Quantization Of The Electromagnetic Field Research Articles

The electromagnetic field’s quantum nature and spin

The interaction of the electron (charge) with the electromagnetic field is investigated using the minimum coupling prescription, in which the electron’s spin appears in the Dirac and Pauli–Schrödinger equations. However, we analyze the use of this approach in Maxwell’s equations and the repercussions here. The application of this prescription resulted in novel electrodynamics associated with spin. The complex Maxwell’s equations are discovered to describe a massless electromagnetic field governed by the Dirac equation interacting with a particle with a double charge whose wavefunction is the electromagnetic field’s energy density. The conservation equation for electromagnetic field helicity (spin) is obtained directly from the new electrodynamics. We demonstrated that the quantization of the electromagnetic field (Maxwell’s equations) interacting with a photon field results in the electromagnetic field’s quantization. The energy density of the electromagnetic field is shown to fulfill a massless Dirac equation. This indicates the material nature of the electromagnetic field (photon). As a result, the electromagnetic spin charge and current densities are determined. We obtained the Josephson energy–current relationship from the spin current produced by an electromagnetic field. A quantum nonlinear Ohm’s law is developed. Quantum power and force associated with the quantum nature of the electromagnetic field are proposed.

Quantum extension to the semiclassical theory of electrical susceptibility

The semiclassical expressions for susceptibilities are inadequate when quantum aspects of light become prominent. A more appropriate formulation of the electrical susceptibility theory requires the quantization of the electromagnetic field. We show here a way to do this up to the second order, within a low-dimensional system. The generalized expressions reveal that susceptibilities depend on the quantum state of light. As experimentally verifiable consequences of this, we predict new phenomena such as photon number-dependent refractive index, even in the linear regime, and non-quadratic second harmonic generation, which should be appreciable when very low-intensity coherent light is involved. We also deduce when the semiclassical expressions, which happen to be particular cases of those derived here, can be used confidently.

Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges

We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert space followed by an imposition of—what effectively amounts to—a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach–Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlarges the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.Quanta 2023; 12: 164–170.

Measurement-induced collective vibrational quantum coherence under spontaneous Raman scattering in a liquid

Spontaneous vibrational Raman scattering is a ubiquitous form of light–matter interaction whose description necessitates quantization of the electromagnetic field. It is usually considered as an incoherent process because the scattered field lacks any predictable phase relationship with the incoming field. When probing an ensemble of molecules, the question therefore arises: What quantum state should be used to describe the molecular ensemble following spontaneous Stokes scattering? We experimentally address this question by measuring time-resolved Stokes–anti-Stokes two-photon coincidences on a molecular liquid consisting of several sub-ensembles with slightly different vibrational frequencies. When spontaneously scattered Stokes photons and subsequent anti-Stokes photons are detected into a single spatiotemporal mode, the observed dynamics is inconsistent with a statistical mixture of individually excited molecules. Instead, we show that the data are reproduced if Stokes–anti-Stokes correlations are mediated by a collective vibrational quantum, i.e. a coherent superposition of all molecules interacting with light. Our results demonstrate that the degree of coherence in the vibrational state of the liquid is not an intrinsic property of the material system, but rather depends on the optical excitation and detection geometry.

Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Białynicki-Birula (BB) approach, based on the Riemann–Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approaches are completely equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the BB quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell’s equations.

