Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031022 (2017)
Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?
Paul Erker,1,2 Mark T. Mitchison,3,4 Ralph Silva,5 Mischa P. Woods,6,7 Nicolas Brunner,5 and Marcus Huber8 1Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain 2Faculty of Informatics, Università della Svizzera italiana, Via G. Buffi 13, 6900 Lugano, Switzerland 3Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom 4Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany 5Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland 6University College London, Department of Physics & Astronomy, London WC1E 6BT, United Kingdom 7QuTech, Delft University of Technology, Lorentzweg 1, 2611 CJ Delft, Netherlands 8Institute 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. Further distribution of this work must maintain attribution to system to an external observer. The pointer is designed to the author(s) and the published article’s title, journal citation, produce a sequence of signals, which are then recorded by 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.
It follows that there is an asymmetric flow of information is characterized by (i) its resolution, i.e., how frequently the between the two parts of the clock, which makes the clock ticks, and (ii) its accuracy, i.e., how many ticks the process irreversible (and singles out a direction for the flow clock provides before its uncertainty becomes greater than of time). This naturally connects the problem to the second the average time between ticks. We find that a given law of thermodynamics [20] because irreversibility is resolution and accuracy can be simultaneously achieved associated with the generation of entropy. One therefore only if the rate of entropy production is sufficiently large; expects that the suitability of a system for measuring time otherwise, a trade-off exists whereby the desired accuracy implies a corresponding propensity to produce entropy. can only be attained by sacrificing some resolution, or vice However, a precise relationship between entropy produc- versa. Furthermore, in the regime where the resolution is tion and clock performance has not yet been demonstrated. arbitrarily low, the accuracy is still bounded by the entropy In fact, we show that such a relationship unavoidably production, suggesting a quantitative connection between becomes apparent when considering a more general ques- entropy production and the clock’s arrow of time. Note that tion: What are the minimal resources required to maintain here the relevant entropy production is not associated with a quantum clock? In order to answer this question, we measurements or erasure of the register but rather with the consider an autonomous quantum clock, i.e., a self- evolution of the pointer system itself. In the following, we contained device working without any external control illustrate this behavior by explicitly calculating the dynam- or timing. The clock must be an isolated system evolving ics of a simple clock model. We then present a conjecture, according to a time-independent Hamiltonian [19]. backed up by general thermodynamic arguments, that such Moreover, the resources powering the clock should not trade-offs are exhibited by any implementation of an themselves require another clock to be prepared. autonomous clock. 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. In particular, our model makes explicit the physical Our objective is to find the fundamental limits on mechanism of the clock’s operation, including its initial- quantum clocks. To that end, we consider autonomous 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.
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An autonomous clock evolves under a time-independent energy gap Eh. The second qubit is connected to a cold bath Hamiltonian, such that a steady stream of ticks is recorded at temperature Tc
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 Zv → −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 IV. PERFORMANCE OF THE CLOCK renewal process; i.e., the time between any pair of 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 ttick and the Δ related to the entropy production. Specifically, we consider standard deviation ttick. The resolution of the clock is then here the heat dissipated into the cold bath per tick of the ν 1 clock, tick ¼ =ttick; ð4Þ
Qc ¼ðd − 1ÞEc: ð3Þ i.e., the average number of ticks the clock provides per 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 [Qh ¼ðd − 1ÞEh], represents the funda- equal to the average time between ticks. Since the waiting mental minimum energy expenditure associated with one times are independent,ffiffiffi the uncertainty in the time of the nth p Δ tick of the clock. This is because, in principle, a large part tick is simply n ttick, and therefore of the energy Eγ carried away by the emitted photon could 2 be captured and recycled (e.g., dumped back into the hot ttick N ¼ Δ : ð5Þ bath). Consequently, the dissipated heat (3) is associated ttick with an irreversible entropy production of at least βcQc per tick. Figure 2 illustrates the intimate relationship between the ν The performance of our autonomous clock is quantified accuracy N and the resolution tick versus the dissipated by the resolution and accuracy of its ticks. By resolution, energy Qc, 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 Qc, for various values of the resolution νtick. At low energy, the accuracy increases linearly with the dissipated energy, independently of the resolution. However, for higher energies, the accuracy saturates. (b) Resolution νtick as a function of dissipated heat per tick Qc, 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 kBTc ¼ Ew, kBTh ¼ 1000Ew and g ¼ ℏγ ¼ ℏΓ ¼ 0.05Ew, while the ladder dimension d and cold qubit energy Ec are varied independently. Note that d ≥ 10 for all of the plotted points; thus, kBTc ¼ Ew ≪ Eγ ¼ðd − 1ÞEw, and we can safely ignore the absorption of a photon (i.e., the reverse of the decay process).
