Rydberg Atoms

The expectation values of hydrogen carry factors of $n$ that were only bookkeeping at low $n$. Push the principal quantum number to $n\sim 50$ or $100$ and those factors dominate everything: the atom becomes a hundred thousand times larger than its ground state, binds its electron by a few millielectronvolts, lives for microseconds rather than nanoseconds, and responds to a laboratory electric field a billion times more strongly than a normal atom. These are Rydberg atoms, and their exaggerated scaling makes them both a clean laboratory for one-electron physics and the working medium of neutral-atom quantum technology.

For an alkali the valence electron at high $n$ sees the core as a point charge, so the quantum-defect description holds with an effective quantum number $n^{\ast }=n-{\delta }_{\ell }$. For high-$\ell$ states ${\delta }_{\ell }\to 0$ and the atom is hydrogenic to high precision. Every scaling law below is written in $n^{\ast }$; for hydrogen or for high-$\ell$ alkali states, read $n^{\ast }=n$.

The scaling laws

All the exaggerated properties descend from two hydrogenic facts: the size grows as $n^{\ast 2}$ and the binding energy falls as $n^{\ast -2}$. Everything else is a power of those.¹

Each exponent has a short reason.

• Size $\sim n^{\ast 2}$. Directly $⟨r⟩=\frac{a_0}{2}[3n^2-\ell (\ell +1)]$; the leading term is $\frac{3}{2}{n}^2{a}_0$.
• Binding $\sim n^{\ast -2}$. The Rydberg formula $E=-R_M/n^{\ast 2}$.
• Level spacing $\sim n^{\ast -3}$. The derivative $\mathbf{d}E/\mathbf{d}n=2R_M/n^{\ast 3}$; adjacent Rydberg levels crowd together.
• Dipole matrix element $\sim n^{\ast 2}$. Set by $⟨r⟩$ between neighbouring states, so a Rydberg atom carries a giant transition dipole.
• Lifetime $\sim n^{\ast 3}$. The spontaneous rate scales as ${\omega }^3|d{|}^2\sim (n^{\ast -3}{)}^3(n^{\ast 2}{)}^2=n^{\ast -3}$ summed over the low-lying decay channels, so $\tau \sim n^{\ast 3}$ for low-$\ell$ states (and as steeply as $n^{\ast 5}$ for circular states, which can decay only by slow $\Delta n=1$ steps).
• Polarizability $\sim n^{\ast 7}$. Second-order perturbation theory gives $\alpha \sim |d{|}^2/\Delta E\sim (n^{\ast 2}{)}^2/n^{\ast -3}=n^{\ast 7}$; the huge dipole and the tiny level spacing multiply.

Numbers at n = 50

The scaling laws are dramatic once evaluated. For a sodium Rydberg state near $n=50$ (taking $n^{\ast }\approx n$ for a high-$\ell$ level), the leading term of the mean radius is

a hundred thousand times the ground-state radius. The binding energy and the spacing to the next level are

A field of only a few hundred volts per centimetre ionizes such a state, and a microwave photon bridges adjacent levels. The static polarizability, scaling as $n^7$, is some ten orders of magnitude above the ground-state value, which is why Rydberg atoms are exquisite electric-field sensors.

The levels crowd toward the ionization limit

Because the binding falls as $n^{\ast -2}$ and the spacing as $n^{\ast -3}$, the Rydberg levels pile up just below the ionization threshold. An infinite number of them fit into the last fraction of an electronvolt below $E=0$, forming a quasi-continuum that merges smoothly into the true continuum above it. The transition from discrete bound states to free states is where quantum-defect theory connects to electron-ion scattering.

The Rydberg blockade

The giant transition dipole makes two Rydberg atoms interact enormously strongly. Two atoms each in a Rydberg state, separated by $R$, shift their combined energy by a dipole-dipole or van der Waals interaction that inherits the $n^{\ast }$ scaling of the dipoles. In the van der Waals regime the shift is

the steepest scaling in atomic physics. The consequence is the Rydberg blockade: if one atom is already excited to the Rydberg state, the interaction shifts the doubly-excited state so far off resonance that a laser tuned to the single-atom transition can no longer excite a second atom within a blockade radius $R_b$, set by $C_6/R_b^6\approx \hslash \Omega$ with $\Omega$ the Rabi frequency of the drive.

The blockade turns a pair of neutral atoms into a conditional two-qubit gate: the state of one atom controls whether the other can be excited. This is the mechanism behind neutral-atom quantum processors, and it also underlies deterministic single-photon sources and strongly interacting photonic media.

Where Rydberg atoms appear

The same exaggerated scaling that makes Rydberg atoms useful in the laboratory makes them observable across physics.

• Quantum computing and simulation. The blockade implements entangling gates between neutral atoms held in optical-tweezer arrays, and the tunable van der Waals interaction realizes spin models for quantum simulation.
• Metrology and field sensing. The $n^7$ polarizability makes Rydberg atoms primary standards for electric-field and microwave-field measurement, with a calibration traceable to atomic structure.
• Radio astronomy. Transitions between adjacent high-$n$ levels of hydrogen, $n\to n-1$ near $n\sim 100$, fall in the radio band; these radio recombination lines are emitted by ionized interstellar gas as electrons cascade down the Rydberg ladder after recombining.
• Precision tests. Circular Rydberg states, with their long lifetimes and clean single-electron structure, are used in cavity quantum electrodynamics and in tests of the hydrogenic energy formula.

The Rydberg atom is the hydrogen problem read at its extreme limit. The wave functions, energies, and expectation values are the ones derived earlier in this module; only the value of $n$ has changed, and with it every observable by a definite power. The next module leaves the gross structure behind and resolves each level into its fine structure, the small ${\alpha }^2$ corrections that the exact Coulomb solution omitted.