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Optics Express

Simulation of spatiotemporal light dynamics based on the time-dependent Schrödinger equation

Maria Richter, Felipe Morales, Serguei Patchkovskii, Anton Husakou

Published Oct 25, 2023• 1 citation
Publisher

Abstract

We establish a first-principle model for the simulation of spatiotemporal light pulse dynamics based on the combination of the time-dependent Schrödinger equation and the unidirectional propagation equation. The proposed numerical scheme enables computationally efficient simulation while being stable and accurate. We use the new model to examine self-focusing of a short pulse in atomic hydrogen and show that an accurate description of the excited-levels dynamics can only be achieved by a propagation model with an ab-initio description of the light-matter interaction, which accounts for the laser-dressed multilevel structure of the system, including bound and free states, and its sub-cycle response.

DOI
10.1364/oe.499406

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•Jan 1, 2016•0 citations
We establish a first-principle model for the simulation of spatiotemporal light pulse dynamics based on the combination of the time-dependent Schrödinger equation and the unidirectional propagation equation. The proposed numerical scheme enables computationally efficient simulation while being stable and accurate. We use the new model to examine self-focusing of a short pulse in atomic hydrogen and show that an accurate description of the excited-levels dynamics can only be achieved by a propagation model with an ab-initio description of the light-matter interaction, which accounts for the laser-dressed multilevel structure of the system, including bound and free states, and its sub-cycle response.
We show that, for the case of resonant media, the available models for unidirectional propagation of short pulses can face serious challenges with respect to numerical efficiency, accuracy, or numerical artifacts. We propose an alternative approach based on a propagator operator defined in the time domain. This approach enables precise simulations using short time windows even for resonant media and facilitates coupling of the propagation equation with first-principle methods such as the time-dependent Schödinger equation. Additionally, we develop a numerically efficient recipe to construct and apply such a propagator operator.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
A scheme of designing laser pulses for controlling discrete quantum states has been proposed, relying on the recently developed highly accurate Maxwell-Schr\"odinger hybrid simulation. A single electron confined in a quasi-one-dimensional nanoscale potential well has been used as an illustrative example and a control pulse transferring completely the probability density from the ground state to the first excited state has been designed by the present Maxwell-Schr\"odinger hybrid scheme and by the conventional one that solves the time-dependent Schr\"odinger equation only without accounting for feedback from the electron system to the external electromagnetic field. The resultant pulses obtained by these two methods can be different largely owing to the modification of the laser field by the locally strong radiation from the excited electron. The present study demonstrates that light control pulses designed by the conventional method may need to be rectified for practical implementation in experiments.
The stabilization of a one-dimensional delta potential model atom under the influence of a high frequency, short laser pulse is investigated, through both an ab initio numerical integration of the time-dependent Schrödinger equation and a Floquet calculation in the single-dressed-state approximation. Provided the intensity is not too high, the variation of the photodetachment probability is explained by adiabatic stabilization and its oscillating behaviour by that of the Floquet rate of photodetachment.
This dissertation demonstrates the usefulness of exactly solvable quantum models in the investigation of light-matter interaction phenomena associated with the propagation of ultrashort laser pulses through gaseous media. This work fits into the larger research effort towards remedying the weaker portions of the standard set of medium modeling equations commonly used in simulations. The ultimate goal is to provide a self-consistent quantum mechanical description that can integrate Maxwell and Schrödinger systems and provide a means to realistically simulate nonlinear optical experiments on relevant scales. The study of exactly solvable models begins with one of the simplest quantum systems available, one with a 1D Dirac-delta function potential plus interaction with the light field. This model contains, in the simplest form, the most important "ingredients" that control optical filamentation, i.e. discrete and continuum electronic states. The importance of both states is emphasized in the optical intensity regime in which filaments form, where both kinds of electronic states simultaneously play a role and may not even be distinguishable. For this model atom, an analytical solution for the time-dependent light-induced atomic response from an arbitrary excitation waveform is obtained. Although this system is well-known and has been studied for decades, this result is probably the most practically useful and general one obtained thus far. Numerical implementation details of the result are also given as the task is far from trivial. Given an efficient implementation, the model is used in light-matter interaction simulations and from these it is apparent that even this toy model can qualitatively reproduce many of the nonlinear phenomena seen in experiments. Not only does this model capture the basic physics of optical filamentation, but it is also well-suited for high harmonic generation simulations. Next, a theoretical framework for using Stark resonant states (or metastable states) to represent the medium's polarization response is presented. Researchers have recognized long ago the utility of Gamow resonant states as a description of various decay processes. Even though a bound electron experiences a similar decay-like process as it transitions into the continuum upon ionization, it was unclear whether field-induced Stark resonant states carry physically relevant information. It is found that they do, and in particular it is possible to use them to capture a medium's polarization response. To this end, two quantum systems with potentials represented by a 1D Dirac-delta function and a 1D square well are solved, and all the necessary quantities for their use as medium models are presented. From these results it is possible to conjecture some general properties that hold for all resonance systems, including systems that reside in higher than one dimensional space. Finally, as a practical application of this theory, the Metastable Electronic State Approach (MESA) is presented as a quantum-based replacement for the standard medium modeling equations.