The clock that is pulled into step

The frequency that gets an answer drove a passive oscillator — a mass on a spring, damped, doing nothing until pushed — and found it answering most strongly at its own frequency, a quarter cycle behind the push. Every later essay on resonance, from the swing pumped rather than pushed to the resonance that refuses to leak, kept that picture: an oscillator whose motion is supplied from outside and dies away when the supply stops.

A clock is a different kind of oscillator. A pendulum clock’s escapement feeds its pendulum a little energy each swing, a laser’s gain medium feeds its light, a heart’s pacemaker cells recharge themselves between beats. Each keeps oscillating on its own at a characteristic amplitude and frequency, and a disturbance to its amplitude dies away while its phase — where in its cycle it is — is free, with no restoring force at all. What happens when such an oscillator is pushed weakly and periodically is not resonance in the earlier sense. The push barely changes how strongly it swings. It pulls on its phase, and it can pull it into step.

A frequency taken over #

In 1946 Robert Adler, an engineer working on oscillators for radio, wrote down the equation that describes this. If an oscillator’s natural frequency differs from the drive’s by Δω, and the drive is weak enough to leave its amplitude alone, the phase difference φ between them changes as

$$\frac{d\varphi}{dt} = \Delta\omega - K\sin\varphi.$$

The first term is the oscillator running ahead of the drive at its own pace. The second is the drive’s pull, which depends on where in the cycle the push arrives: pushing slightly before the oscillator’s peak advances it, slightly after retards it, and the net effect over a cycle is a correction proportional to the sine of the phase difference. The strength $K$ grows with the drive’s amplitude.

If the detuning is smaller than $K$, the equation has a fixed point: a phase difference, $\arcsin(\Delta\omega/K)$, at which the pull exactly cancels the oscillator’s tendency to run ahead. The oscillator settles there and stays. It now runs at exactly the drive’s frequency, not its own, however long the two are watched. The drawing plots the frequency the oscillator actually runs at against its natural frequency, and the result is a flat plateau: everything within $\pm K$ of the drive has been taken over completely. Outside the plateau the oscillator runs free, but even there the drive pulls it closer than it would otherwise be: its frequency offset is not Δω but $\sqrt{\Delta\omega^2 - K^2}$.

This is injection locking, and it is not a small effect of a weak push. A push of one per cent of the oscillator’s own amplitude can hold its frequency to the drive’s to any number of decimal places, across a range of natural frequencies that may be wide. What is small is the range, not the effect within it.

Why the pull goes as a sine #

The form of the pull is worth understanding, because it is what makes the equation universal. Push a clock’s pendulum a little as it passes through the bottom of its swing, in the direction it is already moving, and it swings slightly higher but arrives at the end of its swing at almost the same moment: the push changes its amplitude, which the escapement then restores, and does nothing lasting to its phase. Push it the same amount near the end of its swing and it turns round slightly early: its phase is advanced. Push it just after it has turned and it is retarded. How a push shifts the phase depends on where in the cycle it arrives, and for a nearly sinusoidal oscillator the dependence is itself a sinusoid.

A periodic drive delivers such pushes every cycle, arriving at a phase that drifts if the two frequencies differ. Averaged over a cycle, the pushes give a net phase shift proportional to the sine of the phase difference: zero when the drive arrives in step or exactly out of step, largest a quarter cycle either side. That is the $K\sin\varphi$ of Adler’s equation, and it does not depend on what the oscillator is made of — only on its being self-sustained, with an amplitude it restores and a phase it does not.

The contrast with passive oscillators is instructive. Two pendulums coupled by a spring do not lock: they exchange their energy back and forth for ever, because neither has a mechanism that restores its own amplitude, and their motion is a sum of two normal modes beating against each other. Give each pendulum an escapement that holds its amplitude, and the beating is replaced by locking; the same coupling that made them swap makes them agree.

Slipping in jumps #

Just outside the locking range the equation has no fixed point, and the phase must keep advancing. But it does not advance uniformly. Near a quarter turn, where the drive’s pull is almost enough to hold it, the phase moves very slowly — the ghost of the fixed point that exists a little further inside the range. Elsewhere in the cycle the pull is weak or works the other way, and the phase rushes round. The result is a staircase: long plateaus near a quarter turn, separated by rapid slips of a whole turn.

