AC Stark Effect Travis Beals Physics 208A UC Berkeley Physics (picture has nothing whatsoever to do with talk)
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)