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An analytic approach to 2D electronic PE spectra of molecular systems
Chemical Physics 383 (2011) 86–92
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An analytic approach to 2D electronic PE spectra of molecular systems V. Szöcs Institute of Chemistry, Comenius University, Mlynská dolina CH2, 842 15 Bratislava, Slovakia
a r t i c l e
i n f o
Article history: Received 3 December 2010 In final form 20 April 2011 Available online 29 April 2011 Keywords: 2D electronic photon echo Spectrogram RWA
a b s t r a c t The three-pulse photon echo (3P-PE) spectra of finite molecular systems and simplified line broadening models is presented. The Fourier picture of a heterodyne detected three-pulse rephasing PE signal in the d-pulse limit of the external field is derived in analytic form. The method includes contributions of one and two-excitonic states and allows direct calculation of Fourier PE spectrogram from corresponding Hamiltonian. As an illustration, the proposed treatment is applied to simple systems, e.g. 2-site two-level system (TLS) and n-site TLS model of photosynthetic unit. The importance of relation between Fourier picture of 3P-PE dynamics (corresponding to nonzero population time, T) and coherent inter-state coupling is emphasized. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The optical 2D PE technique is a unique short-pulse tool to analyze excitonic electronic states of different molecular systems including photosynthetic complexes, quantum wells, etc. [1,2]. The 2D PE Fourier-Transform (FT) spectroscopy is an extension in performing Fourier spectrograms of heterodyne-detected PE signals within a three-pulse set-up [3–6]. The generated signal after the last, third pulse, is mixed with a short-time local oscillator (LO) pulse and a FT is performed in coherence time (i.e. the time between first and the second pulse) and in signal time (the time after the last pulse). Due to the physical set-up, the 2D-PE-FT of the signal is a direct FT of induced polarization and preserves appropriate inter-state coherence. In what follows, we call these FT forms as spectrograms. In contrast to 1D picture (e.g. time-resolved fluorescence), the 2D spectrogram enables to extract different physical properties of the studied system [3,5], including homogeneous/inhomogeneous line broadening (HLB/IHLB) deconvolution and to display the coherent coupling in energy-transfer systems, etc. [7,8,13]. The aim of the present work is to provide a simple theoretical view of electronic 2D-PE for general finite molecular system. The accent is placed in derivation of analytical results for 2D-PE FT signals with possibility of analysis of peak positions in relation to the problem set-up. In view of this fact, we choose most simple models of dissipation: global HLB (due to an inter-state coherence decay) and IHLB (source is a statistical average of inter-state energy fluctuations [8]). In spite of the fact that such a dissipative model is a drastic oversimplification of system relaxation [9], it enables us to move from 2D time domain to 2D FT picture in an analytic form. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.04.019
The approach enables to determine the positions and heights of peaks in 2D-PE spectrograms without numerical FT. An importance of the approach can be directly manifested for any general finite level molecular system constructed from other basic units where the electronic inter-state coupling plays a crucial role. Application of this approach to electronically coupled dimers partially resembles the wavefunction description by Seibt et al., [10], where local vibrations and finite pulse width of exciting pulses are included and to some extend also to the doorway-window approach [11]. Our aim is to derive analytical results for 2D-PE FT signals for a complex, finite state electronic system no matter of the importance of vibronic spectra [10]. Generally, for coupled electro-vibrational system, the numerical FT procedure can hardly be avoided [12] and this statement is also true for exciting pulses with nonzero width. This fact directed us to final simplification. Beside exclusion of vibrational influence we use a d-pulse description of external light pulses. The first deficiency is compensated by the fact that our approach can handle a general electronic, optically active, finite state system. The d-pulse description does not present any drawback due to the contemporary experimental set-up, when the electronic relaxation times are much longer than the time width of applied pulses. Finally, in order to manage it analytically, we use the frame of the rotating wave approach (RWA) [3]. The structure of the paper consists of few Sections and Appendices. Section 2 describes the RWA approach to the general timedependent 3P PE signal. This result is used in Section 3 to perform a direct FT in order to obtain a general PE spectrogram. Section 4 illustrates the PE spectrogram for some simple examples. Appendices cover details of the applied theoretical methods. Appendix A reviews the interaction of a system with a sequence of d-pulses and Appendix B deals with the RWA in a common electronic system where all e-bands are involved. Relations presented in
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Appendices A–C result in a general form of 3D-PE correlation function. Finally, on the basis of Appendix D, the final FT of time-dependent PE signal is performed. 2. Method The system is represented by discrete electronic states forming one ground state set (g-band) with Hamiltonian Hg and n-excited state bands with Hamiltonians He1 ; He2 ; . . . ; Hen . Number of electronic states in each set is arbitrary. The electronic levels between sets a and a + 1 (here a = {g, e1, . . .}) are energetically separated by positive inter-state optical transition energies xa,a+1 (in units h = 1),
H ¼ Hg þ xg;e1 Pe1 þ He1 þ xe1 ;e2 Pe2 þ He2 þ þ xen1 ;en Pen þ Hen ;
ð1Þ
where P ek is a projector into ek-band. We do not specify the internal structure of local Hamiltonians (even they can have an electronvibrational origin as in [15]). In general, we only require that the energy width of every Hek is much smaller than any electronic bandgap. Finally, if the central frequencies of external light pulses are of the order of electronic band-gaps, the applicability of the RWA (together with usual Franck–Condon approach) is justified. In order to describe the system dynamics, driven by an external electric field Hint(t), we suppose nonzero dipole transitions only between neighboring bands (e.g. g M e1, e1 M e2, etc.). Besides RWA, we assume short-time interaction picture [16], which is sketched in Appendix A. Within the RWA approach (see Appendix B for details), an uni ðtÞ (see Eq. (B.3)) evolves according the tary-transformed DM q Liouville Equation (LE) 0
=@t ¼ iLRWA ðtÞq i½Hint;RWA ðtÞ; q @q
ð2Þ
with equilibrium initial g-state DM, qb, and RWA Liouvillian
¼ ½H; q iP q : L0RWA ðtÞq
ð3Þ
Here, according to [14], we introduce electronic dephasing ðPÞ between neighboring bands and,
H¼H
n X
kxe Pek
ð4Þ
k¼1
is the RWA Hamiltonian (see also Appendix C, Eq. (C.1)). The auxiliary energy xe is of the order of electronic interstate bang-gap [14,15]. The semi-classical interaction for external electric field [16] with np pulses has in the RWA an explicit form (see also Appendix B for detail),
Hint;RWA ðtÞ ¼
np X
fp ðt t p ÞfXp exp½iðDp t ~ kp~ r up Þ þ H:c:Þg;
ð5Þ
p¼1
where the terms containing exp½iðxLP þ xe Þt are neglected. In Eq. (5),
Dp ¼ xLP xe
ð6Þ
means a detuning from pth pulse central frequency xLP . Moreover, each pulse is characterized by the dimensionless real envelope ! fp(t tp) centered around pulse interaction time tp, where k p is the pulse wave vector and up its phase. Finally, Xp stands for Rabi frequency operator (scalar product of the field amplitude and dipole transition moment, see Eq. (B.7)). For simplicity we suppose isotropic materials while extension towards anisotropic systems is straightforward. A final note deserves also the electronic dephasing P in Eq. (3) [14], which is proportional to the inverse of the corresponding electronic dephasing time between neighboring electronic bands. In order to simplify the model, we choose a single dephasing
time for all neighboring sets. In spite of this oversimplified approach, it enables us to perform the FT of PE signal (see Appendix D) and obtain analytical results. The interaction part of the Liouvillian, (2), for an interaction term in the form of (5), has a form resembling the short-pulse interaction (see Appendix A, (A.1)). Namely, the interaction part (in (2)) is,
¼ exp½iðDp t ~ þ H:c: Mp q kp~ r up Þ½Xp ; q
ð7Þ
For physical d-pulses (i.e. for fp(t tp) = d(t tp), see Appendix A), the dynamics of DM for times after the last pulse interaction is directly solved. The three-pulse PE is a result of acting three short pulses at times t1 = s T, t2 = T and t3 = 0 having corresponding wave vectors and phases ð~ ki ; ui ; i ¼ 1; 2; 3Þ. The time smeans coherence and T population time (for rephasing PE we use s, T P 0). The LE evolution of the DM according to Eq. (A.5) for rephasing PE has a form
q ð3Þ ðt; T; sÞ ¼ ðiÞ3 exp½ið~ kS~ r þ uS Þ þ iðD1 ðT þ sÞ iD2 TÞrðt; T; sÞ þ H:c:
ð8Þ
Here
~ kS ¼ ~ k1 ~ k2 ~ k3 ;
ð9Þ
uS ¼ u1 u2 u3 is the wave-vector/phase matching condition for corresponding entities and,
