Performance of a ²²⁹Thorium solid-state nuclear clock

The isomer transition energy of ²²⁹Th is currently derived from indirect measurements of the γ-ray spectrum resulting from the decay of ²³³uranium. In 1989–1993 Helmer and Reich performed the first measurements using high-quality germanium detectors (resolution from 300 to 900 eV). They estimated the ²²⁹Th isomer state energy to be E = 3.5 ± 1.0 eV [18, 19]. This unnaturally low value triggered a multitude of investigations, both theoretical and experimental, trying to determine the transition energy precisely and to specify other properties of the isomer state of ²²⁹Th (such as the lifetime and the magnetic moment). However, searches for direct photon emission from the low-lying excited state have failed to report an unambiguous signal [20–23]. In 2005, Guimãraes-Filho and Helene re-analyzed the old experimental data and reported the value of E = 5.5 ± 1.0 eV [24]. New indirect measurements with an advanced x-ray microcalorimeter (resolution from 26 to 30 eV) were carried out by Beck et al in 2007 [25]. They published a new value for the transition energy E = 7.6 ± 0.5 eV for the isomer nuclear transition, shifting it into the VUV domain. This shift probably explains the absence of signatures of the transition in previous experiments. In 2009, Beck et al [6] reviewed their results taking into account the nonzero probability of the 42.43 keV → ^229mTh transition (estimated as 2%) and published a revised version E = 7.8 ± 0.5 eV. This value is currently the most accepted one in the community but cannot be considered definite until a direct measurement is made successfully. The dominant uncertainty in the prediction of Beck et al [25] is connected to the value of the branching ratio b = 1/13 from the 29.19 keV level to the ground state [26]. Estimations of this value in various references differ. In [26], two alternative values for this branching ratio are mentioned: 25% that would result in E = 9.3 ± 0.6 eV and 51% that would result in E = 14.0 ± 1.0 eV. In the present analysis, we use the value of 7.8 ± 0.5 eV; scaling the obtained results to different transition energies is straightforward.

While a direct measurement of the isomer transition energy in ²²⁹Th is outstanding, the decay rate γ of the isomer state also remains unknown. The most recent estimations of the half-life of the isomer state for a bare nucleus are based on theoretical calculations of the matrix element of the magnetic moment [27]. The theory was verified by a comparison with experimental data for transitions at higher energies. It predicts a half-life of T_1/2 = (10.95 h)/(0.025E³) for the isomer transition, where E is given in eV. For E = 7.8 eV, this yields T_1/2 = 55 min, which corresponds to a spontaneous decay rate of γ_s = 2.09 × 10⁻⁴ s⁻¹. In [28], it is predicted that the probability of spontaneous magnetic dipole emission in a transparent non-magnetic dielectric medium with refractive index n is n³ times higher than that in vacuum. The reason for this factor is the renormalization of the density of emitted photon states. For a CaF₂ crystal, n = 1.55 at a wavelength λ = 160 nm (corresponding to E = 7.8 eV); the enlargement factor is 3.7 for the spontaneous decay rate of the isomeric state. Therefore, if the crystal contains N_e excited thorium nuclei, the number dN_ph of 160 nm photons emitted during a short time interval dt can be estimated as

$$\begin{equation} \mathrm{d}N_{\mathrm{ph}}=N_{\mathrm{e}}\gamma_{\mathrm{s}} n^3 \,\mathrm{d}t. \end{equation} \tag{ 1 }$$

The total decay rate γ is determined not only by the spontaneous decay of the nucleus; it can be enhanced by interactions between the nucleus and the electron shell in electronic bridge processes and bound internal conversion effects [29, 31, 32]. Such a decay can be accompanied by the emission of several low-energy photons with widely distributed energies [29]. This would make it difficult to separate decay events from optical noise sources (i.e. from scintillation or radioluminescence).

There are a number of theoretical calculations on electronic bridge processes in the thorium system. In an isolated neutral atom the thorium isomer state lifetime is expected to be about 4.5 min for E = 3.5 eV and about 10⁻⁵ s for E = 7.6 eV [29]⁶ . In the Th³⁺ ion for E = 7.6 eV, the expected lifetime is the same as that in a bare nucleus described above if the valence electron is in the ground 5f_5/2 state. It decreases by a factor of 20 if the valence electron is in the metastable 7s state [31]. In the Th⁺ ion the lifetime decreases 10²–10³ times for E = 3.5 and 5.5 eV [32]; there are not sufficient data about the Th⁺ ion electronic structure to calculate the lifetime at E = 7.8 eV. Similar calculations for the Th⁴⁺ ion, as will occur in the solid-state approach presented here, are yet to be performed.

