Autonomous quantum clocks: does thermodynamics limit our ability to measure time?

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.


I. INTRODUCTION
The passage of time is a fundamental prerequisite for our existence. Hence time has always been at the heart of natural sciences, from Greek philosophy [1][2][3] to modern physics [4][5][6]. Nevertheless, while it appears in all physical theories and at all energy scales, time remains one of the most mysterious concepts in physics, in particular in the context of quantum mechanics (even leaving aside relativistic considerations). This arguably represents one of the main obstacles towards the unification of physical theories.
In order to make progress on these questions, and eventually find out what time really is [7,8], it is a necessary first step to find out what can we say about time? To that end it is of fundamental importance to study how well we can measure time and what are the ultimate limits of such a measurement process.
The concept of time in quantum theory has been explored following various directions. The relation between time and energy, the physical quantity that is time invariant, has led to fundamental limitations in the form of quantum speed limits [9][10][11][12][13]. Another approach aimed at promoting time from being a mere classical variable to a fully quantum description [14][15][16]. Notably quantum evolution is here captured via the notion of correlations. Finally, various models of quantum systems aimed at measuring time, i.e. quantum clocks, were proposed; see e.g. [17][18][19][20]. These models typically consider a specific degree of freedom of a quantum system, prepared in a judiciously chosen initial state, then subjected to a unitary evolution, and finally measured. The measurement result is then interpreted as a time interval measurement, the precision of which can then be related to the properties of the clock (e.g. its dimension [18,19]).
However, the procedures of the state preparation and the measurement are usually not discussed explicitly. These models thus allow one to measure a time interval, e.g. for implementing a given unitary operation (by timing an interaction), but cannot be considered a complete model of a quantum clock.
Indeed, a crucial feature of complete quantum clock is to continuously provide a time reference to an external observer. It is thus essential that the clock model explicitly specifies the process of information read-out. This naturally leads to the fact that a clock must consist of two parts [21], as shown in Fig. 1. The first part is the pointer, i.e. the physical system, the internal dynamics of which will be effectively dictated by the passage of time. The second part is the register, which will mediate the transfer of information from the system to an external observer. The register will thus store classical information obtained about the evolution of the pointer. In other words the register records the ticks of the clock.
There is thus an asymmetric flow of information between the two parts of the clock, which makes the process irreversible (and thus singles out a direction for the flow of time). This naturally connects the problem to the foundations of statistical physics and thermodynamics, in particular to the second law [22], which implies that the evolution of physical systems towards equilibrium is universal (as rigorously proven for various notions of thermalisation [23]). In the present context, this immediately implies that physical systems in equilibrium will be unsuitable for building a clock, as they are invariant under time evolution. Thus for any system to serve as a clock it has to be out of equilibrium. Now the main question is: how to create a clock from an out-of-equilibrium system, and which are the minimal resource required to maintain the clock? Importantly, the resources consumed by the clock should not contain any reference to time. This is to ensure a fair bookkeeping, i.e. that the resources consumed by the clock do not require a measurement of time to be produced.
Below we will describe simple autonomous quantum clocks, powered by what is arguably the minimum out-ofequilibrium resource: two heat baths at different temperatures. This model captures the fundamental relation between the performance of the clock, in terms of accuracy and resolution, and an irreversible entropy production. This will allow us to explore the fundamental limitations that any (non-relativistic) clock experiences through the very laws of quantum mechanics, and to characterize the resources required to maintain a working clock.

II. AUTONOMOUS QUANTUM CLOCKS
The model of an autonomous quantum clock is illustrated in Fig. 2. We consider a device producing a sequence of signals (ticks) registered by a classical memory. The device is thus genuinely bipartite, and consists of a pointer and a register. The pointer is a physical system interacting with the register, and thus inducing a signal then recorded onto the register.
Importantly we require the clock to work autonomously, that is without any form of external control or timing. Here we consider what is arguably the simplest possible resource to maintain the pointer out of equilibrium: two heat baths at different temperatures.
