Beating Signals in 2D Spectroscopy: Electronic or ... - ACS Publications

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Beating Signals in 2D Spectroscopy: Electronic or Nuclear Coherences? Application to a Quantum Dot Model System Joachim Seibt* and Tõnu Pullerits Department of Chemical Physics, Lund University, Box 124, SE-2100, Lund, Sweden S Supporting Information *

ABSTRACT: In 2D electronic spectroscopy, oscillatory signals have recently received increased attention. The key issue is the origin of such beating whether it is vibrational or electronic. In analogy to the distinction between rephasing and nonrephasing contributions to 2D spectra, we separate signal components with positive and negative coherence beating frequency via a third Fourier transformation and consider two-dimensional cuts at the respective positions. We apply this approach to a model system for the description of quantum dots (QDs) and analyze the possibility to distinguish between vibrational effects associated with the longitudinal optical (LO) phonon mode and electronic fine structure splitting.

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ground state.22 Recently an approach was introduced enabling quantification of vibrational effects in coherence beatings of molecular aggregates.23 A thorough comparison of the 2D spectroscopic features caused by a vibronic molecule or an excitonic dimer was recently carried out,24 the signatures of vibrations-induced quantum beats in 2D spectra were investigated,25 and the distinctive character of electronic and vibrational coherences in disordered molecular aggregates was discussed.26 Inspired by concepts from photosynthesis,27 quantum dots (QDs) can be used for light harvesting in devices like QDsensitized solar cells,28,29 where their role is analogous to the photosynthetic antenna pigments. The size-dependent spectroscopic properties of quantum dots are reported in the literature.30−34 In CdSe, the energy gaps between the fine structure states are similar to the longitudinal optical (LO) phonon frequency: about 200 cm−1. A slight indication of the beatings in 2D spectroscopy of CdSe quantum dots with such frequency has been reported.35 A question does rise: is this oscillation due to the LO mode or due to the electronic fine structure states? In this work we carry out a comparison of oscillatory signals from these two origins. Note that it has been reported that the exciton−phonon coupling of the LO phonon mode strongly depends on the excitonic state.36 However, we consider the effects of fine structure splitting and vibrations associated with the LO mode separately in our investigation. We adapt the approach by Butkus et al.,24 but concentrate on the oscillatory components of the 2D spectra. In analogy to the rephasing and nonrephasing signals, we separate the positive

INTRODUCTION The capability of spreading spectral information in multiple dimensions can reveal details, which are not available by onedimensional spectroscopic techniques. The general experimental and theoretical background of coherent multidimensional spectroscopy in the optical spectral range has been described extensively in the literature.1−4 Specific ways of calculating 2D spectra5−10 and modeling of 2D spectroscopy for the detailed analysis of excitation processes in molecular dimers11,12 have been discussed. The concept of 2D spectroscopy can be expanded by applying a Fourier transformation with respect to the remaining parametric time dependence so that a 3D representation is obtained.13−16 By choosing proper phase-matching conditions, the double quantum coherence signals17 can be excluded and the third frequency spectral components fall to the energy range typical for vibrational oscillations or excitonic coherence beatings. Such beating signals of electronic 2D spectroscopy have received particularly large attention in the context of the question whether vibrational and electronic coherences can be distinguished. The respective investigations were motivated by the observation of long-lived coherences in so-called Fenna− Matthews−Olson (FMO) antenna complexes.18 Also, the phase differences between beatings of 2D spectral features have been considered. A phase shift of 90° was interpreted as a signature of electronic quantum transport.19 Based on the vibronic exciton model,20 it was shown that strong enhancement of weak vibronic transitions takes place in FMO. This allows to explain the long lifetime of the beatings.21 An analogous enhancement of vibronic transitions was shown to occur as a result of nonadiabatic coupling between vibronic levels. The model included vibrationally excited levels, and the beatings were related to vibrational coherences in the electronic © 2013 American Chemical Society

Received: June 20, 2013 Revised: August 14, 2013 Published: August 30, 2013 18728

dx.doi.org/10.1021/jp406103m | J. Phys. Chem. C 2013, 117, 18728−18737

The Journal of Physical Chemistry C

Article

and negative frequency components in respect of the additional frequency axis. This article is organized as follows: In the Theoretical Methods, we present the background of the calculation of 3D spectra and recapitulate properties of the quantum dot model13 as far as they are relevant for the current study. We initiate the discussion of our results with a prediction of the peak positions in two-dimensional cuts of 3D spectra taken at a positive or negative frequency. Two cases are analyzed: the features either originate from a LO mode or the same fine structure splitting as the vibrational frequency. With this background, the effects appearing in two-dimensional cuts at a positive or negative frequency position from rephasing and nonrephasing 3D spectra are discussed under different assumptions about the quantum dot model system, aiming at a distinction between effects caused by vibrations and electronic fine structure splitting. The findings are summarized in Conclusions.

