1. adiabatic transfer processes
TECHNISCHE UNIVERSITÄT
KAISERSLAUTERN
Lecture 7
Coherent light-matter interaction:
Optically driven
adiabatic transfer processes
K. Bergmann
Lecture course - Riga, fall 2013

2. summary of 6th lecture: spectral properties of 2 + 1system
|a+> = c+,g |g> + c+,e |e>
|a-> = c-,g |g> + c-,e |e>
|f> e f g S P
bare states
weak
strong
J = 7
population may reach level f via
two different paths
interference structure
Ref. 71

3. summary of 6th lecture: Rate equation (incoherent radiation) for 3-LS
e.g. Gaussian pulse shape 2 1 3 P S 4
loss to other levels
50%
33%
coincident pulses
maximum transfer: 33%
reached only without loss
through spontaneous emission
50%
33%
delayed pulses
maximum transfer: 25 %
reached only without loss
through spontaneous emission

4. S P repetition: electromagnetically induced transparency
Lorentzian profile
amplitude of transition dipole moment:
<1| |a>  <1||3>  <1||2>
File: Lorentz0 …50_2xSUM needed
bare P 1 3 S 2
adiabatic AT
fluorescence
two transition dipole moments:
180o out of phase
Intensity ≠ 0 ?
P – laser frequency

5. spectral profile probed on |1> -- |2> transition
repetition: electromagnetically induced transparency
spectral profile probed on |1> -- |2> transition
File: EIT_Final-1
as coupling
 of
|2> -- |3>
increases
intensity
start EIT/AT
probe laser frequency (|1> -- |2>)
 = decay rate of level |2>

6. repetition: electromagnetically induced transparency (EIT)
 (much) smaller than :
( = natural line width)
interference structure
(narrow) observed,
which is called: EIT
also zero (EIT) between
AT-components
 (much) larger than :
two separate features
(of width ≈ ) observed,
called AT-splitting

7. summary of 6th lecture: EIT
bare states P S
dressed states P S
Ne* 2 1 3 P S
Experiment
transition dipole moments
|1>  |a+> and |1>  |a->
are 180o out of phase
transition amplitude zero
when frequency tuned to the
centre between the states
Ref. 24
exactly zero

8. understanding the relevance of the “zero-energy” eigenvalue
the goals for today
understanding the structure of the 3-level RWA Hamiltonian in analogy to the 2-level case.
understanding the structure of the 3-level eigenvalues und eigenfunctions
understanding the relevance of the “zero-energy” eigenvalue
understanding the basics of the STIRAP process for very efficient, lossless transfer op population between levels
seeing examples for successful applications of the STIRAP method

9. 3.3 The 3-level RWA Hamiltonian
3.3.1 Basics
3.3.2 Adiabatic energies and adiabatic states
3.3.3 The relevance of the zero-energy eigenstate
3.3.4 Variation of |<1|ao>|2 with laser overlap
3.3.5 Preview of the STIRAP process

10. 3.3.1 3-level system: basics -- parameters
reminder: some basic facts 2
Rabi frequency ik = ik EL / S
 E2,3
 E1,2 P
2,3
ik = <i|er|k>
transition dipole moment
23
12
1,2 3 1
ik << E1,3
 E1,3
each field couples one pair of levels only
selectivity by frequency and/or polarization
J = 0, M = 0
M = 0,
J = 1
M = -1
M = +1
 +
 - 
large pulse area:  T > 
many Rabi cycles in two-level systems
adiabatic evolution

11. 3.3.1 3-level system: basics -- relevant detunings2 1 3 P S
P = S  0
 E1,2 2 1 3 P S
P = S = 0
one- and two-photon resonance
only two-photon resonance
P also quoted es 1, with index „1“ for one-photon detuning
 E1,2 2 1 3 P S P
P - S
 E1,2 2 1 3 P S P S
1 =
same thing
differently drawn
2 =
no resonance
no resonance

