Two-photon induced ultrafast coherence decay of highly excited states in single molecules

Coherence is a key aspect of a large variety of processes, ranging from the coherent delocalisation of excitation energy, which is important for energy transfer in supramolecular nanostructures, to coherence between electronic states of a single quantum system, which is essential for quantum optical applications. Coherent control schemes exploit this quantum mechanical property by actively manipulating the outcome of dynamical processes. Moreover, this technique allows measuring dynamical processes under the influence of dephasing. However, going beyond the ensemble averaged situation, i.e. working on the level of single quantum systems, is highly challenging for quantum systems embedded in a solid matrix at elevated temperature. Since interactions between the quantum system and its specific local environment are a priori unknown, this requires a reliable approach to retrieve the relevant parameters governing the ultrafast coherent dynamics. Here, we present measurements of the ultrafast coherence decay of two-photon accessible excited states in single organic molecules embedded in a disordered environment at room temperature. We combine this experimental approach with a quantum dynamics identification procedure, which yields a minimum three-level model to describe the obtained data with very good agreement. In particular, we are able to retrieve the ultrafast (coherent) excited state dynamics in single molecules and demonstrate its sensitivity to the local nanoenvironment from molecule to molecule. This work provides a robust approach to measure and analyse ultrafast quantum dynamics in complex nanosystems.


Introduction
To investigate as well as to manipulate ultrafast dynamical processes in nanoscale objects, e.g. single molecules or (in)organic nanostructures, coherent control methods provide capabilities beyond conventional spectroscopic methods [1][2][3][4][5][6][7][8][9]. Coherent control refers to the active steering of quantum mechanical processes towards a final desired state. This approach involves either the suppression or enhancement of transition probabilities within the system investigated by quantum mechanical interference. Hence, quantum processes that can be addressed with coherent control schemes take place on time scales shorter than the effective dephasing time T 2 [7,8,10]. Since in condensed phase at room temperature these time scales are often of the order of some tens of femtoseconds, ultrashort pulsed excitation schemes are required. A further difficulty arises, because in condensed phase the local environment, and thus the dephasing time, is specific for each individual molecule or nanostructure. To understand the intrinsic coherent dynamics on a nanoscale level, ensemble averaging has to be avoided to obtain information about the decoherence properties of each individual system. A combination of coherent control schemes with single-molecule techniques is therefore required [11]. Most coherent control experiments are based on multi-photon processes [12][13][14], for which high photostability of the system is a prerequisite owing to relatively high excitation powers. It is thus crucial to develop a reliable technique to induce multi-photon processes in single complex quantum systems, measure them reproducibly, and furthermore analyse the results with high confidence.
Very recently, we have demonstrated that photoluminescence-detected two-photon coherent control schemes can be applied to single organic molecules embedded in a disordered matrix at room temperature [5]. We combined a coherent control experiment on single molecules, exploiting phase-shaped single-pulse excitation schemes, with a quantum dynamics identification (QDI) procedure. This approach allowed us to retrieve the excited state energy landscape for each molecule. Moreover, we were able to resolve transitions into a 'hidden' highly excited state that is not connected to a photoluminescent channel.
Here, we focus on the ultrafast coherence dynamics of a two-photon accessible excited state in single molecules at room temperature. We employ amplitude-shaped double-pulse sequences to measure the twophoton induced coherence decay of highly excited states of the conjugated polymer methyl-substituted laddertype poly(para-phenylene), MeLPPP. Analysing these data with the QDI procedure, we find that in single molecules the high-lying excited state features a rapid coherence decay with an average dephasing time of 75 fs. In addition, we demonstrate that from this two-photon allowed state rapid population relaxation within about 150 fs occurs into the photoluminescent lowest excited singlet state. The distributions of both times are widely spread due to varying local interactions between each molecule and its specific disordered matrix.

