Visualizing Hidden Ultrafast Processes in Individual Molecules by

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Visualising Hidden Ultrafast Processes in Individual Molecules by Single-Pulse Coherent Control Kevin Wilma, Chuan-Cun Shu, Ullrich Scherf, and Richard Hildner J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b08674 • Publication Date (Web): 22 Oct 2018 Downloaded from http://pubs.acs.org on October 23, 2018

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Journal of the American Chemical Society

Wilma et al, Coherent Control of Single Molecules

Visualising Hidden Ultrafast Processes in Individual Molecules by Single-Pulse Coherent Control Kevin Wilmaa,*, Chuan-Cun Shub,c,1,*, Ullrich Scherf d, Richard Hildnera,2 a Soft Matter Spectroscopy, University of Bayreuth, 95440 Bayreuth, Germany b Institute of Super-Microstructure and Ultrafast Process in Advanced Materials, School of Physics and Electronics, Central South University, Changsha 410083, China c School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia d Fachbereich C – Mathematik und Naturwissenschaften and Institut für Polymertechnologie, Universität Wuppertal, 42097 Wuppertal, Germany

1,2 Corresponding Authors: [email protected] [email protected] * These authors contributed equally

Abstract Coherent control of single quantum systems in complex environments has great potential to manipulate and understand photoinduced chemical and biological processes on a molecular level. However, heterogeneous environments usually impede full control and complicate interpretation. Here, we demonstrate photoluminescence-detected ultrafast phase-only coherent control on single organic molecules in a disordered matrix at room temperature. Combined with a multi-parameter quantum

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dynamics identification procedure we reconstruct multi-photon processes and energy landscapes for each molecule. We find strong phase dependencies of the corresponding transitions into highly excited states. Importantly, also transitions into hidden states, which are not connected to photoluminescent channels, are monitored and controlled. Our combined approach provides a general toolbox to manipulate and understand ultrafast photoinduced processes in single quantum systems, which is a prerequisite to control chemical and biological function.

Introduction Quantum coherent control is a powerful technique that has been applied to many different quantum systems, including ions, molecules, semiconductors, and plasmonic nanostructures.1–12 This technique aims at controlling photoinduced processes by specifically tailored electro-magnetic fields.1,2 These fields are shaped in their spectral phase and amplitude, and/or polarisation to exploit quantum interference of multi-photon transitions between competing pathways towards the desired outcome. For instance, photochemical reactions,13–15 energy transfer16,17 and ionic currents in biological systems were controlled;18 moreover, in scattering experiments specific constituents (electronic or molecular) were selectively detected and addressed.6,19 A great challenge in this field is to realise coherent control of single quantum systems that are embedded in a complex environment, such as polymers, proteins, or living tissue. This approach will allow to gain unprecedented insights into quantum coherent phenomena beyond the ensembleaveraged response. Ultimately, the control of chemical and biological function by external stimuli may


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become possible on a truly molecular level. Since coherent control schemes are usually based on multi-photon processes,20–22 which require relatively strong electro-magnetic fields, a high stability of the quantum systems is essential. Recently, coherent control of non-resonant two-photon transitions has been shown on single quantum dots.11 A genetic algorithm, using photoluminescence as feedback, optimised the phase distribution of the laser field for maximum signal. For single organic molecules at room temperature, however, the demonstration of phase-sensitive multi-photon processes is hampered so far by rapid photobleaching,23 an irreversible photochemical reaction, which severely limits observation times. From a theoretical point of view a highly challenging issue is that a single molecule surrounded by a complex environment represents a ”black box”: For each molecule the interactions with its particular local surrounding are a priori unknown; these interactions modify intramolecular dynamics and excited-state energy landscapes, and thus change interactions with externally applied electro-magnetic fields. Although quantum tomography approaches can be employed to extract unknown quantum processes,24–27 the identification of the relevant parameters from single molecules embedded in a (disordered) environment is a challenging and fundamental task still to be solved. Here, we demonstrate photoluminescence-detected ultrafast coherent control of single organic molecules embedded in a disordered, solid matrix at room temperature. Using phase-shaped femtosecond laser pulses, we manipulate multi-photon transition probabilities in a methyl-substituted ladder-type poly(para-phenylene), MeLPPP (see Fig. 2 for the chemical structure). This -conjugated polymer is photochemically stable, possesses an extraordinary two-photon absorption cross section,28 and features reasonably high two-photon induced photoluminescence (PL),29 which allows for its detection on the single-molecule level. The experimentally retrieved phase-dependent PL signals are reconstructed using a novel quantum dynamics identification (QDI) procedure. We are thus able to unravel the underlying ultrafast photoexcitation dynamics as well as the excited state energy



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landscape of each molecule in its specific local environment, and ultimately to identify, explore, and control ultrafast multi-photon processes in single organic molecules at room temperature.

