The width that is a lifetime

There are two ways to characterise a resonator and they look like different experiments. Hit it and watch it ring down, timing how many cycles it takes to fade. Or drive it at a series of frequencies and plot the response, measuring how narrow the peak is.

The two numbers agree to two hundredths of a per cent, and the residual is the spacing of the frequency grid rather than anything physical. They agree because they are the same measurement.

Why they cannot differ #

The argument has one moving part.

A linear system is completely described by what it does to an impulse. Its response to any input at all is that impulse response convolved with the input, and convolution in time is multiplication in frequency — so the system’s response to a steady drive at frequency $\omega$ is the Fourier transform of the impulse response, evaluated at $\omega$.

That is the whole of it. The ringdown is the impulse response; the resonance curve is its transform. Measuring one and measuring the other are the same act performed in two coordinate systems, and the quality factor is a property of the function rather than of either view of it.

For a damped oscillator the impulse response is $e^{-\zeta\omega_0 t}\cos\omega_d t$, whose transform has magnitude squared close to a Lorentzian of half-width $\zeta\omega_0$. The amplitude falls by $e$ in a time $1/\zeta\omega_0$; the peak’s full width at half power is $2\zeta\omega_0$; and the ratio of centre frequency to width is $1/2\zeta$, which is also $\pi$ times the number of cycles to $1/e$. One parameter, two appearances.

The two measurements, in practice #

Both are made routinely and they fail in different ways, which is the practical reason for having both.

From the decay, count cycles to some fraction of the starting amplitude. This is easy when the resonator rings for a long time and impossible when it does not: a system with $Q = 3$ has essentially finished after one cycle, and there is nothing to count. It also needs a clean impulse, which is harder than it sounds — anything that excites more than one mode gives a decay that is a sum of exponentials and does not have a single time constant at all.

From the width, sweep the drive and find the half-power points. This works at any $Q$ and is the only option at low $Q$. Its difficulty is the opposite one: at very high $Q$ the peak is narrower than the sweep can resolve, the drive itself has a linewidth, and the sweep must be slow enough that the resonator reaches its steady state at every point — which takes $Q$ cycles at each step, so a sharp resonance is slow to measure by exactly the factor that makes it sharp.

That last point is the useful trade and it is worth stating as a rule. A resonator’s selectivity and its speed are the same number. A filter that passes a band $\Delta f$ wide takes about $1/\Delta f$ to settle. There is no design that evades this, because it is not a design constraint; it is the statement that a narrow function has a wide transform, which is the same theorem that fixes how far along its own path a wave stays in step with itself and the same one that fixes what a wave packet must give up in position to be sharp in wavelength.

The width of a packet against the width of its own spectrum is the same reciprocal relation as the one between a ringdown and a linewidth, and there is no mechanism in either. Only the transform: a function narrow in one variable is broad in its conjugate, and the two widths multiply to a constant that depends on the shape and on nothing physical. That is why the relation holds for a bell, a laser, a nucleus and a circuit without any of them having anything else in common.

The third definition, which is the one that is not a convention #

There is a definition of the quality factor that mentions neither a decay nor a width, and it is the one that makes the other two provably equal:

$$Q = 2\pi \times \frac{\text{energy stored}}{\text{energy lost per cycle}}$$

Read it as a bookkeeping statement about one cycle. If a resonator loses a fraction $2\pi/Q$ of its energy each cycle, then after $Q/2\pi$ cycles it has lost most of it — which is the ringdown, since energy goes as amplitude squared and the amplitude falls by $e$ in $Q/\pi$ cycles. And if the drive has to supply that same fraction each cycle to hold the amplitude steady, then a drive slightly off resonance, whose phase slips by $2\pi/Q$ over a cycle, has stopped being able to supply it — which is the linewidth.

The reason this definition is worth having is that it applies to things with no obvious decay and no obvious sweep. A resonant cavity full of standing waves, a planet’s tidal bulge lagging behind the moon that raises it, a rubber ball’s hysteresis loop: each has an energy stored and an energy lost per cycle, and the ratio behaves like a $Q$ in every calculation even where no ringdown has ever been observed. It is also the definition that makes $Q$ dimensionless without any argument about which frequency to divide by.

The identity behind all three is that a damped resonator is a pole of the response in the complex frequency plane, at $\omega_0(\sqrt{1-\zeta^2} - i\zeta)$. The real part is what the peak sits at, the imaginary part is what the decay rate is, and the ratio of the two is $Q$ up to a factor of two. Every statement in this essay is a statement about the position of one point.

Where the identity breaks, and why that is useful #

The equality holds for one damped mode. Every interesting measurement in physics is one where it fails, and the failure is the information.

Suppose a sample contains many resonators whose frequencies are not quite the same — atoms moving at different speeds, nuclei in slightly different local fields, oscillators with slightly different masses. Driving the collection and sweeping gives a broad response, because the individual peaks sit at different places. But hitting the collection and watching it ring gives a decay set by each resonator’s own damping, which may be far slower than the broad width suggests.

