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Analysis of vibrational coherences in homodyne and two-dimensional heterodyne photon-echo spectra of Nile Blue
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Chemical Physics 341 (2007) 113–122 www.elsevier.com/locate/chemphys
Analysis of vibrational coherences in homodyne and two-dimensional heterodyne photon-echo spectra of Nile Blue Dassia Egorova b
a,* ,
Maxim F. Gelin b, Wolfgang Domcke
a
a Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany Department of Chemistry and Biochemistry, College Park, University of Maryland, MD 20742-4454, USA
Received 16 February 2007; accepted 6 July 2007 Available online 17 July 2007 This work is dedicated to Douwe Wiersma.
Abstract We propose a simple model which allows a comprehensive interpretation of the available experimental photon-echo data on Nile Blue. Homodyne signals such as two- and three-pulse photon echo, transient grating and three-pulse peak shift are addressed. An explanation of their time evolution is provided with a single parameter set. A new interpretation of two-dimensional (2D) heterodyne experimental spectra is proposed which emphasizes the strong electron–vibrational coupling of the 590 cm1 mode of Nile Blue. The model is used for predictions of 2D correlation and relaxation signals monitored with improved resolution (low temperature, short pulse durations). It is argued that such measurements could provide the first experimental detection of the signatures of electron–vibrational coupling in 2D photon-echo spectra. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Electron–vibrational coupling; 2D electronic photon-echo pectroscopy; Nile Blue
1. Introduction Ultrafast photon-echo (PE) spectroscopy belongs to the realm of the four-wave mixing (FWM) techniques. The observed spectra are thus determined by the third-order laser-induced polarization. The third-order polarization depends on the system dynamics, as well as carrier frequencies, durations, and relative time delays of the laser pulses, and obeys the corresponding phase-matching condition [1,2]. If a density-matrix description is employed for the system dynamics, then PE spectroscopy can be shown to reflect the dynamics of both populations and coherences of the density matrix in real time (in contrast to pump– probe and fluorescence spectroscopy, which primarily deliver information about the population dynamics).
*
Corresponding author. Tel.: +49 89 28913618; fax: +49 89 28913622. E-mail address: [email protected] (D. Egorova).
0301-0104/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.07.010
Two types of detection schemes of PE signals can be distinguished [1,2]: In the homodyne or time-integrated measurement, the signal intensity is recorded as a function of the delays between the pulses, whereas within heterodyne detection (time-gated PE), the dependence of the signal on the detection time can be monitored as well. Homodyne PE techniques include three-pulse PE, two-pulse PE, transient grating and peak-shift measurements. The latter is the most popular and frequently employed method, since the peak-shift dynamics is believed to provide information on solvation mechanisms [2–6]. Two-color peak-shift PE spectroscopy, which exploits the dependence of the peak shift on the carrier frequencies of the pulses, has been suggested and implemented as well [7,8]. As for the heterodyne detection, the so-called two-dimensional (2D) or Fourier-transformed PE [9,10] has gained particular attention in recent years. This approach, which has proven invaluable for the determination of vibrational couplings in the infrared [11], has recently been successfully implemented in the
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optical domain, providing a direct information on electronic couplings in the seven-chromophore Fenna–Matthews–Olson protein [12,13]. In the standard perturbative approach of nonlinear optics, the third-order optically induced polarization is expressed in terms of the nonlinear response functions of the system [1]. As the material systems under investigation become more and more complex (that is, multidimensional), the evaluation of the polarization in terms of the nonlinear response functions often is no longer possible and alternative strategies such as a nonperturbative treatment of the system–field interaction [14,15] may be more advantageous. A drawback of the nonperturbative method is that the obtained total polarization must be further analyzed in order to single out a specific combination of wave vectors responsible for a particular FWM signal. There exist several realizations of the nonperturbative calculation of the threepulse PE polarization [16–21]. In the present work, we use a recently developed method [22,23] which is computationally more efficient than the other available schemes. Apart from the choice of a computational method for evaluation of the third-order polarization, an appropriate model for the underlying system dynamics is necessary for the interpretation of the experimentally observed PE signals. The (multimode) Brownian oscillator model has become the working horse in theoretical optical spectroscopy. It provides an adequate description provided a material system under study can be described within the Born– Oppenheimer approximation by a few electronic states which are coupled to a collection of vibrational modes. This model has frequently been employed for simulations of integrated PE signals [1–6,8,24]. However, with a few notable exceptions [23,25,26], only the electronic degrees of freedom have explicitly been considered in theoretical descriptions of 2D PE spectroscopy, while the vibrations were taken into account as a phonon bath [9,12,13,27–29,19]. If the signals exhibit oscillatory behaviors in the time domain which are induced primarily by a single Franck– Condon active vibrational mode, an underdamped Brownian oscillator may be a sufficient first-order approximation. We have recently used this model to study the influence of electron–vibrational coupling, finite pulse durations, and broadening mechanisms on optical PE signals [22,23]. In particular, it has been shown that the vibrational mode strongly coupled to the electronic transition has to be taken into account explicitly in the interpretation of 2D electronic spectra [23]. It is thus of interest to investigate whether these predictions can be experimentally verified for a real physical system. Nile Blue (NB) dye is a perfect candidate for this purpose. The strong electron–vibrational coupling in this molecule has been revealed already in a pump–probe experiment by Fragnito et al. [30] and in a PE study by Moshary et al. [31]. The recent PE experiments on NB in various solvents include both homodyne [3,4,8,24,32] and 2D heterodyne [26] techniques. The transients measured via the homodyne PE detection exhibit vibrational coher-
ences with a period of about 60 fs [24,32] and look very similar to those obtained in our previous simulations within an underdamped Brownian oscillator model [22]. Oscillations with the same period have been resolved in transient absorption measurements [30,33] and the pump–probe spectra of Ref. [30] could be reasonably well modeled within a free displaced oscillator [34]. On the other hand, 2D PE spectra of NB in acetonitrile measured by Brixner et al. [26] at room temperature exhibit rather broad structures and are not very informative. The observed line shapes and their modification by various relaxation mechanisms are not well understood. It is not unreasonable to assume that the active mode of NB may have a strong effect on the shapes of the 2D PE spectra. In this paper, we summarize the recent experimental PE data on NB and propose a simple and elucidating physical model for their interpretation. We start with the time-integrated PE and show that the model is capable to reproduce and explain major features in the observed PE signals. In the second part, we employ the same model for simulations of 2D electronic spectra reported in Ref. [26]. In order to interpret the observed line shapes, we present the predictions of our model for 2D spectra measured under improved resolution conditions (low temperature and short pulse durations). In contrast to previous simulations [8,24,26], which take into account 40 Franck–Condon active modes of NB [35], we do not attempt to reproduce quantitatively the different optical PE signals of NB. Our goal is the development of a simple and transparent model, which allows to understand what particular features of the system dynamics and relaxation processes are responsible for specific signatures in the PE signals, both in the time and in the frequency domain. 2. Method A detailed description of the employed method is given elsewhere [22,23] and we only briefly review it here. We start from the kinetic equation for the reduced density matrix r (h = 1) ot rðtÞ ¼ i½H þ H int ðtÞ; rðtÞ ðR þ }ÞrðtÞ;
ð1Þ
where H is the system Hamiltonian and Hint(t) describes the field–matter interaction. Environmental degrees of freedom have been traced out; their influence is accounted for by the vibrational relaxation operator R and pure electronic dephasing (optical dephasing) }. In this work, RrðtÞ is described within Redfield theory [36] and electronic dephasing is introduced phenomenologically via }rðtÞ neg P g rðtÞP e þ H:c:;
ð2Þ
neg being the optical dephasing rate. We assume that the material system Hamiltonian can be expressed as the sum of the ground state (jgi) and excited state (jei) contributions H ¼ Hg þ He
ð3Þ
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
and write the system–field interaction in the electric dipole and rotating-wave approximations as H int ðtÞ ¼
3 X
V a ðtÞ expðika rÞ þ H:c:;
ð4Þ
a¼1
where V a ðtÞ ¼ ka Ea ðt sa Þ expfixa tgX :
ð5Þ
ka, ka, Ea(t sa), sa, and xa denote wave vector, amplitude, dimensionless envelope, central time and carrier frequencies of the pulses and the transition dipole moment operator is defined as l ¼ X þ X y;
X ¼ jgihej:
ð6Þ
The total polarization is calculated via P ðtÞ ¼ hlrðtÞi:
ð7Þ
The specific PE contribution to the polarization obeys the phase-matching condition k = k1 + k2 + k3. As has been shown in Refs. [22,23], within the rotating-wave approximation (RWA) the PE polarization PPE can be calculated by performing three simultaneous propagations of the auxiliary density matrices ri(t) (i = 1, 2, 3) P PE ðtÞ ¼ expfiðk1 þ k2 þ k3 ÞrghX ðr1 ðtÞ r2 ðtÞ r3 ðtÞÞi þ c:c:;
ð8Þ
ot r1 ðtÞ ¼ i½H V 1 ðtÞ V y2 ðtÞ V y3 ðtÞ; r1 ðtÞ ðR þ }Þr1 ðtÞ; ot r2 ðtÞ ¼ i½H V 1 ðtÞ V y2 ðtÞ; r2 ðtÞ ðR þ }Þr2 ðtÞ; ot r3 ðtÞ ¼ i½H V 1 ðtÞ V y3 ðtÞ; r3 ðtÞ ðR þ }Þr3 ðtÞ: ð9Þ The reduced Hamiltonian H is obtained by extracting the energy gap ee between the jgi and jei states from the excited-state contribution He of the system Hamiltonian (3). The Va(t) are defined as in Eq. (5), but with the reduced a instead of xa, carrier frequencies x a ¼ xa ee : x
ð10Þ
Eqs. (8) and (9) are valid in third order in the system–field interaction, allow for any pulse durations and automatically account for pulse-overlap effects. The numerical effort for the calculation of PPE via Eq. (9) is less demanding than that required by other nonperturbative methods [16,18–20,37]. It should be noted, in particular, that the PE polarization is obtained with a single propagation of the auxiliary density matrices r1, r2, r3 (for each pulse configuration). The construction of multi-time nonlinear response functions (which becomes prohibitively complicated for nonadiabatic and non-separable multi-mode systems) is thus avoided. The PE polarization depends not only on the detection time t, but also on the delay between the pulses, which are referred to as coherence time s = s2 s1 (delay between the first and the second pulse) and population time T = s3 s2 (delay between the second and the third pulse).
