Quantum defect theory for alkali Rydberg states, effective quantum number, binding energy,
orbital radius, radiative lifetime, and approximate van der Waals blockade radius.
Rydberg atoms are atoms promoted to states with very high principal quantum number (n ≫ 1), where the valence electron is weakly bound and orbits at a distance of n²a₀, up to hundreds of nanometres from the core. These states have extraordinary properties: lifetimes scaling as n³ (reaching ~hundreds of μs at n~70), electric polarizabilities scaling as n⁷, and most importantly, van der Waals interactions scaling roughly as n¹¹ away from resonances. That steep scaling makes Rydberg–Rydberg interactions billions of times stronger than ground-state interactions, enabling the Rydberg blockade, the mechanism behind neutral-atom two-qubit gates. This calculator uses quantum defect theory (the Rydberg–Ritz formula) for single-atom state properties and an n*¹¹ scaling model for quick blockade estimates.
Validity note: binding energies and n* values are reliable quick estimates;
C₆ values are order-of-magnitude estimates. Real blockade strengths depend on magnetic sublevel,
polarization, quantization axis, Förster defects, pair-state mixing, and electric fields. Use ARC,
PairInteraction, or paper-specific C₆ tables before designing an experiment.
Rydberg State Selector
Select a state above to see Rydberg properties.
Effective quantum number
—
n* = n − δ₀ − δ₂/(n−δ₀)²
Binding energy
—
— cm⁻¹
Orbital radius ⟨r⟩
—
— nm
≈ n*² a₀ (mean radius, l=0)
Radiative lifetime
—
τ ≈ τ₀ × n*³
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)²
gives 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 Calculator
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
Logarithmic scale: 0.01 – 100 MHz
💡 Typical experiment: Tweezer arrays use atom spacings of 3–10 μm with
Rydberg Rabi frequencies of 0.1–10 MHz. For a 2-qubit gate you need r < Rb.
|C₆| approx
—
GHz · μm⁶
Blockade radius Rb
—
μm
Interaction at r
—
MHz
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.
🔬
Hood Lab context:
We excite 87Rb to nS1/2 or nD5/2 states (n = 60–80) via two-photon excitation:
780 nm (D2 line) + 480 nm UV light. At n = 70 the blockade radius is ~10 μm with Ω/2π = 1 MHz,
comfortably exceeding our tweezer array spacing of 4–6 μm.
The 480 nm laser is frequency-doubled from a 960 nm ECDL and locked to a transfer cavity.
We use the D-state series
to access larger C₆ coefficients and longer blockade radii.
Blockade radius Rb vs. principal quantum number n, Rb nS₁/₂
Shaded region: Rb for Ω/2π = 0.1 – 10 MHz. Dashed line: current Ω. Dot: selected n.
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:
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: 780 nm (D2) + 480 nm → nS/nD; 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.
🔬
Hood Lab gate fidelity target:
For a Rb tweezer processor we target CZ gate fidelity ≥ 99.5%. The dominant errors at n = 70 are
spontaneous emission (εse ∝ ΓRyd·tgate ~ 5×10−3 at Ω/2π = 1 MHz)
and atom loss from the tweezer during the gate sequence. Increasing Ω to 5 MHz reduces tgate
by 5× and εse by 5× — but demands higher 480 nm power (∝ Ω²) and tighter focusing.
Use the Gate Fidelity Budget tool
to explore the full error budget.
Key Papers — Rydberg Physics & Neutral-Atom Gates
[1]Jaksch et al., Fast Quantum Gates for Neutral Atoms, PRL 85, 2208 (2000) — original Rydberg blockade gate proposal.
DOI
[2]Urban et al., Observation of Rydberg blockade between two atoms, Nature Physics 5, 110 (2009) — first experimental blockade.
DOI
[3]Gaëtan et al., Observation of collective excitation of two individual atoms in the Rydberg blockade regime, Nature Physics 5, 115 (2009) — concurrent blockade observation (Browaeys group).
DOI
[4]Levine et al., Parallel implementation of high-fidelity multiqubit gates with neutral atoms, PRL 123, 170503 (2019) — first >97% Rydberg CZ gate.
DOI
[5]Evered et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023) — 99.5% two-qubit gate fidelity (Harvard/QuEra).
DOI
[6]Saffman, Walker & Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010) — comprehensive review of blockade physics and gates.
DOI
References & Tools
Source confidence note.
Quick blockade-radius outputs here are useful order-of-magnitude estimates. For publishable
values of \(C_6\), pair potentials, Stark maps, or Förster defects, use ARC or a dedicated
pair-state calculation.
rough estimate