Variations on the Theme of Stimulated Raman Adiabatic Passagr

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Chapter 2

Variations on the Theme of Stimulated Raman Adiabatic Passagr: Control of Chemical Reactions Suhail P. Shah, Vandana-Kurkal and Stuart A. Rice The Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637

We describe the exploitation of adiabatic transfer of population between selected states of a molecule as a means for the control of product selectivity in a branching unimolecular reaction. The particular procedures employed to achieve that adiabatic transfer of population depend on the character of the spectrum of states of the molecule; the ones we consider are either extensions of, or are illuminated by, the STIRAP method. The results of studies of efficient selective population of one of a pair of nearly degenerate states in the thiophosgene molecule, and of efficient promotion of the H C N --> C N H isomerization reaction, are reported. In the former case the adiabatic population transfer is achieved by use of the Kobrak-Rice extended STIRAP method, while in the latter case it is achieved by use of consecutive STIRAP processes. In both cases the efficiency of selective population transfer is shown to be insensitive to radiative coupling between background states and the subset of active states, and to radiative coupling between the background states. We also report the results of a study of adiabatic population transfer to one of a pair of degenerate states when the STIRAP condition (existence of a field-matter state with zero eigenvalue) cannot be satisfied by the subset of active states in the molecular spectrum of states. The criteria for efficient and selective population transfer in this case are discussed.

16

© 2002 American Chemical Society

Bandrauk et al.; Laser Control and Manipulation of Molecules ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

17

Introduction A c t i v e control o f product selectivity in a chemical reaction with optical fields, first suggested on the basis o f theoretical arguments, has been experimentally demonstrated for several small molecule fragmentation and ionization reactions [1]. The key step in achieving active control is the exploitation o f quantum interference effects to generate sensibly complete selective transfer o f population between specified states o f a molecule. O f the currently available methods for efficiently transferring population within a subset o f quantum states selected from the full manifold o f states o f a molecule, those that rely o n adiabatic passage are very appealing because they satisfy important criteria for experimental implementation. F o r example, the stimulated R a m a n adiabatic passage ( S T I R A P ) process [2] has been shown to be an efficient means o f generating population transfer in three state and multi-state systems. A n important feature o f the S T I R A P process is that the "counter-intuitive" ordering o f pulses used (see below) is remarkably robust with respect to variation in the overlap between the pulses and to their respective shapes, provided the criteria for adiabatic passage are met. The S T I R A P process involves the creation o f stationary states o f a coupled coherent field-matter system. In its simplest form this scheme relies on two suitably timed coherent pulses o f light coupling a three state system that has non vanishing transition moments that connect state | 1> with state |2> and state |2> with state |3>, but has zero transition moment between states |1> and |3>. The fields associated with the pump pulse Q that couples the initial state |1> to the intermediate state |2>, and the Stokes pulse Q that couples |2> to the final state |3>, are large enough to generate many cycles o f Rabi oscillation between |1> and |2>, and between |2> and |3>. The coherent field-matter eigenstates can be represented as linear superpositions o f the states |1>, |2> and |3>, and the system wavefunction as a linear combination o f the coherent field-matter eigenstates. Selective population transfer from state |1> to state |3> is generated by varying the contributions to the system wavefunction o f the coherent field-matter eigenstates; this is a c c o m p l i s h e d through c o n t r o l o f the ratio o f R a b i frequencies. A d i a b a t i c transfer o f population from state |1> to state |3> accompanies this variation when there is counterintuitive ordering o f the pulses, with the Stokes pulse preceding but overlapping the pump pulse. Criteria for variation o f the Rabi frequencies that satisfy the condition o f adiabatic transfer o f population can be found in the literature; they contain constraints on both the strengths and the rates o f change o f the Stokes and pump fields. Several extensions o f the original S T I R A P method o f population transfer have been reported [3-11]. Thus, it is now possible to design efficient population transfers between initial and final states in systems in w h i c h the intermediate states are autoionizing or replaced by unstructured continua. It is now accepted that in a ladder-like distribution o f energy states, provided the lifetimes o f the target states are longer than the pulse durations, complete population transfer from an arbitrary initial state to a selected target state is possible using a suitable p

