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Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15870–15879
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The double dark resonance in a cold gas of Cs atoms and molecules

ZhiFang Feng, WeiDong Li, LianTuan Xiao, and SuoTang Jia  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15870-15879 (2008)
http://dx.doi.org/10.1364/OE.16.015870


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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

1. Introduction

“Double dark” resonance [1

1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

], 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

2. S. Jin, S. Gong, R. Li, and Z. Xu, “Coherent population transfer and superposition of atomic states via stimulated Raman adiabatic passage using an excited-doublet four-level atom,” Phys. Rev. A 69, 023408 (2004). [CrossRef]

, 3

3. Y. Niu, S. Gong, R. Li, and S. Jin, “Creation of atomic coherent superposition states via the technique of stimulated Raman adiabatic passage using a Λ-type system with a manifold of levels,” Phys. Rev. A 70, 023805 (2004). [CrossRef]

] and applied to nonlinear optics [4

4. S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003). [CrossRef]

, 5

5. B. K. Dutta and P. K. Mahapatra, “Nonlinear optical effects in a doubly driven four-level atom,” Phys. Scr. 75, 345–353 (2007). [CrossRef]

], Doppler-free resonance [6

6. C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A 65, 043805 (2002). [CrossRef]

, 7

7. E. S. Fry, M. D. Lukin, T. Walther, and G. R. Welch, “Four-level atomic coherence and cw VUV lasers,” Opt. Commun. 179, 499–504 (2000). [CrossRef]

], high efficiency four-wave mixing [8

8. W. F. Yang, S. Q. Gong, Y. P. Niu, S. Q. Jin, and Z. Z. Xu, “Enhancement of four-wave mixing induced by interacting dark resonances,” J. Phys. B: At. Mol. Opt. Phys. 38, 2657–2663 (2005). [CrossRef]

, 9

9. Y. P. Niu, R. X. Li, and S. Q. Gong, “High efficiency four-wave mixing induced by double-dark resonances in a five-level tripod system,” Phys. Rev. A 71, 043819 (2005). [CrossRef]

, 10

10. H. J. Li and G. X. Huang, “Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime,” Phys. Rev. A 76, 043809 (2007). [CrossRef]

] and group velocity controlling [11

11. E. Paspalakis and P. L. Knight, “Transparency, slow light and enhanced nonlinear optics in a four-level scheme,” J. Opt. B: Quantum and Semiclass Opt. 4, S372–S375 (2002). [CrossRef]

, 12

12. M. Mahmoudi, R. Fleischhaker, M. Sahrai, and J. Evers, “Group velocity control in the ultraviolet domain via interacting dark-state resonances,” J. Phys. B: At. Mol. Opt. Phys. 41, 025504 (2008). [CrossRef]

].

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 Boson-Einstein 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

13. M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat , and P. Pillet, “Optical Pumping and Vibrational Cooling of Molecules,” Science 321, 232 (2008). [CrossRef] [PubMed]

]. The magnetic Feshbach resonance [14

14. H. Feshbach, E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, “Conditions for Bose-Einstein condensation in magnetically trapped atomic cesium,” Phys. Rev. A 46, R1167 (1992). [CrossRef]

, 15

15. E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, “Feshbach resonances in atomic Bose-Einstein condensates,” Phys. Rep. 315, 199–230 (1999). [CrossRef]

, 16

16. D. Kleppner, “Professor Feshbach and His Resonance,” Phys. Today 57, 12–14 (2004). [CrossRef]

], with high conversion efficiencies, is restricted to the creation of molecules in the higher ro-vibrational level, due to energy conservation [17

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). [CrossRef]

, 18

18. K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, J. H. Denschlag, G. R. Jin, C. K. Kim, and K. Nahm, “Electromagnetically induced transparency in an atom-molecule Bose-Einstein condensate,” condmat/0603094.

].

Since the first experimental realization of cold Cs 2 molecules through PA [19

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). [CrossRef]

], the detailed information of the spectroscopy of cold Cs 2 molecules have been explored in a series of works [20

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). [CrossRef]

, 21

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

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). [CrossRef]

, 23

23. M. Vatasescu and 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). [CrossRef]

]. 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+6P 3/2), 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 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)= 2πh̄/(Ev+1-Ev), termed as vibrational period in [23

23. M. Vatasescu and 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). [CrossRef]

]. 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” (Tint,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+6P 3/2) are coupled by the tunnelling mechanism, and the free atoms and the ground molecular states are two long-living 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

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). [CrossRef] [PubMed]

] 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

1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

, 25

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. A66, 015802 (2002). [CrossRef]

, 26

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). [CrossRef]

].

