The wait that sets the precision

The answer that was not there before found that a quantum measurement does not read off a value that was already there: it forces a two-state system into one of two answers, with probabilities set by the state, and the answer did not exist until it was asked for. The value a gentle measurement returns softened the asking, and found that a weak enough question disturbs the system little and learns little. Both essays were about what one measurement can say about a state. This one is about what many measurements can say about a frequency, which is the question every clock asks, and about the arrangement of drive and wait that answers it best.

An atom with two energy levels separated by $\Delta E$ has a natural frequency $\nu_0 = \Delta E/h$, and the second is defined, since 1967, as 9,192,631,770 periods of one such frequency in caesium. To use an atom as a clock, a laboratory oscillator has to be tuned to the atom’s frequency and kept there, and the atom has to tell the oscillator when it is off. The way it tells is by responding — by making a transition from the lower level to the upper — when driven at its own frequency and not otherwise. How sharply it discriminates is the whole of a clock’s precision.

One long pulse #

The straightforward method was Isidor Rabi’s, developed in the 1930s for measuring the magnetic moments of nuclei. Send atoms through a region where an oscillating field drives them for a time $\tau$. If the field is exactly at the atomic frequency and its strength is right, every atom is turned from the lower state to the upper. If the field is off by a small amount, the atom’s own oscillation and the drive’s slip out of step during the pulse and the transfer is incomplete. The response, plotted against the drive frequency, is a peak whose width is about $0.8/\tau$: to resolve a frequency difference of one hertz, the atom must be driven for most of a second.

That is the Fourier limit on any measurement of a frequency — sharpness in frequency has to be paid for in time — and it cannot be beaten. What Ramsey noticed in 1949 is that the long drive is not the efficient way to pay. During the pulse the drive is doing two jobs at once: comparing frequencies, and rotating the atom from one state to the other. The comparison benefits from length; the rotation does not, and the long drive makes the rotation sensitive to everything that varies along the way, such as the field’s strength, which in a long beam apparatus could not be kept uniform.

Two short pulses and a wait #

Ramsey’s method separates the two jobs. A short pulse of strength and length chosen to turn the atom a quarter of the way — a $\pi/2$ pulse — puts it in an equal superposition of the two states. Then the atom is left alone for a time $T$. Then a second $\pi/2$ pulse, from the same oscillator, is applied and the atom’s state is measured.

On resonance the two quarter turns add to a half turn and every atom ends in the upper state. Off resonance, something happens during the wait, and the response becomes a set of fringes: as the drive frequency is varied, the fraction of atoms ending in the upper state oscillates between nothing and everything, with a period in frequency of $1/T$. The central fringe, the one that marks the atomic frequency exactly, is about $1/2T$ wide. The figure computes the full response from the exact evolution of a two-level atom, including what the short pulses do off resonance, and the central fringe is a third narrower than the peak a single pulse lasting the same total time gives.

The improvement over Rabi’s method in a real apparatus was larger than that factor, because the method made the measurement insensitive to how the field varied between the pulses. In a long beam machine the atoms might be driven by fields of slightly different frequency at different places, and Rabi’s single long drive would average the differences into a broadened, shifted line. Ramsey’s atoms are not driven at all in between; they only have to feel the same oscillator twice.

What the wait does #

The wait does the comparison, and the way it does it is worth seeing on the sphere of possible states.

The state of a two-level atom is a point on a sphere, with the lower state at the south pole and the upper at the north. The first $\pi/2$ pulse tips it from the pole to the equator. On the equator the state is an equal superposition of the two levels, and it does not stand still: the relative phase of the two components advances at the atomic frequency, which on the sphere is a steady turning round the equator. The drive’s own phase advances at the drive’s frequency. Watched in a frame turning with the drive, the atom’s state turns round the equator at the difference, the detuning $\delta$, and after the free time $T$ it has turned through the angle $\delta T$.

The second pulse reads that angle. If the state is still where the first pulse put it, the second pulse completes the half turn to the north pole. If it has turned a quarter of the way round, the second pulse tips it to a point on the equator and the atom is found in either state with equal odds. If it has turned half way round, the second pulse sends it back to the south pole. The upper-state fraction is $(1 + \cos\delta T)/2$, which is exactly the fringe pattern.

