Light in a dielectric medium moves slower than in vacuum. The corresponding electromagnetic field equations are then no longer invariant under ordinary Lorentz transformations, but only under such transformations corresponding to this reduced velocity. Based on this physical symmetry, an effective theory for low-energy electromagnetic phenomena in dielectrics is constructed. It has none of the problems of the old formulations of Minkowski and Abraham. Dispersion in the optical regime and the Kerr effect arise in a natural way from higher order interaction terms. The effective field theory is quantized by standard methods and quantum corrections can be calculated in a systematic way. Thus it relates many different classical and quantum optical phenomena into a unified description.
By modeling a dielectric medium with two independent reservoirs, i.e., electric and magnetic reservoirs, the electromagnetic field is quantized in a linear dielectric medium consistently. A Hamiltonian is proposed from which using the Heisenberg equations, not only the Maxwell equations but also the structural equations can be obtained. Using the Laplace transformation, the wave equation for the electromagnetic vector potential is solved in the case of a homogeneous dielectric medium. Some examples are considered showing the applicability of the model to both absorptive and nonabsorptive diele…
Canonical quantization of electromagnetic field inside the time--spatially dispersive inhomogeneous dielectrics is presented. Interacting electromagnetic and matter excitation fields create the closed system, Hamiltonian of which may be diagonalized by generalized polariton transformation. Resulting dispersion relations coincide with the classical ones obtained by the solution of wave equation, the corresponding mode decomposition is, however, orthogonal and complete in the enlarged Hilbert space.
We present a Gupta-Bleuler quantization scheme for the electromagnetic field in time-dependent dielectric media. Starting from the Maxwell equations, a generalization of the Lorentz gauge condition adapted to time varying dielectrics is derived. Using this gauge, a Gupta-Bleuler approach to quantize all polarizations of the radiation field and the corresponding constraint condition are introduced. This new approach is different from the quantized electromagnetic field in vacuum in the sense that here the contributions of unphysical photons cannot be thoroughly eliminated, which further lead to…
A new mathematical and computational technique for calculating quantum vacuum expectation values of energy and momentum densities associated with electromagnetic fields in bounded domains containing inhomogeneous media is discussed. This technique is illustrated by calculating the mode contributions to the difference in the vacuum force expectation between opposite ends of an inhomogeneous dielectric non-dispersive medium confined to a perfectly conducting rigid box.
Based on calculations involving an idealized boundary condition, it has long been assumed that the stress on a spherical conducting shell is repulsive. We use the more realistic case of a Drude dielectric to show that the stress is attractive, matching the generic behavior of Casimir forces in electromagnetism. We trace the discrepancy between these two cases to interactions between the electromagnetic quantum fluctuations and the dielectric material.
We derive an expression for the Casimir force between slabs with arbitrary dielectric properties characterized by their reflection coefficients. The formalism presented here is applicable to media with a local or a non-local dielectric response, an infinite or a finite width, inhomogeneous dissipative, etc. Our results reduce to the Lifshitz formula for the force between semi-infinite dielectric slabs by replacing the reflection coefficients by the Fresnel amplitudes.
A quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive and absorbing dielectric medium is developed. The classical Maxwell equations with spatially varying and Kramers-Kronig consistent permittivity are regarded as operator-valued field equations, introducing additional current- and charge-density operator fields in order to take into account the noise associated with the dissipation in the medium. It is shown that the equal-time commutation relations between the fundamental electromagnetic fields $\hat …
The local-field renormalization of the spontaneous emission rate in a dielectric is explicitly obtained from a fully microscopic quantum-electrodynamical, many-body derivation of Langevin-Bloch operator equations for two-level atoms embedded in an absorptive and dispersive, linear dielectric host. We find that the dielectric local-field enhancement of the spontaneous emission rate is smaller than indicated by previous studies.
The stress tensor for the quantized electromagnetic field is calculated in the region between a pair of dispersive, dielectric half-spaces. This generalizes the stress tensor for the Casimir energy to the case where the boundaries have finite reflectivity. We also include the effects of finite temperature. This allows us to discuss the circumstances under which the weak energy condition and the null energy condition can be violated in the presence of finite reflectivity and finite temperature.
Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as MEMS embedded in dielectric media, Casimir induced friction in nano-machinery, micro-fluidics and molecular electronics. Computation of such stresses based on cavity QED generally require numerical analysis based on a regularization process. A new scheme is described that has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating n…
The Hamiltonian of a polariton model for an inhomogeneous linear absorptive dielectric is diagonalized by means of Fano's diagonalization method. The creation and annihilation operators for the independent normal modes are explicitly found as linear combinations of the canonical operators. The coefficients in these combinations depend on the tensorial Green function that governs the propagation of electromagnetic waves through the dielectric. The time-dependent electromagnetic fields in the Heisenberg picture are given in terms of the diagonalizing operators. These results justify the phenomen…
The electromagnetic field is canonically quantized in the presence of a linear, dispersive and dissipative medium that is in uniform motion. Specifically we calculate the change in the normal modes of the coupled matter-field system and find a Hamiltonian that contains negative energy normal modes. We interpret these modes as the origin of phenomena such as quantum friction, and find that a detector initially in its ground state and coupled to the electromagnetic field in the vicinity of, or within a uniformly moving medium has a non-zero probability of excitation at T=0K.
We study the scattering of the quantized electromagnetic field from a linear, dispersive dielectric using the scattering formalism for quantum fields. The medium is modeled as a collection of harmonic oscillators with a number of distinct resonance frequencies. This model corresponds to the Sellmeir expansion, which is widely used to describe experimental data for real dispersive media. The integral equation for the interpolating field in terms of the in field is solved and the solution used to find the out field. The relation between the in and out creation and annihilation operators is found…
An ad hoc quantization scheme for the electromagnetic field in a weakly dispersive, transparent dielectric leads to the definition of canonical and kinetic forms for the momentum of the electromagnetic field in a dispersive medium. The canonical momentum is uniquely defined as the operator that generates spatial translations in a uniform medium, but the quantization scheme suggests two possible choices for the kinetic momentum operator, corresponding to the Abraham or the Minkowski momentum in classical electrodynamics. Another implication of this procedure is that a wave packet containing a s…
The quantization of the electromagnetic field in a three-dimensional inhomogeneous dielectric medium with losses is carried out in the framework of a damped-polariton model with an arbitrary spatial dependence of its parameters. The equations of motion for the canonical variables are solved explicitly by means of Laplace transformations for both positive and negative time. The dielectric susceptibility and the quantum noise-current density are identified in terms of the dynamical variables and parameters of the model. The operators that diagonalize the Hamiltonian are found as linear combinati…
More recently, Feigel has considered the quantum vacuum contribution to the momentum of electromagnetic media. However, in Feigel's treatment he did not take into account the relativistic transformation of the optical constants (electric permittivity and magnetic permeability) of moving media. Here it is shown that the effect arising from such a transformation will also provide a quantum vacuum contribution to the velocity of media, in addition to the one derived by Feigel himself.
We consider the electric and magnetic field fluctuations in the vacuum state in the region external to a half-space filled with a homogeneous non-dissipative dielectric. We discuss an appropriate limit to an ideal metal and concentrate our interest on the renormalized field fluctuations, or equivalently to renormalized electric and magnetic energy densities, in the proximity of the dielectric-vacuum interface. We show that surface divergences of field fluctuations arise at the interface in an appropriate ideal conductor limit, and that our limiting procedure allows to discuss in detail their s…
A canonical relativistic formulation is introduced to quantize electromagnetic field in the presence of a polarizable and magnetizable moving medium. The medium is modeled by a continuum of four vectors in a phenomenological way. The covariant wave equation for the vector potential and the covariant constitutive equation of the medium are obtained as the Euler- Lagrange equations using the Lagrangian of the total system. A tensor of the third rank which couples the electromagnetic field with the medium is introduced and the susceptibility tensor of the medium is obtained in terms of the coupli…
In a recent work, S.A.R. Horsley and coworkers [Nature Photon 9, 436-439 (2015)] showed rather interestingly that planar dielectric media, for which the real and imaginary parts of the dielectric permittivity are related by spatial Kramers-Kronig (KK) relations, are one-way reflectionless, whatever the angle of incidence. Such a fascinating property, besides of extending our comprehension of the fundamental phenomenon of reflection, may offer new ways for the design of antireflection surfaces and thin materials with efficient light absorption. However, KK dielectric media are generally describ…
We develop a dual description of quantum optics adapted to dielectric systems without magnetic property. Our formalism, which is shown to be equivalent to the standard one within some dipolar approximations discussed in the article, is applied to the description of polaritons in dielectric media. We show that the dual formalism leads to the Huttner-Barnett equations [B. Huttner, S. M. Barnett, Phys. Rev. A \textbf{46}, 4306 (1992)] for QED in dielectric systems. More generally, we discuss the role of electromagnetic duality in the quantization procedure for optical systems and derive the struc…
A general theory of the interaction of the quantized electromagnetic field with atoms in the presence of dispersing and absorbing dielectric bodies of given Kramers--Kronig consistent permittivities is developed. It is based on a source-quantity representation of the electromagnetic field, in which the electromagnetic-field operators are expressed in terms of a continuous set of fundamental bosonic fields via the Green tensor of the classical problem. Introducing scalar and vector potentials, the formalism is extended in order to include in the theory the interaction of the quantized electroma…
A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium in thermal equilibrium, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a …
We derive effective photon modes that facilitate an intuitive and convenient picture of photon dynamics in a structured Kramers-Kronig dielectric in the limit of weak absorption. Each mode is associated with an effective line-width which determines the temporal decay rate of the photon. These results are then applied to obtain an expression for the Beer-Lambert-Bouguer law absorption coefficient for unidirectional propagation in structured media consisting of dispersive, weakly absorptive dielectric materials.