s


18 set o f overlapping pulses. A n d , it is possible to use an extended S T I R A P method for selective population transfer from an initial state to one o f two degenerate final states, w h i c h thereby permits control o f product selectivity in a branching unimolecular reaction. There are several different ways o f i m p l e m e n t i n g o p t i c a l control o f molecular dynamics, all derived from the same basic principles. A l t h o u g h these method are independent o f the size o f the system to be controlled, accurate prediction o f the character o f the control field becomes more and more difficult as the n u m b e r o f degrees o f freedom increases. F o r e x a m p l e , the m u l t i d i m e n s i o n a l i t y and c o m p l i c a t e d topography o f the potential energy surfaces o f a large polyatomic molecule makes it very difficult to use pulse t i m i n g control o f product selection in a unimolecular reaction because the evolution o f a wavepacket on a complicated potential energy surface is difficult to visualize and follow. The multipath interference control o f product selection in a unimolecular reaction o f a large polyatomic molecule becomes more difficult as the number o f states per unit energy interval increases because identifying the suitable pairs o f pathways between the same initial and the final states requires detailed knowledge o f the character o f a l l the states i n v o l v e d . Hence it is desirable to have a reduced states formalism o f control process. A m o n g s t its other attributes, we regard S T I R A P and extended S T I R A P methods o f control o f population transfer to be examples o f reduced states descriptions o f molecular dynamics. In this paper we briefly discuss three issues: (1) H o w sensitive is the selectivity o f adiabatic population transfer between specified initial and final states to the presence o f background states that are radiatively coupled to the subset o f active states and are radiatively coupled to each other? (2) C a n adiabatic population transfer be used for control o f product formation in an isomerization reaction involving large atomic displacements? (3) C a n adiabatic population transfer be used to selectively populate one o f a pair o f degenerate states when the conditions for S T I R A P are not met? T o address (1) we have examined the K o b r a k - R i c e [10,11] extended S T I R A P method for selective population o f one o f a pair o f nearly degenerate states in thiophosgene, taking account o f radiative transitions between the active subset o f states and a large set o f background states, and radiative transitions between the background states. The selectivity o f population transfer is shown to be insensitive to the presence o f those background states. T o address (2) we have studied the H C N —> C N H i s o m e r i z a t i o n r e a c t i o n . T h e use o f straightforward S T I R A P excitation with the ground vibrational state o f H C N the initial state, a state just above the barrier to isomerization the intermediate state, and the ground vibrational state o f H N C the final state is not feasible, as it involves very high intensity pulses due to the very s m a l l transition dipole moments between the states mentioned. However, the use o f successive S T I R A P excitations to generate the product C N H is feasible, and is shown to be very efficient even when account is taken o f radiative transitions between the active


19 subset o f states and a large set o f background states, and radiative transitions between the b a c k g r o u n d states. T o address (3) w e have studied selective adiabatic population transfer to one o f a pair o f degenerate states in a subsystem o f states with the f o l l o w i n g characteristics: T h e ground state |1> is radiatively coupled to two degenerate states, |2> and |3>, and these are i n turn radiatively coupled to a fourth (branch) state |4>. C o n d i t i o n s for complete transfer o f population from state |1> to state |2> or state |3> are determined for a w i d e range o f ratios o f the transition moments between the states. It is shown that both selectivity and efficiency o f population transfer can be achieved even when the S T I R A P condition is not satisfied.

