The double dark resonance in a cold gas of Cs atoms and molecules

The double dark resonance in a cold gas of Cs atoms and molecules ZhiFang Feng , WeiDong Li∗ , LianTuan Xiao, SuoTang Jia Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China [email protected]

Abstract: We theoretically investigated the properties of the effective four-level stimulated Raman adiabatic passage scheme in a cold gas of Cs atoms and molecules, where exists the tunnelling coupling between two excited molecular states due to the 0 − g (6S, 6P3/2) double well structure. The double dark resonance is predicted in the absorption spectrum when the tunnelling coupling strength is large enough. The double dark resonance not only reveals the formation of the ultra-cold molecules, but also provides further evidence for the tunnelling as one effective coupling mechanism between the two excited molecular states. The effect of the various experimental conditions on this phenomena has been discussed. © 2008 Optical Society of America OCIS codes: (270.1670) Coherent optical effects; (020.2070) Effects of collisions; (290.5910) Scattering, stimulated Raman; (300.1030) Absorption

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Received 7 Jul 2008; revised 4 Sep 2008; accepted 4 Sep 2008; published 22 Sep 2008

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15. E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman,“Feshbach resonances in atomic Bose–Einstein condensates,” Phys. Rep. 315, 199-230 (1999). 16. D. Kleppner,“Professor Feshbach and His Resonance,” Phys. Today 57, 12-14 (2004). 17. H. Y. Ling, H. Pu, and B. Seaman,“Creating a Stable Molecular Condensate Using a Generalized Raman Adiabatic Passage Scheme,” Phys. Rev. Lett. 93 , 250403 (2004). 18. K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, and J. H. Denschlag,“Atom-Molecule Dark States in a Bose-Einstein Condensate,” Phys. Rev. Lett. 95, 063202 (2005); G. R. Jin, C. K. Kim, and K. Nahm,“Electromagnetically induced transparency in an atom-molecule Bose-Einstein condensate,” condmat/0603094. 19. A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet,“Formation of Cold Cs2 Molecules through Photoassociation,” Phys. Rev. Lett. 80, 4402-4405 (1998). 20. A. Fioretti, D. Comparat, C. Drag, C. Amiot, O. Dulieu, F. Masnou-Seeuws, and P. Pillet,“Photoassociative spectroscopy of the Cs2 0− g long-range state,” Eur. Phys. J. D 5, 389-403 (1999). 21. B. Laburthe Tolra, C. Drag and P. Pillet,“Observation of cold state-selected cesium molecules formed by stimulated Raman photoassociation,” Phys. Rev. A 64, 061401(R) (2001). 22. M. Vatasescu, O. Dulieu, C. Amiot, D. Comparat, C. Drag, V. Kokoouline, F. Masnou-Seeuws, and P. Pillet,“Multichannel tunneling in the Cs2 0− g photoassociation spectrum,” Phys. Rev. A 61, 044701 (2000). 23. M. Vatasescu, F. Masnou-Seeuws,“Time-dependent analysis of tunneling effect in the formation of ultracold molecules via photoassociation of laser-cooled atoms,” Eur. Phys. J. D 21, 191-204 (2002). 24. R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen,“Molecules in a Bose-Einstein Condensate,” Science 287, 1016-1019 (2000). 25. E. Paspalakis and P. L. Knight,“Transparency and parametric generation in a four-level system † Reviewing of this paper was handled by a member of the Editorial Board,” J. Mod. Opt. 49, 87-95 (2002); “Electromagnetically induced transparency and controlled group velocity in a multilevel system,”Phys. Rev. A 66, 015802 (2002). 26. G. Wasik, W. Gawlik, J. Zachorowski, and Z. Kowal,“Competition of dark states: Optical resonances with anomalous magnetic field dependence,” Phys. Rev. A 64, 051802(R) (2001). 27. M. Mackie, R. Kowalski, and J. Javanainen,“Bose-Stimulated Raman Adiabatic Passage in Photoassociation,” Phys. Rev. Lett. 84, 3803-3806 (2000). 28. Jie Ma, Lirong Wang, Yanting Zhao, Liantuan Xiao, and Suotang Jia,“Absolute frequency stabilization of a diode laser to cesium atom-molecular hyperfine transitions via modulating molecules,” Appl. Phys. Lett. 91, 161101 (2007). 29. M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge, (1997).