There have been several experimental attempts to determine the ^229mTh lifetime without knowledge of the exact isomer energy. Inamura and Haba [33] measured the temporal variation of the α decay activity in a sample of ²²⁹Th after excitation in a hollow-cathode electric discharge. They proposed a value of T_1/2 = 2 ± 1 min for the half-life of the isomeric state, which corresponds to a total excited state decay rate of γ = 6 ⁺⁶₋₂ × 10⁻³ s⁻¹. Kikunaga et al [34] studied the α spectrum of chemically 'fresh' ²²⁹Th produced in the α decay of ²³³U. The authors obtain T_1/2 < 2 h with 3σ confidence. The latter experiment was not suited to measuring half-lives of the order of minutes because of the long chemical preparation process. The analysis of both the experiments is significantly complicated by the presence of various chemical compositions of thorium, mainly ThO⁺, Th and Th⁺ in [33] and hydroxide and chloride complexes in [34]. These chemical structures determine the energies of electronic levels and hence the rates of electronic bridge processes. Therefore the different results obtained in [33, 34] may be explained by the different chemical structures and the times of preparation and measurement.

In this work, we assume the band gap of the thorium-doped CaF₂ material to be sufficiently large to neglect electronic bridge processes and contributions from low-energy photons to the measured fluorescence count rates. In other words, we expect no electronic levels near 7.8 eV in the thorium-doped CaF₂ crystal. This assumption will be substantiated in a future publication [16]. We perform our analysis using the theoretical value for the bare nucleus lifetime γ = γ_s·n³ = 7.8 × 10⁻⁴ s⁻¹, enhanced by the refractive index, and assume the relaxation via electronic bridge processes to be negligible. Again, as for the energy, scaling our results to different values of the ^229mTh lifetime is straightforward.

In this work, we consider the particular system of thorium nuclei doped into calcium fluoride single crystals. CaF₂ is a standard material in UV optics; it is routinely grown and doped in many laboratories and companies worldwide using various techniques (Czochralski, Bridgman–Stockbarger and micro-pulling down). Standard procedures for cutting, cleaving, polishing and coating are established; CaF₂ is well characterized concerning radiation damage, both concerning UV laser radiation and radiation related to the nuclear decay chain of ²²⁹Th.

CaF₂ has a wide band gap of about 12 eV; the values in the literature range from 11.6 eV [14] to 12.1 eV [15]. The valence band of CaF₂ consists of 2p levels of the fluorine ions, while the bottom of the conduction band originates from the 4s and 3d orbitals of the calcium ions. Optical transmission of better than 70% in the 160 nm wavelength regime is routinely achieved in high-purity single crystals.

The simple cubic lattice structure and the fact that only the Ca²⁺ site qualifies as a doping site make thorium-doped CaF₂ amenable to full electronic structure simulations [16], which is currently out of reach for much more complex crystal structures (i.e. LiCaAlF₆ as proposed in [12]).

Thorium atoms in a solid-state crystal lattice are confined in the Lamb–Dicke regime; the recoil energy E_rec = (7.8 eV)²/(2M_Thc²) ≃ 1.4 × 10⁻¹⁰ eV is far below the energy required for creating a phonon with a recoil momentum p_rec = 7.8 eV/c in the lattice. Therefore the only process that could transfer the recoil energy to the lattice vibrations must include the scattering of optical phonons. However, optical modes are expected to be frozen out even at room temperature [12]. Therefore, internal and external degrees of freedom are decoupled and there is no sensitivity of the nuclear transition to recoil or first-order Doppler effects. However, the interaction between the thorium nuclear levels and the crystal lattice environment can cause some inhomogeneous effects which will affect the performance of a solid-state nuclear clock.

We assume the ²²⁹Th ions to be chemically bound into a VUV transparent ionic crystal. Therefore the fine interaction of the thorium nuclei with the crystal lattice environment is absent because of the absence of unpaired electronic spins. The remaining interaction is the hyperfine interaction, where the Hamiltonian $\skew3\hat {H}'=\skew3\hat {H}_{\mathrm {HFS}}$ can be represented in a multipole expansion $\skew3\hat {H}_{\mathrm {HFS}}=H_{E0}+H_{M1}+H_{E2}+\cdots$ [12]. In the following, we will discuss the individual terms in more detail.