So the wait converts a frequency error into an angle, and the angle grows in proportion to the wait. A detuning of a tenth of a hertz is invisible in a drive lasting a tenth of a second, and after a free time of five seconds it is a half turn — the difference between every atom up and every atom down. This is the same structure as an interferometer: the first pulse is a beam splitter that divides the atom between two states, the wait is the two arms in which the components accumulate different phases, and the second pulse is the beam splitter that recombines them. A Ramsey sequence is an interferometer whose arms are energy levels rather than paths.

Width set by the wait #

The line’s width depends on nothing about the atom, only on the time it is left alone, and so the history of atomic clocks is largely a history of making that time longer. In a caesium beam clock the atoms fly between two driving regions a metre or so apart at a couple of hundred metres a second, and the free time is several milliseconds, giving a line some tens of hertz wide. Jerrold Zacharias proposed in 1953 to throw the atoms upward through a single driving region and let gravity bring them back through it: the same field felt twice, with the atoms’ rise and fall as the wait. His fountain failed, because there were too few slow atoms in a thermal beam to make a signal. It worked once laser cooling could prepare clouds of atoms at a few microkelvin, and caesium fountains, operating since the 1990s, now define the second in a dozen national laboratories with free times of about half a second and lines about a hertz wide.

Other atoms bought their free time differently. The hydrogen maser, built by Ramsey’s group in 1960, stores hydrogen atoms in a Teflon-coated bulb for about a second, where they bounce off the walls without losing the phase of their 1420-megahertz hyperfine transition, and lets them radiate into a cavity. The wall coating was the trick: an atom striking bare glass forgets its phase, and one striking Teflon mostly does not. Masers are still the most stable oscillators over hours, and they are the flywheels that keep a laboratory’s time between the slower, more accurate readings of its fountain.

Optical clocks use transitions at hundreds of terahertz, where a line a hertz wide is a part in $10^{14}$ of the frequency rather than a part in $10^{10}$. Atoms held in a lattice of laser light can be interrogated for a second or more, and some clocks use Ramsey’s sequence, others a single long pulse, depending on which systematic effects matter most. Their precision has reached a part in $10^{18}$, enough to measure the difference in the rate of time between two clocks a centimetre apart in height.

When the atoms are not alike #

The wait cannot be made arbitrarily long, and the reason is not only that atoms fall out of the apparatus. A Ramsey measurement on a cloud of atoms is the sum of many single-atom measurements, and if the atoms do not all have the same frequency, their fringes do not line up.

Each atom’s fringes are centred on its own frequency. If the atoms’ frequencies differ — because the magnetic field varies across the cloud, because the atoms collide, because they move and see the drive Doppler-shifted — the cloud’s signal is an average of shifted fringe patterns, and once the spread is comparable to the fringe spacing $1/T$, the fringes cancel. In the figure, a spread one-tenth of the fringe spacing already removes a quarter of the contrast. Lengthening the wait narrows each atom’s fringes but leaves the spread as it was, so the averaging gets worse, and beyond some point a longer wait loses contrast faster than it gains sharpness.

That is the same loss of phase that blurs the signal an MRI scanner reads, where protons in slightly different fields precess at slightly different rates and the total signal fades. Erwin Hahn found the cure in 1950: halfway through the wait, flip every spin over with a $\pi$ pulse, so that those that had gained phase now lose it at the same rate, and they come back into step at the end. The spin echo removes the effect of any spread that stays constant during the measurement, and it works in clocks too, though there it also removes the very detuning the clock is trying to measure, so it is used to diagnose the spread rather than inside the clock’s own cycle.

The floor under the answer #

Even with identical atoms and a long wait, the frequency cannot be found exactly, because the answer each atom gives is a single yes or no.

On the side of a fringe, where the clock is steered, each atom is in the upper state with probability one half, and measuring it gives a definite answer with no information beyond the answer. With $N$ atoms, the fraction found in the upper state scatters about one half by about $1/2\sqrt N$ from run to run, and a frequency error is distinguishable from that scatter only if it moves the fringe by more. This is quantum projection noise, named by Wayne Itano and colleagues in 1993, and it puts a floor under every clock built from independent atoms: the frequency is located to about $1/(2\pi T\sqrt N)$ per measurement, improving as the square root of the number of measurements averaged.