Creating large Fock states and massively squeezed states in optics using systems with nonlinear bound states in the continuum

The quantization of the electromagnetic field leads directly to the existence of quantum mechanical states, called Fock states, with an exact integer number of photons. Despite these fundamental states being long-understood, and despite their many potential applications, generating them is largely an open problem. For example, at optical frequencies, it is challenging to deterministically generate Fock states of order two and beyond. Here, we predict the existence of an effect in nonlinear optics, which enables the deterministic generation of large Fock states at arbitrary frequencies. The effect, which we call an n-photon bound state in the continuum, is one in which a photonic resonance (such as a cavity mode) becomes lossless when a precise number of photons n is inside the resonance. Based on analytical theory and numerical simulations, we show that these bound states enable a remarkable phenomenon in which a coherent state of light, when injected into a system supporting this bound state, can spontaneously evolve into a Fock state of a controllable photon number. This effect is also directly applicable for creating (highly) squeezed states of light, whose photon number fluctuations are (far) below the value expected from classical physics (i.e., shot noise). We suggest several examples of systems to experimentally realize the effects predicted here in nonlinear nanophotonic systems, showing examples of generating both optical Fock states with large n (n > 10), as well as more macroscopic photonic states with very large squeezing, with over 90% less noise (10 dB) than the classical value associated with shot noise.

Kazimirov efekat

Introduction/purpose: The quantization of the electromagnetic field gives rise to quantum fluctuations which in turn produce a force on macroscopic boundaries. This phenomenon is called the Casimir effect. Method: The second quantization of the electromagnetic field is employed. The Zeta function regularization technique has been applied. Results: Because of the electromagnetic field quantization, a force on macroscopic boundaries is observed. Conclusions: Vacuum fluctuations due to quantum effects give macroscopic results.

Canonical Quantization of the Electromagnetic Field in Arbitrary ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}-Gauge

We carry out the canonical quantization of the electromagnetic field in arbitrary ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}-gauge and compute its propagator. In this way we fill a gap in the literature and clarify some existing confusion about Feynman iϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{i}\\epsilon $$\\end{document} prescription for the propagator of the electromagnetic field. We also discuss the BRST quantization and investigate the apparent singularities present in the theory when the gauge parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} takes the value −1. We find that this is a mere artifact due to the choice of basic modes and show that in the appropriate basis the commutation relations and the BRST transformation are, in fact, independent of the gauge parameter. The latter only appears as the coefficient of a BRST exact term in the Hamiltonian, which constitutes an extremely simple proof of the independence of any physical process on the gauge parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}.

Minimal length phenomenology and the black body radiation

The generalized uncertainty principle (GUP) modifies the uncertainty relation between momentum and position giving room for a minimal length, as predicted by candidates theories of quantum gravity. Inspired by GUP, Planck’s distribution is derived by considering a new quantization of the electromagnetic field. We elaborate on the thermodynamics of the black body radiation obtaining Wien’s law and the Stefan–Boltzmann law. We show that such thermodynamics laws are modified at Planck-scale.

Experimental test of the third quantization of the electromagnetic field

Each mode $j$ of the electromagnetic field is mathematically equivalent to a harmonic oscillator with a wave function ${\ensuremath{\psi}}_{j}({x}_{j})$ in the quadrature representation. An approach was recently introduced in which ${\ensuremath{\psi}}_{j}({x}_{j})$ was further quantized to produce a field operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}_{j}({x}_{j})$ [J. D. Franson, Phys. Rev. A 104, 063702 (2021)]. This approach allows a generalization of quantum optics and quantum electrodynamics based on an unknown mixing angle $\ensuremath{\gamma}$ that is somewhat analogous to the Cabibbo angle or the Weinberg angle. The theory is equivalent to conventional quantum electrodynamics for $\ensuremath{\gamma}=0$, while it predicts an inelastic photon scattering process for $\ensuremath{\gamma}\ensuremath{\ne}0$. Here we report the results of an optical scattering experiment that set an upper bound of $|\ensuremath{\gamma}|\ensuremath{\le}1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ rad at the 99% confidence level, provided that the particles created by the operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}_{j}({x}_{j})$ have negligible mass. High-energy experiments would be required to test the theory if the mass of these particles is very large.