The electric monopole shift (H_E0) arises from contact interaction between the electron cloud and the finite nuclear volume. As the ground state and the excited nuclear state can have different sizes, this effect leads to a shift of the nuclear transition relative to a bare nucleus. Typical monopole shifts are about 1 GHz [35]; they will depend on the specific choice of the host crystal. However, this effect will be identical (up to temperature effects) for all nuclei having the same position within the lattice and hence deteriorate the solid-state nuclear clock accuracy, but not the stability. Temperature effects appear because the electron density at a given lattice site is temperature dependent. This temperature dependence is estimated as 10 kHz K⁻¹ [12], which leads to a broadening of about 10 Hz for a temperature stability of ΔT ≈ 1 mK, that is attainable by standard techniques. It should be noted that a certain contribution to the decoherence rate can come from crystal lattice vibrations. These vibrations will modulate the electronic density in the vicinity of the nucleus and thus broaden the transition line. A detailed study of this broadening effect is beyond the scope of this paper. We expect that this effect, if relevant at all, can be effectively suppressed by cooling the crystal lattice.

The electric quadrupole shift (H_E2) appears due to the electric quadrupole moment Q of the ²²⁹Th nucleus in the presence of a non-vanishing electric field gradient at the position of the nucleus. Reported experimental values for the quadrupole moment in the ground state are Q_g = 3.149(32) eb [36] and Q_g = 3.11(16) eb [37]; the value Q_is of the quadrupole moment in the isomer state is still unknown; estimations [38] give Q_is ≈ 1.8 eb. If the electric field gradient in the principal axes has two equal components, ϕ_xx = ϕ_yy = −ϕ_zz/2, the matrix element of the quadrupole interaction term can be written as [39]

$$\begin{equation} \langle Im|\skew3\hat{H}_{E2}|Im'\rangle =\delta_{mm'}\frac{Q \phi_{zz}}{4I (2I-1)}\left[3m^2-I(I+1)\right], \end{equation} \tag{ 2 }$$

where the z-axis is the axis of symmetry of the electric field, ϕ_zz = ∂²φ/∂z² and I and m are the nuclear angular moment and its projection on the axis of symmetry of the electric field, respectively. In a Th-doped CaF₂ crystal, the thorium ion will most probably replace a Ca²⁺ ion in a Th⁴⁺ state. To recover an ionic (and hence transparent) crystal, the two remaining electrons will be localized by a charge compensation mechanism, either by introducing two additional fluorine (F⁻) interstitials or by creating a vacancy by leaving a neighboring Ca²⁺ site empty. First simulations indicate a preference for charge compensation by interstitials which will be the main contribution to the electric field gradient [40].

Consider the charge compensation by two F⁻ interstitials as shown in figure 1(b). We expect the geometrical lattice distortion around the dopant to be negligible. Using the Ca²⁺ lattice constant a = 5.6 Å and the quadrupole antishielding factor γ_∞ = −177.5 [41], we estimate ϕ_zz ≈ −5.1 × 10¹⁸ V cm⁻². This leads to quadrupole splittings of the order of a few hundreds of MHz, which can be easily resolved in laser spectroscopy (see figure 2). Note that this splitting could also be directly measured experimentally using the standard NMR techniques, which would give valuable information on the details of the charge compensation mechanism.

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The quadrupole interaction of the nucleus with phononic modes of the crystal lattice leads to inhomogeneous broadening and mixing between different quadrupole sublevels as the typical distances between the quadrupole sublevels are much shorter than characteristic phonon energies. Comparison with experimental data for the quadrupole spin-lattice relaxation of the halogen nuclei in CsBr and CsI crystals [42] and scaling the theory of Van Kranendonk [43] and Yoshida and Moriya [44] to the CaF₂ crystal yield a relaxation and mixing rate of the order of 2π·(5–100) Hz at room temperature. This rate scales as T², leading to about 2π·(0.3–7) Hz at liquid nitrogen temperature. The temperature of the crystal should, however, not be so low as to entirely suppress the mixing, as populations accumulating in quadrupole sublevels not interacting with the laser field could lead to a loss of signal or necessitate additional repump lasers.