So the two measurements disagree, and the ratio between them says how much of the observed width is inhomogeneous — a spread of centres rather than a loss.

This is the whole basis of a family of techniques. In magnetic resonance the width of the line gives a time called $T_2^*$ and the decay of a spin echo gives $T_2$; the difference between them is the static spread of local fields, and the echo exists precisely to remove it. In spectroscopy the same distinction separates Doppler broadening from natural linewidth. In mechanical testing it separates a genuinely lossy material from a batch of slightly different specimens.

The general lesson is one this collection keeps meeting: two measurements that must agree are worth making twice, because the size of the disagreement is a measurement of the assumption that they should.

The phase carries the same information a second time. A resonance is a pole, and a pole fixes both the magnitude and the phase of the response — so the two are not independent measurements and a system whose phase and magnitude do not correspond is a system with more than one thing going on in it. That correspondence is the practical test for whether a measured line is a single resonance or two overlapping ones, and it is sharper than any fit to the magnitude alone.

What sets the number in real objects #

The quality factors of physical resonators span more than fifteen orders of magnitude, and the range is worth a paragraph because it says what the limit is made of.

The upper end is limited by something other than craftsmanship, and the lower end is a design choice. A car’s suspension is deliberately built at about $Q = 1$: it should return to level without ringing, which is the boundary between the oscillatory and non-oscillatory regimes and the reason a shock absorber is chosen rather than tolerated. A guitar string is a few hundred. A quartz crystal is a hundred thousand, which is why a wristwatch keeps time. A superconducting microwave cavity reaches $10^{11}$, and an isolated atomic transition — where the only loss is spontaneous emission — reaches $10^{15}$.

At the top of that range the loss is no longer a property of a material. It is the coupling to whatever the resonator is talking to, which for an atom is the electromagnetic field itself, and cannot be reduced without changing the field’s available modes. That is why the highest quality factors are obtained by isolating rather than by improving: a mirror suspended on the thinnest fibre that will hold it, a cavity cooled until its walls stop absorbing, an ion held where nothing can touch it.

The instrument this becomes #

Turn the argument round and the resonator becomes a sensor for anything that changes either half of the pair.

A frequency shift measures a mass. A quartz microbalance is a crystal whose resonant frequency drops when something lands on it, and the shift is resolvable to a fraction of the linewidth — so the mass resolution is set by $Q$. Depositing a nanogram on a square centimetre is routine.

A width change measures a loss. The same crystal in a liquid has a much lower $Q$, and the broadening measures the viscosity of what it is in — the crystal is sampling how far sideways momentum gets in a cycle, which is a length, and the length is what the damping reports.

And a ringdown measures an absorption without needing a stable source. Cavity ring-down spectroscopy fills an optical cavity, switches the light off, and times the decay. Because the answer is a time rather than an intensity, fluctuations in the source cancel out completely — a fifty-per-cent variation in how much light went in changes nothing about how fast it leaks out. That is a rare and valuable property, and it comes directly from the fact that the quantity of interest lives in the decay rather than in the amplitude.

A ringdown measurement assumes the decay is exponential, and it is — in the middle. At long times there are departures, because a strictly exponential decay would require a spectrum with no lower bound on its energy, and every real system has one. A system whose late behaviour is not exponential has no single lifetime to report, and its lineshape is not a Lorentzian either: the two failures are the same failure seen through the transform.

Lifetimes nobody could have timed #

The most extreme uses of the identity are the ones where only one side of it is available, and the other side is inferred.

A spectral line’s natural width is an excited state’s lifetime. An atom in an excited state radiates and the state decays; the emitted line therefore has a width, and the width is the decay rate. Nobody times the decay of an ordinary allowed transition directly — it lasts a few nanoseconds — but the line is measurable and the lifetime follows. The relation is the same one drawn at the top of this page, applied to a system that has never been hit with a hammer.

A forbidden transition gives a linewidth too small to measure and a lifetime that can be watched. The 21 cm line of hydrogen is a transition so improbable that a given atom waits about ten million years for it, giving a natural width of about $10^{-15}$ hertz against a frequency of $1.4$ gigahertz — a quality factor of $10^{24}$, and a line whose observed width is entirely Doppler and entirely about the gas rather than the atom. Every measurement of that line is a measurement of motion, and the identity is what guarantees it.

And the Mössbauer effect turned a gamma ray into a resonator with $Q$ near $10^{12}$. A nucleus in a crystal emitting a gamma ray without recoil produces a line whose width is the excited state’s own, and a shift of one part in $10^{12}$ becomes measurable by moving the source a few millimetres a second. That resolution is what made it possible to weigh a photon’s fall down a tower and confirm that a clock lower down runs slow over a height of twenty-two metres.

In each case the two halves of the pair are wildly different in how easy they are to observe, and which half is the experiment depends entirely on the numbers. That is the practical value of an identity: it lets a quantity be obtained from whichever of its two faces happens to be turned outward.