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

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). [CrossRef]

], 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~300µK, has been experimentally observed in 1998 [19

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). [CrossRef]

]. This efficient scheme is attributed to the double-well structures in the excited 0-g (6S+6P 3/2) potential curves, that provides an efficient mechanism (tunnelling) for transferring the population to the inner well (R<15a 0), where spontaneous emission may lead to formation of cold molecules in low vibrational levels of the electronic state a 3+ u (6s,6s) [22

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). [CrossRef]

, 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+ u (6s,6s) are coupled by a pump laser. Then we construct a STIRAP scheme [17

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). [CrossRef]

, 27

27. M. Mackie, R. Kowalski, and J. Javanainen, “Bose-Stimulated Raman Adiabatic Passage in Photoassociation,” Phys. Rev. Lett. 84, 3803–3806 (2000). [CrossRef] [PubMed]

] 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 states of 0-g (6S+6P 3/2) (denoted by |b 2,1〉) and the lower triplet state a 3+ 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 Ω1, by which the excited molecules are formed in a precise vibrational level of the outer well in 0-g (6S+6P 3/2) (denoted as |b 1〉) from the ultra-cold atoms |a〉. This process is reasonable by considering the relative larger energy spacing (around 4.8 GHz) lying between -2.98 cm -1 and -2.82 cm -1 in the outer well of 0-g (6S+6P 3/2) [23

23. M. Vatasescu and 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). [CrossRef]

], compared with the high resolution (around MHz) of laser spectroscopy in current experimental condition [28

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). [CrossRef]

]. The second laser field with Rabi frequencies Ω2, 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 σ12. As shown in [23

23. M. Vatasescu and 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). [CrossRef]

], the tunnelling coupling strength depends on the intensity of the PA laser and its detuning with |b 2〉. To simplify our analysis and to emphasize on the resonance tunnelling coupling between the two excited states |b 1〉 and |b 2〉, 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

29. M. O. Scully and M. S. Zubairy, Quantum Optics , (Cambridge University Press, Cambridge, (1997).

])

H=H0+HI+Ht,
(1)

where

H0=h¯(δ+Δ)b1b1+h¯(δ+Δ)b2b2+h¯δgg,
HI=h¯2(Ω1b1aa+H.c.)h¯2(Ω2gb2+H.c.),
Ht=h¯σ12(b2b1+b1b2).
(2)
Fig. 1. The level scheme. Δ and δ denote the detunings. The inner and outer molecular states |b 1,2〉 spontaneously decay with a rate γb 1,b 2 to levels outside this scheme. The molecular ground state |g〉 is contributed with a decay rate γg, which phenomenologically takes into account losses through inelastic collisions.

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 ρ^ where the density matrix element (ρ^µν) describes the probabilities of being in the µ states for µ=ν and the polarization for µν. And the corresponding master equations, governed by Hamiltonian (1), are

tρ̂μν=γμνρ̂μνih¯[ρ̂μν,H],
(3)

where γµν(µν) is the transverse decay rate from state µ to ν(µ,ν=a,b 1,b 2,g) and is defined as γµν≡(γµ+γν)/2. In the absence of molecular decaying, i.e., γb 1,b 2,g=0, the total particle number is conserved and ρaa+2(ρb 1 b 1+ρb 2 b 2 +ρgg)= 1, where ρµµ(µ=a,b 1,2) 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 ρb 1 a as the function of the probe laser detuning (δ), which performs the properties of the absorption spectrum [29

29. M. O. Scully and M. S. Zubairy, Quantum Optics , (Cambridge University Press, Cambridge, (1997).

]. Substituting the Hamiltonian (1) into Eq. (3) and taking the mean values for the density matrix operator, ρµν=〈ρ^µν〉, we have found the equations of motion for the density matrix elements ρb 1 a, ρb 2 a and ρga

tρb1a=[i(+Δ)γab1]ρb1aiσ12ρb2aiΩ12(4ρb1b1ρaaρaa2),
tρb2a=[i(δ+Δ)γab2]ρb2a2iΩ1ρb2b1ρaa+iΩ22ρgaiσ12ρb1a,
tρga=(iδγag)ρga+iΩ22ρb2a2iΩ1ρgb1ρaa.
(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., Ω1 is small enough (two order smaller (to be shown in Fig. 5) than the unit, which is taken as the spontaneous emission line-width (γab 1~50MHz) 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

29. M. O. Scully and M. S. Zubairy, Quantum Optics , (Cambridge University Press, Cambridge, (1997).

], we arrive at

ρb1a=iΩ1{Ω224+(iδ+γag)[i(Δ+δ)+γab2]}12Ω22[i(Δ+δ)+γab1]+2(iδ+γag)[σ122+(iΔ+iδ+γab1)(iΔ+iδ+γab2)].
(5)

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 Ω2 and the first order in probe Rabi frequencies Ω1 in our calculation.