h h ii rðt; T; sÞ ¼ Gðt þ T þ sÞ X3 ðT þ sÞ; X2 ðsÞ; Xþ1 qb is an inter-state dipole moment correlation function. The h i Xi ðtÞ ¼ exp iL0RWA t Xi is RWA time evolution of Rabi operator (Eq. (B.7))
corresponding to the ith interaction. The h i 0 GðtÞ ¼ HðtÞ exp iLRWA t is the free system propagator (see Appendix B). In obtaining the above result we have used the fact that
qb Xþ1 ¼ 0 (the operator Xþ1 is of the form je1ihgj whereas the ground-state DM qb has a jgihgj structure). In the heterodyne PE experiment, the field-induced signal is subsequently mixed with an external LO mode. For d-pulse LO model the heterodyne detected PE signal is simply proportional to the created polarization, S(t, T, s) / P(3)(t, T, s), where ð3Þ ðt; T; sÞVðtÞÞ is a trace of DM and dipole matrix Pð3Þ ðt; T; sÞ ¼ Trðq operator in the RWA representation [8]. The time-domain picture of the rephasing PE signal is thus proportional to the dipole correlation function
SC ðt; T; sÞ ¼ hXðT þ sÞXþ ðt þ T þ sÞXðsÞXþ ib þ hXðsÞXþ ðt þ T þ sÞXðT þ sÞXþ ib hXðsÞXðT þ sÞXþ ðt þ T þ sÞXþ ib ;
ð10Þ
where the thermal average over ground-state DM is introduced,
h ib Trð. . . qb Þ: In Eq. (10) the known correlation functions can be retraced. For simplified, general TLS (only one e-band), the first part corresponds to the R2(t, T, s) function [3] and the second part is simply R3(t, T, s). The third part has no TLS analogy (is zero for general TLS) [7] and corresponds to the double-particle correlation function [7]. After a simple algebra we can reveal that the correlation function SC(t, T, s) is proportional to a sum of known functions [3]
RC ðt; T; sÞ ¼ R2 ðt; T; sÞ þ R3 ðt; T; sÞ exp½iDE tR2E ðt; T; sÞ;
ð11Þ
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V. Szöcs / Chemical Physics 383 (2011) 86–92
where the first two contributions depend solely on Hg and He1 [3], þ
þ
R2 ðt; T; sÞ ¼ hXðT þ sÞX ðt þ T þ sÞXðsÞX ib
ð12Þ
R3 ðt; T; sÞ ¼ hXðsÞXþ ðt þ T þ sÞXðT þ sÞXþ ib :
Sðt; T; sÞ ¼ Fðt; T; sÞ ðFðt; T; sÞÞ with
g;e1 ðt sÞ r2 ðt sÞ2 =2 ðt Fðt; T; sÞ ¼ exp½ix þ sÞ=T 2 SC ðt; T; sÞ:
The third function, þ
þ
R2E ðt; T; sÞ ¼ hXðsÞXðT þ sÞX ðt þ T þ sÞX ib
ð13Þ
contains interference with band e2 [7] with the band-gap difference,
DE ¼ xe1 ;e2 xg;e1
ð14Þ
and indicates a presence of two-excitation processes. In models where two-exciton states are constructed from smaller, identical local one-excitation states (e.g. energy-transfer schemes constructed from local TLS models as in photosunthetic unit (PSU) models), the prefactor exp[iDEt] simply equals to 1(DE = 0). As it can be seen, we preserve the general case of nonzero DE ([7,8]). The picture becomes much simpler when all external pulses have the same amplitudes and central frequencies (i.e. Xi X; xLP xL for all pulses). The same is true for LO mode (xLO = xL). The last step represents a direct inclusion of IHLB and HLB. As regards the IHLB, we suppose [7] that it originates from a statistical fluctuation of inter-state energy frequencies xg;e1 . Supposing a Gaussian distribution of xg;e1 around an average fre g;e1 it gives quency x
g;e1 ðt sÞ exp½r2 ðt sÞ2 =2: hexp½ixg;e1 ðt sÞi ! exp½ix Here, r has a meaning of a global IHLB factor. Finally, we use a gross HLB concerning a decay of DM coherences (formally a result of electronic dephasing operator P in Eq. (3), [14]). We take it in a most simple form: a global decay time T2 for all coherences. Finally, we obtain the rephasing time-dependent electronic PE signal [8] in a form
Sðt; T; sÞ ¼ exp½iw
ð15Þ
g;e1 ðt sÞ r2 ðt sÞ2 =2 exp½ix ðt þ sÞ=T 2 RC ðt; T; sÞ þ c:c:; where (see also [3]) we have introduced a phase-shift factor w. Eq. (15) together with (11) is the starting point for Fourier picture analysis of the electronic PE signal (we note that we use times where t, s, T P 0, see Appendix A). As it can be traced up in our derivation, the Franck–Condon principle plays a substantial role. The semiclassical approach ensures that the extension toward analysis of electron-vibrational models of PE in the time domain using (15) together with (11) is possible, too. Nevertheless, the rigid electronic states (with neglected vibrations) and a common IHLB/HLB broadening enable us to perform the FT analytically. A final note is related to possible inclusion of non-rephasing PE terms. The procedure is similar to the above steps used for non~ 1 þ~ rephasing contribution at ~ kNR ¼ k k2 ~ k3 and phase uN= u + u u . The obtained time-signal resembles the form 1 2 3 R of (15) with complex-conjugate exponential term and a dipole correlation function (analogical to (11) containing the R1, R4 correlation functions [3] and an additional 2E part). In order to show the possibility of obtaining analytic FT of PE signals, in what follows we disregard non-rephasing contribution [10]. 3. Fourier-Transform of the PE-signal
Sðxt ; T; xs Þ ¼
0
1
Z
1
expðixt :tÞ expðixs :sÞSðt; T; sÞdtds;
ð18Þ
According Appendix D, the FT of PE signal of the form of Eq. (17) can be calculated using the FT of Fðt; T; sÞ alone. The resulting spectrogram incorporates a common HLB/IHLB broadening. Combining the results of Appendices D and C with the above equations, we obtain a final result for the spectrogram in frequency plane {xs, xt} g;e1 ; þx g;e1 g around the central point fx
X
Sðxt ; T; xs Þ ¼
ag ;be1 ;cg ;de1
h i exp½iTðEag Ecg Þ þ exp½iTðEbe Ede1 Þ 1
ð1Þ ð1Þ c1 ðag ; be1 ; cg ; de1 ÞLTr2 XA þ xA ; XD þ xD X exp½iTðEbe1 Ede1 Þc2 ðag ; be1 ; ce2 ; de1 ÞLTr2 ag ;be1 ;ce2 ;de1
ð2Þ ð2Þ XA þ xA ; XD þ xD
ð19Þ
where the weight of one-excitation contributions are defined as
c1 ðag ; be1 ; cg ; de1 Þ ¼ hag jXg;e1 jbe1 ihbe1 jXþe1 ;g jcg ihcg jXg;e1 jde1 i E hde1 jXþe1 ;g qb
ð20Þ
and for coefficient of double-excitations holds,
c2 ðag ; be1 ; ce2 ; de1 Þ ¼ hag jXg;e1 jbe1 ihbe1 jXe1 ;e2 jce2 ihce2 jXþe2 ;e1 jde1 i E hde1 jXþe1 ;g qb :
ð21Þ
Here, the indexes ag, cg (for g-state), be1 ; de1 (for e1-states), and ce2 (for e2-states) are eigenvalue-indexes of corresponding bands with electronic eigen-energies En (see Appendix C). The coefficients c1 ðag ; be1 ; cg ; de1 Þ (or c2 ðag ; be1 ; ce2 ; de1 Þ) are one- (two-) exciton dipole correlation coefficients and characterize the corresponding Liouville pathways [3]. The IHLB/HLB influence is mirrored in the line-shape function LTr2 ðxA ; xD Þ having, for the simple model, the same form for all peaks (see Eq. (D.6) of Appendix D, a TLS lineshape function). This is due to the choice of model IHLB/HLB processes. Involving different excited state lifetimes for individual states (i.e. adding imaginary part to the eigen-energies as in the model of discrete states, [3]) can modify the resulting spectrograms. The first part of Eq. (19) contains one-excitation correlations with frequencies
Xð1Þ A ¼
Eag Ecg 2
Xð1Þ D ¼
þ
Eag þ Ecg 2
Ebe Ede1 1
2 þ
;
ð22Þ
Ebe1 þ Ede1 2
and two-excitation part with
Xð2Þ A ¼
Eag þ Ece
Xð2Þ D ¼
2
2
Ede1 þ
Eag Ece
2
2
þ
DE ; 2
ð23Þ
DE : 2
Here, according to Appendix D, we introduced diagonal (xD) and anti-diagonal (xA) frequencies having back-transformation
The FT spectrogram of time-dependent 3 pulse PE signal S(t, T, s) is generally defined as
Z
ð17Þ
ð16Þ
0
where a common convention w = p/2 is used [3]. The time-dependent PE signal is then of the form
xA ¼ ðxt þ xs Þ=2; g;e1 þ ðxt xs Þ=2: xD ¼ x
ð24Þ
The one-excitation poles of S(xt, T, xs), P1, are at frequencies XAð1Þ þ xA ¼ 0 and Xð1Þ D þ xD ¼ 0. Using the above results we obtain the poles positions at
V. Szöcs / Chemical Physics 383 (2011) 86–92
xPs1 ¼ xPs0 Ebe1 þ Ecg ;
ð25Þ
xPt 1 ¼ xPt 0 þ Ede1 Eag point P0 at
g;e1 ; xPs0 ¼ x P0 g;e1 : xt ¼ x
ð26Þ
The strength of P1 poles is 2c1 ðag ; be1 ; cg ; de1 Þ (see (20)). On the {xs, xt} frequency plane (and around the central point n o xPs0 ; xPt 0 ), the frequencies of one-excitation poles fulfill the condition
xPs1 xPs0 þ ðxPt 1 xPt 0 Þ ¼ Ede1 Ebe1 þ Ecg Eag
ð27Þ
i.e. they are positioned at differences of g e1transition energies. We see that for identical e-states (i.e. for be1 ¼ de1 ) and degenerate g-state the one-excitation poles are situated on the diagonal of the {xs, xt} plane with strength equal to 2c1 ðag ; be1 ; ag ; be1 Þ, i.e. the diagonal cut of S(xt, T, xs) replicates the absorption of the system. As can be easily seen, there are N 2g N 2e1 one-excitation poles (Ng means the number of states of the g-band and likewise Ne1 for the e1-band). As an example we mention the coupled TLS model previously analyzed ([8], N g ¼ 1; N e1 ¼ 2) resulting in four one-excitation poles positioned not only on the diagonal {xs, xt} plane but also on anti-diagonal of frequency plane. The two-excitation poles P2 of S(xt, T, xs) fulfill the conditions ð2Þ Xð2Þ A þ xA ¼ 0 and XD þ xD ¼ 0, resulting in
xPs2 ¼ xPs0 þ Ede1 Ece2 DE ; P2 t
ð28Þ
P0 t
x ¼ x þ Ede1 Eag : We have N g N e1 N e2 two-excitation poles (N e2 means the number of states of e2-band). It should be noticed that the energy band-gap difference DE shifts the central point of two-excitation poles. From (28) we obtain the relation
xPs2 xPs0 þ DE þ xPt 2 xPt 0 ¼ Eag þ Ece : 2
ð29Þ
