--- title: Rydberg Atoms module: The Quantum Hydrogen Atom moduleNumber: 2 lessonNumber: 7 order: 207 summary: > A Rydberg atom is an atom excited to a very high principal quantum number, and every hydrogenic property becomes exaggerated by a power of n. Size grows as n², binding falls as n⁻², radiative lifetime lengthens as n³, and the static polarizability explodes as n⁷. The levels crowd toward the ionization limit, and the enormous dipole interaction between two Rydberg atoms produces the blockade that underlies neutral-atom quantum computing. topics: [The Quantum Hydrogen Atom] sources: - book: Foot ref: "Ch. 3 — The Alkalis; §3.3 Rydberg Atoms; Ch. 9 — Cold Atoms" - book: Demtröder ref: "Ch. 5 — Atoms with More Than One Electron; Rydberg states and their properties" - book: Bransden & Joachain ref: "Ch. 3 — The Hydrogen Atom; the n-scaling of hydrogenic expectation values" draft: false --- The [expectation values](/atomic-physics/quantum-hydrogen-atom/expectation-values-virial) 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](/atomic-physics/quantum-hydrogen-atom/quantum-defects-alkali-spectra) 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.[^foot-33] > **Definition (Rydberg scaling laws).** For a state of effective quantum number > $n^\ast$, the leading dependences are > $$ > \langle r\rangle \sim n^{\ast 2}a_0, > \qquad > E_{\text{bind}} \sim \frac{R_M}{n^{\ast 2}}, > \qquad > \Delta E_{n,n+1} \sim \frac{2R_M}{n^{\ast 3}}, > $$ > $$ > \tau_{\text{rad}} \sim n^{\ast 3}, > \qquad > \alpha_{\text{pol}} \sim n^{\ast 7}, > \qquad > d_{n,n\pm1} \sim n^{\ast 2}ea_0. > $$ Each exponent has a short reason. - **Size $\sim n^{\ast 2}$.** Directly $\langle r\rangle = \tfrac{a_0}{2}[3n^2-\ell(\ell+1)]$; the leading term is $\tfrac32 n^2 a_0$. - **Binding $\sim n^{\ast -2}$.** The Rydberg formula $E = -R_M/n^{\ast 2}$. - **Level spacing $\sim n^{\ast -3}$.** The derivative $\d E/\d n = 2R_M/n^{\ast 3}$; adjacent Rydberg levels crowd together. - **Dipole matrix element $\sim n^{\ast 2}$.** Set by $\langle r\rangle$ 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. $$ % caption: The exaggerated size of a Rydberg atom. At n = 50 the mean radius is % about 2500 a₀ ≈ 0.13 μm, drawn here far larger than the ground-state atom on % the left. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % ground-state atom (tiny) \fill[black!75] (0,0) circle (1.2pt); \draw[black, thick] (0,0) circle (0.18); \node[black, anchor=north] at (0,-0.45) {ground state}; % Rydberg atom (huge) \fill[black!75] (4.6,0) circle (1.2pt); \draw[acc, very thick] (4.6,0) circle (2.4); \draw[acc, dashed] (4.6,0) -- (6.3,1.7); \node[acc, anchor=west] at (6.4,1.8) {$n^2 a_0$}; \node[acc, anchor=north] at (4.6,-2.6) {Rydberg state}; \end{tikzpicture} $$ ## 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 $$ \langle r\rangle \approx \tfrac{3}{2}n^2 a_0 = \tfrac{3}{2}(2500)(0.0529~\text{nm}) \approx 0.20~\mu\text{m}, $$ a hundred thousand times the ground-state radius. The binding energy and the spacing to the next level are $$ E_{\text{bind}} = \frac{13.6~\text{eV}}{50^2} \approx 5.4~\text{meV}, \qquad \Delta E_{50,51} = \frac{2(13.6~\text{eV})}{50^3} \approx 0.22~\text{meV}. $$ 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. | property | scaling | value at $n=50$ | | --- | --- | --- | | mean radius $\langle r\rangle$ | $n^{\ast 2}$ | $\sim 0.2~\mu\text{m}$ | | binding energy | $n^{\ast -2}$ | $\sim 5.4~\text{meV}$ | | level spacing | $n^{\ast -3}$ | $\sim 0.22~\text{meV}$ | | radiative lifetime | $n^{\ast 3}$ | $\sim 30~\mu\text{s}$ | | geometric cross-section | $n^{\ast 4}$ | $\sim 10^{5}$ times larger | $$ % caption: The Rydberg scaling laws on a log-log plot: each property is a % straight line in log n whose slope is its power. Positive powers (radius, % lifetime, polarizability) rise; the binding energy (slope −2) falls. