Probing Multiple Electric Dipole Forbidden Optical Transitions in Highly Charged Nickel Ions
Shi-Yong Liang, Ting-Xian Zhang, Hua Guan, Qi-Feng Lu, Jun Xiao, Shao-Long Chen, Yao Huang, Yong-Hui Zhang, Cheng-Bin Li, Ya-Ming Zou, Ji-Guang Li, Zong-Chao Yan, Andrei Derevianko, Ming-Sheng Zhan, Ting-Yun Shi, Ke-Lin Gao
PProbing Multiple Electric Dipole Forbidden Optical Transitions in Highly ChargedNickel Ions
Shi-Yong Liang,
1, 2, 7, ∗ Ting-Xian Zhang,
1, 7, ∗ Hua Guan,
1, 2, † Qi-Feng Lu, Jun Xiao, † Shao-LongChen,
1, 2, 8
Yao Huang,
1, 2
Yong-Hui Zhang, Cheng-Bin Li, † Ya-Ming Zou, Ji-Guang Li, Zong-Chao Yan,
5, 1
Andrei Derevianko, Ming-Sheng Zhan, Ting-Yun Shi, and Ke-Lin Gao
1, 2, † State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China Key Laboratory of Atomic Frequency Standards,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China Shanghai EBIT Laboratory, Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Department of Physics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 Department of Physics, University of Nevada, Reno, Nevada 89557, USA University of Chinese Academy of Sciences, Beijing 100049, China Max-Planck-Institut f¨ur Kernphysik, Heidelberg 69117, Germany (Dated: January 27, 2021)Highly charged ions (HCIs) are promising candidates for the next generation of atomic clocks, ow-ing to their tightly bound electron cloud, which significantly suppresses the common environmentaldisturbances to the quantum oscillator. Here we propose and pursue an experimental strategy that,while focusing on various HCIs of a single atomic element, keeps the number of candidate clock tran-sitions as large as possible. Following this strategy, we identify four adjacent charge states of nickelHCIs that offer as many as six optical transitions. Experimentally, we demonstrated the essentialcapability of producing these ions in the low-energy compact Shanghai-Wuhan Electron Beam IonTrap. We measured the wavelengths of four magnetic-dipole ( M
1) and one electric-quadrupole ( E α needed for fundamental physics applications. We argue that all the six transitions innickel HCIs offer intrinsic immunity to all common perturbations of quantum oscillators, and oneof them has the projected fractional frequency uncertainty down to the remarkable level of 10 − . I. INTRODUCTION
Quantum metrology of atomic time-keeping has seendramatic improvements over the past decade with novelapplications spanning from chronometric geodesy [1, 2] tofundamental physics, such as dark matter searches [3, 4]and multi-messenger astronomy [5]. Currently, opticalatomic clocks using neutral atoms or singly charged ionshave demonstrated fractional frequency uncertainties atthe level of 10 − or even 10 − [6–9]. These uncertain-ties refer to the ability to protect the quantum oscillatorfrom environmental perturbations, such as stray mag-netic and electric fields. As these existing technologiesmature, they are reaching the stage where one needs tounderstand numerous sources of environmental pertur- ∗ These authors contributed equally to this work. † Email: [email protected], xiao [email protected],[email protected], [email protected] bations in greater detail. In some cases, the perturba-tions cannot be fully eliminated and one needs to de-vote significant efforts to measuring and characterizingthe perturbations; these lead to non-universal system-atic corrections to the clock frequency that are specificto experimental realization of the clock.Novel classes of atomic clocks must start with quan-tum oscillators that offer a much more improved inherentimmunity to environmental perturbations than the moremature technologies. One of such systems is the nuclearclock based on the unique property of the