This means that for complex systems the anti-diagonal cut of the {xs, xt} plane contains shifted two-excitation poles. For coupled TLS model previously mentioned ([8], N g ¼ 1; N e1 ¼ 2; N e2 ¼ 1) the two P2 poles are situated exactly on the anti-diagonal (D E = 0). The same holds for any complex aggregate made from identical TLS parts (e.g. cyclic model of PSU – see later). As regards an influence of population time T, we see from Eq. (19), that the FT spectrogram dynamics reproduces mainly the e1 spectrum/dynamics having anti-diagonal waving when Ebe1 Ede1 – 0. It is noticeable that both the one- and twoexcitation contributions of FT spectrogram (Eq. (19)) follow the same T-envelope. The energetic ‘‘structure’’ of He1 plays a crucial role in spectrogram dynamics (related to time T). 4. Simple examples The most simple model to check our method is a 3-level system with single g, e1 and e2 states. The Hamiltonian has a simple form: Hg = 0, the first one-excitation state with He1 ¼ 0 lies above the g g;e1 and the e2 single-state with state with inter-state frequency x Hamiltonian He2 ¼ 0 has a relative frequency DE with respect to g;e1 . According to our previous results, the spectrogram x S(xt, T, xs) in the {xs, xt} plane consists of two peaks: the first one with a strength 2jd1 j4 ðd1 dg;e1 Þ is centered around the central g;e1 ; x g;e1 g and in fact corresponds to the TLS spectropoint fx gram. The second peak is shifted in the xs-direction and has a weight jd1j2jd2j2 (now d2 de1 ;e2 ). Due to the single state of He1 ,
89
there is no influence of the population time. In case that jd2j2/ jd1j2 1, this ‘‘shifted’’ two-excitation peak disappears. Let us compare spectrograms of two other simple models having nearly the same absorption spectrum (i.e. diagonal frequency cut). The first example is known as a V-model: simple one-excitation system with common g-state (Hg = 0), two discrete e1-states (He1 ¼ ffDE; 0g; f0; þDEgg, which, in Mathematica notation [17], represents a matrix) centered around mean inter-state energy fre g;e1 . The corresponding dipole transfer vector is quency x V g;e1 ¼ fd1 ; d2 g. Using the symbolic software Mathematica [17], a straightforward application of methods presented in previous section, leads to the spectrogram S(xt, T, xs) having four poles in the {xs, xt} plane. The poles are centered around the point g;e1 ; x g;e1 g. The spectrogram contains four one-excitation poles. fx Two diagonal poles (first one at point { DE/2, DE/2} with normalized strength 2jd1j4 and the second at {+DE/2, +DE/2} with strength 2jd2j4) and two anti-diagonal poles at points { DE/2, ±DE/2} with equal strength 2jd1j2jd2j2. It is important to note that different dipole transfer moments (d1 – d2) lead to different absorption peak heights (diagonal cut), but for V-model preserve the strength of anti-diagonal peaks. The spectrogram symmetry with respect to the diagonal frequency cut for T = 0 is preserved for nonzero population times, too (see Fig. 1, column I). The second model is a coherently-coupled symmetric TLS with inter-site coupling J. We analyzed this model in our previous papers [7,8] with a result showing that the spectrogram consists of a square of peaks with energy distance 2jJj centered around the g;e1 ; x g;e1 g (see Fig.2a of paper [8]). We can now central point fx reproduce this result in a much simpler way. The common ground state jgi = jgIijgIIi with Hamiltonian Hg = 0 is a direct product of local (I and II) states jgIi and jgIIi. The Hilbert space of the first excited state e1 consist of products jeIijgIIi and jgIijeIIi, with Hamiltonian He1 ¼ ff0; Jg; fJ; 0gg and dipole transfer vector V g;e1 ¼ fdI ; dII g (note that the dipole moments, dI and dII, have the same absolute value, but different orientation). Finally, the two-excitonic space e2 is equal to jeIijeIIi with Hamiltonian He2 ¼ 0 and dipole transfer vector V e1 ;e2 ¼ fdII ; dI gþ . The construction of Hilbert space and Hamiltonian of the problem leads directly to the condition DE = 0 (see Eq. (14)), i.e. equal one and two-excitation energy band-gap. Using Eq. (19) together with definition of coefficients c1 and c2 (Eqs. (20) and (21)), we reproduce our previous results for the limit T = 0 (note that in our papers, [7,8], we finally averaged the coefficients c1 and c2 over different site orientation of dipole moments dI and dII when preserving their relative angle). Here, we do not show the analytic form of spectrogram S(xt, T, xs) for nonzero population time T. Rather, we analyze the poles. First, the diagonal poles are of one-excitation origin. For J < 0, the lowest diagonal pole has a strength jdI + dIIj4/2, the strength of the second diagonal pole is jdI dIIj4/2. The non-diagonal poles consist of one and two-excitation contributions. The strength of the left - upper pole is 2 jdI dII j2 =2 ½jdI þ dII j4 ðdI dII Þ2 ðdI þ dII Þ2 =4. The first part reflects the one-excitation contribution whereas the second part is of two-excitation origin. Finally, the right-lower pole strength is 2 equal to jdI dII j2 =2 ½jdI dII j4 ðdI þ dII Þ2 ðdI dII Þ2 =4 with the excitation-origin decomposition as mentioned above. We simply notice, that the spectrogram is non-symmetric around the diagonal cut (e.g. for dI dII the last non-diagonal pole disappears). For J > 0 the role of diagonal poles interchanges (the same holds for antidiagonal poles). In figure, Fig. 1, we compare the spectrograms of a V-model (column I) and of a coupled TLS (column II) for two different population times: T = 0 (first row) and TjJj = p/4 (second row). We choose the model parameters in a such way that the diagonal cuts are nearly equal (for V-model: d2/d1 = 0.4 and for symmetric TLS the angle between dI and dII is equal to p/4). The IHLB and HLB parameters (r and 1/T2) are the same for both cases. The energy scale is chosen in a way as to compare the two models,
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V. Szöcs / Chemical Physics 383 (2011) 86–92
Fig. 1. The 2D PE spectra (absolute value of real part of S(xt, T, xs)) of the 1 + 2E V-system (left column I) vs. spectra of coupled TLS (column II). The population time T is set to: (a) T = 0; (b) TjJj = p/4.
i.e. we choose DE = 2jJj(J < 0). We notice, that the two models have nearly the same absorption spectrum (same diagonal cuts). Moreover, they have similar, square-like, pole structure. What is different is the diagonal symmetry and population-time dynamics of anti-diagonal peaks. For V-model these peaks are symmetric around the diagonal not only for T = 0 but also for T > 0. The antidiagonal peaks of symmetric TLS behave differently. They have different height and their asymmetry is preserved for nonzero population time T. The diagonal symmetry of the spectrogram with increasing population time is an indication how to distinguish between different excitation state structures. At the end of this paragraph, we shortly analyze a ring-like structure with N coupled TLS, i.e. with 2N ‘‘local’’ energy levels. This is a most simple model of a primary PSU. As regards the inter-site coherent coupling between one-excitation states, we choose nonzero coupling J only between nearest neighbor sites. The joined system consist of a common ground state Q j gi ¼ j g i i ði ¼ 1; 2; . . . NÞ with Hamiltonian Hg = 0. The N-times degenerate one-excitation space is constructed from local-site g;e1 . The Hamiltonian excitations and has a mean ‘‘gap’’ frequency x He1 has a form of a band matrix with a cyclic closure (ðHe1 Þi;j ¼ Jdi;j1 and ðHe1 Þ1;N ¼ ðHe1 ÞN;1 ¼ J). Finally, the N(N 1)/2 degenerate two-excitation states (two local states are excited) e1 ;e2 ¼ x g;e1 i.e. DE = 0, see have a Hamiltonian He2 ¼ 0 (note that x Eq. (14)). Note that for N = 2 the coupled symmetric TLS model is reproduced. Due-to the rotational structure of the one-excitation Hamiltonian He1 , it can be easily diagonalized. The one-excitation eigenvalues are
g;e1 ; x g;e1 g {xs, xt} plane centered around the central point fx exhibits N2 one-excitation peaks at points {J cos j1, J cos j2} and N two-excitation peaks at {J cos j, J cos j} (see Eq. (28)). Thus, the two-excitation peaks are placed exactly on the anti-diagonal cut (similar to the case of coupled symmetric TLS). The whole spectrogram is characterized by a square of overall length 2jJj. The global width of the spectrogram quadrate is a direct indication of coherent inter-site coupling between TLS constituents.
5. Summary and conclusions A simple method for calculation of Fourier spectrograms in finite electronic systems has been presented. Our implementation is based on the application of the RWA and Franck–Condon approach. Using simple form of HLB/IHLB dissipation mechanisms, we have been able to perform analytically the FT of time-dependent PE signal. Eq. (19) possesses a direct approach for calculation the 3P PE spectra in any discrete n-level electronic system. In spite of the fact that the included LB processes are global, inclusion of different local state lifetimes can be performed [3]. Our analysis also shows that the one-excitation Hamiltonian spectrum directs both the one and two-excitation contribution of the spectrogram dynamics.It was illustrated for twin models (i.e. V-system vs. J-coupled TLS) with similar diagonal cuts of the spectrogram. In these models, also the spectrogram diagonal asymmetry for a J-coupled TLS, in contrast to the simple local V-system, is preserved for nonzero T.
Ej ¼ J cos j; Acknowledgements
where
j ¼ 2p
nj ; N
nj ¼ 0; 1 . . . N 1
designates the quantum numbers of one-excitation states. According to our analysis (Eq. (25)),the spectrogram S(x t, T, xs) in the
This work was supported by the Slovak Ministry of Education ˇ acky´ and (Vega project 1/0046/08). The author thanks Dr.P.Ban Dr.T.Pálszegi for critical reading of the article and valuable comments.