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If homodyne detection is employed, an integration over the detection time t is performed (time-integrated PE) and the signal is given by Z 2 hom ð11Þ S PE ðs; T Þ dtjP PE ðt; s; T Þj : This signal is sometimes referred to as ‘‘two-dimensional’’ [32], since it depends both on coherence and population times. We do not use this designation in order to avoid confusion with the heterodyne Fourier-transformed PE technique described below. Instead, we refer to the signal (11) as three-pulse PE. It reduces to the two-pulse PE if the second and the third pulses act simultaneously (T = 0). The transient-grating signal can be obtained from S hom PE ðs; T Þ as a scan along the population time T for a fixed coherence time s (normally, s = 0). In a peak-shift measurement, the coherence time at which the signal (11) reaches its maximum value is recorded as a function of the population time. The heterodyne detection (time-gated PE) preserves the time resolution by mixing of the signal with an additional heterodyne pulse (local-oscillator field). In this case, the spectrum can be evaluated as Z S het ðt; s; T Þ dsELO ðs tÞ cosðxLO s wLO ÞP PE ðs; s; T Þ; PE ð12Þ where ELO, wLO, and xLO denote the envelope, phase, and frequency of the local oscillator. The two-dimensional (2D) PE spectrum is obtained as the Fourier transform of the heterodyne signal (12) with respect to detection time t and coherence time s Z Z S PE ðxs ; xt ; T Þ ds dt expðixs sÞ expðixt tÞS het PE ðt; s; T Þ: ð13Þ The frequency xs, which is related to the delay between the first and the second pulse, is the so-called absorption frequency. The frequency xt is connected to the detection time t and is referred to as emission frequency. The case T = 0 corresponds to the two-pulse photon echo and is often referred to as 2D correlation spectrum. Measurements with finite T allow the monitoring the population dynamics and relaxation and are referred to as 2D relaxation spectra. In the limit of ideal detection, the local-oscillator envelope in Eq. (12) can be replaced by a Dirac delta-function, ELO(s t) = d(s t), and the 2D signal becomes Z Z ideal S PE ðxs ; xt ; T Þ ds dt expðixs sÞ expðixt tÞP PE ðt; s; T Þ; ð14Þ i.e., the double Fourier transform of the PE polarization. One can easily show that the finite duration of the localoscillator pulse can be taken into account by multiplying the ideal signal (14) with the Fourier transform of the envelope ELO(t). It is possible to record both the real and imaginary parts of SPE(xs, xt, T). As in our previous study [23], we restrict ourselves to the discussion of the real part, since it can be
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easier rationalized in terms of absorption and emission. The plots of the imaginary parts can be found in the supplementary material. 3. Model The resonant Raman (RR) experiment of Lawless and Mathies [35] has revealed that NB possesses about 40 Franck–Condon active modes. However, the main part of the RR intensity is provided by only few modes and the most prominent peaks correspond to the 590 cm1 and 1640 cm1 modes [35]. Vibrational coherences due to these two modes have been detected in the pioneering PE experiment by Becker et al. [38] (pulses of 6 fs duration), while the Franck–Condon activity and a pronounced excited-state displacement of the 590 cm1 mode have been revealed in a pump–probe study by Fragnito et al. [30] and in a PE experiment by Moshary et al. [31]. The recent homodyne PE experiments [24,32] report long-lived oscillations with a period of about 60 fs along the population time axis, while the signal is rather narrow along the coherence time axis and coherent recurrences in this variable can be observed only at low temperature [24]. Oscillations with the same period have also been resolved in peak-shift measurements [24,32] and it can be concluded that ultrafast excitation initializes coherent vibrational motion in the excited state of NB. The vibrational period of the 590 cm1 RR-active mode is about 57 fs, which implies that the vibrational excitation of this particular mode is usually detected in the PE experiments. The linear absorption spectrum of NB in acetonitrile [3,8] peaks at about 635 nm (15,748 cm1), while measurements in methanol [32] and poly(methyl mathacrylate) (PMMA) [24] yield a value of about 625 nm (16,000 cm1) for the absorption maximum. The fluorescence occurs at lower frequencies [8,24,32]. The Stokes shift can be estimated as 880 cm1 in acetonitrile and 855 cm1 in methanol. It is thus reasonable to assume that only the 590 cm1 mode is coherently excited by pulses with carrier frequencies in the vicinity of the absorption maximum if the pulse durations are about 25 fs [24,32] or longer. For a transparent interpretation of the measured signals, we consider separately the contributions of the strongly displaced 590 cm1 mode and of the remaining weakly coupled modes to the PE spectra. The approach is realized in a simplified manner: the 590 cm1 mode is assumed to be the only mode coupled to the electronic transition, while the electron–vibrational couplings of the remaining modes are neglected. The Franck–Condon active mode is represented as a displaced oscillator, and the corresponding system Hamiltonian is thus written as H g ¼ jgihg hgj; hg ¼ Xfby b þ 1=2g; ð15Þ D H e ¼ jeiðhe þ eÞhej; he ¼ X by b þ 1=2 pffiffiffi ðby þ bÞ : 2 ð16Þ
The second quantization representation has been adopted for the vibrational parts hg and he. The same vibrational frequency X is assumed for both electronic states (which is justified for the 590 cm1 mode by the RR experiment [35]), D is the dimensionless displacement of the excitedstate equilibrium geometry from that of the ground state and e is the vertical excitation energy. The vibrational relaxation (RrðtÞ in Eq. (1)) is introduced via a linear coupling of the system oscillator mode to a harmonic bath, characterized by an Ohmic spectral function J(xb) = gxb exp(xb/xc) [39], where g is a dimensionless system–bath coupling parameter and xc is the bath cut-off frequency. In this regime, the employed Redfield theory is equivalent to the damped Markovian harmonic oscillator model [1,40]. The assumption that the 590 cm1 mode only is responsible for the observed Stokes shift leads to reorganization energy of about 725–735 cm1 along this coordinate. For the calculations we choose D = 1.58 and the damping parameters xc = X, g = 0.1 (such weak system–bath coupling can be accurately described by Redfield theory [36,41]), which results in total reorganization energy of about 737 cm1. The value of D is twice higher than the displacement of the 590 cm1 mode reported in Ref. [35]. This is due to our wish to neglect a number of NB modes with small displacements D 6 0.3, but obtain a reasonable value for the reorganization energy. Note also that the value of D = 1.5 has been assumed in Ref. [34] for an adequate modeling (displaced oscillator) of the NB pump–probe spectra [30]. For a separate discussion of PE signals in the absence of strong electron–vibrational coupling, we put D 0 in Eq. (16) so that the model reduces to a two-level system (TLS). There is no vibrational relaxation in this case and the signal is controlled by the rate of the optical dephasing only. The rate of optical dephasing neg (Eq. (2)) depends on temperature and has been chosen as (130 fs)1 for low temperature (10 K) and (33 fs)1 for room temperature (300 K) (this choice leads to a good agreement with the available experimental data [8,24,32,42]). Note that since the overdamped low-frequency modes modify the material system relaxation by increasing the optical dephasing neg, their influence is taken into account in our model. All three laser pulses are assumed to be produced by the same source and have equal amplitudes (k1 = k2 = k3 = k), carrier frequencies (x1 = x2 = x3 = x) and durations. We assume Gaussian envelopes 2
EðtÞ ¼ expfðCtÞ g:
ð17Þ
The pulse duration at FWHM pffiffiffiffiffiffiffiffi (full width at half maximum) is thus given by 2 ln 2=C. Since the reduced Hamiltonian and RWA are employed, the vertical excitation energy e and carrier frequencies x of the pulses do not enter the calculations directly. Of importance is only the detuning between x and e, which will be specified for the particular examples considered.
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
4. Simulated photon echo 4.1. Integrated signal In this section, we give an overview of the available experimental data on homodyne PE spectra of NB and demonstrate the capability of our model to reproduce the experimentally observed trends. We separately discuss contributions of the strongly displaced 590 cm1 mode and of the other modes with small excited-state displacements and address several particular aspects observed in experiments in more detail. Recent homodyne PE measurements of NB include a study in PMMA by Nagasawa et al. [24], in acetonitrile by Larsen et al. [3] and Prall et al. [8] and in methanol by Dietzek et al. [32]. The study in PMMA addressed the temperature dependence of the signal. Three-pulse PE (3PPE) signals as a function of both coherence and population time as well as scans of the signal along the coherence time s for fixed population time T, transient-grating (TG) profiles and peak-shift (PS) dynamics have been reported for various temperatures ranging from 9 K to 295 K. The behavior of the 3PPE signal along the population time axis is characterized by long-lived oscillations with a period of about 60 fs and remains practically unchanged with increasing temperature. For the scans along s at fixed T, a coherent beating with the same period could be resolved at low temperatures only. The PS dynamics exhibits 60 fs oscillations as well, which are superimposed on a fast initial decay. At higher temperature, a decrease of the amplitude of the oscillations and of the initial (T = 0) value of the PS has been observed. The study in methanol [32] has yielded the 3PPE as a function of s and T, as well as TG and PS profiles at room temperature. The observed dynamical features of the signals are the same as in Ref. [24]. The measurements in acetonitrile [3,8] are somewhat less spectacular, since relatively long pulses have been employed (40–50 fs at FWHM, compared to 25 fs in Refs.
a
117
[24,32]). A comparison of the measurements of Refs. [24,3,8] confirms our predictions for the dependence of the PS dynamics on the pulse durations [22]: the peak shift values at short population times are larger if longer pulses are employed. Here we are concerned with the shorter-pulses results of Refs. [24,32]. In accordance with the experiments (pulses of 625–635 nm wavelength and 26–28 fs duration), we put C = 0.6X for the envelopes (Eq. (17)) and employ resonant excitation (x = e). As initial condition, we assume a Boltzmann distribution for the vibrational states in the electronic ground state. We start by discussing the temperature dependence of the integrated PE signal and the interpretation of the experimental results on NB in PMMA [24]. The observed temperature effects are most pronounced along the coherence time axis (i.e., for the cuts of the spectrum (11) at fixed values of population time T). Here, we consider the case T = 0 (zero delay between the second and the third pulses), i.e. two-pulse (2P) PE profiles. Note that the coherence time at which this signal reaches its maximum gives the initial value of PS. The experimentally observed signal modifications with increasing temperature exhibit two dominant features: (i) a coherent oscillation observed at low temperature disappears and (ii) the overall signal shape narrows and becomes more symmetric. Fig. 1 displays the 2PPE signal calculated at low temperature (10 K) and room temperature (300 K) for (a): the strongly displaced 590 cm1 mode in the system Hamiltonian, and (b): for the two-level system, i.e., with D 0 in Eq. (16). The signal calculated with the displaced system mode exhibits recurrences with the vibrational period at 10 K (dashed line), which are strongly suppressed at room temperature (solid line). The damping of the oscillation at higher temperature is the result of both a more efficient vibrational relaxation and increase of the dephasing between the two electronic states. The overall shape of the first peak is essentially independent of temperature.
b
2PPE
2PPE
10K 10K
300K
300K
0
50
coherence time τ, fs
100
150
0
50
100
150
coherence time τ, fs
Fig. 1. Two-pulse photon-echo (2PPE) signal calculated (a) with the displaced 590 cm1 mode and (b) for the two-level system. Solid and dashed lines correspond to calculations at room temperature (300 K) and low temperature (10 K), respectively.