Th nucleus– the existence of a nuclear transition in a laser-accessiblerange [10, 11]; unfortunately, despite substantial world-wide efforts [12, 13], this transition is yet to be observeddirectly. The suppression of environmental perturba-tions for the nuclear oscillator comes from the nuclearsize being ∼ times smaller than the size of a neutralatom. Alternative novel systems are highly charged ions(HCIs) [14, 15]. Similar to the nuclear clock, here theoscillator size is also substantially reduced, owing to theelectronic cloud size shrinking as 1 /Z with the increasing a r X i v : . [ phy s i c s . a t o m - ph ] J a n ion charge Z . HCIs were proposed as promising candi-dates for the next generation of atomic clocks [14]. Inaddition, beyond the improved time-keeping, HCIs openintriguing opportunities for probing new physics beyondthe standard model of particle physics [16, 17].Compared to the single, yet to be spectroscopicallyfound nuclear transition, there is a plethora of suitableHCIs (see a review [18] for a sample of proposals). Adetailed analysis [19] indicates that with certain HCIs,atomic clocks “can have projected fractional accuraciesbeyond the 10 − − − level for all common systematiceffects, such as blackbody radiation, Zeeman, ac-Stark,and quadrupolar shifts”. Moreover, compared to the nu-clear clock, where the direct observation of the clock tran-sition remains elusive, the clock transitions in HCIs canbe found with conventional spectroscopy or from atomic-structure computations. Indeed, here we report spectro-graphic measurements of wavelengths for five clock tran-sitions with an accuracy of several ppm (see Table I),setting up the stage for the more accurate laser spec-troscopy.Despite the lack of suitable electric-dipole ( E
1) tran-sitions for direct laser cooling, recent successes in sym-pathetic cooling and quantum logic spectroscopy of HCIshave paved way for precision spectroscopic measurementswith HCIs [20, 21]. It is worth emphasizing that thesenewly demonstrated technologies can be applied univer-sally to a wide range of HCIs. The multitude of suitableclock HCI candidates is a “blessing in disguise”, as oneneeds to commit to building the infrastructure for a spe-cific ion. As with any new endeavor, one would like tomitigate potential problems with picking a “wrong” ion.Here we propose and pursue a straddling strategy thatwould allow one to explore several clock transitions usingnot only the same HCI production system but also ionsof the same atomic element.A suitable HCI has to possess a number of propertiesenabling precision spectroscopy and compatibility withoperating an atomic clock. Generally, one may distin-guish between three classes of visible or near visible op-tical forbidden transitions in HCIs that can be used fordeveloping optical clocks:1. Magnetic-dipole ( M
1) transitions between twohyperfine-structure levels of the same electronicstate [19, 22].2. Forbidden transitions between level crossing elec-tronic states, which tend to be sensitive to variationof the fine structure constant [16, 23–25].3. Forbidden transitions between the ground-statefine structure levels [14, 19, 26, 27].Type 1 transitions occur in few-electron heavy HCIs [19]that are challenging to produce and trap. Type 2 transi-tions involve a complex energy structure that can impedeinitialization and read-out of the clock states. Here we fo-cus on type 3 transitions that offer simplicity in both pro-ducing the ions and clock operation. More specifically we choose HCIs of nickel of various charge states [19, 26, 27]:Ni , Ni , Ni , and Ni . The clock transitionsare shown in Fig. 1. All the traditional clock perturba-tions are strongly suppressed for these ions due to thecharge scaling arguments [14, 19, 28]. As pointed outin Ref. [14], the major issue with HCI clocks is the so-called quadrupolar shift of the clock transition, when thequadrupole ( Q ) moment of the clock state couples to thealways existing E -field gradients in ion traps. While the Q -moment of an electronic cloud does scale as 1 /Z , thisreduction is not sufficient to suppress the quadrupolarshift below the desired level of accuracy. Thus, it isbeneficial to select clock states with either vanishing orstrongly suppressed Q -moments.There are four M E , Ni , Ni , and Ni that offer the desiredflexibility. These ions have varying number of electrons inthe 3 p shell, see Fig. 1. The clock transitions are betweenthe fine structure components of the ground electronicstate. There are four stable isotopes Ni, Ni, Ni, and Ni without nuclear spin; these can be used to searchfor new physics with isotope shift