0198506961.pdf

144 The interaction of atoms with radiation

7.7 The a.c. Stark effect or light shift

In addition to its effect on the populations, the perturbing radiation also

changes the energy of the levels and we calculate thislight shiftin this

section. We can write eqns 7.41 for ̃c 1 and ̃c 2 in matrix form as

i

d

dt

(

̃c 1

̃c 2

)

=

(

δ/2Ω/ 2

Ω/ 2 −δ/ 2

)(

̃c 1

̃c 2

)

. (7.89)

This has solutions of the form

(

̃c 1

̃c 2

)

=

(

a

b

)

e−iλt. (7.90)

The equation for the eigenvaluesλis

∣

∣

∣

∣

δ/ 2 −λ Ω/ 2

Ω/ 2 −δ/ 2 −λ

∣

∣

∣

∣=λ

(^2) −

(

δ

2

) 2

−

(

Ω

2

) 2

=0. (7.91)

Henceλ=±

(

δ^2 +Ω^2

) 1 / 2

/2. For Ω = 0 the unperturbed eigenvalues

areλ=±δ/2, corresponding to two levelsδapart, as shown in Fig. 7.8.

This result of time-dependent perturbation theory closely resembles the

equations for a time-independent perturbation of two states with an

energy separation ofδ(see Appendix A); the two states are the excited

state atE 2 and a level at energyE 1 +
ω, corresponding to the ground

state plus a photon of the radiation field, see Fig. 7.9. This system of

atom plus photon is called a ‘dressed atom’—see Cohen-Tannoudjiet al.

(1992) for a much more in-depth treatment.

Normally light shifts are most important at large frequency detuning

where the effect of absorption is negligible; in this case|δ|Ωandthe

eigenvalues are

λ±

(

δ

2

+

Ω^2

4 δ

)

. (7.92)

The states are shifted from their unperturbed eigenfrequencies by the

light shift±Ω^2 / 4 δ. From eqn 7.89 we see that the amplitude ̃c 1 is

associated with the state with unperturbed energy +δ/2 and lies above

the other state whenδ>0. This state with amplitude ̃c 1 has a light

shift of

∆ωlight=

Ω^2

4 δ

. (7.93)

Fig. 7.8The treatment of the inter-
action of a two-level atom with radi-
ation by time-dependent perturbation
theory leads to eqn 7.89, that looks sim-
ilar to a time-independent perturbation
(proportional to Ω) of two energy lev-
els with an energy separation ofδ.The
light shift is the difference between the
unperturbed energies and energy eigen-
values of the system when it is per-
turbed by the radiation.

Unperturbed

Perturbed

Light

shift