Calculation of higher-order corrections to the light shift of the 5 s 2 1 S 0 -- 5s5p 3 P o 0 clock transition in Cd
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Calculation of higher-order corrections to the light shift of the s S - s p P o clocktransition in Cd S. G. Porsev , and M. S. Safronova , Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina, Leningrad District, 188300, Russia Joint Quantum Institute, NIST and the University of Maryland, College Park, Maryland 20742, USA (Dated: April 1, 2020)In the recent work [A. Yamaguchi et. al , Phys. Rev. Lett. , 113201 (2019)] Cd has beenidentified as an excellent candidate for a lattice clock. Here, we carried out computations neededfor further clock development and made an assessment of the higher-order corrections to the lightshift of the 5 s S - 5 s p P o clock transition. We carried out calculations of the magnetic dipoleand electric quadrupole polarizabilities and linear and circular hyperpolarizabilities of the 5 s S and 5 s p P o clock states at the magic wavelength and estimated uncertainties of these quantities.We also evaluated the second-order Zeeman clock transition frequency shift. I. INTRODUCTION
The Cd 5 s S - 5 s p P o transition has several de-sirable attributes for the development of a lattice clock.This clock has more than an order of magnitude smallerblackbody radiation (BBR) shift (a Stark shift resultingfrom the thermal radiation of the atoms environment,which is generally at 300 K temperature) in comparisonwith Sr and Yb [1–3]. The size of a BBR shift is a prop-erty of the specific atomic transition used as a frequencystandard and an uncertainty in the BBR shift is knownto be one of the limiting systematic uncertainties in theclock uncertainty budget [4, 5]. Short of cryogenic cool-ing, it cannot be suppressed and need to be quantifiedwith high accuracy.Two isotopes, Cd and
Cd, both with 12% naturalabundance, have a nuclear spin of 1/2, which precludestensor light shifts from the lattice light, another advan-tageous feature. Cd has the narrow 5 s S - 5 s p P o intercombination transition allowing Doppler cooling to1.58 µ K and simplifying a control of higher-order lat-tice light shifts [1]. The light for all of the transitionsneeded for the Cd clock, including the magic lattice, canbe generated by direct, or frequency-doubled or quadru-pled semiconductor lasers [1].In 2019, Cd clock magic wavelength was measured tobe 419 . . × − at 300 K in Ref. [1],in agreement with Ref. [3]. Recent progress opens a path-way to a rapid progress in Cd clock development and callsfor a detailed investigation of the clock systematic effects.In this work we calculated properties needed to quan-tify higher-order light shifts: magnetic dipole and electricquadrupole polarizabilities and linear and circular hyper-polarizabilities of the 5 s S and 5 s p P o clock statesat the magic wavelength and estimated uncertainties of these quantities. We also evaluated the second-order Zee-man clock transition frequency shift in the presence of aweak magnetic field.The paper is organized as follows. The general for-malism and main formulas are presented in Section II.In Section III, we briefly describe the method of calcula-tion. Section IV is devoted to a discussion of the resultsobtained, and Section V contains concluding remarks. II. GENERAL FORMALISM
We consider the Cd atom in a state | i (with the totalangular momentum J = 0) placed in a field of the latticestanding wave with the electric-field vector given by E = 2 E cos( kx ) cos( ωt ) . (1)Here k = ω/c , ω is the lattice laser wave frequency, c is the speed of light, and the factor 2 accounts for thesuperposition of forward and backward traveling alongthe x-axis waves. The atom-lattice interaction leads tothe optical lattice potential for the atom that at | kx | ≪ U ( ω ) ≈ − α E ( ω )(1 − k x ) E − { α M ( ω ) + α E ( ω ) } k x E − β ( ω )(1 − k x ) E . (2)Here α E , α M , and α E are the electric dipole, magneticdipole, and electric quadrupole