1. Introduction “Double dark” resonance [1] , as a novel spectral feature appearing in a system with multiple coherent interacted quantum superposition states, has been shown as a powerful mechanism to coherently control the adiabatic passage [2, 3] and applied to nonlinear optics [4, 5], Dopplerfree resonance [6, 7], high efficiency four-wave mixing [8, 9, 10] and group velocity controlling [11, 12]. To form a sample of ultra-cold ground-state (or even selected excited-state) molecules is one of the central issues in the ultra-cold molecules field. There are two experimentally demonstrated and efficient tools to create ultra-cold molecules from ultra-cold (even from the BosonEinstein condensates) atoms. The photo-association (PA), where two cold atoms absorb a photon to create an excited molecule and then a stable ground molecule is formed by spontaneous emission, is a successful method but with low conversion efficiencies from atoms to molecules. Recently, a shaped broadband femtosecond laser source is applied to improve the conversion efficiency after the PA process in Cs atom-molecules system [13] . The magnetic Feshbach resonance [14, 15, 16], with high conversion efficiencies, is restricted to the creation of molecules in the higher ro-vibrational level, due to energy conservation [17, 18]. Since the first experimental realization of cold Cs 2 molecules through PA [19], the detailed information of the spectroscopy of cold Cs 2 molecules have been explored in a series of works [20, 21, 22, 23]. Compared with other alkali dimers, a universal feature of the experimental spectrum is reported and interpreted as a consequence of the double-well shape of the 0 − g (6S + 6P3/2 ), separated by a potential barrier at distance R ≈ 15a 0. When by PA of two cold cesium atoms an excited level of the outer well (R > 15a 0) is populated, tunnelling is suggested as an #98199 - $15.00 USD

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efficient mechanism for transferring the population to the inner well (R < 15a 0 ), which provides a rather efficient channel in spontaneous emission for the creation of cold molecules in low vibrational levels of the a 3 ∑+ u (6s + 6s) electronic state. Further dynamical information can be extracted from the corresponding characteristic times for the vibration dynamics, T vib (Ev ) = 2π h¯ /(Ev+1 − Ev ), termed as vibrational period in [23]. Due to a small level spacing (around 0.6 cm−1 ) in the outer well, the characteristic time is in the range of 200 − 250 ps, while the relative larger level spacing (around 9.5 cm −1 ) in inner well results in a small characteristic time 3.5 ps. Meanwhile, the “beating time” (T int,ext ∼ 400 − 450ps ), determined by the tunnelling between the outer well and inner well, has been reported. All of these results suggested this PA process can be understood as an effective four-level structure, where the two separated excited molecule levels in 0 − g (6S + 6P3/2 ) are coupled by the tunnelling mechanism, and the free atoms and the ground molecular states are two longliving stable states. Replacing the spontaneous emission with a stimulated emission, where the inner excited molecular state and the ground molecular state a 3 ∑+ u (6s + 6s) are coherently coupled by a pump laser, we arrive at an effective four-level stimulated Raman adiabatic passage (STIRAP) scheme. The STIRAP [24] has been demonstrated as a robust and efficient process to convert the ultra-cold atoms into a molecular ground state. The central prerequisite for the STIRAP for four-level system is the double dark resonance, which can be observed by measuring the probe absorption spectrum consisting of two electromagnetically induced transparency (EIT) windows separated by a sharp absorption peak [1, 25, 26]. In this paper, we will investigate the properties of the effective four-level STIRAP in a cold gas of Cs atom-molecule. We present our scheme in the first section and the equations of motion for density matrix element in the second section. The probe absorption spectrum is calculated under the weak probe field condition in the third section. Some conclusions and experimental proposals are presented in the last section. 2. Four-level model In this section, we introduce a stimulated emission into the experimentally proved process, formation of ultra-cold molecules through PA [19], and present an effective four levels STIRAP scheme. To investigate the properties of the absorption spectrum, we find an analytical expression for the probe absorption under the weak probe field condition. 2.1. An effective four-level model The formation of the ultra-cold Cs 2 in their lowest electronic triplet state a 3 ∑+ u , at the temperature T ∼ 300μ K, has been experimentally observed in 1998 [19]. This efficient scheme is attributed to the double-well structures in the excited 0 − g (6S + 6P3/2 ) potential curves, that provides an efficient mechanism (tunnelling) for transferring the population to the inner well (R < 15a0), where spontaneous emission may lead to formation of cold molecules in low vibrational levels of the electronic state a 3 Σ+ u (6s, 6s) [22, 23]. As mentioned in the Introduction, this spontaneous emission is replaced by a stimulated emission, where the inner excited molecular state and the ground molecular state a 3 Σ+ u (6s, 6s) are coupled by a pump laser. Then we construct a STIRAP scheme [17, 27] for the formation of ultra-cold Cs 2 (illustrated by the spectra in Fig. 1). As shown in Fig. 1, the free Cs atom (denoted by |a), the inner and outer excited molecular 3 + states of 0− g (6S + 6P3/2 ) (denoted by |b 2,1 ) and the lower triplet state a ∑u (|g) construct an effective four-level system. The first laser field, called as the probe laser in the following, actually is a PA laser with Rabi frequencies Ω 1 , by which the excited molecules are formed in a precise vibrational level of the outer well in 0 − g (6S + 6P3/2) (denoted as |b 1 ) from the ultracold atoms |a. This process is reasonable by considering the relative larger energy spacing #98199 - $15.00 USD

