† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant No. 2012CB821305) and the National Natural Science Foundation of China (Grant Nos. 91336211 and 11504094).
The dynamic polarizabilities of the 7s and 6d states of Ra+ are calculated using a relativistic core polarization potential method. The magic wavelengths of the 7s1/2−6d3/2,5/2 transitions are identified. Comparing to the common radio-frequency (RF) ion traps, using the laser field at the magic wavelength to trap the ion could suppress the frequency uncertainty caused by the micromotion of the ion, and would not affect the transition frequency measurements. The heating rates of the ion and the powers of the laser for the ion trapping are estimated, which would benefit the possible precision measurements based on all-optical trapped Ra+.
The singly charged radium ion Ra+ has prominent applications in the fields of the precision measurement of time and frequency as well as the test of the fundamental theory of physics.[1–8] Possessing two ultra-narrow forbidden electronic quadrupole transitions 7s1/2−6d3/2 and 7s1/2−6d5/2 in the optical regime, Ra+ is a promising candidate for the optical atomic clock.[3–5] The precision experiment has been executed for the study of the variation of the fine structure constant α by comparing 27Al+ and 199Hg+ clocks.[9] Having two optical clock transitions makes Ra+ a good alternative candidate for testing the variation of α.[5] On the other hand, as the heaviest alkaline earth metal ion, Ra+ is a good candidate for the study of the atomic parity-nonconservation (PNC). Theoretical works have shown that the 7s 2S1/2−6d 2D3/2 transition in Ra+ is sensitive to the PNC effects.[6–8]
To evaluate the accuracies of the potential optical clock, the dipole polarizability is essential for the determination of the energy level shifts due to the stray electromagnetic fields. Theoretically, the determination of the dipole polarizability depends on the electric dipole (E1) matrix elements, while the determination of PNC amplitudes also depends on E1 matrix elements.[1] In the case of Ra+, E1 matrix elements can be calculated using the modern atomic structure theory methods.[1,7,10] However, there are no experimental reports on the E1 matrix elements of Ra+.[6]
Nevertheless, the knowledge of the magic wavelengths of the atomic systems could be used to extract out with high precision the transition oscillator strengths,[11] which are connected to the transition matrix elements. The magic wavelength for a transition is the wavelength at which the ac Stark shift of the transition energy is zero.[12] It means that, for the upper and lower energy level of the transition concerned, the difference of the dynamic polarizabilities is zero. The transition frequency measurement at the magic wavelength does not rely on a precision determination of the intensity of the laser field. A notable application of the magic wavelengths is to the optical lattice clocks operated on the neutral atoms.[13] Recent experiments have demonstrated the ability to trap a single ion in the optical lattice.[14,15] Theoretical work pointed out that, compared to the radio-frequency ion traps, the experiments could benefit from the effectively micromotion-free trapping of the ion in an optical trap.[16] It implies that, by choosing a proper laser at the magic wavelength for the clock transition to build the optical trap, it is promising to establish the all-optical trapped ion clocks. Actually, the magic wavelengths for the 4s1/2−3d5/2 transition in Ca+ have been observed, and meanwhile the ratio of the matrix elements for the 4s1/2−4p1/2 and 4s1/2−4p3/2 has been determined.[17]
For these motivations, we calculated the dynamic polarizabilities of the 7s1/2, 6d3/2, and 6d5/2 states in Ra+ using the relativistic core polarization potential method. A brief introduction of the computational details will be given in the second section. The magic wavelengths for the 7s1/2−6d3/2 and 7s1/2−6d5/2 transitions are identified. The possibilities of taking advantage of the magic wavelengths to get the transition matrix elements, as well as trapping Ra+ using the lasers at the wavelengths, are discussed in the third section. Finally, a summary is given in the fourth section. Atomic units are used in this work.
