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Feedback-Induced Advantage in Quantum Clockworks

Jakob Miller, Paul Erker

Abstract

Atomic frequency standards have achieved steadily increasing precision over the past seventy years, enabled in part by feedback mechanisms that stabilise their output. In parallel, the timekeeping capabilities of quantum systems have been explored within the recently developed ticking-clock framework, which models clocks as dynamical systems producing a stochastic sequence of ticks. However, a theoretical description that unifies these perspectives and incorporates feedback into autonomous quantum clocks has been lacking. We introduce a framework for feedback-controlled clockworks in which classical information extracted from the tick sequence is used to influence the subsequent dynamics of the clock. We show that such feedback preserves the core structural features of self-timing and clockwork independence that characterise autonomous ticking clocks. We further identify the signal-to-noise ratio $\mathfrak{S}$ as the fundamental figure of merit for assessing the performance of feedback-controlled clocks. Applying our framework to two representative architectures, we prove that classical clockworks cannot surpass the optimal signal-to-noise ratio achievable without feedback. In contrast, for quantum clockworks we present numerical evidence that feedback can provide a genuine performance enhancement, improving the maximal attainable signal-to-noise ratio. These results establish feedback as a potentially essential ingredient in pushing the fundamental limits of timekeeping in the quantum regime.

Feedback-Induced Advantage in Quantum Clockworks

Abstract

Atomic frequency standards have achieved steadily increasing precision over the past seventy years, enabled in part by feedback mechanisms that stabilise their output. In parallel, the timekeeping capabilities of quantum systems have been explored within the recently developed ticking-clock framework, which models clocks as dynamical systems producing a stochastic sequence of ticks. However, a theoretical description that unifies these perspectives and incorporates feedback into autonomous quantum clocks has been lacking. We introduce a framework for feedback-controlled clockworks in which classical information extracted from the tick sequence is used to influence the subsequent dynamics of the clock. We show that such feedback preserves the core structural features of self-timing and clockwork independence that characterise autonomous ticking clocks. We further identify the signal-to-noise ratio as the fundamental figure of merit for assessing the performance of feedback-controlled clocks. Applying our framework to two representative architectures, we prove that classical clockworks cannot surpass the optimal signal-to-noise ratio achievable without feedback. In contrast, for quantum clockworks we present numerical evidence that feedback can provide a genuine performance enhancement, improving the maximal attainable signal-to-noise ratio. These results establish feedback as a potentially essential ingredient in pushing the fundamental limits of timekeeping in the quantum regime.
Paper Structure (22 sections, 8 theorems, 68 equations, 4 figures, 1 algorithm)

This paper contains 22 sections, 8 theorems, 68 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Clock precision.
  2. Reference point.
  3. Summary.
  4. Classical Markovian systems
  5. Analyzing clock performance: exact calculation and upper bounds
  6. Exact calculation using full counting statistics
  7. Vectorization of density matrices and superoperators.
  8. Calculating the average current and noise.
  9. The generalized inverse $\widehat{\mathcal{L}}_{}{^+}$.
  10. Hyperaccurate currents and uncertainty relations
  11. Kinetic uncertainty relation.
  12. Clock uncertainty relation.
  13. Quantifying clock precision
  14. $\mathfrak{S}$ versus $\mathfrak{N}$.
  15. Implicit assumptions in (constant) feedback policies
  16. ...and 7 more sections

Key Result

Theorem 2

Consider $G$ classical clockworks of dimension two together with a feedback policy $\Pi$ such that for all $a=1,\ldots,G$ we have $\mathcal{G} =\tilde{ \mathcal{G} }\times\tilde{ \mathcal{G} }$, for some $\{0\}\neq\tilde{ \mathcal{G} }\subset\mathbb{R}_{\geq0}$, and let the joint system

Figures (4)

  • Figure 1: Schematic representation of our feedback framework. Jumps in the (quantum) clockworks are observed by detectors, connected to the classical control unit. Based on the observed pattern of jumps, the control unit applies feedback to the clockworks, according to the feedback policy $\Pi$ (see \ref{['def:feedback_policy']}), and calculates an estimate of the time $N(t)$, that is displayed in the tick register.
  • Figure 2: Data from a numerical simulation in Python PythonGithub of the signal-to-noise ratio of the qubit clockwork and an integrated current, counting each jump in the clockwork, plotted versus the control parameters $E^{(a)}$ and $\phi^{(a)}$. The red star marks the parameters that achieve the maximum SNR $\mathfrak{S}^*/\Gamma\approx1.19$ for a single qubit clockwork. The red triangle and circle mark the parameters between which the optimal feedback policy for two copies of a qubit clockwork, constructed in \ref{['app:go_example_qubit_clockwork']}, switches.
  • Figure 3: We evaluate and plot the analytical expression for the signal-to-noise ratio $\mathfrak{S}/\Gamma$ using Mathematica MathematicaGithub.
  • Figure 4: Data from a numerical simulation in Python PythonGithub, showing the signal-to-noise ratio of an integrated current counting every jump from two qubit clockworks, under energy-based feedback, with equal weights. The parameters $\alpha_1$ and $\alpha_2$ determine at which energy the first clockwork is operated at, after a tick from qubit clockwork one or two occurred, and vice versa for the energy of qubit clockwork two. The red square marks the parameters maximizing the SNR.

Theorems & Definitions (16)

  • Definition 1: Feedback policy
  • Theorem 2
  • Corollary 2
  • Proposition 3
  • proof
  • Corollary 4
  • Definition 9
  • Theorem 9
  • proof
  • Corollary 9
  • ...and 6 more