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arXiv:2103.00596 · Physical Review A

Third quantization of the electromagnetic field

J. D. Franson

Published Dec 1, 2021• 3 citations
arXivHTMLPDF

Abstract

We consider an approach in which the usual wave function ${\ensuremath{\psi}}_{j}({x}_{j})$ in the quadrature representation of mode $j$ of the electromagnetic field is further quantized to produce a field operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}_{j}({x}_{j}).$ 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 Schr\"odinger 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.

DOI
10.1103/physreva.104.063702

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We consider an approach in which the usual wave function \ensuremathψj(xj){\ensuremath{\psi}}_{j}({x}_{j})\ensuremathψj​(xj​) in the quadrature representation of mode jjj of the electromagnetic field is further quantized to produce a field operator \ensuremathψ\ifmmode^\elseˆ\fij(xj).{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}_{j}({x}_{j}).\ensuremathψ\ifmmode^\elseˆ\fi​j​(xj​). 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 Schr"odinger 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.
Each mode jjj 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.
In a concise, but accessible for the first acquaintance form the procedure for the quantization of linear oscillator is set out. By analogy with this procedure the procedure of quantization (second quantization) of classical Maxwell’s electrodynamics is set up. The physical sense of the wave functions arguments of transverse electromagnetic field and its Fourier transformation is set up. One pay attention as for quantum coherent (almost a classic) states of the electromagnetic field and for photonics Fock states. Attention is drawn to the fact of absence the power of the universal content of such concepts as field amplitude, phase and number of particles (photons), which are used by experimenter’s to describe the states of a quantized field. The semi quantitative description the interaction processes of a quantum electromagnetic field with substance is set up. Specified situations are shown in which the discrepancy between the predictions of classical and quantum electrodynamics is noticeable at the macroscopic level.
Introduction Up to now the electromagnetic field has been treated as a classical field, describable by c -number functions. The great success of classical electromagnetic theory in accounting for a variety of optical phenomena, particularly those connected with wave propagation, interference and diffraction, amply justifies the classical approach. Moreover, as we have seen in the preceding chapters, in some cases the classical wave theory also gives a good account of itself in the treatment of the interaction of electromagnetic fields. For example, it is able to describe such seemingly non-classical effects as photoelectric bunching and the photo-electric counting statistics. It might almost seem that there is little justification for going beyond the domain of classical wave theory in optics. On the other hand, it can be argued that optics lies well and truly in the quantum domain, in the sense that we often encounter situations in which very few quanta or photons are present. In the microwave region of the electromagnetic spectrum, and at still longer wavelengths, the number of photons in each mode of the field is usually very large, and we are justified in treating the system classically. However, in the optical region the situation is usually just the opposite. As we show in Section 13.1, for light produced by practically all sources other than lasers, the average number of photons per mode is typically much less than unity.
The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.
The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.