Squeezing and measurement of a mechanical quadrature via PID feedback
Alberto Hijano, Tero T. Heikkilä
Abstract
Proportional-Integral-Derivative (PID) control is used for automatically regulating a measurable quantity to a desired setpoint. It is widely used in different types of classical control electronics. Here, we show how extending the feedback theory in quantum systems to include the derivative and integral parts influences both the transient and steady-state behavior of the amplitude and squeezing of a mechanical quadrature in an optomechanical system. We show that, in contrast to standard proportional feedback, derivative feedback affects both the conditional and unconditional squeezing. Furthermore, we demonstrate how feedback may be employed to drive a mechanical quadrature to track a desired reference signal. Our findings offer new routes for an improved quantum state control and measurement precision.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper extends the theory of measurement-based quantum feedback in optomechanical systems from proportional-only control to full PID (Proportional-Integral-Derivative) control. The authors analyze how each component of the PID controller—individually and in combination—affects both the conditional and unconditional squeezing of a mechanical quadrature, as well as the ability to drive a quadrature to track a reference signal. The key technical novelty lies in incorporating the derivative term within the SLH quantum stochastic framework, which requires a non-trivial modification of the coupling operators since the derivative of a stochastic process falls outside the standard formalism. The paper builds directly on the foundational work by Clerk, Marquardt, and Jacobs (2008) on back-action evasion and proportional feedback squeezing, and on Gough's (2020) quantum Kalman filter-based PID controller framework.
Methodological Rigor
The theoretical treatment is rigorous within its scope. The authors employ the SLH formalism consistently, derive the Belavkin-Kushner-Stratonovich (BKS) equations for the filtered estimates under PID feedback, and provide complete analytical expressions for both conditional and unconditional variances in the steady state [Eqs. (10-11)]. The treatment of the derivative feedback term—transferring it from the Hamiltonian to the coupling operators via Eq. (S.15-S.17)—is a technically careful maneuver that maintains consistency with the quantum stochastic calculus framework.
The analytical expressions are derived in the good-cavity limit (G, γ ≪ κ), which is well-motivated physically but does limit generality. The full dynamical equations are solved numerically, and code is made publicly available, supporting reproducibility. The supplemental material is thorough, providing all necessary equations.
However, several methodological limitations deserve mention. The paper assumes a phase-sensitive amplifier deep in the quantum limit that adds no appreciable noise—a strong idealization. There is no analysis of robustness to realistic imperfections such as detection inefficiency, classical noise in the feedback loop, or finite bandwidth of the PID controller. Additionally, the rotating-wave approximation is employed throughout without discussion of its validity regime for the feedback terms, particularly the derivative term which could have fast timescale components.
Key Findings
1. Derivative feedback affects both conditional and unconditional variances, unlike proportional feedback which only affects the unconditional variance. While αD increases conditional variance, it reduces excess noise, yielding net unconditional squeezing.
2. Integral feedback produces transient squeezing that vanishes in steady state (in the absence of a setpoint), because the integral of the autocorrelation σ vanishes asymptotically.
3. Combined PD feedback can achieve faster and enhanced squeezing compared to either action alone, though the effects are not independent—αD suppresses P-induced squeezing [Eq. (11)].
4. PI control is effective for driving a quadrature to a reference value, eliminating steady-state offset (a well-known property of integral action in classical control). The derivative action is counterproductive for setpoint tracking in this system.
5. Transfer function analysis [Eq. (12)] enables systematic PID tuning using classical control theory tools.
Potential Impact
The paper provides a conceptual bridge between classical PID control theory—one of the most widely deployed control strategies in engineering—and quantum feedback control. This is a natural and useful extension, though not deeply surprising. The practical impact depends critically on experimental realizability. As the authors note, measurement-based feedback squeezing has not yet been demonstrated experimentally, though ground-state feedback cooling has been achieved. The framework could be relevant to:
The transfer function approach to tuning feedback parameters is practically useful and accessible to experimentalists familiar with classical control theory.
Timeliness & Relevance
The work is timely given recent experimental advances in measurement-based feedback cooling of mechanical oscillators (Rossi et al. 2018, Whittle et al. 2021, Rej et al. 2025). As these experiments mature toward feedback-induced squeezing, having a complete PID framework ready is valuable. The connection to force sensing and quantum gravity tests provides additional motivation.
Strengths
Limitations
Overall Assessment
This is a solid theoretical contribution that systematically extends quantum feedback squeezing to PID control. The mathematical treatment is careful and complete, and the results are clearly presented. However, the conceptual advance is incremental—translating well-known classical PID properties to the quantum domain with relatively predictable outcomes. The impact would be significantly enhanced by analysis under realistic experimental conditions or by demonstration of qualitatively new phenomena unique to the quantum setting.