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {$\log n$}; \draw[->, black] (0,0) -- (0,3.8) node[above, black!70] {$\log$ property}; % polarizability slope 7 (steepest) \draw[black, very thick, densely dotted] (0.5,0.25) -- (4.3,3.5); \node[black, anchor=west] at (4.3,3.5) {polarizability}; % lifetime slope 3 \draw[black, very thick, dashed] (0.5,0.25) -- (5.2,2.4); \node[black, anchor=west] at (5.2,2.4) {lifetime}; % radius slope 2 \draw[acc, very thick] (0.5,0.5) -- (5.2,1.6); \node[acc, anchor=west] at (5.2,1.6) {radius}; % binding slope -2 (falling) \draw[black, very thick, dash dot] (0.5,3.2) -- (5.2,0.7); \node[black, anchor=west] at (5.2,0.7) {binding}; \end{tikzpicture} $$ ## 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. $$ % caption: The Rydberg series converging on the ionization limit: energies % −R/n*² accumulate below E = 0, with the spacing shrinking as n*⁻³ so infinitely % many levels crowd the last sliver of binding. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % ionization limit \draw[black, thick] (0,3.2) -- (5.6,3.2); \node[black!70, anchor=west] at (5.6,3.2) {ionization}; % converging levels: y = 3.2 - 3.0/n^2, n=3..12 \foreach \n in {3,4,5,6,7,8,10,14}{ \pgfmathsetmacro{\yy}{3.2 - 8.0/(\n*\n)} \draw[black, thick] (0,\yy) -- (4.6,\yy); } \node[black, anchor=east] at (-0.1,-0.28) {low $n$}; \node[black, anchor=east] at (-0.1,3.0) {high $n$}; % shading of quasi-continuum near limit \node[black, anchor=west] at (4.7,2.85) {crowding}; \end{tikzpicture} $$ ## 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 $$ \Delta E_{\text{vdW}}(R) = \frac{C_6}{R^6}, \qquad C_6 \sim n^{\ast 11}, $$ 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 $C_6/R_b^6 \approx \hbar\Omega$ with $\Omega$ the Rabi frequency of the drive. > **Definition (Rydberg blockade).** Within a radius $R_b$ of an already-excited > Rydberg atom, the van der Waals shift exceeds the excitation linewidth, so at > most one atom in the region can be excited. A laser pulse then drives the > collective state > $$ > |gg\rangle \to \frac{1}{\sqrt{2}}\big(|rg\rangle + |gr\rangle\big), > $$ > a single shared excitation, rather than two independent ones. 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. $$ % caption: The Rydberg blockade. An excited atom shifts every doubly-excited % state within the blockade radius R_b off resonance, so a second atom inside % that radius cannot be excited; one outside can. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % blockade sphere \draw[acc, thick, dashed] (0,0) circle (2.0); \node[acc, anchor=south] at (0,2.05) {blockade radius}; % excited atom at center \fill[acc] (0,0) circle (3pt); \node[acc, anchor=north] at (0,-0.3) {excited}; % blocked atom inside \draw[black, thick] (1.1,0.5) circle (5pt); \node[black, anchor=west] at (1.4,0.5) {blocked}; % free atom outside \draw[black, thick] (3.2,-0.6) circle (5pt); \node[black, anchor=west] at (3.5,-0.6) {can excite}; \end{tikzpicture} $$ ## 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](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession), the small $\alpha^2$ corrections that the exact Coulomb solution omitted. [^foot-33]: **Foot**, _Atomic Physics_, §3.3 and Ch. 9 — the $n$-scaling of Rydberg-atom size, binding, lifetime, and polarizability, the convergence of the series on the ionization limit, and the van der Waals blockade between two Rydberg atoms.