Optical N-Wave-Mixing Spectroscopy with Strong ... - ACS Publications

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Optical N-Wave-Mixing Spectroscopy with Strong and Temporally Well-Separated Pulses: The DoorwayWindow Representation Maxim F. Gelin,† Dassia Egorova,‡ and Wolfgang Domcke*,† † ‡

Department of Chemistry, Technische Universit€at M€unchen, D-85747 Garching, Germany Institute of Physical Chemistry, Christian-Albrechts-Universit€at zu Kiel, D-24098 Kiel, Germany ABSTRACT: We have extended the doorwaywindow representation of optical pumpprobe spectroscopy with weak pulses toward N-wave-mixing spectroscopy with temporally well-separated pulses of arbitrary strength. The expressions for the signals in the strong-pulse doorwaywindow representation are derived in the framework of the nonperturbative theory of N-wave-mixing spectroscopy. The strongpulse doorwaywindow representation is complementary to the equation-of-motion phase-matching approach. The latter fully accounts for pulse-overlap effects in signals induced by weak pulses but is computationally more expensive. The performance of the doorwaywindow approximation for temporally well-separated strong pulses is illustrated for an electronic two-level system with an underdamped Condon-active vibrational mode.

1. INTRODUCTION The doorwaywindow (DW) representation of optical pump probe spectroscopy has been introduced and developed by Yan, Fried, and Mukamel.1,2 The DW representation has provided a firm basis for the description of pumpprobe signals in terms of vibrational wavepackets in the ground and excited electronic states. The pump pulse creates the doorway state D, that is, the initial wavepackets in the ground state and in the excited state. The D state evolves according to the intrinsic (without external fields) system dynamics and is projected on the window state W, which is created by the probe pulse, yielding the signal.3 The DW picture has several important advantages. Conceptually, it facilitates the interpretation and the understanding of the measured transients. Computationally, the DW approximation substantially simplifies calculations of optical signals induced by temporally well-separated pulses in comparison with the perturbative nonlinear response function approach37 or various nonperturbative methods.818 The DW approximation can straightforwardly be applied to any master equation describing the system dynamics. It provides the basis for the introduction of semiclassical or classical approximations.14,1923 The (semi)classically evaluated DW functions can be combined with classical molecular dynamics to simulate pumpprobe signals.22,23 The DW framework can be used to approximately calculate9,24 or simulate25 the nonlinear response functions themselves. The DW approximation has been generalized to overlapping pulses (via time-dependent D operators26,27) and sequential three-pulse photon-echo signals.28 The DW description is usually applied in the context of the third-order pumpprobe spectroscopy, so that the D and W operators are proportional to the intensities of the pump and probe pulses, respectively. Following recent trends in developing ultrafast nonlinear spectroscopy, the present work extends the weak-pulse DW representation of pumpprobe spectroscopy in r 2011 American Chemical Society

two directions. We address N-wave-mixing signals, which are induced by temporally well-separated pulses, and we consider pulses of arbitrary strength. To our knowledge, the only attempt in this direction has been undertaken by Tanimura and Mukamel,29,30 who developed a formal scheme for the description of the D and W wavepackets for harmonic oscillators beyond the weak-pulse limit, taking into account higher-order contributions to the nonlinear response functions. Recently, Schweigert and Mukamel31 improved the δ-pulse description of nonlinear responses by retaining finite pulse envelopes. In ref 32, fielddependent nonlinear response functions were introduced for describing strong-pulse effects in rotational four-wave mixing. The extension of the DW approximation to account for strong pulses is motivated by recent progress in spectroscopy with intense pulses10 and optimal control of nonlinear optical responses.3338 The phenomena include, for example, the dependence of the transient absorption3941 and fluorescence quantum yield42,43 of light-harvesting complexes on the intensity of the pump pulse, manipulation of electronic coherences of the system density matrix,44,45 creation of nonlinear heterodyning in femtosecond rotational four-wave mixing,32 pulse-strength-dependent threepulse photon-echo peak shift of semiconducting single-walled carbon nanotubes,46,47 manipulation of Liouville pathways in multiply enhanced odd-order wave-mixing spectroscopy,48,49 and observation of triexciton quantum coherences in quantum wells.50 Very recently, a strong femtosecond infrared pump pulse and an attosecond extreme-ultraviolet probe pulse were employed to Special Issue: Shaul Mukamel Festschrift Received: December 19, 2010 Revised: February 21, 2011 Published: March 22, 2011 5648

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monitor spin wavepackets in Krþ ions.51 Given the complexity of the evolution of nontrivial systems driven by strong laser pulses, it is highly desirable to have a method in the theoretical toolbox that allows one to disentangle pulse-overlap-induced transient effects from the wavepacket dynamics of the material system. The extension of the DW approximation toward many pulses is motivated by the development of six-wave-mixing spectroscopy5260 and pulse-train spectroscopy.61,62 Time- and frequency-resolved four-wave-mixing techniques have revolutionized the field of timeresolved spectroscopy of complex systems, providing coherent dynamic responses rather than featureless spectra. Six-wave-mixing spectroscopy opens the world of nonequilibrium processes and reactions to detailed spectroscopic observation by the realization of four-wave-mixing experiments on systems that have been prepared in a nonequilibrium state by an extra pair of pulses. It also allows for the resolution of those spectral signatures that normally are buried under featureless responses delivered by lower-order spectroscopies. On the other hand, six-wave-mixing spectroscopies reflect the system dynamics indirectly, through the transients that depend on the carrier frequencies and time delays of the pulses involved. The interpretation of these signals is almost impossible without theoretical support. The DW approximation for describing N-wave-mixing signals induced by strong pulses is a highly efficient computational tool, which can straightforwardly be applied to the calculation of multipulse responses of any material system whose time evolution is described by a master equation. The strong-pulse DW approximation complements the recently developed equation-of-motion phase-matching approach (EOM-PMA).11,63 While the EOMPMA allows us to compute N-wave-mixing signals induced by weak pulses with full account of pulse-overlap effects, it is computationally much more expensive than the DW approximation. The present paper is structured as follows. In section 2, we define the problem and introduce the basic equations. In section 3, we formulate the strong-pulse DW representation, present (without derivation) the explicit expressions for two-pulse and N-pulse signals, and discuss their physical content. In section 4, the strong-pulse DW formulas are specialized for the one-pump, one-probe (2P) and two-pump, one-probe (3P) signals. Section 5 contains proof-of-principle calculations of these signals for a two-state displaced harmonic oscillator model. Section 6 is the conclusion. Detailed derivations and technicalities are deferred to the Appendices A, B, and C.

