Initial state laser control of curvecrossing reactions using the Rayleigh–Ritz
variational procedure
Peter Gross, Ashish K. Gupta, Deepa B. Bairagi, and Manoj K. Mishra
Citation: J. Chem. Phys. 104, 7045 (1996); doi: 10.1063/1.471421
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Initial state laser control of curve-crossing reactions using
the Rayleigh–Ritz variational procedure
Peter Gross
Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
Ashish K. Gupta, Deepa B. Bairagi, and Manoj K. Mishra
Department of Chemistry, Indian Institute of Technology, Powai, Bombay 400 076, India
~Received 21 June 1995; accepted 18 January 1996!
A new two-step procedure for laser control of photodissociation is presented. In the first step of the
procedure, we show that control of photodissociation product yields can be exerted through
preparation of the initial wave function prior to application of the photodissociation field in contrast
to previous laser control studies where attention has focused on the design of the field which induces
dissociation. Specifically, for a chosen channel from which maximum product yield is desired and
a given photodissociation field, the optimal linear combination of vibrational eigenstates which
comprise the initial wave function is found using a straightforward variational calculation. Any
photodissociation pulse shape and amplitude can be assumed since the Schrödinger equation is
solved directly. Application of this method to control of product yields in the photodissociation of
hydrogen iodide is demonstrated. The second step of the control procedure involves the preparation
of the coherent superposition of discrete levels obtained from the previous step; design of the
preparatory field can be done analytically for two or three level systems as demonstrated here or
with other well-studied iterative field design methods. © 1996 American Institute of Physics.
@S0021-9606~96!01516-1#
I. INTRODUCTION
In recent years there has been a renewed interest in laser
control of chemical reactions. From the theoretical side, a
variety of approaches or schemes for laser field design have
emerged including the optimal control method introduced by
Rabitz and co-workers,1,2 the pump-and-dump scheme of
Rice and Tannor,3,4 and the coherent control method advocated by Brumer and Shapiro.5 The latter method has in particular attracted attention because, unlike the optimal control
method for example, the fields employed are simple multicolor continuous wave ~cw! fields, and it has also been attempted in the laboratory with some success.6 Although the
coherent control method and its variations have relied on
perturbation theory ~therefore resulting in presumably small
total yields although control of relative product yields, or the
branching ratio, may be very good!, recently these authors
have published a study of two-color control of diatomic dissociation in intense fields, i.e., the molecule–field interaction
is treated nonperturbatively.7
In this work we develop a very simple extension of the
perturbative coherent control method whereby an optimal
~small! superposition of vibrational eigenstates is prepared in
the ground electronic potential. This superposition is defined
as the initial wave function which is then subjected to the
photodissociation pulse which results in selective dissociation out of a desired channel. The ‘‘control part’’ of this
scheme is entirely encompassed in the preparation of the
initial wave function; the photodissociation pulse is assumed
to be fixed. Finding the optimal superposition state for a
desired product yield objective is done using the Rayleigh–
Ritz variational procedure. Although our method is similar in
spirit to one of the Brumer and Shapiro coherent control
J. Chem. Phys. 104 (18), 8 May 1996
variants,8 our method is more general in that it can accommodate any photodissociation pulse shape or intensity.
The underlying motivation for this study is the fact that
photodissociation processes have been shown to be influenced ~sometimes dramatically! by the initial state prior to
dissociation. Examples include photodissociation of HCl
~Refs. 9, 10! and HI,11 where the product branching ratios
were found to be highly sensitive to the initial vibrational
state of the ground electronic curve. This vibrationallymediated photodissociation technique has been exploited in
bond-selective photodissociation of HOD where the relative
yields of the resulting products ~H1OD or OH1D! can be
controlled to a large degree through selective IR excitation of
certain vibrational modes of the ground surface before application of the UV photodissociation pulse.12–15 Also, a theoretical control study has been presented for the HOD system
where a non-stationary vibrational state is prepared with an
intense IR pulse and then subsequently dissociated with a
weak UV pulse.16
Like the studies mentioned above, our control method of
photodissociation reactions is through manipulation of the
vibrational states prior to application of the photodissociation
pulse. In particular, we advocate preparation of a nonstationary vibrational state as in Ref. 16. However, our
method is more general and systematic in that one determines the best nonstationary state for a given control objective. This nonstationary state is comprised of only those
eigenstates which we choose to be a part of the superposition. Usually, only a few ~two or three! eigenstates will be
included in the initial wave function prior to photodissociation; experimental preparation of this coherent two or three
level superposition may be possible in the laboratory.
