# Numerical Method for Nonlinear Optical Spectroscopies: Ultrafast Ultrafast Spectroscopy

## TL;DR

The paper introduces UF$^2$, a new numerical method for efficiently calculating nonlinear optical spectroscopic signals, offering significant computational speed-ups while maintaining accuracy.

## Contribution

UF$^2$ is a novel, simple, and efficient numerical approach for computing nonlinear wavepackets in optical spectroscopy, outperforming existing methods.

## Key findings

- UF$^2$ achieves identical spectra to standard methods.
- The method offers substantial computational speed-ups.
- It is easy to implement and understand.

## Abstract

We outline a novel numerical method, called Ultrafast Ultrafast (UF$^2$), for calculating the $n^\text{th}$-order wavepackets required for calculating n-wave mixing signals. The method is simple to implement, and we demonstrate that it is computationally more efficient than other methods in a wide range of use cases. Resulting spectra are identical to those calculated using the standard response function formalism but with increased efficiency. The computational speed-ups of UF$^2$ come from (a) non-perturbative and costless propagation of the system time-evolution (b) numerical propagation only at times when perturbative optical pulses are non-zero and (c) use of the fast Fourier transform convolution algorithm for efficient numerical propagation. The simplicity of this formalism allows us to write a simple software package that is as easy to use and understand as the Feynman diagrams that organize the understanding of $n$-wave mixing processes.