Effect of background coupling on the selectivity of population transfer in thiophosgene The thiophosgene m o l e c u l e ( S C C 1 ) has three stretching (V1V2V3) and three bending ( v V 5 V ) vibrational degrees o f freedom. T h e vibrational spectrum has been very carefully studied by Gruebele and co-workers [12]. Consequently the frequencies, deviations from harmonic behaviour and the energies o f the states are k n o w n , w h i c h facilitates the c a l c u l a t i o n o f transition dipole moments between a l l states up to 21000 cm" above the vibrationless ground state. T h e data w e used for the energies and the v i b r a t i o n a l states were taken from reference 12. 2

4

6

1

The K o b r a k - R i c e extended S T I R A P technique for controlled population transfer is based o n a subset o f five states: an initial state, an intermediate state, a branch state and a pair o f degenerate target states. T h e initial state and the intermediate states are coupled by a resonant pump pulse and a Stokes pulse resonantly couples the intermediate state and the degenerate target states. A third pulse, longer than the pump and Stokes pulses and on throughout their duration, resonantly couples the degenerate target states and the branch state. These active states are represented by thick lines F i g , 1. The rotating wave approximation to the Hamiltonian for this system o f five states and three fields is f

0

0

0

0

0

0

0

0

0

" 4,

n* ,

W

3

where

Q = 27i/u e (t)f h, £ 2 / ;

l2

p

S(f)

^

0

0

0

0

= 2Kji e {t)lh 2i

s

0

;

and Q

h(j)

=2nn e (t)/h 5l

b

are

the time dependent R a b i frequencies c o u p l i n g the states indicated. T h e field intensities chosen to provide sufficient splitting o f the bare states manifold to


20

1210020.

|17>

|201020>

116>

|300002>

|7>

|100040>

|18>

|310000>

|200020> |100031> |201011>

(Branch Level for selective population transfer to f3>)

Figure 1 : Schematic diagram of all the vibrational states of thiophosgene molecule considered in the present work.


21 facilitate the adiabatic transfer o f population from the ground state to the selected target state are shown below 2

Pump pulse Stokes pulse Branch pulse

Intensity ( W / c m ) 1.21 x I 0 1.46 x 10 9.30 x l O 9

8

1 1

/ (ps) 85 0 215 0

FWHM(ps) 85 143 215

The Schrodinger equation was solved in a basis o f bare matter eigenstates using a fourth order Runge-Kutta integrator. The results [13] indicate that almost 100% o f the population o f the initial state is transferred to the desired target state using this subset o f five states. It is important to note that the vibrational spectrum o f thiophosgene is rich in the energy range from w h i c h the five levels are chosen. W e now show that the selectivity in the population transfer to the desired target states is not affected by radiative coupling o f the subset o f active states to the background states and radiative coupling between the background states. The background states considered for this w o r k are displayed as light lines in F i g . 1. The selective population transfer achieved w i t h counterintuitive and intuitive ordering o f the pump and Stokes pulses is displayed in Figs. 2a and 3b as a function o f detuning o f the branch frequency. A s expected, our results show that the ratio o f populations transferred to the target states with counterintuitive excitation has an inverse dependence on the ratio o f the magnitudes o f the transition moments connecting the branch state and the target states. O u r calculations also show that the selectivity and magnitude o f the population transfer to a target state are sensibly independent o f the nonresonant coupling with the background states. The effect o f detuning o f the branch frequency from resonance is o f the same magnitude as when the background states are not considered for the case o f counterintuitive ordering o f the pump and Stokes pulses. W h e n intuitive ordering o f the pump and Stokes pulses is employed, there is an oscillatory dependence o f the population transfer on detuning, and a reduction in the absolute y i e l d in the target state. These oscillations in the population arise from interference effects.

Successive STIRAP excitation control of the HCN - » CNH reaction A c t i v e control o f product formation in an isomerization reaction is usually very difficult because o f the occurrence o f large atomic displacements in the reaction and because there is non negligible non-linear coupling o f the molecular degrees o f freedom along the reaction path. However, because o f these complications, isomerization reactions provide a useful vehicle for testing the control fields calculated using different reduced representations o f molecular dynamics. W e have used the H C N - » C N H isomerization for this purpose, since the path o f the



DC

iS

>

32 g

CP

oo

0

o o

b

1

100

T3 CD

50

|310000>

|302000>

CP3

d

as

>

>

-100

0

0.2

0.4

0.6 °

cP^P

|302000>

PcP

|310000> 0.8

|310000>

|302000>

Figure 2 : (a) Variation in the population transfer to target states \310000> and \302000> with \300000> as the branch state, calculated with inclusion of all of the radiative couplings between the 21 states shown in Fig. 1 ; (b) Same as (a), but with \013000> as the branch state.

