Self-Analysis of Coherent Oscillations in Time-Resolved Optical Signals

Article pubs.acs.org/JPCA

Self-Analysis of Coherent Oscillations in Time-Resolved Optical Signals Dassia Egorova* Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany ABSTRACT: The specific origin of oscillations in time-resolved optical signals, in particular, for complex systems with nontrivial interstate couplings and nonseparable electron−nuclear motion, is often difficult to assign. Here, we show that coherent oscillations in two-dimensional photon-echo are capable of self-analysis; their beating maps provide a tool to tell apart ground-state bleach (GSB), stimulated emission (SE), and excited-state absorption (ESA) contributions to the oscillatory signal component. Because GSB carries information on ground-state coherence while SE and ESA reflect the excited-state coherence, the observed oscillations can be unambiguously assigned to ground-state or excited-state coherent motion. The findings prove especially advantageous for systems with dense detectable manifolds of states pertaining to each electronic state. An analogous analysis for frequency-resolved (dispersed) pump−probe spectroscopy is discussed briefly.

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INTRODUCTION Time-resolved nonlinear spectroscopy has become an indispensable tool for studies of molecular dynamics and structure, and its applications cover a huge variety of physical, chemical, and biological systems. Most recently, coherent oscillations in time-resolved signals have afresh driven significant attention. Already in the pioneering applications, it has been recognized that femtosecond laser pulses often induce coherent vibrational motion coupled to the electronic excitations, but the latter-day reports on coherence in biological systems1−9 remain a subject of intensive discussions and sometimes controversial interpretations because the coherence origin and function are hard to identify. Independently of their specific origin, coherent oscillations in time-resolved signals carry information on the system eigenstates accessible for the laser excitation and involved in the initiated ultrafast dynamics. Also, very rapidly depopulated states can be accessed and investigated in this way. As in linear stationary spectroscopy, the dipole strengths of the involved transitions are crucial for the eigenstates’ detection. Multiple field−matter interactions in time-resolved nonlinear techniques are able to provide, however, much higher resolution and sensitivity as compared to the linear signals since several dipoles contribute, in general, to the detected signatures, specific dipole combinations may significantly improve the detection of weak transitions. In principal, not only the location but also the nature of eigenstates can be potentially addressed by timeresolved nonlinear techniques. In order to obtain this information, a detailed understanding of interconnection between the system properties and the recorded signals is necessary. Here, we consider two experimental techniques: the focus is on two-dimensional (2D) photon-echo (PE) spectroscopy in the optical domain (also referred to as 2D electronic © 2014 American Chemical Society

spectroscopy, 2D ES), while frequency-resolved (dispersed) pump−probe (PP) spectroscopy is addressed only briefly. Although PP remains the most frequently used technique, 2D ES has become the most popular and versatile experimental method for monitoring coherent oscillations. Its most recent advance is to exploit the arising coherence for the spectroscopic detection of dark states6,10,11 immediately involved in the excited-state dynamics and responsible for such fundamental processes as photoprotection, electron transfer, or singlet fission. As any four-wave mixing scheme, 2D PE and PP signals represent a superposition of ground-state and excited-state contributions. The coherent dynamics of interest occurs, usually, in the excited electronic state(s). Therefore, the first step on the way to the detection of the relevant eigenstates is the separation of the ground-state and excited-state coherent dynamics and the exclusion of the ground-state contribution when analyzing the signal coherences. In the following, we present a universal scheme for analysis of coherent signatures in 2D ES. We explore the ability of the technique to separate oscillatory components spectrally, that is, to trace back what specific transitions lead to the observed coherences. We demonstrate that coherent oscillations detected in the signal are capable of self-analysis; they provide an opportunity to tell apart ground-state bleach (GSB), stimulated emission (SE), and excited-state absorption (ESA) oscillatory contributions to the signal. Because oscillations in GSB carry information on ground-state coherence while SE and ESA reflect the excited-state coherence, an unambiguous assignment of the observed oscillations to ground-state or to excited-state motion becomes possible. The frequency-resolved PP signal Received: September 24, 2014 Revised: October 7, 2014 Published: October 7, 2014 10259