0021-9606/96/104(18)/7045/7/$10.00
© 1996 American Institute of Physics
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7045
Gross et al.: Laser control of curve-crossing reactions
7046
We emphasize that the method presented here has redefined the control problem from designing the photodissociation pulse which results in the desired products to preparing
a coherent superposition of a few discrete ~bound! levels. In
many cases this might be preferable to direct design of the
photodissociation pulse because the traditional control problem, i.e., the problem of finding the field which produces a
particular objective, now focuses solely on the preparation of
a well-defined coherent superposition of a few discrete levels. Iterative control methods for field design, such as optimal control theory,2 would obviously be far more computationally tractable for designing fields involving a few bound
discrete level system rather than photodissociation systems
involving continuum states and/or multidimensional potential energy surfaces. Furthermore, the dynamics of these systems under the influence of laser fields ~including analytical
solutions for two and three level systems! and controllability
of discrete level systems has been well-studied.17–21
The balance of the paper is as follows: In Sec. II, the
nonperturbative initial state control method is described and
its relation to the Rayleigh–Ritz variational principle emphasized. In Sec. III, the method is applied to photodissociation
of hydrogen iodide which possesses two dissociation pathways leading to either ground state or spin–orbit excited
iodine. Curve-crossing systems have been used in the past
for testing control schemes because of their relative simplicity from both a theoretical,22,23 and experimental
perspective.24 Also in this section are some calculations to
get a ‘‘feel’’ for the sensitivity of product branching ratios
and yields in HI photodissociation as a function of initial
vibrational state. We then present results for multicolor photodissociation pulses and compare product yields using unoptimized and optimized initial wave functions. It is shown
that, by using the variational procedure to obtain the optimal
combination of vibrational states, significant enhancement of
product yields from either the I~2P 3/2! or I*~2P 1/2! channels is
produced as desired. In Sec. IV, we present an analytical
noniterative method for determining the preparatory field
which provides the necessary coherent superposition of ~here
three! vibrational eigenstates. This method, which invokes
the rotating wave approximation ~RWA! in order to solve the
time-dependent Schrödinger equation ~TDSE!, determines
the amplitudes and phases of two cw pulses which induce
transitions between the relevant vibrational states. Finally,
Sec. V provides a summary and future extensions.
where Û(T,0) is the ~not necessarily unitary! propagator and
c~0! is the initial wave function to which the photodissociation pulse is applied. The time-integrated flux, which is directly related to the product yield, is
E
T
0
dt ^ ĵ & t 5
E
T
0
dt ^ c ~ t ! u ĵ u c ~ t ! & ,
where the flux operator is defined as
ĵ5
1
@ p̂ d ~ r2r d ! 1 d ~ r2r d ! p̂ # .
2m
E
T
0
dt ^ ĵ & t 5
E
T
0
dt ^ c ~ 0 ! u Û † ~ t,0! ĵÛ ~ t,0! u c ~ 0 ! & ,
~5!
or
E
T
0
dt ^ ĵ & t 5 ^ c ~ 0 ! u F̂ u c ~ 0 ! & ,
~6!
where the operator F̂ is defined as
F̂5
E
T
0
dtÛ † ~ t,0! ĵÛ ~ t,0! .
~7!
As mentioned previously, control over photodissociation
product yields is sought through preparation of the initial
wave function c~0! as a coherent superposition of eigenstates
of the ground electronic potential. ~Since rotational motion is
ignored in the present work, eigenstates of the ground potential refer only to vibrational eigenfunctions.! Thus, the
propagator Û(T,0) in the previous equations is predetermined for all time t50 to t5T; we can only manipulate c~0!
in order to maximize *T0 dt ^ ĵ & t and therefore the product
yield. To do this, the Rayleigh–Ritz variational method is
employed to find the extreme eigenvalues of the operator F̂
defined in Eq. ~7!. Note that F̂ is bounded from below and
from above; if the initial wave function is normalized such
that ^c~0!uc~0!&51.0, then the lower and upper extremes are
0.0 and 1.0. We proceed by first expanding c~0! in a basis of
~M11! vibrational eigenfunctions,
M
c~ 0 !5 ( c mf m .
