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arXiv:2506.21702 · Journal of Chemical Theory and Computation

Focused Sampling for Low-Cost and Accurate Ehrenfest Modeling of Cavity Quantum Electrodynamics

Ming‐Hsiu Hsieh, Alex Krotz, Roel Tempelaar

Published Nov 28, 2025• 3 citations
arXivHTMLPDF

Abstract

An economic modeling approach for cavity quantum electrodynamics is provided by mean-field dynamics, wherein the optical field is described classically while a self-consistent interaction with quantum emitters is incorporated through the Ehrenfest theorem. However, conventional implementations of mean-field dynamics are known to suffer from a catastrophic leakage of zero-point energy, to lose accuracy in the short-cavity limit, and to require large numbers of trajectories to be sampled. Here, we address these three shortcomings within a single integrated approach. This approach builds on our recently proposed modification of the Ehrenfest theorem, referred to as decoupled mean-field (DC-MF) dynamics, in combination with a focused sampling scheme that enforces zero-point energy at the single-trajectory level. The approach is shown to yield high accuracy in both short and long-cavity limits while reaching convergence within a minimal amount of trajectories.

DOI
10.1021/acs.jctc.5c01639

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An economic modeling approach for cavity quantum electrodynamics is provided by mean-field dynamics, wherein the optical field is described classically while a self-consistent interaction with quantum emitters is incorporated through the Ehrenfest theorem. However, conventional implementations of mean-field dynamics are known to suffer from a catastrophic leakage of zero-point energy, to lose accuracy in the short-cavity limit, and to require large numbers of trajectories to be sampled. Here, we address these three shortcomings within a single integrated approach. This approach builds on our recently proposed modification of the Ehrenfest theorem, referred to as decoupled mean-field (DC-MF) dynamics, in combination with a focused sampling scheme that enforces zero-point energy at the single-trajectory level. The approach is shown to yield high accuracy in both short and long-cavity limits while reaching convergence within a minimal amount of trajectories.
A much-needed solution for the efficient modeling of strong coupling between matter and optical cavity modes is offered by mean-field mixed quantum-classical dynamics, where a classical cavity field interacts self-consistently with quantum states of matter through Ehrenfest's theorem. We previously introduced a modified mean-field approach, referred to as decoupled mean-field (DC-MF) dynamics, wherein vacuum fluctuations of the cavity field are decoupled from the quantum-mechanical ground state as a means to resolve an unphysical drawing of energy from the vacuum fluctuations by a two-level atom. Here, we generalize DC-MF dynamics for an arbitrary number of (nondegenerate) atomic levels and show that it resolves an unphysical lack of emission from a three-level atom predicted by conventional mean-field dynamics. We furthermore show DC-MF to provide an improved description of reabsorption and (resonant) two-photon emission processes.
We describe vacuum fluctuations and photon-field correlations in interacting quantum mechanical light-matter systems by generalizing the application of mixed quantum classical dynamics techniques. We employ the multitrajectory implementation of Ehrenfest mean-field theory, traditionally developed for electron-nuclear problems, to simulate the spontaneous emission of radiation in a model quantum electrodynamical cavity-bound atomic system. We investigate the performance of this approach in capturing the dynamics of spontaneous emission from the perspective of both the atomic system and the cavity photon field through a detailed comparison with exact benchmark quantum mechanical observables and correlation functions. By properly accounting for the quantum statistics of the vacuum field, while using mixed quantum classical (mean-field) trajectories to describe the evolution, we identify a surprisingly accurate and promising route towards describing quantum effects in realistic correlated light-matter systems.