Results and Discussion The general concept of our approach is illustrated in Fig. 1: Single MeLPPP molecules are embedded in a polystyrene film at room temperature. The excitation source is a Ti:Sapphire oscillator, which provides pulses with a Fourier-limited width of about 40 fs at a centre energy of ħ ~ 13000 cm-1 (corresponding to ~ 770 nm, Fig. 2). For MeLPPP this centre energy is far off-resonant for the transition from the electronic ground state S0 to the first electronically excited singlet state S1 (Fig. 2). However, highly efficient non-resonant two-photon transitions from S0 into the second excited singlet state S2 can be induced (see Ref.

29).

Importantly, our excitation conditions are chosen such that we

do not induce transitions into vibrational levels of S2 (Fig. 2) to simplify the theoretical description. The population created in S2 rapidly relaxes by internal conversion within ~ 150 fs to the first excited state S1.30 State S1 then decays radiatively via PL with a characteristic lifetime of ≥ 100 ps back to the ground state.31 Hence, the PL intensity yields the relative efficiency of two-photon-induced population transfer from S0 to S2. For our PL-detected coherent control experiments we exploit this strongly allowed two-photon transition in MeLPPP. We apply a quadratic spectral phase function ()  (  0)2/2 to the exciting field (see Methods section, equation 2) with the linear spectral chirp  as control parameter. The spectral amplitude A is kept unchanged. The two-photon induced PL signal as a function of  is monitored by a single-photon sensitive photodiode. We then employ our QDI procedure to reconstruct the quantum dynamics of each single molecule within a minimal quantum mechanical model. This procedure does in principle not require prior knowledge about the molecular system and solves a quantum multi-parameter optimisation problem by minimising the residual between the experimentally measured and theoretically calculated PL signals. Details of our


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experiment and the mathematical formulation of the QDI procedure can be found in the Methods section as well as in the Supporting Information. The chirp-dependent two-photon induced PL signal of a single MeLPPP molecule is shown in Fig. 3a. The PL features a maximum for a transform-limited pulse ( = 0 fs2), and a symmetric decrease for increasing magnitude of the spectral chirp . This response is consistent with the established dependence of non-resonant two-photon transition probabilities between two electronic states on the spectral phase of the incident laser field (see SI, Theoretical analysis and simulations).6,21 The chirpdependent relative phase between all frequency components of the laser pulse controls quantum interference between all pathways, that contribute to the two-photon S0 → S2 transition. To extract the intrinsic quantum dynamical processes from this PL trace we start our QDI procedure with the minimal three-level model outlined above, comprising three electronic states Si with corresponding energies Ei (i = 0,1,2), see Fig. 3b. The model involves four free parameters, the two-photon Rabi-frequency 2P coupling states S0 and S2, the energy E2 and the dephasing rate 2 of S2, as well as the relaxation rate 21 for the internal conversion from S2 to S1. The energy of the ground state S0 is set to E0 = 0 cm-1, and that of the first excited state S1 is fixed at E1 = 22000 cm-1 (Fig. 2), because the latter state is not involved in laser-induced transitions. This three-level model with a set of optimal parameters reproduces the experimental data with high quality (Fig. 3, solid line; see also the parameter optimisation details in Supplementary Fig. S2). This good agreement also implies that vibrational levels in S2 can indeed be neglected (see Fig. 2 and SI, Ensemble spectroscopy). For the specific molecule in Fig. 3 we find that the energy of state S2 is E2 = 25940 cm-1, which provides nearly perfect spectral overlap with the laser spectrum for the two-photon S0 → S2 transition (Fig. 2). Consequently, the two-photon Rabi frequency is also relatively large with 2P = 531 cm-1, and thus a substantial population of more than 50 % is transferred from the ground state into the excited states for small chirps (Fig. 3, inset, and Supplementary Fig. S3). The dephasing