2. SETTING THE STAGE

are the projection operators on the ground and excited electronic states, respectively. The interaction of the system with N laser pulses is written in the dipole approximation, the Condon approximation, and the rotating wave approximation (RWA) as follows HN ðtÞ ¼

H ¼ Hg þ He

Hg ¼ P g hg

ð2Þ

He ¼ P e ðεe þ he Þ

ð3Þ

Here hg and he represent the corresponding vibrational Hamiltonians, εe is the vertical excitation energy of the excited electronic state |eæ, and P g jgæÆgj

P e jeæÆej

ð4Þ

X † jeæÆgj

X jgæÆej

E a ðtÞ ¼ λa Ea ðt τa Þ expfiðk a r ωa tÞg

ð5Þ

ð6Þ ð7Þ

λa, ka, ωa, Ea(t), and τa denote the amplitude, wave vector, frequency, dimensionless envelope, and central time of the pulses. The RWA amounts to the omission of the counterrotating terms (∼exp{(i(εeþωa)t}) while retaining the corotating terms (∼exp{(i(εeþωa)t}) in the responses to the applied fields. We therefore use the reduced carrier frequencies ωa f ωa εe

ð8Þ

throughout the paper. We write the master equation for the reduced density matrix as (p = 1) Dt FðtÞ ¼ i½H þ HN ðtÞ, FðtÞ þ ðR þ p ÞFðtÞ

ð9Þ

The term p FðtÞ ¼ ξeg P g FðtÞP e þ H:c: describes optical dephasing, rate. The operator

27,64

ð10Þ

ξeg being the corresponding

R ¼ P gR g þ P eR e

ð11Þ

accounts for vibrational relaxation in the ground and excited electronic states.65,66 For simplicity of notation, R is written as a time-independent operator. Upon the substitution R F(t) f Rt 0 0 0 0 dt R (t t )F(t ) or R F(t) f R (t)F(t), all of the derived formulas remain valid. B. Nonlinear Polarization and Phase Matching. The system’s response to the applied field is determined by the total complex nonlinear polarization (angular brackets indicate the trace) PðtÞ ÆX † FðtÞæ

ð12Þ

which contains contributions corresponding to all possible values of the wave vector

ð1Þ

Adopting bra-ket notation to denote the electronic ground state |gæ and the excited state |eæ, we write

Here

A. Hamiltonian, Master Equation, and RWA. We write the

molecular Hamiltonian H as the sum of an electronic groundstate Hamiltonian, Hg, and an excited-state Hamiltonian, He,

N

ðX † E a ðtÞ þ XE a ðtÞÞ ∑ a¼1

k ¼

N

∑ la ka a¼1

ð13Þ

In general, la can be arbitrary integer numbers. If the pulses are temporally well-separated, then la can only take the values 0, (1, (2 (see Appendix A). In applications, we have to extract a particular contribution Pk ðtÞ ∼ expfikrg

ð14Þ

which obeys a specific phase-matching condition, that is, eq 13 with specific values of la. The extraction of Pk(t) from P(t) is considered in Appendix A. The phase-matching polarization 5649

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Pk(t) can be measured, for example, by heterodyning with a local oscillator (LO) field Z ¥ Sk ¼ dt E LO ðtÞPk ðtÞ ð15Þ ¥

yielding the N-wave-mixing signal (E LO (t) is defined analogously to E a (t)).

3. STRONG-PULSE DOORWAYWINDOW (DW) APPROXIMATION A. Two-Pulse Signals. Consider two pulses (N = 2) in the interaction Hamiltonian eq 8. Let T τ2 τ1 be the interpulse delay and Ta (a = 1,2) be the characteristic pulse durations. If the pulses are temporally well-separated, we can neglect the transient terms ∼Ea(t) for t . Ta. This is the only assumption made to obtain the strong-pulse DW approximation. The detailed derivations can be found in Appendix B. The final formula for the twopulse signal reads

Sk ðTÞ ¼ ÆW GðTÞD æ

ð16Þ

Here G(T) is the field-free evolution operator, which corresponds to the master eq 9 with no external pulses (HN(t) = 0). The D and W operators are explicitly defined as follows D ¼ Q 1 G1 ðδ1 , δ1 ÞFeq Z W ¼

¥

δ2

ð17Þ

dt E LO ðtÞX † G2 ðt, δ2 ÞQ 2

ð18Þ

Here, δa = χT a , χ ≈ 1 being a numerical factor.67 Ga(t,t0 ) is the time evolution operator which corresponds to the master eq 9 with the single external pulse #a. Feq is the equilibrium Boltzmann distribution, eq B2. The projection operators Q a are defined in terms of P g and P e. They are specific to the particular kind of spectroscopy and account for the phase-matching conditions (see Appendix A). B. N-Pulse Signals. The strong-pulse DW approximation can be extended to N temporally well-separated pulses. The derivations are given in Appendix C. The final expression for the N-wave-mixing signal is given by the intuitively straightforward extension of eq 16 * T> W N

