Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?

Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time? Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031022 (2017) 1,2 3,4 5 6,7 5 8 Paul Erker, Mark T. Mitchison, Ralph Silva, Mischa P. Woods, Nicolas Brunner, and Marcus Huber Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Faculty of Informatics, Università della Svizzera italiana, Via G. Buffi 13, 6900 Lugano, Switzerland Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland University College London, Department of Physics & Astronomy, London WC1E 6BT, United Kingdom QuTech, Delft University of Technology, Lorentzweg 1, 2611 CJ Delft, Netherlands Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, A-1090 Vienna, Austria (Received 7 November 2016; revised manuscript received 7 May 2017; published 2 August 2017) 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. DOI: 10.1103/PhysRevX.7.031022 Subject Areas: Quantum Physics, Quantum Information, Statistical Physics I. INTRODUCTION and finally measured. The result is interpreted as a time- interval measurement, whose precision can be related to Although quantum systems provide the most accurate the properties of the clock (e.g., its dimension [16]). measurements of time [1–3], the concept of time in However, the procedures of the state preparation and the quantum theory remains elusive. This issue has been measurement are usually not discussed explicitly. These explored in several directions. The relation between time models thus allow one to measure a time interval, e.g., for and energy, the physical quantity that is time invariant in implementing a given unitary operation (by timing an closed systems, has led to fundamental limitations in interaction). This functionality is analogous to a stop- the form of quantum speed limits [4–8]. Another approach watch, but it cannot be considered a complete model of a has aimed to promote time from a mere classical quantum clock. parameter to a fully quantum description [9–13]. Indeed, a crucial feature of a clock (as opposed to a Notably, quantum evolution is captured here via the stopwatch) is to continuously provide a time reference to an notion of correlations. Finally, various models of quantum external observer. It is thus essential that any complete systems designed to measure time, i.e., quantum clocks, model of a quantum clock explicitly specifies the process of have been proposed; see, for example, Refs. [14–17]. information read-out. This leads us to consider a clock as a These models typically consider a specific degree of bipartite system [18,19], shown in Fig. 1(a). The first part freedom of a quantum system, prepared in a judiciously of the clock is the pointer, i.e., a subsystem whose internal chosen initial state, then subjected to a unitary evolution, dynamics are effectively dictated by the passage of time. The second part is the register, which stores classical information obtained about the evolution of the pointer, Published by the American Physical Society under the terms of thereby mediating the transfer of information from the the Creative Commons Attribution 4.0 International license. system to an external observer. The pointer is designed to Further distribution of this work must maintain attribution to produce a sequence of signals, which are then recorded by the author(s) and the published article’s title, journal citation, and DOI. the register as ticks. 2160-3308=17=7(3)=031022(12) 031022-1 Published by the American Physical Society PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) (a) (c) (b) FIG. 1. (a) A pointer system generates a time-ordered sequence of events that are recorded and displayed by the register. (b) We consider a pointer comprising a two-qubit heat engine that drives a thermally isolated load up a ladder, whose highest-energy state undergoes radiative decay back to the ground state. Photons are thus repeatedly emitted and registered by a photodetector as ticks of the clock. (c) A virtual qubit is a pair of states in the engine’s two-qubit Hilbert space whose energy splitting is resonant with the ladder. The thermal baths drive population into the virtual qubit’s higher-energy state and out of its lower-energy one, creating a population inversion described by a negative virtual temperature. Hence, placing the virtual qubit in thermal contact with the ladder forces the load upwards, thereby performing work. is characterized by (i) its resolution, i.e., how frequently the It follows that there is an asymmetric flow of information clock ticks, and (ii) its accuracy, i.e., how many ticks the between the two parts of the clock, which makes the clock provides before its uncertainty becomes greater than process irreversible (and singles out a direction for the flow the average time between ticks. We find that a given of time). This naturally connects the problem to the second resolution and accuracy can be simultaneously achieved law of thermodynamics [20] because irreversibility is only if the rate of entropy production is sufficiently large; associated with the generation of entropy. One therefore otherwise, a trade-off exists whereby the desired accuracy expects that the suitability of a system for measuring time can only be attained by sacrificing some resolution, or vice implies a corresponding propensity to produce entropy. versa. Furthermore, in the regime where the resolution is However, a precise relationship between entropy produc- arbitrarily low, the accuracy is still bounded by the entropy tion and clock performance has not yet been demonstrated. production, suggesting a quantitative connection between In fact, we show that such a relationship unavoidably entropy production and the clock’s arrow of time. Note that becomes apparent when considering a more general ques- here the relevant entropy production is not associated with tion: What are the minimal resources required to maintain a quantum clock? In order to answer this question, we measurements or erasure of the register but rather with the evolution of the pointer system itself. In the following, we consider an autonomous quantum clock, i.e., a self- illustrate this behavior by explicitly calculating the dynam- contained device working without any external control ics of a simple clock model. We then present a conjecture, or timing. The clock must be an isolated system evolving backed up by general thermodynamic arguments, that such according to a time-independent Hamiltonian [19]. trade-offs are exhibited by any implementation of an Moreover, the resources powering the clock should not autonomous clock. themselves require another clock to be prepared. Specifically, we discuss a natural class of autonomous clocks driven by minimal nonequilibrium resources, II. AUTONOMOUS QUANTUM CLOCKS namely, the flow of heat between two thermal reservoirs. Our objective is to find the fundamental limits on In particular, our model makes explicit the physical quantum clocks. To that end, we consider autonomous mechanism of the clock’s operation, including its initial- ization and power supply. We make use of thermodynam- clocks, i.e., those which are complete and self-contained. In ical concepts in order to analyze the clock as an particular, the operation of the device should not require autonomous thermal machine [21–23], with the goal of any time-dependent control that would necessitate another producing a series of regular ticks. external clock. This allows all resources needed for time- This approach allows us to show that the clock’s keeping to be carefully accounted for. In this section, we irreversible entropy production dictates fundamental discuss some of the general features of autonomous clocks, limits on its performance. The performance of the clock before specifying a particular model in Sec. III. 031022-2 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) An autonomous clock evolves under a time-independent energy gap E . The second qubit is connected to a cold bath Hamiltonian, such that a steady stream of ticks is recorded at temperature T <T and has energy gap E <E . The c h c h at the register, as depicted in Fig. 1. The process by which engine delivers work to a load, represented by a system information is transferred from pointer to register should be with d equally spaced energy levels, i.e., a discrete ladder, effectively irreversible, in order to ensure the unidirectional with energy spacing E ¼ E − E . w h c flow of time as recorded by the register. In addition, this The temperature difference between the two baths process should occur spontaneously, i.e., without any induces a heat current in the system from the hot qubit external intervention or time-dependent coupling between to the cold one. This flow of heat delivers energy to the the pointer and register. To ensure that the probability of load, causing it to “climb” the ladder. The action of the this spontaneous process is larger than that of its time machine can be understood in terms of the resonant reverse, the free energy of the pointer must decrease. exchange of energy between the load and a virtual qubit Therefore, in order to continue producing ticks, the clock [23]. This virtual qubit is a special pair of states in the needs a source of free energy driving it out of equilibrium. engine’s Hilbert space that are coupled to the ladder, In principle, any nonequilibrium quantum system could illustrated in Fig. 1(c). Assuming that the engine-ladder provide the free energy needed to power a clock. However, coupling is weak, the populations of the virtual qubit states a large class of nonequilibrium states is difficult to prepare are thermally distributed at the virtual temperature in practice unless a clock is already available, e.g., so that a E − E resonant driving field can be applied for a known period of h c k T ¼ ; ð1Þ B v time. We exclude such resource states in order to ensure fair β E − β E h h c c bookkeeping, i.e., the resources’ initial preparation should where β ¼ 1=k T . In other words, the virtual qubit’s not itself require time measurements. It is also clearly c;h B c;h −β E v w states are occupied in the ratio p =p ¼ e , where p desirable—yet inessential—that such resources be natu- 1 0 1 (p ) denotes the population of the state with higher (lower) rally abundant or otherwise easy to generate. energy and β ¼ 1=k T . Therefore, whenever the virtual Here we argue that the minimal nonequilibrium resource v B v qubit has a negative temperature, i.e., a population inver- consists of two thermal reservoirs at different temperatures. sion, the load moves up the ladder as it “thermalizes” with Indeed, the presence of one heat bath is unavoidable since the virtual qubit. The virtual temperature is conveniently this represents the environment at ambient temperature T . parametrized by the virtual qubit’s population bias Furthermore, a second reservoir at temperature T >T h c can be prepared deterministically without detailed under- p − p 0 1 standing of the bath’s internal structure and without any Z ¼ ¼ tanhðβ E =2Þ; ð2Þ v v w p þ p 0 1 well-timed operations. This is because the thermal state represents a condition of minimal knowledge [24] towards which plays a central role in characterizing the performance which generic quantum systems (i.e., those not integrable of our clock, as we show below. nor many-body localized) equilibrate [25]. In this sense, the To complete the description of our clock, we must minimal out-of-equilibrium resource is an equilibrated specify how the pointer interacts with the register. The (thermalized) resource with a higher average energy con- top level of the ladder is assumed to be unstable, and it tent than the environment. Any other potential