Glauber and Lewenstein's 1991 quantum optics of dielectric media, formulated for an idealized lossless, frequency-independent permittivity, left open the extension of its quantization and consequences (eigenmode-plane-wave operator mapping, altered vacuum fluctuations, modified spontaneous emission, transition-radiation photon statistics, photodetection) to realistic media and to a microscopic account of embedded probes and detectors. Macroscopic quantum electrodynamics has since resolved the core of that program: dispersive and absorbing media are quantized (Huttner-Barnett 1992; Matloob, Loudon, Barnett and Jeffers 1995; Dung, Knoll and Welsch 1998), exact non-Weisskopf-Wigner (beyond-exponential) decay in bounded lossy media is available, and anisotropic, magnetodielectric and nonlinear generalizations exist. What survives unresolved is the microscopic interface between the macroscopically quantized field and embedded quantum probes: no first-principles, material-specific derivation determines the local-field correction for emitters in real, absorbing, inhomogeneous or nanostructured dielectrics - the real-cavity (Glauber-Lewenstein), virtual-cavity (Lorentz) and competing models still disagree and are discriminated empirically, host by host. No microscopic quantum model of a photodetector embedded in a dielectric, with local-field coupling and back-action, has been formulated either; that detector question is retained below only as an optional related direction.

Background

In 1991, Glauber and Lewenstein constructed the quantum electrodynamics of an idealized linear dielectric - lossless, with a frequency-independent permittivity ϵ(r)\epsilon(\mathbf{r}) that may vary arbitrarily with position. They gave two equivalent quantization schemes: an expansion of the field in single-frequency eigenmodes of the medium, and an expansion in vacuum plane waves, related by scattering theory, with the corresponding photon operators connected by a unitary transformation that mixes creation and annihilation operators. With these tools they showed that the ground state of a dielectric-filled cavity is a squeezed state when analyzed in terms of vacuum photons - a state whose fluctuations in one quadrature (one field-amplitude component) fall below the vacuum level while the conjugate component's rise above it - so that field components inside a dielectric can be super- or sub-fluctuant relative to vacuum; they computed the resulting changes in spontaneous emission (decay of an excited atom by photon emission) for electric- and magnetic-dipole transitions of atoms inside or near the medium; they showed that transition radiation - radiation emitted when a charged particle crosses a boundary between media of different permittivity - has Poissonian photon statistics; and they clarified which field operators correspond to photons that a photodetector can actually register.

The theory rested on explicit idealizations. The permittivity was real and frequency-independent, excluding material dispersion (frequency dependence of the refractive index) and absorption; the medium was coupled in adiabatically (slowly in time); spontaneous emission was treated in the Weisskopf-Wigner approximation, the standard perturbative treatment yielding purely exponential decay; and the local field - the microscopic electric field acting at the position of an embedded atom, which differs from the macroscopic field averaged over many atoms - was modeled by cutting an empty spherical hole out of the dielectric around the atom, in the spirit of the classical Clausius-Mossotti relation between molecular polarizability and dielectric constant. This 'real-cavity' prescription competes with the older 'virtual-cavity' (Lorentz) prescription; the two give different correction factors for emission rates and disagree for real materials. The paper left open how its results survive in realistic media - dispersive and absorbing, anisotropic, magnetically susceptible or nonlinear, of finite extent; its photodetection discussion identified which field operators a detector can register but did not model a realistic detector embedded in the medium.

