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Physical Review A

‘‘Squeezed states’’ in Helmholtz optics

Kurt Bernardo Wolf, E.V. Kurmyshev

Published Apr 1, 1993• 11 citations
Publisher

Abstract

Squeezed states have been described in optics with the mathematics of quantum mechanics. Yet, electromagnetic fields of a single color obey the Helmholtz equation, of which the Schr\"odinger equation is but the paraxial (parabolic) approximation. We extend the ordinary squeezed states to solutions of the Helmholtz equation that contain them as their paraxial approximation. Third-order aberration expansion and asymptotic analyses are performed on these states.

DOI
10.1103/physreva.47.3365

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Squeezed states have been described in optics with the mathematics of quantum mechanics. Yet, electromagnetic fields of a single color obey the Helmholtz equation, of which the Schr"odinger equation is but the paraxial (parabolic) approximation. We extend the ordinary squeezed states to solutions of the Helmholtz equation that contain them as their paraxial approximation. Third-order aberration expansion and asymptotic analyses are performed on these 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.
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.
A companion paper Beetle and Jongewaard de Boer (The paraxial approximation in quantum optics I: henochromatic modes of a scalar field. J Mod Opt. 2025) has argued that the best way to associate single-particle quantum states of a scalar field to the modes of a narrowly collimated beam of classical radiation modelled in the paraxial approximation uses the ‘henochromatic’ states previously introduced by Sudarshan et al. (Paraxial-wave optics and relativistic front description. II. The vector theory. Phys Rev A. 1983;28:2921–2932. doi: 10.1103/PhysRevA.28.2921, Paraxial-wave optics and relativistic front description. I. The scalar theory. Phys Rev A. 1983;28:2933–2942. doi: 10.1103/PhysRevA.28.2921). This paper extends that result to Maxwell fields, again emphasizing the central role of unitarity in defining the association. The principal new technical element in the present discussion has to do with the intertwining of polarization and spatial degrees of freedom in the resulting single-photon states.
This paper examines how best to associate quantum states of a single particle to different modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation. Our analysis stresses the importance of the relationship between two inner products naturally arising in the problem. These are the inner product used to expand a general beam as a superposition of orthogonal modes in the paraxial approximation, on the one hand, and the canonical inner product on which the statistical interpretation of quantum (field) theory is founded, on the other. While several candidates for the sort of association between beam modes and single-particle quantum states have been proposed in the literature, here we argue that one of them is uniquely well suited to the task. Specifically, the mapping of beam modes to the ‘henochromatic’ quantum states previously introduced by Sudarshan, Simon and Mukunda [Paraxial-wave optics and relativistic front description. I. The scalar theory. Phys Rev A. 1983;28:2921–2932; Paraxial-wave optics and relativistic front description. II. The vector theory. Phys Rev A. 1983;28:2921–2932.] is unique within a large class of similar mappings in that it is unitary in a mathematically precise sense.