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Appendix A. Short-time pulse interaction According to [16], the dynamics of the external field-driven system described by the LE, @ q/@ t = iL(t)q ( h = 1), for d-field pulses can be solved analytically. If the Liouvillian consist of a free Liouvillian, L0, and the system is illuminated by np number of d-field pulses acting at times ti(t1 < t2. . . < tnp), i.e. for
LðtÞ ¼ L0
np X
Mp dðt t p Þ
We suppose that the total Liouvillian consist of parts corresponding to electronic Hamiltonian (H) system-field interaction (Hint(t)) and electronic dephasing (P) [14],
Lq ¼ ½H þ Hint ðtÞ; q iP q: Using the fact that the electronic dephasing operator commutes with the operator O(t), we obtain the LE for the reduced DM =@t ¼ iLR ðtÞq with extended Liouvillian @q
ðA:1Þ ¼ ½H þ Hint ðtÞ; q iP q : LR ðtÞq
p¼1
then for times after the last interaction (t > tnp), solution for DM is
qðtÞ ¼ G0 ðt tnp Þ expðiMnp ÞG0 ðtnp tnp1 Þ
Here, H is a RWA electronic Hamiltonian
H ¼ OðtÞþ HOðtÞ P;
expðiMnp1 Þ . . . G0 ðt 2 t1 Þ expðiM1 Þqb ; where
G0 ðtÞ ¼ HðtÞ exp½iL0 t
ðB:5Þ
ðA:2Þ
which, due properties of projectors acting on sub-bands Hek in Eq. (1), i.e. P ei Hek P el ¼ dei el dei ek Hek , results in Eq. (4). The interaction with ! the semiclassical external field E ðtÞ, ! !
is the free propagator (H(t) is the Heaviside step function). Here we have used the fact that the system was initially in thermal equilibrium qb, i.e. L0qb = 0. Using the above result, for three pulses acting in times t1 = s T, t2 = T and t3 = 0(s, T P 0) and for t > 0 we obtain the DM evolution
qðt; T; sÞ ¼ G0 ðtÞ expðiM3 ÞG0 ðTÞ expðiM2 ÞG0 ðsÞ expðiM1 Þqb : The 3rd order expansion in external interaction results in 3
ð3Þ
q ðt; T; sÞ ¼ ðiÞ G0 ðtÞM3 G0 ðTÞM2 G0 ðsÞM1 qb :
ðA:3Þ
3
q ðt; T; sÞ ¼ ðiÞ G0 ðt þ T þ sÞM3 ðT þ sÞM2 ðsÞM1 qb
ðA:4Þ
ðA:5Þ
Our system of interest consists of a set of (electronically) distinct states, one g-band and n-distinct e-bands. We suppose that the energy spectrum of each state-band is much less than the energy band-gap (xa,a+1 and a 2 fg; fek gnk¼1 gg) which is of the order of external field central frequency (i.e. xg;e1 xe1 ;e2 . . . xLP ). We define a RWA unitary operator
kxe Pek :
ðB:1Þ
k¼1
Here xe is an arbitrary energy of the order of inter-band transition, Pek is a projector on the state-band ek and the summation is over all e-bands. Using the projector properties of operators we can simply prove that
OðtÞ ¼ P g þ
n X
!
!
V ¼ X þX þ ;
X
!
X¼
!
V ;
fa;aþ1g2fg;ei g
ðB:6Þ
a;aþ1
where the sum in auxiliary operator X covers only upward transfers ({a, a + 1} = {g, e1};{e1, e2}; . . .{en1 , en}). Utilizing the transformation ! ! properties of operators X and X þ it can be proven that !
Appendix B. RWA for distinct electronic states
OðtÞ ¼ expðiPtÞ; P ¼
Supposing the dipole transfer moments to be nonzero!only between neighboring sub-bands, the dipole moment operator V can be properly splitted
!
OðtÞþ X OðtÞ ¼ expðixe tÞ X ;
and will be used as a base for PE calculation.
n X
Hint ðtÞ ¼ OðtÞþ Hint ðtÞOðtÞ:
!
Eq. (A.3) can be rewritten in a natural form ð3Þ
is transformed accordingly
!
Introducing an interaction representation for Mi in Eq. (A.1)
M i ðtÞ ¼ exp½iL0 tMi exp½iL0 t;
Hint ðtÞ ¼ V E ðtÞ;
!
!
OðtÞþ X þ OðtÞ ¼ expðixe tÞX þ : !
Finally, using a standard form of external field E ðtÞ as a sum timeseparated pulses [3] and neglecting RWA terms containing expðiðxLP þ xe ÞtÞ (xLP means a carrier frequency of pulse number p, see the main text) we obtain the interaction Hint ðtÞ in form of Eq. (5), where !!
Xp ¼ X E 0ðpÞ
ðB:7Þ
has a meaning of a Rabi frequency operator. Appendix C. The correlation functions The general correlation function RC(t, T, s) in Eq. (11), in eigen ¼ 0 in Eq. (3)), has a simple form. Then value representation (P q (see Eq. (4))
H¼H
n X
k xe P ek
k¼1
¼ Hg þ He1 þ Dg;e1 P e1 þ He2 þ ðDg;e1 þ De1 ;e2 ÞPe2 þ
ðC:1Þ
where
P ek expðikxe tÞ:
ðB:2Þ
k¼1
Dg;e1 ¼ xg;e1 xe ; De1 ;e2 ¼ xe1 ;e2 xe :
ðtÞ by a unitary The DM q(t) is connected with the reduced DM q transformation
The decomposed RWA Hamiltonian for each subspace (we need only g, e1 and e2 sub-spaces) can be written as
qðtÞ ¼ OðtÞq ðtÞOðtÞþ :
ðB:3Þ
Hg ¼ Hg ; He1 ¼ He1 þ Dg;e1 P e1 ; He2 ¼ He2 þ ðDg;e1 þ De1 ;e2 ÞPe2 :
ðB:4Þ
The RWA time-picture of Rabi operators entering Eqs. (12) and (13) has nonzero elements only between nearest bands and with simple time evolution
=@t ¼ iLR q , where The original LE, @ q/@t = iLq, results in @ q
LR ¼ OðtÞþ LOðtÞ ½P; . . .; and [ , ] means a commutator.
Xa;aþ1 ðtÞ ¼ exp½iHa tXa;aþ1 exp½iHaþ1 t
ðC:2Þ
ðC:3Þ
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V. Szöcs / Chemical Physics 383 (2011) 86–92
and h.c. form for Xþ aþ1;a ðtÞ (we note that here a = {g, e1, e2}). Explicitly,
X
R2 ðt; T; sÞ ¼
exp½isðEag Ede1 Þ exp½itðEbe1 Ecg Þ
ag ;be1 ;cg ;de1
c1 ðag ; be1 ; cg ; de1 Þ
exp½iTðEag Ecg Þ
ðC:4Þ
t ¼ t s; t þ 2 h0; 1i; t 2 h1; 1i
for the first correlation function and
X
R3 ðt; T; sÞ ¼
and new frequencies, called diagonal (xD) and anti-diagonal (xA) [8], are introduced
exp½isðEag Ede1 Þ exp½itðEbe1 Ecg Þ
ag ;be1 ;cg ;de1
exp½iTðEbe1 Ede1 Þ
c1 ðag ; be1 ; cg ; de1 Þ
ðC:5Þ
for the second function. Here, Eag is an eigenvalue of Hg with eigenstates jagi and Ebe1 means the same for He1 with states j be1 i (indexes ag, cg for g-state and be1 ; de1 for e1-state). The common dipole correlation coefficient has a form in eigenvalue representation. The double-excitation correlation function follows from Eq. (13)
X
R2E ðt; T; sÞ ¼
exp½isðEag Ede1 Þ exp½itðEce Ede1 Þ 2
ag ;be1 ;ce2 ;de1
exp½iTðEbe1 Ede1 Þ
g;e1 . In case that the energy width of each e-band is much x0s ¼ x g;e1 and for frequencies {xs, xt}, which are around smaller than x g;e1 ; x g;e1 ; g, we can eliminate the second part the central point fx of contribution in Eq. (D.3), SC⁄(xt, T, xs)[8]. Finally, we have to accomplish the FT in Eq. (D.4) (we note that t, s P 0). In doing so, the following transformation {t, s} ? {t+, t}can be performed,
c2 ðag ; be1 ; ce2 ; de1 Þ
ðC:6Þ
g;e1 þ xA þ xD ; xs ¼ x g;e1 þ xA xD : xt ¼ x
In order to perform the FT of S (t, T, s), we use the following result for a contribution of exp½ixat t exp½ixbs s part [17]
Z
1
Z
0
1 0
g;e1 ðt sÞ r2 ðt sÞ2 =2 expðixt :tÞ expðixs :sÞ exp½ix
ðt þ sÞ=T 2 exp½ixat t exp½ixbs s xat þ xbs xa xbs ¼ LTr2 þ xA ; t þ xD : 2 2 Here,
pffiffiffiffiffiffiffi
2p exp x2D =2r2
where Ece is the eigenvalue of He2 with eigenstate j ce2 i (indexes ag 2 for g -state and be1 ; de1 for e1-state and ce2 for e2 state).
LTr2 ðxA ; xD Þ ¼
Appendix D. The Fourier-Transform of a time signal
denotes a general TLS line-shape function.
If a time signal is an imaginary part of another signal, i.e. f ðtÞ ¼ Im½FðtÞ, then
References
f ðtÞ ¼ ðFðtÞ F ðtÞÞ=2i:
ðD:1Þ
For the FT defined by
f ðxÞ ¼
Z
expðixtÞf ðtÞdt;
ðD:2Þ
for real values of t, x by means of Eq. (D.1) we get,
f ðxÞ ¼ ðFðxÞ F ðxÞÞ=2i: Thus, for PE signal of the form of Eq. (17) together with Eq. (18) we are able to write (using a suitable signal phase-shift in Eq. (15), i.e. leaving out the complex unit i)
Sðxt ; T; xs Þ ¼ SC ðxt ; T; xs Þ SC ðxt ; T; xs Þ; C
ðD:3Þ
C
where S (xt, T, xs) is the FT of S (t, T, s)
SC ðxt ; T; xs Þ ¼
Z
0
1
Z
1
expðixt tÞ expðixs sÞ;
ðD:4Þ
0
g;e1 ðt sÞ r2 ðt sÞ2 =2 ðt þ sÞ=T 2 RC ðt; T; sÞdtds exp½ix and SC⁄(xt, T, xs) is the complex conjugate of SC(xt, T, xs). We g;e1 and note, that SC(xt, T, xs) is centered around frequencies x0t ¼ x
ðD:5Þ
C
r
T 1 2 ixA
ðD:6Þ
[1] E.L. Read, G.S. Schlau-Cohen, G.S. Engel, J. Wen, R.E. Blankenship, G.R. Fleming, Biophys. J. 95 (2008) 847. [2] K. Gundogdu, K.W. Stone, D.B. Turner, K.A. Nelson, Chem. Phys. 341 (2007) 89. [3] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York Oxford, 1995. [4] S. Mukamel, Ann. Rev. Phys. Chem. 51 (2000) 691. [5] D.M. Jonas, Ann. Rev. Phys. Chem. 54 (2003) 425. [6] M. Cho, Chem. Rev. 108 (2008) 1331. [7] V. Szöcs, T. Palszegi, A. Tortschanoff, H.F. Kauffmann, J. Chem. Phys. 116 (2002) 8218. [8] V. Szöcs, T. Pálszegi, V. Lukeš, J. Sperling, F. Milota, W. Jakubetz, H.F. Kauffmann, J. Chem. Phys. 124 (2006) 124511. [9] Y. Tanimura, J. Phys. Soc. Jpn. 75 (2006) 082001. [10] J. Seibt, K. Renziehausen, D.V. Voronine, V. Engel, J. Chem. Phys. 130 (2009) 134318. [11] M.F. Gelin, D.S. Kosov, Chem. Phys. 347 (2008) 177. [12] T. Brixner, T. Mancˇal, I.V. Stiopkin, G.R. Fleming, J. Chem. Phys. 121 (2004) 4221. [13] W. May, O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Weinheim, 2004. [14] M.F. Gelin, D. Egorova, W. Domcke, J. Chem. Phys. 123 (2005) 164112. [15] D. Egorova, M.F. Gelin, W. Domcke, J. Chem. Phys. 126 (2007) 074314. [16] E. Fick, G. Sauermann, The Quantum Statistics of Dynamic Processes, Springer, 1990. [17] S. Wolfram, The Mathematica Book, Fifth ed., Addison-Wesley, 2003.
