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arXiv:2304.08164 · Physical Review A

Lie-algebraic quantum phase reduction based on heterodyne detection

Wataru Setoyama, Yoshihiko Hasegawa

Published Jan 2, 2025• 1 citation
arXivHTMLPDF

Abstract

Measurement backaction inherently alters observed dynamics in quantum physics. In the realm of quantum synchronization, this backaction induces a phase bias, making the assessment of synchronization critically dependent on the choice of the observables. In this study we extend the quantum phase reduction approach [W. Setoyama and Y. Hasegawa, Phys. Rev. Lett. 132, 093602 (2024)] into heterodyne detection, offering a comprehensive theoretical framework for analyzing quantum synchronization dynamics through uniform continuous measurement over all possible quadrature observables. This method averages out the backaction, allowing for unbiased evaluation of synchronization between quantum oscillators while avoiding measurement-induced phase bias. Furthermore, by defining the phase and limit-cycle solution independently of specific observables, our proposed method consistently adapts to the scenario where the observables are freely modified during the time evolution. Through simulations of noise-induced synchronization, our method reveals that the number of phase clusters between oscillators is restricted by their bosonic levels.

DOI
10.1103/physreva.111.012202

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Measurement backaction inherently alters observed dynamics in quantum physics. In the realm of quantum synchronization, this backaction induces a phase bias, making the assessment of synchronization critically dependent on the choice of the observables. In this study we extend the quantum phase reduction approach [W. Setoyama and Y. Hasegawa, Phys. Rev. Lett. 132, 093602 (2024)] into heterodyne detection, offering a comprehensive theoretical framework for analyzing quantum synchronization dynamics through uniform continuous measurement over all possible quadrature observables. This method averages out the backaction, allowing for unbiased evaluation of synchronization between quantum oscillators while avoiding measurement-induced phase bias. Furthermore, by defining the phase and limit-cycle solution independently of specific observables, our proposed method consistently adapts to the scenario where the observables are freely modified during the time evolution. Through simulations of noise-induced synchronization, our method reveals that the number of phase clusters between oscillators is restricted by their bosonic levels.
Limit-cycle oscillators are the basic building blocks for synchronization; yet, the notion of a quantum limit cycle has remained unclear. Here, we study quantum limit cycles and synchronization in the presence of continuous heterodyne measurement. The resulting quantum trajectories, i.e., time evolutions of the quantum state conditioned on the measurement outcome, make the quantum limit cycles apparent. We focus on the paradigmatic model of the quantum van der Pol oscillator and on two-level systems. Our work provides insights into limit cycles in quantum systems, emphasizing their similarity to classical limit cycles subject to noise. Additionally, we connect theoretical measures of quantum synchronization to quantities experimentally accessible via heterodyne detection.
It is well known from classical physics that weakly coupled self-sustained oscillators may spontaneously lock their phases. Just like classical synchronization is known to break down due to noise induced phase slips, we show here how the synchronization of continuous variable quantum systems breaks down by proliferation of quantum phase slips. Within a Keldysh path integral formulation of limit cycles, we analyze the phase dynamics and show how, in spite of strong phase correlations, quantum phase slips degrade the actual phase locking. This approach also allows us to address non-Markovian effects on the synchronization of limit cycles, which we illustrate explicitly for superconducting resonators coupled via a voltage biased double quantum dot.
We introduce a general framework of phase reduction theory for quantum nonlinear oscillators. By employing the quantum trajectory theory, we define the limit-cycle trajectory and the phase according to a stochastic Schrödinger equation. Because a perturbation is represented by unitary transformation in quantum dynamics, we calculate phase response curves with respect to generators of a Lie algebra. Our method shows that the continuous measurement yields phase clusters and alters the phase response curves. The observable clusters capture the phase dynamics of individual quantum oscillators, unlike indirect indicators obtained from density operators. Furthermore, our method can be applied to finite-level systems that lack classical counterparts.
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Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, due to backaction arising from quantum fluctuations in the measurement field. This standard quantum limit can be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as backaction-evading measurement. This technique has not been implemented using optical interferometers to date. Here we demonstrate, in a cavity optomechanical system operating in the optical domain, a continuous two-tone backaction-evading measurement of a localized gigahertz-frequency mechanical mode of a photonic-crystal nanobeam cryogenically and optomechanically cooled close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of backaction-evading measurements in nanomechanical resonators for optical ultrasensitive measurements of motion and force.