Extreme Measurement

If, for example, I say, “That train arrives here at seven o’clock,” I mean something like this: “The pointing of the small hand of my watch to seven and the arrival of the train are simultaneous events.”

Thirty-three centimetres

In 2010, in a laboratory at Boulder, Colorado, a group of physicists of NIST — the heir of that office of standards the fourth chapter saw born among rigged scales and the black body — performed the experiment that opens the last chapter of this story, and that in a century will stand in the books beside Richer’s pendulum. The ingredients: two nearly identical clocks, among the best ever built, each founded on the oscillations of a single aluminium ion suspended in vacuum. Aluminium had been chosen because its clock transition is extraordinarily deaf to the disturbances of the world — but it has a defect: it does not let itself be interrogated with the available lasers, it is a perfect pendulum that does not answer questions. The solution, awarded a Nobel of its own, was to borrow an idea from quantum computers: beside the aluminium one traps a second ion of another, more talkative species, and the two are quantum-coupled so that the second translates — the interpreter reads the state of the pendulum and reports it to the laser. A clock with two atoms: one that knows the time and one that knows how to speak. The two instruments, in the same room at Boulder — which with its federal and university laboratories has for decades been the world capital of time — beat in agreement to the seventeenth figure. Then the experimenters raised one of the two by thirty-three centimetres — the height of a stool — and set themselves to listen. The raised clock began to run faster: by very little, four parts in a hundred thousand billion billion, but cleanly, stably, measurably. No defect: physics. General relativity has predicted for a century that time runs more slowly where gravity is more intense — lower, nearer the mass of the Earth — and that every step of a building therefore inhabits a time slightly different from that of the step below. Einstein had deduced it thinking of stars and planets; half a century before Boulder, to see it, a tower had been needed — the famous Harvard experiment of 1959, twenty-two metres of height difference and gamma rays — and the effect had surfaced with difficulty from the noise. The Boulder experiment read it between the floor and a stool, in plain view. In the same work, to complete the task, the two clocks measured also the other relativistic effect: set in motion at the speed of a leisurely cyclist, one ion slowed its own time exactly by as much as prescribed by special relativity — the twin paradox verified at ten kilometres an hour, the travelling twin younger by fractions of a femtosecond and science fiction demoted to a laboratory exercise [Chou et al. 2010] .

It is worth pausing a moment on the threshold of this result, for it marks a change of regime in the story we are telling. For all the book, relativity has been, for measure, a correction: something the designers of GPS must remember to apply, a refinement for specialists. From the experiment of the thirty-three centimetres onward, the relation reverses: the clocks have become so fine that the relativistic structure of spacetime is no longer a disturbance to be corrected but the first thing they see — the very landscape in which they work. There no longer exists, not even in principle, “the exact time” of a laboratory: there exists the time of this table, slightly different from the time of that table, and the difference is measure, not error. The reader understands why this chapter had to close the book: when the instrument becomes sensitive to the shape of the universe, the history of measure and the history of fundamental physics return to being — as in the days of Maxwell and the coils — the same history.

The pendulum of light

To understand how one came to feel a stool, one must open the clock. The principle is the one it has always been, and the reader knows it from Huygens and Harrison: a clock is an oscillator plus a counter — something that beats as regularly as possible, and something that counts the beats. The caesium of 1967 had replaced the balance wheel with an atom, and its beat — nine billion and some oscillations a second, in the microwave band — is the metronome on which the world has lived for half a century. The optical clocks take the next step, and the logic is the clockmaker’s: for a pendulum of equal quality, a faster one divides time into finer notches. The optical transitions of atoms — those that emit visible light rather than microwaves — oscillate hundreds of thousands of billions of times a second, tens of thousands of times faster than caesium: a dial with proportionally finer notches, and so, for the same skill in reading it, a clock potentially better by as many orders of magnitude [Ludlow et al. 2015] .

