Differentiation of True Nonlinear and Incoherent Mixing of Linear

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Differentiation of True Nonlinear and Incoherent Mixing of Linear Signals in Action-Detected 2D Spectroscopy Alex Arash Sand Kalaee, Fikeraddis Damtie, and Khadga Jung Karki J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01129 • Publication Date (Web): 18 Apr 2019 Downloaded from http://pubs.acs.org on April 18, 2019

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The Journal of Physical Chemistry

Differentiation of True Nonlinear and Incoherent Mixing of Linear Signals in Action-Detected 2D Spectroscopy Alex Arash Sand Kalaee,∗,† Fikeraddis Damtie,† and Khadga Jung Karki∗,‡ †Mathematical Physics and NanoLund, Lund University, Box 118, 22100 Lund, Sweden ‡Chemical Physics Lund University, P.O. Box 124, 22100 Lund, Sweden E-mail: [email protected]; [email protected]

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Abstract Phase-modulation and phase-cycling schemes have been commonly used in electronic two-dimensional (2D) spectroscopy where the observables are incoherent signals such as fluorescence or photo-current. Although the methods have distinct advantages compared to the coherent signal detected 2D spectroscopy in sensitivity, possibility to measure spectra from isolated quantum systems and direct visualization of the contributions from the different states to the action signals, ambiguities in interpreting the spectra have emerged. Recent reports have shown that apart from the nonlinear signals from the four pulse interactions, mixing of the linear signals due to nonlinear population dynamics during the long measurement time of the action signals can also contribute to the measured 2D spectra. Exciton-exciton annihilation has been considered to play a major role in the mixing of the linear signals. Thus, it has become important to further characterize the origin of the measured signals. Here, using a non-perturbative simulation of the 2D spectra based on the time evolution of the density matrix in the Lindblad form, we show that the exciton-exciton annihilation contributes to the measured signal only if the quantum yields of the different excited states are not the same. In these cases, the mixed signals can be distinguished from the true nonlinear signals if the phases are measured with respect to the linear signals. In an action-detected 2D spectra, the mixed signals have a π phase shift relative to the true nonlinear signals. A detailed discussion on the experimental implementations of the schemes used in the simulations is also provided.

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Introduction. Since its first implementation, 1,2 different phase modulation schemes 1,3,4 have been used in the Fourier transform spectroscopy of molecules, 5,6 molecular aggregates 7,8 and photoactive devices. 9–12 Electronic 2D spectroscopy 2,7–10 based on four phase modulated beams has recently become popular. Action signals, such as fluorescence and photo-current, have been used in most of the measurements. Similarly, phase cycling schemes 13 have been used in 2D spectroscopy in conjunction with the detection of photo-ions 14 and molecular fragments. 15 One of the far reaching advantages of action-detected 2D spectroscopy is that the method is technically capable of measuring in-situ ultra-fast dynamics in operational devices, 10 isolated systems and even single molecules. However, the research field is still in infancy and relatively few experimental results have been reported. Most of the reported results have shown changes in the 2D spectra at characteristic relaxation time scales of the excited states indicating that the signals originate from nonlinear interactions between the sequence of the four laser pulses and the sample. 5,7,8,10,14,15 Nevertheless, some of the reports have only shown 2D spectra at a single population time, 9 and others have reported no change in the spectra along the population time, 12 which seems to suggest that the applicability of the method to investigate the ultra-fast dynamics may be specific to appropriate systems. In phase modulated 2D spectroscopy, see Fig 1, the phases of four collinear pulsed beams are modulated at four frequencies, φ1 , φ2 , φ3 and φ4 , by using acousto-optic modulators in a dual Mach-Zehnder interferometer. 2,7,10 The interaction of the beams with the sample modulates incoherent signals at frequencies given by ±n1 φ1 ± n2 φ2 ± n3 φ3 ± n4 φ4 , where ni are integers. Out of all the modulated signals, the signals at φ1 − φ2 − φ3 + φ4 and −φ1 + φ2 − φ3 + φ4 , which are also known as the rephasing and the non-rephasing signals, result from the nonlinear interactions with the four beams. Amplitudes and phases of these signals are measured as a function of the inter-pulse time delays to obtain 2D spectra in the time domain. The Fourier transform of the time domain data yields 2D spectra in the frequency domain. 4


