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arXiv:quant-ph/9806077 · Physical Review A

Quantum-state transformation by dispersive and absorbing four-port devices

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

Published Jun 1, 1999• 53 citations
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Abstract

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.

DOI
10.1103/physreva.59.4716

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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.
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.
The problem of decorrelation of entangled (squeezed-vacuum-type) light pulses of arbitrary shape passing through dispersive and absorbing four-port devices of arbitrary frequency response is studied, applying recently obtained results on quantum-state transformation (L Knöll et al 1999 Phys. Rev. A 59 4716). The fidelity and indices of (quantum) correlation based on the von Neumann entropy are calculated, with special emphasis on the dependence on the mean photon number, the pulse shape and the frequency response of the devices. In particular, it is shown that the quantum correlations can decay very rapidly due to dispersion and absorption, and the degree of degradation intensifies with increasing mean photon number.
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.