Anyone who has tuned an old radio past a strong station, or listened to two organ pipes nearly in unison, has heard this. The beat between two nearly equal frequencies should be a smooth rise and fall; near locking it becomes a series of pulses, long silences broken by quick bursts, and the pulses come more slowly than the difference in frequency would suggest. The average slip rate is $\sqrt{\Delta\omega^2 - K^2}$, which the drawing checks by integrating the equation, and it goes to zero at the edge of the range as the square root of the distance from it — the signature of a fixed point being born.

How wide the range is #

Adler also found how the locking strength depends on the oscillator. The half-width of the locking range, as a fraction of the frequency, is the drive’s amplitude over the oscillator’s own, divided by twice the oscillator’s quality factor. It grows in proportion to the push, so the locking region in the plane of detuning and push strength is a wedge opening from zero — the simplest example of what are called Arnold tongues, after Vladimir Arnold’s study of the regions where one oscillation locks to another.

The factor of $Q$ is the interesting part. A resonance’s width is its lifetime, and a high-$Q$ oscillator is one whose own resonance holds its frequency tightly: it has a long memory of its phase and a small pull costs it little. Such an oscillator is hard to lock, which is exactly what is wanted of a good clock and not wanted of an oscillator meant to follow a reference. A quartz crystal, with a $Q$ of a million, can be pulled only by parts per million; an electronic oscillator with a $Q$ of ten follows a push a hundred thousand times more readily.

That trade is used deliberately in lasers. A powerful laser with a noisy, poorly defined frequency can be locked to a weak, stable one by injecting a small amount of the stable laser’s light into it: within the locking range the powerful laser takes on the weak one’s frequency and phase while keeping its own power. The locking range, set by the power ratio and the powerful laser’s cavity $Q$, decides how closely the two must be tuned by hand before the physics takes over.

A body clock and the daylight #

Every body carries a self-sustained oscillator whose natural period is not quite a day — about twenty-four hours and ten or twenty minutes in most people, measured in volunteers kept away from any clue to the time. It stays in step with the day only because it is locked: daylight, especially morning light, pushes its phase each day, and the natural period lies well within the locking range. The drawing applies Adler’s equation to that clock with the day as the drive, allowing the daylight to pull it by up to two hours a day, and asks what happens after a flight that shifts local time suddenly.

The clock is then far from its locked phase and returns along the equation’s trajectory: quickly at first, slowing as it approaches. A three-hour shift is corrected to within a quarter of an hour in about five days; a six-hour shift in six and a half — not twice as long, because the pull is stronger further from the locked phase. But near a twelve-hour shift the pull vanishes: the clock sits near the equation’s unstable point, where the daylight pushes it as much one way as the other, and it lingers there before sliding. An eleven-and-a-half-hour shift takes nearly twice as long to recover from as a six-hour one. The real body clock is more complicated — it responds differently to light at different times of day, and advancing it is harder than delaying it, which is why flying east is usually worse — but the lingering near half a day is a feature of the equation that it shares.

Clocks that pull each other #

Huygens’s clocks had no master. Each pulled on the other through the slight give of the beam they hung from, and the question was whether they would settle. The drawing gives each of two oscillators a pull on the other of equal strength and different natural frequencies. The difference between their phases obeys Adler’s equation with twice the coupling, so they lock if their frequencies differ by less than twice the pull, at a fixed phase difference, and run at the average of their natural frequencies. If their frequencies differ by more, they slip.

Huygens’s clocks locked in anti-phase, swinging in opposite directions — a detail that depends on how the beam moves, not on the equation of phases, and which was explained fully only in the last few decades with models of the pendulums and the beam together. The general result is the one the drawing shows: mutual pulling, however weak, locks oscillators whose frequencies are close enough, and it does so without any of them being in charge.

