Nonlinear Clock-Pulling Mechanisms

• Nonlinear clock pulling is defined as a process where oscillator timing is corrected by state-dependent feedback, such as delayed, nonlinear transduction of amplitude and phase.
• This mechanism enables advanced synchronization and frequency stabilization, outperforming linear correction methods over longer intervals.
• Its applications span atomic clocks, quantum oscillators, mechanical systems, and wireless power transfer, showcasing its versatility in diverse timing control scenarios.

Searching arXiv for relevant papers on nonlinear clock-pulling and related mechanisms. Nonlinear clock-pulling mechanism denotes a class of phase- and frequency-regulation processes in which a clock, oscillator, or clock-like subsystem is not merely corrected by linear detuning compensation, but is dynamically “pulled” through nonlinear state dependence, delayed feedback, dissipative coupling, or nonlinear transduction between amplitude, phase, and frequency. Across the literature, the term does not refer to a single universal model. Instead, it appears in several technically distinct settings: nonlinear time synchronization in distributed systems (Wang et al., 2019), cavity-pulling transduction in Ramsey-operated atomic clocks (Gozzelino et al., 2018), delayed quantum self-oscillators (Liu et al., 2023), environment-mediated synchronization in quantum “Huygens” models (Tyagi et al., 2024), nonlinear phase maps for mechanically coupled clocks (Buescu et al., 2024), relational quantum dynamics with non-ideal clocks (Mendes et al., 2021), autonomous optomechanical pendulum clocks (Brunelli et al., 12 Jun 2025), and nonlinear PT-symmetric wireless power transfer, where the phrase is used explicitly for a feedback-controlled frequency-selection mechanism (Hao et al., 15 Jul 2025). In contrast, some clock-pulling phenomena remain effectively linear but strongly suppressed, as in bad-cavity active optical clocks (Xu et al., 2014).

1. Conceptual scope and definition

In the broadest usage supported by the literature, “clock pulling” describes a mechanism by which an oscillator’s effective phase, frequency, or timing trajectory is driven toward a preferred state by coupling to a reference, a feedback loop, a shared environment, or an internal nonlinear resonance condition. What makes the mechanism nonlinear is not merely the presence of oscillation, but the fact that the restoring action depends nontrivially on the system state: pulse area, cavity detuning, delayed state variables, higher-order correlations, or nonlinear gain.

A useful contrast appears between linear and nonlinear formulations of clock regulation. In software synchronization, the linear approximation assumes constant skew over each synchronization interval,

$s(t)=\beta,$

leading to

$T_c(t) = (1+\beta)t + (o+t_0),$

whereas the nonlinear model assumes

$s(t) = at + \beta,$

which yields

$T_c(t) = at^2 + (1+\beta)t + (o+t_0).$

Here the quadratic term $at^2$ captures frequency shift over time and is the operative nonlinear correction pathway (Wang et al., 2019). This formulation makes explicit that clock pulling can mean learning how the rate itself evolves, rather than repeatedly correcting offset.

In atomic-clock physics, the mechanism can instead be a nonlinear transduction from microwave pulse area to frequency shift. In Ramsey-operated compact clocks, the atomic signal depends on both detuning and pulse area as

$$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$

so the cavity-pulling shift inherits a nontrivial dependence on $\theta$ through the atomic coherence and the cavity response (Gozzelino et al., 2018). In that setting, clock pulling is nonlinear because amplitude fluctuations modulate the phase shift and hence the clock frequency.

A different usage arises in nonlinear PT-symmetric wireless power transfer, where the “clock” is the oscillation frequency of coupled resonators and “pulling” is a phase-sensitive feedback action implemented by a phase-locked loop. There, nonlinear clock pulling means that a feedback loop continuously nudges the oscillation frequency until the system locks to a desired steady state, even when that state is not the minimum-gain branch of the conventional PT-symmetric picture (Hao et al., 15 Jul 2025).

2. Nonlinear pulling in synchronization and control

In distributed time synchronization, the nonlinear model in (Wang et al., 2019) is explicitly presented as an alternative to purely linear correction. The paper argues that time synchronization should not be modeled as a purely linear correction process when the clock’s frequency itself drifts over the synchronization interval. The general clock model is

$T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$

with $s(t)$ the skewness or frequency deviation (Wang et al., 2019). The nonlinear mechanism consists of learning a time-varying skew and then correcting the slave clock along a curved trajectory that matches the actual drift.

This mechanism is operationalized as a three-step workflow: the system collects timestamp data, learns the clock system, and corrects the clock time (Wang et al., 2019). The method is designed for two-way timestamp exchange using the usual four timestamps $t_1,t_2,t_3,t_4$, and the paper suggests learning the first and second derivatives of clock discrepancy with methods such as high-order SVM. The slope and offset correspond to skew and offset, and the curvature corresponds to higher-order frequency shift (Wang et al., 2019).

