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arXiv:2506.10666 · Physical Review A

Quantum mechanical pendulum clock

Matteo Brunelli, Mohammad Mehboudi, Nicolas Brunner, Patrick P. Potts

Published Apr 20, 2026• 0 citations
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Abstract

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.

DOI
10.1103/hb36-7m2r

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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.
We introduce a quantum thermodynamic system that can operate as a thermal clock in analogy with the thermal clock first introduced by Ernst Mach in which the temperature difference between cooling bodies in contact can be used as a kind of relational time. Our model is based on an optomechanical system in which photons are transferred irreversibly between two cavities due to the modulation of the cavity cooling rate by a mechanical system coupled to a heat bath. We contrast the ensemble average view with a single system view using the theory of continuous measurement and we show that, by making a weak continuous measurement of the energy difference between the systems, a natural Mach thermal clock arises in an appropriate semiclassical limit. We investigate how quantum fluctuations modify this result.
Time remains one of the least well-understood concepts in physics, most notably in quantum mechanics.A central goal is to find the fundamental limits of measuring time.One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly.Here, we explore these questions by introducing a model of time measurements that is complete and autonomous.Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium-a prerequisite for any system to function as a clock-powered by minimal resources, namely, two thermal baths at different temperatures.Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock's performance in terms of its accuracy and resolution.Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power.More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.
Why are classical theories often sufficient to describe the physics of our world even though everything around us is entirely composed of microscopic quantum systems? The boundary between these two fundamentally dissimilar theories remains an unsolved problem in modern physics. Position measurements of small objects allow us to probe the area where the classical approximation breaks down. In quantum mechanics, Heisenbergâ s uncertainty principle dictates that any measurement of the position must be accompanied by measurement induced back-action---in this case manifested as an uncertainty in the momentum. In recent years, cavity optomechanics has become a powerful tool to perform precise position measurements and investigate their fundamental limitations. The utilization of optical micro-cavities greatly enhances the interaction between light and state-of-the-art nanomechanical oscillators. Therefore, quantum mechanical phenomena have been successfully observed in systems far beyond the microscopic world. In such a cavity optomechanical system, the fluctuations in the position of the oscillator are transduced onto the phase of the light, while fluctuations in the amplitude of the light disturb the momentum of the oscillator during the measurement. As a consequence, correlations are established between the amplitude and phase quadrature of the probe light. However, so far, observation of quantum effects has been limited exclusively to cryogenic experiments, and access to the quantum regime at room temperature has remained an elusive goal because the overwhelming amount of thermal motion masks the weak quantum effects. This thesis describes the engineering of a high-performance cavity optomechanical device and presents experimental results showing, for the first time, the broadband effects of quantum back-action at room temperature. The device strongly couples mechanical and optical modes of exceptionally high quality factors to provide a measurement sensitivity ∼ ⁣104\sim\!10^4∼104 times below the requirement to resolve the zero-point fluctuations of the mechanical oscillator. The quantum back-action is then observed through the correlations created between the probe light and the motion of the nanomechanical oscillator. A so-called â variational measurementâ , which detects the transmitted light in a homodyne detector tuned close to the amplitude quadrature, resolves the quantum noise due to these correlations at the level of 10\% of the thermal noise over more than an octave of Fourier frequencies around mechanical resonance. Moreover, building on this result, an additional experiment demonstrates the ability to achieve quantum enhanced metrology. In this case, the generated quantum correlations are used to cancel quantum noise in the measurement record, which then leads to an improved relative signal-to-noise ratio in measurements of an external force. In conclusion, the successful observation of broadband quantum behavior on a macroscopic object at room temperature is an important milestone in the field of cavity optomechanics. Specifically, this result heralds the rise of optomechanical systems as a platform for quantum physics at room temperature and shows promise for generation of ponderomotive squeezing in room-temperature interferometers.