(

Sk ðT1 , ... , TN 1 Þ ¼ NY 1

)

GðTa ÞQ a Ga ðδa , δa Þ GðT1 ÞD

+

(i) As their weak-pulse counterparts, the strong-pulse D and W operators (for specific values of the carrier frequencies of the pulses) are evaluated only once (see Appendices B and C). Once the D and W operators are known, the time evolution of the two-pulse signal is determined by a single T propagation of the D operator and its subsequent convolution with the W operator (eq 16). To evaluate the N-pulse signal, N 1 sequential propagations are required (eq 19). (ii) The strong-pulse DW approximation allows us to extract the N-wave-mixing signal Sk for any specific phase-matching direction k from the total nonlinear polarization P(t) without performing extra calculations. The necessary phase selection is carried out by the projection operators Q a. For example, the D and W operators for pumpprobe spectra correspond to electronic populations of the density matrix (section 4). The D and W operators for photon-echo spectra, on the other hand, correspond to electronic coherences (Appendixes B and C). The strong-pulse DW approximation allows us to compute the two-pulse signal Sk(T) by performing (a) a single short-time (during the action of the pump pulse) propagation of the master eq 9 to obtain the D operator (eq 17), (b) a few short-time (during the action of the probe pulse) propagations of the master eq 9 to obtain the W operator (eq 18), and (c) a single field-free propagation of the master eq 9 for the time interval T to obtain Sk(T) (eq 16). This should be compared with the exact evaluation of the signal via eq A7, which requires nT nΘ propagations of the master eq 9 in the presence of both fields (nT is the number of points necessary for the discretization of the time interval T, and nΘ g 4 is the number of points in the discrete Fourier transform, which is needed to select the phase-matched polarization). For N pulses, the strong-pulse DW approximation allows us to compute the signal Sk(T1,...,TN1) in any phasematching direction k at particular values of interpulse delays Ta by performing a single time propagation of the master eq 9. The exact calculation of the same signal, on the other hand, requires (nΘ)N1 propagations of the master eq 9. The strong-pulse DW approximation accounts for the system bath coupling during the action of the pulses automatically. However, if the systembath coupling is weak and the pulses are not too long, we may neglect the systembath coupling during the action of the pulses. This “no-bath” approximation amounts to the replacement of the evolution operators Ga(δa,δa) and GN(t,δN) in eqs 16 and 19 by their bath-free counterparts

ð19Þ

0 0 GNB a ðt, t Þ Ga ðt, t ÞjR ¼ 0

a¼2

Here, Ta τaþ1 τa (a = 1, 2, ..., N1) are the interpulse separations, T> ensures the appropriate time ordering of the operators, the D operator is given by eq 17, and the W operator is defined as Z ¥ dt E LO ðtÞX † GN ðt, δN ÞQ N ð20Þ WN ¼ δN

The propagators G(Ta), Ga(δa,δa), and the projection operators Q a are determined analogously to their two-pulse counterparts. C. General Remarks on the Strong-Pulse DW Approximation. The strong-pulse DW approximation has two significant computational advantages.

ð21Þ

The no-bath approximation is usually invoked in the weak-pulse DW description of pumpprobe and spontaneous emission signals.20,26,6870 The strong-pulse DW approximation can be used to establish the domain of validity of the no-bath approximation. Within the present paper, we limit ourselves to the consideration of systems with two electronic (ground and excited) states. However, the strong-pulse DW approximation can be extended to more complicated systems. First, we can consider the replacement of the excited electronic state |eæ by several narrowly spaced intramolecularly coupled electronic states |eiæ. In this case, all of the derived formulas remain true if we adopt the diabatic representation for the excited-state Hamiltonian in eq 3, He = ∑i,j |eiæÆej|(εiδij þ hij), where εi ≈ εe, define the projection 5650



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operator to the excited states (eq 4) as P e = ∑i |eiæÆei|, and write the transition dipole moment operator of eq 6 as X = ∑i Vgi|gæÆei|, V2gi being the oscillator strength of the corresponding electronic transition. Second, we can remove the restriction εi ≈ εe and include not only radiative transitions from the ground state to the excited states but also radiative transitions among the excited states, X = ∑i Vgi|gæÆei| þ ∑ij Vij|eiæÆej|. In this case, the formal expressions for the two-pulse and N-pulse signals, eqs 1618 and 1920, remain the same, but the analysis of the phase dependence of the strong-pulse DW operators (Appendix A) has to be modified.

4. STRONG-PULSE DW APPROXIMATION OF PUMPPROBE SIGNALS Here, we concentrate on pumpprobe signals. The standard one-pump, one-probe (2P) transient absorption signal is considered in subsection 4A. The two-pump, one-probe (3P) signal, in which an extra pump pulse is used, is considered in subsection 4B. A. 2P Signal. The 2P signal is described by a slight modification of the expression of eq 15 Z ¥ ð1, 2Þ ð2Þ dt E 2 ðtÞðPk ðtÞ Pk ðtÞÞ ð22Þ S2P ðTÞ ¼ Im ¥

The notation P(1,2) (t) means that the polarization is evaluated via k the master eq 9 in the presence of both the pump (a = 1) and the probe (a = 2) pulses, while P(2) k (t) is induced solely by the probe pulse. The 2P phase-matching condition in eq 13 reads l1 ¼ 0

l2 ¼ 1

ð23Þ

To calculate D according to eq 17, the evolution operator G1(δ1,δ1) must act on the equilibrium distribution Feq. The latter corresponds to the electronic population of the density matrix. Therefore, we must use eq A9 to determine the phase vector dependence of D . Because l1 = 0, the 2P D operator is independent of the phase factors exp{(ik1r} and D 2P Q 1 D P g DP g þ P e DP e Feq

