Using localized double-quantum-coherence spectroscopy to reconstruct the two-exciton wave function of coupled quantum emitters

In four-wave mixing experiments the sample is excited by a sequence of three pulse envelopes

$$\begin{eqnarray*} &&\fl E(r,t) = E^1(r,t-t_3-t_2-t_1) \mathrm{e}^{\mathrm{i}\omega_{\mathrm{l}}(t-t_3-t_2-t_1)+\mathrm{i}\varphi_1} + E^2(r,t-t_3-t_2) \mathrm{e}^{\mathrm{i}\omega_{\mathrm{l}}(t-t_3-t_2)+\mathrm{i}\varphi_2}\\ &&+E^3(r,t-t_3) \mathrm{e}^{\mathrm{i}\omega_{\mathrm{l}}(t-t_3)+\mathrm{i}\varphi_3}+\mathrm{c.c.} \end{eqnarray*}$$

separated by variable time delays t₁, t₂ and t₃ with different phases φ₁, φ₂ and φ₃ and laser frequency ω_l (illustrated in figure 1(a)). Since we treat only a single nanostructure, a dissection of quantum pathways using the direction of the incoming fields is not possible. Therefore, a phase cycling technique is applied. Here, the simulated experiment is carried out several times with different phase contributions φ₁, φ₂ and φ₃. This allows us to extract a specific signal, described by the linear combination φ_S = lφ₁ + mφ₂ + nφ₃ of the phases using postprocessing algorithms (where l, n and m are integers). To extract a certain polarization P_l,m,n from the detected polarization $P(t,\varphi _1, \varphi _2, \varphi _3) = \sum _{l,m,n}P_{l,m,n}(t)\mathrm {e}^{\mathrm {i}(l\varphi _1+m\varphi _2+n\varphi _3)}$, the experiment is repeated for sufficient phase combinations [42] (determined by the matrix e^{i(lφ₁+mφ₂+nφ₃)}), so that the polarization P_l,m,n can be extracted by inverting the matrix e^{i(lφ₁+mφ₂+nφ₃)}.

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In this paper, we focus on double-quantum-coherence spectroscopy (DQCS), where the extracted phase combination is φ_III = φ₁ + φ₂ − φ₃. Two Liouville pathways contribute to this signal for a three-band system as depicted in figure 1(b). The pathways are shown in figure 2, presented by double-sided Feynman diagrams [29]: in both quantum pathways, the first pulse creates a coherence between the ground state and a single exciton, and the second pulse creates a coherence between a two-exciton and the ground state. The third pulse creates in pathway (i) a single exciton to ground state coherence, whereas in pathway (ii) a two-exciton to single exciton coherence is created. The fourth pulse depicted in figure 1(a) is the local oscillator for heterodyne detection [24]. This allows a detection of the signal phase.

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In the following, QDs—as an example of a system of coupled nanostructures—are examined in the 2D spectroscopy experiment. We will demonstrate that the DQCS can reveal the energy of single and two-exciton states of the coupled QDs [24, 43].

As a model system, we examine n = 3 coupled QDs that are spatially close enough to couple via Förster coupling but spatially separated enough to have no wave function overlap. Such a coupling is even possible by assuming that the wave functions do not overlap.

The local single exciton basis elements of n two-level QDs are the n basis elements |i〉, where |i〉 means that only QD i is excited and the other QDs are in the ground state. The n(n − 1)/2 two-exciton basis elements are |ij〉, where |ij〉 means that only QD i and j are in excited states and all other QDs are in ground states. We restrict the discussion to a maximum of two excitations, which is sufficient for spectroscopy in third order of the optical field. The ground state of the whole nanostructure is denoted by |g〉. The Hamiltonian H = H₀ + H_C + H_e−l consists of three parts: H₀, which corresponds to the energy of the electron in the isolated uncoupled system, the field–matter interaction H_e−l, which describes the dipole coupling to a classical electric field, and the Coulomb interaction H_C. They are given by

$$\begin{equation} H_0 = \epsilon_0 |{g}\rangle\langle{g}|+ \sum_i^n \epsilon_i |{i}\rangle \langle{i}| + \sum_{i < j} (\epsilon_i+\epsilon_j) |{ij\rangle}\langle{ij}|, \end{equation} \tag{ 1a }$$

$$\begin{equation} H_{\mathrm{C}} = \sum_{i<j} V_{ij} |{ij}\rangle\langle{ij}| + \sum_{i,j\neq i} V_{ij}^{\mathrm{F}} |{i}\rangle\langle{j}|+\sum_{k, i\neq k, j\neq k} V_{ij}^{\mathrm{F}} |{ki}\rangle\langle{kj}|, \end{equation} \tag{ 1b }$$

$$\begin{equation} H_{\mathrm{e-l}} = \sum_i^n \mu_{gi}\cdot E(t)|{g}\rangle\langle{i}| + \sum_{i, j \neq i}^n \mu_{ij}\cdot E(t) |{i}\rangle\langle{ij}|+ \mathrm{h. a.} \end{equation} \tag{ 1c }$$

Coulomb coupling H_C between the QDs leads to the formation of delocalized states. These delocalized single exciton |e〉 and two-exciton states |f〉 can be written as a linear combination of the local basis

$$\begin{eqnarray} |{e}\rangle &=& \sum_i^n c_i^e|{i}\rangle, \quad |{f}\rangle = \sum_{i, j \neq i}^n c_{ij}^f|{ij}\rangle, \end{eqnarray} \tag{ 2 }$$

for single exciton states and two-exciton states, respectively. These states are obtained by diagonalizing the Hamiltonian H₀ + H_C including the coupling between the nanostructures (see figure 1(b)).

The pure electronic part of the Hamiltonian reads in delocalized basis

$$\begin{equation*} H_0+H_{\mathrm{C}}=\hbar \omega_g|{g}\rangle\langle{g}|+\sum_e \hbar\omega_e|{e}\rangle\langle{e}|+\sum_f\hbar \omega_f|{f}\rangle\langle{f}| \end{equation*}$$

with _g, _e and _f as the eigenenergy for the ground state, the single-exciton state and the double-exciton state, respectively. For rewriting the field–matter part of equation (1c) in a delocalized basis, we use the inverse transformation for single excitons $|{i}\rangle =\sum _e c_i^{e*}|{e}\rangle$ and two-exciton states $|{ij}\rangle = \sum _f c_{ij}^{f*}|{f}\rangle$. With these relations, the electron light coupling Hamiltonian in delocalized basis reads

$$\begin{eqnarray} &&\fl H_{\mathrm{e-l}} = \sum_i^n \sum_e c_i^e\mu_{gi}\cdot E(t)|{g}\rangle\langle{e}|+ \sum_{i, j\neq i}^n \sum_{e, f} c_i^{e*} c_{ij}^f \mu_{gj}\cdot E(t) |{e}\rangle\langle{f}| +\mathrm{h.\,a.}\nonumber\\&&\ms\ms = \sum_e \mu_{ge}\cdot E(t)|{g}\rangle\langle{e}| + \sum_{e,f} \mu_{ef}\cdot E(t) |{e}\rangle\langle{f}| +\mathrm{h.\,a.} \end{eqnarray} \tag{ 3 }$$

with the dipole moments in delocalized basis

$$\begin{eqnarray} &&\mu_{ge} = \sum_i^n c_i^e\mu_{gi} \quad\mathrm{and}\quad \mu_{ef} = \sum_{i, j \neq i}^n c_i^{e*} \mu_{gj} c_{ij}^f \end{eqnarray} \tag{ 4 }$$

for ground state to single exciton and single exciton to two-exciton transitions, respectively.