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Figure 1: Schematic of a setup for action-detected 2D spectroscopy. A sequence of four pulses whose phases are modulated at frequencies φ1 , φ2 , φ3 and φ4 interact with the sample imparting modulations in the amplitude of incoherent responses such as fluorescence and photo-current. The modulated response from the sample at the frequencies φ1 − φ2 − φ3 + φ4 and −φ1 + φ2 − φ3 + φ4 are recorded as a function of inter-pulse time delays t21 , t32 and t43 . The Fourier transform of the data along t21 and t43 provides the 2D spectra at different intervals of t32 . The nonlinear response from a system induced by four-wave mixing of light fields can be described by three different interaction pathways commonly called as the ground state bleach (GSB), stimulated emission (SE), and excited state absorption (ESA). In action-detected 2D spectroscopy using phase modulation, 7,10 the ESA consists of two pathways, ESAI and ESAII, with different signs such that the total amplitude of the ESA signal depends on the quantum yields of the signals from the different states. This contrasts with the 2D spectroscopy based on the detection of coherent response, where only one ESA pathway contributes. 16–18 Note that the ESAI pathway corresponds to the bleaching of population in state |1i in model b of Fig 2 (see Methods below) and ESAII corresponds to promoting the population to state |3i in the same model. Particularly in fluorescence detected 2D spectroscopy of excitonic systems with monomolecular recombination kinetics, the quantum yields from both the pathways are often the same such that they cancel each other and the final signal is a sum of the GSB and SE pathways. This is generally not the case in photo-current detected 2D spectroscopy. 10 It is important to consider that the linear signals at frequencies φ21 = φ2 − φ1 and φ43 = φ4 − φ3 , which originate from two-field interactions, are orders of magnitude stronger than the nonlinear signals from four-field interactions. As

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the action-signals are recorded over a significantly longer time compared to the typical ultrafast dynamics in the system, other slow nonlinear population dynamics can incoherently mix the linear signals. The mixed signals appear at the same modulation frequencies as the true nonlinear signals. It has been shown that such effects can dominate the photo-current detected 2D spectra in semiconductors that have high carrier mobility. 19 Similar effects can prevail in any action-detected nonlinear spectroscopy of semiconductors, metal particles and molecular aggregates. Thus, there is an urgent need to develop a methodology to identify if the mixed signals have dominant contributions in the spectra. 19 Here, by explicitly simulating an experiment, we show that the nonlinear signals in the 2D spectra are phase shifted by π with respect to the mixed signals. The phase shifts can be unambiguously identified in an experiment if the linear as well as the nonlinear signals relevant for the 2D spectra are recorded simultaneously without applying any phase correction during the lock-in detection.

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Methods. We consider two different models of an excitonic system that consist of two closely spaced excited states, see Fig. 2. The energy levels at 1.55 eV and 1.46 eV are similar to the levels 6


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corresponding to the outer and inner rings, respectively, of bacteriochlorophylls in the lightharvesting complex of purple bacteria. 20 Recent measurements of fluorescence-detected 2D spectra have shown peaks across the diagonal, 5,21 the origin of which has been debated. It has been suggested that either a correlated excitation of the two rings 21 or exciton-exciton annihilation after the energy transfer from the outer ring to the inner ring could result in the off-diagonal peaks. 22 We have used model a (Fig. 2 a) to simulate the 2D spectra that includes contributions from the exciton-exciton annihilation as a representative process that can mix the linear signals. The model has two independent systems, each with three eigenstates that are denoted by |0i, |1i and |2i, and |3i, |4i and |5i, respectively. The transition dipole matrix elements are adjusted such that µkl E0 = 8 meV for all the included elements, E0 is the electric field strength, and the corresponding relaxation rates are τ10 = τ43 = 1 ns and τ21 = τ54 = 100 fs. Similarly, we have used model b (Fig. 2 b) with four eigenstates, |0i, |1i, |2i and |3i, to simulate the spectra of correlated excitation of the two rings. The model assumes that all the states can be excited from a common ground state. The transition matrix elements have the same size as in a and the relaxation rates are τ10 = 160 ps, τ21 = 160 fs and τ32 = 100 fs. The density matrix, ρˆ(t), is propagated in time by using the Lindblad Master Equation. 23 The simulations are done following the method developed by Damtie et al. 24 and the details are given in the Supplementary Information (SI).