dt1

∫0

∞

dt 3

(3)

and σ2D,NR (ωτ , t 2 , ωt )=

∫0

∞

dt1

∫0

∞

dt 3

(3) exp(iωτ t1) exp(iωt t3)PNR (t1 , t 2 , t3),

(4)

for the rephasing and nonrephasing contributions, respectively. The Fourier transformations are only taken half-sided from zero to infinity, as the response functions vanish for negative time arguments. 3D spectra can be obtained by applying an additional Fourier transformation to 2D spectra, which still exhibit a parametric time dependence on t2:

THEORETICAL METHODS For the calculation of 2D spectra we are using the conventional perturbative expansion of the system density matrix in terms of system-field interaction. The bath is added via cumulant expansion.37 In this way response functions with dependence on the time intervals t1, t2, and t3 between system-field interaction points can be derived.3 The response functions R3g(t1, t2, t3) and R4g(t1, t2, t3) describe ground state bleaching (GSB) processes, R1g(t1, t2, t3) and R2g(t1, t2, t3) account for stimulated emission (SE), and R1f* (t1, t2, t3) and R2f* (t1, t2, t3) are related to excited state absorption (ESA). The subscripted characters g and f indicate whether in the underlying excitation processes only the electronic ground state or also a doubly excited state is involved besides the singly excited state.1 Due to causality, the response functions are nonzero only if all time arguments are larger than zero.1 We use the impulsive limit assuming δ-shaped pulses. We follow the notation where the time delay between the first two pulses is τ, the delay between the second and the third pulse is T, and the time t accounts for the period until the signal is detected after excitation by the third pulse. Under the assumption that the detection direction is chosen as ks⃗ = −k1⃗ + k2⃗ + k3⃗ , where ki⃗ denotes the wavevector of the pulse i, the response functions can be assigned to the rephasing (P(3) R ) or (3) nonrephasing (PNR ) part of the third-order polarization, depending on whether they are nonzero for positive or negative τ values. Negative τ values mean that pulse 2 arrives before pulse 1. For the rephasing part, one finds

σ3D(ωτ , ωT , ωt ) =

∫0

∞

dt 2 exp( −iωT t 2)σ2D(ωτ , t 2 , ωt ) (5)

For the evaluation of the 3D spectra by considering twodimensional cuts, the amplitude A(σ3D(ωτ,ωT = ω̃ ,ωt)) and the phase ϕ(σ3D(ωτ,ωT = ω̃ ,ωt)) are taken at a selected frequency position ωT = ω̃ . When only the real or imaginary part of the 2D spectrum is Fourier transformed, the amplitudes of the two-dimensional cuts of the 3D spectrum at ωT = ω̃ and ωT = −ω̃ are identical. In the case of Fourier transformation of the real part, the sign of the phase is changed when the sign of ωT is inverted, as also in the case of Fourier transformation of the imaginary part. However, in the latter an additional phase shift by π appears. The 3D spectrum from Fourier transformation of the real part of the 2D spectrum σ3D, Re(ωτ , ωT , ωt ) = =

∫0

∞

dt 2 exp(− iωT t 2)Re(σ2D(ωτ , t 2 , ωt ))

1 * (ωτ , − ωT , ωt )) (σ3D(ωτ , ωT , ωt ) + σ3D 2 (6)

consists of a symmetric combination of 3D spectra calculated according to eq 5 with different signs of ωT, where the term containing the negative value appears in conjugate complex form. The 3D spectrum from Fourier transformation of the imaginary part of the 2D spectrum σ3D, Im(ωτ , ωT , ωt ) =

PR(3)(t1 , t 2 , t3) = R 2g(t1 , t 2 , t3) + R3g(t1, t 2 , t3)

=

∫0

∞

dt 2 exp(− iωT t 2)Im(σ2D(ωτ , t 2 , ωt ))

1 * (ωτ , − ωT , ωt )) (σ3D(ωτ , ωT , ωt ) − σ3D 2 (7)