The signal of 2D spectroscopy is measured in dependence of the three delay times t₁, t₂ and t₃ between the pulses (cf figure 1(a)). The spectroscopic signal is measured using heterodyne detection, where the polarization induced by the three pulses is mixed with a local oscillator [24]:

$$\begin{eqnarray} &&S^{(3)}_{\varphi_{\mathrm{III}}}(t_1,t_2,t_2)=\int_{-\infty}^{\infty} \mathrm{d}t P^{(3)}_{\varphi_{\mathrm{III}}}(t) E^{4*}(t)\mathrm{e}^{\mathrm{i} \omega_{\mathrm{l}} t} \end{eqnarray} \tag{ 5 }$$

to be able to extract the real and imaginary parts of the signal. Phase cycling ensures that we only retrieve the contributions from the polarization with phase combination ϕ_III = ϕ₁ + ϕ₂ − ϕ₃. The polarization connected to ϕ_III generated by the three pulses can be described using response functions [24]:

$$\begin{eqnarray} &&\fl P^{(3)}_{\varphi_{\mathrm{III}}}(t)=\int_0^\infty \mathrm{d}\tau_3 \int_0^\infty \mathrm{d}\tau_2 \int_0^\infty \mathrm{d}\tau_1 R^{(3)}_{\varphi_{\mathrm{III}}}(t,t-\tau_3,t-\tau_3-\tau_2,t-\tau_3-\tau_2-\tau_1).\qquad \end{eqnarray} \noindent \tag{ 6 }$$

For the case of our three-band model, only the processes depicted in the two Liouville diagrams in figure 2 contribute, leading to

$$\begin{eqnarray} &&\fl R^{(3)}_{\varphi_{\mathrm{III}}}(t,\tilde{t}_3,\tilde{t}_2,\tilde{t}_1)=R^{(3)}_{i}(t,\tilde{t}_3,\tilde{t}_2,\tilde{t}_1)+R^{(3)}_{ii}(t,\tilde{t}_3,\tilde{t}_2,\tilde{t}_1) \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\fl R^{(3)}_{i}(t,\tilde{t}_3,\tilde{t}_2,\tilde{t}_1)=- \left(\frac{\mathrm{i}}{\hbar}\right)^3\mathrm{e}^{\mathrm{i} \omega_{\mathrm{l}} (\tilde{t}_1+\tilde{t}_2-\tilde{t}_3-t_1-2t_2-t_3)}\nonumber\\ &&\sum_{ee'f} \mu_{e'f} \mu_{ge'}\cdot E^{3*}(\tilde{t}_3-t_3) \mu_{fe}\cdot E^2(\tilde{t}_2-t_3-t_2)\mu_{eg}\cdot E^1(\tilde{t}_1-t_3-t_2-t_1) \nonumber\\ &&\mathrm{e}^{-\mathrm{i} \xi_{fe'}(t-\tilde{t}_3)-{\mathrm{i}} \xi_{fg}(\tilde{t}_3-\tilde{t}_2)-{\mathrm{i}} \xi_{eg} (\tilde{t}_2-\tilde{t}_1)}\end{eqnarray} \noindent \tag{ 8 }$$

$$\begin{eqnarray} &&\fl R^{(3)}_{ii}(t,\tilde{t}_3,\tilde{t}_2,\tilde{t}_1)= \left(\frac{\mathrm{i}}{\hbar}\right)^3\mathrm{e}^{\mathrm{i} \omega_{\mathrm{l}} (\tilde{t}_1+\tilde{t}_2-\tilde{t}_3-t_1-2t_2-t_3)} \nonumber\\ &&\sum_{ee'f} \mu_{ge'} \mu_{e'f}\cdot E^{3*}(\tilde{t}_3-t_3) \mu_{fe}\cdot E^2(\tilde{t}_2-t_3-t_2)\mu_{eg}\cdot E^1(\tilde{t}_1-t_3-t_2-t_1)\nonumber\\ &&\mathrm{e}^{-\mathrm{i} \xi_{e'g}(t-\tilde{t}_3) -{\mathrm{i}} \xi_{fg}(\tilde{t}_3-\tilde{t}_2)-{\mathrm{i}} \xi_{eg} (\tilde{t}_2-\tilde{t}_1)}. \end{eqnarray} \tag{ 9 }$$

Here, we introduced the frequencies ω_ij = ω_i − ω_j, with the exciton energies ℏω_i, and ξ_ij = ω_ij + iγ_ij with the dephasing γ_ij. The first pulse induces in both pathways a coherence between the ground state and a single exciton state. The second pulses then create a quantum coherence between the ground state and a two-exciton state from the ground state to single exciton coherence, giving the method its name double-quantum coherence. After the third pulses both pathways differ having either a ground state to single exciton coherence or a single exciton to two-exciton coherence.

A Fourier transform of the signal S_φIII(t₁,t₂,t₃) with respect to two of the three time intervals at a fixed third time interval retrieves the 2D spectrum:

$$\begin{eqnarray} &&S^{(3)}_{\varphi_{\mathrm{III}}}(\Omega_1,\Omega_2,t_3)=\int_0^\infty \mathrm{d} t_1 \int_0^\infty \mathrm{d} t_2 \,\mathrm{e}^{\mathrm{i} \Omega_1 t_1+{\mathrm{i}} \Omega_2 t_2} S^{(3)}_{k_{\mathrm{III}}}(t_1,t_2,t_3). \end{eqnarray} \tag{ 10 }$$

For the DQCS, we take the Fourier transform with respect to the time intervals t₁ and t₂, so that the signal depends on the single and two-exciton frequencies Ω₁ and Ω₂, respectively. The far-field double-quantum-coherence signal reads

$$\begin{equation} S_{\varphi_{\mathrm{III}}}^{(3)}(\Omega_1, \Omega_2, t_3)=S_{\mathrm{i}}^{(3)}(\Omega_1, \Omega_2, t_3)+S_{\mathrm{ii}}^{(3)}(\Omega_1, \Omega_2, t_3) \end{equation} \tag{ 11 }$$

with

$$\begin{eqnarray*} &&\fl S_{\mathrm{i}}^{(3)}(\Omega_1, \Omega_2, t_3)=\frac{1}{\hbar^3} \sum_{e,e',f} \mu_{e'f}\cdot E_4^{*}(\omega_{fe'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g}) \mu_{ef}^*\cdot E_2(\omega_{fe})\\ &&\times \mu_{ge}^*\cdot E_1(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{fe'}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})} \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\fl S_{\mathrm{ii}}^{(3)}(\Omega_1, \Omega_2, t_3)=-\frac{1}{\hbar^3} \sum_{e,e',f} \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'f}\cdot E_3^{*}(\omega_{fe'})\\ &&\times \mu_{ef}^*\cdot E_2(\omega_{fe})\ \mu_{ge}^*\cdot E_1(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})}, \end{eqnarray*}$$

with E_k(ω) as the Fourier transformation of the electric field envelopes for the pulses k = 1,2,3,4. The signal consists of the contributions of two Liouville pathways for the case of the three-band system formed from ground state, single excitons and two excitons.

