A single trapped ion is used as an optical clock and as a qubit

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Radiative transitions, external fields and collisions

Deriving the Spectrum, Ch Radiative transitions and lifetimes

Problem

A single trapped

\mathrm{Ca^+}

ion is used as an optical clock and as a qubit. Its valence electron has the ground level

4s\,{}^2S_{1/2}

, and its

3d

levels lie below

4p

. The only levels below

3d\,{}^2D_{5/2}

are therefore

3d\,{}^2D_{3/2}

60.8\,\mathrm{cm^{-1}}

lower, and the ground level, reached by the

729.1\,\mathrm{nm}

line

3d\,{}^2D_{5/2}\to4s\,{}^2S_{1/2}

. ((a)) Show that the

729.1\,\mathrm{nm}

line can be neither electric-dipole nor magnetic-dipole, and that it is allowed as an electric-quadrupole line. ((b)) Estimate its rate as the chapter did for oxygen's green line: an electric-dipole reference rate at the same frequency with

|\vec d|=ea_0

, cut by

(ka_0)^2

. Give the reference rate,

(ka_0)^2

and

A_{E2}

. ((c)) Find the resulting lifetime of

3d\,{}^2D_{5/2}

and the natural width

\Delta\nu

of the line, and compare the lifetime with the measured

1.17\,\mathrm{s}

. ((d)) The step

3d\,{}^2D_{5/2}\to3d\,{}^2D_{3/2}

is a magnetic-dipole line. Estimate its rate from

A_{M1}=\mu_0\omega^3|\vec\mu_{21}|^2/(3\pi\hbar c^3)

with

|\vec\mu_{21}|^2\approx\mu_{\rm B}^2

, and decide whether it competes with the quadrupole decay.

Related theory

Quantum Mechanics

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