Coherent Multidimensional Optical Probes for Electron Correlations and Exciton Dynamics : From NMR to X-rays

Linear-spectroscopy is one-dimensional (1D); the absorption spectrum provides information about excitation energies and transition dipoles as projected into a single frequency axis. In contrast, multidimensional optical spectroscopy uses sequences of laser pulses to perturb or label the electronic degrees of freedom and watch for correlated events taking place during several controlled time intervals. The resulting correlation plots can be interpreted in terms of multipoint correlation functions that carry considerably more detailed information on dynamical events than the two-point functions provided by 1D techniques1–7. Correlations between spins have been routinely used in NMR to study complex molecules. The Nobel prize was awarded to Richard Ernst8 for inventing the technique and to Kurt Wuthrich9 for developing pulse sequences suitable for large proteins. Optical analogues of 2D NMR techniques first designed to study vibrational dynamics by Raman or infrared pulses1 and later extended to resonant electronic excitations in chromophore aggregates10 have been made possible thanks to the development of stable femtosecond laser sources with controlled phases11. In an ideal heterodyne-detected 2D experiment (Fig. 1) 3 laser pulses with wavevectors k1, k2, k3 interact sequentially with the molecules in the sample to create a polarization with wavevector k4 given by one of the linear combinations ±k1 ±k2 ±k3. In all other directors the polarization vanishes due to the random phases of contributions from different molecules. The coherent signal is generated in directions close to the various possible k4. The missmatch caused by frequency variation of the index of refraction is optimized (“phase matched”) to generate an intense signal detected by interference with a 4th pulse at the desired wavevector k4. When the radiation field is described quantum mechanically the entire process can be viewed as a concerted 4 photon process. The signal S(t3,t2,t1) depends parametrically on the time intervals between pulses which constitute the primary control-parameters. Other parameters include the direction k4, pulse polarizations, envelope shapes, and even the phases. Figure 1 Scheme of the time-resolved four-wave-mixing experiment. All calculations are given for the three-band scheme shown on the bottom left. We shall illustrate the power of 2D techniques and how they work using the three-band model system shown in Fig. 1 which has a ground state (g), a singly excited manifold (e) and a doubly excited manifold (f ). The dipole operator can induce transitions between g to e and e to f . All transitions in the system are stimulated: spontaneous emission is neglected. This three-band model represents electronic excitations in the various physical systems covered in the this article. Multidimensional signals monitor the dynamics of the system’s density matrix during the time intervals between pulses. Diagonal elements of this matrix ρnn represent populations of various states, while the off diagonal elements ρnm (n ≠ m), known as coherences, carry additional valuable phase information. These signals can be described intuitively using the Feynman diagrams shown in Fig. 2 which display the Liouville space pathways: sequences of interactions with the various fields and the relevant elements of the density matrix during the controlled intervals between interactions6. The two vertical lines represent the ket (left) and the bra (right) of the density-matrix. Figure 2 Feynman diagrams for two 2D techniques with wavevectors kI and kIII. Incoming and outgoing arrows represent the interaction events, labels indicate states of the system during various intervals between interactions. ESA - excited state absorption, GSB ... Time runs from bottom to top and the labels mark the density matrix elements during the evolution periods between interactions. The arrows represent interactions with photons and are labelled by their wavevectors. Photon absorption is accompanied by a molecular excitation (g to e or e to f transition) whereas photon emission induces deexcitation (e to g or f to e). Our discussion will focus on two signals: the photon-echo SkI with kI = −k1+k2+k3 and the double-quantum-coherence SkIII with kIII =+k1+k2−k3. We first present the Feynman diagrams and the quantum pathways relevant for the two techniques for the generic exciton model of Fig. 1. Simulated signals are then presented for three different physical systems: Wannier excitons in semiconductor quantum wells12–15, Frenkel excitons in photosynthetic complexes6,7, and soft x-ray core excitons in molecules16–19. We demonstrate that