Advances in trapping and laser control of a single ion^[1–3] began a new era for the frequency standards in the optical range, which are aimed to achieve a fractional accuracy of 10⁻¹⁶–10⁻¹⁸. A number of ions, such as ¹⁹⁹Hg⁺,^{[4] 171}Yb⁺,^{[5–9] 115}In⁺,^{[10] 88}Sr⁺,^{[11,12] 40}Ca⁺,^{[13,14] 27}Al⁺,^[15] etc. have been undertaken in the experiments to attain such promising optical frequency standards. Among them, ¹⁷¹Yb⁺ is unique in the sense that it has three potential optical transitions that can be used for the clocks.^[16] Out of these, there are two narrow 6s ²S_1/2(F = 0, m_F = 0) → 5d ²D_3/2(F = 2, m_F = 0),^[17,18] 6s ²S_1/2(F = 0, m_F = 0) → 5d ²D_5/2(F = 2, m_F = 0) ^[19] quadrupole (E2) transitions and an ultra-narrow 6s ²S_1/2(F = 0, m_F = 0) → 4f¹³6s^2 2F_7/2(F = 3, m_F = 0) octupole (E3) transition^[9,20,21] with their respective wavelengths at 435.5 nm, 411 nm, and 467 nm. The transitions at the wavelengths 435.5 nm and 467 nm with low systematic shifts are the most suitable ones for precision frequency standards owing to their extremely small natural line-widths 3.02 Hz and 1 nHz, respectively. Their precisely measured transition frequencies ν_o have already been reported as 688358979309307.82(36) Hz^[18] and 688358979309308.42(42) Hz^[6] for the E2 transition and 642121496772645.36(39) Hz^[9] and 642121496772644.91(37) Hz^[6] for the E3 transition. These two transitions are endorsed by the international committee for weight and measures (CIPM) for the secondary representation of the standard international (SI) second owing to their being least sensitive to the external electromagnetic fields. Other than being a potential candidate for the frequency standards, Yb⁺ is also being considered for studying the parity non-conservation effect,^[22,23] violation of the Lorentz symmetry,^[24] searching for possible temporary variation of the fine structure constant,^[25,26] etc.

In an atomic clock, the measured frequency of the interrogated transition is always different than its absolute value due to the systematics. The net frequency shift depends upon many environmental factors and experimental conditions, which may or may not be canceled out at the end. Thus, they need to be accounted to establish accurate frequency standards. Careful design of the ion trap is also very important for minimizing the systematics caused by the environmental factors.^[27] Electric quadrupole shift is one of the major systematics when the states associated with the clock transition have finite quadrupole moments (Θs). In the ¹⁷¹Yb⁺ ion, the experimentally measured Θ value of the 5d ²D_3/2 state^[28] is 50 times larger than the 4f¹³6s^2 2F_7/2 state. A recent theoretical study suggested a more precise Θ value of the 5d ²D_3/2 state.^[16] Three independent theoretical investigations^[16,29,30] showed very large disagreements with the measured Θ value of the 4f¹³6s^2 2F_7/2 state.^[8] Therefore, it is imperative to carry out further investigations to attain more precise and reliable values of Θs and probe the reason for the anomalies between the calculated and the experimental results. Here we also perform another calculation using the relativistic coupled-cluster (RCC) method to evaluate the Θ values of the 5d ²D_3/2 and 4f¹³6s^2 2F_7/2 states to use them in the present analysis. We also analyze in detail the suitable confining potentials, electric fields, and field gradients that can create a nearly ideal quadrupole trap condition for carrying out precise measurements of the Θ values. Using these inputs, we estimate typical values of the quadrupole shifts of the 6s ²S_1/2(F = 0, m_F = 0) → 5d ²D_3/2(F = 2, m_F = 0) and 6s ²S_1/2(F = 0, m_F = 0) → 4f¹³6s^2 2F_7/2(F = 3, m_F = 0) clock transitions considering a number of ion trap geometries and discuss their possible pros and cons to make an appropriate choice. This analysis identifies a suitable geometry of the end cap ion trap to measure Θs of the 5d ²D_3/2 and 4f¹³6s^2 2F_7/2 states of Yb⁺, which is being developed at the National Physical Laboratory (NPL), India.^[31,32]