Third quantization of the electromagnetic field

We consider an approach in which the usual wave function in the quadrature representation of mode j of the electromagnetic field is further quantized to produce a field operator. Since the electromagnetic field is already second quantized, this corresponds to an additional or third quantization. The third-quantization approach can be used to perform certain quantum optics calculations in the Heisenberg picture that could only be performed in the Schrodinger picture when using the conventional second-quantized theory. This approach also allows an interesting generalization of quantum optics and quantum electrodynamics that is analogous to symmetry breaking in elementary particle theory. The predictions of the generalized theory could be tested using a proposed photon scattering experiment.

Impact of gauge fixing on angular momentum operators of the covariantly quantized electromagnetic field

Covariant quantization of the electromagnetic field imposes the so-called gauge-fixing modification on the Lagrangian density. As a result of that, the total angular momentum operator receives at least one gauge-fixing-originated contribution, whose presence causes some confusion in the literature. The goal of this work is to discuss in detail why such a contribution, having no classical interpretation, is actually indispensable. For this purpose, we divide canonical and Belinfante-Rosenfeld total angular momentum operators into different components and study their commutation relations, their role in the generation of rotations of quantum fields, and their action on states from the physical sector of the theory. Then, we examine physical matrix elements of operators having gauge-fixing-related contributions, illustrating problems that one may encounter due to careless employment of the resolution of identity during their evaluation. The resolution of identity, in the indefinite-metric space of the covariantly quantized electromagnetic field, is extensively discussed because it takes a not-so-intuitive form if one insists on explicit projection onto states from the physical sector of the theory. Our studies are carried out in the framework of the Gupta-Bleuler theory of the free electromagnetic field. Relevant remarks about interacting systems, described by covariantly quantized electrodynamics, are given.

Conductor's elastic response to the vacuum-field radiation pressure

The momentum transfer from an electromagnetic field to reflecting and absorbing surfaces was asserted by the Maxwell equations. This phenomenon, with important implications for the development of the cosmos, has also been investigated in the discretized energy scale using optomechanical devices. With the quantization of the electromagnetic field, it was discovered that the vacuum field may influence the dynamics of some physical systems, such as in the Casimir effect and the radiative decay of an atom. Here, the effect of the radiation pressure by the electromagnetic vacuum on the surface of a compressible conductor is analyzed, and the model based on the Born-Markov master equation predicts a harmonic strain and momentum analog to the classical counterpart. A fundamental difference observed is the oscillating purity of the deformation state. In addition, it was demonstrated that the time-averaged force originated from the elastic reflection of the vacuum-field modes is comparable to the Casimir force for two ideal metallic plates separated by a distance proportional to the reflectivity cutoff wavelength.

Testing the quantization of electromagnetic field in a quarter-wavelength transmission line resonator by traveling-wave scattering measurements

A half-wavelength cavity, wherein the electromagnetic field is quantized as standing-wave photons, is a typical artificial quantum system that has widely been used to implement quantum information processing. In this paper, we propose an approach to confirm the quantization of the electromagnetic field in another type of cavity, namely, a quarter-wavelength transmission line resonator (TLR), by using traveling-photon scattering measurements. Furthermore, we show that the number of photons in a TLR can be destructively detected by using the usual IQ-mixed technique. We hope that quarter-wavelength TLRs may also serve as quantum devices to implement on-chip solid-state quantum information processing.

Quantum optics and moving dissipative media: A phenomenological approach

Within the framework of a phenomenological quantization scheme, we present the quantization of the electromagnetic field in the presence of a moving absorptive and dispersive magneto-dielectric slab (MDS) with uniform velocity in the direction parallel to its interface. We derive the quantum input-output relations for the case that quantum states propagate perpendicularly to the moving MDS.We thoroughly investigate the impact of the motion of the movingMDS on quantum properties of the incident states. To illustrate this, by modeling the dispersive and dissipative effects of the slab by the Lorentz model in its rest frame, we compute the quadrature squeezing and the Mandel parameter for the transmitted state when the incident states from left and right sides are, respectively, the coherent and the quantum vacuum states. It is shown that how quantum features of the incident state are degraded when it is transmitted through the moving MDS.

