The Thermodynamic Cost of Measuring Time

The Thermodynamic Cost of Measuring Time VIEWPOINT The Thermodynamic Cost of Measuring Time A simple model of an autonomous quantum clock yields a quantitative connection between the clock's thermodynamic cost and its accuracy and resolution. by Anthony J. Short he ability to precisely measure time has had a dra- matic impact on society, from the marine chronome- ters that facilitated navigation in the 18th century, T to the satellite-borne atomic clocks that enable the Global Positioning System (GPS) today. But what fun- damental limitations does physics place on our ability to measure time? In a new paper, Paul Erker from the Au- tonomous University of Barcelona, Spain, and colleagues [1] argue that thermodynamics plays a key role in such limita- tions. Considering a simple model of a quantum clock, they establish a quantitative connection between two of its fea- Figure 1: A schematic of Erker and colleagues' autonomous quantum clock [1]. Heat ows from a hot bath into a cold bath tures—accuracy and resolution—and the “thermodynamic through a heat engine composed of two quantum systems, for cost” of running it in terms of its heat dissipation and en- example, two atoms that can be in an excited or ground state. tropy increase. Thermodynamics also lies at the heart of our Some of the engine's energy is extracted to raise a load system up perception that time flows inexorably forwards from the past a ``ladder'' of energy states. Whenever the load reaches the top of to the future. These results therefore link our ability to mea- the ladder, it rapidly decays to the bottom, emitting a photon into a sure time with the flow of time itself. detector that yields a ``tick'' of the clock. (APS/Alan Stonebraker) Quantum theory underpins almost all contemporary physics and has been used to design the most accurate clocks A model of a ticking autonomous clock similar to the one existing today. To investigate the fundamental limits of considered here was also presented in a previous paper [9], measuring time, Erker and colleagues therefore focused on but an advantage of Erker and colleagues’ model is that it quantum clocks. To ensure that all aspects were fully ac- explicitly includes the physical workings and power supply counted for, they considered an autonomous clock, which of the clock. operates without any external control or power supply, gen- In particular, the new clock is comprised of the smallest erating a sequence of “ticks” that are transmitted to the possible heat engine [10], consisting of just two quantum outside world. A number of other interesting models of au- systems, each with two energy levels. The engine is con- tonomous quantum clocks have been considered (see, for nected to two thermal baths, one hot and one cold. In example, Refs. [2–8]), the performance of which do not ap- addition, it is connected to a load system, which has a finite pear to be influenced by thermodynamics. However, these “ladder ” of possible states with equally spaced energies. As do not tick in the same sense as Erker and co-workers’ the clock evolves, heat flows from the hot bath to the cold, model, and they generally involve either unphysical systems raising the load up the ladder. When the load reaches the that have no minimum energy or initial states whose prepa- top of the ladder, it rapidly decays to the bottom, emitting a ration would arguably require precise timings provided by photon into a detector that yields a tick of the clock, and the an additional clock. A key advantage of Erker and co- process begins again (Fig. 1). The performance of the clock workers’ model is that the initial state does not need to be is described in terms of its resolution (the frequency of suc- fine-tuned nor does it require any precise timings to prepare. cessive ticks) and its accuracy (the number of ticks before the timing of the clock is uncertain by approximately one tick). H. H. Wills Physics Laboratory, University of Bristol, Tyndall Av- Interestingly, the team discovered that the achievable res- enue, Bristol BS8 1TL, United Kingdom olution and accuracy of the clock depend on the amount of physics.aps.org 2017 American Physical Society 02 August 2017 Physics 10, 88 heat dissipated into the cold bath, with more dissipated heat state or their longevity (the time after which their accuracy leading to improved performance. Furthermore, by altering and resolution notably decrease). Note that in the latter case, the parameters of the clock, such as the number of steps in thermodynamics also appears to play a crucial role, as an the ladder, this improvement in performance can be applied autonomous clock cannot continue to operate once it has to either the accuracy or the resolution, with a nonlinear reached a timeless equilibrium state. trade-off between the two quantities. The heat dissipated in the cold bath is closely related to the entropy increase of This research is published in Physical Review X. the clock, and the authors show that in a particular limit (in which the ladder has many steps and is weakly coupled to the engine), the accuracy of the clock is equal to half of its REFERENCES entropy increase—a surprisingly simple connection between [1] P. Erker, M. T. Mitchison, R. Siva, M. P. Woods, N. Brunner, two very different quantities. The increasing entropy of the and M. Huber, ``Autonomous Quantum Clocks: Does Thermo- Universe, as it evolves towards more disordered states, is dynamics Limit our Ability to Measure Time?'' Phys. Rev. X 7, believed to give rise to the asymmetry we perceive in time. 031022 (2017). For example, an object dropped on the floor can dissipate [2] A. Peres, ``Measurement of Time by Quantum Clocks,'' Am. J. its energy as heat, but we do not see the reverse, in which Phys. 48, 552 (1980). an object is propelled off the floor by absorbing heat. These [3] D. N. Page and W. K. Wootters, ``Evolution without Evolution: results therefore provide a quantitative connection between Dynamics Described by Stationary Observables,'' Phys. Rev. the irreversible flow of time suggested by thermodynamics D. 27, 2885 (1983). and our ability to measure it. [4] Y. Aharonov and T. Kaufherr, ``Quantum Frames of Reference,'' The team argues that a similar connection will apply to Phys. Rev. D 30, 368 (1984). [5] V. Buºek, R. Derka, and S. Massar, ``Optimal Quantum other autonomous quantum clocks and that an increase in Clocks,'' Phys. Rev. Lett. 82, 2207 (1999). entropy is a necessary component of such models. A key [6] J. Lindkvist, C. Sabín, G. Johansson, and I. Fuentes, ``Motion argument is that an autonomous quantum clock operates and Gravity Effects in the Precision of Quantum Clocks,'' Sci. as a machine with finite power, and this is believed to be Rep. 5, 10070 (2015). impossible without an increase in entropy—essentially any [7] A. S. L. Malabarba, A. J. Short, and P. Kammerlander, ``Clock- perfectly efficient clock would tick infinitely slowly. Al- Driven Quantum Thermal Engines,'' New J. Phys. 17, 045027 though this is a strong argument, it would be interesting to (2015). explore other models of thermodynamical clocks and to un- [8] M. Woods, R. Silva, and J. Oppenheim, ``Autonomous Quan- derstand the general case in more detail, both conceptually tum Machines and Finite Sized Clocks,'' arXiv:1607.04591. and quantitatively. [9] S. Rankovic, Y.-C. Liang, and R. Renner, ``Quantum Clocks and Their SynchronisationThe Alternate Ticks Game,'' It would also be good to determine the relationship arXiv:1506.01373. between thermodynamical quantum clocks and the more [10] N. Brunner, N. Linden, S. Popescu, and P. Skrzypczyk, ``Virtual mechanical quantum clocks considered previously [2–8] in Qubits, Virtual Temperatures, and the Foundations of Thermo- which entropy does not appear to play a central role. Fi- dynamics,'' Phys. Rev. E 85, 051117 (2012). nally, one could study additional properties of autonomous quantum clocks, such as their sensitivity to the precise initial 10.1103/Physics.10.88 physics.aps.org 2017 American Physical Society 02 August 2017 Physics 10, 88 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physics American Physical Society (APS)