3. The double dark resonance

The probe absorption spectrum can be obtained by investigating the behavior of the Im(ρb 1 a) when changing the probe detuning δ. Based on the Eq. (5) and weak probe field condition, we will study the feature of the probe absorption spectrum in a cold gas of Cs atom-molecule system.

Fig. 2. Probe absorption spectra for different tunnelling rates. The parameters are Ω1=0.01γab1, Ω2=2γab1, γab2=0.2γab1, γag=0 and Δ=0. (a) σ12=0, (b) σ12=0.25γab1, (c) σ12ab1, (d) σ12=10γab1, (e) σ12=20γab1, (f) σ12=40γab1

By increasing the tunnelling coupling strength, the typical double dark resonance spectra [1

1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

] are observed in the case of a large enough σ12. This is because the tunnelling coupling modifies our scheme from an effective two-level structure (σ12=0) to four-level structure. A pair of the absorption peaks around the resonance point δ/γ ab 1=0 are emerging at a weak tunnelling coupling σ12~0.25γab 1 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

1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

]. 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 rate γab 2=γag=0 and γab 1=1. Then from Eq. (5), we write the imaginary parts of ρb 1 a as

Im(ρb1a)=2Ω1[Ω224δ(Δ+δ)2][Ω2222δ(Δ+δ)]2+[Ω222(Δ+δ)+2δσ1222δ(Δ+δ)2]2.
(6)

Obviously, there are two zero probe absorptions for

δ=Δ±Δ2+Ω222,
(7)

and for Δ=0, two transparency points appear at detuning δ=±Ω2/2 (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

1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

]. On the other hand, the two absorption peaks are located at

δ=122(2σ122+Ω222)
(8)

Considering the conditions in Fig. 2, Ω2=2γab 1, we have δγab1=±σ122+1. These simple results are found in Fig. 2 for the weak tunnelling coupling (σ12~0), δ/γab1~±1 (see Fig. 2(b) and (c)); while for strong tunnelling coupling, i.e. σ12/γab 1≫1, δ/γab 1~±σ12 (see Fig. 2(d),(e),(f)).

Fig. 3. (Color online). Probe absorption spectra for different driving detuning, with Δ=0 (solid line), Δ=-γab 1 (dashed line), and Δ=-4γab 1 (dotted line) for Ω1=0.01γab 1, Ω2=2γab 1, γab 2=0.2γab 1, γag=0, σ12=γab 1.

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 |b 2〉 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 |b 2〉. 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 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 proves the prediction of Eq. (8), the distance between the two absorption peaks is 2(2σ122+Ω222).

Fig. 4. (Color online). Probe absorption spectra for different pump frequencies and tunneling rate, with Ω2=2γab 1 (solid lines); Ω2=5γab 1 (dashed lines); Ω2=10γab 1 (dot lines). other parameters are Ω1=0.01γab 1, Δ=0, γab 2=0.2γab 1, γag=0. (a) σ12=γab 1, (b) σ12=4γab 1.

Same as the dark resonance in the atom-molecule system [18

18. K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, J. H. Denschlag, G. R. Jin, C. K. Kim, and K. Nahm, “Electromagnetically induced transparency in an atom-molecule Bose-Einstein condensate,” condmat/0603094.

], 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/γab 1 in Fig. 6. Compared with the results in [18

18. K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, J. H. Denschlag, G. R. Jin, C. K. Kim, and K. Nahm, “Electromagnetically induced transparency in an atom-molecule Bose-Einstein condensate,” condmat/0603094.

], the single dark peak is split into two peaks around δ/γab 1=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.

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 Δ=0, γab 2=0.2γab 1, γag=0, σ12=γab 1, and (a) Ω1=0.01γab 1, Ω2=10γab 1, (b) Ω1=0.01γab 1, Ω2=2γab 1, (c) Ω1=0.1γab 1, Ω2=2γab 1.

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 a 3+ u (6s,6s). The effective four-level structure is composed due to the double-well shape of the 0-g (6S+6P 3/2), 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

23. M. Vatasescu and 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). [CrossRef]

]. Since the energy spacing (4.8GHz between two excited vibrational levels in the outer well [23

23. M. Vatasescu and 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). [CrossRef]

]) is much lager than the laser spectroscopy resolution (MHz [28

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). [CrossRef]

]), 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 symmetrical breaking of the absorption spectra and the adjustable transparency windows. The frequency difference between |b 1〉 and |b 2〉, 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

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). [CrossRef]

], and the relatively larger energy spacing in the inner well [23

23. M. Vatasescu and 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). [CrossRef]

] makes it easy to introduce the stimulated Raman laser, coupling the molecular ground state a 3+ 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.