(1)

corresponds to the respective antisymmetric combination. Therefore, the two-dimensional cuts at positive and negative ωT only yield complementary information if the full complex numbered 2D spectrum is Fourier transformed. This kind of evaluation is preferable, because it allows to separate features related to coherence beatings which evolve with the same frequency but with a conjugate complex phase evolution relative to each other during the interval t2 by choosing a positive or negative frequency value. Instead of considering two-dimensional cuts of the 3D spectrum at a frequency

whereas the nonrephasing part reads (3) PNR (t1 , t 2 , t3) = R1g(t1 , t 2 , t3) + R 4g(t1 , t 2 , t3)

− R 2*f (t1 , t 2 , t3)

∞

exp( −iωτ t1) exp(iωt t3)PR(3)(t1 , t 2 , t3),

■

− R1*f (t1 , t 2 , t3)

∫0

σ2D,R (ωτ , t 2 , ωt )=

(2)

2D spectra with respect to the conjugated frequencies of the time variables τ and t and a parametric dependence on T = t2 are obtained according to the formulas 18729



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position with inverted sign, one can get an equivalent result by changing the sign in the argument of the complex exponential in eq 5, that is, take the inverse Fourier transformation (regardless of a prefactor). In this sense, an analogy can be drawn to eqs 3 and 4, where the direction of the Fourier transformation with respect to the ωτ axis is chosen depending on whether rephasing or nonrephasing contributions are considered. The peak intensities from the different response function contributions depend on the appearing products of four transition dipole moments corresponding to the system-field interactions. If vibrational effects are taken into account, the transition dipole moments contain overlaps between vibrational eigenfunctions, the so-called Franck−Condon (FC) amplitudes. A density matrix formulation of the response functions in the basis of the vibrational eigenfunctions38 allows to derive these FC amplitudes. For the different response functions Rn(t1, t2, t3) = ∑i,j,k,lCn;i,j,k,l R̃ n;i,j,k,l(t1, t2, t3) containing the product of transition dipole moments Cn;i,j,k,l and a time-dependent component R̃ n;i,j,k,l(t1, t2, t3), one finds C1; i , j , k , l = ⟨χg , l |χe , j ⟩⟨χe , k |χg , l ⟩⟨χg , i |χe , k ⟩⟨χe , j |χg , i ⟩

(8)

C2; i , j , k , l = ⟨χg , l |χe , k ⟩⟨χe , j |χg , l ⟩⟨χe , k |χg , i ⟩⟨χg , i |χe , j ⟩

(9)

C3; i , j , k , l = ⟨χg , k |χe , l ⟩⟨χe , l |χg , i ⟩⟨χe , j |χg , k ⟩⟨χg , i |χe , j ⟩

(10)

C4; i , j , k , l = ⟨χg , i |χe , l ⟩⟨χe , l |χg , k ⟩⟨χg , k |χe , j ⟩⟨χe , j |χg , i ⟩

(11)

C5; i , j , k , l = ⟨χe , j |χf , l ⟩⟨χf , l |χe , k ⟩⟨χe , k |χg , i ⟩⟨χg , i |χe , j ⟩

(12)

C6; i , j , k , l = ⟨χe , k |χf , l ⟩⟨χf , l |χe , j ⟩⟨χg , i |χe , k ⟩⟨χe , j |χg , i ⟩

(13)

Figure 2. Dependence of the FC amplitudes χ between the lowest vibrational eigenstates of two displaced harmonic oscillators on the Huang−Rhys factor S.

functions. If not specified otherwise, the chosen parameters of the line shape functions are the same as in our previous article.13 As additional model assumptions, we either take one vibrational mode in each electronic state into account in terms of line shape functions including a Lorentzian spectral density contribution with central frequency ω0,L = 200 cm−1 to account for vibrational effects or we assume a system without this vibrational contribution, but with a splitting30 of the singly excited state into fine structure levels |e1⟩ and |e2⟩ with an energy gap ωe1e2 = 200 cm−1 instead. We take the energy difference of ground- and singly excited state as ωeg = 17000 cm−1. This value corresponds to the vertical transition energy and thus slightly differs from electronic excitation energy. The binding energy is chosen as εf = 200 cm−1. In the case of an electronic fine structure splitting, we assume the electronic excitation energy ωe2g to be equal to ωeg, while ωe1g is shifted to higher energies. Even if the Lorentzian spectral density contribution is neglected, still a Debye spectral density contribution enters. While part of the homogeneous line broadening is contained in the Lorentzian spectral density and, thus, neglected, the Debye spectral density contribution with the reduced cutoff frequency ωc = 5 cm−1 still leads to homogeneous broadening and also influences the way inhomogeneous broadening, quantified by the parameters given at the end of this paragraph, appears in the 3D spectrum. This influence will be addressed in Results and Discussion. The spectrum of the exponentially damped Brownian oscillator related to the Debye spectral density with the given parameters consists of a broadened, structureless feature with maximum far below 200 cm−1, so that the contribution of the Debye spectral density to the 3D spectrum in the energetic region close to 200 cm−1 is kept small. The transition energy ωeg varies due to the size distribution of the QDs leading to inhomogeneous broadening. This is represented by a parameter Δ, which accounts for a shift of the transition energy from the average transition energy. We take a Gaussian distribution of Δ around zero with a standard deviation corresponding to FHWM(Δ) = 800 cm−1 into account. The vibrational frequency ω0,L and the energy splitting between the fine structure levels ωe1e2 depend on the QD size. We assume a linear dependence of ω0,L and ωe1e2 on the parameter Δ. The choice of these parameter values results in a variation of ω0,L and ωeg from 240 cm−1 to 160 cm−1 when Δ