resource for decays to the ground state by emitting a photon at energy the clock would feature lower entropy at equal energies and E ¼ðd − 1ÞE . This photon is then detected at the γ w thus additional knowledge or control to prepare. In the register, which in turn makes the clock tick. Note that following, we base our quantitative analysis on clocks the presence of the decay channel also allows, in principle, driven by thermal baths. However, we emphasize that the for the reverse process. However, we assume that the notion of an autonomous clock is more general and could background temperature satisfies k T ≪ E so that such B c γ be extended to various different scenarios and resource processes are negligible. states. In summary, the flow of heat through the engine drives the load up the ladder, which eventually reaches the top III. MINIMAL THERMAL CLOCK MODEL level and decays back to the ground state while emitting a We now specialize to a concrete model of an autonomous photon. The process is repeated, thus generating a steady quantum clock where the pointer is driven by the heat flow stream of photons that are recorded by the register as ticks between two thermal baths. For simplicity, we base our of the clock. Importantly, the evolution of the ladder’s model on the smallest quantum heat engine that was energy is probabilistic, leading to a stochastic sequence of introduced in Ref. [23] (see Appendix A for a detailed ticks. The distribution of ticks depends, in particular, on the description). dimension of the ladder d and the bias Z . Intuitively, if the The machine consists of two qubits, each coupled to an bias is small (Z negative but close to zero), the probability for the load to move up is only marginally larger than its independent thermal bath, as depicted in Fig. 1(b). The first qubit, connected to the hot bath at temperature T , has probability of going down. The probability distribution 031022-3 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) over the levels of the ladder thus rapidly becomes quite For our model of the autonomous clock, we assume that broad, which makes the clock tick slowly and at irregular after each spontaneous emission event, the entire pointer is time intervals. On the other hand, if Z → −1, i.e., the reset to its initial state—specifically, a product state with virtual qubit has essentially complete population inversion, the ladder in its ground state and the engine qubits in then the probability for the ladder population to move equilibrium with their respective baths. This approximation downward is negligible, resulting in shorter and more is valid in the weak-coupling limit, where the engine qubits regular time intervals between ticks. are minimally perturbed by their interaction with the ladder. The ticks of the clock can therefore be described as a renewal process; i.e., the time between any pair of IV. PERFORMANCE OF THE CLOCK consecutive ticks is statistically independent from, and In order for the clock to deliver ticks, the engine must identically distributed to, the time between any other pair of raise the ladder’s energy and necessarily dissipate energy consecutive ticks. into the cold bath. Our goal now is to relate the performance Now, let the distribution of waiting times between two of the clock to this dissipated energy, which is closely consecutive ticks be characterized by the mean t and the tick related to the entropy production. Specifically, we consider standard deviation Δt . The resolution of the clock is then tick here the heat dissipated into the cold bath per tick of the clock, ν ¼ 1=t ; ð4Þ tick tick Q ¼ðd − 1ÞE : ð3Þ i.e., the average number of ticks the clock provides per c c second. The accuracy is the number of ticks N such that the Note that this quantity, rather than the heat supplied to the uncertainty (standard deviation) of the Nth tick time is machine per tick [Q ¼ðd − 1ÞE ], represents the funda- equal to the average time between ticks. Since the waiting h h mental minimum energy expenditure associated with one times are independent, the uncertainty in the time of the nth pffiffiffi tick of the clock. This is because, in principle, a large part tick is simply nΔt , and therefore tick of the energy E carried away by the emitted photon could be captured and recycled (e.g., dumped back into the hot tick N ¼ : ð5Þ bath). Consequently, the dissipated heat (3) is associated Δt tick with an irreversible entropy production of at least β Q c c per tick. Figure 2 illustrates the intimate relationship between the The performance of our autonomous clock is quantified accuracy N and the resolution ν versus the dissipated tick by the resolution and accuracy of its ticks. By resolution, energy Q , calculated by numerical solution of the equa- we refer to the average number of ticks the clock provides tions of motion (see Appendix B). We find that, for a given per unit time. The ticks are not distributed regularly, and we amount of dissipated energy, there is a trade-off between characterize the accuracy by the number of ticks provided accuracy and resolution. In other words, engineering a before the next tick is uncertain by the average time interval good clock featuring both high accuracy and high reso- between ticks [26]. lution requires a large amount of energy to be dissipated (a) (b) (c) FIG. 2. Illustration of the fundamental trade-off between the dissipated heat and the achievable accuracy and resolution. (a) Accuracy N as a function of dissipated heat per tick Q , for various values of the resolution ν . At low energy, the accuracy increases linearly c tick with the dissipated energy, independently of the resolution. However, for higher energies, the accuracy saturates. (b) Resolution ν as a tick function of dissipated heat per tick Q , for various values of the accuracy N. The resolution first increases with dissipated energy but then quickly saturates to a maximal value. (c) Trade-off between accuracy and resolution when the energy dissipation rate is fixed. The data are computed for fixed values of k T ¼ E , k T ¼ 1000E and g ¼ ℏγ ¼ ℏΓ ¼ 0.05E , while the ladder dimension d and cold qubit B c w B h w w energy E are varied independently. Note that d ≥ 10 for all of the plotted points; thus, k T ¼ E ≪ E ¼ðd − 1ÞE , and we can c B c w γ w safely ignore the absorption of a photon (i.e., the reverse of the decay process). 031022-4 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) the simplifying assumptions that the clock ticks as soon as the load reaches the top of the ladder and that d is large enough for reflections from the boundaries of the ladder to be negligible. Under the foregoing approximations, the resolution is given by p − p ↑ ↓ ν ¼ : ð6Þ tick Quite intuitively, the resolution is inversely proportional to the dimension d, corresponding to the “height” of the ladder, but it is proportional to the difference of transition FIG. 3. Accuracy N versus dissipated energy Q for various rates p − p , which quantify the “speed” at which the load ↑ ↓ values of the dimension d of the ladder, according to the climbs. approximation (8) with the same bath temperatures as in Fig. 2. On the other hand, as demonstrated in Appendix C, the accuracy is given by and thus a higher production of entropy per tick. This is nicely illustrated in Fig. 2(c), which showcases the nature N ¼ djZ j; ð7Þ of entropy production as a resource. The curves for different entropies are clearly ordered; i.e., more entropy which is entirely independent from the clock’s overall implies that either more resolution or more accuracy can be dynamical time scale, set by the rates p . Instead, the ↑;↓ achieved. It is interesting to note, however, that the accuracy depends only on the dimensionless quantities Z relationship between the two is nontrivial and the trade- and d. In turn, the bias Z encapsulates the dependence of off features nonlinear dependencies. the clock’s accuracy on the dissipated heat. In the case Finally, we note that in the regime of low-energy of our model, using Eqs. (B5) and (B7), the accuracy is dissipation, the relationship between accuracy and entropy given by production at fixed resolution is directly proportional, as ðβ − β ÞQ − β E seen in Fig. 2(a). In the next section, we recover this c h c h γ N ¼ d tanh : ð8Þ behavior analytically in the weak-coupling regime. 2d Note, however, that the relation between Z and the heat V. ACCURACY IN THE WEAK-COUPLING LIMIT exchanged with the two baths is more general than the We now investigate the relationship between accuracy model considered here [27] (see Appendix E for a dis- and dissipated power by an alternate approximate analysis, cussion). It follows that the accuracy in the weak-coupling valid when the interaction between the engine and the limit depends on the amount of dissipated heat but not on ladder is weak. In this regime, the accuracy is limited by the dissipation rates. the dissipated power and the dimension of the ladder, while The behavior described by Eq. (8) is illustrated in Fig. 3, the resolution is not focused upon. This is in contrast to where we plot the accuracy versus the dissipated energy for Fig. 2(a), where the resolution is fixed, and the dimension is fixed dimension. We observe that the accuracy first allowed to vary. In particular, we show that the accuracy is increases linearly but eventually saturates to its maximum essentially independent of the details of the clock’s dynam- value N ¼ d. Indeed, increasing Q leads to a stronger bias ics, being determined only by the bias of the virtual qubit in the virtual qubit, saturating at jZ j → 1 as Q → ∞. v c Z and the ladder dimension d. Thus, the accuracy is limited by both the dimension d and Focusing on the ladder, its evolution can be approxi- the dissipated energy Q . Hence, achieving a certain mated by a biased random walk, induced by the interaction accuracy requires a minimum dimension as well as a with the virtual qubit. This is easily understood by the fact minimum dissipated energy per tick. that the resonant interaction with the virtual qubit cannot Even if the dimension is unbounded, we find that the induce any coherence on the ladder. Moreover, the reso- dissipated energy still imposes a fundamental limitation. nance is exactly at the energy of a transition of one step up Taking the limit d → ∞, the accuracy is linearly dependent or down, and independent of the ladder’s position. The rates on the dissipated heat: at which the ladder population moves upwards (p )or −β E v w downwards (p ) satisfy p =p ¼ e as a consequence ↓ ↑ ↓ ðβ − β ÞQ − β E c h c h γ N → : ð9Þ of detailed balance. This description of the clock is derived in Appendix C as a perturbative approximation to the two-qubit engine, which becomes exact in the limit of Noting that Q ¼ Q þ E , we can recast the above in h c γ vanishingly small engine-ladder coupling. We also make the illustrative form 031022-5 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) β Q − β Q ΔS regime where the machine works reversibly. A finite power, c c h h tick N → ¼ ; ð10Þ however, is essential for the resolution of any autonomous 2 2 clock: A clock working at Carnot efficiency ticks infinitely where ΔS is the increase in the entropy of the clock in a tick slowly. Hence, even in the rather artificial regimes of single tick. We may interpret the regularity of each tick as T → 0 or T → ∞, the requirement of a finite resolution c h representative of the strength of the arrow of time. Thus, implies a minimal dissipated heat and thus a minimal Eq. (10) quantifies, in a concrete manner, the connection entropy production. between the arrow of time of a clock and its irreversibility. It is also possible to consider more general nonequili- brium resources