An example is given on the left in figure 3(a) for the absolute value of the spectrum. Here, the detection frequencies are given as detuning around the single and the double gap frequency, respectively. The right plot in figure 3(a) shows the corresponding imaginary part. We assumed that the spectral bandwidth of the exciting pulses is much larger than the bandwidth of the exciton energy and therefore we assume a constant E_i(ω) = E_i.

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In these 2D spectra, we see several resonances that connect the single-exciton state at energy ω_eig to the two-exciton states at ω_fig. The strength is a qualitative indicator of which delocalized single states contribute to which two-exciton state. For example, from figure 3(a), we see that f₁ is mainly connected to e₃ and e₁, whereas f₂ is mainly connected to e₂ and e₃, and f₃ to e₁ and e₂. Note that the strength of the resonances is only a hint of the connection between these states due to interference effects.

Note that in most experimental realizations, a Fourier transform of the second and third delay times is recorded. Since the Fourier transformation of the last third delay time can be recorded using a spectrometer, this makes the experiment more feasible. The latter presented method for extracting the two-exciton coefficients will also work in this configuration, since the algorithm relies on the second delay time. The method for extracting the single exciton wave function needs to be used together with a photon echo φ_I with a 2D spectra of the first and third delay times, since it relies on a localization of the pulse at the first delay time.

The objective for our localized spectroscopy is controlling optical excitations in such a way that just one emitter in a coupled nanostructure becomes locally excited. Realizing field localization on a nanometer length scale can be fulfilled using different methods. The presented protocols in this work do not depend on the particular method. It is only important that one emitter of the coupled quantum system can be excited about a factor of 10 higher than the others in terms of the electric field in order to achieve a good reconstruction of the delocalized exciton states.

In addition to near-field spectroscopy with coated [1] and uncoated fiber tips [15] or metal tips [12, 16, 17], there exist also metal nano-antennas [18–23] for controlling fields on the nanoscale.

In this section, we apply a localization technique as motivated in section 3.1. We replace one or two pulses of the pulse sequence with pulses localized, so that it excites only a single QD. All other pulses still excite all dots equally. We will start with a modified description of the signal. Then we present calculated spectra.

In order to describe this in our framework, we have to adapt the Hamiltonian for localized excitation. In the local basis, it is obvious that we have to take the electric field at the position of the QD:

$$\begin{eqnarray} &&H_{\mathrm{el-L}}=\sum_i \mu_{gi}\cdot E(r_i,t) |g \protect\rangle \protect\langle i|+\sum_{ij} \mu_{gi}\cdot E(r_i,t) |j \protect\rangle \protect\langle ij| +\mathrm{h.a.} \end{eqnarray} \tag{ 12 }$$

In order to express H_el−L in the delocalized states, we can insert the expansion of the local states in terms of the delocalized exciton states:

$$\begin{eqnarray} &&H_{\mathrm{el-L}}=\sum_{ie} c^{e}_i\mu_{gi}\cdot E(r_i,t) |g \protect\rangle \protect\langle e|+\sum_{i<j ef} {c^{e}_i}^* \mu_{gi}c^{f}_{ij}\cdot E(r_i,t) |e \protect\rangle \protect\langle f| +\mathrm{h.a.} \end{eqnarray} \tag{ 13 }$$

We note that we cannot formulate the dipole moments in terms of delocalized dipole moments for near-field distributions. This can only be done in far-field excitation assuming constant electric fields over the whole nanostructure. Note that the far-field Hamilton-operator equation (3) can be obtained from the localized form by assuming a spatial constant electric field over the whole nanostructure.

For the first modification of the DQCS signal, we use a localization of the first pulse of the sequence. This will control the QDs involved in the ground state to single exciton transition. For this purpose, equation (11) can be rewritten using the Hamiltonian reformulated in localized fields for the first interaction. Assuming a perfect local excitation, the spatial field distribution of the localized first pulse can be written as E^loc_1,i(r_k) = E^loc_1,iδ_ik if QD i is the excited dot. Thus, the sum over the QD number k vanishes and the simplified signal reads

$$\begin{equation} S^{\mathrm{loc}}_{E_1}(i, \Omega_1, \Omega_2, t_3)=S^{\mathrm{loc}}_{E_1,\mathrm{i}}(i, \Omega_1, \Omega_2, t_3)+S^{\mathrm{loc}}_{E_1,\mathrm{ii}}(i, \Omega_1, \Omega_2, t_3) \end{equation} \tag{ 14 }$$

with

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1,\mathrm{i}}(i, \Omega_1, \Omega_2, t_3)=\frac{1}{\hbar^3} \sum_{e,e',f} \mu_{e'f}\cdot E_4^{*}(\omega_{fe'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g})\\ &&\times \mu_{ef}^*\cdot E_2(\omega_{fe})\ c_i^{e*}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{fe'}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})} \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1,\mathrm{ii}}(i, \Omega_1, \Omega_2, t_3)=-\frac{1}{\hbar^3} \sum_{e,e',f} \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'f}\cdot E_3^{*}(\omega_{fe'})\\ &&\times \mu_{ef}^*\cdot E_2(\omega_{fe})\ c_i^{e*}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})}. \end{eqnarray*}$$

In figure 4(a), we calculated the signal for the localization of the first pulse at different QDs. This leads to a decomposition of the imaginary part obtained for far-field excitation (figure 3(a), right). We can use this information to see which resonance gains contributions from which QD at the ground state to single exciton transition during the first pulse. Since this transition is proportional to the single exciton expansion coefficient c^e_i∝S^loc_E1,i(i,Ω₁,Ω₂,t₃)/(μ*_gi·E^loc_1,i(ω_eg)), cf equation (14), we can conclude which QD contributes most to which single exciton state. For instance, the resonances at e₂ mainly arise due to the first QD. The single exciton state e₂ has only optical transitions to f₃ and f₂. Similarly, the single exciton e₃ arises mostly from QD 2, having two transitions to two excitons, a strong one at f₁ and the smaller one at f₂. But there are also contributions from the second dot to the exciton state e₂: a small for the strong resonance with f₃ and a smaller peak connecting e₂ with f₂.

