Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.22926 · Physical Review Applied

Using coherent feedback for a periodic clock

Stefan Zeppetzauer, Leonardo Assis Morais, Xin He, Gerard J Milburn, Arkady Fedorov

Published Sep 16, 2025• 1 citation
arXivHTMLPDF

Abstract

A driven linear oscillator and a feedback mechanism are two necessary elements of any classical periodic clock. Here, we introduce a fully quantum clock using a driven oscillator in the quantum regime and coherent quantum feedback. We show that if we treat the model semiclassically, this system supports limit cycles, or self-sustained oscillations, as needed for a periodic clock. We then analyze the noise of the system quantum mechanically and prove that the accuracy of this clock is higher compared with the clock implemented with the classical measurement feedback. We experimentally implement the model using two superconducting cavities with incorporated Josephson junctions and microwave circulators for the realization of the quantum feedback. We confirm the appearance of the limit cycle and study the clock accuracy both in frequency and time domains. Under specific conditions of noisy driving, we observe that the clock oscillations are more coherent than the drive, pointing towards the implementation of a quantum autonomous clock.

DOI
10.1103/b9hw-r9gk

Related papers

A quantum ticking self-oscillator using delayed feedback

Yanan Liu, William J. Munro, Jason Twamley

Abstract Self-sustained oscillators (SSOs) is a commonly used method to generate classical clock signals and SSOs using delayed feedback have been developed commercially which possess ultra-low phase noise and drift. Research into the development of quantum self-oscillation, where one can also have a periodic and regular output tick, that can be used to control quantum and classical devices has received much interest and quantum SSOs so far studied suffer from phase diffusion which leads to the smearing out of the quantum oscillator over the entire limit cycle in phase space seriously degrading the system’s ability to perform as a self-oscillation. In this paper, we explore quantum versions of time-delayed SSOs, which has the potentials to develop a ticking quantum clock. We first design a linear quantum SSO which exhibits perfect oscillation without phase diffusion. We then explore a nonlinear delayed quantum SSO but find it exhibits dephasing similar to previously studied non-delayed systems.

arXiv:2307.14567•New Journal of Physics•Nov 29, 2023•0 citations

Experimental realization and synchronization of a quantum van der Pol oscillator

Y. Li, Zihan Xie, Xiaodong Yang +8 more

Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.

arXiv:2504.00751•Science Advances•Oct 10, 2025•17 citations

Feedback-Induced Advantage in Quantum Clockworks

Jakob Miller, Paul Erker

Atomic frequency standards have achieved steadily increasing precision over the past seventy years, enabled in part by feedback mechanisms that stabilise their output. In parallel, the timekeeping capabilities of quantum systems have been explored within the recently developed ticking-clock framework, which models clocks as dynamical systems producing a stochastic sequence of ticks. However, a theoretical description that unifies these perspectives and incorporates feedback into autonomous quantum clocks has been lacking. We introduce a framework for feedback-controlled clockworks in which classical information extracted from the tick sequence is used to influence the subsequent dynamics of the clock. We show that such feedback preserves the core structural features of self-timing and clockwork independence that characterise autonomous ticking clocks. We further identify the signal-to-noise ratio $\mathfrak{S}$ as the fundamental figure of merit for assessing the performance of feedback-controlled clocks. Applying our framework to two representative architectures, we prove that classical clockworks cannot surpass the optimal signal-to-noise ratio achievable without feedback. In contrast, for quantum clockworks we present numerical evidence that feedback can provide a genuine and robust performance enhancement, improving the maximal attainable signal-to-noise ratio. We provide additional numerical evidence, suggesting that the magnitude of the improvement strongly depends on the amount and type of the stored, classical information. These results establish feedback as a potentially essential ingredient in pushing the fundamental limits of timekeeping in the quantum regime.