to power the clock. In order to satisfy the VI. FUNDAMENTAL LIMITS OF GENERAL requirement of autonomy, such resources should not AUTONOMOUS CLOCKS themselves need any well-timed control in order to be produced. In principle, it is conceivable that such a resource The simple thermal clock model we discuss above could allow the clock to achieve higher efficiency than is illustrates the fact that our ability to accurately and possible with thermal driving. However, an autonomous precisely measure time necessarily generates an increase clock that does not generate any entropy but nonetheless of entropy (via heat dissipation). Equivalently, this implies has finite resolution would constitute an autonomous an intrinsic work cost for measuring time. It is natural to ask machine operating at finite power with unit efficiency. whether the connection between clock performance and Therefore, if the performance of autonomous quantum entropy production is a specific aspect of our model or, on clocks is not always associated with a fundamental entropy the contrary, a universal feature of any procedure for production, then the prospect of quantum machines is far measuring time. Below, we argue in favor of the latter: more revolutionary than is widely believed at present. Any autonomous clock must increase entropy. Finally, it is also worth pointing out that, while we focus The core insight underlying our argument is that, as here on a specific source for the entropy production of the discussed in Sec. II, the ticks of any autonomous clock clock (namely, the heat dissipated by the thermal machine involve a spontaneous and effectively irreversible transition driving the clock), there will generally be additional energy in a pointer system, thus inducing a corresponding change costs required for operating the clock. In particular, the in the register to which it is coupled. In order to bias the preparation (and reset) of the initial state of the register forward transition in favor of its time reverse (i.e., to avoid will generate entropy due to Landauer’s erasure princi- the clock ticking “backwards”), the transition must reduce ple [28,29]. the free energy of the pointer. Hence, for the clock to run Even if the qualitative bound (10) derived in our work continuously, it needs access to a system out of thermal represents a fundamental limit for any clock, it still equilibrium that can replenish the free energy of the pointer. underestimates the necessary costs of running the best Now, the essential question is whether it is possible for the clocks available today. For instance, a typical atomic clock clock to convert this free energy into ticks with perfect [30] runs at resolutions of the order of 10 Hz, and an efficiency, i.e., without increasing entropy. accuracy of 10 seconds before being off by a second. Let us first discuss this question in the context of clocks Equation (10) would imply a minimal power consumption driven by thermal baths. It is clear that beyond the specific for such a clock of the order of about 50 μW. In practice, model we have studied, one could consider more general the real costs are orders of magnitude higher. This is similar designs for the thermal machine. The basic necessary to the case of information erasure: Even though Landauer’s ingredient is simply the ability to move the population principle is the only known fundamental limit, current of the pointer out of equilibrium so that an unstable level erasure techniques operate far less efficiently. generates a tick. This transition is biased in the forward direction so long as the unstable level is much higher in VII. CONCLUSION AND OUTLOOK energy than the thermal background. Such a mechanism can indeed work for a variety of physical implementations Our work represents a first step towards rigorously of the pointer (i.e., with a more complex level structure). characterizing the necessary resources and limitations of The ladder could comprise multiple levels which trigger a the process of timekeeping. In a nutshell, we introduced the decay, while the machine could feature more than two concept of autonomous quantum clocks to discuss these qubits. questions, and we argued that the measurement of time Nonetheless, all these possible extensions and more inevitably leads to an increase in entropy. Moreover, we sophisticated designs will still have to comply with the explicitly discussed a simple model of an autonomous basic laws of thermodynamics. In particular, the efficiency quantum clock and found that the amount of entropy of the conversion of energy to a tick is fundamentally produced represents an actual resource for measuring time. bounded by the Carnot efficiency η ¼ 1 − T =T . Every unit of heat dissipated can be spent to increase either C c h Moreover, this maximal efficiency can only be achieved the accuracy or the resolution of the clock. Additionally, the in a limit where the power vanishes, corresponding to the dimension of a key constituent of the clock (the ladder) 031022-6 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) National Science Foundation (SNF) through the project imposes a limit on the achievable accuracy and resolution, independently of the amount of dissipated heat. In other “Information and Physics”, and the National Centres of words, in analogy to the findings of Refs. [16,19], the Competence in Research Quantum Science and Hilbert space dimension imposes a fundamental constraint Technology (QSIT). on the performance of the clock. Reaching this optimal P. E., M. T. M., and R. S. contributed equally to this regime requires a minimal rate of entropy production. This work. provides a quantitative basis for the intuitive connection between the second law of thermodynamics and the arrow of time (see, for example, Refs. [31,32]). In order to APPENDIX A: DESCRIPTION OF THE measure how much time has passed, we inevitably need TWO-QUBIT HEAT ENGINE to increase the entropy of the Universe from the perspective Here, we give a detailed description of the two-qubit heat of the register. engine of Ref. [23], which represents the pointer of the Here, these considerations only concern the scenario of autonomous quantum clock. The machine consists of two minimal autonomous clocks, i.e., where the resources qubits, each one connected to a thermal bath. The first qubit exploited to operate the clock are simply two thermal with energy gap E is connected to the bath at T . The h h baths at different temperatures. While these arguably second qubit is connected to the bath at T and has energy represent the most abundant resources found in nature gap E <E . The engine is connected to a d-dimensional c h [25], it would be interesting to consider other quantum ladder, featuring equally spaced energy levels (with spacing systems, e.g., with multiple conserved quantities [33–36]. E ), which is not connected to any heat bath. The free More broadly, the relevant question is to what extent our Hamiltonian of the total system (two qubits and ladder) is choice of free resources impacts our ability to measure thus given by time. For instance, one could consider more general passive states [37], which would commute with the system d−1 X X Hamiltonian and thus satisfy the requirement of autonomy. H ¼ E j1i h1jþ kE jki hkj; ðA1Þ 0 j w j w Thermal clock models can furthermore be used to work out j¼h;c k¼0 the thermodynamic cost of controlling other quantum systems [16,38,39] in an autonomous fashion, i.e., imple- where j1i denotes the excited state of qubit j ¼ h, c, and menting locally apparent time-dependent Hamiltonians by jki denotes the state of the kth level of the ladder. As a coupling to an autonomous thermal clock. Moreover, design constraint, we take operating two clocks in parallel could lead to a drastic enhancement of the clock’s performance. While classical E ¼ E þ E : ðA2Þ h c w clocks running in parallel would not offer any fundamental improvement, one could consider quantum resources that Hence, the following energy levels of the total system are feature coherence or entanglement [40,41]. Could these degenerate in energy: j0i j1i jki and j1i j0i jk þ 1i . c h w c h w genuine quantum phenomena be used to increase our This allows for energy to be exchanged between the qubits ability to measure time? We look forward to future research and the ladder. Specifically, we consider the interaction in this direction. Hamiltonian d−1 ACKNOWLEDGMENTS H ¼ g ðj1i j0i jk þ 1i h0j h1j hkj þ H:c:Þ: ðA3Þ int c h w c h w We are grateful to Ämin Baumeler, Nicolas Gisin, k¼0 Patrick Hofer, Daniel Patel, Sandu Popescu, Gilles Pütz, The machine will be operated in the weak-coupling regime, Sandra Rankovic Stupar, Renato Renner, Christian Klumpp, and Stefan Wolf for fruitful discussions. M. H. i.e., g ≪ E , E . Note that our design constraint on the c w acknowledges funding from the Swiss National Science energies (A2) ensures that H has a significant effect even int Foundation (AMBIZIONE PZ00P2_161351) and the in the weak-coupling regime. Henceforth, we refer to the Austrian Science Fund (FWF) through the START joint system of ladder and engine as the pointer since it will Project No. Y879-N27. M. W. and M. T. M. acknowledge be the system from which the register will derive informa- funding from the UK research council EPSRC. R. S. and tion reflecting the passage of time. N. B. acknowledge the Swiss National Science Foundation The functioning of the engine can be understood (Starting Grant DIAQ, Grant No. 200021_169002, and intuitively as follows. The temperature difference between QSIT). P. E. acknowledges funding by the European the baths induces a heat flow from the first qubit (at T )to Commission (STREP RAQUEL), the Spanish MINECO, the second (at T ). This heat flow is made possible by our Projects No. FIS2008-01236 and No. FIS2013-40627-P, design constraint (A2). Specifically, a quantum of energy with the support of FEDER funds, the Generalitat de E from the first qubit can be transferred to a quantum of Catalunya CIRIT, Project No. 2014-SGR-966, the Swiss energy E in the second qubit, while the remaining energy 031022-7 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) E − E ¼ E is transferred to the ladder. This process The rates γ determine the overall time scale of the h c w h;c corresponds to the first term in the interaction Hamiltonian dissipative processes acting on the two engine qubits. (A3). Indeed, the reverse process is also possible, repre- In addition, the ladder system couples to a reservoir of sented by the second term in Eq. (A3). For the engine to electromagnetic-field modes at temperature T . The ladder deliver work (i.e., to raise the energy of the ladder), we need is designed so that only the highest energy transition to ensure that the first process is more likely than the jd − 1i → j0i couples significantly to the electromag- w w second. This can be done by judiciously choosing the netic field. This transition is associated with the emission of parameters (energies and temperatures) as we will see now. a photon having energy ðd − 1ÞE , while Γ is the sponta- We follow the approach of Ref. [23], which captures, in neous emission rate. A photodetector registers the emitted simple and intuitive terms, the effect of the two-qubit photon, producing a macroscopically measurable “tick”. engine on the ladder [42]. In order to bias the transition in The detector is assumed to work with perfect efficiency and the direction negligible time delay. Furthermore, the background temper- ature T is assumed to be low enough that we can ignore the j0i j1i jki → j1i j0i jk þ 1i ; ðA4Þ reverse transition j0i → jd − 1i , wherein the ladder c h w c h w w w absorbs a photon while in the ground state; i.e., we require