Measuring the Thermodynamic Cost of Timekeeping

All clocks, in some form or another, use the evolution of nature toward higher entropy states to quantify the passage of time. Because of the statistical nature of the second law and corresponding entropy flows, fluctuations fundamentally limit the performance of any clock. This suggests a deep relation between the increase in entropy and the quality of clock ticks. Indeed, minimal models for autonomous clocks in the quantum realm revealed that a linear relation can be derived, where for a limited regime every bit of entropy linearly increases the accuracy of quantum clocks. But can such a linear relation persist as we move toward a more classical system? We answer this in the affirmative by presenting the first experimental investigation of this thermodynamic relation in a nanoscale clock. We stochastically drive a nanometer-thick membrane and read out its displacement with a radio-frequency cavity, allowing us to identify the ticks of a clock. We show theoretically that the maximum possible accuracy for this classical clock is proportional to the entropy created per tick, similar to the known limit for a weakly coupled quantum clock but with a different proportionality constant. We measure both the accuracy and the entropy. Once nonthermal noise is accounted for, we find that there is a linear relation between accuracy and entropy and that the clock operates within an order of magnitude of the theoretical bound.

Popular Summary

Clocks are essential building blocks of modern technology, from computers to GPS receivers. They are also essentially engines, irreversibly consuming resources in order to generate accurate ticks. But what resources have to be expended to achieve a desired accuracy? Here, we answer this question experimentally by measuring, for the first time, the entropy generated by a minimal clock.

Our clock consists of a vibrating membrane integrated into an electronic circuit: Each oscillation of the membrane provides one tick. The resources that drive the clock are the heat supplied to the membrane and the electrical work used to measure it. In operation, the clock converts these resources to waste heat, thus generating entropy. By measuring this entropy, we can therefore deduce the amount of resources consumed.

Our experiment tests the clock under different operating conditions, measuring its timekeeping error and the entropy generation in each case. In both experiment and theory, we find a fundamental relationship, at least for this type of clock, between the resources consumed and the accuracy. Surprisingly, this relationship is similar for several clock designs, both classical and quantum.

This work suggests that clocks, like other engines such as motors and computers, obey a fundamental performance limit determined by the laws of thermodynamics. There is no such thing as a free minute—at least if you want to measure it.

Figure 1

(a) For timekeeping, the clockwork consumes resources, part of which are lost as waste. The hands of the clock register the clock’s ticks. (b) Simple mechanical clock. Here, a mass is suspended from a spring and the heat from the environment excites the mass’ motion at frequency $f_0$. These vibrations are probed by a signal of power $P_{\mathrm{cav}}$. This system (the clockwork) generates a periodic signal, which is registered to identify the clock’s ticks. (c) A schematic of our electromechanical system acting as a clock. A nanometer-thick membrane is driven by a white-noise signal of power $P_{\mathrm{WN}}$. The membrane’s vibrations are probed by a rf cavity driven with a signal of power $P_{\mathrm{cav}}$. The cavity output signal, and thus the clock tick’s, are registered by an oscilloscope.

Figure 2

(a) Experimental setup. A metalized silicon nitride membrane is suspended over two metal electrodes, forming a capacitor $C_C$. One of the electrodes is connected to a rf tank circuit which acts as a readout cavity. Electrode 2 is grounded. The tank circuit is formed from a 223 nH inductor $L$, and two 10 pF capacitors $C_D$ and $C_M$. Parasitic capacitances contribute to $C_M$ and parasitic losses in the circuit are parametrized by an effective resistance $R$. The cavity can be probed by injecting a rf signal at port 1 via a directional coupler. The output signal is measured at port 2 using a vector network analyzer or a spectrum analyzer. The membrane’s motion can be excited by injecting a signal at port 3. Bias resistors allow a dc voltage $V_{\mathrm{dc}}$ to be applied to electrode 1. Red (blue) arrows indicate resources (waste) for our system. (b) $|S_{21}|$ as a function of probe frequency $f_p$. (c) One of the mechanical sidebands observed in the spectrum of the cavity output signal when an excitation tone at frequency $f_E$ is injected at port 3 and swept in frequency while the cavity is driven at its resonant frequency via port 1. The sideband power grows when $f_E$ coincides with the resonance frequency of the membrane $f_0$. (d) Demodulated readout signal $V(t)$, as a function of time, for $P_{\mathrm{cav}}=14\mathrm{dBm}$. $P_{\mathrm{WN}}=0.25\mathrm{W}$ and $P_{\mathrm{WN}}=0.063\mathrm{W}$ for the red and blue traces, respectively. The inset shows the demodulation circuit. (LO, local oscillator; BPF, bandpass filter).