The Thermodynamic Cost of Measuring Time

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Abstract

VIEWPOINT The Thermodynamic Cost of Measuring Time A simple model of an autonomous quantum clock yields a quantitative connection between the clock's thermodynamic cost and its accuracy and resolution. by Anthony J. Short he ability to precisely measure time has had a dra- matic impact on society, from the marine chronome- ters that facilitated navigation in the 18th century, T to the satellite-borne atomic clocks that enable the Global Positioning System (GPS) today. But what fun- damental limitations does physics place on our ability to measure time? In a new paper, Paul Erker from the Au- tonomous University of Barcelona, Spain, and colleagues [1] argue that thermodynamics plays a key role in such limita- tions. Considering a simple model of a quantum clock, they establish a quantitative connection between two of its fea- Figure 1: A schematic of Erker and colleagues' autonomous quantum clock [1]. Heat ows from a hot bath into a cold bath tures—accuracy and resolution—and the “thermodynamic through a heat engine composed of two quantum systems, for cost” of running it in terms of its heat dissipation and en- example, two atoms that can be in an excited or ground state. tropy increase. Thermodynamics also lies at the heart of our Some of the engine's energy is extracted to raise a load system up perception that time flows inexorably forwards from the past a ``ladder'' of energy states. Whenever the load reaches the top of to the future. These results therefore link our ability to mea- the ladder, it rapidly decays to the bottom, emitting a photon into a sure time with the flow of time itself. detector that yields a ``tick'' of the clock. (APS/Alan Stonebraker) Quantum theory underpins almost all contemporary physics and has been used to design the most accurate clocks A model of a ticking autonomous clock similar to the one existing today. To investigate the fundamental limits of considered here was also presented in a previous paper [9], measuring time, Erker and colleagues therefore focused on but an advantage of Erker and colleagues’ model is that it quantum clocks. To ensure that all aspects were fully ac- explicitly includes the physical workings and power supply counted for, they considered an autonomous clock, which of the clock. operates without any external control or power supply, gen- In particular, the new clock is comprised of the smallest erating a sequence of “ticks” that are transmitted to the possible heat engine [10], consisting of just two quantum outside world. A number of other interesting models of au- systems, each with two energy levels. The engine is con- tonomous quantum clocks have been considered (see, for nected to two thermal baths, one hot and one cold. In example, Refs. [2–8]), the performance of which do not ap- addition, it is connected to a load system, which has a finite pear to be influenced by thermodynamics. However, these “ladder ” of possible states with equally spaced energies. As do not tick in the same sense as Erker and co-workers’ the clock evolves, heat flows from the hot bath to the cold, model, and they generally involve either unphysical systems raising the load up the ladder. When the load reaches the that have no minimum energy or initial states whose prepa- top of the ladder, it rapidly decays to the bottom, emitting a ration would arguably require precise timings provided by photon into a detector that yields a tick of the clock, and the an additional clock. A key advantage of Erker and co- process begins again (Fig. 1). The performance of the clock workers’ model is that the initial state does not need to be is described in terms of its resolution (the frequency of suc- fine-tuned nor does it require any precise timings to prepare. cessive ticks) and its accuracy (the number of ticks before the timing of the clock is uncertain by approximately one tick). H. H. Wills Physics Laboratory, University of Bristol, Tyndall Av- Interestingly, the team discovered that the achievable res- enue, Bristol BS8 1TL, United Kingdom olution and accuracy