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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 d Quite intuitively, the resolution is inversely proportional to the dimension d, corresponding to the “height” of the FIG. 3. Accuracy N versus dissipated energy Q for various ladder, but it is proportional to the difference of transition c − “ ” values of the dimension d of the ladder, according to the rates p↑ p↓, which quantify the speed at which the load approximation (8) with the same bath temperatures as in Fig. 2. climbs. 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 djZvj; 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 Zv relationship between the two is nontrivial and the trade- and d. In turn, the bias Zv 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 seen in Fig. 2(a). In the next section, we recover this ðβ − β ÞQ − β Eγ N ¼ d tanh c h c h : ð8Þ behavior analytically in the weak-coupling regime. 2d
V. ACCURACY IN THE WEAK-COUPLING LIMIT Note, however, that the relation between Zv and the heat 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 c in the virtual qubit, saturating at jZvj → 1 as Qc → ∞. Zv 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 Qc. 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 −βvEw downwards (p↓) satisfy p↑=p↓ ¼ e as a consequence β − β Q − β Eγ → ð c hÞ c h of detailed balance. This description of the clock is derived N 2 : ð9Þ in Appendix C as a perturbative approximation to the two-qubit engine, which becomes exact in the limit of Noting that Qh ¼ Qc þ Eγ, we can recast the above in vanishingly small engine-ladder coupling. We also make the illustrative form
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β Q − β Q ΔS regime where the machine works reversibly. A finite power, N → c c h h ¼ tick ; ð10Þ 2 2 however, is essential for the resolution of any autonomous clock: A clock working at Carnot efficiency ticks infinitely Δ where Stick is the increase in the entropy of the clock in a slowly. Hence, even in the rather artificial regimes of single tick. We may interpret the regularity of each tick as Tc → 0 or Th → ∞, the requirement of a finite resolution 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 10 [30] runs at resolutions of the order of 10 Hz, and an efficiency, i.e., without increasing entropy. 16 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 ηC ¼ 1 − Tc=Th. Every unit of heat dissipated can be spent to increase either 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) imposes a limit on the achievable accuracy and resolution, National Science Foundation (SNF) through the project 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 of the register. Here, we give a detailed description of the two-qubit heat Here, these considerations only concern the scenario of engine of Ref. [23], which represents the pointer of the minimal autonomous clocks, i.e., where the resources autonomous quantum clock. The machine consists of two exploited to operate the clock are simply two thermal qubits, each one connected to a thermal bath. The first qubit baths at different temperatures. While these arguably with energy gap Eh is connected to the bath at Th. The represent the most abundant resources found in nature second qubit is connected to the bath at Tc and has energy [25], it would be interesting to consider other quantum gap Ec
