Optimal control theory for a target state distributed in time:
Optimizing the probe-pulse signal of a pump-probe-scheme
Andreas Kaiser
a)
and Volkhard May
Humboldt-Universita
¨
t zu Berlin, Institut fu
¨
r Physik, Newtonstraße 15, D-12489 Berlin, Germany
͑Received 5 April 2004; accepted 17 May 2004͒
Optimal control theory ͑OCT͒ is formulated for the case of a two-color pump-probe experiment. The
approach allows to calculate the pump-pulse shape in such a way that the probe-pulse absorption
signal is maximized. Since the latter quantity is given by the time-averaged expectation value of a
time dependent operator ͑the probe-pulse field-strength times the dipole operator͒ a version of OCT
has to be used where the target state is distributed in time. The method is applied to a molecular
three-level system with the pump-pulse driving the transition from the electronic ground state into
the first-excited electronic state and the probe-pulse connecting the first-excited state with a higher
lying electronic state. Depending on the probe-pulse duration, the vibrational wave packet becomes
localized or at least highly concentrated in the Franck-Condon window for the transition into the
higher-excited state. The dependence on the probe-pulse duration and on the delay time between the
optimized pump-pulse and the probe-pulse is discussed in detail. The whole study demonstrates the
feasibility of laser pulse induced temporal wave packet localization and the use of spectroscopic
quantities as target states in experiments on femtosecond laser pulse control.
© 2004 American Institute of Physics. ͓DOI: 10.1063/1.1769370͔
I. INTRODUCTION
Among the various attempts undertaken to achieve fem-
tosecond laser pulse control of molecular dynamics those
experiments are of particular interest which also take an op-
tical signal as the observable to be maximized in a feedback-
controlled self-learning loop.
1–3
As in other cases these
learning loops follow the general suggestion of Ref. 4 and
consist of a pulse shaper, a device to measure the signal to be
optimized, and an evolutionary algorithm. The latter notices
the measured signal and realizes a feedback by iteratively
optimizing the exciting laser pulse.
A transient absorption signal from the S
1
level into a
higher-excited singlet level of a carotenoid has been used in
Ref. 1 to control energy flow pathways in the light-
harvesting antenna complex LH2. Selected vibrational mode
excitation of crystalline polydiacetylene was demonstrated in
Ref. 2 with the feedback signal derived from that of a coher-
ent anti-Stokes Raman scattering setup. Luminescence radia-
tion has been taken in Ref. 3 to maximize the population of
a long-lived charge-transfer state in a charge-transfer coordi-
nation complex. All these approaches demonstrated the fea-
sibility of optical detection within optical control and repre-
sent an alternative to the maximization of, for example, an
observable derived from mass spectroscopy ͑cf., e.g., Ref. 5͒.
It is the aim of this paper to demonstrate that spectro-
scopic signals can also be incorporated into the theoretical
tool used to study femtosecond laser pulse experiments, i.e.,
into the optimal control theory ͑OCT͒.
6–9
To begin with we
will concentrate on a sufficient simple reference example and
study a pump-probe scheme in a molecular system with the
pump beam driving population into the first-excited state and
the probe beam testing the resulting vibrational wave packet
motion in this first-excited state via a transition into a higher-
excited state ͑cf. Fig. 1͒. The control task to be addressed
will be the search for the optimal pump-pulse which maxi-
mizes the probe-pulse signal.
10
In particular it will be of
interest in which manner the probe-pulse duration influences
the wave packet formation by the pump pulse. As it is al-
ready obvious at this point such a control task has to be
based on OCT for the case of a target state distributed in time
͑see Refs. 11–17 and Appendix A, by the way, this is just
the point which essentially improves the investigations of
Ref. 10͒.
In order to determine the probe-pulse signal in a pump-
probe scheme one usually calculates
18
S
pr
ϭϪ
͵
d
ץ
E
pr
͑
͒
ץ
P
pr
͑
͒
, ͑1͒
with E
pr
and P
pr
denoting the field strength of the probe pulse
and the probe-pulse induced polarization, respectively. The
latter quantity has to be properly deduced from the expecta-
tion value of the dipole operator
ˆ given at the absence of
dissipation and for zero temperature as
͗
⌿(
)
͉
ˆ
͉
⌿(
)
͘
where the state vector ͉⌿͑
͒͘ follows from the solution of the
time-depending Schro
¨
dinger equation including the applied
fields ͑we will comment on this in more detail below͒.
If one takes S
pr
as the quantity to be maximized in a
control scheme it becomes immediately obvious that the first
part of the standard control functional has to be generalized
to
J
0
͑
E
͒
ϭ
͵
d
͗
⌿
͑
;E
͒
͉
f
͑
͒
O
ˆ
͉
⌿
͑
;E
͒
͘
. ͑2͒
a͒
Electronic mail: andreas.kaiser@physik.hu-berlin.de
JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 6 8 AUGUST 2004
25280021-9606/2004/121(6)/2528/8/$22.00 © 2004 American Institute of Physics