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arXiv:cond-mat/0205360 · Physical Review A

Input-output theory for fermions in an atom cavity

Christopher P. Search, Sierk Pötting, W. Zhang, Pierre Meystre

Published Oct 25, 2002• 17 citations
arXivHTMLPDF

Abstract

We generalize the quantum optical input-output theory developed for optical cavities to ultracold fermionic atoms confined in a trapping potential, which forms an ``atom cavity.'' In order to account for the Pauli exclusion principle, quantum Langevin equations for all cavity modes are derived. The dissipative part of these multimode Langevin equations includes a coupling between cavity modes. We also derive a set of boundary conditions for the Fermi field that relate the output fields to the input fields and the field radiated by the cavity. Starting from a constant uniform current of fermions incident on one side of the cavity, we use the boundary conditions to calculate the occupation numbers and current density for the fermions that are reflected and transmitted by the cavity.

DOI
10.1103/physreva.66.043616

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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.
We give a rigorous quantum-field theoretical derivation of the Langevin-equations approach to the problem of determining the radiation field inside and outside a resonator-like cavity by starting from Maxwell's equations with field- matter interaction and taking into account the presence of a cavity with a finite Q value by modeling its input/output port by a dielectric (multislab) wall.1,2 We show that if the transparency of the wall is small compared with unity, in the coarse-grain approximation the field generated and/or amplified by the active medium inside the cavity may be described by quantum Langevin equations. The coarse-grain time is determined by the time of flight of a light signal through the cavity.
The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\ensuremath{\delta}$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.
We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultra-strong light-matter coupling regime, in which the vacuum Rabi frequency is comparable to the electronic transition frequency and the anti-resonant 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. Observable spectra (reflection, absorption, transmission and electroluminescence) are calculated analytically in the case of a time-independent vacuum Rabi frequency.
We derive the quantum mechanical description of light-matter interactions in optical cavities characterized solely by radiative decay. Unique to radiative decay is the conservation of photon number and coherence, in stark contrast to absorptive losses. This prohibits the description of such cavities by traditional means, e.g., coupling it to a bath of harmonic oscillators into which energy is dissipated. Here, we propose a description of cavities with radiative decay by introducing cavity and noise operators in terms of scattering modes. A multimode input-output formalism to predict measurable far field quantities arises naturally. We apply our general model to the special case of the single excitation regime. We find dynamics reminiscent of the dissipative Jaynes-Cummings model, but with vanishing backaction and a rich temporal and spectral structure of the output modes.