1. Introduction

Recently, there have been some extensive studies in the production and manipulation of the ultracold molecules, which have shown significantly promising applications in areas ranging from precision mmeasurement to quantum information storage and processing [1–3]. Photoassociation (PA) of the ultracold atoms has been established as a versatile tool in the formation of the ultracold molecules [4]. The PA process is dependent on the collisional interaction properties. On the other hand, PA spectroscopy (PAS) delivers the detail knowledge of the structure of the formed molecules. Specially, PAS provides a unique opportunity to obtain information on the molecular long-range state. In particular, PAS has been used vastly to perform precise measurements of the high-lying vibrational levels that are essential to determine the molecular parameters near the dissociation limits and often difficult to access by the traditional bound-bound molecular spectroscopy.

Numerous experimental studies have been reported about the ultracold Cesium molecule (Cs₂) formed by PA, for instance, resonant four-body interaction in cold Cesium Rydberg atoms [5], absolute frequency stabilization to Cesium atom molecular hyperfine transitions [6], and precise measurement of Cesium atomic triplet state scattering length [7]. As PA spectra deeply covers abundant rotational data, reasonable investigations are required to precisely determine the rotational constants of the long-range molecules, which are crucial in predicting the unexplored levels and obtaining the precise potential energy curve [8]. Recently, some efforts have been made to obtain the rotational constants of the low-lying vibrational levels in the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state [9]. However, the rotational constants of the high-lying vibrational levels, which are important to precisely define the near-dissociation part of the potential energy curves, are difficult to determine by conventional direct measuring method, i.e. using a high resolution wavelength meter. The small frequency intervals between neighbouring rotational levels preclude the accurate measurements. An available method is to employ a Fabry-Perot reference cavity to monitor the changes of the relative frequency. However, the sensitivity is not sufficiently high. In particular, the measuring accuracy of a marker interferometer is greatly suppressed by the low frequency 1/f noise which results from the long scanning time in PA experiments. An active locking could be carried out to remedy the noise for some situations (with sub-MHz accuracy), nevertheless, powerful calibration schemes with quick and expedient implement still need to be widely researched.

In this paper, we develop a robust technique, namely “double PA spectroscopy”, for frequency measurement with high sensitivity and demonstrate the experimental determination of the rotational constants for the high-lying vibrational levels in the molecular long-range state. The rotational constants of the vibrational levels [v_D] – v = 15 ~42 in the Cs₂ 0_g⁻ long-range state below the (6S_1/2 + 6P_1/2) limit have been measured.

2. Experimental section

The experimental setup is schematically depicted in Fig. 1(a), basically consists of three parts: laser system, double pass and PA system. PA is induced by a cw widely tunable Ti: Sapphire laser (Coherent MBR-110, ~1 W, line-width ~0.1 MHz) driven by a 10W Verdi pump laser. The long-time frequency drift of Ti: Sapphire laser system is less than 500 kHz by locking to its self-reference cavity. The absolute laser frequency measurement is performed and monitored by a wavelength meter (HighFinesse-Angstrom WS/7R, absolute accuracy ~60 MHz). The wavelength meter is calibrated against the Cesium atomic hyperfine resonance transition, 6S_1/2 (F = 4) → 6P_3/2 (F’ = 5), corresponding to wavenumber of 11732.176 cm⁻¹.

 

Fig. 1 (a). Experimental setup. M: high reflective mirror; OI: optical isolator; H: half wave plate; PBS: polarization beam splitter; L: lens; S: Shutter; Q: quarter wave plate; AOM: acousto-optic modulator; BT: beam trap; BPF: band pass filter; APD: Avalanche photodiode; PD: photodiode; CCD: Charge-coupled Device. Figure 1 (b). Schematic diagram of the double PA spectroscopy. The dashed inset demonstrates the sequence diagram. t_s, ~10 ms, represents the response time in which the laser beam can be turned on/off completely.

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Ultracold Cesium atoms are produced in a typical vapour-loaded MOT [10]. The temperature is measured to be about 120 μK by time-of-flight method. Atomic number is measured to be about 5 × 10⁷ using absorption method, and the dimension (FWHM) of the atomic cloud is ~800 µm, yielding a peak density on the order of 10¹¹ cm⁻³. Both the trapping laser and the repumping laser are frequency stabilized diode laser systems (Toptica DL100, ~160 mW with line-width 0.8 MHz).

Formation of the ultracold molecules by PA and the double PA Spectroscopy are schematically demonstrated in Fig. 1(b). Initially an excitation from a free scattering state of two atoms to a bound excited molecular level is driven by resonant laser with frequency ν. This excited molecule spontaneously decays into ground state. PAS directly illustrates the number of produced excited molecules as a function of ν. PA spectra below the dissociation limit (6S_1/2 + 6P_1/2) are recorded by monitoring the atomic fluorescence as PA laser scans. Modulation technology is implemented to improve the signal-to-noise ratio of PA spectra [10].

