The atoms that measure better together

The wait that sets the precision built an atomic clock out of a pair of short pulses and a long, undisturbed wait, and found that the width of the line it measures depends on nothing about the atom and everything about the wait. It ended with a floor that no wait can lower. Each atom, at the end of the sequence, is asked whether it is in the upper state, and on the side of the fringe where the clock is steered each answers yes or no with equal probability. The answer that was not there before is the reason: a measurement does not read off a value the atom already held, it produces one. So the count of atoms answering yes scatters from run to run like the count of heads in a hundred tosses, and a frequency error shows only when it moves that count by more than its scatter.

That scatter — quantum projection noise — falls as the square root of the number of atoms, and for thirty years it was treated as the limit of an atomic measurement in the same way that the shot noise of light was treated as the limit of an optical one. The noise pushed below the floor showed that for light the limit belongs to one shape of noise and can be beaten by reshaping it. The same is true of atoms, and the way it is done says something about what the floor was. The atoms’ answers are independent because nothing correlated them. Correlate them — entangle them — and the scatter of the total need not grow as the square root of their number.

A hundred atoms as one spin #

Each atom in a clock is a two-level system, and a two-level system has the mathematics of a spin of one half: its state is an arrow on a sphere, with the upper state at the north pole, the lower at the south, and the equal superpositions round the equator. The first pulse of the clock sequence puts every atom on the equator, pointing the same way. A hundred such atoms together are described by their total spin, of length fifty, pointing along that same direction — an arrow of length N/2 for N atoms, the angular momenta adding as the angular momentum that is not a rotation found they must.

The total spin cannot point exactly. Its components across its own direction do not commute with each other, so by the rule that the questions that can be asked together derived from that fact, their uncertainties are bounded below: the product of the two spreads is at least half the length of the spin. For independent atoms the bound is met equally in both directions, each transverse component having a variance of N/4, and the uncertainty is a round disk of radius $\sqrt{N}/2$ seen end-on — the left panel of the first figure. A phase error δφ in the clock tips the whole spin through that angle and moves its tip sideways by (N/2)δφ. Against a disk of radius $\sqrt{N}/2$, that is visible only when δφ is larger than about $1/\sqrt{N}$. The square root in the projection-noise limit is the ratio of the disk’s radius to the spin’s length.

The uncertainty principle fixes the disk’s area, not its shape. A disk of the same area could be an ellipse, narrow in one direction and wide in the other, and if the narrow direction lay along the tip’s motion a smaller δφ would show.

Shearing the disk #

The way Masahiro Kitagawa and Masahito Ueda proposed in 1993 to make the ellipse is called one-axis twisting. It needs an interaction among the atoms with energy proportional to the square of the total spin’s component along one axis — the vertical, say. Under such an interaction, a part of the spin state with a given vertical component precesses about the vertical at a rate proportional to that component. The top of the uncertainty disk, which has a slightly positive vertical component, turns one way; the bottom turns the other; the middle stays where it is. The disk is sheared, as a deck of cards is sheared by pushing its top sideways, and an ellipse results, tilted from the horizontal.

The figure shows the result of the best amount of twisting for a hundred atoms. The ellipse’s variance is 1.22 in its narrow direction and 535 in its long one, so the product of the two spreads has, if anything, grown; nothing has been gained in total. What has changed is that in one direction the spread is now a fifth of what it was. Rotate the whole state so that that direction lies along the tip’s motion under a phase error — a single further pulse — and the phase is read against the narrow width.

The twisting also costs something. Shearing a disk that sits on a sphere does not keep it flat, and the spread of directions it creates shortens the average spin by 11.8 per cent. A shorter spin moves its tip less for the same phase, so the useful measure, introduced by David Wineland and colleagues in 1992, compares the narrow spread with the length that remains. Measured that way, the best twisted state of a hundred atoms reads a phase with 12.0 decibels less variance than independent atoms: a sixteenth.

The calculation is exact. The second-order moments of a twisted state have closed forms, and the figures check those formulas against a direct evolution of a twelve-atom state, component by component.

The same error read against two noises #

What the ellipse buys is clearest in the count itself. A phase error of a tenth of a radian, read with a hundred atoms, moves the expected number of atoms found in the upper state from fifty to fifty-five. For independent atoms the count is spread by five either way, so the two distributions overlap almost entirely: a single reading, compared with a threshold halfway between them, lands on the wrong side nearly a third of the time, and telling the two apart reliably takes many repetitions. After twisting, the count is spread by 1.25, the two distributions barely touch, and one reading in forty-four is wrong.

