Determination of the 5d6p ³F₄–5d^{2 3}F transition probabilities of Ba I

Abstract

Whether the transitions between 6s5d ³D and 5d6p ³F can be used for laser cooling of barium heavily depends upon the transition probabilities of 5d6p ³F–5d^{2 3}F. Taking the transition 6s5d ³D₃–5d6p ³F₄ as a scale, the leakage rate of 5d6p ³F₄–5d^{2 3}F was used to evaluate the transition probabilities. 706 nm laser pulses with different durations were applied to a barium atomic beam for 6s5d ³D₃–5d^{2 3}F₄ optical pumping, and the remaining percentages in 6s5d ³D₃ were measured. After exponential fitting, the transition probability of 5d6p ³F₄–5d^{2 3}F_3,4 was determined to be 2.1(4)×10⁴ s^-1, which is in agreement with theoretical calculations using the scaled Thomas–Fermi–Dirac method.

Graphical abstract

Keywords

laser cooling / leakage rate / transition probability / scaled Thomas–Fermi–Dirac method

1 Introduction

Laser cooling and trapping has already been achieved in many alkaline-earth neutral atoms, and the stability of optical frequency standard based on these trapped atoms (e.g. Sr) have reached a very high level [1]. However, because of its complicated energy structure, laser cooling and trapping of barium was not realized until 2008 [2]. In Willman’s group at the University of Groningen (Netherlands), three cooling and five trapping lasers of different frequencies were used. Since the 6s5d ³D states were estimated to have long lifetimes [3] and easily populated, the 6s5d ³D–5d6p ³F transitions will possibly be expected to be part of a simpler and more effective scheme for barium cooling and trapping. Also, the ultra-narrow linewidth of the 6s5d ³D₂–6s^{2 1}S₀ transition makes barium a promising candidate for a new active atomic clock.

Especially for the 6s5d ³D₃–5d6p ³F₄ transition, due to selection rules, the atoms in 5d6p ³F₄ cannot decay into other fine levels of 6s5d ³D states except ³D₃, and the 5d6p ³F₄–6s5d ¹D₂ transition is also forbidden, so the only theoretically possible leaking channel is 5d6p ³F₄–5d^{2 3}F_3,4. However, because the energy level spacing between 5d6p ³F and 5d^{2 3}F is very small, the transition probabilities were expected to be rather small. As can be estimated, if the branching ratio of 5d6p ³F₄–5d^{2 3}F_3,4 were less than 10^-5, the transition 6s5d ³D₃–5d6p ³F₄ could be considered to be quasi-closed, and laser cooling and trapping of barium could be realized by a single 706 nm laser. The transition probability of 6s5d ³D₃–5d6p ³F₄ was experimentally determined to be 5.0×10⁷ s^-1 [4]. However, at present, there is no experimental data available about the transition probabilities of 5d6p ³F–5d^{2 3}F.

It is difficult to measure the transition probabilities of 5d6p ³F–5d^{2 3}F directly because of the long wavelength and the weakness of the transition. Therefore, since 5d6p ³F₄–5d^{2 3}F_3,4 is the only leaking channel of the 6s5d ³D₃–5d6p ³F₄ transition, we carried out a measurement of the 5d6p ³F₄–5d^{2 3}F_3,4 transition indirectly by measuring the leakage rate of the 6s5d ³D₃–5d6p ³F₄ transition. We applied a counter-propagating 706 nm laser pulse to the atomic beam where the atoms had been populated to 6s5d ³D₃. The number of the 6s5d ³D₃–5d6p ³F₄ transition cycles was adjusted by varying the duration of the pulse, and the remaining percentages of the atoms in ³D₃ represented by the relative intensity of the fluorescence were finally recorded.

Before the experiment, the scaled Thomas–Fermi–Dirac (STFD) method [5] was used to theoretically calculate the transition probabilities of 5d6p ³F–5d^{2 3}F. In the following section, the results are listed and compared with our experimental data.