For three decades this advantage stayed theoretical, for a reason of desolating banality: no electronic device can count four hundred thousand billion oscillations a second. The oscillator was there, the counter was missing — a clock without gears, a perfect pendulum that beat too fast for anyone. The solution arrived at the turn of the millennium and won the Nobel for its makers: the frequency comb, the spectrum of a laser of very short pulses which, seen at the frequency-meter, appears as millions of perfectly equidistant teeth — a ruler of light. Resting the optical frequency of the clock on one of the teeth, and anchoring the spacing of the teeth to a countable frequency, the comb acts as a gear-down: the missing gear between light and electronics, the transmission chain Harrison would have recognized at a glance under the quantum disguise.

There remained to perfect the pendulum, and here the atomic physics of recent decades has given its masterpieces. The atoms must be stopped — thermal motion smears the spectral lines, it is the Doppler effect of a passing ambulance — and they are stopped with light. The trick, awarded yet another Nobel of this story, exploits the very Doppler it would combat: one tunes the lasers just below the frequency the atom would absorb at rest, so that only the atoms running toward the beam — and seeing it raised by their own motion — absorb photons, taking each time a small contrary push. Whichever way the atom flees, it finds a wind of light in its face: the physicists call the result “optical molasses,” atoms floundering in a honey of photons and cooling there to millionths of a degree above the zero of Thomson. As for the laser that then interrogates the stopped atoms, it must be quieter than the pendulum it measures: it is disciplined by anchoring it to cavities of special glass suspended in vacuum and isolated from every vibration, within which the light bounces hundreds of thousands of times between near-perfect mirrors — and the ultimate limit of these metronomes of light is today the thermal noise of the mirrors themselves, the Brownian quiver of the atoms of the glass: to make the best clock in the world one must first listen to, and then silence, the thermal agitation of a mirror. Regnault could not have imagined a more extreme homage to his discipline. Then they must be kept still without touching them, and here the schools are two. The first imprisons a single ion in a cage of electric fields: a single atom, controlled almost perfectly — it is the road of the Boulder aluminium. The second captures thousands in an optical lattice: a standing wave of light whose crests act as egg-cartons for atoms, each in its little hollow. The light of the trap, however, disturbs the atomic levels — and here lies one of the most elegant ideas of the discipline: there exists, for certain species like strontium and ytterbium, a “magic wavelength” at which the two levels of the clock are shifted exactly in the same way, so that the transition between them does not notice the cage. Thousands of identical pendulums, each in its niche of light, interrogated in chorus by a laser made in turn quiet beyond belief on a cavity of ultrastable glass: this is an optical lattice clock, and it is with such instruments that one listens to stools [Ludlow et al. 2015] .

The functioning, reduced to the bone, is a dialogue that repeats forever: one prepares the atoms, interrogates them with the laser, counts how many have answered — that is, how many have jumped a level — and from the count one understands whether the laser was beating a breath above or a breath below the pendulum; one corrects, and again, cycle after cycle, with the laser held on a leash by the atoms as the steering wheel is held in lane by the one who drives. The clock is not a thing that beats: it is a conversation that never breaks off. And there governs, over everything, a law of patience that is pure mathematics of waves: the fineness with which one locates the centre of a line grows with the duration of the listening — to feel fine one must listen long — so that the frontier passes also through coherence, the art of making the dialogue between laser and atom last intact for whole seconds. Even at the quantum bottom of measure, time is bought with time.

Taming the last systematics

The numbers of these instruments must be stated with the propriety the discipline demands, for “precision” is too generic a word. The figure that counts is the fractional systematic uncertainty: how much, of all the known effects that can shift the beat — and after six chapters the reader knows that the enemy is the systematic, not the tremor — remains after each has been measured and corrected, in proportion to the frequency itself. For the best optical clocks this figure has fallen below one part in a billion billion: ten to the minus eighteen. The order of magnitude deserves its canonical image: a clock like that, started at the Big Bang, would today be wrong by less than a second. And it is a gain of a factor of a hundred over the best caesium, which was already the most precise instrument ever built by humanity [Ludlow et al. 2015] .