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
An analytic approach to 2D electronic PE spectra of molecular systems V. Szöcs Institute of Chemistry, Comenius University, Mlynská dolina CH2, 842 15 Bratislava, Slovakia
a r t i c l e
i n f o
Article history: Received 3 December 2010 In final form 20 April 2011 Available online 29 April 2011 Keywords: 2D electronic photon echo Spectrogram RWA
a b s t r a c t The three-pulse photon echo (3P-PE) spectra of finite molecular systems and simplified line broadening models is presented. The Fourier picture of a heterodyne detected three-pulse rephasing PE signal in the d-pulse limit of the external field is derived in analytic form. The method includes contributions of one and two-excitonic states and allows direct calculation of Fourier PE spectrogram from corresponding Hamiltonian. As an illustration, the proposed treatment is applied to simple systems, e.g. 2-site two-level system (TLS) and n-site TLS model of photosynthetic unit. The importance of relation between Fourier picture of 3P-PE dynamics (corresponding to nonzero population time, T) and coherent inter-state coupling is emphasized. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The optical 2D PE technique is a unique short-pulse tool to analyze excitonic electronic states of different molecular systems including photosynthetic complexes, quantum wells, etc. [1,2]. The 2D PE Fourier-Transform (FT) spectroscopy is an extension in performing Fourier spectrograms of heterodyne-detected PE signals within a three-pulse set-up [3–6]. The generated signal after the last, third pulse, is mixed with a short-time local oscillator (LO) pulse and a FT is performed in coherence time (i.e. the time between first and the second pulse) and in signal time (the time after the last pulse). Due to the physical set-up, the 2D-PE-FT of the signal is a direct FT of induced polarization and preserves appropriate inter-state coherence. In what follows, we call these FT forms as spectrograms. In contrast to 1D picture (e.g. time-resolved fluorescence), the 2D spectrogram enables to extract different physical properties of the studied system [3,5], including homogeneous/inhomogeneous line broadening (HLB/IHLB) deconvolution and to display the coherent coupling in energy-transfer systems, etc. [7,8,13]. The aim of the present work is to provide a simple theoretical view of electronic 2D-PE for general finite molecular system. The accent is placed in derivation of analytical results for 2D-PE FT signals with possibility of analysis of peak positions in relation to the problem set-up. In view of this fact, we choose most simple models of dissipation: global HLB (due to an inter-state coherence decay) and IHLB (source is a statistical average of inter-state energy fluctuations [8]). In spite of the fact that such a dissipative model is a drastic oversimplification of system relaxation [9], it enables us to move from 2D time domain to 2D FT picture in an analytic form. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.04.019
The approach enables to determine the positions and heights of peaks in 2D-PE spectrograms without numerical FT. An importance of the approach can be directly manifested for any general finite level molecular system constructed from other basic units where the electronic inter-state coupling plays a crucial role. Application of this approach to electronically coupled dimers partially resembles the wavefunction description by Seibt et al., [10], where local vibrations and finite pulse width of exciting pulses are included and to some extend also to the doorway-window approach [11]. Our aim is to derive analytical results for 2D-PE FT signals for a complex, finite state electronic system no matter of the importance of vibronic spectra [10]. Generally, for coupled electro-vibrational system, the numerical FT procedure can hardly be avoided [12] and this statement is also true for exciting pulses with nonzero width. This fact directed us to final simplification. Beside exclusion of vibrational influence we use a d-pulse description of external light pulses. The first deficiency is compensated by the fact that our approach can handle a general electronic, optically active, finite state system. The d-pulse description does not present any drawback due to the contemporary experimental set-up, when the electronic relaxation times are much longer than the time width of applied pulses. Finally, in order to manage it analytically, we use the frame of the rotating wave approach (RWA) [3]. The structure of the paper consists of few Sections and Appendices. Section 2 describes the RWA approach to the general timedependent 3P PE signal. This result is used in Section 3 to perform a direct FT in order to obtain a general PE spectrogram. Section 4 illustrates the PE spectrogram for some simple examples. Appendices cover details of the applied theoretical methods. Appendix A reviews the interaction of a system with a sequence of d-pulses and Appendix B deals with the RWA in a common electronic system where all e-bands are involved. Relations presented in
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V. Szöcs / Chemical Physics 383 (2011) 86–92
Appendices A–C result in a general form of 3D-PE correlation function. Finally, on the basis of Appendix D, the final FT of time-dependent PE signal is performed. 2. Method The system is represented by discrete electronic states forming one ground state set (g-band) with Hamiltonian Hg and n-excited state bands with Hamiltonians He1 ; He2 ; . . . ; Hen . Number of electronic states in each set is arbitrary. The electronic levels between sets a and a + 1 (here a = {g, e1, . . .}) are energetically separated by positive inter-state optical transition energies xa,a+1 (in units h = 1),
H ¼ Hg þ xg;e1 Pe1 þ He1 þ xe1 ;e2 Pe2 þ He2 þ þ xen1 ;en Pen þ Hen ;
ð1Þ
where P ek is a projector into ek-band. We do not specify the internal structure of local Hamiltonians (even they can have an electronvibrational origin as in [15]). In general, we only require that the energy width of every Hek is much smaller than any electronic bandgap. Finally, if the central frequencies of external light pulses are of the order of electronic band-gaps, the applicability of the RWA (together with usual Franck–Condon approach) is justified. In order to describe the system dynamics, driven by an external electric field Hint(t), we suppose nonzero dipole transitions only between neighboring bands (e.g. g M e1, e1 M e2, etc.). Besides RWA, we assume short-time interaction picture [16], which is sketched in Appendix A. Within the RWA approach (see Appendix B for details), an uni ðtÞ (see Eq. (B.3)) evolves according the tary-transformed DM q Liouville Equation (LE) 0
=@t ¼ iLRWA ðtÞq i½Hint;RWA ðtÞ; q @q
ð2Þ
with equilibrium initial g-state DM, qb, and RWA Liouvillian
¼ ½H; q iP q : L0RWA ðtÞq
ð3Þ
Here, according to [14], we introduce electronic dephasing ðPÞ between neighboring bands and,
H¼H
n X
kxe Pek
ð4Þ
k¼1
is the RWA Hamiltonian (see also Appendix C, Eq. (C.1)). The auxiliary energy xe is of the order of electronic interstate bang-gap [14,15]. The semi-classical interaction for external electric field [16] with np pulses has in the RWA an explicit form (see also Appendix B for detail),
Hint;RWA ðtÞ ¼
np X
fp ðt t p ÞfXp exp½iðDp t ~ kp~ r up Þ þ H:c:Þg;
ð5Þ
p¼1
where the terms containing exp½iðxLP þ xe Þt are neglected. In Eq. (5),
Dp ¼ xLP xe
ð6Þ
means a detuning from pth pulse central frequency xLP . Moreover, each pulse is characterized by the dimensionless real envelope ! fp(t tp) centered around pulse interaction time tp, where k p is the pulse wave vector and up its phase. Finally, Xp stands for Rabi frequency operator (scalar product of the field amplitude and dipole transition moment, see Eq. (B.7)). For simplicity we suppose isotropic materials while extension towards anisotropic systems is straightforward. A final note deserves also the electronic dephasing P in Eq. (3) [14], which is proportional to the inverse of the corresponding electronic dephasing time between neighboring electronic bands. In order to simplify the model, we choose a single dephasing
time for all neighboring sets. In spite of this oversimplified approach, it enables us to perform the FT of PE signal (see Appendix D) and obtain analytical results. The interaction part of the Liouvillian, (2), for an interaction term in the form of (5), has a form resembling the short-pulse interaction (see Appendix A, (A.1)). Namely, the interaction part (in (2)) is,
¼ exp½iðDp t ~ þ H:c: Mp q kp~ r up Þ½Xp ; q
ð7Þ
For physical d-pulses (i.e. for fp(t tp) = d(t tp), see Appendix A), the dynamics of DM for times after the last pulse interaction is directly solved. The three-pulse PE is a result of acting three short pulses at times t1 = s T, t2 = T and t3 = 0 having corresponding wave vectors and phases ð~ ki ; ui ; i ¼ 1; 2; 3Þ. The time smeans coherence and T population time (for rephasing PE we use s, T P 0). The LE evolution of the DM according to Eq. (A.5) for rephasing PE has a form
q ð3Þ ðt; T; sÞ ¼ ðiÞ3 exp½ið~ kS~ r þ uS Þ þ iðD1 ðT þ sÞ iD2 TÞrðt; T; sÞ þ H:c:
ð8Þ
Here
~ kS ¼ ~ k1 ~ k2 ~ k3 ;
ð9Þ
uS ¼ u1 u2 u3 is the wave-vector/phase matching condition for corresponding entities and,