ð24Þ

probe pulse and is thus independent of δ2 (provided δ2 . τ2). However, W 2P is not Hermitian because we calculate the complex nonlinear polarization in eq 12. Therefore, Re W 2P reduces to the weak-pulse W operator. Despite these differences, the weak-pulse and strong-pulse DW approximations yield identical 2P signals for weak pulses. Because D 2P (eq 24) is determined by the electronic populations of the density matrix, the optical dephasing operator (eq 10) does not affect it (p D2P 0). The dissipation operator R (eq 11) does not mix ground and excited electronic states. Therefore, the 2P signal can be represented as the sum of the electronic ground-state contribution (bleaching) and the excited electronic state contribution (stimulated emission) 2P 2P 2P S2P ðTÞ ¼ ImÆW 2P gg Ggg ðTÞD gg þ W ee Gee ðTÞD ee æ

Here, the D and W operators in the ground and excited electronic states are given by eqs 24 and 25, respectively. The propagators in the ground state (Ggg(T)) and those in the excited electronic state (Gee(T)) are described by the master eq 9 with no external fields and no optical dephasing (HN(t) = p = 0). B. 3P Signal. The 3P signal can be defined as Z ¥ ð1, 2, 3Þ S3P ðT1 , T2 Þ ¼ Im dt E 2 ðtÞðPk ðtÞ ¥ ð1, 3Þ

Pk

W 2P Q 2 W P g WP g þ P e WP e

ð25Þ

Because the 2P local oscillator field coincides with the probe field, the W operator in eq 25 assumes the form Z δ2 W ¼ dt E 2 ðtÞX † G2 ðt, δ2 Þ ð26Þ δ2

Equations 16, 24, and 25 provide the strong-pulse DW formulas for the description of 2P signals. If the pulses are weak, the 2P D and W operators can be evaluated in the leading (second) order in the pump and probe pulse amplitudes, respectively. However, the strong-pulse D 2P and W 2P operators do not coincide with the weak-pulse DW operators in this limit. The reason is that D 2P depends on the time interval δ1 (see eq 17). If δ1 increases, a part of the pulse-free evolution is included in D 2P. As to W 2P (eq 18), its value is controlled by the

ð2, 3Þ

ðtÞ Pk

ð3Þ

ðtÞ þ Pk ðtÞÞ

ð28Þ

The notation P(1,2,3) (t) means that the polarization is evaluated k via the master eq 9 in the presence of three pulses. The 3P phasematching condition eq 13 reads l1 ¼ 0

l2 ¼ 0

l3 ¼ 1

ð29Þ

Starting from eq 19, the strong-pulse DW formula for the 3P signal can be obtained analogously to that for the 2P signal S3P ðT1 , T2 Þ ¼ ÆW 3P GðT2 ÞQ 2 fG2 ðδ2 , δ2 Þ 1gGðT1 ÞD 3P æ ð30Þ Here

P(2) k (t).

The last term in this expression stems from To calculate W according to eq 18, the evolution operator G2(t,δ2) must act on the transition dipole moment X†. The latter corresponds to the electronic coherence of the density matrix. Therefore, we must use eq A9 to determine the phase vector dependence of W . Because l2 = 1, the 2P W operator is ∼exp{ik2r}:

ð27Þ

D 3P P g DP g þ P e DP e Feq

ð31Þ

W 3P P g WP g þ P e WP e

ð32Þ

Q 2 F P g FP g þ P e FP e

ð33Þ

Analogously to the 2P signal (eq 27), the 3P strong-pulse DW signal can be decomposed into the electronic ground-state and excited-state contributions.

5. ILLUSTRATIVE APPLICATION A. Model System. To illustrate and test the strong-pulse DW approximation, we calculate 2P and 3P signals for an electronic two-state system with a single underdamped vibrational mode. The vibrational Hamiltonians are

hg ¼ ΩðP2 þ Q 2 Þ=2

ð34Þ

he ¼ ΩfðP2 þ Q 2 Þ=2 ΔQ g

ð35Þ

Here, Q and P are the dimensionless coordinate and momentum. The excited-state vibrational Hamiltonian is a shifted harmonic oscillator, and Δ= 2 is the dimensionless horizontal 5651



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Figure 1. 2P signal S2P(T) induced by weak (λa = 0.01 eV, (a)) and strong (λa = 0.1 eV, (b)) short (Γa = Ω) pulses. Full black lines, exact calculation; full gray lines, strong-pulse DW approximation. Dotted and dashed lines depict the ground-state and excited-state contributions to the strong-pulse DW signal.

displacement of the minimum of the excited-state potential surface from the minimum of the electronic ground-state surface. The frequency Ω is 0.05 eV, so that the fundamental vibrational period is 2π/Ω = 83 fs. The vibrational relaxation operator R in the master eq 9 is treated within the multilevel Redfield formalism.71,72 The bath spectral function is taken in the Ohmic form J(ω) = ηω exp{ω/ ωc} with the cutoff frequency ωc = Ω. The dimensionless parameter η controls the dissipation strength. The master eq 9 is converted into matrix form by an expansion in terms of the eigenstates of the system Hamiltonian. The fieldmatter interaction is treated numerically exactly. The fourth-order RungeKutta scheme is used for propagations of the reduced density matrices and calculations of the D and W operators (see ref 64 for computational details). Strictly speaking, the Redfield operator becomes field-dependent when the field strength increases.73,74 This effect is negligible, however, for the pulse strengths employed in the present work (see, e.g., ref 44). The pulse envelopes are assumed to be Gaussian Ea ðtÞ ¼ expfðΓa tÞ2 g