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The signal calculated for the two-level system (Fig. 1b) changes its shape and narrows significantly for 300 K (solid line) compared to 10 K (dashed line). This effect is due to the faster optical dephasing. In contrast, to the impulsive limit, the TLS signal maximum is not at s = 0, but delivers a certain nonzero PS value for T = 0. Final pulse durations and the pulse overlap effect may thus influence significantly the PE profiles and lead to a meaningful PS dynamics even in the absence of vibrational excitation (see discussion below). Significant effects can be observed only if the pulse carrier frequencies are close to the transition frequency of the TLS. In the following, we interpret such modifications as contributions due to the modes with small excited-state displacements. The overall shape of the experimental 2PPE signal at low temperature can be understood as the superposition of the two contributions displayed in Fig. 1. At high temperatures, these two contributions can be hardly distinguished and the presence of the vibrationally excited mode cannot be resolved in the 2PPE signal (and other signal scans at fixed population times T). The narrowing of the experimental signal and the shift of its maximum (PS at T = 0) to smaller s occur due to the modes with negligible excited-state displacements. In order to illustrate the role of electron–vibrational coupling on the echo signal, the results of calculations at 10 K with various excited-state displacements are contrasted in Fig. 2 (the signals are scaled to obtain comparable intensity). The inset shows the initial (T = 0) peak shift as a function of D. The increase of D leads to the appearance of prominent oscillative structures in the 2PPE and to an exponential-like decrease of the peak-shift value at T = 0. The signals calculated with jDj 6 0.4 have shapes similar to the two-level system echo, although a weak vibrational excitation can be recognized for D = 0.4
Δ 0.5
1
1.5
2 25
2PPE, arbitary units (scaled)
20 15 10 5 0
—30
0
30
60
τ, fs
90
Peak Shift (T=0), fs
0
120
Fig. 2. Two-pulse photon-echo (2PPE) signal calculated for the 590 cm1 mode with the excited-state displacements D = 0.4 (dashed line) and D = 0.8 (dotted-dashed line) at low temperature (10 K). The curves of Fig. 1 are the solid lines. The inset shows the initial PS value (T = 0) as a function of D.
(dashed line in Fig. 2). The qualitative change to the coupled-mode shape takes place for jDj P 0.6 so that the 2PPE profiles obtained with D = 0.8 (dotted-dashed line in Fig. 2) and D = 1.58 look rather similar. The experiments prove that in contrast to the 2PPE signal, the TG technique and PS measurements are capable to visualize vibrational excitation of the 590 cm1 mode at room temperature as well [24,32]. The PS dynamics and TG profiles calculated at 300 K are shown in Fig. 3a and b, respectively. If no vibrational motion is initialized upon excitation (two-level system, solid lines), an exponentiallike decay of the PS is observed (from 14 fs at T = 0 to zero at T P 80 fs). The TG signal reaches its maximum at the same time scale, which is to be expected, since the TG scan is recorded at s = 0. The calculations with the displaced mode (dashed lines in Fig. 3) illustrate that vibrational coherences can be well captured by both PS and TG measurements. One can observe a decrease in the oscillation amplitude which is due to relaxation processes and a very slow overall decay of the signals. As has been proposed for the 2PPE signal, the experimental PS and TG dynamics [24,32] can be interpreted as the superposition of the two contributions. The weakly displaced modes are responsible for the PS decay and increase of the TG signal at short population times, whereas the oscillations are due to the strong electron–vibrational coupling of the 590 cm1 mode which also determines the signal dynamics at long T. The simulations of the PS dynamics at lower temperature (Fig. 4) result in a larger amplitude of the oscillations for the coupled mode and a larger initial value (T = 0) and slower decay for the two-level system, which is in agreement with the experimentally observed behaviour [24]. To clarify its dependence on electron–vibrational coupling we have also calculated the PS for the 590 cm1 mode with the small excited-state displacement D = 0.4 (dashed line in Fig. 4, cf. dashed line in Fig. 2). At longer population times (T P 100–150 fs), the difference between the PS dynamics obtained with D = 0.4 and the standard D = 1.58 is only in the oscillations amplitude. The short-time dynamics obtained with D = 0.4 is qualitatively different and proves that the weakly displaced modes are responsible for the initial decay of the PS. We underline that the PS decay of the TLS can be observed exclusively due to resonant pulses of final durations and does not exist in the impulsive limit. The calculations suggest that the influence of the electron–vibrational coupling on the PS evolution is two-fold: the initialized coherent vibrational motion can be ‘‘destructive’’ at short population times (relatively small PS values), but it definitely allows for a longer survival of the non-stationary PS dynamics. Fig. 5 compares the integrated 3PPE signals as a function of the both delay times s and T (see Refs. [24,32] for the experimental records). The signal calculated with D = 0 is shown Fig. 5a up to T = 100 fs (no changes are observed for longer population times). The decay of the peak shift (cf. Fig. 3a) is accompanied by a broadening
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
15
b Transient grating signal (scaled)
a
peak shift, fs
10
5
0
—5
119
0
100
200
300
400
0
100
200
300
400
population time T, fs
population time T, fs
Fig. 3. Peak-shift (a) and transient-grating (b) profiles at 300 K calculated for the two-level system (solid lines) and the strongly displaced 590 cm1 mode (dashed lines) in the system Hamiltonian.
dynamics (dashed lines in Fig. 3), the oscillations of the width of the signal pointed out in Ref. [32] must have the same period as observed in the PS and TG profiles.
24 20
peak shift, fs
16
4.2. Fourier-transformed two-dimensional PE
12
In contrast to the integrated PE measurements, the 2D heterodyne experiment on NB provided relatively little dynamical information [26]. It rather served as a demonstration of feasibility of the measurement. Only one peak with a width of about 800–850 cm1 has been resolved, and its dynamics has been recorded up to T = 100 fs with time steps of 5–10 fs. A theoretical analysis within the multimode Brownian oscillator has been performed and it has been argued that a higher-lying excited electronic state has to be taken into account. However, the particular peak shapes of the experimental spectrum have not been accurately reproduced by the simulations and their interpretation remains unclear. Considering homodyne PE spectra of NB, it is reasonable to assume that the Franck–Condon activity of the 590 cm1 mode manifests itself in the 2D heterodyne spectra as well, and may be responsible for the peculiar shapes of the peaks. In this section we propose an analysis of the experimental correlation and relaxation
8 4 0 —4 —8
0
100
200
300
population time T, fs Fig. 4. Peak-shift profiles at 10 K calculated with the same system parameters as in Fig. 3a (solid lines). The dashed line represents PS dynamics obtained for the 590 cm1 mode with D = 0.4.
of the signal along coherence time s. Fig. 5b and c demonstrate the modulations of the signal due to the presence of the underdamped vibrational mode (up to T = 100 fs in (b) and up to T = 400 fs in (c)). As can be recognized from Fig. 5b and c as well as by comparison of PS and TG 60
40 20 0 —20 —40 20
40
60
80
population time T, fs
100
60
coherence time , fs
coherence time , fs
coherence time , fs
60
40 20 0 —20 —40 20
40
60
80
population time T, fs
100
40 20 0 —20 —40 0
100
200
300
400
population time T, fs
Fig. 5. Contour plots of the three-pulse photon-echo (3PPE) signal at 300 K. Simulations for the two-level system are shown in (a) up to T = 100 fs, since no changes are observed at longer time scale. The signal evolution obtained with the displaced 590 cm1 mode is presented in (b) and (c) for up to T = 100 fs and up to T = 400 fs, respectively.
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1400 1600 1800 2000 2200 2400
Fig. 6. Two-dimensional (a) correlation (T = 0) and (b,c) relaxation (T = 20, 50 fs) spectra calculated with the displaced 590 cm1 mode in the system Hamiltonian at 300 K.
spectra employing the simplified model of NB introduced above. The most pronounced deviation of the experimental correlation spectrum (T = 0) from the shape predicted by the simulations [26] is a strong distortion from the diagonal orientation in the spectral range where emission and absorption frequencies exceed the carrier frequency of the pulses, xt, xs > x. In this region, the spectrum is elongated parallel to the absorption-frequency (xs) axis. This shoulder along increasing xs observed at T = 0 gradually disappears with increasing T (relaxation spectra) and a slight elongation to larger xt develops. Other particular experimental features are a displacement of the signal maximum from the diagonal (xt = xs) to somewhat larger xs for all T, and an overall intensity decrease with increasing population time [26]. Pulses of 41 fs duration (FWHM) and center wavelength of 595 nm (16,807 cm1) have been employed in the experiment. The linear absorption spectrum [3,8] peaks at about 635 nm (15,748 cm1), i.e., the system has been excited about 1100 cm1 above the absorption maximum, which corresponds to about two vibrational quanta of the strongly displaced 590 cm1 mode. Most probably, the 41 fs pulses prepare a superposition of a couple higher-lying vibrational states of this mode. The probability of vibrational excitation of the higher-frequency strong RR mode (1640 cm1), on the other hand, remains negligible. The weakly displaced modes are not expected to significantly contribute to the spectra in the considered case of strongly off-resonant excitation (the intensity of linear absorption at 595 nm is 50% lower than at the maximum). We thus limit the discussion to 2D spectra calculated with the strongly coupled 590 cm1 mode in the system Hamiltonian. Fig. 6 shows 2D correlation (T = 0) and relaxation (T = 20 fs, 50 fs) spectra calculated with parameters corresponding to the experimental conditions: the pulse durations are determined by C = 0.366X, the carrier frequencies of the pulses are x = e + 2X, and temperature is 300 K.1 The actual absorption and emission frequencies 1
To facilitate a comparison of our 2D PE spectra with those measured by Brixner et al. [26], note that 100 cm1 = 0.019 fs1 and that the orientations of the frequency axes in Ref. [26] is opposite to ours.
