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arXiv:quant-ph/0502164 · Physical Review A

Exact quantization of a paraxial electromagnetic field

Andrea Aiello, J. P. Woerdman

Published Dec 30, 2005• 22 citations
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Abstract

A nonperturbative quantization of a paraxial electromagnetic field is achieved via a generalized dispersion relation imposed on the longitudinal and the transverse components of the photon wave vector. This theoretical formalism yields a seamless transition between the paraxial- and the Maxwell-equation solutions. This obviates the need to introduce either ad hoc or perturbatively defined field operators. Moreover, our (exact) formalism remains valid beyond the quasimonochromatic paraxial limit.

DOI
10.1103/physreva.72.060101

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A nonperturbative quantization of a paraxial electromagnetic field is achieved via a generalized dispersion relation imposed on the longitudinal and the transverse components of the photon wave vector. This theoretical formalism yields a seamless transition between the paraxial- and the Maxwell-equation solutions. This obviates the need to introduce either ad hoc or perturbatively defined field operators. Moreover, our (exact) formalism remains valid beyond the quasimonochromatic paraxial limit.
A quantum version of the classical paraxial approximation is established. After expanding the classical free-field wave equation in powers of \ensuremath{\theta} (the characteristic opening angle of the paraxial rays), the familiar time-dependent paraxial equation appears in zeroth order. The arguments employed in deriving this approximation scheme are used in turn to identify the subspace of the photon Fock space consisting of paraxial states of the field. The quantum version of the paraxial approximation is then obtained by restricting the Maxwell field operators to this domain. Exact equal-time commutation relations are recovered as expansions in \ensuremath{\theta}. In zeroth order, the theory yields a quantized analog of the classical paraxial wave equation, and formally resembles a nonrelativistic many-body theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same true-number operator. In addition, it is shown that the O(${\mathrm{\ensuremath{\theta}}}^{2}$) difference between them vanishes for experiments described by stationary coherent states.
We introduce a generalized angular spectrum representation for quantized light beams. By using our formalism, we are able to derive simple expressions for the electromagnetic vector potential operator in the case of: a) time-independent paraxial fields, b) time-dependent paraxial fields, and c) non-paraxial fields. For the first case, the well known paraxial results are fully recovered. c ○ 2008 Optical Society of America OCIS codes: 000.1600, 270.0270. Propagation of nonclassical states of the electromagnetic field is an issue of growing interest in quantum optics for both fundamental and technological purposes. 1 Consider, for instance, the relevance of propagation of entangled photons to quantum cryptographic systems. 2 The purpose of the present Letter is to provide a novel perfectly general formalism for the representation of quantized light beams which can be used in any regime of propagation. Such objective is achieved by using a dispersion relation that some of us have recently introduced in Ref., 3 and a generalized angular spectrum representation for the field operators. 4 The usefulness of our approach5 becomes manifest whenever one deals with quantum systems for which both paraxial and nonparaxial regimes of propagation may be relevant as, e.g., down-converted photon pairs. 6 Consider the plane-wave expansion of the positivefrequency part of the electromagnetic vector potential operator Â(r, t) = Â(+) (r, t) + H.c. in the Coulomb gauge7 Â (+) ∫ (r, t) = d 3 (
A three-dimensional electromagnetic field quantization scheme is developed using the Green's-function method in a homogeneous medium that shows loss and dispersion. Special attention is paid to the decomposition of the quantized fields into their longitudinal and transverse parts. The characteristic nature of these two parts is discussed. To demonstrate the generality of this formulation, the field quantization in lossless dispersive media and in free space is obtained by imposing appropriate limits. We also calculate the vacuum fluctuations of the field for the transverse and longitudinal parts separately.
The notion that the electromagnetic field is quantized is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantization 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 quantization. By contrast, here we quantize 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.
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.
Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.