dx.doi.org/10.1021/jp509657u | J. Phys. Chem. A 2014, 118, 10259−10267

The Journal of Physical Chemistry A

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This allows one to establish the guidelines for discriminating ground-state and excited-state coherent motion. We emphasize and demostrate below that a separate consideration of rephasing and nonrephasing contributions to the signal is very advantageous and helpful for the analysis. Because we work in the stick spectrum limit, the location of the peaks is determined exactly by the transition energies, and their intensity depends only on the dipole moments involved. The rules discussed here provide, therefore, a foundation for investigations of the eigenstate origin by 2D ES. The “self-analysis” of coherent oscillations in 2D PE and PP signals outlined below holds for any physical origin of the states described by eq 1. To make the discussion more specific, we address a class of molecular systems where the manifold of the electronic ground state consists of vibrational states. As for the electronically excited states, their multilevel structure may be of vibrational origin as well or may correspond to the so-called vibronic states, that is, the system eigenstates of mixed electronic and vibrational character. This kind of mixing arises whenever coupled electron−nuclear motion is initiated in the system. The most well-known examples are conical intersections, but also dipole−dipole interactions in oligomers can lead to vibronic manifolds. Under these conditions, coherent motion in the electronic ground state (if induced in the experiment) has exclusively vibrational origin, while excitedstate coherences are coherences between the system eigenstates of either vibrational or vibronic character. Recently, much effort has been invested to find strategies to distinguish coherences of vibrational and electronic origin in 2D ES.12−18 Here, we aim to distinguish ground-state and excited-state coherences (this was also attempted in ref 15 but by different means). This achieved, the ground-state coherent contribution (that correspond to purely vibrational motion in most cases) can be excluded, and the analysis of the remaining excited-state coherences (which represents a complex nontrivial issue if coupled electron− nuclear motion is involved) can be considerably facilitated. 2D ES Beating Amplitudes. The explicit expressions for the beating amplitudes of physical systems described by eq 1, provided that the field−matter interaction is determined by the dipole operator, eq 2, read (the expressions for GSB, SE and ESA components are given)

can be viewed as a realization of 2D PE, where the resolution over one variable (the so-called excitation frequency) is integrated out. Therefore, the same ideas are employed to clarify if and how the three contribution can be told apart in this technique.

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METHODS Stick Spectrum Limit and Beating Maps. When employing the 2D PE technique to access the system eigenstates by decoding coherent oscillations, it appears natural to chose the stick spectrum limit as the first approximation. We combine this limit12 and the recently introduced beating maps analysis13 to establish clear and possibly universal guidelines that allow one to distinguish between ground-state and excitedstate coherent motion in the recorded signals. The presented analysis is especially advantageous for systems with dense detectable manifolds of eigenstates. We consider a general model describing physical systems with three distinct manifolds of eigenstates coupled by the laser field. The interaction with the laser field leads to electronic excitations and provides the only weak interaction between the manifolds. The system Hamiltonian can be written as H=

∑ |g ⟩Eg ⟨g | + ∑ |e⟩Ee⟨e| + ∑ |d⟩Ed⟨d| g

e

d

(1)

where |g⟩ denote eigenstates pertaining to the electronic ground state and |e⟩ and |d⟩ denote eigenstates corresponding to single and double electronic excitations, respectively. The weak laser field optically couples the manifold |g⟩ and the manifold |e⟩, as well as the manifold |e⟩ and the manifold |d⟩; the corresponding dipole operator (linear dipole approximation is assumed hereafter) reads μ=

∑ |g ⟩μge ⟨e| + ∑ |e⟩μed ⟨d| + H.c. ge

ed

(2)

where μij denote the transition dipole strengths between the eigenstates. The foundation of the analysis presented here is the ability of 2D ES to provide information on what particular transitions are involved in the formation and detection of each coherence. The recently introduced beating maps13 are indispensable for this purpose. Each beating map shows the amplitude of a particular oscillation (beating mode) found in the signal. It is recorded as a function of the two signal variables, the excitation and probe frequencies ωτ and ωt. In this way, beating maps visualize and spectrally locate every particular oscillation out of all simultaneously present in the signal. As the signal itself, each beating mode amplitude can be viewed as a superposition of the three contributions, GSB, SE, and ESA. While hardly separable in the conventional signal, the three contributions exhibit quite distinct behavior in the beating maps and do not necessarily overlap. The explicit expressions for the maps in the stick spectrum limit are given by eqs 3−8 below. They have been derived assuming that the oscillations with the opposite phase contribute to the same map. In principle, a more precise analysis is possible if also the phase information is kept. However, one has to be aware that the signal phase may be influenced by such instrumental effects like duration, intensity, and temporal overlap of the laser pulses employed. The discussion of the phase effects is, therefore, avoided here. As can be seen from eqs 3−8, the location and intensity of GSB, SE, and ESA contributions are quite distinct for each map.

