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Quantum Optics Journal of the European Optical Society Part B

Exact theory of spontaneous decay of a twofold excited non-linear oscillator

Alexander E. Kozhekin, V. I. Rupasov

Published Jun 1, 1992• 4 citations
Publisher

Abstract

The interaction of a quantized electromagnetic field with a non-linear oscillator is described by a model which is analogous to the Dicke model. The exact two-particle eigenstates of the 'oscillator+field' system are obtained and these states are used to solve the problem of spontaneous decay of a twofold excited state of a non-linear oscillator. The shape of radiation lines and the 'linear oscillator' paradox are discussed. The authors also hope the exact solution of the problem will be useful for investigations of two-photon correlations in a non-linear medium for all cases when multiphoton correlations are insignificant.

DOI
10.1088/0954-8998/4/3/005

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The interaction of a quantized electromagnetic field with a non-linear oscillator is described by a model which is analogous to the Dicke model. The exact two-particle eigenstates of the 'oscillator+field' system are obtained and these states are used to solve the problem of spontaneous decay of a twofold excited state of a non-linear oscillator. The shape of radiation lines and the 'linear oscillator' paradox are discussed. The authors also hope the exact solution of the problem will be useful for investigations of two-photon correlations in a non-linear medium for all cases when multiphoton correlations are insignificant.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
In this paper we investigate the dynamics of a model for a three-level atom in interaction with a radiation field. The exact solution to the spontaneous-emission problem is derived using methods developed earlier by the authors, and expressions are obtained for the probabilities of the atom’s being in the first or second excited states at any time t. For the special case that the strengths of the coupling between each of the excited states and the λth mode of the field are proportional, detailed conclusions can be drawn concerning the effects of such factors as system size, coupling function, and level splitting on the temporal evolution of the system. The evolution of excited quantum systems having one versus two modes of decay to the ground state is also compared, and similarities and differences in the temporal behavior are noted. Finally the relevance of the theory presented in this paper to experimental problems in radiation chemistry and physics is indicated in our concluding remarks.
The spontaneous emission of a set of NNN identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.
Starting from the quantized version of Maxwell's equations for the electromagnetic field in an arbitrary linear Kramers-Kronig dielectric, the Heisenberg equations of motion for a two-level atom resonantly coupled to the radiation field in the presence of dispersive and absorbing dielectric bodies are derived. The theory is applied to the problem of spontaneous decay of excited atoms in absorbing media. The decay rate is calculated for the (Glauber-Lewenstein) real-cavity model, and a comparison with the recently studied (Clausius-Mosotti) virtual-cavity model [S. Scheel, L. Kn\"oll, D.-G. Welsch, and S. M. Barnett, Phys. Rev. A 60, 1590 (1999)] is given. It is shown that owing to nonradiative decay associated with absorption, the rate of spontaneous decay sensitively depends on the cavity radius, particularly when the atomic transition frequency approaches an absorption band of the medium. Only when the effect of absorption is fully disregarded, is the familiar local-field correction factor recovered.
We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its n=1n=1n=1 state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.