Electric quadrupole shift Δν_Q to an atomic state with angular momentum F arises due to the interaction of the quadrupole moment Θ(γ,F) with an applied external electric field gradient ∇E, where γ represents the other quantum numbers of the state. A non-zero atomic angular momentum results in a non-spherical charge distribution, thus the atom acquires higher order moments. Following this, it is advantageous to choose states with J < 1 or F < 1 in a clock transition for which Θ = 0. However, the excited states of the 6s ²S_1/2(F = 0, m_F = 0) → 5d ²D_3/2(F = 2, m_F = 0) and 6s ²S_1/2(F = 0, m_F = 0) → 4f¹³6s^2 2F_7/2(F = 3, m_F = 0) clock transitions have J = 3/2, F = 2 and J = 7/2, F = 3, resulting in nonzero quadrupole shifts. These shifts can be estimated by calculating the expectation value of the Hamiltonian^[33]

To calculate the atomic state wave functions for the determination of the Θ(γ,J) values, we adopt the Bloch approach.^[36] Following this approach, we express the wave function |Ψ_v〉 of the 5d ²D_3/2 state with the valence orbital v in the 5d_3/2 orbital as

In a perturbative procedure, Ω_v/a can be expressed as

In the RCC theory framework, the wave functions of the considered states are expressed as (e.g., see Refs. [16] and [37])

After obtaining amplitudes of the MBPT and RCC operators using the equations described above, the Θ values of the considered states are evaluated using the expression

We present in Table 1 the Θ values of the 4f¹⁴5d ⁵D_3/2 and 4f¹³6s^2 2F_7/2 states of Yb⁺ obtained from various methods described above. We also compare our results with the other calculation results and available experimental values. We consider the results from the CCSD_ex method as our recommended values as this method accounts for the more physical effects. We have also estimated uncertainties to the CCSD_ex results by estimating the neglected contributions due to truncation in the basis functions and from the omitted correlation effects that could mainly arise through the triply excited configurations. We had also presented these values using the CCSD⁽⁴⁾ method in our previous work.^[16] Differences in the results from the CCSD⁽⁴⁾ and the CCSD methods indicate that the higher non-linear terms are insignificant for the 4f¹⁴5d ²D_3/2 state, but they contribute slightly to the 4f¹³6s^2 2F_7/2 state. As said before, these series are solved iteratively to include infinity numbers of terms in this work. Again, we have removed uncertainties due to the calculated energies that enter into the amplitude solving Eq. (18) of the CCSD method by using the experimental energies.

Table 1.

Table 1.

Table 1. Demonstration of trends in the Θ values (in a.u.) from lower to higher order methods. Our results are compared with the values available from other calculations and the experimental results. The estimated final values are shown in bold fonts. Uncertainties to these values are also quoted separately. . Method 4f145d 5D3/2 4f136s2 2F7/2 DHF 2.504 −0.258 MBPT(2) 2.049 −0.344 LCCSD 2.028 −0.230 CCSD(2) 2.060 −0.208 CCSD(4) 2.068 −0.216 CCSD 2.061 −0.223 CCSDex 2.079(8) −0.224(10) Others 2.174a −0.22b 2.157c −0.20d 2.068(12)e −0.216(20)e Experiment 2.08(11)f −0.041(5)g a [39] b [29] c [40] d [30] e [16] f [28] g [8].

Table 1. Demonstration of trends in the Θ values (in a.u.) from lower to higher order methods. Our results are compared with the values available from other calculations and the experimental results. The estimated final values are shown in bold fonts. Uncertainties to these values are also quoted separately. .

We plan to employ a modified Paul trap^[42] of end cap geometry as shown in Fig. 1(a). In reality, such traps are not capable of producing pure quadrupole potential Φ⁽²⁾ due to the geometric modifications of the hyperbolic electrode, machining inaccuracies, and misalignments. On the other hand, for precision measurements with ions stored in a non-ideal trap, the anharmonic components of the potential Φ^(k>2)(x,y,z) are non-negligible due to the fact that they change the ion dynamics and also affect the systematics. For minimizing such effects several groups, such as MPQ Germany,^[43] NRC Canada,^[11] PTB Germany,^[44] and European Metrology Research Programme on ion clocks,^[45] have come up with different end cap trap designs for establishing single ion frequency standards. Here we aim to identify a new end cap trap geometry in which the trap induced quadrupole shift can be minimized. This trap can also add minimum anharmonicity to the confining potential and small micromotions.^[46] In a cylindrically symmetric trap as shown in Fig. 1(a), only the even order multipoles contribute. Here, in order to estimate the quality of the trap potentials, we consider k up to 10 since the amplitudes of Φ^(k) fall drastically at higher k. The tensor components of ∇E for each multipole potential Φ^(k) are opted from their electric field components E_x,y,z. The corresponding fractional quadrupole shift Δν_Q/ν₀ at each k is estimated from Eq. (2). The variation of Δν_Q/ν₀ for all multipole potentials up to k = 10 at two different distances from the trap center are estimated for the E2 and E3 clock transitions, which are shown in Fig. 2. The reported experimental values of the quadrupole shifts for these two transitions are also depicted in the same figure for comparison.