ð36Þ

Ta = 2(ln 2)1/2/Γa being the pulse durations (full width at halfmaximum). In all calculations, the reduced carrier frequencies from eq 8 are taken as ωa = 4Ω (a = 1, 2, 3), so that the pulses are in resonance with the transition from the first vibrational level in the electronic ground state to the forth vibrational level in the excited

Figure 2. Same as in Figure 1, but for long pulses (Γa = 0.3Ω). The inset in (b) shows the oscillatory structures on an enlarged scale.

electronic state. A moderate vibrational dissipation (η = 0.3) is assumed. To emphasize coherent effects, the temperature is taken to be 0. B. 2P Signal. To test the strong-pulse DW approximation, we compare the 2P signals obtained by the evaluation of eq 27 with those obtained by the numerically exact evaluation of eq A6. Figure 1a shows the 2P signal excited and detected by weak (λa = 0.01 eV) and short (Ta = 18 fs) pulses. Full black lines correspond to the exact calculation, and gray lines correspond to the strong-pulse DW approximation. The dotted and dashed lines depict the ground-state and the excited-state contributions to the strong-pulse DW signal. The strong-pulse DW approximation (gray line) is seen to reproduce the exact signal (black line) after T > 70 fs. The electronic ground-state contribution (dotted line) is weakly dependent on T, while the electronic excited-state contribution (dashed line) exhibits pronounced damped beatings, which are responsible for the oscillatory behavior of the entire signal. These beatings with the period of 2π/Ω = 83 fs reflect the vibrational wavepacket dynamics in the excited electronic state, which is caused by a substantial (Δ = 2) displacement of the potential energy surfaces. Figure 1b depicts the same transients as those in Figure 1a but for pulses which are 10 times stronger (λa = 0.1 eV). In this case, the weak-pulse DW description breaks down because higherorder systemfield interactions contribute significantly to the signal. The strong-pulse DW approximation cannot describe the Rabi oscillations (the so-called coherent artifact) for |T| < 50 fs when the pump and probe pulses overlap. However, for T > 70 fs, the strong-pulse DW (gray line) and exact (black line) signals become identical. As for weak pulses (Figure 1a), the electronic 5652



Figure 3. 2P signal S2P(T) induced by weak (λa = 0.01 eV, (a)) and strong (λa = 0.1 eV, (b)) short (Γa = Ω) pulses. Full lines, strong-pulse DW approximation; dotted lines, no-bath DW approximation.

ground-state (dotted line) and excited-state (dashed line) contributions are of comparable magnitude. However, the former is now more oscillatory than the latter. This reflects the redistribution of vibrational populations in the ground and excited electronic states due to multiple Rabi cyclings. The physical effect arising from the excited-state potential energy surface (coherent vibrational wavepacket motion) thus becomes visible in the ground-state signal (compare with refs 44 and 75). Figure 2a shows weak-pulse transients as depicted in Figure 1a but for relatively long pulses (Ta = 61 fs). Because the pulse duration is now comparable with the vibrational period, the exact 2P signal (black line) is almost featureless. After a rise during the pump pulse, it decays to a constant value, which is predominantly determined by the electronic ground-state contribution (dotted line). The electronic excited-state contribution (dashed line) is much smaller. The strong-pulse DW description (gray line) is essentially exact for T > 180 fs. Figure 2b is the strong-pulse (λa = 0.1 eV) counterpart of Figure 2a. When the pulses are temporally well-separated (T > 180 fs), the strong-pulse DW (gray line) and exact (black line) signals coincide. The strong-pulse 2P signal exhibits vibrational beatings for T > 180 fs (see inset in Figure 2b), although the corresponding weak-pulse 2P signal (Figure 2a) is featureless. The reason is that strong and relatively long pulses not only induce coherent system dynamics and thus create pronounced vibrational wavepackets in both electronic states but also allow one to detect these wavepackets.76,77 In the no-bath approximation, eq 21, vibrational relaxation during the action of the pulses is neglected. The strong-pulse DW representation, on the other hand, accounts for dissipation

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Figure 4. 3P signal S3P(T1,T2) induced by weak (λa = 0.01 eV, (a)) and strong (λa = 0.1 eV, (b)) short (Γa = Ω) pulses. The time delay between the first two pulses is fixed at T1 = 79 fs. Full black lines, exact calculation; full gray lines, strong-pulse DW approximation. Dotted and dashed lines depict the ground-state and excited-state contributions to the strongpulse DW signal.

during the action of the pulses. It is therefore of interest to compare the no-bath DW approximation with the strong-pulse DW approximation. As expected, the agreement between the two is perfect for very short (Ta , 2π/Ω) pulses (not shown). If the pulses are longer (Γa = Ω, Ta = 18 fs), the agreement is reasonable for both weak (Figure 3a) and strong (Figure 3b) pulses. If the pulse duration is further increased (e.g., Γa = 0.3Ω), the no-bath DW approximation fails for both weak and strong pulses (not shown). As has been demonstrated by Tanimura and Mukamel29,30 for a multimode harmonic oscillator model, the no-bath approximation is accurate if the system vibrations are overdamped and the temperature is high, so that quantum fluctuations and correlations can be neglected. The present analysis shows that the nobath approximation works quite well also in the opposite limit of underdamped system dynamics, provided that the pulses (both weak or strong) are relatively short on the time scale of the system dynamics. C. 3P Signal. In 3P signals, an extra pump pulse is employed to induce the system dynamics. We compare here the strong-pulse DW signals computed via eq 30 with the exact signals obtained by the evaluation of eq A5. Figure 4 shows 3P signals excited and detected by short and weak (panel a) and short and strong (panel b) pulses. The pulse parameters are the same as those for Figure 1. The signals are plotted as a function of the time delay T2 between the 5653



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contribution (dotted line) both for weak and strong pulses. However, the overall behavior of the signals in Figure 5a and b differs significantly. For weak pulses, the signal is almost featureless (Figure 5a). For well-separated (T2 > 250 fs) strong pulses, the 3P signal shows vibrational beatings (see inset in Figure 5b). The 2P signal exhibits qualitatively similar behavior (compare Figures 2b and 5b).