polarizabilities, respec-tively, and β is the hyperpolarizability defined below.The ac 2 K -pole polarizability of the | i state with theenergy E is expressed (we use atomic units ~ = m = | e | = 1) as [7] α λK ( ω ) = K + 1 K K + 1[(2 K + 1)!!] ( α ω ) K − × X n ( E n − E ) |h n || T λK || i| ( E n − E ) − ω , (3)where λ stands for electric, λ = E , and magnetic, λ = M , multipoles and h n || T λK || i are the reduced matrixelements of the multipole operators, T E ≡ D , T M ≡ µ ,and T E ≡ Q .The expression for the hyperpolarizability of the | i state depends on the polarization of the lattice wave. Be-low we consider the cases when the lattice wave is linearlyor circularly polarized, and the 4th order correction to anatomic energy is determined by the linear or circular hy-perpolarizability, respectively.The expression for the linear hyperpolarizability β l ( ω )is given by [6] β l ( ω ) = 19 Y ( ω ) + 245 Y ( ω ) , (4)with the quantities Y ( ω ) and Y ( ω ) determined as Y ( ω ) ≡ X q R ( qω, qω, qω )+ X q,q ′ [ R ′ ( qω, , q ′ ω ) − R ( q ′ ω ) R ( qω, qω )] ,Y ( ω ) ≡ X q R ( qω, qω, qω ) + X q ′ R ( qω, , q ′ ω ) , and q, q ′ = ± β c ( ω ) can be writtenas β c = 19 X ( ω ) + 118 X ( ω ) + 115 X ( ω ) , (5)where X ( ω ) ≡ X q,q ′ [ R ′ ( qω, , q ′ ω ) − R ( q ′ ω ) R ( qω, qω )] ,X ( ω ) ≡ X q,q ′ ( − ( q + q ′ ) / R ( qω, , q ′ ω ) ,X ( ω ) ≡ X q R ( qω, qω, qω ) + 16 X q ′ R ( qω, , q ′ ω ) , and R J m J n J k ( ω , ω , ω ) ≡ X γ m ,γ n ,γ k h γ J k d k γ m J m i h γ m J m k d k γ n J n i h γ n J n k d k γ k J k i h γ k J k k d k γ J i ( E m − E − ω ) ( E n − E − ω ) ( E k − E − ω ) , (6) R J m ( ω ) ≡ X γ m |h γ J || d || γ m J m i| E m − E − ω , R J k ( ω, ω ) ≡ X γ k |h γ J || d || γ k J k i| ( E k − E − ω ) . (7)The notation R ′ , i.e., the prime over R , means that theterm | γ n i = | γ i (where γ n includes all other quantumnumbers except J ) should be excluded from the summa-tion over γ n in Eq. (6).The properties of the lattice potential for the Cd atomin its ground and excited clock states are determinedby Eq. (2) and depend on the frequency. Below weanalyze these properties at the experimentally deter-mined magic wavelength λ ∗ = 419 . ω ∗ , corresponding to this wavelength,is ω ∗ ≈ − ≈ . . u . .At the magic frequency the electric dipole polarizabil-ities of the clock 5 s S and 5 s p P o states are equal toeach other, i.e., α E S ( ω ∗ ) = α E P o ( ω ∗ ). These polarizabil-ities were calculated in Ref. [1] to be 63 . .
9) a.u..Using the formulas given above, we calculated the M E β l,c of the clock states at the magic fre-quency ω ∗ , found respective differential polarizabilitiesand hyperpolarizabilities, and determined uncertaintiesof these values. III. METHOD OF CALCULATION
We carried out calculations in the framework of high-accuracy relativistic methods combining configurationinteraction (CI) with (i) many-body perturbation the-ory (CI+MBPT method [8]) and (ii) linearized coupled-cluster (CI+all-order method) [9]. In these methods theenergies and wave functions are found from the multipar-ticle Schr¨odinger equation H eff ( E n )Φ n = E n Φ n , (8)where the effective Hamiltonian is defined as H eff ( E ) = H FC + Σ( E ) . (9)Here, H FC is the Hamiltonian in the frozen core ap-proximation and Σ is the energy-dependent correction,which takes into account virtual core excitations in thesecond order of the perturbation theory (the CI+MBPTmethod) or in all orders of the perturbation theory (theCI+all-order method).To accurately calculate the valence parts of the po-larizabilities and hyperpolarizabilities, we solve the in-homogeneous equation using the Sternheimer [10] orDalgarno-Lewis [11] method following formalism devel-oped in Ref. [12]. We use an effective (or “dressed”)electric-dipole operator in our calculations that includesthe random-phase approximation (RPA). To calculatesuch complicated quantities as R J m J n J k and carry outaccurately three summations over intermediate states, wesolve the inhomogeneous equation twice. A detailed de-scription of this approach is given in Ref. [6]. IV. RESULTS AND DISCUSSION