~8!
We then substitute the above basis set expansion into Eq. ~6!
and construct the matrix elements,
II. FORMULATION
The solution of the time-dependent Schrödinger equation,
~1!
at some time T can be expressed as
c ~ T ! 5Û ~ T,0! c ~ 0 ! ,
~4!
Here m is the reduced mass along the reaction coordinate, r d
is the grid point where the flux is evaluated, and p̂ is the
momentum operator. Using Eq. ~2!, the time-integrated flux
in Eq. ~3! becomes
m50
]
c 5Ĥ ~ t ! c ,
i\
]t
~3!
~2!
F kl 5 ^ f k u F̂ u f l & ,
~9!
of the F matrix. Diagonalization of F provides the desired
maximum eigenvalue F max which is equal to the maximum
product yield ~flux!. The corresponding eigenvector, cmax , is
the set of coefficients defined in Eq. ~8! which define the
initial wave function c~0! which leads to the product yield
F max . Since for experimental and practical reasons one
would restrict the basis set expansion in Eq. ~8! to only the
J. Chem. Phys., Vol. 104, No. 18, 8 May 1996
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Gross et al.: Laser control of curve-crossing reactions
7047
lowest vibrational eigenstates, the matrix F will be quite
small in most cases of interest and therefore computationally
trivial to diagonalize.
It is important to note that to obtain the matrix elements
F kl it is not necessary to compute the entire propagator
Û(T,0). To see this, we first replace the integral in Eq. ~7!
with the following summation ~as we do in actual computations!:
Nt
F̂5Dt
(
n50
~10!
Û † ~ nDt,0! ĵÛ ~ nDt,0! ,
where the total time of propagation is divided up into N t 11
equally spaced time points and N t •Dt5T. Using Eq. ~10!,
the matrix elements become @cf. Eq. ~9!#
FIG. 1. Diabatic potential energy curves of HI.
Nt
F kl 5Dt
(
n50
^ f k u Û † ~ nDt,0! ĵÛ ~ nDt,0! u f l & ,
~11!
or
^ck~nDt!u ĵucl~nDt!&5
Nt
F kl 5Dt
(
can be evaluated as
n50
^ c k ~ nDt ! u ĵ u c l ~ nDt ! & ,
~12!
where c k (nDt) and c l (nDt) are solutions of the TDSE with
the chosen photodissociation pulse assuming the initial conditions c~0!5f k , f l , i.e., the initial wave function is a vibrational eigenstate. Thus, if the expansion in Eq. ~8! contains three eigenstates, then we need only propagate the
TDSE from t50 to t5T three times rather than construct the
entire propagator Û(T,0).
The nonperturbative initial state control procedure can
be outlined as follows:
~1! Choose a photodissociation pulse and time interval T for
which the TDSE is to be solved. Note that the length of
the pulse and the total propagation time T need not be
the same; if desired, one may propagate the TDSE after
the pulse has turned off in case there remains some molecules with sufficient energy which have not yet dissociated when the pulse is shut off. There are no restrictions on the field intensity or shape of the field since the
TDSE is solved numerically.
~2! Choose the eigenstates of the ground potential surface
which will be included in the variational calculation. As
in other similar calculations, the larger the basis set expansion, the ‘‘better’’ the results. In the initial state control method, ‘‘better’’ results means better product
yields. However, in practice only a few ~two or three!
eigenstates can reasonably be included in the expansion
due to experimental constraints of preparing a large coherent superposition of eigenstates prior to application of
the photodissociation pulse.
~3! Propagate the TDSE M 11 times from t50 to t5T for
each initial condition c~0!5fm , m50,1...,M in the basis set expansion @Eq. ~8!#. During propagation, the matrix elements F kl must be accumulated according to Eq.