A (cm")

-50

fep^eo

-100

0.2

0.4

0.6

0.8

• counterintuitive o Intuitive

23

• (6,0,2) (19528.57/cm) (175J4.40/cm)

(5,0,1) Resonant Pump2 (2,0,1)

(3,0,2)

(14p4.3/cm)

(13702.24/cni Resonant Stokes 1

(2,0,0) (12l39.9/cm)

/ - (3,0,1) (11674.4S/cm) (1,0,1) (2,0,1) (8585187/cm)

(10^51.9/cm) Resonant Stokes2

(IAD (5393.|70/cm) Resonant Pump 1

(0,0,0) (0,0,0)

(0.0/cm) (Initial State)

(5023.2/em) (Target State)

CNH

HCN Figure 3 : A schematic diagram of the vibrational states of HCN and CNH considered in this work.


24 reaction is simple (largely determined by the bending motion). The barrier to this isomerization reaction is large, w i t h the obvious consequence that the Franck-Condon factors that connect the ground vibrational state o f H C N w i t h its v i b r a t i o n a l l y e x c i t e d states decrease r a p i d l y w i t h increasing v i b r a t i o n a l excitation and are sensibly zero for a transition from the ground vibrational level to a delocalized state o f the molecule with energy just above the barrier. Hence the population in the ground v i b r a t i o n a l state o f H C N cannot be easily transferred to the ground vibrational state o f C N H by, say, delayed pulses (as in the T a n n o r - R i c e control method). A different approach to c o n t r o l l i n g the isomerization is based on the use o f successive vibrational transitions. One possible path o f that type is invoked when an optimal shaped field is used to m a x i m i z e the formation o f C N H . The optimal field is found to generate successive transitions up and down the ladder o f vibrational states o f H C N and C N H respectively. Shah and Rice [14,15] reported a study o f the accuracy o f an o p t i m a l f i e l d for the H C N —> C N H i s o m e r i z a t i o n calculated u s i n g the Z h a o - R i c e reduced space representation [16]. They showed that at each level o f approximation to the reaction path (determined by the number o f degrees o f freedom included i n the calculation), an optimal field that generates h i g h product y i e l d can be found. However, the field calculated at a particular level o f approximation does not give any insight for the calculation o f the optimal field at the next level o f approximation, or to the optimal field when all the degrees o f freedom o f the H C N molecule are taken into account. A s already indicated in the Introduction, the use o f straightforward S T I R A P excitation to control the isomerization reaction by using the ground vibrational state o f H C N for the initial state, a state just above the barrier to isomerization for the intermediate state, and the ground vibrational state o f H N C for the final state is not feasible, as it involves very high intensity pulses [17] due to the very small transition dipole moments between the states mentioned. W e now show that an alternative approach, that uses successive S T I R A P excitations to generate the product C N H , is feasible and very efficient. The three dimensional potential energy surface for non rotating H C N / C N H is very w e l l studied [18,19]. The molecular degrees o f freedom (dominated by the C - H , N - H , C - N stretching motions and the C N H bending motion) are strongly coupled. Experimental values for the vibrational energy levels o f H C N used in the present study were taken from reference 18. The transition dipole moments associated with the transitions between these states were obtained from the IR intensities o f the vibrational transitions reported by Bostchwina et al [19]. W e seek two successive efficient transfers o f population, g —>ii—> i and i -> i ->f, that have the property that the transfer o f population from the ground state o f H C N to the ground state o f C N H is sensibly complete without the use o f excessively large Stokes and pump fields. W e refer to the first set o f pulses as p u m p l ( p i ) and Stokes 1 ( S I ) , and to the second set o f pulses as pump 2 (p2) and Stokes (S2). The states that we considered for the successive S T I R A P excitations are shown in F i g . 4. T h e first S T I R A P process is associated w i t h the states H C N states g =(0,0,0) and h = (2,0,1) and the delocalized state h = (5,0,1). The Stokes 1 pulse is i n resonance w i t h the transition between (2,0,1) o f H C N and the delocalized state ( 5 , 0 , 1 ) ; the pump 1 pulse is resonant w i t h the transition (0,0,0) - » (2,0,1). 2

