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Physical Review A

Paraxial quantum propagation

Ivan Deutsch, J. C. Garrison

Published Mar 1, 1991• 89 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.43.2498

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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(\ensuremathθ2{\mathrm{\ensuremath{\theta}}}^{2}\ensuremathθ2) difference between them vanishes for experiments described by stationary coherent states.
The classical concept of a paraxial wave is extended to quantum optics as a condition on the state vector, and the family of vectors obeying this condition defines a paraxial Hilbert space. Field operators acting on this space are approximated by the product of a slowly-varying envelope operator and a classical carrier wave. This permits the discussion of Gaussian pulses and paraxial wave packets. This treatment is extended to obtain an expansion of state vectors and field operators in powers of the opening angle of the cone of paraxial rays. The leading term in the expansion yields the component of the angular momentum along the propagation axis as a sum of well defined orbital- and spin-parts. With the same technique, it is shown that approximate localization of photons is possible for localization volumes much larger that the mean wavelength in the paraxial state.
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 (
The premise of this submission is to identify the role of the Wigner function [1] in paraxial wave optics, where it can be naturally introduced on the basis of the analogy between the paraxial wave equation and the Schrodinger equation of nonrelativistic quantum mechanics. Also known as the Wolf function, it describes the cross section of a paraxial beam using a real-valued function of displacement from axis and transverse components of the wave vector. This can be perceived as a quasi-probability distribution over the system's phase space, following the dynamics of a state space density of an ensemble of free noninteracting particles. In a (2+1)–dimensional picture this corresponds to the propagation of straight linear rays where the Wolf function describes the local phase space intensity. There is no notion of relative phase but negative values can be present, providing an alternative basis for description of interference phenomena. This corresponds exactly to the concept of pencils of positive and negative rays, developed by E. C. G. Sudarshan in the late 70’s and 80’s [2, 3].
Paraxial light and atom optics are compared. To lowest order the slowly varying amplitude of a light field in a dielectric medium with a spatially dependent refractive index satisfies an equation which has the form of a Schr\"odinger equation: the ``optical Schr\"odinger equation.'' The unsystematic procedure of neglecting certain second-order derivatives is replaced by a systematic expansion which allows the calculation of consecutive corrections. The general theory is applied to harmonic motion in a graded index fiber and to tunneling between coupled fibers. The physical relations between wave evolution of massive particles and paraxially propagating waves are elucidated.
The manipulation of distinct degrees of freedom of photons plays a critical role in both classical and quantum information processing. While the principles of wave optics provide elegant and scalable control over classical light in spatial and temporal domains, engineering quantum states in Fock space has been largely restricted to few-photon regimes, hindered by the computational and experimental challenges of large Hilbert spaces. Here, we introduce ``Fock-space optics", establishing a conceptual framework of wave propagation in the quantum domain by treating photon number as a synthetic dimension. Using a superconducting microwave resonator, we experimentally demonstrate Fock-space analogues of optical propagation, refraction, lensing, dispersion, and interference with up to 180 photons. These results establish a fundamental correspondence between Schrödinger evolution in a single bosonic mode and classical paraxial wave propagation. By mapping intuitive optical concepts onto high-dimensional quantum state engineering, our work opens a path toward scalable control of large-scale quantum systems with thousands of photons and advanced bosonic information processing.