Illustrated Physics
physics, drawn
Concept

Atomic clock — where it appears

A clock whose rate is set by the frequency of a transition between two energy levels of an atom rather than by a swinging or vibrating object. The second is defined by caesium's 9,192,631,770 Hz hyperfine line, and optical clocks now keep time to a part in 10¹⁸.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The clock that gains going one way and loses going the other. The rate at which a flown clock gains on a clock left at 30° latitude, in nanoseconds per hour, against the aeroplane's ground speed, with east taken as positive. Two terms are drawn and then their sum. Height alone gives 3.5 nanoseconds an hour at 9 km and does not care which way the aircraft is pointed. Motion costs time, and because the ground is already moving eastward at 402 metres a second, flying east adds to that speed and flying west subtracts from it — so the kinematic term is much larger going east and can change sign going west. The sum crosses zero at 180 metres a second eastward, which is the ground speed at which an aeroplane's clock keeps the time of the airfield it left. Over the two flights Hafele and Keating actually made, this simple model gives -61 nanoseconds eastward and +304 westward, against their own predictions of -40 and +275 and their measurements of -59 and +273. The model here uses one average altitude, one average speed and one latitude, where the real prediction integrated the flight logs; getting the signs and the rough sizes out of three lines of arithmetic is the point, and the last twenty per cent is what the logs are for. What no amount of arithmetic supplies is the thing the experiment settled: that the effect is real, that it acts on a caesium clock in a passenger seat, and that a difference of a few hundred nanoseconds after two days is measurable.

The two clocks that flew in opposite directions

Two caesium clocks were flown round the world in 1971, one each way, and came back disagreeing with the clock left behind — one having lost 59 nanoseconds and the other gained 273. Height alone would have made both gain. The sign flip comes from the ground already moving eastward at 400 metres a second before the aircraft took off.

relativity · Time dilation How much slower a warm atom ticks. The fraction by which a free atom's clock runs slow on average because of its thermal motion, (3/2)kT/mc², against temperature on logarithmic axes, for five atoms used in clocks or tests of relativity: hydrogen 4.1 × 10⁻¹¹ at 300 K, aluminium ion 1.5 × 10⁻¹² at 300 K, strontium 4.8 × 10⁻¹³ at 300 K, caesium 3.1 × 10⁻¹³ at 300 K, mercury ion 2.1 × 10⁻¹³ at 300 K. Every line falls in proportion to the temperature and to one over the mass. A clock aiming at one part in 10¹⁸ must hold strontium below 0.63 mK and hydrogen below 7.3 µK — or know their temperature well enough to correct for it. Pound and Rebka's gravitational shift across 22.5 m, 2.46 × 10⁻¹⁵, is marked for scale.

The clock that runs slow because it is warm

Every atom in a gas is moving, and every moving clock runs slow. The first-order Doppler shifts of a hot gas cancel on average; the time dilation does not, because it has only one sign. A hydrogen maser at 40 °C runs slow by four parts in a hundred billion from this alone, a nucleus in a crystal runs slow even at absolute zero, and the first test of gravitational time dilation nearly drowned in a one-kelvin temperature difference.

relativity · Time dilation Ramsey fringes: two short pulses and a long wait. The fraction of atoms found in the upper state against the detuning of the drive from the atoms' frequency, in cycles per pulse length, for two π/2 pulses of length τ separated by a free time T = 10τ (solid), computed from the exact two-level evolution. Fringes of period 1/T ride under an envelope set by the short pulses (dotted, the response to one pulse of length 2τ). The central fringe is 0.0444 cycles per τ wide at half height, close to 1/2T. A single long pulse covering the same total time, 12τ (dashed), gives a peak 0.0666 wide: 1.50 times wider, and with no way of telling a large detuning from a small one by its side fringes alone.

The wait that sets the precision

To measure an atom's frequency, the obvious method is to drive it and look for the response, and the sharpness of the answer is set by how long the drive lasts. Norman Ramsey found in 1949 that most of the drive is wasted. Two short pulses with a long wait between them do better than one long pulse, because during the wait the atom keeps its own time undisturbed, and any difference from the drive's time piles up as an angle the second pulse can read. Every primary clock in the world runs on that idea, and its precision is set by two numbers: how long the atoms can be left alone, and how many of them there are.

quantum · Measurement The tick an elliptical orbit adds to a clock. The periodic part of the time a satellite clock gains or loses against one on the ground, after its constant rate offset is removed, against hours from closest approach, for the two Galileo satellites stranded in 2014 on orbits of eccentricity 0.156 and period 12.9 hours, a GPS satellite at 0.01 and a Galileo satellite on its intended near-circular orbit. The stranded satellites' clocks run ahead and behind by ±367 ns every orbit: slow near closest approach, where they are deepest and fastest, fast near the far point. A GPS clock wobbles by ±23 ns, and a satellite on a circular orbit by almost nothing. The wobble is −2√(GMa) e sin E/c², fixed by the orbit alone, which is what makes it a test.

The satellites launched into the wrong orbit

In August 2014 two European navigation satellites were left on the wrong orbit by a frozen fuel line in their rocket's upper stage: an ellipse instead of a circle. For navigation they were nearly useless. For relativity they were a gift. A clock on an elliptical orbit dips deeper into the Earth's gravity and speeds up as it falls, then climbs and slows, and both effects make it lose time near the bottom of the ellipse and gain it back near the top — a wobble of 367 nanoseconds every 12.9 hours, set by the orbit alone. Watching that wobble for three years checked the gravitational slowing of clocks five times more closely than anything had in the forty years before.

relativity · Time dilation A round uncertainty sheared into a thin one. The uncertainty in the direction of 100 atoms' combined spin, seen along the spin, drawn out to one standard deviation in each direction. Left: the atoms each in an independent superposition — a disk of variance N/4 = 25 in every transverse direction, so a rotation shows only once it moves the spin by √N/2 = 5.0 units. Right: after one-axis twisting, which turns the spin about the vertical axis at a rate proportional to its height, so that the upper half of the disk is pushed one way and the lower half the other (arrows). The disk becomes an ellipse with variances 1.22 and 535, its long axis tilted 12.2° from the horizontal. Its narrow direction, turned to lie along the rotation to be measured, is 12.0 dB quieter than the disk, allowing for the 11.8% the twisting costs in the length of the spin.

The atoms that measure better together

A clock read with a hundred independent atoms locates its phase no better than the scatter of a hundred coin tosses allows, because each atom answers the final question by itself. Let the atoms interact for a moment before they are read and that scatter can be shaped: the round uncertainty of their combined spin is sheared into a thin ellipse, and a phase error invisible against the coin-toss noise becomes four standard deviations wide. The gain grows with the number of atoms, and it is real only if the detector can count atoms nearly one by one.

quantum · Measurement

Named alongside it

The objects these essays reach for when they reach for this one.

Time dilationBloch sphereEquivalence principleGravitational redshiftQuantum projection noiseSatellite navigationBinomial distributionDebye modelEccentricityEntanglementGravitational potentialHafele keating

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