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Changes in relaxation times and strength constant and its eﬀect on four-wave mixing spectra Jose Luis Paz,* T. Cusati, and A.J. Hern andez Departamento de Quımica, Universidad Sim on Bolıvar, Apartado 89000, Caracas 1080A, Venezuela Received 15 April 2003; in revised form 12 June 2003

Abstract In the present work we have studied modiﬁcations in the four-wave mixing (FWM) signal with independent simultaneous changes in the ratio between the longitudinal and transversal relaxation times T1 =T2 and with the strength constant of the crossed harmonic potential curves that describe the two-levels system employed in this study, which is measured by the quotient between the ~ 0 =x0 . In the formalism employed, the permanent dipole moments of the states resonance frequencies of the harmonic curves d ¼ x in the uncoupled basis have been included and the rotating wave approximation is neglected in order to observe the processes out of the resonance region. We have observed changes in the shape and intensity of the FWM signal spectra with changes in the ratio T1 =T2 and changes in the intensity and positions of the lines by modifying the parameter d. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Intramolecular coupling; Four-wave mixing; Relaxation times; Potential curves; Strength constant

1. Introduction The study of radiation–matter interaction is considered the principal aim of nonlinear optics. Depending of the physical situation, phenomena such as multiphotonic absorption and emission, harmonic generation, parametric eﬀects, stimulated dispersion, nonstationary and highly coherent radiation generation, optical bistability, phase conjugation, and photon and multiphoton saturation can arise. The study of these phenomena leads to diﬀerent applications in laser technology, such as: laser spectroscopy, photochemistry, photophysics, and material processing [1]. The nonlinear optical interactions are used in a routinely way to the experimental determination of some materials properties. The measurement of fast relaxation processes and decay rates is a very important but diﬃcult task, thus, many nonlinear techniques have been developed to this end [2]. The relaxation process can be described by two phenomenological parameters: the longitudinal relaxation time T1 , representing the decay * Corresponding author. Fax: +58-2-9063961. E-mail address: [email protected] (J.L. Paz).

0022-2852/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-2852(03)00205-4

of energy (or populations) associated with population relaxation mechanisms in a Boltzman type regime, and the transversal relaxation time T2 , representing the phase memory (or polarization) associated with lost of coherence mechanisms of the system in absence of the incident ﬁeld. In the past years, the dynamic of two-level systems under intense electromagnetic ﬁelds has been well studied, where one of the principal aims was the measure of the longitudinal T1 and transversal T2 relaxation times associated with the energy dissipation mechanisms of the system. These relaxation times are relatively small for some dyes and they are determined by nonlinear spectroscopy (e.g., by four-wave mixing technique) [3]. FWM represents a nonlinear spectroscopic technique involving mixing of waves in a medium to produce a dispersion of signals of diﬀerent frequencies. In the present study, the signal is generated starting from the interaction of two incident beams of diﬀerent intensity, frequency, and diﬀerent propagation directions: the pump beam of frequency x1 and wave vector k1 and the probe beam of frequency x2 and wave vector k2 , which generate a dispersion of signals of diﬀerent frequencies

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Fig. 1. (a) Diabatic and (b) adibatic representations of the crossed harmonic potential curves.

and propagation directions. The signal of interest in this study have frequency x3 ¼ 2x1 x2 and propagation direction ~ k3 2~ k1 ~ k2 [4]. There exist in the literature studies that employ twolevel molecular systems without internal characteristics. However, it is always possible to include a simple vibrorotational internal structure to these models by incorporating, in the calculation of the intramolecular coupling phenomenon, two electronic states which are coupled through the avoided crossing rule. In this case, the intramolecular coupling phenomenon can be described by two crossed harmonic potential curves, with minima displaced in energy and nuclear coordinate. Moreover, these curves can hold diﬀerent strength constants, which imply they have diﬀerent associated frequencies.

In the present work, we study modiﬁcations of the FWM spectra with changes in the longitudinal and traversal relaxation times T1 and T2 , respectively, with the quotient T1 =T2 , as well as with changes in the strength constants of the crossed potential harmonic curves employed to deﬁne our molecular system, where the permanent dipole moments of the states in the uncoupled basis are also considered explicitly. Here, the strength constant is measured by the parameter ~ 0 =x0 , which represents the frequencies quotient of d¼x the two potential curves, and the present model is applied to the organic dye Malachite Green. Changes of the FWM spectra with modiﬁcations of the characteristic parameters of the intramolecular coupling, v and V0 , for ﬁxed and equal relaxation times and for ﬁxed strength of the constants used to characterize the

Fig. 2. General spectrum for the FWM signal by using v ¼ 0:51 and V0 ¼ 0:1.

