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Cooperative Lamb shift in dense ensembles of point quasi-stationary atoms

Ли Ичао

Published Sep 8, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.18720/spbpu/3/2026/vr/vr26-5918

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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.
This work analyzes lifetimes of collective states in dense ensembles of point quasi-stationary atoms using the finite complex spectrum of a two-atom reradiation matrix. The objective of the work is to derive the finite interatomic part of the two-atom reradiation matrix and use its complex spectrum to calculate collective decay rates and lifetimes. The tasks of the work are to formulate the microscopic atom-field model, specify the atomic level scheme and transition dipoles, include the virtual one-photon sector needed in the perturbative derivation, identify the ultraviolet problem of the off-diagonal Green-matrix elements, 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 interatomic Green-matrix elements, parameter differentiation, and numerical diagonalization of the 6×6 reradiation matrix. The main result is a finite complex interaction kernel and numerical curves showing how superradiant and subradiant lifetime branches depend on interatomic separation. MATLAB was used for numerical matrix construction, eigenvalue calculation, lifetime extraction, branch classification, and plotting.
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.
According to quantum electrodynamics, the exchange of virtual photons in a system of identical quantum emitters causes a shift of its energy levels. Such shifts, known as cooperative Lamb shifts, have been studied mostly in the near-field regime. However, the resonant electromagnetic interaction persists also at large distances, providing coherent coupling between distant atoms. Here, we report a direct spectroscopic observation of the cooperative Lamb shift of an optical electric-dipole transition in an array of Sr(+) ions suspended in a Paul trap at inter-ion separations much larger than the resonance wavelength. By controlling the precise positions of the ions, we studied the far-field resonant coupling in chains of up to eight ions, extending to a length of 40 μm. This method provides a novel tool for experimental exploration of cooperative emission phenomena in extended mesoscopic atomic arrays.
For pt. I, see ibid., vol. 6, 1348 (1973). The authors extend the perturbation theory of super-radiant emission to derive a number of results on the level shifts of n atom systems. These also divide into 'coherent' and 'incoherent' parts. The incoherent part divides further into a generalized Lamb shift which is not cooperative, and an interatomic term which is. The former proves to be the only shift is not cooperative, and an interatomic term which is. The former proves to be the only shift which can depend on the presence of ambient free photons. These results for the generalized Lamb shift support the view that a dynamical neoclassical theory of the Lamb shift is unacceptable. The theory is compared with the pseudoboson theories of linear dielectrics. The pseudoboson theory of an inverted dielectric (the 'amplifier') should be intrinsically unstable unlike the pseudoboson theory of the refractive index which is concerned with the 'attenuator'.