Magic Wavelengths for Optical-Lattice Based Cs and Rb Active Clocks

: Active clocks could provide better stabilities during initial stages of measurements over passive clocks, in which stabilities become saturated only after long-term measurements. This unique feature of an active clock has led to search for suitable candidates to construct such clocks. The other challenging task of an atomic clock is to reduce its possible systematics. A major part of the optical lattice atomic clocks based on neutral atoms are reduced by trapping atoms at the magic wavelengths of the optical lattice lasers. Keeping this in mind, we ﬁnd the magic wavelengths between all possible hyperﬁne levels of the transitions in Rb and Cs atoms that were earlier considered to be suitable for making optical active clocks. To validate the results, we give the static dipole polarizabilities of Rb and Cs atoms using the electric dipole transition amplitudes that are used to evaluate the dynamic dipole polarizabilities and compare them with the available literature values.


Introduction
The future definition of second in the International System of Units (SI) requires the realization of ultraprecise, narrow and concise form of clocks. Active clocks are expected to achieve better short-term stabilities [1] against the available optical clocks when a hybrid clock system consisting of the proposed active clock and an appropriate passive clock are designed together. Such a hybrid system is expected to cater both the short-and long-term stabilities to an indigenous frequency standard. Therefore, active clocks can serve as conducive time keeping standard at the short-term time scale. The output frequency of an active clock is determined by an atomic transition in perturbation-free environment. Proposed more than a decade ago, active optical clocks thus offer a possible solution to the bottleneck of accomplishing high stabilities in clocks [2][3][4][5].
In an active clock, the atomic system itself acts as an oscillator and generates the radiation with the clock frequency, which is then simply received and converted into an output signal [6], whereas, in passive clocks, which work in good-cavity regime, a local oscillator which is generally a laser source whose output frequency is stabilized to the atomic signal, is attached to the atomic system. The generated frequency is monitored after the system is prepared in certain quantum state. Once the local oscillator is synchronized with the frequency of atomic transition, the frequency is measured and converted into time. The active optical clocks follow the basic principle of construction of an

Method of Evaluation
The shift in the nth energy level of an atom placed in an oscillating electric field with amplitude E is given by [30][31][32][33] where α F n n (ω) is the dynamic dipole polarizability for a hyperfine level F n of state n, and is given by Here, δE nk = E 0 n − E 0 k with E 0 i denoting the unperturbed energies of the corresponding states for i = n, k and ks are intermediate states to which transitions from state n are possible in accordance to the dipole selection rules. (p) kn = Ψ k |D|Ψ n is the E1 matrix element between the states |Ψ n and |Ψ k .
α J n n0 (ω) and α J n n2 (ω) in the above equations are the scalar and tensor components of atomic dipole polarizability of state with angular momentum J n and magnetic projection M J and are of the following forms and α J n n2 (ω) = 2 Here, | J n ||D||J k | are reduced matrix elements (reduced using Wigner-Eckart theorem) with J k being angular momentum of intermediate state k. The term in curly bracket refers to 6-j symbols. The total J-dependent dynamic polarizability for linearly polarized light is given as: For a suitable choice of the electric field polarization and level F n or J n , λ magic can be determined for different magnetic sublevels M F or M J in a transition. The differential ac Stark shift of a transition is the difference between the ac Stark shifts of the states involved in the transition and can be formulated as Here, '∆E' and '∆E ' represent the energy shift in two different states involved in the transition with K = F n or J n . Our main aim is to find the values of ω and λ at which δ(∆E)(ω) will be zero. Dipole polarizability of any atom with closed core and one electron in outermost shell can be estimated by evaluating the core, core-valence and valence correlation contributions. i.e., [27] where α (ω) are the core, core-valence and valence correlation contributions, respectively. The subscript '0' in α (c) 0 (ω) refers to contributions from the inner core orbitals without the valence orbital. Our valence contribution (α (val) n (ω)) to the polarizability is divided into two parts, Main and Tail, in which the first few dominant and