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arXiv:2112.04952 · Journal of Physics B Atomic Molecular and Optical Physics

Quantifying the breakdown of the rotating-wave approximation in single-photon superradiance

Mads Anders Jørgensen, Martijn Wubs

Published Aug 15, 2022• 11 citations
arXivHTMLPDF

Abstract

Abstract We study quantitatively the breakdown of the rotating-wave approximation (RWA) when calculating collective light emission by quantum emitters, in particular in the weak-excitation limit. Our starting point is a known multiple-scattering formalism where the full light–matter interaction leads to induced inter-emitter interactions described by the classical Green function of inhomogeneous dielectric media. When making the RWA in the light–matter interaction, however, these induced interactions differ from the classical Green function, and for free space we find a reduction of the interatomic interaction strength by up to a factor of two. By contrast, for the corresponding scalar model the relative RWA error for the inter-emitter interaction even diverges in the near field. For two identical emitters, the errors due to the RWA in collective light emission will show up in the emission spectrum, but not in the sub- and superradiant decay rates. In case of two non-identical emitters, also the collective emission rates will differ by making the RWA. For three or more identical emitters, the RWA errors in the interatomic interaction in general affect both the collective emission spectra and the collective decay rates. Ring configurations with discrete rotational symmetry are an interesting exception. Interestingly, the maximal errors in the collective decay rates due to making the RWA occur for finite emitter separations.

DOI
10.1088/1361-6455/ac89be

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Abstract We study quantitatively the breakdown of the rotating-wave approximation (RWA) when calculating collective light emission by quantum emitters, in particular in the weak-excitation limit. Our starting point is a known multiple-scattering formalism where the full light–matter interaction leads to induced inter-emitter interactions described by the classical Green function of inhomogeneous dielectric media. When making the RWA in the light–matter interaction, however, these induced interactions differ from the classical Green function, and for free space we find a reduction of the interatomic interaction strength by up to a factor of two. By contrast, for the corresponding scalar model the relative RWA error for the inter-emitter interaction even diverges in the near field. For two identical emitters, the errors due to the RWA in collective light emission will show up in the emission spectrum, but not in the sub- and superradiant decay rates. In case of two non-identical emitters, also the collective emission rates will differ by making the RWA. For three or more identical emitters, the RWA errors in the interatomic interaction in general affect both the collective emission spectra and the collective decay rates. Ring configurations with discrete rotational symmetry are an interesting exception. Interestingly, the maximal errors in the collective decay rates due to making the RWA occur for finite emitter separations.
We present results of our theoretical research on collective emission of photons (super- and subradiance) and photon propagation in photonic environments.Just like the single-atom emission, collective emission depends on the photonic environment. For quantum emitters in layered media, we study whether surface plasmons can enhance collective emission, especially for emitters at distances smaller than an optical wavelength.A common approximation in quantum optics is the rotating-wave approximation in the light-matter interaction. We show that in general multi-atom emission rates depend on the approximation, but exceptions exist. Interatomic interactions induced by the electromagnetic field are given by the classical Green tensor, but only when not making the rotating-wave approximation.As a final example of quantum electromagnetics, we describe photons propagating through a layered medium with alternating loss and gain, so-called loss-compensated media. There is no analogous compensation of the quantum noise associated with loss and gain media.
Strong and ultra-strong light-matter coupling are remarkable phenomena of quantum electrodynamics arising when a matter excitation interacts so strongly with the electromagnetic field that the two systems cannot be distinguished as in usual perturbation theory. This is generally achieved by coupling an excitation with large oscillator strength to the confined electromagnetic mode of an optical microcavity. In this work we report on a phenomenon of non-perturbative spontaneous emission taking place in the absence of optical confinement, when a collective electronic excitation superradiantly decays into free space radiation. By using a quantum model based on the input-output formalism, we calculate the linear optical properties of a dense electron gas displaying superradiant excitations. We show that the commonly used rotating-wave and Markov approximations yield unphysical results when the radiative broadening is comparable to the matter excitation energy, i.e. in the ultra-strong coupling regime. Our formalism also allows computing the electron gas thermal emission, showing that Kirchhoff's law applies in all the regimes of light-matter interaction.
Collective emission of light from distributions of two-level systems was first predicted in 1954 by Robert Dicke, who showed that when N quantum emitters absorb photons, their collective radiative decay rate can be significantly enhanced (superradiance) or suppressed (subradiance) relative to the single-emitter decay rate. In this work, we derive analytical expressions for the collective decay rates and collective Lamb shifts for the coherent interaction of a single photon with topologically one-dimensional, continuous distributions of quantum two-level systems: an infinite line and an infinite helix. We compare our solution with arrangements where the distribution of transition dipole vector emitters is discrete rather than continuous, and when short- ( 1 / r 3 ), intermediate- ( 1 / r 2 ), and long-range ( 1 / r ) interaction terms are included. We conduct a detailed comparison of our results averaging out polarization effects, with eigensolutions of similar systems maintaining the dot product between transition dipole vectors and the electric field, finding that the thermally averaged collective decay rate is enhanced in both approaches. Our analytical solution for the helix is then used to make order-of-magnitude estimates of the maximally superradiant state, thermally averaged collective decay rate, and percentage of trapped states in helical architectures of molecular quantum emitters in protein fibers. Given the inclusion of short- and intermediate-range interaction terms in our numerical models for realistic protein fiber architectures and the averaging out of polarization effects in our analytical solution, these results show excellent agreement for sparse arrangements of emitters. Our work thus bridges the theoretical gap between different formalisms for treating superradiant matter distributions, aids the engineering of helical devices to harness quantum optical effects for computing with superradiant error correction and subradiant memories, and motivates the discovery and creation of flexible platforms for quantum information processing using the intrinsic helical geometries of biomatter.
The dynamics of a collection of resonant atoms embedded inside an inhomogeneous nondispersive and lossless dielectric is described with a dipole Hamiltonian that is based on a canonical quantization theory. The dielectric is described macroscopically by a position-dependent dielectric function and the atoms as microscopic harmonic oscillators. We identify and discuss the role of several types of Green tensors that describe the spatio-temporal propagation of field operators. After integrating out the atomic degrees of freedom, a multiple-scattering formalism emerges in which an exact Lippmann-Schwinger equation for the electric field operator plays a central role. The equation describes atoms as point sources and point scatterers for light. First, single-atom properties are calculated such as position-dependent spontaneous-emission rates as well as differential cross sections for elastic scattering and for resonance fluorescence. Secondly, multiatom processes are studied. It is shown that the medium modifies both the resonant and the static parts of the dipole-dipole interactions. These interatomic interactions may cause the atoms to scatter and emit light cooperatively. Unlike in free space, differences in position-dependent emission rates and radiative line shifts influence cooperative decay in the dielectric. As a generic example, it is shown that near a partially reflecting plane there is a sharp transition from two-atom superradiance to single-atom emission as the atomic positions are varied.
The spectrum of cooperative light emission from ensembles of multilevel atoms is studied in optical vector theory and without applying the rotating-wave approximation. The effects of counter-rotating terms are included using a unitary transformation method. The spectra are analyzed and interpreted in terms of the radiative eigenmodes of the atom ensemble. We further show how the qualitative features arise from the structure of the underlying two-particle dipole-dipole interaction induced by the the vacuum field. We predict that for a suitable modification of the ensemble properties, the sign of the cooperative Lamb shift can be reversed, while still maintaining strong superradiant emission. Finally, we discuss the effects of finite detection resolution and of averaging over many realizations of the random distribution of atoms for given ensemble parameters.