we simply demand that the probability p of occupying the 1 that k T ≪ ðd − 1ÞE . B c w state j0i j1i is larger than the probability p of occupying c h 0 To quantify the ticks of the clock, in principle, one would the state j1i j0i ; recall that the ladder is only weakly c h have to keep track of the density operator of the pointer ρðtÞ connected to the ambient heat bath. As the machine works for all times t. However, as argued in the main text, in the in the weak-coupling regime, these probabilities basically weak-coupling regime, the qubit states do not change depend only on the baths’ temperatures and the qubits’ appreciably from the thermal states corresponding to energies, the state of each qubit being close to a thermal equilibrium with their respective reservoirs. Each tick is state at the temperature of the corresponding bath. Hence, therefore independent of the previous ticks, and one can the transition (A4) is biased, assuming that study the relevant quantifiers of the clock (i.e., resolution and accuracy) from the probability distribution in time of a E E h c single tick. < : ðA5Þ T T h c We describe the dynamics of the clock in the “no-click” subspace, i.e., the subensemble ρ ðtÞ conditioned on no The effect of the engine on the ladder is determined by the spontaneous emission having occurred up to time t.We two states j0i j1i and j1i j0i , which define the machine’s c h c h assume that the pointer begins in the normalized state virtual qubit. The engine simply places the load in thermal † † contact with the virtual qubit, which has energy gap −β E σ σ −β E σ σ h h h c c c c e e E − E ¼ E , hence resonant with the ladder’s energy ρ ð0Þ¼ ⊗ ⊗ j0i h0j; ðB3Þ h c w 0 w Z Z h c spacing, and virtual temperature determined by the pop- −β E v w ulation ratio p =p ¼ e . The load will thus effectively 1 0 where Z are the partition functions necessary for c;h “thermalize” with the virtual qubit. This causes the load to normalization. Equation (B3) describes the situation where climb the ladder so long as the bias (2), or equivalently the the qubits are in equilibrium with their respective reser- virtual temperature (1), is negative. Indeed, one can voirs, and the ladder has just decayed and been reset into immediately check that the condition (A5) is satisfied the ground state (i.e., the register has just ticked). The whenever the virtual qubit has a negative bias. subsequent evolution of the conditional density operator ρ ðtÞ follows from the master equation (ℏ ¼ 1): APPENDIX B: DYNAMICS OF THE CLOCK dρ In order to model the dynamics of the pointer and ¼ iðρ H − H ρ Þþ L ρ þ L ρ ; ðB4Þ 0 eff eff 0 h 0 c 0 dt compute the distribution of ticks, we use the following master equation formulation. The effect of each reservoir where the effective non-Hermitian Hamiltonian is given by on its corresponding qubit is represented by the super- H ¼ H þ H þ H , with spontaneous emission eff 0 int se operator described by the contribution −β E j j L ¼ γ D½σ þ γ e D½σ ; ðB1Þ j j j j iΓ H ¼ − jd − 1i hd − 1j: ðB5Þ se for j ¼ h, c. Here, we defined the qubit lowering operators σ ¼j0i h1j, and the dissipator in Lindblad form As a result of the non-Hermitian contribution, ρ ðtÞ does j 0 not stay normalized. The trace of the conditional density operator P ðtÞ¼ Tr½ρ ðtÞ corresponds to the probability † † 0 0 D½Lρ ¼ LρL − fL L; ρg: ðB2Þ 2 that a tick has not yet occurred. The probability density 031022-8 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) WðtÞ of the waiting time between two consecutive ticks μðtÞ¼ nqðn; tÞ; ðC2Þ then follows from n 2 2 dP σ ðtÞ¼ (n − μðtÞ) qðn; tÞ: ðC3Þ WðtÞ¼ − : ðB6Þ dt The speed of the ladder is determined by a simple For our purposes, we need only the mean and variance of calculation, the waiting time, which are given by dμðtÞ dqðn; tÞ t ¼ dττWðτÞ; ðB7Þ ¼ n ¼ p − p : ðC4Þ tick ↑ ↓ dt dt 2 2 ðΔt Þ ¼ dτðτ − t Þ WðτÞ: ðB8Þ The variance may be similarly calculated from tick tick 2 X dσ ðtÞ dqðn; tÞ ¼ ((n − μðtÞ) dt dt APPENDIX C: BIASED RANDOM WALK dμðtÞ APPROXIMATION − 2(n − μðtÞ) qðn; tÞ): ðC5Þ dt In this appendix, we determine the accuracy of the autonomous clock from a stochastic model of the pointer’s Using Eqs. (C2) and (C3), the second term can be shown to evolution. Specifically, we make two simplifying assump- vanish, while the first term simplifies to tions. First, the evolution of the pointer is simplified to a continuous biased random walk of the ladder, with rates dσ ðtÞ ¼ p þ p : ðC6Þ ↑ ↓ controlled by the populations of the virtual qubit of the dt two-qubit engine. In other words, the ladder has a rate We are now in a position to find the relevant quantifiers per unit time to move upward and a rate to move down, and of the clock. The average time between ticks is taken to be the ratio of the rates is given by the ratio of populations of the time for the ladder to travel from the bottom to the top the virtual qubit. This is an accurate description in the of its spectrum of d eigenvalues, regime where the thermal couplings are much larger than the interaction between the engine and the ladder d d and the spontaneous emission rate (see the following t ¼ ¼ ; ðC7Þ tick dμðtÞ=dt p − p section for details). Under this assumption, the density ↑ ↓ operator of the ladder is diagonal and can be replaced by a where, for simplicity, we replace d − 1 by d since the vector of populations of the energy levels. The second dimension of the ladder has been assumed to be large. assumption is that the dimension of the ladder is large The resolution ν , i.e., the number of ticks per unit time, is enough so that, for most of its evolution, the population tick the inverse of t , distribution does not feel the boundedness of the ladder tick Hamiltonian. p − p ↑ ↓ From the preceding arguments, the state of the ladder can ν ¼ ; ðC8Þ tick be described by a time-dependent probability distribution on a grid of integers (that label the energy levels) qðn; tÞ, corresponding to Eq. (6). where n ∈ Z, qðn; tÞ > 0, and qðn; tÞ¼ 1. The evolu- In the time taken for a single tick, the variance of the tion is determined by the forward rate p per unit time of ↑ ladder will have increased by jumping to the next integer, together with the backward rate p of jumping to the previous integer. An equation of dσ ðtÞ p þ p ↓ ↑ ↓ Δσ ¼ t ¼ d : ðC9Þ tick motion of the distribution can thus be constructed: dt p − p ↑ ↓ dqðn; tÞ Assuming the decay mechanism is good enough that the ¼ p qðn − 1;tÞþ p qðn þ 1;tÞ ↑ ↓ dt uncertainty in a single tick is determined solely by the uncertainty in when the ladder reaches the top (i.e., − ðp þ p Þqðn; tÞ: ðC1Þ ↑ ↓ the variance), then the uncertainty in the time interval In order to characterize the resolution and accuracy, we between consecutive ticks is simply must understand how quickly the position of the ladder sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi moves up, as well as how much it spreads on the way. We σðt ¼ t Þ d p þ p tick ↑ ↓ Δt ¼ ¼ : ðC10Þ denote the mean and variance of the distribution by μ and tick dμðtÞ=dt p − p p − p 2 ↑ ↓ ↑ ↓ σ , respectively, 031022-9 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) The accuracy N is defined as the number of ticks until the H ρ ¼ iðρH − H ρÞ; ðD3Þ se se se clock is uncertain by a single tick. This implies that the variance of the load has grown to the size of the entire and similarly for H . 2 2 int ladder, σ ¼ d . It follows that We transform the density operator to a dissipative −L t interaction picture defined by ρ~ ðtÞ¼ e ρ ðtÞ. The time p − p 0 0 ↑ ↓ N ¼ d ; ðC11Þ dependence of superoperators is given, in this picture, by p þ p ↑ ↓ −L t L t ~ 0 0 ~ H ðtÞ¼ e H e and H ðtÞ¼ H . Following the int int se se standard perturbative argument [43], we obtain which is equivalent to Eq. (7) since p =p ¼ p =p . ↑ ↓ 1 0 APPENDIX D: DERIVATION OF THE BIASED dPρ~ 0 0 0 ~ ~ ¼H Pρ~ ðtÞþ dt PH ðtÞH ðt ÞPρ~ ðt Þ; ðD4Þ RANDOM WALK MODEL se 0 int int 0 dt Treating the pointer as a stochastic system is motivated by our understanding that the core of the machinery lies in the valid to second order in the small quantities g and Γ.Wenow coupling of the ladder to the engine’s virtual qubit, whose apply the Born-Markov approximation to the t integral main effect is to create a bias such that the ladder’s energy is above, extending the lower integration limit to negative morelikelytoincreasethandecrease.Inthissection,weplace infinity, and making the replacement ρ~ ðt Þ → ρ~ ðtÞ. These 0 0 this (essentially classical) description of the pointer on a steps are justified by the assumption that γ ≫ g, Γ, so the firmer footing, deriving it from the two-qubit engine model integrand decays rapidly to zero compared to the time scale detailed above, working in the regime where the engine- over which Pρ~ ðtÞ changes appreciably. ladder coupling g and the spontaneous emission rate Γ are Equation (D4) is then simplified by expanding the both small in comparison to the thermal dissipation rates γ . c;h commutators, tracing over the engine qubits, and then In the limit of γ ≫ g, Γ, we use the Nakajima-Zwanzig transforming back to the Schrödinger picture. The resulting projection operator technique to derive an evolution equa- master equation decouples the evolutions of the populations tion for the conditional reduced density operator of the and coherences when ρ ðtÞ is expressed in the eigenbasis of ladder, ρ ðtÞ¼ Tr ½ρ ðtÞ. We introduce the projector w h;c 0 B . Since, by assumption, there is no initial coherence [see Eq. (22)], we quote only the result for the populations Pρ ðtÞ¼ ρ ðtÞ ⊗ ρ ⊗ ρ ; ðD1Þ 0 w h c where ρ denotes a local thermal state of the hot or cold h;c dρ qubit, for j ¼ h, c, ¼ p D½B ρ þ p D½B ρ ↓ w w ↑ w dt −β E σ σ Γ j j j ρ ¼ e ; ðD2Þ j − ðjd − 1i hd − 1jρ þ ρ jd − 1i hd − 1jÞ: w w w w ðD5Þ −β E j j while Z ¼ 1 þ e is the corresponding partition func- tion. Writing Eq. (23) as dρ =dt ¼ Lρ , we decompose the 0 0 Liouvillian as L ¼ L þ H þ H , where we defined the Introducing the probability vector q with elements 0 se int Hamiltonian superoperator q ðtÞ¼ Tr½ρ ðtÞjni hnj, we have dq=dt ¼ Aq, with n w w 0 1 −p p ↑ ↓ B C B C p −ðp þ p Þ ↑ ↑ ↓ B C B C B C A ¼ . : ðD6Þ B C B C B C B −ðp þ p Þ p C ↑ ↓ ↓ @ A p −ðp þ ΓÞ ↑ ↓ This is equivalent to Eq. (C1) for the probabilities † † 2 iE t p ¼ 2g dte hσ ðtÞσ ð0Þσ ðtÞσ ð0Þi; ðD7Þ ↓ h c q ðtÞ¼ qðn; tÞ, but with an additional term proportional to Γ describing spontaneous decay from the upper level. The forward and backward rates are Laplace-transformed ∞ † † 2 −iE t p ¼ 2g dte hσ ðtÞσ ð0Þσ ðtÞσ ð0Þi; ðD8Þ correlation functions of the engine qubits, ↑ h h c 031022-10 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) where the angle brackets denote an average with respect to spreading as much as would be expected from a simply ρ ⊗ ρ , while the operator time dependence is given by stochastic model, which in turn would lead to a higher h c L t σ ðtÞ¼ e σ , where L is the adjoint Liouvillian accuracy. Clocks that are even more coherent (while not h;c h;c defined by Tr½QL ðPÞ ¼ Tr½L ðQÞP for arbitrary necessarily autonomous) have been observed [16] to spread 0 0 operators P and Q. Explicitly, we have σ ðtÞ¼ much less than thermal clocks. 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Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?