The eletromagnetic field in a linear absorptive dielectric medium, is quantized in the framework of the damped polarization model. A Hamiltonian containing a reservoir with continuous degrees of freedom, is proposed. The reservoir minimally interacts with the dielectric polarization and the electromagnetic field. The Lagevin-Schrodinger equation is obtained as the equation of motion of the polarization field. The radiation reaction electromagnetic field is considered. For a homogeneous medium, the equations of motion are solved using the Laplace transformation method.
It has very recently been suggested that asymmetric coupling of electromagnetic fields to thermal reservoirs under nonequilibrium conditions can produce unexpected oscillatory behavior in the local photon statistics in layered structures. Better understanding of the predicted phenomena could enable useful applications related to thermometry, noise filtering, and enhancing optical interactions. In this work we briefly review the field quantization and study the local steady state temperature distributions in optical cavities formed of lossless and lossy media to show that also local field tempe…
Starting from a Lagrangian, the electromagnetic field is quantized in the presence of a body rotating along its axis of symmetry. Response functions and fluctuation-dissipation relations are obtained. A general formula for rotational friction and power radiated by a rotating dielectric body is obtained in terms of the dyadic Green's tensor. Hamiltonian is determined and possible generalizations are discussed. As an example, the rotational friction and power radiated by a spherical dielectric in the vicinity of a semi-infinite dielectric plane is obtained and discussed in some limiting cases.
Quantum-statistical effects occur during the propagation of electromagnetic (EM) waves inside the dielectric media or metamaterials, which include a large class of nanophotonic and plasmonic waveguides with dissipation and noise. Exploiting the formal analogy between the Schrodinger equation and the Maxwell equations for dielectric linear media, we rigorously derive the effective Hamiltonian operator which describes such propagation. This operator turns out to be essentially non-Hermitian in general, and pseudo-Hermitian in some special cases. Using the density operator approach for general no…
The phenomenological Maxwell field is quantized for arbitrarily space- and frequency-dependent complex permittivity. The formalism takes account of the Kramers--Kronig relation and the dissipation-fluctuation theorem and yields the fundamental equal-time commutation relations of QED. Applications to the quantum-state transformation at absorbing and amplifying four-port devices and to the spontaneous decay of an excited atom in the presence of absorbing dielectric bodies are discussed.
We determine the commutation relations satisfied by the quantized electromagnetic field in the presence of macroscopic dielectrics and conductors, with arbitrary dispersive and dissipative properties. We consider in detail the case of two plane-parallel material slabs, separated by an empty gap, and we show that at all points in the empty region between the slabs, including their surfaces, the electromagnetic fields always satisfy free-field canonical equal-time commutation relations. This result is a consequence of general analyticity and fall-off properties at large frequencies satisfied by …
We derive expressions for the quantum electromagnetic field in a dispersive and dissipative dielectric medium, treating the medium as a continuum. We compare the Langevin approach with the Fano diagonalization procedure for the coupled system composed of the electromagnetic field, the dielectric medium, and the reservoir. In particular, we show that the quantized electric and magnetic fields obtained in both methods have exactly the same form. More generally, in thermal equilibrium, correlation functions computed in both methods are identical.