Materials and methods
The detailed concept of our experimental and theoretical approach was already introduced in [5]. Singlemolecule samples were prepared by dissolving MeLPPP (molecular weight M n =55300 Da) in toluene (Sigma-Aldrich, 99.7%) at a concentration of 10 −9 M containing 5 mg ml −1 polystyrene. Spincoating of this solution onto a freshly cleaned quartz cover slip yields samples with thickness of several 100 nm and spatially wellisolated individual MeLPPP-molecules. In our single-molecule experiment we exploit that, owing to the laddertype backbone, MeLPPP possesses a very rigid structure. Thus a high degree of symmetry is preserved even in a solid matrix. Hence, the electronic states of MeLPPP possess alternating even and odd parity, resulting in mutually exclusive one-photon and two-photon allowed transitions from the electronic ground state S 0 into excited electronic states [15,16]: One-photon allowed (but two-photon forbidden) optical transitions occur into the lowest excited state S 1 , and two-photon (but one-photon forbidden) transitions can be induced with high efficiency into the higher lying excited state S 2 , respectively (see figure 1(a)). After two-photon excitation into S 2 rapid population relaxation into S 1 takes place within typically 150 fs, from where photoluminescence can be detected, i.e. the photoluminescence from single MelPPP molecules is directly proportional to the two-photon transition probability from S 0 into S 2 .
We use a Ti:Sapphire oscillator (Griffin-10-WT, KMLabs) to generate ultrashort pulses with a Fouriertransform limited width of about 40 fs (400 cm −1 or 20 nm spectral bandwidth). Its output spectrum is centred at a photon energy ω 0 =13000 cm −1 (770 nm), which allows to induce non-resonant two-photon transitions into the second excited state S 2 of MeLPPP located at about 26400 cm −1 . A pulse shaper (MIIPS-HD, Biophotonic Solutions) is used to compensate dispersions induced by the optical elements [17,18], see appendix A for the pulse characterisation. A pair of identical time-delayed, phase-locked pulses is created via simultaneous amplitude and phase shaping by a liquid crystal display spatial light modulator using homewritten modulation masks. The applied modulation reads A cos 0.5 }|for the amplitude and 0.5 sgn cos 0.5 for the phase mask, where sgn is the sign function, Δf=0 and the time delay τ was varied between 0 and 300 fs in 6 fs steps. For each measurement the time-averaged excitation power at the sample plane was constant at 2.5 mW for all delays. These pulse pairs are directed to the sample and focussed onto single MeLPPP molecules by an objective with low numerical aperture (NA=0.5). The photoluminescence signal is collected in transmission by a second objective (NA=0.85) and separated from the excitation light by two BG 39 colour glass filters (Schott) and one shortpass filter (filter edge at 492 nm, AHF). Finally, the delay-dependent photoluminescence is detected with a single-photon sensitive photodiode (PDM, Micro Photon devices). The coherence decay was measured by varying the delay from 0 fs to 300 fs, while keeping the inter-pulse phase difference constant at Δf=0 rad. All experiments were carried out at room temperature under ambient conditions.