ACKNOWLEDGMENTS Xd−1 1 0 1 0 1 Hint ¼ g ðj icj ihjk þ iwh jch jhhkjw þ H:c:Þ: ðA3Þ We are grateful to Ämin Baumeler, Nicolas Gisin, k¼0 Patrick Hofer, Daniel Patel, Sandu Popescu, Gilles Pütz, Sandra Rankovic Stupar, Renato Renner, Christian The machine will be operated in the weak-coupling regime, Klumpp, and Stefan Wolf for fruitful discussions. M. H. i.e., g ≪ Ec, Ew. Note that our design constraint on the acknowledges funding from the Swiss National Science energies (A2) ensures that Hint has a significant effect even 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 Th)to Commission (STREP RAQUEL), the Spanish MINECO, the second (at Tc). 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 Eh from the first qubit can be transferred to a quantum of Catalunya CIRIT, Project No. 2014-SGR-966, the Swiss energy Ec in the second qubit, while the remaining energy
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Eh − Ec ¼ Ew is transferred to the ladder. This process The rates γh;c determine the overall time scale of the 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 Tc. The ladder deliver work (i.e., to raise the energy of the ladder), we need is designed so that only the highest energy transition − 1 → 0 to ensure that the first process is more likely than the jd iw j iw couples significantly to the electromag- 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 − 1ÞEw, 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 Tc is assumed to be low enough that we can ignore the 0 1 → 1 0 1 0 → − 1 j icj ihjkiw j icj ihjk þ iw; ðA4Þ reverse transition j iw jd iw, wherein the ladder absorbs a photon while in the ground state; i.e., we require we simply demand that the probability p1 of occupying the that k T ≪ ðd − 1ÞE . 0 1 B c w state j icj ih is larger than the probability p0 of occupying To quantify the ticks of the clock, in principle, one would 1 0 the state j icj ih; recall that the ladder is only weakly 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 : ðA5Þ single tick. Th Tc We describe the dynamics of the clock in the “no-click” subspace, i.e., the subensemble ρ0ðtÞ conditioned on no The effect of the engine on the ladder is determined by the spontaneous emission having occurred up to time t.We 0 1 1 0 ’ two states j icj ih and j icj ih, which define the machine s 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 e h h h h e c c c c E − E ¼ E , hence resonant with the ladder’s energy ρ 0 ⊗ ⊗ 0 0 h c w 0ð Þ¼ Z Z j iwh j; ðB3Þ spacing, and virtual temperature determined by the pop- h c −β E ulation ratio p1=p0 ¼ e v w . The load will thus effectively where Z are the partition functions necessary for “thermalize” with the virtual qubit. This causes the load to c;h 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 ρ0ðtÞ follows from the master equation (ℏ ¼ 1): APPENDIX B: DYNAMICS OF THE CLOCK dρ0 In order to model the dynamics of the pointer and ρ † − ρ L ρ L ρ ¼ ið 0Heff Heff 0Þþ h 0 þ c 0; ðB4Þ compute the distribution of ticks, we use the following dt 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- Heff ¼ H0 þ Hint þ Hse, with spontaneous emission operator described by the contribution −β † L ¼ γ D½σ þγ e jEj D½σ ; ðB1Þ Γ j j j j j − i − 1 − 1 Hse ¼ 2 jd iwhd j: ðB5Þ for j ¼ h, c. Here, we defined the qubit lowering operators σ 0 1 ρ j ¼j ijh j, and the dissipator in Lindblad form As a result of the non-Hermitian contribution, 0ðtÞ does not stay normalized. The trace of the conditional density 1 † † P0 t ρ0 t D½L ρ ¼ LρL − fL L; ρg: ðB2Þ operator ð Þ¼Tr½ ð Þ corresponds to the probability 2 that a tick has not yet occurred. The probability density