We carried out calculations of the M E A. Linear and circular hyperpolarizabilities of the S and P o clock states In calculating quantities given by Eqs. (6) and (7) amain contribution comes from valence electrons. Thecore electrons contribution is much smaller and we in-cluded it only to R ( ω ) terms.Indeed, as follows from Eq. (7), the quantity R ( ω, ω )can be treated as the derivative of R ( ω ) over ω , i.e., R ( ω, ω ) = ∂ R ( ω ) ∂ω = lim ∆ → R ( ω + ∆) − R ( ω )∆ . Since the core contribution to R ( ω ) is rather insensitiveto ω and ∆ is small, the core contributions to R ( ω + ∆)and R ( ω ) are practically identical and cancel each otherin the expression for R ( ω, ω ).Taking into account the uncertainty of our results forthe S and P o hyperpolarizabilities, we assume thatthe core contribution to the R Jn ( ω , ω , ω ) terms isalso negligible. This assumption is based on the cal-culation of the static hyperpolarizability for the Sr ground state that was found to be 62.6 a.u. [13]. Thisis negligibly small compared to valence contribution to R Jn ( ω , ω , ω ) in case of the quite similar 5 s S and5 s p P o clock states in Sr [6].The results of calculation of the linear and circularhyperpolarizabilities of the S and P o clock states arepresented in Table I. Our recommended value of differ-ential linear hyperpolarizability, ∆ β l ( ω ∗ ) = − . × a . u . , is two orders of magnitude smaller (in absolutevalue) than analogous differential hyperpolarizability forSr, ∆ β l = − . × a.u. [6]. In case of Cd, theabsolute values of the contributing terms are generally TABLE I: Contributions to the linear and circular hyperpolar-izabilities β l,c (5 s S ) and β l,c (5 s p P o ) (in a.u.) calculatedin the CI+all-order (labeled as “CI+All”) and CI+MBPT (la-beled as “CI+PT”) approximations at the magic frequency ω ∗ = 0 . β l,c ≡ β l,c ( P o ) − β l,c ( S ) is thedifference of the “Total” P o and S values. Numbers inbrackets represent powers of 10. The uncertainties are givenin parentheses. 5 s S s p P o Contrib. CI+All CI+PT CI+All CI+PT β l Y ( ω ) 3.61[4] 2.71[4] -5.30[5] -5.37[5] Y ( ω ) 5.64[4] 5.08[4] 4.37[5] 4.81[5]Total 9.24[4] 7.80[4] -9.23[4] -5.61[4]∆ β l -1.85[5] -1.34[5]Recommended − . × Ref. [2] − . × β c X ( ω ) -1.98[4] -1.88[4] -6.03[5] -5.95[5] X ( ω ) 41 34 7.21[6] 6.61[6] X ( ω ) 6.21[4] 5.53[4] -1.45[6] -1.11[6]Total 4.23[4] 3.66[4] 5.15[6] 4.90[6]∆ β c . × Ref. [2] 3 . × smaller than in Sr, and there are significant cancellationsbetween them.The circular hyperpolarizability of the P o state is twoorders of magnitude larger in absolute value than thecircular hyperpolarizability of the S state and the linearhyperpolarizability of the P o state. This is explained asfollows: the main contribution to β c ( P o )( ω ∗ ) comes fromthe term R ( ω ∗ , , ω ∗ ) ≡ X γ m ,γ n ,γ k h P o || d || γ m J m = 1 ih γ m J m = 1 || d || γ n J n = 1 ih γ n J n = 1 || d || γ k J k = 1 ih γ k J k = 1 || d || P o i ( E m − E P o − ω ∗ )( E n − E P o )( E k − E P o − ω ∗ ) . In the sum over γ n there is the intermediate state 5 s p P o separated from P o by the fine-structure inter- TABLE II: The dynamic M E s S and 5 s p P o states at the magic frequency,calculated in the CI+MBPT (labeled as “CI+MBPT”) andCI+all-order (labeled as “CI+All”) approximations. Therecommended value of ∆ α QM is given in the line “Recom.∆ α QM ”. The uncertainties are given in parentheses.Polariz. CI+MBPT CI+All α M ( S ) 1.5 × − × − α M ( P o ) -4.0 × − -3.9 × − ∆ α M -5.5 × − -5.5 × − α E ( S ) 2.29 × − × − α E ( P o ) 8.97 × − × − ∆ α E × − × − ∆ α QM × − × − Recom. ∆ α QM × − Ref. [2] 3.13 × − val. In this case the energy denominator E P o − E P o ≈
