AC Stark EffectTravis BealsPhysics 208A

UC Berkeley Physics

(picture has nothing whatsoever to do with talk)

What is the AC Stark Effect?

Caused by time-varying (AC) electric field, typically a laser.

Shift of atomic levels

Mixing of atomic levels

Splitting of atomic levels

(another pretty but irrelevant picture)

DC Stark Shift

Constant “DC” electric field

Usually first-order (degenerate) pert. theory is sufficient

DC Stark Effect can lift degeneracies, mix states

H′

stark = p · E

= −ezE = −eEr cos θ

|2, 0, 0〉 |2, 1, 0〉 |2, 1,+1〉|2, 1,−1〉

|2, 1,+1〉|2, 1,−1〉

|2, 1, 0〉 − |2, 0, 0〉√2

|2, 1, 0〉 + |2, 0, 0〉√2

Hydrogen n=2 levels

AC/DC: What’s the difference?

AC →time-varying fieldsAttainable DC fields typically much smaller (105 V / cm, versus 1010 V / cm for AC)

AC Stark Effect can be much harder to calculate.

(highly relevant picture)

One-level Atom

Monochromatic variable field

Atom has dipole moment d, polarizability α. Thus, interaction has the following form:

Now, we solve the following using the Floquet theorem:

Hint = −dF cos ωt −1

2αF

2cos

2ωt

idΨ

dt= HintΨ

One-level Atom (2)Get solution:

AC Stark energy shift is Ea, kω’s correspond to quasi-energy harmonics

Ψ(r, t) = exp(−iEat)k=∞∑

k=−∞

Ck(r) exp(−ikωt)

Ea(F ) = −

1

4αF

2

Ck =∞∑

S=−∞

(−1)kJS

(αF 2

8ω

)Jk+2S

(dF

ω

)with ,

➊

➋

One-level Atom (3)

Weak, high frequency field:

Arguments of Bessel functions in ➋ are small, so only the k=S=0 term in ➊ is significant.

Quasi-harmonics not populated, basically just get AC Stark shift Ea

dF << ω, αF 2 << ω

One-level Atom (4)

Strong, low-frequency field:

Bessel functions in ➋ kill all terms except S=0, and k=±dF/ωOnly quasi-harmonics with energies ±dF are populated, so we get a splitting of the level into two equal populations

dF >> ω, αF 2 << ω

One-level Atom (5)Very strong, very low-frequency field:

Only populated quasi-energy harmonics are those with

Thus, have splitting of levels, get energies

dF >> ω, αF 2 >> ω

k ! ±dF

ω±

αF 2

4ω

E(F ) = ±dF ±αF 2

4−

αF 2

4

Multilevel AC Stark Effect

∆Ei =3πc2Γ

2ω30

I∑ c2

ij

δij

intensity

electronic ground

state |gi> shift

transition co-efficient: μij = cij ||μ||

detuning: δij = ω - ωijexcited state energy: ħω0

width of excited state

Assumptions & RemarksUsed rotating wave approximation (e.g. reasonably close to resonance)

Assumed field not too strong, since a perturbative approach was used

Can use non-degen. pert. theory as long as there are no couplings between degen. ground states

In a two-level atom, excited state shift is equal magnitude but opposite sign of ground state shift

AC Stark in Alkalis

Udip(r) =πc2Γ

2ω30

(2 + PgF mF

∆2,F

+1 − PgF mF

∆1,F

)I(r)

!,

FS

21P

2

P2

21

21

23

21

0

L=0

L’=1

(b)

J’=

J’=

(c)

J =

HFS!

HFS!

,

F=2

F=1

F’=2

F’=1

(a) F’=3

23

2

S

"

(Figure from R Grimm et al, 2000)

I = 3/2

AC Stark in Alkalis (2)

Udip(r) =πc2Γ

2ω30

(2 + PgF mF

∆2,F

+1 − PgF mF

∆1,F

)I(r)

F, mF are relevant ground state quantum numbers

laser polarization0: linear, ±1: σ± Landé factor

detuning between 2S1/2,F=2 and 2P3/2

detuning between 2S1/2,F=1 and 2P1/2

What good is it?

Optical traps

Quantum computing in addressable optical lattices — use the shift so we can address a single atom with a microwave pulse

References

N B Delone, V P Kraĭnov. Physics-Uspekhi 42, (7) 669-687 (1999)

R Grimm, M Weidemüller. Adv. At., Mol., Opt. Phys. 42, 95 (2000) or arXiv:physics/9902072

A Kaplan, M F Andersen, N Davidson. Phys. Rev. A 66, 045401 (2002)