Spectroscopic properties of lithium like ions: Prospective elements for quantum computation

Rydberg physics is the most exquisite playground for the exploration of quantum technologies. Therefore, in this spectral investigation based on weakest bound electron potential model theory (WBEPMT). The quantum defects and Rydberg energies of low-lying and high-lying spectral Rydberg energy sequence in the following configurations: 1s2ns2Se1/2 and 1s2np2Po1/2 in Lithium-like ions (Z=4-6) Beryllium (Be II), Boron (B III) and Carbon (C IV) are computed with their radial expected values. Total 43 low-lying energies and quantum defects are computed and compared with 25 low lying radial expected values. The results are in good agreement with the previously published experimental / theoretical results. Based on the good agreement of low-lying data, total 257 new values of high-lying Rydberg energies and quantum defects are computed with 575 radial expected values currently not listed at NIST. The deviation in both sets of data doesn’t exceed 1 cm-1 .


Introduction
The recent advancements in quantum technologies are based on a new highly tuneable, reproducible and 'mother's nature' most versatile building block for quantum systems called Rydberg atoms. That can be utilized as the platform for quantum computation. Especially, in the present scenario most important developing applications based on Rydberg atoms are quantum bits, quantum sensing and quantum imaging, and quantum optics. In the last decade the use of Rydberg atoms for quantum devices is developing at much faster pace than ever. However, at the moment we are very sure that as the field grows and develop, more and more quantum devices based on Rydberg atoms are available for commercial systems that will soon revolutionise the society and becomes helpful to solve a wide range of societal problems. correspondingly, a rapid evolution occurs in experimental techniques, such as Stark effect spectroscopy [1], millimetre wave technique [2], two or three-photon Doppler-free spectroscopy techniques [3] optical double-resonance and level-crossing techniques [4] etc. Regarding theoretical techniques various methods developed [5][6][7]. The exquisite theoretical techniques are self-consistentfield theory (SCFT) [8], many-body perturbation theory (MBPT) [9] and quantum defect theory (QDT) [10]. For the computation of spectral properties of many valence electrons atoms, these theoretical techniques show high complexity in computation. The solution provided by Zheng et.al [11] by put forward a semi-empirical method called weakest bound electron potential model theory (WBEPMT). This theory based on a main idea of separating the ionic-core with the one most loosely bound electron called weakest bound electron (WBE). The ionic-core containing all remaining electrons called non-weakest bound electrons (NWBE's) and nucleus, in any atomic or ionic system. In weakest bound electron potential model (WBEPM) theory the ideas of quantum defects, foreign level configuration mixing, spectral Rydberg energy series and dependency of Rydberg states series on azimuthal quantum number was also presented. Based on the hypothetical ideas present in WBEPM theory various analytical investigations were performed for different elements neutral and ionic states successfully [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].
Within the limits of WBEPMT, Lithium like ions (Be II, B III and C IV) behave as hydrogen atom and single electron in their outer most orbit treated as WBE and the only difference is ionic-core containing nucleus and two NWBE's having a net charge of +1. In this way Lithium like ions (Be II, B III and C IV) WBE resides and moves in the potential of ionic-core.
The interpretation of the energy levels of the valence electron in Li like-ionic systems is peculiarly simple in a way that the WBE resides in the nuclear charge coulombic field, shielded by the two electrons of the Kshell. Electron's interaction effect can be neglected, and the main adjustment to the asymptotic coulomb potential is created by core-polarization effect. Thus, the problem can be solved by utilizing two different approaches; firstly, a model potential that is formulate to compute the observed quantum defects for the states of the WBE, and secondly, coupled equations that includes electron of S e ground state and a P o pseudo-state. These S e and P o utilized to write the Schrodinger equation of one electron system [32]. The results obtained within the frame work of WBEPM theory are essentially good and they have deviations of less than 1cm -1 for the allowed transitions discussed in previous literature published [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].
Another underlying purpose of this investigation lies in a particular observation during the beam foil spectroscopy (BFS). This observation offers another field of research in atomic spectroscopy called 'doubly excited states' [33]. Specially observe in the BFS of Lithium-like and Beryllium-like atoms [34].
For the doubly excited states, in three-electron systems, single excitation allows 2S e -doublet states with a single excited electron outside the 1s 2 1S e closed shell core, whereas double excitation allows 4S e -quartet states where the three-electron spins are all aligned. On the experimental side, transitions between the doubly excited-states have been observed in the Li atom [35], and the beam-foil spectroscopy [36] has shown strong excitation of the preceding transitions. On the basis of these experimental observations since 1959, extensive theoretical investigations of the lithium isoelectronic series have been the objective of physicists. Among the methods applied to the ground state properties of Li-like ions are the wave functions method used by Patil [37], the Hylleraas-type variational method, and the 1/Z expansion perturbation theory used by Yan et al. [38], the Hylleraas coordinates employed by Perkins [39], Ho [40], and King [41], the full core plus correlation method utilized by Chung and Zhu [42], and the quasi-projection operator technique developed by Temkin et al. [43]. But, the treatment of the properties in three-electron systems remains a very complex physical problem. As a result, it is necessary to overcome such physical problems in the framework of simple analytical models. For this purpose, we have presented a new analytical model named the WBEPMT [44]. This method permits one to succeed in obtaining simple expressions of the Rydberg energies and Radial expected values in connection with an understanding of the importance of WBE-NWBE's correlation effects in atoms. The advantages of the WBEPM theory have been demonstrated previously [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. In this paper, this method is applied to the computation of the radial expectation values of Lithiumlike ions (Be II, B III and C IV) with their quantum defects and low-lying and high-lying spectral Rydberg energy sequence in 1s 2 ns 2 S e 1/2 and 1s 2 np 2 P o 1/2 up to n=50 electron's state.

