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arXiv:2409.02857 · New Journal of Physics

Physically constrained quantum clock-driven dynamics

Lea Lautenbacher, Giovanni Spaventa, Dario Cilluffo, Susana F. Huelga, Martin Bodo Plenio

Published Oct 21, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract Thermal machines are physical systems designed to convert thermal energy into practical work through cyclic state transformations. A key component in such a machine is a clock-equipped control element that dictates which interaction Hamiltonian governs the system-reservoir interactions at specific times, while itself remaining unaffected. However, in the context of quantum dynamics, it is well known that maintaining perfect isolation is practically impossible, except under highly idealized conditions. In this study, we begin with such an idealized model for a clock and systematically relax its main assumptions to develop a more realistic framework. Our approach yields a simplified yet physically consistent description of clock dynamics, enabling the analysis of deviations from ideal time-keeping, which we interpret as clock degradation. We introduce a continuous time operator to derive a lower bound on this degradation, grounded in a generalized time-energy uncertainty relation. These results highlight trade-offs between clock performance and energy uncertainty in physically realizable control schemes.

DOI
10.1088/1367-2630/ae15b7

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Abstract Thermal machines are physical systems designed to convert thermal energy into practical work through cyclic state transformations. A key component in such a machine is a clock-equipped control element that dictates which interaction Hamiltonian governs the system-reservoir interactions at specific times, while itself remaining unaffected. However, in the context of quantum dynamics, it is well known that maintaining perfect isolation is practically impossible, except under highly idealized conditions. In this study, we begin with such an idealized model for a clock and systematically relax its main assumptions to develop a more realistic framework. Our approach yields a simplified yet physically consistent description of clock dynamics, enabling the analysis of deviations from ideal time-keeping, which we interpret as clock degradation. We introduce a continuous time operator to derive a lower bound on this degradation, grounded in a generalized time-energy uncertainty relation. These results highlight trade-offs between clock performance and energy uncertainty in physically realizable control schemes.
We consider an isolated autonomous quantum machine, where an explicit quantum clock is responsible for performing all transformations on an arbitrary quantum system (the engine), via a time-independent Hamiltonian. In a general context, we show that this model can exactly implement any energy-conserving unitary on the engine, without degrading the clock. Furthermore, we show that when the engine includes a quantum work storage device we can approximately perform completely general unitaries on the remainder of the engine. This framework can be used in quantum thermodynamics to carry out arbitrary transformations of a system, with accuracy and extracted work as close to optimal as desired, while obeying the first and second laws of thermodynamics. We thus show that autonomous thermal machines suffer no intrinsic thermodynamic cost compared to externally controlled ones.
We consider an isolated autonomous quantum machine, where an explicit quantum clock is responsible for performing all transformations on an arbitrary quantum system (the engine), via a time-independent Hamiltonian. In a general context, we show that this model can exactly implement any energy-conserving unitary on the engine, without degrading the clock. Furthermore, we show that when the engine includes a quantum work storage device we can approximately perform completely general unitaries on the remainder of the engine. This framework can be used in quantum thermodynamics to carry out arbitrary transformations of a system, with accuracy and extracted work as close to optimal as desired, while obeying the first and second laws of thermodynamics. We thus show that autonomous thermal machines suffer no intrinsic thermodynamic cost compared to externally controlled ones.
No physical clock is free: every clock is a quantum system that interacts with what it measures. This note studies the exact relationship between clock precision and the decoherence induced by universal time-dilation coupling in solvable constrained quantum systems. Using a clock-system interaction determined by energy coupling, the work derives the exact coherent-overlap decay law associated with time-dilation decoherence and shows how relational quantum dynamics emerges from constrained systems. The global time translation remains a gauge symmetry, while the time evolution recovered by conditioning on the clock exhibits the corresponding deformed relational dynamics. The main result is an exact resolution-decoherence tradeoff: the product of the clock's time resolution and the induced decoherence rate is independent of the clock state and depends only on the coupling strength and the system energy difference. This shows that improving the precision of a quantum clock necessarily increases the disturbance it causes when universally coupled to the system it measures. The result provides an exact model-level realization of a fundamental limitation of physical reference frames: a reference system that allows more precise access to information must pay an unavoidable quantum cost. The work is formulated within finite constrained quantum models with declared assumptions: idealized clocks, a specified universal dilation coupling, and no dynamical gravitational field. The results are accompanied by verification and reproducibility materials.
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.
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.
Precise and autonomous clocks are of fundamental interest and central importance to both foundational studies and practical applications. Here, we construct a blueprint for a quantum clock governed by time-independent interactions. By carefully-engineered coherent transport in dissipative spin chains, we achieve a scaling exponent at the precision-resolution trade-off fundamental bound, bringing this within reach of physically realistic and experimentally accessible systems. We further introduce a sudden-quench protocol that enables repeated operation through a simple initialization and detachment mechanism. Remarkably, the protocol is robust to imprecise detachment timing, implying that high-precision timekeeping can be achieved even when driven by a clock with much lower precision.