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Fig. 1. The level scheme. Δ and δ denote the detunings. The inner and outer molecular states b1,2 spontaneously decay with a rate γb ,b to levels outside this scheme. The molecular 1 2 ground state |g is contributed with a decay rate γg , which phenomenologically takes into account losses through inelastic collisions.

(around 4.8 GHz) lying between −2.98 cm −1 and −2.82 cm −1 in the outer well of 0 − g (6S + 6P3/2 ) [23], compared with the high resolution (around MHz ) of laser spectroscopy in current experimental condition [28]. The second laser field with Rabi frequencies Ω 2 , called as the pump laser, provides a stimulated emission, coupling the inner excited molecules |b 2 with deeply bound molecular state |g. The coupling between the outer and inner excited molecular states is realized by tunnelling mechanism and the tunnelling rate is denoted by σ 12 . As shown in [23], the tunnelling coupling strength depends on the intensity of the PA laser and its detuning with |b2 . To simplify our analysis and to emphasize on the resonance tunnelling coupling between the two excited states |b 1 and |b2 , we neglect their frequency difference, which plays a similar role as the detuning (Δ) discussed in the section III. The dynamics of the system are governed by the following Hamiltonian (in the rotating wave approximation [29]) H = H0 + HI + Ht ,

(1)

where H0 HI Ht

= h¯ (δ + )b†1 b1 + h¯ (δ + )b†2 b2 + h¯ δ g† g, h¯ h¯ = − Ω1 b†1 aa + H.c. − Ω2 g† b2 + H.c. , 2 2

(2)

= h¯ σ12 b†2 b1 + b†1 b2 .

The operators a and a † are annihilation and creation operators for atoms in state |a, similarly for b1,2 (b†1,2 ) and g (g† ): annihilating (creating) molecules in states |b 1,2 and |g, respectively. We have assumed that only |b 1 −→ |a and |b2 −→ |g transitions are dipole allowed [29], while the transitions |b 1 −→ |b2 is provided by tunnelling mechanism. When the tunnelling coupling between the two excited molecular states is so strong that these two states are melted #98199 - $15.00 USD

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into one excited state, our model can be reduced to the normal Λ type three-level scheme, which is a more popular model to study the quantum dynamics in atom-molecule system [17]. But as shown in [23], this requirement can not be satisfied for Cs, since it is impossible for the resonance tunnelling condition and the higher PA efficient rate to be synchronously fulfilled in the current experimental condition. This is why we do not consider this special case in the following parts. 2.2. The equations of motion of the density matrix and weak probe field condition To investigate the atom-molecule coherence dynamics, it is helpful to introduce 4 × 4 dimensional density matrix operator ρˆ where the density matrix element ( ρˆ μν ) describes the probabilities of being in the μ states for μ = ν and the polarization for μ = ν . And the corresponding master equations, governed by Hamiltonian (1), are ∂ i ρˆ μν = −γμν ρˆ μν − ρˆ μν , H , ∂t h¯