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The single exciton state e₁ is mainly connected to the third QD. State e₁ has strong transitions to f₁ and f₃. This way, we already gain new information, which would not be possible without localization of the exciting pulses.

We can also localize the second pulse for the application of our algorithms; therefore we insert for this interaction the local form of the electron light interaction Hamiltonian, equation (13). For the additional localized field at QD j for the second pulse, we use again perfect localization E^loc_2,j(r_l) = E^loc₂δ_j,l and obtain

$$\begin{equation} \fl S^{\mathrm{loc}}_{E_1, E_2}(i, j, \Omega_1, \Omega_2, t_3) = S^{\mathrm{loc}}_{E_1, E_2, \mathrm{i}}(i, j, \Omega_1, \Omega_2, t_3) + S^{\mathrm{loc}}_{E_1, E_2, \mathrm{ii}}(i, j, \Omega_1, \Omega_2, t_3) \end{equation} \tag{ 15 }$$

with

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1, E_2, \mathrm{i}}(i, j, \Omega_1, \Omega_2, t_3)=\frac{1}{\hbar^3} \sum_{e,e',f}\sum_{k\neq j}^n \mu_{e'f}\cdot E_4^{*}(\omega_{fe'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g})\\ &&\times c_k^e c_{kj}^{f*} \mu_{gj}^*\cdot E^{\mathrm{loc}}_{2,j}(\omega_{fe})\ c_i^{e*}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{fe'}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})} \end{eqnarray*} \noindent$$

and

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1, E_2, \mathrm{ii}}(i, j, \Omega_1, \Omega_2, t_3)=-\frac{1}{\hbar^3} \sum_{e,e',f}\sum_{k\neq j}^n \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'f}\cdot E_3^{*}(\omega_{fe'})\\ &&\times c_k^e c_{kj}^{f*} \mu_{gj}^*\cdot E^{\mathrm{loc}}_{2,j}(\omega_{fe})\ c_i^{e*}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})}, \end{eqnarray*} \noindent$$

still under the assumption of a localization of the first pulse at QD i.

Using this localization scheme, we can decompose each of the three plots with a localization of the first pulses of figure 4(a) in three plots with also the second pulse localized. In figure 5, we can see the contribution of the QDs to the single exciton to two-exciton transition μ_ef, connected to their contribution to the ground state to single exciton transition. If we focus on the case with the first pulse mainly exciting QD 1 (first row), we see that the peak at e₂ with f₃ is mainly visible for a second pulse at QD 3 (and much smaller for a localization at QD 2). On the other hand, with the first pulse mainly exciting QD 3 we see that the peak at e₁ and f₃ is mainly visible for a localization of the second pulse at QD 1. Therefore, we can conclude f₃ that the mainly contributing local two exciton is a state with an excited QD 1 and QD 3. Also the second and the third row show which QD is connected to which single exciton to two-exciton transition of the corresponding spectra of figure 4(a).

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In principle, we can see that, in the case of not too strong coupling and thus low delocalization, figure 5 shows from which local single excitons the two excitons |ij〉 localized at QD i and j are formed.

Next, we will show that for reconstructing the two-exciton wave function (see section 3.6.3) it is sufficient to localize the second pulse only. Then, the signal reads

$$\begin{equation} S^{\mathrm{loc}}_{E_2}(j, \Omega_1, \Omega_2, t_3) = S^{\mathrm{loc}}_{E_2, \mathrm{i}}(j, \Omega_1, \Omega_2, t_3) + S^{\mathrm{loc}}_{E_2, \mathrm{ii}}(j, \Omega_1, \Omega_2, t_3) \end{equation} \noindent \tag{ 16 }$$

with

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_2, \mathrm{i}}(j, \Omega_1, \Omega_2, t_3) =\frac{1}{\hbar^3} \sum_{e,e',f}\sum_{k\neq j}^n \mu_{e'f}\cdot E_4^{*}(\omega_{fe'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g})\\ &&\times c_k^e c_{kj}^{f*} \mu_{gj}^*\cdot E^{\mathrm{loc}}_{2,j}(\omega_{fe})\ \mu_{ge}^*\cdot E_1(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{fe'}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})} \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_2, \mathrm{ii}}(j, \Omega_1, \Omega_2, t_3) =-\frac{1}{\hbar^3} \sum_{e,e',f}\sum_{k\neq j}^n \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'f}\cdot E_3^{*}(\omega_{fe'})\\ &&\times c_k^e c_{kj}^{f*} \mu_{gj}^*\cdot E^{\mathrm{loc}}_{2,j}(\omega_{fe})\ \mu_{ge}^*\cdot E_1(\omega_{eg})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}t_3}}{(\Omega_2-\xi_{fg})(\Omega_1-\xi_{eg})}. \end{eqnarray*} \noindent$$

The first pulse excites all QDs equally in a far field like the third and fourth pulses.

In figure 4(b), spectra calculated with equation (16) are presented. They represent—opposite to figure 4(a)—the decompositions of the imaginary part using a localization of the second pulse of the far-field spectrum. Here, the two-exciton states cannot be assigned to a certain QD in such a simple way as in the exciton case, since the signal is proportional to c^e_kc^f*_kjμ*_gj·E^loc_2,j(ω_fe) and thus to a product of the single and two-exciton wave functions. As expected already from the spectrum with two localized pulses, the main part of the resonance at e₂ with f₃ originates from QD 3, whereas the main part of the resonance at e₁ with f₃ originates from QD 1. Thus f₃ is mainly composed of e₁ and e₂, which are primarily located at QD 3 and QD 1, respectively. In a similar way, we see that f₁ has contributions from e₁ and e₃ mainly localized at QD 2 and QD 3.

Localized spectroscopy allows several new applications, since spectroscopic data can be used to obtain more information about coupled nanostructures. In this section, we briefly review a protocol to reconstruct the wave functions of single exciton states, first introduced in [26], and extend this to two-exciton states in coupled nanosystems. Section 3.7 presents a filtering method, which is important for enhancing the algorithms used for the reconstruction and supplements this first introductory section.

If the expansion coefficients c^e_i (single exciton wave functions) and c^f_ij (two-exciton wave functions) of equation (2), respectively, are known, it is possible to reconstruct the many particle wave functions |e〉 and |f〉. Additionally, the wave functions of the excited state |i〉 of the isolated emitter i must be known for a full reconstruction. The amplitude and phase of the coefficients c represent how the delocalized wave function is spread over the different QDs. The main idea behind the reconstruction of the wave functions is that spatiotemporal control of electronic fields enables us to eliminate the sum over the QDs i. 2D DQCS, on the other hand, enables us to select specific single and two-exciton states e and f spectrally to remove the summation over e and f. These two techniques allow us to find the coefficients c^e_i for a single exciton state e for QD i and c^f_ij for a two-exciton state f for QDs i and j. In the following, we present a calculation and measurement instruction for reconstructing the wave function for a certain single exciton state  $\skew4\bar{e}$ and a certain two-exciton state  $\skew8\bar{f}$.