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potential curves employed, have been studied previously [5,6].

2. Characteristics of the system Intramolecular coupling has received special attention due to its application in optical collisions in ultra cold atoms and in optical suppression [7]. This coupling is related to potential energy crossing due to interaction among the electronic and nuclear motions in polyatomic molecules. The molecular model used in the present work is described by two crossing harmonic potential curves. Each curve carries its own fundamental vibrational level with diﬀerent associate frequencies, making possible to measure changes in the strength constants by using diﬀerent frequency quotient among them. Both curves

are displaced horizontally in nuclear coordinate and vertically in energy, as depicted in Fig. 1. The coupling between the electronics states is induced by the inclusion of a perturbation in the total Hamiltonian represented by the residual spin–orbit term H 0 ðrÞ, which depends weakly on the electronic coordinates. In consequence, the characteristic parameters of our model are: the coupling parameter (v), the energy diﬀerence between the minima of the curves (V0 ) and the energy height (S) at which the coupling occurs. The vibronic states in the coupled basis: W ðr; RÞ ¼ h

jV00 j

i1=2 2 jV00 j2 þ ðE10 E Þ ðE10 E Þ w1 ðr; RÞU1j ðRÞ w2 ðr; RÞU2k ð RÞ V00 ð1Þ

Fig. 3. FWM spectra with intramolecular coupling parameters v ¼ 0:51 and V0 ¼ 0:1 for the ratio between relaxation times T1 =T2 : (a) 0.01, (b) 0.1, (c) 1, (d) 10, and (e) 100.

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are built using a variational trial wave function, expressed as a linear combination of the wave function of each of the states involved, and by solving the respective secular determinant [8,9]. The energy values of these states are expressed as: 2 i 1 1h 2 E ¼ ðE10 þ E20 Þ ðE10 E20 Þ þ 4Vjk ; 2 2

ð2Þ

where 1 E10 ¼ ; 2

1 E20 ¼ d þ V0 ; V00 ¼ vhu10 ju20 i and 2 1=4 i ð4dÞ Sh 1 exp 1 ð1 þ dÞ : hu10 ju20 i ¼ 1=2 2 ð 1 þ dÞ The newly generated coupled-basis states have different dipole moments than those of the uncoupled

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basis. The expressions that deﬁne these dipole moments are given by the following equations: ( ) 2 jV00 j laa ¼ 2jV00 j2 þ ðE10 Ea ÞðE10 E20 Þ ( i 1 h 2 a ð E E Þ ð E E Þ þ V m22 j j 10 10 20 00 2 jV00 j ) 2ðE10 Ea Þm12 þm11 ; ð3Þ v ( lþ ¼

2 jV00 j

)1=2

2 4jV00 j2 þ ðE10 E20 Þ ðE10 E20 Þ ðm11 m22 Þ m12 ; v

ð4Þ

Fig. 4. FWM spectra with intramolecular coupling parameters v ¼ 0:01 and V0 ¼ 0:01 for the ratio between relaxation times T1 =T2 : (a) 0.01, (b) 0.1, (c) 1, (d) 10, and (e) 100.

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where a ¼ þ or ), mii are the permanent dipole moments and mij correspond to the transition dipole moments of state in the uncoupled basis. As it is known, temporal evolution of the system can be described by means the Liouville–von Neumann equation: i hðdqðtÞ=dtÞ ¼ ½H ; qðtÞ, where the total Hamiltonian H is described by H ¼ H0 þ H 0 , being H0 the unperturbed Hamiltonian and H 0 the intramolecular coupling perturbation. Moreover, the system-ﬁeld interaction Hamiltonian is described by an electric dipole interaction in the semiclassical approximation and the system–reservoir interaction Hamiltonian is introduced in a phenomenological way. The reservoir structure is not included in our formalism and the solvent is considered transparent to the radiation. The response of the system to its interaction with the electromagnetic ﬁeld is measured by the macroscopic polarization, which in turns is responsible for the system response [10]. In terms of density matrix

