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Physical Review A

Spontaneous emission from the bare ground state of an optically driven three-level atom: Perturbation theory and energy conservation

J. L. Cohen, P. R. Berman

Published Aug 1, 1996• 3 citations
Publisher

Abstract

We present a perturbative derivation of the fluorescence spectrum for a three-level atom driven by a classical radiation field. The energies of the three atomic levels are equal to \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathrm{i}}$ (i=1, 2, and 3) with ${\mathrm{\ensuremath{\omega}}}_{3}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\omega}}}_{2}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{32}$\ensuremath{\gg}${\mathrm{\ensuremath{\omega}}}_{21}$ (${\mathrm{\ensuremath{\omega}}}_{\mathit{ij}}$=${\mathrm{\ensuremath{\omega}}}_{\mathit{i}}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{j}}$). The radiation field, having a frequency \ensuremath{\Omega}\ensuremath{\gtrsim}${\mathrm{\ensuremath{\omega}}}_{31}$, can be considered to drive the 2-3 transition only. Spontaneous emission from the ground state, via a normally counter-rotating atom-vacuum interaction term in the Hamiltonian, conserves energy when a photon is subsequently absorbed from the laser field. The spectrum is explained in terms of a Raman process, and in terms of an interaction between the laser field and the coupled atom-vacuum field system. Connection is made with a dressed-atom description of the problem. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1686

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We present a perturbative derivation of the fluorescence spectrum for a three-level atom driven by a classical radiation field. The energies of the three atomic levels are equal to \ensuremath{\Elzxh}\ensuremathωi{\mathrm{\ensuremath{\omega}}}_{\mathrm{i}}\ensuremathωi​ (i=1, 2, and 3) with \ensuremathω3{\mathrm{\ensuremath{\omega}}}_{3}\ensuremathω3​\ensuremath{\gtrsim}\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​\ensuremath{\gtrsim}\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​ and \ensuremathω32{\mathrm{\ensuremath{\omega}}}_{32}\ensuremathω32​\ensuremath{\gg}\ensuremathω21{\mathrm{\ensuremath{\omega}}}_{21}\ensuremathω21​ (\ensuremathωij{\mathrm{\ensuremath{\omega}}}_{\mathit{ij}}\ensuremathωij​=\ensuremathωi{\mathrm{\ensuremath{\omega}}}_{\mathit{i}}\ensuremathωi​-\ensuremathωj{\mathrm{\ensuremath{\omega}}}_{\mathit{j}}\ensuremathωj​). The radiation field, having a frequency \ensuremath{\Omega}\ensuremath{\gtrsim}\ensuremathω31{\mathrm{\ensuremath{\omega}}}_{31}\ensuremathω31​, can be considered to drive the 2-3 transition only. Spontaneous emission from the ground state, via a normally counter-rotating atom-vacuum interaction term in the Hamiltonian, conserves energy when a photon is subsequently absorbed from the laser field. The spectrum is explained in terms of a Raman process, and in terms of an interaction between the laser field and the coupled atom-vacuum field system. Connection is made with a dressed-atom description of the problem. \textcopyright{} 1996 The American Physical Society.
This paper is concerned with two rather basic phenomena: the incoherent fluorescence spectrum of an atom driven by an intense laser field and the coupling of the atom to the (empty) modes of the radiation field. The sum of the many-photon processes gives rise to the inelastic part of the atomic fluorescence, which, for a two-level system, has a well-known characteristic three-peak structure known as the Mollow spectrum. From a theoretical point of view, the Mollow spectrum finds a natural interpretation in terms of transitions among laser-dressed states which are the energy eigenstates of a second-quantized two-level system strongly coupled to a driving laser field. As recently shown, the quasi-energies of the laser-dressed states receive radiative corrections which are nontrivially different from the results which one would expect from an investigation of the coupling of the bare states to the vacuum modes. In this article, we briefly review the basic elements required for the analysis of the dynamic radiative corrections, and we generalize the treatment of the radiative corrections to the incoherent part of the steady-state fluorescence to a three-level system consisting of 1S, 3P and 2S states.
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.
This study concerns the shape of the resonance fluorescence spectrum emitted by a V-type three-level atom driven by two coherent fields which are resonant with the transitions connecting the ground state and the second and third level, respectively. We are especially interested in monitoring the spontaneous emission from the top level to the ground state. If the field connecting levels 1 and 2 is absent or small, the spontaneous emission from level 3 is characterized by the standard Mollow spectrum. If instead the 1-2 transition is saturated, the spectrum of resonance fluorescence from the top level is characterized by the following signatures: (i) the appearance of sidebands due to Rabi splitting effects, (ii) a reduction of its integrated area because of coherent population trapping1, and (iii) a significant reduction of its linewidth if the natural lifetime of level 2 is longer than level 3. The latter effect is especially interesting and rather unexpected, as the decay properties of an optical transition are being transferred to another by controlling the relative magnitudes of the driving fields. We have carried out our calculations with the help of the regression theorem and also developed a simple physical interpretation in terms of appropriate dressed states2.
We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.
The radiative emission of strongly driven, two-level-like barium atoms into the modes of a confocal optical cavity is studied as a function of barium density. As the barium density is increased, the spatial profile, intensity, and spectrum of light emitted out the ends of the confocal cavity display dramatic changes indicative of a transition from fluorescence to lasing. In this novel two-level system, laser action can be explained in terms of an inversion between the dressed atom-field states or in terms of a type of stimulated hyper-Raman scattering in which atoms lase from the ground to the excited atomic state. A simple dressed-atom rate-equation analysis of this two-level-atom laser is presented. It is pointed out that this same system may provide a means of realizing two-photon-like laser action.
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.