The FC amplitudes consist of overlap integrals between vibrational eigenstates χα,ν, where α denotes the electronic state and ν is the vibrational quantum number. They are ordered according to the time sequence of the respective transitions from right to left. The coefficients C5;i,j,k,l and C6;i,j,k,l are related to the response functions R1f* and R2f* . The dependencies of the FC amplitudes between the first two eigenstates of displaced harmonic oscillators39 on the Huang−Rhys factor S are shown in Figure 2.

Figure 1. Schemes for a model system with two vibrational levels in each electronic state (left-hand side) and for a purely electronic model system with fine structure splitting in the singly excited state (righthand side).

For the calculation of 3D spectra of quantum dots we choose a model system including the electronic ground state |g⟩, the singly excited state |e⟩ and the doubly excited state |f⟩ with electronic excitation energies ωeg and ωfe = ωeg − εf, where εf denotes the binding energy of the doubly excited state. We calculate the response functions by using expressions which stem from a cumulant expansion and contain line shape 18730



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varies from −2σΔ to 2σΔ with σΔ = 340 cm−1. The energy shift of singly and doubly excited state energy levels is assumed as Δ and 2Δ, respectively.

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RESULTS AND DISCUSSION To explain the calculated 3D spectral features for the quantum dot model system, we start with a Feynman diagram analysis under simplified model assumptions. We either take two vibrational levels in each electronic state into account or assume purely electronic levels, as sketched in Figure 1. In both cases, we assume a level spacing of 200 cm−1, which corresponds to the central vibrational frequency ω0,L in the Lorentzian spectral density contribution of the quantum dot model. We do not include inhomogeneous broadening, neglect a binding energy shift of the doubly excited state (εf = 0 cm−1) and assume a high enough frequency resolution, so that all appearing peaks can be clearly identified. The peak positions in two-dimensional cuts of 3D spectra taken at ωT = 200 cm−1 or ωT = −200 cm−1 for the system with vibrations are indicated in the upper diagrams of Figures 3 and 4 for the rephasing and nonrephasing cases, respectively. Peaks from SE, GSB, and ESA processes are indicated as blue, green, and orange symbols, respectively. The corresponding diagrams for a purely electronic system are shown at the top of Figures 5 (rephasing) and 6 (nonrephasing). All peaks are obtained from the evaluation of the corresponding Feynman diagrams, which are subject to the condition that during t2 a coherence between different sublevels of the same electronic state (g,e,f) appears. They are displayed below the peak position diagrams in the respective figures. In the Feynman diagrams for the system with vibrations, the expressions |ϵ; nε⟩ denote the vibrational eigenstate with quantum number nε within the electronic state ε, which can be the electronic ground state g, the singly excited state e, or the doubly excited state f. The ωτ and ωt positions are related to the energy difference between initial and final state of the first and the final transition. All diagrams start from the electronic ground state g. In thermal equilibrium at room temperature, both the lowest and the first excited vibrational state can be the starting population. Nevertheless, peaks related to processes which start from the lowest vibrational level are more intensive. In the peak position diagrams, the respective symbols are surrounded by black lines. As discussed in Theoretical Methods, four system− field interactions are involved in the processes, which lead to the peaks. Consequently, the peak intensities depend on the products of four FC amplitudes (overlaps between vibrational wave functions) given in eqs 8−13. The dependence of the FC amplitudes on the Huang−Rhys factor is shown in Figure 2. For moderate Huang−Rhys factors S < 1, peaks that stem from excitation processes, including two zero−zero transitions are most intensive. In Figures 3 and 4, the respective peaks are indicated by a black cross. Note that the FC amplitude χ01 has a positive sign and χ10 has a negative sign. Under the assumption that the equilibrium positions of ground and doubly-excited state potentials are not shifted relative to each other, the signs of the FC amplitudes for transitions between singly and doubly excited states become opposite compared to those for transitions between ground and singly excited states. Positive and negative signs of the peaks are indicated by the choice of circle and square symbols for the respective peaks. The signs are both determined by the prefactors from the multiplication of the involved FC amplitudes (see eqs 8−13) and the sign of the response function in the combination of all rephasing or