The significance of the nonlinear pulling mechanism in this context is that it permits much longer synchronization intervals than linear methods. The numerical tests use 2000 communications per synchronization interval, Gaussian measurement noise with rms $T_c(t) = (1+\beta)t + (o+t_0),$0, and nonlinear learning up to 2nd order (Wang et al., 2019). The nonlinear method is tested from 2 seconds to 100 seconds and is reported to converge quickly in only a few steps, while even a 100-second interval still converges (Wang et al., 2019). A hybrid strategy—start with 2 seconds, then switch to 200 seconds—converges within about 20 seconds and avoids early overshoot spikes (Wang et al., 2019). By contrast, the linear model works reasonably well at 2 seconds but shows large error at 10 seconds (Wang et al., 2019).

This suggests that, in synchronization theory, nonlinear clock pulling is best understood as frequency-aware correction. The key claim is not that the clock is instantaneously forced to the master time, but that the clock is pulled by a learned model of skew evolution. A plausible implication is that this formulation shifts the dominant estimation target from offset to frequency dynamics.

3. Atomic and optical clock realizations

In Ramsey-operated compact clocks, the nonlinear clock-pulling mechanism is associated with cavity pulling mediated by microwave amplitude fluctuations. The interrogating microwave pulse has area $T_c(t) = (1+\beta)t + (o+t_0),$1, and the clock frequency shifts by

$T_c(t) = (1+\beta)t + (o+t_0),$2

where $T_c(t) = (1+\beta)t + (o+t_0),$3 is an extra phase acquired during free evolution because the atomic coherence emits field back into a detuned cavity (Gozzelino et al., 2018). The cavity-pulling shift depends on cavity detuning $T_c(t) = (1+\beta)t + (o+t_0),$4, loaded quality factor $T_c(t) = (1+\beta)t + (o+t_0),$5, and pulse area through a nontrivial function $T_c(t) = (1+\beta)t + (o+t_0),$6 that crosses zero near $T_c(t) = (1+\beta)t + (o+t_0),$7 (Gozzelino et al., 2018).

The paper derives a drift decomposition

$T_c(t) = (1+\beta)t + (o+t_0),$8

with representative coefficients at $T_c(t) = (1+\beta)t + (o+t_0),$9,

$s(t) = at + \beta,$0

Using a measured fractional amplitude fluctuation of roughly $s(t) = at + \beta,$1 per day, the open-loop model predicts a frequency-aging rate on the order of $s(t) = at + \beta,$2/day, dominated by the $s(t) = at + \beta,$3 term (Gozzelino et al., 2018). The proposed mitigation is a four-point interrogation sequence that constructs an amplitude error signal

$s(t) = at + \beta,$4

and applies a pure integrator correction

$s(t) = at + \beta,$5

Experimentally, the open-loop slope of $s(t) = at + \beta,$6 versus pulse-area offset was about 1300 per cent; a deliberate negative amplitude step produced a fractional clock frequency jump of $s(t) = at + \beta,$7, recovered with a time constant of about 25 s; and long-term drift improved from about $s(t) = at + \beta,$8/day to about $s(t) = at + \beta,$9/day over more than ten days (Gozzelino et al., 2018).

In contrast, the cesium active optical clock in the bad-cavity regime exhibits suppressed cavity pulling rather than a nonlinear pulling law. The governing relation is

$T_c(t) = at^2 + (1+\beta)t + (o+t_0).$0

with $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$1 (Xu et al., 2014). For the reported values $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$2, $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$3, and $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$4, the pulling fraction is about $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$5 (Xu et al., 2014). Measured cavity detunings of 140.8 MHz and 281.6 MHz produced frequency shifts of 3.69 MHz and 6.80 MHz, corresponding to suppression factors 38.2 and 41.4, respectively (Xu et al., 2014). The paper explicitly treats this as linear in detuning but strongly reduced.

The atomic-clock literature therefore distinguishes two cases. One is genuinely nonlinear transduction, where pulse-area fluctuations enter the cavity-pulling pathway (Gozzelino et al., 2018). The other is a linear pulling law with a suppression factor set by the bad-cavity condition (Xu et al., 2014). A common misconception is that all cavity-pulling effects are nonlinear; the cited results show that strong sensitivity reduction can coexist with an explicitly linear pulling formula.

4. Delayed and dissipative quantum pulling

The delayed quantum self-oscillator of (Liu et al., 2023) provides a different perspective. The system is a ring cavity with delayed amplified feedback. In the linear case, the expectation value obeys

$T_c(t) = at^2 + (1+\beta)t + (o+t_0).$6

which is the quantum analogue of the classical delay differential equation

$T_c(t) = at^2 + (1+\beta)t + (o+t_0).$7

For suitable parameters, especially with $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$8, the paper reports that $T_c(t) = at^2 + (1+\beta)t + (o+t_0).$9 oscillates indefinitely with no visible decay and no apparent phase diffusion, although the mean energy grows with time (Liu et al., 2023).