031022-8 AUTONOMOUS QUANTUM CLOCKS: DOES … PHYS. REV. X 7, 031022 (2017) X WðtÞ of the waiting time between two consecutive ticks μðtÞ¼ nqðn; tÞ; ðC2Þ then follows from n X 2 2 dP0 σ ðtÞ¼ (n − μðtÞ) qðn; tÞ: ðC3Þ WðtÞ¼− : ðB6Þ dt n For our purposes, we need only the mean and variance of The speed of the ladder is determined by a simple the waiting time, which are given by calculation, Z ∞ μ X ττ τ d ðtÞ dqðn; tÞ ttick ¼ d Wð Þ; ðB7Þ ¼ n ¼ p↑ − p↓: ðC4Þ 0 dt dt Z n ∞ Δ 2 τ τ − 2 τ ð ttickÞ ¼ d ð ttickÞ Wð Þ: ðB8Þ The variance may be similarly calculated from 0 2 X dσ ðtÞ dqðn; tÞ ¼ ((n − μðtÞ)2 dt dt APPENDIX C: BIASED RANDOM WALK n APPROXIMATION dμðtÞ − 2(n − μðtÞ) qðn; tÞ): ðC5Þ In this appendix, we determine the accuracy of the dt ’ 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 dσ2ðtÞ 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 per unit time to move upward and a rate to move down, and We are now in a position to find the relevant quantifiers the ratio of the rates is given by the ratio of populations of of the clock. The average time between ticks is taken to be the virtual qubit. This is an accurate description in the the time for the ladder to travel from the bottom to the top regime where the thermal couplings are much larger of its spectrum of d eigenvalues, than the interaction between the engine and the ladder and the spontaneous emission rate (see the following d d ttick ¼ ¼ ; ðC7Þ section for details). Under this assumption, the density dμðtÞ=dt p↑ − p↓ operator of the ladder is diagonal and can be replaced by a vector of populations of the energy levels. The second where, for simplicity, we replace d − 1 by d since the assumption is that the dimension of the ladder is large dimension of the ladder has been assumed to be large. ν enough so that, for most of its evolution, the population The resolution tick, i.e., the number of ticks per unit time, is distribution does not feel the boundedness of the ladder the inverse of ttick, Hamiltonian. p↑ − p↓ ν From the preceding arguments, the state of the ladder can tick ¼ ; ðC8Þ be described by a time-dependent probability distribution d on a grid of integers (that labelP the energy levels) qðn; tÞ, corresponding to Eq. (6). ∈ Z 0 1 where n , qðn; tÞ > , and nqðn; tÞ¼ . 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 2 σ p↑ p↓ p↓ of jumping to the previous integer. An equation of Δσ2 d ðtÞ þ ¼ ttick ¼ d : ðC9Þ 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;tÞþp↓qðn þ 1;tÞ dt uncertainty in a single tick is determined solely by − ðp↑ þ p↓Þqðn; tÞ: ðC1Þ the uncertainty in when the ladder reaches the top (i.e., 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 pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi moves up, as well as how much it spreads on the way. We σ p↑ p↓ Δ ðt ¼ ttickÞ d þ denote the mean and variance of the distribution by μ and ttick ¼ ¼ : ðC10Þ 2 dμðtÞ=dt p↑ − p↓ p↑ − p↓ σ , respectively,
031022-9 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017)
H ρ ρ † − ρ The accuracy N is defined as the number of ticks until the se ¼ ið Hse Hse Þ; ðD3Þ clock is uncertain by a single tick. This implies that the variance of the load has grown to the size of the entire 2 2 and similarly for H . ladder, σ ¼ d . It follows that int We transform the density operator to a dissipative ρ~ −L0tρ p↑ − p↓ interaction picture defined by 0ðtÞ¼e 0ðtÞ. The time N ¼ d ; ðC11Þ dependence of superoperators is given, in this picture, by p↑ þ p↓ H~ −L0tH L0t H~ H intðtÞ¼e inte and seðtÞ¼ se. Following the which is equivalent to Eq. (7) since p↑=p↓ ¼ p1=p0. standard perturbative argument [43], we obtain
