University of Groningen

Ultrafast solvation dynamics explored by nonlinear optical spectroscopyde Boeij, Wilhelmus Petrus ; Wiersma, D. A.

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217

Chapter 7 Liouville-space pathways interference in transient four-wave mixing and six-wave mixing spectroscopy

Part of the work as presented in this chapter is covered by the following papers:

Maxim S. Pshenichnikov, Wim P. de Boeij, and Douwe A. Wiersma, ‘Coherent control over Liouville-space pathways interference in transient four-wave mixing spectroscopy’, Phys. Rev. Lett. 76, 4701(1996).

Maxim S. Pshenichnikov, Wim P. de Boeij, and Douwe A. Wiersma, ’Phase-locked stimulated photonecho: Coherent control over Liouville-space pathways’, in ‘Femtochemistry, Ultrafast Chemical andPhysical Processes in Molecular systems’, edited by M. Chergui, (World Scientific, Singapore, 1996),p.501.

Maxim S. Pshenichnikov, Wim P. de Boeij, and Douwe A. Wiersma, ‘Coherent control in Ultrafastphoton echo spectroscopy: Liouville-space pathways interference’, Springer Series in ChemicalPhysics, Vol. 62, Ultrafast Phenomena X, Edited by P.F. Barbara, J.G. Fujimoto, W.H. Knox, and W.Zinth, (Springer Verlag, Berlin, 1996) pp. 215-216.

Chapter 7

218

Abstract

A novel interference effect in transient four-wave mixing is demonstrated. Thephenomenon is based on phase-controlled Liouville-space pathways interference andobserved in the heterodyne-detected stimulated photon echo. The theoretical third-orderperturbative treatment of the heterodyne detected photon echo is presented, and expressionsfor the HSPE signals are derived. In a similar treatment, the results for the fifth-order signalsare given. HSPE experiments are reported on the infrared dye molecule DTTCI in the solventethylene glycol. For the low-intensity HSPE results it is demonstrated that interference effectsoccur between separate contributions, echo and virtual echo. With the ability to change thephase relations between the excitation pulses, a precise control over this interference can beexerted. Changing the phase-difference between the first two excitation pulses from B/2 to0 leads from no signal to maximum echo signal. Experiments using higher excitation energies,reveal the existence of fifth-order scattering processes leading to additional interference effectbetween the third-order signals and fifth-order signals. A Brownian oscillator dynamicalmodel is successfully used to analyse the effect and simulate the experimental data.

E1, ω1

E3, ω3E2, ω2

E4, ω4

E1 E2∗

E3

E1

E2∗

E3

|a⟩⟨a|

|b⟩

|a⟩

|d⟩

|c⟩

|c⟩⟨c|

|a⟩⟨a|

|c⟩⟨c|(a) (b)

Liouville-Pathways Interference

219

Fig. 7.1. (a) Energy level scheme and electric fields interactions for the coherent Raman scattering(PIER-4) experiment. E -E are the incident electric fields, and E denotes the signal field. (b) Relevant1 3 4

interfering Feynman diagrams for the PIER-4 experiment.

7.1 Introduction

One of the most remarkable phenomena in cw four-wave mixing spectroscopy concernsthe effect of dephasing on an initially unpopulated excited state resonance. In this[1-3]

experiment the coherent Raman signal is monitored as a function of frequency differencebetween the exciting light fields. The relevant energy level scheme for this experiment isdepicted in Figure 7.1a. When the frequency difference equals an excited-state splitting, onemight expect intuitively the Raman signal to reflect the combined resonance. However, thisturns out not to be the case, unless the system is perturbed by collisions, which inducedephasing on the optical transitions. The effect, denoted as PIER-4 (pressure-induced extra-resonance in four-wave mixing) was first reported by Bloembergen and co-workers onsodium in the gas phase and later also demonstrated for condensed phase molecular systems[1]

(DICE, dephasing-induced coherent emission) by Hochstrasser and co-workers.[2]

Dephasing-induced resonances can be grasped most easily in terms of interferencebetween different Liouville-space pathways that describe the generation of the relevant third-order optical polarization responsible for the four-wave mixing signal. For the PIER-4 case,the relevant diagrams are depicted in Figure 7.1b. It can be shown that, in the absence ofcollisions, the different excitation pathways interfere destructively with one another and thatno net polarization is generated. When, with an increase in pressure or temperature,[1]

dephasing effects become important, the Liouville-space pathways are no longer equivalentand as a result a finite nonlinear optical polarization is generated.