The dispersive/absorbing core was closed by what is now called macroscopic quantum electrodynamics (macroscopic QED). Huttner and Barnett (1992) quantized the homogeneous absorbing dielectric through a damped Hopfield model (the medium's polarization oscillators are coupled to a dissipative reservoir, yielding damped polariton, i.e. mixed light-matter, modes); Matloob, Loudon, Barnett and Jeffers (1995) quantized absorbing dielectrics with Langevin noise sources - operator-valued noise currents that the fluctuation-dissipation theorem requires, without which the field commutators would decay in time; Dung, Knoll and Welsch (1998) gave the full three-dimensional quantization for inhomogeneous, dispersive, absorbing media whose complex permittivity obeys the Kramers-Kronig relations (the analyticity relations linking dispersion and absorption), with the electromagnetic Green tensor - the classical response to a point current source - carrying the mode structure; and the review of Knoll, Scheel and Welsch consolidated the framework, including exact (non-Weisskopf-Wigner) decay theory, lossy microcavities, and extensions to amplifying, magnetic and nonlinear media. Anisotropic and magnetodielectric media were quantized canonically by Philbin (2010), Judge, Steel, Sipe and de Sterke (2013), Raabe and Welsch (2005) and Sambale, Welsch, Dung and Buhmann (2009); absorbing nonlinear media by Crosse and Scheel (2010, 2011); and direct descendants of the Glauber-Lewenstein mode scheme now treat dispersive objects, dielectric spheres (Lorenz-Mie quantization, i.e., quantizing the classical Lorenz-Mie theory of light scattering by a sphere) and time-varying media (Forestiere and Miano; Maurer, Gonzalez-Ballestero and Romero-Isart; Prabhu, Parra-Contreras, Goldschmidt and Sinha; Miano, Cangemi and Forestiere).

What macroscopic QED deliberately does not fix is the field at a point inside the medium. It takes the macroscopic permittivity as input, so the connection between the macroscopic field and the microscopic local field acting on an embedded atom - the local-field correction - remains an input phenomenology rather than a derived quantity. This is where the source question survives. Scheel, Knoll and Welsch (1999) showed that once absorption is included, the real-cavity decay rate depends sensitively on the assumed cavity radius near absorption bands, and the familiar correction factors are recovered only when absorption is ignored. Microscopic theories for model systems - dense atomic gases (Fleischhauer), dilute gases (Fu and Berman), random media (Donaire) - disagree with one another in the absorbing case, and the treatment itself has been contested (Crenshaw and Bowden; Crenshaw). Experiments on luminescent centers such as rare-earth ions and quantum dots discriminate among the models host by host rather than validating a first-principles theory (Duan, Reid and Wang; Duan and Reid; Dolgaleva and Boyd), and local-field corrections of up to 80 percent arise in computed van der Waals potentials of atoms in magnetodielectric multilayers (Sambale et al.). On the detection side, general quantum-photodetector frameworks including back-action exist (Young, Sarovar and Leonard; Biswas and van Enk), and coincidence count rates in absorbing media have been computed within macroscopic QED (Crosse and Scheel), but a microscopic model of a detector embedded in a dielectric - its atoms coupled to the local field, with detection back-reacting on the field - does not yet exist.

Problem Statement

This problem is the surviving open core of the program left open by Glauber and Lewenstein (1991) in the quantum optics of dielectric media: give a microscopic, material-specific foundation for how an embedded quantum probe couples to the quantized electromagnetic field of a real dielectric. Concretely, the problem has two parts: (i) Start from an explicit microscopic model of the material - polarizable constituents with their statistics, spatial correlations and absorption, consistent with the Kramers-Kronig relations - and derive the local field (the microscopic electric field at the emitter's position, as distinct from the macroscopic field averaged over many atoms) acting on an embedded emitter, together with the resulting electric- and magnetic-dipole spontaneous-emission rates, radiative lifetimes, level shifts and vacuum-fluctuation spectra, valid for absorbing, inhomogeneous and structured media, including interfaces, multilayers and photonic or metamaterial structures where no spherical-hole picture applies. (ii) Determine under which material conditions the standard phenomenological corrections - the real-cavity (Glauber-Lewenstein), virtual-cavity (Lorentz) and competing factors - are recovered, thereby reconciling the conflicting predictions of the existing microscopic theories and yielding quantitative, testable predictions that discriminate among the candidate models. Both parts serve one objective: replacing the empirical, host-by-host choice of a local-field model with a derived, material-specific one. A microscopic quantum model of photodetection for absorbing detector atoms embedded in such media - their coupling to the local field and the back-action of detection on the field - remains undeveloped and is noted here as an optional related direction; it is not a required deliverable, because the source's photodetection discussion was confined to identifying the field operators a detector can register. The quantization of dispersive and absorbing media, exact non-Markovian decay in bounded lossy media, and the anisotropic, magnetodielectric and nonlinear generalizations - also left open by the source - are excluded from this problem as resolved by macroscopic QED.