measurements [29–31]and for initial spectroscopic measurements. These spin-0isotopes, however, will be susceptible to the quadrupolarshifts for clock transitions. However, these shifts canbe suppressed by using the Ni isotope (nuclear spin I = 3 / • P / F = 0 and P / F = 1 or F = 2 for Ni and Ni , • P F = 1 / P F = 1 / , • P F = 1 / P F = 3 / , • P F = 3 / P F = 1 / and Ni .This selection is based on the following reasoning [19]: Q -moments (rank 2 tensors) of the F = 0 and F = 1 / P / F =1 , P F = 3 / Q -momentsvanish due to selection rules for the electronic angularmomentum J . Thereby, the Q -moments are determinedby the nuclear Q -moments or hyperfine mixing [11] and,as such, are strongly suppressed. As an indication of at-tainable accuracy, Refs. [26, 27] evaluated relevant prop-erties of the clock transitions in Ni and Ni andestimated common systematic uncertainties to be below10 − , in line with the more general estimates of Ref. [19].The second-order Doppler shift induced by the excess mi-cromotion of the trapped ion is expected to be suppressedto below 10 − by compensating the stray electric fieldto a level below 0.1 V/m [32, 33]. In a cryogenic trap, theheating rate of the trapped ions caused by the collisionswith the background gas and the anomalous motionalheating is reduced, and hence the second-order Dopplershift induced by the secular motion is also expected to TABLE I. Observed and calculated wavelengths of magnetic-dipole ( M
1) and electric-quadrupole ( E
2) transitions, where thelines a through f are candidate clock transitions, in nm.Line Ion Transition Type NIST This workVacuum Air (observed) Vacuum Theory a Ni s p P / − P / M b Ni s p P − P M c Ni s p P − P M d Ni s p P / − P / M e Ni s p P − P E f Ni s p P − P E g Ni s p P − P M d, M1361 nmc, M1670 nmf, E2365 nme, E2498(3) nmb, M1512 nm F=3F=2F=1F=0F=2F=1² P ² P Ni Ni s ²3 p ² P F=7/2F=5/2F=3/2F=1/2F=1/2F=3/2F=5/2F=3/2³ P ³ P ³ P s ²3 p ⁴ ³ P F=3/2F=1/2F=3/2F=5/2F=7/2F=5/2F=3/2F=1/2³ P ³ P ³ P s ²3 p ² ³ P Ni s ²3 p ⁵ ² P a, M1423 nmF=3F=2F=1F=0F=2 Ni ² P ² P F=1
FIG. 1. Partial energy-level diagrams for highly chargednickel ions. Clock transitions are explicitly drawn. Magnetic-dipole ( M
1) transitions are shown in magenta and electric-quadrupole ( E
2) transitions in green. The labeling of transi-tions is the same as in Table I. be sufficiently small [18]. Based on these arguments, weexpect the attainable fractional systematic uncertaintyof all the six clock transitions in Ni HCIs to be 10 − .As the first essential step towards realizing the Ni HCIclocks, we produced the target ions at our newly builtlow-energy compact Shanghai-Wuhan Electron Beam IonTrap (SW-EBIT) [34]. The wavelengths of four M E M b , c , and g (listed in Table I) in Ni and Ni are observed and characterized for the firsttime in the laboratory. We also carried out calculationsfor these ions using an ab initio relativistic method of atomic structure, the multi-configuration Dirac-Hartree-Fock (MCDHF) method [35, 36]. We evaluated relevantspectroscopic properties, such as transition wavelengthsand natural linewidths. We also estimated the sensitivityto the hypothetical variation of the fine structure con-stant α and found that all considered clock transitions inNi HCIs are more susceptible to the variation than mostof the commonly employed singly charge ions or neutralatoms. Thus, Ni HCIs can be used for placing stringentconstraints on the spatial or temporal variation of α . II. EXPERIMENTAL METHOD AND RESULTSA. Production of Ni HCIs
To produce Ni HCIs, we injected the Ni(C H ) (nick-elocene) molecular beam into the trap center. Then thecharge-state distribution of Ni HCIs was measured usingthe electron-beam current of 6 mA and the electron-beamenergy of 500 eV, which is higher than the ionization en-ergies 319.5 eV, 351.6 eV, 429.3 eV, and 462.8 eV neededfor Ni , Ni , Ni , and Ni , respectively. The ex-traction period was 0.3 s and the magnetic flux densitywas 0.16 T. As shown in Fig. 2, the target ions Ni ,Ni , Ni , and Ni were produced, and the ionsof two distinct isotopes, Ni and Ni, were resolved.The techniques for measuring charge-state distributionare described in Ref. [34].
