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Journal of Physics B Atomic and Molecular Physics

Radiation damping of atomic states in the presence of an external time dependent potential

Mark P. Silverman, F. M. Pipkin

Published Dec 1, 1972• 7 citations
Publisher

Abstract

The problem of an atom interacting simultaneously with the quantized modes of the vacuum electromagnetic field and with a classical time dependent field is investigated utilizing the general theory of damping devised by Heitler. Spectral lineshapes are obtained as a function of the variables of the classical field and are studied in detail for the case of spontaneous emission from coupled levels of dissimilar lifetimes. In the limit that the time dependent coupling vanishes, the resulting spectral line is the same as that derived by Weisskopf and Wigner.

DOI
10.1088/0022-3700/5/12/018

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 28, 2009•0 citations
The problem of an atom interacting simultaneously with the quantized modes of the vacuum electromagnetic field and with a classical time dependent field is investigated utilizing the general theory of damping devised by Heitler. Spectral lineshapes are obtained as a function of the variables of the classical field and are studied in detail for the case of spontaneous emission from coupled levels of dissimilar lifetimes. In the limit that the time dependent coupling vanishes, the resulting spectral line is the same as that derived by Weisskopf and Wigner.
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 calculate exactly the lineshape for a model of an excited two-level atom in interaction with a continuous spectrum of radiation for the problem of spontaneous emission. Specifically, for the case of a d=1 radiation field, we use the exact results reported in our earlier work [J. Math. Phys. 14, 414, 423 (1973)] for the probability ρ(τ) of the atom’s being in the excited state at time τ to obtain an analytic expression for the lineshape of the emitted radiation (i.e., the photon). We also calculate the lineshape using two Wigner–Weisskopf style approximations and the results, for a given choice of coupling function, are compared numerically with the profile generated using the exact solution. These comparisons show convincingly the success with which these two approximations can be expected to reproduce the qualitative and quantitative features of the exact lineshape for values of the coupling constant α ranging between 0.01 and 0.3. Finally, we calculate explicitly the scattering operator corresponding to the above spontaneous emission problem and work out exactly the probability of forward and backward scattering of a photon by the atom at and away from resonance.
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 time-dependent electric energy density surrounding a two-level atom fixed at r = 0 is studied, the atom being taken in its excited state at t = 0 and the field being initially in the vacuum state. The atom-field coupling includes both rotating and counter-rotating terms. The energy density of the spontaneously emitted field in the rotating wave approximation is shown to behave non-causally, while in the presence of the complete coupling it is shown to vanish outside a sphere of radius r = ct centred on the atom. The deviations of atomic dynamics from the exponential Wigner-Weisskopf behaviour during spontaneous decay are shown to be deeply influenced by the counter-rotating terms. It is concluded that the virtual photons induced by the counter-rotating terms in the atom-field coupling are essential in order to ensure causality and cannot be neglected in any accurate treatment of spontaneous emission.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
The nonperturbative theory of the cooperative spontaneous emission from a two level atom trapped in one-dimensional damped nanocavity with a single resonance mode is presented. The time-dependent spectral properties and nonlinear dynamics of a separate photon emission by the macroolecular-like system "excited atom coupled to a resonance decaying mode" have been analyzed. The investigation has been carried out by solving the Schrödinger equation in the interaction picture with the help of the Green functions method in the Heitler-Ma's form . The formalism was supplemented with the novel algorithm in operating causal singular functions and with fundamentals of the theory of quantum quasi-stationary systems. The proposed theory accounts automatically of both reabsorptions of emitted photon and its simultaneous escaping out of the cavity. Solutions of the wave equation were found without using intermediate virtual states and series expansions. In accordance with the theory of quasistationary systems the field of mode decaying exponentially in the empty nanocavity was represented with the Lorenz-shaped packet of stationary photonic states (quasi-modes). The electro-dipolar interaction between the atom and the mode field was adopted to be switched on suddenly. The expressions and plots of emission spectral densities probabilities together with photon emission probability dynamics as functions of time for various ratios Γ/4g. For Γ/4g<1 the transient emission spectrum reveals the presence of two symmetrical side-bands and the central peak interconnected with each other in the area of interaction with the atom. Since the central component oscillates, decaying simultaneously in time at two rates infinity Γ/2 and ~ Γ/4, in the area of interaction the emission is a triplet with satellites oscillating in the interaction area and being stable outside of it. So the final spectrum is a doublet outside of nanocavity. On the contrary for Γ/4g ≥1 the spectrum is a singlet, and the emission occurs in exponentially decaying way.