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AC Stark Effect
Travis Beals
Physics 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
|2, 0, 0!
|2, 1, −1"
|2, 1, 0!
|2, 1, +1!
Constant “DC” electric
field
Usually first-order
(degenerate) pert.
theory is sufficient
DC Stark Effect can
lift degeneracies, mix
states
!
Hstark
=p·E
= −eẑE = −eEr cos θ
|2, 1, 0! − |2, 0, 0!
√
2
|2, 1, −1"
|2, 1, +1!
|2, 1, 0! + |2, 0, 0!
√
2
Hydrogen n=2 levels
AC/DC: What’s the
difference?
(highly relevant picture)
AC →time-varying fields
Attainable DC fields typically much smaller
(105 V / cm, versus 1010 V / cm for AC)
AC Stark Effect can be much harder to
calculate.
One-level Atom
Monochromatic variable field
Atom has dipole moment d, polarizability α.
Thus, interaction has the following form:
1
2
2
Hint = −dF cos ωt − αF cos ωt
2
Now, we solve the following using the Floquet
theorem:
dΨ
i
= Hint Ψ
dt
One-level Atom (2)
Get solution:
➊
Ψ(r, t) = exp(−iEa t)
k=∞
!
Ck (r) exp(−ikωt)
k=−∞
➋
1
2
with Ea (F ) = − αF ,
4
"
#
"
#
∞
2
!
αF
dF
k
Ck =
(−1) JS
Jk+2S
8ω
ω
S=−∞
AC Stark energy shift is Ea, kω’s
correspond to quasi-energy harmonics
One-level Atom (3)
Weak, high frequency field:
dF << ω, αF 2 << ω
Arguments of Bessel functions in ➋ are
small, so only the k=S=0 term in ➊is
significant.
Quasi-harmonics not populated, basically just
get AC Stark shift Ea
One-level Atom (4)
Strong, low-frequency field:
2
dF >> ω, αF << ω
Bessel functions in ➋ kill all terms except
S=0, and k=±dF/ω
Only quasi-harmonics with energies ±dF are
populated, so we get a splitting of the level
into two equal populations
One-level Atom (5)
Very strong, very low-frequency field:
2
dF >> ω, αF >> ω
Only populated quasi-energy harmonics are
those with
2
dF
αF
k!±
±
ω
4ω
Thus, have splitting of levels, get energies
αF 2
αF 2
E(F ) = ±dF ±
−
4
4
Multilevel AC Stark Effect
width of excited state
transition co-efficient: μij = cij ||μ||
2
!
c
3πc Γ
ij
I
∆Ei =
3
2ω0
δij
2
electronic ground
state |gi> shift
excited state energy: ħω0
intensity
detuning: δij = ω - ωij
Assumptions & Remarks
Used 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
(a)
2
P3
F’=3
,
! HFS
2
0
,
! FS
2
P1
F’=2
F’=1
2
"
I = 3/2
F=2
2
S1
! HFS
2
F=1
(Figure from R Grimm et al, 2000)
(b)
2
πc Γ
Udip (r) =
2ω03
!
(c)
2 +L’=1
PgF mF
1 − PgF m
J’= F
+
J’=
∆2,F
∆1,F
3
1
L=0
J=
1
2
2
2
"
I(r)
AC Stark in Alkalis (2)
laser polarization
0: linear, ±1: σ±
2
πc Γ
Udip (r) =
2ω03
Landé factor
!
2 + PgF mF
1 − PgF mF
+
∆2,F
∆1,F
"
I(r)
detuning between 2S1/2,F=2 and 2P3/2
detuning between 2S1/2,F=1 and 2P1/2
F, mF are relevant ground state quantum numbers
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)