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arXiv:2305.01865

Collective Lamb Shift and Modified Linewidth of An Interacting Atomic Gas

H. Ma, Susanne F. Yelin

Published May 3, 2023• 0 citations
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Abstract

Finding a comprehensive and general description of the collective Lamb shift and cooperative broadening in a radiatively interacting system is a long-standing open question. Both energy levels and linewidth of individual atoms are modified by the exchange of real and virtual photons making up the dipole-dipole interaction. We introduce a method to theoretically study weakly-driven, low-excited ensembles of two-level atoms, and obtain an analytic description of the collective Lamb shift and linewidth via a self-consistent formalism including infinite order of correlations which stem from only two-body interactions. We predict the dependency of these quantities, as measurables, on system parameters: the number density of the ensemble, the detuning of an external probe field, and the geometry of the sample.

DOI
10.48550/arxiv.2305.01865

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Finding a comprehensive and general description of the collective Lamb shift and cooperative broadening in a radiatively interacting system is a long-standing open question. Both energy levels and linewidth of individual atoms are modified by the exchange of real and virtual photons making up the dipole-dipole interaction. We introduce a method to theoretically study weakly-driven, low-excited ensembles of two-level atoms, and obtain an analytic description of the collective Lamb shift and linewidth via a self-consistent formalism including infinite order of correlations which stem from only two-body interactions. We predict the dependency of these quantities, as measurables, on system parameters: the number density of the ensemble, the detuning of an external probe field, and the geometry of the sample.
A landmark in the development of quantum electrodynamics was the discovery that emission–reabsorption of virtual photons modifies the value of energy levels in an atom from those computed by using Dirac’s equation (Lamb shift). An early result of statistical quantum electrodynamics was that the exchange of virtual photons in an ensemble of identical atoms leads as well to a change in the frequency of the radiation emitted from this system (cooperative Lamb shift). Dicke’s discovery that coherence effects lead to the shortening of the emission lifetime from a small sample by a factor equal to the number of atoms in the ensemble (superradiance or cooperative decay rate) was an early landmark in quantum optics. Both cooperative decay rate and cooperative Lamb shift were shown to have the same physical origin—the exchange of virtual photons, a process described by the Lienard–Wiechert dipole–dipole interaction. This effective potential is the kernel of the integral equation describing the dynamics of the system. This complex long-range kernel gives, for both cooperative quantities, strong dependence on the geometry of the atomic cloud. I summarize the known expressions for the initial cooperative decay rate and the cooperative Lamb shift in different geometries. The results for both the scalar photon and the vector photon (electrodynamics) theories for experimentally realizable systems of either uniform or phased polarization are given.
This work studies the cooperative Lamb shift in dense ensembles of point quasi-stationary atoms through a microscopic two-atom model. The objective of the work is to derive the finite interatomic part of the two atom reradiation matrix and use it to calculate distance-dependent cooperative frequency shifts. The tasks of the work are to formulate the atom-field Hamiltonian, specify the atomic level scheme and transition dipoles, include the one-photon virtual sector needed in the perturbative derivation, identify the ultraviolet problem of the off diagonal Green-matrix elements responsible for the cooperative Lamb shift, compare three derivation routes, and diagonalize the two-atom matrix for axial and arbitrary-coordinate geometries. The study uses analytical derivation in the electric-dipole and p·A gauges, direct principal-value analysis of the off-diagonal interaction integral, parameter differentiation, and numerical diagonalization of the 6×6 reradiation matrix. The main result is a consistent finite working formula for the off-diagonal interaction kernel and numerical curves showing how cooperative Lamb-shift branches depend on interatomic separation. MATLAB was used for numerical matrix construction, eigenvalue calculation, branch classification, and plotting.
Abstract We investigate the spectral shift known as the collective Lamb shift in forward scattering for a cold dense atomic cloud. The shift results from resonant dipole–dipole interaction mediated by real and virtual photon exchange, forming many-body states displaying various super- and subradiant behaviour. However, the scattering spectrum reflects the overall contributions from these states but also averages out the radiative details associated with the underlying spin orders, causing ambiguity in determination and raising controversy on the scaling property of this shift. We employ a Monte–Carlo simulation to study how the collective states contribute to emission. We thus distinguish two kinds of collective shift that follow different scaling laws. One results from dominant occupation of the near-resonant collective states. This shift is usually small and insensitive to the density or the number of participating atoms. The other comes from large spatial correlation of dipoles, associated with the states of higher degree of emission. This corresponds to larger collective shift that is approximately linearly dependent on the optical depth. We further demonstrate that the spatial spin order plays an essential role in superradiant emission. Our analysis provides a novel perspective for understanding collective scattering and cooperative effects.
We systematically investigate the far-zone fluorescence from a number of laser-driven two-level atoms with the cooperative effects stemming from the virtual exchange of photons between identical atoms. Due to the large interaction-induced energy level shifts, i.e. the collective Lamb shifts, both atomic excitation and photon emission do not maximize at the resonant light-atom coupling. In addition, the long-range dissipative interactions among atoms lead to the disproportion between the atomic excitation and the photon emission. Moreover, we discuss the temporal properties of the far-field fluorescence light from the atomic ensemble. The first-order correlation function indicates a damped light-atom dynamics combined with a strong background of the elastic photon scattering. The second-order correlation function illustrates that in the weak-excitation limit the scattered light exhibits the nonclassical photon antibunching behavior, while the bunched photons arrive at the photodetector in the saturated-excitation limit.
When atoms are coupled to a common electromagnetic environment, the exchange of photons through dipole-dipole interactions leads to the emergence of cooperative effects. As a particular example, superradiance arises from spontaneous emission when this exchange leads to constructive interference of the emitted photons. Here, we introduce an integrated method for studying cooperative radiation in many-body systems. This method, which allows to study extended systems with arbitrarily large number of particles, can be formulated by an effective, nonlinear, two-atom master equation that describes the dynamics using a closed form which treats single- and many-body terms on an equal footing. We apply this method to a homogeneous gas of initially inverted two-level atoms, and demonstrate the appearance of both superradiance and subradiance, identifying a many-body coherence term as the source of these cooperative effects. We describe the many-body induced broadening -- which is analytically found to scale with the optical depth of the system -- and light shifts, and distinguish spontaneous effects from induced ones. In addition, we theoretically predict the time-dependence of subradiance, and the phase change of the radiated field during the cooperative decay.
The coherent photon scattering of atoms in an atomic array is studied. It is found that due to cooperative photon exchanges, the excitation probability of an atom in an array parallel to the direction of laser propagation ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{k}}$) will either grow or decay along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{k}}$, depending on the detuning of the laser. The symmetry of the system for atomic separations of $a=j\ensuremath{\lambda}/2$ causes the excitation distribution and scattered radiation to abruptly become symmetric about the center of the array, where jjj is an integer and $\ensuremath{\lambda}$ is the transition wavelength. For atomic separations of $a<\ensuremath{\lambda}/2$, a collection of nonradiating states ($\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}0$) disrupts the described trend. In order to interpret this surprising result, a band-structure calculation in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is conducted, where the decay rates and the collective Lamb shifts of the eigenmodes versus quasimomentum are obtained. This calculation shows that the collective exchange of photons in an array strongly affects its scattered radiation, allowing one to easily manipulate the collective Lamb shift and directly excite either superradiant or subradiant eigenmodes by correctly choosing the angle of the driving laser.