3.6.1. Single exciton states

In this section, we summarize briefly the method introduced in [26]. For reconstructing the wave function $c_i^{\skew4\bar{e}}$, we have to select only the contribution from the spectroscopic signal of a specific single exciton state $\skew4\bar{e}$. We achieve this by extracting the DQCS signal around the frequencies $\Omega _1\approx \skew4\bar{\Omega }_1\approx \omega _{\skew4\bar{e} g}$ and $\Omega _2\approx \skew4\bar{\Omega }_2\approx \omega _{\skew8\bar{f} g}$, where $\skew8\bar{f}$ is chosen such that the signal has only a strong contribution connected to  $\skew4\bar{e}$. For this, the spectral contributions of other delocalized states are well separated in the DQCS signal from this resonance. Then, the only contributions of the sum over e and f are terms with $\skew4\bar{e}$ and $\skew8\bar{f}$. We look at the localized signal for $\skew4\bar{e}$ at a fixed time interval T₃:

$$\begin{equation} S^{\mathrm{loc}}_{E_1}(i, \skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3) = S^{\mathrm{loc}}_{E_1, \mathrm{i}}(i, \skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3) + S^{\mathrm{loc}}_{E_1, \mathrm{ii}}(i, \skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3) \end{equation} \noindent \tag{ 17 }$$

with

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1, \mathrm{i}}(i, \skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3) =\frac{1}{\hbar^3} \sum_{e'} \mu_{e' \skew8\bar{f}}\cdot E_4^{*}(\omega_{\skew8\bar{f} e'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g})\\ &&\times \mu_{\skew4\bar{e} \skew8\bar{f}}^*\cdot E_2(\omega_{\skew8\bar{f} \skew4\bar{e}})\ c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{\skew4\bar{e} g})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{\skew8\bar{f} e'}T_3}}{(\skew4\bar{\Omega}_2-\xi_{\skew8\bar{f} g})(\skew4\bar{\Omega}_1-\xi_{\skew4\bar{e} g})} \end{eqnarray*} \noindent$$

and

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_1, \mathrm{ii}}(i, \skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3) =-\frac{1}{\hbar^3} \sum_{e'} \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'\skew8\bar{f}}\cdot E_3^{*}(\omega_{\skew8\bar{f}e'}) \\ &&\times \mu_{\skew4\bar{e} \skew8\bar{f}}^*\cdot E_2(\omega_{\skew8\bar{f} \skew4\bar{e}})\ c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{\skew4\bar{e} g})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}T_3}}{(\skew4\bar{\Omega}_2-\xi_{\skew8\bar{f} g})(\skew4\bar{\Omega}_1-\xi_{\skew4\bar{e} g})}. \end{eqnarray*}$$

The only part of this signal that depends on the selective excitation of QD i is $c_i^{\skew4\bar{e} *}\mu _{gi}^*\cdot E^{\mathrm {loc}}_{1,i}(\omega _{\skew4\bar{e}g})$. Since both Liouville diagrams share the same interaction at the first step, all other parts do not depend on i. Thus, for the coefficient c^e_i a proportionality relation⁵ holds:

$$\begin{equation} c_i^{\skew4\bar{e} *} A = S^{\mathrm{loc}}_{E_1}(i,\skew4\bar{\Omega}_1,\skew4\bar{\Omega}_2, T_3)/(\mu_{gi}^*\cdot E_{1,i}^{\mathrm{loc}}(\omega_{\skew4\bar{e} g})). \end{equation} \tag{ 18 }$$

Assuming that all dipole moments μ*_gi for the single exciton transitions and the localized field $E^{\mathrm {loc}}_{1,i}(\omega _{\skew4\bar{e}g})$ are known, we can extract from equation (18) all coefficients $c_i^{\skew4\bar{e} *}$ except for a proportionality factor A. From the normalization of the wave functions we find that $\sum _i^n |{c_i^{\skew4\bar{e}}}|^2=1$, so that the coefficient can be determined (except for an arbitrary phase) and the three wave function coefficients are thus reconstructed. To summarize the method for reconstructing the single exciton wave function, we need as input information the dipole moments μ_gi for all QDs i. In the measurement, we first select frequencies $\skew4\bar{\Omega }_1$ and $\skew4\bar{\Omega }_2$ with a main contribution by the single exciton state $\skew4\bar{e}$ to be reconstructed. Then DQCS signal $S^{\mathrm {loc}}_{E_1}(i,\skew4\bar{\Omega }_1,\skew4\bar{\Omega }_2, T_3)$ is recorded for all QDs i. In postprocessing the expansion coefficient of the exciton wave function $c_i^{\skew4\bar{e}}$ are obtained $c_i^{\skew4\bar{e}}=S^{\mathrm {loc}}_{E_1}(i, \skew4\bar{\Omega }_1,\skew4\bar{\Omega }_2, T_3)/\left (\mu _{gi}^*\cdot E_{1,i}^{\mathrm {loc}}(\omega _{\skew4\bar{e} g}) A\right )$, where A is found by the normalization of the wave function $|{\skew4\bar{e}}\rangle$: $\sum _i |c_i^{\skew4\bar{e}}|^2$.

3.6.2. Determining and enhancing the quality of the reconstructed single exciton states

In table 1, all original single exciton states that belong to the chosen example can be compared with the reconstructed values. We find that e₁ is the state with the best reconstruction. Furthermore, the best reconstruction is reached choosing the energy of the two-excitonic state f₃ to ground state transition for Ω₂. This is because that peak has the lowest influence from other resonances (cf the spectrum in figure 3(a)). For the same reason, we choose for the reconstruction of e₂ the resonance with f₃ and of e₃ the resonance with f₁.

Table 1. Comparison of the original (O) with the reconstructed (R) single exciton wave function coefficients for a shifted measurement point using a perfect localization: The phase is written in multiples of 2π. The error is calculated via $\sqrt {\sum _{i=1}^n\Vert{{c_i^{\mathrm {rec}}}|^2-|{c_i^{\mathrm {org}}}|^2}|^2}$. Additionally, the results of a measurement point at the center of the resonance are given (C). A reconstruction after filtering e₂ is marked with (F) and a reconstruction after filtering states e₁ and e₂ (FF).

