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Good presentation on the AC stark effect
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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)