# A Comparison of Different Classical, Semiclassical and Quantum Treatments of Light-Matter Interactions: Understanding Energy Conservation

## TL;DR

This paper compares classical, semiclassical, and quantum methods for simulating light-matter interactions, highlighting the accuracy of the Ehrenfest+R approach in energy conservation and correct optical signal prediction.

## Contribution

It demonstrates that the Ehrenfest+R method accurately reproduces quantum optical signals and energy conservation, outperforming traditional approaches in semiclassical light-matter interaction simulations.

## Key findings

- Ehrenfest dynamics are accurate only in the linear response regime.
- Maxwell-Bloch and dielectric theory predict incorrect signals due to double-counting.
- Ehrenfest+R approach correctly balances self-interaction and quantum fluctuations.

## Abstract

The optical response of an electronic two-level system (TLS) coupled to an incident continuous wave (cw) electromagnetic (EM) field is simulated explicitly in one dimension by the following five approaches: (i) the coupled Maxwell-Bloch equations, (ii) the optical Bloch equation (OBE), (iii) Ehrenfest dynamics, (iv) the Ehrenfest+R approach and (v) classical dielectric theory (CDT). Our findings are as follows: (i) standard Ehrenfest dynamics predict the correct optical signals only in the linear response regime where vacuum fluctuations are not important; (ii) both the coupled Maxwell-Bloch equations and CDT predict incorrect features for the optical signals in the linear response regime due to a double-counting of self-interaction; (iii) by exactly balancing the effects of self-interaction versus the effects of quantum fluctuations (and insisting on energy conservation), the Ehrenfest+R approach generates the correct optical signals in the linear regime and slightly beyond, yielding, e.g., the correct ratio between the coherent and incoherent scattering EM fields. As such, Ehrenfest+R dynamics agree with dynamics from the quantum OBE, but whereas the latter is easily applicable only for a single TLS in vacuum, the former should be applicable to large systems in environments with arbitrary dielectrics. Thus, this benchmark study suggests that the Ehrenfest+R approach may be very advantageous for simulating light-matter interactions semiclassically.