Spectroscopic effects due to interference between different excitation pathways are alsoknown for other frequency domain non-linear optical spectroscopic techniques. For[4,5,6,7]

instance the disappearance of a signal in multi-photon ionization and in spontaneous[5]

emission, as well as in one- or three photon absorption have been demonstrated. In[6] [5,7]

Figure 7.2(a) and (b) two examples of interfering pathways are depicted. The intriguing

(a) (b) (c) (d)

SE SVE S(3) S(5)

|1⟩

|2⟩

|1⟩

|2⟩|3⟩

|1⟩

|2⟩ |4⟩

|1⟩

|2⟩

|3⟩|4⟩

S(1) S(3)S-(2) S+

(2)

Chapter 7

220

Fig. 7.2. Liouville pathway interferences for different coherent processes. (a) Parametric second-orderprocesses S and S . (b) One photon absorption S and three photon absorption S . (c) Conventional- +

(2) (2) (1) (3)

echo S and virtual echo S . (d) Third-order (four photon) S and fifth-order (six photon) S processes.E VE(3) (5)

The observable in the experiment (a) and (b) is the final population *2,+2*, probed for instance bymonitoring the emission or multi-photon ionization yield. In the experiments (c) and (d), the observableis the induced polarization (*1,+2* and *2,+1*).

feature of the effect is that the system in spite of being driven in a non-linear fashion does notrespond to the external fields under certain conditions. The light travels unimpeded throughthe medium as a result of the specific interaction with the optical dipole. That is why thisprocess is often referred to as a ‘bleaching’ effect. [4]

Interestingly enough, when the coherent Raman experiment (PIER-4) is performed in thetime-domain using two time-coincident short optical pulses of different colour - shortcompared to the inverse transition line width - the excited state resonance is always found,irrespective of whether the system experiences pure dephasing or not. The essential[8,9]

difference with the cw-case is that for pulsed excitation the system has no time to evolveappreciably within the pulse duration. Relaxation towards the steady state is thus essential[10]

for destructive interference effects in cw coherent Raman spectroscopy. In view of this factit seems unlikely that dephasing-induced interference effects play a role in time-domain four-wave mixing spectroscopy.

In this chapter we report on the observation of a novel dephasing-induced interferenceeffect in heterodyne detected phase-locked stimulated photon echo (HSPE). The presence[11-16]

of multiple pathways in the HSPE experiment allows for the possibility to exert interferenceeffects. Higher excitation energies can result in the generation of higher order (fifth)[14-16]

effects. Four photon versus four photon interferences (Fig. 7.2(c)) and four photon versus[14,15]

six photon interferences (Fig. 7.2(d)) will be demonstrated on a dye solution at roomtemperature. For such a non-Markovian dynamical system, Liouville-pathway interferenceeffects are important in generating the nonlinear polarization. In Section 7.2 the third-orderheterodyne-detected stimulated photon echo experiment is reviewed, followed by a discussionon the fifth-order terms. In the numerical Section 7.3 the fifth-order signals are calculated fora model system in the spectral diffusion limit. For this system, exact expressions can be

P (3)(t) ' iS

3

m4

0

dt3m4

0

dt2m4

0

dt1 µ40 R(3)(t3,t2,t1) E3(t&t3) E2(t&t2&t3) E1(t&t1&t2&t3)

E1*

Es*

E3 E2

E2*

Es*

E1

E3

E2*

Es*

E3

E1

RIVRIIIRI

E1*

Es*

E2

E3

RII

SHSPE( t34,t23,t12)%Re RA( t34,t23,t12) cos[N12&N34]% Im RA( t34,t23,t12) sin[N12&N34]%Re RB( t34,t23,t12) cos[N12%N34]& Im RB(t34,t23,t12) sin[N12%N34]

Liouville-Pathways Interference

221

(7.1)

Fig. 7.3. Double sided Feynman diagrams describing the evolution of the reduced density matrixelements. The vertices indicate interactions with the electric fields. Aside the vertices the correspondingelectric fields are depicted.

(7.2)

derived for third and fifth-order non-linear signals. After a short discussion of theexperimental setup, in Section 7.4 the results are presented on the interference among thedifferent third-order signals as well as between the third-order and fifth-order contributions.Constructive and fully destructive interferences will be demonstrated. Higher order signalsin the phase-locked pump-probe (PLPP) experiment will be reported. The chapter isconcluded in Section 7.5.

7.2 Theoretical Background

Application of the perturbative expansion results in the following expression for thethird-order nonlinear polarization: [13]

In Eq.(7.1), R is the third-order nonlinear response function, and µ denotes the transition(3)0

dipole strength. The relevant Liouville space pathways for the HSPE experiment representingthe time-domain evolution of the density matrix in combination with the perturbing electricfields can be graphically represented by the double sided Feynman diagrams and are givenin Figure 7.3. Following the treatment presented in Chapter 4 (Eq.(4.7) to Eq.(4.15)), we findthe following result for the HSPE signal:

where the conventional echo signal is governed by R and is equal to R =R +R , and R[17]A A II III B

describing the virtual echo is given by R =R +R . Eq.(7.2) represents the most general[18,19]B I IV

case for the phase-settings N and N . The flexibility to adjust the phases of the incident12 34

P (5)(t) ' i£

5

m4

0

dt5m4

0

dt4m4

0

dt3m4

0

dt2m4

0

dt1 µ60 R(5)(t5,t4,t3,t2,t1) E5(t&t5)E4(t&t4&t5) ×

E3(t&t3&t4&t5)E2(t&t2&t3&t4&t5)E1(t&t1&t2&t3&t4&t5)

R(5)(t5,t4,t3,t2,t1) ' j,1,,2,,3,,4

R(5),1,2,3,4

(t5,t4,t3,t2,t1)

R(5),1,2,3,4

(t5,t4,t3,t2,t1)'exp(Q (5),1,2,3,4

(t5,t4,t3,t2,t1))