(3)

where γ μν ( μ = ν ) is the transverse decay rate from state μ to ν (μ , ν = a, b 1 , b2 , g) and is defined as γ μν ≡ (γμ + γν )/2. In the absence i.e., γ b1 ,b2 ,g = 0, the total of molecular decaying, particle number is conserved and ρ aa + 2 ρb1 b1 + ρb2b2 + ρgg = 1, where ρ μ μ (μ = a, b1,2 ) are the particle population in state |μ . The coherent dynamics can be experimentally observed by measuring the absorption spectrum. Theoretically, one calculates the imaginary part of ρ b1 a as the function of the probe laser detuning (δ ), which performs the properties of the absorption spectrum [29]. Substituting the Hamiltonian (1) into Eq. (3) and taking the mean values for the density matrix operator, ρμν = ρˆ μν , we have found the equations of motion for the density matrix elements ρ b1 a , ρb2 a and ρga

∂ ρb a ∂t 1 ∂ ρb a ∂t 2 ∂ ρga ∂t

Ω1 2 4ρb1 b1 ρaa − ρaa , −i (δ + ) − γab1 ρb1 a − iσ12 ρb2 a − i 2 Ω2 = −i (δ + ) − γab2 ρb2 a − 2iΩ1 ρb2 b1 ρaa + i ρga − iσ12 ρb1 a , 2 Ω2 = (−iδ − γag ) ρga + i ρb2 a − 2iΩ1ρgb1 ρaa . 2 =

(4)

Generally, it is not easy to have the analytical solution for Eq. (4 ). But we can analytically solve Eq. (4) in the case of the weak probe field, i.e., Ω 1 is small enough (two order smaller (to be shown in Fig. 5) than the unit, which is taken as the spontaneous emission line-width (γab1 ∼ 50MHz) of excited Cs molecules through this paper). Starting with the case when all atoms are prepared in the atomic ground state |a and following the procedures in [29], we arrive at

iΩ1 Ω22 /4 + (iδ + γag ) i ( + δ ) + γab2 . (5) 2 ρ b1 a = 1 2 2 Ω2 i ( + δ ) + γab1 + 2 (iδ + γag ) σ12 + i + iδ + γab1 i + iδ + γab2 The long time limit has been involved in order to find the solution Eq. (5). Due to the weak probe field condition, we keep all orders in pump Rabi frequencies Ω 2 and the first order in probe Rabi frequencies Ω 1 in our calculation. 3. The double dark resonance The probe absorption spectrum can be obtained by investigating the behavior of the Im(ρ b1 a ) when changing the probe detuning δ . Based on the Eq. (5) and weak probe field condition, #98199 - $15.00 USD

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Fig. 2. Probe absorption spectra for different tunnelling rates. The parameters are Ω1 = 0.01γab1 , Ω2 = 2γab1 , γab2 = 0.2γab1 , γag = 0 and = 0. (a) σ12 = 0, (b) σ12 = 0.25γab1 , (c) σ12 = γab1 , (d) σ12 = 10γab1 , (e) σ12 = 20γab1 , (f) σ12 = 40γab1