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Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031022 (2017) 1,2 3,4 5 6,7 5 8 Paul Erker, Mark T. Mitchison, Ralph Silva, Mischa P. Woods, Nicolas Brunner, and Marcus Huber Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Faculty of Informatics, Università della Svizzera italiana, Via G. Buffi 13, 6900 Lugano, Switzerland Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland University College London, Department of Physics & Astronomy, London WC1E 6BT, United Kingdom QuTech, Delft University of Technology, Lorentzweg 1, 2611 CJ Delft, Netherlands Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, A-1090 Vienna, Austria (Received 7 November 2016; revised manuscript received 7 May 2017; published 2 August 2017) 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. DOI: 10.1103/PhysRevX.7.031022 Subject Areas: Quantum Physics, Quantum Information, Statistical Physics I. INTRODUCTION and finally measured. The result is interpreted as a time- interval measurement, whose precision can be related to Although quantum systems provide the most accurate the properties of the clock (e.g., its dimension [16]). measurements of time [1–3], the concept of time in However, the procedures of the state preparation and the quantum theory remains elusive. This issue has been measurement are usually not discussed explicitly. These explored in several directions. The relation between time models thus allow one to measure a time interval, e.g., for and energy, the physical quantity that is time invariant in implementing a given unitary operation (by timing an closed systems, has led to fundamental limitations in interaction). This functionality is analogous to a stop- the form of quantum speed limits [4–8]. Another approach watch, but it cannot be considered a complete model of a has aimed to promote time from a mere classical quantum clock. parameter to a fully quantum description [9–13]. Indeed, a crucial feature of a clock (as opposed to a Notably, quantum evolution is captured here via the stopwatch) is to continuously provide a time reference to an notion of correlations. Finally, various models of quantum external observer. It is thus essential that any complete systems designed to measure time, i.e., quantum clocks, model of a quantum clock explicitly specifies the process of have been proposed; see, for example, Refs. [14–17]. information read-out. This leads us to consider a clock as a These models typically consider a specific degree of bipartite system [18,19], shown in Fig. 1(a). The first part freedom of a quantum system, prepared in a judiciously of the clock is the pointer, i.e., a subsystem whose internal chosen initial state, then subjected to a unitary evolution, dynamics are effectively dictated by the passage of time. The second part is the register, which stores classical information obtained about the evolution of the pointer, Published by the American Physical Society under the terms of thereby mediating the transfer of information from the the Creative Commons Attribution 4.0 International license. system to an external observer. The pointer is designed to Further distribution of this work must maintain attribution to produce a sequence of signals, which are then recorded by the author(s) and the published article’s title, journal citation, and DOI. the register as ticks. 2160-3308=17=7(3)=031022(12) 031022-1 Published by the American Physical Society PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) (a) (c) (b) FIG. 1. (a) A pointer system generates a time-ordered sequence of events that are recorded and displayed by the register. (b) We consider a pointer comprising a two-qubit heat engine that drives a thermally isolated load up a ladder, whose highest-energy state undergoes radiative decay back to the ground state. Photons are thus repeatedly emitted and registered by a photodetector as ticks of the clock. (c) A virtual qubit is a pair of states in the engine’s two-qubit Hilbert space whose energy splitting is resonant with the ladder. The thermal baths drive population into the virtual qubit’s higher-energy state and out of its lower-energy one, creating a population inversion described by a negative virtual temperature. Hence, placing the virtual qubit in thermal contact with the ladder forces the load upwards, thereby performing work. is characterized by (i) its resolution, i.e., how frequently the It follows that there is an asymmetric flow of information clock ticks, and (ii) its accuracy, i.e., how many ticks the between the two parts of the clock, which makes the clock provides before its uncertainty becomes greater than process irreversible (and singles out a direction for the flow the average time between ticks. We find that a given of time). This naturally connects the problem to the second resolution and accuracy can be simultaneously achieved law of thermodynamics [20] because irreversibility is only if the rate of entropy production is sufficiently large; associated with the generation of entropy. One therefore otherwise, a trade-off exists whereby the desired accuracy expects that the suitability of a system for measuring time can only be attained by sacrificing some resolution, or vice implies a corresponding propensity to produce entropy. versa. Furthermore, in the regime where the resolution is However, a precise relationship between entropy produc- arbitrarily low, the accuracy is still bounded by the entropy tion and clock performance has not yet been demonstrated. production, suggesting a quantitative connection between In fact, we show that such a relationship unavoidably entropy production and the clock’s arrow of time. Note that becomes apparent when considering a more general ques- here the relevant entropy production is not associated with tion: What are the minimal resources required to maintain a quantum clock? In order to answer this question, we measurements or erasure of the register but rather with the evolution of the pointer system itself. In the following, we consider an autonomous quantum clock, i.e., a self- illustrate this behavior by explicitly calculating the dynam- contained device working without any external control ics of a simple clock model. We then present a conjecture, or timing. The clock must be an isolated system evolving backed up by general thermodynamic arguments, that such according to a time-independent Hamiltonian [19]. trade-offs are exhibited by any implementation of an Moreover, the resources powering the clock should not autonomous clock. themselves require another clock to be prepared. Specifically, we discuss a natural class of autonomous clocks driven by minimal nonequilibrium resources, II. AUTONOMOUS QUANTUM CLOCKS namely, the flow of heat between two thermal reservoirs. Our objective is to find the fundamental limits on In particular, our model makes explicit the physical quantum clocks. To that end, we consider autonomous mechanism of the clock’s operation, including its initial- ization and power supply. We make use of thermodynam- clocks, i.e., those which are complete and self-contained. In ical concepts in order to analyze the clock as an particular, the operation of the device should not require autonomous thermal machine [21–23], with the goal of any time-dependent control that would necessitate another producing a series of regular ticks. external clock. This allows all resources needed for time- This approach allows us to show that the clock’s keeping to be carefully accounted for. In this section, we irreversible entropy production dictates fundamental discuss some of the general features of autonomous clocks, limits on its performance. The performance of the clock before specifying a particular model in Sec. III. 031022-2 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) An autonomous clock evolves under a time-independent energy gap E . The second qubit is connected to a cold bath Hamiltonian, such that a steady stream of ticks is recorded at temperature T <T and has energy gap E <E . The c h c h at the register, as depicted in Fig. 1. The process by which engine delivers work to a load, represented by a system information is transferred from pointer to register should be with d equally spaced energy levels, i.e., a discrete ladder, effectively irreversible, in order to ensure the unidirectional with energy spacing E ¼ E − E . w h c flow of time as recorded by the register. In addition, this The temperature difference between the two baths process should occur spontaneously, i.e., without any induces a heat current in the system from the hot qubit external intervention or time-dependent coupling between to the cold one. This flow of heat delivers energy to the the pointer and register. To ensure that the probability of load, causing it to “climb” the ladder. The action of the this spontaneous process is larger than that of its time machine can be understood in terms of the resonant reverse, the free energy of the pointer must decrease. exchange of energy between the load and a virtual qubit Therefore, in order to continue producing ticks, the clock [23]. This virtual qubit is a special pair of states in the needs a source of free energy driving it out of equilibrium. engine’s Hilbert space that are coupled to the ladder, In principle, any nonequilibrium quantum system could illustrated in Fig. 1(c). Assuming that the engine-ladder provide the free energy needed to power a clock. However, coupling is weak, the populations of the virtual qubit states a large class of nonequilibrium states is difficult to prepare are thermally distributed at the virtual temperature in practice unless a clock is already available, e.g., so that a E − E resonant driving field can be applied for a known period of h c k T ¼ ; ð1Þ B v time. We exclude such resource states in order to ensure fair β E − β E h h c c bookkeeping, i.e., the resources’ initial preparation should where β ¼ 1=k T . In other words, the virtual qubit’s not itself require time measurements. It is also clearly c;h B c;h −β E v w states are occupied in the ratio p =p ¼ e , where p desirable—yet inessential—that such resources be natu- 1 0 1 (p ) denotes the population of the state with higher (lower) rally abundant or otherwise easy to generate. energy and β ¼ 1=k T . Therefore, whenever the virtual Here we argue that the minimal nonequilibrium resource v B v qubit has a negative temperature, i.e., a population inver- consists of two thermal reservoirs at different temperatures. sion, the load moves up the ladder as it “thermalizes” with Indeed, the presence of one heat bath is unavoidable since the virtual qubit. The virtual temperature is conveniently this represents the environment at ambient temperature T . parametrized by the virtual qubit’s population bias Furthermore, a second reservoir at temperature T >T h c can be prepared deterministically without detailed under- p − p 0 1 standing of the bath’s internal structure and without any Z ¼ ¼ tanhðβ E =2Þ; ð2Þ v v w p þ p 0 1 well-timed operations. This is because the thermal state represents a condition of minimal knowledge [24] towards which plays a central role in characterizing the performance which generic quantum systems (i.e., those not integrable of our clock, as we show below. nor many-body localized) equilibrate [25]. In this sense, the To complete the description of our clock, we must minimal out-of-equilibrium resource is an equilibrated specify how the pointer interacts with the register. The (thermalized) resource with a higher average energy con- top level of the ladder is assumed to be unstable, and it tent than the environment. Any other potential resource for decays to the ground state by emitting a photon at energy the clock would feature lower entropy at equal energies and E ¼ðd − 1ÞE . This photon is then detected at the γ w thus additional knowledge or control to prepare. In the register, which in turn makes the clock tick. Note that following, we base our quantitative analysis on clocks the presence of the decay channel also allows, in principle, driven by thermal baths. However, we emphasize that the for the reverse process. However, we