Results and Discussion. In previous simulations, the fluorescence signal has been calculated by summing the recombination events. 24 This method, however, cannot be used to simulate the mixing of linear signals that is proportional to the product of the population densities in the low lying excited states of model a, Sx ∝ ρ11 × ρ44 . 25 Thus, in order to obtain the contributions of such processes in the 2D spectra, we have used the modulations in the density of populations imparted by the train of phase modulated pulses. In order to rigorously show that the two

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signals are equivalent, we compare the 2D spectra obtained from them. Fig. 3 shows the spectra obtained for model a from the recombination signals Rel10 (a) and Rel43 (b) with the spectra obtained from the corresponding population density signals ρ11 (c) and ρ44 (d). Only the analysis of real part of the total correlation spectra, i.e. the sum of the rephasing and non-rephasing spectra, are presented throughout this work. Apart from a difference in scale, the shape and sign of the spectral surface plot for the two pairs of spectra share same topological characteristics, which indicates similar dynamics. Hence, we consider the modulations in the density of populations as a viable proxy for the externally measured signals including the fluorescence.

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Figure 3: Comparison of the signals for (a) relaxations Rel10 and (b) Rel43 with the corresponding population densities (c) ρ11 and (d) ρ44 . The mixing of linear signals due to exciton-exciton annihilation in a molecular aggregate

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shown in model a can be understood as a two-step process where the first step is an energy transfer from state |1i to state |4i, which at the same time de-excites the first molecule and excites the second molecule to the high lying state |5i as shown by the solid red arrows in Fig. 4. The excitation transferred to |5i is given by S5x ∝ ρ11 × ρ44 . As the populations in the states |1i and |4i are modulated at frequencies φ21 and φ43 , S5x exhibits modulations at the frequencies φ43 ± φ21 , which are identical to the modulations in the non-rephasing and rephasing nonlinear signals. Note that the population in |5i is transferred from |4i, a process which also imparts modulations in the population density of |4i, S4x ∝ ρ11 × ρ44 , at φ43 ±φ21 . However, the modulations in |4i and |5i are in anti-phase (see Fig.4). Similar to the two ESA pathways, whether the exciton-exciton annihilation contributes to the measured signal depends on the quantum yields, η5 and η4 , of the states |5i and |4i, respectively. Some authors have wrongly assumed a universal contribution from the annihilation process in fluorescence detected 2D spectra of molecular dimers by analyzing only the population of |5i, 22 which is not based on an experimental observable. For example, typical quantum yields of fluorescence from both the states in a molecular system is the same such that η5 − η4 = 0; consequently the annihilation does not contribute to the measured nonlinear signals. On the other hand, quantum yields of photo-current or photo-ion from the two states are generally not equal and in most of the cases η5 > η4 . Therefore, the contributions of the annihilation in such signals are given by Sx ∝ ηS5x , where 0 ≤ η. Fig. 5 (a) shows the 2D spectra of the mixed signal Sx from model a obtained by setting η = 1 and (b) shows the corresponding spectra obtained from the nonlinear population signal ρ11 from model b. Considering Fig. 5 (a), we note strong off-diagonal peaks from exciton-exciton annihilation. As model b only allows a single exciton to be present, excitonexciton annihilation is not possible. Hence, the obtained signal is the true non-linear signal, which provides a reference for the mixed signal and its spectral line shape. Comparison with Fig. 5 (b) reveals that the inclusion of exciton-exciton annihilation shifts the phase of the signal by π such that the mixed 2D spectra has an opposite sign. Similarly, the amplitudes

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Figure 4: Schematic depiction of mixing of linear transitions in model a. In exciton-exciton annihilation energy is transferred from |1i to |4i, which de-excites the first molecule and excites the second molecule to the high lying state |5i. The second molecule subsequently relaxes back to |4i via fast non-radiative vibrational relaxation. The modulations in the population densities at |4i and |5i at the mixed frequencies are in anti-phase. of the peaks in (a) are significantly smaller than in (b). From the relative sizes of the offdiagonal peaks in the two spectra we observe that if the quantum yield of Sx is about an order of magnitude larger than that of the true non-linear signal then the annihilation signals dominate in the off-diagonal peaks of the total spectrum. Thus, in a typical measurement of a system where Sx and true non-linear are of similar magnitude, the resulting 2D spectra has dominant diagonal peaks and possibly weak off-diagonal peaks with an opposite sign. 0

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Figure 5: Comparison of (a) combined signals Sx and (b) population density ρ11 for model b. Finally we outline a procedure that can be used in experiments to distinguish between the mixed and the nonlinear signals. This distinction is not possible in the current im-

