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Lifetimes of collective states in dense ensembles of point quasi-stationary atoms

Ли Хэсюй

Published Sep 8, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.18720/spbpu/3/2026/vr/vr26-5927

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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.
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.
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In this paper we analyze the energy eigenvalues of the Dicke Hamiltonian which describes the interaction of NNN two-level atoms with an electromagnetic field. Using as a starting point the classification of the collective states proposed by Tavis and Cummings, we first derive approximate analytic expressions for the energy eigenvalues which are appropriate for special values of the collective parameters. We next evaluate a first-order correction which allows the application of our results to a wider class of collective states. An accurate description of the energy spectrum for arbitrary values of the collective parameters is provided using a method first introduced by Scharf for the case of resonant interaction, suitably generalized to account for possible frequency detuning between the radiation and the atomic resonant frequency. The limit of validity of the approximation is analyzed and the results compared with the numerical diagonalization of the Dicke Hamiltonian. With the exception of asymptotically small ranges of values of the parameters, the analytic expression for the density of energy eigenvalues derived in this paper compares very well with the results of the numerical diagonalization.
We analytically and numerically study collective emission from an ensemble of atoms in the weak-excitation regime with the initial condition of uniform excitation of the ensemble. We show that under certain conditions, subradiance at the later stages of the collective decay does not necessarily need to be accompanied by early-time superradiance. We analyze the conditions where such subradiance-only decay occurs for ordered and disordered ensembles in one-, two-, and three-dimensional spatial configurations.
We present a collective description of density matrix for identical multi-level atoms, which are either excited initially or driven coherently or pumped incoherently. We consider density matrix as expectation value of projection or transition operators in a basis of atom's product states and then identify identical matrix elements with several integers, which specify uniquely the involved transition or projection operators. To remove the redundancy of these identical elements, we treat them as single quantity and derive the equation for this quantity by mapping the transition or projection operators to a single vector specified with the integers in Liouville space and work out the actions of the master equation constitutions (as superoperators) on these vectors. As a result, the number of necessary elements increases polynomially with the number of atoms rather than exponentially as expected if the redundancy is not removed. This numerical advantage is explored to study superradiance from hundreds of two-level atoms. Our simulation identifies the conditions of observing the superradiance with and without dipole involved.
The objective of this thesis is to expand our understanding of collective light-scattering phenomena, which manifest themselves in the strong suppression of radiation from the scattering medium. In particular, we are interested in subradiance, a collective effect in spontaneous emission which, as a result of the dissipative coupling of atoms through common electromagnetic modes, creates long-lived excitation modes in an ensemble of cold atoms, with lifetimes much longer than those of individual, isolated atoms. Previously, subradiance had been well-characterized for dilute cold-atom scattering media and the low-intensity limit of the driving laser (linear-optics regime). In this thesis, we study this collective phenomenon beyond the linear-optics and dilute regimes.To study experimentally subradiance beyond the linear-optics regime, we varied the saturation parameter of the excitation laser up to relatively large values and we have observed the increase of the subradiant population with increasing saturation parameter. More importantly, we have identified the super-linear scaling of the subradiant population beyond the linear-optics regime, due to a process similar to optical pumping via multi-excitation superradiant states. This study showed that the increase in the intensity of the driving laser beyond the linear-optics regime can overcome the weak coupling of subradiant modes with the external world.Moreover, we have numerically studied subradiance using the vectorial coupled-dipole model, with a focus on understanding the effect of the near-field dipole-dipole interaction. For that, we extensively compared the vectorial and scalar (which does not contain near-field interaction terms) models and studied the subradiant lifetime as a function of the density and optical thickness of the cold-atom cloud. We observed that the near-field terms are detrimental for subradiance, which was seen in the reduction of the subradiant lifetimes with increasing density. This detrimental character of near-field terms can be interpreted as being due to an effective inhomogeneous broadening induced by the near-field interaction, which reduces the subradiant lifetimes as the density increases.The detrimental character of near-field interaction has also been previously numerically identified for Anderson localization of light waves (Anderson localization is a coherent multiple-scattering wave phenomenon in a strongly scattering medium that results in a cessation of a diffusive wave transport; it also yields long-lived modes): the scalar coupled-dipole model predicts Anderson localization of light waves, but not the vectorial one. However, adding a strong magnetic field allows one to recover localized modes for vectorial light. In this context, we numerically studied a possible signature of Anderson localization of light in cold atoms, based on the measurement of intensity fluctuations of the scattered light. We have shown that at the transition to the localized regime, the fluctuations are enhanced. In particular, for vectorial light, we have demonstrated the absence and recovery of enhanced intensity fluctuations without and with the applied magnetic field, which further corroborates that the intensity fluctuations could serve as an experimental way to probe the localized states.