assume that the notion of an autonomous clock is more general and could background temperature satisfies k T ≪ E so that such B c γ be extended to various different scenarios and resource processes are negligible. states. In summary, the flow of heat through the engine drives the load up the ladder, which eventually reaches the top III. MINIMAL THERMAL CLOCK MODEL level and decays back to the ground state while emitting a We now specialize to a concrete model of an autonomous photon. The process is repeated, thus generating a steady quantum clock where the pointer is driven by the heat flow stream of photons that are recorded by the register as ticks between two thermal baths. For simplicity, we base our of the clock. Importantly, the evolution of the ladder’s model on the smallest quantum heat engine that was energy is probabilistic, leading to a stochastic sequence of introduced in Ref. [23] (see Appendix A for a detailed ticks. The distribution of ticks depends, in particular, on the description). dimension of the ladder d and the bias Z . Intuitively, if the The machine consists of two qubits, each coupled to an bias is small (Z negative but close to zero), the probability for the load to move up is only marginally larger than its independent thermal bath, as depicted in Fig. 1(b). The first qubit, connected to the hot bath at temperature T , has probability of going down. The probability distribution 031022-3 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) over the levels of the ladder thus rapidly becomes quite For our model of the autonomous clock, we assume that broad, which makes the clock tick slowly and at irregular after each spontaneous emission event, the entire pointer is time intervals. On the other hand, if Z → −1, i.e., the reset to its initial state—specifically, a product state with virtual qubit has essentially complete population inversion, the ladder in its ground state and the engine qubits in then the probability for the ladder population to move equilibrium with their respective baths. This approximation downward is negligible, resulting in shorter and more is valid in the weak-coupling limit, where the engine qubits regular time intervals between ticks. are minimally perturbed by their interaction with the ladder. The ticks of the clock can therefore be described as a renewal process; i.e., the time between any pair of IV. PERFORMANCE OF THE CLOCK consecutive ticks is statistically independent from, and In order for the clock to deliver ticks, the engine must identically distributed to, the time between any other pair of raise the ladder’s energy and necessarily dissipate energy consecutive ticks. into the cold bath. Our goal now is to relate the performance Now, let the distribution of waiting times between two of the clock to this dissipated energy, which is closely consecutive ticks be characterized by the mean t and the tick related to the entropy production. Specifically, we consider standard deviation Δt . The resolution of the clock is then tick here the heat dissipated into the cold bath per tick of the clock, ν ¼ 1=t ; ð4Þ tick tick Q ¼ðd − 1ÞE : ð3Þ i.e., the average number of ticks the clock provides per c c second. The accuracy is the number of ticks N such that the Note that this quantity, rather than the heat supplied to the uncertainty (standard deviation) of the Nth tick time is machine per tick [Q ¼ðd − 1ÞE ], represents the funda- equal to the average time between ticks. Since the waiting h h mental minimum energy expenditure associated with one times are independent, the uncertainty in the time of the nth pffiffiffi tick of the clock. This is because, in principle, a large part tick is simply nΔt , and therefore tick of the energy E carried away by the emitted photon could be captured and recycled (e.g., dumped back into the hot tick N ¼ : ð5Þ bath). Consequently, the dissipated heat (3) is associated Δt tick with an irreversible entropy production of at least β Q c c per tick. Figure 2 illustrates the intimate relationship between the The performance of our autonomous clock is quantified accuracy N and the resolution ν versus the dissipated tick by the resolution and accuracy of its ticks. By resolution, energy Q , calculated by numerical solution of the equa- we refer to the average number of ticks the clock provides tions of motion (see Appendix B). We find that, for a given per unit time. The ticks are not distributed regularly, and we amount of dissipated energy, there is a trade-off between characterize the accuracy by the number of ticks provided accuracy and resolution. In other words, engineering a before the next tick is uncertain by the average time interval good clock featuring both high accuracy and high reso- between ticks [26]. lution requires a large amount of energy to be dissipated (a) (b) (c) FIG. 2. Illustration of the fundamental trade-off between the dissipated heat and the achievable accuracy and resolution. (a) Accuracy N as a function of dissipated heat per tick Q , for various values of the resolution ν . At low energy, the accuracy increases linearly c tick with the dissipated energy, independently of the resolution. However, for higher energies, the accuracy saturates. (b) Resolution ν as a tick function of dissipated heat per tick Q , for various values of the accuracy N. The resolution first increases with dissipated energy but then quickly saturates to a maximal value. (c) Trade-off between accuracy and resolution when the energy dissipation rate is fixed. The data are computed for fixed values of k T ¼ E , k T ¼ 1000E and g ¼ ℏγ ¼ ℏΓ ¼ 0.05E , while the ladder dimension d and cold qubit B c w B h w w energy E are varied independently. Note that d ≥ 10 for all of the plotted points; thus, k T ¼ E ≪ E ¼ðd − 1ÞE , and we can c B c w γ w safely ignore the absorption of a photon (i.e., the reverse of the decay process). 031022-4 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) the simplifying assumptions that the clock ticks as soon as the load reaches the top of the ladder and that d is large enough for reflections from the boundaries of the ladder to be negligible. Under the foregoing approximations, the resolution is given by p − p ↑ ↓ ν ¼ : ð6Þ tick Quite intuitively, the resolution is inversely proportional to the dimension d, corresponding to the “height” of the ladder, but it is proportional to the difference of transition FIG. 3. Accuracy N versus dissipated energy Q for various rates p − p , which quantify the “speed” at which the load ↑ ↓ values of the dimension d of the ladder, according to the climbs. approximation (8) with the same bath temperatures as in Fig. 2. On the other hand, as demonstrated in Appendix C, the accuracy is given by and thus a higher production of entropy per tick. This is nicely illustrated in Fig. 2(c), which showcases the nature N ¼ djZ j; ð7Þ of entropy production as a resource. The curves for different entropies are clearly ordered; i.e., more entropy which is entirely independent from the clock’s overall implies that either more resolution or more accuracy can be dynamical time scale, set by the rates p . Instead, the ↑;↓ achieved. It is interesting to note, however, that the accuracy depends only on the dimensionless quantities Z relationship between the two is nontrivial and the trade- and d. In turn, the bias Z encapsulates the dependence of off features nonlinear dependencies. the clock’s accuracy on the dissipated heat. In the case Finally, we note that in the regime of low-energy of our model, using Eqs. (B5) and (B7), the accuracy is dissipation, the relationship between accuracy and entropy given by production at fixed resolution is directly proportional, as ðβ − β ÞQ − β E seen in Fig. 2(a). In the next section, we recover this c h c h γ N ¼ d tanh : ð8Þ behavior analytically in the weak-coupling regime. 2d Note, however, that the relation between Z and the heat V. ACCURACY IN THE WEAK-COUPLING LIMIT exchanged with the two baths is more general than the We now investigate the relationship between accuracy model considered here [27] (see Appendix E for a dis- and dissipated power by an alternate approximate analysis, cussion). It follows that the accuracy in the weak-coupling valid when the interaction between the engine and the limit depends on the amount of dissipated heat but not on ladder is weak. In this regime, the accuracy is limited by the dissipation rates. the dissipated power and the dimension of the ladder, while The behavior described by Eq. (8) is illustrated in Fig. 3, the resolution is not focused upon. This is in contrast to where we plot the accuracy versus the dissipated energy for Fig. 2(a), where the resolution is fixed, and the dimension is fixed dimension. We observe that the accuracy first allowed to vary. In particular, we show that the accuracy is increases linearly but eventually saturates to its maximum essentially independent of the details of the clock’s dynam- value N ¼ d. Indeed, increasing Q leads to a stronger bias ics, being determined only by the bias of the virtual qubit in the virtual qubit, saturating at jZ j → 1 as Q → ∞. v c Z and the ladder dimension d. Thus, the accuracy is limited by both the dimension d and Focusing on the ladder, its evolution can be approxi- the dissipated energy Q . Hence, achieving a certain mated by a biased random walk, induced by the interaction accuracy requires a minimum dimension as well as a with the virtual qubit. This is easily understood by the fact minimum dissipated energy per tick. that the resonant interaction with the virtual qubit cannot Even if the dimension is unbounded, we find that the induce any coherence on the ladder. Moreover, the reso- dissipated energy still imposes a fundamental limitation. nance is exactly at the energy of a transition of one step up Taking the limit d → ∞, the accuracy is linearly dependent or down, and independent of the ladder’s position. The rates on the dissipated heat: at which the ladder population moves upwards (p )or −β E v w downwards (p ) satisfy p =p ¼ e as a consequence ↓ ↑ ↓ ðβ − β ÞQ − β E c h c h γ N → : ð9Þ of detailed balance. This description of the clock is derived in Appendix C as a perturbative approximation to the two-qubit engine, which becomes exact in the limit of Noting that Q ¼ Q þ E , we can recast the above in h c γ vanishingly small engine-ladder coupling. We also make the illustrative form 031022-5 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) β Q − β Q ΔS regime where the machine works reversibly. A finite power, c c h h tick N → ¼ ; ð10Þ however, is essential for the resolution of any autonomous 2 2 clock: A clock working at Carnot efficiency ticks infinitely where ΔS is the increase in the entropy of the clock in a tick slowly. Hence, even in the rather artificial regimes of single tick. We may interpret the regularity of each tick as T → 0 or T → ∞, the requirement of a finite resolution c h representative of the strength of the arrow of time. Thus, implies a minimal dissipated heat and thus a minimal Eq. (10) quantifies, in a concrete manner, the connection entropy production. between the arrow of time of a clock and its irreversibility. It is also possible to consider more general nonequili- brium resources to power the clock. In order to satisfy the VI. FUNDAMENTAL LIMITS OF GENERAL requirement of autonomy, such resources should not AUTONOMOUS CLOCKS themselves need any well-timed control in order to be produced. In principle, it is conceivable that such a resource The simple thermal clock model we discuss above could allow the clock to achieve higher efficiency than is illustrates the fact that our ability to accurately and possible with thermal driving. However, an autonomous precisely measure