Q (5),1,2,3,4

(t5,t4,t3,t2,t1)'&iTeg(,1t1%,3t3%t5)&g&,1

(t1)&g,2,3

(t3)&g,4%

(t5)

&,1,3[g,1,2(t2)&g

,1,2(t2%t3)&g

&,1(t1%t2)%g

&,1(t1%t2%t3)]

&,1[g,1,2(t2%t3%t4)&g

,1,2(t2%t3%t4%t5)&g

&,1(t1%t2%t3%t4)%g

&,1(t1%t2%t3%t4%t5)]

&,3[g,3,4(t4)&g

,3,4(t4%t5)&g

,2,3(t3%t4)%g

,2,3(t3%t4%t5)]

g&&

(t) ' g%%

(t) ' g(

!%(t) ' g(

%!(t) ' g(t).

Chapter 7

222

(7.3)

(7.4)

(7.5)

(7.6)

(7.7)

pulses results in the possibility to exhibit control over phase of the interfering contributions(R and R ). A condition of destructive interference (the phases of the individual contributionsA B

are opposite) or constructive interference (the contributions are in phase) can be controlledby the experimentalist.

When increasing the excitation energies, fifth-order contributions start to contribute tothe final signal. Recently there has been a tremendous interest in higher order techniques.[20-22]

For instance Fleming et al. have proposed and demonstrated that fifth-order non-linearexperiments can be used to separate the different contributions present in the systemdynamics. Furthermore, several fifth-order (Kerr-like) experiments have been performed[20]

to disentangle nuclear motions in solvent. Seventh-order experiment in the form of[21,22]

Raman-echoes were also utilized to attack similar problems. [23]

In this section, the fifth-order effects are addressed that are encountered in the HSPEexperiment. Extending the perturbative expansion up to the fifth-order term gives: [24]

Where the fifth-order response function is given by the sum over all possible pathways as:[24]

The the indices , label the interactions in the evolution through the Liouville space (,=+1i i

for *e,+e* or *e,+g*, and ,=-1 for *g,+g* and *g,+e*). Figure 7.4 lists all 16 possible Liouvillei

pathways one has to consider in the HSPE experiment. The fifth-order nonlinear responsefunctions are given by:

where Q is related to the lineshape function g (t) as follows:±±

The lineshape functions g (t) are defined as:±±

R(5)' R(5)

A (1,1,1,2,3)% R(5)

B (1,1,2,2,3)% R(5)

C (1,2,2,2,3)%

R(5)D (1,2,3,3,3)

% R(5)E (1,2,3,3,4)

% R(5)F (1,2,3,4,4)

S(5)HSPE( t34,t23,t12)% j

,1,2,3,4

R(5)A (1,1,1,2,3),1,2,3,4

( t34,t23,t12) exp[i,3N12%iN34]%

R(5)B (1,1,2,2,3),1,2,3,4

( t34,t23,t12) exp[iN34]%

R(5)C (1,2,2,2,3),1,2,3,4

( t34,t23,t12) exp[i,1N12%iN34]%

R(5)D (1,2,3,3,4),1,2,3,4

( t34,t23,t12) exp[i,1N12%iN34]%

R(5)E (1,2,3,4,4),1,2,3,4

( t34,t23,t12) exp[i,1N12]%

R(5)F (1,1,1,2,3),1,2,3,4

( t34,t23,t12) exp[i,1N12%i,3N34]

Liouville-Pathways Interference

223

(7.8)

Fig. 7.4. Double sided Feynman diagrams describing the evolution of the reduced density matrixelements up to fifth-order. The vertices indicate interactions with the electric fields.

(7.9)

As a result of the degeneracy in the applied excitation pulses (four pulses govern fiveinteractions) each of the 16 relevant fifth-order Liouville pathways (Fig. 7.4) can beconfigured in six different ways, resulting in six response functions (labelled A to F):

where the interaction order of the pulses is denoted by the sequence of numbers (betweenbrackets), representing the electric fields of the interactions along the system evolution. Thesixfold permutation in the pulses, and the 16 separate Liouville pathways leads to 6×16=96pathways. Note that each of these pathways shows a specific dependence on the initial relativephase-settings of the incident electric fields. Summation over all possible pathways gives thefollowing expression for the fifth-order signal:

In Appendix 7A, a full listing of the possible permutation of pulse interactions as well as thecomplete expressions for the individual response functions R to R (via Q as in Eq.(7.5))A F

(5)

is given. In the remaining part of this section we address the phase-locked pump-probe (PLPP)

experiment. In this experiment the fourth pulse coincides with the third pulse and can[11,12,25]

S(3)PLPP(t12) % exp[&gRe(t12)] cos[gIm(t12)&N12]

S(5)PLPP(t12) % &1 & 5×exp[&gRe(t12)] cos[gIm(t12)&N12]

S(5)tot (t34,t23,t12) %

&32 exp(&½)2t 212&½)2t 2

34%)2t12t34M(t23))×

cos(8t34(1&M(t23)))×cos(8 t34(1&M(t23))&N12%N34)%

&32 exp(&½)2t 212&½)2t 2

34&)2t12t34M(t23))×

cos(8t34(1&M(t23)))×cos(8 t34(1&M(t23))%N12%N34)%

&16 exp(&½)2t 234)×

cos(8t34(1&M(t23))) cos(8t34(1&M(t23))%N34)%

&16 exp(&½)2t 212)×cos(N12)