12.  H ~ 3.3.1 3-level system: the RWA Hamiltonian 2 2  1  E1,2  E1,2P S 1 2
unitary transformation 
2-level system
3-level system
 RWA Hamiltonian
H ~ 

13. the adiabatic energies for X = 0
level system: the RWA Hamiltonian
 E1,2 2 1 3 P S 1 2
choice of
zero-point
of energy
no coupling
between
levels 1 and 3
„reading“ the 3-level
RWA Hamiltonian
coupling of level 1
to all other levels 
coupling of level 2
to all other levels
coupling of level 3
to all other levels
the adiabatic energies for X = 0

14. P S eff = (2P + 2S) 1/2 1 = 0, eff = 0  1 = 0, eff  0 
level system: basics 2 1 3 P S E
eff = (2P + 2S) 1/2
Interaction Hamiltonian:
two photon resonance
P = S
preview:
energy of adiabatic states:
1 = 0, eff = 0 
1 = 0, eff  0 
eff

15. S P eff = (2P + 2S) 1/2 eff  1 = 0,  = 0 ε Ω t
level system: basics 2 1 3 P S E
eff = (2P + 2S) 1/2
Interaction Hamiltonian:
two photon resonance
P = S
preview: Ω ΩS ΩP t ε ε+ ε0 ε-
eff
energy of adiabatic states:
1 = 0,  = 0 

16. 3.3.2 3-level system: adiabatic energies and adiabatic states
time-independent Schroedinger equation: (TISE)
|a> = (a1, a2, a3) is a vector in a three-dimensional Hilbert space
TISE in matrix form (and RWA):
three linear independent
(mutually orthogonal)
solutions |ao,> with
energies wanted
rewritten and with indices for solutions added:

17. WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states
special case:
index omitted 1 3 2 WP WS 4
solution, when determinant = 0, yields eigenvalues,
for one- and two-photon resonance: !
two-photon resonance
one-photon resonance
one solution,
two more solution:

18. WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states ½
the three-state adiabatic energies
(for one- and two-photon resonance) are: 1 3 2 WP WS 4
new feature for 3-level system
eff
P = S = 0
P and/or S  0

19. 3.3.2 3-level system: adiabatic energies and adiabatic states
detuning i  0 allowed 2 1 WS
2 = 0 WP 3
solution when determinant = 0 yields eigenvalues 4 1 !
leads, in general, to a 3rd power
polynomial which may not have a
(physically meaningful) solution
two-photon resonance X X
is again a solution,
IF 2 = 0
2 = 0

20. WS WP 3.3.2 3-level system: adiabatic energies and adiabatic states 1
adiabatic energies (eigen values) with 1 0 allowed WS
2 = 0 WP 
exists, whenever 2 = 0 3 4 1
eff = 0 1
for 1 = 0:
o, - and +
degenerate when eff = 0
for 1 > 0: o and - degenerate when eff = 0
for 1 < 0: o and + degenerate when eff = 0

21. o = 0 WP WS P = 0  = 0 P = S  = 45o S = 0  = 90o
level system: adiabatic energies and adiabatic states 1 3 2 WP WS 4
the RWA Hamiltonian, 1 = 0:
o = 0
to be shown:
verify for yourself:
the adiabatic state eigenfunctions:
special !
P = 0  = 0
P = S  = 45o
S = 0  = 90o
independent of S
independent of P

22. 3.3.2 3-level system: adiabatic energies and adiabatic states
verification that
is the eigenfunction with eigenvalue  = 0 for !
= 0 !
o = 0 !
verified

23. o = 0 WP WS 1 = 0   = 45o  cos  = sin  =
level system: adiabatic energies and adiabatic states 1 3 2 WP WS 4
the RWA Hamiltonian, 1  0:
reported only:
o = 0
the adiabatic state eigenfunctions, 1≠ 0 allowed:
special solution remains !
1 = 0   = 45o  cos  = sin  =

