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CRC Press eBooks

Effects of Atomic Broadening on Spontaneous Emission in an Optical Microcavity

Kikuo Ujihara

Published Jul 9, 2020• 0 citations
Publisher

Abstract

In this chapter, the authors develop a consistent theory of spontaneous emission in an optical micro cavity from a broadened atom. They derive the atomic broadening from first principles by assuming the presence of reservoirs coupled to the atoms. The simplest model of the atom is the so-called two-level atom, which has only two energy levels, the excited and the ground level. However, realistic atoms are surrounded by media that affect the atoms, giving them homogeneous or inhomogeneous broadening. The important parameter of coupling ratio of spontaneous emission to a desired quasi mode cannot be exactly treated by the present mathematical model of the cavity, although it is easily seen qualitatively that the ratio will decrease with increasing atomic broadening. The authors assume that each atom emits independently and ignored the possible interaction among atoms during spontaneous emission. Solving the master equation for the density matrix elements, the readers have derived formulas for spontaneous emission rate and emission spectrum.

DOI
10.1201/9780138738938-3

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Quantum theory of spontaneous emission from a two-level atom in a one-dimensional cavity with output coupling

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A nonperturbative fully quantum-theoretical analysis describing the transient spontaneous emission of an initially excited two-level atom in a one-dimensional cavity with output coupling is presented. A delay-differential equation for the probability amplitude of the upper atomic state is strictly derived from the Schrodinger equation. With this equation, both the time evolution of the atomic state and the emission spectrum are calculated. Furthermore, the mode function makes the calculations of emitted fields both inside and outside the cavity possible. The calculated result of the vacuum Rabi oscillation in an underdamped cavity and the result of enhanced spontaneous emission rate in an overdamped cavity are presented. A retardation-governed revival of the upper atomic state is predicted. The calculated results also show the inhibition of spontaneous emission by detuning or by setting the atom at a node.>

IEEE Journal of Quantum Electronics•Jan 1, 1989•18 citations
In this chapter, the authors develop a consistent theory of spontaneous emission in an optical micro cavity from a broadened atom. They derive the atomic broadening from first principles by assuming the presence of reservoirs coupled to the atoms. The simplest model of the atom is the so-called two-level atom, which has only two energy levels, the excited and the ground level. However, realistic atoms are surrounded by media that affect the atoms, giving them homogeneous or inhomogeneous broadening. The important parameter of coupling ratio of spontaneous emission to a desired quasi mode cannot be exactly treated by the present mathematical model of the cavity, although it is easily seen qualitatively that the ratio will decrease with increasing atomic broadening. The authors assume that each atom emits independently and ignored the possible interaction among atoms during spontaneous emission. Solving the master equation for the density matrix elements, the readers have derived formulas for spontaneous emission rate and emission spectrum.
Starting from first principles we simulate numerically the dynamics of a system of two-level atoms interacting with a multimode electromagnetic field in a one-dimensional cavity. In particular, we investigate the spontaneous emission of an excited two-level atom in the cavity. We show how the decay depends on the mode spectrum and on the position of the atom in the cavity. We study how the spontaneous emission is modified when the initially excited atom is surrounded by a material medium modeled as a system of two-level atoms. We also study the propagation of photon wave packets emitted by the atom within the material media. Our microscopic model provides us with a deeper understanding of the atom-field interaction and offers us a framework for a systematic investigation of the transition from a microscopic to a macroscopic (phenomenological) description of the spontaneous decay in material media (e.g., dielectrics).
A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from NNN identical two-level atoms is investigated further. It is shown that the equation of motion for the p(<N)p(<N)p(<N)-atom density matrix is coupled to the (p+1p+1p+1)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time ttt can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of 1N\frac{1}{N}N1​. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of 2N2N2N coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.
Spontaneous emission competes with stimulated emission in many interactions of light with matter. In the usual analyses which describe the interaction of an atom with a coherent optical field, the spontaneous emission characteristics, e.g., probability and spectral distribution, are not determined. The spontaneous emission from an atom which interacts with a coherent light wave is considered. The competition between coherent photons and spontaneous photons is treated in detail for a system consisting of a stationary atom, an open cavity and spatial fields. In the model chosen, a multilevel atom which spontaneously decays by interacting with spatial fields has two nondegenerate states coupled by an interaction with a single mode of the cavity. The Laplace-transformed Schrodinger equation is solved for specified initial conditions of the system. It is found that the interaction with the coherent field modifies the spectral distribution of spontaneous radiation from the atom. For spontaneous transitions involving an atomic state which interacts with the coherent field, the spectral distributions can no longer be described by Lorentzian functions. The new distributions exhibit a broadening and splitting for strong interactions between the atom and the coherent field. It is shown that the qualitative features of these distributions can be predicted from the energy-level diagram of the atom-cavity system. The net probability of the system gaining a coherent or cavity photon is calculated by integrating over the emitted spontaneous frequencies. The equivalence of this approach to the method of computing probabilities by integrating over time is demonstrated by using Parseval's theorem.
A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).
This chapter gives an overview of the field as it stands at the time of writing in mid 1995. It shows how the picture radically changes when there is more than one atom in the cavity. The simplest light-atom interaction one can think of is the coupling between a single atom and a single mode of the electromagnetic field. This system provides considerable insight into the fundamental physics of the light-atom coupling but the reader is warned that the more general many mode-many atom interactions are much richer in their dynamics. An atom in free space couples not only to one, but to a continuum of modes. The mathematics of the Weisskopf-Wigner theory of spontaneous emission tends to obscure the relatively simple physics of the emission process. An alternative, shorter way of calculating the single atom spontaneous emission rate is to use Fermi’s Golden rule, which is a prescription for calculating general transition rates to first order in perturbation theory.
A nonperturbative fully quantum-theoretical analysis describing the transient spontaneous emission of an initially excited two-level atom in a one-dimensional cavity with output coupling is presented. A delay-differential equation for the probability amplitude of the upper atomic state is strictly derived from the Schrodinger equation. With this equation, both the time evolution of the atomic state and the emission spectrum are calculated. Furthermore, the mode function makes the calculations of emitted fields both inside and outside the cavity possible. The calculated result of the vacuum Rabi oscillation in an underdamped cavity and the result of enhanced spontaneous emission rate in an overdamped cavity are presented. A retardation-governed revival of the upper atomic state is predicted. The calculated results also show the inhibition of spontaneous emission by detuning or by setting the atom at a node.>