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Physical Review

The Decay of Resonance Radiation by Spontaneous Emission

Emil J. Hellund

Published Mar 1, 1953• 45 citations
Publisher

Abstract

A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.

DOI
10.1103/physrev.89.919

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A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law \ensuremathΣ\ensuremathμ,\ensuremathνC\ensuremathμ\ensuremathνt\ensuremathλ\ensuremathμe\ensuremath−\ensuremathβ\ensuremathνt,{\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},\ensuremathΣ\ensuremathμ,\ensuremathν​C\ensuremathμ\ensuremathν​t\ensuremathλ\ensuremathμ​e\ensuremath−\ensuremathβ\ensuremathν​t, where the \ensuremathλ\ensuremathμ{\ensuremath{\lambda}}_{\ensuremath{\mu}}\ensuremathλ\ensuremathμ​ and \ensuremathβ\ensuremathν{\ensuremath{\beta}}_{\ensuremath{\nu}}\ensuremathβ\ensuremathν​ are complex constants lying in the right half-plane and the C\ensuremathμ\ensuremathν{C}_{\ensuremath{\mu}\ensuremath{\nu}}C\ensuremathμ\ensuremathν​ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
The phenomena which are principally responsible for the interaction of radiation with matter in everyday life are absorption and spontaneous emis­ sion. Light falling on matter is absorbed, leaving atoms in excited states which spontaneously emit radiation with a spread of frequency inversely proportional to the decay time. The situation is modified when the atomic decay times are sufficiently long and the radiation sufficiently intense, for then radiation may fall on excited atoms, resulting in stimulated emission rather than absorption. We saw in Chap. 2 that this third process was required to obtain the Planck blackbody radiation law in Einstein’s derivation. It is stimulated emission which enables an atomic medium to amplify incident radiation, and therefore is an essential ingredient in light amplifiers and oscillators. In fact, the word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
The excitation and decay probabilities for resonance scattering of radiation from an atom with two coupled excited states in an external static field are calculated as a function of time and frequency. Various oscillatory terms are found in the probabilities. These oscillations depend on the frequency of radiation, the external coupling and the energy difference between the excited states. Two special cases are investigated in detail. In the first case where one of the excited states does not decay a 'hole' appears in the emission line at a frequency equal to the frequency difference between the ground state and the unperturbed non-decaying excited state. In the second case where the two excited states decay with the same rate, one of the two lines in the emission spectrum is suppressed and the other is enhanced.
The spontaneous decay of an excited atom by photon emission is one of the most common and elementary physical process present in nature and in laboratories. The decay is random in time with constant probability density, as it can be inferred by the exponential law observed experimentally. Despite the simplicity of the process, in Quantum Mechanics the decay itself is considered a law of nature which is not further analyzed or explained. However it is legitimate to ask for the reason of its randomness and for the dynamics of the atom around the decay. The decay process of an isolated atom is usually assumed to be instantaneous, the so called Quantum jumps. Particular experimental arrangements can widen the duration of the transition from an excited state to the ground state to a finite time. In general this is due to the quantum back action of the detector. The development of Quantum Optics has enormously enriched the possibilities to study in detail the process, and new illuminating insights have been obtained, but the question is still valid and its answer is still elusive. In this paper we analyze the spontaneous decay within the new prospect introduced by a recent model that complete standard Quantum Mechanics in a formalism which is able to describe the dynamics of measurement and of spontaneous decay. Comparison is made with more phenomenological theories, the Quantum Mechanics of open system and the Stochastic wave function model. In the new model the stochastic photon emission by an excited atom is triggered by the vacuum fluctuations. It is suggested how to re-analyze three types of experiments, already realized in many laboratories, in order to reveal the presence and the effect of these fluctuations.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.