GSB A reph (ωT = |Eg ′ g | , ωτ , ωt )

∼

∑ μge μg ′ e μge′μg ′ e′[αgδ(Eeg − ωτ)δ(Ee′ g ′ − ωt ) ee ′

+ αg ′δ(Eeg ′ − ωτ )δ(Ee ′ g − ωt )]

(3)

SE A reph (ωT = |Ee ′ e| , ωτ , ωt )

∼

∑ αgμge μge′μeg ′μe′ g ′[δ(Eeg − ωτ )δ(Ee′ g ′ − ωt ) gg ′

+ δ(Ee ′ g − ωτ )δ(Eeg ′ − ωt )]

(4)

ESA A reph (ωT = |Ee ′ e| , ωτ , ωt )

∼

∑ αgμge μge′μed μe′ d [δ(Eeg − ωτ )δ(Ede − ωt ) gd

+ δ(Ee ′ g − ωτ )δ(Ede ′ − ωt )]

(5)

for the rephasing maps and 10260



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1D PP Beating Maps Amplitudes. One-dimensional (1D) “maps” of the frequency-resolved PP signal can be obtained from eqs 10−12 by dropping the ωτ-dependent δ function

GSB (ωT = |Eg ′ g | , ωτ , ωt ) A nonreph

∼

∑ μge μge′μg ′ e μg ′ e′[αgδ(Eeg − ωτ) ee ′

× δ(Ee ′ g − ωt ) + αg ′δ(Eeg ′ − ωτ )

GSB APP (ωT = |ωg ′ g | , ωt )

× δ(Ee ′ g ′ − ωt )] SE A nonreph (ωT

∼ (∑ μge ′μg ′ e ′)

(6)

e′

= |Ee ′ e| , ωτ , ωt ) ∼

×

∑ αgμge μge′μg ′ e μg ′ e′[δ(Eeg − ωτ)δ

e

gg ′

+ αg ′{δ(ωeg − ωt ) + δ(ωeg ′ − ωt )}]

(Eeg ′ − ωt ) + δ(Ee ′ g − ωτ )δ(Ee ′ g ′ − ωt )]

∼ (∑ αg μge μge ′) ∑ μeg ′μe ′ g ′[δ(ωe ′ g ′ − ωt )

ESA A nonreph (ωT = |Ee ′ e| , ωτ , ωt )

g

∑ αgμge μge′μed μe′ d [δ(Eeg − ωτ)

× δ(Ede ′ − ωt ) + δ(Ee ′ g − ωτ ) × δ(Ede − ωt )]

∼ (∑ αg μge μge ′) ∑ μed μe ′ d [δ(Ede ′ − ωt ) g

for the nonrephasing maps. The beating frequency is denoted as ωT, all three frequencies (ωt, ωτ, ωT) are determined by the eigenstate structure, Eij = Ei − Ej. The thermal prefactors αg arise from the initial condition

∑ αg |g ⟩⟨g | g

g

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+ δ(Ede − ωt )]

d

(15)

RESULTS AND DISCUSSION: RULES OF SELF-ANALYSIS Model. In order to introduce the rules, we chose a possibly simple but illustrative eigenstate structure shown in Figure 1. The ground-state manifold is represented by the initial state with E0  0 and vibrationally excited states with energies Eg and Eg′. Single electronic excitation populates the states with energies E1, E2, and E3. For double electronic excitations, two

(9)

The sum of rephasing and nonrephasing contributions determines the amplitudes of the beating maps of the total signal GSB A total (ωT = |ωg ′ g | , ωτ , ωt )

∼

(14)

ESA APP (ωT = |Ee ′ e| , ωt )

(8)

e−Eg / kT αg = ∑′ e−E′g / kT

g′

+ δ(ωeg ′ − ωt )]

gd

ρ0 =

(13)

SE APP (ωT = |Ee ′ e| , ωt )

(7)

∼

∑ μge μg ′ e [αg {δ(ωeg ′ − ωt ) + δ(ωeg − ωt )}

∑ μge μg ′ e μge′μg ′ e′[αgδ(ωeg − ωτ) ee ′

× {δ(ωe ′ g ′ − ωt ) + δ(ωe ′ g − ωt )} + αg ′δ(ωeg ′ − ωτ ){δ(ωe ′ g − ωt ) + δ(ωe ′ g ′ − ωt )}]

(10)

SE A total (ωT = |Ee ′ e| , ωτ , ωt )