The nonlinear version adds two-photon absorption and yields

$at^2$0

The authors emphasize that

$at^2$1

so the nonlinear quantum dynamics is not a simple quantum analogue of classical cubic damping (Liu et al., 2023). Numerically, the nonlinear delayed system exhibits dissipative oscillation, the decay is “primarily due to quantum phase diffusion,” and no parameter choices were found that allow indefinite oscillation of $at^2$2 without phase diffusion (Liu et al., 2023).

This establishes an important negative result: nonlinear delayed feedback does not automatically improve clock pulling in the quantum regime. The linear delayed oscillator is closer to an ideal ticking clock, whereas the nonlinear delayed oscillator loses long-term phase coherence (Liu et al., 2023). A plausible implication is that quantum higher-order correlations obstruct the straightforward transfer of classical saturation-based stabilization mechanisms.

A related but conceptually distinct quantum mechanism appears in the quantum analogue of Huygens’ clock (Tyagi et al., 2024). There, two qubits synchronize through a shared noisy environment, with the cross-dissipator

$at^2$3

appearing multiplied by the environmental correlation coefficient $at^2$4 (Tyagi et al., 2024). The environment acts as a shared escapement: correlated noise ($at^2$5) pulls phases toward synchronization, anti-correlated noise ($at^2$6) toward antisynchronization, and $at^2$7 gives no expected phase synchronization (Tyagi et al., 2024). In this setting, the pulling mechanism is dissipative and collective rather than controller-based.

5. Mechanical, optomechanical, and discrete-map mechanisms

The study of three aligned Huygens clocks develops a nonlinear discrete phase-pulling model for mechanically coupled limit-cycle oscillators (Buescu et al., 2024). With nearest-neighbor impacts, the phase-difference dynamics is reduced to the planar map

$at^2$8

The nonlinearity is entirely in the sine dependence of the phase differences. The map has fixed points $at^2$9, $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$0, $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$1, and $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$2 modulo $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$3-translations, with $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$4 a sink and $$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$5 a source (Buescu et al., 2024). The synchronized outcome is a phase-opposed arrangement: the outer clocks asymptotically differ by half a cycle relative to the central clock (Buescu et al., 2024).

The pulling mechanism is therefore geometric: weak once-per-cycle perturbations generate invariant lines, saddles, heteroclinic curves, and a basin structure that funnels almost all initial conditions toward the phase-opposed attractor (Buescu et al., 2024). This is not a continuous frequency servo, but a nonlinear phase map whose invariant geometry performs the effective pulling.

The optomechanical pendulum clock in (Brunelli et al., 12 Jun 2025) realizes a still different mechanism. A mechanical oscillator plays the role of the pendulum, a three-level emitter in an optical cavity provides the escapement, and the full Hamiltonian includes radiation-pressure coupling

$$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$6

The clock operates autonomously using thermal baths rather than coherent drive (Brunelli et al., 12 Jun 2025). The essential feedback loop is: mechanical motion tunes the cavity-emitter detuning; resonance permits photon emission; emitted photons kick the mechanics; and the resulting oscillation resets the resonance condition. The paper states that this self-consistent feedback generates a limit cycle (Brunelli et al., 12 Jun 2025).

At the semiclassical level, the mean-field equations include

$$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$7

and the equations are closed by factorizing third-order cumulants, yielding a nonlinear mean-field description (Brunelli et al., 12 Jun 2025). The paper explicitly identifies the nonlinear clock pulling as the fact that the timing of photon release is pulled toward the mechanical phase where resonance occurs, with the tick phase-locked to the mechanical motion with a small delay (Brunelli et al., 12 Jun 2025). Ticks are monitored through the jump operator

$$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$8

and performance is characterized by accuracy

$$S(\omega_0,\theta)\propto \sin^2\theta \left[1-\cos\!\left((\omega_0-\omega_{12})T\right)\right],$$9

and resolution

$\theta$0

The paper further reports that the Allan variance obeys the long-time form

$\theta$1

for the filtered clock (Brunelli et al., 12 Jun 2025).

These mechanical examples show that nonlinear clock pulling can arise from discrete impacts, nonlinear phase maps, or state-dependent resonance gating. In each case, the mechanism is phase-selective and self-consistent rather than merely dissipative.

