Entropic costs of extracting classical ticks from a quantum clock

Abstract

We experimentally realize a quantum clock by using a charge sensor to count charges tunneling through a double quantum dot (DQD). Individual tunneling events are used as the clock's ticks. We quantify the clock's precision while measuring the power dissipated by the DQD and, separately, the charge sensor in both direct-current and radio-frequency readout modes. This allows us to probe the thermodynamic cost of creating ticks microscopically and recording them macroscopically. Our experiment is the first to explore the interplay between the entropy produced by a microscopic clockwork and its macroscopic measurement apparatus. We show that the latter contribution not only dwarfs the former but also unlocks greatly increased precision, because the measurement record can be exploited to optimally estimate time even when the DQD is at equilibrium. Our results suggest that the entropy produced by the amplification and measurement of a clock's ticks, which has often been ignored in the literature, is the most important and fundamental thermodynamic cost of timekeeping at the quantum scale.

Figures (11)

• Figure 1: Concept versus experiment. (a) A DQD with charge occupation states $\ket{0},\ket{L},\ket{R}$ stochastically exchanging charges with thermal environments can be used as a clock by identifying quantum jumps as ticks (experimentally $\sim 60 k_B$ per tick, see value estimated as a maximum from Fig. \ref{['fig:panel2']}(a)). Classically recording ticks to make them redundantly accessible for macroscopic agents requires measuring the quantum state and turning it into redundant information (right side). We found the cost of extraction to lie between $\sim 10^{9}\,k_B$ and $\sim 10^{11} k_B$ per tick in our experiment, depending on the measurement method, see respective values shown in Fig. \ref{['fig:panel2']}(b,c). Nevertheless readout dissipation is several orders of magnitude smaller than macroscopic dissipation of the order $\gtrsim 10^{23}k_B$. (b) Scanning electron microscope image of a device similar to the one used in this experiment. Blue circles represent a DQD and the eye represents a charge sensor dot used for classical readout. $V_{L}$ and $V_{R}$ are the voltages applied to gate electrodes that control the number of charges within the left and right dot respectively. Charge transport can occur from either source to drain, drain to source, or any intermediary process (black arced arrows), influenced by an applied source-drain voltage $V_{\rm DQD}$ across the DQD. (c) Sketch of a charge stability diagram. Intersection of charge occupation states $\ket{0}$, $\ket{L}$ and $\ket{R}$ is a triple point. Fluctuations about this point cause a cycling of states (black arrow) which correspond to a forward clock tick. (d) The simplest case of a forward clock tick measured with current in relation to lab time. The trace forms a discrete, three-level telegraph signal that corresponds to the charge occupation states in (c). (e) Schematic representation of the same forward clock tick as (d) in terms of charge transport. Initially, a charge enters from the source to the left dot $\ket{0} \rightarrow \ket{L}$, moves to the right dot $\ket{L} \rightarrow \ket{R}$, and finally moves to the drain $\ket{R} \rightarrow \ket{0}$. (f) Readout trace from both dc (top) and rf reflectometry (bottom), with respect to lab time. Black trace highlights the result of our level-identification algorithm. Shaded regions indicate ticks: red for forward and blue for backward.
• Figure 2: Clock precision and entropy trade-off (empirical data and theory comparison). (a) Here, the precision $\mathcal{S}$ of the clock defined by the net number of charges $\Theta_{\rm net} = \nu^{-1} N(t)$ transported across the DQD is plotted against the entropy dissipated per tick (scale on the right). Furthermore, the normalized precision of the optimal estimator $\Theta_{\rm opt}$ from eq. \ref{['eq:Theta[i]']} is shown, also as a function of entropy per charge transport (scale on the left). The dots indicate empirical estimates for the precision (SM supp Sec. \ref{['supp:empirical_precision']}), and stars the theory prediction from the Markovian model (SM supp Sec. \ref{['supp:theoretical_precision']}). Drifts in the device can lead to changes in the transition rates and ultimately deviations between theoretical and empirical values. (b) Data obtained with the dc measurement of the sensor dot showing how precision $\mathcal{S}$ for $\Theta_{\rm net}$ and $\Theta_{\rm opt}$ scales as a function of the dissipation rate in the sensor dot. (c) Data and dissipation for the reflectometry method, exhibiting three orders of magnitude lower dissipation than the dc method. Error bars show the estimated standard error as calculated in the SM supp.
• Figure 3: (Extended Data) Measurement traces of length $10\,{\rm s}$ using both the dc (b) and the rf reflectometry method (d). The trace is taken for a DQD bias $V_{\rm DQD}=0\,{\rm mV}$, a sensor dot bias $V_{\rm cs}=0.345\,{\rm mV}$, and frequency $f_{\rm rf}=114\,{\rm MHz}$ for the rf reflectometry drive. (a) Histogram $p(I)$ of the current values for the full $30\,{\rm min}$ data trace showing the emergence of three peaks corresponding to the DQD charge occupation states, shown as the dashed lines. (c) Analogously, the histogram $p(V)$ for the demodulated Y-component of rf sensor signal is shown here (details in Sec. \ref{['supp:details_reflectometry']} of the SM supp). (e) The discretized signal $S(t)$ for both the dc and rf reflectometry methods, obtained from the measurement traces by identifying one of the three states from the noisy signal. For most times, the discretized signals from the two readout methods agree with each other. In general, rf is more sensitive than dc sensing, leading to instances where the two discretized signals are not exactly identical. Nevertheless, as can be seen, this difference does not significantly affect level identification.
• Figure 4: (a) Charge stability diagram. Colormap indicates the current flowing through the sensor dot as a function of $V_{L}$ and $V_{R}$. Indices m and n indicate the number of charges in the left and right dot, respectively. Note the charges are holes, not electrons. Boundaries of the honeycomb pattern correspond to the transitions between the charge occupation states of the DQD. The intersection of three boundaries indicates a triple point. (b) Closer view of a triple point in the latched regime.
• Figure 5: dc and rf sensor circuit. The device is mounted on a printed circuit board containing a lumped element rf cavity with an integrated bias tee. Gray wavy lines show rf signal propagation and gray straight lines the dc current path. The rf input signal is generated by a Zurich Instruments UHFLI lock-in amplifier. The reflected signal is amplified at 4K using a CITLF2 low-noise cryogenic amplifier. Dashed purple lines indicate the different stages of the dilution refrigerator. Dual-phase demodulation of the reflected rf signal occurs within the lock-in amplifier (blue shaded region). LPF stands for low pass filter.