quant-pharXiv:2603.04556•Mar 6, 2026•0 citations

Quantum clocks driven by measurement

Adil A. Gangat, Gerard J Milburn

In classical physics, clocks are open dissipative systems driven from thermal equilibrium and necessarily subject to thermal noise. We describe a quantum clock driven by entropy reduction through measurement. The mechanism consists of a superconducting transmon qubit coupled to an open co-planar resonator. The cavity and qubit are driven by coherent fields and the cavity output is monitored with homodyne detection. We show that the measurement itself induces coherent oscillations, with fluctuating period, in the conditional moments. The clock signal can be extracted from the observed measurement currents and analysed to determine the noise performance. The model demonstrates a fundamental principle of clocks at zero temperature: good clocks require high rates of energy dissipation and consequently entropy generation.

arXiv:2109.05390•Sep 12, 2021•1 citation

Quantum synchronization in presence of shot noise

Florian Höhe, Lukas Danner, Ciprian Padurariu +3 more

Abstract Synchronization is a widespread phenomenon encountered in many natural and engineered systems with nonlinear classical dynamics. How synchronization concepts and mechanisms transfer to the quantum realm and whether features are universal or platform specific are timely questions of fundamental interest. They can be studied in superconducting electrical circuits which provide a well-established platform for nonlinear quantum dynamics. Here, we consider a Josephson-photonics device, where a dc-biased Josephson junction creates (non-classical) light in a microwave cavity. The combined quantum compound constitutes a self-sustained oscillator: a system susceptible to synchronization. This is due to the inherent effect of an in-series resistance, which realizes an autonomous feedback mechanism of the charge transport on the driving voltage. Accounting for the full counting statistics of transported charge not only yields phase diffusion, but allows us to describe phase locking to an ac-signal and the mutual synchronization of two such devices. Thereby one can observe phase stabilization leading to a sharp emission spectrum as well as unique charge transport statistics revealing shot noise induced phase slips. Two-time perturbation theory is used to obtain a reduced description of the oscillators phase dynamics in form of a Fokker–Planck equation in generalization of classical synchronization theories.

arXiv:2306.15292•New Journal of Physics•Feb 1, 2025•4 citations

Quantum dynamics of a mechanical resonator driven by a cavity

Andrew D. Armour, Denzil A. Rodrigues

We explore the quantum dynamics of a mechanical resonator whose position is coupled to the frequency of an optical (or microwave) cavity mode. When the cavity is driven at a frequency above resonance the mechanical resonator can gain energy and for sufficiently strong coupling this results in limit-cycle oscillations. Using a truncated Wigner function approach, which captures the zero-point fluctuations in the system, we develop an approximate analytic treatment of the resonator dynamics in the limit-cycle regime. We find that the limit-cycle oscillations produced by the cavity are associated with rather low levels of energy fluctuations in the resonator. Compared to a resonator at the same temperature which is driven by a pure harmonic drive to a given average energy, the cavity-driven oscillations can have much lower energy fluctuations. Furthermore, at sufficiently low temperatures, the cavity can drive the resonator into a non-classical state which is number-squeezed.

