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we demonstrated a high sensitive trap-loss spectroscopy technique by modulating fluorescence of cold atoms in magneto-optical trap, which allow a direct spectroscopy detection of the rovibrational levels with a very weak transition probability. The low-lying vibrational spectroscopy of υ = 3~17 of Cs2 0g - pure long-range state have been observed with rotational structures, which are well resolved up to J = 8. The rotational constants are obtained by fitting experimental data to a nonrigid rotation model.
©2010 Optical Society of America
1. Introduction
Photoassociation of ultracold atoms has become a convenient and powerful technique to obtain high resolution molecular spectroscopy [1], which provides information on the molecular excited state levels which is essential for determination of the molecular parameters, and therefore for potential curves. For homonuclear alkali dimers, there has been especially interest in the pure long-range 0g - state [2], whose double-well structure offers an efficient formation rate of ultracold ground molecules in the photoassociation process. The spectral data of the 0g - state associated had been obtained for H2 [3], Li2 [4], Na2 [5], K2 [6], Rb2 [7–9] and Cs2 [10–12], from which long-range molecular coefficient Cn and scattering length had been derived. Furthermore, very recently the precise data of the rovibrational levels of Cs2 0g - state had been employed to achieve absolute frequency stabilization of a diode laser to atoms-molecules hyperfine transitions [13] and construct a two-color photoassociation system for sensitively testing the variation of the electron-to-proton mass ratio [14].
Photoassociation spectroscopy of Cs2 0g - pure long-range excited state had been obtained by Pillet and associates using ionization detection technique in 1999 [9]. Despite advantages of sensitivity and zero background, ionization detection unavoidably destructed the formated cold molecules, which is often necessary for the further study. Furthermore, ion spectroscopy can’t directly provide the information about photoassociation transition intensity due to the intervention of the extra ionization laser [15]. A feasible technique to directly detect the excited molecular state levels is trap-loss detection by monitoring the fluorescence yield from the trapped atoms. Pillet’s group had obtained trap-loss spectroscopy of Cs2 0g - state below the 6S1/2 + 6P3/2 dissociation limit (P3/2), and determined the vibrational states of υ = 25~80 [16]. Wester et al. also obtained partial trap-loss spectroscopy in an optical dipole trap [17]. Recently, the vibrational data of Cs2 0g -(P3/2) state had extended down to υ = 18 [11,18]. Based on these spectroscopic data, the long-range coefficient of 0g -(P3/2) state and Cs 6P3/2 atomic radiative lifetime had been determinated [19]. Howerve, trap-loss spectroscopy for more low-lying vibrational states (υ<18) were yet not observed due to the decreased photoassociation rate.
In this paper, we demonstrated a high sensitive trap-loss spectroscopy technique by modulating fluorescence of cold atoms in a magneto-optical trap, which allow a direct detection of the low-lying vibrational states of long-range molecular states. The vibrational states υ = 3~17 of Cs2 0g - pure long-range state have been observed with rotational structures, which are well resolved up to J = 8. And the rotational constants are obtained by fitting experimental data to a nonrigid rotation model.
2. Experiments
Our experiment setup is schematically shown in Fig. 1 . The sample of ~107 cold Cs atoms was produced in a conventional vapor cell magneto-optical trap (MOT). The temperature of sample was ~200 μK measured by the time of flight method. This trap consists of three pairs of mutually orthogonal laser beams with opposite σ+/− polarizations of intensity nearly 3mW/cm2. A magnetic field for MOT is generated by a pair of anti-Helmholtz coils with a typical gradient of 15 Gauss/cm along the axis of the coils. The trapping laser is a frequency-stabilized diode laser (DL100) of which the line-width is less than 1MHz by using a standard saturated absorption technology.
Fig. 1 Experimental setup. The repumping laser is not shown here. The trapping laser (DL100) frequency has been locked to about two natural line-widths red detuned from the 6S1/2 (F = 4)→6P3/2 (F’ = 5) atomic transition by saturation absorption spectroscopy. WLM: wavelength meter; OI: optical isolator; SP: shaping prism; L: lens; M: mirror; PC: personal computer.
