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Transfer-of-coherence-enhanced stimulated emission and electromagnetically induced absorption in Zeeman split Fg \ Fe = Fg − 1 atomic transitions R. Meshulam, T. Zigdon, A. D. Wilson-Gordon,* and H. Friedmann Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel *Corresponding author:
[email protected] Received April 30, 2007; accepted May 27, 2007; posted July 3, 2007 (Doc. ID 82315); published August 2, 2007 The probe absorption spectra in single and multiple tripod systems formed when a weak polarized pump and a tunable polarized probe interact with a Zeeman split Fg → Fe = Fg − 1 atomic transition are characterized by two interfering stimulated Raman features separated by an electromagnetically induced absorption (EIA) peak at the line center. These Raman features can appear as either sharp stimulated emission peaks or electromagnetically induced transparency windows. In the multitripod systems, the EIA and stimulated emission peaks derive from the combined effects of interference between the stimulated Raman features and transfer of coherence from the excited to ground states. © 2007 Optical Society of America OCIS codes: 190.5650, 270.1670.
The probe absorption spectrum of an atomic degenerate two-level system interacting with a strong pump and weak probe can exhibit narrow features near the two-photon Raman resonance, where the pump and probe have equal frequencies [1]. When the probe absorption spectrum is characterized by a sharp dip, the phenomenon is called electromagnetically induced transparency (EIT) [2], whereas when it is characterized by a sharp peak, it is called electromagnetically induced absorption (EIA) [3]. EIA has been studied extensively, both experimentally and theoretically. In the original experiments, EIA was obtained for pump and probe lasers with both parallel and perpendicular polarizations interacting with a cycling degenerate two-level transition in which Fe = Fg + 1 and Fg ⬎ 0 [3,4]. It was also observed in the Hanle configuration [5]. For the case of perpendicularly polarized pump and probe lasers, EIA has been shown to be due to transfer of coherence (TOC) via spontaneous emission from the excited state to the ground state [6]. In our earlier work on EIA [7], we explained that the TOC that leads to EIA in degenerate systems can only take place when ground-state population trapping is incomplete, that is, when Fe = Fg + 1. However, incomplete population trapping and hence TOC can also be achieved when Fe = Fg − 1, with Fg ⬎ 1, when the degeneracy is lifted by applying a weak magnetic field [see Figs. 1(a) and 1(b) for the Fg = 2 → Fe = 1 transition in the D2 line of 87Rb, without and with a magnetic field]. In this Letter, we show that the probe absorption spectra in multitripod systems formed when a weak polarized pump and a tunable polarized probe laser interact with an Fg → Fe = Fg − 1 transition in the presence of a weak magnetic field are characterized by two interfering stimulated Raman features separated by an EIA peak at the line center (see Fig. 2). At moderate pump intensities, the stimulated Raman features resemble EIT windows as observed by Lezama 0146-9592/07/162318-3/$15.00
et al. (see Fig. 8 of [4]) and Fuchs et al. (see Fig. 4 of [8]). However, at higher pump Rabi frequencies, they appear as stimulated emission peaks centered near the Raman frequencies, which gradually become weaker and move towards the line center as the pump becomes more intense. We will show that both the central absorption peak and the stimulated emission peaks result from the combined effects of interfering stimulated Raman features and TOC. To understand the origin of the stimulated emission peaks, we return to a simple near-degenerate ⌳ system such as the one shown in Fig. 1(c), formed when a near-degenerate Fg = 1 → Fe = 0 transition is
Fig. 1. (Color online) Energy-level scheme for (a) and (b) Fg = 2 → Fe = 1 in the D2 line of 85Rb, (c) and (d) Fg = 1 → Fe = 0 in the D2 line of 87Rb. In (a) and (b), the populations of the Zeeman sublevels are written on the multitripod schemes for the parameters V1 / ⌫ = 1, ⌫gg / ⌫ = 0.0005, ␥ / ⌫ = 0.001, and N = 1012 atoms cm−3, with B = 0 in (a) and B = 1 G in (b). In (c), the pump is + polarized (⌳ system), whereas in (d) the pump is polarized (tripod system). © 2007 Optical Society of America
August 15, 2007 / Vol. 32, No. 16 / OPTICS LETTERS
Fig. 2. (Color online) Probe absorption spectra for Fg = 2 → Fe = 1 transition in the D2 line of 85Rb, as a function of pump–probe detuning ␦ = 2 − 1 for various pump Rabi frequencies, V1 / ⌫ = 1 (solid curve), V1 / ⌫ = 2 (dashed curve), and V1 / ⌫ = 3 (dotted–dashed curve). Other parameters are as in Fig. 1(b).