time necessarily generates an increase clock that does not generate any entropy but nonetheless of entropy (via heat dissipation). Equivalently, this implies has finite resolution would constitute an autonomous an intrinsic work cost for measuring time. It is natural to ask machine operating at finite power with unit efficiency. whether the connection between clock performance and Therefore, if the performance of autonomous quantum entropy production is a specific aspect of our model or, on clocks is not always associated with a fundamental entropy the contrary, a universal feature of any procedure for production, then the prospect of quantum machines is far measuring time. Below, we argue in favor of the latter: more revolutionary than is widely believed at present. Any autonomous clock must increase entropy. Finally, it is also worth pointing out that, while we focus The core insight underlying our argument is that, as here on a specific source for the entropy production of the discussed in Sec. II, the ticks of any autonomous clock clock (namely, the heat dissipated by the thermal machine involve a spontaneous and effectively irreversible transition driving the clock), there will generally be additional energy in a pointer system, thus inducing a corresponding change costs required for operating the clock. In particular, the in the register to which it is coupled. In order to bias the preparation (and reset) of the initial state of the register forward transition in favor of its time reverse (i.e., to avoid will generate entropy due to Landauer’s erasure princi- the clock ticking “backwards”), the transition must reduce ple [28,29]. the free energy of the pointer. Hence, for the clock to run Even if the qualitative bound (10) derived in our work continuously, it needs access to a system out of thermal represents a fundamental limit for any clock, it still equilibrium that can replenish the free energy of the pointer. underestimates the necessary costs of running the best Now, the essential question is whether it is possible for the clocks available today. For instance, a typical atomic clock clock to convert this free energy into ticks with perfect [30] runs at resolutions of the order of 10 Hz, and an efficiency, i.e., without increasing entropy. accuracy of 10 seconds before being off by a second. Let us first discuss this question in the context of clocks Equation (10) would imply a minimal power consumption driven by thermal baths. It is clear that beyond the specific for such a clock of the order of about 50 μW. In practice, model we have studied, one could consider more general the real costs are orders of magnitude higher. This is similar designs for the thermal machine. The basic necessary to the case of information erasure: Even though Landauer’s ingredient is simply the ability to move the population principle is the only known fundamental limit, current of the pointer out of equilibrium so that an unstable level erasure techniques operate far less efficiently. generates a tick. This transition is biased in the forward direction so long as the unstable level is much higher in VII. CONCLUSION AND OUTLOOK energy than the thermal background. Such a mechanism can indeed work for a variety of physical implementations Our work represents a first step towards rigorously of the pointer (i.e., with a more complex level structure). characterizing the necessary resources and limitations of The ladder could comprise multiple levels which trigger a the process of timekeeping. In a nutshell, we introduced the decay, while the machine could feature more than two concept of autonomous quantum clocks to discuss these qubits. questions, and we argued that the measurement of time Nonetheless, all these possible extensions and more inevitably leads to an increase in entropy. Moreover, we sophisticated designs will still have to comply with the explicitly discussed a simple model of an autonomous basic laws of thermodynamics. In particular, the efficiency quantum clock and found that the amount of entropy of the conversion of energy to a tick is fundamentally produced represents an actual resource for measuring time. bounded by the Carnot efficiency η ¼ 1 − T =T . Every unit of heat dissipated can be spent to increase either C c h Moreover, this maximal efficiency can only be achieved the accuracy or the resolution of the clock. Additionally, the in a limit where the power vanishes, corresponding to the dimension of a key constituent of the clock (the ladder) 031022-6 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) National Science Foundation (SNF) through the project imposes a limit on the achievable accuracy and resolution, independently of the amount of dissipated heat. In other “Information and Physics”, and the National Centres of words, in analogy to the findings of Refs. [16,19], the Competence in Research Quantum Science and Hilbert space dimension imposes a fundamental constraint Technology (QSIT). on the performance of the clock. Reaching this optimal P. E., M. T. M., and R. S. contributed equally to this regime requires a minimal rate of entropy production. This work. provides a quantitative basis for the intuitive connection between the second law of thermodynamics and the arrow of time (see, for example, Refs. [31,32]). In order to APPENDIX A: DESCRIPTION OF THE measure how much time has passed, we inevitably need TWO-QUBIT HEAT ENGINE to increase the entropy of the Universe from the perspective Here, we give a detailed description of the two-qubit heat of the register. engine of Ref. [23], which represents the pointer of the Here, these considerations only concern the scenario of autonomous quantum clock. The machine consists of two minimal autonomous clocks, i.e., where the resources qubits, each one connected to a thermal bath. The first qubit exploited to operate the clock are simply two thermal with energy gap E is connected to the bath at T . The h h baths at different temperatures. While these arguably second qubit is connected to the bath at T and has energy represent the most abundant resources found in nature gap E <E . The engine is connected to a d-dimensional c h [25], it would be interesting to consider other quantum ladder, featuring equally spaced energy levels (with spacing systems, e.g., with multiple conserved quantities [33–36]. E ), which is not connected to any heat bath. The free More broadly, the relevant question is to what extent our Hamiltonian of the total system (two qubits and ladder) is choice of free resources impacts our ability to measure thus given by time. For instance, one could consider more general passive states [37], which would commute with the system d−1 X X Hamiltonian and thus satisfy the requirement of autonomy. H ¼ E j1i h1jþ kE jki hkj; ðA1Þ 0 j w j w Thermal clock models can furthermore be used to work out j¼h;c k¼0 the thermodynamic cost of controlling other quantum systems [16,38,39] in an autonomous fashion, i.e., imple- where j1i denotes the excited state of qubit j ¼ h, c, and menting locally apparent time-dependent Hamiltonians by jki denotes the state of the kth level of the ladder. As a coupling to an autonomous thermal clock. Moreover, design constraint, we take operating two clocks in parallel could lead to a drastic enhancement of the clock’s performance. While classical E ¼ E þ E : ðA2Þ h c w clocks running in parallel would not offer any fundamental improvement, one could consider quantum resources that Hence, the following energy levels of the total system are feature coherence or entanglement [40,41]. Could these degenerate in energy: j0i j1i jki and j1i j0i jk þ 1i . c h w c h w genuine quantum phenomena be used to increase our This allows for energy to be exchanged between the qubits ability to measure time? We look forward to future research and the ladder. Specifically, we consider the interaction in this direction. Hamiltonian d−1 ACKNOWLEDGMENTS H ¼ g ðj1i j0i jk þ 1i h0j h1j hkj þ H:c:Þ: ðA3Þ int c h w c h w We are grateful to Ämin Baumeler, Nicolas Gisin, k¼0 Patrick Hofer, Daniel Patel, Sandu Popescu, Gilles Pütz, The machine will be operated in the weak-coupling regime, Sandra Rankovic Stupar, Renato Renner, Christian Klumpp, and Stefan Wolf for fruitful discussions. M. H. i.e., g ≪ E , E . Note that our design constraint on the c w acknowledges funding from the Swiss National Science energies (A2) ensures that H has a significant effect even int Foundation (AMBIZIONE PZ00P2_161351) and the in the weak-coupling regime. Henceforth, we refer to the Austrian Science Fund (FWF) through the START joint system of ladder and engine as the pointer since it will Project No. Y879-N27. M. W. and M. T. M. acknowledge be the system from which the register will derive informa- funding from the UK research council EPSRC. R. S. and tion reflecting the passage of time. N. B. acknowledge the Swiss National Science Foundation The functioning of the engine can be understood (Starting Grant DIAQ, Grant No. 200021_169002, and intuitively as follows. The temperature difference between QSIT). P. E. acknowledges funding by the European the baths induces a heat flow from the first qubit (at T )to Commission (STREP RAQUEL), the Spanish MINECO, the second (at T ). This heat flow is made possible by our Projects No. FIS2008-01236 and No. FIS2013-40627-P, design constraint (A2). Specifically, a quantum of energy with the support of FEDER funds, the Generalitat de E from the first qubit can be transferred to a quantum of Catalunya CIRIT, Project No. 2014-SGR-966, the Swiss energy E in the second qubit, while the remaining energy 031022-7 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) E − E ¼ E is transferred to the ladder. This process The rates γ determine the overall time scale of the h c w h;c corresponds to the first term in the interaction Hamiltonian dissipative processes acting on the two engine qubits. (A3). Indeed, the reverse process is also possible, repre- In addition, the ladder system couples to a reservoir of sented by the second term in Eq. (A3). For the engine to electromagnetic-field modes at temperature T . The ladder deliver work (i.e., to raise the energy of the ladder), we need is designed so that only the highest energy transition to ensure that the first process is more likely than the jd − 1i → j0i couples significantly to the electromag- w w second. This can be done by judiciously choosing the netic field. This transition is associated with the emission of parameters (energies and temperatures) as we will see now. a photon having energy ðd − 1ÞE , while Γ is the sponta- We follow the approach of Ref. [23], which captures, in neous emission rate. A photodetector registers the emitted simple and intuitive terms, the effect of the two-qubit photon, producing a macroscopically measurable “tick”. engine on the ladder [42]. In order to bias the transition in The detector is assumed to work with perfect efficiency and the direction negligible time delay. Furthermore, the background temper- ature T is assumed to be low enough that we can ignore the j0i j1i jki → j1i j0i jk þ 1i ; ðA4Þ reverse transition j0i → jd − 1i , wherein the ladder c h w c h w w w absorbs a photon while in the ground state; i.e., we require we simply demand that the probability p of occupying the 1 that k T ≪ ðd − 1ÞE . B c w state j0i j1i is larger than the probability p of occupying c h 0 To quantify the ticks of the clock, in principle, one would the state j1i j0i ; recall that the ladder is only weakly c h have to keep track of the density operator of the pointer ρðtÞ connected to the ambient heat bath. As the machine works for all times t. However, as argued in the main text, in the in the weak-coupling regime, these probabilities