The lines a warm vapour hides

The spectrum is a subtraction found that a spectral line is the difference between two energy levels, and the line that is really two found that sodium’s yellow line splits into a pair when the electron’s magnetic moment is counted. The line that takes eleven million years went a level further down, to the hyperfine splitting made by the nucleus’s own magnet. Every one of those structures is measured by looking at light from atoms, and every one assumed that the measurement could resolve it. In a gas at room temperature it usually cannot. The atoms are moving, and their motion blurs their lines far more than the structures being looked for separate them.

The blurring can be beaten without cooling the gas or slowing a single atom. The trick, invented in the early 1970s, uses two beams of laser light going opposite ways through the same gas, and it works by choosing, out of all the atoms, only those that happen to be standing still.

A hump eighty times too wide #

Rubidium’s strong red line at 780 nanometres has, from its ground level marked F = 2, three transitions to three closely spaced upper levels, F′ = 1, 2 and 3, split by the interaction between the electron and the nucleus’s magnetism. They lie 423.6 and 266.65 megahertz below the highest, and each is 6.07 megahertz wide — the width that the width that is a lifetime found is fixed by how long the upper level lasts, 26 nanoseconds.

In a glass cell at room temperature the atoms move at a few hundred metres a second in every direction. An atom moving towards the laser at speed $v$ sees the light’s frequency raised by $v/\lambda$ — the note that changes on approach — and so absorbs when the laser is tuned below the line by that amount. The spread of velocities along the beam, 169 metres a second either way, spreads each line over hundreds of megahertz. The three transitions merge into one smooth hump 511 megahertz wide, eighty-four times wider than any of them, and the excited state’s structure, the very thing a spectroscopist wants, is invisible in it.

Cooling helps only slowly: the Doppler width goes as the square root of the temperature, so cooling the vapour to liquid-nitrogen temperature narrows the hump by a factor of two, and the vapour would then hardly exist. The structure has to be dug out of the hump with the atoms still moving.

Two beams that agree only on atoms at rest #

Send a weak probe beam through the cell, and a stronger pump beam of exactly the same frequency through it in the opposite direction. A moving atom sees the two beams Doppler-shifted oppositely: the beam it moves towards is shifted up, the one it moves away from is shifted down.

At any detuning, the probe is absorbed by the atoms whose velocity brings it into resonance, and the pump by atoms with the opposite velocity. The pump is strong enough to keep the atoms it addresses largely excited — to saturate them — so they are no longer in the ground level to absorb anything else. Usually the pump and the probe are talking to different atoms, and the pump has no effect on what the probe sees. But when the laser is tuned exactly to a transition, both beams are resonant with the same atoms: those moving neither towards nor away from either beam. The pump has emptied the ground level of exactly the atoms the probe needs, and the probe passes through with less absorption.

So the probe’s absorption, plotted against the laser frequency, shows a narrow dip at each transition, the Lamb dip, named after Willis Lamb, who predicted the related dip in a gas laser’s output in 1964. The dip’s width is the width of the class of atoms selected, which is the natural width of the line, broadened slightly by the pump’s own strength — not the Doppler width.

How strong the pump must be is a compromise. A weak pump empties too few atoms to make a visible dip; a strong one empties them thoroughly but broadens the very thing being measured, because an atom driven hard by light responds over a wider range of frequencies — the line is power-broadened by the square root of one plus the saturation, the ratio of the pump’s intensity to the intensity that would excite half the atoms. For rubidium that saturation intensity is a few milliwatts per square centimetre, so a pump of a few milliwatts spread over a beam a few millimetres across sits near saturation, and the dips in the figures, at a saturation of one, are 8.6 megahertz wide rather than the natural 6.07. The probe must be weak by comparison, ten or a hundred times, so that it reads the hole without digging one of its own. Every practical arrangement, a beam splitter taking a tenth of the light for the probe and sending the rest back through the cell as the pump, is a setting of that one ratio.

Two kinds of width #

The method rests on a distinction that runs through the whole of spectroscopy. A line can be broad because every atom in the sample has a broad response — each excited atom decays quickly, so each one’s line is wide — or because every atom has a narrow response but they are not all at the same frequency. The first is homogeneous broadening: nothing can be done about it without changing the atoms. The second is inhomogeneous: the sample is a crowd of narrow lines at slightly different places, and the breadth belongs to the crowd, not to anyone in it.