2

3


25 1.1

-

I

(5,0,1)

a. | E

cg

0.7

Jo

0.5 [•

3 Q. O

(0,0,0) of HNC

0.3 [• t

0.1

t-

(0,0,0) of HCN

-0.1

200

400

600

800

1000 1200

Time (ps)

Figure 4 : Variation of population with respect to time in the presence of background states The second set of pulses involves transitions between the delocalized state i = (5,0,1), and the C N H states h = (2,0,1) and / = (0,0,0). The Stokes 1 pulse precedes the pumpl pulse and, after the population transfer to state / , the Stokes 2 pulse and the pump 2 pulse are applied. The Hamiltonian for this system is, using the rotating wave approximation, 0 0 0> 0 2

2

(

0 H =

0

0

u 0

^> and ^ 5 2

0

0

0 0

0

0

0

0 j are the Rabi frequencies associated with the pump 1,

Stokes 1, pump 2 and Stokes 2 pulses, respectively. The field parameters associated with these pulses are shown below. 2

Stokes 1 Pump 1 Stokes 2 Pump 2

Intensity (W/cm ) 8.40 x 10 9.30 x 10 9.30 x 10 3.38 x l O 11

11

10

1 1

t (ps) 133 194 423 484 0

Frequency (a.u.) 0.04095 0.03912 0.04160 0.01558

F W H M (ps) 85 85 85 85

The states considered in the above discussion are now coupled to a set of background states taken from the manifold of vibrational states. The radiative couplings between all the states (shown in Fig. 3) are incorporated in our study


26 the effect o f background c o u p l i n g on the conversion o f H C N to C N H . The variation o f population w i t h respect to time i n the presence o f the two sets o f pulses is shown i n F i g . 4. W e find, from this calculation, that 95.06% o f the population initially i n the (0,0,0) state o f H C N is transferred to the (0,0,0) state o f C N H w i t h negligible transfer to the intermediate state and to the background states. B a s e d on these results, we infer that even i n the presence o f a substantial number o f background states that are radiatively coupled to the active states o f the subset and to each other, very efficiently transfer o f population to a selected target state is possible.

Optimized Coherent Population Transfer to Degenerate Final States in Multi-Level Systems The efficiency and completeness o f the population transfer generated by a S T I R A P process suggests that it may be related to population transfer generated by an optimal control process. A l t h o u g h the efforts to show that the S T I R A P sequence o f Stokes and pump pulses arise naturally from optimal control theory have been unsuccessful thus far, there are intriguing indications o f a theoretical analog between the two methodologies. F o r example, M a l i n o v s k y and Tannor [5] have shown that for N-state systems with a ladder-like distribution o f states, the S T I R A P two pulse field can be obtained from an optimization algorithm. U s i n g a variant o f optimal control theory, they studied the effect o f " l o c a l optimization" o f an initial field guess to maximize the ratio o f the population o f a selected target state to that o f a fully populated initial state. The control field obtained exactly matched the S T I R A P process field, w h i c h in the iV-level case consists o f N-l overlapping pulses, each resonant w i t h transitions between adjacent states o f the distribution o f states. This section addresses the question whether optimization algorithms that, in principle, y i e l d the set o f all control fields that guide the transfer o f population from the initial state to the target state, are able to generate optimal fields in systems in w h i c h the final states are degenerate. In particular, we shall focus our attention on a simple four-level system for w h i c h the conditions for a S T I R A P process are not met. Specifically, we examine a four state system o f states with a " d i a m o n d " arrangement o f nonzero transition moments [20]. This system has a manifold o f dressed eigenstates that does not contain a state whose projections to intermediate states are identically zero. The optimization scheme we use is derived from techniques developed previously in the theory o f control o f product formation in a chemical reaction. C o n s i d e r a system w i t h a pair o f degenerate target states, \t> and \t]>, embedded within a subsystem o f N states. These states are radiatively coupled by M (M < N) radiation fields. In the interaction representation, the Hamiltonian for the system represents the interaction energy o f the field-matter system, and the effect o f the field is fully characterized by the set o f Rabi frequencies o f the radiation fields, {Q} and their respective detunings from resonance { A } . A s s u m i n g the field interacts with the system between an initial time t = 0 and final time f , the condition for optimized population transfer may be written as max