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rate 2 = 1/61 fs-1 is high indicating a rapid loss of coherence in S2. This number is in good agreement with our measurement of the envelope of the optical free induction decay of state S2 of single MeLPPP molecules (Supplementary Fig. S1). The relaxation rate from S2 to the photoluminescent level S1 was found to be 21 = 1/190 fs-1, which is in line with previous ensemble pump-probe data30. We also successfully applied this minimal model to other single molecules featuring a symmetric chirpdependent response (see Supplementary Fig. S4 and Table S1 for further examples). From those data sets we retrieved different values for the free parameters. This variability demonstrates the strong electronic heterogeneity from molecule to molecule at room temperature. Each individual molecule features its distinct intrinsic excited state energy landscape and dephasing/relaxation dynamics, determined by electrostatic interactions with the specific local environment. In this context, we emphasise that in our proof-of-principle study the aim is not to provide a large statistics of parameters from many molecules, because these parameters are already known for our model system MeLPPP, as discussed above. Our experiment combined with theoretical simulations aims at showing that phase-sensitive quantum coherent phenomena in single organic molecules are observed and that the underlying non-resonant two-photon transitions can be identified with a minimal model. Remarkably, for a considerable fraction of single molecules we recorded chirp-dependent PL traces that are distinctly different from the symmetric ones shown in Fig. 3 and Supplementary Fig. S4. These particular traces are clearly asymmetric, see Fig. 4a-c for representative examples from three different molecules. In Fig. 4a the maximum of the PL appears around  = 0 fs2, yet, the decrease of the PL is steeper for negative chirp . The traces in Figs. 4b, c exhibit the opposite behaviour with a steeper decrease for positive values of . Additionally, the PL signal in Fig. 4c features its maximum at negative chirp (at about  = -240 fs2). Reproducing such asymmetric traces with the three-level model leads to clear systematic deviations between data and simulations, see Supplementary Fig. S7. We therefore suggest that further quantum dynamical processes beyond two-photon transitions play a role.


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We therefore extend our model to allow for transitions into a highly excited singlet state, labelled S3, with an energy of E3 ~ 39000 cm-1, which is accessible from the two-photon allowed state S2 via a further one-photon transition (Fig. 4g). That is, resonance-mediated (2+1)-photon excitations32 from the ground state S0 to S3 may take place. Excellent agreement between experiment and simulation is obtained, see Figs. 4a-c (solid lines), by introducing only three additional free parameters: The energy E3 and the dephasing rate 3 of the highly excited state S3, as well as the one-photon Rabi-frequency 1P coupling states S2 and S3. From PL quantum yield measurements we found that state S3 is deactivated essentially entirely by fast nonradiative internal conversion processes directly to the ground state (see Methods section). Hence, population of state S3 represents a loss channel for the PL signal, and we can neglect the population relaxation from S3 to S2 and S1 in this extended model. For the data shown in Fig. 4 all parameters retrieved from the simulations are listed in the Supplementary Table S2. We find that the energy of the two-photon accessible state S2 is distributed around 25827 cm-1, which is similar to the molecules with symmetric response. The energy of the highly-excited electronic state S3 is about 38796 cm-1, which is nearly exactly the energy of one laser photon above state S2. Thus, both resonant and near-resonant (2+1)-photon transitions are possible due to the broad bandwidth of the laser pulse, and interference between all possible pathways can occur. This leads to substantial averaged one-photon Rabifrequencies 1P of 285 cm-1 for the S2 → S3 transition, and consequently to reasonably high population probabilities of up to 30 % for the highly excited state S3 (Figs. 4d-f). The dephasing rates for states S2 and S3 are very similar and distributed around an average value of 1/71 fs-1, and the relaxation rates from the two-photon state S2 to the photoluminescent level S1 are spread around 1/154 fs-1, in agreement with the numbers determined for molecules with symmetric response. To gain deeper insight into the processes that give rise to the peculiar asymmetry in the PL traces, we depict in Figs. 4d-f the simulated chirp-dependent populations of all involved electronic states. Notably,



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the electronic ground state is depopulated symmetrically around the maximum depletion at 0 fs2 (black curves). The very first step in the light-molecule interaction is therefore a non-resonant two-photon transition into state S2, in analogy to the molecules with a symmetric response. The populations of the excited states (blue-green and purple curves in Figs. 4d-f), however, exhibit a mutually opposite asymmetry. After the chirp-dependent two-photon transition into S2, further chirp-dependent population transfer into the highly excited state S3 takes place (which is then deactivated non-radiatively). This latter population transfer depends sensitively on the interplay between the instantaneous frequency of the

driving

field,

the

energy

mismatch

E3 – E2, as well as the one-photon Rabi-frequency 1P coupling S2 and S3 (Supplementary Fig. S5).

