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Journal of Qinghai Normal University

Quantisation of the Electromagnetic Field and the Number State Representation for the Electromagnetic Field

Yuan Hai-liang

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

From Maxwell equations for free space and by converting the amplitudes in Fourier expansion of the vector potential into operators according to quantisation correspondence,this paper obtains Hamiltonian operator,which quantises the electromagnetic field,and introduces the number state representation for the field.

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The notion that the electromagnetic field is quantised is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantisation of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantisation. By contrast, here we quantise the electromagnetic field in a less mathematically and more physically-motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly-used notion of a fictitious cavity that applies boundary conditions to the field.
In this chapter, the authors prove that the free electromagnetic field is equivalent to an infinite set of uncoupled one-dimensional harmonic oscillators, one oscillator being associated with each value of the propagation vector k and polarization mode α. In order to quantize the electromagnetic field the authors shall have to express the field equations in Hamiltonian form and impose the canonical commutation rules on the canonically conjugate variables. Because the electromagnetic field possesses an infinite number of degrees of freedom it is also possible to formulate the theory in terms of a very different set of dynamical variables, the field operators themselves. To be more precise, the field operators only have off-diagonal matrix elements in the representation in which the photon occupation numbers are diagonal. In a state with a definite number of photons the expectation value of any field strength vanishes, and it is only meaningful to discuss fluctuations in the field strengths.
In chapter 2, following the Maxwell description, we discuss vector potential, the continuity equation, and energy and momentum corresponding to optical fields. Based on the Maxwell depiction, quantization of electromagnetic fields is obtained. Quantized electromagnetic fields, such as Fock states, coherent states, squeezed states, and Schrödinger cat states, provide a general understanding of the quantum nature of the field. Thereby, we build the necessary foundation-level knowledge required to understand electromagnetic fields in the quantum domain.
In this paper it is shown that Maxwell's theory of the electromagnetic field in vacuum can be stated in a form closely parallel to Dirac's theory of the electron. The electromagnetic field is described by a three-by-one column matrix whose elements are linear combinations of the three independent spinor components of the field. The Maxwell equations take a form similar to the Dirac equation for a free electron but involving three-by-three matrices. In terms of a wave function normalized so that the integral-square is the number of photons, the classical expressions for the energy, momentum, and angular momentum in the field are related to expected values of the Hamiltonian, momentum, and angular momentum operators. The angular momentum operator consists of an orbital part plus a three-by-three spin-one matrix part. As in Dirac's theory, the contributions from the states of negative Hamiltonian eigenvalue must be subtracted---these are states of opposite circular polarization.
In a recent paper [Southall et al., Locally-acting mirror Hamiltonians, arXiv:1908.07597 (2021)], we showed that it is possible to design locally-acting mirror Hamiltonians which only affect incoming but do not change the dynamics of outgoing wave packets. To better justify our approach, this paper presents a systematic quantisation of the electromagnetic field in position space. Starting from the assumption that the basic building blocks of the electromagnetic field in one dimension are localised bosonic wave packets with a clear direction of propagation -- so-called bosons localised in position (BLiPs) -- we identify the relevant Schroedinger equation and construct Lorentz covariant electric and magnetic field observables. Our description contains the standard description of the quantised electromagnetic field which is shown to apply to a subspace of states.
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.