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

|2, 0, 0 |2, 1, 0 |2, 1,+1|2, 1,1

|2, 1,+1|2, 1,1

|2, 1, 0 |2, 0, 02

|2, 1, 0+ |2, 0, 02

Hydrogen n=2 levels

AC/DC: Whats 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 . Thus, interaction has the following form:

Now, we solve the following using the Floquet theorem:

Hint = dF cost1

2F

2cos

2t

id

dt= Hint

One-level Atom (2)Get solution:

AC Stark energy shift is Ea, ks correspond to quasi-energy harmonics

(r, t) = exp(iEat)k=

k=

Ck(r) exp(ikt)

Ea(F ) = 1

4F

2

Ck =

S=

(1)kJS

(F 2

8

)Jk+2S

(dF

)with ,

One-level Atom (3)

Weak, high frequency field:

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

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

dF

One-level Atom (4)

Strong, low-frequency field:

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

dF >> , F 2

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 >> , F 2 >>

k ! dF

F 2

4

E(F ) = dF F 2

4F 2

4

Multilevel AC Stark Effect

Ei =3pic2

230

I c2ij

ij

intensity

electronic ground

state |gi> shift

transition co-efficient: ij = cij ||||

detuning: ij = - ijexcited state energy: 0

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) =pic2

230

(2 + PgF mF

2,F+

1 PgF mF1,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) =pic2

230

(2 + PgF mF

2,F+

1 PgF mF1,F

)I(r)

F, mF are relevant ground state quantum numbers

laser polarization0: linear, 1:

Land 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 Kranov. Physics-Uspekhi 42, (7) 669-687 (1999)

R Grimm, M Weidemller. 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)