∼

∑ αgμge μge′μeg ′μe′ g ′[δ(ωeg − ωτ) gg ′

× {δ(ωe ′ g ′ − ωt ) + δ(ωeg ′ − ωt )} + δ(ωe ′ g − ωτ ){δ(ωeg ′ − ωt ) + δ(ωe ′ g ′ − ωt )}]

(11)

ESA A total (ωT = |Ee ′ e| , ωτ , ωt )

∼

∑ αgμge μge′μed μe′ d [δ(Eeg − ωτ)

Figure 1. Eigenstates of the model system chosen as a realization of eq 1. The solid vertical lines indicate the five detectable beating frequencies: ground-state frequencies Ωg and Ωg′ and excited-state frequencies Ω12, Ω23, and Ω13. The dashed vertical lines indicate some of those possible probe frequencies ωt, which differ from the three excitation frequencies ωτ = E1, E2, and E3.

gd

× {δ(Ede ′ − ωt ) + δ(Ede − ωt )} + δ(Ee ′ g − ωτ ){δ(Ede − ωt ) + δ(Ede ′ − ωt )}] (12) 10261



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states with energies Ed and Ed′ are accessible. In the majority of applications, one is usually interested in the excited-state dynamics completely determined by the states of manifold |e⟩. However, in the four-wave mixing schemes like 2D PE and PP spectroscopies considered here, the manifolds |g⟩ and |d⟩ become involved and are crucial for the signal analysis. The employed notation for the states of the three manifolds aims to distinguish between the physically relevant manifold |e⟩ (states denoted by indices 1, 2 and 3) and “auxiliary” manifolds |g⟩ and |d⟩. 2D PE signal is recorded as the function of the so-called excitation and probe frequencies ωτ and ωt and of the waiting time. For the systems described by eq 1, it is expected to show peaks at excitation frequencies corresponding to the energy of states optically accessible upon a single electronic excitation, ωτ = Ee (provided that only one, the lowest state, is initially populated and its energy is chosen as the origin). For the model considered, the possible excitation frequencies are, therefore, ωτ = E1, E2, and E3. As for the probe frequencies, GSB and SE processes visualize all possible ωt = Eeg = Ee − Eg, and ESA may arise at every ωt = Ede = Ed − Ee. The three |g⟩ levels and the two |d⟩ levels of the model lead to fifteen possibilities for ωt (five per each ωτ); some possible probe frequencies, distinct from ωτ, are indicated in Figure 1. Oscillations with waiting time, which reflect the system coherences, may arise with frequencies Ωg = Eg, as well as with all possible excited-state frequencies Ωee′ = |Ee − Ee′| (e ≠ e′). 2D PE signal of the model may, therefore, show oscillations with frequencies Ωg = Eg, Ωg′ = Eg′, Ω12 = E2 − E1, Ω13 = E3 − E1, and Ω23 = E3 − E2 (solid vertical lines in Figure 1). The oscillations with frequencies Ωg and Ωg′ can be referred to as ground-state coherences and arise due to GSB. The oscillations with frequencies Ω12, Ω13, and Ω23 reflect excited-state coherences and appear due to SE and ESA processes. Ground-State Coherence in 2D ES Beating Maps. Rephasing and nonrephasing beating maps of the ground-state mode Ωg are shown in Figure 2a. They arise exclusively due to GSB contribution (eqs 3 and 6), which is represented by blue squares. The corresponding Feynman diagrams can be found in Figure 5. First of all, we note that the rephasing map of the groundstate mode looks exactly as the nonrephasing one shifted to lower probe frequencies ωt by the beating mode frequency Ωg. This is a general rule valid for any ground-state mode: the location of the peaks in the nonrephasing GSB maps does not depend on the mode frequency, whereas in the rephasing maps, the shift to the lower probe frequencies is unique for each particular mode because it is determined by the mode itself. The number and the location of the peaks along both frequencies are determined by the number and energetic location of the excited states accessible for the bleach (the states with energies E1, E2, and E3 in Figure 1). The next observation concerns the relative intensity of the observed peaks: the maps are diagonally symmetric. In the nonrephasing map, the symmetry axis is located at the diagonal ωτ = ωt, whereas it is shifted to the diagonal ωτ = ωt − Ωg in the rephasing map. The intensity of the peaks depends on the transition dipoles between the involved |g⟩ states and |e⟩ states. Diagonal peaks scale as μ20eμ2ge (e = 1, 2, 3), off-diagonal peaks are determined by μ0eμ0e′μgeμge′ (e,e′ = 1, 2, 3; e ≠ e′). Different sizes of the squares in Figure 2 represent various dipole strengths; quite intense μ201μ2g1 = μ203μ2g3, and much weaker μ202μ2g2 have been chosen for the considered example. This

