## 4 Discussion: Probing the Coherent Intradot and Interdot Dynamics in Dimer Ensembles  
The coherences built by the first pulse can be probed by transient absorption (TA) spectra using a second pulse. We compute the transient absorption spectrum as the linear response function, \(S(\omega)\), of the time-dependent oscillating dipole resulting from the coherent superposition of states that has been built by the pump pulse [86; 87]. In the eigenstate basis, the time-dependent dipole of each dimer takes the following form:  
\[\mu(t)=\text{Tr}[\hat{\mu}\rho(t)]=\sum_{k,l}c_{k}^{*}(t)c_{l}(t)\mu_{kl}=\sum _{k,l}\rho_{kl}(t)\mu_{kl} \tag{10}\]  
where \(\mu_{kl}\) are the transition dipoles in the eigen exciton basis and the \(c_{k}(t)\)'s are the time-dependent amplitudes on the eigenstates (Equation (8)). Because of the electronic coherences, \(\rho_{kl}(t)=c_{k}^{*}(t)c_{l}(t)\)  
Figure 5: Dynamic of populations and coherences computed for a single heterodimer (D\({}_{\text{A}}\) = 2.7 nm and D\({}_{\text{B}}\) = 3.5 nm, level structure shown in Figure 4B) upon excitation by a sequence of the UV and the IR pulse with a delay time of 100 fs. (**A**). Population in the zero order excitons. (**B**) Population in the eigenexcitons. (**C**). Real part of the two intradot coherences. (**D**). Real part of the six interdot coherences between local excitons. GS, ground state. the dipole is non stationary and beats in time with periods that correspond to the coherence periods. The transient absorption spectrum, \(S(\omega)\), is defined as follows [82; 83]:  
\[S(\omega)=-2\mathrm{Im}[\mu(\omega)E^{*}(\omega)] \tag{11}\]  
where \(\mu(\omega)\) is the Fourier transform (FT) of the dipole given by Equation (10), including the interaction with the pump and the probe pulse; and \(E(\omega)\) is the complex conjugate of the FT of the time profile of the electric field of the two pulses, both computed for positive frequencies, \(\omega\). Because \(\mu(t)\) and \(E(t)\) are real, their FTs obey the symmetry properties \(\mu(-\omega)=\mu^{*}(\omega)\) and \(E(-\omega)=E^{*}(\omega)\). A positive value of the response function, \(S(\omega)\), corresponds to an absorption process, while a negative value corresponds to emission. Coherences with beating times spanning several orders of magnitude can be probed in the pump-probe transient absorption spectra. The beating frequencies of the coherences are defined by the energy differences between each pair of eigenstates, \(E_{i}-E_{j}\). The beating of the phase of the coherence can be followed through the absorption/emission signals recorded when scanning the delay time between the pump and the probe pulses. The transient absorption spectrum computed for an ensemble of 1000 hetero dimers QD\({}_{\mathrm{A}}\)-QD\({}_{\mathrm{B}}\) (mean D\({}_{\mathrm{A}}\) = 2.7 nm, mean D\({}_{\mathrm{B}}\) = 3.5 nm, size dispersion 5%) and L = 0.2 nm is shown in Figure 6, superimposed with the energy distribution of the eigen excitons of the ensemble shown in Figure 4B. It is computed for a value of the delay time between the pump and the probe pulses, \(\tau\) = 100 fs. The majority of the emission appears in the low energy side of each exciton band. When two identical VIS pulses (2.37 eV/570 THz) are used (panel A), the high transition frequencies in the 2 to 2.6 eV range, which corresponds to coherences between the local excitons and the GS, are resonant with the probe pulse and efficiently probed. However, the low transition frequencies between excitons are not resonant with the probe VIS pulse. An IR pulse (0.41 eV/100 THz) allows probing the exciton-exciton coherences, as shown in the inset of panel B. Because the carrier frequencies of the pump and the probe pulses used for computing the transient spectrum shown in panel B are markedly different, the signal in the high frequency range is the transient absorption spectrum induced by the pump, while the low frequency range shown in the inset is results from the probe. In panel A, the spectra attributable to the pump and the probe overlap.