Chiral Polaritonics: Analytical Solutions, Intuition, and Use

Abstract

Preferential selection of a given enantiomer over its chiral counterpart has become increasingly relevant in the advent of the next era of medical drug design. In parallel, cavity quantum electrodynamics has grown into a solid framework to control energy transfer and chemical reactivity, the latter requiring strong coupling. In this work, we derive an analytical solution to a system of many chiral emitters interacting with a chiral cavity similar to the widely used Tavis–Cummings and Hopfield models of quantum optics. We are able to estimate the discriminating strength of chiral polaritonics, discuss possible future development directions and exciting applications such as elucidating homochirality, and deliver much needed intuition to foster the newly flourishing field of chiral polaritonics.

Toward Molecular Chiral Polaritons

“…Bare matter Ĥ M and field Ĥ L Hamiltonian feature the bare matter ℏω m and cavity ℏω cav excitation energy, respectively. The corrections collected in Ĥ L M c are essential for gauge-invariance, realistic descriptions, and large interaction strength . Importantly, the dipolar moments are defined with respect to the center of mass of each individual molecule.…”

“…Importantly, the dipolar moments are defined with respect to the center of mass of each individual molecule. Following ref , we can obtain an intuitive solution for the interaction between N strongly simplified chiral emitters and the standing chiral field. In a simplified form, the analytic solution reads: ω ± ≈ 1 2 ω cav 2 + ω italicm 2 + 8 ξ λ g 2 ± false[ ω cav 2 − ω m 2 false] 2 + 16 g 2 ( ω cav + ω m ξ λ ) ( ω cav ξ λ + ω m ) where λ is the eigenvalue of the helicity operator describing either a LH (λ = +1) or a RH (λ = −1) cavity mode, ξ is the matter chirality parameter, and g is the fundamental coupling strength.…”

“…(a) Spectrum of polaritonic eigenvalues ω ± of the chiral Hopfield model with a left-handed cavity mode as a function of the cavity frequency ω cav in relation to the matter excitation ω m and chiral factor ξ. The eigenvalues are calculated with typical values for dye molecules and account for corrections via Ĥ L M c (see ref for details). (b) Same as (a) but as a series of 1D plots for different fixed chirality parameters ξ.…”

“…zero-point or ground-state) energy of ℏω ± /2. The vacuum energy of the entire system therefore is U vac = ℏ (ω + + ω – )/2, where ω ± are the two polaritonic transition energies. , Recent independent theoretical efforts suggest that the changes in the ground state energy will differ for each enantiomer and increase with the molecular concentration as N or N for small or large hybridization, respectively. , …”

“…Chiral polaritonics might provide here a unique possibility. The optical fields inside the chirality-preserving cavity undergo vacuum fluctuations that break the chiral symmetry of free space . How strong the electromagnetic environment varies inside the cavity from the free-space conditions can be well controlled by changing size, shape and quality of the optical mirrors.…”