Excitonic Wave Function Reconstruction from Near-Field Spectra Using Machine Learning Techniques

A general problem in quantum mechanics is the reconstruction of eigenstate wave functions from measured data. In the case of molecular aggregates, information about excitonic eigenstates is vitally important to understand their optical and transport properties. Here we show that from spatially resolved near field spectra it is possible to reconstruct the underlying delocalized aggregate eigenfunctions. Although this high-dimensional nonlinear problem defies standard numerical or analytical approaches, we have found that it can be solved using a convolutional neural network. For both one-dimensional and two-dimensional aggregates we find that the reconstruction is robust to various types of disorder and noise.

Figure 1

Top: Sketch of the setup. The red arrows represent the transition dipole moments $\overrightarrow{\mu }$ of the molecules. We use distance $a$ as our unit of length. The aggregate is located in the $x\text{−}y$ plane. Bottom: Left: spatially and frequency resolved near-field spectrum stemming from a Hertzian dipole located $2a$ above the aggregate and with dipole pointing along the $z$ axis. Middle: spatially resolved spectra for fixed frequency are fed into a CNN. Right: extracted wave functions corresponding to the respective spectra.

Figure 2

Examples of eigenstate wave functions and corresponding spectra $\mathcal{A}({\overrightarrow{R}}_{\mathrm{dip}})$ used to train the CNN for a 1D linear chain with $N=20$ molecules (top) and a 2D array with $N=10\times 5$ molecules (bottom). The excitation-dipole is located at a distance $z_{\mathrm{dip}}=2a$ above the aggregate (the distance $a$ is defined in Fig. 1). The wave functions corresponding to the spectra are shown in the insets in the 1D case (together with the predictions of the CNN) and in the upper panel in the 2D case (the arrows indicate the orientation of the molecular transition dipoles). In all cases shown here the loss is ${10}^{-2}$. Further examples, including the disorder-free case, can be found in the Supplemental Material [37].

Figure 3

Distribution of loss-values for different values of ${\sigma }_{\mathrm{d}}$ as indicated in each panel for the 1D (top) and the 2D (bottom) cases. For each ${\sigma }_{\mathrm{d}}$ the number of realizations is 10 000 (1D case) and 1000 (2D case). We take all eigenstates into account.

Figure 4

Top: Distribution of the loss-values when noise is added to the spectra of the 1D disorder-free system. In each panel a specific value ${\sigma }_{\mathrm{n}}$ for the relative strength of the noise has been used, which is provided in the panel. For all cases the CNN has been trained with ${\sigma }_{\mathrm{n}}=0.1$. Bottom: For the ${\sigma }_{\mathrm{n}}$ values of the top row we show one example of how the noise alters the spectrum. In all these cases the same noise free spectrum is used.