We model a collection of NNN two-level systems (TLSs) coupled to a multimode cavity via Meyer-Miller-Stock-Thoss (MMST) dynamics, sampling both electronic and photonic zero-point energies (ZPEs) and propagating independent trajectories in Wigner phase space. By investigating the ground-state stability of a single TLS, we use MMST dynamics to separately study both electronic ZPE effects (which would naively lead to the breakdown of the electronic ground state) and photonic ZPE effects (which would naively lead to spontaneous absorption). By contrast, including both effects, i.e., sampling both electronic and photonic ZPEs, leads to the dynamical stability of the electronic ground state. Therefore, MMST dynamics provides a practical way to identify the contributions of self-interaction and vacuum fluctuations. More importantly, we find that MMST dynamics can predict accurate quantum dynamics for both electronic populations and electromagnetic field intensity in the high saturation limit. For a single TLS in a cavity, MMST dynamics correctly predicts the initial exponential decay of spontaneous emission, Poincar\'e recurrences, and the positional dependence of a spontaneous emission rate. For an array of NNN equally spaced TLSs with only one TLS excited initially, MMST dynamics correctly predicts the modification of spontaneous emission rate as a function of the spacing between TLSs. Finally, MMST dynamics also correctly models Dicke's superradiance and subradiance, i.e., the dynamics when all TLSs are excited initially, including the correct quantum statistics for the delay time (as found by counting trajectories, for which a full quantum simulation is hard to achieve). Therefore, this work raises the possibility of simulating large-scale collective light-matter interactions with methods beyond mean-field theory.
A single two-level emitter coupled to a single mode of an electromagnetic resonator constitutes the most fundamental instance of light--matter interaction. Realizations of this type of system are investigated in the field of cavity quantum electrodynamics (cavity QED). Particularly important are schemes in which the coupling is strong enough to exceed the spontaneous emission rate as well as cavity damping. In this case, the coherent exchange of excitation between source and photon is the dominating process, providing the unique possibility to deterministically control the quantum evolution of the system. Great progress has recently been accomplished, technologically as well as conceptually, and the focus section in this issue of Journal of Optics B: Quantum and Semiclassical Optics features five articles highlighting important recent developments in the field. Strong coupling between single atoms and the electromagnetic cavity field have been achieved in two distinct regimes, the optical and the microwave region. The latter is represented by the micromaser, a high-Q microwave cavity pumped by a beam of highly excited Rydberg atoms. It has proved to be an almost ideal system for the generation of non-classical field states. A constraint in these experiments is the fact that the field inside the cavity can only be examined indirectly, by analysing the occupation of atoms leaving the cavity. While only two atomic states were previously identified for this purpose, V A Reshetov shows in his article (pages 119--126) that more information on the field may be extracted by taking advantage of the Zeeman substructure of the atomic levels. In the optical domain of cavity QED, by contrast, the state of the resonator field may be probed directly by measuring the light leaking from the cavity using photodetectors. In order to minimize damping, high-quality cavities must be employed. In most optical experiments these are Fabry--Perot resonators with ultra-high reflectivity mirrors, with losses as low as a few parts per million being state of the art. At the same time, the coupling of atoms and photons is enhanced by reducing the length of the cavity, so that strong coupling conditions are reached. A very attractive application of this type of cavity QED setup is the possibility of exerting control over the quantum dynamics of the system. An impressive example is the implementation of quantum feedback, reported in the paper by W P Smith and L A Orozco (pages 127--134). Using information gained from detecting a single photon escaping from the cavity, active feedback is applied to halt the dynamical evolution of the system. This is the first time active feedback has been used to modify the quantum evolution of a system at the single-photon level. The authors discuss how detunings between the subsystems affect the quantum feedback, an issue particularly relevant for ensembles of atoms with a spread in detuning. Perfect quantum feedback in this case requires the use of additional parameters to control the system. The majority of optical cavity QED experiments performed so far have used as a medium atoms traversing the cavity on random trajectories, such as, for example, thermal atomic beams. In this case, the observation of quantum effects is degraded in two ways. The Doppler effect and the finite transit time lead to line broadening, while the spatial structure of the cavity mode results in atoms in different locations experiencing different coupling with the photonic field. J E Reiner et al (pages 135--142) investigate the impact of these effects on quantum fluctuations in their cavity QED system. In order to overcome line broadening, they have developed a continuous cold atomic beam, extracted from a magneto-optical trap. An even better control of broadening effects and fluctuating coupling strength is provided by trapping the atoms inside the cavity and cooling them to