basically weak-coupling regime, the qubit states do not change depend only on the baths’ temperatures and the qubits’ appreciably from the thermal states corresponding to energies, the state of each qubit being close to a thermal equilibrium with their respective reservoirs. Each tick is state at the temperature of the corresponding bath. Hence, therefore independent of the previous ticks, and one can the transition (A4) is biased, assuming that study the relevant quantifiers of the clock (i.e., resolution and accuracy) from the probability distribution in time of a E E h c single tick. < : ðA5Þ T T h c We describe the dynamics of the clock in the “no-click” subspace, i.e., the subensemble ρ ðtÞ conditioned on no The effect of the engine on the ladder is determined by the spontaneous emission having occurred up to time t.We two states j0i j1i and j1i j0i , which define the machine’s c h c h assume that the pointer begins in the normalized state virtual qubit. The engine simply places the load in thermal † † contact with the virtual qubit, which has energy gap −β E σ σ −β E σ σ h h h c c c c e e E − E ¼ E , hence resonant with the ladder’s energy ρ ð0Þ¼ ⊗ ⊗ j0i h0j; ðB3Þ h c w 0 w Z Z h c spacing, and virtual temperature determined by the pop- −β E v w ulation ratio p =p ¼ e . The load will thus effectively 1 0 where Z are the partition functions necessary for c;h “thermalize” with the virtual qubit. This causes the load to normalization. Equation (B3) describes the situation where climb the ladder so long as the bias (2), or equivalently the the qubits are in equilibrium with their respective reser- virtual temperature (1), is negative. Indeed, one can voirs, and the ladder has just decayed and been reset into immediately check that the condition (A5) is satisfied the ground state (i.e., the register has just ticked). The whenever the virtual qubit has a negative bias. subsequent evolution of the conditional density operator ρ ðtÞ follows from the master equation (ℏ ¼ 1): APPENDIX B: DYNAMICS OF THE CLOCK dρ In order to model the dynamics of the pointer and ¼ iðρ H − H ρ Þþ L ρ þ L ρ ; ðB4Þ 0 eff eff 0 h 0 c 0 dt compute the distribution of ticks, we use the following master equation formulation. The effect of each reservoir where the effective non-Hermitian Hamiltonian is given by on its corresponding qubit is represented by the super- H ¼ H þ H þ H , with spontaneous emission eff 0 int se operator described by the contribution −β E j j L ¼ γ D½σ þ γ e D½σ ; ðB1Þ j j j j iΓ H ¼ − jd − 1i hd − 1j: ðB5Þ se for j ¼ h, c. Here, we defined the qubit lowering operators σ ¼j0i h1j, and the dissipator in Lindblad form As a result of the non-Hermitian contribution, ρ ðtÞ does j 0 not stay normalized. The trace of the conditional density operator P ðtÞ¼ Tr½ρ ðtÞ corresponds to the probability † † 0 0 D½Lρ ¼ LρL − fL L; ρg: ðB2Þ 2 that a tick has not yet occurred. The probability density 031022-8 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) WðtÞ of the waiting time between two consecutive ticks μðtÞ¼ nqðn; tÞ; ðC2Þ then follows from n 2 2 dP σ ðtÞ¼ (n − μðtÞ) qðn; tÞ: ðC3Þ WðtÞ¼ − : ðB6Þ dt The speed of the ladder is determined by a simple For our purposes, we need only the mean and variance of calculation, the waiting time, which are given by dμðtÞ dqðn; tÞ t ¼ dττWðτÞ; ðB7Þ ¼ n ¼ p − p : ðC4Þ tick ↑ ↓ dt dt 2 2 ðΔt Þ ¼ dτðτ − t Þ WðτÞ: ðB8Þ The variance may be similarly calculated from tick tick 2 X dσ ðtÞ dqðn; tÞ ¼ ((n − μðtÞ) dt dt APPENDIX C: BIASED RANDOM WALK dμðtÞ APPROXIMATION − 2(n − μðtÞ) qðn; tÞ): ðC5Þ dt In this appendix, we determine the accuracy of the autonomous clock from a stochastic model of the pointer’s Using Eqs. (C2) and (C3), the second term can be shown to evolution. Specifically, we make two simplifying assump- vanish, while the first term simplifies to tions. First, the evolution of the pointer is simplified to a continuous biased random walk of the ladder, with rates dσ ðtÞ ¼ p þ p : ðC6Þ ↑ ↓ controlled by the populations of the virtual qubit of the dt two-qubit engine. In other words, the ladder has a rate We are now in a position to find the relevant quantifiers per unit time to move upward and a rate to move down, and of the clock. The average time between ticks is taken to be the ratio of the rates is given by the ratio of populations of the time for the ladder to travel from the bottom to the top the virtual qubit. This is an accurate description in the of its spectrum of d eigenvalues, regime where the thermal couplings are much larger than the interaction between the engine and the ladder d d and the spontaneous emission rate (see the following t ¼ ¼ ; ðC7Þ tick dμðtÞ=dt p − p section for details). Under this assumption, the density ↑ ↓ operator of the ladder is diagonal and can be replaced by a where, for simplicity, we replace d − 1 by d since the vector of populations of the energy levels. The second dimension of the ladder has been assumed to be large. assumption is that the dimension of the ladder is large The resolution ν , i.e., the number of ticks per unit time, is enough so that, for most of its evolution, the population tick the inverse of t , distribution does not feel the boundedness of the ladder tick Hamiltonian. p − p ↑ ↓ From the preceding arguments, the state of the ladder can ν ¼ ; ðC8Þ tick be described by a time-dependent probability distribution on a grid of integers (that label the energy levels) qðn; tÞ, corresponding to Eq. (6). where n ∈ Z, qðn; tÞ > 0, and qðn; tÞ¼ 1. The evolu- In the time taken for a single tick, the variance of the tion is determined by the forward rate p per unit time of ↑ ladder will have increased by jumping to the next integer, together with the backward rate p of jumping to the previous integer. An equation of dσ ðtÞ p þ p ↓ ↑ ↓ Δσ ¼ t ¼ d : ðC9Þ tick motion of the distribution can thus be constructed: dt p − p ↑ ↓ dqðn; tÞ Assuming the decay mechanism is good enough that the ¼ p qðn − 1;tÞþ p qðn þ 1;tÞ ↑ ↓ dt uncertainty in a single tick is determined solely by the uncertainty in when the ladder reaches the top (i.e., − ðp þ p Þqðn; tÞ: ðC1Þ ↑ ↓ the variance), then the uncertainty in the time interval In order to characterize the resolution and accuracy, we between consecutive ticks is simply must understand how quickly the position of the ladder sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi moves up, as well as how much it spreads on the way. We σðt ¼ t Þ d p þ p tick ↑ ↓ Δt ¼ ¼ : ðC10Þ denote the mean and variance of the distribution by μ and tick dμðtÞ=dt p − p p − p 2 ↑ ↓ ↑ ↓ σ , respectively, 031022-9 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017) The accuracy N is defined as the number of ticks until the H ρ ¼ iðρH − H ρÞ; ðD3Þ se se se clock is uncertain by a single tick. This implies that the variance of the load has grown to the size of the entire and similarly for H . 2 2 int ladder, σ ¼ d . It follows that We transform the density operator to a dissipative −L t interaction picture defined by ρ~ ðtÞ¼ e ρ ðtÞ. The time p − p 0 0 ↑ ↓ N ¼ d ; ðC11Þ dependence of superoperators is given, in this picture, by p þ p ↑ ↓ −L t L t ~ 0 0 ~ H ðtÞ¼ e H e and H ðtÞ¼ H . Following the int int se se standard perturbative argument [43], we obtain which is equivalent to Eq. (7) since p =p ¼ p =p . ↑ ↓ 1 0 APPENDIX D: DERIVATION OF THE BIASED dPρ~ 0 0 0 ~ ~ ¼H Pρ~ ðtÞþ dt PH ðtÞH ðt ÞPρ~ ðt Þ; ðD4Þ RANDOM WALK MODEL se 0 int int 0 dt Treating the pointer as a stochastic system is motivated by our understanding that the core of the machinery lies in the valid to second order in the small quantities g and Γ.Wenow coupling of the ladder to the engine’s virtual qubit, whose apply the Born-Markov approximation to the t integral main effect is to create a bias such that the ladder’s energy is above, extending the lower integration limit to negative morelikelytoincreasethandecrease.Inthissection,weplace infinity, and making the replacement ρ~ ðt Þ → ρ~ ðtÞ. These 0 0 this (essentially classical) description of the pointer on a steps are justified by the assumption that γ ≫ g, Γ, so the firmer footing, deriving it from the two-qubit engine model integrand decays rapidly to zero compared to the time scale detailed above, working in the regime where the engine- over which Pρ~ ðtÞ changes appreciably. ladder coupling g and the spontaneous emission rate Γ are Equation (D4) is then simplified by expanding the both small in comparison to the thermal dissipation rates γ . c;h commutators, tracing over the engine qubits, and then In the limit of γ ≫ g, Γ, we use the Nakajima-Zwanzig transforming back to the Schrödinger picture. The resulting projection operator technique to derive an evolution equa- master equation decouples the evolutions of the populations tion for the conditional reduced density operator of the and coherences when ρ ðtÞ is expressed in the eigenbasis of ladder, ρ ðtÞ¼ Tr ½ρ ðtÞ. We introduce the projector w h;c 0 B . Since, by assumption, there is no initial coherence [see Eq. (22)], we quote only the result for the populations Pρ ðtÞ¼ ρ ðtÞ ⊗ ρ ⊗ ρ ; ðD1Þ 0 w h c where ρ denotes a local thermal state of the hot or cold h;c dρ qubit, for j ¼ h, c, ¼ p D½B ρ þ p D½B ρ ↓ w w ↑ w dt −β E σ σ Γ j j j ρ ¼ e ; ðD2Þ j − ðjd − 1i hd − 1jρ þ ρ jd − 1i hd − 1jÞ: w w w w ðD5Þ −β E j j while Z ¼ 1 þ e is the corresponding partition func- tion. Writing Eq. (23) as dρ =dt ¼ Lρ , we decompose the 0 0 Liouvillian as L ¼ L þ H þ H , where we defined the Introducing the probability vector q with elements 0 se int Hamiltonian superoperator q ðtÞ¼ Tr½ρ ðtÞjni hnj, we have dq=dt ¼ Aq, with n w w 0 1 −p p ↑ ↓ B C B C p −ðp þ p Þ ↑ ↑ ↓ B C B C B C A ¼ . : ðD6Þ B C B C B C B −ðp þ p Þ p C ↑ ↓ ↓ @ A p −ðp þ ΓÞ ↑ ↓ This is equivalent to Eq. (C1) for the probabilities † † 2 iE t p ¼ 2g dte hσ ðtÞσ ð0Þσ ðtÞσ ð0Þi; ðD7Þ ↓ h c q ðtÞ¼ qðn; tÞ, but with an additional term proportional to Γ describing spontaneous decay from the upper level. The forward and backward rates are Laplace-transformed ∞ † † 2 −iE t p ¼ 2g dte hσ ðtÞσ ð0Þσ ðtÞσ ð0Þi; ðD8Þ correlation functions of the engine qubits, ↑ h h c 031022-10 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) where the angle brackets denote an average with respect to spreading as much as would be expected from a simply ρ ⊗ ρ , while the operator time dependence is given by stochastic model, which in turn would lead to a higher h c L t σ ðtÞ¼ e σ , where L is the adjoint Liouvillian accuracy. Clocks that are even more coherent (while not h;c h;c defined by Tr½QL ðPÞ ¼ Tr½L ðQÞP for arbitrary necessarily autonomous) have been observed [16] to spread 0 0 operators P and Q. Explicitly, we have σ ðtÞ¼ much less than thermal clocks. The possibility of achieving expð−iE t − γ Z t=2Þσ for j ¼ h, c, implying that more accurate clocks via the use of stronger couplings and j j j j coherence is thus an important direction for future work. 2 −β E c c 4g e p ¼ ; ðD9Þ Z Z ðγ Z þ γ Z Þ h c h h c c 2 −β E [1] T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, h h 4g e p ¼ ; ðD10Þ B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S. Z Z ðγ Z þ γ Z Þ h c h h c c Safronova, G. F. Strouse, W. L. Tew, and J. Ye, Systematic −18 Evaluation of an Atomic Clock at 2 × 10 Total from which one readily verifies that p =p ¼ ↑ ↓ Uncertainty, Nat. Commun. 6, 6896, 2015. −ðβ E −β E Þ −β E h h c c v w e ¼ e . Self-consistency of the Born-Mar- [2] N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, kov approximation requires that p , p ≪ γ . ↓ ↑ j N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and −18 A. D. Ludlow, An Atomic Clock with 10 Instability, Science 341, 1215 (2013). APPENDIX E: MODEL-INDEPENDENT LIMITS [3] C. 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