the other less dominant transitions of Equations (6) and (7) are included, respectively. The Main term is calculated by using single-double (SD) all-order method, which is described in Refs. [34,35]. Briefly, in SD method, the wave function of the valence electron n can be represented as an expansion: where |Φ v is the lowest-order wave function of the state which can be obtained as Here, |Φ 0 is the Dirac-Hartree-Fock (DHF) wave function for the closed core and the terms a † i and a i are the creation and annihilation operators. The indices m and n represent the excited states. a, and b refer to the occupied states and index v designates the valence orbital. The terms ρ ma and ρ mv are ascribed as the single core and valence excitation coefficients, whereas ρ mnab and ρ mnva are the double core and valence excitation coefficients. The partial triple excitations (SDpT) are also included for obtaining SDpT matrix elements where triple excitations were expected to contribute significantly. In all-order SDpT approximation, an additional term (linear triple excitation term) is added to the calculated SD wave function and resulting wave function becomes We solve the all-order equations using a finite basis set consisting of single-particle states which are linear combinations of 70 B-splines set. The large and small components of Dirac wave function are defined on a nonlinear grid and are constrained to a large cavity of radius R = 220 a.u. The cavity radius is chosen in such a way that it can accommodate as many transitions as practically possible to reduce the uncertainty. The E1 matrix element for the transition between states |Ψ v and |Ψ w can be obtained as In the case of SD approximation, the resulting expression for the numerator of Equation (14) consists of the sum of the DHF matrix element z wv and twenty other terms, which are linear or quadratic functions of the excitation coefficients [36,37]. In the present work, only two terms have dominant contributions to the transition matrix elements, and they are given as: and Here,ρ mnab = ρ mnab − ρ nmab and z wv are lowest order matrix elements of dipole operator. There are obviously some missing correlations to this term. To estimate some of these omitted correlation corrections and assess the uncertainties associated with these matrix elements, we carried out the scaling of the single excitation coefficients. These missing correlations can be compensated by adjusting the single valence excitation coefficients ρ mv [38] to the known experimental value of valence correlation energy as These modified ρ mv coefficients can be utilized to recalculate the matrix elements. Here, δE expt v refers to the difference between the experimental energy [39] and lowest order DF energy and δE theory v is the correlation energy due to single double excitations. In SDpT approximation, this correlation energy is due to single, double and partial triple excitations. Thus, one needs to calculate the scaling differently for SD and SDpT correlations. These modified E1 matrix elements are referred to as SD sc and SDpT sc E1 matrix elements respectively. We utilize the value of ratio R = Z (c) /Z (a) to establish the recommended set of E1 matrix elements and their uncertainties. If R > 1, then SD sc are regarded as Z Final value, otherwise SD results are used as Z Final value. To evaluate uncertainties in the recommended values of E1 matrix elements, we take the maximum difference between Z Final recommended value and other three all order values of SD, SDpT, SD sc and SDpT sc . The tail and core contribution and α (cv) n (ω) are calculated by using DHF approximation. To improve the precision of results for polarizabilities, these E1 matrix elements are combined with experimental energies from the National Institute of Science and Technology Atomic Database (NIST AD) [39].

Results
Accurate determination of polarizability of a state plays a key role in predicting λ magic precisely. Using the E1 matrix elements of the transitions involving the low-lying states up to 8P, 8S and 8D calculated using the all-order SD method, we first evaluate the static polarizabilities of the 7S 1/2 and 6P 1/2,3/2 states of the Cs and 6S 1/2 and 5P 1/2,3/2 states of the Rb atoms and compare them with the previously available experimental and theoretical results in Tables 1 and 2, respectively. We explicitly list corresponding E1-matrix elements for the allowed transitions in both atoms. To provide the most precise set of known data for these transitions, we replace few all-order theoretical values of the E1 matrix elements by the experimental ones, where high-precision values are already available. Further, we use excitation energies from the measurements as listed in NIST database for further improvement of our results.

