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

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 balancingthe 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 1 arXiv:1812.03265v2 [physics.optics] 16 Jan 2019 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.

Dissipative Equation of Motion for Electromagnetic Radiation in Quantum Dynamics

“…Most implementations of these methods have been limited to just a few-often just two-level systems. An overview and comparative analysis of these approaches have been recently contributed by Nitzan, Subotnik, and co-authors [34][35][36]. In particular, these researchers have proposed a correction to Ehrenfest dynamics that, by construction, reproduces spontaneous emission according to the radiative decay rate deduced from Fermi's golden rule (FGR) [37].…”

Quasiclassical modeling of cavity quantum electrodynamics

“…Recently, attempts have been made to include quantum EM-field effects even when evolving classical EM fields [29,30]. For example, in the recently proposed Ehrenfest+R approach [28,[31][32][33], our research group included vacuum fluctuations by propagating a swarm of augmented Maxwell-Schrödinger equations. Nevertheless, because Ehrenfest+R was developed in free space, the performance of the method in cavities is unknown.…”

Incorporating Lindblad Decay Dynamics into Mixed Quantum-Classical Simulations

“…We want to emphasize the connections and differences between the current L-MFE method and previous approaches that try to accomplish Lindblad dynamics through stochastic wavefunction approaches, such as Monte Carlo wave-function methods [35][36][37][38] and the Ehren-fest+R approach. [10][11][12]39,40 The Monte Carlo wavefunction method 35,36 expresses the stochastic time evolution of the system's wavefunction as follows 34,56 d|ψ…”

“…In particular, we give the derivation of a method to propagate the coefficients of the MFE method that fully agrees with Lindblad dynamics. The coefficient-propagation method derived using this framework is inspired by the Ehrenfest+R method; [10][11][12]39,40 however, our approach exactly recovers Lindblad dynamics whereas the Ehren-fest+R approach does not. Additionally, we derive a simple fix to the approximations present in the Ehren-fest+R method and give rigorous justifications for particular algorithmic choices.…”