References

  1. R. J. Glauber and M. Lewenstein, 'Quantum optics of dielectric media', Physical Review A 43, 467-491 (1991). DOI: 10.1103/PhysRevA.43.467, https://doi.org/10.1103/PhysRevA.43.467
  2. B. Huttner and S. M. Barnett, 'Quantization of the electromagnetic field in dielectrics', Physical Review A 46, 4306-4322 (1992). DOI: 10.1103/PhysRevA.46.4306, https://doi.org/10.1103/PhysRevA.46.4306
  3. R. Matloob, R. Loudon, S. M. Barnett and J. Jeffers, 'Electromagnetic field quantization in absorbing dielectrics', Physical Review A 52, 4823-4838 (1995). DOI: 10.1103/PhysRevA.52.4823, https://doi.org/10.1103/PhysRevA.52.4823
  4. Ho Trung Dung, L. Knoll and D.-G. Welsch, 'Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics', Physical Review A 57, 3931-3942 (1998). DOI: 10.1103/PhysRevA.57.3931, arXiv:quant-ph/9711039, https://arxiv.org/abs/quant-ph/9711039
  5. L. Knoll, S. Scheel and D.-G. Welsch, 'QED in dispersing and absorbing media', arXiv:quant-ph/0006121 (2000); published in 'Coherence and Statistics of Photons and Atoms', ed. J. Perina, Wiley (2001). https://arxiv.org/abs/quant-ph/0006121
  6. S. Scheel, L. Knoll and D.-G. Welsch, 'Spontaneous decay of an excited atom in an absorbing dielectric', Physical Review A 60, 4094-4104 (1999). DOI: 10.1103/PhysRevA.60.4094, arXiv:quant-ph/9904015, https://arxiv.org/abs/quant-ph/9904015
  7. Ho Trung Dung, L. Knoll and D.-G. Welsch, 'Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity', Physical Review A 62, 053804 (2000). DOI: 10.1103/PhysRevA.62.053804, https://doi.org/10.1103/PhysRevA.62.053804
  8. M. Fleischhauer, 'Spontaneous emission and level shifts in absorbing disordered dielectrics and dense atomic gases: A Green's-function approach', Physical Review A 60, 2534 (1999). DOI: 10.1103/PhysRevA.60.2534, arXiv:quant-ph/9902076, https://arxiv.org/abs/quant-ph/9902076
  9. M. E. Crenshaw and C. M. Bowden, 'Effects of Local Fields on Spontaneous Emission in Dielectric Media', Physical Review Letters 85, 1851 (2000). DOI: 10.1103/PhysRevLett.85.1851, arXiv:quant-ph/0004101, https://arxiv.org/abs/quant-ph/0004101
  10. M. E. Crenshaw, 'Comparison of Quantum and Classical Local-field Effects on Two-Level Atoms in a Dielectric', Physical Review A 78, 053827 (2008). DOI: 10.1103/PhysRevA.78.053827, arXiv:0805.2134, https://arxiv.org/abs/0805.2134
  11. H. Fu and P. R. Berman, 'Microscopic Theory of Spontaneous Decay in a Dielectric', Physical Review A 72, 022104 (2005). DOI: 10.1103/PhysRevA.72.022104, arXiv:quant-ph/0507210, https://arxiv.org/abs/quant-ph/0507210
  12. M. Donaire, 'Dipole emission and coherent transport in random media I', arXiv:0811.0323 (2008). https://arxiv.org/abs/0811.0323
  13. C.-K. Duan, M. F. Reid and Z. Wang, 'Local field effects on the radiative lifetime of emitters in surrounding media: Virtual- or real-cavity model?', Physics Letters A 343, 474-480 (2005). DOI: 10.1016/j.physleta.2005.06.037, arXiv:cond-mat/0505587, https://arxiv.org/abs/cond-mat/0505587
  14. C. K. Duan and M. F. Reid, 'Macroscopic Models for the Radiative Relaxation Lifetime of Luminescent Centers Embedded in Surrounding Media', Spectroscopy Letters 40, 237-246 (2007). DOI: 10.1080/00387010701247340, https://doi.org/10.1080/00387010701247340