B. Spectral measurements
The spectra of the trapped Ni HCIs were observedby a Czerny-Turner spectrograph (Andor Kymera 328i)equipped with an Electron Multiplying Charge-CoupledDevice (EMCCD, Andor Newton 970, pixel: 1600 × µ m) and a 1200 l/mm grating blazed at500 nm. To maximize the number of the Ni HCIs of a spe-cific charge state, different electron-beam energies wereused, i.e.
370 eV, 400 eV, 500 eV, and 540 eV for Ni ,Ni , Ni , and Ni , respectively. As illustrated inFig. 3, the fluorescence emitted from the Ni HCIs wasfocused by a single N-BK7 Bi-Convex lens (focal length
110 120 130 140 150 1600123 I n t en s i t y ( a r b . un i t s ) Wien filter voltage (V) Ni Ni
11+ 58 Ni Ni
12+ 58 Ni Ni
13+ 58 Ni
14+ 58 Ni Ni Ni C Ni Ni FIG. 2. Charge-state distribution of the Ni HCIs, obtainedby averaging 3 measurements. f = 10 cm at 633 nm) on the spectrograph entranceslit. The distance between the trap (DT2, drift tube 2 inSW-EBIT [34]) center and the front principal plane of thelens remained fixed at 197 mm, which was about twicethe focal length. Before setting up the spectrograph, aCharge-Coupled Device (CCD) was placed on the imageplane to image the two inner edges of DT2 (1 mm slitwidth) that were illuminated by the hot cathode. In or-der to ensure that the lens was aligned with the opticalaxis, we adjusted the angle and position of the lens untilthe edge image became mirror-symmetric. Because of thedispersion of the lens, to ensure that the spectrograph slitwas always precisely located on the image plane, the dis-tance L between the slit of the spectrograph and the backprincipal plane of the lens was calculated and adjusted forevery central wavelength of the measured spectra. Thegrating was set to zero-order to image the inner edges ofDT2 through spectrograph with its maximum slit widthand minimum iris aperture behind the slit. Similarly,the angle and position of the spectrograph were adjusteduntil the image of the edges became mirror-symmetricto ensure the spectrograph alignment with the opticalaxis. A one inch aperture was placed before the lensto block the stray light. For calibration, a conjugatedoptical system was installed on the opposite side of thespectrograph. A diffuser attached by a 0.5 mm slit wasplaced on the object plane. A low-pressure gas-dischargelamp filled with Kr illuminated the slit, and the slit wasimaged to the trap center to overlap with the trappedion cloud. During the spectral exposure time of 10 to60 minutes, the Kr lamp as the calibration light sourceflashed at a period of 1 to 3 minutes. The slit of thespectrograph was set to 30 µ m, and the iris aperture inthe spectrograph was set to 40 steps to obtain the F/7.6aperture. The focal length of the spectrograph was tunedto minimize the linewidth in each spectral range.All the spectra were binned to a non-dispersive direc-tion after removing the cosmic ray noise, as shown inFig. 4 (a). The dispersion function was obtained by FIG. 3. Scheme of observation and calibration of the mea-sured lines. The Ni HCIs are trapped at the center of DT2 inSW-EBIT. fitting the NIST-tabulated Ritz in-the-air wavelengthsof the calibration lines to a cubic polynomial, againsttheir column numbers of the line centroids. The residu-als of the calibration lines and the 1- σ fitting confidenceband are shown in Fig. 4 (b). To determine the linecentroids, the measured lines and calibration lines werefitted to a Gaussian or a multi-Gaussian profile, as shownin Fig. 4 (c). C. Observed wavelengths
