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Journal of Optics B Quantum and Semiclassical Optics

Derivation of a Hamiltonian for photon decay in a cavity

S. M. Dutra, Gerard Nienhuis

Published Sep 6, 2000• 19 citations
Publisher

Abstract

We present a rigorous first principles derivation of the heuristic approach behind most of cavity QED: the quantization of the field in a high-Q cavity as if the mirrors were perfect with losses modelled by a coupling with the outside. We derive the Hamiltonian describing this coupling, thereby allowing us to avoid the coarse-graining approximation implicit in the usual Langevin or master equation treatments of cavity damping. We obtain an explicit expression for the coupling constant that describes the non-Markovian nature of the reservoir in the regime beyond the validity of the coarse-graining approximation.

DOI
10.1088/1464-4266/2/5/305

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A Hamiltonian for cavity decay

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We present a rigorous first principles derivation of the heuristic approach behind most of cavity QED: the quantization of the field in a high-Q cavity as if the mirrors were perfect with losses modelled by a coupling with the outside. We derive the Hamiltonian describing this coupling, thereby allowing us to avoid the coarse-graining approximation implicit in the usual Langevin or master equation treatments of cavity damping. We obtain an explicit expression for the coupling constant that describes the non-Markovian nature of the reservoir in the regime beyond the validity of the coarse-graining approximation.
Summary form only given. The conventional way to treat cavity dissipation in quantum optics is by applying the system-reservoir approach. The small system consists of the quantized discrete modes of a fictitious perfect cavity. The reservoir is the external field described by continuous annihilation and creation operators for the modes of free-space. This conventional approach has been used since the early days of the laser. The Hamiltonian that describes this approach became known as the Gardiner-Collett Hamiltonian. Despite being useful in describing several phenomena in cavities with a high Q factor, it is well known that this approach breaks down when the Q decreases. We have derived the Gardiner-Collett Hamiltonian from a rigorous modes of the universe expansion, as an approximation for high-Q, and obtained the coupling strength between the discrete cavity annihilation and creation operators (small system) and the continuous annihilation and creation operators of free-space (reservoir). We have found that the coupling strength depends on the frequency even when the reflectivity is frequency independent. The physical consequence of this frequency dependence is that for short times compared to one cavity round-trip time, the reservoir (free-space) is no longer Markovian. Our explicit expression for this frequency dependent coupling strength unlocks this fast time non-Markovian regime that had been previously inaccessible within the Gardiner-Collett Hamiltonian. Finally, we also show how to generalize the Gardiner-Collett Hamiltonian to arbitrary quality factors. This generalized Hamiltonian can be applied to develop a full quantum theory of excess noise in lasers.
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.
Starting from the inhomogeneous quantum-mechanical Maxwell equations we give a rigorous quantum-field theoretical derivation of the Langevin equations for a high-Q cavity field and specify the conditions of validity without borrowing from the standard relaxational treatment. In particular, we show that the use of Langevin equations means a loss of information about field changes during times of the order of the time of flight of a light signal through the cavity, which may be crucial in picosecond and femtosecond spectroscopy.
Phenomenological approaches to photon loss have long been the workhorse of cavity-QED, but prove inadequate in the presence of sufficiently broadband light-matter interactions. We present a rigorous and ab initio derivation of a quantum master equation for a quantized optical cavity mode coupled to a dipole, using a quasinormal mode (QNM) quantization procedure for plasmonic and dielectric open-system cavity-QED, which is valid in broadband light-matter interaction regimes, including ultrastrong coupling (USC). The theory supports general three-dimensional resonators with arbitrary dispersion and loss, and thus can be applied to a wide range of realistic open cavities. Our ab initio and gauge-invariant approach fully recovers a recent result for the spectral density of a quantized cavity with a single dipole and reveals important departures from previous heuristic assumptions about system-reservoir coupling. We identify a new criterion for what we term the "broadband dissipative" regime of cavity-QED, where phenomenological models require corrections in accordance with the intrinsic and spatially-dependent complex phase of the QNM, and also shed light on fundamental limits to single-mode models in extreme coupling regimes. Using experimentally motivated plasmonic and dielectric cavity examples, we show validity ranges of our QNM master equation and spectral USC calculations, and discuss prospects for near-term experimental observation of the broadband dissipative effects.
Effective models to describe the dynamics of an open cavity have been extensively discussed in the literature. In many of these models the cavity leakage to the outside is treated as a loss introduced phenomenologically. In contrast to these, we focus here on characterizing the outgoing photon using a novel approach where the outside is treated as part of the system. In such a global system, in order to separately characterize the photon inside and outside cavity, we demonstrate a first-principle derivation of a coherent cavity-reservoir coupling function for cavities with mirrors consisting of a stack of dielectric layers. In particular, we show that due to the effects induced by the multilayer nature of the cavity mirror, even in the standardly defined high-finesse cavity regime, the cavity-reservoir system description might differ from the one where the structure of the mirror is neglected. Based on this, we define a generalized cavity response function and a cavity-reservoir coupling function, which account for the longitudinal geometric structure of the cavity mirror. This allows us to define an effective reflectivity for the cavity with a multilayer mirror as if it was sitting in a well-defined effective mirror plane. We estimate the error of such a definition by considering cavities of different lengths and mirror structures. Finally, we apply this model to characterize the dynamics of a single photon produced in such a cavity and propagating outside.
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.