1. adiabatic transfer processes
TECHNISCHE UNIVERSITÄT
KAISERSLAUTERN
Lecture 9
Coherent light-matter interaction:
Optically driven
adiabatic transfer processes
K. Bergmann
Lecture course - Riga, fall 2013

2. Informal voluntary test (2) for “Optically driven adiabatic transfer processes” Nov. 22, 2013
Identification (Name or any code – for anonymity): ………………………………………
(1) Name key conditions (properties of the radiation field, such as intensity, bandwidth,
detuning, pulse shape) required for observing the phenomenon called “coherent
population return (CPR)” ?
coherence, detuning larger than natural linewidth, smooth pulse shape and sufficiently
high intensity to guarantee adiabatic following
(2) What physical mechanism causes the phenomenon of electro-magnetically induced
transparency (EIT) ?
Interference of amplitudes of transition dipole moment to the two adiabatic states related
to the coupling of two bare states by the (strong) radiation field
Write down the RWA-Hamiltonian for a three
level system interacting with two lasers field
(P- and S-laser) for the case of two-photon
resonance. 1 
Calculate the eigenvalues of the 3-level RWA
Hamiltonian, see (4), for the case of
one-photon resonance, i.e. P = S = 0.

3. down-chirp, best with intensity peak
(5) Consider the multi-level system shown below. The ground state level can be coupled by a
frequency-variable laser to three excited state. The transform limited bandwidth of the laser is large enough (i.e. the pulse length is short enough) to be able to excite all three level at once. What needs to be done when only the upper one of the three levels shall be populated ? Explain the process is the frame of adiabatic energy, with needed frequency, pulse timing and pulse duration shown.
down-chirp, best with intensity peak
at the time when the crossing with
the upper state occurs  E

4. signing up for the exam today and on Friday
comment regarding the exam
90 minutes about 8 – 10 problems
passing grade guaranteed if 1/3 of maximum points reached
suggestions for the preparation:
check the questions for each lecture
get acquainted with the determination of eigenvalues and eigenfunctions
work in small teams and learn with and from each other
signing up for the exam today and on Friday

5. summary of 8th lecture – the five phases of STIRAP
The five phases of the
transfer process
Rabi frequencies 1 3 2 WP WS 4
eigenvalues
mixing angle
populations
AT by S
P = 0
EIT for P
by S
transfer AP
EIT for S
by P
AT by P
S = 0

6. summary of 8th lecture – the five phases of STIRAP
The five phases of the
transfer process
Rabi frequencies 1 3 2 WP WS 4
eigenvalues
<a±|ao>  
maximum here ∙
mixing angle
populations
largest rate of change of mixing angle
= highest risk of non-adiabatic losses
relevant for understanding optimal delay of pulses for maximum transfer
largest AT splitting = best protection
against non-adiabatic losses

7. summary of 8th lecture - adiabaticity criterion∙
„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 d
|a+>
|ao>
d|ao>
Comment:
d|ao> is a vector of arbitrarily short length.
therefore both |ao> and ao> + d|ao> are of length „unity“ , as required.
alternatively, one may consider the angle between ao> and d|ao> to be 90o - ,
with  being an arbitrarily small angle

8. adiabaticity criterion for transfer within an open 3-level system
the goals for this lecture
adiabaticity criterion for transfer within an open 3-level system
criterion for pulsed lasers (in terms of energy)
modification for non-transforme limited radiation
dependence on geometry
examples
what determines the optimum pulse delay ?
further experimental results
Lecture 10: extension to multilevel systems (e.g. matter wave mirrors)
plus: the new concept of SCRAP (Stark chirped rapid adiabatic passage)
including the concept of “adiabatic elimination” ( 3-LS  2-LS)

9. 2eff T >>>(1/T)
4.2 Conditions for adiabatic evolution – rewritten in terms of pulse energy
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 !
(see next lecture: adiabatic elimination)
energy required for „adiabatic evolution“ increases as 1/T
if e.g. E = 1 J is o.k. for a 10 ns pulse, E = 1 J needed for 10 fs pulse
for a shorter pulse, stronger coupling is required to guarantee adiabatic
following (AF)
higher pulse energy is needed
higher pulse energy may lead to undesirable multi-photon processes
AF condition is difficult (if not impossible) to meet for e.g. fs-pulses