Static Polarizabilities
To demonstrate role of various contributions to α J n (0), we provide individual contributions from different E1 matrix elements to "Main" (α contributions explicitly along with the net results in the evaluation of static polarizabilities (ω = 0) in atomic units (a.u.) for the 7S 1/2 , 6P 1/2 and 6P 3/2 states of Cs atom in Table 1. As shown in Table 1, our calculated α J n n0 (0) value of 6257(41) a.u. for the 7S 1/2 state of the Cs atom is comparable with the α J n n0 (0) values of 6238(41) and 6140 a.u., which were calculated by Tchoukova et al. [30] and Wijngaarden et al. [40], respectively. Tchoukova et al. [30] employed the relativistic all-order method for calculating the results, whereas Wijngaarden et al. [40] used calculated oscillator strengths using the method of Bates and Damgaard [41] to evaluate the results for polarizabilities. We also found agreement of our calculated results with experimental results of α J n n0 (0) = 6238(6) measured by Bennett et al. using laser spectroscopy. The value of scalar polarizability α J n n0 (0) for the 6P 1/2 state is estimated to be 1339(18) a.u., which agrees with the results given by Arora et al. [25] and Wijngaarden et al. [40] as 1338 and 1290 a.u., respectively. This result also matches very well with experimental result of 1328.4(6) a.u. measured by Hunter et al. [42]. Both scalar and tensor contributions for the 6P 3/2 state along with the uncertainties are provided in the same table. The values for the static scalar and tensor polarizabilities of the 6P 3/2 state in Cs are obtained as 1648 (35) and −262(9) a.u., which are again in reasonable agreement with the values given by Arora et al. [25] and Wijngaarden et al. [40]. Further, our calculated results for static scalar and tensor polarizabilities of the 6P 3/2 state are very much comparable with the results measured by Tanner et al. [43] calculated by employing crossed beam laser spectroscopy.
In Table 2, we tabulate our calculated values for total static dipole polarizabilities for the states 6S 1/2 , 5P 1/2 and 5P 3/2 of the Rb atom. It is observed that out of all the considered transitions, contributions from 6S 1/2 − 6P 3/2 , 5P 1/2 − 4D 3/2 and 5P 3/2 − 4D 5/2 transitions are maximum. As can be seen in the table, the obtained α J n n0 (0) value 5124(59) a.u. of static polarizability for the 6S 1/2 state for Rb atom is in close agreement with the other theoretical calculation, which is 5110 a.u. by Wijngaarden [44] who employed Coulomb approximation to evaluate the results. The value of static scalar polarizability of 5P 1/2 is estimated to be 813(14) a.u., which is in agreement with the polarizability results calculated by Arora et al. [25] and Zhu et al. [45]. Zhu et al. estimated polarizability using many-body perturbation theory. Our calculated result of polarizability for 5P 1/2 state matches very well with the value of 810.6(6) measured by Miller et al. [46]. The calculated values for the static scalar and tensor polarizabilities of 5P 3/2 state in Rb are obtained as 875 (14) and -168(4) a.u. These values match very well with the values calculated by Arora et al. [25] and Zhu et al. [45]. The experimental value of scalar and tensor polarizabilities for the 5P 3/2 state of Rb atom was obtained by Krenn et al. [47] and are in accord with our calculations. There is a reasonable agreement between our calculations and the values reported by other theoretical calculations using a variety of many-body methods and experimental measurement, ascertaining that our static values of polarizabilities are reliable. Correspondingly, we expect that the dynamic polarizabilities evaluated in our calculations will also be accurate enough to determine λ magic for the 7S 1/2 − 6P 1/2,3/2 transitions in Cs and 6S 1/2 − 5P 1/2,3/2 transitions in Rb.  −261 [25] 6140 [40] 1290 [40] 1600 [40] −233 [40] Experiment 6238 (6) (7) 2.967 (7) 5P 1/2 − 7D 3/2 0.79 (6) 2.6(4)

Magic Wavelengths