6. Feedback selection, symmetry breaking, and relational clocks

The most explicit use of the phrase “nonlinear clock-pulling mechanism” appears in nonlinear PT-symmetric wireless power transfer (Hao et al., 15 Jul 2025). The system is a coupled-resonator dimer with nonlinear gain in the transmitter. In the generalized effective Hamiltonian, the gain $\theta$2 depends on oscillation amplitude, so the steady state must be determined self-consistently (Hao et al., 15 Jul 2025). In the PT-symmetric limit $\theta$3, the conventional real eigenfrequencies are

$\theta$4

and there is also the special mode

$\theta$5

which requires

$\theta$6

Conventional nonlinear PT-WPT literature treats $\theta$7 as unstable because it lies at the largest required gain (Hao et al., 15 Jul 2025).

The paper’s revision is that the gain landscape has a relative extremum at $\theta$8, and a phase-locked loop can render that extremum dynamically stable. The idealized phase relation is

$\theta$9

Around the negative-resistance operating point, the stable equilibrium is restricted to

$T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$0

and the paper states that the system frequency increases continuously when $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$1 and decreases continuously when $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$2 (Hao et al., 15 Jul 2025). This is the restoring polarity that pulls the oscillation frequency toward $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$3.

The claimed consequence is forced symmetry breaking inside the PT-symmetry phase: the feedback loop actively steers the system to the high-gain branch $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$4, which corresponds to the highest transfer efficiency among the available steady states (Hao et al., 15 Jul 2025). This usage is noteworthy because the mechanism is neither atomic nor synchronization-theoretic in the usual sense; it is a control-theoretic stabilization of a nonlinear frequency state.

A more abstract form of clock-induced pulling appears in the Page–Wootters framework with interaction and quasi-ideal clocks (Mendes et al., 2021). There the clock is not a classical oscillator but a finite-dimensional quantum clock coupled gravitationally to the system via

$T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$5

Conditioning on quasi-ideal clock states produces an effective mixed-state evolution containing explicit dependence on the initial system state,

$T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$6

The right-hand side is therefore not closed in $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$7 alone (Mendes et al., 2021). The paper interprets this as a clock-induced modification of relational evolution: the non-ideal clock actively alters the system’s effective dynamics, producing nonlinearity and initial-condition dependence (Mendes et al., 2021). This is a very different notion of clock pulling, but it preserves the central feature that the clock is no longer a passive time label.

7. Comparisons, misconceptions, and boundaries of the concept

The literature supports several distinctions that clarify the scope of nonlinear clock pulling.

Setting	Pulling variable	Mechanism type
Distributed synchronization (Wang et al., 2019)	Clock trajectory	Nonlinear skew learning
Ramsey compact clocks (Gozzelino et al., 2018)	Clock frequency	Nonlinear amplitude-to-phase transduction
Delayed quantum SSO (Liu et al., 2023)	Oscillator phase	Delayed feedback with quantum dephasing
Quantum Huygens model (Tyagi et al., 2024)	Relative phase	Correlated dissipative pulling
Three aligned clocks (Buescu et al., 2024)	Phase differences	Nonlinear discrete map
Quantum pendulum clock (Brunelli et al., 12 Jun 2025)	Tick timing	Resonance-gated optomechanical feedback
PT-WPT dimer (Hao et al., 15 Jul 2025)	Oscillation frequency	PLL-based nonlinear state selection

One common misconception is that any clock-pulling effect is necessarily nonlinear. The cesium active optical clock provides a direct counterexample: its cavity-pulling law is linear in detuning but reduced by a suppression factor $T_c(t) = t_0 + \int_0^t (1+s(t))\,dt + o,$8 in the deep bad-cavity regime (Xu et al., 2014). Another misconception is that adding nonlinearity automatically improves phase coherence. The delayed quantum self-oscillator shows the opposite: the nonlinear delayed system exhibits dephasing and damped oscillations, while the linear delayed system is the one that supports perfect oscillation without apparent phase diffusion (Liu et al., 2023).

The term “clockwork” must also be distinguished from “clock pulling.” The geometric-phase-based atomic clockwork proposed in (Munshi et al., 2011) concerns coupling femtosecond and nanosecond clock ticks via phase-dependent energy shifts, but the available material does not provide the mechanism beyond the abstract. Likewise, the high-energy-theory “clockwork mechanism” analyzed in (Craig et al., 2017) is a theory of exponentially localized zero modes and is not a timing-control mechanism. The paper explicitly states that clockwork is an intrinsically abelian phenomenon and that a nonlinear “clock-pulling” mechanism is not possible in the symmetry-protected sense used there (Craig et al., 2017). These usages are terminologically adjacent but conceptually separate.

Taken together, the literature suggests that “nonlinear clock-pulling mechanism” is best treated as a family resemblance term rather than a single canonical construction. The shared structure is a nonlinear relation between state and timing correction: a clock or oscillator is pulled because the restoring action depends on amplitude, phase, delay, environmental correlation, nonlinear gain, or finite-clock back-reaction. The specific mathematics, however, varies sharply across synchronization theory, atomic metrology, quantum dissipative dynamics, nonlinear control, and autonomous clock models.