Time keeping and searching for new physics using metastable states of Cu, Ag and Au

Cu has two long-lived metastable states ($N=2,3$ in Tab. 1), Ag has one ($N=4$), and Au has one ($N=2$). Only the states of Cu and Au have lifetimes substantially larger than 1 s, comparable to those of the currently used Sr and Yb lattice clocks. The 0.2 s lifetime of the Ag upper clock state may turn out to be a competitive disadvantage for achieving top performance. In the following we consider the clock transitions between these four states and their respective ground states. Note that the two clocks states of Cu are very similar and therefore in most of cases we present the data for only the ²D_5/2 state.

The atoms considered here all exhibit hyperfine structure in the ground state, in the clock state and in the excited state addressed in laser cooling. The nuclear spins are given in Tab. 3. The hfs splitting is given by Landau

The total angular momentum is $\mathbf{F}=\mathbf{J}+\mathbf{I}$, where $I$ is nuclear spin. $A$ and $B$ are magnetic dipole and electric quadrupole hfs constants, respectively. They are reported in Tab. 4.

For example, the hyperfine structure of Au has been studied experimentally with high precision in the 1960s, and has also been calculated Itano ; Bieron . The hyperfine splitting between $F=1,2$ in the ground state amounts to 6.10 GHz Dahmen . The splittings in ²D_5/2 are Au-hfs

$F=1\leftrightarrow F=2$: 1.00 GHz,

$F=2\leftrightarrow F=3$: 0.71 GHz, and

$F=3\leftrightarrow F=4$: 0.52 GHz.

$F=0\leftrightarrow F=1$: 1.11 GHz,

$F=1\leftrightarrow F=2$: 1.31 GHz, and

$F=2\leftrightarrow F=3$: 0.31 GHz.

Knowledge of the atomic polarizabilities for both states of the clock transition is important for estimation of the frequency shift caused by black-body radiation and for finding the so called magic frequency of the lattice laser field, i.e. the frequency at which the dynamic polarizabilities of both states are equal, causing no frequency shift.

The static scalar polarizability $\alpha_{v}(0)$ of an atom in state $v$ is given by

where $D$ is the electric dipole operator with the RPA correction (see the previous section), and the summation goes over the complete set of excited many-electron states.

Static scalar polarizabilities of the ground states of Cu, Ag and Au are known from a number of calculations and measurements pol0 . Table 5 presents the recommended values taken from Ref. pol0 . In contrast, to the best of our knowledge, there is no similar data for the upper clock states of Cu, Ag and Au. Therefore, we performed the calculations using two different approaches.

In the first approach we stay within the CIPT method and calculate twenty odd-parity states for each value of the total angular momentum $J$ which satisfies the electric dipole selection rules for the transitions from the ground and clock states ($J=1/2,3/2,5/2,7/2$). Then we use the formula (6) to perform the calculations for both states. These calculations show three important things: (a) there is good agreement with other data for the ground state, (b) there is good saturation of the summation in (6), (c) the summation for the clock states is strongly dominated by the transitions to the states of the $5d^{9}6s6p$ configuration (we use the Au atom as an example).

The last fact implies that a different approach can be used, previously suggested for atoms with open $f$-shells AKozlov . In this second approach we use the fact that the sum (6) is dominated by the $6s$ - $6p$ transitions while the open $5d^{9}$ subshell remains unchanged. Therefore, the open $d$-shell is attributed to the core and treated as a closed shell with an occupational number of 0.9. The atom is treated as a system with two external electrons above the closed-shell core and an appropriate CI+MBPT CI+MBPT method is used (see Ref. AKozlov for more details). The advantage of this approach is the efficient completeness of the basis with two electron excitations. The shortcoming is the omission of the transition amplitudes involving excitations from the $d$-shell. In contrast, the CIPT approach includes all amplitudes; however the summation in (6) is truncated much earlier.

In the end, both approaches give similar results. The results for the clock states are presented in Table 5 together with estimated uncertainties. For these estimations we used a comparison of the two approaches for the clock states as well as comparison of the CIPT calculation with other data for the ground states.