elements, the macroscopic polarization can be expressed by: D*E ~ P ðx3 Þ ¼ N l h* i * ¼ N l þ qþ ðx3 Þ þ l þ qþ ðx3 Þ ~ dic qD ðx3 Þ : ð5Þ Here, N corresponds to the absorbent molecules density, qij are the coherence density matrix elements, qD corresponds to the diﬀerence between the populations density matrix elements, lþ and lþ are the transition dipole moments and dic represents the diﬀerence between the permanent dipole moments. The rotating wave approximation is neglected in this model, which allows us to study the processes occurring outside the resonance region. We have also considered a homogeneously broadened two-level system including a simpliﬁed version of the vibrational structure associated to the intramolecular coupling and we have used a per-

Fig. 5. Modiﬁcations of the resonances in the FWM spectrum under the changes in the ratio T1 =T2 for the coupling parameter case v ¼ 0:51 and V0 ¼ 0:1: (a) peak 1, (b) peak 2, and (c) peak 3.

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turbative treatment to third order in the total ﬁeld (second order in the pump ﬁeld and ﬁrst order in the probe and signal ﬁelds). At this point, it is possible to establish the Optical Bloch equations that include the resonance frequency of the system x0 and the longitudinal and transversal relaxation times T1 and T2 , respectively: dqþ i i ¼ Hþ qD qþ ½H Hþþ h h dt 1 þ ix0 qþ ; T2 dqþ i i ¼ Hþ qD þ qþ ½H Hþþ h h dt 1 ix0 qþ ; T2 1

dqD 2i Hþ qþ qþ Hþ ¼ qD q0D : h T1 dt

ð6Þ

quency of the FWM signal x3 that includes the intramolecular coupling parameters: 1 1 1 1 *4 ð0Þ~2~ ~ P ðx3 Þ ¼ NiqD E1 E2 2l þ þ D ðD C Dþ Þ 3 2 3 1 1 1 1 1 1 þ þ þ þ þ þ D1 ðD2 Þ ðD1 Þ k D ðDþ 1 1Þ 1 1 1 *2 þ~ dic2 l þ þ b ðDþ ðD Þ Þ 3 D ðD2 Þ 1 1 þ þ þ þ þ þ ðD1 Þ ðDD Þ ðD1 Þ ðD5 Þ 1 1 1 1 1 þ þ þ ; þ D b D D D 3 1 DD 1 D5 D D2 ð9Þ

ð7Þ ð8Þ

Substituting the corresponding expressions derived for the density matrix elements qþ ðx3 Þ, qþ ðx3 ), and qD ðx3 ), it is also possible to obtain an expression for the local macroscopic polarization as a function of the fre-

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where the parameters are deﬁned as follows: * * ~ dic ¼ l l þþ ; qD ¼ qþþ q ; D ¼ x1 x2 ; i 1 1 iðx1 x2 Þ; b ¼ ix3 ; k ¼ 2ix1 ; T1 T1 T1 1 1 þ i x0 xj ; D þ iðx0 2x1 Þ: D j ¼ 5 ¼ T2 T2 ð10Þ

C¼

Fig. 6. Modiﬁcations of the resonances in the FWM spectrum under the changes in the ratio T1 =T2 for the coupling parameter case v ¼ 0:51 and V0 ¼ 0:1: (a) peak 4, (b) peak 5, and (c) peak 6.

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Fig. 7. Modiﬁcations of the resonances in the FWM spectrum under the changes in the strength constant parameter d for the coupling parameter case v ¼ 0:51 and V0 ¼ 0:1: (a) peak 1, (b) peak 2, and (c) peak 3.

3. Calculations It is known that the FWM signal intensity is proportional to the square of the macroscopic polarization. In our model, the intensity is obtained as the quotient between j~ P ðx1 ; x2 ; dÞj2 and the polarization evaluated at the resonance frequency j~ P ðx1 ¼ x2 ¼ x0 ; dÞj2 . Three-dimensional graphs, representing the FWM signal intensity as a function of the pump and probe detuning (in frequency space), was obtained. These spectra were deﬁned in the frequency space range x0 < x1 < x0 and 2x0 < x2 < 2x0 . In this study, we have chosen the following conditions: (a) resonance frequency of the uncoupled states x0 ¼ 3:0628 1015 s1 ¼ 16 280 cm1 , corresponding to the frequency of the Malachite Green dye, (b) m12 ¼ m21 ¼ 0:1, m11 ¼ 1:0, and m22 ¼ 1:3 D, respectively, for the transition and permanent dipole moments of the uncoupled states (giving a value of d ¼ 0:3 D for the diﬀerence in permanent dipolar moments). As in previous works, we have studied the eﬀect of changing the vibronic coupling parameters v and V0 ,