2 Experimental measurement

The low lying fine structure and transitions of barium about this work are shown in Fig. 1, and the schematic experimental setup is shown in Fig. 2. A barium beam was produced by a resistively heated oven at 1070(20) K. After leaving the oven, the atoms were perpendicularly excited by a 791.35 nm laser beam and a following 728.23 nm laser beam with the diameters 5 mm for both and the optical power 7 mW and 6 mW respectively. During the first interaction with the 791 nm laser, the atoms were excited to the 6s6p ³P₁ state and spontaneously decayed to longlife metastable states 6s5d ³D₁ and ³D₂. Subsequently, the atoms in 6s5d ³D₂ were optically pumped to the 6s5d ³D₃ state through the 728 nm stimulated transition. After two steps of optical pumping, about 40\% atoms had reached 6s5d ³D₃ state. Another counter-propagating pulsed 706.19 nm laser resonant with the 6s5d ³D₃–5d6p ³F₄ transition was switched by an acousto–optical modulator (AOM) and imposed on the atomic beam to drive the transition cycles. The laser was with the power 5 mW and the beam was slightly focused toward the oven orifice. At the end of the setup, to detect the number of the atoms in 6s5d ³D₃, a 650.06 nm probe laser, with the optical power 1 mW and the beam diameter 5 mm, crossed the atomic beam at a angle of 45^°. The atoms were excited to 5d6p ³D₃, and the fluorescence during the spontaneous radiation was collected and focused to a photomultiplier tube (PMT).

The time-varying fluorescence signals corresponding to 100 µs and 400 µs 706 nm laser pulses are shown in Fig. 3. as examples. The horizontal axis represents the time zeroed at the starting point of the pulse, the durations of the pulses are also shown. The vertical axis shows the intensity of the fluorescence in arbitrary units. As can be seen from the curves, the intensity of the fluorescence decreased over time, reached a minimum, and recovered. Because of the Doppler effect, the atoms excited by the counter-propagating laser had almost the same velocities, therefore the horizontal coordinates also represent the relative spatial position of atoms which interacted with the laser for different time spans. In this experiment, the minimum value of the curves decreased when the duration of the pulse increased. However, the minimum did not continue to decrease when the duration was longer than about 350 µs, and the minimum fluorescence intensity for 400 µs was equal to the level caused by the CW laser. Since the Doppler shift caused by the cooling effect in such a short time was much smaller than the natural line width, for durations longer than 400 µs or CW, all of the atoms resonant with the 706 nm laser were optically pumped to the 5d^{2 3}F states finally. The diameter of the 706 nm laser was a little smaller than that of the 650 nm probe laser and the atomic beam at the probing region, which made the intensity of the fluorescence stop decreasing to absolute zero. The power broadening of the 650 nm probe laser, which made the atoms with adjacent velocities also excited when probing, was another important reason. Accordingly, the CW background would not affect our measurement and data analysis.

According to the measurement of Garcia [4], the transition probability of 6s5d ³D₃–5d6p ³F₄ is 5.0×10⁷ s^-1. Given the intensity of the pumping laser, we can calculate the average number of transition cycles that the atoms experienced during the laser pulse,

\begin{array}{}\text{(1)}& n=\mathrm{\Delta }t{A}_{ik}\cdot {\displaystyle \frac{I/I_s}{1+I/I_s}}\end{array}

where \mathrm{\Delta }t is the duration of the laser pulse, A_{ik} is the transition probability between the lower and the upper energy levels, I is the intensity of the laser field and I_s is the saturation intensity. For the transition 6s5d ³D₃–5d6p ³F₄, A_{ik} is 5.0×10⁷ s^-1, I_s is 2.95 mW/cm², and the typical I of 706 nm laser is about 50 times of the I_s. We tried different durations of the laser pulse, and the percentage of the remaining atoms in 6s5d ³D₃ were recorded. Theoretically, the percentage has exponential relation to the cycling times, where the base of the exponent is the difference between 1 and the leakage rate. The data, plotted in Fig. 4. was fitted with the exponent function,

\begin{array}{}\text{(2)}& x/\%={(1-a)}^n\end{array}

where n is the number of transition cycles and a is the leakage rate to 5d^{2 3}F. Finally, we find a=4.1E-4 with the fitting error 1E-5 and the averaged random standard error 8E-5.

