Real-time simulation makes understanding nonlinear quantum dynamics much easier

In everyday life, many systems obey simple, linear rules: if you pull twice as hard on a spring, it stretches twice as far. However, when we introduce very large forces or complicated interactions, that linear rule breaks down into a "nonlinear" regime. In quantum systems, these nonlinear responses not only deepen our fundamental understanding of quantum dynamics, but also guide the design of functional materials and future quantum technologies.

"Some of the most interesting behaviors of quantum materials and other physical systems are revealed when they're exposed to unusual conditions," remarks Atsushi Ono, Assistant Professor of Physics at Tohoku University and author of the research paper.

Physicists have long computed nonlinear response functions in the frequency domain. In this approach, one builds so-called multipoint correlation functions--calculations that combine information at different points in space and time--and then employs perturbation theory to assemble the final result. While well established, these methods become extremely involved when particles in a material are strongly correlated, or when significant dissipation occurs. In those cases, carrying out all the required multipoint correlation calculations demands a large number of steps and complex diagrams.

"The previous method required a complicated juggling act of dozens of correlation functions," explains Ono. "This new framework allows for extracting information about nonlinear responses without the need for explicit multipoint correlations. It simplifies the process."

To achieve this, the new method applies carefully designed external fields and directly tracks how observables evolve in time. By applying a numerical functional-derivative procedure to that time-evolution data, one can extract the nonlinear response functions at each desired order. In practice, the only requirement is the ability to propagate the system forward in time under the chosen field.

Because this framework relies solely on time-evolution, it can be integrated into virtually any real-time simulation tool. As a result, systems that were previously hard to treat with standard frequency-domain approaches can now be studied efficiently with a unified procedure.

The findings in this study validate the framework's reliability and applicability, paving the way for exploring nonlinear phenomena across a wide variety of dynamical systems and enabling novel spectroscopy techniques, improved materials design, and the discovery of unexpected behaviors. This framework may serve as a useful guide that helps us develop next-generation quantum devices, such as quantum computers and atomic optics.

This study was published in the journal Physical Review Letters on July 9, 2025.