For a caesium fountain with a million atoms, half a second of free time and a measurement every second and a half, the floor is about $4\times10^{-14}$ after one second of averaging. Fountains operate close to it, which is why their performance after a day is limited by how long they run. For an optical clock the same arithmetic gives a floor tens of thousands of times lower, because the fringe width is the same in hertz and the frequency is fifty thousand times higher — the whole case for optical clocks in one ratio. Optical clocks do not reach their projection-noise floor at short times, because the laser that drives them is itself noisy, but the best now come within a factor of a few of it.

The floor assumes the atoms answer independently. If they are entangled, their answers can be correlated so that the scatter of the total is smaller than $\sqrt N$ — the noise pushed below the floor for light has its counterpart for atoms, called spin squeezing — and clocks using squeezed states of a few hundred atoms have beaten the projection-noise limit by a few decibels.

Where the method stops #

The analysis treats each atom as two levels and the pulses as perfect rotations, which real atoms only approximate. Caesium’s clock transition is between two of sixteen sublevels, and the others have to be kept empty or out of resonance. The atoms in a fountain move, slowly, and a moving clock runs slow by an amount set by its speed squared, which in a fountain is a few parts in $10^{17}$ and in a thermal beam clock was a correction of parts in $10^{13}$, one of the reasons fountains replaced beams. They feel a slight shift from the light of the room, from collisions with each other, from the residual magnetic field and from gravity’s effect on the clock rate itself; every one of these is measured and corrected for at the level of parts in $10^{16}$, and the correction budgets of fountain clocks run to dozens of entries.

The second pulse must be phase-coherent with the first. An oscillator whose own phase wanders during the wait adds its wandering to the atoms’, and since the clock reads the difference between the two, it cannot tell its own noise from the atoms’. That is why a clock’s free time is ultimately bounded by the quality of the local oscillator as well as by the atoms, and why the lasers driving optical clocks are among the most stable oscillators ever built.

What the pictures cannot show #

The fringe figure is computed in units of the pulse length, for a free time only ten times the pulse length, so that the fringes and the envelope can be seen together. In a fountain the free time is several thousand times the pulse length, and the fringes under the envelope are so dense that only the central few are ever used. The spread figure assumes a Gaussian distribution of frequencies that stays fixed during the measurement; real spreads change with time and position, and their effect on a clock is a systematic shift as well as a loss of contrast.

And the noise figure draws a floor that real clocks approach from above. Every number on it is a limit, not a performance.

The domain of the method is a set of identical two-level systems that can be left undisturbed between two phase-coherent pulses. Inside it, the fringe width is $1/2T$ and the precision is $1/(2\pi T\sqrt N)$ per measurement. Outside it — systems that decay during the wait, that interact, or that cannot be left alone — the advantage of separating the pulses is lost.

Still open: how far entanglement can take a clock #

Projection noise falls as $1/\sqrt N$ for independent atoms; with suitably entangled atoms it could in principle fall as $1/N$, the Heisenberg limit. Squeezed clocks have demonstrated improvements of several times in the short-term noise, but entangled states are fragile: they lose their advantage faster than independent atoms lose their coherence, and a clock that averages for hours must keep its advantage over many cycles. Whether entanglement can improve the long-term precision of the best optical clocks, rather than only their short-term noise, and which kinds of entangled state survive the systematic effects a real clock must correct, are being tested now with arrays of hundreds of atoms in optical lattices and tweezers.

The method itself rests on one exchange. A single long drive spends its time both rotating the atom and comparing frequencies; two short π/2 pulses with a free time T between them leave the comparison to the wait, during which a detuning δ grows into an angle δT that the second pulse reads as (1 + cos δT)/2 — so the line is 1/2T wide, set by how long the atoms are left alone, and N atoms locate it to $1/(2\pi T\sqrt{N})$. The second is defined by a measurement whose precision comes from doing nothing to the atoms for half a second, as carefully as possible.