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Physical Review A

Inhibited spontaneous emission and energy oscillation between two-level atoms

A. Shalom

Published Jan 1, 1992• 3 citations
Publisher

Abstract

The theoretical model presented here is motivated by experimental works demonstrating the suppression of spontaneous emission by atoms placed inside a cavity with a fundamental frequency above that for the atomic transition. In my model, two distinguishable two-level atoms, one of them initially excited, the other in the ground state, are coupled to a nonresonant radiation mode, with no direct interaction between the atoms. The exchange of energy between the atoms is investigated, and it is found that it oscillates from one atom to the other with angular frequency \ensuremath{\Omega} related to the radiative lifetime \ensuremath{\tau} by \ensuremath{\Omega}\ensuremath{\tau}=(3/2${\mathrm{\ensuremath{\pi}}}^{2}$)(${\mathrm{\ensuremath{\omega}}}_{0}$/${\mathrm{\ensuremath{\omega}}}_{0}$-\ensuremath{\omega})(${\mathrm{\ensuremath{\omega}}}_{0}$/\ensuremath{\omega}${)}^{3}$, with ${\mathrm{\ensuremath{\omega}}}_{0}$ and \ensuremath{\omega}, the frequencies for the radiation mode and the atomic transition, respectively.

DOI
10.1103/physreva.45.443

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The theoretical model presented here is motivated by experimental works demonstrating the suppression of spontaneous emission by atoms placed inside a cavity with a fundamental frequency above that for the atomic transition. In my model, two distinguishable two-level atoms, one of them initially excited, the other in the ground state, are coupled to a nonresonant radiation mode, with no direct interaction between the atoms. The exchange of energy between the atoms is investigated, and it is found that it oscillates from one atom to the other with angular frequency \ensuremath{\Omega} related to the radiative lifetime \ensuremath{\tau} by \ensuremath{\Omega}\ensuremath{\tau}=(3/2\ensuremathπ2{\mathrm{\ensuremath{\pi}}}^{2}\ensuremathπ2)(\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​/\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​-\ensuremath{\omega})(\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​/\ensuremath{\omega})3{)}^{3})3, with \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ and \ensuremath{\omega}, the frequencies for the radiation mode and the atomic transition, respectively.
Summary form only given. The feasibility of cavity-controlled spontaneous emission of a motionless two level atom is analyzed quantum mechanically and nonperturbatively. It is proposed to control the emission by putting the excited atom into the combined field of two lossless modes with their frequencies /spl omega//sub 2,1/ detuned symmetrically about the atomic one /spl omega//sub /spl alpha//, i.e. /spl omega//sub 2,1/=/spl omega//sub /spl alpha///spl plusmn//spl Delta/. The controlling /spl omega//sub 2,1/ modes are isolated from the free radiation field by an one-dimensional ideal cavity (or by two crossed ones) at zero temperature. These modes are taken to be linearly polarized in the same plane. Because the wavelength and cavity dimensions are assumed to be of the same order, the retardation effects are neglected. The spontaneous emission is treated as a radiative decay of the macromolecule-like "atom+/spl omega//sub 2,1/ modes"-subsystem (the H/sub s/-one) under its interaction with the free H/sub k/-field.
It is well known that spontaneous emission is not an intrinsic atomic property, but rather results from the coupling of the atom to the vacuum modes of the electromagnetic field. As such, it can be modified by tailoring the electromagnetic environment into which the atom can radiate. This was already realized by Purcell, who noted that the spontaneous emission rate can be enhanced if the atom placed inside a cavity is resonant with one of the cavity is resonant with one of the cavity modes, and by Kleppner, who discussed the opposite case of inhibited spontaneous emission. It has also been recognized that spontaneous emission need not be an irreversible process. Indeed, a system consisting of a single atom coupled to a single mode of the electromagnetic field undergoes a periodic exchange of excitation between the atom and the field. This periodic exchange remains dominant as long as the strength of the coupling between the atom and a cavity mode is itself dominant. 23 refs., 6 figs.
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.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}r21​ can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}O2​ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2{A}_{2}A2​ and A2{A}_{2}A2​ radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.