Having evaluated the static dipole polarizability of Cs and Rb atoms, we adopt a similar procedure for calculation of dynamic polarizabilities of these atoms. We anticipate similar accuracies in the dynamic polarizability values as their corresponding static values except at wavelength values in close vicinity of any resonant transition for the considered S 1/2 or P 1/2,3/2 states We determine the λ magic values of the (m + 1)S 1/2 − mP 1/2 and (m + 1)S 1/2 − mP 3/2 transitions in Cs (m = 6) and Rb (m = 5) atoms using these dynamic polarizabilities. Before we proceed, we would like to clarify that λ magic for both (J n ,M J ) and (F n , M F ) levels of the considered transition for Cs and Rb have been calculated. Since α J n = α F n for S 1/2 and P 1/2 states, the magic wavelengths among (J n ,M J ) and (F n , M F ) levels for (m + 1)S 1/2 -mP 1/2 transition are the same and are shown by the intersection of α J polarizability curves for (m + 1)S 1/2 and nP 1/2 states of Cs and Rb atom in Figures 1 and 2, respectively. To acquire magic wavelength among the (J n ,M J ) levels of the (m + 1)S 1/2 → mP 3/2 transitions, we plot total dynamic dipolar polarizabilities for the (m + 1)S 1/2 and mP 1/2,3/2 states for Cs and Rb atoms in Figures 3 and 4, respectively. Similar plots for (F n , M F ) levels of (m + 1)S 1/2 → mP 3/2 transitions are shown in Figures 5 and 6 for Cs and Rb atom, respectively. The results for the (J n ,M J ) and (F n ,M F ) levels are tabulated separately in Tables 3 and 4 for Cs atom and Tables 5 and 6 for Rb atom for the ease of pulling out exact values of the magic wavelengths. The resonant wavelengths which are crucial in locating the magic wavelengths are also tabulated.    Table 3. Magic wavelengths (λ magic s) (in nm) with corresponding polarizabilities (α J (ω)s) (in a. u.) for the 7S 1/2 − 6P 1/2 and 7S 1/2 − 6P 3/2 transitions in the Cs atom with the linearly polarized lights along with the resonant wavelengths (λ res s) (in nm).  Table 4. λ magic s (in nm) and their corresponding polarizabilities (in a.u.) in Cs (I = 7/2) of the transitional states involved in 7S 1/2 | F n , M F → 6P 3/2 | F n , M F transition are listed.  −1 1130 (7) −47 1386 (2) −1320 1392.0(2) −3469

Cs Atom
As shown in Figure 1 and Table 3, we were able to locate nine λ magic s for the 7S 1/2 − 6P 1/2 transition in Cs within 900-1500 nm. Out of these, six magic wavelengths were located within 900-1100 nm and support blue detuned traps as indicated by negative polarizability values at these wavelengths. The other three located towards higher wavelength range offer red-detuned traps. One would have expected magic wavelengths within 1098-1378 nm where 7S 1/2 − 9P 1/2 , 7S 1/2 − 6P 1/2 and 7S 1/2 − 8P 3/2 resonant transitions occur; however, the opposite nature of 7S 1/2 and 6P 1/2 polarizabilities around 7S 1/2 − 6P 1/2 resonance at 1359 nm prohibits crossing of the 7S 1/2 and 6P 1/2 polarizability curves. On the other hand, in Table 3 and Figure 3, we identify seven λ magic s for the 7S 1/2 − 6P 3/2 (M J = ±1/2) transition placed between six different resonances all supporting blue detuned traps. The polarizability of the 6P 3/2 state has two important resonant transitions (6P 3/2 − 6D 3/2 and 6P 3/2 − 6D 5/2 ) in the wavelength range considered in this work. In contrast, the polarizability of the 7S 1/2 state can have significant contributions from several resonant transitions in the considered wavelength range. Thus, they are expected to cross with the polarizability of the 6P 3/2 states in between these resonant transitions. We find this trend in locating magic wavelengths in between two resonances, except for a few cases where the λ magic s are missing. We also find that three λ magic s were located between 7S 1/2 − 10P 1/2 and 7S 1/2 − 9P 3/2 resonant transitions. Similarly, eight magic wavelengths were found for 7S 1/2 − 6P 3/2 (M J = ±3/2) transition in the considered wavelength range. It can be noticed from the results in Table 3 that all the magic wavelengths favor red-detuned traps.
