The atom the size of a bacterium

The spectrum is a subtraction builds hydrogen’s lines from the energies $-13.6/n^2$ electronvolts and draws the familiar series crowding onto their limits. Almost everything written about atoms concerns the bottom of that sequence: the ground state, the first few excited levels, the visible and ultraviolet lines between them. The sequence does not stop there. Levels with $n$ of a hundred or several hundred exist, can be made in the laboratory with lasers, and are found in interstellar space, where the lines between them are seen by radio telescopes.

An atom with its electron in such a level is called a Rydberg atom, and it is an exaggeration of everything an ordinary atom is. It is enormous, fragile, slow, and — in its most circular states — astonishingly long-lived. None of that needs new physics. Every one of its properties is the ground state’s value multiplied by a power of $n$, and the powers explain why these atoms have become one of the most flexible instruments in atomic physics.

An atom made of powers of n #

The radius grows as $n^2$: the Bohr radius times ten thousand at $n = 100$, half a micrometre. The binding energy falls as $1/n^2$: 1.36 millielectronvolts, about sixteen kelvin in temperature units, so a single collision with a molecule at room temperature can knock the electron off. The spacing between neighbouring levels falls faster, as $1/n^3$, which puts transitions between them in the microwave band: 6.6 gigahertz at $n = 100$. The orbital period grows as $n^3$, a tenth of a nanosecond. The lifetime of the circular state, where the electron’s orbital angular momentum is as large as it can be, grows as $n^5$. And the electric field needed to pull the electron away falls as $1/n^4$, to three volts per centimetre.

The powers come straight from the Bohr orbits. The velocity of the electron is the ground state’s divided by $n$; the radius is multiplied by $n^2$; everything else is a combination. The log-log drawing turns each property into a straight line whose slope is its power of $n$, and the spread of the lines is the story of this essay: between $n = 1$ and $n = 300$ the lifetime changes by two million million and the ionising field by eight thousand million, in opposite directions.

An atom the size of a bacterium #

At $n = 30$ a hydrogen atom is larger than a small virus. At $n = 100$ it is twice the wavelength of the light that would be needed to see it. At $n = 300$, a level produced in laboratory beams and seen in the radio spectra of interstellar gas, it is larger than a bacterium — a single proton and a single electron occupying a sphere ten micrometres across. Such an atom survives only where nothing touches it: in a good laboratory vacuum, or in the thin gas of an interstellar cloud, where the next atom is centimetres away.

In interstellar space these atoms announce themselves. Hydrogen recombining in the ionised gas around hot stars captures its electron into high levels and cascades down, emitting lines between neighbouring levels at every step. The lines between levels near $n = 100$ fall at a few gigahertz, and radio telescopes see them as the “radio recombination lines” that map ionised regions through dust that hides them optically — the same trick the line that takes eleven million years plays with neutral hydrogen, played with the other end of the atom’s sequence of levels.

How an atom is made that large #

Hydrogen is the cleanest case to calculate and the hardest to excite: its ground state is 13.6 electronvolts deep, an ultraviolet photon away from the top. Laboratories almost always use an alkali atom instead — rubidium or caesium — whose single outer electron sits only four electronvolts below the ionisation limit and can be lifted there by two or three visible and near-infrared laser photons in succession, each tuned to a real intermediate level. The last photon selects $n$: tuning it by a few gigahertz moves the atom from one Rydberg level to the next, and the levels are so closely spaced near the top that the laser must be narrow enough to resolve them.

A laser photon carries one unit of angular momentum, so laser excitation produces states of low angular momentum — the elongated, non-circular orbits that pass close to the nucleus. Making a circular state needs a further step: the atom is placed in carefully controlled electric and magnetic fields and swept with radio-frequency radiation, adding one unit of angular momentum at a time, sometimes fifty in succession, until the electron’s orbit is as round as its level allows. The field an atom calls strong shows how a field reorganises an atom’s levels once it competes with the atom’s internal energies; in a Rydberg atom those energies are so small that fields of a volt per centimetre are strong, and the reorganisation is the tool that builds the circular state.

The core that the orbit remembers #

An alkali Rydberg atom is not quite hydrogen. Its outer electron orbits a core of the nucleus and all the inner electrons, which has a net charge of one and so looks like a proton from far away. But an orbit of low angular momentum is a long ellipse that dives close to the nucleus once per revolution, and inside the core the electron feels more than one unit of charge. It is pulled harder there, and its energy is lowered.