542 cm − is small and, respectively, the contribution ofthis term is large, leading to much larger hyperpolariz-ability for the circular polarization.We compare our results with those obtained in Ref. [2]in Table I. There is a reasonable agreement for differentialcircular hyperpolarizability while our differential linearhyperpolarizability is 5 times smaller in absolute valuethan that found in Ref. [2]. B. M and E polarizabilities at the magicfrequency To accurately calculate the valence part of the E Q in the right hand side. As inthe case of hyperpolarizability, we calculated these quan-tities using both the CI+all-order and CI+MBPT meth-ods, including the RPA corrections to the operator Q .The core contributions were calculated in the RPA. Forthe M α QM ≡ ∆ α E + ∆ α M , where∆ α M ≡ α M ( P o ) − α M ( S ) , ∆ α E ≡ α E ( P o ) − α E ( S ) . (10)To determine the uncertainty of ∆ α QM we note thatthe α M ( S ) polarizability is very small and we can neglect it. The α M ( P o ) polarizability is more thanthree orders of magnitude larger in absolute value than α M ( S ), but still an order of magnitude smaller than∆ α E . Therefore, the uncertainty of ∆ α QM is mostly de-termined by the uncertainty in ∆ α E and we estimate itto be 4%. Comparing our recommended value for ∆ α QM with the result obtained in Ref. [2], we see that there isa fair agreement between them. C. Second order Zeeman shift
In this section we consider a systematic effect due tosecond order Zeeman shift which both clock states expe-rience in the presence of a weak external magnetic field.If an atom is placed in a such magnetic field B , the in-teraction of the atomic magnetic moment µ with B isdescribed by the Hamiltonian H = − µ · B . (11)The atomic magnetic moment µ is mostly determined bythe electronic magnetic moment and can be written as µ = − µ ( J + S ) , (12)where J and S are the total and spin angular momentaof the atomic state and µ is the Bohr magneton definedas µ = | e | ~ / (2 mc ).Directing the external magnetic field B along the z -axis ( B = B z ≡ B ), we calculate the second order Zee-man shift, ∆ E , (in the absence of hyperfine interaction)as ∆ E = − α M1 B , (13)where α M1 is the magnetic-dipole polarizability. For astate | J = 0 i it is reduced to the scalar polarizability,given by α M1 = 23 X n |h n || µ || J = 0 i| E n − E . (14)To estimate the second order Zeeman shift for the clocktransition ∆ ν ≡ ∆ E ( P o ) − ∆ E ( S ) h we note that the α M ( S ) polarizability is negligiblysmall compared to α M ( P o ), so we can write ∆ ν ≈ ∆ E ( P o ) /h .For an estimate of α M ( P o ) we take into account thatthe main contribution to this polarizability comes fromthe intermediate state 5 s p P o . Then, from Eq. (14) weobtain α M1 ( P o ) ≈ h P o || µ || P o i E P o − E P o . (15)Using for an estimate |h P o || µ || P o i| ≈ √ µ (16)and substituting it to Eq. (13) we find∆ E ( P o ) ≈ − µ E P o − E P o B (17)in agreement with the result obtained in Ref.[14].Using the experimental value of energy difference E P o − E P o ≈
542 cm − , we arrive at∆ ν ≈ − B , where ∆ ν is in mHz and the magnetic field B is in G. V. CONCLUSION
We carried out calculations of the magnetic dipoleand electric quadrupole polarizabilities as well as lin-ear and circular hyperpolarizabilities of the clock 5 s S and 5 s p P o states at the magic wavelength and com-pared them with other available data. We also evaluated the second-order Zeeman shift for the clock transitionfrequency. These values are required for an assessmentof the higher-order corrections to the light shift of the5 s S - 5 s p P o clock transition. We have demon-strated that the linear differential hyperpolarizability forthe clock transition for Cd is two orders of magnitudesmaller than for Sr and Yb. We also found the circu-lar hyperpolarizability to be much larger than the linearhyperpolarizability and explained the source of this dif-ference. A knowledge of the multipolar polarizabilitiesand hyperpolarizabilities at different polarizations of thelattice wave is needed for further Cd clock developmentand selection of the lattice configurations to minimize thehigher-order light shifts. VI. ACKNOWLEDGEMENTS
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