Apart from the random measuring error which mainly came from the drifting of the laser frequency and power and the fluctuation of the intensity of atomic beam, the simplification of the energy level structure model would also make systematic errors. Taking the lifetime of the lower state 6s5d ³D₃ into account, we found the corresponding uncertainty about a was 8E-10.

Since 5d6p ³F₄–5d^{2 3}F_3,4 is the only permitted leakage channel of the 6s5d ³D₃–5d6p ³F₄ transition, the leakage rate is the ratio between the transition probabilities of 5d6p ³F₄–5d^{2 3}F_3,4 and 5d6p ³F₄–6s5d ³D₃. Using Garcia’s data, the sum of the probabilities of 5d6p ³F₄–5d^{2 3}F₃ and 5d6p ³F₄–5d^{2 3}F₄ was determined to be 2.1(4)×10⁴ s^-1.

3 Calculation with scaled Thomas–Fermi–Dirac method

We also calculated the transition probabilities of all the seven lines between 6p5d ³F and 5d^{2 3}F states as reference. The line strength is

\begin{array}{}\text{(3)}& \begin{array}{cc}\mathcal{S}\hfill & =〈\left[(L_c,l)L,S\right]J\left|\left|P\right|\right|\left[(L_c,l^{\prime })L^{\prime },S\right]J^{\prime }〉\hfill \\ \hfill & =D_{line}^2{D}_{mult}^2{P}_{nl,n^{\prime }{l}^{\prime }}^2\hfill \end{array}\end{array}

in which

\begin{array}{}\text{(4)}& D_{\mathrm{line}}={(-1)}^{(L+S+J^{\prime }+1)}{[J,J^{\prime }]}^{1/2}\left\{\begin{array}{ccc}L& S& J\\ J^{\prime }& 1& L^{\prime }\end{array}\right\}\end{array}

where

\begin{array}{}\text{(5)}& [J,J^{\prime }]=(2J+1)(2J^{\prime }+1)\end{array}

And for the situation l^w-l^{w-1}{l}^{\prime }, the active atom whose n and l quantum number are identical with other atoms in the initial or final configurations.

\begin{array}{}\text{(6)}& \begin{array}{cc}{D}_{mult}=\hfill & \sqrt{w}〈l^n{v}_{c}SL\{|l^{n-1}{v}_c{S}_c{L}_c〉\hfill \\ \hfill & \cdot {(-1)}^{L_c+L+l^{\prime }+1}{[L,L^{\prime }]}^{1/2}\left\{\begin{array}{ccc}{l}_c& l& L\\ 1& L^{\prime }& l^{\prime }\end{array}\right\}\hfill \end{array}\end{array}

⟨l^n{v}_{c}SL\{|l^{n-1}{v}_c{S}_c{L}_c⟩ is the coefficient of fractional parentage [6], which has value 1 for the configuration of double electrons,

\begin{array}{}\text{(7)}& \begin{array}{cc}{P}_{nl,n^{\prime }{l}^{\prime }}\hfill & =⟨nl\Vert r\Vert n^{\prime }{l}^{\prime }⟩\hfill \\ \hfill & ={(-1)}^l{[l,l^{\prime }]}^{1/2}\left(\begin{array}{ccc}l& 1& l^{\prime }\\ 0& 0& 0\end{array}\right){\displaystyle {\int }_0^{\mathrm{\infty }}}{R}_{nl}r{R}_{n^{\prime }{l}^{\prime }}dr\hfill \end{array}\end{array}