~12!. The most efficient way to do this is to propagate all
M 11 wave functions c~0!5f0 ,f1 ,...,fM simultaneously. At each time step nDt, the matrix elements F kl
F
]c*
i\
k ~nDt !
cl~nDt!
2m
]r
2c*
k ~nDt !
G
]cl~nDt!
,
]r
r
~13!
d
where the r d subscript indicates that the wave functions
and their derivatives are evaluated at the flux indicator
point r d located in the asymptotic region of the potential.
In practice, the derivatives were evaluated with a ninepoint differentiation formula. Since the F matrix is Hermitian, it is necessary to compute only the upper or
lower triangular elements of F.
~4! Diagonalize F and pick the largest eigenvalue and corresponding eigenvector which represents the superposition of eigenstates that provides the highest desired product yield as discussed earlier.
III. APPLICATION TO PHOTODISSOCIATION OF HI
The hydrogen iodide system ~HI! used in this work is
modeled as a rotationless oscillator. Field–molecule interaction is within the semiclassical dipole approximation and the
laser field is assumed to be linearly polarized along the molecular axis. The five relevant potential energy curves are
shown in Fig. 1. All potential energy parameters, diabatic
couplings, and transition dipoles are taken from Ref. 25.
~Note that the diabatic coupling strength listed in Table I of
Ref. 25 between the 1P1 and 3S1 states is an order of magnitude too large, although the correct value is shown in Fig.
2 of the same reference.! The TDSE is set up exactly as in
Ref. 11 except here the ground electronic state potential is
included in the electronic Hamiltonian matrix. The TDSE
was solved numerically using the split-operator fast Fourier
transform ~FFT! method.26 A total spatial grid length of 9.0
spanning from 1.0 to 10.0 ~atomic units used unless otherwise noted! was used and divided into 256 equally spaced
grid points. ‘‘Ramp’’ ~linear! optical potentials were included at the ends of the grid ranging from r58.5 to r510.0
with a maximum height of 0.05 in order to avoid spurious
wave packet reflection. The flux point r d on all curves was
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7048
Gross et al.: Laser control of curve-crossing reactions
FIG. 2. Photodissociation branching ratio @I*~2P 1/2!/I~2P 3/2!# vs. laser excitation energy ~v!. The field employed is A cos ( v t), where A50.01 atomic
units and the pulse length is 100 optical cycles.
set at 8.0. The vibrational eigenfunctions of the ground state
potential were computed using the Fourier grid Hamiltonian
~FGH! method.27
In order to test our program and to demonstrate the sensitivity of the photodissociation branching ratio with respect
to initial vibrational state, we repeated the calculations in
Ref. 11 with the difference that we do not assume weak-field
conditions whereby the initial wave function can simply be
projected ‘‘upstairs’’ onto the upper surfaces and propagated
without explicit inclusion of the time-dependent field in the
Hamiltonian. Figure 2 presents the product branching ratio
@I*~2P 1/2!/I~2P 3/2!# vs frequency ~excitation energy! for
c~0!5f0 , f1 , and f2 initial conditions. The photodissociation pulse here is cw with amplitude 0.01 and a length of 100
optical cycles. The branching ratio is determined by computing the total time-integrated fluxes out of channels leading to
I*~2P 1/2! ~3S1 and 3P0! and I~2P 3/2! ~1S0 , 3P1 , and 1P1!.
@Note that in the multicurve problem here, flux must be
evaluated for all five components of the wave function corresponding to the five potential curves. Thus, the flux operator ĵ defined in the previous section refers to flux out of the
I~2P 3/2! channel ~J 1! or the I*~2P 1/2! channel ~J 2!.# Our results qualitatively agree with those in Ref. 11 although they
do not match exactly because the field strength used here is
outside the perturbation regime. Note also, as in Ref. 11,
there is strong evidence of sensitivity to initial vibrational
quantum number.
FIG. 3. Product yield ~flux! from ~a! I~2P 3/2! and ~b! I*~2P 1/2! channels vs v
~same field used for Fig. 2 results employed!. Results are shown assuming
the initial wave function c~0! is one of the three lowest vibrational eigenstates f0 , f1 , f2 .