More precisely, the clock is in a laboratory at temperature T c . The only free resource is thus the Gibbs state (thermal state) of any constituent of the lab. Since the thermal state does not evolve in time, it is fundamentally time-invariant, and thus cannot induce any dynamics in the clock. To operate the clock, it is thus necessary to use some out-of-equilibrium resource state. The simplest possibility here consists of using a thermal state at a different temperature. This provides the necessary source of free energy. Importantly, this resource can be prepared deterministically without any knowledge about the internal state of the system and without any well timed operations (unitaries). We thus suppose that the clock, in fact just the pointer, has access to a heat bath at a temperature T h > T c .
The connection to the two heat baths at different temperature will induce a heat flow through the pointer, which is accompanied by an irreversible entropy production. This motivates the use of thermodynamical concepts in order to discuss the clock as an autonomous thermal machine, with the goal of producing a regular sequence of signals.
Below we first introduce an explicit model of an autonomous quantum clock. We take advantage of conceptual and technical notions recently developed for the study of autonomous quantum thermal machines [24][25][26]. The performance of the clock will be explored in particular with respect to entropy production. We will be focusing on two central figures of merit for discussing the performance of the clock. For one, it should be able to resolve as small units of time as possible. The clock's resolution can thus be defined as the number of ticks per second (as compared to an absolute clock). The second is accuracy, i.e. the average time (measured in seconds) until the clock is off by one second (again with respect to an absolute clock). The simulation of the clock model elucidates the quantitative trade-off relation that connects these figures of merit to the dissipated heat. In the final chapter we present general arguments that show that these trade-offs are inevitably exhibited by any implementation of thermal clocks.

III. MODEL
We now discuss a simple model of an autonomous quantum clock. Again, the clock is autonomous in the sense that it does not require any external control or timing. It functions only via thermal contact with two heat baths at different temperatures.
The basis of our model is a small autonomous quantum heat engine discussed in Ref. [26]. The machine consists of two qubits, each one connected to a thermal bath. The first qubit, connected to the hot bath at temperature T h , has energy gap E h . The second qubit is connected to a cold bath at temperature T c and has energy gap E h > E c . The engine delivers work to a load, represented by a system with d equally spaced energy levels, i.e. a (bounded) discrete ladder. The ladder is not connected to any heat bath, and its (constant) energy gap is denoted E w . The free Hamiltonian of the total system (two qubits and ladder) is thus given by where |1 j denotes the excited state of qubit j = h, c, and |k w denotes the state of k-th level of the ladder. As a design constraint we take that Hence the following energy levels of the total system are degenerate in energy: |0 c |1 h |k w and |1 c |0 h |k + 1 w . This allows for energy to be exchanged between the qubits and the ladder. Specifically, we consider the interaction Hamiltonian The machine will be operated in the weak coupling regime, i.e. g E c , E w . Note that our design constraint on the energies (2) ensures that H int has a significant effect even in the weak-coupling regime. To complete the model, we describe the interaction of each qubit with its heat bath via a standard thermalization model (see Appendix A).
Henceforth we refer to the joint system of ladder and (heat-)engine as pointer, since it will be the system from which the register will derive information reflecting the passage of time.
The functioning of the engine can be understood intuitively as follows. The temperature difference between the two bath will induce a heat flow in the system, from the first qubit (at T h ) to the second (at T c ). This heat flow is made possible by our design constraint (2). Specifically, a quantum of energy E H from the first qubit can be transferred to a quantum of energy E c in the second qubit, while the remaining energy E h − E c = E w is transferred to the ladder. This process corresponds to the first term in the interaction Hamiltonian (3). Indeed the reverse process is also possible, represented by the second term in (3). For the engine to deliver work (i.e. to raise the energy of the ladder), we need to ensure that the first process is more likely than the second. This can be done by judiciously choosing the parameters (energies and temperatures) as we will see now.