Locking everywhere #

Once the shape of the equation is recognised, it turns up in a startling range of places. The electrical grid of a continent is thousands of generators, each a spinning machine with its own natural speed, locked together by the grid itself into a single frequency — fifty or sixty hertz, the same to parts per million from one end of the continent to the other — and a generator that slips out of step is disconnected automatically before it can damage itself or its neighbours. A synchronous motor is the same arrangement run the other way: its magnetised rotor locks to a field that turns with nothing turning and rides round at exactly the field’s speed, lagging by an angle that grows with the load, until a load past the limit — the edge of its locking range — makes it slip poles and stall. The cells of the heart’s natural pacemaker are each self-sustained oscillators with slightly different rates, and they fire together because they pull on one another electrically. Fireflies in parts of south-east Asia flash in unison across whole trees, each adjusting its next flash to its neighbours’. Electronic circuits called phase-locked loops, which contain an Adler equation by design, sit in every radio, mobile telephone and computer, holding a local oscillator to a reference.

For a crowd of oscillators with a spread of natural frequencies, each pulling on all the others, Yoshiki Kuramoto showed in 1975 that synchrony appears suddenly: below a critical coupling set by the spread nothing locks, and above it a growing fraction of the crowd falls into step. The single-pair locking of this essay is the smallest case of that transition.

A junction that counts in steps #

The most precise use of locking is in the definition of the volt. A Josephson junction, two superconductors separated by a thin barrier, is a self-sustained oscillator in a precise sense: a steady voltage across it makes its superconducting phase advance at a rate proportional to the voltage, a voltage that is a frequency, 483.6 gigahertz per millivolt. Irradiate it with microwaves and its phase locks to the microwave drive — not only at the drive’s frequency but at its harmonics — and while it is locked the voltage across it is fixed at exactly the drive frequency, or a multiple, times Planck’s constant over twice the electron charge.

The current–voltage curve of such a junction is therefore a staircase of perfectly flat steps, one for each harmonic the junction can lock to: the plateau of the first drawing, repeated. Because the steps depend only on a frequency, which can be measured to parts in $10^{15}$, and on fundamental constants, arrays of thousands of such junctions are the world’s voltage standards. The flatness of a locking plateau, which in a laser or a body clock is a convenience, here defines a unit.

Where Adler’s equation stops #

Weak driving. The equation assumes the drive leaves the oscillator’s amplitude alone and acts only on its phase. A strong drive changes the amplitude too, and at strong enough driving the oscillator stops being a clock at all and simply follows the drive as a passive resonator would. The transition between the two regimes is where the dynamics become complicated.

Harmonics and subharmonics. An oscillator can lock not only to a drive near its own frequency but to one near twice or half of it, or any rational ratio, each with its own, narrower tongue. With all the tongues drawn together they form a structure — the devil’s staircase — whose steps at every rational frequency ratio fill the axis, and at strong driving the tongues overlap and the motion can become chaotic.

Noise. Inside the locking range, noise jostles the phase about its fixed point and occasionally kicks it over the barrier into a slip. Near the edge of the range the barrier is low and slips become frequent, so the boundary of locking is not sharp for a noisy oscillator; the rate of noise-induced slips is itself an Arrhenius-like escape over a barrier set by $K$ and Δω.

What the phases do not show #

The drawings show phases and frequencies and nothing of the waveform. A locked oscillator runs at the drive’s frequency with its own waveform, and the drive’s influence may be visible only in timing. They also leave out the energy: an injection-locked laser gets its power from its own gain medium, not from the weak injected light, which supplies only the phase. That is what makes locking so useful — a small signal controls a large one — and it is the same division of labour as in a faint signal made measurable by a strong one, with the roles reversed.

Still open: synchrony in the brain #

Neurons fire rhythmically and influence one another, and large populations of them fall into synchronised oscillations — the brain rhythms seen in recordings from the scalp. Whether this synchrony is a mechanism, part of how the brain binds information or routes it between regions, or a side effect of coupled activity with no function of its own is an open question in neuroscience, argued with experiments that disrupt the rhythms and models that treat neurons as coupled phase oscillators. The mathematics of locking is not in doubt; what the brain uses it for is.

The habit worth carrying away is to ask whether an oscillator keeps its own time before asking how it answers a push. A passive resonator answers with an amplitude; a self-sustained oscillator answers with a phase, and a push too weak to change how it swings can still take over when it swings — within a range of frequencies that grows with the push and shrinks with the oscillator’s own sharpness.