Previously, these five M a and d havebeen also measured in Tokamak [38, 39], but experimen-tal observation of the other three lines b , c , and g hasnot been reported in the literature. In this work, we ob-served and identified all five M d , our result of 360.105(2) nm substantially deviatesfrom the value of 360.123(2) nm observed from Tokamakplasma by Hinnov et al. [39]. To test our result, two linesfrom Ar + were measured without any change of the opti-cal system comparing to the measurement of line d , andthe measured wavelengths in air were 357.660(2) nm and358.843(2) nm, which were in good agreement with theRitz wavelengths in NIST database, i.e. , 357.661538 nmand 358.844021 nm. For the E e and f , the tran-sition rates are too small to be observable by our tech-nique. However, we deduced the wavelength of line f inTable I from those of lines c and g via the Rydberg-Ritzcombination principle. D. Measurement uncertainties
Line centroid uncertainty.
The line centroids ofthe measured lines and their calibration lines were de-termined by the centers of the fitted Gaussian profiles.Since the statistical uncertainty of the line centroid wasmainly caused by the low signal-to-noise ratio, we evalu- line aNi C oun t s Column number (a)
400 600 800 1000 1200-1.00.01.0 (b) R e s i dua l ( p m ) (c) Raw counts Gaussian fitColumn number C oun t s FIG. 4. (a) A spectrum of line a from Ni and its calibration lines from Kr atom whose approximate wavelengths are labeledin the figure in nm. (b) Residuals of cubic polynomial fits of the calibration lines. The gray band is a 1- σ confidence band.The uncertainties in the calibration lines are dominated by the line centroid uncertainties of the Gaussian fits. (c) Spectrumof line a and its Gaussian fit. ated the statistical uncertainty by performing at least 20measurements on the line of interest, as shown in Fig 5.For all five measured lines, this uncertainty was smallerthan 0.4 pm. The systematic uncertainty of the line cen-troid is mainly caused by the non-ideal Gaussianity ofthe line because of the optical aberration and the Zee-man components. In this work, since the measured linesand their calibration lines shared a similar profile, theoptical aberration effect was largely offset. In the trapcenter, the magnetic flux density was ∼ ∼ ∼
90 pm linewidth. Further-more, the Zeeman effect would not alter the line centroidbecause the Zeeman components were symmetrically dis-tributed; in addition, the Zeeman effect was negligible forthe Kr lamp due to the low magnetic field of 0.4 mT.
Dispersion function uncertainty.
The statisticaluncertainty for the dispersion function was caused bythe centroid statistical uncertainties of calibration lines,which were reduced by the statistics of the line centroids.The systematic dispersion function uncertainty of a mea-sured line was estimated by averaging the absolute val-ues of the fitted residuals of its calibration lines of all themeasured spectra.
Calibration systematic uncertainty.
Since the im-age of the calibration light source might not be over-lapped exactly with the trapped ion cloud, the spatialdeviation and misalignment could cause wavelength off-set between the measured lines and their calibration lines.In this work, a spatial deviation of less than 2 mm wouldresult in a wavelength uncertainty of less than 1 pm. The W a v e l eng t h ( n m ) Measurement sequence
FIG. 5. The calibrated wavelengths of line a in air derivedfrom a series of 26 measurements. The wavelength uncer-tainty of each single spectrum was calculated as the quadra-ture of the line centroid uncertainty and the 1- σ confidenceinterval of the fitted dispersion function. The weighted aver-age wavelength is represented by the solid purple line and itsuncertainty is represented by lilac band. misalignment could cause a wavelength uncertainty ofless than 1 pm, which was estimated from five measure-ments of the Ar
553 nm line by resetting the opticalsystem every time. Thereby, the overall systematic un-certainty caused by our calibration scheme was expectedto be less than 2 pm.
