## 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} (6*S*,6*P*
_{3/2}) 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], 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], Doppler-free 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 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]. 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} (6*S*+6*P*
_{3/2}), separated by a potential barrier at distance *R*≈15*a*
_{0}. When by PA of two cold cesium atoms an excited level of the outer well (*R*>15*a*
_{0}) is populated, tunnelling is suggested as an efficient mechanism for transferring the population to the inner well (*R*<15*a*
_{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}* (6

*s*+6

*s*) electronic state. Further dynamical information can be extracted from the corresponding characteristic times for the vibration dynamics,

*T*

*(*

_{vib}*Ev*)= 2

*πh̄*/(

*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

*, while the relative larger level spacing (around 9.5 cm*

_{ps}^{-1}) in inner well results in a small characteristic time 3.5

*. Meanwhile, the “beating time” (*

_{ps}*T*~400–450

^{int,ext}*), determined by the tunnelling between the outer well and inner well, has been reported.*

_{ps}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} (6*S*+6*P*
_{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}* (6

*s*+6

*s*) 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*~300

*µK*, has been experimentally observed in 1998 [19]. This efficient scheme is attributed to the double-well structures in the excited 0-

_{g}(6

*S*+6

*P*

_{3/2}) potential curves, that provides an efficient mechanism (tunnelling) for transferring the population to the inner well (

*R*<15

*a*

_{0}), where spontaneous emission may lead to formation of cold molecules in low vibrational levels of the electronic state

*a*

^{3}∑

^{+}

*(6*

_{u}*s*,6

*s*) [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}∑

^{+}

*(6*

_{u}*s*,6

*s*) 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 states of 0-_{g} (6*S*+6*P*
_{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 Ω

_{1}, by which the excited molecules are formed in a precise vibrational level of the outer well in 0-

_{g}(6

*S*+6

*P*

_{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}(6

*S*+6

*P*

_{3/2}) [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 |

*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])

where

$${H}_{I}=-\frac{\overline{h}}{2}({\Omega}_{1}{b}_{1}^{\u2020}\mathit{aa}+H.c.)-\frac{\overline{h}}{2}({\Omega}_{2}{g}^{\u2020}{b}_{2}+H.c.),$$

$${H}_{t}=\overline{h}{\sigma}_{12}\left({b}_{2}^{\u2020}{b}_{1}+{b}_{1}^{\u2020}{b}_{2}\right).$$

The operators *a* and *a*
^{†} are annihilation and creation operators for atoms in state |*a*〉, similarly for *b*
_{1,2}(*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 |*b*
_{2}〉→|*g*〉 transitions are dipole allowed [29], while the transitions |*b*
_{1}〉→|*b*
_{2}〉 is provided by tunnelling mechanism. When the tunnelling coupling between the two excited molecular states is so strong that these two states are melted 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 **$\widehat{\rho}$** where the density matrix element (*$\widehat{\rho}$ _{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

where *γ _{µν}*(

*µ*≠

*ν*) is the transverse decay rate from state

*µ*to

*ν*(

*µ*,

*ν*=

*a*,

*b*

_{1},

*b*

_{2},

*g*) and is defined as

*γ*≡(

_{µν}*γ*+

_{µ}*γ*)/2. In the absence of molecular decaying, i.e.,

_{ν}*γb*

_{1},

*b*

_{2},g

*=*0, the total particle number is conserved and

*ρ*+2(

_{aa}*ρb*

_{1}

*b*

_{1}+

*ρb*

_{2}

*b*

_{2}+

*ρ*)= 1, where

_{gg}*ρ*(

_{µµ}*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]. Substituting the Hamiltonian (1) into Eq. (3) and taking the mean values for the density matrix operator, *ρ _{µν}*=〈

*$\widehat{\rho}$*〉, we have found the equations of motion for the density matrix elements

_{µν}*ρb*

_{1}

*a*,

*ρb*

_{2}

*a*and

*ρ*

_{ga}$$\frac{\partial}{\partial t}{\rho}_{{b}_{2}a}=\left[-i\left(\delta +\Delta \right)-{\gamma}_{{\mathrm{ab}}_{2}}\right]{\rho}_{{b}_{2}a}-2i{\Omega}_{1}{\rho}_{{b}_{2}{b}_{1}}{\rho}_{\mathrm{aa}}+i\frac{{\Omega}_{2}}{2}{\rho}_{\mathrm{ga}}-i{\sigma}_{12}{\rho}_{{b}_{1}a},$$

$$\frac{\partial}{\partial t}{\rho}_{\mathrm{ga}}=\left(-i\delta -{\gamma}_{\mathrm{ag}}\right){\rho}_{\mathrm{ga}}+i\frac{{\Omega}_{2}}{2}{\rho}_{{b}_{2}a}-2i{\Omega}_{1}{\rho}_{g{b}_{1}}{\rho}_{\mathrm{aa}}.$$

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 (γ*ab*
_{1}~50*MHz*) 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

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*(*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.

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(*ρb*
_{1}
*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 400~450*ps* in [23]. If we take the decay rate of the outer excited molecular levels (*γ ab*
_{1}) 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 0-50*γ ab*
_{1}. Considering the weak probe filed condition, we take Ω_{1}=0.01*γab*
_{1}, 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 ab*
_{1}=0 are emerging at a weak tunnelling coupling σ_{12}~0.25*γab*
_{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]. 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

Obviously, there are two zero probe absorptions for

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

Considering the conditions in Fig. 2, Ω_{2}=2*γab*
_{1}, we have
$\delta \u2044{\gamma}_{{\mathrm{ab}}_{1}}=\pm \sqrt{{\sigma}_{12}^{2}+1}$. These simple results are found in Fig. 2 for the weak tunnelling coupling (σ_{12}~0), *δ*/*γab*1~±1 (see Fig. 2(b) and (c)); while for strong tunnelling coupling, i.e. σ_{12}/*γab*
_{1}≫1, *δ*/*γab*
_{1}~±σ_{12} (see Fig. 2(d),(e),(f)).

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 Ω_{2} 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
$\sqrt{2(2{\sigma}_{12}^{2}+{\Omega}_{2}^{2}\u20442)}$.

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 ab*
_{1}). Imposing the same initial condition as the weak probe field approximation case, *ρaa*(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}=0.1

*γab*

_{1}. 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*/*γab*
_{1} in Fig. 6. Compared with the results in [18], 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.

## 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}* (6

*s*,6

*s*). The effective four-level structure is composed due to the double-well shape of the 0-

_{g}(6

*S*+6

*P*

_{3/2}), separated by a potential barrier at the distance

*R*≈15

*a*

_{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 (4.8

*GHz*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 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], 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

*a*

^{3}∑

^{+}

*(6*

_{u}*s*+6

*s*) 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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