xs and xt have been reduced by the energy of the zero–zero transition and the reduced carrier frequencies of the pulses 1950 cm1. are x We succeeded to reproduce the particular shape of the experimental spectrum at T = 0 and its dynamical changes with T with good accuracy. The intensity loss with increasing T can be recognized and is easily explained as a fast vibrational relaxation of the population of the higher-lying levels initially excited. A further analysis is required, however, for an interpretation of the observed line shapes. Assuming that our simple model is sufficient to capture the vibronic dynamics of NB, we can try to gain more insight and understanding by improving ‘‘experimental’’ resolution. First, we get rid of the spectral broadening induced by relaxation and dephasing processes and simulate the signal with the same parameters as in Fig. 6, but for lower temperature. Fig. 7 shows 2D correlation (T = 0) and relaxation (T = 20 fs, 50 fs) spectra calculated at 10 K. As can be recognized from the correlation spectrum (Fig. 7a), two vibrational states with quantum numbers m = 3 and m = 4 are initially populated since two peaks can be distinguished at absorption frequencies xs 3X = 1770 cm1 and xs 4X = 2360 cm1. The peak corresponding to the transition to the higher level m = 4 (xs = 4X) is much weaker than that at xs = 3X. This level is populated less than the m = 3 level since the reduced carrier frequency of the pulses is closer to 3X, and it decays faster due to vibrational relaxation. That is why the elongation of the spectrum towards higher values of xs gradually disappears with increasing population time T. A prominent negative wing at higher emission frequencies, which is present at T = 0, decreases with increasing population time, and an effective elongation of the relaxation spectra towards larger values of xt is observed. The shift of the peak maximum from the diagonal to larger values of absorption frequency at high temperatures is caused by the averaging of the two transitions resolved at 10 K. At T = 50 fs, the maximum of the 300 K-signal (Fig. 6c) moves to somewhat smaller values of xs as compared to T = 0, 20 fs due to fast depopulation of the fourth (m = 4) excited vibrational level, but remains offdiagonal.
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
c 2600
2400
2400
2400
2200
2200
2200
t
t
b 2600
t
a 2600
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2000
2000
1800
1800
1800
1600
1600
1600
1400 1600 1800 2000 2200 2400
121
1400 1600 1800 2000 2200 2400
1400 1600 1800 2000 2200 2400
Fig. 7. Two-dimensional (a) correlation (T = 0) and (b,c) relaxation (T = 20, 50 fs) spectra calculated with the same parameters as in Fig. 6 but at low temperature (10 K).
the coupling cross peaks at xs = 2X, xt = 3X and at xs = 3X, xt = 4X. This effect arises due to a decrease of the negative contribution compared to T = 0. If one compares low-temperature simulations of Figs. 7 and 8, it may appear that the effect of longer pulses (Fig. 7) is just to cut out the part which is resonant with the pulse carrier frequency from the spectra obtained with shorter pulses. However, this is not really true, since shorter pulses also initiate a different (more coherent) system dynamics. On the other hand, the finite duration of the local-oscillator pulse extracts the spectral part close to its frequency from the ideally-detected signal (limit of an infinitely short local-oscillator pulse). The ideal detection can thus be experimentally realized if the 2D PE spectrum along the absorption and emission frequencies fits the corresponding laser pulse bandwidths. Alternatively, the ideally-detected signal may be obtained by a tuning of the frequency of the local-oscillator pulse. 5. Conclusions We have proposed a simplified model picture of NB which provides a comprehensive description of available PE experimental data. The measured integrated PE signals have been interpreted as a superposition of two contributions. One contribution arises from the strongly displaced underdamped 590 cm1 mode, which is coherently excited by the ultrashort pulses. The other one is monotonous in time and is due to the presence of the modes with no or
2500
2500
2250
2250
2000
2000 t
t
The spectral range accessible with 41 fs pulses is relatively narrow. More information on the underlying dynamics can be gained if shorter pulses are employed. Since pulses of the 25 fs duration seem to be routinely available in PE experiments [24,32], we have performed a simulation with C = 0.6X. The obtained 2D correlation (T = 0) and relaxation (T = 50 fs) spectra are shown in Fig. 8 (the other parameters are as in Fig. 7). As expected, a broader spectral range is covered, although the most intense peaks correspond to the transition closest to the carrier frequency and are at the same position as in Fig. 7. Several cross peaks separated by the vibrational frequency can be resolved. As has been outlined previously [23], two types of cross peaks can be distinguished in 2D spectra of the displaced oscillator: ‘‘emission peaks’’, which represent emission from each excited vibrational level to the electronic ground-state manifold, and ‘‘coupling peaks’’, which arise due to coupling of transitions with different frequencies through the common ground state. The cross peaks above the diagonal xt = xs correspond to the coupling peaks, while the cross peaks under the diagonal represent a superposition of the coupling peaks with emission peaks. Both types of cross peaks may experience intensity modifications with T due to vibrational motion and relaxation, i.e., due to changes in populations of different vibrational levels in the excited electronic state. An overall intensity loss is expected at T = 50 fs due to fast vibrational relaxation of the initially excited levels. Indeed, the peaks in Fig. 8b are less intense than the peaks in the correlation spectrum in Fig. 8a, with the exception of
1750
1750
1500
1500
1250
1250
1250 1500 1750 2000 2250 2500
1250 1500 1750 2000 2250 2500
Fig. 8. Two-dimensional (a) correlation (T = 0) and (b) relaxation (T = 50 fs) spectra calculated with the same parameters as in Fig. 7 but with shorter pulses, C = 0.6X (25 fs at FWHM).
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negligible excited-state displacements, provided resonant pulses with finite durations are employed. We have discussed the dynamics and temperature dependence of the 2PPE, analyzed TG and PS profiles and presented contour plots of the 3PPE signal as a function of both coherence time and population time. We have addressed modifications of homodyne PE spectra due to the electron–vibrational coupling and pointed out the role of the pulse overlap effect in the PS dynamics. For the 2D heterodyne spectra, only the contribution corresponding to the displaced 590 cm1 mode has been considered, since off-resonant (with respect to the absorption maximum) pulses have been employed in the experiment and the contribution of the weakly displaced modes is therefore not expected to be significant. We have provided a new interpretation of the experimental 2D line shapes and have performed simulations which predict 2D correlation and relaxation spectra of NB for improved resolution conditions. If the strongly displaced 590 cm1 mode could be experimentally resolved (as peaks separated by the corresponding vibrational frequency), this would be the first experimental detection of electron–vibrational coupling in 2D electronic PE spectra. The study demonstrates the capability of PE spectroscopy to monitor underdamped vibrational dynamics in the excited electronic state and to provide information on relaxation time scales. 2D heterodyne spectra can be very informative and directly interpreted if the broadening is minimized and the employed pulses are short compared to the time scale of vibrational motion. Pulse durations and frequencies can be varied in order to address different dynamical features and spectral ranges. Under good resolution conditions, the 2D technique appears to be more advantageous than conventional one- and two-color PS measurements. However, the interpretation of cross peaks in 2D spectra requires some caution. If underdamped coherent vibrational motion is initiated by the excitation (this can be clarified by performing, e.g., a TG measurement), the modifications of the spectra due to electron–vibrational coupling have to be taken into account. Acknowledgement DE is grateful to Michael Thoss and Tobias Brixner for inspiring discussions and to DFG for financial support. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemphys. 2007.07.010. References [1] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, University Press, Oxford, 1995.
