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arXiv:quant-ph/9905098

Resonance flourescence in atomic coherent systems: spectral features

S. N. Sandhya

Published May 28, 1999• 0 citations
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Abstract

We study resonance flourescence in a four level ladder system and illustrate some novel features due to quantum interference and atomic coherence effects. We find that under three photon resonant conditions, in some region of the parameter space of the rabi frequencies $Ω_1,Ω_2,Ω_3$, emission is dominantly by the level 4 at the line center even though there is an almost equal distribution of populations in all the levels. As one increases $Ω_3$ with $Ω_1 and Ω_2$ held fixed, the four level system 'dynamically collapses' to a two level system. The steady state populations and the the resonance flourescence from all the levels provide adequate evidence to this effect.

DOI
10.48550/arxiv.quant-ph/9905098

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A theoretical study is made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level-atom system in a ladder configuration, which is subject to spontaneous-emission decay to the electromagnetic-field vacuum. Two cases are examined: in the equispaced (ES) level case, the two atomic transition frequencies are nearly equal and two photon absorption processes occur from a single coherent laser field that couples both transitions and has a small detuning from each; in the non- equispaced (NES) level case, the atomic transition frequencies are rather different and two photon absorption processes occur from two coherent laser fields, each coupled to a single transition and in near resonance with it. In both cases, a situation of small two-photon detuning occurs. Optical Bloch equations for the atomic density matrix in rotating frames are given and matrix expressions for determining the steady-state populations and coherences are obtained. Analytic expressions are also given for special cases. The fluorescent intensity is obtained from the populations of the intermediate and upper states, with the normally ordered variance (NOV) for quadrature components of the fluorescent field involving, in addition, the atomic coherences. The fluorescent intensities are shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases.In both cases, the fluorescent intensities show the well known resonances from the upper state at zero two-photon detuning and from the intermediate state at zero one-photon detuning for the lower transition. However, further resonances are also found. The intermediate-state intensity displays a resonance at zero two-photon detuning. Also, the upper-state intensity shows a resonance at zero one-photon detuning for the upper transition, but only for the NES case; evidently, the transfer rate from upper to lower transition coherences destroys this resonance for the ES case. The time-averaged NOV is also shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. For the ES case, the quadrature frequency is chosen as the single laser frequency and for the NES case, it is chosen as either the average of the two laser frequencies or the lower transition laser frequency. Squeezing occurs near zero two-photon detuning for both ES and NES cases, though only for the quadrature frequency equal to the average laser frequency in the latter case. It also occurs near zero detuning for the lower transition for both cases, though only for the quadrature frequency equal to the lower transition laser frequency in the NES case. The squeezing minima show a splitting effect for large Rabi frequencies corresponding to two-photon or one-photon Rabi splitting of the dressed atom levels for the situation near two-photon or one-photon resonance, respectively. The large squeezing near two-photon resonance for moderate Rabi frequencies and large one-photon detuning corresponds to essentially pure three-level squeezing, while that near one-photon resonance for weak Rabi frequencies involves two-level squeezing.
The resonance fluorescence from a four-level atom driven by two coherent fields is studied. If the dipole moments between the two upper levels and each of the two lower levels are parallel, the spontaneous emission paths from the upper levels to the lower levels will have strongly or even completely destructive quantum interference. We show that this quantum interference will result in a complete inhibition of the fluorescence when one of the driving fields is tuned at the middle point of the upper levels. The quantum interference will lead to the dependence of atomic populations at steady state on the initial condition, when both of the driving fields are tuned to the middle point of the upper levels. When the dipole moments are close to but not exactly parallel, or none of the driving fields is tuned at the middle point of the upper levels and there is a small difference between detunings of the two driving fields to the uppermost level, ultranarrow lines at the spectral center appear because of the destructive quantum interference.
