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Japanese Journal of Applied Physics

Nonresonant Absorption of One Photon by One Atom and Resonant Absorption of Two Photons by Two Atoms

Masataka Mizushima

Published Nov 1, 1990• 1 citation
Publisher

Abstract

When a radiation field of frequency ω1 interacts with atoms, etch of which has a transition frequency ω b a =(E b -E a )/\hbar, with ω1-ω b a =Δ≠0, nonresonant absorption can take place with probability P 1 inversely proportional to Δ 2 (a pressure broadening). When another radiation field of frequency ω2, such that ω1+ω2=2 ω b a , interacts simultaneously with the gas a resonant two-photon absorption can take place in addition to the nonresonant absorption. The probability of this two-photon absorption process, P 2, is found to be inversely proportional to Δ 4. If \varOmega=| <n-1|E|n ><b| µ|a >| is the Rabi frequency of the transition, it is found that P 2/(P 1(Δ)+P 1(-Δ)) is given by 12{ \varOmega(-Δ)\varOmega(-Δ)}2/{Δ 2(\varOmega(-Δ)2+\varOmega(Δ)2)}.

DOI
10.1143/jjap.29.l2121

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Physical Review A•Jun 1, 1996•80 citations
When a radiation field of frequency ω1 interacts with atoms, etch of which has a transition frequency ω b a =(E b -E a )/\hbar, with ω1-ω b a =Δ≠0, nonresonant absorption can take place with probability P 1 inversely proportional to Δ 2 (a pressure broadening). When another radiation field of frequency ω2, such that ω1+ω2=2 ω b a , interacts simultaneously with the gas a resonant two-photon absorption can take place in addition to the nonresonant absorption. The probability of this two-photon absorption process, P 2, is found to be inversely proportional to Δ 4. If \varOmega=| <n-1|E|n ><b| µ|a >| is the Rabi frequency of the transition, it is found that P 2/(P 1(Δ)+P 1(-Δ)) is given by 12{ \varOmega(-Δ)\varOmega(-Δ)}2/{Δ 2(\varOmega(-Δ)2+\varOmega(Δ)2)}.
Two-photon absorption is theoretically analyzed within the semiclassical formalism of radiation-matter interaction. We consider an ensemble of inhomogeneously broadened three-level atoms subjected to the action of two counterpropagating fields of the same frequency. By concentrating in the limit of large detuning in one-photon transitions, we solve perturbatively the Bloch equations in a non-usual way. In this way we derive an analytical expression for the width of the two-photon resonance that makes evident sub-Doppler two--photon spectroscopy. We also derive an analytical expression for the Stark shift of the two-photon resonance.
We consider two separate atoms interacting with a single-mode optical or microwave resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counterrotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with a probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission process can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.
We consider two separate atoms interacting with a single-mode optical resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counter-rotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission processes can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
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{)}^{3})3, with ${\mathrm{\ensuremath{\omega}}}_{0}$ and \ensuremath{\omega}, the frequencies for the radiation mode and the atomic transition, respectively.
We analyze the fluorescence and absorption spectra of a two-level atom driven by a bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, separated by ${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$=2\ensuremath{\delta}, and Rabi frequencies (at resonance) 2${\mathrm{\ensuremath{\Omega}}}_{1}$ and 2${\mathrm{\ensuremath{\Omega}}}_{2}$ such that their ratio \ensuremath{\alpha}=${\mathrm{\ensuremath{\Omega}}}_{2}$/${\mathrm{\ensuremath{\Omega}}}_{1}$1. We focus on the case of ${\mathrm{\ensuremath{\omega}}}_{1}$ close to the atomic frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ near the Rabi sideband frequency ${\mathrm{\ensuremath{\omega}}}_{1}$+2${\mathrm{\ensuremath{\Omega}}}_{1}$; the detunings are denoted by ${\mathrm{\ensuremath{\Delta}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\Delta}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{1}$+2${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$. We find that the spectra depend critically on the detuning ${\mathrm{\ensuremath{\Delta}}}_{2}$: For large ${\mathrm{\ensuremath{\Delta}}}_{2}$, the fluorescence spectrum consists of the well known Mollow triplet, centered at ${\mathrm{\ensuremath{\omega}}}_{1}$; for smaller (but nonzero) ${\mathrm{\ensuremath{\Delta}}}_{2}$, the spectrum is composed of a triplet at ${\mathrm{\ensuremath{\omega}}}_{1}$ together with doublets near the sideband frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2${\mathrm{\ensuremath{\Omega}}}_{1}$. However, when ${\mathrm{\ensuremath{\Delta}}}_{2}$=0 (and \ensuremath{\alpha}\ensuremath{\ll}1), the spectrum consists of a doublet centered at ${\mathrm{\ensuremath{\omega}}}_{1}$ and triplets at ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2${\mathrm{\ensuremath{\Omega}}}_{1}$: there is then no fluorescence at ${\mathrm{\ensuremath{\omega}}}_{1}$. As \ensuremath{\alpha} increases, additional triplet structures appear in the spectrum at frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2n${\mathrm{\ensuremath{\Omega}}}_{1}$ with intensities proportional to ${\mathrm{\ensuremath{\alpha}}}^{2(\mathit{n}\mathrm{\ensuremath{-}}1)}$, n\ensuremath{\gtrsim}1, and a line reappears at ${\mathrm{\ensuremath{\omega}}}_{1}$, with intensity proportional to ${\mathrm{\ensuremath{\alpha}}}^{4}$. The absorption by the system of a weak probe beam is also strongly dependent on the detuning, and the spectrum is composed of emission-dispersion-absorption features located near ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2n${\mathrm{\ensuremath{\Omega}}}_{1}$. An analysis in the dressed-atom picture is presented which explains the physical origin of all these features, in both fluorescence and absorption. \textcopyright{} 1996 The American Physical Society.