h h ii rðt; T; sÞ ¼ Gðt þ T þ sÞ X3 ðT þ sÞ; X2 ðsÞ; Xþ1 qb is an inter-state dipole moment correlation function. The h i Xi ðtÞ ¼ exp iL0RWA t Xi is RWA time evolution of Rabi operator (Eq. (B.7))
corresponding to the ith interaction. The h i 0 GðtÞ ¼ HðtÞ exp iLRWA t is the free system propagator (see Appendix B). In obtaining the above result we have used the fact that
qb Xþ1 ¼ 0 (the operator Xþ1 is of the form je1ihgj whereas the ground-state DM qb has a jgihgj structure). In the heterodyne PE experiment, the field-induced signal is subsequently mixed with an external LO mode. For d-pulse LO model the heterodyne detected PE signal is simply proportional to the created polarization, S(t, T, s) / P(3)(t, T, s), where ð3Þ ðt; T; sÞVðtÞÞ is a trace of DM and dipole matrix Pð3Þ ðt; T; sÞ ¼ Trðq operator in the RWA representation [8]. The time-domain picture of the rephasing PE signal is thus proportional to the dipole correlation function
SC ðt; T; sÞ ¼ hXðT þ sÞXþ ðt þ T þ sÞXðsÞXþ ib þ hXðsÞXþ ðt þ T þ sÞXðT þ sÞXþ ib hXðsÞXðT þ sÞXþ ðt þ T þ sÞXþ ib ;
ð10Þ
where the thermal average over ground-state DM is introduced,
h ib Trð. . . qb Þ: In Eq. (10) the known correlation functions can be retraced. For simplified, general TLS (only one e-band), the first part corresponds to the R2(t, T, s) function [3] and the second part is simply R3(t, T, s). The third part has no TLS analogy (is zero for general TLS) [7] and corresponds to the double-particle correlation function [7]. After a simple algebra we can reveal that the correlation function SC(t, T, s) is proportional to a sum of known functions [3]
RC ðt; T; sÞ ¼ R2 ðt; T; sÞ þ R3 ðt; T; sÞ exp½iDE tR2E ðt; T; sÞ;
ð11Þ
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where the first two contributions depend solely on Hg and He1 [3], þ
þ
R2 ðt; T; sÞ ¼ hXðT þ sÞX ðt þ T þ sÞXðsÞX ib
ð12Þ
R3 ðt; T; sÞ ¼ hXðsÞXþ ðt þ T þ sÞXðT þ sÞXþ ib :
Sðt; T; sÞ ¼ Fðt; T; sÞ ðFðt; T; sÞÞ with
g;e1 ðt sÞ r2 ðt sÞ2 =2 ðt Fðt; T; sÞ ¼ exp½ix þ sÞ=T 2 SC ðt; T; sÞ:
The third function, þ
þ
R2E ðt; T; sÞ ¼ hXðsÞXðT þ sÞX ðt þ T þ sÞX ib
ð13Þ
contains interference with band e2 [7] with the band-gap difference,
DE ¼ xe1 ;e2 xg;e1
ð14Þ
and indicates a presence of two-excitation processes. In models where two-exciton states are constructed from smaller, identical local one-excitation states (e.g. energy-transfer schemes constructed from local TLS models as in photosunthetic unit (PSU) models), the prefactor exp[iDEt] simply equals to 1(DE = 0). As it can be seen, we preserve the general case of nonzero DE ([7,8]). The picture becomes much simpler when all external pulses have the same amplitudes and central frequencies (i.e. Xi X; xLP xL for all pulses). The same is true for LO mode (xLO = xL). The last step represents a direct inclusion of IHLB and HLB. As regards the IHLB, we suppose [7] that it originates from a statistical fluctuation of inter-state energy frequencies xg;e1 . Supposing a Gaussian distribution of xg;e1 around an average fre g;e1 it gives quency x
g;e1 ðt sÞ exp½r2 ðt sÞ2 =2: hexp½ixg;e1 ðt sÞi ! exp½ix Here, r has a meaning of a global IHLB factor. Finally, we use a gross HLB concerning a decay of DM coherences (formally a result of electronic dephasing operator P in Eq. (3), [14]). We take it in a most simple form: a global decay time T2 for all coherences. Finally, we obtain the rephasing time-dependent electronic PE signal [8] in a form
Sðt; T; sÞ ¼ exp½iw
ð15Þ
g;e1 ðt sÞ r2 ðt sÞ2 =2 exp½ix ðt þ sÞ=T 2 RC ðt; T; sÞ þ c:c:; where (see also [3]) we have introduced a phase-shift factor w. Eq. (15) together with (11) is the starting point for Fourier picture analysis of the electronic PE signal (we note that we use times where t, s, T P 0, see Appendix A). As it can be traced up in our derivation, the Franck–Condon principle plays a substantial role. The semiclassical approach ensures that the extension toward analysis of electron-vibrational models of PE in the time domain using (15) together with (11) is possible, too. Nevertheless, the rigid electronic states (with neglected vibrations) and a common IHLB/HLB broadening enable us to perform the FT analytically. A final note is related to possible inclusion of non-rephasing PE terms. The procedure is similar to the above steps used for non~ 1 þ~ rephasing contribution at ~ kNR ¼ k k2 ~ k3 and phase uN= u + u u . The obtained time-signal resembles the form 1 2 3 R of (15) with complex-conjugate exponential term and a dipole correlation function (analogical to (11) containing the R1, R4 correlation functions [3] and an additional 2E part). In order to show the possibility of obtaining analytic FT of PE signals, in what follows we disregard non-rephasing contribution [10]. 3. Fourier-Transform of the PE-signal
Sðxt ; T; xs Þ ¼
0
1
Z
1
expðixt :tÞ expðixs :sÞSðt; T; sÞdtds;
ð18Þ
According Appendix D, the FT of PE signal of the form of Eq. (17) can be calculated using the FT of Fðt; T; sÞ alone. The resulting spectrogram incorporates a common HLB/IHLB broadening. Combining the results of Appendices D and C with the above equations, we obtain a final result for the spectrogram in frequency plane {xs, xt} g;e1 ; þx g;e1 g around the central point fx
X
Sðxt ; T; xs Þ ¼
ag ;be1 ;cg ;de1
h i exp½iTðEag Ecg Þ þ exp½iTðEbe Ede1 Þ 1
ð1Þ ð1Þ c1 ðag ; be1 ; cg ; de1 ÞLTr2 XA þ xA ; XD þ xD X exp½iTðEbe1 Ede1 Þc2 ðag ; be1 ; ce2 ; de1 ÞLTr2 ag ;be1 ;ce2 ;de1
ð2Þ ð2Þ XA þ xA ; XD þ xD
ð19Þ
where the weight of one-excitation contributions are defined as
c1 ðag ; be1 ; cg ; de1 Þ ¼ hag jXg;e1 jbe1 ihbe1 jXþe1 ;g jcg ihcg jXg;e1 jde1 i E hde1 jXþe1 ;g qb
ð20Þ
and for coefficient of double-excitations holds,
c2 ðag ; be1 ; ce2 ; de1 Þ ¼ hag jXg;e1 jbe1 ihbe1 jXe1 ;e2 jce2 ihce2 jXþe2 ;e1 jde1 i E hde1 jXþe1 ;g qb :
ð21Þ
Here, the indexes ag, cg (for g-state), be1 ; de1 (for e1-states), and ce2 (for e2-states) are eigenvalue-indexes of corresponding bands with electronic eigen-energies En (see Appendix C). The coefficients c1 ðag ; be1 ; cg ; de1 Þ (or c2 ðag ; be1 ; ce2 ; de1 Þ) are one- (two-) exciton dipole correlation coefficients and characterize the corresponding Liouville pathways [3]. The IHLB/HLB influence is mirrored in the line-shape function LTr2 ðxA ; xD Þ having, for the simple model, the same form for all peaks (see Eq. (D.6) of Appendix D, a TLS lineshape function). This is due to the choice of model IHLB/HLB processes. Involving different excited state lifetimes for individual states (i.e. adding imaginary part to the eigen-energies as in the model of discrete states, [3]) can modify the resulting spectrograms. The first part of Eq. (19) contains one-excitation correlations with frequencies
Xð1Þ A ¼
Eag Ecg 2
Xð1Þ D ¼
þ
Eag þ Ecg 2
Ebe Ede1 1
2 þ
;
ð22Þ
Ebe1 þ Ede1 2
and two-excitation part with
Xð2Þ A ¼
Eag þ Ece
Xð2Þ D ¼
2
2
Ede1 þ
Eag Ece
2
2
þ
DE ; 2
ð23Þ
DE : 2
Here, according to Appendix D, we introduced diagonal (xD) and anti-diagonal (xA) frequencies having back-transformation
The FT spectrogram of time-dependent 3 pulse PE signal S(t, T, s) is generally defined as
Z
ð17Þ
ð16Þ
0
where a common convention w = p/2 is used [3]. The time-dependent PE signal is then of the form
xA ¼ ðxt þ xs Þ=2; g;e1 þ ðxt xs Þ=2: xD ¼ x
ð24Þ
The one-excitation poles of S(xt, T, xs), P1, are at frequencies XAð1Þ þ xA ¼ 0 and Xð1Þ D þ xD ¼ 0. Using the above results we obtain the poles positions at
V. Szöcs / Chemical Physics 383 (2011) 86–92
xPs1 ¼ xPs0 Ebe1 þ Ecg ;
ð25Þ
xPt 1 ¼ xPt 0 þ Ede1 Eag point P0 at
g;e1 ; xPs0 ¼ x P0 g;e1 : xt ¼ x
ð26Þ
The strength of P1 poles is 2c1 ðag ; be1 ; cg ; de1 Þ (see (20)). On the {xs, xt} frequency plane (and around the central point n o xPs0 ; xPt 0 ), the frequencies of one-excitation poles fulfill the condition
xPs1 xPs0 þ ðxPt 1 xPt 0 Þ ¼ Ede1 Ebe1 þ Ecg Eag
ð27Þ
i.e. they are positioned at differences of g e1transition energies. We see that for identical e-states (i.e. for be1 ¼ de1 ) and degenerate g-state the one-excitation poles are situated on the diagonal of the {xs, xt} plane with strength equal to 2c1 ðag ; be1 ; ag ; be1 Þ, i.e. the diagonal cut of S(xt, T, xs) replicates the absorption of the system. As can be easily seen, there are N 2g N 2e1 one-excitation poles (Ng means the number of states of the g-band and likewise Ne1 for the e1-band). As an example we mention the coupled TLS model previously analyzed ([8], N g ¼ 1; N e1 ¼ 2) resulting in four one-excitation poles positioned not only on the diagonal {xs, xt} plane but also on anti-diagonal of frequency plane. The two-excitation poles P2 of S(xt, T, xs) fulfill the conditions ð2Þ Xð2Þ A þ xA ¼ 0 and XD þ xD ¼ 0, resulting in
xPs2 ¼ xPs0 þ Ede1 Ece2 DE ; P2 t
ð28Þ
P0 t
x ¼ x þ Ede1 Eag : We have N g N e1 N e2 two-excitation poles (N e2 means the number of states of e2-band). It should be noticed that the energy band-gap difference DE shifts the central point of two-excitation poles. From (28) we obtain the relation
xPs2 xPs0 þ DE þ xPt 2 xPt 0 ¼ Eag þ Ece : 2
ð29Þ