We follow the approach of Ref. [26], which captures in simple and intuitive terms the effect of the two-qubit engine on the ladder [27]. In order to bias the transition in the direction we simply demand that the probability p 1 of occupying the state |0 c |1 h is larger than the probability p 0 of occupying the state |1 c |0 h ; recall that the ladder is only weakly connected to the ambient heat bath. As the machine works in the weak-coupling regime, these probabilities basically depend only on the baths temperatures and the energies, the state of each qubit being close to a thermal state at the temperature of the bath. We arrive at the condition The effect of the engine on the ladder is thus captured by the two states |0 c |1 h and |1 c |0 h which define a 'virtual qubit'. The engine simply places the pointer in thermal contact with the virtual qubit, which has energy gap E h − E c = E w , hence resonant with the pointer. The pointer will thus effectively "thermalize" with the virtual qubit. Thus the main quantity of interest here is the bias of the virtual qubit where defines the temperature of the virtual qubit. Indeed, one can immediately check that the condition (5) is satisfied whenever the virtual qubit has a negative bias (or equivalently a negative temperature), i.e. a population inversion. The quantity Z v will play a central role for characterizing the performance of our clock, as we will see below.
To complete the description of our clock we must describe how the pointer interacts with the register. The top level of the ladder |d − 1 w is assumed to be unstable, and decays to the ground state |0 w by emitting a photon at energy E γ = (d − 1)E w . This photon is then detected at the register, which in turn makes the clock tick. Note that the presence of the decay channel also allows in principle for the reverse process. However, we assume here that the background temperature T c is low enough in order to make such processes negligible.
The functioning of the clock can now be summarized simply as follows. Consider the ladder to be initially in the ground state |0 w . The energies and temperatures are set such that the engine generates a virtual qubit with negative bias Z v < 0. Hence the load (or equivalently the average energy of the pointer) goes up. After some time, the ladder reaches the top level |d − 1 w , which triggers a photon emission and makes the clock tick. The load is reset to the ground state and the process is repeated.
Importantly, the movement of the ladder energy is probabilistic, and the amount of time ∆t taken for it to reach the top level will vary. The distribution of ∆t will depend on the number of levels of the ladder d, and on the bias Z v . Intuitively if the bias is small (Z v negative but close to zero), the probability for the ladder to move up is only marginally larger than its probability of going down. The probability distribution of the ladder over the levels thus rapidly becomes quite broad, which makes the clock tick at very irregular time intervals. On the other hand, if Z v → −1, i.e. the virtual qubit has essentially complete population inversion, then the probability for the ladder population to move downward is negligible, minimizing ∆t and resulting in more regular time intervals between ticks.

IV. PERFORMANCE OF THE CLOCK
To investigate the performance of our autonomous quantum clock, we simulate numerically the evolution of the above model. In particular, we are interested in the following quantities which are relevant figures of merit for characterizing the clock's performance. The resolution of the clock, ν tick , is defined as the average number of ticks the clock provides per second. Note that such a definition is standard, and used e.g. for assessing the performance of atomic clocks. Equivalently one can consider the average time interval t tick = 1/ν tick between two ticks. The accuracy of the clock is defined as the average number of ticks N before the clock is off by one tick.
