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Encyclopedia of Physical Science and Technology EN003H-565 June 13, 2001 20:37

226 Coherent Control of Chemical Reactions

D. Coherent Population Transfer φ 1 ,φ 2 , and φ 3 are the wave functions of the initial, inter-
mediate, and final states, respectively, (denoted by |1 , |2 ,
So far, we have considered various theoretical treatments
and |3 in Fig. 22), subscripts P and S label the pump and
of time-dependent wave packets controlled by laser pulses
Stokes processes, respectively, and P ( S ) denotes the
to produce a desired product in a chemical reaction. An-
detuning of the pump (Stokes) laser. In the discussion that
othertypeofproblem,basedonadiabaticbehaviorofwave
follows, the two laser fields are assumed to be pulsed.
functions, is the transfer of population from one state to
The interaction matrix elements of the total
another.
Hamiltonian are expressed in the three-state model as
Suppose that a laser pulse interacts adiabatically with a
 
molecular system. By the term “adiabatic” is meant that −2 P (t) 0
an eigenstate ψ (t) at time t satisfies the time-independent h  (t) 0 S (t) , (57)
∗

P
Schroedinger equation: 2  
∗
0 (t) −2
S
H(t)ψ (t) = E (t)ψ (t). (56) where P (t) is the Rabi frequency of the pump pulse,
S (t) and is the Rabi frequency of the Stokes pulse. By
In the adiabatic limit, t is considered to be a parameter,
diagonalizing the determinant of this interaction matrix,
and ψ (t) is called an adiabatic state. One of the inter-
the adiabatic states ψ 0 and ψ ± , with eigenfrequencies ω 0
esting properties of this limit is that a population can be
and ω ± , are analytically derived as
inverted by evolving the system adiabatically. This pro-
cess is called adiabatic passage. Population transfer in- ψ 0 (t) = cos[!(t)]φ 1 − sin[!(t)]φ 3 (58)
duced by a laser is generally called “coherent population
with eigenfrequency
transfer.” For a two-level system, the complete population
inversion is produced by a π-pulse or by adiabatic rapid ω 0 (t) =− (t), (59)
passage.
Population transfer in a three-level system can be ψ + (t) = sin[η(t)] sin[!(t)]φ 1 + cos[η(t)]φ 2
achieved by using one laser (known as the “pump laser,”
+ sin[η(t)] cos[!(t)]φ 3 (60)
which may be either continuous wave or pulsed) to con-
nect the ground and intermediate levels, and a second laser with eigenfrequency
(the “Stokes laser”) to connect the intermediate and final

1 2 2 2
levels. This method, known as stimulated Raman adiabatic ω + (t) =− (t) − | P (t)| +| S (t)| + (t) ,
2
passage or STIRAP, is illustrated in Fig. 22. In this exam-
(61)
ple, the three levels have a -type configuration, where
and
ψ − (t) = cos[η(t)] sin[!(t)]φ 1 − sin[η(t)]φ 2
(62)
+ cos[η(t)] cos[!(t)]φ 3
with eigenfrequency

1
2
2
2
ω − (t) =− (t) + | P (t)| +| S (t)| + (t) .
2
(63)
The mixing angle !(t) in Eqs. (58), (60), and (62) is given
by
| P (t)|
(64)
sin[!(t)] =
2 2
| P (t)| +| S (t)|
| S (t)|
, (65)
cos[!(t)] =
2 2
| P (t)| +| S (t)|
and

2 2
| P (t)| +| S (t)|
FIGURE 22 Three-level excitation scheme used for STIRAP. tan[η(t)] = 2 2 2 .
[Reproduced with permission from Bergmann, K., Theuer, H., and | P (t)| +| S (t)| + (t) + (t)
Shore, B. W. (1998). Rev. Mod. Phys. 70, 1003.] (66)