This means that for complex systems the anti-diagonal cut of the {xs, xt} plane contains shifted two-excitation poles. For coupled TLS model previously mentioned ([8], N g ¼ 1; N e1 ¼ 2; N e2 ¼ 1) the two P2 poles are situated exactly on the anti-diagonal (D E = 0). The same holds for any complex aggregate made from identical TLS parts (e.g. cyclic model of PSU – see later). As regards an influence of population time T, we see from Eq. (19), that the FT spectrogram dynamics reproduces mainly the e1 spectrum/dynamics having anti-diagonal waving when Ebe1 Ede1 – 0. It is noticeable that both the one- and twoexcitation contributions of FT spectrogram (Eq. (19)) follow the same T-envelope. The energetic ‘‘structure’’ of He1 plays a crucial role in spectrogram dynamics (related to time T). 4. Simple examples The most simple model to check our method is a 3-level system with single g, e1 and e2 states. The Hamiltonian has a simple form: Hg = 0, the first one-excitation state with He1 ¼ 0 lies above the g g;e1 and the e2 single-state with state with inter-state frequency x Hamiltonian He2 ¼ 0 has a relative frequency DE with respect to g;e1 . According to our previous results, the spectrogram x S(xt, T, xs) in the {xs, xt} plane consists of two peaks: the first one with a strength 2jd1 j4 ðd1 dg;e1 Þ is centered around the central g;e1 ; x g;e1 g and in fact corresponds to the TLS spectropoint fx gram. The second peak is shifted in the xs-direction and has a weight jd1j2jd2j2 (now d2 de1 ;e2 ). Due to the single state of He1 ,
89
there is no influence of the population time. In case that jd2j2/ jd1j2 1, this ‘‘shifted’’ two-excitation peak disappears. Let us compare spectrograms of two other simple models having nearly the same absorption spectrum (i.e. diagonal frequency cut). The first example is known as a V-model: simple one-excitation system with common g-state (Hg = 0), two discrete e1-states (He1 ¼ ffDE; 0g; f0; þDEgg, which, in Mathematica notation [17], represents a matrix) centered around mean inter-state energy fre g;e1 . The corresponding dipole transfer vector is quency x V g;e1 ¼ fd1 ; d2 g. Using the symbolic software Mathematica [17], a straightforward application of methods presented in previous section, leads to the spectrogram S(xt, T, xs) having four poles in the {xs, xt} plane. The poles are centered around the point g;e1 ; x g;e1 g. The spectrogram contains four one-excitation poles. fx Two diagonal poles (first one at point { DE/2, DE/2} with normalized strength 2jd1j4 and the second at {+DE/2, +DE/2} with strength 2jd2j4) and two anti-diagonal poles at points { DE/2, ±DE/2} with equal strength 2jd1j2jd2j2. It is important to note that different dipole transfer moments (d1 – d2) lead to different absorption peak heights (diagonal cut), but for V-model preserve the strength of anti-diagonal peaks. The spectrogram symmetry with respect to the diagonal frequency cut for T = 0 is preserved for nonzero population times, too (see Fig. 1, column I). The second model is a coherently-coupled symmetric TLS with inter-site coupling J. We analyzed this model in our previous papers [7,8] with a result showing that the spectrogram consists of a square of peaks with energy distance 2jJj centered around the g;e1 ; x g;e1 g (see Fig.2a of paper [8]). We can now central point fx reproduce this result in a much simpler way. The common ground state jgi = jgIijgIIi with Hamiltonian Hg = 0 is a direct product of local (I and II) states jgIi and jgIIi. The Hilbert space of the first excited state e1 consist of products jeIijgIIi and jgIijeIIi, with Hamiltonian He1 ¼ ff0; Jg; fJ; 0gg and dipole transfer vector V g;e1 ¼ fdI ; dII g (note that the dipole moments, dI and dII, have the same absolute value, but different orientation). Finally, the two-excitonic space e2 is equal to jeIijeIIi with Hamiltonian He2 ¼ 0 and dipole transfer vector V e1 ;e2 ¼ fdII ; dI gþ . The construction of Hilbert space and Hamiltonian of the problem leads directly to the condition DE = 0 (see Eq. (14)), i.e. equal one and two-excitation energy band-gap. Using Eq. (19) together with definition of coefficients c1 and c2 (Eqs. (20) and (21)), we reproduce our previous results for the limit T = 0 (note that in our papers, [7,8], we finally averaged the coefficients c1 and c2 over different site orientation of dipole moments dI and dII when preserving their relative angle). Here, we do not show the analytic form of spectrogram S(xt, T, xs) for nonzero population time T. Rather, we analyze the poles. First, the diagonal poles are of one-excitation origin. For J < 0, the lowest diagonal pole has a strength jdI + dIIj4/2, the strength of the second diagonal pole is jdI dIIj4/2. The non-diagonal poles consist of one and two-excitation contributions. The strength of the left - upper pole is 2 jdI dII j2 =2 ½jdI þ dII j4 ðdI dII Þ2 ðdI þ dII Þ2 =4. The first part reflects the one-excitation contribution whereas the second part is of two-excitation origin. Finally, the right-lower pole strength is 2 equal to jdI dII j2 =2 ½jdI dII j4 ðdI þ dII Þ2 ðdI dII Þ2 =4 with the excitation-origin decomposition as mentioned above. We simply notice, that the spectrogram is non-symmetric around the diagonal cut (e.g. for dI dII the last non-diagonal pole disappears). For J > 0 the role of diagonal poles interchanges (the same holds for antidiagonal poles). In figure, Fig. 1, we compare the spectrograms of a V-model (column I) and of a coupled TLS (column II) for two different population times: T = 0 (first row) and TjJj = p/4 (second row). We choose the model parameters in a such way that the diagonal cuts are nearly equal (for V-model: d2/d1 = 0.4 and for symmetric TLS the angle between dI and dII is equal to p/4). The IHLB and HLB parameters (r and 1/T2) are the same for both cases. The energy scale is chosen in a way as to compare the two models,
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V. Szöcs / Chemical Physics 383 (2011) 86–92
Fig. 1. The 2D PE spectra (absolute value of real part of S(xt, T, xs)) of the 1 + 2E V-system (left column I) vs. spectra of coupled TLS (column II). The population time T is set to: (a) T = 0; (b) TjJj = p/4.
i.e. we choose DE = 2jJj(J < 0). We notice, that the two models have nearly the same absorption spectrum (same diagonal cuts). Moreover, they have similar, square-like, pole structure. What is different is the diagonal symmetry and population-time dynamics of anti-diagonal peaks. For V-model these peaks are symmetric around the diagonal not only for T = 0 but also for T > 0. The antidiagonal peaks of symmetric TLS behave differently. They have different height and their asymmetry is preserved for nonzero population time T. The diagonal symmetry of the spectrogram with increasing population time is an indication how to distinguish between different excitation state structures. At the end of this paragraph, we shortly analyze a ring-like structure with N coupled TLS, i.e. with 2N ‘‘local’’ energy levels. This is a most simple model of a primary PSU. As regards the inter-site coherent coupling between one-excitation states, we choose nonzero coupling J only between nearest neighbor sites. The joined system consist of a common ground state Q j gi ¼ j g i i ði ¼ 1; 2; . . . NÞ with Hamiltonian Hg = 0. The N-times degenerate one-excitation space is constructed from local-site g;e1 . The Hamiltonian excitations and has a mean ‘‘gap’’ frequency x He1 has a form of a band matrix with a cyclic closure (ðHe1 Þi;j ¼ Jdi;j1 and ðHe1 Þ1;N ¼ ðHe1 ÞN;1 ¼ J). Finally, the N(N 1)/2 degenerate two-excitation states (two local states are excited) e1 ;e2 ¼ x g;e1 i.e. DE = 0, see have a Hamiltonian He2 ¼ 0 (note that x Eq. (14)). Note that for N = 2 the coupled symmetric TLS model is reproduced. Due-to the rotational structure of the one-excitation Hamiltonian He1 , it can be easily diagonalized. The one-excitation eigenvalues are
g;e1 ; x g;e1 g {xs, xt} plane centered around the central point fx exhibits N2 one-excitation peaks at points {J cos j1, J cos j2} and N two-excitation peaks at {J cos j, J cos j} (see Eq. (28)). Thus, the two-excitation peaks are placed exactly on the anti-diagonal cut (similar to the case of coupled symmetric TLS). The whole spectrogram is characterized by a square of overall length 2jJj. The global width of the spectrogram quadrate is a direct indication of coherent inter-site coupling between TLS constituents.
5. Summary and conclusions A simple method for calculation of Fourier spectrograms in finite electronic systems has been presented. Our implementation is based on the application of the RWA and Franck–Condon approach. Using simple form of HLB/IHLB dissipation mechanisms, we have been able to perform analytically the FT of time-dependent PE signal. Eq. (19) possesses a direct approach for calculation the 3P PE spectra in any discrete n-level electronic system. In spite of the fact that the included LB processes are global, inclusion of different local state lifetimes can be performed [3]. Our analysis also shows that the one-excitation Hamiltonian spectrum directs both the one and two-excitation contribution of the spectrogram dynamics.It was illustrated for twin models (i.e. V-system vs. J-coupled TLS) with similar diagonal cuts of the spectrogram. In these models, also the spectrogram diagonal asymmetry for a J-coupled TLS, in contrast to the simple local V-system, is preserved for nonzero T.
Ej ¼ J cos j; Acknowledgements
where
j ¼ 2p
nj ; N
nj ¼ 0; 1 . . . N 1
designates the quantum numbers of one-excitation states. According to our analysis (Eq. (25)),the spectrogram S(x t, T, xs) in the
This work was supported by the Slovak Ministry of Education ˇ acky´ and (Vega project 1/0046/08). The author thanks Dr.P.Ban Dr.T.Pálszegi for critical reading of the article and valuable comments.