Figure 3. First row: Peak positions in two-dimensional cuts of the rephasing 3D spectra taken at ωT = 200 cm−1 (left-hand side) and ωT = −200 cm−1 (right-hand side) for the case of two vibrational levels with oscillation frequency ω0,L = 200 cm−1 in each electronic state. The colors blue, green, and orange are related to SE, GSB, and ESA contributions. Discs and squares indicate positive and negative sign, respectively. The peaks are marked by black surrounding lines if the initial state of the excitation sequence is the lowest vibrational level of the electronic ground state. Black crosses indicate the involvement of two zero−zero transitions in the excitation sequence. Second to fourth row: Feynman diagrams of rephasing response functions R2g, R3g, and R1f* .

nonrephasing contributions (see eqs 3 and 4), where ESA response functions appear with a negative sign. Opposite signs of the prefactors of different peaks are equivalent to a phase shift of π if amplitude and phase of the 3D spectrum are evaluated. Note that in eqs 1 and 2 the ESA response functions enter with a negative sign, which is included in the determination of the resulting sign of the different vibrational peaks from ESA processes. We point out that, in the case of large Huang−Rhys factors, the pathways, including vibrational 18731



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Figure 5. First row: Peak positions in two-dimensional cuts of the rephasing 3D spectra taken at ωT = 200 cm−1 (left-hand side) and ωT = −200 cm−1 (right-hand side) in the case that two electronic levels with splitting energy ωe1e2 = 200 cm−1 in the singly and excited state are taken into account and that vibrations are neglected. The blue and orange symbols are related to SE and ESA contributions. Discs and squares indicate positive and negative sign, respectively. Second to third row: Feynman diagrams of rephasing response functions R2g and R1f* .

rectangular alignment, where the spacing in both ωτ and ωt directions corresponds to the difference between vibrational eigenstates. While the GSB contributions in the cut of the rephasing 3D spectrum at ωT = 200 cm−1 correspond to the excitation sequences starting from the lowest vibrational level of the electronic ground state, at ωT = −200 cm−1 the GSB peaks are related to excitation sequences starting from the first excited vibrational level of the electronic ground state. Therefore, the latter are less intensive at room temperature. At zero temperature, these peaks disappear completely. For nonrephasing GSB contributions, one finds the opposite assignment between initial state and sign of the ωT frequency. In the corresponding cuts of 3D spectra of the purely electronic system at ωT = 200 cm−1 and ωT = −200 cm−1 (Figure 5), no GSB contribution appears, as no coherence with appropriate beating frequency can be created if only one level of the electronic ground state is taken into account. Different from the GSB contributions, SE and ESA peaks related to an initial population of the lowest and the first excited vibrational state of the electronic ground state appear both at ωT = 200 cm−1 and at ωT = −200 cm−1 in rephasing and nonrephasing 3D spectra. If the initial state is vibrationally excited, the peaks are located at lower ωτ values. The pattern of

Figure 4. First row: Peak positions in two-dimensional cuts of the nonrephasing 3D spectra taken at ωT = 200 cm−1 (left-hand side) and ωT = −200 cm−1 (right-hand side) for the case of two vibrational levels with oscillation frequency ω0,L = 200 cm−1 in each electronic state. The colors blue, green, and orange are related to SE, GSB, and ESA contributions. For the explanation of the symbols, see Figure 3. Second to fourth row: Feynman diagrams of rephasing response functions R1g, R4g, and R*2f .

level 2 and higher, may give significant contributions to the signal. We do not include these pathways in the Feynman diagram analysis, but the effect is included in calculations with cumulant expansion expressions. In the following, specific aspects of the peak structure in the two-dimensional cuts of 3D spectra will be addressed, as far as they are relevant for the later discussion of our results from model calculations for quantum dots. The vibrational peaks in the cuts of the rephasing 3D spectra at ωT = 200 cm−1 and ωT = −200 cm−1 are located at positions with ωτ ≥ ωt and ωτ ≤ ωt, respectively. In the nonrephasing cuts, the peak positions are symmetric with respect to the diagonal. From each type of process (SE, GSB, and ESA), four peaks appear with a 18732



