Exponential gain in clock precision using quantum correlated ticks

TL;DR

This paper introduces a quantum-clock concept where autonomous, correlated ticks arise from quantum correlations in a chain of coupled qubits with biased leads. By solving a free-fermion mapping and optimizing boundary couplings, the authors demonstrate an exponential improvement in clock precision with system size, achieving a diffusion-to-current ratio scaling of ${D/J \sim N^{-1.86}}$ and a waiting-time variance ${Var}[T_n] \sim \log n$. The clock’s timing emerges from a shifted Fermi-sea state ${\rm FS}^+$ and exhibits universal sine-kernel statistics, linking the results to random-matrix theory and full counting statistics. The approach remains robust to realistic imperfections and is amenable to implementation on current quantum platforms, offering a path toward ultra-fast, thermodynamically efficient clocks driven by intrinsic quantum correlations rather than external feedback loops.

Abstract

Creating precise timing devices at ultra-short time scales is not just an important technological challenge, but confronts us with foundational questions about timekeeping's ultimate precision limits. Research on clocks has either focused on long-term stability using an oscillator stabilized by a level transition, limiting precision at short timescales, or on making individual stochastic ticks as precise as possible. Here, we prove the viability of a conceptually different avenue: the autonomous self-correction of consecutive ticks by quantum correlations. This provides a new paradigm that integrates the advantages and insights from quantum transport theory to operate clocks at ultra-short timescales. We fully solve a model of coupled quantum systems and show how the emergent Pauli exclusion principle correlates the clock at the quantum level yielding an exponential advantage in precision. We furthermore demonstrate through simulations with realistic imperfections that this remarkable gain in precision remains stable providing a roadmap for implementation with contemporary quantum technologies.

Figures (8)

• Figure 1: Figures of merit and setup. a) Stochastic events define clock 'ticks' and for a clock to be precise the tick distribution should be narrow. For uncorrelated ticks, errors add up (see a), whereas correlations can suppress error accumulation (see b). c) An exemplary model based on a spin chain connected to biased leads where the current leaving on the right defines the ticks. d) The couplings between the sites are carefully chosen to exponentially suppress error accumulation. It turns out that the optimal couplings are constant in the chain's bulk but increase symmetrically towards the end to match the boundary dissipation. e) The transmission function shows the probability with which an excitation traverses the chain, which approaches a boxcar as the number of sites goes to infinity. f) Random realizations of the tick number trajectories are sketched to illustrate the linear relation between the uncertainty in the tick number and the uncertainty in the tick time.
• Figure 2: Current and waiting time fluctuations. a) For $N=20$ sites histogram data is shown for the waiting time probability $p[T_1]$ between subsequent ticks. The circles show the unconditional distribution, triangles (diamonds) show the distribution conditioned on the previous tick being faster (slower) than the expectation value. Vertical lines indicate the averages, and the solid red curve a Wigner--Dyson fit to $p[T_1]$. b) The relative current noise $D/J\sim N^{-1.86}$ in log-log scale as it decays with increasing chain length $N$. c) A Monte Carlo approximations for the variance of $T_n$, the time of the $n$th tick, is shown as a function of $n$, for different chain lengths $N$ in semi-log scale. The inset shows a log-log scale with the same data. For comparison, theoretical prediction and the diffusive scaling are shown.
• Figure 3: Number variance bulk vs. boundary for $N=40$ sites. a) Short time fluctuations are shown in a log-linear scale. The dots show results from the Monte-Carlo stochastic Schrödinger equation (SSE) evolution, obtained through histogram binning of the tick times. Error bars show the sample error. In addition, the numerically exact results for the classical current fluctuations from Eq. \ref{['eq:VarNt_COV_matrix']} are shown in dashed lines ($k=0$). The fluctuations of the coherent current at the boundary, (Eq. \ref{['eq:VarNtcoh_k']} with $k=1$, green line), closely match the fluctuations of the classical current. In contrast, the fluctuations in the bulk ($k=\lfloor N/2\rfloor$) are shown to be approximately half as large as the fluctuations at the boundary. For comparison $0.5$ times the SSE result is shown (diamond symbol). The red solid line is the analytical prediction for the bulk fluctuations in the limit of $N\rightarrow\infty$ sites (Eq. \ref{['eq:VarNT_FINALFULL']}). b) The number variance is shown over a longer time scale. In addition to part a), it can be seen here that while the bulk fluctuations ($k=\lfloor N/2\rfloor$) at short times are approximately half of those at the boundary; after very long times, they increase and match the boundary fluctuations. The $N\rightarrow\infty$ analytics maintain the logarithmic scaling indefinitely because of the unbounded chain length.
• Figure 4: Dispersion relation and occupation probabilities. Left $y$-axis: we show the dispersion relation as a function of the bulk lattice momentum. Right $y$-axis: numerical results for the occupation probabilities of the momentum modes in the stationary regime are shown for $N=20,40$ and $100$ together with the shifted Fermi sea ansatz for the limit $N\rightarrow\infty$.
• Figure 5: Scaling cross-over with imperfections. Here we show the variance of the waiting time for the $n$th tick as a function of $n$, for a chain of length $N=40$. The dots show results from a stochastic Monte Carlo simulation with $10^5$ ticks. In order of increasing variance: first, the case where the numerically optimized couplings are chosen like in the main text; second, in the presence of coupling disorder of $0.5\%$; and third, in the presence of non-zero temperature environments where the entropy has been chosen as $\Sigma = 5.5 k_B$. Dashed diagonal lines indicate the theoretical prediction for the asymptotic diffusive scaling $Dn/J^3$, the vertical lines with the star symbol indicate the cross-over time $t^*$ between the logarithmic and linear scaling as predicted by the theory.