of the clock depend on the amount of physics.aps.org 2017 American Physical Society 02 August 2017 Physics 10, 88 heat dissipated into the cold bath, with more dissipated heat state or their longevity (the time after which their accuracy leading to improved performance. Furthermore, by altering and resolution notably decrease). Note that in the latter case, the parameters of the clock, such as the number of steps in thermodynamics also appears to play a crucial role, as an the ladder, this improvement in performance can be applied autonomous clock cannot continue to operate once it has to either the accuracy or the resolution, with a nonlinear reached a timeless equilibrium state. trade-off between the two quantities. The heat dissipated in the cold bath is closely related to the entropy increase of This research is published in Physical Review X. the clock, and the authors show that in a particular limit (in which the ladder has many steps and is weakly coupled to the engine), the accuracy of the clock is equal to half of its REFERENCES entropy increase—a surprisingly simple connection between [1] P. Erker, M. T. Mitchison, R. Siva, M. P. Woods, N. Brunner, two very different quantities. The increasing entropy of the and M. Huber, ``Autonomous Quantum Clocks: Does Thermo- Universe, as it evolves towards more disordered states, is dynamics Limit our Ability to Measure Time?'' Phys. Rev. X 7, believed to give rise to the asymmetry we perceive in time. 031022 (2017). For example, an object dropped on the floor can dissipate [2] A. Peres, ``Measurement of Time by Quantum Clocks,'' Am. J. its energy as heat, but we do not see the reverse, in which Phys. 48, 552 (1980). an object is propelled off the floor by absorbing heat. These [3] D. N. Page and W. K. Wootters, ``Evolution without Evolution: results therefore provide a quantitative connection between Dynamics Described by Stationary Observables,'' Phys. Rev. the irreversible flow of time suggested by thermodynamics D. 27, 2885 (1983). and our ability to measure it. [4] Y. Aharonov and T. Kaufherr, ``Quantum Frames of Reference,'' The team argues that a similar connection will apply to Phys. Rev. D 30, 368 (1984). [5] V. Buºek, R. Derka, and S. Massar, ``Optimal Quantum other autonomous quantum clocks and that an increase in Clocks,'' Phys. Rev. Lett. 82, 2207 (1999). entropy is a necessary component of such models. A key [6] J. Lindkvist, C. Sabín, G. Johansson, and I. Fuentes, ``Motion argument is that an autonomous quantum clock operates and Gravity Effects in the Precision of Quantum Clocks,'' Sci. as a machine with finite power, and this is believed to be Rep. 5, 10070 (2015). impossible without an increase in entropy—essentially any [7] A. S. L. Malabarba, A. J. Short, and P. Kammerlander, ``Clock- perfectly efficient clock would tick infinitely slowly. Al- Driven Quantum Thermal Engines,'' New J. Phys. 17, 045027 though this is a strong argument, it would be interesting to (2015). explore other models of thermodynamical clocks and to un- [8] M. Woods, R. Silva, and J. Oppenheim, ``Autonomous Quan- derstand the general case in more detail, both conceptually tum Machines and Finite Sized Clocks,'' arXiv:1607.04591. and quantitatively. [9] S. Rankovic, Y.-C. Liang, and R. Renner, ``Quantum Clocks and Their SynchronisationThe Alternate Ticks Game,'' It would also be good to determine the relationship arXiv:1506.01373. between thermodynamical quantum clocks and the more [10] N. Brunner, N. Linden, S. Popescu, and P. Skrzypczyk, ``Virtual mechanical quantum clocks considered previously [2–8] in Qubits, Virtual Temperatures, and the Foundations of Thermo- which entropy does not appear to play a central role. Fi- dynamics,'' Phys. Rev. E 85, 051117 (2012). nally, one could study additional properties of autonomous quantum clocks, such as their sensitivity to the precise initial 10.1103/Physics.10.88 physics.aps.org 2017 American Physical Society 02 August 2017 Physics 10, 88

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PhysicsAmerican Physical Society (APS)

Published: Aug 2, 2017

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