27 V

[lfl.WI -lfl,(0l -Sla,.(0l ] 2

i n M 4 |

2

J

;=1

=0 ,

= t i m

where I a,(t) I denotes the time-dependent population o f the desired target state, 2

I a (t) I denotes the time-dependent population o f the state degenerate to the 2

n

,V-2

target state, and

2

^\a.(t)\

is the sum o f populations o f other states in the

»=i

system. The computational algorithm performs a gradient search for the incident fields that m a x i m i z e the population o f the target state w h i l e simultaneously m i n i m i z i n g the population o f the other states in the system [13,14], where the populations taken are those at the end o f the interaction with the field. The four state system we have examined is shown in F i g . 5. In a molecule, these states may be an initially fully populated ground vibrational state o f the ground electronic potential energy surface, a pair o f degenerate vibrational (target) states near the continua o f different exit channels o f an electronically excited potential energy surface, and a fourth (branch) state w h i c h may be a rovibronic excited state. A pump pulse radiatively couples the ground state to the two degenerate states and a Stokes pulse radiatively couples the two degenerate states to the branch state. The Schrodinger equation for the four state system

a,(0" a (t)

_

1

0

0

a,(r)

~

2

0

0

2

(

0

ia£l e p

lA

''

0

\

chit)

J

0 _a (r) 0 { T h e differences between the transition moments and the c o r r e s p o n d i n g differences in the Rabi frequencies have been subsumed into four parameters a , P, 5 and y that play the role o f normalized transition dipole moments; they are m u l t i p l i e r s w h i c h permit representation o f the several R a b i frequencies connecting states |1> and |2>, |1> and |3>, |2> and |4> and |3> and |4> in the form aQ, , /3£l , }Qs and SQs, respectively, subject to the constraints of + /3~=T and 8 + y = 1. a (0. 4

p

2

4

p

2

W e seek to selectively populate one o f the two degenerate target states using relatively simple pulses. T o be useful, the population transfer scheme must be efficient even when the initial state and the target state are coupled by an extremely small transition dipole moment. In our calculations we used a variety o f different ratios o f transition dipole moments connecting the ground state to each o f the two degenerate target states. The field is optimized to m a x i m i z e the population in the selected target state w h i l e simultaneously m i n i m i z i n g the population in the other target state and the branch state. For a two-pulse field, the R a b i frequencies o f the pump and Stokes pulses are u n k n o w n s . T h e o p t i m i z a t i o n c r i t e r i o n leaves a large number o f c o m b i n a t i o n s o f R a b i frequencies w h i c h achieve the population transfer goal [20], W e find that it is sufficient to fix the shapes o f the pump and Stokes pulses to be Gaussian w i t h optimizable time delays. T h i s choice provides a direct comparison w i t h the locally optimized fields, studied previously, as w e l l as being simpler to generate in the laboratory. In our calculations we assumed the Gaussian envelope o f the


Figure 5 : Four-level diamond system with pump and Stokes pulses. The dipole moment factors a, [3, yand Sare related by ot + /3 = 1 and f + 8 = 1. 2