Fig. 1.

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	Fig. 1. (a) Our proposed electrode assembly of the end-cap trap with 2z0 = 0.7 mm, 2z2 ≈ 1.0 mm, 2r1 = 1 mm, 2r2 = 1.4 mm, 2r3 = 2 mm, θi = 10°, and θo = 45°. (b) Variation of the trap depth as a function of dc voltage U for a fixed radio-frequency ωrf and an ac voltage V.

Fig. 2.

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Fig. 2. Fractional electric quadrupole shifts due to the multipoles up to k = 10 for the (a) E2 and (b) E3 transitions. The ion is off centered by a distance of 0.1 μm (green) and 10 μm (black). The harmonic potential gives a spatially independent |ΔνQ/νo|. The reported fractional accuracies lie within the grey bands and so far the best accuracies are 1.1 × 10−16[8] and 5 × 10−17[18] for the E2 and E3 transitions, respectively.

The dominating perturbation of Φ⁽²⁾ arises from the octupole term Φ⁽⁴⁾, which may largely affect the quadrupole shift of the trapped ion frequency standards due to the wrong choice of electrode geometry. As an example, we simplify the analysis considering Φ(x,y,z) ≃ Φ⁽²⁾ + Φ⁽⁴⁾ since other higher orders are less significant for the quadrupole shift (Fig. 2). In the absence of any asymmetries, the trap potential can be written as

The strengths of the multipole potentials depend on c_k as given in Eq. (23). The magnitudes of c_k depend on the geometric parameters of the trap electrodes, such as radius r_o, angle θ_i of the electrode carrying rf (that is the inner electrode), inside and outside radii r₁ and r₂, angle θ_o of the dc carrying electrode (that is the outer electrode) which is coaxial to the inner one and also their mutual tip-to-tip separations 2z₀ and 2z₁, respectively, as shown in Fig. 1(a). We have obtained the trap potentials for various choices of these geometric factors by carrying out numerical simulations using the boundary element method.^[48] Then, we have characterized the multipole components in it by fitting ∑_k Φ^(k) for k up to 10. The potentials are obtained for various combinations of θ_i, θ_o, 2r₀, and 2z₀ but at the fixed values of 2r₁ = 1.4 mm, 2r₂ = 2 mm, and 2z₁ = 1.16 mm. We fix these parameters keeping in mind that the laser beams from the three orthogonal directions can impinge on the ion without any blockage as described in Ref. [31]. These three laser beams will be used for detecting the micromotions in all the three directions independently.^[49,50] After studying a series of trap geometries, we find that the diameters of the outer electrode have a weak influence on Φ^(k) which is below the accuracy that is expected from the machining tolerances. In Fig. 3, the variation of the quadrupole shift at the center of the trap with the diameter and the tip-to-tip separation of the inner electrode is shown, keeping the θ_i and θ_o values fixed. When placing the inner electrodes further away from each other, it is necessary to operate the trap at a larger voltage to obtain the required trap depth. On the other hand, placing them close to each other or having larger diameters can introduce optical blockage for the three orthogonal laser beams. Also micromotions of the ions can increase at large 2r_o and 2z₀.^[46] We obtain the optimized values as 2r₀ = 1 mm and 2z₀ = 0.7 mm for which Δν_Q is reduced but not minimized. Further attempt to minimize the quadrupole shift causes increase in anharmonicity and micromotions. The dependences of the quadrupole shift on θ_i and θ_o are shown in Figs. 3(e)–3(f). These clearly show that the quadrupole shift increases at large θ_i but it has a relatively weak influence on θ_o. A pair of inner electrodes with flat surfaces will introduce minimum shift. However θ_i = 0 gives optical blockage at our optimized 2z₀ = 0.7 mm for impinging three orthogonal Gaussian laser beams of waist ∼ 30 μm on the ion and overlapping them with the other laser beams. To avoid the optical blockage, we have optimized the values of θ_i and θ_o at 10° and 45°, respectively. These would produce an insignificant number of scattered photons from the tails of the Gaussian laser beams, which will propagate along the three mutually orthogonal directions in our described design reported in Ref. [31]. For our trap geometry, the coefficients c₂/2R² and c₄/2R⁴ are estimated to be 0.93 × 10⁶ m⁻² and 0.11 × 10¹² m⁻⁴, respectively.