Fig. 6. (Color online). Atom population as a function of probe detuning δ/γab 1 for different tunnelling rate, with σ12=0.5γab 1 (dashed line) and σ12=γab 1 (solid line). other parameters are Ω1=0.01γab 1, Ω2=2γab 1, γab 2=0.2γab 1,γag=0, Δ=0.

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.

References and links

1.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]

2.

S. Jin, S. Gong, R. Li, and Z. Xu, “Coherent population transfer and superposition of atomic states via stimulated Raman adiabatic passage using an excited-doublet four-level atom,” Phys. Rev. A 69, 023408 (2004). [CrossRef]

3.

Y. Niu, S. Gong, R. Li, and S. Jin, “Creation of atomic coherent superposition states via the technique of stimulated Raman adiabatic passage using a Λ-type system with a manifold of levels,” Phys. Rev. A 70, 023805 (2004). [CrossRef]

4.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003). [CrossRef]

5.

B. K. Dutta and P. K. Mahapatra, “Nonlinear optical effects in a doubly driven four-level atom,” Phys. Scr. 75, 345–353 (2007). [CrossRef]

6.

C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, “Unexpected Doppler-free resonance in generalized double dark states,” Phys. Rev. A 65, 043805 (2002). [CrossRef]

7.

E. S. Fry, M. D. Lukin, T. Walther, and G. R. Welch, “Four-level atomic coherence and cw VUV lasers,” Opt. Commun. 179, 499–504 (2000). [CrossRef]

8.

W. F. Yang, S. Q. Gong, Y. P. Niu, S. Q. Jin, and Z. Z. Xu, “Enhancement of four-wave mixing induced by interacting dark resonances,” J. Phys. B: At. Mol. Opt. Phys. 38, 2657–2663 (2005). [CrossRef]

9.

Y. P. Niu, R. X. Li, and S. Q. Gong, “High efficiency four-wave mixing induced by double-dark resonances in a five-level tripod system,” Phys. Rev. A 71, 043819 (2005). [CrossRef]

10.

H. J. Li and G. X. Huang, “Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime,” Phys. Rev. A 76, 043809 (2007). [CrossRef]

11.

E. Paspalakis and P. L. Knight, “Transparency, slow light and enhanced nonlinear optics in a four-level scheme,” J. Opt. B: Quantum and Semiclass Opt. 4, S372–S375 (2002). [CrossRef]

12.

M. Mahmoudi, R. Fleischhaker, M. Sahrai, and J. Evers, “Group velocity control in the ultraviolet domain via interacting dark-state resonances,” J. Phys. B: At. Mol. Opt. Phys. 41, 025504 (2008). [CrossRef]

13.

M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat , and P. Pillet, “Optical Pumping and Vibrational Cooling of Molecules,” Science 321, 232 (2008). [CrossRef] [PubMed]

14.

H. Feshbach, E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, “Conditions for Bose-Einstein condensation in magnetically trapped atomic cesium,” Phys. Rev. A 46, R1167 (1992). [CrossRef]

15.

E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, “Feshbach resonances in atomic Bose-Einstein condensates,” Phys. Rep. 315, 199–230 (1999). [CrossRef]

16.

D. Kleppner, “Professor Feshbach and His Resonance,” Phys. Today 57, 12–14 (2004). [CrossRef]

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). [CrossRef]

18.

K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, J. H. Denschlag, 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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

23.

M. Vatasescu and 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). [CrossRef]

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). [CrossRef] [PubMed]

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. A66, 015802 (2002). [CrossRef]

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). [CrossRef]

27.

M. Mackie, R. Kowalski, and J. Javanainen, “Bose-Stimulated Raman Adiabatic Passage in Photoassociation,” Phys. Rev. Lett. 84, 3803–3806 (2000). [CrossRef] [PubMed]

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). [CrossRef]

29.