There is a tempting misreading of this figure: that the twisted atoms know their answers better, or that each has become more definite. Neither is true. Each atom, examined alone, is no more certain than before; it is still as likely to be found up as down. What has changed is the correlation between them. In the twisted state an atom found up makes it more likely that another is found down, so the total fluctuates less than a sum of independent answers would. The correlations are entanglement, of the kind the correlation no instructions produce found between two particles, spread here thinly over a hundred: no pair of atoms is strongly entangled, and the effect is collective. What two have they cannot give a third explains why that has to be so — entanglement shared among many cannot be concentrated in any one pair — and it is also why a squeezed state is fragile. Lose a single atom, and the correlations that atom held with the rest are gone.

Too little twisting, and too much #

A little twisting gains a little, and the gain grows as the shear increases — up to a point. The third figure follows the gain as the twist is turned up. For a hundred atoms it peaks at 12.0 decibels and then falls, crossing zero at a twist about two and a third times as strong and plunging below. Two things go wrong at once. The ellipse is drawn on the surface of a sphere, and as it lengthens it begins to wrap round the sphere’s curve; its narrow direction is no longer a straight line, and no single rotation can lay it along the motion of the tip. And the spin keeps shortening: by the time the curves cross zero the average length has fallen so far that the shorter lever arm wastes whatever the narrow spread had gained.

The curves for more atoms rise higher and peak sooner. With ten thousand atoms the best twist gives 26.2 decibels, a factor of four hundred in variance, and is reached at about half the scaled twist that a hundred atoms need. A larger sphere, relative to the size of the uncertainty on it, is flatter on the scale of the ellipse, so the ellipse can be stretched further before the curvature spoils it.

How fast the gain grows #

The best gain from one-axis twisting grows as the two-thirds power of the number of atoms, so the phase error falls as N^(−5/6) rather than N^(−1/2). That is a large improvement and not the largest possible. Quantum mechanics bounds the variance of any phase estimate made with N two-level systems at 1/N² — the phase error at 1/N — and that bound is called the Heisenberg limit. In the figure it is the dashed line, falling as 1/N relative to independent atoms. It can be reached in principle with more elaborate interactions, or with a state in which all the atoms are either all up or all down together, whose phase turns N times as fast as a single atom’s. Such states exist in laboratories for a few tens of ions or qubits, and they are so fragile that losing one particle destroys the whole advantage. The twisted state is the compromise: much of the gain, and an advantage that degrades gracefully when an atom is lost.

The limit has a form that connects it to something else. The fastest a state can change into an orthogonal one, which the fastest a state can stop being itself found is set by its spread in energy, is the same bound viewed from the other side: a phase accumulates at a rate set by the energy, and a state whose energy spreads over N times the range of a single atom’s can turn N times as fast, and so register a phase N times as finely.

Squeezing by looking #

Twisting is not the only way to make the ellipse, and the other way belongs to this subject more directly. Measure the number of atoms in the upper state gently — with a beam of light that passes through the cloud and is shifted in phase by an amount proportional to that number, without telling which atoms are up — and the result is a count with an uncertainty set by the light, not by the atoms. If the light’s resolution is better than the $\sqrt{N}/2$ spread of the count, then after the measurement the count is known to that resolution. The atoms have not been individually examined, and each is still in its superposition, but their total is now pinned more tightly than any set of independent atoms could have it.

That is a squeezed state, made by a measurement rather than an interaction, and the arithmetic is the ordinary arithmetic of combining a prior spread with a reading. For ten thousand atoms, whose count spreads by fifty, a gentle reading good to five atoms leaves a spread of just under five: a hundredfold reduction in variance, twenty decibels, before any losses are counted. The value a gentle measurement returns described what such a weak reading can and cannot say about individual particles; here the point is that it says a great deal about their sum, because the sum is the one quantity the light was coupled to. The squeezing comes from the information the reading gained, and the rotation that follows, laying the narrow direction along the clock’s phase, is then the same as before.

The price is again in the light. Each photon that probes the atoms has a chance of being scattered by one of them, which knocks that atom out of its superposition and shortens the collective spin; reading more precisely needs more photons and scatters more atoms. The best squeezing comes from an optical cavity that makes each photon pass the atoms many times, increasing the phase shift it acquires faster than the chance of a scatter, and that is the arrangement in which the largest gains have been reported.