Chapter 7

224

(7.10)

(7.11)

(7.12)

thus be regarded as a degenerate HSPE experiment. For the third-order phase-locked pump-probe case (t =0, N =0) we find the following expression for the signal (Eq.(4.26)): 34 34

where g (t) and g (t) denote the real and imaginary part of the lineshape function g(t). UponRe Im

the substitution of the conditions t =0 and N =0 in Eq.(7.9) and the equations for the six34 34

different response functions (Appendix 7A), we find:

The fifth-order PLPP signal consists of a negative baseline and a signal similar (but oppositein phase (factor -1)) to the third-order response.

7.3 Numerical calculations

In the numerical modelling of the system dynamics, we follow the procedure as waspresented in Section 4.3.2, where the third-order HSPE signal is calculated in the spectraldiffusion limit. The signal expression was already given in Eq.(4.33), and the signal shapesand dependence on the initial phase settings (N and N ) were given in Figure 4.3. Similar12 34

as for the third-order approach, this model system can be used to predict the signal shapes forthe fifth-order contributions. Upon the substitutions of the relations Eq.(4.28) and Eq.(4.29)in the fifth-order signal expression given by Eq.(7.9) we find:

Four different terms can be distinguished in the final expression for the fifth-order signal(Eq.(7.12)). The first and the second term have the same mathematical form as the result

-50 -25 0 25 50

t34 [fs]

-0.5

0.0

-0.5

0.0

0

1000

2000

t23 [fs]

φ 12 = 0; φ 34 = 0

-50 -25 0 25 50

t34 [fs]

-0.2

-0.1

0.0

-0.2

-0.1

0.0

0

1000

2000

t23 [fs]

φ 12 = π/2; φ 34 = 0(a) (b)

-50 -25 0 25 50

t34 [fs]

-0.1

0.0

0.1

-0.1

0.0

0.1

0

1000

2000

t23 [fs]

φ 12 = 0; φ 34 = π/2(d)(c)

-50 -25 0 25 50

t34 [fs]

-0.4

-0.2

0.0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

0

1000

2000

t23 [fs]

φ 12 = π/2; φ 34 = π/2

-50 -25 0 25 50

t34 [fs]

-0.2

-0.1

0.0

-0.2

-0.1

0.0

0

1000

2000

t23 [fs]

φ 34 = 0(f)(e)

-50 -25 0 25 50

t34 [fs]

-0.1

0.0

0.1

-0.1

0.0

0.1

0

1000

2000

t23 [fs]

φ 34 = π/2

Liouville-Pathways Interference

225

Fig. 7.5. Numerical calculation for the heterodyne detected fifth-order non-linear signals measured underfour specific phase settings: a): N =0, N =0, b): N =B/2, N =0, c): N =0, N =B/2, d): N =B/2,12 34 12 34 12 34 12

N =B/2. The system coupling parameter ) is given by )= 60 THz. The correlation function is given34

by: M(t) = exp(-7t) where 7=1 THz. Panel (e) and (f) depict the separate contribution from thepopulation term (third term in Eq. 7.12)) for phase settings N =0 and N =B/2, respectively.34 34

Chapter 7

226

obtained for the third-order signal (Eq.(4.33)). The first term presents the echo-likebehaviour, whereas the second term in the expression is identical to the third-order virtualecho response. The third term depends only on the separation between the third and the fourthpulse (t ). During the time interval t the system finds itself in a diagonal population state.34 12

With the assumption of a population decay time much larger than any experimental delay, nodependence on the time t is observed. The fourth term in the expression only depends on the12

delay between the first and the second pulse (t ). During this time an off-diagonal density12

matrix element is encountered. Both during the times t and t , the system is in a population23 34

state. Figure 7.5 (panels (a),(b),(c),(d)) shows the result of the calculations for the fifth-order

signals from the model system, with parameters identical to the third-order case (Sect. 4.3.2).The panels (e) and (f) in Figure 7.5 show the contributions that are attributed to the term thathas an evolution through the diagonal population state during the time t . (third term in12

Eq.(7.12)) The contributions leading to a baseline in the signal are not included in thecalculations (fourth term in Eq.(7.12)). Note that the fifth-order signals have a sign oppositeto the third-order result. Within the applied model, the fifth-order signals as depicted inFigure 7.5 are simply the sum of the previously calculated third-order results (figure 4.3, anddiscussion in Section 4.3.2) and the contributions arising from population terms (Figure7.5(e) and 7.5(f)). This simple relation results from the fact that a degenerate pulse sequenceis applied in the experiment is in fact a degenerate experiment in fifth order. Several delaysare intrinsically zero, which results in a significant simplification in the fifth-order signalexpression. Given the relative amplitudes of the contributing terms, one can conclude that thecontribution arising from the population term presents a significant part of the total signal.Note that this population term only depends on the phase setting N . 34

7.4 Results

The interferences in the heterodyne detection of the third-order as well as the fifth-ordersignals will be demonstrated on the signals from the infrared dye molecule DTTCI, dissolvedin ethylene glycol. Heterodyne detection of the photon echo in the low intensity limit (third-order perturbative terms) as well as in the high intensity limit (inclusion of the fifth-orderprocess) will be reported. Along with the HSPE results, the fifth-order effects in the phase-locked pump-probe signals will be presented. For a description of the experimental setup, thereader is referred to section 4.4. To induce the fifth-order effects, higher pulse energies areemployed.