both techniques provide new insights into the structure and exciton dynamics in semiconductor nanostructures and molecular aggregates and are highly sensitive to the separation between core-shells and the localization of the core-excited states. The three contributions to the SkI signal depicted in Fig. 2 are known as ground state bleaching (GSB), excited state stimulated emission (ESE) and excited state absorption (ESA)6. In the GSB pathway the system returns to the ground state (and described by the density matrix element ρgg), during the second interval t2, after interacting with the first two pulses. The third interaction is affected by the decrease of the ground state population which reduces (bleaches) the subsequent photon absorption. In the ESE pathway, the system resides in the singly-excited (e) manifold during t2 and the third interaction brings it back to the ground state by stimulated emission. The ESA pathway shares the same t1 and t2 history of the ESE, however the third interaction now creates a doubly-excited state f . The SkI signal is usually displayed as a frequency/frequency correlation plot SkI (Ω3,t2,Ω1) obtained by a double Fourier transform with respect to the time delays t1 and t3, holding t2 fixed. Ω3 and Ω1 reveal the various resonance transitions, as can be anticipated from the diagrams. Only single-exciton ωeg resonances corresponding to optical coherences ρeg show up during t1 and are projected onto the Ω1 axis. The Ω3 axis shows either ωe′g resonances (ESE, GSB) or ωfe (ESA). The t2 evolution reflects exciton populations ρee and intra-band single-exciton coherences ρee′. Population transport, coherence oscillations and spectral diffusion dominate this interval in the ESE and ESA paths6. Since the molecular frequencies during t1 (ωge) are negative and during t3 (ωeg and ωfe) are positive, the Ω1 frequency axis is reversed in the 2D plots. With this convention uncoupled excitons only show diagonal peaks. Off diagonal cross-peaks are markers of some kind of communication between various excitations which causes their resonance frequency to be different during t1 and t3. This can be attributed either to exciton delocalization or to population transport. A simple interpretation of the signals is possible by using a basis of states localized on the various chromophores. Since the dipole is localized on each chromophore and can only excite one chromophore at a time, cross peaks only appear when the chromophores are coupled. NMR spectra are similarly interpreted in terms of the couplings of localized spin states8. The couplings of chromophores can always be formally eliminated by diagonalizing the single-exciton Hamiltonian and switching to the delocalized exciton basis. However in this representation the dipole operator matrix elements will depend on the details of the eigenstates, which prevents the simple intuitive interpretation of the signal. The SkIII technique has two ESA-type contributions (Fig. 2). 2D spectra is obtained by either correlating t1 → Ω1 with t2 → Ω2, SkIII (Ω1, Ω2, t3), or t2 → Ω2 with t3 → Ω3, SkIII (t1,Ω2,Ω3). The density matrix evolution during t1 and t2 is identical for the ESA1 and ESA2 diagrams: single-exciton resonances corresponding to ρeg show up during t1. During t2 the system is in a coherent superposition (coherence) ρfg between the doubly-excited state f and the ground-state g. Two-exciton double-quantum-coherence resonances corresponding to the different doubly excited states f are then projected onto Ω2. The t3 evolution is very different: In ESA1 the system is in a coherence between f and e′ (ρfe′) which results in resonances at Ω3 = ωfe′, corresponding to all possible transitions between doubly- and singly- excited states. For ESA2 the system is in a coherence between e′ and g (ρe′g) and reveals single- exciton resonances at Ω3 = ωe′g as t3 is scanned. When the single-exciton states e and e′ do not interact (e.g. when they belong to two uncoupled chromophores), the corresponding two-exciton state is given by a direct product |f〉 = |ee′〉 and the double-excitation energy is the sum ef = ee + ee. In that case ωeg =ωfe′ =ee, the two diagrams exactly cancel and the signal vanishes! The entire SkIII signal is thus induced by correlations and its peak pattern provides a characteristic fingerprint for the correlated doubly excited wavefunctions. This conclusion goes beyond the present simple model. SkIII vanishes for uncorrelated many electron systems described by the Hartree Fock wavefunction and thus provides an excellent background-free probe for electron correlations12,20. The (Ω2,Ω3) correlation plots spread the two-exciton (f state) information along both axes, thus improving the resolution of the two-exciton manifold.