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Journal of Modern Optics

Quantum Optics in Oscillator Media

L. Knöll, Ulf Leonhardt

Published Jun 1, 1992• 36 citations
Publisher

Abstract

A quantum theory of the optical field in presence of a linear dispersive medium with losses representing a passive optical device is given. The medium is modelled by quantum oscillators coupled to thermal reservoirs. Time-ordered correlation functions of the optical field can be expressed by integrals of time-ordered source operators multiplied by classical field propagators which are influenced by dispersion and loss effects. Additionally correlation functions of thermal source quantities are needed. For a dielectric layer the propagator is derived, and some simple applications are discussed.

DOI
10.1080/713823539

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A quantum theory of the optical field in presence of a linear dispersive medium with losses representing a passive optical device is given. The medium is modelled by quantum oscillators coupled to thermal reservoirs. Time-ordered correlation functions of the optical field can be expressed by integrals of time-ordered source operators multiplied by classical field propagators which are influenced by dispersion and loss effects. Additionally correlation functions of thermal source quantities are needed. For a dielectric layer the propagator is derived, and some simple applications are discussed.
A quantum optical formulation of the action of passive, lossless optical instruments on quantum light fields is developed. The quantum Maxwell equations are formally solved and the commutation relations for various combinations of field operators at different times are studied. General relationships between field correlation functions and correlation functions of source-quantity operators and free-field operators are derived. Formulas are presented for the case when the free field is the vacuum field. Furthermore, the mixing of source-field light with coherent free-field light is treated. The theory is applied to the calculation of the photocount distribution. The theory presented renders it possible to take into account the effects of light propagation through the optical system. It is shown that the effect of the optical instruments may be taken into account by introducing an apparatus function in a way which formally corresponds to that of classical optics. However, the calculation of the corresponding convolution integrals is governed by operator ordering rules, which are essential in the case of quantum light fields and which can give rise to substantial differences between classical and quantum optics.
We derive the macroscopic electromagnetic-field and medium operators for a linear dispersive medium with a microscopic model. As an alternative to the previous treatments in the literature, we show that the canonical momentum for the macroscopic field can be chosen to be −∊0Ê instead of −Dˆ with the standard minimal-coupling Hamiltonian. We find that, despite the change in the field operator normalization constants, the equal-time commutators among the macroscopic electric-field, magnetic-field, and medium operators have the same values as their microscopic counterparts under a coarse-grained approximation. This preservation of the equal-time commutator is important from a fundamental standpoint, such as the preservation of micro-causality for macroscopic quantities. The existence of more than one normal frequency mode at each k vector in a realistic causal-response medium is shown to be responsible for the commutator preservation. The process of macroscopic averaging is discussed in our derivation. The macroscopic field operators we derive are valid for a wide range of frequencies below, above, and around resonances. Our derivation covers the lossless, slightly lossy, and dispersionless as well as dispersive regimes of the medium. The local-field correction is also included in the formalism by inclusion of dipole–dipole interactions. Comparisons are made with other derivations of the macroscopic field operators. Using our theory, we discuss the questions of field propagation across a dielectric boundary and the decay rate of an atom embedded in a dielectric medium. We also discuss the question of squeezing in a linear dielectric medium and the extension of our theory to the case of a nonuniform medium.
The quantum optics of metamaterials starts with the question of whether the same effective-medium theories apply as in classical optics. In general, the answer is negative. For active plasmonics but also for some passive metamaterials, we show that an additional effective-medium parameter is indispensable besides the effective index, namely, the effective noise-photon distribution. Only with the extra parameter can one predict how well the quantumness of states of light is preserved in the metamaterial. The fact that the effective index alone is not always sufficient and that one additional effective parameter suffices in the quantum optics of metamaterials is both of fundamental and practical interest. Here, from a Lagrangian description of the quantum electrodynamics of media with both linear gain and loss, we compute the effective noise-photon distribution for quantum light propagation in arbitrary directions in layered metamaterials, thereby detailing and generalizing our previous work. The effective index with its direction and polarization dependence is the same as in classical effective-medium theories. As our main result, we derive both for passive and for active media how the value of the effective noise-photon distribution too depends on the polarization and propagation directions of the light. Interestingly, for s-polarized light incident on passive metamaterials, the noise-photon distribution reduces to a thermal distribution, but for p-polarized light it does not. We illustrate the robustness of our quantum optical effective-medium theory by accurate predictions both for power spectra and for balanced homodyne detection of output quantum states of the metamaterial.
The isolation usually encountered in optics between the part of a field that is of interest and its source motivates the consideration of a radiation mode weakly coupled to a quantum-mechanical source. After the introduction of some refinements into the quantum mechanics of a damped radiation mode, the field is expressed as the sum of two parts, one due to the source (the source field) and the other due to the loss mechanism (the "vacuum" field). The characteristic function for the field is calculated up to second order in perturbation theory. This function is then compared with the characteristic function for the field in the presence of a classical source. A method is exhibited by which a classical source can be found such that the two characteristic functions are identical when averaged over a half cycle. In particular, the two sources yield the same expectation values for the instantaneous amplitude and energy of the field. The description of the equivalent classical source must be given in statistical terms, in general, and requires only a knowledge of 〈S(0)(t)〉〈{S}^{(0)}(t)〉〈S(0)(t)〉 and 〈S(0)(t1)S(0)(t2)〉〈{S}^{(0)}({t}_{1}){S}^{(0)}({t}_{2})〉〈S(0)(t1​)S(0)(t2​)〉, where S(0){S}^{(0)}S(0) is the dipole-moment operator of the quantum-mechanical source unperturbed by the mode under consideration (but otherwise arbitrarily complex, with the possibility of strong coupling to other modes). The theory is illustrated by a consideration of several simple sources---a two-level system, a harmonic oscillator, and a blackbody---for which equivalent classical sources are found. The two-time correlation functions for the field obtained with the two types of sources are compared and are shown to be the same up to first order in $\ensuremath{\xi}\ensuremath{\tau}$, where $\ensuremath{\tau}$ is the difference between the two times and $\ensuremath{\xi}$ is the inverse of the field relaxation time; the physical meaning of the second-order difference in the correlation functions is discussed. A limiting process, in which both the coupling to the source and the damping become small, is suggested as a method of adapting the results to free fields, but it is pointed out that for discussion of a single mode, a free field is physically less satisfactory than a damped field. It is concluded that, within a reasonable approximation scheme, the source field may be described classically (the "vacuum" field furnishing all the necessary quantum-mechanical properties of the total field).
We present a canonical quantum theory of radiation in nonlinear media taking into account the effects of linear dispersion and absorption in a consistent way. We start from a microscopic model and apply recently developed concepts of the quantization of radiation in dispersive and absorbing linear media and extend the theory in order to include nonlinear optical processes. We derive propagation equations in space and time for the quantized radiation field, with special emphasis on the propagation of quantum-light pulses in Kerr media. In particular, the method enables us to derive systematically the noise sources that arise naturally in the nonlinear terms.
A microscopic quantum description of the interaction between a radiation field coupled to sources and linear passive optical systems is presented. The quantum input-output relations for correlation functions that determine the quantum statistical properties are established under general conditions. In particular the influence of the vacuum field is taken into account. The treatment has advantageously been performed with the help of Green function techniques, which allow one to include loss mechanisms and dispersion of the linear medium. To illustrate the general results two interesting examples, the diminishing of squeezing due to a lossy Fabry-Perot interferometer and the action of a black screen, are given.