In the classic Chinese novel Journey to the West, the Monkey King has a special power that enables him to be in two or more places simultaneously. Normally, an amazing thing such as this could never happen in real life. However, in the quantum world, where the motions of objects obey the rules of quantum mechanics, this could happen. These weird quantum rules usually only apply in the microscopic world to atoms and molecules, because quantum properties are generally very fragile in the macroscopic world. However, scientists have recently been making efforts to push macroscopic and mesoscopic mechanical systems into the quantum world, i.e. they are putting conventional mechanical systems into the quantum mechanical regime. Spring oscillators, swings, and pendulum clocks are typical mechanical systems that we often see in daily lives, and they can be described by a basic physical model: the harmonic oscillator. In the quantum regime, harmonic oscillators have some quantum properties, including discrete energy spectra and zero-point motion. To observe these quantum properties, the environmental thermal noise must be suppressed so that the energy of the quantum motion dominates over the energy of the thermal motion. To this end, a generic model of a cavity optomechanical system is used, as shown in Fig. 1. The system consists of an optical cavity with a fixed mirror and a movable mirror, where the latter is attached to a spring and undergoes harmonic oscillation. A laser beam is launched into the cavity and is reflected multiple times between the two mirrors, and the cavity field thus builds up, resulting in a greatly enhanced optical field, which then exerts a considerable force on the movable mirror. The interaction between the optical field and the mechanical motion can reduce the thermal motion noise to enable the fragile quantum properties to be observed. The study of such systems has produced an emerging field called cavity optomechanics [1–3]. The pioneering work was conducted by Braginsky and co-workers in the 1960s [4]. In recent years, various experimental systems have been proposed and investigated, including Fabry–Pérot cavities [by groups including MIT/Caltech LIGO Laboratory, the Mavalvala group at MIT, the Aspelmeyer group at the University of Vienna, the Heidmann group at the University of Pierre and Marie Curie, and the Bouwmeester group at the University of California, Santa Barbara (UCSB)], whispering-gallery cavities (by groups including the Vahala group at Caltech, the Kippenberg group at École Polytechnique Fédérale de Lausanne, the Wang group at the University of Oregon, the Lipson group at Cornell University, the Carmon group at the University of Michigan, the Bowen group at the University of Queensland, and the Xiao group at Peking University), photonic crystal cavities (by groups including the Painter group at Caltech, the Wong group at Columbia University, and the Tang group at Yale University), separated mechanical oscillators inside or near a cavity (by groups including the Harris group at Yale University, the Regal group at Joint Institute for Laboratory Astrophysics (JILA), the Favero group at Paris Diderot University, and the Weig group at the University of Munich), superconducting circuits (by groups including the Schwab group at Caltech, the Lehnert group at JILA, the Cleland group at UCSB, and the Teufel group at National Institute of Science and Technology, Boulder), and cold atoms (by groups including the Kimble group at Caltech, the Stamper-Kurn group at the University of California, Berkeley, and the Treutlein group at the University of Basel). Illustration of a generic cavity optomechanical system. The left mirror is fixed and the right mirror is attached to a spring. Cavity optomechanics has been featured as the most recent milestone in photon history in Nature. The research in cavity optomechanics is important to both fundamental physics studies and the applied sciences. First, cavity optomechanics provides a unique platform for the study of fundamental quantum physics, including macroscopic quantum phenomena, decoherence, and quantum–classical transitions. The field offers the best test bed to study effects like gravity-induced decoherence, which is important to the understanding of macroscopic quantum phenomena. Second, cavity optomechanics is promising for high-precision measurements of small forces, masses, displacements, and accelerations, and is considered to be capable of surpassing the standard quantum limit. Third, cavity optomechanics provides resources for both classical and quantum information processing. For instance, optomechanical devices can serve as information storage devices and act as interfaces between light beams with different wavelengths or even microwaves. In the last few years, researchers have made considerable efforts to put mechanical systems into their quantum ground states [5]. Recent efforts have demonstrated optomechanically induced transparency [6], normal-mode splitting [7], quantum-coherent coupling [8], wavelength conversion [9], and measurements performed below the standard quantum limit [10]. Future developments will aim to integrate different quantum systems to form hybrid quantum devices, e.g. hybrid optomechanical and electromechanical systems. In this way, we can enable phonons, photons, and electrons to work together in the quantum world. Recent years have seen rapidly growing interest in the field of cavity optomechanics. As shown in Fig. 2, the publications and citations in this field have grown exponentially. The now booming development in this field will turn the dream of manipulating macroscopic mechanical systems in a quantum manner into reality. Numbers of publications (a) and citations (b) per year in the period from 2005 to 2013 with the keyword ‘cavity optomechanics’, retrieved from the Web of Science.
Time remains one of the least well-understood concepts in physics, most notably in quantum mechanics.A central goal is to find the fundamental limits of measuring time.One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly.Here, we explore these questions by introducing a model of time measurements that is complete and autonomous.Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium-a prerequisite for any system to function as a clock-powered by minimal resources, namely, two thermal baths at different temperatures.Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock's performance in terms of its accuracy and resolution.Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power.More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.
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.