numerical study on strong-laser-induced interference in the B state of I2.

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Physical Chemistry Chemical Physics

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Published on 24 January 2014. Downloaded by Lomonosov Moscow State University on 31/01/2014 14:00:52.

strong-laser-induced interference in the B state of I2

Yukiyoshi Ohtsukia* a

Department of Chemistry, Graduate School of Science, Tohoku University Sendai 980-8578, Japan

Haruka Goto,b Hiroyuki Katsuki,b,c† and Kenji Ohmorib,c,d b

Institute for Molecular Science, National Institutes of Natural Sciences Myodaiji, Okazaki 444-8585, Japan c

The Graduate University for Advanced Studies (SOKENDAI) Shonan Village, Hayama, Kanagawa 240-0193, Japan d

CREST, Japan Science and Technology Agency Kawaguchi, Saitama 332-0012, Japan

* Corresponding author: [email protected] †Present address: Graduate School of Materials Science, Nara Institute of Science and Technology, Ikoma 630-0192, Japan

1

Physical Chemistry Chemical Physics Accepted Manuscript

Theoretical/numerical study on


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In the B state of I2, strong-laser-induced interference (SLI) was recently observed in the population of each vibrational eigenstate within a wave packet, which was initially prepared by a pump pulse and then strongly modulated by an intense femtosecond near-infrared (NIR) laser pulse.

It was suggested that the interference as a function of the time delay occurs between the

eigenstate reached by Rayleigh scattering and that by Raman scattering.

To verify this

mechanism and further discuss its characteristics, we theoretically/numerically study the SLI by adopting a two-electronic-state model of I2.

Numerical simulation reasonably reproduces the

experimental signals and confirms the theoretical consequences, which include the  -phase shifts between Stokes and anti-Stokes transitions and (practically) no contribution from the energy shifts induced by the NIR pulse.

2


Abstract

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Introduction

A vibrational quantum beat originates from a superposition of vibrational eigenstates,


i.e., a vibrational wave packet [1, 2].

It is one of the examples of quantum beats that are

generally observed in any physical systems, provided that the observed signal monitors the total probability that is expressed as the square of the sum of the amplitudes associated with multiple distinguishable processes [3, 4].

To observe the vibrational quantum beat, a pair of weak laser

pulses that have a shorter temporal width than a typical vibrational period are used in a typical experiment. state.

The first (pump) pulse creates a vibrational wave packet in an electronic excited

Then, the second (probe) pulse that appears with a specified time delay excites a portion

of the wave packet to a higher electronic state.

Reflecting the vibrational wave-packet motion,

the quantum beat as a function of the time delay is observed, e.g., in the fluorescence from the higher electronic state.

If the probe pulse selectively detects the vibrational eigenstates of the

wave packet, the quantum beat would disappear because the so-called “which-path” information would be specified, analogous to Young’s double-slit experiment.

In the “standard”

pump-probe experiment, we do not observe a “vibrational-eigenstate-resolved” quantum beat from the wave packet. Recently, Goto et al. [5] proposed a new concept called strong-laser-induced interference (SLI), which involves the introduction of an intense laser pulse that strongly modulates a wave packet.

As schematically illustrated in Fig, 1, in the experiment, a

vibrational wave packet is created in the B state of I2 by a weak pump laser pulse.

This wave

packet interacts with an intense near-infrared (NIR) laser pulse after a specified time delay through stimulated Raman scattering.

The induced transitions open up additional pathways

that connect the initial state with the eigenstates within the wave packet.

It was suggested that

this obscures the “which-path” information regarding how to reach the eigenstates within the 3


1


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eigennstate as a funnction of thee time delay [[5,6].

As th he SLI is acccompanied byy a certain degree


of poopulation trannsfer, it can also a be regard rded as a possible tool forr manipulatinng coherent dynam mics, which include the coherent c conntrol of chem mical reaction ns [7-12] andd quantum inform mation proceessing particu ularly by usiing moleculees [13-20].

Figure 1

Potenntial-energy curves of X and B statees of I2 with h a schematicc illustrationn of the exciitation proceesses.

A weeak pump pulse initiallyy creates a wave w packet in the B staate, which iss then

moduulated by ann intense NIR R laser pulsee.

The inseet illustrates the excitatioon processess with

time delay,  d .

merically stud dy the SLI inn order to disccuss Our purrpose here is to theoreticaally and num the exxperimentallly observed quantum q beaats [5] in detaail, focusing on its fundam mental aspeccts. In Seection 2, we derive d analyttical expressiions that quaalitatively describe the m major featuress of the SLI beat signaals.

In Secttion 3, we seemi-quantitattively reproduce the expeerimental signals

4


wavee packet, resuulting in a qu uantum beat tthat appears in the population of eachh vibrational

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We discuss and summarize the results in


Sections 4 and 5, respectively.

2

Theory

We consider a two-electronic-state model that corresponds to the X and B electronic states of I2 molecules to describe the vibrational wave-packet dynamics.

This molecular

system interacts with a weak pump pulse, Epump (t ) , and an intense NIR laser pulse, E NIR (t   d ) , that appears with time delay,  d [5].

The former and latter pulses mainly

induce the electronic and stimulated Raman transitions, respectively.

In the present study, we

consider the situation in which the two pulses are linearly polarized and parallel to each other, and there is no temporal overlap between the two pulses.