The obtained delay-dependent photoluminescence signals are analysed by our QDI procedure, which yields information about the energy landscape of each molecule and the ultrafast relaxation and dephasing dynamics of the quantum states involved. In this present work, the simulations yield a minimum three-level model, that consists of S 0 , S 1 and S 2 with corresponding energies E 0 , E 1 and E 2 . To describe the underlying quantum processes we model single molecules as open quantum systems, including the dissipative influence of a non-Markovian local environment. In this situation the time-dependent evolution of the density matrix of the three states is calculated by the Lindblad master equation [19] t where D 12 =D 21 denotes the two-photon transition dipole moment. The first summand on the right-hand side in this master equation reflects the interaction of the external time-dependent electric field with the two-photon dipole moment of the quantum system. The time-dependent electric laser field E(t) can be obtained by with the experimentally measured amplitude A(ω) of the laser spectrum, which leads to the two-photon Rabi frequency Ω 2P =D 2 12 ε 2 0 with the peak strength of the electric field ε 0 . The Lindblad operator L 21 = Γ 21 (ρ 22 |1><1|−1/2{|2><2, ρ}) describes the population relaxation from S 2 to S 1 with a rate of Γ 21 , which is typically of the order of 100 fs for highly excited states in organic molecules [20]. The superoperator D 2 =γ 2 (ρ 22 |2><2|−1/2{|2><2|, ρ}) describes dephasing of state S 2 with a rate of γ 2 , which is usually some tens of femtoseconds at room temperature due to interaction with the disordered local environment [11]. The brackets |1> and |2> represent the eigenvectors of S 1 and S 2 , respectively. Since the local surrounding of single molecules varies from molecule to molecule, the corresponding variations in interactions with the environment will lead to a distribution of the parameters Ω 2P, E 1 , E 2 , Γ 21 and γ 2 . To this end, a multi-parameter optimisation problem is solved to identify a set of these free parameters within the minimal three-level model. More details of the QDI procedure can be found in [5]. To calculate the coherence decays, a snapshot of the populations and coherences is taken 400 fs after the second pulse in each sequence. In analogy to our recent work [5], in figures 2(a)-(c) the solid lines represent the sum of the populations in states S 2 and S 1 owing to the fast population relaxation from S 2 to S 1 .

Results and discussion
The idea for the detection of quantum coherence between the electronic ground state S 0 and the two-photon accessible, second excited state S 2 of single MeLPPP molecules is shown in figure 1(b). We exploit constructive quantum interference between the excitation pathways generated by the sequence of phase-locked pulse pairs (Δf=0) [7]. The interaction with the first excitation pulse induces population transfer from S 0 to S 2 via a twophoton process and creates quantum coherence, i.e. a superposition of wave functions of the states S 0 and S 2 . The phase of the wave function describing this superposition state is determined by that of the first excitation pulse. In the time interval between the pulses, this coherence will decay (the phase memory will be lost) due to electronic dephasing processes caused by interactions with the local surrounding. If for a given delay τ coherence still persists, the second, phase-locked pulse increases the excited state population of S 2 by constructive quantum mechanical excitation pathway interference. If the delay exceeds the coherence time 1/γ 2 , however, phase coherence in the system is completely lost, excitation pathway interference does not take place any more, and the excited state population converges to the incoherent level. This decay of coherence and thus of population transfer into state S 2 as function of the delay is then reflected in a decay of the detected PL signal of each molecule. Analysis of this delay-dependent decay of the PL signal allows to determine ultrafast electronic dephasing and population relaxation dynamics in highly excited states of single molecules at room temperature.
Examples of measured coherence decays are shown in figures 2 (a)-(c) for three different single MeLPPP molecules 1, 2, and 3, respectively. The detected PL-signal decays with increasing delay to a constant value after τ=50-100 fs. This observation indicates a rapid loss of coherence between S 0 and S 2 due to the dephasing influence of the environment combined with fast population relaxation from S 2 to the photoluminescent state S 1 [21,22] (see also below). To demonstrate that the recorded delay-dependent PL traces can indeed be attributed to the coherence decay, i.e. to molecular dynamics, we further plot the envelope of the collinear intensity autocorrelation of our pulse (grey filled area in figure 2(a)), which is clearly shorter than the measured PL decay (see the collinear autocorrelation function in appendix A, figure A1). A further control experiment is described in appendix B (figure B1), where we depict the phase-dependent variation of the two-photon induced PL of MeLPPP at a constant delay time of 70 fs, i.e. after complete decay of the pulse autocorrelation function.