we will study the feature of the probe absorption spectrum in a cold gas of Cs atom-molecule system. Since our final molecular state is the ground molecular state, it is reasonable to assume |g is long-living lowest molecular state. Given the system parameters (in the caption of the Fig. 2), we plotted the Im(ρ b1 a ) as the function of the probe detuning δ in Fig. 2 for different tunnelling coupling rates σ 12 . Considering the PA process, the character tunnelling time (“beating time” in [23]) related to the transfer of the population from the outer well to the inner well can be 400 ∼ 450ps in [23]. If we take the decay rate of the outer excited molecular levels (γ ab1 ) as the unit, which is in inverse proportion to the lifetime (about 10ns) for Cs atoms in the excited vibrational level, then the range for the tunnelling coupling will be in the range of 0 − 50γ ab1 . Considering the weak probe filed condition, we take Ω 1 = 0.01γab1 , which belongs to the weak PA coupling region [23]. By increasing the tunnelling coupling strength, the typical double dark resonance spectra [1] are observed in the case of a large enough σ 12 . This is because the tunnelling coupling modifies our scheme from an effective two-level structure (σ 12 = 0) to four-level structure. A pair of the absorption peaks around the resonance point δ /γ ab1 = 0 are emerging at a weak tunnelling coupling σ 12 ∼ 0.25γab1 in Fig. 2(b) and then becoming clearer by enhancing the tunnelling coupling in Fig. 2(c-f). This is nothing but the feature of double-dark resonance, and an expected signal for the four coherently interacted atom-molecule states in a cold gas of Cs atoms and molecules with STIRAP [1] . This indicates that there are interference effects generated by the interactions of two dark states, which are composed of |a, |b 2 and |g. To quantify the properties of double-dark resonance, we assume that the transverse decay

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rate γab2 = γag = 0 and γab1 = 1. Then from Eq. (5), we write the imaginary parts of ρ b1 a as

Im ρb1 a =

Ω22 2

2

2 Ω 2Ω1 42 − δ ( + δ )

2 .

2 2 Ω 2 − 2δ ( + δ )2 − 2δ ( + δ ) + 22 ( + δ ) + 2δ σ12

Obviously, there are two zero probe absorptions for −Δ ± Δ2 + Ω22 , δ= 2

(6)

(7)

and for Δ = 0, two transparency points appear at detuning δ = ±Ω 2 /2 (see Fig. 2), i.e. the original dark resonance is split into a pair of dark lines. This is the typical signal for the two dark resonances [1]. On the other hand, the two absorption peaks are located at Ω22 1 2 δ =± 2 2σ12 + . (8) 2 2 2 + 1. These simple Considering the conditions in Fig. 2, Ω 2 = 2γab1 , we have δ /γab1 = ± σ12 results are found in Fig. 2 for the weak tunnelling coupling ( σ 12 ∼ 0), δ /γab1 ∼ ±1 (see Fig. 2(b) and (c)); while for strong tunnelling coupling, i.e. σ 12 /γab1 1, δ /γab1 ∼ ±σ12 (see Fig. 2(d),(e),(f)).

Fig. 3. (Color online). Probe absorption spectra for different driving detuning, with = 0 (solid line), = −γab1 (dashed line), and = −4γab1 (dotted line) for Ω1 = 0.01γab1 , Ω2 = 2γab1 , γab2 = 0.2γab1 , γag = 0, σ12 = γab1 .

The symmetry of the absorption spectrum can be broken by any small pump field detuning (Δ), as shown in Fig. 3. In our simplified model, the frequency difference between |b 1 and |b2 has been neglected, but should play the similar role as Δ and break the symmetry of the absorption spectrum. Therefore, our simplified model can be realized by properly adjusting the pump field detunning to eliminate the frequency difference between |b 1 and |b2 . By increasing the pump field detuning Δ, the center of the absorption spectra is shifted to the right. When the left absorption peak is decreasing, the right one is increasing. Meanwhile, the absorption line is turned into a transparency line close to the point of two-photon resonance, δ ≈ 0. Increasing #98199 - $15.00 USD

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Fig. 4. (Color online). Probe absorption spectra for different pump frequencies and tunneling rate, with Ω2 = 2γab1 (solid lines); Ω2 = 5γab1 (dashed lines); Ω2 = 10γab1 (dot lines). other parameters are Ω1 = 0.01γab1 , = 0, γab2 = 0.2γab1 , γag = 0. (a) σ12 = γab1 , (b) σ12 = 4γab1 .