Generated Apr 20, 2026
Comparison History (39)
Paper 1 has higher potential impact due to major novelty and foundational significance: a 3D geometrically local stabilizer Hamiltonian enabling exponentially long-lived passive quantum memory at nonzero temperature addresses a central bottleneck in fault-tolerant quantum computing and condensed-matter theory. If correct and physically realizable, it would influence quantum error correction, topological phases, and quantum hardware design broadly. Paper 2 is timely and practically useful for optomechanics/control, but extends existing feedback frameworks (PID) with more incremental conceptual advance and narrower cross-field reach.
Paper 1 demonstrates a striking experimental advance: a milligram-scale levitated superconducting oscillator at mK temperatures with <0.8 μHz linewidth and >110 h ring-down, enabling ultra-isolated macroscopic dynamics and femtonewton-level force/drag measurements. This is highly novel, methodologically demanding, and broadly enabling for precision sensing, low-temperature physics, and foundational tests (quantum-classical boundary, collapse models, quantum gravity-inspired effects). Paper 2 is a useful theoretical/control extension (PID in quantum feedback) with applications in optomechanics, but is more incremental and likely narrower in near-term impact than the demonstrated hardware milestone.
Paper 2 offers a broadly applicable computational/mathematical method: exact tridiagonalization for two major bosonic model families (Bose–Hubbard and optomechanics), enabling scalable diagonalization and efficient symplectic split-operator propagators with favorable complexity. This can unlock larger, more accurate simulations across condensed matter, AMO, quantum optics, and quantum information, with clear real-world relevance for modeling quantum devices. Paper 1 is timely and useful for quantum control in optomechanics, but its impact is narrower (specific feedback extensions and squeezing behavior) and more system-specific than a general-purpose algorithmic advance.
Paper 2 addresses the timely and rapidly growing field of quantum batteries, proposing a practical solution to two fundamental challenges (charging efficiency degradation from counter-rotating terms in ultrastrong coupling and decoherence during storage). The combination of dynamical modulation for efficiency improvement and bath engineering for noise immunity is novel and has broader potential impact across quantum energy storage, quantum thermodynamics, and quantum technologies. Paper 1 extends classical PID control to quantum optomechanics, which is incremental compared to existing feedback control methods, with a narrower scope of impact.
Paper 1 bridges a ubiquitous classical control technique (PID) with quantum systems, offering new methods for quantum state control and precision measurement. Its findings are broadly applicable to various quantum technologies and optomechanical systems. Paper 2, while methodologically rigorous, focuses on a more specialized area of quantum lattice simulations and QCD, likely resulting in a narrower scope of impact compared to the widespread utility of quantum feedback control.
Paper 2 combines machine learning with Casimir force measurements to create a novel material characterization technique, bridging quantum physics, materials science, and AI. This interdisciplinary approach has broader potential impact: it introduces a fundamentally new measurement paradigm using vacuum fluctuations as a broadband source, addresses an inverse problem with practical applications in thin-film characterization, and demonstrates a creative intersection of quantum phenomena with modern ML methods. Paper 1, while technically sound, extends existing PID control theory to quantum optomechanics in a more incremental fashion with narrower scope.
Paper 2 addresses a fundamental trade-off in quantum photonics (purity vs. brightness of nonclassical light) with a novel mechanism—interference-engineered many-body interactions in cavity-coupled atomic arrays—achieving four orders of magnitude improvement in antibunching. It offers broader impact across quantum technologies (single-photon sources, photon-pair generation), introduces a scalable and programmable platform, and connects many-body physics with quantum photonics. Paper 1 extends classical PID control to quantum optomechanics, which is incremental compared to the transformative potential of Paper 2's approach to on-demand quantum light generation.
Paper 2 extends classical PID control to quantum systems, providing a fundamental advancement in quantum feedback theory. This has broad applicability across various quantum technologies for state control and precision measurement. Paper 1, while highly relevant for specific applications like smart grids, focuses on a narrower use case of Rydberg electric field sensing. The foundational nature and broader generalizability of Paper 2 suggest a higher overall scientific impact across multiple fields.
Paper 1 has higher likely impact due to stronger methodological grounding and clearer experimental relevance: extending quantum feedback theory to full PID (including integral/derivative) in optomechanics directly affects achievable squeezing and precision measurement, with immediate applications in quantum sensing and control. It builds on a mature platform where such feedback can be implemented and benchmarked. Paper 2 is timely and cross-disciplinary, but “quantum-resistant teleportation” largely secures a classical side-channel with PQC; the core physics is not fundamentally new and practical security claims depend heavily on assumptions (memory coherence, threat model, leakage models), making real-world uptake less certain.