Why a clock is a sharp resonance and nothing else #

The quantity a clock needs is exactly the one this essay is about, and stating it that way explains a change that has happened in timekeeping over the last twenty years.

A clock works by locking an oscillator to a resonance. How well it can be locked depends on how finely the centre of the peak can be located, and a peak of width $\Delta f$ centred at $f$ can be split by a factor depending on the signal-to-noise ratio — so the fractional accuracy available goes as $\Delta f/f$ divided by that ratio. The first factor is $1/Q$.

Caesium’s microwave transition sits at 9.19 gigahertz, and an atom interrogated for about a second gives a line about a hertz wide, so $Q$ is around $10^{10}$. That number, together with how many atoms can be counted, is the whole of a caesium clock’s performance, and it is why the second was defined on that transition for half a century.

An optical transition offers the same trick at a frequency fifty thousand times higher. Strontium’s clock transition is at 429 terahertz and is intrinsically about a millihertz wide, which is a quality factor near $10^{17}$ — seven orders of magnitude better than caesium’s, from nothing but putting the same linewidth under a much larger frequency.

What limits the achieved $Q$ is the trade this essay names. Resolving a millihertz line requires interrogating for a thousand seconds, and the laser asking the question has to stay coherent for that long — so the practical linewidth is the laser’s rather than the atom’s, and the effort of the field has gone into ultra-stable cavities that hold a laser’s frequency steady for minutes. Selectivity and time are one number, and an optical clock is an apparatus for buying the time.

The planet that rings for weeks #

The other end of the range is worth naming because the resonator is unusually large.

A great earthquake sets the whole Earth vibrating in its own normal modes — the gravest of them a football-shaped oscillation with a period of about fifty-four minutes, in which the planet alternately elongates along one axis and the other. The mode’s quality factor is about five hundred.

Put that into the ringdown relation and the amplitude falls by a factor of $e$ in $Q/\pi$ cycles, which is about a hundred and sixty periods, or six days. After the largest earthquakes such modes have been tracked for weeks, as a slow beat in the readings of gravimeters all over the world.

The measurement is made both ways, exactly as this essay describes. Long records are transformed and the peaks’ widths are read; short ones are fitted as decaying sinusoids. The two agree, which is the homogeneity check — and where they disagree, the difference reports that a single mode has been split by the Earth’s rotation and its departure from a sphere into a cluster of slightly different frequencies, which is the inhomogeneous broadening of the section above, produced by a planet’s oblateness rather than by a spread of atoms.

And the $Q$ itself is the quantity of interest. It measures how much energy the mantle absorbs per cycle, which depends on its temperature and on whether any of it is molten — so a number derived from how long a planet rings after an earthquake is a thermometer for the inside of it.

What the picture cannot show #

The transform assumes the system is linear and time-invariant. Everything above follows from the impulse response existing, and nothing above survives a resonator whose stiffness depends on amplitude. A nonlinear resonator’s response curve leans over, has two stable branches over a range of drive frequencies, and cannot be read as a linewidth at all — and its ringdown changes frequency as it decays.

The spectrum drawn is of a finite record. Transforming a decay of finite length convolves the true Lorentzian with the transform of the window, which puts a floor under any measured width. The record here is ninety periods, which is ample for $Q = 25$ and would be hopeless for $Q = 10^6$; the general condition is that the record outlast the ringdown, and a measurement that does not is measuring its own window.

A Lorentzian is an approximation even for one mode. The exact transform of a damped cosine has a second pole at negative frequency, and near resonance the contribution from it is small but not zero. It matters at low $Q$, which is exactly where the whole framework is least useful.

The energy definition and the pole definition agree only at high $Q$. The three definitions above coincide when the resonance is sharp and separate by factors of order $1/Q^2$ when it is not, so a quoted $Q$ near one is a rough description rather than a number. That is not a defect of the definitions; it is that a system which barely oscillates barely has a resonance, and the boundary case has no oscillation at all.

And “half power” is a convention. The full width at half maximum is one of several definitions of a width, differing by factors of order one, and quality factors quoted from different traditions are not always the same quantity. The definition that is not a convention is $2\pi$ times the energy stored divided by the energy lost per cycle, and it is the one that makes the relation to the ringdown exact.

The ladder from here #

Later rungs on this anchor: the nonlinear resonator, whose response curve folds over and whose ringdown drifts in frequency; coupled resonators and mode splitting, where two peaks and one decay are the same object; the fluctuation–dissipation theorem, which says that the same damping controlling the width also fixes how much the resonator jiggles when nothing is driving it at all; parametric amplification, where the loss is cancelled rather than reduced; and the Q of a mode in an open system, where the “loss” is radiation and the linewidth is a leak rather than a friction.

The neighbouring ladders are the response of a driven oscillator, which is the rung this one stands on, and the three ways of coming to rest, which is where the free decay is classified and where the identity here runs out. Coherence length is the same reciprocal relation applied to a source instead of a resonator.