Generalized noise terms for the quantized fluctuational electrodynamics

Abstract

The quantization of optical fields in vacuum has been known for decades, but extending the field quantization to lossy and dispersive media in nonequilibrium conditions has proven to be complicated due to the position-dependent electric and magnetic responses of the media. In fact, consistent position-dependent quantum models for the photon number in resonant structures have only been formulated very recently and only for dielectric media. Here we present a general positiondependent quantized fluctuational electrodynamics (QFED) formalism that extends the consistent field quantization to describe the photon number also in the presence of magnetic field-matter interactions. It is shown that the magnetic fluctuations provide an additional degree of freedom in media where the magnetic coupling to the field is prominent. Therefore, the field quantization requires an additional independent noise operator that is commuting with the conventional bosonic noise operator describing the polarization current fluctuations in dielectric media. In addition to allowing the detailed description of field fluctuations, our methods provide practical tools for modeling optical energy transfer and the formation of thermal balance in general dielectric and magnetic nanodevices. We use QFED to investigate the magnetic properties of microcavity systems to demonstrate an example geometry in which it is possible to probe fields arising from the electric and magnetic source terms. We show that, as a consequence of the magnetic Purcell effect, the tuning of the position of an emitter layer placed inside a vacuum cavity can make the emissivity of a magnetic emitter to exceed the emissivity of a corresponding electric emitter.

Quantized fluctuational electrodynamics for three-dimensional plasmonic structures

“…Here we have additionally used the constitutive relations D = ε 0 εE + δP and B = µ 0 (µH + δM), where ε 0 and µ 0 are the permittivity and permeability of vacuum, ε = ε r + iε i and µ = µ r + iµ i are the relative permittivity and permeability of the medium with real and imaginary parts denoted by subscripts r and i, and the polarization and magnetization fields δP and δM denote the polarization and magnetization that are not linearly proportional to the respective field strengths [41]. In the context of the fluctuational electrodynamics and the present work, δP and δM describe the small thermal fluctuations of the linear polarization and magnetization fields [34]. For the remainder of this work, the current density of free charges J f is also included in the electric permittivity for notational simplicity.…”

“…They relate the electric field strength E, the magnetic field strength H, the electric flux density D, and the magnetic flux density B to the free electric charge density ρ f and the free electric current density J f . In the frequency domain, Maxwell's equations are written for positive frequencies as [34]…”

“…From Maxwell's equations in Eqs. (1)-(4) it follows that the electric field obeys the well-known equation [34]…”

“…where the terms J e = −iωδP and J m = −iωµ 0 δM represent the polarization and magnetization currents that act as field sources in the noise operator theory [34] and in the classical fluctuational electrodynamics [42,43]. The electric term J e includes contributions from both the electric currents due to free charges and polarization terms associated with dipole currents and thermal dipole fluctuations.…”

“…We recently solved the cavity commutation relation anomaly and photon-number problem by introducing a quantized fluctuational electrodynamics (QFED) model to describe photon number and showed that the expectation values of the properly normalized annihilation and creation operators result in a meaningful photon-number model and thermal balance conditions [31][32][33][34]. This far, our models have been strictly one-dimensional and limited to normal incidence in planar structures, which has provided an adequate framework for describing the fundamental properties of cavity fields.…”