2


29 a 1.0000 0.1961 0.9806 0.1961 0.9806

P 0.0100 0.9806 0.1961 0.9806 0.1961

Y 0.6690 0.1961 0.1961 0.9806 0.9806

5 0.7433 0.9806 0.9806 0.1961 0.1961

Optimal Target Yield 0.9995 0.9610 0.9806 0.5359 0.0384

t -ts p

1.270 -2.560 -0.810 -3.290 -2.560

Rabi frequencies of each pulse to have a maximum at 32GHz, which represents the minimum required for the algorithm to converge upon the optimum solution. Our calculations have covered a considerable range of values of the ratios a/p and y/8. The results are summarized in the table below. In the case that a//5 = 100/1 and y/8 = 9/10, the target state is practically "dark" with respect to transitions from the initial state. The "counter-intuitive" ordering of pulses (t - t > 0) when full transfer of population to a target state occurs bears a resemblance to the mechanism of population transfer via adiabatic passage. However, the population dynamics indicate that with the optimized pulses incident on the system, there are often large fluctuations of the populations of states |2> and |4> which eventually vanish at the end of the interaction with the field. The level-switching behavior of STIRAP [2] is clearly absent here. In the case where a/j5= 1/5 and y/8 = 1/5, the optimal sequence of pulses is "intuitive", with the pump pulse preceding the Stokes pulse, and with a significant overlap between them. The large population transferred is a consequence of the fact that the magnitudes of the transition dipole moments connecting the target state to the initial and branch states are much larger than those that connect state |2> with the initial and branch states. Conversely, when a//i= 5/1 and y/8= 5/1, the maximum population transferable is found to be no greater than about 4 percent. Again, the optimal sequence of pulses is in the intuitive order, indicating that the mechanism of population transfer resembles a stimulated emission pumping from the initial to the final states. When a/fi = 1/5 and y/8 = 5/1, we find that the optimal field depends only on the magnitude of the difference between the centers of the pump and Stokes pulses, not their sign. In other words, the ordering of the pulses is immaterial. The optimum difference in time of the centers of the pulses listed in the table effects the same population transfer irrespective of whether the pump pulse precedes the Stokes pulse or vice versa [20]. While in this case an optimized pair of pulses transfers nearly 50 percent of the population into the target state, one may opt to use a single pump pulse resonant between the initial state and the two degenerate states to exploit the larger dipole moment transition between the initial state and the final state. Conversely, when o//3 = 5/1 and y/8 = 1/5, the optimum sequence of pulses almost fully populates the target state at the end of the interaction with the pulses. Once again, we find that the factor determining the efficiency of population transfer is the magnitude of the time delay between the pump and Stokes pulses, not their order. The results we have obtained indicate that in cases where the target state has a close to vanishing dipole moment transition with the initial state, Complete (or nearly complete) population transfer is still achievable provided one finds a p

s


30 suitable branch state in the system that has a dipole moment transition to the target state w h i c h is slightly greater than the dipole moment transition to the other degenerate state. Within-the range o f dipole moments studied, in a l l but one case one may find a suitable sequence o f pulses w h i c h may be generated by the experimentalist that effects a population transfer w h i c h is S T I R A P - l i k e in its efficiency, but different from S T I R A P in the dynamics o f the transfer process.

Conclusions O n e class o f methods for o v e r c o m i n g the difficulty that arises i n a c h i e v i n g active control o f the molecular dynamics o f a polyatomic molecule, arising from the dependence o f the evolution o f a state o f the molecule on a large number o f coupled degrees o f freedom, is based on taking advantage o f adiabatic transfer o f population between states. This can be accomplished by use o f a S T I R A P , m o d i f i e d S T I R A P , or sequence o f S T I R A P processes, as w e l l as by use o f l o c a l l y o p t i m i z e d fields. T h e method o f choice depends on details o f the structure o f the manifold o f states o f the molecule.

Acknowledgements The research was supported by a grant from the N a t i o n a l Science Foundation (CHE-9807127)

References 1.

2.

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