  15. K. Dolgaleva and R. W. Boyd, 'Local-field effects in nanostructured photonic materials', Advances in Optics and Photonics 4, 1 (2012). DOI: 10.1364/AOP.4.000001, https://doi.org/10.1364/AOP.4.000001
  16. A. Sambale, D.-G. Welsch, Ho Trung Dung and S. Y. Buhmann, 'Local-field-corrected van der Waals potentials in magnetodielectric multilayer systems', Physical Review A 79, 022903 (2009). DOI: 10.1103/PhysRevA.79.022903, arXiv:0901.0634, https://arxiv.org/abs/0901.0634
  17. A. C. Judge, M. J. Steel, J. E. Sipe and C. M. de Sterke, 'Canonical quantization of macroscopic electrodynamics in a linear, inhomogeneous magnetoelectric medium', Physical Review A 87, 033824 (2013). DOI: 10.1103/PhysRevA.87.033824, arXiv:1209.4401, https://arxiv.org/abs/1209.4401
  18. T. G. Philbin, 'Canonical quantization of macroscopic electromagnetism', New Journal of Physics 12, 123008 (2010). DOI: 10.1088/1367-2630/12/12/123008, arXiv:1009.5005, https://arxiv.org/abs/1009.5005
  19. C. Raabe and D.-G. Welsch, 'Casimir force acting on magnetodielectric bodies embedded in media', Physical Review A 71, 013814 (2005). DOI: 10.1103/PhysRevA.71.013814, arXiv:quant-ph/0408075, https://arxiv.org/abs/quant-ph/0408075
  20. J. A. Crosse and S. Scheel, 'Quantum electrodynamics in absorbing nonlinear media', Physica Scripta T140, 014041 (2010). DOI: 10.1088/0031-8949/2010/T140/014041, https://doi.org/10.1088/0031-8949/2010/T140/014041
  21. J. A. Crosse and S. Scheel, 'Coincident count rates in absorbing dielectric media', Physical Review A 83, 023815 (2011). DOI: 10.1103/PhysRevA.83.023815, arXiv:1012.0771, https://arxiv.org/abs/1012.0771
  22. S. M. Young, M. Sarovar and F. Leonard, 'Fundamental Limits to Single-Photon Detection Determined by Quantum Coherence and Backaction', Physical Review A 97, 033836 (2018). DOI: 10.1103/PhysRevA.97.033836, arXiv:1710.09512, https://arxiv.org/abs/1710.09512
  23. S. M. Young, M. Sarovar and F. Leonard, 'General modeling framework for quantum photodetectors', Physical Review A 98, 063835 (2018). DOI: 10.1103/PhysRevA.98.063835, arXiv:1811.08018, https://arxiv.org/abs/1811.08018
  24. S. Biswas and S. J. van Enk, 'The Heisenberg picture of photodetection', Physical Review A 102, 033705 (2020). DOI: 10.1103/PhysRevA.102.033705, arXiv:2007.05444, https://arxiv.org/abs/2007.05444
  25. C. Forestiere and G. Miano, 'Operative approach to quantum electrodynamics in dispersive dielectric objects based on a polarization-mode expansion', Physical Review A 106, 033701 (2022). DOI: 10.1103/PhysRevA.106.033701, arXiv:2108.03668, https://arxiv.org/abs/2108.03668
  26. P. Maurer, C. Gonzalez-Ballestero and O. Romero-Isart, 'Quantum electrodynamics with a nonmoving dielectric sphere: quantizing Lorenz-Mie scattering', Journal of the Optical Society of America B 40, 3137 (2023). DOI: 10.1364/JOSAB.498540, arXiv:2106.07975, https://arxiv.org/abs/2106.07975
  27. A. Prabhu, J. Parra-Contreras, E. A. Goldschmidt and K. Sinha, 'Quantum Electrodynamics with Time-varying Dielectrics', arXiv:2310.13878 (2023). https://arxiv.org/abs/2310.13878
  28. G. Miano, L. M. Cangemi and C. Forestiere, 'Quantum emitter interacting with a dispersive dielectric object: a model based on the modified Langevin noise formalism', Nanophotonics 14, 4019-4027 (2025). DOI: 10.1515/nanoph-2024-0703, arXiv:2410.10730, https://arxiv.org/abs/2410.10730