Figure 5. 3P signal S3P(T1,T2) induced by weak (λa = 0.01 eV, (a)) and strong (λa = 0.1 eV, (b)) relatively long (Γa = Ω) pulses. The time delay between the first two pulses is fixed at T1 = 263.3 fs. Full black lines, exact calculation; full gray lines, strong-pulse DW approximation. Dotted and dashed lines depict the ground-state and excited-state contributions to the strong-pulse DW signal. The inset in (b) shows the oscillatory structures on an enlarged scale.

second and third pulses. The time delay between the first two pulses is fixed at T1 = 79 fs to make sure that the pulses are temporally well-separated. The signals exhibit pronounced coherent vibrational dynamics, which is excellently reproduced by the strong-pulse DW approximation for T2 > 100 fs (compare full black and gray lines). On the other hand, the 3P (Figure 4) and 2P (Figure 1) signals differ considerably, demonstrating the possibility to manipulate the signals (and, therefore, populations and coherences of the system density matrix) by an extra pump pulse. As found for the 2P signal (Figure 1a), the ground-state contribution to the weak-pulse 3P signal (Figure 4a) is almost featureless (dotted line). The T2 dependence of the signal is predominantly determined by the excited-state contribution (dashed line). For strong pulses, the situation is the opposite (Figure 4b). The excited-state contribution (dashed line) is now much smaller than the ground-state contribution (dotted line). The latter dominates the signal behavior, exhibiting pronounced vibrational beatings. Figure 5 shows 3P signals excited and detected by relatively long weak (panel a) and strong (panel b) pulses. The pulse parameters are the same as those for Figure 2. The time delay between the first two pulses is fixed at T1 = 263.3 fs to make sure that the pulses are temporally well-separated. The strong-pulse DW approximation (gray line) reproduces the exact signal (black line) for T2 > 250 fs. The excited-state contribution (dashed line) is much smaller than the ground-state

6. CONCLUSIONS We have extended the weak-pulse DW representation to the description of signals obtained with temporally well-separated multiple pulses of arbitrary strength. In addition, we have extended the formulation to describe general N-wave-mixing signals in the DW approximation. The theory is applicable whenever the master equation, which determines the dynamics of the material system, can be solved either analytically or by numerical propagation of the density matrix in time. By simulating 2P and 3P signals for a two-state single vibrational mode model, we have shown that the strong-pulse DW approximation is exact for pulses of arbitrary strengths and durations, provided that the pulses are temporally well-separated. For this model system, the strong-pulse DW calculations are faster than the exact calculations by a factor of about 100. The strong-pulse DW approximation is appropriate for the simulation and interpretation of N-wave-mixing signals induced by strong and nonoverlapping pulses. The strong-pulse DW approximation is complementary to the equation-of-motion phase-matching approach,11,63 which is suitable for the computation of N-wave-mixing signals induced by weak pulses. The equation-of-motion phase-matching approach fully accounts for pulse-overlap effects, but is computationally more expensive than the DW approximation. ’ APPENDIX A: EXTRACTION OF THE PHASE-MATCHING POLARIZATION In general, the extraction of a certain N-pulse induced phasematching polarization Pk(t) (eq 14) from the total polarization P(t) (eq 12) for arbitrary pulse strengths requires the N-fold Fourier transform of the total polarization P(t).13 Within the RWA, an (N1)-fold Fourier transform is sufficient. To show this, consider the N-pulse master eq 9 and introduce the operator ~ P g þ P e eik1 r O

ðA1Þ

P g and P e (eq 4) being the projection operators on the electronic ground state and the excited electronic state. We define the new density matrix ~ † FðtÞO ~ FðtÞ O

ðA2Þ

which obeys the same master eq 9, but with ka f ka ka k1

ðA3Þ

in the interaction Hamiltonian eq 7. This transformation brings the phase-matching condition of eq 13 to the form k ¼ 5654

N

∑ la ka a¼2

ðA4Þ



ARTICLE

(note that the summation in eq A4 starts from a = 2). If jha kar are the new phase angles, the phase-matching polarization Pk(t) is extracted from the total polarization P(t) as follows Z 2π N 1 dj ... dj expfi la ja gPðtÞ Pk ðtÞ ¼ 2 N ð2πÞN 1 0 a¼2

∑

ðA5Þ The k1 dependence of Pk(t) is determined by the transformation eq A2. For example, the 2P polarization P2P(t) obeys the phasematching condition in eq 23. Therefore, it can be computed by the formula Z 1 2π dj2 expfij2 gPðtÞ ðA6Þ P2P ðtÞ ¼ 2π 0 Equations A5 and A6 are valid for arbitrary pulse strengths. If the pulses are weak, there exist more efficient methods for the extraction of two-pulse,8 three-pulse,11, and N-pulse63 phasematching polarizations. In practice, the integrals in eqs A5 and A6 are evaluated by discretization, which is the essence of the so-called phase-cycling procedure.78,79 For example 1 nΘ 1 expfimΘgPk ðmΘ; tÞ P2P ðtÞ ¼ nΘ m ¼ 0