The photoassociation is induced by a CW tunable Ti:sapphire laser system (MBR110) with a typical line-width of less than 100 kHz and output power up to ~600mW. The photoassociation laser is focused to a diameter of 500μm, resulting in available intensities up to ~300W/cm2. The absolute laser frequency was measured by a commercial wavelength meter (HighFinesse-Angstrom WS/7R) with an accuracy of 60MHz. Comparing with the absolute accuracy of 600MHz in Ref [10], the spectral accuracy was effectively improved. Thus it became possible to make the more accurate analysis for the spectral data. The relative frequency changes were measured by monitoring the transmission signal through a confocal Fabry-Perot cavity with a free spectral range of 750 MHz.
The fluorescence yield from the trapped cold atoms is collected by a convex lens and is detected by an avalanche photodiode (APD Hamamatsu S3884) with 850nm band-pass filters.
In order to improve the sensitivity of detection, we employed lock-in detection technique by modulating fluorescence of cold atoms. In our experiment, the trapping laser wavelength was modulated by a high precise function generator. For sine wavelength modulation [20,21], the instantaneous laser frequency is
where ν0 is the optimal trapping frequency, i.e. two natural line-widths red detuned from the 6S1/2 (F = 4)→6P3/2 (F’ = 5) Cs atomic transition, m is the laser wavelength modulation index, and f is the laser modulation frequency. The frequency variation results in the instantaneous fluorescence of the trapped atoms to be modulated with the same frequency [22]. The fluorescence of trapped atoms is modulated,Where I 0 is the fluorescence of trapped atoms without modulation and δI is the modulation amplitude of fluorescence dependent on modulation index m.The modulated fluorescence signal is demodulated using a lock-in amplifier (SR830) with an integration time of 300ms, the equivalent noise bandwidth is 31 mHz. In order to eliminate the background caused by modulation, the automatic phase tracking and low-pass filter is used in the process of demodulation.
Results
Figure 2 shows three typical spectrum for vibrational quantum number 17, 10 and 4, whose detunings comparing with 6P3/2 dissociation limit are ~51cm−1, ~61cm−1 and ~71cm−1, respectively. The maximum signal-to-noise ratio of spectral signal detected by the improved trap-loss spectroscopy technique is up to 26, while that is down to 3 by the traditional trap-loss spectroscopy technique in our experiment. The maximum loss ratio of signal is up to 36%. It is exciting that we observe rotational progression up to J = 8 in this way. Those large rotational levels can be rationalized due to additional angular momentum contributions, such as from orbital angular moment due to p-wave or d-wave [10,15].
Fig. 2 Rotationally resolved rap-loss spectra for υ = 17, 10 and 4 of Cs2 0g - pure long-range state. Rotational progressions are observed for each vibrational atate up to J = 8. The rotational structure becomes broader for lower values of υ.
All of observed spectra show similar intensity undulation, which is the result of the variation of the Franck-Condon factors for the photoassociation transitions between the initial state of the two cold free atoms and the final rovibrtional levels of the long-range states [23]. In our experiment, trap loss spectroscopy in the range of red detuning from 50cm−1 to 80cm−1 below the 6P3/2 dissociation limit have been measured carefully. By comparing the calculated vibrational energy spacings [16], the obtained spectral data can be assigned to low-lying vibrational progressions v = 3~17 of 0g - pure long-range state.
It is worth noting that some of the vibration (v = 5、7 and 9) spectra are not detected. The reason is mainly that the Franck-Condon factor of these levels is too small. The small transition probabilities in photoassociating from atoms to molecules in these levels results in the loss of atoms is very limited. By changing the demodulation parameters, such as increasing sensitivity, some rotational energy levels (such as J = 2 and 4) can be observed in our experiment.