In the frame of the relativistic core polarization potential method, Ra+ is simplified as a frozen core part and the valence electron part. The first step is the Dirac–Fock (DF) calculation of the frozen core part based on B-spline basis sets to obtain the core orbital functions, the details of which can be found in our previous work.[11,18] Then a semi-empirical core polarization potential is introduced to approximate the correlation interaction between the core and valence electron and with the functional form
![](cpb_25_9_093101/cpb160793ieqn1.gif)
![](cpb_25_9_093101/cpb160793ieqn2.gif)
In the present work, the basis set of 100 B-splines of k = 7 order which are defined on the finite cavity [0, 220 a.u.] is used. A fermi nuclear model is adopted. The potential parameters are shown in Table
![]() | Table 1. The core polarizabilities and cut-off parameters for Ra+ used in the present work. . |
In the calculations, the electron multipole transition operator should include a polarization correction. The modified transition operator is
![]() | Table 2. Comparison of the electric dipole reduced matrix elements of Ra+. RCCSD refers to the RCC method taking into account single and double excitations. RCCSD(T) refers to the RCC method taking into account single, double, and important triple excitations. . |
For an ion in a single-model laser field, the energy shift of a given atomic state i can be written as[21]
The dynamic dipole polarizability αi(ω) can be written as
![](cpb_25_9_093101/cpb160793ieqn5.gif)
![](cpb_25_9_093101/cpb160793ieqn6.gif)
![](cpb_25_9_093101/cpb160793ieqn7.gif)
![](cpb_25_9_093101/cpb160793ieqn8.gif)
The static polarizabilities of the 7s1/2, 6d3/2, and 6d5/2 states in Ra+ are presented in Table
![]() | Table 3. The static scalar and tensor polarizabilities of the 7s1/2, 6d3/2, and 5d5/2 states in Ra+. . |
The dynamic polarizabilities of the 7s1/2 and 6d3/2 states are shown in Fig.
![]() | Fig. 1. The dynamic polarizabilities of the 7s1/2 and 6d3/2 states in Ra+. The magic wavelengths are identified by arrows. |
![]() | Table 4. The contributions of the clock transition 7s1/2−6d3/2 at magic wavelengths within the photon energy range from 0.0 to 0.15 a.u. for Ra+. The numbers in parentheses are uncertainties in the digits calculated by assuming certain matrix elements have ±2% uncertainties. . |
It can be noted from Table
Figure
![]() | Fig. 2. The dynamic polarizabilities of the 7s1/2 and 6d5/2 states in Ra+. The magic wavelengths are identified by arrows. |
![]() | Table 5. The contributions of the clock transition 7s1/2−6d5/2 at magic wavelengths within the photon energy range from 0.0 to 0.15 a.u. for Ra+. The numbers in parentheses are uncertainties in the digits calculated by assuming certain matrix elements have ±2% uncertainties. . |
In our previous work on the dynamic polarizabilities of Ca+,[11] it was suggested that the magic wavelengths of the 4s1/2−3d5/2 transition near 395 nm were measured for the determination of the matrix elements of the 4s−4pJ transitions. This is because the contributions to the polarizabilities from 4s−4pJ transitions are at least two orders of magnitude higher than the contributions from other transitions. But, in the case of Ra+, the situation is different and more complicated than Ca+. By measuring two or three magic wavelengths, one might not deduce certain matrix elements with high precision.
Ra+ can be laser cooled by taking advantage of the 7s1/2−7p1/2 resonance. The theoretical predicted lifetime of the 7p1/2 state is 8.72 ns.[7] Then the Doppler cooling limit is about 0.4 mK. With the sideband cooling technology,[22] the temperature of the ion could be of the order of μK. Trapping a single ion Ra+ by using the laser at the magic wavelengths would benefit the precision measurement on the frequencies of the 7s1/2−6d3/2 and 7s1/2−6d5/2 transitions, since the effectively micromotion-free trapping of the ion in an optical trap would suppress the frequency uncertainty of the measurement, comparing to the common radio frequency Paul traps.[16,23] However, when the ion is trapped in the optical dipole trap, it would be heated by the scattering of the trap photon. The heating rate is associated with the laser intensity and the dynamic polarizability of the quantum state and has the form[24]
![]() | Table 6. The intensity required to build a 4 mK optical dipole trap and the heating rate using the laser at the magic wavelength. . |
It is noted from Table
The dynamic polarizabilities of the ground state 7s1/2 and the excited 6d3/2,5/2 states in Ra+ are calculated by the relativistic core polarization method, and the magic wavelengths of the 7s1/2−6d3/2 and 7s1/2−6d5/2 transitions are identified. Unlike the case of Ca+, the contributions to the polarizabilities from individual transitions are more complicated in Ra+. It might not deduce the transition matrix elements with high precision by the measurements of the magic wavelengths predicted in this work. The heating rates of Ra+ in the laser field at the magic wavelengths are estimated. The magic wavelength of the 7s1/2−6d5/2 transition near 1824 nm is preferable for the optical trapping of Ra+. Optical trapping of the ion is effectively micromotion-free and the precision measurement of the frequency would benefit from it.
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