S. Scheel

Spontaneous decay of an excited atom in an absorbing dielectric

Starting from the quantized version of Maxwells equations for the electromagnetic field in an arbitrary linear Kramers-Kronig dielectric, spontaneous decay of the excited state of a two-level atom embedded in a dispersive and absorbing medium is studied and the decay rate is calculated. The calculations are performed for both the (Clausius-Mosotti) virtual cavity model and the (Glauber-Lewenstein) real cavity model. It is shown that owing to nonradiative decay associated with absorption the rate of spontaneous decay sensitively depends on the cavity radius when the atomic transition frequency approaches an absorption band of the medium. Only when the effect of absorption is fully disregarded, then the familiar local-field correction factors are recovered.

QED commutation relations for inhomogeneous Kramers-Kronig dielectrics

Recently a quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive, and absorbing dielectric medium has been developed and applied to a system consisting of two infinite half-spaces with a common planar interface (H.T. Dung, L. Knoll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998)). Here we show that the scheme, which is based on the classical Green-tensor integral representation of the electromagnetic field, applies to any inhomogeneous medium. For this purpose we prove that the fundamental equal-time commutation relations of QED are preserved for an arbitrarily space-dependent, Kramers-Kronig consistent permittivity. Further, an extension of the quantization scheme to linear media with bounded regions of amplification is given, and the problem of anisotropic media is briefly addressed.

Entanglement generation and degradation by passive optical devices

The influence of losses in the interferometric generation and the transmission of continuous-variable entangled light is studied, with special emphasis on Gaussian states. Based on the theory of quantum-state transformation at absorbing dielectric devices, the amount of entanglement is quantified by means of the relative-entropy measure. Upper bounds of entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement. In particular, they do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure.

Equivalence of the Langevin and auxiliary-field quantization methods for absorbing dielectrics

Recently two methods have been developed for the quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics. The first is based upon the introduction of a quantum Langevin current in Maxwells equations [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); Ho Trung Dung, L. Kn{o}ll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998); S. Scheel, L. Kn{o}ll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)], whereas the second makes use of a set of auxiliary fields, followed by a canonical quantization procedure [A. Tip, Phys. Rev. A 57, 4818 (1998)]. We show that both approaches are equivalent.

OUANTUM-STATE TRANSFORMATION BY DISPERSIVE AND ABSORBING FOUR-PORT DEVICES

The recently derived input-output relations for the radiation field at a dispersive and absorbing four-port device [T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661 (1996)] are used to derive the unitary transformation that relates the output quantum state to the input quantum state, including radiation and matter and without placing frequency restrictions. It is shown that for each frequency the transformation can be regarded as a well-behaved SU(4) group transformation that can be decomposed into a product of U(2) and SU(2) group transformations. Each of them may be thought of as being realized by a particular lossless four-port device. If for narrow-bandwidth radiation far from the medium resonances the absorption matrix of the four-port device can be disregarded, the well-known SU(2) group transformation for a lossless device is recognized. Explicit formulas for the transformation of Fock-states and coherent states are given.