respectively, while keeping the relaxation times equal to the value T1 ¼ T2 ¼ 1:3 1013 s and the strength constant d ¼ 1 [5,6]. We showed previously, that it is always possible to reproduce the case of a two-level system without intramolecular coupling for the particular case when the coupling parameters are v ¼ 0:51, V0 ¼ 0:1, and S ¼ 0:1 (normalized to hx0 , to make them dimensionless) [11]. Other coupling parameters values have been also included in the present work due to their physical importance. These cases are: (i) v ¼ 0:01 and V0 ¼ 0:01 (nearly degenerate and weak coupling case) and (ii) v ¼ 0:5 and V0 ¼ 1. Fig. 2 shows the FWM spectrum corresponding to the coupling parameters v ¼ 0:51 and V0 ¼ 0:1, for the case where the relaxation times are equal (T1 ¼ T2 ) and the potential curves have the same frequency, i.e., d ¼ 1. In this ﬁgure, the resonances are identiﬁed with numbers; the peaks numbered from 7 to 12 are the images of the ﬁrst six resonances (the symmetry is obtained under the change of xi by xi ) [5,6], thus we have used this symmetry to reduce our study to only six peaks in the spectrum represented in Fig. 2.

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Fig. 8. Modiﬁcations of the resonances in the FWM spectrum under the changes in the strength constant parameter d for the coupling parameter case v ¼ 0:51 and V0 ¼ 0:1: (a) peak 4, (b) peak 5, and (c) peak 6.

In general, the values for the parameters of the relaxation times and strength constants taken in this study were all chosen from the values employed in previous works, as reported in the literature [8,9]. The present work is divided in two separate case studies as described below. 3.1. Relaxation time changes The results presented in this section are related to modiﬁcations in the FWM spectra by changes in T1 and T2 , speciﬁcally in the ratio T1 =T2 , which only aﬀects the expressions related to the FWM macroscopic polarization but does not have any relation to the molecular structure of the dye. The particular ratios values taken in this study are: (a) T1 =T2 ¼ 0:01, (b) T1 =T2 ¼ 0:1, (c) T1 =T2 ¼ 1:0, (d) T1 =T2 ¼ 10, and (e) T1 =T2 ¼ 100. The FWM spectra for the case when v ¼ 0:51 and V0 ¼ 0:1 is depicted in Fig. 3. The case v ¼ 0:01, V0 ¼ 0:01 is depicted in Fig. 4. As a result of the changes in the relaxation times, modiﬁcations in the shape of some resonances and in the

intensity of other ones are observed. Speciﬁcally, the peaks 1, 2, and 4 change their shape and the peaks 3, 5, and 6 change their intensity. These changes on the peaks are related to their dependence with diﬀerent elements of the density matrix that conform the global expression for the polarization. Reduced polarization expressions for each peak in the spectrum have been obtained in previous work [12]. These resonance changes can be observed explicitly in Figs. 5 and 6. As mentioned above, peaks 1, 2, and 4 change their shape, speciﬁcally, they become wider by increasing the ratio T1 =T2 , while peaks 3, 5, and 6 decrease their intensity in two orders of magnitude for each order of magnitude increase on the T1 =T2 ratio. These changes are related to the reduced polarization expressions that characterizes each peak [12]. Peaks 1, 2, and 4 depend on the populations element of density matrix and, after substituting the respective resonant elements in the corresponding reduced polarization expressions, no dependence in the longitudinal relaxation time T1 is observed. Their change in shape is associated with the modiﬁcations that suﬀers the term b in the reduced polarization expression as a function of

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erate changes in the ratio T1 =T2 , which can be of various orders of magnitude [13]. 3.2. Strength constant changes

Fig. 9. FWM spectra with changes in the strength constants d and intramolecular coupling parameters v ¼ 0:01 and V0 ¼ 0:01 for the following cases: (a) d ¼ 1, (b) d ¼ 0:9, and (c) d ¼ 0:2.