10. Further comments on Stimulated Raman Adiabatic Passage (STIRAP)
further examples
decay rate („leak“) through |a+> and |a-> states
adiabatic conditions involving laser pulses
the role of geometry/shape of the laser beam profiles

11. WS WP 3.3.5 STIRAP features - examples
reminder: the „signature“ of the STIRAP process for on resonance tuning and short-lived state 2
Ne* 1 3 2 WP WS 4
lifetime of population in
level 2 << interaction time

12. STIRAP-population transfer between
Niveau- und Kopplungsschema
STIRAP-population transfer between
Praseodym-states in a Pr:YSO crystal
Experimental verification – in the solid state (Praseodym)
4,6 MHz
4,8 MHz
preparation
opt. pumping
pump
Stokes
probe
electronic lifetime >> T
l = 606 nm
17,3 MHz
10,2 MHz
T  4 K
PP,S  80 mW,   300 m, P,S  2  0.7 MHz,
duration of laser pulse T  20 s
Ref. 77
eff T  90

13. 4.4.3 Experimental verification – in the solid state (Praseodym)
frequency generator
AOM
travelling acoustic wave acts as a phase
grating
AOM
the frequency of light diffracted into the first diffraction order is shifted by AOM.
diffraction efficiency is high (50 – 90%) under Bragg condition:
Ref. 77
Bragg: diffraction angle = „reflection angle“

14. 4.4.3 Experimental verification – in the solid state (Praseodym)
lifetime of population in level 2 > interaction time
transfer via the dark state
transfer via the bright states
|ao(t)> = cos (t) |1> – sin (t) |3>
|a±(t)> = sin (t) |1> ± |2> + cos (t) |3>
S before P
 = 0o initially
S after P
 = 90o initially
Ref. 77
eff T  90
PRL 99, (2007)

15. when lifetime of population in level 2 << interaction time:
Experimental verification - transfer via „bright“ state
M = 0
J = 1
J = 2
M =0
J = 0
probe laser
Na2
400 MHz
400 MHz off resonance
S prior P
transfer via
dark state
S after P
transfer via
bright state
the maximum D
dissappears for
on-resonance
tuning
when lifetime of population in level 2 << interaction time:
reducing the „leak“ for transfer via the „bright“ states by detuning from resonance
Ref. 3

16. WP WS 3.3.2 3-level system: adiabatic states off one-photon resonance1 3 2 WP WS 4 1
 = 90o transfer
via bright states
|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 ~
leak rate increases with eff
and decreases with |1|

17. 4.2 Conditions for adiabatic evolution: variation with geometry
Three different geometries
eff  >> 1
elliptical
(vertical)
circular
elliptical
(horizontal)
particles cross the laser beam near the axis x z
power of the lasers in all three cases the same
Questions:
(a) Is the transfer efficiency (quality of adiabaticity criterion)
in all cases the same (identical laser power !) …
…. or is one geometry to be preferred ? If so, which one ?
(b) Two circular laser beams: how would the transfer efficiency change, if the diameter of the laser beams is doubled (power unchanged): improve – unaffected – become worse ?

18. 4.2 Conditions for adiabatic evolution: variation with delay and width
|<1|ao>| < 1
S not >> P
good coupling
|<1|ao>| = 1, S >> P
|<1|ao>| = 1
good coupling and thus good adiabatic following
weaker coupling
|<1|ao>| < 1
good coupling
|<1|ao>| = 1
S not >> P
weaker coupling

19. 2eff T >>> 2eff T > ½ (1 + N2)
Conditions for adiabatic evolution: consequence of fluctuations
pulses in the „real world“ of ns-lasers:
the phase is not constant but
characterized by the size and
frequency of phase jumps:
see autocorrelation function
the field amplitude doesn´t vary smoothly; characterized by the power spectrum (ns-pulses)
transforme limited pulse
real or simulated pulse
real or simulated pulse
transforme limited pulse
reported only
2eff T >>>
 T > 10
2eff T > ½ (1 + N2)
= measure for (phase) fluctuations
Ref. 8