Secondary Quantization of the Electromagnetic Field in Inhomogeneous Structures Based on Formalism of Scattering Matrix

Secondary Quantization of the Electromagnetic Field in Inhomogeneous Structures Based on Formalism of Scattering Matrix

The numerical investigation of colliding optical solitons as an all-optical-gate using the method of Lines

The bidirectional propagation and collision of counter-propagating optical solitons in a two-level resonant medium is analyzed. For sufficiently short duration of the pulses and sufficiently high power levels, self-induced transparency allows the pulses to propagate without losses at anomalously low velocities. Here, we use an extended semiclassical approach, called Maxwell–Bloch model. The quantization of the electromagnetic field is neglected, however, a full quantum mechanical formulation is used to describe the behavior of the matter. The numerical simulations are performed by using the method of lines. For improved accuracy, spline interpolation of the fields is introduced. Results are presented to demonstrate the use of soliton collision for all-optical switching, in particular, as OR- and XOR-gates.

Photons in the presence of parabolic mirrors

We present a vectorial analysis of the behavior of the electromagnetic field in the presence of boundaries with parabolic geometry. The relevance of the use of symmetries to find explicit closed expressions for the electromagnetic fields is emphasized. Polarization and phase related angular momenta of light have an essential role in the proper definition of the generator $\mathfrak{A}_3$ of a symmetry transformation that distinguishes the parabolic geometry. Quantization of the electromagnetic field in terms of the resulting elementary modes is performed. The important case of a boundary defined by an ideal parabolic mirror is explicitly worked out. The presence of the mirror restricts the eigenvalues of $\mathfrak{A}_3$ available to the electric and magnetic fields of a given mode via compact expressions. Modes previously reported in the literature are particular cases of those described in this work.

Radiative heat transfer between two magnetodielectric metamaterial slabs with finite thickness in non-equilibrium situation

In this paper, we consider a system including two maghnetodielectric slabs with different temperature that are placed in vacuum at zero temperature and very short separation distances from each other. Based on the canonical quantization of the electromagnetic field in the presence of dissipative media, we investigate the radiative heat transfer arising from thermal and quantum fluctuations in out of thermal equilibrium but stationary situation. For this purpose, we calculate the ensemble average of Poynting vector by driving quantum correlation relations between noise polarization and magnetization operators and extracting the electromagnetic Green tensor of the system. Finally, we analyze the transferred radiation of aforementioned system by employing the numerical result of the Poynting vector.

Quantization of the Electromagnetic Field in Three-Dimensional Photonic Structures on the Basis of the Scattering Matrix Formalism (S Quantization)

A technique for quantization of the electromagnetic field in photonic nanostructures with three-dimensional modulation of the dielectric constant is developed on the basis of the scattering matrix formalism (S quantization in the three-dimensional case). Quantization is based on equating the eigenvalues of the scattering matrix to unity, which is equivalent to equating each other the sets of Fourier expansions for the fields of the waves incident on the structure and propagating away from the structure. The spatial distribution of electromagnetic fields of the modes in a photonic nanostructure is calculated on the basis of the R and T matrices describing the reflection and transmission of the Fourier components by the structure. To calculate the reflection and transmission coefficients of individual real-space and Fourier-space components, the structure is divided into parallel layers within which the dielectric constant varies as a function of two-dimensional coordinates. Using the Fourier transform, Maxwell’s equations are written in the form of a matrix connecting the Fourier components of the electric field at the boundaries of neighboring layers. Based on the calculated reflection and transmission vectors for all polarizations and Fourier components, the scattering matrix for the entire structure is formed and quantization is carried out by equating the eigenvalues of the scattering matrix to unity. The developed method makes it possible to obtain the spatial profiles of eigenmodes without solving a system of nonlinear integro-differential equations and significantly reduces the computational resources required for calculating the probability of spontaneous emission in three-dimensional systems.