The results of this second section are concerned with changes in the structure of the molecular system, which are generated by changes in the strength constants of the harmonic potential curves employed. These changes are induced by modifying the parameter d, which is related to the quotient between the frequencies of both curves as ~ 0 =x0 . The values investigated in this case given by d ¼ x are: (a) d ¼ 1, (b) d ¼ 0:9, and (c) d ¼ 0:2, with the coupling parameters as described already in Section 3.1, and by and taking T1 =T2 ¼ 1. Results show changes in the intensity and displacement of the resonances in the FWM spectrum by modiﬁcating parameter d, which is related with changes in the width of the potential curves and the consequent frequency modiﬁcation of the coupled system. Because of the results are not clearly observed in the global spectra, we have presented resonances in Figs. 7 and 8, which show changes in intensity and displacement in the resonances to diﬀerent pump and probe detuning values by changing the value of d, for the particular case v ¼ 0:51, V0 ¼ 0:1. Firstly, we can observe a decreasing in the intensity and displacement to lower pump and probe detuning values by decreasing the value of d. These results can be extended to other cases of v and V0 . However, for the case depicted in Fig. 9 (v ¼ 0:01 and V0 ¼ 0:01), we can observe the opposite tendency in the global spectra, i.e., that there is a decreasing in the signal intensity but a displacement to higher pump and probe detuning values by decreasing the value of d. These changes are related to modiﬁcations in the transition dipolar moment between the diﬀerent electronic levels involved in the coupled basis, which in turns are generated by modiﬁcations in the overlap integral induced by changes in the parameter d associated with the structure of the molecule employed for the study (e.g., see Eqs. (3) and (4)).

4. Final comments T1 and x3 in the area around the maximum intensity. On other hand, the peaks 3, 5, and 6 suﬀer changes in the intensity values that depend on the coherence elements of the density matrix. The magnitude of the intensity is based in the reduced polarization expressions which is inversely proportional to the square of longitudinal relaxation time. For each order of magnitude that the ratio T1 =T2 is modiﬁed, the intensity change in two orders of magnitude. Finally, we should add that it has been shown experimentally that changes in the solvent can also gen-

In this study we have presented the behavior of the FWM signal when the relaxation times T1 and T2 and the strength constant d are modiﬁed. We can distinguish two cases. In the ﬁrst case, we can observe changes in the shape and intensity of the resonances with changes in the T1 =T2 ratio. These changes can be associated to the reduced macroscopic polarization expressions that characterize each peak in the spectrum, which in turns, allow us to identify the physical aspects involved in their shape and localization [12]. Using the reduced macroscopic polarization expressions obtained here, it was

J.L. Paz et al. / Journal of Molecular Spectroscopy 221 (2003) 106–115

possible to know in detail the physical aspects involved in the localization and shape of the peaks involved in the FWM phenomenon, speciﬁcally, it is possible to study the dependence of this expression with the parameters under consideration. A detailed study of the topological modiﬁcations of each resonance with the change in the T1 =T2 ratio necessarily involves the solvent. In our particular case of the Malachite Green dye, diﬀerent longitudinal/transversal relaxation times ratios are observed depending on the solvent employed [13]. Moreover, for solvents with slow viscosity only one relaxation time is observed, which changes to two diﬀerent relaxation times when the solvent viscosity is increased. In the second case, we can observe changes in the intensity and localization of the resonances with changes strongly related to the molecular structure, which generates modiﬁcations in the resonance frequency of the states in the coupled base. In this case, the vibrational levels change their relative distance, i.e., changes in the resonance frequency induce changes in the position of the peaks in the spectrum of the system. Speciﬁcally, changes in the parameters d, produce modiﬁcation in the behavior of the dipolar moments of the coupled states. These quantities are directly related with the parameter d and aﬀect strongly the polarization behaviour, generating changes in the magnitude of intensity and localization of the resonances in the FWM spectrum. Work including more detailed aspects related to the study of relaxation times and strength constant, including stochastically the solvent and its eﬀects on T1 and T2 will be published elsewhere.

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Acknowledgments The present work was supported by the Consejo Nacional de Investigaciones Cientıﬁcas y Tecnol ogicas (CONICIT) (Grant G-97000593) and by the Decanato de Investigaciones of the Universidad Sim on Bolıvar (Grant GID-13).

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