20. 4.3 Conditions for adiabatic evolution: consequence of fluctuations
100 %
population evolution for
a pulse which is:
< 100 %
smooth
fluctuating
statistics of transfer efficiency
variation of transfer
with Rabi frequency:
fluctuations can be
overcome by stronger
coupling
Ref. 8

21. 4.3 Conditions for adiabatic evolution2 1 3 P S
If one of the frequencies fluctuates (or both):
deviation from two-photon resonance
no  = 0 eigenvalue no dark state
loss by spontaneous emission
However: if the fluctuations are (perfectly) correlated – and identically – in time:
deviation from 1-photon resonance will still allow for 2  0 (2-photon resonance)
fluctuation of 1 will lead to some non-adiabatic coupling, which decreases if the Rabi frequency
increases 2 1 3 P S

22. 4.3 Conditions for adiabatic evolution: consequence of fluctuations
It is difficult (if not impossible)
to guarantee perfect phase-
correlation for radiation from
two different laser sources 2 1 3 P S 2 1 3 P S
It is possible to achieve perfect
phase-correlation for radiation
derived from a single laser source
two-photon resonance
maintained despite
fluctuations, but ….
M = -1
M = 0
M = +1
„white light STIRAP“ + -
time dependence of 1
leads to some loss
by non-adiabatic coupling
Ref. 82
M. Auzinsh et.al. Phys.Rev. A78, (2008)

23. WP WS D2 = 0 P = S  = 45o tan = P / S = 1
The transfer process in the state-vector picture 1 3 2 WP WS 4
equal Rabi frequencies
(or ratio ≠ 0 and ≠ )
D2 = 0
composition of |ao>
P = S  = 45o
tan = P / S = 1
no component |2>
50% of the population in dark (or trapping) state |ao>
component |2>
25% of the population in each one of the bright states |a±>
loss of population through optical pumping (decay to other levels)

24. 4.3 The transfer process in the state-vector picture
adiabatic state wave functions
shown as state vectors
referenced to the bare states
P = 0  = 0
P = S  = 45o
S = 0  = 90o

25. the solution of an optical puzzle: dark lines
reminder
Electromagnetically induced transparency – trapped states
the solution of an optical puzzle: dark lines B x
„dark resonance“ or
„population trapping“
inhomogeneous
B-field across cell
vapor cell, sodium
multimode laser
spectral components of
multimode laser couple
m-states and form dark
resonances
observation
in Pisa 1976
at specific
location(s):
B-field splitting
= mode spacing
gradient of B-field E
Ref. 83

26. solution of the three-level puzzle
Electromagnetically induced transparency – trapped states
reminder
M = 0
solution of the three-level puzzle
induced by linearly polarized coherent radiation
decay  depletion by optical pumping
2/3 of population remains in initial level z z
J = 1  J = 0 transition
as an example
change of quantization axis E y y
decomposition: linear polarization
 circular-right circular-left x
M = ± 1
„dark states“ solve the puzzle: 50% of the population in M = -1 and M = + 1 remain in a dark state: ½ (the „missing“) 1/3
still: decay  depletion by optical pumping
but now: 1/3 of population remains in initial level ??

27. Questions related to the topics discussed in lecture 9
(9.1) What are the conditions for efficient population transfer via a bright
adiabatic state ?
(9.2) Consider two circular laser beams (P and S): how would the transfer
efficiency change, if the diameter of the laser beams is doubled (power
unchanged), improve – unaffected – become worse ?
(9.3) Related to (7.2): Consider elliptical shape of the laser (same laser power)
and analyze the quality of global adiabaticity criterion. The long axis of the
ellipse may be perpendicular or parallel to the direction of travel of the
atoms (axis of the atomic beam). Compared to circular beams: is an elliptical
beam in one or both of the orientations to be preferred to optimize the
transfer efficiency ?
(9.4) Influence of ratio of maximum Rabi frequency o on transfer: Given is a
fixed width (in space or time) of the laser diameter and a suitable displace-
ment or delay of the interation. You know that we have good transfer for
Pmax  Smax. Would the transfer efficiency change (if yes: why and how)
for Pmax >> Smax.

28. end of 9th lecture Coherent light-matter interaction:
Optically driven adiabatic transfer processes
end of 9th lecture