The results of the calculations indicate that the values of the polarizabilities of the clock states of Cu, Ag and Au are similar to those of the ground state. This is a non-standard situation. More often, the polarizabilities of excited states are larger. Indeed, the higher is the state on the energy scale, the smaller is the energy denominator in (6). Present results can be explained by the fact that summation in (6) is dominated by the states of the $5d^{10}np$ configurations for the ground state (we use Au again as an example) and by the states of the $5d^{9}6s6p$ configuration for the clock state. The later states are higher on the energy scale.

The BBR shift is given by (see, e.g. BBR )

where $\alpha$ is the fine structure constant, $T$ is temperature, $\alpha_{c}(0)$ and $\alpha_{g}(0)$ are static scalar polarizabilities of the clock and ground states. For simplicity, we do not include the dynamic correction to the BBR shift. For the more complete formula see, e.g. BBR .

The BBR shift is proportional to the difference between the polarizabilities of the two states. The similarity of the polarizabilities implies a substantial cancellation of the black-body radiation (BBR) frequency shift, a very favourable effect. Since the differences are smaller or close to the uncertainty of both numbers we can only give upper limits for the BBR shifts. The results are presented in Table 5. These limits are lower than the shifts in the standard ytterbium and strontium lattice clocks. More accurate estimations might be possible if the polarizabilities are measured or calculated to higher accuracy.

Magic frequencies can be found in the vicinity of every resonance for one of the polarizabilities, i.e. when the frequency of the lattice laser field is approximately equal to the excitation energy (energy denominator in (6)). The first magic frequency is near the first resonance for the ground state polarizability, i.e. $\hbar\omega_{m}\simeq 30535\,{\rm cm}^{-1}$ for Cu, $\hbar\omega_{m}\simeq 29552\,{\rm cm}^{-1}$ for Ag and $\hbar\omega_{m}\simeq 37359\,{\rm cm}^{-1}$ for Au. Note that since the clock states have large values of the total angular momentum ($J=5/2$), the magic frequencies would also depend on the quadrupole contribution to the polarizabilities. The current level of computational accuracy does not allow to find accurate values of the magic frequencies. Having more experimental data may help. In the vicinity of a resonance or a few resonances a semi-empirical formula can be used

where $\alpha^{\prime}_{a}(0)$ is chosen in such a way that $\alpha_{a}(\omega=0)$ is equal to known (e.g. experimental) static polarizability of state $a$. Summation in (8) goes over close resonances. If the static polarizability is known to sufficient accuracy and amplitudes $A_{ab}$ of E1-transitions are extracted from experimental data or from accurate atomic calculations, then (8) can be used to find magic frequencies.

Interaction of atomic electrons with external electric field and its gradient lead to Stark and electric quadrupole shifts of transition frequencies. These shifts are tiny in optical lattice clocks ¹¹1On the inner surface of a metallic vacuum chamber there can be spatial variations of the electrostatic potential of order 0.1 V. The typical size of a vacuum chamber may be 10 cm. Thus, the electric field gradient is smaller than 0.1 V/(10 cm$)^{2}$. Assuming a typical quadrupole moment $Q\sim 1$ a.u. leads to a negligible quadrupole shift $\sim 10^{-5}$ Hz. The corresponding Stark shift is $\sim 10^{-7}$ Hz.. We consider the shifts in more details in the Appendix.

The linear Zeeman shift is given by the expression

where $g_{F}$ is the $g$-factor of a particular hfs state. It is related to the electron $g_{J}$-factor by

Electron $g$-factors have approximate values $g_{1/2}\approx 2$, $g_{3/2}\approx 0.8$, $g_{5/2}\approx 1.2$. More accurate values for Cu, Ag and Au can be found in NIST tables NIST . For a clock state with $J=5/2$ and $F=2$ we have $g_{2}=(11/12)g_{5/2}=1.1$. For a clock state with $J=3/2$ and $F=2$ we have $g_{2}=(1/2)g_{3/2}=0.4$.

The linear Zeeman shift can be avoided if only transitions between states with $F_{z}=0$ are considered, as sug- gested in the past for clock operation. Alternatively, one can average over the transition frequencies with positive and negative $F_{z}$ in order to cancel the linear shift. However, the large individual shifts will make it difficult to achieve an accurate cancellation.