The double PA spectroscopy directly offers a precise reference of the frequency difference and enables measuring small frequency interval of the adjacent rotational levels for high-lying vibrational states. As shown in the dashed inset of Fig. 1(b), the double PA spectrum is obtained through alternately illuminating the cold atomic sample by two PA laser beams (I and II). Partial output from the PA laser is frequency shifted by a double-passed acousto-optic modulator (AOM, MT110-B50A1-IR, AA Optoelectronic) to provide a calibration beam with an fixed offset of Δ₀ = ν₁ – ν₂ = 220 MHz. The modulated laser beam (Beam II) is diffracted from the double pass setup and well overlapped in space with the main output of the PA Laser (Beam I). Beam I at first induces trap loss of the cold atoms and a ro-vibrational spectrum including rotational levels from J = 0 to J = n is acquired. Later, Beam I is quickly switched off while simultaneously Beam II interacts with the atoms in turn. A selected spectral feature containing rotational level J = n is thus scanned twice (denoted as J’ = n). A series of the double PA spectra are therefore recorded. As PA laser scanning with stringent linearity and unvaried speed, frequency interval between the resonance peaks J = n and J’ = n exactly equals to Δ₀ = 220 MHz, i.e. the fixed frequency difference between Beam I and Beam II. This known precise separation of Δ₀ is used to calibrate the double PA spectra, frequency intervals Δυ_J (J = 0, 1, …, n-1) of the neighboring rotational levels are accurately deduced. It should be noted that the intensities of Beam I and II are set identical in order to avoid effects of laser induced frequency shifts [11]. The continuity and linearity of the frequency scans are assured by monitoring the PA laser frequency real time with a wavelength meter (WS/7R). Ultimately, the rotational constants are obtained by fitting the data of Δυ_J to the rotor model [12].

3. Results and discussion

Trap-loss spectroscopy [13] conveniently maps out the binding energies of the vibrational levels. The vibrational spectroscopy of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state has been reported in the previous works based on the direct trap-loss detection [9]. However, the ro-vibrational spectrum of the high-lying vibrational levels (specially [v_D] – v < 23) with enhanced resolution has still remain to be achieved by direct trap-loss detection. Here, we employ a lock-in method based on modulating the fluorescence of the ultracold atoms in order to availably improve sensitivity of the trap-loss detection [14]. A typical PA ro-vibrational spectrum (red line), which specifies the high-lying vibrational level ([v_D] – v = 17) of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state, is acquired as the atomic sample is illuminated only by Beam I with an intensity of 43.5 W/cm², as shown in Fig. 2(a). As the response time of the atoms against the PA Beam is typically long (~2s), the PA laser scanning speed should be small to enhance the resolution. The PA Laser is slowly scanned at a rate of 5.0 MHz/s. In addition, the response time (~10 ms) for switching the two shutters (S₁ and S₂) is shorter than the loading time (~1 s) of MOT, thus the resonance peaks are clearly demonstrated without affecting each other. The resolved rotational structure with rotational progressions J as high as 6 is therefore clearly observed. It should be noted that the frequency intervals Δυ_J for the neighbouring rotational progressions of the high-lying vibrational levels (such as [v_D] – v = 17) are rather small comparing to those for the low-lying levels. As the value of the fixed frequency offset can be set through the AOM, whose accuracy is on the order of kHz, the slight Δυ_J can be directly measured using this precise reference,.

 

Fig. 2 (a) PA spectrum (red curve) vs. double PA spectrum (green curve) for the high-lying vibrational level ([v_D] – v = 17) of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state. The PA spectrum is directly inverted in order to make curves comparable. PA laser frequency is tuned near 11174.91cm⁻¹. (b) Dependence of the frequency intervals Δυ_J on J for [v_D] – v = 17 of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state. The circle symbols represent experimental data, while the lines are the fits to the nonlinear non-rigid and linear rigid rotor models.

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In order to obtain the rotational constants for the high-lying vibrational levels, we investigated the dependence of Δυ_J on J for high-lying vibrational level ([v_D] – v = 17) of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state, as shown in Fig. 2(b). The non-linear relationship between Δυ and J is clearly exhibited. The main reason for the non-linearity is that the long-range interaction between the two constituent atoms of the long-range molecules is rather weak comparing with the ordinary chemical bond. The red curve in Fig. 2(b) is the fit plotted in accordance with a non-rigid rotor model [12]. The rotational frequency interval is described as Δυ_J = 2B(J + 1)-4D(J + 1)³, where B is the rotational constant and D is the centrifugal distortion constant. In order to test the appropriateness of this nonlinear model, the fitting results (blue curve) using rigid rotor model are also given. The adjusted R² for the non-rigid and rigid fittings are 0.99162 and 0.97249, respectively. Clearly, the non-rigid rotor model is more precise than the rigid rotor model to obtain accurate rotational constants B_{v = 17} = 25.73 ± 3.80 MHz and centrifugal distortion constants D_{v = 17} = 0.07 ± 0.02 MHz.