∑

2π Θ ðA7Þ nΘ

According to ref 8, nΘ = 4 yields the exact P2P(t) for weak, but overlapping, pump and probe pulses. This choice is also exact for temporally well-separated pulses of any intensity. For the exact calculations discussed in section 5, nΘ = 8 was sufficient to obtain converged results. If the pulses are temporally well-separated, we can consider one pulse at a time. In this situation, the transformation of eq A2 allows us to obtain the explicit phase vector dependence of the density matrix. Indeed, applying eq A2 to the single-pulse (N = 1) master eq 9 yields ~† ~ fG ~ 1 ðt, t 0 ÞFðt 0 ÞgO FðtÞ ¼ O

ðA8Þ

~ 1(t,t0 ) is k1-independent; Due to eq A3, the evolution operator G it is described by the master eq 9 with k1 = 0. Therefore, the entire k1 dependence of the density matrix F(t) is determined by the operators O~ and the initial value F(t0 ). Equation A8 allows one to find this dependence. It is convenient to consider two cases separately. If F(t0 ) is initially in the electronic population state (F(t0 ) P gF(t0 )P g þ P eF(t0 )P e), then Fh(t0 ) = F(t0 ) and P g FðtÞP g ∼ 1 P e FðtÞP e ∼ 1 P e FðtÞP g ∼ e

ik1 r

ik 1 r

P g FðtÞP e ∼ e

ðA9Þ

If F(t0 ) is initially in the electronic coherence state (F(t0 ) = P eF(t0 )P g), then Fh(t0 ) = F(t0 ) exp{ik1r} and ik 1 r

P g FðtÞP g ∼ e ik1 r

P e FðtÞP e ∼ e

P e FðtÞP g ∼ 1

P g FðtÞP e ∼ ei2k1 r

ðA10Þ

These results extend the wavefunction-based analysis of ref 8 to density matrices. To evaluate the DW signal in the phase-matching direction of interest, we must select the appropriate contribution to the density matrix, which, according to eqs A9 and A10, can be

proportional to e(i2k1r, e(ik1r, or 1. This can conveniently be done by introducing the projection operator Q 1, such that Q 1 FðtÞ

ðA11Þ

retains the desired contribution, while all of the others are cancelled. The explicit form of Q 1 (and, in general, Q a, a = 1, ..., N) is specified by the phase-matching conditions in eq 14, and its action on the density matrix is determined by eqs A9 and A10.

’ APPENDIX B: STRONG-PULSE DW APPROXIMATION FOR TWO-PULSE SIGNALS Let us introduce the evolution operator G(2)(t,t0 ), which transforms the density matrix F(t0 ) into the density matrix F(t) for t g t0 according to the two-pulse (N = 2) master eq 9 FðtÞ Gð2Þ ðt, t 0 ÞFðt 0 Þ

ðB1Þ

Suppose that in the infinite past (t0 = ¥) both pulses are switched off, and the system is in its ground electronic state and in vibrational equilibrium Fg ¼ Z1 g expfhg =ðkB Teq Þg

Feq ¼ P g Fg

ðB2Þ

(Zg is the partition function, kB is the Boltzmann constant, and Teq is the temperature). The total complex polarization is therefore PðtÞ ÆX † Gð2Þ ðt, ¥ÞFeq æ

ðB3Þ

Assume that the pulses #1 and #2 are temporally well-separated, that is, the time interval between the pulses, T τ2 τ1, is much longer than the pulse durations Ta (a = 1, 2). Then, we introduce the times τða ( Þ ¼ τa ( δa (δa = χTa , and χ ≈ 1 is a numerical factor67) and split the time interval ]¥,¥[ into the following five sub-intervals ðþÞ

¥, τ2

ðþÞ

, ½τ2

ðÞ

, τ2

ðÞ

, ½τ2

ðþÞ

, τ1

ðþÞ

, ½τ1

ðÞ

, τ1

ðÞ

, ½τ1

, ¥½ ðB4Þ

The choice of δa guarantees that there is no pulse #2 during (þ) () (þ) [τ() 1 ,τ1 ], there is no pulse #1 during [τ2 ,τ2 ], and there are no pulses during the remaining three intervals. Furthermore Gð2Þ ðt, t 0 Þjλ1 ¼ λ2 ¼ 0 Gðt t 0 Þ Gðt t 0 ÞFeq Feq

ðB5Þ

G(t t0 ) being the stationary field-free evolution operator for the master eq 9. We can thus rewrite eq B3 as ðÞ

PðtÞ ÆX † G2 ðt, τ2

ðÞ

ÞGðτ2

ðþÞ

τ1

ðþÞ

ÞG1 ðτ1

ðÞ

, τ1

ÞFeq æ ðB6Þ

Here, Ga(t,t0 ) are the evolution operators for the master eq 9 with a single pulse #a. Equation B6 then yields Z ¥ dt E LO ðtÞPðtÞ ¼ ÆWGðTÞDæ ðB7Þ S ¥

Here, we have introduced the doorway operator D ¼ G1 ðδ1 , δ1 ÞFeq 5655

ðB8Þ


The Journal of Physical Chemistry B and the window operator Z ¥ W ¼ dt E LO ðtÞX † G2 ðt, δ2 Þ δ2

ARTICLE

ðB9Þ

Because l2 = 2, the 2PE W operator is proportional to exp{i2k2r}. According to eqs A10 and A11, we obtain W 2PE Q 2 W P g WP e

Three important technical points should be emphasized here. (i) In eqs B8 and B9, all pulses are centered at τa = 0, and τLO is the time delay between the second pulse and the local oscillatorR pulse. These time shifts R δa are justified by the þδa f(t τ ) dt = identity ττaaδ a δa f(t) dt, which holds a for any operator f(t). Upon these time shifts, each pulse Ea(t τa) exp{(i(kar ωat)} transforms into Ea (t τa) exp{(i(kar ωa(t þ τa))}, acquiring an extra phase factor of exp{-iωaτa}. Fortunately, we can ignore these extra phase factors because they mutually cancel in eq B7 due to the RWA. (ii) The W operator is determined by the quantity X†(τ) X†G2(τ,δ2). Thus, X†(τ) must be propagated backward in time from τ = t to τ = δ2, starting from the initial condition X†(t) X†. Such a backward evolution is governed by the master equation80