A second-order Zeeman shift is unavoidable. Therefore, it is important to know its value. If we consider transitions between definite hfs components, then the shift is strongly dominated by transitions within the same hfs multiplet. The total shift is the difference between the second-order shifts in the clock and in the ground state. Both shifts are given by

Here $\Delta E_{\rm{HFS}}(F,F^{\prime})=E(F\,I\,J)-E(F^{\prime}\,I\,J)$ is the hfs interval. It has a different sign depending on whether this is an up or down transition.

It follows from (III.2) that

and

Using experimental values for $A$ and $B$ (see Table 4) we calculate the second-order Zeeman shift for Cu, Ag and Au. The results are presented in Tables 6 and 7. The shift for Ag was studied before AgZ2 . Our result differs from theirs; this may be due to a simple calculational error. Table 6 presents separate contributions from the shifts in the ground and excited states. One can see that the disagreement may come from the sing error in a particular contribution. Different signs are caused by energy denominators. For example, when we move from first to the second line of the table, the sign of the energy denominator for the ground state contribution changes and so does the contribution itself. Since other contribution remains the same, the total shift must change.

Table 7 shows the second-order Zeeman shift for ⁶³Cu and ¹⁹⁷Au. As in the case of ¹⁰⁷Ag the shift is small. Note that Cu has one clock transition with both tiny quadratic shift coefficient and no linear shift. By measuring two or more $F_{gz}=0\rightarrow F_{ez}=0$ Zeeman components and taking appropriate combinations of the corresponding transition frequencies the second-order shift may be substantially reduced.

The quadratic shift vanishes in the considered approximation for transitions between states with maximum value of $F$ and its projection $F_{z}$ (see bottom lines of Table 7). This is because there are no terms in (11) which would satisfy the selection rules. Note also that the (non-zero) numbers in Table 7 should be considered as rough estimations only. This is because of uncertainties of the experimental data for the electric quadrupole hfs constant $B$, in particular for Cu Cu-hfs1 . The numbers can change several times depending on which set of data is used.

It is convenient to present to parametrise the $\alpha$-dependence of atomic frequencies by the formula $\omega=\omega_{0}+q[(\frac{\alpha}{\alpha_{0}})^{2}-1]$ CJP , where $\alpha_{0}$ and $\omega_{0}$ are present-day values of the fine structure constant and the frequency of the transition, $q$ is the sensitivity coefficient which comes from the calculations. To monitor possible frequency change one atomic frequency is measured against another. Then

The value $K=2q/\omega$ is called an enhancement factor. It shows that if $\alpha$ changes in time then $\omega$ changes $K$ times faster. Calculated values of $q$ and $K$ for different optical clock transitions are presented in Table 8. Note that we include not all known clock transitions but only those which are sensitive for $\alpha$-variation searches. There are six transitions where $|K|>1$. The largest values of $|K|$ correspond to the smallest values of transition frequency $\omega$. It would be wrong to say that all these transitions are good for searching for $\alpha$-variation. This is because the accuracy of the measurements is equally important (see also discussion in Ref. Yb-DFS ). The true figure of merit is the ratio of the relative frequency shift due to variation of $\alpha$ and the fractional uncertainty of the measurements, $(q/\omega)/(\delta\omega/\omega)=q/\delta\omega$. This ratio does not depend on $\omega$. Therefore, looking for a large value of $K$ caused by the small value of $\omega$ brings no benefit. The value of the relativistic energy shift $q$ is more important. Comparing the values of $q$ for different clock transitions (see Table 8) we see that the E2 clock transition in Au is essentially as good as the recently proposed new transitions in neutral ytterbium and only 30% smaller than the octupole transition in the ytterbium ion (Yb II).

In the standard model extension, the term in the Hamiltonian responsible for the LPI violation can be presented in the form (see, e.g., Ref. EEP )

where $c_{00}$ is the unknown parameter characterizing the magnitude of the LPI violation, $U$ is the gravitational potential, $c$ is the speed of light, $\hat{K}=c\gamma_{0}\gamma^{j}p_{j}/2$ is the relativistic operator of kinetic energy in which $\gamma_{0}$ and $\gamma^{j}$ are Dirac matrices, and $\mathbf{p}=-i\hbar\mathbf{\nabla}$ is electron momentum operator.