The second-order Doppler effect leads to a shift and broadening of the isomer transition due to vibrations of the nuclei. Using a classical harmonic oscillator model, we find the shift S = −ω〈v²〉/(2c²) = −3ωk_BT/(2M_Thc²) and the broadening $\Delta \omega =\omega \langle \Delta v^2 \rangle /(2 c^2) = \sqrt {6}\,\omega k_{\mathrm {B}} T/(2M_{\mathrm {Th}}c^2)$. Here ω is the isomer transition frequency, v the velocity of the thorium nuclei, $\langle \Delta v^2 \rangle =\sqrt {\langle v^4 \rangle - \langle v^2 \rangle ^2}$. This leads to a shift of about 2π·1.14·T  (in Hz) and to a broadening of about 2π·0.93·T  (in Hz) where T is the temperature in kelvins. At liquid nitrogen temperature, we expect a shift of about 2π·88 Hz and a broadening of about 2π·72 Hz.

The term H_M1, i.e. the magnetic dipole interaction with randomly oriented spins of surrounding nuclei, also contributes to broadening. The characteristic energies of spin–spin interactions correspond to the temperature range of the order of 1 μK; the broadening is expected to be temperature independent for experimentally relevant conditions, in contrast to all other broadenings discussed above, which decrease with decreasing temperature. A detailed evaluation (see appendix A) yields a contribution to the optical relaxation rate from ∼2π·84 Hz for the $|I=5/2,m={\pm } 3/2\rangle \leftrightarrow |I'=3/2,m'={\pm } 1/2\rangle$ optical transition to ∼2π·251 Hz for the $|I=5/2,m={\pm } 5/2\rangle \leftrightarrow |I'=3/2,m'={\pm } 3/2\rangle$ optical transition.

To proceed, we expect a global uniform shift of about 1 GHz dominated by the electric monopole shift compared to the bare nucleus isomer transition frequency (assuming identical doping complexes and charge compensation). The main sources of inhomogeneous broadening are magnetic dipole interaction and the second-order Doppler effect, leading to an ensemble linewidth of 2π·(84–251) Hz and 2π·0.93 Hz · T, respectively, where T is the temperature in kelvins. Taking the ensemble linewidth of 2π·150 Hz, we expect a quality factor Q = 1.26 × 10¹³ for the solid state clock transition. Note that crystal field shifts and broadenings (and their temperature dependence) can be directly studied with narrow band lasers, transferring the Mößbauer spectroscopy into the optical regime.

To tackle these problems, we propose an interrogation scheme based on counting the spontaneous nuclear decay fluorescence photons after illuminating the quantum discriminator with the spectroscopy laser. The (n, n + 1) interrogation cycle consists of four time intervals, as depicted in figure 4. In the first interval θ between t_n and t'_n, the crystal sample is illuminated by laser radiation with the frequency detuned by δ_m to the blue side of the nominal frequency ω_n of the generator. Then a laser shutter is closed and the fluorescence photons are counted by a photodetector during the second time interval θ' between t'_n and t_n+1. Then these two operations are repeated, but with the laser frequency shifted to the red side by δ_m of the nominal value. After the last measurement, we determine the frequency offset f = ω_n − ω and make a correction to the nominal frequency. As no further state initialization, repumping or sample preparation is required, the next interrogation cycle can start immediately.

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We will now consider a single interrogation cycle in more detail. Solving equation (14), we express the excited state population ρ_exc at time t'_n as

$$\begin{eqnarray} &&\fl \rho_{\mathrm{exc}}(t'_n)=\rho_{\mathrm{exc}}(t_n)\exp\left[-\theta\cdot\left(\gamma+\frac{5R/2}{1+(\delta_m+f)^2/\Gamma'\,^2}\right)\right]+\frac{R}{\gamma[1+(\delta_m+f)^2/\Gamma'\,^2]+5R/2}\nonumber\\ &&\times \left\{1- \exp\left[-\theta\cdot\left(\gamma+ \frac{5R/2}{1+(\delta_m+f)^2/\Gamma'\,^2}\right)\right] \right\}. \end{eqnarray} \tag{ 16 }$$

The number N_n of nuclear fluorescence photons counted by the photodetector during time θ' between t'_n and t_n+1 is

$$\begin{equation} N_{n}=N_{\mathrm{eff}}\cdot \rho_{\mathrm{exc}}(t'_n)\cdot(1-\mathrm{e}^{-\gamma \theta'}), \end{equation} \tag{ 17 }$$

where

$$\begin{equation} N_{\mathrm{eff}}=N_{\mathrm{Th}}\cdot \frac{\gamma_{\mathrm{s}} n^3}{\gamma}\cdot\frac{k\Omega}{4\pi} \end{equation} \tag{ 18 }$$

is the product of the number N_Th of interrogated thorium nuclei, the ratio γ_sn³/γ of the rate of radiative decay of the isomer state to the total decay rate enhanced by possible electronic bridge processes, and the probability of detecting the emitted photon. Here k is the quantum efficiency of the photodetector and Ω is the effective solid angle covered by the detector.