[2] W.P. de Boeij, M.S. Pshenichnikov, D.A. Wiersma, Annu. Rev. Phys. Chem. 49 (1998) 99. [3] D. Larsen, K. Ohta, Q. Xu, M. Cyrier, G.R. Fleming, J. Chem. Phys. 114 (2001) 8008. [4] K. Ohta, D. Larsen, M. Yang, G.R. Fleming, J. Chem. Phys. 114 (2001) 8020. [5] W.P. de Boeij, M.S. Pshenichnikov, D.A. Wiersma, Chem. Phys. Lett. 253 (1996) 53. [6] W.P. de Boeij, M.S. Pshenichnikov, D.A. Wiersma, J. Phys. Chem. 100 (1996) 11806. [7] R. Agarwal, B.S. Prall, A.H. Rizvi, M. Yang, G.R. Fleming, J. Chem. Phys. 116 (2002) 6243. [8] B.S. Prall, D.Y. Parkinson, G.R. Fleming, J. Chem. Phys. 123 (2005) 054515. [9] S. Mukamel, Annu. Rev. Phys. Chem. 51 (2000) 691. [10] D.M. Jonas, Annu. Rev. Phys. Chem. 54 (2003) 425. [11] S. Woutersen, P. Hamm, J. Phys.: Condens. Matter 14 (2002) R1035. [12] T. Brixner, J. Stenger, H.M. Vaswani, M. Cho, R.E. Blankenship, G.R. Fleming, Nature 434 (2005) 625. [13] M. Cho, H.M. Vaswani, T. Brixner, J. Stenger, G.R. Fleming, J. Phys. Chem. B 109 (2005) 10542. [14] L. Seidner, G. Stock, W. Domcke, J. Chem. Phys. 103 (1995) 3998. [15] W. Domcke, G. Stock, Adv. Chem. Phys. 100 (1997) 1. [16] S. Meyer, V. Engel, Appl. Phys. B 71 (2000) 293. [17] T. Renger, V. May, O. Ku¨hn, Phys. Rep. 343 (2001) 137. [18] T. Kato, Y. Tanimura, Chem. Phys. Lett. 341 (2001) 329. [19] S. Mukamel, D. Abramavicius, Chem. Rev. 104 (2004) 2073. [20] T. Mancˇal, A.V. Pisliakov, G.R. Fleming, J. Chem. Phys. 124 (2006) 234504. [21] P. Kjellberg, T. Pullerits, J. Chem. Phys. 124 (2006) 024106. [22] M.F. Gelin, D. Egorova, W. Domcke, J. Chem. Phys. 123 (2005) 164112. [23] D. Egorova, M.F. Gelin, W. Domcke, J. Chem. Phys. 126 (2007) 019706. [24] Y. Nagasawa, K. Seike, T. Muromoto, T. Okada, J. Phys. Chem. A 107 (2003) 2431. [25] S.M. Gallagher Faeder, D.M. Jonas, J. Phys. Chem. A 103 (1999) 10489. [26] T. Brixner, T. Mancˇal, I.V. Stiopkin, G.R. Fleming, J. Chem. Phys. 121 (2004) 4221. [27] D. Keusters, H.-S. Tan, W.S. Warren, J. Phys. Chem. A 103 (1999) 10369. [28] C.N. Borca, T. Zhang, X. Li, S.T. Cundiff, Chem. Phys. Lett. 416 (2005) 311. [29] X. Li, T. Zhang, C.N. Borca, S.T. Cundiff, Phys. Rev. Lett. 96 (2006) 057406. [30] H.L. Fragnito, J.Y. Bigot, P.C. Becker, C.V. Shank, Chem. Phys. Lett. 160 (1989) 101. [31] F. Moshary, M. Arend, R. Friedberg, S.R. Hartmann, Phys. Rev. A 46 (1992) R33. [32] B. Dietzek, N. Christensson, P. Kjellberg, T. Pascher, T. Pullertis, A. Yartsev, Phys. Chem. Chem. Phys. 9 (2007) 701. [33] J. Hauer, T. Buckup, M. Motzkus, J. Chem. Phys. 125 (2006) 061101. [34] W. Pollard, H.L. Fragnito, J.-Y. Bigot, C.V. Shank, R.A. Mathies, Chem. Phys. Lett. 168 (1990) 239. [35] M.K. Lawless, R.A. Mathies, J. Chem. Phys. 96 (1992) 8037. [36] D. Egorova, M. Thoss, H. Wang, W. Domcke, J. Chem. Phys. 119 (2003) 2761. [37] H. Wang, M. Thoss, Chem. Phys. Lett. 389 (2004) 43. [38] P.C. Becker, H.L. Fragnito, J.Y. Bigot, C.H. Brito Cruz, R.L. Fork, C.V. Shank, Phys. Rev. Lett. 63 (1989) 505. [39] U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1999. [40] T. Takagahara, E. Hanamura, R. Kubo, J. Phys. Soc. Jpn. 44 (1978) 728. [41] A.G. Redfield, Adv. Magn. Reson. 1 (1965) 1. [42] S. Nakanishi, H. Ohta, N. Makimoto, H. Itoh, Phys. Rev. B 45 (1992) 2825.
Chemical Physics 341 (2007) 113–122 www.elsevier.com/locate/chemphys
Analysis of vibrational coherences in homodyne and two-dimensional heterodyne photon-echo spectra of Nile Blue Dassia Egorova b
a,* ,
Maxim F. Gelin b, Wolfgang Domcke
a
a Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany Department of Chemistry and Biochemistry, College Park, University of Maryland, MD 20742-4454, USA
Received 16 February 2007; accepted 6 July 2007 Available online 17 July 2007 This work is dedicated to Douwe Wiersma.
Abstract We propose a simple model which allows a comprehensive interpretation of the available experimental photon-echo data on Nile Blue. Homodyne signals such as two- and three-pulse photon echo, transient grating and three-pulse peak shift are addressed. An explanation of their time evolution is provided with a single parameter set. A new interpretation of two-dimensional (2D) heterodyne experimental spectra is proposed which emphasizes the strong electron–vibrational coupling of the 590 cm1 mode of Nile Blue. The model is used for predictions of 2D correlation and relaxation signals monitored with improved resolution (low temperature, short pulse durations). It is argued that such measurements could provide the first experimental detection of the signatures of electron–vibrational coupling in 2D photon-echo spectra. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Electron–vibrational coupling; 2D electronic photon-echo pectroscopy; Nile Blue
1. Introduction Ultrafast photon-echo (PE) spectroscopy belongs to the realm of the four-wave mixing (FWM) techniques. The observed spectra are thus determined by the third-order laser-induced polarization. The third-order polarization depends on the system dynamics, as well as carrier frequencies, durations, and relative time delays of the laser pulses, and obeys the corresponding phase-matching condition [1,2]. If a density-matrix description is employed for the system dynamics, then PE spectroscopy can be shown to reflect the dynamics of both populations and coherences of the density matrix in real time (in contrast to pump– probe and fluorescence spectroscopy, which primarily deliver information about the population dynamics).
*
Corresponding author. Tel.: +49 89 28913618; fax: +49 89 28913622. E-mail address: [email protected] (D. Egorova).
0301-0104/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.07.010
Two types of detection schemes of PE signals can be distinguished [1,2]: In the homodyne or time-integrated measurement, the signal intensity is recorded as a function of the delays between the pulses, whereas within heterodyne detection (time-gated PE), the dependence of the signal on the detection time can be monitored as well. Homodyne PE techniques include three-pulse PE, two-pulse PE, transient grating and peak-shift measurements. The latter is the most popular and frequently employed method, since the peak-shift dynamics is believed to provide information on solvation mechanisms [2–6]. Two-color peak-shift PE spectroscopy, which exploits the dependence of the peak shift on the carrier frequencies of the pulses, has been suggested and implemented as well [7,8]. As for the heterodyne detection, the so-called two-dimensional (2D) or Fourier-transformed PE [9,10] has gained particular attention in recent years. This approach, which has proven invaluable for the determination of vibrational couplings in the infrared [11], has recently been successfully implemented in the
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optical domain, providing a direct information on electronic couplings in the seven-chromophore Fenna–Matthews–Olson protein [12,13]. In the standard perturbative approach of nonlinear optics, the third-order optically induced polarization is expressed in terms of the nonlinear response functions of the system [1]. As the material systems under investigation become more and more complex (that is, multidimensional), the evaluation of the polarization in terms of the nonlinear response functions often is no longer possible and alternative strategies such as a nonperturbative treatment of the system–field interaction [14,15] may be more advantageous. A drawback of the nonperturbative method is that the obtained total polarization must be further analyzed in order to single out a specific combination of wave vectors responsible for a particular FWM signal. There exist several realizations of the nonperturbative calculation of the threepulse PE polarization [16–21]. In the present work, we use a recently developed method [22,23] which is computationally more efficient than the other available schemes. Apart from the choice of a computational method for evaluation of the third-order polarization, an appropriate model for the underlying system dynamics is necessary for the interpretation of the experimentally observed PE signals. The (multimode) Brownian oscillator model has become the working horse in theoretical optical spectroscopy. It provides an adequate description provided a material system under study can be described within the Born– Oppenheimer approximation by a few electronic states which are coupled to a collection of vibrational modes. This model has frequently been employed for simulations of integrated PE signals [1–6,8,24]. However, with a few notable exceptions [23,25,26], only the electronic degrees of freedom have explicitly been considered in theoretical descriptions of 2D PE spectroscopy, while the vibrations were taken into account as a phonon bath [9,12,13,27–29,19]. If the signals exhibit oscillatory behaviors in the time domain which are induced primarily by a single Franck– Condon active vibrational mode, an underdamped Brownian oscillator may be a sufficient first-order approximation. We have recently used this model to study the influence of electron–vibrational coupling, finite pulse durations, and broadening mechanisms on optical PE signals [22,23]. In particular, it has been shown that the vibrational mode strongly coupled to the electronic transition has to be taken into account explicitly in the interpretation of 2D electronic spectra [23]. It is thus of interest to investigate whether these predictions can be experimentally verified for a real physical system. Nile Blue (NB) dye is a perfect candidate for this purpose. The strong electron–vibrational coupling in this molecule has been revealed already in a pump–probe experiment by Fragnito et al. [30] and in a PE study by Moshary et al. [31]. The recent PE experiments on NB in various solvents include both homodyne [3,4,8,24,32] and 2D heterodyne [26] techniques. The transients measured via the homodyne PE detection exhibit vibrational coher-
ences with a period of about 60 fs [24,32] and look very similar to those obtained in our previous simulations within an underdamped Brownian oscillator model [22]. Oscillations with the same period have been resolved in transient absorption measurements [30,33] and the pump–probe spectra of Ref. [30] could be reasonably well modeled within a free displaced oscillator [34]. On the other hand, 2D PE spectra of NB in acetonitrile measured by Brixner et al. [26] at room temperature exhibit rather broad structures and are not very informative. The observed line shapes and their modification by various relaxation mechanisms are not well understood. It is not unreasonable to assume that the active mode of NB may have a strong effect on the shapes of the 2D PE spectra. In this paper, we summarize the recent experimental PE data on NB and propose a simple and elucidating physical model for their interpretation. We start with the time-integrated PE and show that the model is capable to reproduce and explain major features in the observed PE signals. In the second part, we employ the same model for simulations of 2D electronic spectra reported in Ref. [26]. In order to interpret the observed line shapes, we present the predictions of our model for 2D spectra measured under improved resolution conditions (low temperature and short pulse durations). In contrast to previous simulations [8,24,26], which take into account 40 Franck–Condon active modes of NB [35], we do not attempt to reproduce quantitatively the different optical PE signals of NB. Our goal is the development of a simple and transparent model, which allows to understand what particular features of the system dynamics and relaxation processes are responsible for specific signatures in the PE signals, both in the time and in the frequency domain. 2. Method A detailed description of the employed method is given elsewhere [22,23] and we only briefly review it here. We start from the kinetic equation for the reduced density matrix r (h = 1) ot rðtÞ ¼ i½H þ H int ðtÞ; rðtÞ ðR þ }ÞrðtÞ;
ð1Þ
where H is the system Hamiltonian and Hint(t) describes the field–matter interaction. Environmental degrees of freedom have been traced out; their influence is accounted for by the vibrational relaxation operator R and pure electronic dephasing (optical dephasing) }. In this work, RrðtÞ is described within Redfield theory [36] and electronic dephasing is introduced phenomenologically via }rðtÞ neg P g rðtÞP e þ H:c:;
ð2Þ
neg being the optical dephasing rate. We assume that the material system Hamiltonian can be expressed as the sum of the ground state (jgi) and excited state (jei) contributions H ¼ Hg þ He
ð3Þ
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
and write the system–field interaction in the electric dipole and rotating-wave approximations as H int ðtÞ ¼
3 X
V a ðtÞ expðika rÞ þ H:c:;
ð4Þ
a¼1
where V a ðtÞ ¼ ka Ea ðt sa Þ expfixa tgX :
ð5Þ
ka, ka, Ea(t sa), sa, and xa denote wave vector, amplitude, dimensionless envelope, central time and carrier frequencies of the pulses and the transition dipole moment operator is defined as l ¼ X þ X y;
X ¼ jgihej:
ð6Þ
The total polarization is calculated via P ðtÞ ¼ hlrðtÞi:
ð7Þ
The specific PE contribution to the polarization obeys the phase-matching condition k = k1 + k2 + k3. As has been shown in Refs. [22,23], within the rotating-wave approximation (RWA) the PE polarization PPE can be calculated by performing three simultaneous propagations of the auxiliary density matrices ri(t) (i = 1, 2, 3) P PE ðtÞ ¼ expfiðk1 þ k2 þ k3 ÞrghX ðr1 ðtÞ r2 ðtÞ r3 ðtÞÞi þ c:c:;
ð8Þ
ot r1 ðtÞ ¼ i½H V 1 ðtÞ V y2 ðtÞ V y3 ðtÞ; r1 ðtÞ ðR þ }Þr1 ðtÞ; ot r2 ðtÞ ¼ i½H V 1 ðtÞ V y2 ðtÞ; r2 ðtÞ ðR þ }Þr2 ðtÞ; ot r3 ðtÞ ¼ i½H V 1 ðtÞ V y3 ðtÞ; r3 ðtÞ ðR þ }Þr3 ðtÞ: ð9Þ The reduced Hamiltonian H is obtained by extracting the energy gap ee between the jgi and jei states from the excited-state contribution He of the system Hamiltonian (3). The Va(t) are defined as in Eq. (5), but with the reduced a instead of xa, carrier frequencies x a ¼ xa ee : x
ð10Þ
Eqs. (8) and (9) are valid in third order in the system–field interaction, allow for any pulse durations and automatically account for pulse-overlap effects. The numerical effort for the calculation of PPE via Eq. (9) is less demanding than that required by other nonperturbative methods [16,18–20,37]. It should be noted, in particular, that the PE polarization is obtained with a single propagation of the auxiliary density matrices r1, r2, r3 (for each pulse configuration). The construction of multi-time nonlinear response functions (which becomes prohibitively complicated for nonadiabatic and non-separable multi-mode systems) is thus avoided. The PE polarization depends not only on the detection time t, but also on the delay between the pulses, which are referred to as coherence time s = s2 s1 (delay between the first and the second pulse) and population time T = s3 s2 (delay between the second and the third pulse).