Figure 3 shows six typical double PA spectra for vibrational levels ([v_D] – v = 15, 20, 22, 29, 39 and 41) of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state, whose detunings are ~2.24 cm⁻¹, ~6.04 cm⁻¹, ~7.88 cm⁻¹, ~19.64 cm⁻¹, ~41.76 cm⁻¹, and ~47.45 cm⁻¹ below the dissociation limit 6S_1/2 + 6P_1/2 (~11178.151 cm⁻¹), respectively. The resolved rotational structure (J = 0 ~5) is clearly observed for all vibrational levels ([v_D] – v = 15 ~43). In fact, some spectra are rich in rotational structures (J = 0 ~7). Those large rotational levels can be rationalized due to additional angular momentum contributions, such as from orbital angular momentum due to p-wave or d-wave. All of the observed spectra demonstrate similar intensity undulation, which is the result of the variation of the Franck-Condon factors for the photoassociation transitions from initial scattering state of two free cold atoms to the final ro-vibrtional levels of the long-range states.

 

Fig. 3 Six typical double PA spectra with the reference of frequency difference are provided for typical vibrational levels of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state. The intensity of the PA laser beams (I and II) is 43.5 W/cm⁻² and keeps unchanged. The Ti:sapphire laser is slowly scanned at a rate of 5.0 MHz/s. The colorful curves are the multipeak Lorentzian fitting results.

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For the higher-lying vibrational levels ([v_D] – v < 15) of the Cs₂ 0_g⁻ long-range state which are very closely located near the dissociation limit, the hyperfine structures and the perturbation effects [15] play a devastating role that adversely hindered us from clearly observing and distinguishing the ro-vibrational structure in the PA spectra. Thus, here we focus on the vibrational levels ([v_D] – v = 15 ~42), specially the high-lying vibrational levels ([v_D] – v = 15 ~22) which have not fully investigated in the previous experimental reports using traditional trap loss detection [9]. In current research, the double PA spectra for [v_D] – v = 15 ~22 of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state with explicitly resolved rotational structure have been directly obtained based on the high-sensitive modulation spectral technique, which facilitates reaching improved signal-to-noise ratio (SNR) as high as 21.5 (for [v_D] – v = 29). By fitting the data of Δυ_J to the non-rigid rotor model, all the values of rotational constants B and centrifugal distortion constants D for [v_D] – v = 15 ~22 of Cs₂ 0_g⁻ long-range state are listed in Table 1.

Table 1. Rotational Constants and Centrifugal Distortion Constants of Vibrational Levels ([v_D] – v = 15 ~22)

View Table

Figure 4 depicts the rotational constants for the vibrational levels ([v_D] – v = 15 ~43) of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state as a function of the vibrational quantum number [v_D] – v. In particular, the rotational constants for the high-lying vibrational levels ([v_D] – v = 15 ~22) are illustrated in the blue region as a “new extended region” comparing to the previous experimental report [9]. Here, the uncertainty is mainly due to a possible statistical error in the process of multi-peak fitting (~2 – 3 MHz) as well as the error in the determination of the resonant peak position (~1 MHz). It is valuable to find that the rotational constants decrease gradually as [v_D] – v increasing in the case of high-lying vibrational levels and reach almost zero at the range of the dissociation limit. This can be understood by considering the molecular inter-nuclear separation as described in Ref. 16. In addition, the rotational constants of the high-lying vibrational levels ([v_D] – v = 15 ~22) are consistent with those of the low-lying vibrational levels. The blue curve is the linear regression result. The adjusted R² of fitting is 0.982 and the regression coefficient is 2.67 MHz. For comparison we also present in Fig. 4 the previous experimental values [9] of the rotational constants for the different vibrational levels ([v_D] – v = 23 ~42). The current experimental results demonstrate a similar trend with the previous values.

 

Fig. 4 Dependence of rotational constants B (blue balls) on vibrational quantum number [v_D] –v of the Cs₂ 0_g⁻ (6S_1/2 + 6P_1/2) long-range state. The red circles represent the experimental values of the rotational constants for different vibrational states listed in Ref. 9.

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4. Conclusions

In conclusion, a double PA spectroscopy technique has been proposed that provides a precise frequency reference to experimentally determine the rotational constants of the high-lying vibrational levels of the molecular long-range state. A linear dependence of the rotational constants on the vibrational quantum number is found. The proposed technique allows one to reach all the data of the available experimental ro-vibrational levels below the dissociation limit. Nevertheless, at the present stage of the experiment, the interpretation of the results remains limited, due to not considering the perturbation effect that stems from the mixing with the triplet sigma ungerade levels. Our scheme has several advantages. First, the rotational constants of the high-lying vibrational levels of Cs₂ long-range states have been directly measured for the first time. Furthermore, a high quality cavity that necessitates elaborate controls and additional expenditure is not required. Finally, the simple and robust scheme can be adapted to investigate a number of other long-range molecular species.