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time intervals play a role. The correlations are accounted for by the second-order cumulant expansion, where the calculation of the response functions is given in terms of an exponential containing line shape functions with dependencies on the time delays t1, t2, and t3 and their combinations. The combinations of time intervals are not captured by a Feynman diagram description. The line shape functions are calculated from the spectral density. If under the model assumption of vibrations described by a Lorentzian spectral density component, the central frequency ω0,L differs from the chosen ωT position, the peak intensity distribution within the vibrational structure in the two-dimensional cut at ωT depends on the difference between ω0,L and ωT. Under the influence of exponentially damped environmental fluctuations entering in terms of the Debye spectral density component, the vibrational coherence beating frequency can be shifted. To obtain a frequency component corresponding to the chosen value, the difference between ωT and the vibrational frequency needs to be compensated by such frequency shifts, which is possible by various combinations of shifts of the vibrational levels involved in the coherence during t2. Therefore, the difference between ωT and ω0,L influences a relatively large energetic range in the order of the vibrational spacing. Examples for the resulting changes of the peak intensity distribution in the (ωτ,ωt)coordinate plain are shown in the Supporting Information. Also, under the model assumption of electronic fine structure splitting instead of vibrations, where the splitting between the fine structure levels ωe1e2 is chosen close to 200 cm−1 and the Lorentzian spectral density component is neglected, environmental fluctuations from the Debye spectral density contribution can compensate the difference between ωT and ωe1e2. However, as no second-order correlation effects between the fine structure levels with respect to the delay times between the electronic transitions appear, only an energy range within the (ωτ,ωt)-coordinate plain in the order of the compensated frequency difference between ωT and ωe1e2 is influenced by the involvement of environmental fluctuations from the Debye spectral density component. In the calculations for the quantum dot model, the parameters and assumptions specified in Theoretical Methods are used. We start with a discussion of the results which are obtained under the model assumption of vibrational effects by including a Lorentzian spectral density contribution. A dependence of the vibrational frequency on the electronic excitation energy shift parameter Δ leads to modified intensity distributions within the underlying, however unresolved vibrational structure for different values of Δ. The twodimensional cuts of 3D spectra taken at ωT = 200 cm−1 for the rephasing response functions R2g, R3g, R1f* and their sums are displayed in Figure 7. The sum of all rephasing contributions at ωT = −200 cm−1 is shown in the upper panel of Figure 8. In the contributions of all rephasing response functions, inhomogeneous broadening effects due to the Gaussian distribution of Δ lead to elongated peak structures, which are tilted relative to the diagonal line of the two-dimensional cut. Elongated peak structures are obtained as a consequence of constructive superposition of peak components from different QD sizes and, thus, different values of Δ. If the directions of the gradient of the peak component positions with respect to Δ and the phase gradient are perpendicular, constructive superposition effects appear. A change in the electronic excitation energy depending on the Gaussian-distributed quantum dot size leads to a shift of

Figure 6. First row: Peak positions in two-dimensional cuts of the nonrephasing 3D-spectra taken at ωT = 200 cm−1 (left-hand side) and ωT = −200 cm−1 (right-hand side) in analogy to Figure 5. Second to third row: Feynman diagrams of nonrephasing response functions R1g and R*2f .

SE peaks with respect to ωt is related to transitions from a selected vibrational level of the singly excited state to the electronic ground state. Therefore, an emission-like progression toward smaller ωt values appears, which reflects the vibrational structure of the electronic ground state. In contrast, the absorption-like progression of ESA peaks toward larger ωt stems from transitions from a selected vibrational level of the singly excited state to the doubly excited state. For the purely electronic system, one peak from both SE and ESA processes is found at a position with ωτ > ωt or ωτ ≤ ωt in the cuts of rephasing 3D spectra at ωT = 200 cm−1 and ωT = −200 cm−1, respectively. The SE peaks appear at off-diagonal positions in cuts of rephasing 3D spectra, whereas in the cuts of nonrephasing 3D spectra they appear on the diagonal. The ESA peaks are located at the same ωτ position as the SE peaks, but their ωt position is lowered by the difference between the fine structure levels. As spectral densities with finite width enter in the calculation of the response functions, both for the model with vibrations and for the purely electronic model, non-negligible peak intensity can also be obtained if the difference between vibrational or fine structure sublevels does not exactly match the chosen ωT position of a two-dimensional cut (in our case, ωT = ±200 cm−1). Then, however, the spectral features with respect to the ωτ and ωt axes depend on the selected ωT value, as correlations between the evolutions during the conjugated 18733