The two virtues of the clockmaker, for that matter, remain distinct as on the target of the sixth chapter. There is stability — how little the beat fluctuates from one interrogation to the next — and here the lattices have the structural advantage of the crowd: with thousands of atoms interrogated in chorus, the law of the root of Gauss works for them, and the average converges in minutes where a single ion asks for days. And there is accuracy — how close the average beat is to that of the unperturbed atom — which is not bought with patience but only with the hunt for systematics, item by item. Steady hand and straight sight, again: the two coins with which precision is paid have not changed since the days of the chronometers, only the minting has changed. And the rival species — the aluminium of the ions, the strontium and the ytterbium of the lattices — exchange the records by blows of publications, in a competition that has the collateral merit of multiplying the cross-comparisons: the ratios between the frequencies of different species, measured by now to the eighteenth figure as well, are pure numbers that no redefinition can touch — the network of mutual checks ready for the day of the vacant throne.

The hunt for the last systematic spectres is an involuntary summary of this whole book. There is the thermal spectrum: every wall at room temperature radiates — it is the black body of Charlottenburg, returned in the guise of a disturbance — and that radiation shifts the atomic levels imperceptibly, so that to correct the clock one must know the temperature of the room with a thermometrist’s care: the old science of Regnault enlisted in the service of the new, the fifth and eighth chapters shaking hands. The most radical have chosen not to correct but to suppress: whole clocks cooled in cryogenic environments, where the walls, gelid, have almost ceased to radiate — the pendulum shut in an artificial thermal night so the room should not weigh on its beat. There is the gravitational spectrum: at ten to the minus eighteen, a centimetre of height difference is seen — so the height of every clock above the geoid must be measured, and the redshift within the laboratory, between one end and the other of the cloud of atoms, must be put on the balance sheet. And there are the collisions among the atoms, the residual electric and magnetic fields, even the pressure of the interrogating light: each measured, modelled, corrected, and declared in the table of uncertainties that accompanies every clock as the certificate accompanied the copies of the metre. Méchain would have wept with relief before one of these tables: everything that killed him — the unexplained deviation, the nameless systematic — figures in it as an ordinary item of accounting.

The clock as altimeter

But it is the geodetic consequence that gives the cleanest vertigo. If time runs differently at different heights, and if the clocks feel the centimetre, then a clock is an altimeter — and of a new kind: it does not measure the distance from a conventional level, it measures directly the gravitational potential, that is, the magnitude that decides where water flows and what “higher” means. The geoid — that ideal surface the levellings of the nineteenth century pursued across the continents with centuries of expeditions — becomes legible by comparing clocks: two linked laboratories that beat the same stand at the same level of potential, and every disagreement is a difference in height. How much this chronometric levelling serves, an anecdote the engineers hand down as a warning makes clear. The national heights start from different “zeros” — each country anchors its own sea level to a historical tide gauge: the North Sea for Germany, the Mediterranean for Switzerland — and the zeros do not coincide: when, in the early 2000s, the two halves of a bridge over the Rhine at Laufenburg approached the meeting point, the difference between the two zeros, well known and equal to a good twenty-odd centimetres, had been applied with the wrong sign: the carriageways arrived offset by over half a metre, and one abutment had to be lowered in the course of the works. Classical levelling, which propagates the heights for thousands of kilometres accumulating errors at every station, is the horse-borne telegraph of this craft; a network of clocks that read directly the gravitational potential is its electrical version — a single, physical, planetary zero, without chains of levelling to drag across the continents. The latest-generation lattice clocks have demonstrated exactly the performance this levelling requires, with stability and accuracies of sub-centimetre geodesy [McGrew et al. 2018] ; and the networks to compare them already exist, because optical time does not travel in a suitcase — it travels in fibre. Not out of snobbery: comparisons by satellite, which sufficed for caesium, stop two orders of magnitude above the necessary, and a clock at ten to the minus eighteen compared by GPS is a violin heard through an intercom. The dedicated fibres, with the light relayed and cleaned station by station, bring the comparison to the level of the clocks themselves — provided one tames the fibre itself, which breathes with the traffic and with the temperature of the ground: one does it by sending part of the light back along the same path and dividing by two the disturbance of the journey, which is, to the letter, the synchronization protocol of Poincaré, summed up in photonics a century later. The effects of height difference, meanwhile, are already daily accounting of the world’s time: the international scale is referred by convention to the geoid, and every laboratory corrects its own contribution for the height at which it lives — Boulder, on its plateau a kilometre and a half above the seas, ages with respect to Paris by a minute but perfectly known fraction, and discounts it at every report. General relativity has entered the double-entry bookkeeping of the accountants of time. Europe has stitched in recent years dedicated optical-fibre backbones that link the metrological institutes across the frontiers, with Italy among the protagonists: a backbone that starts from Turin — her again, the heir of the table of the decisive — and carries the institute’s time toward the laboratories and the radio telescopes of the country. On these networks the clocks of the capitals compare themselves continuously, and the old Magnetic Union of Gauss, the observatories that raised their eyes at the same minute, lives again in a photonic version; while the geodetic community has been working for years on the institutional consequence, a world reference system of heights that should pension off, once and for all, the Babel of national zeros [McGrew et al. 2018] .

