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arXiv:2609.16121

The Transient Counting Statistics of Autonomous Quantum Clocks

Oscar Arandes, Sreenath K. Manikandan

quant-phSubmitted Sep 14, 2026• Announced Sep 16, 2026• 0 citations
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Abstract

At the heart of every atomic clock is a quantum coherent emitter, the laser, that could in principle be treated autonomously. While their frequency or phase stability is typically studied in the semiclassical regime of a large number of emitted quanta, recent works on autonomous quantum clocks suggest that the few-quanta regime can offer statistical information in the temporal domain. This provides complementary means to probe the fundamental limits to precise timekeeping in the quantum regime. Motivated by these works, we consider simple models for two- and three-level quantum emitters subject to an athermal fueling through dephasing, which generically results in coherences in the steady state. A large deviation principle with finite-time contributions is used to identify the transient counting statistics and their dependence on the initial conditions. We conclude by discussing the implications for precise timekeeping and quantum sensing using autonomous quantum emitters as clocks in their transient regime.

Comments
21 pages, 8 figures

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At the heart of every atomic clock is a quantum coherent emitter, the laser, that could in principle be treated autonomously. While their frequency or phase stability is typically studied in the semiclassical regime of a large number of emitted quanta, recent works on autonomous quantum clocks suggest that the few-quanta regime can offer statistical information in the temporal domain. This provides complementary means to probe the fundamental limits to precise timekeeping in the quantum regime. Motivated by these works, we consider simple models for two- and three-level quantum emitters subject to an athermal fueling through dephasing, which generically results in coherences in the steady state. A large deviation principle with finite-time contributions is used to identify the transient counting statistics and their dependence on the initial conditions. We conclude by discussing the implications for precise timekeeping and quantum sensing using autonomous quantum emitters as clocks in their transient regime.
Here we explore the possibility of precise time keeping in quantum systems using athermal resources. We show that quantum measurement engineered reservoirs can be used as athermal resources to drive the ticks of a quantum clock. Two- and three-level quantum systems act as transducers in our model, converting the quantum measurement-induced noise to produce a series of ticks. The ticking rate of the clock is maximized when the measured observable maximally noncommutes with the clock's Hamiltonian. We use the large deviation principle to characterize the statistics of observed ticks within a given time period and show that it can be sub-Poissonian---quantified by Mandel's Q parameter---alluding to the quantum nature of the clock. We discuss the accuracy and efficiency of the clock, and extend our framework to include hybrid quantum clocks fueled by both measurements, and thermal resources. We make comparisons to relatable recent proposals for quantum clocks, and discuss alternate device implementations harvesting the quantum measurement engineered nonequilibrium conditions, beyond the clock realization.
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Counting processes provide a fundamental description of stochastic events ranging from photon detection to clock ticks. A central question is how accurately such events can be timed when only finite memory resources are available. Here, we investigate this problem within a general framework of finite-dimensional classical and quantum counting processes. We derive a rigorous finite-memory variance bound obeyed by every classical ddd-state counting process, which is tight and saturated by a discrete Erlang-type ladder process. Through numerical optimization, we identify quantum counting processes that violate this classical bound, achieving smaller first-tick fluctuations than any classical process with the same memory size and mean tick time. For the qubit case, we further derive an analytical large-mean bound within a single-Kraus no-tick family, showing that the quantum advantage persists asymptotically within this class. The optimized quantum processes exhibit coherent conditioned dynamics and approach a continuous-time quantum-jump description as the mean increases. Our results establish a finite-memory quantum advantage in temporal precision and connect discrete-time counting processes with continuous-time quantum timekeeping.
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.
Measuring time means counting the occurrence of periodic phenomena. Over the past centuries a major effort was put to make stable and precise oscillators to be used as clock regulators. Here we consider a different class of clocks based on stochastic clicking processes. We provide a rigorous statistical framework to study the performances of such devices and apply our results to a single coherently driven two-level atom under photodetection as an extreme example of non-periodic clock. Quantum Jump MonteCarlo simulations and photon counting waiting time distribution will provide independent checks on the main results.
Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to ≈4.5 dB\approx 4.5 \text{ dB}≈4.5 dB compared with the two-level model. Furthermore, we identify an even greater stability enhancement of ≈5.4 dB\approx 5.4 \text{ dB}≈5.4 dB for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an 27Al+^{27}\text{Al}^{+}27Al+ optical ion clock.