The time-dependent Schrödinger

equation is expressed as

i

  (r ) E (t )    B (t )     B (t )   H B (t )     , H X (t )    X (t )   t   X (t )     (r ) E (t )

where  X (t ) respectively.

and  B (t )

(1)

are the vibrational wave packets in the X and B states,

The total electric field is given by E (t )  Epump (t )  ENIR (t   d ) .

transition moment function is  (r ) , with r being the inter-nuclear distance.

The

Taking into

account the lowest-order induced dipole interaction (polarization interaction), we adopt the following Hamiltonian of the N state (N=X, B):

5


with the help of a set of line search procedures.


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(2)

where H N0 , VN (t ) , and  N ( r ) are the field-free Hamiltonian, the polarization interaction, and the polarizability function, respectively.

Here, we neglect the contribution of the pump

pulse to the polarization interaction because it is much less intense than the NIR pulse.

In the

present study, we treat the energy shifts due to the polarization interactions by introducing the time-dependent zero-order Hamiltonian, H N0 (t ) .

It is defined by the following eigenvalue

equation with an eigenstate, vN , and a time-dependent eigenvalue,   N v (t ) :

1   H N0 (t ) v N   N v  v N  N (r )[ ENIR (t   d )]2 v N  v N  N v (t ) v N 2  

where   N v is the energy eigenvalue of H N0 .

,

(3)

It should be noted that the major contribution

from the Rayleigh scattering is included in the energy shift.

Because of the definition of

H N0 (t ) , the Raman transitions are induced by the interaction,

VN (t )  H N (t )  H N0 (t ) .

(4)

In the following analytical treatment, we assume that the electronic transitions and the Raman transitions are induced by the pump and NIR pulses, respectively, although no such assumption is imposed in the numerical analyses (Section 3).

When the molecule is initially in

the lowest vibrational state of the X state, 0  v X  0 , the eigenvalue of which is set to zero,

6



1 H N (t )  H N0  VN (t )  H N0   N (r ) [ ENIR (t   d )]2 , 2

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respect to   (r ) Epump (t ) :

i   B (t )   dt1 U B (t , t1 )  (r ) Epump (t1 ) 0  

.

(5)

Here, the time evolution operator, U N (t2 , t1 ) , (N=X, B), is, in general, defined by

 i t  U N (t2 , t1 )  T exp    2 ds H N ( s )    t1 

with the time-ordering operator, T .

(6)

We also introduce the zero-order time evolution operator,

U N0 (t2 , t1 ) , associated with the Hamiltonian, H N0 (t ) , in Eq. (3). It is expressed as the product of the field-free and energy-shift parts such that

0  i t  U N0 (t2 , t1 )  exp    2 ds H N0 ( s )   e iH N (t2 t1 )  ei N (t2  d , t1  d ) , t   1 

(7)

where

 N (t2 , t1 )   vN  N v (t2 , t1 ) vN

(8)

vN

with

7


the wave packet,  B (t ) , after the pulses can be approximated by the first-order solution with


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t2



ds [ ENIR ( s )]2 .

(9)

t1

 In Eq. (5), we then expand U B (t2 , t1 ) in a perturbation series about VB (t ) .

As we

consider the NIR pulse that appears after the pump pulse in the present study, the lowest-order  solution with respect to VB (t ) is given by

0 i  i   dt1 U B0 (t , t1 ) (r ) Epump (t1 ) 0  eiH B t  ei B (, ) B   

 B(0) (t )

.

(10)

Here, the initial excited wave packet, B , is defined by

B 





0

dt1 eiH B t1  Epump (t1 )  (r ) 0  Epump ( H B0  )  (r ) 0

.

(11)



with Epump ( H B0  ) being the Fourier-component operator of the pump pulse [10].

Note that

the wave packet,  B(0) (t ) , describes the time evolution including the energy shifts due to the intense NIR pulse non-perturbatively, although it will be referred to as a zero-order solution for convenience. Similarly, we have the first-order solution:

 B(1) (t ) 

1 

2





0

0

dt1 eiH B (t t1 )  ei  B (, t1  d ) VB (t1 ) eiH B t1  ei  B (t1  d , ) B



8

.

(12)



1  N v (t2 , t1 )  v N  N (r ) v N 2

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To obtain the SLI signals, we calculate the population of each vibrational eigenstate


and normalize it with respect to the corresponding population in the absence of the NIR pulse. It should be noted that the experimental SLI signal in Ref. [5] is normalized by its intensities averaged over the negative time delays of the NIR pulse; not by its intensity in the absence of the NIR pulse. We assume, however, that these experimental normalized signals could be directly compared with the theoretical ones in the present study. If we consider the v B -th populations

with 2

i vB B 

 vB  B(0) (t )

PBv ( d )

S Bv ( d ) 

and

PB(0) v

without 2

the

NIR

pulse,

PBv ( d )  vB  B (t )

2

and

 PB(0) v , respectively, the signal, S Bv ( d ) , is expressed as

 1  S B(1)v ( d )  S B(2) v ( d )  

,

(13)

where the superscripts represent the power with respect to the intensity of the NIR pulse.