’ APPENDIX C: STONG-PULSE DW APPROXIMATION FOR N-PULSE SIGNALS The approach developed in Appendix B can be extended to any N-pulse response. We assume that the pulses are temporally well-separated, that is, the time intervals between the pulses Ta τ a þ 1 τ a

ðB11Þ

Here D Q 1D

W Q 2W

ðC1Þ

τða ( Þ ¼ τa ( δa (δa = χTa, and χ ≈ 1 is a numerical factor67) and split the time interval ]¥,¥[ into the following 2N þ 1 sub-intervals ðþÞ

¥, τN

ðB10Þ

Sk ðTÞ ÆW GðTÞD æ

a ¼ 1, 2, ... , N 1

are much longer than the pulse durations Ta. We introduce the times

Dt FðtÞ ¼ i½H þ HN ðtÞ, FðtÞ ðR b þ p ÞFðtÞ It differs from its standard (forward) counterpart in eq 9 in two respects. First, the sign in front of the dissipation operator is reversed. Second, the forward relaxation operator R is replaced by its backward (superscript b) analogue R b. The latter can readily be constructed from the former by the following simple rule: if R F(t) = ∑j AjF(t)Bj (Aj and Bj being certain operators in the Hilbert space), then R bF(t) = ∑jBjF(t)Aj. (iii) Strictly speaking, we should write G(Tδ1δ2) instead of G(T) in eq B7. However, this correction leads to a (small) shift in the time domain and does not have any physical significance. It can safely be neglected if the system does not evolve significantly during the action of the pulse. Equations B7B9 do not yet yield the final strong-pulse DW formula because S in eq B7 is determined by the total polarization P(t) rather than by the phase-matching polarization Pk(t). To obtain Sk(T), it is essential that the D operator B8 is solely determined by the pulse #1, the W operator B9 be solely determined by the pulse #2, and the propagator G(T) be fieldindependent. It is thus necessary to establish the k1 dependence of the D operator and the k2 dependence of the W operator. As is explained in Appendix A, this can be done by the appropriate projection operators Q 1 and Q 2

ðB15Þ

,

N

ðþÞ f½τða þ Þ , τða Þ , ½τða Þ , τa 1 g ∑ a¼1 ðþÞ

τ0

¥

ðC2Þ

The total N-pulse induced polarization is thus given by the expression ) + * 8 N X Gðτa þ 1 τa ÞGa ðτa , τa Þ Feq :a ¼ 1 ðÞ

ðC3Þ

τN þ 1 t

(T> ensures the appropriate time ordering of the operators). The N-wave-mixing signal eq 15 thus reads Z ¥ S dt E LO ðtÞPðtÞ ¥

* (

¼ T> W

NY 1

) GðTa ÞGa ðδa , δa Þ GðT1 ÞD

+ ðC4Þ

a¼2

Here, the D operator is given by eq B8 and the W operator is defined as Z ¥ W ¼ dt E LO ðtÞX † GN ðt, δN Þ ðC5Þ δN

To account for the phase-matching condition in eq 14, we introduce the appropriate projection operators Q a. Thus, the Nwave-mixing signal in the phase-matching direction k is given by the expression

ðB12Þ

*

Consider the two-pulse photon-echo (2PE) signal as an example. The signal obeys the phase-matching condition

T>

Sk ðT1 , ... , TN 1 Þ ¼ ) + NY 1 Q NW GðTa ÞQ a Ga ðδa , δa Þ GðT1 ÞQ 1 D (

a¼2

ðB13Þ

ðC6Þ

Because l1 = 1, the 2PE D operator is proportional to exp{ik1r}. According to eqs A9 and A11, we get

This formula might be useful for the evaluation of multi-time quantum measurement correlation functions,81 for describing pulse-train-induced optical responses,61 and in (possibly strongpulse) quantum interferometry.82

l1 ¼ 1

l2 ¼ 2

D 2PE Q 1 D P e DP g

ðB14Þ

5656



ARTICLE

Equation C6 should be evaluated step by step. The D operator yields the density matrix after the first pulse is over. Its explicit k1 dependence is established by the projection operator Q 1. The soobtained quantity, DQ 1 Q 1D, is propagated via the field-free Q evolution operator G(T1), yielding DQ 1 (T1) = G(T1)D1 . SubseQ quently, D1 (T1) is propagated in the presence of the second pulse, resulting in D2 = G2(δ2,δ2)DQ 1 (T1), and its appropriate k2 dependence is established, DQ 2 Q 2D2. The procedure is Q (TN1) is convoluted with WQ repeated, and the final DN1 Q NW, yielding the desired N-wave-mixing signal. In practice, it may be more efficient to evaluate the phase-matching polarization Pk(t) directly from the N-pulse master eq 9, inserting the projection operators Q a between neighboring pulses. As an illustration, consider the three-pulse photon-echo (3PE), which obeys the phase-matching condition l1 ¼ 1

l2 ¼ 1

l3 ¼ 1

ðC7Þ

Using eqs A9 and A10, we obtain S3PE ðT1 , T2 Þ ¼ ÆW 3PE GðT2 ÞQ 2 G2 ðδ2 , δ2 ÞGðT1 ÞD 3PE æ ðC8Þ Here D 3PE P e DP g

ðC9Þ

W 3PE P g WP g þ P e WP e

ðC10Þ

Q 2 F P g FP g þ P e FP e

ðC11Þ

The strong-field effects in the 3PE signal have been discussed in ref 47.

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