S( φ

12

= 0

, φ3

4 =

0)

[arb

.uni

ts] 0.0

1.0

1.0

0.0

1.0

0.0

1.0

0.0

S( φ

12 = π

/2, φ3

4 = 0) [arb.units]

0.5

0.0

0.5

0.0

0.5

0.0

0.5

0.0

0.0

0.5

0.0

0.5

0.0

0.5

S( φ

12

= 0

, φ3

4 =

π/2

) [a

rb.u

nits

]

-100 -50 0 50 100-0.5

0.0

0.5

0.0

0.6

0.0

0.6

0.0

0.6

S( φ

12 = π

/2, φ3

4 = π/2) [arb.units]

-50 0 50 100-0.6

0.0

0.6

t13 = 210 fs

Delay between pulses 3 & 4, t 34 [fs]

(a) (b)

(c) (d)

t13 = 2 ps

t13 = 200 ps

t13 = 20 ps

t13 = 210 fs

t13 = 2 ps

t13 = 20 ps

t13 = 200 ps

Liouville-Pathways Interference

227

Fig. 7.6. Heterodyne detected stimulated photon echo signals for DTTCI in EG, measured under fourspecific phase settings: a): N =0, N =0, b): N =B/2, N =0, c): N =0, N =B/2, d): N =B/2, N12 34 12 34 12 34 12 34

=B/2. Four different settings of t are indicated. The setting of the delay t is t =55fs. In this13 12 12

experiment, the delay t is scanned. Excitation energy amounts 150 pJ per pulse. The experimental34

results are depicted by solid circles, the fit to the data based on the MBO model is included in each paneland depicted by the solid line.

S(3)E(φ12,φ34)

S(3)VE(φ12,φ34)

|i⟩⟨i| |f⟩⟨f|

Chapter 7

228

Fig. 7.7. Schematic diagram of the four-photon Liouville pathway interference between the conventionalS and virtual echo S . *i,+i* and *f,+f* denote the initial and final state, respectively.(3) (3)

E VE

7.4.1 Four-photon Liouville pathway interferences

Figure 7.6 displays the HSPE signals for four different phase settings at different waitingtimes. In this experiment, the delay t between pulses E and E is scanned, while the delay34 3 4

t between pulses E and E is fixed to t =55fs. The nonlinear polarization induced by the12 1 2 12

excitation pulses can be described using the four basic Liouville-space diagrams (Fig. 7.3).In a similar fashion to the pressure-induced resonance case interference occurs between twopathways that drive the system from the ground state *g,+g* to the excited state *e,+e* via acoherent superposition *g,+e* (R Fig. 7.3) or *e,+g* (R in Fig. 7.3). Since the photon echoI II

is a coherent four-wave mixing effect, the evolution of the system in the ground state needsto be taken into account as well (R and R in Fig. 7.3). Figure 7.7 schematically depicts theIII IV

two interfering pathways in the system evolution from the initial state *i,+i* to the final state*f,+f*. The echo signal S (represented by the response function R =R +R ) constitutes oneE A II III

path, whereas the virtual echo S (R =R +R ) constitutes the other path. Because theVE B I IV

conventional and virtual echoes are emitted in the same direction, they can, in principle,interfere. The particular setting of the relative phases of the excitation (and local oscillator)pulses determines whether the interference between the different pathways is constructive ordestructive. If N =0 these contributions add constructively (Fig. 7.7, left panels) and an12

enhanced echo signal results. When the relative phase between the first and second pulses isset to N =B/2, a phase shift of B occurs between the conventional and virtual echo signals.12

The resulting destructive interference leads to suppression of the echo (Fig. 7.7, right panels).The echo signal completely vanishes at t=200 ps (Fig. 7.7, right panels). By this time the13

real and virtual echoes have merged in time, which happens when the system has lost allphase memory. Two ‘echoes’ have become indistinguishable and complete destructiveinterference occurs between equal amplitude but oppositely phased Liouville pathways. Incontrast, for shorter waiting times, the interference is incomplete because the real and virtualecho fields peak at different times. Waiting-time dependent HSPE signals for N =B/2 thus12

clearly exhibit the transition from a partly inhomogeneously to a homogeneously broadenedsystem with increasing time. The observed narrowing of the signals with time is caused by thesame effect. Just by changing the phase difference between the first two pulses by B/2, one

S(3)(φ12,φ34)

S(5)(φ12,φ34)

|i⟩⟨i| |f⟩⟨f |

Liouville-Pathways Interference

229

Fig. 7.8. Schematic representation of the Liouville pathways interferences between four-photon S and(3)

six-photon S processes. *i,+i* and *f,+f* denote the initial and final state, respectively. (5)

switches from the case of full constructive to complete destructive interference. This presentsa prime example of coherent control of a nonlinear optical polarization.