As a

“minimal” model, we consider the lowest-order term that consists of the interference between

 B(0) (t )

and  B(1) (t ) , and is expressed as SB(1)v ( d )  2Re QBv ( d )  , where

QBv ( d ) 

i  2 v '(  v)

vB  B (r ) v 'B

v 'B B  I Bv, Bvʹ exp  i(Bv  Bvʹ ) d  vB B

(14)

with

IBv, Bvʹ 







dt1 exp  i Bv (t1 , )  i Bvʹ (t1 , )  i(Bv  Bvʹ )t1 [ ENIR (t1 )]2 .

9

(15)


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This interference term qualitatively captures the characteristics of the SLI signal.

For example,


we see from Eq. (14) that the beat frequencies correspond to the energy separations of the v B -th vibrational eigenstate and its neighboring states, in good agreement with the power spectrum of the beat signal in Fig. 3 of Ref. [5].

The visibility of the signals is proportional to the

intensity of the NIR pulse through the Fourier integral in Eq. (15).

As these features are

consistent with the experimental observations, we reach the conclusion that the dominant contribution to the beat signals is attributed to the interference between the eigenstates reached by Rayleigh scattering and that by Raman scattering.

As for the higher-order signals, a brief

discussion is given in Appendix A.

3

Numerical results

We numerically reproduce the SLI signals to confirm the mechanisms discussed in Section 2 and to examine their characteristics in detail.

The experimental signals are

accompanied by small oscillations/fluctuations, that may be originating from experimental noise. As this drawback imposes a limitation on the numerical analyses, we attempt to semi-quantitatively reproduce the signals. We take the RKR (Rydberg-Klein-Rees) parameters of the X- and B-state potentials from Refs. [21] and [22], respectively, and reconstruct the potential curves by the cubic spline interpolation (Fig. 1).

The range is set to [2.1 Å, 6.0 Å] , which is uniformly divided into 29

grid points. Then, we calculate the vibrational eigenvalues and eigenstates by using a standard diagonalization procedure.

By comparing the calculated eigenvalues with the experimental

values [21-23], we see that the difference in value is typically smaller than 0.1 cm-1 (not shown). 10


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(1) by using the second-order split-operator method in combination with the fast Fourier


transform (FFT), in which the time grid is set to 3.3 102 fs . The optical interaction between the X and B states is calculated by using Pauli matrices.

The overlap integral between the

wave packet and each vibrational eigenstate in the B state provides the population, which leads to the signal as a function of the time delay,  d , after the normalization defined in Eq. (13). In the present study, we restrict ourselves to the vibronic dynamics and ignore the rotational motion, which can be validated if we focus on few-ps dynamics.

The

angle-dependent parts of the optical interactions are replaced by averaged values instead of taking the statistical average.

Specifically, the operators, cos and cos 2  , are replaced

with their averaged values,  cos  and  cos 2   , respectively, where  is the angle between the polarization vector of the laser pulses and the molecular axis.

For example, the

transition dipole function in Eq. (1) is given by  (r )   BX (r )  cos  , where  BX (r ) is taken from Ref. [24], while the value of  cos  is unknown. roughly estimate the averaged values.

For reference, here we

From the experimental conditions, the rotational

temperature is estimated to be ~2 K [5, 25], which roughly leads to the average rotational quantum number J  5 .

(The rotational constant of I2 is Be  0.037 cm1 .)

If we further

assume that the absolute value of the average magnetic quantum number, M  J 2  2.5 , we would obtain  cos  0.45 and  cos2   0.40 .

These values will be chosen as the

initial values in the search procedure (Appendix B) as shown later. Similarly, the polarizability of the N-electronic state (N=X, B) is approximated by

 Nmol (r )   N (r )  ( cos2   1 3) N (r ) , where  N ( r ) and  N (r ) are the mean and anisotropic polarizability components, respectively. 11

As for the X-state polarizability,  Xmol ( r ) ,


Using these potentials, we numerically solve the time-dependent Schrödinger equation in Eq.


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 cos2   0.40 .

By assuming a static polarizability, Ref. [26] provides the parameters 4


associated with  X ( r ) and  X (r ) , in which we have  Xmol (r )    X(  ) (r  rXe ) as a  0

power series of the inter-nuclear coordinate around the equilibrium distance, rXe  2.66 Å . Similarly, we express the unknown B-state polarization function as a power series around

rBe  3.02 Å ,

4

 Bmol (r )    B(  ) (r  rBe ) ,

(16)

 0

where {  B( ) } are the parameters to be determined by a search procedure.

To remove the

unrealistic behavior of  Nmol ( r ) at a long distance, we introduce the atomic polarizability,

 Natom , which is connected with  Nmol (r ) using a switching function defined by

g N (r ) 

1 1  tanh[ N (r  rNsw )] . 2 2

(17)

Here, rNsw and  N specify the switching distance and the width of the interval over which the switching occurs, respectively.

The latter values are chosen as  X   B  1.8 Å 1 .

We then

express the polarizability function in Eq. (2) as

 N (r )  [1  g N (r )] Nmol (r )  2 g N (r ) Natom .

(18)

12


which plays a minor role in reproducing the experimental signals, we use a fixed value,

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As for the X state, we assume fixed values of  Xatom  32.9 au [23] and rXsw  3.96 Å , the


latter of which corresponds to twice the length of the van der Waals radius of the I atom.

On

the other hand, the parameters,  Batom and rBsw , are determined by a search procedure (Appendix B).

