl}$ , a joint nonlinear measurement is required. The authors demonstrate that an SU(1,1) interferometer scheme, which utilizes a time-reversal evolution (re-interaction with the nonlinear medium) and joint measurement of the pump and signal, successfully achieves Heisenberg scaling.

Significance and Claims
The paper claims to resolve the debate regarding the viability of quantum advantage under realistic pump conditions. The authors assert that:

Quantum advantage persists despite pump quantization and depletion, but it is not guaranteed by the mere presence of squeezing.
The optimal operating point for quantum sensing shifts from the regime of maximum squeezing to the regime of maximum flux/depletion when the pump is treated as a quantum object.
Measurement strategies must evolve alongside the source dynamics. The transition from a squeezing-optimized regime to a flux-optimized regime necessitates a shift from linear detection (photon counting) to nonlinear, joint measurements (e.g., SU(1,1) interferometry) to realize the full quantum potential.
The framework highlights that the fundamental limits of phase estimation are dictated not just by the properties of the generated signal, but by the joint coherence and entanglement of the entire pump-signal system.

The work suggests that designing optimal quantum sensors requires a holistic approach that accounts for the full cycle of the quantum resource—from its creation by a quantized pump to its final readout in a correlated, nonlinear measurement. This perspective extends beyond optical sensing to other many-body quantum dynamics, including Tavis-Cummings models, Brillouin scattering, and circuit QED.

Technical Summary: Pump Quantization Changes Optimal Measurements for Quantum Sensing

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