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V. Szöcs / Chemical Physics 383 (2011) 86–92
Appendix A. Short-time pulse interaction According to [16], the dynamics of the external field-driven system described by the LE, @ q/@ t = iL(t)q ( h = 1), for d-field pulses can be solved analytically. If the Liouvillian consist of a free Liouvillian, L0, and the system is illuminated by np number of d-field pulses acting at times ti(t1 < t2. . . < tnp), i.e. for
LðtÞ ¼ L0
np X
Mp dðt t p Þ
We suppose that the total Liouvillian consist of parts corresponding to electronic Hamiltonian (H) system-field interaction (Hint(t)) and electronic dephasing (P) [14],
Lq ¼ ½H þ Hint ðtÞ; q iP q: Using the fact that the electronic dephasing operator commutes with the operator O(t), we obtain the LE for the reduced DM =@t ¼ iLR ðtÞq with extended Liouvillian @q
ðA:1Þ ¼ ½H þ Hint ðtÞ; q iP q : LR ðtÞq
p¼1
then for times after the last interaction (t > tnp), solution for DM is
qðtÞ ¼ G0 ðt tnp Þ expðiMnp ÞG0 ðtnp tnp1 Þ
Here, H is a RWA electronic Hamiltonian
H ¼ OðtÞþ HOðtÞ P;
expðiMnp1 Þ . . . G0 ðt 2 t1 Þ expðiM1 Þqb ; where
G0 ðtÞ ¼ HðtÞ exp½iL0 t
ðB:5Þ
ðA:2Þ
which, due properties of projectors acting on sub-bands Hek in Eq. (1), i.e. P ei Hek P el ¼ dei el dei ek Hek , results in Eq. (4). The interaction with ! the semiclassical external field E ðtÞ, ! !
is the free propagator (H(t) is the Heaviside step function). Here we have used the fact that the system was initially in thermal equilibrium qb, i.e. L0qb = 0. Using the above result, for three pulses acting in times t1 = s T, t2 = T and t3 = 0(s, T P 0) and for t > 0 we obtain the DM evolution
qðt; T; sÞ ¼ G0 ðtÞ expðiM3 ÞG0 ðTÞ expðiM2 ÞG0 ðsÞ expðiM1 Þqb : The 3rd order expansion in external interaction results in 3
ð3Þ
q ðt; T; sÞ ¼ ðiÞ G0 ðtÞM3 G0 ðTÞM2 G0 ðsÞM1 qb :
ðA:3Þ
3
q ðt; T; sÞ ¼ ðiÞ G0 ðt þ T þ sÞM3 ðT þ sÞM2 ðsÞM1 qb
ðA:4Þ
ðA:5Þ
Our system of interest consists of a set of (electronically) distinct states, one g-band and n-distinct e-bands. We suppose that the energy spectrum of each state-band is much less than the energy band-gap (xa,a+1 and a 2 fg; fek gnk¼1 gg) which is of the order of external field central frequency (i.e. xg;e1 xe1 ;e2 . . . xLP ). We define a RWA unitary operator
kxe Pek :
ðB:1Þ
k¼1
Here xe is an arbitrary energy of the order of inter-band transition, Pek is a projector on the state-band ek and the summation is over all e-bands. Using the projector properties of operators we can simply prove that
OðtÞ ¼ P g þ
n X
!
!
V ¼ X þX þ ;
X
!
X¼
!
V ;
fa;aþ1g2fg;ei g
ðB:6Þ
a;aþ1
where the sum in auxiliary operator X covers only upward transfers ({a, a + 1} = {g, e1};{e1, e2}; . . .{en1 , en}). Utilizing the transformation ! ! properties of operators X and X þ it can be proven that !
Appendix B. RWA for distinct electronic states
OðtÞ ¼ expðiPtÞ; P ¼
Supposing the dipole transfer moments to be nonzero!only between neighboring sub-bands, the dipole moment operator V can be properly splitted
!
OðtÞþ X OðtÞ ¼ expðixe tÞ X ;
and will be used as a base for PE calculation.
n X
Hint ðtÞ ¼ OðtÞþ Hint ðtÞOðtÞ:
!
Eq. (A.3) can be rewritten in a natural form ð3Þ
is transformed accordingly
!
Introducing an interaction representation for Mi in Eq. (A.1)
M i ðtÞ ¼ exp½iL0 tMi exp½iL0 t;
Hint ðtÞ ¼ V E ðtÞ;
!
!
OðtÞþ X þ OðtÞ ¼ expðixe tÞX þ : !
Finally, using a standard form of external field E ðtÞ as a sum timeseparated pulses [3] and neglecting RWA terms containing expðiðxLP þ xe ÞtÞ (xLP means a carrier frequency of pulse number p, see the main text) we obtain the interaction Hint ðtÞ in form of Eq. (5), where !!
Xp ¼ X E 0ðpÞ
ðB:7Þ
has a meaning of a Rabi frequency operator. Appendix C. The correlation functions The general correlation function RC(t, T, s) in Eq. (11), in eigen ¼ 0 in Eq. (3)), has a simple form. Then value representation (P q (see Eq. (4))
H¼H
n X
k xe P ek
k¼1
¼ Hg þ He1 þ Dg;e1 P e1 þ He2 þ ðDg;e1 þ De1 ;e2 ÞPe2 þ
ðC:1Þ
where
P ek expðikxe tÞ:
ðB:2Þ
k¼1
Dg;e1 ¼ xg;e1 xe ; De1 ;e2 ¼ xe1 ;e2 xe :
ðtÞ by a unitary The DM q(t) is connected with the reduced DM q transformation
The decomposed RWA Hamiltonian for each subspace (we need only g, e1 and e2 sub-spaces) can be written as
qðtÞ ¼ OðtÞq ðtÞOðtÞþ :
ðB:3Þ
Hg ¼ Hg ; He1 ¼ He1 þ Dg;e1 P e1 ; He2 ¼ He2 þ ðDg;e1 þ De1 ;e2 ÞPe2 :
ðB:4Þ
The RWA time-picture of Rabi operators entering Eqs. (12) and (13) has nonzero elements only between nearest bands and with simple time evolution
=@t ¼ iLR q , where The original LE, @ q/@t = iLq, results in @ q
LR ¼ OðtÞþ LOðtÞ ½P; . . .; and [ , ] means a commutator.
Xa;aþ1 ðtÞ ¼ exp½iHa tXa;aþ1 exp½iHaþ1 t
ðC:2Þ
ðC:3Þ
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V. Szöcs / Chemical Physics 383 (2011) 86–92
and h.c. form for Xþ aþ1;a ðtÞ (we note that here a = {g, e1, e2}). Explicitly,
X
R2 ðt; T; sÞ ¼
exp½isðEag Ede1 Þ exp½itðEbe1 Ecg Þ
ag ;be1 ;cg ;de1
c1 ðag ; be1 ; cg ; de1 Þ
exp½iTðEag Ecg Þ
ðC:4Þ
t ¼ t s; t þ 2 h0; 1i; t 2 h1; 1i
for the first correlation function and
X
R3 ðt; T; sÞ ¼
and new frequencies, called diagonal (xD) and anti-diagonal (xA) [8], are introduced
exp½isðEag Ede1 Þ exp½itðEbe1 Ecg Þ
ag ;be1 ;cg ;de1
exp½iTðEbe1 Ede1 Þ
c1 ðag ; be1 ; cg ; de1 Þ
ðC:5Þ
for the second function. Here, Eag is an eigenvalue of Hg with eigenstates jagi and Ebe1 means the same for He1 with states j be1 i (indexes ag, cg for g-state and be1 ; de1 for e1-state). The common dipole correlation coefficient has a form in eigenvalue representation. The double-excitation correlation function follows from Eq. (13)
X
R2E ðt; T; sÞ ¼
exp½isðEag Ede1 Þ exp½itðEce Ede1 Þ 2
ag ;be1 ;ce2 ;de1
exp½iTðEbe1 Ede1 Þ
g;e1 . In case that the energy width of each e-band is much x0s ¼ x g;e1 and for frequencies {xs, xt}, which are around smaller than x g;e1 ; x g;e1 ; g, we can eliminate the second part the central point fx of contribution in Eq. (D.3), SC⁄(xt, T, xs)[8]. Finally, we have to accomplish the FT in Eq. (D.4) (we note that t, s P 0). In doing so, the following transformation {t, s} ? {t+, t}can be performed,
c2 ðag ; be1 ; ce2 ; de1 Þ
ðC:6Þ
g;e1 þ xA þ xD ; xs ¼ x g;e1 þ xA xD : xt ¼ x
In order to perform the FT of S (t, T, s), we use the following result for a contribution of exp½ixat t exp½ixbs s part [17]
Z
1
Z
0
1 0
g;e1 ðt sÞ r2 ðt sÞ2 =2 expðixt :tÞ expðixs :sÞ exp½ix
ðt þ sÞ=T 2 exp½ixat t exp½ixbs s xat þ xbs xa xbs ¼ LTr2 þ xA ; t þ xD : 2 2 Here,
pffiffiffiffiffiffiffi
2p exp x2D =2r2
where Ece is the eigenvalue of He2 with eigenstate j ce2 i (indexes ag 2 for g -state and be1 ; de1 for e1-state and ce2 for e2 state).
LTr2 ðxA ; xD Þ ¼
Appendix D. The Fourier-Transform of a time signal
denotes a general TLS line-shape function.
If a time signal is an imaginary part of another signal, i.e. f ðtÞ ¼ Im½FðtÞ, then
References
f ðtÞ ¼ ðFðtÞ F ðtÞÞ=2i:
ðD:1Þ
For the FT defined by
f ðxÞ ¼
Z
expðixtÞf ðtÞdt;
ðD:2Þ
for real values of t, x by means of Eq. (D.1) we get,
f ðxÞ ¼ ðFðxÞ F ðxÞÞ=2i: Thus, for PE signal of the form of Eq. (17) together with Eq. (18) we are able to write (using a suitable signal phase-shift in Eq. (15), i.e. leaving out the complex unit i)
Sðxt ; T; xs Þ ¼ SC ðxt ; T; xs Þ SC ðxt ; T; xs Þ; C
ðD:3Þ
C
where S (xt, T, xs) is the FT of S (t, T, s)
SC ðxt ; T; xs Þ ¼
Z
0
1
Z
1
expðixt tÞ expðixs sÞ;
ðD:4Þ
0
g;e1 ðt sÞ r2 ðt sÞ2 =2 ðt þ sÞ=T 2 RC ðt; T; sÞdtds exp½ix and SC⁄(xt, T, xs) is the complex conjugate of SC(xt, T, xs). We g;e1 and note, that SC(xt, T, xs) is centered around frequencies x0t ¼ x
ðD:5Þ
C
r
T 1 2 ixA
ðD:6Þ
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