Page 157 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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

Coherent Control of Chemical Reactions 219

In the weak field limit, the time evolution of wave
packets in pump–probe experiments can be evaluated
by second-order time-dependent perturbation theory. The
second-order solution of Eq. (24) is expressed as
1 t t 2
(2)
iH g
X (t) =− dt 1 exp − (t − t 2 )
g 2 dt 2
h 0 0 h

iH e
× µ ge (R)exp − (t 2 − t 1 ) µ eg (R)
h

iH g t 1
× exp − |X g0 (R) ε(t 2 )ε(t 1 ). (30)
h
If both the pump and probe pulses are assumed to be δ
functions, Eq. (30) can be expressed as

iH g
(2) (2)

X (t) = exp − (t − τ) X (τ) (31)
g
g
h
where τ is the delay time between the pump and probe
(2)
pulses, and |X (τ) , the ground-state wave packet cre-
g
ated just after irradiation by the probe pulse, has the
form

iε 0 iH e τ
(2) (1)
e
g
X (τ) = µ ge (R)exp − X (0) . (32)
2h h
In the discussion so far, instantaneous excitation or de-
excitation by a δ function pulse has been assumed to trans-
fer wave packets from one electronic state to another state.
For realistic pulses, the wave packets may be obtained by
FIGURE 13 A pump–dump control scheme, used to control the
numerically integrating Eqs. (25) and (30). branching ratio of the dissociation of a triatomic molecule, ABC.
B. Controlling Wave Packets In order to control a wave packet with tailored laser
with Tailored Laser Pulses pulses,weintroduceatargetoperator W,whichisaprojec-
2
1. Perturbative Treatment tion operator (W = W) that localizes the wave packet at
a target position on an electronically excited potential en-
An intuitive method for controlling the motion of a wave ergy surface. The target operator is one of the fundamental
packet is to use a pair of pump–probe laser pulses, as quantities in control problems. There are several kinds
shown in Fig. 13. This method is called the pump–dump of target operators, depending on the type of object that
control scenario, in which the probe is a controlling pulse is to be controlled. For population control, if a vibronic
that is used to create a desired product of a chemical re- state X ef is chosen as the target, then its target operator
action. The controlling pulse is applied to the system just is expressed as W =|X ef
X ef |. For wave packet shap-
at the time when the wave packet on the excited state po- ing, if a Gaussian wave packet characterized by its aver-
tential energy surface has propagated to the position of age position R and average momentum P is placed on
¯
¯
the desired reaction product on the ground state surface. an electronically excited state e, its target operator is ex-
In this scenario the control parameter is the delay time τ. pressed as W G =|X eG
X eG |, with a coordinate represen-
This type of control scheme is sometimes referred to as tation expressed as
R|W G |R =
R|X eG
X eG | R . Here
the Tannor–Rice model.
R | X eG is given as
There are many other variables in addition to τ that

¯ ¯ 2
may be used to control the reaction products by manipu-
R | X eG = (2πa ) 1 4 exp i P (R − R) − (R − R) ,
¯
2 −
lating the motion of wave packets. These include the time- h 4a 2
dependent frequency, amplitude, and phase functions of (33)
the laser pulse. The use of tailored laser fields to alter
the shape of a wave packet is a very general method for where a, the square root of the variance, is the uncertainty
controlling the outcome of a chemical reaction. in the position of the wave packet.