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

Coherent Control of Chemical Reactions 223

Equation (52) has the same structure as that of Eq. (41).
An expression for the optimal control pulse in a mixed
case can therefore be obtained as

ε(t) =−2A(t)Im

(t)| ˆµ|ρ(t) , (53)
where the time-dependent multiplier (t) satisfies the Li-
ouville equation,
∂
†
ih | (t) = L (t) | (t) , (54)
∂t
with the final condition | (t f ) = |W .
Afundamentallimitationtocoherentpopulationcontrol
FIGURE 18 Time variation of the optimized electric field. [Re- is that it is impossible to transfer 100% of the population in
produced with permission from Sugawara, M., and Fujimura, Y. a mixed state. That is, the maximum value of the popula-
(1994). J. Chem. Phys. 100, 5646. Copyright American Institute
tion transferred cannot exceed the maximum of the initial
of Physics.]
population distribution of a system without any dissipa-
tive process such as spontaneous emission. This result can
propagation can be constructed by time reversal, because be simply verified using the unitary property of the den-
the time-dependent Schr¨odinger equation is unitary in the sity operator, ρ(t) = U(t, t 0 )ρ(t 0 )U (t, t 0 ), where ρ(t 0 )is
†
case of no dissipation. the diagonalized density operator at t = t 0 , U(t, t 0 )isthe
So far we have treated only the case of wave packets time-evolution operator given by
constructed from pure states. Consider now the control of a t
molecularsysteminamixedstateinwhichtheinitialstates ˆ i
U(t, t 0 ) = T exp − dt V I (t ) , (55)
are distributed at a finite temperature. The time evolution h t 0
of the system density operator ρ(t) is determined by the
ˆ

Liouville equation, T is a time-ordering operator, and V I (t ) is the interac-
tion between the molecules and the controlling pulses
∂
ih ρ(t) = L(t)ρ(t), (48) in the interaction representation. The eigenvalues of ρ(t)
∂t are thus invariant with respect to unitary transformation.
where the Liouville operator L(t)isgivenby The population of a target state |k at time t,
k|ρ(t)|k ,
satisfies the condition that the minimum eigenvalue of
L(t)ρ(t) = H(t)ρ(t) − ρ(t)H(t). (49)
ρ(t 0 ) ≤
k|ρ(t)|k ≤ the maximum eigenvalue of ρ(t 0 ).
It is convenient to introduce a Liouville space, or double That is, the maximum population in a target state at time
space, that is a direct product of cap and tilde spaces. In t f is equal to the maximum eigenvalue of ρ(t 0 ). Therefore,
Liouville space, operators are considered to be vectors and in the mixed state case, one must choose a target operator
Hilbert-space commutators are considered to be operators. appropriate for this restriction.
Equation (48) is then expressed as
∂ 3. Experimental Examples of Wave
ih |ρ(t) = L(t)|ρ(t) , (50)
∂t Packet Control
where |ρ(t) is a vector, and The key technological advance that has made optical pulse
shaping widely available is the pulse modulator depicted
L(t) = L 0 − Mε(t) (51)
in Fig. 19. For a Gaussian laser pulse the product (full-
ˆ
˜
is an operator in Liouville space. Here L 0 = H 0 − H 0 , width at half-maximum) of duration τ and radial fre-
2
ˆ
˜
and H 0 and H 0 are molecular Hamiltonians in the cap and quency bandwidth δω is 0.44. (For a sech pulse, the
tilde spaces, respectively. Similarly, M = ˆµ − ˜µ. In the Li- product is 0.32.) For such transformed-limited pulses the
ouville representation, the objective function is rewritten group velocity is the same for all frequencies. The prop-
as erties of a laser pulse can be tailored by dispersing the
pulse, filtering the frequency components, and finally re-
2
1 t f |ε(t)| t f

J = h

W G | ρ(t f ) − dt + 2Re i dt 1 constituting the modified pulse. This method is illustrated
2 0 A(t) 0 in Figure 19, where grating G 1 is placed at the focal point

∂ of lens L 1 . A multipixel spatial light modulator (SLM)
×

(t)| ih ρ(t) −

(t)|L(t)|ρ(t) . (52)
∂t placed in the Fourier plane is programmed to alter the