The lowering has a simple form: the level’s energy is $-\text{Ry}/(n - \delta)^2$, as though $n$ were reduced by a constant $\delta$, the quantum defect, which depends on the angular momentum and hardly at all on $n$. For rubidium’s s-states, whose orbits dive deepest, $\delta$ is about three: the $n = 50$ s-state sits where hydrogen’s $n = 47$ would be. For orbits of angular momentum four or more, which never approach the core, $\delta$ is close to zero and the levels are hydrogen’s. The same core penetration is why sodium’s yellow line is where it is — the line that is really two measures the splitting of a p-state that also dips into the core — and in Rydberg atoms it becomes a way of measuring the core itself, from how the defect changes with angular momentum.

The lifetime a warm room takes away #

A circular Rydberg state can decay only one way: to the circular state one level down, since it has the largest angular momentum of its level and each photon can remove only one unit. That makes its lifetime simple to compute and very long.

The spontaneous lifetime here is computed classically: the power a charge on the Bohr orbit radiates, from a charge that turns must glow, divided by the energy of one photon of the orbital frequency. That is the calculation which, applied to the ground state, predicted an atom collapsing in sixteen picoseconds and helped to bury classical physics. Applied to a circular Rydberg state, it gives 32 milliseconds at $n = 51$, and experiments on circular states at that level measure about thirty. The difference between those two outcomes is the subject of the last section; for now it is enough that at high $n$ the classical calculation is right.

The lifetime rises as $n^5$ because the radiated power falls steeply as the orbit grows — the acceleration falls as $1/n^4$ and the power as its square — while the photon energy falls only as $1/n^3$. By $n = 100$ a circular state would live nearly a second.

In practice it does not, and the reason is the room. A body at 300 kelvin glows with thermal radiation that peaks in the infrared and extends down through the microwave band, exactly where transitions between neighbouring Rydberg levels lie. That radiation drives transitions up and down the sequence of levels at a rate proportional to the temperature, and the curve that would not come down supplies the number of photons available at each frequency. At room temperature it limits the lifetime of an $n = 51$ state to about a tenth of a millisecond — three hundred times shorter than its natural lifetime — and it wins above $n = 8$. Cooling the surroundings to four kelvin moves the crossover to $n = 34$ and gives an $n = 51$ atom ten milliseconds. The long lives of Rydberg atoms can be used only in a cold box, which is why every experiment that exploits them is cryogenic.

A controllable environment for one atom #

Because a Rydberg atom radiates in the microwave band, the space around it can be engineered on the scale of its wavelengths — centimetres — and its emission changed at will. The solution that is thrown away recounts the experiment in which Rydberg atoms passed between two parallel plates lived more than twenty times longer, because the plates left no mode of the right wavelength for them to emit into. Put the same atoms in a superconducting cavity whose mirrors bounce a photon back thousands of times, and a single atom and a single photon exchange energy back and forth, coherently; Serge Haroche and colleagues used circular Rydberg atoms in exactly such a cavity to count photons without destroying them, work recognised by the Nobel Prize in 2012.

The long lifetime makes those experiments possible, and the $n^5$ in the drawing is where it comes from. An atom that emitted in nanoseconds would be gone before it crossed the cavity; one that lives for tens of milliseconds crosses it many times over.

The field that pulls a Rydberg electron away #

The weakness of the binding has a use of its own.

An electric field applied to an atom tilts the potential the electron sits in, lowering the barrier on the downhill side. When the barrier falls below the electron’s energy, the electron leaves. For a level of principal quantum number $n$ the field needed goes as $1/n^4$, and it falls from tens of thousands of volts per centimetre at $n = 10$ to three volts per centimetre at $n = 100$. A torch battery with its terminals a centimetre apart would ionise every level above $n = 121$.

That steep dependence turns the atom into its own detector. Ramp an electric field up over a few microseconds and each level ionises at its own field, at its own moment; timing when the freed electrons arrive at a detector reads out which level each atom was in. State-selective field ionisation, as it is called, can tell $n = 50$ from $n = 51$ with near-perfect efficiency, and it is how the photon-counting experiments above read their atoms.