It should be noted that the fitted value of  Batom is a numerical artifact as

explained immediately above Eq. (17) and should not be regarded as an actual atomic excited-state polarizability. Next, we consider the pump and NIR pulses.

As shown in Eq. (11), all information

about the pumping processes is included in the initial excited wave packet through the Fourier-component operator of the pump pulse, Epump ( H B0  ) .

For the sake of concrete

description, we assume a Gaussian pump pulse in the present study

 t2  0 * Epump (t )  Epump exp   2  cos p t   pump (t )   pump (t ) .  2 p   

(19)

0 , plays a minor role in the simulation as long as we restrict ourselves to the The intensity, Epump

weak response regime.

The wavelength associated with the central frequency, p , is chosen

as a fixed value of 540 nm according to the experimental conditions [5].

On the other hand,

the pulse temporal width,  p , will be determined by the search procedure.

For the sake of

generality, we introduce a phase modulation,  () , as a function of frequency,  , whereby the Fourier-component operator of  pump (t ) is expressed as

13


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pump ( H B0

) 

 2

 2 p 0  p exp   Epump  

2   H B0  0  .      ( ) i H  p B   2    

(20)

If we further assume the rotating-wave approximation (RWA) and ignore the counter-rotating part, we have the initial excited wave packet

0

B  pump ( H B0  )  (r )  0  ei  ( H B

where BG

)

BG

,

(21)

represents the wave packet when there is no phase modulation in the pump pulse.

The modulation in Eq. (21) would introduce the phase shift,  (Bv ) , to each vibrational eigenstate involved in B .

As long as we employ the present model with two electronic

states in our search trials, we could not find a pump pulse transform-limited or linearly chirped that gave the NIR beat consistent with the experimental one for vB  25 , whose resonance is located at the edge of the pump-pulse spectrum; the computed beat was always phase-shifted by

 2 from the experimental one. We have, therefore, introduced a step-function to describe the phase modulation at the edge of the spectrum without modifying the present model, and that step-function is given by

 ( ) 



1 , 2 1  exp[ (  c )]

where  c and  respectively.

(22)

specify the switching frequency and width of the switching interval,

In the simulation, a fixed value of   2.2 ps is assumed, while  c is 14


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This search has given the phase shift only at the edge of the

pump-pulse spectrum, as will be shown later (Table 1).


We also assume that the NIR pulse is approximated by a Gaussian pulse

 (t   d )2  0 * ENIR (t   d )  ENIR exp   cos NIR (t   d )   NIR (t   d )   NIR (t   d ) , 2   2  

(23)

0 , and the temporal width,  , are determined by the search, while the where the intensity, ENIR

central frequency, NIR , is set to a fixed value corresponding to NIR  1.4 μm [5].

As

shown in Section 2, the intensity of the NIR pulse, which is well approximated by 2

 NIR (t   d ) , makes a dominant contribution to the beat signals. Thus, even if there are small phase modulations in the NIR pulse, they are expected to make a minor contribution to the beats. Because of this, we simply use the Gaussian NIR pulse in Eq. (23). To search the optimal values of the parameters, we introduce a function that evaluates the residual error between the numerically obtained signals, { SB(cal) v ( d ) }, and the experimentally observed signals, { SB(exp) v ( d ) },

F



v  25,27,29



1 f wv 

(exp) S B(cal) v ( d )  S Bv ( d ) d d ,

(24)

0

where 0 and  f specify the time interval during which the signals are evaluated.

In the

present numerical examples, we set 0 and  f to 300 fs and 2200 fs, respectively, and discretize the time interval,  d [ 0 ,  f ] , into 150 points. 15

The experimental signal curves are


determined by the search.


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The weights,


{ wv }, defined by

wv  SB(exp) v ( d )

max

 SB(exp) v ( d )

min

 d [ 0 ,  f ] ,

are introduced to normalize the beat amplitudes of the signals.

(25)

Summarizing the above

discussion, the function, F , depends on 11 parameters, including  cos  in the transition dipole interaction, {  B( ) (  0, , 4) } [Eq. (16)], rBsw [Eq. (17)],  Batom [Eq. (18)],  p [Eq. (19)],  c [Eq. (22)], and  [Eq. (23)].

When searching the temporal widths of the pulses,

we adjust the pulse intensities so that the total energies of the pulses are unchanged. For operational convenience, we also introduce a positive constant,  , that scales the magnitude of 0 ) 2 . It should be noted that the amplitude of the polarization interaction such that  B (r )( ENIR 0 0 the NIR pulse, ENIR , is not an independent parameter as it always appears as  cos  ENIR 0 (transition dipole interaction) and  B ( r )( ENIR ) 2 (polarization interaction).

The optimal

values are searched by using the procedure explained in Appendix B. The initial values of the parameters, which are chosen by our preliminary calculations as well as by rough estimation, give an initial guess value of F0  0.552 in Eq. (24).

In the

first search, we focus on the three parameters,  p ,  , and  , to roughly estimate more reasonable values.

After the three successive minimization steps, we have F1  0.893 F0 .

In

the second search, we consider the seven parameters associated with the polarizability and find an improved value, F2  0.975 F1 .

In the third search, we consider the five pulse parameters,

 p ,  ,  ,  cos  , and  c , which are not considered in the second search, and have 16


constructed by the cubic spline interpolation using original experimental data.

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As the change in value is so small, we stop the search here.