24. WP WS 3.3.2 3-level system: adiabatic energies and adiabatic states1 3 2 WP WS 4
|1| >> eff then tan 2 ≈ sin 2 ≈ 2 
the adiabatic state eigenfunctions, 1≠ 0 allowed:
decay rate leak through „leak“ (state 2) during the transfer process:
proportional to |<1|a>| and to coefficient of |2>, i.e. proportional sin  cos 
→ for small  (large 1) → leak ~
will be added to the lecture material
for full analytic solutions with both 1 ≠ 0 and 2 ≠ 0 allowed, see
Fewell, Shore, Bergmann: Australian J. Phys. 50, 281 – 308 (1997)

25. Example: three-level system
relevance of the zero-energy eigenvalue 2 1 3 
c  a
Example: three-level system
relevance of the zero-energy eigenvalue
|ao> = co1|1> + co2|2> + co3 |3>
underlines the relevance of
the two-photon resonance
 = c2  0

26. S P 3.3.4 Variation of <1| ao>|2 with pulse overlap 2
The movie shows the variation of the adiabatic state
coordinateswith respect to the bare state coordinates
as a function of the overlap between S and P laser
When we have initially
ΩS≠ 0 and ΩP = 0
then  = 0, therefore
|ao>, |1> and |>
are all parallel:
only the adiabatic state
|ao> is populated
File: STATEVECTORMOVIE100808

27. 3.3.5 STIRAP features - examples
Ne*
3P1
3P0
3,1P1
|1ñ
|3ñ
|2ñ S P
final level population
radiation from |2>
transfer
metastable
levels
radiation from |2>
interaction time >> rad

28. 3.3.5 preview of STIRAP features – examples, the Ne* level scheme
the carrier of a suitable level system: Ne*
intermediate 
transfer
thermal population depleted
by optical pumping
probing ↑ ↑
final, J = 2
initial, J = 0 ←
detection

29. WS WP 3.3.5 preview of STIRAP features - examples Ne*, Na2,
SO2, NO, …
particle beam 1 3 2 WP WS 4
the „signature“ of the
STIRAP process
mixing angle changes
smoothly from 0o to 90o
mixing angle = 45o

30. 4. Stimulated Raman Adiabatic Passage (STIRAP)
4.1 Details of the transfer process
4.2 Conditions for adiabatic evolution
4.3 The transfer process in the state vector picture
4.4 Experimental verification

31. WS WP WS WP 4.1 Details of the STIRAP-transfer process
The building blocks for the STIRAP process
are now on hand:
process to be
completed in a time
short compared to
lifetime of |3>
(e.g. Rydberg states)
3-level-system eigenvalues and eigenfunctions
and: EIT – AT – AP
electro-magn. ind. transp. – Autler Townes – adiabatic passage 1 3 2 WP WS 4
…..putting them together for: 1 3 2 WP WS 4
a lambda
level-system
a ladder
level-system

32. WP WS 4.1 Details of the STIRAP-transfer process Rabi frequencies3 2 WP WS 4
Rabi frequencies
tan  = P / S
adiabatic energies
mixing angle -
populations

33. 4.1 Details of the STIRAP-transfer process2
adiabatic states
degeneracy
bare states 3
tan  = P / S 4 1 -

34. WS WP Wp = 0 4.1 Details of the STIRAP-transfer process Autler-Townes2
Autler-Townes WS WP
Wp = 0 3
tan  = P / S 4 1
|1> -

35. WS WP Wp  0 4.1 Details of the STIRAP-transfer process EIT at work2
EIT at work WS WP
Wp  0 3
tan  = P / S
P radiation NOT absorbed 4 1
|1> -

36. WS WP 4.1 Details of the STIRAP-transfer process transfer - 2 3
tan  = P / S 4 1
|1> -

37. WS WP WS  0 4.1 Details of the STIRAP-transfer process EIT at work2
EIT at work WS WP
WS  0 3
tan  = P / S
S radiation NOT absorbed 4 1
|1> -