We now address the question of how the interference effect in phase-locked echo isrelated to the phenomenon of dephasing-induced resonance in four-wave mixing. (DICE andPIER-4). Of course, both effects are manifestations of interference between differentLiouville-pathways. However, for the rest the effects are quite different. In the dephasing-induced-resonance case the two excitation pathways are, in the absence of collisions,oppositely phased and hence no net polarization is formed. It is only after collisions haverandomly changed the phase between the excitation pathways, that a coherent signal can begenerated. In phase-locked heterodyne-detected echo, dephasing (for instance induced[1-3,10]

by collisions between chromophore and solvent molecules) plays quite a different role in theinterference between excitation pathways. First, whether or not dephasing occurs, the effectof interference between the real and virtual echo signals is always there. Second, the phasebetween the interfering Liouville pathways is not determined by collisions but imposed on thesystem from the outside. In this case dephasing drives the virtual and real echo to merge intime, when the system has lost all phase memory. At that point the pathways can be made tointerfere completely destructively.

7.4.2 Four-photon and six-photon interferences

Besides the interference between equal-order perturbation terms, interferences betweendifferent order processes can occur. In Figure 7.8, the interference between a third-orderprocess and a fifth-order process is schematically depicted. Multiple pathways can bring thesystem from the initial state to the final state. Note that each set of pathways (4 for the third-order process, and 96 for the fifth-order process) includes a different number of intermediatesteps. To demonstrate this effect, the intensity in the HSPE experiment is

S( φ

12

= 0

, φ3

4 =

0)

[arb

.uni

ts] 0.0

1.0

1.0

0.0

1.0

0.0

1.0

0.0

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

S( φ

12 = π

/2, φ3

4 = 0) [arb.units]

-1

-0.5

0

0.5

0.0

0.5

0.0

0.5

0.0

0.5

S( φ

12

= 0

, φ3

4 =

π/2

) [a

rb.u

nits

]

-100 -50 0 50 100-0.5

0.0

0.5

0.0

1.0

0.0

1.0

0.0

1.0

S( φ

12 = π

/2, φ3

4 = π/2) [arb.units]

-50 0 50 100-1.0

0.0

1.0

t13 = 210 fs

Delay between pulses 3 & 4, t 34 [fs]

(a)

(b)

(c) (d)

t13 = 2 ps

t13 = 200 ps

t13 = 20 ps

t13 = 210 fs

t13 = 2 ps

t13 = 20 ps

t13 = 200 ps

Chapter 7

230

Fig. 7.9. Heterodyne detected stimulated photon echo signals for DTTCI in EG, measured under fourspecific phase settings: a): N =0, N =0, b): N =B/2, N =0, c): N =0, N =B/2, d): N =B/2, N12 34 12 34 12 34 12 34

=B/2. Four different settings of t are indicated. The setting of the delay t is t =55fs. In this13 12 12

experiment, the delay t is scanned. The excitation energy amounts 1nJ per pulse. The experimental34

results are depicted by solid circles, the fit to the data based on the MBO model is included in each paneland depicted by the solid line.

P (3)Y i 3

' &iP (5)

Y i 5' %i

P (3)@P(3)

Y i 3× i× i 3' &i

Liouville-Pathways Interference

231

(7.13)

increased to enhance the fifth-order contributions to the final signal. The results of an experiment with excitation energies of 1 nJ are depicted in Figure 7.9.

The same settings and scanning conditions are chosen as in Figure 7.6, only this time a largerpulse energy is applied. Compared to the low-intensity results (Figure 7.6), several changesin the HSPE signals can be seen. The signals detected with the phase setting N =0, show,34

compared to the results presented in Figure 7.6, a symmetric dip in the signal around zerodelay. This additional and negative contribution to the signal is attributed to the fifth-orderterms. Besides the fifth-order processes, one might wonder what the influence of the cascadethird-order processes is, since under certain circumstances one is bound to see these particularsignals. Whether these cascade signals are important in the HSPE experiments depends onseveral factors. The most important factor is the concentration used in the experiment. The(homodyne detected) signal resulting from third-order and fifth-order processes scaleproportional to the square of the concentration whereas the signal resulting from cascadeprocesses scale proportional to the fourth power of the concentration. For the heterodynedetection method, the square and fourth power dependence change to linear and quadraticdependence, respectively. A low concentration as applied in the experiment reported(Fig. 7.9) can effectively reduce the possible appearance of the cascade signal. Furthermorethe application of a heterodyne detection mechanism allows for the discrimination betweena cascade third-order process and a direct fifth-order process. This is based on the resultingphase in the emitted electric field. Although lost in the conventional integrated detectionmethods, the heterodyne method of detection is sensitive to the phase of the signal. The phase-factors of the different terms of interest are:

In the last term, representing the cascade process, the polarization from the first third-orderinteraction acquires an i , and the relation between the induced polarization and the electric3

field associated with this polarization again introduces an i in the phase. The last i , is a result3

from the second (cascade) third-order process. For the final cascade polarization we end upwith i which is equal to -i. In case the fifth-order signal exceeds the cascade signal, a negative7

than rather a positive signal (assuming that the third-order signal is positive) should beobserved. As can be inferred from the large additional negative contribution in the signalpresented in Figure 7.9(a), the fifth-order term thus dominates.

