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Physical Review A

Resonators in quantum optics: A first-principles approach

L. Knöll, W. Vogel, D.‐G. Welsch

Published Jan 1, 1991• 51 citations
Publisher

Abstract

Input-output relations in resonators are studied on the basis of a recently developed field-theoretical approach to the action of passive systems in quantum optics [Kn\"oll, Vogel, and Welsch, Phys. Rev. A 36, 3803 (1987)]. This method allows the rigorous proof of recently adopted quantum stochastic approaches, including the presence of sources that form the active medium in the cavity. Quantum Langevin equations are obtained in a coarse-graining approximation. Moreover, we derive all required commutation relations that have been postulated so far. Correlation functions of the field quantities measured outside the cavity are related to correlation functions of field operators of the intracavity field and the incoming field.

DOI
10.1103/physreva.43.543

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Input-output relations in resonators are studied on the basis of a recently developed field-theoretical approach to the action of passive systems in quantum optics [Kn"oll, Vogel, and Welsch, Phys. Rev. A 36, 3803 (1987)]. This method allows the rigorous proof of recently adopted quantum stochastic approaches, including the presence of sources that form the active medium in the cavity. Quantum Langevin equations are obtained in a coarse-graining approximation. Moreover, we derive all required commutation relations that have been postulated so far. Correlation functions of the field quantities measured outside the cavity are related to correlation functions of field operators of the intracavity field and the incoming field.
We give a rigorous quantum-field theoretical derivation of the Langevin-equations approach to the problem of determining the radiation field inside and outside a resonator-like cavity by starting from Maxwell's equations with field- matter interaction and taking into account the presence of a cavity with a finite Q value by modeling its input/output port by a dielectric (multislab) wall.1,2 We show that if the transparency of the wall is small compared with unity, in the coarse-grain approximation the field generated and/or amplified by the active medium inside the cavity may be described by quantum Langevin equations. The coarse-grain time is determined by the time of flight of a light signal through the cavity.
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.
The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\ensuremath{\delta}$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.
Within the framework of exact quantization of the electromagnetic field in dispersing and absorbing media the input-output problem of a high-QQQ cavity is studied, with special emphasis on the absorption losses in the coupling mirror. As expected, the cavity modes are found to obey quantum Langevin equations, which could be also obtained from quantum noise theories, by appropriately coupling the cavity modes to dissipative systems, including the effect of the mirror-assisted absorption losses. On the contrary, the operator input-output relations obtained in this way would be incomplete in general, as the exact calculation shows. On the basis of the operator input-output relations the problem of extracting the quantum state of an initially excited cavity mode is studied and input-output relations for the sss-parametrized phase-space function are derived, with special emphasis on the relation between the Wigner functions of the quantum states of the outgoing field and the cavity field.
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.
In this paper, we present a very simple approach to input-output relations in optical cavities, limiting ourselves to one- and two-photon states of the field. After field quantization, we derive the nonunitary transformation between inside and output annihilation and creation operators. Then we express the most general two-photon state generated by inside creation operators, through base states generated by output creation operators. After renormalization of coefficients of the inside two-photon state, we calculate the outside photon-number probability distribution in a general case; then we treat with some detail the single-mode and symmetrical cavity case. We found that the statistics of emitted photons may exhibit either quantum or classical behavior, depending both on source properties and on cavity characteristics.