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Figure 8. Two-dimensional cuts of the sum of all rephasing contributions at ωT = −200 cm−1 and the sum of all nonrephasing contributions at ωT = 200 cm−1 and ωT = −200 cm−1. For further explanations, see Figure 7.

components from different quantum dot sizes with respect to Δ. The latter case was shown in our previous article.13 The contributions from the separate response functions are discussed in the Supporting Information. All rephasing response functions share the property that the respective contributions, and therefore also the sum of them, are located below the diagonal in the cut at ωT = 200 cm−1 and above the diagonal in the cut at ωT = −200 cm−1. This finding agrees with the predicted peak positions from the Feynman diagram analysis. Because of the assumption that both the electronic excitation energy shift parameter Δ and the vibrational frequency ω0,L depend on the quantum dot size, the elongated peaks are tilted. In the two-dimensional cut at ωT = 200 cm−1, a tilt toward lower ωt values appears in the lower energy region, as ω0,L increases with decreasing value of Δ. The results at ωT = −200 cm−1 show a peak tilt tendency toward lower ωt values in the higher energy region. As discussed in our previous article,13 in the case of rephasing 3D spectra, a tilt of elongated peaks in the two-dimensional cuts relative to the diagonal allows to draw conclusions about the dependence of vibrational frequency and Huang−Rhys factor of the singly excited state on the parameter Δ and, thus, on the particle size. The relatively large extent of the peak tilt in the results obtained under the model assumption of a vibrational component in terms of a Lorentzian spectral density contribution is caused by the influence of a difference between

Figure 7. Two-dimensional cuts of amplitude (left-hand side) and phase (right-hand side) of the triple Fourier transformed rephasing response functions R2g, R3g, R*1f of the quantum dot model system and their sum taken at ωT = 200 cm−1. Only one electronic level is taken into account in the singly excited state. The oscillation frequency in the Lorentzian spectral density is assumed to decrease with increasing shift of the electronic excitation energies as a linear function. The tilts are indicated by black lines.

the positions of GSB and SE peak components along the diagonal, as the frequencies of the first and final transition are affected by the shift of the electronic excitation energy in the same way. The same effect appears for the ESA contribution under our assumption that the electronic energies of singly and doubly excited state are shifted by Δ and 2Δ, respectively. If both singly and doubly excited states would be shifted by Δ, elongated peaks would appear in the direction of the ωτ axis. In the case that the electronic excitation energy of the doubly excited state is constant, while the one of the singly excited state depends on Δ, no peak elongation and enhancement, but rather destructive superposition effects and a diffuse peak structure, would be found because of parallel directions of the phase gradient and the gradient of the positions of peak 18734



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ωT and the central frequency of the Lorentzian spectral density component ω0,L. If a mismatch between the Δ-dependent value of ω0,L and the chosen ωT appears, it can be compensated by shifts of the involved vibrational levels under the influence of environmental fluctuations from the Debye spectral density component. If one thinks in terms of a coherent superposition of vibrational eigenstates, the resonance profiles determine the relative amplitudes and phases of the involved eigenstates depending on their shifted energetic position due to the influence of environmental fluctuations. The involvement of different vibrational levels influences an energy range in the order of the vibrational spacing. Therefore, under the influence of inhomogeneous broadening, a pronounced peak tilt relative to the diagonal is obtained. In 3D spectra of the nonrephasing contributions R1g, R4g, R2f* and their sums, the phase exhibits a gradient along the diagonal of the two-dimensional cuts. This leads to destructive superposition of signals from different Δ values, so that diffuse peaks with lower intensity compared to the rephasing results are obtained. The cuts taken at ωT = 200 cm−1 and ωT = −200 cm−1 are shown in the middle and lower panel of Figure 8, respectively. Results from calculations without inclusion of a Lorentzian spectral density contribution and, therefore, without a pronounced vibrational frequency component close to 200 cm−1 are shown for two-dimensional cuts of rephasing 3D spectra taken at ωT = 200 cm−1 and ωT = −200 cm−1 in the upper and lower panel of Figure 9, respectively. As discussed in the Feynman diagram analysis, vibrational coherences and coherences between electronic fine structure levels30 both lead to the appearance of peaks from SE and ESA processes in the

