Foundations of the cooperative-mode theory of coherent resonant scattering by an atomic sphere: eigenmode completeness and microscopic relaxation

Prasad and Glauber [Phys. Rev. A 83, 063821 (2011)] developed a vector-multipole theory of coherent resonant scattering by a uniform sphere of two-level atoms, expressing the multipole scattering coefficients as Mittag-Leffler (partial-fraction) expansions over the sphere's cooperative emission modes, i.e. hybrid matter-radiation eigenmodes with complex decay constants. The paper itself acknowledges two foundational gaps: (1) incoherent relaxation enters only phenomenologically, through a postulated damping constant and Langevin force justified by a generic appeal to the fluctuation-dissipation theorem, with no microscopic derivation for any specific mechanism; (2) the spectral expansion rests on the unproven assumption that the cooperative magnetic- and electric-multipole modes form a complete set for each multipole order, exhausting all zeros of the resonance denominators in the complex detuning plane. The theory is formulated in the weak-excitation, linear-response regime in which the number of excitations satisfies $n_{ph}\ll N/2$, so that the $O(1/N)$ correction (equal to $2n_{ph}/N$ in the $n_{ph}$-photon sector) is negligible; this is its stated domain of validity rather than part of the open goal. The open problem is to close the two acknowledged gaps: establish or refute the modal completeness and the domain of validity of the expansion; and derive from first principles the damping operator and the Langevin-force statistics for collisional dephasing, Doppler broadening, radiative transfer to nonresonant levels, and density fluctuations, together with their quantitative modification of mode residues, widths, and spectra.

Background

When light resonant with an atomic transition propagates through an optically thick atomic ensemble, the atoms do not scatter independently: the field radiated by one atom drives the others, so emission and scattering become cooperative (collective). In the extreme Dicke limit, where the whole sample is small compared with the wavelength, cooperative spontaneous emission produces superradiance — an emission burst whose rate scales with the square of the atom number. In extended samples, interference and propagation instead make the relevant enhancement scale with the resonant optical depth. A single-photon version of this physics — the timed Dicke state, i.e. the symmetric single-excitation state phase-matched to the driving wave — decays at a rate of order $N\Gamma_0/(kR)^2$ set by the optical depth ($N$ atoms in a cloud of radius $R$, resonant wave number $k$, single-atom spontaneous decay rate $\Gamma_0$), as analyzed for example by Svidzinsky, Chang, and Scully. Work in the 2000s–2010s established that in dilute cold atoms most of this physics is captured by classical coupled dipoles — the coupled-dipole model, in which each atom is a linear oscillator driven by the incident field plus the fields radiated by all the other atoms.

Prasad and Glauber developed a continuum, analytically controlled version of this theory for simple geometries in a series of four papers on cooperative radiation and scattering by ensembles of two-level atoms, of which the paper considered here (Phys. Rev. A 83, 063821 (2011)) is the last. The medium is a uniform sphere of $N$ two-level atoms (number density $n_0$, transition frequency $\omega_0$, electric-dipole matrix element $\mu$, sphere radius $R$, size parameter $\beta = kR$). In the weak-excitation regime the atomic polarization obeys a linear Maxwell–Lorentz equation equivalent to Maxwell's equations in a medium with the resonant susceptibility $\chi(\omega) = -n_0|\mu|^2/[3\hbar(\omega-\omega_0)]$. The incident plane wave is resolved, using vector spherical harmonics generated by the angular momentum operator $\mathbf{L} = -i\mathbf{r}\times\nabla$, into magnetic-multipole (MM) and electric-multipole (EM) partial waves — the transverse multipole decomposition familiar from Mie scattering of a dielectric sphere — and the problem is solved order by order: the scattering amplitudes in the $\ell$th multipole order are ratios $a_\ell = N_\ell^m(\nu)/D_\ell^m(\nu)$ and $b_\ell = N_\ell^e(\nu)/D_\ell^e(\nu)$ of entire functions of the scaled detuning $\nu$, with total cross sections $\sigma_{\ell m} = 2(2\ell+1)\pi|a_\ell|^2/k^2$ and $\sigma_{\ell e} = 2(2\ell+1)\pi|b_\ell|^2/k^2$.

The crucial analytic step is spectral. The zeros of the denominators $D_\ell^m$ and $D_\ell^e$ in the complex detuning plane coincide with $-i\lambda$, where $\lambda$ are the complex decay constants of the sphere's cooperative modes — the hybrid matter-field eigenmodes of the coupled atom-field system computed in the earlier papers of the series. Assuming these modes form a complete set for each multipole type and order, so that they exhaust all zeros of the denominators, and that the amplitudes vanish at infinity in the complex plane, the meromorphic amplitudes admit a Mittag-Leffler expansion — the classical result of complex analysis that a meromorphic function with suitable growth equals a partial-fraction sum over its poles. The resonant terms then interfere, producing Fano sequences of narrow peaks and dips (asymmetric resonance line shapes arising from interference between a resonant and a background amplitude) and strong anisotropies of the differential cross section even for small spheres. The paper notes that the numerators and denominators are entire functions except possibly at $\nu = 0$ and $\nu = n_0|\mu|^2/(3\hbar)$, where $x_\nu = \gamma_\nu R$ — the argument of the spherical Bessel functions describing the field inside the sphere, with $\gamma_\nu$ the propagation constant in the resonant medium — has branch points; it argues that the resulting sign ambiguity of the numerators and denominators cancels in the ratios, so that the amplitudes are meromorphic in the complex-$\nu$ plane.