Figure 2. (a) Nonrephasing (“NReph”) and rephasing (“Reph”) beating maps of the ground-state mode Ωg. (b) 1D PP map of the same mode. Different sizes of the squares represent various peak intensities of the GSB contribution. The red, orange, and pink arrows indicate the connection between the peak locations and the excitedstate frequencies Ω12, Ω23, and Ω13, respectively. The blue arrows relate the peak locations and the mode frequency Ωg.

choice is random; in every specific application, the relative intensities of the peaks depend, of course, on the system properties, and the overall pattern created by the GSB contribution to the beating maps varies accordingly. Offdiagonal peaks are shown as blue squares in colored frames; each frame color corresponds to a particular combination of e and e′ for the dipoles involved. In the considered example, three such combinations are possible: 1 and 2 (red frame), 1 and 3 (orange frame), and 2 and 3 (pink frame). Below, we show that peaks of the same intensity also arise in the excitedstate maps due to SE. For any ground-state mode, the beating maps obey the same rules as far as the peak location is 10262



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concerned, but the relative intensities of the peaks vary for different modes according to the respective dipole strengths. Excited-State Coherence in 2D ES Beating Maps. Beating maps of excited-state modes are formed by SE (eqs 4 and 7) and ESA (eqs 5 and 8) contributions. The maps of the excited-state modes Ω12 and Ω13 are depicted in Figures 3a and 4, respectively. SE is represented by circles, and ESA is shown by triangles. The Feynman diagrams corresponding to the maps of the mode Ω12 are given in Figure 6. Because we do not differentiate the phase of the oscillations, two pathways leading to the oscillation with the same frequency (and opposite phase) contribute to the same map. Therefore, the maps representing an excited-state mode show peaks at two excitation frequencies ωτ separated by the energy corresponding to the beating mode frequency; it is Ω12 = E2 − E1 in Figure 3a and Ω13 = E3 − E1 in Figure 4. Further, the nonrephasing and rephasing maps transform into each other upon the exchange of the two excitation frequencies. The number of peaks at each excitation frequency (i.e., the number of detectable probe frequencies) depends on the number of levels accessible for the emission to the ground state (SE contribution) and for ESA (|d⟩ states). This leads to five peaks (three of SE and two of ESA origin) at each ωτ. Each of the five peaks at one excitation frequency (in one column) has its counterpart (a peak of the same intensity) at the other ωτ (in the other column). The two peaks of such a pair are shifted along the probe frequency ωt by the beating mode frequency (see red arrows for Ω12 in Figure 3a and orange arrows for Ω13 in Figure 4). SE necessarily leads to such a pair at the diagonal of the nonrephasing map, which becomes diagonally symmetric cross peaks in the rephasing map. The intensity of these peaks (red and orange circles with no colored frame in Figures 3a and 4) is determined by the dipoles μ201μ202 for Ω12 and μ201μ203 for Ω13. If several ground-state levels are detectable, SE also forms peaks at lower detection frequencies: in Figure 3, the blueframed circles are caused by the SE to the level with the energy Eg, while the green-framed circles arise due to the emission to the state with the energy Eg′. The intensity of the blue-framed circles is determined for mode Ω12 (Ω13) by μ01μ02μg1μg2 (μ01μ03μg1μg3); it is the same as that of the red-framed (orange-framed) squares in the map of the mode Ωg in Figure 2. The green-framed circles are determined by the product μ01μ02μg′1μg′2 for mode Ω12 and μ01μ03μg′1μg′3 for mode Ω13. ESA peak pairs of equal intensity must not be diagonal (offdiagonal) in the nonrephasing (rephasing) map. Their location is determined by the location of the |d⟩ states with respect to the two |e⟩ states corresponding to the beating mode considered (E1 and E2 for the mode Ω12; E1 and E3 for the mode Ω12). The multilevel structure of the |d⟩ manifold tends to appear as peaks at higher detection frequencies: the greenframed triangles are due to the presence of the state with the energy Ed′ (they scale as μ01μ02μ1d′μ2d′ in the map of Ω12 and as μ01μ03μ1d′μ3d′ in the map of Ω13), while the black-framed triangles are caused by the coupling to the lowest |d⟩ level (their intensity is determined by μ01μ02μ1dμ2d and μ01μ03μ1dμ3d, respectively). If at least two distinct excited-state frequencies are detected in the signal, a comparison of the beating maps of the different excited-state modes provides an additional hint to tell apart SE and ESA contributions. As predicted by eqs 4, 7, 5, and 8, the comparison of Figures 3a and 4 proves that the location of SE peaks along ωt depends (does not depend) on the beating

