Modeling of light-matter interactions with neural networks

We show that a neural network (NN) can be used for automated generation of computer models of light-matter interaction. Nonlinear input-output maps are created for phase-shaped femtosecond laser pulses in the exemplary cases of second-harmonic generation and molecular fluorescence yield. Simulations and experiments demonstrate that the NN has the capability of generalizing and extrapolating beyond initial training data, by predicting the response of the investigated systems for arbitrary laser pulse shapes. Applications are envisioned in the area of quantum control, specifically for the interpolation and extrapolation of control maps and possibly as a tool for investigating control mechanisms. In a wider scope, neural networks might generally provide effective computer models for light-matter interactions for cases where ab initio calculations are intractable.

Figure 1

(Color online) Correspondence between optical experiments with shaped femtosecond laser pulses and functional input-output maps.

Figure 2

(Color online) Schematic representation of a NN architecture and working principle. Neurons are depicted as spheres, and the flow of information and error correction is indicated by right-pointing and left-pointing arrows, respectively. If the weights $w_i$ are adjusted correctly, the input vector is mapped onto the desired output vector by the underlying functional map.

Figure 3

(Color online) Simulated examples for the pulse shapes used during the training process. The first column displays the spectral phase (dashed black line) and intensity (solid line) while the second column shows the respective temporal phase (dashed black line) and intensity (solid line) profile. (a) Second-order spectral phase of $6000{\mathrm{fs}}^2$, (b) third-order spectral phase of ${10}^5{\mathrm{fs}}^3$, and (c) sinusoidal spectral phase with $A=\pi ∕2$, $\tau =250\mathrm{fs}$, ${\varphi }_0=\pi ∕2$.

Figure 4

(Color online) Experimental setup. Using a 128 pixel LCD pulse shaper, a variety of pulse shapes is first generated, and the corresponding values of observables (SHG signal and NK88 fluorescence) are recorded. This data is then used for the training of a NN. In a second step, an evolutionary optimization is carried out, with the goal of maximizing the SHG signal. Finally, the data gained in this optimization is used for testing the accuracy of the NN predictions for pulse shapes not present in the training data set.

Figure 5

(Color online) (a) Molecular structure of the two isomer configurations of the cyanine dye NK88. Irradiated by light of the proper wavelength, this molecule can undergo trans-cis isomerization. (b) Simplified potential energy surface. The reaction coordinate is the twist angle about the $\mathrm{C}\mathrm{C}$ double bond. The excitation at $400\mathrm{nm}$ and the molecular fluorescence at $500\mathrm{nm}$ are indicated by up and down arrows, respectively. (c) The ground-state absorption spectrum of the trans isomer.

Figure 6

(Color online) Mean square errors, obtained by averaging the squared deviation $(o-t{)}^2$ between the NN output $o$ and the simulated SHG signal $t$ over the training data set for the following input parametrizations: (A) the 128 spectral phases, (B) the sine and cosine of the spectral phases, (C) the real and imaginary parts of $E\left(\omega \right)$, and (D) the envelope of the electric field in the time domain, $E\left(t\right)$.

Figure 7

(Color online) Capability of the NN to reproduce the data presented during training, illustrated for pulses with quadratic spectral phase. The solid lines correspond to NN predictions, the symbols to the actual results of (a) SHG simulation and (b) SHG experiment (circles/upper curve) as well as molecular fluorescence from NK88 in methanol (diamonds/lower curve).

Figure 8

(Color online) Ability of the NN to generalize beyond the data presented during training, illustrated for pulses obtained during an evolutionary optimization. The solid lines correspond to NN predictions, the symbols to the actual results of (a) SHG simulation and (b) SHG experiment (upper curve) as well as molecular fluorescence from NK88 in methanol (lower curve). Signal values are shown for all individuals of all generations of the optimization, hence they do not correspond to a monotonous function.

Figure 9

(Color online) Sample electric field generated by the evolutionary algorithm during the experimental SHG maximization. The electric field of the individuum number 300 is shown.

Figure 10

(Color online) Frequency distribution of the absolute error $(o_1-t_1)$ between $o_1$, the SHG signal predicted by the NN, and $t_1$, the experimentally measured SHG signal.