These quantities are directly related to the photon counting statistics of the clock. Here we assume that after each decay the quantum state of the pointer is reset as follows. Each engine qubit is reset to a thermal state with temperature corresponding to its bath, while the pointer is reset to the ground state. This is a good approximation in the weak-coupling regime, where the machine is minimally perturbed by its coupling to the pointer. Let us define the time delay τ n between the (n − 1) th and n th ticks. It follows from the aforementioned assumption that the time delays τ n are independent, identically distributed (i.i.d.) random variables with probability density function I(τ ), known as the delay function [28]. All properties of the photon counting statistics can be expressed in terms of I(τ ). In particular, the mean and the variance of the time between consecutive ticks is given by Now let us assume that t = 0 coincides with the "zeroth" tick, which starts the clock. The m th tick occurs at time t m = τ 1 +τ 2 +. . .+τ m . Since the τ n are i.i.d. variables, the standard deviation of t m is simply √ m∆t tick . Therefore the number of ticks N before the clock is off by one tick, i.e. the accuracy, is given by Clearly, in order for the clock to deliver ticks, the engine must raise the ladder's energy and thus necessarily dissipate energy into the cold bath. Our goal now is to relate the amount of energy that needs to be dissipated to the performance of the clock. Specifically, we consider here the heat dissipated into the cold bath per tick of the clock Note that this quantity, rather than the heat supplied to the machine per tick (Q h = (d − 1)E h ), represents the fundamental minimum energy expenditure per tick. This is because, in principle, a large part of the energy E γ carried away by the emitted photon could be captured and recycled (e.g. dumped back into the hot bath). Figs. 3 illustrates the relationship between the accuracy N and the resolution ν tick versus the dissipated energy Q c . Note that here the dimension of the pointer d and the energy gap of the cold qubit E c are varied while holding all other parameters fixed. In Fig. 3, the behavior of the accuracy is investigated, while keeping the resolution fixed. We observe that at low energy, the accuracy increases linearly with the dissipated energy, independently of the resolution. However, for higher energies, the accuracy saturates. This maximal accuracy decreases as the resolution increases. Similarly, we investigate the behavior of the resolution while keeping the accuracy fixed. Again we find that the resolution first increases with dissipated energy, but then quickly saturates to a maximal value. This maximal resolution decreases as accuracy increases.
Altogether we find that the three important quantities, i.e. accuracy, resolution and dissipated energy, are intimately related. Specifically, for given dissipated energy, there is a trade-off between accuracy and resolution. In other words, engineering a good clock, featuring both high accuracy and high resolution, requires a large amount of energy to be dissipated.
Next we will see that the performance of our clock model can also be qualitatively captured by a simpler analysis, detailed below (and in greater depth in Appendix B). Focusing on the ladder, its evolution can be approximated by a biased random walk, induced by the interaction with the virtual qubit (6). That is, the probability distribution of the ladder over the levels evolves stochastically. The rates at which the ladder population moves upwards (p ↑ ) or downwards (p ↓ ) are proportional to the occupations of the upper and lower levels of the virtual qubit, p 1 and p 0 respectively. Upon reaching the top level, the ladder is reset to the ground state, and the clock ticks. Thus the tick occurs on average after a time The time distribution of a single tick can be characterized by the spread of the population distribution when the pointer approaches the top level. For sufficiently large dimension d of the ladder, so that the effects of bouncing off the top and bottom can be neglected, the uncertainty of a single tick can be estimated to be From the relation p ↑ p ↓ = p1 p0 and equation (6), we obtain In turn, via equation (10), the accuracy is found to be Thus one concludes that the accuracy of the clock is fundamentally limited by the dimension of the ladder, since |Z v | is bounded by 1. In fact, the bias |Z v | encapsulates the dependence of the clock performance on the dissipated heat. In the case of our model, using (6)- (7), the accuracy is given by where β c,h = (k B T c,h ) −1 . In fact, the relation between Z v and the heat exchanged with the two baths is more general than the model considered here ( [29], see Appendix D for discussion). This behaviour is illustrated in Fig. 4, where we plot the accuracy versus the dissipated energy for fixed dimension. Similarly as before, we observe that the accuracy increases linearly at low energy, but then saturates to a maximum value N → d as seen from equation (16). Indeed, increasing Q c leads to a stronger bias in the virtual qubit, saturating at |Z v | → 1 when Q c → ∞.
Even if the dimension is unbounded, we find that the dissipated energy also imposes a fundamental limitation. Taking the limit d → ∞, the accuracy is linearly dependent on the dissipated heat: The above analysis reveals a strong dependence of the accuracy with respect to both the dimension and the dissipated energy. Indeed, fixing either d or Q c limits the accuracy. Hence, achieving a certain accuracy requires a minimum dimension as well as a minimum dissipated energy.