State	Type	|ce1|	|ce2|	|ce3|	arg(ce1)	arg(ce2)	arg(ce3)	Error
e1	O	0.115	0.104	0.988	0.500	0.500	0.000
	R	0.205	0.081	0.975	0.414	0.190	0.710	0.04
	C	0.247	0.077	0.966	0.433	0.281	0.787	0.06
e2	O	0.977	0.169	0.131	0.500	0.500	0.500
	R	0.949	0.110	0.294	0.836	0.820	0.952	0.09
	F	0.969	0.138	0.205	0.845	0.858	0.650	0.03
	C	0.969	0.136	0.206	0.725	0.680	0.907	0.03
e3	O	0.180	0.980	0.083	0.000	0.500	0.500
	R	0.360	0.858	0.366	0.342	0.792	0.903	0.28
	F	0.381	0.920	0.093	0.316	0.794	0.968	0.16
	FF	0.138	0.987	0.081	0.302	0.796	0.873	0.02

We measured the quality in terms of the squared error $\sqrt {\sum _{i=1}^n\Vert{{c_i^{\mathrm {rec}}}|^2-|{c_i^{\mathrm {org}}}|^2}|^2}$ of the difference of the original wave function coefficient c^org_i and the reconstructed one c^rec_i. This way of error determination does not consider the phase, but it has the most physical meaning since the square of the coefficient gives information about the occupation probability.⁶

We can still improve the quality of reconstruction of ce1i by moving the measurement point spectrally in the direction of higher single exciton energies, since then we decrease the influence of other resonances.7 This works well mostly except for reconstructing state e2. Also, one would assume that moving the measurement point in the negative Ω1 direction improves the reconstruction quality. But in this special case, interferences of positive and negative contributions are destructive on the right-hand side of the strong peak (e2, f3) and so the influence of other resonances decreases moving to higher energies. Another idea for increasing the quality of reconstruction is averaging over an area of measurement points or additional weighting the results with a Gaussian centered at the main resonance peak; however, this is beyond the scope of this paper. Additionally, we compared in table 1 the reconstruction for choosing Ω1, Ω2 at the center of the resonances.

Reconstructing the exciton wave functions shows overall good agreement for both the amplitude and the relative phase. The values of the resonance with the highest oscillator strength are exemplary illustrated in figure 6.

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3.6.3. Two-exciton states

Similar to the procedure for single excitons, the two-exciton wave functions can also be reconstructed. As a prerequisite, the single exciton reconstruction has to be completed and c^e_i must be known for all single excitons (except for a global phase factor) connected to the two-exciton state.

Here, we use the signal with a localized second pulsed (equation (16)). We select a single exciton state $\skew4\bar{e}$ and a two-exciton state $\skew8\bar{f}$; then $S^{\mathrm {loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)$ is the signal with localized second pulses evaluated for frequencies $\skew4\bar{\Omega }_1$ and $\skew4\bar{\Omega }_2$ for which mainly $\skew4\bar{e}$ and $\skew8\bar{f}$ contribute ($\skew4\bar{\Omega }_1\approx \omega _{\skew4\bar{e}g}$ and $\skew4\bar{\Omega }_2\approx \omega _{\skew8\bar{f}\skew4\bar{e}}$). We start with rewriting the signal with localized second pulse (equation (16)) just for one selected single exciton state $\skew4\bar{e}$ and one selected two-exciton state $\skew8\bar{f}$ as explained in section 3.6.1 and obtain

$$\begin{eqnarray*} &&S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) = S^{\mathrm{loc}}_{E_2, \mathrm{i}}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) + S^{\mathrm{loc}}_{E_2, \mathrm{ii}}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) \end{eqnarray*}$$

with

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_2, \mathrm{i}}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) =\frac{1}{\hbar^3} \sum_{e'}\sum_{k\neq j}^n \mu_{e' \skew8\bar{f}}\cdot E_4^{*}(\omega_{\skew8\bar{f} e'})\ \mu_{ge'}\cdot E_3^{*}(\omega_{e'g}) \\ &&\times c_k^{\skew4\bar{e}} c_{kj}^{\skew8\bar{f} *} \mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})\ \mu_{g \skew4\bar{e}}^*\cdot E_1(\omega_{\skew4\bar{e} g})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{\skew8\bar{f} e'}T_3}}{(\skew4\bar{\Omega}_2-\xi_{\skew8\bar{f} g})(\skew4\bar{\Omega}_1-\xi_{\skew4\bar{e} g})} \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\fl S^{\mathrm{loc}}_{E_2, \mathrm{ii}}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) =-\frac{1}{\hbar^3} \sum_{e'}\sum_{k\neq j}^n \mu_{ge'}\cdot E_4^{*}(\omega_{e'g})\ \mu_{e'\skew8\bar{f}}\cdot E_3^{*}(\omega_{\skew8\bar{f}e'}) \\ &&\times c_k^{\skew4\bar{e}} c_{kj}^{\skew8\bar{f} *} \mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})\ \mu_{g \skew4\bar{e}}^*\cdot E_1(\omega_{\skew4\bar{e} g})\ \frac{\mathrm{e}^{-{\mathrm{i}}\xi_{e'g}T_3}}{(\skew4\bar{\Omega}_2-\xi_{\skew8\bar{f} g})(\skew4\bar{\Omega}_1-\xi_{\skew4\bar{e} g})}. \end{eqnarray*}$$

We can use the appearance of the factors $c_k^{\skew4\bar{e}} c_{kj}^{\skew8\bar{f} *} \mu _{gj}^*\cdot E_2^{\mathrm {loc}}(\omega _{\skew8\bar{f} \skew4\bar{e}})$ in the signal as the starting point for being able to derive an equation analogous to equation (18) for the extraction of two-exciton wave functions. We can reformulate this by introducing proportionality factors $B_{\skew4\bar{e}}^{\skew8\bar{f}}$, which do not depend on localization:

$$\begin{equation} \sum_{k\neq j}^n c_k^{\skew4\bar{e}}c_{kj}^{\skew8\bar{f} *}B_{\skew4\bar{e}}^{\skew8\bar{f}}= S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)/(\mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})). \end{equation} \tag{ 19 }$$

Here, the factor $B_{\skew4\bar{e}}^{\skew8\bar{f}}$ combines all parts that do not depend on k or j; the form of equation (19) should again not depend on the dephasing model. But as a substantial difference to the single exciton case, we have several factors $B_{\skew4\bar{e}}^{\skew8\bar{f}}$ multiplied with single and two-exciton wave function coefficients under a sum over k and j.

Therefore, the next step for isolating the two-exciton wave function $c_{kj}^{\skew8\bar{f} *}$ would be isolating the single exciton wave function $c_k^{\skew4\bar{e}}$ on the left-hand side of equation (19). But we have to recognize that the other factors $B_{\skew4\bar{e}}^{\skew8\bar{f}}$ and $c_{kj}^{\skew8\bar{f}*}$ depend also on the single and two-exciton states, so we cannot apply the orthogonality relation like in the single exciton case. Dividing by $B_{\skew4\bar{e}}^{\skew8\bar{f}}$ before applying the orthogonality relation is no alternative, since these factors are unknown and should consequently not appear on the right-hand side. Instead of calculating $B_{\skew4\bar{e}}^{\skew8\bar{f}}$ for every e', we compute the ratio of the factor of two arbitrary exciton states: for this we multiply equation (19) with $c_j^{\tilde {e}}$ and sum over j and obtain

$$\begin{eqnarray*} &&\sum_{j, k\neq j}^n c_k^{\skew4\bar{e}} c_j^{\tilde{e}} c_{kj}^{\skew8\bar{f} *}B_{\skew4\bar{e}}^{\skew8\bar{f}} = \sum_j^n c_j^{\tilde{e}} S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)/(\mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})). \end{eqnarray*} \noindent$$