Theory
The theory based on the concept of one electron problem like hydrogen atoms. In such systems every atomic or ionic system contains a single most loosely bound electron in valence shell termed as WBE and an ioniccore having +(Z-1) charge. The ionic core formed by the combination of leftover electrons other than WBE termed as NWBE's with nucleus. Due to shielding effect ionic core formed a net columbic potential in which WBE may polarize or penetrate towards ionic core and forms a dipole. Many quanta mechanical properties of the atomic or ionic system are depend upon the behaviour of WBE. Therefore, the accurate and precise information about WBE is advantageous. So, by treating WBE as one electron problem. The Hamiltonian function can be written as Eq. 1.
For describing the potential of an ionic core in which WBE reside, the one electron Schrodinger equation can be written as Eq. 2.
Herein, = − * (4) In the above Eq. 3 α is the effective nuclear charge * , is the distance between WBE and the nucleus and l are the azimuthal quantum number of the WBE. Introduction of a is the parameter into the expression of the potential of ionic core need to be computed.
Also, in the potential Eq. 3, the first term gives coulombic potential and the second term gives electric dipole potential due to the dipole formed between ionic core and WBE. Now, by solving Eqs. 1 and 3 for the potential of an ionic core, we get Eq. 6.
The Eq. 6 can be expressed in polar coordinates as Eq. 7.
Now, by utilizing polar coordinates and separating the angular function , ( , ) from the radial function ( ), we obtain Eq. 9.
And, the integral formula concerned with two generalized Laguerre polynomials is given as Eq. 17.
The solution may be used to calculate expectation values. For a transition from the level ( , ) to the level ( , ), the expectation value of is If we let = 1 and = , we can further derive Eq. 24.
Where, < > is the Radial expectation value of the weakest bound electron i.
The negative value of of the weakest bound electron in Eq. 14 equal to the ionization energy of the weakest bound electron that is Eq. 25.
Where, * is the effective principal quantum number. The * and * are unknown values and are calculated by the transformation between Eigen-values of quantum defect theory (QDT) and WBEPM theory, which gives Eq. 26. * * = − (26) Where, Zo is the atomic kernel net charge number and δn is the Quantum defect as a function of principal quantum number n.
During the process of successive ionization there are many possible values of ionization limits, corresponding to different WBE in each residual ion, provided that each residual ion should lie in the ground state. Therefore, the energy of the source atom, in which all electrons and nucleus are infinitely apart from each other, we have Eq. 27.
Where, E is the total electronic energy of source system in its ground state, Ii are the ionization limits at each stage of ionization. Thus, there is no problem of freezing orbits in the ionization procedure [45][46][47]. Now, the WBE of an atom or residual-ion is considered to be rotated around an ionic-core. So, the term energy of spectrum like Rydberg energy levels (E) is equal to the sum of ionization limit (E limit) and Rydberg energy ( ′ ) of WBE.
Where, δ0 is the lowest Rydberg state Quantum defect of the series, coefficients (ai's, i =1,2,3,4) in Eq. 30 are obtained by the method of least-square fitting of the first few low-lying experimental values of the Rydberg energy states of the given series [48].