The absorption profile of a four-level ladder atomic system interacting with three driving fields is studied perturbatively and analytical results are presented. Numerical results where the driving field strengths are treated up to all orders are presented. The absorption features are studied in two regimes—(i) the weak middle transition coupling, i.e. Ω 2 ≪ Ω 1,3 and (ii) the strong middle transition coupling Ω 2 ≫ Ω 1,3 . In case (i), it is shown that the ground-state absorption and the saturation characteristics of the population of level |2⟩ reveal deviation due to the presence of upper level couplings. In particular, the saturation curve for the population of level |2⟩ shows a dip for Ω 1 = Ω 3 . While the populations of levels |3⟩ and |4⟩ show a maxima when this resonance condition is satisfied. Thus the resonance condition provides a criterion for maximally populating the upper levels. A second-order perturbation calculation reveals the nature of these minima (maxima). In the second case, I report two important features: (a) filtering of the Aulter–Townes doublet in the three-peak absorption profile of the ground state, which is achieved by detuning only the uppermost coupling field and (b) control of line-width by controlling the strength of the upper coupling fields. This filtering technique coupled with the control of linewidth could prove to be very useful for high resolution studies.
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Summary form only given. In a three-level system, two resonant optical fields can excite spin coherence on the ground states via two-photon resonance. One of the two-superposition states created by the atom-field interactions is decoupled from the excited state. Thus, the population can be trapped in that state showing non-absorption resonance. The essential feature of this decoupled state is dark resonance. Therefore, the existence of the dark resonance is a basis of non-absorption resonance, coherent population trapping, and electromagnetically induced transparency. In a four-level system, however, the two-laser induced spin coherence can be broken when a third laser is applied. At the same time, new coherence between the third laser and one of the pre-existed lasers is created. This phenomenon is useful for optical switching owing to annihilation and creation of the dark resonance in the four-level system. It is noted that the switching time depends on the dark resonance build-up time, which is inversely proportional to the applied Rabi frequency. Numerical simulations are shown for potentially ultrahigh-speed optical switches. Each diffracted four-wave mixing signal is proportional to the spin coherence strength. For a preliminary experiment, a spectral hole-burning crystal is used. The intensity of the diffracted signals is reversed at line center. This coherence switching effect is based on the dark resonance interactions.
Four-level atom dynamics is studied in a ladder system in the nine parameter space consisting of driving field strengths, detunings and decay constants, ${{\ensuremath{\Omega}}_{1},{\ensuremath{\Omega}}_{2},{\ensuremath{\Omega}}_{3},{\ensuremath{\Delta}}_{1},{\ensuremath{\Delta}}_{2},{\ensuremath{\Delta}}_{3},{\ensuremath{\Gamma}}_{2},{\ensuremath{\Gamma}}_{3},{\ensuremath{\Gamma}}_{4}}$. One can selectively excite or induce two-level behavior between particular levels of ones choice by appropriately tuning the driving field strengths at three-photon resonance. The dynamics may be classified into two main regions of interest (i) small ${\ensuremath{\Omega}}_{2}$ coupling the \ensuremath{\mid}2⟩-\ensuremath{\mid}3⟩ transition and (ii) large ${\ensuremath{\Omega}}_{2}$. In case (i) one sees two-level behavior consisting of adjacent levels and in a particular region in the parameter space, there is an intermittent shelving of the electrons in one of the two subsystems. In case (ii) the levels consist of the ground state and the upper most level. Emission statistics is studied using the delay function approach in both the cases. In case (i), the behavior of the second order correlation function g2(t){g}^{2}(t)g2(t), is similar to that of two-level emission for low ${\ensuremath{\Omega}}_{1}$ coupling the \ensuremath{\mid}1⟩-\ensuremath{\mid}2⟩ transition, and the correlation increases with ${\ensuremath{\Omega}}_{1}$ for smaller time delays. While, in case (ii) when, in addition, ${\ensuremath{\Omega}}_{3}$ coupling the \ensuremath{\mid}3⟩-\ensuremath{\mid}4⟩ transitionis kept low, g2(t){g}^{2}(t)g2(t) shows superpoissonian distribution, which may be attributed to three-photon processes.