We also investigate λ magic between the transitions involving the |7S 1/2 F n , M F = 0 and |6P 1/2,3/2 F n , M F = 0 states. The value of J-dependent polarizability for 7S 1/2 − 6P 1/2 transition is exactly equal to the F-dependent polarizability value due to the absence of tensor part of the polarizability [49]; hence, we have not tabulated F-dependent values for this transition. The magic wavelengths for the allowed transitions between |7S 1/2 F n , M F = 0 and |6P 3/2 F n , M F = 0 states are shown in Figure 5 and listed in Table 4. We choose M F = 0 sublevels in the hyperfine transitions, since, for this particular magnetic sublevel, the first-order Zeeman shift vanishes.
For experimental purposes, it is imperative to choose a magic wavelength which has a low sensitivity towards laser frequency fluctuations and minimum photon scattering. We studied the sensitivity of the magic wavelengths for a 1 nm change in laser wavelength for the 7S 1/2 − 6P 1/2 and 7S 1/2 − 6P 3/2 transitions. We noticed that for the 7S 1/2 − 6P 1/2 transition, magic wavelengths at 973 nm and 1081 nm offer minimum shift in differential stark shifts for a 1 nm change in laser wavelength and are suitable as compared to magic wavelengths at 918.46, 920.101 and 1379.9 nm, which are very sensitive to even a very small change in laser wavelength. The magic wavelengths at 918.46, 920.101 and 1379.9 nm are also very close to the resonance and are expected to have a large photon scattering rate. We recommend the magic wavelength at 1413(2) nm for this transition since it is far detuned and a large value of polarizability value at this wavelength indicates a deep trapping potential. It also offers low sensitivity to laser wavelength fluctuations and small photon scattering rate. For the 7S 1/2 − 6P 3/2 (M J = ±1/2) transition none of the magic wavelengths support a red-tuned trap. λ magic at 1392.4 nm supports a strong blue detuned trap but is very close to the 7S 1/2 − 8P 1/2,3/2 resonances, thus its sensitivity towards the laser fluctuations and photon scattering rate is expected to be very large. On the one hand, the magic wavelength for the 7S 1/2 − 6P 3/2 (M J = ±3/2) transition at 918.677 supports a very strong red detuned trap but at the same time is extremely sensitive to even small laser wavelength fluctuations. Therefore, we recommend use of λ magic at 1213 nm which is far detuned. The recommended magic wavelengths for the F-dependent transitions were investigated in detail by Pan et al. [21] and it was identified that λ magic at 1181 nm for the 7S 1/2 ; F = 4, M F = 0 → |6P 3/2 ; F = 3, M F = 0 is most suitable for experimental purposes since it offers a strong trapping potential not only for the clock states but also for the other two levels involved in the active clock lasing. Moreover, it was found to provide very low sensitivity against variation in the trapping laser wavelength along with a very low photon scattering rate.