Doppler broadening is the purest case of the second kind. Each atom, at its own velocity, has a line exactly as narrow as an atom at rest; the crowd’s velocities spread those narrow lines over half a gigahertz. A strong beam at one frequency picks out the atoms of one velocity and saturates them, burning a hole in the crowd as wide as one atom’s line — the homogeneous width — and the probe reads the hole. The size of the hole, not the size of the crowd, is what the method measures.

The same distinction, and the same trick, appears wherever narrow responses are hidden by a spread. In a magnetic field that varies across a sample, nuclear spins precess at slightly different rates and the signal from all of them fades quickly; a second pulse reverses their phases and they come back into step — the echo that the return a classical cloud never makes found is the one way a spreading crowd can be made to regroup, by a reversal applied from outside. Dyes and ions frozen into a glass each sit in slightly different surroundings and absorb at slightly different frequencies, and a laser burns persistent holes in their broad band — holes once proposed as a way to store data at many frequencies in one spot. In each case the broad line is a census and the narrow line is an individual, and the experiment is designed to address individuals.

The shift hydrogen hid #

Hydrogen shows how much the crowd can conceal. Its atoms are the lightest there are and move fastest, at nearly two kilometres a second at room temperature, and the Doppler width of its red Balmer line is several gigahertz. Inside that width lie the line’s fine-structure components, the splitting that the line that is really two found for sodium, and inside those lies a smaller shift still: the Lamb shift, a displacement of one level by about a gigahertz caused by the electron’s interaction with the fluctuating electromagnetic field of empty space.

In the 1930s several measurements of the Balmer line’s shape hinted that the levels were not where Dirac’s theory put them, and others did not; the Doppler width made the question undecidable by looking at light. Willis Lamb and Robert Retherford settled it in 1947 by a different route entirely, driving the transition between the two nearly degenerate levels directly with microwaves, whose Doppler shifts are negligible. Twenty-five years later Hänsch, Issa Shahin and Schawlow resolved the Lamb shift optically, in a gas discharge at room temperature, with a pump and a probe — the first time the shift that founded quantum electrodynamics had been seen in hydrogen’s own visible light.

Six dips where three lines were #

The spectrum has six dips, not three, and the extra three are a beautiful embarrassment. Two transitions that share the same lower level — here all three, since all start from F = 2 — give a second kind of coincidence. An atom moving at exactly half their frequency difference times the wavelength sees the pump shifted onto one transition and the probe onto the other. The pump empties its ground level through the first transition; the probe, looking for that ground level through the second, finds it emptied. The probe’s absorption dips, at a laser frequency exactly midway between the two transitions, though there is no transition there. These crossover resonances come from atoms moving at definite velocities, 104 metres a second for the pair F′ = 2 and 3, and are as sharp as the true lines.

Subtracting the absorption with the pump off from that with it on removes the hump and leaves the six features cleanly. Their positions are the excited state’s hyperfine splittings, read off a glass cell on a bench to a fraction of a megahertz. The heights in the figure come from a simple model that lets the pump empty the ground level only by exciting atoms. Real rubidium spectra have larger crossovers, often the largest features of all, because an excited atom can decay to the other ground level, F = 1, and be lost to the transition for good — optical pumping, which deepens every hole the pump burns and does so most where two transitions share the work.

Theodor Hänsch, Arthur Schawlow and their colleagues at Stanford made the method a tool between 1971 and 1972, using it to resolve the hyperfine structure of sodium’s yellow lines and the fine structure of hydrogen’s red Balmer line, and to measure the Rydberg constant more precisely than ever before. Schawlow shared the Nobel Prize in 1981 for laser spectroscopy.

The hump as a thermometer #

The Doppler hump that the method works so hard to remove is itself a measurement. Its width goes as the square root of the temperature over the atom’s mass, and it is nothing but the speeds in a still room — Maxwell’s distribution of molecular velocities — written onto the frequency axis by the Doppler effect. Measure the width of a line from a gas whose temperature is known, and the result is the ratio of Boltzmann’s constant to the molecule’s mass.

That was one of the routes by which Boltzmann’s constant was measured before it was fixed by definition in 2019. Groups in Paris, Naples and elsewhere recorded the Doppler profiles of lines in ammonia, water and caesium in cells held at the temperature of melting ice or of water’s triple point, and fitted their widths to parts in ten thousand or better. The hard part was the shape near the line’s wings, where the collisions that saturated-absorption spectroscopy treats as a nuisance distort the Gaussian, and the same detailed models of line shape built to correct those measurements are what tell a saturated-absorption spectrum’s small pedestals apart from its dips. A warm gas’s lines carry two numbers at once, the atom’s structure and the gas’s temperature, and Doppler-free methods are a way of reading the first without the second getting in the way.