The radial wave function R_{nl} was calculated by the scaled Thomas–Fermi–Dirac method [5]. In the radial Schrödinger equation,

\begin{array}{}\text{(8)}& \left[-{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}+{\displaystyle \frac{l(l+1)}{r^2}}+V(r)\right]R(r)=ER(r)\end{array}

V(r) was written by Latter [7] as an approximative conditional function without the radial wave function,

\begin{array}{}\text{(9)}& V(r)=\min (V_{{}_{\mathrm{TFD}}},-{\displaystyle \frac{2}{r}})\end{array}

\begin{array}{}\text{(10)}& V_{{}_{\mathrm{TFD}}}(r)=-\left[{\displaystyle \frac{Z\cdot \phi (r/\mu )}{r}}+{\displaystyle \frac{3\sqrt{2Z\cdot \phi (r/\mu )}}{4\pi \sqrt{r}}}\right]\end{array}

in which \mu =0.8853Z^{-1/3} and Z is the atomic number. \phi is the TF function, which was fitted by Latter as

\begin{array}{}\text{(11)}& \begin{array}{cc}\phi (\mu /r)\hfill & =[1+0.02747{(\mu /r)}^{1/2}+1.243(\mu /r)\hfill \\ \hfill & -\mathrm{\hspace{0.17em}0.1468}{(\mu /r)}^{3/2}+0.02747{(\mu /r)}^2\hfill \\ \hfill & +\mathrm{\hspace{0.17em}0.02747}{(\mu /r)}^{5/2}+0.02747{(\mu /r)}^3]{}^{-1}\hfill \end{array}\end{array}

Eq. (8) was solved by the numerical integration using the method of Numerov [8]. The result was plugged into Eq. (7), and finally the line strength \mathcal{S} could be evaluated. Using the relation between the transition probability A and \mathcal{S},

\begin{array}{}\text{(12)}& \begin{array}{ccc}{A}_{ki}\hfill & =\hfill & {\displaystyle \frac{1}{6}}{\left({\displaystyle \frac{e^2}{\mathrm{\hslash }c}}\right)}^4{\displaystyle \frac{c}{a_0{g}_k}}{\left({\displaystyle \frac{|T_i-T_k|}{R_{\mathrm{\infty }}}}\right)}^3{\displaystyle \frac{S}{e^2{a}_0^2}}\hfill \\ \hfill & =\hfill & {\displaystyle \frac{2.0261\times {10}^{18}}{g_k{\lambda }^3}}\hfill \end{array}\end{array}

the transition probabilities were determined. a₀ is the Bohr radius, T_i, T_k, g_k are the term values and the statistical weight of the upper and the lower levels, and R_{\mathrm{\infty }} is the Rydberg constant. The final results are listed in Table 1, along with the three transition lines previously calculated by Kulaga et al. [9] making use of the “Hartree–Fock with relativistic corrections (HFR) method”. The other four are completely new results. The transition probabilities of 5d6p ³F₄–5d^{2 3}F₃ and 5d6p ³F₄–5d^{2 3}F₄ were determined to be 4.8×10³ s^-1 and 4.5×10⁴ s^-1 respectively. The difference from the experimental data is probably due to configuration mixing.

4 Conclusions

Both experimental measurement and theoretical calculation were used to determine the transition probabilities of 5d6p ³F–5d^{2 3}F for barium. As far as we know, no experiment has been performed on this transition series and only 3 of 7 lines have been calculated. These new data are very important in searching for a more convenient and effective laser cooling method of barium atoms on metastable D state. According to these data, it should be a little difficult to achieve the laser cooling of barium directly with a single 706 nm laser. To overcome the leakage to the 5d^{2 3}F states, some repumping method should be found. Otherwise, another scheme choosing the state 5d6p ³F₂ as the upper level may be taken into account, because the transition probability from 5d6p ³F₂ to ³F–5d^{2 3}F is about one order lower than that of 5d6p ³F₄.

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