Other uncertainties.
In this work, the calibrationlight source and the fluorescence of the trapped ions wereexposed to the spectrograph almost simultaneously, in-dicating that the temperature drift and the mechanical
TABLE II. Uncertainty budget of the measured lines.Source Uncertainty in wavelength (pm)Line a b c d g
Line centroid 0.2 0.2 0.1 0.2 0.3Dispersion function 0.1 0.3 0.3 0.2 0.4Calibration systematic 2 2 2 2 2Total 2 2 2 2 2 drift were canceled out. The shifts due to the Stark effectand collisions can also be neglected at this level of accu-racy. The wavelengths of the selected calibration lines arereliable because their uncertainties in the NIST databaseare all less than 0.3 pm.Table II is the uncertainty budget for the lines a - d and g in air. The total uncertainty was cal-culated as the quadrature of all the uncertainties,which was dominated by the calibration systematicuncertainty. In order to test the reliability of theuncertainty estimation, the wavelengths in-the-air ofthe three lines from Ar HCI were measured, i.e. ,Ar III. THEORETICAL METHOD AND RESULTSA. MCDHF calculations
In the MCDHF method, an atomic wave function Ψis constructed as a linear combination of configurationstate functions (CSFs) Φ of the same parity P , the totalangular momentum J , and its projection M J , i.e. ,Ψ(Γ P JM J ) = N CSF (cid:88) i =1 c i Φ( γ i P JM J ) . (1)Here c i is the mixing coefficient and γ i stands for theremaining quantum numbers of the CSFs. Each CSF it-self is a linear combination of products of one-electronDirac orbitals. Both mixing coefficients and orbitals areoptimized in the self-consistent field calculation. After a set of orbitals is obtained, the relativistic configura-tion interaction (RCI) calculations are used to capturemore electron correlations. In addition to the Coulombinteractions, our RCI calculations also include the Breitinteraction in the low-frequency approximation and thequantum electrodynamic (QED) corrections.In order to obtain high-quality atomic wave functions,we designed an elaborate computational model as follows.In the first step, the self-consistent field (SCF) calcula-tions were performed successively to generate virtual or-bitals. The virtual orbitals were employed to form CSFswhich account for certain electron correlations. Morespecifically, CSFs were produced through single (S)- anddouble (D)-electron excitations from the occupied Dirac-Hartree-Fock orbitals to virtual orbitals, but the doubleexcitation from the atomic core 1 s s p orbitals wereexcluded at this stage. The virtual orbitals were aug-mented layer by layer up to n max = 12 and l max = 6,where n max and l max denote, respectively, the maximumprincipal quantum number and the maximum angularquantum number of the virtual orbitals. In the sec-ond step, the single-reference configuration RCI calcu-lations were performed with the configuration space con-structed from SD excitation from all occupied orbitalsto the set of virtual orbitals. In other words, the corre-lation between electrons in the atomic core, which wereneglected in the first step, were captured. In the last step,we considered part of contributions from the triple- andquadruple-excitation CSFs. In order to limit the numberof CSFs, the MR-SD approach was adopted to producecorresponding CSFs [46, 47]. The multi-reference (MR)configuration sets were created as { s p d , 3 s p d } for Ni , { s p d , 3 s p d , 3 p } for Ni , { s p d ,3 s d , 3 p } for Ni , and { p , 3 s p d , 3 p d } forNi . Additionally, the Breit interaction and the QEDcorrections were included in the RCI computation.Once the atomic wave functions are obtained, we are ina position to calculate the physical quantities under in-vestigation, that is, the reduced matrix elements for cor-responding rank k irreducible tensor operators betweentwo atomic states, i.e. , (cid:104) Ψ(Γ
P J ) (cid:107) O ( k ) (cid:107) Ψ(Γ (cid:48) P (cid:48) J (cid:48) ) (cid:105) . Themagnetic dipole and electric quadrupole transition op-erators are rank 1 and rank 2 operators, respectively.In practice, we performed the calculations using theGRASP2018 package [48].
