Cumulant expansion for the treatment of light–matter interactions in arbitrary material structures

Abstract

Strong coupling of quantum emitters with confined electromagnetic modes of nanophotonic structures may be used to change optical, chemical and transport properties of materials, with significant theoretical effort invested towards a better understanding of this phenomenon. However, a full theoretical description of both matter and light is an extremely challenging task. Typical theoretical approaches simplify the description of the photonic environment by describing it as a single or few modes. While this approximation is accurate in some cases, it breaks down strongly in complex environments, such as within plasmonic nanocavities, and the electromagnetic environment must be fully taken into account. This requires the quantum description of a continuum of bosonic modes, a problem that is computationally hard. We here investigate a compromise where the quantum character of light is taken into account at modest computational cost. To do so, we focus on a quantum emitter that interacts with an arbitrary photonic spectral density and employ the cumulant or cluster expansion method to the Heisenberg equations of motion up to first, second and third order. We benchmark the method by comparing with exact solutions for specific situations and show that it can accurately represent dynamics for many parameter ranges.Light-matter interaction is of paramount importance for unraveling the laws of nature and its deep understanding allows us to control and manipulate physical and chemical systems. In particular, one can modify the properties of a quantum emitter simply by changing its electromagnetic environment, for example by enclosing it within an optical cavity. This may give rise to a change of the decay rate for spontaneous emission in the weak coupling regime, the so-called Purcell effect 1 , or to the appearance of hybrid light-matter states, socalled polaritons, in the strong-coupling regime 2-5 . Over the last decades, it has been shown that strong light-matter coupling can be achieved using a large variety of physical implementations as the "cavity" that provides the electromagnetic field confinement. These include Fabry-Perot cavities consisting of two mirrors 5 , propagating surface plasmon polaritons 6 , plasmonic hole 7 and nanoparticle arrays 8 , isolated plasmonic nanoparticles 9 and nanoparticle-on-mirror geometries 10,11 , as well as hybrid cavities combining plasmonic and dielectric materials [12][13][14] . In many of these systems, the electromagnetic field modes are not well-described by isolated lossy cavity modes, and a correct treatment demands theoretical approaches that are able to deal with the complexity of the electromagnetic field modes and their spectrum.In principle, to treat the problem of light-matter interaction, one can rely on the most general theory that describes light and matter on equal footing, i.e., quantum electrodynamics (QED) 15 . However, treating all light and matter degrees of freedom in the systems described above in a quantum mechanical way is an intractable problem and approx...

Macroscopic QED for quantum nanophotonics: emitter-centered modes as a minimal basis for multiemitter problems

“…Adapting an argument by Sánchez-Barquilla et al [45], here we show that macroscopic QED also enables a straightforward treatment of external incoming EM fields, in particular for the experimentally most relevant case of a classical laser pulse. Assuming that the incoming laser field at the initial time t = 0 has not yet interacted with the emitters (i.e., it describes a pulse localized in space in a region far away from the emitters), it can simply be described by a product of coherent states of the EM modes for the initial wave function, ψ(0)〉 ∏ n |α n (0)〉 ∏ n e α n (0)a † n −α * n (0)a n |0〉, where the index n here runs over all indices of the EM basis (λ, r, ω), and the α n (0) correspond to the classical amplitudes obtained when expressing the laser pulse in the basis defined by these modes.…”

“…Comparison with the dynamics of emitter 1 when emitter 2 is not present (shown as a dashed blue line in Figure 3) furthermore reveals that there is also significant transfer of the population back from emitter 2 to emitter 1. We note that while the system treated here is in the weak coupling regime and no Rabi oscillations are present, the method works equally well under strong lightmatter coupling [45][46][47]. Within the RWA employed here, the emitter populations for long times go to zero.…”

“…We also note that with "minimal basis," we are here referring to a minimal complete basis for the medium-assisted EM field, i.e., this basis contains all the information about the material structure playing the role of the cavity or antenna, and no approximations are made in obtaining it. This then makes it appropriate to serve as a starting point either for numerical treatments [45,46] or for deriving simpler models where, e.g., the full EM spectrum is described by a few lossy modes [47].…”

Ab initio quantum models for thin-film x-ray cavity QED

“…Instead, all the information about the cavity environment is contained in a single function. In this context, we note that effective modes for the macroscopic QED Hamiltonian have been introduced previously [112] and have recently found applications for cavity interactions of multiple atoms [113,114] and strongly coupled light-matter dynamics in complex environments [115]. Alternatively, the Green's function can in principle be approximated in terms of mode parameters [75].…”

“…II B may be advantageous. Alternatively, effective continuum modes [112] based on the Green's function quantization may prove useful, which have been shown to be tractable via cumulant expansions [115] in certain parameter regimes. Following the approach in Refs.…”