APPENDIX D: DERIVATION OF THE BIASED Z dPρ~0 t H Pρ~ 0PH~ H~ 0 Pρ~ 0 RANDOM WALK MODEL ¼ se 0ðtÞþ dt intðtÞ intðt Þ 0ðt Þ; ðD4Þ dt 0 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 t0 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 0 infinity, and making the replacement ρ~0ðt Þ → ρ~0ðtÞ. These this (essentially classical) description of the pointer on a steps are justified by the assumption that γj ≫ 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~0 t changes appreciably. Γ ð Þ ladder coupling g and the spontaneous emission rate are Equation (D4) is then simplified by expanding the γ both small in comparison to the thermal dissipation rates c;h. commutators, tracing over the engine qubits, and then γ ≫ Γ In the limit of j g, , 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 ρwðtÞ is expressed in the eigenbasis of ladder, ρwðtÞ¼Trh;c½ρ0ðtÞ . We introduce the projector Bw. Since, by assumption, there is no initial coherence [see Eq. (22)], we quote only the result for the populations Pρ0ðtÞ¼ρwðtÞ ⊗ ρh ⊗ ρc; ðD1Þ where ρh;c denotes a local thermal state of the hot or cold dρw † qubit, for j ¼ h, c, ¼ p↓D½B ρ þ p↑D½B ρ dt w w w w 1 † −β E σ σ Γ ρ ¼ e j j j j ; ðD2Þ − − 1 − 1 ρ ρ − 1 − 1 j ðjd iwhd j w þ wjd iwhd jÞ: Zj 2
−βjEj ðD5Þ while Zj ¼ 1 þ e is the corresponding partition func- tion. Writing Eq. (23) as dρ0=dt ¼ Lρ0, we decompose the L L H H q Liouvillian as ¼ 0 þ se þ int, where we defined the Introducing the probability vector with elements q ρ q q Hamiltonian superoperator nðtÞ¼Tr½ wðtÞjniwhnj , we have d =dt ¼ A , with
0 1 − B p↑ p↓ C B C B p↑ −ðp↑ þ p↓Þ C B C B . C B . C A ¼ B . C: ðD6Þ B C B − C @ ðp↑ þ p↓Þ p↓ A p↑ −ðp↓ þ ΓÞ
Z ∞ This is equivalent to Eq. (C1) for the probabilities 2 † † 2 iEwt σ σ 0 σ σ 0 p↓ ¼ g dte h hðtÞ hð Þ cðtÞ cð Þi; ðD7Þ qnðtÞ¼qðn; tÞ, but with an additional term proportional 0 Γ to describing spontaneous decay from the upper level. Z ∞ The forward and backward rates are Laplace-transformed 2 − † † 2 iEwt σ σ 0 σ σ 0 correlation functions of the engine qubits, p↑ ¼ g dte h hðtÞ hð Þ cðtÞ cð Þi; ðD8Þ 0
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 ρh ⊗ ρc, while the operator time dependence is given by stochastic model, which in turn would lead to a higher L† † 0t σh;cðtÞ¼e σh;c, where L0 is the adjoint Liouvillian accuracy. Clocks that are even more coherent (while not † defined by Tr½QL0ðPÞ ¼ Tr½L0ðQÞP for arbitrary necessarily autonomous) have been observed [16] to spread operators P and Q. Explicitly, we have σjðtÞ¼ much less than thermal clocks. The possibility of achieving expð−iEjt − γjZjt=2Þσj for j ¼ h, c, implying that more accurate clocks via the use of stronger couplings and coherence is thus an important direction for future work. 2 −β 4g e cEc p↓ ¼ ; ðD9Þ ZhZcðγhZh þ γcZcÞ
2 −β 4g e hEh [1] T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, p↑ ¼ Z Z γ Z γ Z ; ðD10Þ B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S. h cð h h þ c cÞ Safronova, G. F. Strouse, W. L. Tew, and J. Ye, Systematic Evaluation of an Atomic Clock at 2 × 10−18 Total from which one readily verifies that p↑=p↓ ¼ Uncertainty, Nat. Commun. 