Start with the state, then ask whether the interaction is useful.
Rydberg physics is powerful because several quantities scale violently with principal quantum number. The page is now ordered like a lab estimate: pick a state, compute single-atom properties, estimate blockade, then check gate limits and caveats.
Select species, n, l, j and inspect quantum defect, binding energy, radius, and lifetime.
Use the quick C₆/blockade model to set a first-pass tweezer spacing and Rabi frequency.
Compare blockade, Rabi frequency, decay, and expected gate errors before trusting the number.
Magnetic sublevels, Förster defects, electric fields, and pair-state mixing require ARC or paper-specific data.
Rydberg State Selector
Quantum defect formula
In alkali atoms the core electrons shift the effective principal quantum number from n to n* = n − δ(n,l,j). The Rydberg–Ritz formula:
δ(n, l, j) = δ₀ + δ₂ / (n − δ₀)²gives the binding energy E_b = R_y / n*² (in eV) with R_y = 13.6057 eV. Orbital radius scales as ⟨r⟩ ≈ n*² a₀ (mean value for s-wave; accurate to ~10%). Radiative lifetime: τ ∝ n*³ for low-l states, ∝ n*⁵ for high-l circular states.
Rydberg Blockade Lab
Van der Waals blockade radius Rb for two atoms both driven to the Rydberg state. C₆ values are approximations scaled from published data, use ARC for precision calculations.
Blockade Parameters
Van der Waals blockade
Two atoms in |n,l,j⟩ interact approximately via U(r) = −C₆/r⁶ in the van der Waals regime and away from near-degenerate pair-state resonances. When |U(r)| ≫ ℏΩ, double excitation |rr⟩ is energetically forbidden — the Rydberg blockade. The blockade radius is defined by |U(Rb)| = ℏΩ:
R_b = (|C₆| / ℏΩ)^(1/6)C₆ often scales approximately as n*¹¹ within one series, so doubling n roughly multiplies Rb by 2¹¹/⁶ ≈ 3.6. Treat the sign and magnitude as state-specific: S, P, and D states can differ strongly because of angular factors and Förster resonance structure.
Blockade radius Rb vs. principal quantum number n, Rb nS₁/₂
Quantum defects reference, selected species
Values from published literature (Li et al. 2003 for Rb; Goy et al. 1982 / Weber & Sansonetti 1987 for Cs; Lorenzen & Niemax 1983 for Li, Na, K). n* = n − δ₀ − δ₂/(n−δ₀)².
| Species | Series | δ₀ | δ₂ | n*(n=50) |
|---|
C₆ scaling and Förster resonances
For like-state pairs |nℓj, nℓj⟩, C₆ arises from second-order dipole-dipole coupling to nearly degenerate pair states |n'l', n''l''⟩. C₆ diverges near a Förster resonance (exact degeneracy between initial and intermediate pair states), shifting to a C₃/r³ interaction. For P₃/₂ states in Cs at n ≈ 43 a near-perfect Förster resonance gives anomalously large and controllable interactions, exploited in early blockade experiments by the Browaeys group (Gaëtan et al. 2009).
Blackbody radiation limit
At room temperature (T = 300 K, k_BT/h = 6.25 THz), Rydberg states with ℏω_n,n' < k_BT are strongly mixed by blackbody photons. Effective lifetime drops below the spontaneous emission limit for n > 30–40 at 300 K. Cryogenic environments (4 K, 77 K) extend T₁ by orders of magnitude. For n = 50 Rb at 300 K: τ_eff ~ 50 μs (vs ~100 μs spontaneous-only).
Rydberg gate fidelity estimate
A Rydberg CZ gate requires: (1) Ω t_gate = π (π-pulse to |r⟩), (2) blockade U ≫ ℏΩ, (3) gate time t_gate ≪ τ_Rydberg. Key fidelity-limiting errors:
$r = 5\,\mu$m, $R_b = 10\,\mu$m $\Rightarrow \varepsilon_{\rm blk} \sim 10^{-4}$
Two-photon excitation
Direct UV single-photon excitation (e.g., 318 nm for Cs, ~297 nm for Rb) is possible but common experiments use two-photon excitation via an intermediate state (e.g., Rb generic route: 780 nm (D2) + 480 nm → nS/nD; Rb Harvard/QuEra gate route: 420 nm + 1013 nm via 6P; Cs: 852 nm + 509 nm → nP). The effective Rabi frequency is Ω_eff = Ω₁Ω₂ / (2Δ) where Δ is the single-photon detuning. Large Δ reduces photon scattering from the intermediate state.