In fact, a further

search seems to simply focus on the errors that originate from the small oscillations/fluctuations


in the experimental signals without changing the major characteristics.

Table 1 summarizes

the optimal values and the initial guess values. Figure 2 shows the numerically obtained signals together with the experimental signals, in which the numerical results reproduce the oscillating periods and phases.

As the

numerical analyses assume the two-electronic-state model, it means that the Raman transitions within the B state are essential for the SLI beats. As for the v B  25 and v B  29 signals, both the numerical and experimental signals oscillate with almost constant (time-independent) amplitudes.

On the other hand, the visibility of the v B  27 signals increases with time

although there are some differences in the degree of visibility between the numerical and experimental signals.

For reference, we perform the third search by adopting another

residual-error function. This function has the same form as that in Eq. (24) but evaluates only the v B  27 signal while ignoring the v B  25 and v B  29 signals. shown in Fig. 2 by dashed lines.

The results are

Although we see a slight improvement in the visibility of the

v B  27 signal, a further search leads to a larger discrepancy in the other signals (not shown).

17


F3  0.976 F2 .


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0 0  Batom ( ENIR )2  1.48  103 au=1.21[ Batom ( ENIR )2 ]guess 0 0  B(0) ( ENIR )2  1.33  103 au=0.970 [ B(0) ( ENIR )2 ]guess 0 0  B(1) ( ENIR )2  2.18 104 au Å =0.882 [ B(1) ( ENIR )2 ]guess 0 0  B(2) ( ENIR )2  4.86 105 au Å2 =1.08 [ B(2) ( ENIR ) 2 ]guess 0 0  B(3) ( ENIR )2  2.52  105 au Å3 =0.794 [ B(3) ( ENIR )2 ]guess 0 0  B(4) ( ENIR )2  8.34  106 au Å 4 =0.794 [ B(4) ( ENIR )2 ]guess

0 0  cos  ENIR  8.57 104 au  0.978[  cos  ENIR ]guess

c  18380.1 cm 1  (c )guess  23.9 cm 1 2 ln 2 p  49.5 fs  0.990(2 ln 2 p )guess 2 ln 2  146 fs  1.46(2 ln 2 p )guess

18


Table 1 Optimal values of parameters that reproduce the experimental signals

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1.0

Signal intensity (a.u.)


0.9

vB=29

1.1 1.0 0.9

vB=27

1.1 1.0 0.9 0.8

vB=25

500

1000

1500

2000

d (fs)

Figure 2 Numerically calculated and experimental signals associated with the v B  25 , v B  27 , and v B  29 vibrational eigenstates as a function of the time delay,  d .

Red bold and blue

dashed lines represent the results when the residual errors [Eq. (24)] are evaluated for all the three signals and only for the v B  27 signal in the third search, respectively. Open circles and thin solid lines show the original experimental data and their splined curves, respectively [5].

4

Discussion

4.1 Quantum beat signals in Eq. (14)

Based on the analytical (Section 2) and numerical (Section 3) results, we further discuss the characteristics of the SLI beat signals by using the “minimal” model given by Eq. (14) in greater detail.

If the pump pulse is assumed to have a Gaussian profile with phase 19


1.1


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and (21)]

vB B 

  p2 0 Epump  p  Bv X 0 exp   Bv  p 2  2





where Bv X 0  vB  (r ) v X  0 . the matrix elements,



2

  i  (Bv )   ei  (Bv ) vB BG 

, (26)

Substituting Eq. (26) into Eq. (14), we have the product of

vB  B (r ) v 'B  Bvʹ X 0  Bv X 0 , which is an inherently molecular property.

As shown in Section 3, the linear and bilinear components of the polarizability in Eq. (16) make a dominant contribution to the Raman scattering, which means that in vB  B (r ) v 'B , the matrix elements with v 'B  v B  1 and v 'B  vB  2 are important.

In this case, the product

of the matrix elements, { vB  B (r ) v 'B  Bvʹ X 0  Bv X 0 }, always has a negative value.

To

prove this, we emphasize the following points: (i) the vibrational wave functions considered here have large probability amplitudes around the outer turning point, and (ii) the wave functions with similar quantum numbers, v 'B  v B  1 and v 'B  vB  2 , have similar structures.

When the element,

vB  B (r ) v 'B , has a positive value, the probability

amplitudes of the two wave functions have the same signs around the outer turning point. Because of the orthogonal condition, their amplitudes have opposite signs around the inner turning point, leading to the opposite sign between  Bv X 0 and  Bvʹ X 0 .

Thus, the product,

vB  B (r ) v 'B  Bvʹ X 0  Bv X 0 , has a negative value. Similarly, we can show the negative value of the product when the element,

vB  B (r ) v 'B , has a negative value. (We have also

numerically confirmed the negativity although we do not show the results here.) 20


modulations, the probability amplitude of the initial excited wave packet is given by [Eqs. (20)

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1.0



0 vB=29

-

-200

0

vB=27

200

t (fs)

vB=25

0.5

0.0

-200

-100

0

100

200

t (fs)

Figure 3 The phase shifts defined by Eq. (9) are plotted by black dotted ( v B  25 ), red solid ( v B  27 ), and blue dot-dashed ( v B  29 ) lines.

The thin solid line shows the envelope function of the

NIR pulse intensity. The inset shows the phases in the exponent of the integrand in Eq. (15), in which v B  27 and vʹB  26 are chosen as an example.