Fig. 3.

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Fig. 3. Fractional electric quadrupole shifts for the E2 and E3 clock transitions corresponding to the different geometrical parameters: (a), (b) diameter 2ro of the inner electrode; (c), (d) tip-to-tip separation 2z0 of the inner electrodes; and (e), (f) angles θo at which the outer electrodes are machined. Each figure consists of a set of plots for angles of the inner electrode θi, which are 0° (black), 10° (red), 20° (gray), 30° (green), 40° (cyan), 50° (blue), and 60° (purple). The dashed line connects data points for the fixed values of θi and 2ro = 1 mm and 2z0 = 0.7 mm for all of them.

Due to the residual thermal motion of the ion after the laser cooling, it is unlikely to probe the clock transition while the ion is sitting at the center of the trap. Also there is a possibility of shift in the mean position of the ion from the trap center due to imperfect stray field compensation. This results in a spatially dependent Δν_Q in a non-ideal ion trap. At any position, the shift due to potentials for k > 2 increases following a power law of order k – 2 to the separation of the ion from the trap center, as shown in Fig. 2. Figure 4 shows the spatial variation of Δν_Q/ν₀ due to Φ⁽⁴⁾ and compares that with the shift resulting from Φ⁽²⁾. These two figures clearly show that the quadrupole shift due to the dominating anharmonic potential is insignificant for a frequency standard of accuracy ∼ 10⁻¹⁸ when the ion is positioned in the vicinity of the trap center within sub-micron precision and at a tight confinement. The quadrupole shift due to Φ⁽²⁾ can be eliminated by measuring the clock frequency along the three mutually orthogonal orientations in the lab frame and averaging them while quantizing the ion using the magnetic fields of equal amplitudes.^[34,49] Such angular averaging also eliminates the quadrupole shift due to Φ⁽⁴⁾ provided that the trap is perfectly axially symmetric, which is shown in Fig. 4. In practice, the trap can deviate from such an ideal situation, which could lead to inaccuracy in eliminating the quadrupole shift by angular averaging. In such a non ideal trap, the inaccuracies of eliminating the quadrupole shift are generally induced by Φ⁽⁴⁾ but it is below the accuracy of the frequency standard that we are aiming for. The present analysis helps us in building a suitable trap electrode where the effect of Φ⁽²⁾ in the quadrupole shift is reduced.

Fig. 4.

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	Fig. 4. Spatial dependence of the fractional electric quadrupole shifts ΔνQ/νo for the (a) E2 and (b) E3 clock transitions of 171Yb+. The quadrupole trapping potential produces a constant shift (gray). The spatial dependency in the shift along the radial (red) and axial (blue) directions arise from the anharmonic components k > 2 of Φ(k).

The formula for evaluating the quadrupole shift is given by

Fig. 5.

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	Fig. 5. The electric quadrupole shifts obtained in the previously reported works and estimated values from this work are shown in panel (a) for the 6s 2S1/2 → 5d 2D3/2 and panel (b) for the 6s 2S1/2 → 4f136s2 2F7/2 clock transitions.

We have proposed a suitable ion trap geometry for carrying out accurate measurements of the quadrupole shifts of the 6s ²S_1/2 → 5d ²D_3/2 and 6s ²S_1/2 → 4f¹³6s^2 2F_7/2 clock transitions in ¹⁷¹Yb⁺. We have also calculated the Θ values of the 5d ²D_3/2 and 4f¹³6s^2 2F_7/2 states of ¹⁷¹Yb⁺ using the RCC method that is used in our analysis. We have identified an end cap ion trap geometry which can produce nearly an ideal harmonic confinement to minimize the electric quadrupole shift. We also showed that the anharmonic component of the potential adds an insignificant quadrupole shift for a frequency standard with an accuracy ∼ 10⁻¹⁸, when the ion is positioned at the trap center within sub-micron precision. To obtain a figure of merit of our design, we have estimated the quadrupole shifts along with their uncertainties in our proposed setup and compared the result with the previously available values. In our optimized ion trap, we expect to measure the quadrupole moment of the 5d ²D_3/2 state with an accuracy of one part in 10³. We are also aiming to measure the quadrupole moment of the 4f¹³6s^2 2F_7/2 state reliably that could possibly explain the discrepancy between the experimental and theoretical results.