## 2 Materials and Methods: Model Excitonic Hamiltonian  
We consider isolated CdSe QDs with a radius between 1 and 2 nm. For such small QDs, one can assume a spherical symmetry. As the QD size is smaller than the bulk Bohr radius of the exciton, the strong confinement regime limit applies and the optical properties and the dynamics can be described in terms of the transition between quantized levels of holes and electrons in the confining potential. The potential is described as a spherical well of finite depth. For each dot, the EMA and **k.p** parabolic expansion at the \(\Gamma\) point of the Brillouin zone provides a one particle Schrodinger equation for the radial envelope functions. Solving this equation gives the hole and electron energies and wave functions as a function of the QD size. The specific details and values of the parameters of the wells and of the effective mass for the hole and the electron for CdSe QD are given in the Supporting Information (SI, Section 1). We investigate the fast, sub picosecond, dynamical response of large ensembles of QD dimers to sequences of fs laser pulses, focusing on the interdot excitonic coherences. To keep the computational cost reasonable, we use a minimal description of the electronic level structure of each QD. Band mixing [17; 18; 58] for the hole wave functions is not included. The fine structure arising from spin-orbit coupling of the hole wave function and from the crystal field and exchange splitting [17; 18; 20; 21; 58; 74] is also neglected. As we focus on the short time, sub ps optical response, longer time picosecond relaxation processes such as the electron-phonon coupling [27; 46; 54; 75; 76], which is the result of non adiabatic coupling between the electronic and nuclear motion and the coupling to biexcitons, owing to weaker Coulomb interactions and Auger relaxation [30; 33; 34; 35; 36; 37], are not included in our model. We retain three single particle levels and wave functions per dot--see Figure 1A and Figure S1A of Section 1 of the SI--two holes, and one electron. The hole basis states are restricted to the two lowest states of the envelope function, 1S and 2S. The electron wave function is of 1S symmetry. Once the energies and wave functions of the single particle (holes and electron) states have been determined, one can build an effective two particle (electron-hole) excitonic Hamiltonian \(\hat{H}_{he}\big{(}\mathbf{r},\mathbf{r}^{\prime}\big{)}\):  
\[\hat{H}_{he}(\mathbf{r},\mathbf{r}^{\prime})=\hat{H}_{h}(\mathbf{r})+\hat{H}_ {e}(\mathbf{r}^{\prime})+\hat{V}_{Coulomb}\Big{(}\big{|}\mathbf{r}-\mathbf{r }^{\prime}\big{|}\Big{)}+E_{gap} \tag{1}\]  
where the two one particle Hamiltonians \(\hat{H}_{h/e}(\mathbf{r})\) and their parameters are given in Section 1 of SI and \(E_{gap}\) is the value of the bulk band gap, see Figure 1A. Using these parameters, we obtain a good agreement with spectroscopic results in the range of QD diameter from 2 to 4 nm for the 1S exciton of isolated dots; see Figure S1B of SI Section 1. The exciton basis functions are two-particle wave functions; in the strong confinement approximation, they are given as the product of the hole and the electron wave function solutions of \(\hat{H}_{h/e}(\mathbf{r})\):  
\[\phi_{k}^{he}=\phi_{ij}^{he}=\phi_{i}^{hs}(\mathbf{r}_{1},\sigma_{1})\phi_{j }^{e}(\mathbf{r}_{2},\sigma_{2}) \tag{2}\]From two hole and one electron levels per QD, we obtain two local exciton states per QD, the ground state (GS) exciton 1 Se-1 Shy2 and the first excited exciton 1 Se-2 Shy2, where \(i=1\)Se and \(j=1\)Shy3/2, 2 Shy2 in Equation (2). In this convention, the excitons are labelled by their electron and hole functions. In CdSe, the hole state is localized on Se and the lowest 1S and 2S hole states have angular momentum \(J=3/2\). The electron state is localized on Cd. With two exciton levels per QD, one can build eight zero order exciton states for the dimer.