M. O. Scully and M. S. Zubairy, Quantum Optics , (Cambridge University Press, Cambridge, (1997).

OCIS Codes
(020.2070) Atomic and molecular physics : Effects of collisions
(270.1670) Quantum optics : Coherent optical effects
(290.5910) Scattering : Scattering, stimulated Raman
(300.1030) Spectroscopy : Absorption

ToC Category:
Quantum Optics

History
Original Manuscript: July 7, 2008
Revised Manuscript: September 4, 2008
Manuscript Accepted: September 4, 2008
Published: September 22, 2008

Citation
ZhiFang Feng, WeiDong Li, LianTuan Xiao, and SuoTang Jia, "The double dark resonance in a cold gas of Cs atoms and molecules," Opt. Express 16, 15870-15879 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15870


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References

  1. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, "Quantum interference effects induced by interacting dark resonances," Phys. Rev. A 60, 3225-3228 (1999). [CrossRef]
  2. S. Jin, S. Gong, R. Li, and Z. Xu, "Coherent population transfer and superposition of atomic states via stimulated Raman adiabatic passage using an excited-doublet four-level atom," Phys. Rev. A 69, 023408 (2004). [CrossRef]
  3. Y. Niu, S. Gong, R. Li, and S. Jin, "Creation of atomic coherent superposition states via the technique of stimulated Raman adiabatic passage using a Λ-type system with a manifold of levels," Phys. Rev. A 70, 023805 (2004). [CrossRef]
  4. S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, "Nonlinear optics via double dark resonances," Phys. Rev. A 68, 063801 (2003). [CrossRef]
  5. B. K. Dutta and P. K. Mahapatra, "Nonlinear optical effects in a doubly driven four-level atom," Phys. Scr. 75,345-353 (2007). [CrossRef]
  6. C. Y. Ye, A. S. Zibrov, Y. V. Rostovtsev, and M. O. Scully, "Unexpected Doppler-free resonance in generalized double dark states," Phys. Rev. A 65, 043805 (2002). [CrossRef]
  7. E. S. Fry, M. D. Lukin, T. Walther and G. R. Welch, "Four-level atomic coherence and cw VUV lasers," Opt. Commun. 179, 499-504 (2000). [CrossRef]
  8. W. F. Yang, S. Q. Gong, Y. P. Niu, S. Q. Jin, and Z. Z. Xu, "Enhancement of four-wave mixing induced by interacting dark resonances," J. Phys. B: At. Mol. Opt. Phys. 38, 2657-2663 (2005). [CrossRef]
  9. Y. P. Niu, R. X. Li, and S. Q. Gong, "High efficiency four-wave mixing induced by double-dark resonances in a five-level tripod system," Phys. Rev. A 71, 043819 (2005). [CrossRef]
  10. H. J. Li and G. X. Huang, "Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime," Phys. Rev. A 76, 043809 (2007). [CrossRef]
  11. E. Paspalakis and P. L. Knight, "Transparency, slow light and enhanced nonlinear optics in a four-level scheme," J. Opt. B: Quantum and Semiclass Opt. 4, S372-S375 (2002). [CrossRef]
  12. M. Mahmoudi, R. Fleischhaker, M. Sahrai and J. Evers, "Group velocity control in the ultraviolet domain via interacting dark-state resonances," J. Phys. B: At. Mol. Opt. Phys. 41, 025504 (2008). [CrossRef]
  13. M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat and P. Pillet, "Optical Pumping and Vibrational Cooling of Molecules," Science 321, 232 (2008). [CrossRef] [PubMed]
  14. H. Feshbach, Theoretical Nuclear Physics (Wiley, New York, 1992); [CrossRef]
  15. E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, "Conditions for Bose-Einstein condensation in magnetically trapped atomic cesium," Phys. Rev. A 46, R1167 (1992). [CrossRef]
  16. E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, "Feshbach resonances in atomic Bose-Einstein condensates," Phys. Rep. 315, 199-230 (1999). [CrossRef]
  17. D. Kleppner, "Professor Feshbach and His Resonance," Phys. Today 57, 12-14 (2004). [CrossRef]
  18. 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).
  19. 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); [CrossRef]
  20. G. R. Jin, C. K. Kim, and K. Nahm, "Electromagnetically induced transparency in an atom-molecule Bose-Einstein condensate," condmat/ 0603094. [CrossRef]
  21. 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).
  22. 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). [CrossRef]
  23. 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). [CrossRef]
  24. 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). [CrossRef] [PubMed]
  25. 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). [CrossRef]
  26. 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). [CrossRef]
  27. 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); [CrossRef] [PubMed]
  28. "Electromagnetically induced transparency and controlled group velocity in a multilevel system,"Phys. Rev. A 66, 015802 (2002). [CrossRef]
  29. 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).
  30. M. Mackie, R. Kowalski, and J. Javanainen, "Bose-Stimulated Raman Adiabatic Passage in Photoassociation," Phys. Rev. Lett. 84, 3803-3806 (2000).
  31. 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).
  32. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, (1997).

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