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plementations of the experimental setups mainly because of the adjustment of the phase during data processing. The phases of the signals are adjusted such that the real parts are maximized during the overlap of the pulses. This renders the main features of the total correlation spectra to be positive irrespective of whether it originates from the mixed or the true nonlinear signal. As we have mentioned previously, the linear signals at φ21 and φ43 are orders of magnitude stronger than the nonlinear signals at φ43 ± φ21 . Typically in a measurement, the linear signals are filtered out and only the nonlinear signals are recorded by using lock-in amplifiers. These signals, however, are important in assigning the absolute phase of the nonlinear signals. In order to do so, a different method of data acquisition has to be implemented where the signal from the detector is digitized and the data is post-processed using algorithms of fast-Fourier transforms or the generalized lock-in amplifiers. 26–28 Time domain 2D spectra are then constructed from the amplitudes and phases of the linear signals at φ21 and φ43 , and the nonlinear signals at φ43 ± φ21 . Multiplication of the 2D spectra from the linear signals give the reference spectra at φ43 ± φ21 . We have tested the method using the data from the simulations. Fig. 6 compares the 2D spectra of model a (see Fig. 2) obtained from the annihilation signal Sx (a), the true nonlinear signal (b), ESAII pathway (c) and the reference (d). The spectra from the true nonlinear signal in (b) is obtained from the nonlinear modulations of the populations in the lowest excited states ρ11 and ρ44 . These signals arise from the GSB, SE and ESAI pathways. Similarly, the spectra in (c) are obtained from the nonlinear modulations of the populations in the high-lying excited states ρ22 and ρ55 . These signals arise from the ESAII pathways. Although the two spectra have similar features, they have different signs. Note that the spectra measured in the experiments is the sum of (b) and (c), and generally if the quantum yields of the observable from high lying states are the same as that from the lowest excited state, the two ESA pathways cancel resulting in the spectra that has the same sign as in (b). When we compare the reference spectra in (d) with the nonlinear spectra in (b) and (c), we find that the sign of the spectra from the ESAII pathways is the same as the reference spectra. Thus, from the comparison

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we can identify if the measured spectra is dominated by the ESAII pathways or the usual GSB, SE and ESAI pathways. Similarly the sign of the spectra in (a) is similar to that of the reference (d), which also allows us to distinguish the annihilation contributions from the GSB, SE and ESAI pathways. However, the sign alone is not sufficient to distinguish between exciton-exciton annihilation and ESAII. In this case, the prevalence of annihilation can be inferred from the prominent off-diagonal peaks in the spectra. 0.64 0.56

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Figure 6: 2D spectra from the mixed signals due to (a) exciton-exciton annihilation and (b) the true nonlinear signals based on the populations ρ11 + ρ44 , (c) population signal ρ22 + ρ55 that corresponds to ESAII and (d) the reference spectra obtained from the modulations at φ21 =500 kHz and φ43 =800 kHz of the populations ρ11 + ρ44 . See the text for the details.

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Conclusion To conclude, we have shown that exciton-exciton annihilation does not contribute significantly in the 2D spectra of molecular systems where quantum yields of the observable are same for all the excited states. In systems with different quantum yields, the signatures of annihilation can be identified if the signs (or phase) of the signals are compared with the reference spectra constructed from the linear signals. The results address an important and unresolved ambiguity in identifying the origin of the signals measured in action-detected 2D spectroscopy. The methodology we have presented can be applied in any experimental setup of action-detected 2D spectroscopy to distinguish the contributions from the mixing of linear signals, ESAII, and GSB and SE pathways.

Supporting Information Available The time evolution of the system under the Lindblad master equation approach is given in the Supplementary Information. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement Financial support from the Swedish Research Council (VR), the Crafoord Foundation, the Knut and Wallenberg Foundation and NanoLund is gratefully acknowledged. Authors also thank Prof. Andreas Wacker for guidance in implementing the Lindblad Master Equations.

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(28) Jin, A.; Fu, S.; Sakurai, A.; Liu, L.; Edman, F.; Pullerits, T.; Öwall, V.; Karki, K. J. Note: High precision measurements using high frequency gigahertz signals. Rev. Sci. Instrum. 2014, 85 .

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Figure 7

Action-detected 2D spectra depicting exciton-exciton annihilation (bottom-left) and nonlinear ground-state bleach and stimulated emission (bottom-right) from isolated molecular systems with two energy levels (top). The spectra have a relative phase shift of π.

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Differentiation of True Nonlinear and Incoherent Mixing of Linear

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