Comptes Rendus Physique•Apr 23, 2012•22 citations
A driven linear oscillator and a feedback mechanism are two necessary elements of any classical periodic clock. Here, we introduce a fully quantum clock using a driven oscillator in the quantum regime and coherent quantum feedback. We show that if we treat the model semiclassically, this system supports limit cycles, or self-sustained oscillations, as needed for a periodic clock. We then analyze the noise of the system quantum mechanically and prove that the accuracy of this clock is higher compared with the clock implemented with the classical measurement feedback. We experimentally implement the model using two superconducting cavities with incorporated Josephson junctions and microwave circulators for the realization of the quantum feedback. We confirm the appearance of the limit cycle and study the clock accuracy both in frequency and time domains. Under specific conditions of noisy driving, we observe that the clock oscillations are more coherent than the drive, pointing towards the implementation of a quantum autonomous clock.
Abstract Self-sustained oscillators (SSOs) is a commonly used method to generate classical clock signals and SSOs using delayed feedback have been developed commercially which possess ultra-low phase noise and drift. Research into the development of quantum self-oscillation, where one can also have a periodic and regular output tick, that can be used to control quantum and classical devices has received much interest and quantum SSOs so far studied suffer from phase diffusion which leads to the smearing out of the quantum oscillator over the entire limit cycle in phase space seriously degrading the system’s ability to perform as a self-oscillation. In this paper, we explore quantum versions of time-delayed SSOs, which has the potentials to develop a ticking quantum clock. We first design a linear quantum SSO which exhibits perfect oscillation without phase diffusion. We then explore a nonlinear delayed quantum SSO but find it exhibits dephasing similar to previously studied non-delayed systems.
Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.
Atomic frequency standards have achieved steadily increasing precision over the past seventy years, enabled in part by feedback mechanisms that stabilise their output. In parallel, the timekeeping capabilities of quantum systems have been explored within the recently developed ticking-clock framework, which models clocks as dynamical systems producing a stochastic sequence of ticks. However, a theoretical description that unifies these perspectives and incorporates feedback into autonomous quantum clocks has been lacking. We introduce a framework for feedback-controlled clockworks in which classical information extracted from the tick sequence is used to influence the subsequent dynamics of the clock. We show that such feedback preserves the core structural features of self-timing and clockwork independence that characterise autonomous ticking clocks. We further identify the signal-to-noise ratio S\mathfrak{S}S as the fundamental figure of merit for assessing the performance of feedback-controlled clocks. Applying our framework to two representative architectures, we prove that classical clockworks cannot surpass the optimal signal-to-noise ratio achievable without feedback. In contrast, for quantum clockworks we present numerical evidence that feedback can provide a genuine and robust performance enhancement, improving the maximal attainable signal-to-noise ratio. We provide additional numerical evidence, suggesting that the magnitude of the improvement strongly depends on the amount and type of the stored, classical information. These results establish feedback as a potentially essential ingredient in pushing the fundamental limits of timekeeping in the quantum regime.
In classical physics, clocks are open dissipative systems driven from thermal equilibrium and necessarily subject to thermal noise. We describe a quantum clock driven by entropy reduction through measurement. The mechanism consists of a superconducting transmon qubit coupled to an open co-planar resonator. The cavity and qubit are driven by coherent fields and the cavity output is monitored with homodyne detection. We show that the measurement itself induces coherent oscillations, with fluctuating period, in the conditional moments. The clock signal can be extracted from the observed measurement currents and analysed to determine the noise performance. The model demonstrates a fundamental principle of clocks at zero temperature: good clocks require high rates of energy dissipation and consequently entropy generation.
Abstract Synchronization is a widespread phenomenon encountered in many natural and engineered systems with nonlinear classical dynamics. How synchronization concepts and mechanisms transfer to the quantum realm and whether features are universal or platform specific are timely questions of fundamental interest. They can be studied in superconducting electrical circuits which provide a well-established platform for nonlinear quantum dynamics. Here, we consider a Josephson-photonics device, where a dc-biased Josephson junction creates (non-classical) light in a microwave cavity. The combined quantum compound constitutes a self-sustained oscillator: a system susceptible to synchronization. This is due to the inherent effect of an in-series resistance, which realizes an autonomous feedback mechanism of the charge transport on the driving voltage. Accounting for the full counting statistics of transported charge not only yields phase diffusion, but allows us to describe phase locking to an ac-signal and the mutual synchronization of two such devices. Thereby one can observe phase stabilization leading to a sharp emission spectrum as well as unique charge transport statistics revealing shot noise induced phase slips. Two-time perturbation theory is used to obtain a reduced description of the oscillators phase dynamics in form of a Fokker–Planck equation in generalization of classical synchronization theories.
We explore the quantum dynamics of a mechanical resonator whose position is coupled to the frequency of an optical (or microwave) cavity mode. When the cavity is driven at a frequency above resonance the mechanical resonator can gain energy and for sufficiently strong coupling this results in limit-cycle oscillations. Using a truncated Wigner function approach, which captures the zero-point fluctuations in the system, we develop an approximate analytic treatment of the resonator dynamics in the limit-cycle regime. We find that the limit-cycle oscillations produced by the cavity are associated with rather low levels of energy fluctuations in the resonator. Compared to a resonator at the same temperature which is driven by a pure harmonic drive to a given average energy, the cavity-driven oscillations can have much lower energy fluctuations. Furthermore, at sufficiently low temperatures, the cavity can drive the resonator into a non-classical state which is number-squeezed.