B. Calculated wavelengths
As shown in Table I, the calculated wavelengths of the M a through line d and line g agreewith our measured values. The wavelengths of the two E e and line f in Ni and Ni were also calculated. These two lines have not been ob-served yet before. Our calculated wavelengths for thesetwo transitions are in agreement with the NIST recom-mended values. Meanwhile, the calculated wavelengthof line f also agrees with our indirect measurement, seeTable I. C. Properties of the clock transitions
The design of an atomic clock relies on the knowledgeof atomic parameters of the quantum oscillator. Thus,we have computed wavelengths, spontaneous emissionrates A , lifetimes τ , linewidths Γ (2 π Γ = 1 /τ ), and otherparameters for all six candidate clock transitions, andthe results are listed in Table III. As one of the key pa-rameters of clock stability, the quality factor ( Q -factor)is also given in this table. The Q -factor is defined asthe ratio of the clock frequency ν clk to the linewidthΓ of the clock transition, i.e. , Q = ν clk / Γ. Amongthe four M P − P transitionin Ni is the narrowest with its linewidth less than10 Hz, while the linewidths of the other three M Q -factors ofthese four M ∼ . There are twodecay channels from P in Ni and P in Ni tothe lower states. In order to determine the linewidth ofthese E P in Ni , the decayrate is 0.037 s − for the E P − P ) channel and0.011 s − for the M P − P ) channel. For P ofNi , the E P − P ) and M P − P ) transitionrates are 0.03 s − and 22.5 s − , respectively. Therefore,the linewidths for the E . P − P in Ni and 8 mHz for P − P in Ni ,which are respectively smaller than the M c and b , as marked in Fig. 1. This E is particularly attractive for stable clockwork [27],because of its relatively high Q -factor of 7 . × , mean-ing that the statistical uncertainty limited by the quan-tum projection noise [18, 49, 50] of this transition canreach the level of 10 − by averaging over a few days.From the perspective of searching for new physics, weanticipate that by monitoring the Ni HCI clock transi-tion frequencies, stringent constraints could be placedon the possible time variation of the fine structure con-stant α . Following Refs. [17, 52], one can introduce the“sensitivity coefficient” q , defined by ω ( x ) = ω + qx ,where x ≡ ( α/α ) − ω is the clock transi-tion frequency at the nominal value of the fine struc-ture constant α . The sensitivity coefficient q charac-terizes the linear response of the clock frequency ω ( x )to the variation of α , and can be calculated numericallyas q ≈ [ ω (+ x ) − ω ( − x )] / (2 x ). Another commonly usedquantity is the dimensionless enhancement factor [17] K = ∂ ln ω/∂ ln α ≈ q/ω . As shown in Table III,our computed K values for the relevant transitions innickel HCIs are about 2, which is higher than most ofthe current optical clocks. For example [53], Al + has K = 0 . ∼
10 species currently used in theoptical clock community, only the heavy Yb + and Hg + ions have | K | > − , the quan- tum clocks based on the relatively light Ni HCIs will havegreater potential for exploring new physics than most ofthe current atomic clocks. Recently, an improved con-straint of ˙ α/α = 1 . . × − / year was reported basedon the comparison of the S / ( F = 0) − D / ( F = 2)( E K = 1 .
00) and the S / ( F = 0) − F / ( F = 3)( E K = − .