In our control scheme the control objective is not the
branching ratio itself but rather the photodissociation yields
out of each channel. Thus a more relevant frame of reference
for our purposes is a plot of fluxes or yields from both channels as a function of laser frequency. Using the same pulse as
in Fig. 2, we show in Fig. 3 product yields from both channels J 1 and J 2 . Again, as in Fig. 2, there is a significant
sensitivity with respect to initial vibrational quantum number.
As mentioned previously, in our scheme we seek to control the product yields by finding the best superposition of
vibrational eigenstates for the initial wave function c~0!. In
all calculations presented below we include only the lowest
three vibrational eigenfunctions in our basis set expansion
@M 52 in Eq. ~8!#. Instead of employing a single color field
as in Figs. 2 and 3, we employ here multicolor fields of the
form
2
E ~ t ! 5A ~ t !
(
p50
cos~ v 2 v p,0! t,
~14!
where A(t) is the amplitude function, v is the primary photodissociation frequency, and v p,05(E p 2E 0 )/\ is the energy level difference between the pth vibrational energy and
the ground vibrational energy. Although the theory presented
in Sec. II is valid for any field, the above form was chosen
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Gross et al.: Laser control of curve-crossing reactions
FIG. 4. Product yield ~flux! from ~a! I~2P 3/2! and ~b! I*~2P 1/2! channels vs. v
employing a three-color field E(t)5A(t)(2p50 cos@( v 2 v p,0)t# for both unoptimized ~open circles! and optimized ~filled circles! initial wave functions.
Here, constant amplitude fields are used, i.e., A(t)5A.
because all the cw components resonantly excite to the same
energy in the upper electronic continuum states, i.e., v induces a resonant transition from v 50 to energy E in the
continuum; v2v1,0 induces a transition from v 51 to E, etc.
This then allows for control over degenerate states at energy
E which asymptotically correspond to J 1 or J 2 products.8
Here, however, there are no restrictions on the pulse intensity
or envelope, and thus more realistic pulse shapes such as
Gaussians may be employed.
Figure 4 shows results for both unoptimized ~open
circles! and optimized ~filled circles! computed from the initial state control scheme. By unoptimized we mean that we
have chosen the vibrational eigenfunction f0 , f1 , or f2 as
the initial state c~0! which provides the maximum yield for
the product sought. In other words, only one of the three
lowest vibrational states ~the one which provides the best
product yield out the desired channel! is initially populated
in the unoptimized results. For the optimized results we allow for mixing of the lowest three vibrational eigenfunctions
as described above. Figures 4~a! and 4~b! show results assuming a cw field where A(t)5A50.01 in Eq. ~14!. All
results are for the same set of primary frequencies ~v! in the
range 38 000–48 000 cm21. In Fig. 4~a!, J 1 is maximized,
and in Fig. 4~b!, J 2 is maximized. Note that the optimized
results are significantly better than the unoptimized ones, in
some cases almost 50% better.
7049
FIG. 5. Same as Fig. 4 except using a three-color Gaussian field, i.e.,
A(t)5A exp@2a(t2t 0 ) 2#.
In Figs. 5~a! and 5~b! we show similar results as before
except here the amplitude function A(t) is a Gaussian;
A(t)5A exp@2a(t2t 0 ) 2#. The maximum amplitude A is
0.01, the width is a50.000 002, and the mean time t 0 is
~50!~2p/v!. Note that in general the overall yields are lower
in the case of Gaussian pulses ~compare the y-axes of Figs. 4
and 5!. This is largely due to the overall lower field fluence
in the Gaussian case. However, as in the cw field case, improvement using the optimized initial state is still significant.