The presence of the term (13) in the Hamiltonian would manifest itself via a dependence of the atomic frequencies on the time in the year, caused by the changing Sun-Earth distance leading to change of the Sun’s gravitational potential $U$. As in the case of the $\alpha$-variation, at least two clock transitions are needed to measure one clock frequency against the other. The interpretation of the measurements is based on the formula EEP

where $\Delta\omega$ and $\Delta U$ are the change of atomic frequencies and gravitational potential between the measurements, respectively. $R$ in (14) is the relativistic factor which describes the deviation of the kinetic energy $E_{K}$ from the value given by the non-relativistic virial theorem (which states that $E_{K}=-E$, where $E$ is the total energy),

The values of the factor $R$ are calculated in computer codes by varying the value of the kinetic energy operator in the Dirac equation (see Ref. EEP for details).

The results are very sensitive to the many-body effects which means that the effects should be treated very accurately or avoided. Otherwise the results are unstable. A good criterion for the reliability of the results is the achievement of the non-relativistic limit $R=1$. This can be done by setting to zero the value of the fine structure constant $\alpha$ in the computer codes. It turns out that for complicated systems like those considered in present work the best results are obtained by simple estimations based on single-electron consideration. Namely, all clock transitions in Cu, Ag and Au can be considered as a $ns\rightarrow(n-1)d_{5/2}$ single-electron transitions ($n=4,5,6$ for Cu, Ag, Au respectively). Therefore, we just use single-electron energies of theses states in (15). We use the same approach for Hg⁺ and Yb⁺.

The results are presented in Table 8 together with results obtained earlier for other systems. Note, that the results for the transitions involving excitation from the $5d$ shell in Hg⁺ and the $4f$ shell in Yb⁺ are different from what was published before. The old calculations were based on a version of the CI method CI1 ; CI2 which contained a fitting parameter responsible for correct energy interval between states of different configurations. It was assumed that this parameter does not change under variation of the kinetic energy operator. We believe that present results are more reliable because they are free from any assumptions and because they reproduce the non-relativistic limit $R=1$. Note that the values of $R$ for transitions in Yb and Yb⁺, which do not involve excitation from the $4f$ shell are in good agreement with previous calculations ($R=1.12$ and $R=1.42$, see Table 8). This means that present single-electron estimations work well and that accurate many-body calculations are possible for simple systems.

To study the LPI violation one needs to compare two clocks with different values of the relativistic factors $R$ (see formula (14)) over at least half of a year. Table 8 shows that there is a wide range of choices for such clock pairs. In particular, the Au clock is practically as good as the Hg⁺ clock which was used before for this purpose in combination with the Al⁺ clock Hg+clock0 ; EEP . The Cu and Ag clocks, which have $R$ values close to unity, can be used in combination with clocks with large relativistic effects, e.g. Yb or Yb⁺ clocks.

The LLI violation term is a tensor operator

where $C_{0}^{(2)}$ is unknown constant and the relativistic form of the $T_{0}^{(2)}$ operator is given by $T_{0}^{(2)}=c\gamma_{0}(\gamma^{j}p_{j}-3\gamma^{3}p_{3})$.

To study the effect of the LLI violating term (16) one needs long-lived atomic states with large value of the total electron angular momentum $J$, $J>1/2$. All clock states of Cu, Ag and Au satisfy this requirement. This term should cause a dependence of the atomic frequencies on the apparatus orientation in space (e.g., due to Earth rotation). Interpretation of the measurements requires knowing the values of the reduced matrix elements of the operator $T_{0}^{(2)}$ for the clock states. We calculate these matrix elements using the CIPT method to obtain wave functions and the RPA method to obtain effective operator for valence electrons.

The results are presented in Table 8. The results of earlier calculations for Yb II Yb+LLI are also presented for comparison. In contrast to the search of the $\alpha$-variation and LPI violation, one clock state is sufficient for the search of the LLI violation. The comparison of frequencies is done for states with different projections of the total angular momentum $J$ Lia ; Yb+LLI . The large value of the matrix element is important but it is not the most important parameter, e.g. the lifetime of the metastable state is even more important (see, e.g. Yb+LLI for more discussion). The calculations for Cu and Au show that these systems are also suitable for the search of the LLI violation. Ag appears less well suited because its upper clock state has a significantly shorter lifetime.