38. WS WP WS = 0 4.1 Details of the STIRAP-transfer process Autler-Townes2
Autler-Townes WS WP
WS = 0 3
tan  = P / S 4 1
|1>
|3> -

39. | <a+ | ao > | << | o - + |
Conditions for adiabatic evolution
The reasoning follows closely what was presented for the 2-level systems
eigen-states of Hamiltonian are: |ao>, |a+>, |a->
eigen-energies are: o + -
population initially in |ao> and it should stay there
Hamiltonian is time-dependent,
non-adiabatic coupling unavoidable : |ao> |a+> and |a->
quantum mechanics: non-adiabatic (or „diabatic“) coupling is small, if
| <a+ | ao > | << | o - + | ∙

40. | <a+ | ao > | << | + - o |
Conditions
for adiabatic evolution
| <a+ | ao > | << | + - o | ∙ ∙ ∙ - ∙
when
is valid ∙
then also
is valid

41. | <ao | a+ > | << | o - + |
Conditions for adiabatic evolution
| <ao | a+ > | << | o - + | ∙ ∙
see ref. 2 d
|a+>
d |ao> ∙
|ao> ∙
„local“
averaged over pulse duration  T:
< |  | > = ½  / T ∙
<< eff
eff T >> 1
„global“:
Ref. 2

42. adiabatic evolution if: eff  >> 1
Conditions for adiabatic evolution
eff =  P2 + S2 2 WS WP 3
tan  = P / S 4 1
|1>
|3> 
adiabatic evolution if:
eff  >> 1

43. 4.2 a mechanical analog of STIRAP
quantum states
|1> |2> |3>
quantum states
|1> |2> |3>
File: Pendel mit Kopplung
File: Pendel ohne Kopplung
amplitude of vibration corresponds to level population

44. 2eff T >>>(1/T)
Conditions for adiabatic evolution ∙
„local“
eff T >> 1
„global“:
applicable when (phase) fluctuations
during the interaction period are small
The global condition suffices for nearly transforme limited pulses
(constant phase and smooth envelope)
The local condition must be evaluated for pulses with fluctuating phase
(and/or deviation from smooth evolution of the envelope) model needed
Global adiabaticity criterion in terms of pulse energy
pulse energy
eff T >> 1
2eff T2 >>> 1
2eff T >>>(1/T)
problems with large Stark shift !
energy required for „adiabatic evolution“ increases as 1/T
if e.g. E = 1 J is o.k. for a 10 ns pulse, E = J needed for 10 fs pulse

45. 4.2 Transfer process in the state vector picture - adiabatic evolution
evolution is adiabatic :
File: ADIA
observe the
blue arrow

46. 4.2 Transfer process in the state vector picture – nonadiabatic evolution
evolution is NOT adiabatic :
File: NADIA
observe the
blue arrow

47. Questions related to the topics discussed in lecture 7
(7.1) Write down the 3-level RWA Hamiltonian for the case that the
frequencies of the two fields satisfy the conditions for two-photon
resonance, i.e. the energy difference between the P-field and S-field
photons equal the energy difference of the levels 1 and 3.
(7.2) Verify the validity of the eigenfunctions |a+> and |a-> for the three-
level system (two-photon detuning 2 = 0).
(7.3) Calculate and plot the variation of the three-level splitting for a
Gaussian pulse and one-photon detuning 1 smaller, equal, larger
and much larger than the chosen Rabi frequency.
(7.4) How does the contribution of (the initially populated) bare state |1>
to the adiabatic state |ao> vary with the overlap of S and P pulse ?
(7.5) Discuss the „five phases“ of the STIRAP process, each showing
prodeminantly the importance of Auter-Townes splitting, EIT, adiabatic
passage
(7.6) Show that we have a zero-energy eigenvalue, whenever two-photon
resonance is valid, independent of the one-photon detuning

48. end of 7th lecture Coherent light-matter interaction:
Optically driven adiabatic transfer processes
end of 7th lecture