Conclusions Our results demonstrate the first proof of genuinely active coherent control of single organic molecules at room temperature. Specifically, we were able to control population transfer into highly excited states by exploiting quantum interferences of two- and (2+1)-photon transition probabilities, which result in distinctly different shapes of chirp-dependent PL traces from single molecules. This variability in responses result from distinct ultrafast dynamics and energy landscapes due to varying interactions with the specific local environment for each molecule. Although the dynamics and energy landscapes of such molecular systems can in principle also be retrieved e.g. by (PL-detected) 2-dimensional spectroscopy,33–36 this method has not yet reached single-molecule sensitivity and therefore cannot reveal the variability of parameters found here. Notably, the ensemble response measured on an MeLPPP film under experimental conditions, that are identical to those for single molecules, is perfectly symmetric (Supplementary Fig. S6) and might naively be ascribed to pure two-photon S0 → S2 processes. However, in an ensemble the conditions for (2+1)-photon transitions, i.e., for the appearance of asymmetric traces, are always fulfilled for a sub-set of molecules. Hence, our single-


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molecule data clearly demonstrate that in an ensemble asymmetric chirp-dependent behaviours are averaged out, proofing the strength of our single-molecule approach. The combination of ultrafast single-molecule spectroscopy with quantum process identification techniques opens up new avenues to extract and control a priori unknown quantum dynamical processes in single quantum systems embedded in disordered environments. We expect that our study will stimulate further theoretical and experimental work on more complex systems, e.g. to study (ultrafast, coherent) energy transport in single artificial37, biological supramolecular systems38 and quantum dots39, to control photochemical reactions (e.g. in the presence of a catalyst), and to understand and manipulate biologically relevant processes (e.g. photoisomerisation) on a truly molecular level. Moreover, entanglement between molecules can be studied,40,41 because we can access the information from both population dynamics and the evolution of quantum coherences. Despite the very short dephasing times of < 100 fs we are still able to perform rotations on the molecular Bloch-vector (see the oFID in Fig. S1). This can be interpreted as performing quantum logical operations on a single qubit3,9. In this sense, our approach is not only a powerful identification tool but also capable of ultrafast quantum optical operations. Combined with optimal control theory full control of ultrafast light-induced processes in single nanoscale objects may become possible,11,42,43 because shaping of ultrashort pulses allows controlling the interference between different quantum pathways.1,2,14,44

Methods Materials and experimental methods. For the single molecule experiments, MeLPPP (molecular weight Mn = 55300 Da) was dissolved in toluene (Sigma-Aldrich, 99.7%) at a concentration of 10-9 M containing 5 mg/ml polystyrene. Thin films with a thickness of several 100 nm were prepared by spincoating this solution onto a freshly cleaned quartz glass cover slip. The excitation source is a Ti:Sa-