the Rabi frequency of the pump field broadens the transparency windows, as shown in Fig. 4. It is easy to understand that the distance of two transparency points keeps a constant quantity Ω2 and is independent of the tunnelling coupling strength, based on Eq. (7). Furthermore, Fig. 4 also provesthe prediction of Eq. (8), the distance between the two absorption peaks is 2 + Ω2 /2 . 2 2σ12 2 To understand what role the weak probe filed condition plays during the investigation of the double dark resonance in our system, we solve the equations of motion of density matrix elements Eq. (3) with the Runge-Kutta method and long time approximation. In our calculation, we take the evolution time t = 80(1/γ ab1 ). Imposing the same initial condition as the weak probe field approximation case, ρ aa (0) = 1, ρ μν = 0 (for any else μ , ν ), we show the absorption spectra in Fig. 5 for different powers of the pump and probe field. In Fig. 5(a) and (b), we take the weak probe field approximation, where Ω 1 is two orders smaller than the unit. The good consistency with the analytical results (based on Eq. (5)) supports our previous conclusions that the double dark resonance spectra exist in this system. In Fig. 5(c), we take Ω 1 = 0.1γab1 . Since the weak probe filed condition has been broken, we can make out the quiet difference between the analytical results (based on Eq. (5)) and the numerical ones. Even so, Fig. 5(c) shows a typical signal of the double dark resonance though not so clear. Same as the dark resonance in the atom-molecule system [18], the double dark resonance in our system should have some effects on the population of the initial atom state which are able to be experimentally measured. Therefore, we plot the population of the atom state with the probe detuning δ /γ ab1 in Fig. 6. Compared with the results in [18], the single dark peak is split #98199 - $15.00 USD

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Fig. 5. (Color online). Analytical (solid line) and numerical (dotted line) calculations of probe absorptions with different driving field powers and probe field powers. Parameters used for the plot are = 0, γab2 = 0.2γab1 , γag = 0, σ12 = γab1 , and (a) Ω1 = 0.01γab1 , Ω2 = 10γab1 , (b) Ω1 = 0.01γab1 , Ω2 = 2γab1 , (c) Ω1 = 0.1γab1 , Ω2 = 2γab1 .

into two peaks around δ /γ ab1 = 0 and the strong tunnelling coupling will enhance this effect. Therefore, the double dark resonance can be a signal for the cold molecule and also the signal for the tunnelling mechanism for this special double well structure. 4. Conclusion An effective four-level STIRAP scheme for the formation of ultra-cold Cs 2 is proposed with the help of a pump laser coupling the inner excited molecule state with the lowest electronic state a3 Σ+ u (6s, 6s). The effective four-level structure is composed due to the double-well shape of the 0− g (6S + 6P3/2), separated by a potential barrier at the distance R ≈ 15a 0 . The tunnelling is assumed to be an effective mechanism to transfer the population in the outer excited molecular state to the inner one [23]. Since the energy spacing (4.8GHz between two excited vibrational levels in the outer well [23]) is much lager than the laser spectroscopy resolution (MHz [28] ), only one vibrational level in the outer well is considered. With these multiple coherently interacted atom and molecule states, the double dark resonance is a reasonable phenomenon and can be observed in the case of the weak probe laser field and the large tunnelling coupling strength. Adjusting the pump detunings and the Rabi frequencies, we can observe the #98199 - $15.00 USD

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Fig. 6. (Color online). Atom population as a function of probe detuning δ /γab1 for different tunnelling rate, with σ12 = 0.5γab1 (dashed line) and σ12 = γab1 (solid line). other parameters are Ω1 = 0.01γab1 , Ω2 = 2γab1 , γab2 = 0.2γab1 ,γag = 0, = 0.

symmetrical breaking of the absorption spectra and the adjustable transparency windows. The frequency difference between |b 1 and |b2 , neglected in our simplified scheme, results in the symmetry broken for the absorption spectra too. Furthermore, the two peak structures has been found in population of cold atoms as a function of probe detuning, due to these four coherently interacted atom-molecule states. Our proposed scheme is based on the experimental realization of the Cs molecule with PA [19], and the relatively larger energy spacing in the inner well [23] makes it easy to introduce the stimulated Raman laser, coupling the molecular ground state a3 ∑+ u (6s + 6s) with the molecule excited state in the inner well. In a word, we believe that the double dark resonance phenomena can be experimentally observed within the current technology. Acknowledgments WL is supported by the NSF of China (Nos. 10444002, 10674087, 10574084), 973 program (Nos. 2006CB921603, 2008CB317103), SRF for ROCS, SEM, SRF for ROCS, Ministry of Personal of China and SRF for ROCS of Shanxi Province. We gratefully thank Jie liu for the stimulating discussions.

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