pumped by a near-resonant + polarized pump of frequency 1, and probed by a tunable polarized probe of frequency 2. The probe absorption is proportional to the imaginary part of the optical coherence eg2共2兲 between the excited and ground m = 0 Zeeman sublevels [9]:
eg2共2兲 =
V2共g0
2g2
0 − ee 兲 + V 1 g1g2共 2 − 1兲
⌬2 − i␥eg V2共g0 g 2 2
−
0 ee 兲
+ V 1V 2
= 关⌬2 − i␥eg兴 +
冋
g01e共− 1兲 ⌬1 − ⌬2 + i␥gg V12
册
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why such features can be enhanced in the degenerate two-level systems by TOC from the excited state.) Generally, it is assumed [2,10] that all the population is optically pumped into the g2 state so that the second term in the second line of Eq. (1) is negligible, and the probe absorption spectrum is then characterized by an EIT window centered at the Raman resonance frequency ⌬2 = ⌬1. However, as shown in Fig. 3(a), this is not necessarily true for a near-degenerate system. Here, we see that when the pump is sufficiently weak that optical pumping cannot completely overcome the repopulation of state g1 from the reservoir, and the rate of collisions that transfer population between the ground-state Zeeman sublevels ⌫gg Ⰶ ␥, stimulated emission appears at the Raman resonance frequency. When Eq. (1) is generalized to the tripod system [9] obtained by replacing the + polarized pump by one with polarization [see Fig. 1(d)] and the same parameters are used as in Fig. 3(a), we obtain the spectrum shown in Fig. 3(b), which displays two interacting stimulated emission peaks at the Raman resonance frequencies ⌬2 = ± ⌬1, separated by a sharp EIA peak centered at ⌬2 = 0. At higher pump Rabi frequencies 共V1 / ␥eg ⬇ 1兲, all the population is pumped into the mg = 0 state, and the stimulated emission peaks are replaced by two EIT windows at the Raman resonance frequencies [9]. As the pump Rabi frequency is increased further, the mg = ± 1 states are again populated, this time due to coherent population trapping in the ground states. As a result of the interaction between the two dark states, the EIT windows converge towards the line center, thereby narrowing the central absorption peak, which also decreases in amplitude, and eventually coalesce into a single EIT window, which then develops a sharp stimulated emission peak [see Fig. 3(c)].
.
⌬1 − ⌬2 + i␥gg 共1兲
In Eq. (1), 2V1 and 2V2 are the pump and probe Rabi frequencies; ⌬i = egi − i, with i = 1 , 2, are the pump and probe detunings, with ⌬1 = −gFBB in the presence of a magnetic field when 1 = eg2 (assumed throughout); ␥eg = ␥ + ⌫ + ⌫gg and ␥gg = ␥ + ⌫gg are the widths of the optical and Zeeman coherences, respectively, where ␥ is the rate of decay due to time-offlight through the laser beam, ⌫ is the total decay rate of each Zeeman sublevel of the excited state, ⌫gg is the rate of population transfer between the Zeeman sublevels of the ground state, and dephasing col0 are the populations lisions are neglected; g0 g and ee 2 2 0 is the pump of the m = 0 Zeeman sublevels, and eg 1 optical coherence, calculated to all orders in the pump Rabi frequency and to zero order in the probe Rabi frequency, and g1g2共2 − 1兲 is the two-photon coherence. From the first line of Eq. (1), we see that it is the narrow two-photon coherence that leads to the sharp features in the probe spectrum. (This explains
Fig. 3. Probe absorption spectra for Fg = 1 → Fe = 0 in the D2 line of 87Rb. In (a) the pump is + polarized, and in (b) and (c) it is polarized. Parameters are B = 3 G, ⌫gg = 0, ␥ / ⌫ = 0.001, and N = 1012 atoms cm−3, and V1 / ⌫ = 0.1 in (a) and (b) and V1 / ⌫ = 65 in (c). In (a), the populations are ee = 0.0003, g1g1 = 0.135, and g2g2 = g3g3 = 0.432; in (b), they are ee = 0.0005, g1g1 = g3g3 = 0.243, and g2g2 = 0.512; and in (c), they are ee = 0.0004, g1g1 = g3g3 = 0.268, and g2g2 = 0.463.
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Having established that stimulated emission can occur in ⌳ and single tripod systems, we now consider a multitripod system such as the Fg = 2 → Fe = 1 transition, shown in Figs. 1(a) and 1(b), in the absence and presence of a weak magnetic field. This figure shows that lifting the degeneracy of the hyperfine states by applying a magnetic field leads to an increase in the populations of the excited Zeeman sublevels by an order of magnitude, thereby increasing the excited-state two-photon coherences and the transfer of these coherences from the excited to ground hyperfine states. The probe absorption spectrum can be calculated to all orders in the pump Rabi frequency and to first order in the probe Rabi frequency by using the Bloch equations of [7], which include TOC. In Fig. 4(a), we compare the probe absorption spectrum with and without TOC for the Fg = 2 → Fe = 1 transition in the D2 line of 85Rb. We see
that TOC has two effects on the spectrum: first, it sharpens the EIA peak that results from interacting stimulated Raman features, and second, the EIT windows are replaced by stimulated emission peaks. In Fig. 4(b), we consider the same transition in 87Rb, which is a noncycling transition, so that the population in all levels is much reduced and TOC cannot occur. As expected, the spectrum of the open system resembles that of the closed system in the absence of TOC. At high values of V1 / ⌫, a sharp stimulated emission peak appears at the line center. It is, however, weaker than in the single tripod system. In conclusion, we have shown that the nearly degenerate ⌳, single tripod, and multitripod systems are characterized by stimulated Raman features that may appear as either stimulated emission peaks or as EIT windows. The central EIA peak in the single tripod system arises from the interference between the Raman features. In the multitripod system, this interference and TOC together produce a sharp EIA feature flanked by two stimulated Raman features. References
Fig. 4. (Color online) Probe absorption spectra for Fg = 2 → Fe = 1 transition in D2 line of (a) 85Rb (cycling transition) and (b) 87Rb (noncycling transition). In (a), the spectra are calculated with (solid curve) and without (dashed curve) TOC. V1 / ⌫ = 2, and the other parameters are the same as in Fig. 1(b).
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