Exchanging $\skew4\bar{e}$ and $\tilde {e}$ and the index k with j on the left-hand side and using that c^f*_kj = c^f*_jk yields

$$\begin{eqnarray*} &&\sum_{j, k\neq j}^n c_j^{\tilde{e}} c_k^{\skew4\bar{e}} c_{kj}^{\skew8\bar{f} *} B_{\tilde{e}}^{\skew8\bar{f}} = \sum_j^n c_j^{\skew4\bar{e}} S^{\mathrm{loc}}_{E_2}(j, \tilde{e}, \skew8\bar{f}, T_3)/(\mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})), \end{eqnarray*} \noindent$$

so that by dividing the two equations we obtain the ratio

$$\begin{equation} B_{\skew4\bar{e}}^{\skew8\bar{f}}/B_{\tilde{e}}^{\skew8\bar{f}} = \sum_j^n c_j^{\tilde{e}}/c_j^{\skew4\bar{e}}\cdot S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) / S^{\mathrm{loc}}_{E_2}(j, \tilde{e}, \skew8\bar{f}, T_3). \end{equation} \noindent \tag{ 20 }$$

This is the first ingredient in the reconstruction, which can be calculated by the measured or derived quantities for each combination of two arbitrary single exciton states $\skew4\bar{e}$ and $\tilde {e}$. In order to remove factors $B_{\tilde {e}}^{\skew8\bar{f}}$ on the left side, we choose again another arbitrary exciton state $\skew4\hat {e}$ and multiply equation (19) by $B_{\skew4\hat {e}}^{\skew8\bar{f}}$ and divide by $B_{\tilde {e}}^{\skew8\bar{f}}$ to obtain

$$\begin{equation} \sum_{k\neq j}^n c_k^{\skew4\bar{e}}c_{kj}^{\skew8\bar{f} *} B_{\skew4\hat{e}}^{\skew8\bar{f}} = S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)/\left(\mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}})\right) \cdot B_{\skew4\hat{e}}^{\skew8\bar{f}}/B_{\skew4\bar{e}}^{\skew8\bar{f}}. \end{equation} \tag{ 21 }$$

Here the complete right side of equation (21) is known; one contribution is the measurable spectrum, the single exciton coefficients and the ratio $B_{\skew4\hat {e}}^{\skew8\bar{f}}/B_{\tilde {e}}^{\skew8\bar{f}}$, which can be calculated using the measured data and equation (20).

Since $B_{\skew4\hat {e}}^{\skew8\bar{f}}$ does not depend on $\skew4\bar{e}$, we can now calculate the two-exciton wave function by multiplying equation (21) with $c_i^{\skew4\bar{e} *}$ and summing over  $\skew4\bar{e}$. This leads to an orthogonality relation and we obtain

$$\begin{eqnarray*} &&c_{ij}^{\skew8\bar{f} *} B_{\skew4\hat{e}}^{\skew8\bar{f}} = \sum_{\skew4\bar{e}} c_i^{\skew4\bar{e} *} S^{\mathrm{loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)/(\mu_{gj}^*\cdot E_2^{\mathrm{loc}}(\omega_{\skew8\bar{f} \skew4\bar{e}}))\cdot B_{\skew4\hat{e}}^{\skew8\bar{f}}/B_{\skew4\bar{e}}^{\skew8\bar{f}}. \end{eqnarray*}$$

We can now use that the expansion coefficients have been normalized and use $\sum _{i, j < i}^n |{c_{ij}^{\skew8\bar{f}}}|^2=1$ to obtain the two-exciton wave functions of coupled QDs.

After deriving the procedure, we summarize the steps necessary for the reconstruction. As input information we need the dipole moments matrix elements μ_gi for all QDs i and the single excitonic wave function coefficients c^e_i for all single exciton states. As measurement the DQCS signal with a localized second pulse at QD j $S^{\mathrm {loc}}_{E_2}(j,\skew4\bar{e}, \skew8\bar{f}, T_3)$ has to be recorded. In postprocessing, we first calculate the ratios $B_{\skew4\bar{e}}^{\skew8\bar{f}}/B_{\tilde {e}}^{\skew8\bar{f}} = \sum _j^n c_j^{\tilde {e}}/c_j^{\skew4\bar{e}}\cdot S^{\mathrm {loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3) / S^{\mathrm {loc}}_{E_2}(j, \tilde {e}, \skew8\bar{f}, T_3)$ for all combinations of single exciton states. Then $c_{ij}^{\skew8\bar{f}}$ is calculated via $c_{ij}^{\skew8\bar{f}} B_{\skew4\hat {e}}^{\skew8\bar{f}} = \sum _{\skew4\bar{e}} c_i^{\skew4\bar{e} *} S^{\mathrm {loc}}_{E_2}(j, \skew4\bar{e}, \skew8\bar{f}, T_3)/\left (\mu _{gj}^*\cdot E_{2,j}^{\mathrm {loc}}(\omega _{\skew4\bar{e} g})\right )\cdot B_{\skew4\hat {e}}^{\skew8\bar{f}}/B_{\skew4\bar{e}}^{\skew8\bar{f}}$. Afterwards $c_{ij}^{\skew8\bar{f}} B_{\skew4\hat {e}}^{\skew8\bar{f}}$ is normalized, using the two-exciton wave function normalization condition $|B_{\skew4\hat {e}}^{\skew8\bar{f}}|^2=\sum _{ij} |c_{ij}^{\skew8\bar{f}} B_{\skew4\hat {e}}^{\skew8\bar{f}}|^2$. So that we can calculate $c_{ij}^{\skew8\bar{f}}$ up to a global phase.

The results of reconstructing the two-exciton wave functions are shown in table 2. An illustration can be found in figure 6. Again, original states and reconstructed values agree, but not as well as for the exciton state. We can always determine which local two-exciton states contribute most to the delocalized two-exciton state. But crosstalk between the different resonances makes it difficult (cf the reconstruction for f₁) to determine the quantitative contribution of the less contributing local two-exciton states, whereas for the other two two-exciton states the agreement is much more acceptable.

Table 2. Comparison of the original (O) with the reconstructed two-exciton wave function coefficients for a shifted measurement point in (R). The data marked with (F) uses filtered spectra for the reconstruction. Additionally, the results of a measurement point at the center of the resonance for all single exciton states are given (C). For all calculations a perfect localization was assumed. The phase is written in multiples of 2π.