Figure 3. (a) Nonrephasing (“NReph”) and rephasing (“Reph”) beating maps of the excited-state mode Ω12. (b) 1D PP map of the same mode. The SE contribution is indicated by the circles, and the ESA contribution is represented by the triangles. The blue and green arrows indicate the connection between the peak locations and the ground-state frequencies Ωg and Ωg′. The red arrows correspond to the mode frequency Ω12.

frequency in the rephasing (nonrephasing) map, while the situation for ESA peaks is just the opposite. Feynman Diagrams. In order to illustrate the formation of the peaks in terms of Feynman diagrams, we need a slight generalization of the standard description; more specifically, we need to account for the multilevel structure of the electronic states. Figure 5 displays the diagrams corresponding to the GSB 10263



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Figure 6. Nonrephasing (“NReph”) and rephasing (“Reph”) Feynman diagrams corresponding to the beating maps of the excited-state mode Ω12.

the ground-state manifold is relevant for the SE contribution, while for ESA, the multilevel structure of the |d⟩ manifold matters. The number of peaks in Figure 3a is determined by the number of states in these manifolds, that is, by the number of possible x in Figure 6. The diagrams remain valid for any realization of the Hamiltonian given by eq 1. For GSB (Figure 5), g may denote any state belonging to the electronic ground state, and the beating maps would correspond to the energy difference between this state and the initial state. Further, x and y for all states of the excited-state manifold accessible for the bleach must be scanned. The number of all possible combinations of x and y (which is the number of the excited states squared) determines the number of peaks in the beating maps of the ground-state modes. For SE and ESA, the states denoted as 1 and 2 in Figure 6 can be replaced by any pair of the excited states corresponding to a beating mode of interest. Here, the number of peaks at each of the two possible ωτ (corresponding to the energies of the considered state pair) is the sum of all possible x arising from SE and all possible x arising due to ESA. All Contributions in One Map. If the ground-state and excited-state frequencies happen to be very close to each other, and this is often indeed the case if, for example, the excitation of Raman-active modes takes place, GSB, SE, and ESA counterparts will contribute to the same map. The maps for the case Ωg = Ω12 are shown in Figure 7. The prominent signature of the ground-state contribution (GSB) is the presence of peaks at multiple (more than two) excitation frequencies. However, the diagonal symmetry of “pure” GSB maps is destroyed due to the presence of the excited-state contribution (SE and ESA). Further, SE and ESA lead to significant differences between the nonrephasing and rephasing maps (the rephasing map does not simply repeat the nonrephasing one at the lower detection

Figure 4. Nonrephasing (“NReph”) and rephasing (“Reph”) beating maps of the excited-state mode Ω13. The SE contribution is indicated by the circles, and the ESA contribution is represented by the triangles. The blue and green arrows indicate the connection between the peak locations and the ground-state frequencies Ωg and Ωg′. The orange arrows correspond to the mode frequency Ω13.

Figure 5. Nonrephasing (“NReph”) and rephasing (“Reph”) Feynman diagrams corresponding to the beating maps of the ground-state mode Ωg.

beating map of the ground-state mode Ωg shown in Figure 2a. x and y denote the states of the |e⟩ manifold, that is, the states with energies E1, E2, and E3. Diagonal (off-diagonal) peaks in Figure 2a arise if x = y (x ≠ y). The diagrams corresponding to the beating maps of the excited-state mode Ω12 (Figure 3a) are shown in Figure 6. Two pathways leading to the oscillations with the same frequency (and opposite phase) are indicated. The multilevel character of 10264