IV. PREPARATORY FIELD
Here we address the issue of preparing the initial coherent superposition of states resulting from the above proposed
variational calculation. The initial wave function to which
the photodissociation pulse is applied is, for the example
given in the previous section where only the three lowest
vibrational eigenstates were included in the variational calculation,
2
c ~ 0 ! 5 ( c opt
m fm ,
m50
~15!
where c opt
m are the ~complex! optimal coefficients of the vibrational eigenstates fm determined from the diagonalization
of F as described in Sec. II. The problem is to find a preparatory field which gives this superposition. We take the preparatory field form to be
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Gross et al.: Laser control of curve-crossing reactions
7050
E prep~ t ! 52A 1 cos~ v 10t1 d 1 ! 12A 2 cos~ v 21t1 d 2 ! ,
~16!
where the field frequencies v nm 5( e n 2 e m )/\ are chosen
such that the first cw component is resonant with the
v 50→ v 51 transition, and the second cw component is
resonant with the v 51→ v 52 transition. The amplitudes are
assumed to be sufficiently weak such that the conditions
um10uA 1/\!v10 and um21uA 2/\!v21 are fulfilled ~here
m nm 5 ^ f n u m̂ u f m & 5 m mn are the transition dipole matrix elements!. Furthermore, the cw components should not appreciably
excite
nonresonant
transitions
@v 5m→ v
5n:(m,n)Þ(0,1) or ~1,2!#, i.e., the condition
~ m nm ! 2 ~ A 2 ! 2
~ m nm ! 2 ~ A 1 ! 2
!1
2
2
2, 2
\ ~ v nm 2 v 21! 1 ~ m nm ! ~ A 2 ! \ ~ v nm 2 v 10! 2 1 ~ m nm ! 2 ~ A 1 ! 2
~17!
2
must be satisfied. Roughly, pulse lengths in the picosecond
regime and cw amplitudes of '1024 a.u. will fulfill the
above conditions for HI and other similar systems. With
these assumptions, the RWA may be invoked in solving for
the dynamics of the discrete level system whereby all nonresonant transitions are ignored.
The goal is to determine the field parameters A 1 , A 2 , d1 ,
and d2 which provide the required initial state wave function
c~0!. We define the optimal coefficients determined from the
variational procedure c opt
m [ c m ( t ), i.e., the coefficients of the
lowest three vibrational eigenstates resulting from application of the preparatory laser pulse given by Eq. ~16! for a
time duration t. With the aforementioned weak-field assumptions and invoking the RWA, the solution of the TDSE
in the eigenstate basis for the three coefficients for a pulse of
length t is, assuming that before application of the preparatory field E prep(t) all vibrational population is in the ground
vibrational state @uc 0~0!u251.0#,
c 0 ~ t ! 5exp~ 2i e 0 t /\ !
SD
c 1 ~ t ! 5exp~ 2i e 1 t /\ !
S D
c 2 ~ t ! 5exp~ 2i e 2 t /\ !
1
$ a 1 u a u 2 @ cos~ Aat /\ ! 21 # % ,
a
~18!
ia *
Aa
sin~ Aat /\ ! ,
S D
a *b *
$ cos~ Aat /\ ! 21 % ,
a
~19!
~20!
b5A 2m21 exp~i d 2!,
and
where
a5A 1m10 exp~i d 1!,
a5u a u 2 1 u b u 2 . Using the fact that ^c~0!u F̂ u c~0!& in Eq. ~6!
does not depend on the overall phase factor of the initial
wave function @i.e., c~0!→e i b c~0! does not change the expectation value# and performing the necessary algebra, Eqs.
~18!–~20! can be solved for the required cw field amplitudes
and phases,
A 15
A 25
\
u m 10u t A11M 2
cos21 $ 12 @~ 11M 2 ! /M # u c 2 ~ t ! u % ,
u m 10u
M A1 ,
u m 21u
~22!
p
e 1t
2 u 1 1q 1 ,
d 15 2 h 12 b 2
2
~21!
\
~23!
d 25 p 2 d 12 h 12 h 22 b 2
e 2t
2 u 2 1q 1 1q 2 .
\
~24!
Here, M 5 u c 2 ( t ) u /[12 u c 0 ( t ) u ], b52@u01e0t/\#, and u0 ,
u1 , and u2 are the phases of the desired coefficients c opt
0
iu1
[ c 0 ( t ) 5 u c 0 ( t ) u e i u 0 , c opt
, and c opt
1 [ c 1( t ) 5u c 1( t )ue
2
[ c 2 ( t ) 5 u c 2 ( t ) u e i u 2 , respectively. The remaining parameters are defined as follows: h150 for m10.0 or h15p for
m10,0, h250 for m21.0 or h25p for m21,0, q 15p for
Re$c 1~t!%,0 or q 150 for Re$c 1~t!%.0, and q 25p for
Re$c 2~t!%,0 or q 250 for Re$c 2~t!%.0. Finally, if
Re$c 0~t!%,0 then all coefficients should be multiplied by 21
prior to solving for the field parameters in Eqs. ~21!–~24!.
