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Physical Review A

Shifts from a distant neighboring resonance for a four-level atom

Marko Horbatsch, Eric A. Hessels

Published Sep 16, 2011• 36 citations
Publisher

Abstract

In a recent paper [Phys. Rev. A 82, 052519 (2010)], the systematic shifts of a resonance due to quantum-mechanical interference from a distant neighboring resonance were derived. In that paper, the simplest three-level closed system was used to calculate analytic expressions for shifts in the resonant line centers. Here, we extend the analysis to the more relevant four-level system, which consists of two ground states and two excited states and which incorporates the physics of dark states. The shifts are shown to depend on the type of experiment performed and can be much larger than the shifts for the three-level system. The analytic formulas obtained are applied to the $2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}S$-to-$2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}P$ transitions in atomic helium, where significant shifts are found.

DOI
10.1103/physreva.84.032508

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Shifts from a distant neighboring resonance

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The line center of a resonance is affected by the presence of neighboring resonances due to quantum interference between the two resonant processes. Such shifts must be accounted for in high-precision measurements but are easily overlooked. In the present work, we develop an analytic formulation for the effect in the simplest case of a three-level atom. The shifts in this model system are large enough to be of concern for precision measurements.

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In a recent paper [Phys. Rev. A 82, 052519 (2010)], the systematic shifts of a resonance due to quantum-mechanical interference from a distant neighboring resonance were derived. In that paper, the simplest three-level closed system was used to calculate analytic expressions for shifts in the resonant line centers. Here, we extend the analysis to the more relevant four-level system, which consists of two ground states and two excited states and which incorporates the physics of dark states. The shifts are shown to depend on the type of experiment performed and can be much larger than the shifts for the three-level system. The analytic formulas obtained are applied to the 23S2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}S23S-to-23P2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}P23P transitions in atomic helium, where significant shifts are found.
The quantum interference arising from neighboring resonance is investigated in a four-level system. This effect is evaluated through two methods: one is the direct numerical solution of the optical Bloch equation, and the other considers the impact of neighboring resonances as perturbations on other equations, i.e., using a simplified approximate optical Bloch equation for simulation. The results show that the energy shift caused by quantum interference is correlated with the spectral separation of two resonances, and is most significant when the interval is comparable to the spontaneous emission rate [Formula: see text]. For spectral separation that is sufficiently large, with spacing greater than [Formula: see text], the approximate optical Bloch equation method proves to be a straightforward and efficient approach. In addition, two formulas are given, which can be used to evaluate the contributions of neighboring resonances at other positions once the contribution of one is determined. According to the formulas, we evaluated the result of 4He atom as about [Formula: see text] kHz, which is consistent with the value given in the paper [Phys. Rev. A 86 (2012) 040501(R)]. Utilizing these methods can streamline intricate energy level structures, and thus facilitate the evaluation of quantum interference in high-precision spectroscopic measurements.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.
We calculate the line shift in the three-photon resonance-absorption experiment performed by Ferrel, Payne, and Garrett [Phys. Rev. A 36, 81 (1988)] and find it large, to the violet, and close to experiment. In comparison, the calculated single-photon resonance shift in absorption is a factor of 30 times smaller and to the red. The three-photon resonance shift depends on the relative propagation directions of the excitation beams, and we present a formula for the case where only two excitation beams are present.
Optical dark resonance in systems with arbitrary multilevels and multiphotons is studied theoretically. When the system configuration is treelike (the term of graph theory), all the coherence terms in the system can be categorized into several groups on dark resonance. It is proved for generic system configurations with arbitrary multilevels that the resonance conditions can be obtained successively by setting the determinant of the submatrix of the Hamiltonian with respect to each group to zero. We calculate, as one of the practical examples, all the conditions of dark resonance of potassium atoms with two laser fields and one rf field mixing one excited state and eight ground states in the hyperfine structure. Although the system configuration is very complicated, the condition is easily obtained in an analytic form and verified by the result of a numerical simulation. Furthermore, our theory can reveal the fine structure of dark lines with power broadening in the observed light-induced-fluorescence spectra and numerical calculations. In addition, the dark lines with the complicated splitting pattern observed in experiments can be assigned to the relevant level configuration and transitions by the present theory. \textcopyright{} 1996 The American Physical Society.
Based on the vanishing-damping continued-fraction method which I developed previously for the analysis of the resonance behavior of a two-level atom in a two-frequency field with equal intensities in the two field modes [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], I analyze here the related problem in which the two field intensities are unequal. Several analytical results are derived, namely, expressions for the lowest-order Bloch-Siegert shifts of all resonances, and an expression for the positions of the resonances in the limit of one strong and one weak field. It is shown that, in this limit, the resonances exhibit level crossings of the field-atom system.