Rb Atom
At least nine λ magic s are located for the 6S 1/2 − 5P 1/2 transition lying systematically between twelve resonant transitions in the considered wavelength range. The magic wavelengths are pictorially presented in Figure 2 and listed in Table 5. Comparing with the results published by Xiao-Run et al. [50], we ascertain that our results are in reasonable agreement with them; however, we have not considered magic wavelengths below 900 nm as we encountered a large number of resonances for the 6S 1/2 state in this wavelength range. Xiao-Run et al. [50] quoted magic wavelengths at 1342 and 1421 nm for the 6S − 5P 1/2 transition, which agree well with our results at 1342.4(2) and 1421.8 (7) nm. According to the evaluations, it is analyzed that Rb favors the formation of blue detuned traps for most of the magic wavelengths except for magic wavelengths near 1342 and 2771 nm for the 6S 1/2 − 5P 1/2 transition. Similarly, the magic wavelengths for the 6S 1/2 − 5P 3/2 are shown in Figure 6 and listed in Table 5. The figure and table clearly show that none of the evaluated magic wavelengths lying in the range 900-1500 nm for 6S 1/2 − 5P 3/2 (M J = ±1/2) and 6S 1/2 − 5P 3/2 (M J = ±3/2) transitions offer a scope for a red detuned trap; however, while exploring the wavelengths in far infrared region, we found possibility of red detuned traps at 2771 nm for these transitions. In the case of 6S − 5P 3/2 transitions, our results at 1336.5(2) and 1453.3(8) nm match well with the results published in [50]. Similarly in Figure 6 and Table 6 , we list the λ magic values for all allowed |6S 1/2 F n , M F = 0 → |5P 3/2 F n , M F = 0 transitions. The F-dependent polarizabilities values at the respective λ magic s are listed as well. For this wavelength range, 34 λ magic s are located, out of which 30 λ magic s support blue detuned traps.
For the 6S 1/2 − 5P 1/2 transition, λ magic s at 1342.4 and 1421.8 nm are recommended for a red-detuned and a blue-detuned trap, respectively. The former wavelength offers less sensitivity to the fluctuations in laser wavelengths as compared to the later magic wavelength. Both these magic wavelengths are far-detuned and will provide very small photon scattering. λ magic s at 1461.7 and 1453.3 nm for the 6S 1/2 − 5P 3/2 (M J = ±1/2) and 6S 1/2 − 5P 3/2 (M J = ±3/2) transitions, respectively, are recommended for experiments with blue-detuned traps and are least sensitive to the fluctuations in laser wavelengths.  Table 6. λ magic s (in nm) and their corresponding polarizabilities (in a.u.) in Rb (I = 5/2) of transitional states involved in 6S | F n , M F → 5P 3/2 | F n , M F transition are listed.

Conclusions
We present a list of reliable magic wavelengths for the 7S − 6P 1/2,3/2 and 6S − 5P 1/2,3/2 transitions for Cs and Rb atoms, respectively, considering the linearly polarized light. These magic wavelengths will be useful for the purpose of trapping of atoms in optical traps for constructing active optical clocks. We calculated static dipolar polarizabilities for the 7S 1/2 and 6P 1/2,3/2 states of Cs atom, and for the 6S 1/2 and 5P 1/2,3/2 states of Rb atom. The reliability of these results was verified by comparing the static dipole polarizability values with the other available theoretical and experimental results. The reported electric dipole matrix elements determined for the evaluation of dipole polarizabilities of the above states will be very useful for other studies using the Rb and Cs atoms in the future.
Author Contributions: B.A. conceived the idea of excited state transitions i.e., 7S-6P1/2,3/2 and 6S-5P1/2,3/2 transitions in Cs and Rb atoms respectively for optical-lattice based active clocks proposal. B.A. took the lead of the work and encouraged S.S. and J. to investigate static dipole polarizabilities and magic wavelengths for the considered transitions in Cs and Rb.; S.S. developed the theory and formalism and calculated the results for static dipole polarizabilities for 6S1/2, 5P1/2,3/2 states of Rb and 7S1/2, 6P1/2,3/2 states of Cs. S.S. also evaluated magic wavelengths for the above stated transitions in Rb and Cs, created the tables and designed the figures for the same; J. verified the results for magic wavelengths by independent implementation and contributed in the write-up of manuscript with the support of B.A., J. edited the manuscript and drafted it to the final version by implementing the suggestions of all the authors; B.K.S. and Y.-m.Y. provided critical feedback and helped shape the research, analysis and manuscript, they also aided in interpretation and discussion of the results and commented on the manuscript. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.