## 3 Results
### Coherent Excitonic Quantum Electronic Dynamics in the QD Dimer  
The total Hamiltonian is time-dependent because it includes the dipole interaction between the electronic states of the QD dimer and the electric fields of the sequence of fs laser pulses used to pump and probe the ensemble. In the simulations reported below, we typically use two laser pulses: the first one to steer the coherent dynamics in the excitonic manifold and the second one for probing. The total electronic Hamiltonian is written as the sum of the field free dimer Hamiltonian, \(H_{0}\), given by Equation (1) and the dipole coupling to the laser field:  
\[\hat{H}(t)=H_{0}-\mathbf{E}(t).\hat{\mu} \tag{6}\]  
where \(\mathbf{E}(t)\) is the time-dependent electric field and \(\hat{\mu}\) is the dipole operator. We use as a basis set the eight excitons and the ground state. In the zero order exciton basis, the dipole interaction couples the ground state to all the exciton states, with the largest coupling to the local excitons. We neglect the Coulomb interactions between the GS and the excitons because the energy difference is large, larger than 2 eV, and the Coulomb coupling strengths are far smaller. In \(H_{0}\), all zero order excitons are coupled by the Coulomb interactions (Equations (3)-(5)). We compute the excitonic dynamics by numerically integrating the time-dependent Schrodinger equation (TDSE) using the Hamiltonian, Equation (6), expressed in eigenstates of \(H_{0}\) (the eigenexcitonbasis) for each dimer in the ensemble. These states are stationary when there is no interaction with the laser pulse. In this basis set of nine electronic states, the wave function takes the form:  
\[\left|\Psi(t)\right\rangle=\sum_{i=0}^{8}c_{i}(t)|i\rangle \tag{7}\]  
where \(i\) = 0 is the GS and \(i\) = 1-8 are the eight eigen exciton singlet states of each dimer. To solve the TDSE, we first transform the transition dipole matrix elements to the eigen exciton basis, which leads to a redistribution of the zero order oscillator strengths over all the eigen excitons. Then, the TDSE is integrated numerically:  
\[ih\frac{d\mathbf{c}}{dt}=\mathbf{H}(t)\mathbf{c} \tag{8}\]  
where \(\mathbf{c}\) is the vector of the time-dependent amplitudes, \(c_{i}(t)\) of Equation (7). The time profile of the electric field for each laser pulse is confined by a Gaussian envelope:  
\[\mathbf{E}(t)=\mathbf{E}_{0}\cos(\omega t)\exp\left(-\frac{(t-t_{0})^{2}}{2 \sigma^{2}}\right) \tag{9}\]  
In the simulations reported below, we use two pulses. The pump has a carrier frequency, \(\omega\), of 570 THz (2.36 eV) in the visible ((VIS), tuned to excite the two local excitons h1e\({}_{\text{A}}\) and h1e\({}_{\text{B}}\) of the heterodimer shown in Figure 4B (=2.7 nm and D\({}_{\text{B}}\) = 3.5 nm and L = 0.2 nm and 5% dispersion in diameter) with similar amplitudes and generate an interdot electronic coherence. The pump field strength, \(E_{0}\), is set to 0.05 10\({}^{9}\) V/m (10\({}^{-4}\) au of field, which corresponds to a peak intensity of 3.3 10\({}^{9}\) W/cm\({}^{2}\)). The width of the pulse, \(\sigma\), is 8 fs, which leads to an energy per pulse of about 0.50 nJ for a beam waist of 50 microns. We probe the electronic dynamics either with the same VIS pulse, or with an IR pulse (\(\omega\) = 100 THz (0.41 eV), \(\sigma\) = 8 fs, \(E_{0}\) = 0.1 10\({}^{9}\) V/m (5 10\({}^{-4}\) au of field, peak intensity 8.8 10\({}^{9}\) W/cm\({}^{2}\)). We consider an ensemble of 1000 dimers with a random orientation of the molecular frame of the dimer with respect to the laboratory frame. The exciton quantum dynamics of each dimer in the ensemble is computed solving Equation (8). We begin by showing in Figure 5 the dynamics of a single heterodimer with D\({}_{\text{A}}\) = 2.7 nm and D\({}_{\text{B}}\) = 3.5 nm (see Figure 4B) and a short ligand (L = 0.2 nm) in the zero order and in the eigenexciton basis set.