Red dashed and black dotted lines

show the phases with and without inclusion of the phase shifts, respectively, although we see virtually no difference between them.

We next consider the integral, IBv, Bvʹ , in the interference term, which is defined by Eq. (15). In the exponential part of the integrand, there are extra phases due to the energy shifts, {  B v (t ,  ) }, which are defined by Eq. (9).

Using the parameters in Table 1, we

calculate the phase shifts as a function of time and show some examples in Fig. 3.

In the inset,

we explicitly show one example of the time-dependent phases in the exponent with and without the energy shifts, in which we virtually see no difference between them.

In fact, the vibrational

quantum

is

number

dependence

of

the 21

phase

shift

quite

small,


1.5

phase [mod(2 ) ]

phase shift, Bv(t,-)


DOI: 10.1039/C3CP54023E


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phase-shift terms in the exponent cancel each other so that the integral reduces to a simple


Fourier integral.

As assumed in Eq. (23), when the NIR pulse is approximated by a Gaussian

pulse, the integral IBv, Bvʹ is real and positive. According to the above discussion, we have the following expression of the quantum beat signal within the lowest-order approximation with respect to the NIR pulse intensity,

S Bv ( d )  1 



v '(  v )

ABv, Bvʹ sin  (Bv  Bvʹ ) d   (Bv )   (Bvʹ )

(27)

with a real-valued coefficient

ABv , Bvʹ

1  v B  B ( r ) v 'B 

v 'B BG v B BG

IBv , Bvʹ .

(28)

This shows that the phase of the SLI beat signal can be controlled by the phase of the pump pulse, independent on the Raman-scattering and transition-moment elements. When there is no phase modulation in the pump pulse, Eq. (27) shows that the phase of each beat component is determined by the frequency differences between the vibrational state considered and its neighboring states. As schematically illustrated in Fig. 4, we thus consider the three typical cases of the quantum-beat patterns according to the energies of the vibrational eigenstates in the initial excited wave packet, B . Figure 4(a) [(c)], referred to as the high-energy (low-energy) regime, corresponds to the v=29 (v=25) signal, in which the

22


 B v (,  )   B v 1 (,  )  5  103 ~ 6  103 around vB  27 . Because of this, the two

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Figurre 4 Schem matic illustraation of threee typical caases of SLI, referred to as a (a) high-eenergy regim me, (b) midddle-energy reegime, and (c) low-enerrgy regime.

Solid and d dotted arroows illustratte the

v B  1 and v B  2 traansitions, resspectively.

interfference term is mainly atttributed to S Stokes (anti-Stokes) transsitions.

In tthe middle-eenergy

regim me [Fig. 4(b))] , which is associated w with the v=2 27 signal, th he beat is indduced by bo oth the Stokees and the anti-Stokes a trransitions.

For the sak ke of illustraation, if we consider only the

nsitions, as sh hown in Fig.. 5, the signaals are fitted by b v 'B  v B  1 tran

S B(fit) 0.07sin  (B 29  B 28 ) d  , v= 229 ( d )  1  0

(29)

S B(fit) 6  0.12sin  (B 27  B 228 ) d   sin  (B 27  B26 2 ) d  v= 227 ( d )  0.96  1  1  0.966  0.24sin  (2B 27   B 28  B 26 ) d  cos  (B 28  B 26 ) d   2  2

(30)

and

  S B(fit) 4  0.12sin  ( B 25   B 226 ) d   , v= 225 ( d )  0.94 2  23

(31)



DOI: 10.1039/C3CP54023E


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 2 -phase shift in Eq. (31), which is introduced by the parameter, c , in Table 1. We see It confirms that

the  -phase shifts between the Stokes and anti-Stokes Raman transitions exist, as explained in Eq. (27).

Figure 5 also explains why the visibility of the v=27 signal increases with time.

According to Eq. (30), the beat originates from the sum of the two sine functions with opposite phases, in which the anharmonicity of the potential determines the shape of the signal envelope.

1.1 1.0 0.9

Signal intensity (a.u.)


from Fig. 5 that such simplified expressions well fit the experimental signals.

vB=29

1.1 1.0 0.9

vB=27

1.1 1.0 0.9 0.8

vB=25

0

500

1000

1500

2000

2500

d (fs)

Figure 5 Experimental (splined) signals and analytically derived expressions given in Eqs. (29), (30), and (31) [also see Eq. (27)] are shown by thin black and bold red lines, respectively.

4.2 Difference/similarity between SLI beat signals and those observed in a “standard” pump-probe experiment 24


respectively, where we have slightly adjusted the baselines in Eqs. (30) and (31) and added the

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Eprobe (t ) , that induces the transitions to a higher electronic state, H F0 , through the electric dipole interaction, FB (r)Eprobe (t  d ) , with d being the time delay. As the discussion below will not be affected by the choice of the final electronic state, we do not specify it explicitly.

The signal is defined by the population of the final state,

0 vibrational eigenstate of H F with an eigenvalue   F f .

simplicity.)

f F , which is a

(We assume a single final state for

If the temporal width of the probe pulse is sufficiently short, it cannot

energetically distinguish the transitions from the vibrational eigenstates in the B-state wave packet to the final state,

f F . Because of this uncertainty, there appears a quantum beat in

the population of the final state as a function of the time delay, d .