6, 6896, 2015. −ðβ E −β E Þ −β E e h h c c ¼ e v w . Self-consistency of the Born-Mar- [2] N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, ≪ γ kov approximation requires that p↓, p↑ j. N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, An Atomic Clock with 10−18 Instability, APPENDIX E: MODEL-INDEPENDENT LIMITS Science 341, 1215 (2013). ON THERMALLY RUN CLOCKS [3] C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Frequency Comparison of Two High- We have argued that the accuracy of the autonomous Accuracy Alþ Optical Clocks, Phys. Rev. Lett. 104, 070802, clock is constrained by the amount of heat that the clock 2010. dissipates as it provides ticks. This was obtained by relating [4] W. Pauli, Handbuch der Physik (Springer, Berlin, 1926), – the upwards bias of the ladder’s evolution to the ratio of Vol. 23, pp. 1 278. populations of the virtual qubit, which, for the two-qubit [5] N. Margolus and L. Levitin, The Maximum Speed of Dynamical Evolution, Physica D (Amsterdam) 120, 188 engine, is found to satisfy [23] (1998). β β − β [6] L. Mandelstam and I. Tamm, The Uncertainty Relation wEw ¼ hEh cEc: ðE1Þ Between Energy and Time in Non-relativistic Quantum 9 − 1 Mechanics, J. Phys USSR , 249 (1945). Multiplying by the spectral width of the ladder, d , and [7] I. Marvian, R. W. Spekkens, and P. Zanardi, Quantum Speed since (Ec ¼ Eh þ Ew), we find that Limits, Coherence, and Asymmetry, Phys. Rev. A 93, 052331 (2016). βwEγ ¼ βhðQc þ EγÞ − βcQc: ðE2Þ [8] D. P. Piers, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, Generalized Geometric Quantum Speed This expression may be intuitively understood as follows. Limits, Phys. Rev. X 6, 021031 (2016). Every time the thermal machine prepares the virtual qubit [9] A. Miyake, Entropic Time Endowed in Quantum Correla- tion, arXiv:1111.2855. in the appropriate state that is ready to exchange Ew with [10] Ä. Baumeler and S. Wolf, Causality-Complexity- the external system, it must also absorb Eh from the hot Consistency: Can Space-Time Be Based on Logic and reservoir and dissipate Ec to the cold reservoir. This is, in fact, true not only for the two-qubit engine but Computation?, arXiv:1602.06987. [11] D. Page and W. Wootters, Evolution without Evolution: also for a large class of autonomous quantum thermal Dynamics Described by Stationary Observables, Phys. Rev. machines [27] (in the weak-coupling regime). In other D 27, 2885 (1983). words, while the bias can be tweaked by changing the [12] W. Wootters, Time Replaced by Quantum Correlation, Int. machine design from the two-qubit engine to more com- J. Phys. Sci. 23, 701 (1984). plex constructions, it is always constrained to obey [13] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum Time, Eq. (E1). Therefore, the trade-off between accuracy and Phys. Rev. D 92, 045033 (2015). power for the autonomous clocks is a general feature not [14] A. Peres, Measurement of Time by Quantum Clocks, Am. J. limited to the model in this paper. Phys. 48, 552 (1980). On the other hand, it would be interesting to investigate [15] J. Lindkvist, C. Sabin, G. Johansson, and I. Fuentes, Motion clocks that deviate from weak coupling, as they may be and Gravity Effects in the Precision of Quantum Clocks, Sci. Rep. 5, 10070 (2015). able to outperform stochastic models via the buildup of [16] M. Woods, R. Silva, and J. Oppenheim, Autonomous coherence. Even in the simplest case of the two-qubit Quantum Machines and Finite Sized Clocks, engine, there is some buildup of coherence in the subspace arXiv:1607.04591. of the interaction between engine and ladder, which is [17] V. Buzek, R. Derka, and S. Massar, Optimal Quantum maintained as the ladder moves upward. In Ref. [23], this is Clocks, Phys. Rev. Lett. 82, 2207 (1999). observed to prevent the ladder’s energy distribution from [18] P. Erker, The Quantum Hourglass (ETH, Zürich, 2014).