95) transition of Yb + clock [54]. Theconstraint on the temporal variation of α is expected tobe further improved by comparing two clocks based onthe E and the E + ,because of its larger K value and smaller projected sys-tematic and statistical uncertainties of the E than those of the E + .Nandy and Sahoo [51] determined the sensitivity coef-ficient to the α -variation for the P / − P / transition inNi ion. In their work, the transition rate and the life-time of the P / state were calculated using the relativis-tic coupled-cluster (RCC) method. Yu and Sahoo [26, 27]calculated some atomic parameters for the P / − P / transition in Ni and the P − P transition in Ni with the RCC and MCDHF methods. Their results arealso listed in Table III for comparison. For lines a , d , and e , our calculated values agree well with other theoreticalresults [26, 27, 51], except for a factor of 3 difference forthe sensitivity coefficient q of line d . There is also a factorof 6 difference in the value of the Q -factor of line e , forwhich we traced back to the trivial factor of 2 π missingin the linewidth definition in Ref. [27].Previous theoretical work on nickel HCIs focuses onatomic properties relevant to the emission from the so-lar, astrophysical, and laboratory plasmas. In Tables IVand V, we present a comparison with the literature val-ues for the spontaneous decay rates and lifetimes. Over-all, our MCDHF values agree well with the results fromother theoretical methods, such as the RCC method andthe multi-reference Møller-Plesset perturbation theory.Moreover, the lifetimes of the P / state in Ni ion,the P state in Ni ion, and the P / state in Ni ion were measured at the heavy-ion storage ring [55–57].We found a good agreement between theory and experi-ment. D. Computational uncertainties
The computational uncertainties in our work includethe neglected correlation contributions, such as thetriple- and quadruple-electron excitations involving the1 s orbital. The upper limit on these effects was esti-mated from the double excitations of the core orbitalsin the single-reference configuration RCI calculations.The “truncation” uncertainties due to the finite numberof virtual orbitals were evaluated based on the conver-gence trends in the above-mentioned three steps. Forthe wavelengths, all the uncertainties were summed to-gether in quadrature. For the M TABLE III. Theoretical spectral properties of clock transitions. Here A is the Einstein coefficient for spontaneous decay, τ is the lifetime of the upper clock state, Γ is the natural linewidth, and Q is the transition quality factor. Also, q and K are,respectively, the sensitivity coefficient and enhancement factor for the variation of the fine structure constant. Numbers insquare brackets stand for the powers of 10, i.e. , x [ y ] ≡ x × y .Transition Type A (s − ) τ (ms) Γ (Hz) Q q (cm − ) K Ni s p P / − P / M s p P − P M P − P E P − P E P − P M s p P − P M P − P E P − P M P − P E s p P / − P / M − .Line This work Other theory a b c d e f frequency-dependent Breit interaction contribution asanother source of error. For the E IV. CONCLUSIONS
To reiterate, the quantum clockwork we explored hereprovides an intriguing possibility for achieving high accu-racy on multiple transitions in HCIs of the same element.Our strategy offers an important flexibility in the pursuitof multiple candidate clock transitions. Particularly, the E Ni has projected fractional uncer-tainty 10 − . We demonstrated the key experimentalcapabilities of using our SW-EBIT facility to generateand extract Ni , Ni , Ni , and Ni ions. Wemeasured the wavelengths of four M E M E ACKNOWLEDGMENTS
The authors thank Xin Tong, Jos´e R. Crespo L´opez-Urrutia, and Yan-Mei Yu for helps and for fruitfuldiscussions. This work was supported by the Strate-gic Priority Research Program of the Chinese Academyof Sciences (Grant No. XDB21030300), the Na-tional Natural Science Foundation of China (Grant Nos.11934014, 11622434, 11974382, 11604369, 11974080,11704398, and 11874090), the National Key Researchand Development Program of China under Grant No.2017YFA0304402, the CAS Youth Innovation PromotionAssociation (Grant Nos. Y201963 and 2018364), theHubei Province Science Fund for Distinguished Young
TABLE V. Lifetimes (in ms) of upper clock states in Ni , Ni , Ni , and Ni . Numbers in square brackets stand forthe powers of 10, i.e. , x [ y ] ≡ x × y .Ion State This work Other theory ExperimentNi
11+ 2 P /
12+ 3 P ∗ , 6.50(15) ∗∗ [56]Ni
12+ 3 P
14+ 3 P
14+ 3 P
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