Figure 3

(a) Accuracy $N$ versus white-noise power $P_{\mathrm{WN}}$ for $P_{\mathrm{cav}}$ in the range 8–14 dBm. (b) Accuracy $N$ versus the accuracy predicted from the classical model $N_C$ for $P_{\mathrm{cav}}$ in the range 8–14 dBm. The black dotted line is a guide to the eye showing the slope corresponding to $N=N_C$. The gray dotted line is chosen to approximate the slope of the displayed curves.

Figure 4

Precision $\mathfrak{p}(F)$ of the demodulated signal as a function of mechanical drive power $P_{\mathrm{WN}}$ for different settings of the cavity illumination power $P_{\mathrm{cav}}$. Crosses show precision calculated from the voltage record data using Eqs. (4) and (7). The experimental data are consistent with the limiting cases of the theory. Curves show predictions from Eq. (c31). The free parameter $\alpha$ is set by fitting to the data with $P_{\mathrm{cav}}=8\mathrm{dBm}$, then held fixed to generate the other curves. We separate the data into two figures for cavity drive powers $P_{\mathrm{cav}}$ of 8–14 dBm (for powers 16–20 dBm, see Appendix pp3).

Figure 5

Simple electrothermal clock. (a) The setup. The Johnson noise of resistor $R_0$, in equilibrium with a hot bath at temperature $T_h$, is filtered to pass frequency $f_0$ with bandwidth $f_0/Q_f$. The resulting signal, whose power is $P_0$, is passed to a matched resistor and amplifier at temperature $T_c$. (b) From the noisy voltage record (points) seen by the amplifier, we can generate clock ticks by estimating the zero crossing of each cycle using a sinusoidal fit (lines). Here $\mathrm{\Delta }t$ marks the sampling interval, $t_{\text{tick}}=1/f_0$ is the average tick interval, $\pm t_r$ is the fit range, and $\mathrm{\Delta }{t}_{\text{tick}}$ is the fit uncertainty.

Figure 6

The thermomechanical clock. (a) The setup. An optomechanical circuit consists of an $LC$ tank circuit whose capacitance, and therefore frequency, is modulated by a thermomechanical resonator at temperature $T_h$. To use the vibrations in a clock, the tank circuit is illuminated by a carrier tone $V_{\mathrm{in}}$ at frequency $f_c$, giving rise to a reflected signal $V_{\mathrm{out}}$ which is passed to a cold matched resistor and amplifier. The effect of the vibrations is to modulate the phase of $V_{\mathrm{out}}$. (b) Sketch of the resulting voltage record at the amplifier input (points), with fits (lines) from which each tick is extracted. The modulation envelope is indicated by the shaded background. Inset: power spectrum at the amplifier input, showing uniform noise background, central delta-function peak from the carrier, and two thermomechanical sidebands. In the experiment, a demodulation circuit was applied after the amplifier [as in Fig. 2] because it makes ticks practically easier to identify in the record. However, the demodulator cannot improve the clock accuracy because it cannot increase the timing information present in the signal $V(t)$; in fact, a detailed calculation would show that the accuracy is unchanged. For simplicity, the demodulator is therefore omitted from this model.

Figure 7

Precision $\mathfrak{p}(F)$ of the demodulated signal as a function of mechanical drive power $P_{\mathrm{WN}}$ for different settings of the cavity illumination power $P_{\mathrm{cav}}$ in the range 16–20 dBm. Crosses show precision calculated from the voltage record data using Eqs. (4) and (7). Curves show predictions from Eq. (c31). The free parameter $\alpha$ is set by fitting to the data with $P_{\mathrm{cav}}=8\mathrm{dBm}$, then held fixed to generate the other curves.

Figure 8

(a) Accuracy $N$ of the clock versus the white-noise power $P_{\mathrm{WN}}$ for cavity drive powers in the range 16–20 dBm. (b) Accuracy $N$ of the clock versus $N_C$ for cavity drive powers in the range 16–20 dBm.The black dotted line is a guide for the eye to show the expected gradient should the extracted accuracy and the theoretical accuracy predicted from the entropy production be equal. The dark gray dotted line shows the approximate gradient of the 20 dBm data of $N=N_C/6$ and the light gray line shows the approximate gradient of the 16 and 18 dBm data of $N=N_C/9$.

Figure 9

The demodulated cavity output signal before (a) and after (b) filtering as a function of time for input powers of 8, 10, and 12 dBm to the cavity with 0.25 W white-noise input from port 3. The traces are vertically offset for clarity.

Figure 10

(a) Noise temperature of the system $T_N$ increasing as the cavity drive power and white-noise power increase for cavity drive powers in the range 8–20 dBm. (b) The increase in the power of the sideband with increase in the white-noise power for different cavity drive powers in the range 8–20 dBm. For both (a) and (b) the higher drive powers have dotted lines and the data shown in the main text have full lines.