Entanglement transformation at absorbing and amplifying four-port devices

Abstract Quantum communication schemes widely use dielec-tric four-port devices as basic elements for construct-ing optical quantum channels. Since for causalityrea-sons the permittivity is necessarily a complex func-tion of frequency, dielectrics are typical examples ofnoisy quantum channels in which quantum coherencewill not be preserved. Basing on quantization of thephenomenological electrodynamics, we construct thetransformation relating the output quantum state tothe input quantum state without placing frequencyrestrictions. Knowledge of the full transformed quan-tum state enables us to compute the entanglementcontained in the output quantum state. We applythe formalism to some typical examples in quantumcommunication. 1 Introduction Quantum communication experiments widely use di-electricfour-portdevices suchasbeam splitters orop-tical fibers as basic elements for constructing opticalquantum channels. Since any frequency-dependentdielectric function describing an optical element, byvirtue of the Kramers-Kronig relations, is necessar-ily a complex function of frequency, absorption is al-ways present which leads to well-known phenomenaas decoherence and entanglement degradation. In or-der to study the problem, quantization of the elec-tromagnetic field in the presence of dielectric mediais needed. A consistent formalism of quantum elec-trodynamics in absorbing media is reviewed in [1].It is based on the Green function expansion of theelectromagnetic field with respect to the fundamen-tal variables of the system composed of the field, thedielectric matter and the reservoir. All relevant infor-mation about the dielectric and geometric propertiesare contained in the classical Green function of thecorresponding scattering problem.The formalism is especially suited for derivinginput-output relations of the field at dielectric slabs[2] on the basis of measurable quantities as trans-mission and absorption coefficients. From the input-output relations we can then derive closed formu-las for calculating the output quantum state fromthe (known) input quantum state [3]. That is, thecomplete density matrix after the transformation isknown, which makes the theory most suitable forstudying entanglement properties of quantum statesof light. The theory has also been extended to coveramplifying media.In this article we will proceed as follows. The quan-tum-state transformation at dielectric four-port de-vices is shortly reviewed in Sec. 2. An application toentanglement degradation of Bell states as well as thederivation of separability criteria for the two-modesqueezed vacuum state are given in Sec. 3 followedby a summary in Sec. 4.1

Entanglement degradation of a two-mode squeezed vacuum in absorbing and amplifying optical fibers

Applying the recently developed formalism of quantum-state transformation of light at absorbing dielectric four-port devices [5], we calculate the quantum state of the outgoing modes of a two-mode squeezed vacuum transmitted through optical fibers of given extinction coefficients. Using the Peres-Horodecki separability criterion for continuous variable systems [4], we then compute both the maximal length of transmission of a two-mode squeezed vacuum through an absorbing system for which the transmitted state is still inseparable and the maximal gain for which inseparability can be observed in an amplifying setup. Finally, we estimate an upper bound of the entanglement preserved after transmission through an absorbing system. The results show that the characteristic length of entanglement degradation drastically decreases with increasing strength of squeezing.

Entanglement Transformation At Dielectric Four-Port Devices

Quantum communication schemes widely use dielectric four-port devices as basic elements for constructing optical quantum channels. Dielectrics with complex permittivity are typical examples of noisy quantum channels in which quantum coherence will not be preserved. Basing on quantization of phenomenological electrodynamics, we construct the transformation relating the output quantum state to the input quantum state without placing frequency restrictions. Knowledge of the full transformed quantum state enables us to compute the entanglement contained in the output state.

Interaction of the quantized electromagnetic field with atoms in the presence of dispersing and absorbing dielectric bodies

A general theory of the interaction of the quantized electromagnetic field with atoms, in the presence of dispersing and absorbing dielectric bodies of given Kramers-Kronig consistent permittivities, is developed. It is based on a source-quantity representation of the electromagnetic field, in which the electromagnetic-field operators are expressed in terms of a continuous set of fundamental bosonic fields via the Green tensor of the classical problem. Introducing scalar and vector potentials, the formalism is extended to include in the theory the interaction of the quantized electromagnetic field with additional atoms. Both the minimal-coupling scheme and the multipolar-coupling scheme are considered. The theory replaces the standard concept of mode decomposition, which fails for complex permittivities. It enables us to treat the effects of dispersion and absorption in a consistent way and to give a unified approach to the atom-field interaction, without any restriction to a particular interaction regime in a particular frequency range. All relevant information about the dielectric bodies, such as form and intrinsic dispersion and absorption, is contained in the Green tensor. The application of the theory to the spontaneous decay of an excited atom in the presence of dispersing and absorbing bodies is addressed.

Quantum state transformation at absorbing and amplifying four-port devices

We have developed a quantum theory of the action of attenuating and amplifying four-port devices starting from the canonical quantization of the electromagnetic field in arbitrary linear, causal dielectrics. In particular, we have presented formulas for calculating the quantum state of the outgoing fields from the quantum state of the incoming fields and the device for given complex refractive-index profile without placing frequency restrictions.