Holding a laser on a line #

The dip has a further use that has made it one of the most common measurements in physics. Wobble the laser’s frequency slightly and detect how the probe signal follows the wobble: the result is the slope of the dip.

The slope is zero exactly on the line and changes sign across it, so it tells which way the laser has drifted and by how much. Fed back to the laser’s tuning, it holds the frequency on the line’s centre. Because the dip is narrow — a few megahertz rather than half a gigahertz — the slope at the centre is steep, and a signal measured to a part in a thousand holds the laser to within a few kilohertz, a few parts in $10^{11}$ of its frequency.

Almost every laser used to cool and trap atoms is held on its transition this way, by a centimetre-long glass cell of the same atoms beside the experiment, because the friction made of light works only if the laser sits a precise fraction of a linewidth below the transition and stays there. The atoms that are cooled to microkelvin in the trap owe their temperature to atoms at room temperature in the reference cell, sorted by velocity by a pair of beams.

What survives the selection #

The method removes the first-order Doppler shift, the one proportional to the velocity along the beams, by choosing atoms for which that velocity is zero. It does not remove everything that motion does. Atoms selected for zero velocity along the beams still move sideways across them, at the full thermal speed, and spend only a short time crossing the beams, which broadens the line by the inverse of the crossing time — transit-time broadening, a few hundred kilohertz in a millimetre-wide beam. And every moving atom’s clock runs slow by the second-order Doppler effect, the time dilation that the clock that runs slow because it is warm found has only one sign and so cannot be averaged away. Atoms moving sideways at 169 metres a second have their frequencies lowered by a part in $10^{13}$, a shift no choice of beam directions can remove.

A second Doppler-free method uses that fact as its boundary. An atom that absorbs two photons at once, one from each of two counter-propagating beams, receives a total momentum of zero along the beams and so no first-order Doppler shift at all — every atom contributes, not only those at rest. Hänsch’s group used two-photon absorption to measure hydrogen’s transition from its ground state to the first excited state with a single beam reflected back on itself, and by 2011 had the frequency to four parts in $10^{15}$, where the second-order Doppler shift of the hydrogen atoms, too hot to be ignored, was one of the largest corrections.

Where the two-level model stops #

The figures use a model in which each atom has one ground level and three excited ones, the pump empties the ground level in proportion to its strength, and the dips have Lorentzian shapes broadened by a fixed saturation. Real rubidium has two ground levels and a richer structure: excited atoms decay to either, which is the optical pumping that makes the crossovers strong; the pump’s strength varies across its beam; atoms in different magnetic sublevels respond differently to the light’s polarisation and to stray magnetic fields; and collisions with other atoms change velocities during the measurement, partly refilling the hole the pump burned and adding a broad pedestal under each dip. The relative heights of the dips in a real spectrum depend on all of these, and quantitative fits use full rate equations or density matrices for every sublevel.

The positions are another matter. They depend only on the transition frequencies and on the geometry of the beams, and the model gets them right to a small fraction of the dips’ width. The domain of the argument is a dilute vapour in which atoms keep their velocities while crossing the beams, probed by beams strong enough to saturate and narrow enough not to be broadened much by transit. Within it, a warm gas shows its atoms’ natural lines.

Still open: how close to the natural width #

Saturated-absorption dips are rarely as narrow as the natural width: the pump’s strength broadens them, the beams’ finite size adds transit broadening, stray fields split them, and collisions with the walls and each other blur them. The best Doppler-free references in vapour cells now hold lasers to tens of hertz for seconds, using techniques — modulation transfer, two-photon transitions in hot vapours, polarisation spectroscopy — that pick particular slices of the atoms’ response. Whether a vapour cell the size of a sugar cube, of the kind now made by etching silicon, can provide a frequency reference stable enough to replace a laboratory’s, and what ultimately limits the stability of a reference made from atoms that are moving at hundreds of metres a second, are being worked out by groups building portable optical clocks.

The method’s central idea is a selection, not a cooling. Two counter-propagating beams of one frequency see a moving atom Doppler-shifted oppositely, so they share atoms only when those atoms are at rest along the beams — or moving at exactly ±λΔν/2 between two transitions — and the pump’s emptying of those atoms shows in the probe as dips a few megahertz wide inside a 511 MHz Doppler hump. Rubidium’s three hidden lines and three crossovers come out of a glass cell at room temperature, and a laser locked to one of them can hold its frequency to a few parts in a hundred thousand million.