The coefficients thus obtained with the pulse of the form in
Eq. ~16! using the field parameters defined by Eqs. ~21!–~24!
are the optimal coefficients in Eq. ~15! to within a common
unimportant phase factor @e i b or e i( b 1 p ) #.
The above direct analytical solution can be used only for
preparation of a maximum of three coherent states. @For a
desired coherent superposition of two states, f0 and f1 , just
the first resonant cw component in Eq. ~16! is sufficient to
prepare any superposition, and the solutions for c 0~t! and
c 1~t! as well as the required field parameters, A 1 and d1 , may
be easily derived as a special case of Eqs. ~21!–~24! and Eqs.
~18!–~20! above.# For more than three states, one must resort
to numerical field design methods such as optimal control
theory.2 However, other noniterative numerical methods
have been recently developed which can ‘‘scan’’ through
field parameter space ~cw component amplitudes, frequencies, and relative phases between field components! and determine the wave function quickly for a wide range of field
parameters.28,29 Such methods could be employed for determining the preparatory field if c~0! is comprised of more
than three eigenstates. We note, however, from a practical
point of view, the preparation of more than three coherent
states would entail more complex preparatory fields ~e.g.,
more than two phase-locked cw pulses!, and thus purely
from an experimental perspective one may not want to prepare a superposition of more than three coherent states.
Therefore, if coherent preparation of three states is the upper
limit of experimental viability, then theory provides a complete noniterative solution to the field design problem as outlined in this paper.
J. Chem. Phys., Vol. 104, No. 18, 8 May 1996
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Gross et al.: Laser control of curve-crossing reactions
V. CONCLUSION
We have presented a scheme for laser control of photodissociation processes which entails preparation of the initial
wave function as a coherent superposition of a few bound
vibrational eigenstates before application of a given photodissociation pulse. The optimal linear combination of eigenstates is found using the Rayleigh–Ritz variational procedure. This scheme is extremely simple to use in that it
requires only a standard wave packet propagation routine
and a subroutine to diagonalize small Hermitian matrices.
Preparation of the coherent superposition of states, at least
for two or three levels, can be done using a simple field of
one or two cw pulses, respectively. Furthermore, assuming
the pulses are sufficiently weak so that the RWA is valid, it
has been demonstrated that the required amplitudes and
phases of the pulses which result in the desired superposition
can be found analytically. Thus, assuming a basis of no more
than three eigenstates for which an optimal linear combination is determined by the proposed variational procedure, the
field design prescription is noniterative and direct using the
proposed control scheme. The use of larger basis sets in the
variational calculation, i.e., allowing for a superposition of
four or more vibrational eigenstates in the initial wave function prior to dissociation, will likely require iterative optimization methods for designing the preparatory field, e.g., optimal control. However, solving the TDSE for a few level
discrete system, even many times, will certainly be less
costly than designing the photodissociation pulse itself via
iterative methods since this would entail repeated wavepacket propagations on the relevant dissociative electronic
curves.
Future work will include application of this control
scheme to other curve-crossing systems30 such as IBr and
HCl as well as to reactive systems such as HOD where initial
state preparation prior to photodissociation has been theoretically and experimentally shown to be very influential in
terms of product yields. Finally, in relation to the issue of
experimental preparation of the superposition of vibrational
eigenstates, it would be worthwhile to determine just how
sensitive the results are to minor deviations in the optimal
eigenstate amplitudes. Sensitivity analysis might prove very
7051
useful in determining how much allowance for error can be
reasonably tolerated in an actual experimental preparation of
the initial wave function.
ACKNOWLEDGMENTS
M.K.M. acknowledges support from the Board of Research in Nuclear Sciences ~BRNS Grant No. 37/9/94-R1DII/787! from the Department of Atomic Energy, India.
D.B.B. acknowledges support from CSIR Junior Research
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