State	Type	|cf12|	|cf13|	|cf23|	φ(cf12)	φ(cf13)	φ(cf23)	Error
f1	O	0.093	0.118	0.989	0.500	0.500	0.000
	R	0.089	0.332	0.939	0.349	0.789	0.503	0.14
	F	0.244	0.367	0.898	0.852	0.819	0.525	0.22
	C	0.140	0.377	0.915	0.313	0.709	0.424	0.19
f2	O	0.988	0.113	0.106	0.500	0.500	0.500
	R	0.793	0.586	0.166	0.762	0.508	0.684	0.48
	F	0.845	0.507	0.171	0.628	0.400	0.347	0.36
	C	0.778	0.601	0.184	0.705	0.424	0.257	0.51
f3	O	0.124	0.987	0.106	0.000	0.500	0.500
	R	0.273	0.935	0.225	0.981	0.544	0.358	0.12
	F	0.201	0.946	0.256	0.879	0.580	0.364	0.10
	C	0.297	0.921	0.253	0.879	0.464	0.270	0.15

The overall reason for this is that the two-exciton wave functions are calculated using the single exciton wave functions: the errors in obtaining the single exciton wave function are cumulated in the two-exciton wave function. Furthermore, for the reconstruction of the two-exciton wave function the spectrum localized at the second pulse enters not only for one combination of Ω₁ and Ω₂ but for every single exciton; this also increases the possible errors. This leads to an overall reduced quality of the reconstructed two-exciton wave function.

In [26], it was shown that a 2D spectrum with the first pulses localized allows to remove certain resonances using postprocessing of the spectra. Since most of the errors in the reconstruction is caused by overlapping resonances, this might enhance the reconstruction of the exciton states.

We will recapitulate the procedure, modified for the extraction process of the coefficients. For this, we note that the spectra equations (14), (16) and (15) have the form of a sum over individual contributions h_e for different exciton states e:

$$\begin{eqnarray} &&S^{\mathrm{loc}}_{E_1}(i,\Omega_1, \Omega_2, t_3) = \sum_{e} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(\Omega_1, \Omega_2, t_3), \nonumber\\ &&S^{(3)}_{\varphi_{\mathrm{III}}}(\Omega_1, \Omega_2, t_3) = \sum_{e} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(\Omega_1, \Omega_2, t_3), \nonumber \\ &&S^{\mathrm{loc}}_{E_1,E_2}(i,j,\Omega_1, \Omega_2, t_3) = \sum_{e} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(j,\Omega_1, \Omega_2, t_3), \nonumber\\ &&S^{\mathrm{loc}}_{E_2}(j,\Omega_1, \Omega_2, t_3) = \sum_{e} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(j,\Omega_1, \Omega_2, t_3), \end{eqnarray} \tag{ 22 }$$

where the functions h_e can be calculated from the measured 2D spectrum:

$$\begin{eqnarray} h_{\skew4\bar{e}}(\Omega_1, \Omega_2, t_3)&=& \sum_i^n c_i^{\skew4\bar{e} } S^{\mathrm{loc}}_{E_1}(i,\Omega_1, \Omega_2, t_3)/(\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg})), \nonumber\\ h_{\skew4\bar{e}}(j,\Omega_1, \Omega_2, t_3) &=& \sum_i^n c_i^{\skew4\bar{e} } S^{\mathrm{loc}}_{E_1,E_2}(i,j,\Omega_1, \Omega_2, t_3)/(\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg})). \end{eqnarray} \tag{ 23 }$$

In the construction of $h_{\skew4\bar{e}}$, we exploited a scalar product, which was constructed by multiplying equation  (22) with  $c_i^{\skew4\bar{e} *}$ and summing over i and using the orthogonality of the delocalized states.

Using this formula, we can filter in postprocessing all resonances to which a certain ground state g to single exciton $\skew4\bar{e}$ transition contributes during the first pulse. The filtered spectra can be calculated as

$$\begin{eqnarray} \fl S_{E_1}^{\mathrm{filter,loc}}(\skew4\bar{e},i,\Omega_1, \Omega_2, t_3)&=& S_{E_1}^{\mathrm{loc}}(i,\Omega_1, \Omega_2, t_3) - c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{\skew4\bar{e}}(i,\Omega_1, \Omega_2, t_3)\nonumber\\ \fl &=& \sum_{e\ \neq\ \skew4\bar{e}} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(\Omega_1, \Omega_2, t_3), \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} \fl S_{\varphi_{\mathrm{III}}}^{\mathrm{filter}}(\skew4\bar{e},\Omega_1, \Omega_2, t_3)&=& S_{\varphi_{\mathrm{III}}}^{(3)}(\Omega_1, \Omega_2, t_3) - \sum_i c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{\skew4\bar{e}}(\Omega_1, \Omega_2, t_3)\nonumber\\ \fl &=& \sum_{e\ \neq\ \skew4\bar{e}} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(\Omega_1, \Omega_2, t_3), \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} \fl S_{E_1,E_2}^{\mathrm{filter,loc}}(\skew4\bar{e},i,j,\Omega_1, \Omega_2, t_3)&=& S_{E_1,E_2}^{\mathrm{loc}}(i,j,\Omega_1, \Omega_2, t_3) - c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{\skew4\bar{e}}(j,\Omega_1, \Omega_2, t_3)\nonumber\\ \fl &=& \sum_{e\ \neq\ \skew4\bar{e}} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(j,\Omega_1, \Omega_2, t_3), \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} \fl S_{E_2}^{\mathrm{filter,loc}}(j,\Omega_1, \Omega_2, t_3)&=& S_{E_2}^{\mathrm{loc}}(\skew4\bar{e},j,\Omega_1, \Omega_2, t_3) -\sum_i c_i^{\skew4\bar{e} *}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{\skew4\bar{e}}(j,\Omega_1, \Omega_2, t_3)\nonumber\\ \fl &=& \sum_{e\ \neq\ \skew4\bar{e}} c_i^{e*}\mu_{gi}^*\cdot E^{\mathrm{loc}}_{1,i}(\omega_{eg}) \,h_{e}(j,\Omega_1, \Omega_2, t_3). \end{eqnarray} \tag{ 27 }$$

In figure 3(b), for each resonance a filtered spectrum is presented; we can clearly see that the corresponding resonances are removed from spectra. Therefore it is expected that using the filtered version (out filtering of the already reconstructed states) of the spectrum can clearly enhance further reconstruction of exciton states.

Here, we try to investigate if the filtered spectrum is improving the quality of the reconstructed coefficients. The filtering improves the quality of the reconstructed wave function as can be seen from the error rates for the reconstructed wave function using a filtered spectrum in tables 1 and 2.

As discussed in section 3.6.1, we find the best reconstruction for e₁ and the second best for e₂, since e₁ dominates the spectrum. This is the reason why we used, while reconstructing the exciton state e₂, a spectrum with filtering the state e₁ out. For the exciton state e₃ we filter two states, since the states e₁ and e₂ are now known. We do not get better values for every single QD, but there is still an overall improvement visible. The results show that filtering can reduce the error of the reconstruction of single excitons (see table 1) at least by a factor of 2 and in the case of the weak resonance e₃ by an order of magnitude.

Since one can determine all exciton states, filtering all the single exciton states is possible for reconstructing the two-exciton states. Again, the filtering technique supplies results for most coefficients at a higher quality as can be seen in table 2. But the improvement is not as high as in the case of single excitons, since we have no available filtering for the two-exciton part.