031022-11 PAUL ERKER et al. PHYS. REV. X 7, 031022 (2017)
[19] S. Rankovic, Y.-C. Liang, and R. Renner, Quantum Clocks [33] Y. Guryanova, S. Popescu, A. Short, R. Silva, and P. and Their Synchronisation—The Alternate Ticks Game, Skrzypczyk, Thermodynamics of Quantum Systems with arXiv:1506.01373 Multiple Conserved Quantities, Nat. Commun. 7, 12049 [20] L. Boltzmann, On Certain Questions of the Theory of (2016). Gases, Nature (London) 51, 413 (1895). [34] N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, [21] N. Linden, S. Popescu, and P. Skrzypczyk, How Small Can Microcanonical and Resource-Theoretic Derivations of Thermal Machines Be? The Smallest Possible Refrigerator, the Thermal State of a Quantum System with Noncommuting Phys. Rev. Lett. 105, 130401 (2010). Charges, Nat. Commun. 7, 12051 (2016). [22] A. Levy and R. Kosloff, Quantum Absorption Refrigerator, [35] M. Lostaglio, D. Jennings, and T. Rudolph, Thermodynamic Phys. Rev. Lett. 108, 070604 (2012). Resource Theories, Non-commutativity and Maximum [23] N. Brunner, N. Linden, S. Popescu, and P. Skrzypczyk, Entropy Principles, New J. Phys. 19, 043008 (2017). Virtual Qubits, Virtual Temperatures, and the Foundations [36] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. of Thermodynamics, Phys. Rev. E 85, 051117 (2012). Kuhnert, W. Rohringer, I. Mazets, T. Gasenzer, and J. [24] E. T. Jaynes, Information Theory and Statistical Mechanics Schmiedmayer, Experimental Observation of a Generalized 106 I, Phys. Rev. , 620 (1957). Gibbs Ensemble, Science 348, 207 (2015). [25] C. Gogolin and J. Eisert, Equilibration, Thermalisation, and [37] W. Pusz and S. Woronowicz, Passive States and KMS States the Emergence of Statistical Mechanics in Closed Quantum for General Quantum Systems, Commun. Math. Phys. 58, 79 Systems, Rep. Prog. Phys. , 056001 (2016). 273 (1978). [26] For an operational definition of accuracy that is independent [38] S. Campbell and S. Deffner, Trade-off Between Speed and of background time, see Ref. [19]. Cost in Shortcuts to Adiabaticity, Phys. Rev. Lett. 118, [27] R. Silva, G. Manzano, P. Skrzypczyk, and N. Brunner, 100601 (2017). Performance of Autonomous Quantum Thermal Machines: [39] A. S. L. Malabarba, A. J. Short, and P. Kammerlander, Hilbert Space Dimension as a Thermodynamic Resource, Clock-Driven Quantum Thermal Engines, New J. Phys. 94 Phys. Rev. E , 032120 (2016). 17, 045027 (2015). [28] R. Landauer, Irreversibility and Heat Generation in the 5 [40] R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, Computing Process, IBM J. Res. Dev. , 183 (1961). Quantum Clock Synchronization Based on Shared Prior [29] D. Reeb and M. M. Wolf, An Improved Landauer Principle Entanglement, Phys. Rev. Lett. 85, 2010 (2000). with Finite-Size Corrections, New J. Phys. 16, 103011 [41] P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. (2014). Sorensen, J. Ye, and M. D. Lukin, A Quantum Network [30] R. Li, K. Gibble, and K. Szymaniec, Improved Accuracy of of Clocks, Nat. Phys. 10, 582 (2014). the NPL-CsF2 Primary Frequency Standard: Evaluation of [42] Note that this analysis is based only on the statistics of the Distributed Cavity Phase and Microwave Lensing model and is thus independent of the details of the thermal- Frequency Shifts, Metrologia 48, 283 (2011). ization model for describing the coupling between the qubits [31] L. Maccone, A Quantum Solution to the Arrow-of-Time and the baths. Dilemma, Phys. Rev. Lett. 103, 080401 (2009). [43] H. P. Breuer and F. Petruccione, The Theory of Open [32] L. Mlodinow and T. Brun, Relation Between the Psycho- Quantum Systems (Oxford University Press, Oxford, logical and Thermodynamic Arrows of Time, Phys. Rev. E 89, 052102 (2014). 2007).
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