1. 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 intra-cavity weak feedback with phase coherence information through the stimulated emission inside a bad cavity regime, i.e., the linewidth of cavity mode is much broader than the linewidth of laser gain profile generating
[
5], according to which the phase coherence can provide a much confined ultra-narrow quantum-limited linewidth in comparison to an atomic transition [
7,
8]. This is the underlying concept of active masers which not only offers short-term stability, but also long-term stability extending over period of years. In addition to this, the insensitivity of frequency of active optical clocks to cavity-length noise [
5] can provide optical frequencies with high stabilities [
9]. The active optical clock can be constructed by using any “free medium”, i.e., neutral atoms, ions and molecules [
5], as long as they have the energy-level structures which follow the phenomenon of stimulated emission for lasing action so that they can work as optical masers. Once realized, the active clocks can overcome the challenges of attaining narrower local oscillators as intended for the passive clocks and hence can serve as prospective candidate for the local oscillator of the next generation passive optical clocks. The output signal of an active optical clock is nothing but a frequency reference to an atomic transition, therefore they can serve as frequency standards with an excellent short-term stability. Since proposed, several theoretical investigations have already been carried out to find the most suitable candidate for the realization of active optical clocks [
10,
11,
12,
13,
14]. On the basis of theoretical appraisals, several experimental configurations are being commenced to establish active clocks among two-level, three-level and four-level schemes using various atoms [
1,
9,
15,
16,
17,
18,
19,
20]. Recently, it was studied how a four-level scheme can, in principle, help in realizing continuous output signals by avoiding the pumping of induced light shift which is a common problem with the other two- and three-level schemes [
9,
18,
19,
20,
21], thereby providing an ultimate strategy for optimizing a more reliable active clock. Such an active optical clock can provide a continuous signal with an excellent short-term stability.
Since alkali metal elements are laboratory friendly systems and possess simple energy level structures, it is possible to find a suitable combination of four-levels in one of the alkali atoms for preparing such configurations. Based on the mechanism of active optical clock, it has been proposed that Cs and Rb are appropriate choices [
9] because of the availability of relevant lasers for cooling of these atomic systems. In this scheme, the pumping laser can be used to couple the ground state with a relatively high-lying upper state from which an electron can decay to another excited state consequently accomplishing the lasing clock state. Due to this, the pumping laser will not be able to couple with the clock states directly and hence, a continuous lasing signal can be produced. To reduce the systematic uncertainties further, one can trap the atoms in optical lattices. A fundamental feature of a lattice clock is that it interrogates a transition with controlled atomic motion. However, in this scheme, it is extremely important to make sure that the lattice light does not cause the light frequency shift of the clock transition. The laser induced light shifts can be avoided by trapping the atoms in optical lattice at the magic wavelength
[
22,
23,
24]. This is a well known technique used in passive atomic clocks [
25,
26,
27,
28] where the magic wavelength trapping is constructed for a ground state and an excited state. Theoretical determination of magic wavelengths in these atoms involves calculation of frequency-dependent polarizabilities of the considered states to find the magic wavelengths, where dynamic polarizability of both states participating in the transition is equal, in other words, the differential Stark shift for the states involved in the transition is zero [
24].
The magic trappings between two excited states are very rare since the excited states are generally short lived. In the four-level active clocks, the clock transition occur between two excited states and to the best of our knowledge, there is only one other proposal of magic trapping based on two excited states [
29]. In this work, we attempt to find magic wavelengths for the 6S → 5P
and 7S → 6P
atomic transitions of Rb and Cs atoms, respectively. These magic wavelengths are also given for all possible hyperfine levels of these transition so that it helps experimentalists to select a clock transition depending on the practical conditions. For this purpose, the dynamic electric dipole (E1) polarizabilities of the
,
and
states of Rb atom and the
,
and
states of Cs atom are calculated by assuming linear polarization of the lattice laser light.
2. Method of Evaluation
The shift in the
nth energy level of an atom placed in an oscillating electric field with amplitude
is given by [
30,
31,
32,
33]
where
is the dynamic dipole polarizability for a hyperfine level
of state
n, and is given by
Here, with denoting the unperturbed energies of the corresponding states for and ks are intermediate states to which transitions from state n are possible in accordance to the dipole selection rules. is the E1 matrix element between the states and .
For linearly polarized light, the above expression can be conveniently represented in terms of rank 0 and 2 tensors as [
26,
28,
32]
where
is the magnetic projection of total angular momentum
.
and
are known as the scalar and tensor components, respectively, and are given by:
and
and
in the above equations are the scalar and tensor components of atomic dipole polarizability of state with angular momentum
and magnetic projection
and are of the following forms
and
Here,
are reduced matrix elements (reduced using Wigner–Eckart theorem) with
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 or , can be determined for different magnetic sublevels or 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, ‘’ and ‘’ represent the energy shift in two different states involved in the transition with or . Our main aim is to find the values of and at which 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
,
and
are the core, core–valence and valence correlation contributions, respectively. The subscript ‘0’ in
refers to contributions from the inner core orbitals without the valence orbital. Our valence contribution (
) 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
is the lowest-order wave function of the state which can be obtained as
Here,
is the Dirac–Hartree–Fock (DHF) wave function for the closed core and the terms
and
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
and
are ascribed as the single core and valence excitation coefficients, whereas
and
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
and
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
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,
and
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
[
38] to the known experimental value of valence correlation energy as
These modified
coefficients can be utilized to recalculate the matrix elements. Here,
refers to the difference between the experimental energy [
39] and lowest order DF energy and
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
and
E1 matrix elements respectively. We utilize the value of ratio
to establish the recommended set of E1 matrix elements and their uncertainties. If
, then
are regarded as
value, otherwise SD results are used as
value. To evaluate uncertainties in the recommended values of E1 matrix elements, we take the maximum difference between
recommended value and other three all order values of SD, SDpT,
and
. The tail and core contribution and
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].