## 2 Materials and Methods: Model Excitonic Hamiltonian
The intradot couplings as well as in the interdot local-local and diagonal Coulomb coupling for the CT states fall in the range of 0.1 to 0.01 eV (800 to 80 cm\({}^{-1}\)). The local-local interdot couplings fall in the 10\({}^{-2}\) to 10\({}^{-3}\) eV range depending on QD size and QD surface-to-surface distance. It decreases very little in the small range of center to center interdot distance (\(r_{A}+L+r_{B}\)) spanned for the two values of \(L\) in Figure 3. This interdot coupling is of the order of the Coulomb coupling between zero order excitons localized on the same dot. On the other hand, interdot coupling strengths involving CT and local excitons are smaller, in the 10\({}^{-3}\)-10\({}^{-4}\) eV range, and fall exponentially with the interdot distance (see Figure S8 of the Section 2 of SI). The interdot CT-CT couplings are smaller than the local-CT ones (10\({}^{-4}\) to 10\({}^{-6}\) eV) and decay exponentially with interdot distance as well, except the coupling between states 5 (h1\({}_{\text{A}}\)-e\({}_{\text{B}}\)) and 6 (h2\({}_{\text{A}}\)-e\({}_{\text{B}}\)) and between 7 (h1\({}_{\text{B}}\)-e\({}_{\text{A}}\)) and 8 (h2\({}_{\text{B}}\)-e\({}_{\text{A}}\)), which are much larger. The reason is that, in each exciton involved in these particular coupling terms, the electron transfer goes in the same direction (from A to B for states 5 and 6 or for B to A for states 7 and 8, see Figure 2). In addition, these matrix elements involve an intradot Coulomb interaction between two hole states, which makes them stronger. As the energies of the two CT states involved in each pair differ by the intradot energy difference between the two hole states, even though the magnitude of the matrix element is large, this CT-CT coupling cannot lead to efficient coherent charge transfer in the dynamics. In general, because it is weak, the coupling between CT states leads to electronic coherences with very long periods, in the tens of ps. They are likely to be dephased by coupling to phonons and environmental effects, leading to trapping of charge and energy on a QD. To summarize, interdot electronic coherences with fs periods are essentially the result of Coulomb interactions between two local excitons or between a CT and a local one. We show below that they could be observed at the level of the ensemble, explicitly taking into account the size dispersion. Figure 3: Absolute values of the Coulomb coupling strengths computed for a hetero dimer QD\({}_{\text{A}}\)–QD\({}_{\text{B}}\) (D\({}_{\text{A}}\) = 2.35 nm, D\({}_{\text{B}}\) = 3.85 nm) on a logarithmic scale in eV for an ensemble of 1000 dimers with a size dispersion of 3% in diameter for two values of the surface-to-surface distance, L = 0.2 nm (**A**) and L = 0.55 nm (**B**). The black dots are the mean values and the bars the distribution of coupling strength values arising from the size dispersion. Note the wide range of couplings strengths, spanning about four orders of magnitude going from intradot coupling and interdot local–local couplings to interdot couplings between charge transfer (CT) excitons. ‘Diag’ and ’off diag’ stand for the diagonal and off diagonal intra dot Coulomb matrix elements in the product basis, Equation (2). Interdot coupling also occurs between CT and local excitons and two CT excitons. To accurately describe electronic coherences with longer periods, one needs to take into account the coupling to the strong 200 cm\({}^{-1}\) phonon mode [27; 64; 65; 81; 82; 83] and other dephasing processes. Electronic coherence is built by exciting the ensemble of dimers with fs laser pulses. The excitons are coupled to the ground state (GS) and between them by the transition dipole matrix elements. The transition dipoles are non zero for pairs of excitons that differ at most by one particle wave function. There is a non zero transition dipole from the GS to all excitons, because an exciton corresponds to a singly excited singlet configuration.