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arXiv:2503.12118 · Quantum Science and Technology

Quantum thermodynamics on a limit cycle

Varinder Singh, Euijoon Kwon, Gerard J Milburn

Published Jan 7, 2026• 1 citation
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Abstract

Abstract We consider a periodic quantum clock based on cooperative resonance fluorescence at zero temperature. The semi-classical dynamics exhibit a Hopf bifurcation to a limit cycle. In the quantum case, this system has an exact steady state and the limit cycle appears in conditional quantum dynamics under homodyne detection. We show that the intrinsic quantum phase diffusion on the limit cycle leads to fluctuations in the period. By simulating the stochastic master equation for homodyne detection, we extract the statistical properties of the clock period. We show that the precision of the clock satisfies the quantum-thermodynamic kinetic uncertainty relations. As energy dissipation increases, the clock’s quality improves, fully validating, in a quantum stochastic system, the link between energy dissipation and clock precision.

DOI
10.1088/2058-9565/ae34df

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Abstract We consider a periodic quantum clock based on cooperative resonance fluorescence at zero temperature. The semi-classical dynamics exhibit a Hopf bifurcation to a limit cycle. In the quantum case, this system has an exact steady state and the limit cycle appears in conditional quantum dynamics under homodyne detection. We show that the intrinsic quantum phase diffusion on the limit cycle leads to fluctuations in the period. By simulating the stochastic master equation for homodyne detection, we extract the statistical properties of the clock period. We show that the precision of the clock satisfies the quantum-thermodynamic kinetic uncertainty relations. As energy dissipation increases, the clock’s quality improves, fully validating, in a quantum stochastic system, the link between energy dissipation and clock precision.
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.
All clocks, classical or quantum, are open non equilibrium irreversible systems subject to the constraints of thermodynamics. Using examples I show that these constraints necessarily limit the performance of clocks and that good clocks require large energy dissipation. For periodic clocks, operating on a limit cycle, this is a consequence of phase diffusion. It is also true for non periodic clocks (for example, radio carbon dating) but due to telegraph noise not to phase diffusion. In this case a key role is played by accurate measurements that decrease entropy, thereby raising the free energy of the clock, and requires access to a low entropy reservoir. In the quantum case, for which thermal noise is replaced by quantum noise (spontaneous emission or tunnelling), measurement plays an essential role for both periodic and non periodic clocks. The paper concludes with a discussion of the Tolman relations and Rovelli's thermal time hypothesis in terms of clock thermodynamics.
We theoretically and experimentally study the precision of a quantum clock near zero temperature, explicitly accounting for the effect of continuous measurement. The clock is created by a superconducting transmon qubit dispersively coupled to an open coplanar resonator. The cavity and qubit are driven by coherent fields, and the cavity output is monitored with a quantum-noise-limited amplifier. When the continuous measurement is weak, it induces persistent coherent oscillations (with fluctuating periods) in the conditional moments of the qubit's energy probability distribution, which are manifest in the output of the resonator. On the other hand, strong continuous measurement leads to an incoherent cycle of quantum jumps. We theoretically find an equality for the precision of the clock in each regime. Independently from the equalities, we derive a kinetic uncertainty relation for the precision, and find that both equalities satisfy this uncertainty relation. Finally, we experimentally verify that our quantum clock obeys the kinetic uncertainty relation for the precision, thus making an explicit link between the (kinetic) thermodynamic behavior of the clock and its precision, and achieving an experimental test of a kinetic uncertainty relation in the quantum domain.
We investigate an optomechanical system as a model of an autonomous mechanical pendulum clock in the quantum regime, whose operation relies only on incoherent (thermal) resources. The escapement of the clock, the mechanism that translates oscillatory motion into ticks, is provided by an emitter in the optical cavity and the operation of the clock relies on the existence of a limit cycle. Since the clock is based on an oscillatory degree of freedom, it can overcome the thermodynamic uncertainty relation and is thus more accurate than clocks that rely only on stochastic transitions. Furthermore, by increasing the amount of emitters in the cavity, the clock approaches the behavior expected for a macroscopic pendulum clock, where fluctuations become irrelevant while the clock dynamics becomes completely irreversible. This allows for investigating the quantum-to-classical transition of pendulum clocks.
The thermodynamic uncertainty relation, originally derived for classical Markov-jump processes, provides a tradeoff relation between precision and dissipation, deepening our understanding of the performance of quantum thermal machines. Here, we examine the interplay of quantum system coherences and heat current fluctuations on the validity of the thermodynamics uncertainty relation in the quantum regime. To achieve the current statistics, we perform a full counting statistics simulation of the Redfield quantum master equation. We focus on steady-state quantum absorption refrigerators where nonzero coherence between eigenstates can either suppress or enhance the cooling power, compared with the incoherent limit. In either scenario, we find enhanced relative noise of the cooling power (standard deviation of the power over the mean) in the presence of system coherence, thereby corroborating the thermodynamic uncertainty relation. Our results indicate that fluctuations necessitate consideration when assessing the performance of quantum coherent thermal machines.
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.