115
If homodyne detection is employed, an integration over the detection time t is performed (time-integrated PE) and the signal is given by Z 2 hom ð11Þ S PE ðs; T Þ dtjP PE ðt; s; T Þj : This signal is sometimes referred to as ‘‘two-dimensional’’ [32], since it depends both on coherence and population times. We do not use this designation in order to avoid confusion with the heterodyne Fourier-transformed PE technique described below. Instead, we refer to the signal (11) as three-pulse PE. It reduces to the two-pulse PE if the second and the third pulses act simultaneously (T = 0). The transient-grating signal can be obtained from S hom PE ðs; T Þ as a scan along the population time T for a fixed coherence time s (normally, s = 0). In a peak-shift measurement, the coherence time at which the signal (11) reaches its maximum value is recorded as a function of the population time. The heterodyne detection (time-gated PE) preserves the time resolution by mixing of the signal with an additional heterodyne pulse (local-oscillator field). In this case, the spectrum can be evaluated as Z S het ðt; s; T Þ dsELO ðs tÞ cosðxLO s wLO ÞP PE ðs; s; T Þ; PE ð12Þ where ELO, wLO, and xLO denote the envelope, phase, and frequency of the local oscillator. The two-dimensional (2D) PE spectrum is obtained as the Fourier transform of the heterodyne signal (12) with respect to detection time t and coherence time s Z Z S PE ðxs ; xt ; T Þ ds dt expðixs sÞ expðixt tÞS het PE ðt; s; T Þ: ð13Þ The frequency xs, which is related to the delay between the first and the second pulse, is the so-called absorption frequency. The frequency xt is connected to the detection time t and is referred to as emission frequency. The case T = 0 corresponds to the two-pulse photon echo and is often referred to as 2D correlation spectrum. Measurements with finite T allow the monitoring the population dynamics and relaxation and are referred to as 2D relaxation spectra. In the limit of ideal detection, the local-oscillator envelope in Eq. (12) can be replaced by a Dirac delta-function, ELO(s t) = d(s t), and the 2D signal becomes Z Z ideal S PE ðxs ; xt ; T Þ ds dt expðixs sÞ expðixt tÞP PE ðt; s; T Þ; ð14Þ i.e., the double Fourier transform of the PE polarization. One can easily show that the finite duration of the localoscillator pulse can be taken into account by multiplying the ideal signal (14) with the Fourier transform of the envelope ELO(t). It is possible to record both the real and imaginary parts of SPE(xs, xt, T). As in our previous study [23], we restrict ourselves to the discussion of the real part, since it can be
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easier rationalized in terms of absorption and emission. The plots of the imaginary parts can be found in the supplementary material. 3. Model The resonant Raman (RR) experiment of Lawless and Mathies [35] has revealed that NB possesses about 40 Franck–Condon active modes. However, the main part of the RR intensity is provided by only few modes and the most prominent peaks correspond to the 590 cm1 and 1640 cm1 modes [35]. Vibrational coherences due to these two modes have been detected in the pioneering PE experiment by Becker et al. [38] (pulses of 6 fs duration), while the Franck–Condon activity and a pronounced excited-state displacement of the 590 cm1 mode have been revealed in a pump–probe study by Fragnito et al. [30] and in a PE experiment by Moshary et al. [31]. The recent homodyne PE experiments [24,32] report long-lived oscillations with a period of about 60 fs along the population time axis, while the signal is rather narrow along the coherence time axis and coherent recurrences in this variable can be observed only at low temperature [24]. Oscillations with the same period have also been resolved in peak-shift measurements [24,32] and it can be concluded that ultrafast excitation initializes coherent vibrational motion in the excited state of NB. The vibrational period of the 590 cm1 RR-active mode is about 57 fs, which implies that the vibrational excitation of this particular mode is usually detected in the PE experiments. The linear absorption spectrum of NB in acetonitrile [3,8] peaks at about 635 nm (15,748 cm1), while measurements in methanol [32] and poly(methyl mathacrylate) (PMMA) [24] yield a value of about 625 nm (16,000 cm1) for the absorption maximum. The fluorescence occurs at lower frequencies [8,24,32]. The Stokes shift can be estimated as 880 cm1 in acetonitrile and 855 cm1 in methanol. It is thus reasonable to assume that only the 590 cm1 mode is coherently excited by pulses with carrier frequencies in the vicinity of the absorption maximum if the pulse durations are about 25 fs [24,32] or longer. For a transparent interpretation of the measured signals, we consider separately the contributions of the strongly displaced 590 cm1 mode and of the remaining weakly coupled modes to the PE spectra. The approach is realized in a simplified manner: the 590 cm1 mode is assumed to be the only mode coupled to the electronic transition, while the electron–vibrational couplings of the remaining modes are neglected. The Franck–Condon active mode is represented as a displaced oscillator, and the corresponding system Hamiltonian is thus written as H g ¼ jgihg hgj; hg ¼ Xfby b þ 1=2g; ð15Þ D H e ¼ jeiðhe þ eÞhej; he ¼ X by b þ 1=2 pffiffiffi ðby þ bÞ : 2 ð16Þ
The second quantization representation has been adopted for the vibrational parts hg and he. The same vibrational frequency X is assumed for both electronic states (which is justified for the 590 cm1 mode by the RR experiment [35]), D is the dimensionless displacement of the excitedstate equilibrium geometry from that of the ground state and e is the vertical excitation energy. The vibrational relaxation (RrðtÞ in Eq. (1)) is introduced via a linear coupling of the system oscillator mode to a harmonic bath, characterized by an Ohmic spectral function J(xb) = gxb exp(xb/xc) [39], where g is a dimensionless system–bath coupling parameter and xc is the bath cut-off frequency. In this regime, the employed Redfield theory is equivalent to the damped Markovian harmonic oscillator model [1,40]. The assumption that the 590 cm1 mode only is responsible for the observed Stokes shift leads to reorganization energy of about 725–735 cm1 along this coordinate. For the calculations we choose D = 1.58 and the damping parameters xc = X, g = 0.1 (such weak system–bath coupling can be accurately described by Redfield theory [36,41]), which results in total reorganization energy of about 737 cm1. The value of D is twice higher than the displacement of the 590 cm1 mode reported in Ref. [35]. This is due to our wish to neglect a number of NB modes with small displacements D 6 0.3, but obtain a reasonable value for the reorganization energy. Note also that the value of D = 1.5 has been assumed in Ref. [34] for an adequate modeling (displaced oscillator) of the NB pump–probe spectra [30]. For a separate discussion of PE signals in the absence of strong electron–vibrational coupling, we put D 0 in Eq. (16) so that the model reduces to a two-level system (TLS). There is no vibrational relaxation in this case and the signal is controlled by the rate of the optical dephasing only. The rate of optical dephasing neg (Eq. (2)) depends on temperature and has been chosen as (130 fs)1 for low temperature (10 K) and (33 fs)1 for room temperature (300 K) (this choice leads to a good agreement with the available experimental data [8,24,32,42]). Note that since the overdamped low-frequency modes modify the material system relaxation by increasing the optical dephasing neg, their influence is taken into account in our model. All three laser pulses are assumed to be produced by the same source and have equal amplitudes (k1 = k2 = k3 = k), carrier frequencies (x1 = x2 = x3 = x) and durations. We assume Gaussian envelopes 2
EðtÞ ¼ expfðCtÞ g:
ð17Þ
The pulse duration at FWHM pffiffiffiffiffiffiffiffi (full width at half maximum) is thus given by 2 ln 2=C. Since the reduced Hamiltonian and RWA are employed, the vertical excitation energy e and carrier frequencies x of the pulses do not enter the calculations directly. Of importance is only the detuning between x and e, which will be specified for the particular examples considered.