Results and Discussion
In this investigation spectral properties of Lithium likeions (Be II, B III and C IV) is presented in which 6 series having configurations: 1s 2 ns 2 S e 1/2, 1s 2 np 2 P o 1/2 are computed by utilizing coupled equations developed in semi-empirical method developed by Zheng et al. [44]. Total 43 low-lying Rydberg energy levels are listed at NIST [50] for above mentioned spectral series in lithium like-ions. The program is developed base on the coupled equations (see Eqs. 24, 29 and 30) in the theory [44]. The program first compute and compared quantum defects and Rydberg energies of 43 low lying levels by computing the coefficients of coupled equations (listed in Table 1). The deviation with the previously published expected values listed at NIST is not greater than 1 cm -1 (listed in Tables 2-7). In the second part program utilized the quantum defects determined in the first part of investigation and compute and compared 25 low-lying expected radial values listed in previously published paper [49]. The deviation is not greater than 0.1 cm -1 , except for few values (listed in Tables 2-7). Since the computed data agrees well with the previously published experimental / theoretical data [49]. The program finally computes the quantum defects, Rydberg energy levels and radial expected values up to n = 50.   Table 3 Spectral Rydberg Energy Series with Quantum Defects and Radii of Li iso-electronic sequence in 1s 2 np 2 P o 1/2 (Be II) State Quantum Defects Rydberg Energies (cm -1 ) Ry (NIST)-Ry (cm -1 ) Radii (cm -1 ) n δ(NIST) [50] δ Ry (NIST) [50] Ry < r > < r > [49] Table 4 Spectral Rydberg Energy Series with Quantum Defects and Radii of Li iso-electronic sequence in 1s 2 ns 2 S e 1/2 (B III)

ai's(i
The results compute in the present computation for spectral Rydberg energy series of Lithium like-ions up to n = 50 and Z = 4-6 are listed in Tables 2-6. Due to the constancy of quantum defects and degeneracy of Rydberg states at higher n only few computed values are shown in Tables 2-6. The detail evaluation is shown in Fig. 1. The comparison with previously published data [49][50] shows good agreement with the computed values. The absolute deviation in general is not greater than 1 cm -1 . Also, in the graphical section by plotting graphs of quantum defects and Rydberg levels as a function of principal quantum numbers called n-δ and n-Ry curves. The change in the regularity of the quantum defects and Rydberg energy levels series can be easily observed with the convergence under same ionization limit and nature of series as describe in Table 1. All 1s 2 ns 2 S e 1/2 series in lithium like ions are low-lying core penetration series, converges under same ionization potentials. In 3 n-δ curves, at low n curve first decreases exponentially shows the regular decrease in Quantum defects and at high n the constancy of Quantum defects shows by asymptotic nature. In contrast, in 3 n-Ry curves at low n the curve first increases exponentially shows the regular increase in the Rydberg States energy and at high n the degeneracy of Rydberg energy states shows by asymptotic nature. Similarly, in 1s 2 np 2 P o 1/2 series in Lithium like-ions 2 out of 3 are high-lying core polarization series, except 1 in C IV shows high-lying core-penetration series converges under same ionization potentials. In 3 n-δ curves, at low n curve first increases exponentially shows the regular increase in Quantum defects and at high n the constancy of Quantum defects shows by asymptotic nature. In all n-Ry curves shows the same behaviour as in low-lying core penetration series (see Fig. 1).
Finally, in the graphs of radial expected values as the function of principal quantum numbers called n-<r> curves (see Fig. 2). The computed curves fitted quite well with the curves of previously published data [49]. By curve fitting the general equation of 2 nd order polynomial for each series of Radial expected values are also presented with each graph (see Fig. 2). Within the same series as the n increase the Radial expected value increases exponentially in each case. But as the Z increases the Radial expected values decrease in comparison from Z = 4-6 in both configurations: 1s 2 ns 2 S e 1/2, 1s 2 np 2 P o 1/2. So, it can be concluded that at high Z the probability of finding WBE near an ionic-core increase where it can polarize or penetrate towards an ionic-core forming a strong dipole generating both kind of spectral series, core penetration as well as corepolarization series under same ionization limit.

Conclusion
Within the limits of semi-empirical method, relatively have simple analytical solutions utilizing the coupled equations. The results computed for the quantum defects, Rydberg energy levels and expected radial values for ns and np states for Z = 4-6, up to n = 50 are accurate and shows good accordance with previously published values [49][50].
The graphical description clearly presents the nature and convergence of all series. In general, all ns series belongs to core-penetration and all np series belongs to core-penetration series with an exception in C IV both ns and np series belongs to low and high-lying corepenetration series (see Fig. 1 and Table 1).
The Radial expected values having great importance in computation of transition probabilities are not only agreed well with the previously published data [49]. A general 2 nd order polynomial equation for each series is also generated by curve fitting shows an excellent agreement with both computed and previously published data [49].

Acknowledgement
All authors would like to thanks Dr. Zaheer-Uddin, Department of Physics, University of Karachi for his kind assistance during the research.

Explanation of Tables
δ: Computed values of Quantum defects by Eq. 30.