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oscillator (Griffin-10- WT, KMLabs) with a repetition rate of 80 MHz. The laser spectrum is centered around 770 nm (13000 cm-1) and has a bandwidth of about 20 nm (400 cm-1) corresponding to a Fourier-limited pulse width of 40 fs. Pulse characterisation is provided in the Supporting Information, Fig. S8. Dispersion compensation and spectral phase modulation is achieved by a pulse shaping system (MIIPS HD, Biophotonic Solutions). An air objective (NA 0.6) focusses the excitation into the sample plane. The time-averaged excitation power was around 3 mW. The PL is collected in transmission with a second air objective (NA 0.85) and is filtered by colour glass (BG 39, Schott) and dielectric shortpass filters (SP492, AHF) to suppress excitation light. The PL is then detected with a single-photon sensitive detector (PDM, Micro Photon Devices) and further processed by home-written software. For the measurements in Figs. 3, 4 and Supplementary Fig. S4 the pulse shaper modulated a quadratic spectral phase function φ(ω) = β(ω − ω0)2/2 onto the laser pulse (see eq. (2) below). The linear spectral chirp β was varied from -3000 to 3000 fs2 in steps of 120 fs2, while keeping the spectral amplitude A(ω) unchanged. All experiments were carried out at room temperature under ambient conditions. To demonstrate that for the chosen concentration of 10−9 M for MeLPPP in PS we are indeed at the single-molecule level, we show in Fig. S9 two PL traces that exhibit the characteristic digital blinking between an “on”-level and the background “off”-level. Calculations on laser power fluctuations (∼ 1 % RMS), relative photon shot noise (∝ 1/√N, with N being the number of detected photons) and baseline fluctuations of the PL give rise to an error of ±42 cps. To demonstrate that excitation into the high-lying excited state S3 indeed represents a loss channel for the PL signal, we measured the PL quantum yield of MeLPPP dissolved in toluene upon excitation at 39200 cm-1 (about 255 nm) using an integrating sphere (JASCO FP-8600, ILF- 835). We found a quantum yield of less than 1 %, which clearly demonstrates that S3 is deactivated essentially entirely by fast non-radiative internal conversion processes directly to the ground state.


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Finally, we want to highlight at this point, that all data displayed in this work is real raw data without any background correction. This demonstrates the absolute magnitude of the induced, observed effects in comparison with the background and noise. Although background should not be evident in two-photon measurements, at this very low light level, the dark counts of the detector and weak autofluorescence of the optical elements can not be avoided. Theoretical methods. We develop a quantum dynamics identification (QDI) procedure to extract information from the typically noisy PL-response of single molecules based on a minimal quantum mechanical model. In our procedure, the density matrix ρ(t) of single molecules in an N-dimensional Hilbert space is described by using the Lindblad master equation 45

𝑑 ―𝑖 𝜌(𝑡) = 𝐻 ― 𝐻𝑐(𝑡),𝜌(𝑡) + 𝑑𝑡 ℏ 0

[

]

𝑁―1

𝑁―1

∑ ℒ [𝜌(𝑡)] + ∑ 𝐷 [𝜌(𝑡)], 𝑖𝑗

𝑖≠𝑗=0

𝑖

(1)

𝑖=0

where the molecular Hamiltonian operator reads 𝐻0 = ∑𝐸𝑗|𝑗 >< 𝑗|. The braket |j > stands for the eigenvector of state Sj with eigenenergy Ej, and𝐻𝑐(𝑡)denotes the Hamiltonian operator for the interaction between the laser field and the molecule. Without using the rotating wave approximation (RWA) the two-photon transition from state Si to state Sj is described by 𝐻𝑐2(𝑖,𝑗) = 𝐷2𝑖𝑗ℜ[𝐸2(𝑡)]with the effective two-photon transition dipole moment Dij; the corresponding one-photon transition is described by 𝐻𝑐1(𝑖,𝑗) = 𝜇𝑖𝑗ℜ[𝐸(𝑡)]with the one-photon transition dipole moment μij.46 In the frequency domain the field of the laser pulse reads E(ω) ≡ A(ω)eiφ(ω) with the experimentally measured spectral amplitude A(ω) and the spectral phase φ(ω). For a chirped quadratic spectral phase being used in our experiments, the complex time-dependent electric field of the laser pulse at a given chirp β is calculated from the spectral amplitude as



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1 𝐸(𝑡) = 2𝜋

∫

∞

𝐴(𝜔)𝑒

𝛽 𝑖 (𝜔 ― 𝜔0)2 ―𝑖𝜔𝑡 2

𝑒

(2)

𝑑𝜔

―∞

Furthermore, we define the two-photon Rabi frequency between the states Si and Sj as Ω2P = Dij2 E02 and the one-photon Rabi frequency as Ω1P = μijE0 with the peak strength of the laser field E0 at β =