And the clock has already begun to leave the laboratory. Japanese groups have built transportable lattice clocks, shut in climate-controlled containers, and placed one at the base and one at the panoramic observatory of the great tower of Tokyo: the four hundred and fifty metres of height difference, read as a difference in beat, returned the height of the tower in agreement with the classical levelling — the first skyscraper in history measured with time. The imagined applications range from the surveillance of sea level to the monitoring of the magmatic reservoirs beneath volcanoes — a clock that slows is mass approaching — and are to be taken for what they are, prospects of research and not chronicle; but the conceptual point is already acquired, and it is the philosophical graft of the last chapter. The third chapter had shown that to measure time served to measure space: longitude as a clock running behind. Relativity has settled the debt in a literal sense: time and space are a single fabric, and the instrument that measures the one measures the other — the optical clock is at once the descendant of Harrison’s chronometer and of Delambre’s theodolite, and it has reabsorbed both. When, in 1983, the metre was defined through light and the second, length became officially a daughter of time; the clocks that feel the geoid close the circle from the other side: height too, the most corporeal of measures — the one the first chapter took with the arm and the pace — is today, in the last analysis, a reading of a clock. The most precise instrument ever built by humanity measures the magnitude the philosophers have never managed to define; and it measures it so well that all the other magnitudes, one after another, have gone to live under its roof.

The second to come

At this point the question is inevitable, and the community has been formally asking it for years: if the optical clocks beat caesium by a factor of a hundred, why is the second still defined on caesium? The short answer is that serious metrological revolutions — the reader learned it at Versailles — are not made when the technology permits, but when the whole system is ready: and the transition is under way, with the due liturgical caution. The optical clocks, meanwhile, already work within the system through the service entrance: their frequencies are recognized as “secondary representations” of the second, their data contribute to steering the international time scale, and TAI is in fact already better than the caesium that formally defines it — the canonical situation of the eve, the dauphin who governs while the king reigns. The organs of the Convention have adopted a roadmap that fixes the criteria to be satisfied — more optical clocks, of different species, in different laboratories, with uncertainties and mutual comparisons at prescribed levels, and a regular and demonstrated contribution to the international time scale — and the expected horizon for the redefinition is around 2030: expected, mark it, not decided — the criteria are still under verification, the choice of the transition (or of the combination of transitions) is open, and the date may slip as all wise dates slip. The pretenders to the throne are a small electoral college: the strontium of the lattices with its red line, the ytterbium in both incarnations — lattice and ion — the aluminium of the interpreters, and others besides; and since none detaches the group decisively, there is also being studied in earnest the most modern option of all, a definition founded not on a queen transition but on a weighted average of several transitions — the second as a committee, the last appearance of the old wisdom of the three chronometers. When the choice comes, the manoeuvre will be the one tested in 1967, in 1983, and in 2019: one will measure the ratio between the new frequency and that of caesium to perfect equivalence, then reverse the arrow, and caesium will step down from the throne with all honours, like the Earth’s rotation before it. To whoever needs, meanwhile, the eighteenth figure — GPS lives happily with far fewer — the answer is this book’s perennial one: at the frontiers. The geodesy that wants the centimetre, the fundamental physics that seeks cracks in the constants, the networks that synchronize radio telescopes continents apart: the excess precision of today is, punctually, the infrastructure of tomorrow — no one knew what Harrison’s chronometer was for better than the Admiralty did, that is, little and badly.