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from 2D PE becomes the delay between pump and probe pulses in a PP experiment. The remaining frequency resolution is provided by the detection (probe) frequency ωt. It is straightforward to employ the same strategy as that for 2D ES in order to try to tell apart GSB, SE, and ESA contributions to the PP signal. The explicit expressions for the PP 1D beating “maps” (they depend on ωt only) in the stick spectrum limit are given by eqs 13−15. In PP, one cannot distinguish between rephasing and nonrephasing signals. Still, the behavior of the three contributions in the 1D maps is distinct. If we keep the assumption that only the lowest state is initially populated, eq 13 predicts that the oscillations with the ground-state frequencies due to GSB arise at ωt = Ee and at ωt = Ee − Eg, where Eg is determined by the mode frequency Ωg and all |e⟩ states are scanned. The amplitude is determined by μ0eμge(∑e′ μ0e′μge′). The map of the ground-state mode Ωg of the considered model is shown in Figure 2b. According to eqs 14 and 15, the oscillations with the excited-state frequencies arise at ωt = Ee − Eg and ωt = Ee′ − Eg (SE contribution), as well as at ωt = Ed − Ee and ωt = Ed − Ee′ (ESA contribution), where e ≠ e′, Ee and Ee′ are determined by the mode frequency Ωee′, and all |g⟩ and |d⟩ states are scanned. The amplitude of the SE contribution scales as μ0eμ0e′μgeμge′, and for the ESA, the product μ0eμ0e′μdeμde′ matters. The 1D map of the excited-state mode Ω12 of our model is shown in Figure 3b. Figures 2b and 3b can be obtained from the maps in Figures 2a and 3a by removing the resolution over the excitation frequency ωτ. The lack of information on ωτ makes the analysis of PP signal much less efficient as compared to 2D ES. It becomes especially challenging if ground-state and excited-state frequencies are close and all three contributions are present in the same map. We skip the graphical representation of this situation because it would be too messy (the reader may imagine it by looking at Figure 7 and ignoring the resolution over the excitation frequency ωτ). Still, few rules can be formulated for PP 1D maps and facilitate the analysis of oscillatory signatures. In both ground-state and excited-state maps, peak pairs of the same intensity are found; they are separated by the mode frequency. The number of these pairs and their location are distinct for ground-state and excited-state coherences. For the ground-state modes, the number of the pairs is determined by the number of the excited states that can efficiently scatter and lead to the excitation of the ground-state vibrations. The upper bound of the probe frequency is determined by the energy of the highest excited state involved in the bleach; the lower bound is the energy of the lowest excited state minus the mode frequency. The number of pairs in the maps of the excited-state modes is as that for 2D ES; it is determined by the number of ground-state levels accessible for SE plus the number of higher excited states accessible for ESA. Together with the number of the peaks, the upper and lower bounds of the probe frequency are capable of reflecting the multilevel structure of the ground-state and of the doubly excited-state manifolds.

Figure 7. Nonrephasing (“NReph”) and rephasing (“Reph”) beating maps of the mode Ωg = Ω12. All three contributions are simultaneously present and partially overlap; GSB is shown by blue squares, SE is indicated by circles, and ESA is represented by triangles. Blue and red arrows correspond to the equal mode frequencies Ωg and Ω12, respectively.

frequencies as is the case for GSB). The most prominent difference in the behavior of excited-state and ground-state contributions arises at the lowest ωτ = E1. Here, SE and ESA components move to higher probe frequencies in the rephasing map as compared to the nonrephasing one, while GSB moves in the opposite direction. All other peaks, at ωτ = E2 and at ωτ = E3, move in the same direction if the rephasing and nonrephasing maps are contrasted. As a consequence, the peaks with overlapping GSB and SE components in the nonrephasing map remain the same in the rephasing map at ωτ = E2, while at E1, the origin and intensity of the overlapping peaks are different in the two maps. The differences in the behavior of the ground-state and the excited-state contributions at the lowest excitation frequency suggest that the peaks found here at ωt = ωτ − Ω (Ω denotes a beating frequency considered) are of the ground-state origin in the rephasing map and of the excited-state origin in the nonrephasing map. Note that the findings of ref 13 can be obtained as a particular realization of Figure 7 if the nonrephasing and rephasing contributions are overlapped (plotted together in one graph) and the number of the considered states is reduced to obtain the model discussed by Butkus et al. 1D Maps of Frequency-Resolved PP Signal. The frequency-resolved (dispersed) PP signal can be viewed as the 2D PE signal where the dependence on the excitation frequency ωτ has been integrated out. The waiting time known

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CONCLUSIONS AND OUTLOOK To summarize, GSB, SE, and ESA contributions and, therefore, ground-state and excited-state coherences can be distinguished in the beating maps of 2D PE signals. The analysis is substantially simplified if rephasing and nonrephasing maps are considered separately. Typical signatures of GSB contributions are (i) the detection of more than two ωτ, (ii) lowering of ωt by the mode frequency in the rephasing maps as 10265