Two further limitations are explicit in the source. First, all incoherent relaxation is excluded except for a phenomenological insertion: following the fluctuation-dissipation theorem — the general principle that any linear damping of a degree of freedom must be accompanied by a fluctuating force whose spectrum is fixed by that damping — a damping constant $\gamma$ and a zero-mean Langevin (fluctuating) force $\mathbf{F}(\mathbf{r},t)$ are added by hand to the polarization equation. The concluding remarks state that this approach 'omits addressing any of the physically important details of energy exchange between the coherent and incoherent degrees of freedom of the atomic medium and the field'. Second, the theory is strictly linear in the field: in the $n_{ph}$-photon sector of spatially uniform excitation, the correction to the polarization equation is exactly $2n_{ph}/N$, negligible only when $n_{ph}\ll N/2$, so atomic saturation and many-photon effects lie outside its validity. This weak-excitation condition is the source theory's stated domain of validity.

The problem stated below asks for the missing foundations of this framework. It sits at the intersection of the cooperative/linear optics of cold atoms and the spectral theory of open (non-Hermitian) electromagnetic systems — where a parallel literature on quasinormal modes and the resonant-state expansion has since made substantial progress on completeness questions.

Problem Statement

Establish the Prasad-Glauber cooperative-mode theory of coherent resonant scattering by a uniform spherical medium of $N$ two-level atoms on complete foundations. The common objective uniting the two requested components is a rigorous and microscopic version of the source theory, within its stated weak-excitation domain of validity $n_{ph}\ll N/2$ (where the $O(1/N)$ correction to the polarization equation, equal to $2n_{ph}/N$ in the $n_{ph}$-photon sector of spatially uniform excitation, is negligible), in which the scattering of a monochromatic wave of frequency $\omega$ by the sphere (size parameter $\beta = kR$, scaled detuning $\nu$, multipole scattering coefficients $a_\ell = N_\ell^m/D_\ell^m$ and $b_\ell = N_\ell^e/D_\ell^e$, total cross sections $\sigma_{\ell m} = 2(2\ell+1)\pi|a_\ell|^2/k^2$ and $\sigma_{\ell e} = 2(2\ell+1)\pi|b_\ell|^2/k^2$) is described by the cooperative-mode (Mittag-Leffler) spectral expansion. The two components are: (1) Completeness of the modal basis. Prove, or refute, that the cooperative MM and EM eigenmodes form a complete set for each multipole type and order — i.e. that the cooperative decay constants $\lambda$ of the modes computed in the earlier papers of the series exhaust the zeros of the denominators $D_\ell^m$ and $D_\ell^e$ in the full complex $\nu$ plane with no zeros missing and no additional (spurious) zeros — equivalently, the eigenvalue conditions $D_\ell^m(\nu = -i\lambda) = 0$ and $D_\ell^e(\nu = -i\lambda) = 0$ are satisfied by precisely these decay constants — and establish the domain of validity of the resulting Mittag-Leffler expansion of the scattering amplitudes across the relevant parameter regimes ($\beta$, density, detuning), including the consequences of the branch points at $\nu = 0$ and $\nu = n_0|\mu|^2/(3\hbar)$ of the quantity $x_\nu = \gamma_\nu R$ (with $\gamma_\nu$ the propagation constant inside the resonant sphere, entering the numerator and denominator functions through the spherical Bessel functions of argument $x_\nu$) — the source argues that the resulting double-valuedness of the numerators and denominators cancels in the amplitude ratios, leaving the amplitudes meromorphic, and this cancellation argument itself requires justification — and of the assumed vanishing of the amplitudes at infinity in the complex plane. (2) Microscopic relaxation. Replace the phenomenological damping constant $\gamma$ and Langevin force of the source paper with a first-principles derivation, for the incoherent relaxation mechanisms it lists — collisional dephasing, Doppler broadening from thermal motion, radiative transfer of coherent excitation to nonresonant levels, and density fluctuations of the medium — of (i) the damping operator acting on the coherent atom-field amplitudes and (ii) the statistical properties (mean and correlations) of the accompanying fluctuation force, in a form consistent with the fluctuation-dissipation theorem; and compute quantitatively how each mechanism modifies the residues (mode strengths), widths, and positions of the cooperative modes and hence the scattering cross sections and their Fano peak-dip structure. The two components are posed jointly: the completeness claim and the domain of validity of the expansion apply to both multipole types and every multipole order $\ell \ge 1$, across the parameters $\beta$, density, and detuning rather than at a single parameter point, and the four relaxation mechanisms enter together, each with its quantitative modification of the mode residues, widths, and positions and the resulting cross sections and Fano peak-dip structure to at least first order in that mechanism.

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