Precision is not limited by the second law of thermodynamics

TL;DR

The paper addresses the fundamental question of how clock precision scales with thermodynamic dissipation, proposing a fully autonomous quantum ring clock that leverages coherent transport. By engineering a three-region coupling profile in a spin-chain ring and analyzing both continuum dispersion and boundary-apt orchestrated emission, the authors show that clock precision can scale as $\mathcal{N}_\infty\sim n^{4/3}$ in the infinite-entropy limit, with waiting-time moments $\mathrm{E}[T]\sim n$ and $\mathrm{Var}[T]\sim n^{2/3}$. Importantly, when finite entropy production per tick is allowed, the precision obeys $\mathcal{N}_\Sigma=e^{\Omega(\Sigma_{\mathrm{tick}})}$ with $\Sigma_{\mathrm{tick}}=\beta\log n$ (found to be $\beta=4$ sufficient), indicating an exponential separation between clock performance and dissipation. This demonstrates that coherent quantum dynamics can surpass conventional thermodynamic precision bounds and suggests routes to high-precision, low-dissipation quantum devices and a revised perspective on the thermodynamics of timekeeping.

Abstract

Physical devices operating out of equilibrium are inherently affected by thermal fluctuations, limiting their operational precision. This issue is pronounced at microscopic and especially quantum scales and can only be mitigated by incurring additional entropy dissipation. Understanding this constraint is crucial for both fundamental physics and technological design. For instance, clocks are inherently governed by the second law of thermodynamics and need a thermodynamic flux towards equilibrium to measure time, which results in a minimum entropy dissipation per clock tick. Classical and quantum models and experiments often show a linear relationship between precision and dissipation, but the ultimate bounds on this relationship are unknown. Our theoretical discovery presents an extensible quantum many-body system that achieves clock precision scaling exponentially with entropy dissipation. This finding demonstrates that coherent quantum dynamics can surpass the traditional thermodynamic precision limits, potentially guiding the development of future high-precision, low-dissipation quantum devices.

Figures (9)

• Figure 1: The ring clock. (a) Schematic depiction. The clock consists of a ring of $n$ quantum systems (egg cups) hosting a single excitation which travels around the ring. Upon completing one cycle, the clock ticks by undergoing a biased jump from the last to the first site. (b) Level diagram of a quantum system providing a directional interface between the first and last site of the ring using a thermal gradient. The level diagram is in the single-excitation subspace, i.e., if one of the sites is in an excited state, all others are in the ground state. See Appendix Sec. \ref{['SM:level_scheme']} for the details. (c) Representative trajectory of the number of ticks $N(t)$ counted by the clock as a function of time (solid line). Due to thermal fluctuations, such a counter can jump backwards (highlighted jump). For the clock to be precise, such backwards jumps must be suppressed, using a strong thermal gradient. (d) Numerically optimized couplings, $g_j$, between the nearest-neighbor sites of the ring clock, for a ring of $n=40$ sites. Based on the dependence of the coupling coefficients on the site position in the ring, well approximated by Eq. \ref{['eq:gn']}, we identify three regions. In the initial ramp region of length $\lambda_\ell$, an excitation present in the first site is autonomously shaped into a traveling wave packet. The bulk propagation region is akin to a delay line. Finally, the boundary matching region, of length $\lambda_r$, ensures that the wave packet is absorbed from the last site without reflection.
• Figure 2: Dispersion relation of the ring, $E(k)$, for a continuum of values $k\in[0,2\pi)$ (solid line, right axis), together with a semi-log plot of the momentum space distribution $|\psi_k|^2 = |\braket{\psi_k}{\psi(t)}|^2$ of two wave packets in the bulk of the ring for $n=100$ and $n=1000$ (filled circles, left axis). The wave packets are guaranteed to be in the bulk by choosing the time $t$ as half the expected time taken by the wave packet to travel along the ring, $2gt = n/2$. The momentum space distribution is centered around $k_0=\pi/2$, indicating a strong concentration of the wave packet around the linear part of the dispersion. For larger values of $n,$ this distribution becomes narrower.
• Figure 3: Clock performance vs. ring-length, $n$, and entropy production per tick, $\Sigma_\mathrm{tick}$. In (a), we show how the expected tick time $\mathrm{E}[T]$ and the standard deviation $\mathrm{Var}[T]^{1/2}$ scale with the number of sites for the numerically optimized choice of couplings. The simulation results are in agreement with the prediction that $\mathrm{E}[T] \sim_{\rm th.} n^1$ and $\mathrm{Var}[T]\sim_{\rm th.} n^{2/3}$. Numerical values for the exponents are determined by linear regression and the uncertainty in the exponent is of order $10^{-7}$ and thus not shown in the figure. Panel (b) shows clock precision $\mathcal{N}_\infty$ as a function of ring length $n$ in the fully irreversible case (filled circles). For comparison, we show the classical and quantum bounds that limit precision by the dimension. The precision bound for classical stochastic clocks (dashed line) is linear in the dimension $\mathcal{N}_\infty \leq n$Woods2022 whereas the one for quantum clocks (full line) scales quadratically $\mathcal{N}_\infty\leq O(n^2)$Yang2020Woods2022. Finally, in (c), we visualize the main result of this work showing that clock precision grows exponentially faster than entropy production $\mathcal{N}_\Sigma = e^{\Omega(\Sigma_\mathrm{tick})}$ (filled circles). We compare this to the TUR bound $\mathcal{N}_\Sigma \leq \Sigma_\mathrm{tick}/2$ that holds for classical dissipative systems (dashed lines). For further comparison, the quadratic scaling $\mathcal{N}_\Sigma \sim \Sigma_\mathrm{tick}^2$ as recently found Dost2023, ignoring sub-leading terms and constant factors (full line).
• Figure 4: Plot of the numerically optimized parameters $\mu_\ell,\mu_r,g,\lambda_\ell,\lambda_r$ as a function of the ring length $n$. In (a), we show the ramp lengths $\lambda_\ell$ and $\lambda_r,$ finding that the left ramp grows as $n^{0.35}$ close to the predicted exponent of $1/3.$ The exponent is determined with linear regression and has uncertainty of order $10^{-6}$ and is thus not shown in the figure. The right ramp does not become larger with the ring length, indicating an agreement with previous literature on apodization showing that only a constant number of couplings have to modified in a coupled cavity array to avoid reflection at a dissipative sink Sumetsky2003. In (b), we show how the optimal coupling constants $\mu_\ell,\mu_r$ and $g$ change as a function of $n$ showing that both $g$ and $\mu_r$ quickly approach a constant value with growing $n,$ whereas $\mu_\ell$ only slowly increases, indicating that the depth of the left ramp keeps increasing with $n$.
• Figure 5: In this panel we show the comparison between the true global optimum of coupling parameters $g_j$ and the optimum for the case where the coupling parameters are given by \ref{['eq:gn_SM']}. In (a), we show the difference between the two coupling parameters for the examplary case of $n=50$ ring sites. The difference between the optimal couplings and the ones obtained using the exponential model is at most of order $10^{-2}$ in the bulk and on the boundaries the difference becomes even smaller, i.e., of the order of $10^{-3}.$ In (b), we plot how the clock precision $\mathcal{N}_\infty$ differs for those two models. We see that the relative difference vanishes for large site numbers. Several outliers due to the numerical optimization have been removed for the site numbers between $n=30$ and $n=40.$ A comparison becomes unfeasible for higher site numbers due to computational constraints in finding the global optimum with more than $50$ parameters. In comparison, the exponential model requires the optimization of only $5$ parameters regardless of the ring length.