For a phase setting of N =B/2, the fifth-order signal manifest itself as an antisymmetric34

(dispersive) feature. The sign of the fifth-order contribution opposes that of the third-ordersignal for the phase settings of N =0, N =B/2, and has an equal phase (equal sign) for the12 34

signal measured with phase settings N =B/2, N =B/2. From this we conclude that the fifth-12 34

order signal is dominated by the contribution from the ‘population’ term in the signal (R ),B(5)

S(5)pop(t34,t23,t12) % &exp(&½)2t 2

34)×cos(8 t34(1&M(t23))) cos(8 t34(1&M(t23))%N34)

Chapter 7

232

(7.14)

which has no dependance on the relative phase N . The importance of the population term12

is confirmed by the fact that for a delay setting of t =100fs, where the electronic coherence12

terms have fully decayed, a nearly intense fifth-order term is measured when t is scanned.34

Contrary to the low intensity results, the conditions for destructive interference (leadingsometimes to the complete disappearance of the signal) change as the fifth-order signals arepresent. Where the third-order signal exhibits a full destructive interference (Figure 7.6(b)and (d) t =200ps), now a fifth-order term remains (Figure 7.9(b) and (d) t =200ps). The13 13

cancellation of the complete signal (third- plus fifth-order) now occurs at t =200 ps in panel13

(a) and (c). The exact delay t where the destructive interference is perfect, depends on the13

ratio between the third and fifth-order contribution. Changing the intensity of the applied laserpulses, results in a different position along the t -coordinate where the final signal approaches13

zero. In order to model and quantify the experimental results measured in the HSPE

experiments, the data is fitted to the MBO model. The crucial ingredient of this model is[13]

formed by either the spectral density or the correlation function M(t). In the correlationfunction description several modes will be used to construct the total correlation function.Here we will apply the same result we obtained on the time-resolved and time-integrated echoexperiments and the same DTTCI molecule as was derived from time-integrated and time-gated echo experiments reported in Chapter 3 (Figure 3.12). The correlation function enclosesmultiple dynamical aspects of the system and system bath dynamics, solute-vibrationalproperties and various solute-solvent interactions. In fitting the results from the HSPEexperiment, as shown in Figure 7.6, the only fitting parameter was formed by the globalamplitude. For the fifth-order response, the exact numerical six-dimensional integrationresults in computer times beyond realistic values. For modelling the fifth-order signal depictedin Figure 7.9, we assumed that the signal is dominated by the population terms, and make useof the assumptions as presented in the numerical section (spectral diffusion limit):

Compared to the data presented in Figure 4.8, measured at a delay setting t =40 fs, the12

third-order data measured at a different delay t =55 fs is closely predicted by the same MBO12

approach. The results of a fit to the high intensity fifth-order HSPE signals are depicted inFigure 7.9. Although the fifth-order signals are modelled by the simplified Eq.(7.14), thecalculations provide a good description of the observed signals changes in the high intensitylimit. Both the negative dip in the (N =0) data and the dispersive feature in the (N =B/2) are34 34

reflected in the calculations. Furthermore the destructive interference both in the panels 7.9(a)and 7.9(c) are nicely predicted in this model. A fifth-order description seems thus sufficientto explain the HSPE data for this range of excitation intensities.

StotalPLPP(t12) ' S(3)

PLPP(t12) % S(5)PLPP(t12) %

&1 & (5&")×exp[&gRe(t12)] cos[gIm(t12)]

Liouville-Pathways Interference

233

(7.15)

Experimentally, the influence of seventh and even higher order terms can be evaluatedin case all delays between the pulses are made significant larger that the typical time theelectronic coherence can survive. For the setting of the separations t =t =100fs, and12 34

(positive) scanning t , besides the spike around zero t delay, no apparent signal is observed.13 13

Based on this result, one can conclude that the contributions from interactions higher that thefifth-order are significantly smaller and can therefore be ignored in the experiment. Of course,this statement only holds for the intensity range studied here, increase of the intensity to evenhigher values, one should carefully reestablish this result. Note that the fifth-order effects havealso been predicted in PIER-4 but not reported yet. Recently, interferences between third-[26]

order and fifth-order polarizations in semiconductor doped glasses have been reported. In[27]

this two-colour quasi-cw experiment, beats in the resulting diffracted signal were observedupon scanning the delay between some of the incident pulses. The beat is attributed tointerferences between third-order and fifth-order terms contributing to the final signal. Similarthird-fifth-order interference effects have also been reported by Wu et al. on GaAs bulkmaterials.[28]

7.4.3 Fifth-order effects in phase-locked pump probe spectroscopy

In the phase-locked pump-probe experiment the local oscillator pulse coincides with thethird excitation pulse (t =0). The expression the fifth-order phase-locked pump-probe signal34

presented in Eq.(7.11) yields a similar result as for the third-order signal, be it that in the fifth-order signal a negative sign and a baseline are present. For the total signal the expressionreads:

where " denotes the ratio between the third-order and fifth-order signal. The experimental results for the high intensity phase-locked pump-probe are depicted in

