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)
Top Related