From the second-order

perturbation, we obtain the wave packet in the F-electronic state after the pulses,

i

 F (t )     

2 



0

0

dt1 eiH F (t t1 )  FB (r ) Eprobe (t1   d ) eiH B t1



B

,

(32)



where we have assumed that there is no temporal overlap between the pump and probe pulses. As for the pump and probe pulses, we make similar assumptions to those introduced when deriving Eq. (27).

Specifically, both the pump and probe pulses have Gaussian profiles, while

only the pump pulse is accompanied by phase modulations.

Except for unimportant constants,

the signal, S (ref ) ( d ) , as a function of the time delay,  d , is expressed as

S (ref ) ( d )   (Ff , Bv )2 vB B

2

2 ) Eprobe (Ff  Bv ) SB(ref v ( d ) ,

vB

25

(33)


In a “standard” pump-probe experiment [1], we typically use a weak probe pulse,


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where

) SB(ref v ( d )  1 



vʹB ( vB )

) AB(ref v, Bvʹ cos  (Bv  Bvʹ ) d  (Bv )  (Bvʹ ) .

(34)

) Here, AB(ref v, Bvʹ is a real-valued coefficient without a definite sign, which is given by

) AB(ref v, Bvʹ 

G  Ff , Bvʹ v 'B B Eprobe ( Ff   Bvʹ )

 Ff , Bv

v B BG

Eprobe ( Ff   Bv )

,

(35)

where  Ff , Bv  f F  FB (r ) vB , and Eprobe (Ff  Bv ) represents the Fourier component of the probe pulse. Although Eqs. (27) and (34) may look similar to each other at first glance, there is a considerable difference between them.

In standard quantum beats, all the frequency

differences are measured simultaneously as long as they can be excited and probed by the pulses. In the SLI beat associated with the v B -th eigenstate, on the other hand, the pairs of frequency differences, {Bv  Bvʹ } , are dominant, in which one of the frequencies is fixed by  Bv .

In

addition, we have the following relation for the sum of the SLI signals, provided that the NIR pulse does not induce the B-X transitions, i.e.,  B(0) (t )  B(0) (t )   B (t )  B (t ) ;



S B( nv) ( d ) vB  B(0) (t )

2

 n 0 ,

(36)

vB

26


DOI: 10.1039/C3CP54023E

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Comparing Eq. (27) with Eq. (34), we notice that there is a  2 -phase shift between


) them except for the signs of the coefficients, { AB(ref v, Bvʹ }.

In the standard experiment, the probe

pulse induces the transitions from the B-state to the “F-state” when the wave packet approaches the Condon point that is specified by the probe frequency. detected as the population of the final state.

The transferred components are

On the other hand, the SLI beats originate from

the interference between the eigenstate reached by Rayleigh scattering and that by Raman scattering.

As shown in Eq. (14), the former (latter) is given by the zero-order (first-order)

 solution with respect to VB (t ) in the lowest-order approximation.

The difference in the

order of the interaction leads to the difference in the phase by  2 because each interaction introduces an extra phase through the imaginary number, i . such as PB(1)v  vB  B(1) (t )

2

In fact, the higher-order term,

, which partly contributes to S B(2) v ( d ) [see Eq. (13)], is expressed

as the sum of cosine functions and includes the pairs of frequency differences {Bvʹ  Bvʺ } without  Bv (not shown). Finally, we consider a special case wherein the pump pulse initially prepares an eigenstate instead of a superposition state.

It is apparent that a standard pump-probe

experiment does not show a quantum beat because the eigenstate is spatially constant in time. We also see that there is no quantum beat in the SLI signals although the NIR pulse generates a wave packet.

This is because the final distribution of the vibrational eigenstates does not

depend on the time delay of the NIR pulse.

5

Conclusions

27


where the superscript, n, represents the power of the NIR pulse intensity.


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interference (SLI) assuming a two-electronic-state model of I2. In the theoretical analyses, the


B-state wave packet,  B (t ) , which is initially prepared by a pump pulse, is expanded in a

perturbation series with respect to the polarization interaction associated with the Raman scattering, while the energy shifts are treated non-perturbatively.

We proposed a “minimal”

model that includes the lowest-order interference term and showed that it qualitatively explains the major characteristics of the experimental signals.

By numerically solving the

time-dependent Schrödinger equation, we have semi-quantitatively reproduced the experimental signals, in which the values of unknown parameters are determined by a set of line search procedures.

As a consequence of the theoretical and numerical results, we have shown that the

SLI beat signal of the v B -th eigenstate is approximated by the sum of the sine functions, the arguments of which are the frequency differences between the v B -th state and its neighboring states.

This also explains the  -phase difference between the Stokes and anti-Stokes

transitions, resulting in the different time-evolution patterns of the signal visibilities.

The

amplitude of each quantum beat is dominated by the intensity of the frequency component of the NIR pulse associated with the Raman scattering.

On the other hand, as the energy shifts

induced by the NIR pulse weakly depend on the vibrational quantum number, they make virtually no contribution to the beat signals. In the present study, we have focused on the fundamental aspects of the SLI signals measured in the wave packet in the B state of I2.

The next step would be to study the

possibility of using shaped NIR pulses as a tool for actively manipulating a wave packet. This would be the challenge to be taken up in molecular eigenstate-based engineering.