temperatures at which their position spread is smaller than the optical wavelength. Much recent activity has been devoted to developing such an ultimate cavity QED system. A Vukics et al (pages 143--153) present a theoretical proposal for dynamically trapping and cooling atoms inside a cavity. By driving the atoms with a standing wave injected from the side of the cavity and by weakly exciting the cavity field itself, they predict three-dimensional trapping of the atom for periods around 1 second, an extremely long time on the typical scale of cavity QED dynamics. The essential component of their scheme is the interference between the two driving fields, determining the temperature of the particle and the stable trapping positions. In the quest for better confinement of radiation, lately, an appealing alternative to the Fabry--Perot cavity has been investigated -- the optical microsphere resonator. A sphere with a diameter of a few tens of microns, it supports so-called whispering gallery modes near its circumference, for which quality factors up to 10 10 have been measured. In order to exploit the correspondingly low field damping rates for cavity QED, a method must be found to couple atomic or solid state emitters to these modes. S Göotzinger et al (pages 154--158) report the coupling of single nanocrystals on the surface of a microsphere through the evanescent field of the mode. Selective excitation of a single nano-emitter was accomplished with a scanning confocal microscope. The system is a promising candidate for cavity QED with single-quantum emitters. The ability to fully control the quantum dynamics of atoms and light is beginning to spawn early applications. The most notable example is quantum information processing. This requires the coherent manipulation of single-quantum systems, which is ideally provided by cavity QED interaction in the strong-coupling limit. The development of practical devices will be among the principal future tasks for research in the field. This will include physical systems, which have not been covered in this issue, such as semiconductors, quantum dots as emitters, or photonic crystals as resonator structures. Cavity QED is therefore guaranteed to remain a vigorous field in the years to come.
Cavity Quantum Electrodynamics (Cavity-QED) offers a powerful framework for controlling light-matter interactions, yet first-principles approaches commonly assume ideal, lossless cavity fields. Here, we introduce a minimal extension to the Pauli-Fierz Hamiltonian that accounts for cavity dissipation through the experimentally accessible cavity quality factor Q. The electromagnetic field is described as a damped harmonic oscillator within the very weak damping approximation. In this framework, cavity losses only enter the light-matter interaction and dipole self-energy terms, without introducing additional photonic degrees of freedom. A time-dependent unitary transformation yields a real-valued Hermitian Hamiltonian that can be incorporated into existing ab-initio QED methods without additional computational cost or changes in scaling. We combined the approach to the QED-CCSD method and investigated the effects of cavity dissipation on the optical spectrum of a p-nitroaniline molecule under strong light-matter coupling. We found that, for realistic quality factors, cavity dissipation can reduce the Rabi splitting by more than 20% compared to the ideal-cavity limit, even though the system remains in the strong-coupling regime. Convergence towards the ideal-cavity limit then requires substantially higher quality factors than those needed to establish strong coupling. These results demonstrate that cavity dissipation can significantly affect polaritonic energetics and highlight the importance of incorporating experimentally realistic cavity losses into quantitative ab-initio Cavity-QED simulations to accurately guide experimental protocols.
Mean-field mixed quantum-classical dynamics could provide a much-needed means to inexpensively model quantum electrodynamical phenomena by describing the optical field and its vacuum fluctuations classically. However, this approach is known to suffer from an unphysical transfer of energy out of the vacuum fluctuations when the light-matter coupling becomes strong. We highlight this issue for the case of an atom in an optical cavity and resolve it by introducing an additional set of classical coordinates to specifically represent vacuum fluctuations whose light-matter interaction is scaled by the instantaneous ground-state population of the atom. This not only rigorously prevents the aforementioned unphysical energy transfer but is also shown to yield a radically improved accuracy in terms of the atomic population and the optical field dynamics, generating results in excellent agreement with full quantum calculations. As such, the resulting method emerges as an attractive solution for the affordable modeling of strong light-matter coupling phenomena involving macroscopic numbers of optical modes.