What the detector must be able to do #

The noise squeezing removes is the noise of a count of atoms. If the detector cannot count atoms to better than that noise, the improvement is invisible. For a hundred atoms the twisted scatter is 1.25 atoms, and a detector that misjudges the count by three atoms adds a scatter more than twice as large as the one the twisting left. The last figure puts this into numbers. With ten thousand atoms a perfect detector would read the full 26.2 decibels; a detector good to three atoms keeps 22.1; one good to ten atoms keeps 13.5. With a hundred atoms and a ten-atom detector, the twisted state is worse than independent atoms read perfectly, because the detector’s error is larger than either state’s noise.

This is why the experimental history of spin squeezing has been as much a history of detection as of preparation. The largest gain demonstrated so far, about twenty decibels, was reported in 2016 by Onur Hosten, Mark Kasevich and colleagues using half a million rubidium atoms. In that experiment the atoms sat inside an optical cavity, and light passing through it measured their collective state to a precision of a few atoms — gently enough to leave it squeezed rather than destroyed, so that the same light both prepared the state and read it. Gains of a few decibels have since been shown on the optical transitions of clocks themselves, with a few hundred atoms. The noise every amplifier must add states the analogous constraint for light: a signal cannot be amplified for a noisy detector without adding noise of its own. One response, demonstrated in 2016, runs the twist backwards after the phase has been imprinted, which turns the small displacement of a narrow ellipse into a large displacement of a round disk, so that a magnified signal meets the detector instead of a quiet one.

Where the advantage holds #

The figures treat the atoms as perfect two-level systems that interact only through the twisting, with no losses, no other interactions and no noise in the twisting itself. Each of those assumptions fails in a real experiment. Atoms are lost from traps; the light used to twist them also scatters off them, which both removes atoms and randomises their phases; and the strength of the interaction varies across a cloud. The gain therefore falls below the curves at large N, and the best demonstrated values sit tens of decibels below the ideal.

A clock also has to keep its advantage through the long wait that the wait that sets the precision depends on. During that wait the noise of the laser that drives the clock adds a random phase to every atom alike, and a squeezed state, whose ellipse is narrow in one direction and very long in the other, is more sensitive to a large random rotation than a disk is: rotated far enough, its long axis swings into the direction that is read. Squeezing therefore helps most when the phase error per cycle is small — when the laser is good — and can make a clock worse when it is not.

What the pictures cannot show #

The uncertainty ellipses are drawn as Gaussian shapes, which is accurate for moderate twisting and many atoms, and misleading when the twist is large enough that the state is curved round the sphere. The count distributions in the second figure are drawn as smooth curves for the twisted state, and in fact the count is a whole number, distributed over a handful of integers when the spread is 1.25. And nothing in the figures shows how the twisting interaction is made. In practice it has come from the atoms’ collisions in a Bose–Einstein condensate, from their shared coupling to light in a cavity, from Rydberg states that make neighbouring atoms interact strongly, and from the Coulomb force between trapped ions. Each gives an interaction of roughly the twisting form, with its own imperfections, and the right mechanism for a working clock is not settled.

Still open: whether a clock can keep the gain for a day #

The figures measure the gain of a single reading. A clock’s real performance is its stability after hours or days of averaging, which depends on the laser’s noise, the atoms’ environment and the clock’s systematic shifts, none of which squeezing touches. Squeezed clocks have shown their gain in the short-term noise — at a second, at a minute — and whether the gain can be carried through to the long averaging times at which the best clocks are compared, without being eaten by the extra sensitivity of the ellipse to large rotations, is now being tested in lattice and tweezer clocks with hundreds of atoms. Proposals that adapt the twist from cycle to cycle, or that use several ensembles with different degrees of squeezing to track one another, aim to keep the advantage over the whole run. How large a fraction of the theoretical 26 decibels at ten thousand atoms a working clock can use is unknown.

The habit worth carrying away is to ask where a floor comes from before accepting it. Independent atoms read a phase against a disk of uncertainty $\sqrt{N}/2$ across, because each answers alone; twisting them shears the disk into an ellipse of the same area, 1.22 against 535 for 100 atoms, and laid along the measurement it reads a 0.1 rad error that independent atoms misjudge 31% of the time with a 2.3% error rate — a gain of 12.0 dB at 100 atoms and 26.2 at 10,000, growing as N^(2/3) towards a Heisenberg limit of N, and visible only to a detector that can count the atoms nearly one at a time. The projection-noise floor was a statement about atoms that do not talk to one another, not about atoms.