28


We have theoretically/numerically studied the mechanisms of the strong-laser-induced

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Y.O. acknowledges stimulating discussions with Professor T. Nakajima.

This work was partly


supported by a Grant-in-Aid for Scientific Research (C) (23550004) (YO).

This work was

partly supported by Photon-Frontier-Consortium Project by MEXT of Japan (KO).

Appendix A: Higher-order solutions of Eq. (5)

In the same manner as the derivation of Eqs. (10) and (12), we obtain the second-order solution

3 

 i

 B(2) (t )       

t2

0

e

iH B0 (t2 t1 )





dt1 eiH B (t t2 )  ei B (, t2  d ) VB (t2 )



dt2

(A1)

  i  B (t2  d , t1  d )

e

VB (t1 ) e

iH B0 t1

 i  B (t1  d , )

e

B .

After minor algebra, we have the probability amplitude of the v B -th eigenstate,



3

i vB  B(2) (t )    ei Bv (t  d )  dt2    



v '(  v)





vʺ(  vʹ )

vB

 B (r )

vʹB

2



dt1 ei Bv t2 i  Bv (, t2 )



v 'B [ ENIR (t2 )]2 ei Bvʹ (t2 t1 )i  Bvʹ (t2 , t1 )

 B (r ) 2

t2

(A2)

vʺB [ ENIR (t1 )]2 ei Bvʺ (t1  d )i  Bvʺ (t1 , ) vʺB B .

We see from Eq. (A2) that the time-delay-dependent terms are expressed as an oscillating function of { Bv   Bvʺ } and correspond to the energy difference of the v B -th eigenstate and its neighboring states. The derivation of Eq. (A2) can be generalized to obtain the nth-order solution, which 29


Acknowledgments


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i vB  B( n) (t )     

n1

ei Bv (t  d )





tn

dtn







t2

dtn1   dt1 ei Bv tn i  Bv (, tn ) 

  



vʺ (  vʹ )

(A3)

vʹB

 B (r ) 2

vʺB [ ENIR (t1 )]2 ei Bvʺ (t1  d )i  Bvʺ (t1 , ) vʺB B .

As the quantum beat signals associated with the interference between the Rayleigh and Raman scattering are dominated by the products, {  B(0) (t ) vB vB  B( n) (t ) }, the beat frequencies, { Bv   Bvʺ }, always include  Bv of the v B -th eigenstate.

On the other hand, the other

interference terms originating from combined Raman transitions can include all possible frequency differences, {  Bvʹ   Bvʺ }, provided that one of the frequencies corresponds to the energy of the vibrational eigenstate involved in the initial excited wave packet. Equation (A3) explicitly shows the trivial result that we do not observe quantum beats when the pump pulse initially prepares an eigenstate, i.e., B  v0B .

In this special case, it

is easy to show that Eq. (A3) reduces to

vB  B( n) (t )  e

i ( Bv Bv0 ) d

 ( d -independent terms) ,

(A4)

for an arbitrary order, n, where Bv0 is the energy eigenvalue of v0B . It means that the population of an arbitrary eigenstate,

vB  B (t )

2

, is independent of the time delay as long as

the NIR pulse does not induce electronic transitions.

30



gives the following probability amplitude,

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Appendix B: Algorithm for multidimensional minimization with a set of line searches


We consider a function of k parameters, F ( X ,Y ,, Z ) , the derivatives of which are not available.

In the present search procedure, starting from a set of initial-guess values,

{ X (0) , Y (0) ,, Z (0) } , we optimize their scale factors. We introduce discretized scale factors, (1) for example, {xn (n  0,1,, N x )} , associated with X and search an optimal value, X opt , that

minimizes F .

Using this notation, our procedure is summarized as follows:

1st step: (1) (1)  min{ F ( xn X (0) , Y (0) ,, Z (0) ) : n  0,1,N x } . Find an optimal value, X opt , from Fmin (2) (1) (1) , from Fmin  min{ F ( X opt , ynY (0) ,, Z (0) ) : n  0,1, N y } . Find an optimal value, Yopt

 (k ) (1) (1) (1) , from Fmin  min{ F ( X opt , Yopt ,, zn Z (0) ) : n  0,1, N z } . Find an optimal value, Z opt

2nd step: ( k 1) (1) (1) (1) (2) , from Fmin  min{ F ( xn X opt , Yopt ,, Zopt ) : n  0,1, N x } . Find an optimal value, X opt ( k  2) (2) (1) (1) (2) Find an optimal value, Yopt , from Fmin  min{ F ( X opt , ynYopt ,, Z opt ) : n  0,1, N y } .

 (2 k ) (2) (2) (1) (2) , from Fmin  min{ F ( X opt , Yopt ,, zn Zopt ) : n  0,1, N z } . Find an optimal value, Z opt

3rd step:

 In the present study, we use

x0  y0    0.90 ,

x1  y1    1.0 , and

x2  y2    1.1 ( N x  N y    2 ) except when searching the two parameters,  cos 

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In the case of  cos  , the three scale factors are chosen as 0.95, 1.00, and 1.05.


When searching c  c  c0 ( c0 is an initial-guess value), we simply divide the range

[0.4, 0.4 ] with    Bv  26   Bv  25  79.5 cm 1 into nine points and calculate the function, F , at each point to find the optimal   c .

References

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and  c .

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