We measured photon absorption in dielectric media and proposed the photon-version Beer-Lambert's law to quantify the absorption. We used a Hong-Ou-Mandel interferometer and 810 nm twin-photons. We found that the depth ratio of the null point in the interference patterns of the interferometer agreed with the classical transmittance of the samples. We established a statistical model of the photon absorption process and proposed an information entropy interpretation to understand the meaning of the Beer-Lambert law. Comparisons of the results of the photon absorption experiments with classical ex…
The derivation of Casimir forces between dielectrics can be simplified by ignoring absorption, calculating energy changes due to displacements of the dielectrics, and only then admitting absorption by allowing permittivities to be complex. As a first step towards a better understanding of this situation we consider in this paper the model of a dielectric as a collection of oscillators, each of which is coupled to a reservoir giving rise to damping and Langevin forces on the oscillators and a noise polarization acting as a source of a fluctuating electromagnetic (EM) field in the dielectric. Th…
We study from a critical perspective several quantum-electrodynamic phenomena commonly related to vacuum electromagnetic (EM) fluctuations in complex media. We compute the resonance-shift, the spontaneous emission rate, the local density of states and the van-der-Waals-Casimir pressure in a dielectric medium using a microscopic diagrammatic approach. We find, in agreement with some recent works, that these effects cannot be attributed to variations on the energy of the EM vacuum but to variations of the dielectric self-energy. This energy is the result of the interaction of the bare polarizabi…
The microscopic theory of the Casimir effect in the dielectric is studied in the framework when absorption is realized via a reservoir modeled by a set of oscillators with continuously distributed frquencies with the aim to see if the effects depend on the form of interaction with the reservoir. A simple case of the one-dimensional dielectric between two metallic plates is considered. Two possible models are investigated, the direct interaction of the electromagnetic field with the reservoir and indirect interaction via an intermediate oscillator imitating the atom. It is found that with the s…
We show that in the presence of free charge carriers the definition of the frequency-dependent dielectric permittivity requires additional regularization. As an example, the dielectric permittivity of the Drude model is considered and its time-dependent counterpart is derived and analyzed. The respective electric displacement cannot be represented in terms of the standard Fourier integral. The regularization procedure allowing to circumvent these difficulties is suggested. For purpose of comparison it is shown that the frequency-dependent dielectric permittivity of insulators satisfies all rig…
We determine the energy-level shift experienced by a neutral atom due the quantum electromagnetic interaction with a layered dielectric body. We use the technique of normal-mode expansion to quantize the electromagnetic field in the presence of a layered, non-dispersive and non-absorptive dielectric. We explicitly calculate the equal-time commutation relations between the electric field and vector potential operators. We show that the commutator can be expressed in terms of a generalized transverse delta-function and that this is a consequence of using the generalized Coulomb gauge to quantize…
In this paper we deal with the macroscopic electromagnetic response of a finite size dispersive dielectric object, in unbounded space, in the framework of quantum electrodynamics using the Heisenberg picture. We apply a Hopfield type scheme to account for the dispersion and dissipation of the matter. We provide a general expression of the polarization density field operator as functions of the initial conditions of the matter field operators and of the electromagnetic field operators. It is a linear functional whose kernel is a linear expression of the impulse response of the dielectric object…
We consider the vacuum energy of the electromagnetic field in the background of spherically symmetric dielectrics, subject to a cut-off frequency in the dispersion relations. The effect of this frequency dependent boundary condition between media is described in terms of the {\it incomplete} $ζ$-functions of the problem. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, ...) terms in the Casimir energy. The application of these expressions to the case of a gas bubble immersed in water is discussed, and results consistent with Schwing…
We describe an event-based approach to simulate the propagation of an electromagnetic plane wave through dielectric media. The basic building block is a deterministic learning machine that is able to simulate a plane interface. We show that a network of two of such machines can simulate the propagation of light through a plane parallel plate. With properly chosen parameters this setup can be used as a beam splitter. The modularity of the simulation method is illustrated by constructing a Mach-Zehnder interferometer from plane parallel plates, the whole system reproducing the results of wave th…
In this paper we consider possible analogue Hawking radiation from a normal dielectric and metamaterial composite, having an analogue horizon where the dielectric parameters vanish and change sign upon crossing this transition zone. We follow a complex path analysis to show the presence of an analogue Hawking temperature at the horizon and subsequent photon production from the ambient electromagnetic field. Possibility of experimental observation is also commented upon.