V. FUNDAMENTAL LIMITS OF AUTONOMOUS CLOCKS
The simple thermal clock model we discuss above illustrates the fact that our ability to accurately measure time necessarily generates an increase of entropy (via heat dissipation). Equivalently, this implies an intrinsic work cost for measuring time. This naturally raises the question of whether this is a specificity of our model, or on the contrary a universal feature of any procedure for measuring time. Below, we argue in favor of the latter: any autonomous clock must increase entropy.
First of all, it is clear that beyond the specific model we discussed, one could consider more general designs for the thermal machine. The basic ingredient needed for the machine to operate as a clock is simply the ability to move the population of the pointer out of equilibrium, so that an unstable level generates a tick. In our model, this corresponds to reaching the top level, which triggers a decay, and a tick is then recorded by the register. This mechanism can indeed work for a variety of physical implementations of the pointer (i.e. with a more complex level structure). The ladder could comprise multiple levels which trigger a decay, while the machine could feature more than two qubits.
Nonetheless, all these possible extensions and more sophisticated designs will still have to comply with the basic laws of thermodynamics. In particular, the efficiency of the conversion of energy to a tick is fundamentally bounded by the Carnot efficiency η C = 1− T C T H . Moreover, this maximal efficiency can only be achieved in a limit where the power vanishes, corresponding to the regime where the machine works reversibly. The power, however, is essential for the resolution of any autonomous clock: a clock working at Carnot efficiency ticks infinitely slowly. Hence, even in the rather artificial regimes of T C → 0 or T H → ∞, the requirement of a finite resolution implies a minimal dissipated heat, and thus a minimal entropy production. Therefore, any autonomous clock with a finite resolution must produce entropy. Moreover, our model also illustrates a general connection between accuracy, dissipated heat and the dimension of the ladder. Specifically, increasing the accuracy requires more entropy to be produced. This can be done up to a maximal accuracy, which depends on the dimension of the ladder d [18]. An interesting open question is whether there could exist more general designs of autonomous clocks (in particular of the ladder) where this trade-off is different.
Finally, it is also worth pointing out that, while we focus here on a specific source for the entropy production of the clock (namely the heat dissipated by the thermal machine driving the clock), there will be generally additional energy costs required for operating the clock. In particular, the preparation (and reset) of the initial state of the register will generate entropy due to Landauer's erasure principle [30,31].

VI. CONCLUSION AND OUTLOOK
Our work represents a first step towards rigorously characterizing the necessary resources and limitations of the process of timekeeping. In a nutshell, we introduced the concept of autonomous quantum clocks to discuss these questions, and argued that the measurement of time inevitably leads to entropy increase. Moreover, we discussed explicitly a simple model of an autonomous quantum clock, and found that the amount of entropy produced represents an actual resource for measuring time. Every unit of heat dissipated can be spent to either increase the accuracy, or the resolution of the clock. Additionally, the dimension of a key constituent of the clock (the ladder) imposes a limit on the achievable accuracy and resolution, independently of the amount of dissipated heat. In other words, in analogy to the findings of [18,19], the Hilbert space dimension imposes a fundamental constraint on the performance of the clock. Reaching this optimal regime requires a minimal rate of entropy production. This puts on a quantitative level the intuitive connection between the second law of thermodynamics and the arrow of time (see e.g. Refs. [32,33]). In order to measure how much time has passed, we inevitably need to increase the entropy of the universe from the perspective of the register.