The high resolution vibration spectra with the rich rotational data provide possible for us to study the molecular constants. The long-range interaction between two constituent atoms of the excited molecules formed by photoassociation is weak comparing with ordinary chemical bond, the nonrigid rotation model should be considered here [24]:
where E is rotational eigenvalues, h is Planck's constant, c is the speed of light, B is the rotational constant (unit cm−1), D is the centrifugal distortion constant (unit cm−1) [25] and J is the rotational quantum number. The effect of nonrigidity is to bring the rotational levels closer together for higher energy states. D is typically several orders of magnitude smaller than B.For the same vibrational quantum number, the interval ΔE of the neighboring rotational levels is,
The rotational constant B and centrifugal distortion constant D can be obtained by fitting experiment data into Eq. (4). Figure 3 shows the nice fits for υ = 4, 10 and 17. The clear nonlinearity of the fitting lines implies it is reasonable to consider centrifugal distortion in this model.Fig. 3 Dependence of the level intervals Δν on the rotational quantum number J for υ = 4, 10 and 17 of Cs2 0g - pure long-range state. The symbols represent experimental data, while the lines are fits to the nonlinear Eq. (4), as described in the text. The square of coefficient of correlation (R2) in the fit process is up to 0.999 for each υ.
The values of B and D for the different vibrational states are listed in Table 1 . The uncertainty is mainly due to a possible systematic error in the process of fitting and error in the determination of the resonant line position. For clearly exhibiting dependence, the rotational constant B is plotted as a function of vibrational quantum number υ in Fig. 4 . Our result shows the similar trend with ones obtained by ion spectroscopy in [10], which have not taken into consideration with the rotating centrifugal distortion. In order to comparing we also give the fitting results without D. Based on the front considerations of the centrifugal distortion, it is obvious that we get more accurate results than reference [10].
Table 1. The experimental energies and the molecular constants (B and D) of vibrational levels of the 0g-(P3/2) pure long-range state.
Fig. 4 Dependence of the rotational constant B on the low-lying vibrational quantum number υ of Cs2 0g - pure long-range state. The full dots and triangles represent results derived from this work, while the open circles represent that in reference [10].
As shown in Fig. 4, the rotational constant decreases gradually with increasing υ, which can be understood by considering that B is defined as B = h/8πIc, where I is a molecular constant called the moment of inertia. For a diatomic molecule, I = μr2, and μ and r are, in turn, the reduced mass and the internuclear separation of the molecule. Thus the rotational constant is a function of r, (i.e., B = h/8πcμr2), which increases with υ for molecules of 0g - state, while other terms keeps invariable. As a result, the rotational constant decreases with increasing υ, and reach ultimately zero at the range of dissociation limit.
Although the centrifugal distortion constants have been derived here, dependence of centrifugal distortion constants on the vibrational quantum number have been not observed clearly due to the limitation of accuracy.
Conclusion
We have reported a direct measurement of the low-lying rovibrational levels of the Cs2 0g -(P3/2) pure long-range excited state by modulating trapping laser wavelength. The range of trap-loss detection is extended to the red detuning of ~72cm−1 below the 6S1/2 + 6P3/2 dissociation limit, and 12 low-lying vibrational states of υ = 3~17 are observed. Rotational spectra are observed for each vibrational state, and the maximum resolved rotational progressions is up to J = 8. Rotational constants for the different vibrational states are derived, and dependence of rotational constant on vibrational quantum number is also studied.
Acknowledgement
We thank Dr. D. Comparat (LAC, Orsay, France) for helpful discussions about experiment. The work was supported by the 973 program (Grant No. 2006CB921603 and 2008CB31710), the 863 program(2009AA01Z319), the National Natural Science Foundation of China (Grant No. 10934004, 60978018, 60978001, 60808009 and 10674086), NSFC Project for Excellent Research Team (Grant No.60821004), and Shanxi natural foundation (Grant No. 2009011059-2).
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