The same fragility makes Rydberg atoms exquisite sensors of electric fields. Their polarisability — how far a field distorts them — grows as $n^7$, so a Rydberg atom in a vapour cell responds to a radio-frequency field with a shift in its optical spectrum, and such cells are now used to measure microwave and radio fields traceably to atomic constants, with no metal antenna to disturb what is being measured.

Where a quantum jump becomes an orbit’s glow #

The classical lifetime calculation worked for a reason, and the reason is the oldest idea in quantum mechanics.

An electron dropping from level 2 to level 1 emits a photon at three times the frequency a classical electron would circle the $n = 2$ orbit. There is no resemblance between the quantum jump and the classical glow. At $n = 10$ the ratio is 1.17; at $n = 100$ it is 1.015; and it closes on one as $1 + 3/2n$. High in the sequence of levels, the photon emitted in a jump between neighbours comes out at exactly the frequency a classical charge on the orbit would radiate. Bohr used this requirement in 1913 to fix the constant in his formula for the levels, before there was any quantum mechanics to derive it from.

Where the quantum picture hands back the old one argues that the classical limit is subtler than large quantum numbers alone; a single stationary state, however high, is not an orbiting electron. But the frequencies, the rates and the energy loss of a circular Rydberg state do converge on the classical ones, and that is why the Larmor formula gives its lifetime correctly. A wavepacket built from a spread of neighbouring Rydberg states does move round the orbit like a classical electron, for a while, before it spreads — and, as the return a classical cloud never makes shows, reassembles later in a way no classical ensemble could.

Where hydrogen’s powers of n stop #

Hydrogen, or hydrogen-like. The powers of $n$ are exact for hydrogen. Rydberg atoms of other elements — rubidium and caesium are the usual choices — have an inner core that the electron penetrates when its orbit is not circular, and their low-angular-momentum levels are shifted by the “quantum defect”; their high-angular-momentum levels are hydrogen-like to high precision. The scalings survive; the constants in front of them change.

Field ionisation is an estimate. The $1/16n^4$ threshold is the classical saddle-point estimate. Real levels in a field split, mix and cross, and whether a given state ionises at the saddle-point field or above it depends on how fast the field is ramped and which states it passes through on the way.

Thermal radiation is a formula. The thermal lifetime drawn is the standard estimate for a level surrounded by a blackbody at a single temperature. A real apparatus has windows, openings and surfaces at different temperatures, and cryogenic experiments shield their atoms carefully precisely because the estimate is so unfavourable.

Isolated atoms. The sizes drawn are for atoms with nothing near them. Two Rydberg atoms a few micrometres apart interact strongly, and at the densities of ordinary gases a Rydberg orbit would enclose thousands of other atoms.

Ten thousand states behind each straight line #

The first figure turns each property into a straight line and so hides that the atom is made of levels: at $n = 100$ there are ten thousand states of the same energy in hydrogen, with every value of angular momentum and its orientation, and the circular state is only one of them. Most of the others do not live long — a state of low angular momentum can drop straight to the bottom in one large jump and lives only as $n^3$ — so “the lifetime of a Rydberg atom” depends enormously on which state it is in, and the figures draw only the circular one. Nor does any figure show the interaction between two Rydberg atoms, which grows as a very high power of $n$ and is strong enough that exciting one atom to a Rydberg level can prevent the excitation of any other within several micrometres.

Still open: how far a blockade can be pushed #

That last effect has made Rydberg atoms a leading platform for quantum computing and quantum simulation. Arrays of hundreds of neutral atoms held in optical tweezers a few micrometres apart are excited to Rydberg levels, and the blockade — one excited atom forbidding its neighbours — provides the interaction a quantum gate needs. The limits are the same numbers as in this essay turned against the experimenter: thermal radiation shortens the Rydberg states, the weakness of the binding makes them sensitive to stray fields, and the atoms move during the logic operation itself. How large and how accurate such arrays can become before those limits bind is being tested now, with gate fidelities around 99.5 per cent reported in 2023 and improving.

The habit worth carrying away is to find the power law before worrying about the physics. When a system is characterised by one large number, its properties are powers of that number, and knowing the powers says at once which properties will dominate and where they will cross. A Rydberg atom is a hydrogen atom with $n$ turned up, and the powers of $n$ make it a micrometre wide, a second long-lived, broken by a battery and heard across a room at the frequency of its classical orbit.