two-dimensional cuts of 3D spectra taken at positive and negative values of ωT. GSB contributions are neither expected at ωT = 200 cm−1 nor at ωT = −200 cm−1 for a purely electronic system with a single energy level in the electronic ground state. Nevertheless a non-negligible GSB contribution of R3g is found for the QD model because of the tail of the Debye spectral density. Under the assumption of a splitting of the singly excited state into electronic fine structure levels, the dependence of the respective energy difference on the quantum dot size does not lead to a pronounced tilt of the elongated peak structure in the SE and ESA contributions. For a purely electronic system without any spectral density contribution accounting for fluctuations of the environment, non-negligible peak intensity would only be obtained if the frequency of a beating between the fine structure levels within the distribution exactly matches the chosen ωT value. The inclusion of a Debye spectral density with spectral components mainly located in the energetic region corresponding to the variation range of the singly excited state fine structure splitting allows to bridge the energy gap between the chosen ωT position and the beating frequency by a phonon transition with appropriate energy, so that peak elongation due to inhomogeneous broadening appears. Even though the involvement of such phonon transitions also slightly shifts the peak position within the two-dimensional cut, the tilt effect in combination with inhomogeneous broadening is much smaller than under the assumption of vibrational effects instead of a fine structure splitting with the same level spacing as the vibrational frequency. If the electronic fine structure splitting is varied and the difference between this splitting and the chosen ωT value is compensated by a phonon transition, the position of the respective peak component within the two-dimensional cut is expected to be shifted by the order of magnitude of the respective phonon frequency, that is, by a value within the variation range of the fine structure splitting. Therefore, the tilt effect in the superposition of signal contributions from different quantum dot sizes is much smaller if electronic fine structure splitting is assumed instead of vibrational effects.

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CONCLUSIONS Breaking the third-order coherent signal to rephasing and nonrephasing components has increased the analytical power of 2D spectroscopy. The method is based on the sign of the frequency, or equivalently on the direction of the corresponding Fourier transform, during the first and the third propagation time. In direct analogy to the distinction between rephasing and nonrephasing signals, we prepare and analyze here the signals that correspond to positive and negative frequency components at the second time evolution. By evaluating Feynman diagrams with evolution in a coherence with positive or negative beating frequency during the second time interval, we determined the expected peak positions for different model assumptions. We considered both a system with two vibrational eigenstates in each electronic state and a purely electronic system with fine structure splitting in the singly excited state. We also pointed out the limitations of a description using Feynman diagrams for the system with vibrations. Based on these rather general considerations, we analyzed results of calculations for a model system of quantum dots. We explained how the influence of inhomogeneous broadening leads to constructive or destructive superposition effects, which result in elongated peaks with high intensity in the case of rephasing contributions and diffuse peaks with low intensity in the case of

Figure 9. Two-dimensional cuts of the sum of all rephasing contributions at ωT = 200 cm−1 and ωT = −200 cm−1. Vibrations are not taken into account by neglecting the Lorentzian contribution to the spectral density. The electronic splitting between the included fine structure levels of the singly excited state is assumed to decrease with increasing shift of the electronic excitation energies relative to averaged levels as a linear function. The tilts are indicated by black lines. 18735



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nonrephasing contributions. Under our model assumption that the vibrational frequency depends on the QD size dependent shift of the electronic excitation energies of singly and doubly excited state, the elongated peaks of all rephasing contributions exhibit a tilt. The tendency of the tilt is opposite in the twodimensional cuts taken at positive and negative frequencies. We gave an explanation of positions and orientation of the peaks from rephasing contributions, thereby drawing a connection to the previously discussed aspect of the expected vibrational peak patterns. We finally addressed the question whether the model assumption of a splitting of the singly excited states in terms of fine structure levels instead of a vibrational substructure would lead to different findings. Thereby we assumed that the respective coherence beating frequency depends on the electronic excitation energy shift via a common particle size dependence, as in the case of vibrational effects. We found out that under this assumption elongated peak structures from inhomogeneous broadening effects are still obtained by taking a Debye spectral density contribution into account in the respective calculation, but that then the tilt effect is much less pronounced than under the model assumption of an additional Lorentzian spectral density component. Thus, the extent of the tilt effect under the influence of inhomogeneous broadening can be identified as a quantitative criterion for the distinction between vibrational and electronic coherences in our QD model system. The proposed tilt effect in two-dimensional cuts of 3D spectra can also appear in other systems than QDs under the conditions that inhomogeneous broadening effects outweigh homogeneous broadening and that a coherence evolution during the population time interval is influenced by inhomogeneous broadening effects.

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ASSOCIATED CONTENT

S Supporting Information *

A discussion of the separate response functions for the model assumption of vibrational effects and of results from additional calculations to illustrate the origin of the tilt effect. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support of the Knut and Alice Wallenberg Foundation and the Swedish Energy Agency. We thank Dr. Thorsten Hansen for the valuable discussions. Collaboration within nmC@LU is acknowledged.

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REFERENCES

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