We introduce a simple quantum theory of the lossy beam splitter. When applied to describe a Fabry-P\'erot cavity this leads to apparently anomalous commutation relations for the intracavity operators. We show that these unfamiliar properties are nevertheless consistent with the fundamental canonical commutator for the vector potential and electric field operators. This result is derived as a consequence of causality as applied to the properties of mirror reflection coefficients.
We introduce a simple quantum theory of the lossy beam splitter. When applied to describe a Fabry-P'erot cavity this leads to apparently anomalous commutation relations for the intracavity operators. We show that these unfamiliar properties are nevertheless consistent with the fundamental canonical commutator for the vector potential and electric field operators. This result is derived as a consequence of causality as applied to the properties of mirror reflection coefficients.
Usual quantum-optical operator relations for a beam splitter are shown to lead to an anomalous commutation relation inside a microcavity. The physical origin of this anomaly is identified as self-interference of the mode whose coherence length is longer than the round-trip length of the cavity. Altered spontaneous emission of an excited atom is found to be a direct manifestation of this anomalous commutation relation. The anomalous Heisenberg uncertainty relations, which are derived from the commutation relation according to the Schwartz inequality, cannot be detected by probing the internal field with a beam splitter. The anomalous commutation relation, however, can be related to the change in the effecitve reflectivity of the beam splitter. The similarity and difference between an excited atom and a probe beam splitter are discussed.
It has been previously shown [Phys. Rev. A50, 89 (1994)PLRAAN1050-294710.1103/PhysRevA.50.89] that cavity-based electromagnetic confinement leads to an “anomalous” fields operators commutation relation that is undetectable by probing the cavity with a beam splitter. However, using this commutator in the case of parametric fluorescence (spontaneous parametric down conversion) when it occurs inside an open cavity implies a strong intensification of this process. This prediction can validate, or not, this commutation relation. The ab initio approach used is based entirely on vacuum field fluctuations and does not resort to the concept of density of states. Finally, through a generalization of creation and annihilation operators in the presence of noise, this approach raises fundamental questions about quantum modes. We expect this work to stimulate new theoretical developments and related experiments, which might lead to new applications in quantum nonlinear optics.
We develop quantum-optical input-output theory for resonators with arbitrary coupling strength, and for input fields whose spectrum can be wider than the cavity free-spectral range, while ensuring that the field-operator commutator relations in space-time variables are correct. The cavity-field commutator exhibits a series of space-time ``echoes,'' representing causal connections of certain space-time points by light propagation. We apply the theory to two-photon wave-packet shaping by cavity reflection, which displays a remarkable illustration of dispersion cancellation. We also show that the theory is amenable to inclusion of intracavity absorbing and emitting atoms, allowing, for example, dissipative losses within the cavity to be incorporated in a quantum mechanically correct way.
A quantum theory of the lossless beam splitter is given in terms of the quasi mode theory of macroscopic canonical quantization used to treat problems in cavity quantum electrodynamics and quantum optics. A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multi-time quantum correlation functions. In the application to the beam splitter the Heisenberg equations of motion for the input operators associated with the quasi mode annihilation operators are formally solved in a rotating picture to show that the unitary transform of the incident quasi mode annihilation operator (via the scattering operator) is just a linear combination of the incident and reflected quasi mode annihilation operators, in accordance with assumptions made in previous treatments of the beam splitter. The results depend on conservation of the transverse component of the wave vector, which follows from the form of the quasi mode-quasi mode coupling constants, and on conservation of the unperturbed energy, which follows from scattering theory. The applicability of quantum scattering theory to the beam splitter is justified in the usual situation where integrated one photon and two photon detection rates are finite for incident light field states of interest.
With the exception of Chapter 1, up to this point we have considered cavityless optical systems. With this chapter, instead, we start focussing on the case that the atomic medium is located in an optical cavity. We describe in detail the case of a unidirectional ring resonator, the boundary condition which characterizes it and the relations between its output fields and the intracavity field (Sections 8.1–8.3). We discuss the transmission of an empty cavity (Section 8.4) and of a cavity that contains a linear medium (Section 8.5). Optical cavities An optical cavity or optical resonator is formed by at least two mirrors, appropriately arranged in such a way that the radiation beam follows a closed path. The beam splitters allow the radiation to escape from the cavity (output) and allow one to inject radiation into the cavity (input). In order to confine the radiation transversally with respect to the direction of propagation, the mirrors are usually spherical (see Fig. 1.2). The case of cavities with spherical mirrors will be described in Part III of this volume. On the other hand, an analytical treatment of the problems is possible only within the plane-wave approximation which we will use in the remainder of Part I and in Part II. In accord with this, we assume here that the mirrors are planar. The two most common types of optical resonator are the Fabry–Perot cavity and the ring cavity . The planar Fabry–Perot cavity consists in two beam splitters in a parallel configuration as shown in Fig. 8.1. Within the cavity there are two counterpropagating fields, E F propagating in the forward direction and E B propagating in the backward direction with respect to the input field E I . Because we have derived the Maxwell–Bloch equations for the unidirectional configuration, in this and the following chapters we focus on the case of a ring cavity, which allows unidirectionality. The Fabry–Perot case will be considered in Chapter 14.
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.