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arXiv:quant-ph/0505117 · Physical Review A

QED of lossy cavities: Operator and quantum-state input-output relations

M. Khanbekyan, L. Knöll, D.‐G. Welsch, А. А. Семенов, W. Vogel

Published Nov 18, 2005• 28 citations
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Abstract

Within the framework of exact quantization of the electromagnetic field in dispersing and absorbing media the input-output problem of a high-$Q$ 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 $s$-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.

DOI
10.1103/physreva.72.053813

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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.
The quantized electromagnetic field inside and outside an absorbing high-QQQ cavity is studied, with special emphasis on the absorption losses in the coupling mirror and their influence on the outgoing field. Generalized operator input-output relations are derived, which are used to calculate the Wigner function of the outgoing field. To illustrate the theory, the preparation of the outgoing field in a Schrödinger cat-like state is discussed.
The quantized electromagnetic field inside and outside an absorbing high-QQQ cavity is studied, with special emphasis on the absorption losses in the coupling mirror and their influence on the outgoing field. Generalized operator input-output relations are derived, which are used to calculate the Wigner function of the outgoing field. To illustrate the theory, the preparation of the outgoing field in a Schr\\"{o}dinger cat-like state is discussed.
The problem of the description of absorption and scattering losses in high-Q cavities is studied. The considerations are based on quantum noise theories, hence the unwanted noise associated with scattering and absorption is taken into account by introduction of additional damping and noise terms in the quantum Langevin equations and input--output relations. Completeness conditions for the description of the cavity models obtained in this way are studied and corresponding replacement schemes are discussed.
The problem of the description of absorption and scattering losses in high- Q cavities is studied. The considerations are based on quantum noise theories, hence the unwanted noise associated with scattering and absorption is taken into account by introduction of additional damping and noise terms in the quantum Langevin equations and input-output relations. Completeness conditions for the description of the cavity models obtained in this way are studied and corresponding replacement schemes are discussed.
We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light-matter coupling play an important role. In particular, our model can be applied to the case of intersubband transitions in doped semiconductor quantum wells embedded in a microcavity. The coupling of the intracavity photonic mode and of the electronic polarization to the external, frequency-dependent, dissipation baths is taken into account by means of quantum Langevin equations in the input-output formalism. In the case of a time-independent vacuum Rabi frequency, exact analytical expressions for the operators are obtained, which allows us to characterize the quantum dissipative response of the cavity to an arbitrary initial condition (vacuum, coherent field, thermal excitation). For a vacuum input in both the photonic and electronic polarization modes, the ground state of the cavity system is a two-mode squeezed vacuum state with a finite population in both photonic and electronic modes. These excitations are, however, virtual and cannot escape from the cavity: for a vacuum input, a vacuum output is found, without any trace of the intracavity squeezing. For a coherent photonic input the linear optical response spectra (reflectivity, absorption, transmission) have been studied, and signatures of the ultrastrong coupling have been identified in the asymmetric and peculiar anticrossing of the polaritonic eigenmodes. Finally, we have calculated the electroluminescence spectra in the case of an incoherent electronic input: the emission intensity in the ultrastrong coupling regime results in being significantly enhanced as compared to the case of an isolated quantum well without a surrounding cavity.
Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both E\mathbf{E}E and H\mathbf{H}H and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field E\mathit{E}E=[E,H]T[\mathbf{E},\mathbf{H}]^T[E,H]T, whose first-order Green operator ggg propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation [E,E†]∝Im g[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g[E,E†]∝Img, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.