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
4. Simulated photon echo 4.1. Integrated signal In this section, we give an overview of the available experimental data on homodyne PE spectra of NB and demonstrate the capability of our model to reproduce the experimentally observed trends. We separately discuss contributions of the strongly displaced 590 cm1 mode and of the other modes with small excited-state displacements and address several particular aspects observed in experiments in more detail. Recent homodyne PE measurements of NB include a study in PMMA by Nagasawa et al. [24], in acetonitrile by Larsen et al. [3] and Prall et al. [8] and in methanol by Dietzek et al. [32]. The study in PMMA addressed the temperature dependence of the signal. Three-pulse PE (3PPE) signals as a function of both coherence and population time as well as scans of the signal along the coherence time s for fixed population time T, transient-grating (TG) profiles and peak-shift (PS) dynamics have been reported for various temperatures ranging from 9 K to 295 K. The behavior of the 3PPE signal along the population time axis is characterized by long-lived oscillations with a period of about 60 fs and remains practically unchanged with increasing temperature. For the scans along s at fixed T, a coherent beating with the same period could be resolved at low temperatures only. The PS dynamics exhibits 60 fs oscillations as well, which are superimposed on a fast initial decay. At higher temperature, a decrease of the amplitude of the oscillations and of the initial (T = 0) value of the PS has been observed. The study in methanol [32] has yielded the 3PPE as a function of s and T, as well as TG and PS profiles at room temperature. The observed dynamical features of the signals are the same as in Ref. [24]. The measurements in acetonitrile [3,8] are somewhat less spectacular, since relatively long pulses have been employed (40–50 fs at FWHM, compared to 25 fs in Refs.
a
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[24,32]). A comparison of the measurements of Refs. [24,3,8] confirms our predictions for the dependence of the PS dynamics on the pulse durations [22]: the peak shift values at short population times are larger if longer pulses are employed. Here we are concerned with the shorter-pulses results of Refs. [24,32]. In accordance with the experiments (pulses of 625–635 nm wavelength and 26–28 fs duration), we put C = 0.6X for the envelopes (Eq. (17)) and employ resonant excitation (x = e). As initial condition, we assume a Boltzmann distribution for the vibrational states in the electronic ground state. We start by discussing the temperature dependence of the integrated PE signal and the interpretation of the experimental results on NB in PMMA [24]. The observed temperature effects are most pronounced along the coherence time axis (i.e., for the cuts of the spectrum (11) at fixed values of population time T). Here, we consider the case T = 0 (zero delay between the second and the third pulses), i.e. two-pulse (2P) PE profiles. Note that the coherence time at which this signal reaches its maximum gives the initial value of PS. The experimentally observed signal modifications with increasing temperature exhibit two dominant features: (i) a coherent oscillation observed at low temperature disappears and (ii) the overall signal shape narrows and becomes more symmetric. Fig. 1 displays the 2PPE signal calculated at low temperature (10 K) and room temperature (300 K) for (a): the strongly displaced 590 cm1 mode in the system Hamiltonian, and (b): for the two-level system, i.e., with D 0 in Eq. (16). The signal calculated with the displaced system mode exhibits recurrences with the vibrational period at 10 K (dashed line), which are strongly suppressed at room temperature (solid line). The damping of the oscillation at higher temperature is the result of both a more efficient vibrational relaxation and increase of the dephasing between the two electronic states. The overall shape of the first peak is essentially independent of temperature.
b
2PPE
2PPE
10K 10K
300K
300K
0
50
coherence time τ, fs
100
150
0
50
100
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coherence time τ, fs
Fig. 1. Two-pulse photon-echo (2PPE) signal calculated (a) with the displaced 590 cm1 mode and (b) for the two-level system. Solid and dashed lines correspond to calculations at room temperature (300 K) and low temperature (10 K), respectively.
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The signal calculated for the two-level system (Fig. 1b) changes its shape and narrows significantly for 300 K (solid line) compared to 10 K (dashed line). This effect is due to the faster optical dephasing. In contrast, to the impulsive limit, the TLS signal maximum is not at s = 0, but delivers a certain nonzero PS value for T = 0. Final pulse durations and the pulse overlap effect may thus influence significantly the PE profiles and lead to a meaningful PS dynamics even in the absence of vibrational excitation (see discussion below). Significant effects can be observed only if the pulse carrier frequencies are close to the transition frequency of the TLS. In the following, we interpret such modifications as contributions due to the modes with small excited-state displacements. The overall shape of the experimental 2PPE signal at low temperature can be understood as the superposition of the two contributions displayed in Fig. 1. At high temperatures, these two contributions can be hardly distinguished and the presence of the vibrationally excited mode cannot be resolved in the 2PPE signal (and other signal scans at fixed population times T). The narrowing of the experimental signal and the shift of its maximum (PS at T = 0) to smaller s occur due to the modes with negligible excited-state displacements. In order to illustrate the role of electron–vibrational coupling on the echo signal, the results of calculations at 10 K with various excited-state displacements are contrasted in Fig. 2 (the signals are scaled to obtain comparable intensity). The inset shows the initial (T = 0) peak shift as a function of D. The increase of D leads to the appearance of prominent oscillative structures in the 2PPE and to an exponential-like decrease of the peak-shift value at T = 0. The signals calculated with jDj 6 0.4 have shapes similar to the two-level system echo, although a weak vibrational excitation can be recognized for D = 0.4
Δ 0.5
1
1.5
2 25
2PPE, arbitary units (scaled)
20 15 10 5 0
—30
0
30
60
τ, fs
90
Peak Shift (T=0), fs
0
120
Fig. 2. Two-pulse photon-echo (2PPE) signal calculated for the 590 cm1 mode with the excited-state displacements D = 0.4 (dashed line) and D = 0.8 (dotted-dashed line) at low temperature (10 K). The curves of Fig. 1 are the solid lines. The inset shows the initial PS value (T = 0) as a function of D.
(dashed line in Fig. 2). The qualitative change to the coupled-mode shape takes place for jDj P 0.6 so that the 2PPE profiles obtained with D = 0.8 (dotted-dashed line in Fig. 2) and D = 1.58 look rather similar. The experiments prove that in contrast to the 2PPE signal, the TG technique and PS measurements are capable to visualize vibrational excitation of the 590 cm1 mode at room temperature as well [24,32]. The PS dynamics and TG profiles calculated at 300 K are shown in Fig. 3a and b, respectively. If no vibrational motion is initialized upon excitation (two-level system, solid lines), an exponentiallike decay of the PS is observed (from 14 fs at T = 0 to zero at T P 80 fs). The TG signal reaches its maximum at the same time scale, which is to be expected, since the TG scan is recorded at s = 0. The calculations with the displaced mode (dashed lines in Fig. 3) illustrate that vibrational coherences can be well captured by both PS and TG measurements. One can observe a decrease in the oscillation amplitude which is due to relaxation processes and a very slow overall decay of the signals. As has been proposed for the 2PPE signal, the experimental PS and TG dynamics [24,32] can be interpreted as the superposition of the two contributions. The weakly displaced modes are responsible for the PS decay and increase of the TG signal at short population times, whereas the oscillations are due to the strong electron–vibrational coupling of the 590 cm1 mode which also determines the signal dynamics at long T. The simulations of the PS dynamics at lower temperature (Fig. 4) result in a larger amplitude of the oscillations for the coupled mode and a larger initial value (T = 0) and slower decay for the two-level system, which is in agreement with the experimentally observed behaviour [24]. To clarify its dependence on electron–vibrational coupling we have also calculated the PS for the 590 cm1 mode with the small excited-state displacement D = 0.4 (dashed line in Fig. 4, cf. dashed line in Fig. 2). At longer population times (T P 100–150 fs), the difference between the PS dynamics obtained with D = 0.4 and the standard D = 1.58 is only in the oscillations amplitude. The short-time dynamics obtained with D = 0.4 is qualitatively different and proves that the weakly displaced modes are responsible for the initial decay of the PS. We underline that the PS decay of the TLS can be observed exclusively due to resonant pulses of final durations and does not exist in the impulsive limit. The calculations suggest that the influence of the electron–vibrational coupling on the PS evolution is two-fold: the initialized coherent vibrational motion can be ‘‘destructive’’ at short population times (relatively small PS values), but it definitely allows for a longer survival of the non-stationary PS dynamics. Fig. 5 compares the integrated 3PPE signals as a function of the both delay times s and T (see Refs. [24,32] for the experimental records). The signal calculated with D = 0 is shown Fig. 5a up to T = 100 fs (no changes are observed for longer population times). The decay of the peak shift (cf. Fig. 3a) is accompanied by a broadening
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
15
b Transient grating signal (scaled)
a
peak shift, fs
10
5
0
—5
119
0
100
200
300
400
0
100
200
300
400
population time T, fs
population time T, fs
Fig. 3. Peak-shift (a) and transient-grating (b) profiles at 300 K calculated for the two-level system (solid lines) and the strongly displaced 590 cm1 mode (dashed lines) in the system Hamiltonian.
dynamics (dashed lines in Fig. 3), the oscillations of the width of the signal pointed out in Ref. [32] must have the same period as observed in the PS and TG profiles.
24 20
peak shift, fs
16
4.2. Fourier-transformed two-dimensional PE
12
In contrast to the integrated PE measurements, the 2D heterodyne experiment on NB provided relatively little dynamical information [26]. It rather served as a demonstration of feasibility of the measurement. Only one peak with a width of about 800–850 cm1 has been resolved, and its dynamics has been recorded up to T = 100 fs with time steps of 5–10 fs. A theoretical analysis within the multimode Brownian oscillator has been performed and it has been argued that a higher-lying excited electronic state has to be taken into account. However, the particular peak shapes of the experimental spectrum have not been accurately reproduced by the simulations and their interpretation remains unclear. Considering homodyne PE spectra of NB, it is reasonable to assume that the Franck–Condon activity of the 590 cm1 mode manifests itself in the 2D heterodyne spectra as well, and may be responsible for the peculiar shapes of the peaks. In this section we propose an analysis of the experimental correlation and relaxation
8 4 0 —4 —8
0
100
200
300
population time T, fs Fig. 4. Peak-shift profiles at 10 K calculated with the same system parameters as in Fig. 3a (solid lines). The dashed line represents PS dynamics obtained for the 590 cm1 mode with D = 0.4.
of the signal along coherence time s. Fig. 5b and c demonstrate the modulations of the signal due to the presence of the underdamped vibrational mode (up to T = 100 fs in (b) and up to T = 400 fs in (c)). As can be recognized from Fig. 5b and c as well as by comparison of PS and TG 60
40 20 0 —20 —40 20
40
60
80
population time T, fs
100
60
coherence time , fs
coherence time , fs
coherence time , fs
60
40 20 0 —20 —40 20
40
60
80
population time T, fs
100
40 20 0 —20 —40 0
100
200
300
400
population time T, fs
Fig. 5. Contour plots of the three-pulse photon-echo (3PPE) signal at 300 K. Simulations for the two-level system are shown in (a) up to T = 100 fs, since no changes are observed at longer time scale. The signal evolution obtained with the displaced 590 cm1 mode is presented in (b) and (c) for up to T = 100 fs and up to T = 400 fs, respectively.
