Optical Lattice Clock — Magic Wavelength

Trap Laser

Wavelength λ 813.4 nm

Trap depth U₀ 50 E_r

Include hyperpolarizability (deep-trap residual shift)

Clock Status

CLOCK LOCKED

Differential light shift Δν

0 Hz

Fractional shift Δν / ν₀

Detuning from magic λ

0.0 nm

Polarizability vs. λ (schematic)

α groundα excitedλ (schematic)

How it works

An optical lattice clock traps atoms (e.g. strontium) in the standing wave of an intense laser so they can be probed for seconds without falling or colliding. The trapping light itself perturbs the atomic energy levels through the AC Stark (light) shift:

ΔE_i(λ) = -½ α_i(λ) · I(x)      (i = ground g, excited e)
Clock light shift:  Δν(λ) = [α_e(λ) − α_g(λ)] · I(x) / (2h)

Because the dynamic polarizability α(λ) depends on wavelength, α_e(λ) and α_g(λ) generally differ — the trap light drags the clock frequency around as its intensity fluctuates. But α_g(λ) and α_e(λ) are two different curves of λ, so they can cross. At that crossing — the magic wavelength (813.4 nm for the ⁸⁷Sr ¹S₀→³P₀ clock transition) — α_e = α_g, the differential shift Δν vanishes to first order, and the clock frequency becomes independent of trap depth and of the atom's position inside each lattice well.

Wavelength slider — detunes the trap laser away from 813.4 nm. Because α_g and α_e have different slopes there, Δν grows roughly linearly with the detuning — watch the ground/excited level "sticks" on each trapped atom split apart.
Trap depth slider — scales the light intensity seen by the atoms (deeper wells = tighter confinement but also a larger light shift whenever λ is off the magic point). The lattice is drawn with a Gaussian beam profile, so atoms near the edge of the cloud sit in a shallower well and shift less than atoms at the center — a real inhomogeneity that lattice-clock groups have to manage.
Hyperpolarizability toggle — even exactly at the magic wavelength the cancellation isn't perfect: a small higher-order term scaling as U₀² (multi-photon light shift) survives at very deep traps, which is why real Sr clocks operate at a compromise trap depth (tens of recoil energies) rather than an arbitrarily deep one.

The wavelength axis here spans tens of nanometers so the effect is visible; a working lattice clock actually holds λ within picometers of the magic point, where residual shifts are suppressed to the 10⁻¹⁸ fractional level. The reference clock frequency used for the fractional-shift readout, ν₀ = 429,228,004,229,873.4 Hz, is the CIPM-recommended value for the ⁸⁷Sr optical lattice clock transition.

Trapped atom Ground level (|g⟩) Excited level (|e⟩)

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Optical Lattice Clock — Magic Wavelength Cancellation