0 fs2. The Lindblad superoperators Lij = Γij(ρii|j> 0 in the parameter space (not necessarily to the global minimum); see Supporting Information, Fig. S3, for more information. In the simulations the time-dependent populations are simulated for each chirp β. To calculate the chirp-dependent traces (e.g. Figs. 3, 4), a snapshot of the populations of all states is taken 1000 fs after the excitation pulse (which is then ”compared” to the experimental traces as outlined above). Owing to the fast population relaxation rate of about 1/150 fs-1 from state S2 into the photoluminescent state S1, the initial population created by two-photon processes in S2 already decayed completely after 1000 fs. To emphasise that the initial population in state S2 contributes to the PL signal, we plot in Figs. 3 and 4 and Supplementary Fig. S4 the simulated curve as blue- green lines, i.e., in the colour code for levels S1 and S2. The initially created population in S2 is directly proportional to the detected PL signal, with the proportionality constant being determined e.g. by the PL quantum yield of state S1, the detection efficiency of our setup, and the relative orientation of the polymer backbone to the polarisation of the exciting laser field.

Supporting Information Additional ensemble and single-molecule experimental measurements; theoretical analysis and simulations; chirp characterisation of the laser pulse.



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Acknowledgement C.C.S. greatly appreciates helpful discussions with Yu Pan and Daoyi Dong. K.W. and R.H. acknowledge financial support from the German Research Foundation (DFG) through projects HI1508/3 and GRK1640. R.H. is also supported by the Elitenetzwerk Bayern (ENB, elite study programme Macromolecular Science). C.C.S. acknowledges financial support from the National Natural Science Foundation of China (NSFC) under Grant No. 61803389. K.W. acknowledges the hospitality at the University of New South Wales during his visit.

The authors declare no competing financial interest.


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Figures

Figure 1: Concept of photoluminescence-detected multi-photon coherent control and quantum dynamics identification of single organic molecules. A Ti:Sapphire oscillator is combined with a pulse shaper to excite single methyl-substituted ladder-type poly(para-phenylene), MeLPPP, molecules embedded in a polystyrene (PS) matrix (top left). The efficiency of multiphoton transitions is controlled by varying the spectral phase  of the laser field, while leaving its spectral amplitude A() unchanged (top right). The measured chirp-dependent photoluminescence signals are sent to a quantum dynamics identification (QDI) procedure (bottom left), which allows to retrieve intra-molecular dynamics and energy landscapes for each single molecule (bottom right). Si (i=0,1,2,3) denote the relevant electronic states of MeLPPP. The red arrows indicate multi-photon excitation into higher electronic states, the grey arrows represent nonradiative internal conversion processes, and the green arrows display photoluminescence.



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Figure 2: Spectral characteristics of MeLPPP and the excitation laser. Photoluminescence (green) and linear, one-photon absorption (black solid) of MeLPPP dissolved in toluene. The chemical structure is shown top left (R1: n-hexyl; R2: methyl; R3: 1,4-decylphenyl). Inset: Photoluminescence-detected two-photon excitation spectrum of MeLPPP (black solid) and normalized power spectrum of the excitation pulse (red filled area). The axes are colour coded accordingly: The top red scale represents the absolute energy scale for the excitation pulse. The bottom black scale is the absolute spectral position of the two-photon absorption of MeLPPP (which is the doubled energy scale as that for the excitation pulse).


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Figure 3: Coherent control of two-photon transitions in a single molecule. a: Photoluminescence signal as a function of spectral chirp  (open circles) overlaid with the result of the QDI simulations (blue-green line). Inset: Chirp-dependent populations (Pop.) of the ground state S0 (black) and of the excited states S1 and S2 (blue-green line). cps: counts per second; the error of the measurement is ± 42 cps. b: Schematic illustration of the three-level model used for the QDI simulations. 2P denotes the two-photon Rabi-coupling between S0 and S2, 2 is the electronic dephasing rate in S2, and 21 is the non-radiative relaxation rate from S2 to the photoluminescent (PL) state S1.



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Figure 4: Coherent control of multi-photon transitions into highly excited electronic states. a-c: Chirp-dependent PL signals for three different single molecules (open circles) and the theoretical QDI simulations (blue-green lines). cps: counts per second; the error of the measurement is ± 42 cps. Dashed lines: guide to the eye to demonstrate the asymmetry of the PL. d-f: Simulated chirp-dependent populations of all involved electronic states. g: Schematic illustration of the extended four-level model used for the QDI simulations. Here, 1P is the onephoton Rabi-frequency for the transition S2 → S3,  the dephasing rate of state S3. The population relaxation from S3 to S2 (and S1) is ignored in the simulations, see text.


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Table of Contents Graphics


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