Let the reader note the serenity of the scene: no Méchain torments himself, no Harrison ages in an antechamber — the redefinition of a unit has become an administered procedure, with public criteria and prudent calendars. It is perhaps the surest sign of the maturity of a discipline: when even its revolutions have their paperwork.

One challenge, however, is genuinely new, and it keeps the laboratories busy as much as the clocks themselves: what use is a second of eighteen figures if one cannot deliver it? The fibres cover the continents but not the oceans, and the final customer of time — the navigation satellite, the radio telescope, the stock exchange — often lives where the fibre does not reach. For this reason dissemination has become research of the front rank again, and even spatial: in 2025 an ensemble of European atomic clocks was installed on the International Space Station, to compare from orbit the clocks of the continents and test the links of the future — time travelling again, as in the days of the Greenwich ball, only that the tide gauge is in the sky. When the redefinition comes, for the user nothing will change, and the reader of the seventh chapter already knows why: successful metrological revolutions are those no one feels pass.

At the quantum frontiers

There remains the last frontier, the one beyond which no engineering avails: the limits that quantum mechanics imposes on measure as such. An atomic clock interrogates its atoms and reads the answer; but the answer of a single atom is quantum, that is, probabilistic — yes or no, with the right frequencies — and with $N$ independent atoms the reading fluctuates as the averages of Gauss fluctuate: the uncertainty falls as the root of $N$. It is an old acquaintance, but here it changes state: the law of the root, which in the sixth chapter was the mathematics of human mistakes, becomes the “quantum projection noise,” a limit that comes not from the imperfection of the instrument but from the grain of reality itself — the price nature asks for every question. And it is a negotiable price, within margins the theory fixes exactly: by preparing the atoms in entangled states, correlated among themselves so that their answers are no longer independent, one can compress the noise below the root of $N$ — the “squeezed” states, in which the uncertainty is squeezed away from the variable that interests and accumulated in the one that does not, as one shifts the air in a mattress. It is not speculation: it is laboratory practice, and it already has a monumental application. The interferometers that listen to gravitational waves — arms of kilometres in which the passage of a ripple of spacetime, departed from black holes in collision billions of light-years away, lengthens and shortens the light by a minuscule fraction of the diameter of a proton — gave physics in 2015 its most celebrated triumph of the century; and for years they have worked with squeezed light injected into their mirrors, gaining from the quantum trick a measurable slice of their reach: the finest measure ever performed by our species is already, in the technical sense, a measure beyond the classical limit. On the clock front, meanwhile, the candidate of the next generation is being prepared: thorium-229, the one atomic nucleus known with a transition mild enough to be excited with a laser. A nuclear clock would beat in a territory almost immune to the electric and magnetic fields that torment the electrons — the pendulum withdrawn into the last citadel of matter — and the enterprise, pursued for decades, passed in the mid-twenties its decisive step, with the transition at last anchored to the laser in the European and American laboratories. If the history of this book teaches anything, in a few decades a chapter like this will open with a stool beneath a thorium clock.

What remains to be measured, after all this, is a question for the preface more than for the farewell; but two items of the inventory deserve the last page, for they close two accounts opened by the book. The first is a salutary humiliation: the universal gravitational constant — Newton’s G, the most ancient constant of physics — is barely known to two parts in a hundred thousand, ten orders of magnitude worse than the beat of the clocks: the laboratories that measure it obtain values that do not agree among themselves more than their respective uncertainties allow, in a season of discrepancies the reader of the seventh chapter will recognize without need of comment. Among the attempts to break the stalemate there is also an Italian road that reknits the threads of the book: at Florence, where Méchain died in Spain pursuing his arc, G has been measured by dropping not spheres of metal but clouds of cold atoms — interferometers in which every atom, split like a wave along two paths, feels the gravity of the test masses and tells it in its own fringes: the same quantum tools as the clocks, turned toward the oldest question of physics. Gravity, the force that gave the book Richer’s pendulum, the meridian, and the geoid, remains nonetheless the worst measured of all: nature keeps its locked rooms even in the best-charted palace.