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b 2600
c 2600
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1400 1600 1800 2000 2200 2400
1400 1600 1800 2000 2200 2400
Fig. 6. Two-dimensional (a) correlation (T = 0) and (b,c) relaxation (T = 20, 50 fs) spectra calculated with the displaced 590 cm1 mode in the system Hamiltonian at 300 K.
spectra employing the simplified model of NB introduced above. The most pronounced deviation of the experimental correlation spectrum (T = 0) from the shape predicted by the simulations [26] is a strong distortion from the diagonal orientation in the spectral range where emission and absorption frequencies exceed the carrier frequency of the pulses, xt, xs > x. In this region, the spectrum is elongated parallel to the absorption-frequency (xs) axis. This shoulder along increasing xs observed at T = 0 gradually disappears with increasing T (relaxation spectra) and a slight elongation to larger xt develops. Other particular experimental features are a displacement of the signal maximum from the diagonal (xt = xs) to somewhat larger xs for all T, and an overall intensity decrease with increasing population time [26]. Pulses of 41 fs duration (FWHM) and center wavelength of 595 nm (16,807 cm1) have been employed in the experiment. The linear absorption spectrum [3,8] peaks at about 635 nm (15,748 cm1), i.e., the system has been excited about 1100 cm1 above the absorption maximum, which corresponds to about two vibrational quanta of the strongly displaced 590 cm1 mode. Most probably, the 41 fs pulses prepare a superposition of a couple higher-lying vibrational states of this mode. The probability of vibrational excitation of the higher-frequency strong RR mode (1640 cm1), on the other hand, remains negligible. The weakly displaced modes are not expected to significantly contribute to the spectra in the considered case of strongly off-resonant excitation (the intensity of linear absorption at 595 nm is 50% lower than at the maximum). We thus limit the discussion to 2D spectra calculated with the strongly coupled 590 cm1 mode in the system Hamiltonian. Fig. 6 shows 2D correlation (T = 0) and relaxation (T = 20 fs, 50 fs) spectra calculated with parameters corresponding to the experimental conditions: the pulse durations are determined by C = 0.366X, the carrier frequencies of the pulses are x = e + 2X, and temperature is 300 K.1 The actual absorption and emission frequencies 1
To facilitate a comparison of our 2D PE spectra with those measured by Brixner et al. [26], note that 100 cm1 = 0.019 fs1 and that the orientations of the frequency axes in Ref. [26] is opposite to ours.
xs and xt have been reduced by the energy of the zero–zero transition and the reduced carrier frequencies of the pulses 1950 cm1. are x We succeeded to reproduce the particular shape of the experimental spectrum at T = 0 and its dynamical changes with T with good accuracy. The intensity loss with increasing T can be recognized and is easily explained as a fast vibrational relaxation of the population of the higher-lying levels initially excited. A further analysis is required, however, for an interpretation of the observed line shapes. Assuming that our simple model is sufficient to capture the vibronic dynamics of NB, we can try to gain more insight and understanding by improving ‘‘experimental’’ resolution. First, we get rid of the spectral broadening induced by relaxation and dephasing processes and simulate the signal with the same parameters as in Fig. 6, but for lower temperature. Fig. 7 shows 2D correlation (T = 0) and relaxation (T = 20 fs, 50 fs) spectra calculated at 10 K. As can be recognized from the correlation spectrum (Fig. 7a), two vibrational states with quantum numbers m = 3 and m = 4 are initially populated since two peaks can be distinguished at absorption frequencies xs 3X = 1770 cm1 and xs 4X = 2360 cm1. The peak corresponding to the transition to the higher level m = 4 (xs = 4X) is much weaker than that at xs = 3X. This level is populated less than the m = 3 level since the reduced carrier frequency of the pulses is closer to 3X, and it decays faster due to vibrational relaxation. That is why the elongation of the spectrum towards higher values of xs gradually disappears with increasing population time T. A prominent negative wing at higher emission frequencies, which is present at T = 0, decreases with increasing population time, and an effective elongation of the relaxation spectra towards larger values of xt is observed. The shift of the peak maximum from the diagonal to larger values of absorption frequency at high temperatures is caused by the averaging of the two transitions resolved at 10 K. At T = 50 fs, the maximum of the 300 K-signal (Fig. 6c) moves to somewhat smaller values of xs as compared to T = 0, 20 fs due to fast depopulation of the fourth (m = 4) excited vibrational level, but remains offdiagonal.
D. Egorova et al. / Chemical Physics 341 (2007) 113–122
c 2600
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Fig. 7. Two-dimensional (a) correlation (T = 0) and (b,c) relaxation (T = 20, 50 fs) spectra calculated with the same parameters as in Fig. 6 but at low temperature (10 K).
the coupling cross peaks at xs = 2X, xt = 3X and at xs = 3X, xt = 4X. This effect arises due to a decrease of the negative contribution compared to T = 0. If one compares low-temperature simulations of Figs. 7 and 8, it may appear that the effect of longer pulses (Fig. 7) is just to cut out the part which is resonant with the pulse carrier frequency from the spectra obtained with shorter pulses. However, this is not really true, since shorter pulses also initiate a different (more coherent) system dynamics. On the other hand, the finite duration of the local-oscillator pulse extracts the spectral part close to its frequency from the ideally-detected signal (limit of an infinitely short local-oscillator pulse). The ideal detection can thus be experimentally realized if the 2D PE spectrum along the absorption and emission frequencies fits the corresponding laser pulse bandwidths. Alternatively, the ideally-detected signal may be obtained by a tuning of the frequency of the local-oscillator pulse. 5. Conclusions We have proposed a simplified model picture of NB which provides a comprehensive description of available PE experimental data. The measured integrated PE signals have been interpreted as a superposition of two contributions. One contribution arises from the strongly displaced underdamped 590 cm1 mode, which is coherently excited by the ultrashort pulses. The other one is monotonous in time and is due to the presence of the modes with no or
2500
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2250
2250
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2000 t
t
The spectral range accessible with 41 fs pulses is relatively narrow. More information on the underlying dynamics can be gained if shorter pulses are employed. Since pulses of the 25 fs duration seem to be routinely available in PE experiments [24,32], we have performed a simulation with C = 0.6X. The obtained 2D correlation (T = 0) and relaxation (T = 50 fs) spectra are shown in Fig. 8 (the other parameters are as in Fig. 7). As expected, a broader spectral range is covered, although the most intense peaks correspond to the transition closest to the carrier frequency and are at the same position as in Fig. 7. Several cross peaks separated by the vibrational frequency can be resolved. As has been outlined previously [23], two types of cross peaks can be distinguished in 2D spectra of the displaced oscillator: ‘‘emission peaks’’, which represent emission from each excited vibrational level to the electronic ground-state manifold, and ‘‘coupling peaks’’, which arise due to coupling of transitions with different frequencies through the common ground state. The cross peaks above the diagonal xt = xs correspond to the coupling peaks, while the cross peaks under the diagonal represent a superposition of the coupling peaks with emission peaks. Both types of cross peaks may experience intensity modifications with T due to vibrational motion and relaxation, i.e., due to changes in populations of different vibrational levels in the excited electronic state. An overall intensity loss is expected at T = 50 fs due to fast vibrational relaxation of the initially excited levels. Indeed, the peaks in Fig. 8b are less intense than the peaks in the correlation spectrum in Fig. 8a, with the exception of
1750
1750
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1250 1500 1750 2000 2250 2500
1250 1500 1750 2000 2250 2500
Fig. 8. Two-dimensional (a) correlation (T = 0) and (b) relaxation (T = 50 fs) spectra calculated with the same parameters as in Fig. 7 but with shorter pulses, C = 0.6X (25 fs at FWHM).
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D. Egorova et al. / Chemical Physics 341 (2007) 113–122
negligible excited-state displacements, provided resonant pulses with finite durations are employed. We have discussed the dynamics and temperature dependence of the 2PPE, analyzed TG and PS profiles and presented contour plots of the 3PPE signal as a function of both coherence time and population time. We have addressed modifications of homodyne PE spectra due to the electron–vibrational coupling and pointed out the role of the pulse overlap effect in the PS dynamics. For the 2D heterodyne spectra, only the contribution corresponding to the displaced 590 cm1 mode has been considered, since off-resonant (with respect to the absorption maximum) pulses have been employed in the experiment and the contribution of the weakly displaced modes is therefore not expected to be significant. We have provided a new interpretation of the experimental 2D line shapes and have performed simulations which predict 2D correlation and relaxation spectra of NB for improved resolution conditions. If the strongly displaced 590 cm1 mode could be experimentally resolved (as peaks separated by the corresponding vibrational frequency), this would be the first experimental detection of electron–vibrational coupling in 2D electronic PE spectra. The study demonstrates the capability of PE spectroscopy to monitor underdamped vibrational dynamics in the excited electronic state and to provide information on relaxation time scales. 2D heterodyne spectra can be very informative and directly interpreted if the broadening is minimized and the employed pulses are short compared to the time scale of vibrational motion. Pulse durations and frequencies can be varied in order to address different dynamical features and spectral ranges. Under good resolution conditions, the 2D technique appears to be more advantageous than conventional one- and two-color PS measurements. However, the interpretation of cross peaks in 2D spectra requires some caution. If underdamped coherent vibrational motion is initiated by the excitation (this can be clarified by performing, e.g., a TG measurement), the modifications of the spectra due to electron–vibrational coupling have to be taken into account. Acknowledgement DE is grateful to Michael Thoss and Tobias Brixner for inspiring discussions and to DFG for financial support. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemphys. 2007.07.010. References [1] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, University Press, Oxford, 1995.
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