We now use the QDI procedure with the minimal three-level model to reproduce these measured delaydependent traces in figures 2(a)-(c). To perform the simulations, the energy of the electronic ground state S 0 was set to E 0 =0 cm −1 and that of state S 1 was kept constant at E 1 =22000 cm −1 , which is justified because S 1 is not optically addressed by the laser pulses. Initial values in the QDI procedure for the free parameters were γ 2 =1/75 fs −1 for the dephasing rate, Γ 21 =1/150 fs −1 for the population relaxation, and E 2 =26000 cm −1 for the energy of S 2 . Moreover, the QDI was carried out for a Rabi-coupling fixed at Ω 2P =120 cm −1 for the S 0 →S 2 transition, because we found that the simulations are not sensitive to the precise value of Ω 2P . Minimising the residuals between data and simulations the QDI procedure identifies the optimised parameters for this minimal model. We find very good agreement between the measured PL and the output of the QDI procedure for the optimised values that are given in the caption of figures 2(a)-(c) for each molecule. For molecules 1 and 2 dephasing rates are around 1/75 fs −1 , relaxation rates of approximately 1/150 fs −1 were retrieved, and we find near-resonance of E 2 with the doubled excitation energy of 26 000 cm −1 . Molecule 3, in contrast, exhibits dynamics considerably different with a faster dephasing rate of 1/52 fs −1 and a slower relaxation rate of 1/264 fs −1 . These examples show that all free parameters strongly vary between different measurements, and that the disordered local surrounding of each molecule has a strong impact on the ultrafast response.
Since within the QDI procedure the molecular dynamics is described by the Lindblad master equation, we have direct access to all time dependent density matrix elements ρ 00 and ρ 22 (populations), as well as ρ 02 and ρ 20 (coherences) for states S 0 and S 2 . The coherence term, defined here as y i , 20 02 ) is shown as function of the delay τ in figures 2(d)-(f). For all molecules the decay of the coherence resembles the decay of the excited state population with a rate of γ 2 (see figure caption). This behaviour further corroborates that the loss of coherence between the S 0 and S 2 wave functions is the predominant process causing the PL decay shown in figures 2(a)-(c). Population relaxation from S 2 to S 1 takes place on slower timescales and thus has a much smaller effect on the overall shape of the PL signals.
The time-dependent information about the density matrix elements allows us to visualise the molecular dynamics by the motion of the Bloch vector on the Bloch sphere during interaction of the molecule with a pair of time-delayed, phase-locked pulses. Using the coherence y introduced above and the population term, defined as z , 22 00 )the position of the tip of the Bloch vector projected onto the y z plane can be calculated. The z-component describes the population of state S 2 and dynamics parallel to the y-axis (at constant z) are due to dephasing effects. For molecule 1 this Bloch vector trace is plotted in figure 2(g) for two different delays τ=0 fs and 120 fs (marked with the coloured arrows in figure 2(a)). At a delay of τ=0 fs the trace of the Bloch vector tip shows that a maximum of 20% of population is transferred from the ground state to S 2 (turquoise curve). At τ=120 fs dephasing in the time interval between the pulses already affects the overall population transfer from S 0 to S 2 , see the intermediate 'horizontal' motion of the Bloch vector tip in the magenta curve, inset figure 2(g). Hence, the population transferred from state S 0 to state S 2 decreases with increasing delay due to the loss of coherence in the time interval between the pulses.
We note, however, that the Bloch plot cannot reproduce all details, e.g. the measured contrast of 2 between the PL signal (and thus S 2 population) at 0 fs delay (turquoise) and e.g. at 120 fs (magenta). This discrepancy is due to the fact, that the populations shown in (a)-(c) are the sum of populations of states S 1 and S 2 , because all population created in S 2 will relax into photoluminescent state S 1 within the excited state lifetime of S 1 (> 100 ps). In contrast, the Bloch plot reflects the dynamics only of state S 2 , and efficient and fast S 2 →S 1 relaxation with time constants of ∼150 fs is not reflected. Nevertheless, this Bloch representation clearly demonstrates that two-photon induced quantum optical operations on femtosecond time scales on a single molecule are feasible.