Atomic interferometry, it must be said, is by now a family of instruments in full right, the twin branch of the clocks in the tree of quantum metrology. The principle reverses that of the lattice: there light acts as a cage for the atoms, here the atoms behave as waves — as de Broglie dared propose a century ago — and they split, travel along two paths, and recombine interfering, exactly as light in the interferometers of Michelson. A wave of matter that falls feels gravity in its own phase: the falling-atom gravimeters by now beat the best classical instruments and are beginning to leave the laboratories, toward the monitoring of water tables, of volcanoes, and of the subsoil; and atomic antennas on kilometre baselines are being designed, shafts and tunnels in which clouds of free-falling atoms would act as a new generation of mirrors for gravitational waves at frequencies the optical interferometers do not feel. The atom, which in this chapter has played the pendulum, knows how to play the plumb line too: quantum metrology is a single box of tools, and we are still opening it. The second item is a promotion: the networks of clocks, born to give the hour, have become instruments of fundamental physics. Comparing year after year different atomic species — which feel the constants of nature with different sensitivities — it has already been established that the fine-structure constant, if it drifts at all in time, does so by less than one part in a billion billion a year: the eternity of the physical laws, promoted from a metaphysical assumption to an experimental result with an error bar. And the cross-comparisons are scrutinized in search of breaths of dark matter crossing the planet and making the clocks start in sequence, or of cracks in relativity: the instrument built to maintain a convention has become the most sensitive detector of what might escape the convention — measure, once again, pays its debt to knowledge with interest. There is finally, beneath everything, the lesson of Heisenberg that quantum metrology has ceased to suffer and learned to administer: to measure is to interact, observation perturbs the observed, and the old dream of the invisible observer is dead forever — but the disturbance, like error two centuries ago, has proved quantifiable, negotiable, even shiftable to where it does least harm. The personal equation of the nineteenth century calibrated the astronomer’s eye; that of the two-thousands calibrates the quantum price of the question. It is the same civilization of the declared limit, pushed to the bottom of things.

The pharaoh’s arm and the atom

The book can now close its circle, and it is right to do so where it all began: at the market. The reader remembers the gesture — the draper unrolling the cloth, the buyer distrusting, the iron arm walled in beside the palace door: measure as a promise between strangers, guaranteed by a third term that both accept. Eight chapters later, everything has changed: the standard has migrated from the king’s body to the Earth, from the Earth to platinum, from platinum to the atom, from the atom to the constants of the cosmic fabric; trust has been depersonalized into protocols, distributed in networks, certified in tables of uncertainty; error has passed from sin to science, convention from secret shame to declared foundation. And yet whoever has followed the thread knows that the gesture has not changed at all. When two institutes compare their optical clocks across a fibre that vaults the Alps, they do exactly what the two contracting parties did before the walled arm: they accept a third term — today an atom of strontium, yesterday a bar of iron — and before it they declare themselves equal. The chain that binds the telephone in the reader’s pocket to the optical lattice of a laboratory, and the lattice to the constants fixed at Versailles, is incomparably longer than the chain that bound the cloth to the standard of the commune: but it is made of the same links — instruments, protocols, institutions, and at every joint an act of trust.

For this reason the history of measure, which we have crossed as a history of science, is at bottom a civil history — perhaps the civil history par excellence. To measure is the way in which strangers manage not to be enemies: the market, the laboratory, the international treaty are variations on the same theme, and every advance of precision has been also a widening of the circle of who can trust whom. The humanity that fixed the Planck constant by acclamation in a congress hall is the same that gathered sixteen men on the church steps to average their feet: infinitely more skilled, not a millimetre different. Protagoras, quoted at the start of the journey, had said that man is the measure of all things; the journey suggests the amendment he lacked: the measure of all things is the agreement among men — and the instruments, from the granite cubits to the clocks that feel a stool, are the monuments that agreement builds itself in order to last. The arm on the door has never been taken down from the wall. We have only, tirelessly, refined it.