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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Quantum state transformation at absorbing and amplifying four-port devices

Stefan Scheel, Eduard Schmidt, L. Knöll, D.‐G. Welsch

Published Jan 20, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1109/cleopr.1999.817881

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QED in the presence of kramers-kronig dielectric media

Stefan Scheel, D.‐G. Welsch, L. Knöll

The macroscopic Maxwell field is quantized for arbitrarily space- and frequency-dependent complex permittivity. The formalism takes account of the dissipation-fluctuation theorem and yields the fundamental equal-time commutation relations of QED. Applications to the quantum-state transformation at absorbing and amplifying four-port devices and the spontaneous decay of an excited atom near absorbing dielectric bodies are discussed.

•Jun 1, 1999•0 citations
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.
Dielectric four-port devices play an important role in optical quantum information processing. Since for causality reasons the permittivity is a complex function of frequency, dielectrics are typical examples of noisy quantum channels, which cannot preserve quantum coherence. To study the effects of quantum decoherence, we start from the quantized electromagnetic field in an arbitrary Kramers-Kronig dielectric of given complex permittivity and construct the transformation relating the output quantum state to the input quantum state, without placing restrictions on the frequency. We apply the formalism to some typical examples in quantum communication. In particular we show that for Fock-entangled qubits the Bell-basis states $|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉$ are more robust against decoherence than the states $|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉.$
The recently derived input-output relations for radiation 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 component the transformation can be regarded as a U(4) group transformation, which can directly be calculated from the underlying complex refractive-index profile of the device without additional postulates. If for narrow-bandwidth radiation far from the medium resonances the absorption matrix of the four-port device can be disregarded, the well-known U(2) group transformation for a lossless device is recognized. Explicit formulas for the transformation of Fock states, coherent states, and phase-space functions are given.
After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.
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.