The QDI analysis was carried out on the measured coherence decays of 30 MeLPPP molecules and the obtained results are summarised in the histograms in figure 3. The maximum of the energy E 2 of the two-photon accessible state S 2 is spread around 26 000 cm −1 with a width of 400 cm −1 . Note that the width of this histogram does clearly not reflect that of the actual distribution of energies of S 2 , but rather the laser bandwidth. The actual distribution of E 2 can be expected to be much broader with a FWHM of around 800 cm −1 as estimated from ensemble two-photon spectra [15]. The narrow distribution about the doubled laser energy is reasonable, however, because in single-molecule experiments usually only those emitters will be detected with the highest PL signal, which requires most efficient excitation conditions, i.e. we detect only the 'most resonant' molecules. Furthermore, we find that the dephasing time is spread between 50 and 120 fs with a maximum around 75 fs. These dephasing times are in agreement with those obtained by pure phase-dependent measurements of the same conjugated polymer as we have reported recently. Moreover, these are in line with the line widths of single MeLPPP molecules at room temperature [21], although this line width was determined from the PL spectra from state S 1 and not for state S 2 . Finally, the distribution of time constants for the relaxation from S 2 to the photoluminescent state S 1 ranges from 100 to 260 fs, which is distributed around the value of 150 fs determined by ensemble pump-probe experiments [22]. This population relaxation-a process referred to as internal conversion-depends on the overlap of vibrational wave functions of the vibrationless state S 2 , which we directly excite by the two-photon process, with an accepting, isoenergetic vibrational mode in S 1 , and on the energy difference between S 2 and S 1 (the so-called energy gap law) [20]. The observed large variations in relaxation times are therefore in line with the distribution of the energy of state S 2 .

Conclusion
We demonstrated the first ultrafast, two-photon induced measurements of the coherence decay of individual organic molecules that are embedded in a complex, disordered environment at room temperature. We note that based on the ergodic principle the coherence decay detected here for single molecules can be identified with the envelope of the optical free-induction decay, as discussed in detail in [7]. Our data were analysed using a QDI procedure, which revealed a strong variation of photophysical parameters from molecule to molecule. In particular, we revealed the dephasing time of a higher excited state as well as the relaxation time into the lower lying photoluminescent state. The broad distribution of dephasing and relaxation times found here is largely determined by the strongly varying interactions between each molecule investigated with its local, disordered environment. These changes in interactions strongly impact on the energy landscape as well as on the ultrafast intra-molecular dynamics. Our combined experimental-theoretical approach represents a powerful, reliable tool to study ultrafast dynamics of nanostructured systems in complex surroundings under ambient conditions. In order to proof the existence of coherence inherent to MeLPPP we show in figure B1 the ensemble coherence decay of MeLPPP dissolved in toluene at room temperature. The PL intensity rapidly decays within the first 50 fs and exhibits a strong periodic modulation on top of the trace for delays up to 200 fs (Δf=0 rad). This periodic modulation is due to the detuning between the doubled centre frequency of the excitation pulses (2·13000 cm −1 ) and the ensemble absorption of the two-photon allowed state S 2 (26 380 cm −1 [15]). The existence of this beating at delays longer 50 fs is already a strong indication for the long living coherence. This is further corroborated by the measurement shown in the inset, for which we varied the inter-pulse phase difference Δf in steps of 0.12 π at a fixed delay time of τ=70 fs. These data feature a clear contrast of almost 40% between the maximum and the minimum signal (note that the non-sinusoidal variation stems from the detuning). At a delay of 70 fs possible artefacts due to intensity autocorrelation of the excitation can be excluded ( figure A1(c)) and all observed effects can be ascribed to intra-molecular properties.
The red dashed lines in figure B1 represent the normalised time-averaged excitation power for each measurement that was monitored simultaneously.