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from the purple bacteria, refs 6 and 10, which have been assigned to the ground-state vibrational motion, contain wellresolved signatures of the excited-state coherence. As for the conventional 2D PE signal, a notion of the location of the GSB, SE, and ESA contributions specific for each mode helps to chose most suitable excitation and probe frequencies when performing a measurement of the dependence on waiting time. In ref 19, for example, the oscillations have been measured in the region where ESA dominates; this makes the assignment of the detected modulation and its frequency to the excited-state coherence very reliable. On the other hand, the measurements on the FMO complex1,3,20 have been performed in the spectral region where GSB may have a significant contribution. Therefore, the detected oscillations may represent a superposition of the excited-state and of the ground-state coherent motion. Multiple oscillations resolved in marine algae2,21 contain most probably both ground-state and excited-state components; a more specific assignment could be possible using the beating maps analysis proposed here. We conclude that the reported rules for the analysis of oscillatory components in 2D PE signals are the way to the unambiguous assignment of the resolved oscillations and elucidating coherent dynamics. The beating maps can be used for the reconstruction of the eigenstate structure and enable the resolution of close-lying eigenstates, hardly detectable in the conventional 2D signals. Applications to systems with dense state manifolds, as, for example, vibronic manifolds in the vicinity of conical intersections, are especially promising. The nature of eigenstates leading to specific beating maps patterns is encoded in the dipoles of the involved transition. The relations established in this work make possible the extraction of this information and a detailed microscopic understanding of couplings and interactions relevant for ultrafast multilevel dynamics in the excited states.

compared to the nonrephasing ones (otherwise the rephasing map is the replica of the nonrephasing one), and (iii) diagonal symmetry in the intensity of the peaks (the symmetry axis is at ωt = ωτ in the nonrephasing maps and at ωt = ωτ − Ω in the rephasing maps). Further, if several ground-state modes are detected, the location of the peaks in the nonrephasing maps of these modes is the same and determined by excited-state energies, but the intensities are mode-dependent. For the peaks of the excited-state origin, (i) only two values of ωτ are possible for one mode; they are determined by the mode frequency; (ii) peaks come in pairs of equal intensity; the two peaks in the pair are separated by the mode frequency along ωt; and (iii) rephasing and nonrephasing maps transform into each other upon the exchange of the two excitation frequencies. If groundstate and excited-state manifolds contain similar frequencies (e.g., if Raman-active modes are present and excited in the experiment), a superposition of all three contributions (GSB, SE, ESA) in one beating map is very probable. In this case, the rules can be applied to determine the origin of the peaks in the same map. A similar scheme can be applied to PP spectroscopy. The resulting 1D maps are more difficult to interpret but can still provide valuable hints that facilitate a proper assignment of oscillatory signatures. In this way, the analysis improves the sensitivity and selectivity of PP spectroscopy. Although performed in the stick spectrum limit, the presented analysis is of considerable practical importance and impact because the outlined rules can be directly applied to decode the recorded experimental data. The intensities of the peaks in the real-life beating maps are influenced by dephasing during the waiting time and by finite durations of the employed femtosecond pulses. The stick spectrum results can be postprocessed to account for these effects. An exponential dephasing with a constant rate Γ would lead to a correction of the peak intensities by a constant factor 1/Γ, that is, it will not influence the relative peak intensities in the map of one particular mode. However, it must be taken into account if several coherences with close-lying frequencies are detected in the same map. Finite pulse durations can be accounted for by a convolution of the stick spectrum predictions along ωτ and ωt with two Gaussians in the frequency domain, provided the temporal envelopes of the pulses are well described by a Gaussian. Alternatively, the dipole strengths of the involved transitions can be modified to account for a realistic laser spectrum. As for unavoidable limitations, the approach is not capable of capturing couplings between populations and coherences as well as coherence transfer by the environment. Still, it can be helpful to establish if these effects are present. Further, the analysis is not applicable, strictly speaking, to conical intersections involving the electronic ground state because the separation between the ground-state and excited-state manifolds is not possible in this case. So far, we have taken advantage of the “self-analysis rules” when interpreting 2D PE beating maps of pentacene films.11 We strongly believe that the presented guidelines can help to better rationalize the majority of the reported experimental results on oscillatory signatures in 2D ES. In particular, the beating maps of bacteriochlorophyll dimer reported in ref 8 exhibit clear GSB signatures for the 546 and 735 cm−1 modes, that is, a significant contribution of the ground-state coherence. The same is true for the maps of the 251 and 339 cm−1 modes of the photosystem II reaction center reported in ref 9. The beating maps of the peripheral light-harvesting complex LH2

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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 I am deeply grateful to Alex Chin, Sarah Morgan, and Artem Bakulin for inspiring discussions during our work on the pentacene project.

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REFERENCES

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Self-Analysis of Coherent Oscillations in Time-Resolved Optical Signals

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