These considerations only concern here the scenario of minimal autonomous clocks, i.e. where the resources exploited to operate the clock are simply two thermal baths at different temperatures. While these represent the minimal (and arguably the most abundant) resources found in nature [23], it would be interesting to consider more general quantum systems, e.g. with multiple conserved quantities [34][35][36][37]. More generally, the relevant question is to what extent our choice of free resources impacts our ability to measure time. For instance, one could consider more general passive states [38], that would commute with the system Hamiltonian and thus satisfy the requirement of autonomy. Thermal clock models can furthermore be used to work out the thermodynamic cost of controlling other quantum systems [19,39,40] in an autonomous fashion, i.e. implementing locally apparent time-dependent Hamiltonians by coupling to an autonomous thermal clock. Moreover, operating two clocks in parallel could lead to a drastic enhancement of the clock's performance. While classical clocks running in parallel would not offer any fundamental improvement, one could consider quantum resources that feature coherence or entanglement [41,42]. Could these genuine quantum phenomena be used to increase our ability to measure time? We are looking forward to future research in this direction of time. Acknowledgements.
We are grateful toÄmin Baumeler, Nicolas Gisin, Sandu Popescu, Sandra Rankovic Stupar, Gilles Pütz, Renato Renner, Christian Klumpp and Stefan Wolf for fruitful discussions. MH acknowledges funding from the Swiss National Science Foundation (AMBIZIONE PZ00P2 161351) and the Austrian Science Fund (FWF) through the START project Y879-N27. MTM was financed by EPSRC via the Controlled Quantum Dynamics CDT. MW acknowledges funding from the UK research council EPSRC. RS and NB acknowledge the Swiss National Science Foundation (Starting grant DIAQ and QSIT). PE acknowledges funding by the European Commission (STREP RAQUEL), the Spanish MINECO, projects FIS2008-01236 and FIS2013-40627-P, with the support of FEDER funds, the Generalitat de Catalunya CIRIT, project 2014-SGR-966, the Swiss National Science Foundation (SNF) through the project 'Information and Physics' and the National Centres of Competence in Research Quantum Science and Technology (QSIT).
where we defined a lowering operator for the ladder as The hot and cold qubits are coupled to independent thermal reservoirs at temperatures T h = 1/(k B β h ) and T c = 1/(k B β c ) respectively, where T c < T h . The effect of each reservoir on its corresponding qubit is represented by the superoperator The rates γ h,c determine the overall time scale of the dissipative processes acting on the two engine qubits. In addition, the ladder system couples to a reservoir of electromagnetic field modes at temperature T c . The ladder is designed so that only the highest energy transition |d − 1 w → |0 w couples significantly to the electromagnetic field. This transition is associated with the emission of a photon having energy (d−1)E w , while Γ is the spontaneous emission rate. A photo-detector registers the emitted photon, producing a macroscopically measurable "tick". The detector is assumed to work with perfect efficiency and negligible time delay. Furthermore, the background temperature T c is assumed to be low enough that we can ignore the reverse transition |0 w → |d − 1 w , wherein the ladder absorbs a photon while in the ground state.
To quantify the ticks of the clock, in principle one would have to keep track of the density operator of the pointer ρ(t) for all time. However, as argued in the main text, in the weak-coupling regime, the qubit states do not change appreciably from the thermal states corresponding to equilibrium with their respective reservoirs. Each tick is thus independent of the previous ticks, and one can study the relevant quantifiers of the clock (i.e. resolution and accuracy) from the probability distribution in time of a single tick. We thus assume the pointer begins in the state where Z c,h are the partition functions necessary to normalize the state. Eq. (A5) describes the situation where the qubits are in equilibrium with their respective reservoirs, and the ladder has just decayed and been reset into the ground state (i.e. the register has just ticked). We may describe the dynamics of the clock in the "no-click" subspace, i.e. the subensemble ρ 0 (t) conditioned on no spontaneous emission having occurred up to time t. The conditional density operator ρ 0 (t) satisfies the master equation ( = 1) where the effective non-Hermitian Hamiltonian is given by H eff = H 0 +H int +H se , with spontaneous emission described by the contribution As a result of the non-Hermitian contribution, ρ 0 (t) does not stay normalized, and its trace norm corresponds to the probability that a tick has not yet occurred, from which the probability density of the time between ticks, referred to as the delay function in Sec. IV, can be calculated: Appendix B: Heuristic analysis of a Stochastic clock In this appendix we determine the accuracy of the autonomous clock from a stochastic model. Specifically, we make two simplifying assumptions. Firstly, the evolution of the pointer is simplified to a continuous biased random walk of the ladder, modulated by the populations of the virtual qubit of the two-qubit engine. That is, the ladder has a rate per unit time to move upward and a rate to move down, and the ratio of the rates is given by the ratio of populations of the virtual qubit. This is an accurate description in the regime where the thermal couplings are much larger than the interaction between the engine and the ladder and the spontaneous emission rate (see Appendix C for details). Under this assumption, the density operator of the ladder is diagonal, and can be replaced by a vector of populations of the energy levels.