Figure 7.10. Here we have only depicted the in-phase setting between the first and secondexcitation pulses. Note that at particular settings of the delay t (t =56 fs) the total PLPP12 12

signal can be made zero, the third-order response fully cancels the fifth-order signal. For thet scan at this setting of t , no pump-probe signal is observed. As a result of the non-linear13 12

interaction with the optical dipole, the light travelling through the sample thus experiencesno pump-pulse induced modulation. Slight changes in the position of the delay t , result in12

either positive or negative signals corresponding to a third-order signal that is larger or

-100 -50 0 50 100

t 12 [fs]

0.0

0.2

0.4

0.6

0.8

1.0

-200 0 200 400 600

t 13 [fs]

0.0

0.2

0.4

0.6

0.8

1.0α

βγ

δ

α

β

γ

δ

Chapter 7

234

Fig. 7.10. Phase-locked pump-probe signals. (a) PLPP signal for a t scan at four different settings of13

(") t =0fs, ($) t =50fs, (() t =56fs, (*) t =100fs. (b).PLPP signal where t is scanned, t is set to12 12 12 12 12 13

t =2ps. The t -scans (panel (a)), are indicated with arrows (",$,(,*).13 13

smaller that the fifth-order response.

7.5 Conclusion

Summarizing, a theoretical base was developed for the description of the interferenceeffects in heterodyne detection of the stimulated photon echo. In the modelling approach, thethird-order and fifth-order perturbative terms contributing to the final HSPE signal were takeninto consideration. HSPE experiments were successfully performed on an infrared dyemolecule DTTCI in solution. Low intensity measurement could be fully explained within theanalysis up to third-order. However, as the intensity is increased, fifth-order terms must betaken into account for the description of the observed features. A multimode Brownianoscillator description allows for a good approximation of the observed signals. The novelinterference effect in phase-locked stimulated photon echo relies on the interplay betweendifferent excitation pathways. Active control of the relative phases of the excitation pulsesallows for switching between constructive and destructive interference. This technology isalso of a special interest to coherent control experiments in chemistry, where Liouville-spacepathway interference is an important option.[29]

Q (5)A

,1,2,3,4

(t,t23,t12)'&iTeg(,3t12%t)&g,2,3

(t12)&g,4%

(t)&,1,3[&g,1,2

(t12)%g&,1

(t12)]

&,1[g,1,2(t12%t23)&g

,1,2(t12%t23%t)&g

&,1(t12%t23)%g

&,1(t12%t23%t)]

&,3[g,3,4(t23)&g

,3,4(t23%t)&g

,2,3(t12%t23)%g

,2,3(t12%t23%t)]

Q (5)B

,1,2,3,4

(t,t23,t12)'&iTeg(t)&g,4%

(t)

&,1[g,1,2(t12%t23)&g

,1,2(t12%t23%t)&g

&,1(t12%t23)%g

&,1(t12%t23%t)]

&,3[g,3,4(t23)&g

,3,4(t23%t)&g

,2,3(t23)%g

,2,3(t23%t)]

Q (5)C

,1,2,3,4

(t,t23,t12)'&iTeg(,1t12%t)&g&,1

(t12)&g,4%

(t)

&,1[g,1,2(t23)&g

,1,2(t23%t)&g

&,1(t12%t23)%g

&,1(t12%t23%t)]

&,3[g,3,4(t23)&g

,3,4(t23%t)&g

,2,3(t23)%g

,2,3(t23%t)]

Q (5)D

,1,2,3,4

(t,t23,t12)'&iTeg(,1t12%t)&g&,1

(t12)&g,4%

(t)

&,1[g,1,2(t23)&g

,1,2(t23%t)&g

&,1(t12%t23)%g

&,1(t12%t23%t)]

&,3[&g,3,4

(t)%g,2,3

(t)]

Q (5)E

,1,2,3,4

(t,t23,t12)'&iTeg(,1t12)&g&,1

(t12)

Q (5)F

,1,2,3,4

(t,t23,t12)'&iTeg(,1t12%,3t)&g&,1

(t12)&g,2,3

(t)

&,1,3[g,1,2(t23)&g

,1,2(t23%t)&g

&,1(t12%t23)%g

&,1(t12%t23%t)]

Liouville-Pathways Interference

235

(7.16)

E1 t1 E2 t2 E3 t3 E4 t4 E5 t5 E6

A 1 0 1 0 1 t12 2 t23 3 t 4

B 1 0 1 t12 2 0 2 t23 3 t 4

C 1 t12 2 0 2 0 2 t23 3 t 4

D 1 t12 2 t23 3 0 3 0 3 t 4

E 1 t12 2 t23 3 0 3 t 4 0 4

F 1 t12 2 t23 3 t 4 0 4 0 4

Table 7A.1. Overview of the possible pulse permutations and corresponding inter-pulse times for thefifth-order perturbative description. E -E denote the interactions with the consecutive electric fields. E1 5 6

denotes the local oscillator pulse. 1-4 denoted the actual pulse in the experiment (Fig 7.1). Theinteraction times are listed under t to t . A time equal to zero (0) indicates that a single electric field1 5

governs more than one interaction.

Appendix 7ABased on the selection of the pulse interactions as given in Table 7A.1, the following

relations for the separate response functions can be derived:

Chapter 7

236

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238