Paper 2 likely has higher impact: it extends quantum feedback control by incorporating full PID (integral and derivative terms) in optomechanical systems, affecting both transient and steady-state conditional/unconditional squeezing and enabling reference tracking. This is methodologically substantive and broadly relevant to quantum control, precision measurement, and quantum sensing, with clear experimental and technological pathways. Paper 1 is a useful incremental protocol improvement within semi-quantum signatures, but its impact is narrower (cryptographic protocol design) and hinges more on security-model acceptance and deployment maturity.
Paper 1 presents a foundational mathematical tool applicable across quantum mechanics, quantum computing, and quantum information theory. Its dimension-agnostic, closed-form solution for unitary mapping eliminates the need for explicit bases, offering a highly novel and broadly useful theoretical advance. While Paper 2 provides a valuable translation of classical PID control to quantum optomechanics, Paper 1's fundamental nature gives it a higher potential for widespread, cross-disciplinary impact in the rapidly growing field of quantum technologies.
Paper 1 extends foundational PID control theory to quantum systems, offering novel methods for quantum state control and squeezing. This theoretical advance has broad implications across quantum sensing, metrology, and optomechanics. In contrast, Paper 2 presents a valuable but narrower engineering optimization for specific FPGA hardware in Quantum Key Distribution. Paper 1's fundamental contributions to quantum measurement precision give it a much higher potential for broad scientific impact and versatile real-world application.
Paper 2 addresses a practical, pressing challenge in superconducting quantum computing—flux control distortion compensation—with demonstrated experimental results on real quantum hardware. Its digital predistortion framework enables automated calibration of flux control channels, directly impacting gate fidelity and scalability of quantum processors. Paper 1, while theoretically interesting in extending PID feedback to quantum optomechanical systems, remains largely theoretical. Paper 2's immediate applicability to the rapidly growing quantum computing industry, experimental validation, and relevance to improving quantum gate fidelity give it broader and more timely impact.
Paper 2 introduces a ubiquitous classical control paradigm (PID) to quantum systems, offering a fundamental advancement in quantum state control and precision measurement. Its theoretical approach has broad applicability across various quantum platforms (e.g., optomechanics, superconducting circuits), whereas Paper 1 focuses on characterizing a specific material defect (VSi in SiC). The high novelty and cross-disciplinary potential of quantum PID feedback give Paper 2 a broader methodological impact.
Paper 2 introduces and rigorously evaluates a novel trapped-ion platform for quantum computing, addressing major outstanding challenges like high-fidelity operations and large-scale engineering. Its potential to serve as a next-generation qubit gives it a broader and more immediate real-world impact in the rapidly growing field of quantum information processing compared to the more specialized optomechanical PID control explored in Paper 1.
Paper 2 likely has higher impact: it extends quantum feedback control theory by incorporating full PID (integral and derivative) terms and analyzes effects on both conditional and unconditional squeezing and tracking, with broad relevance to optomechanics, quantum metrology, and control. The approach is conceptually novel and timely for quantum control of mechanical systems, potentially influencing multiple platforms. Paper 1 is a useful experimental demonstration (2-photon ODMR of NV centers) with clear applications in 3D imaging/sensing, but it is more incremental within an already active NV/ODMR toolkit and narrower in theoretical reach.
Paper 1 presents a more comprehensive theoretical framework connecting entanglement, thermodynamics, and out-of-equilibrium dynamics in 1D Bose gases, with broader implications for quadratic bosonic models and thermodynamic cycles. Its identification of optimal entanglement witness structures and the unified treatment of thermal and non-equilibrium entanglement offers deeper theoretical novelty. Paper 2, while practically useful in extending PID feedback to quantum systems, represents a more incremental advance in quantum control methodology with a narrower scope of impact.
Paper 2 translates widely used classical PID control into the quantum domain, offering tangible advancements in quantum state control and measurement precision. This highly novel approach bridges control theory and optomechanics, presenting clear real-world applications in developing quantum technologies. In contrast, Paper 1 focuses on foundational, theoretical mathematical physics with narrower immediate applicability.
Paper 2 demonstrates a concrete hardware solution for connecting superconducting quantum processors over tens of meters via cryogenic microwave links, addressing a critical scaling bottleneck in quantum computing. The achievement of a loophole-free Bell test with superconducting circuits and the enabling of distributed quantum computing represent significant experimental milestones. Its breadth of impact spans quantum computing, quantum communication, and quantum networking. Paper 1, while theoretically interesting in extending PID control to quantum systems, represents a more incremental contribution to optomechanical feedback control with narrower immediate impact.
Paper 2 addresses a highly timely and practically relevant problem in quantum communication infrastructure—comparing direct entanglement distribution versus space-based quantum repeaters. It provides comprehensive mission-design-relevant analysis including orbital geometry, memory requirements, and fidelity modeling. This has broader impact across quantum networking, satellite engineering, and policy planning for quantum infrastructure. Paper 1, while technically sound in extending PID control to quantum optomechanics, represents a more incremental advance within a narrower subfield with less immediate real-world applicability.