The second assumption is that the dimension of the ladder is large enough so that for most of its evolution, the population distribution does not feel the boundedness of the ladder Hamiltonian. That is, the effects of bouncing off the top and bottom limits of the ladder may be neglected.
From the preceding arguments, the state of the ladder can described by a time-dependent probability distribution on a grid of integers (that label the energy levels) q(n, t), where n ∈ Z, q(n, t) > 0, and n q(n, t) = 1∀t. The evolution is determined by the forward rate p ↑ per unit time of jumping to the next integer, together with the backward rate p ↓ of jumping to the previous integer. An equation of motion of the distribution can thus be constructed: We are most interested in the ticks of the clock. To characterize the resolution and accuracy, we must understand how quickly the position of the ladder moves up, as well as how much it spreads on the way. We denote the mean and variance of the distribution by µ and σ 2 respectively (c.f. Eqs. (8)-(9) of the main text), µ(t) = n n q(n, t), σ 2 (t) = n (n − µ(t)) 2 q(n, t).
Following a similar procedure for the second and third terms, one arrives at where the angle brackets denote an average with respect to ρ h ⊗ ρ c , while the operator time dependence is given by σ h,c (t) = e L † 0 t σ h,c , where L † 0 is the adjoint Liouvillian defined by Tr[QL 0 (P )] = Tr[L † 0 (Q)P ] for arbitrary operators P and Q. Explicitly, we have σ j (t) = exp(−iE j t − γ j Z j t/2)σ j for j = h, c, implying that from which one readily verifies that p ↑ /p ↓ = e −(β h E h −βcEc) = e −βvEw . Self-consistency of the Born-Markov approximation requires that p ↓ , p ↑ γ j .
Appendix D: Model dependent vs independent limits to thermally run clocks We have argued that the accuracy of the autonomous clock is constrained by the amount of heat that the clock dissipates as it provides ticks (16). This was obtained by relating the upwards bias of the ladder's evolution to the ratio of populations of the virtual qubit, which for the two-qubit engine, is found to satisfy [26] Multiplying by the spectral width of the ladder, d − 1, and since (E c = E h + E w ), we find that This expression may be intuitively understood as follows. Every time the thermal machine prepares the virtual qubit in the appropriate state that is ready to exchange E w with the external system, it must also absorb E h from the hot reservoir, and dissipate E c to the cold reservoir. This is in fact true not only for the two-qubit engine, but has been shown to be the case for a large class of autonomous quantum thermal machines [29] (in the weak-coupling regime). That is, while the bias can be tweaked by changing the machine design from the two-qubit engine to more complex constructions, it is always constrained to obey (D1). Therefore, the trade-off between accuracy and power for the autonomous clocks is a general feature not limited to the model in this paper.
On the other hand, it would be interesting to investigate clocks that deviate from weak coupling, as they may be able to outperform stochastic models via the build-up of coherence. Even in the simplest case of the two-qubit engine, there is some build-up of coherence in the subspace of the interaction between engine and ladder, that is maintained as the ladder moves upward. In Ref. [26], this is observed to prevent the ladder's energy distribution from spreading as much as would be expected from a simply stochastic model, which in turn would lead to a higher accuracy. Clocks that are even more coherent (while not necessarily autonomous) have been observed [19] to spread much less than thermal clocks. The possibility of achieving more accurate clocks via the use of stronger couplings and coherence is thus an important direction for future work.