In turn, the excited state population at time t_n+1 is equal to

$$\begin{equation} \rho_{\mathrm{exc}}(t_{n+1})=\rho_{\mathrm{exc}}(t'_{n})\,\mathrm{e}^{-\gamma \theta'}, \end{equation} \noindent \tag{ 19 }$$

and at time t'_n+1 it is equal to

$$\begin{eqnarray} &&\fl \rho_{\mathrm{exc}}(t'_{n+1})=\rho_{\mathrm{exc}}(t_{n+1})\exp\left[-\theta\cdot\left(\gamma+\frac{5R/2}{1+(f-\delta_m)^2/\Gamma'\,^2}\right)\right]+\frac{R}{\gamma[1+(f-\delta_m)^2/\Gamma'\,^2]+5R/2}\nonumber\\ &&\times \left\{1- \exp\left[-\theta\cdot\left(\gamma+ \frac{5R/2}{1+(f-\delta_m)^2/\Gamma'\,^2}\right)\right] \right\}, \end{eqnarray} \noindent \tag{ 20 }$$

and in turn

$$\begin{equation} N_{n+1}=N_{\mathrm{eff}}\cdot \rho_e(t'_{n+1})\cdot(1-\mathrm{e}^{-\gamma \theta'}). \end{equation} \noindent \tag{ 21 }$$

Let us introduce the 'evolution rate in the presence of laser field'

$$\begin{eqnarray} &&G(\Delta)=\gamma+\frac{5R/2}{1+\Delta^2/\Gamma'^2}, \end{eqnarray} \noindent \tag{ 22 }$$

and the functions

$$\begin{equation} a(\Delta)=\frac{2 N_{\mathrm{eff}}}{5}\cdot \left(1-\mathrm{e}^{-\gamma \theta'} \right)\cdot \left(1-\frac{\gamma}{G(\Delta)}\right)\cdot \left(1-\mathrm{e}^{-G(\Delta) \theta} \right), \end{equation} \tag{ 23 }$$

$$\begin{equation} b(\Delta)=\mathrm{e}^{-G(\Delta) \theta-\gamma \theta'}. \end{equation} \tag{ 24 }$$

We then can rewrite expressions (17) and (21) as

$$\begin{equation} N_{n}=a(f+\delta_m)+b(f+\delta_m)N_{n-1}, \end{equation} \tag{ 25 }$$

$$\begin{equation} N_{n+1}=a(f-\delta_m)+b(f-\delta_m)N_{n}, \end{equation} \noindent \tag{ 26 }$$

where N_n−1 is the number of photon counts collected in the last measurement of the previous interrogation cycle, i.e. between t'_n−1 and t_n.

We suppose that the generator offset f is small, i.e. f ≪ Γ'. We can then expand the functions a and b near the working point δ_m into a Taylor series:

$$\begin{equation} a(f{\pm} \delta_m)=a_0\mp a_1 f, \quad b(f{\pm} \delta_m)=b_0{\pm} b_1 f. \end{equation} \noindent \tag{ 27 }$$

Using equations (25)–(27), we express the frequency offset f as

$$\begin{equation} f=\frac{N_{n+1}-N_n-b_0(N_n-N_{n-1})}{2a_1-b_1(N_n-N_{n-1})}. \end{equation}\noindent \tag{ 28 }$$

Now let us calculate the error δf in determining the frequency offset f. Here we suppose that N_n+1 and N_n are Poissonian random numbers with mean values described by (25) and (26), i.e. the error of N_i is $\sqrt {\bar{N}_i}$. The average value $\bar{N_{i}}$ is

$$\begin{equation} \bar{N}_{i}\simeq a_0+b_0 \bar{N}_{i}\simeq \frac{a_0}{1-b_0}. \end{equation} \tag{ 29 }$$

From (25)–(29), we express the error of the frequency offset as

$$\begin{equation} \delta f=\frac{\sqrt{a_0(1-b_0)(1+b_0+b_0^2)}}{\sqrt{2}\left[a_1(1-b_0)-b_1a_0\right]}. \end{equation} \noindent \tag{ 30 }$$

Substituting explicit expressions for a₀, b₀, a₁ and b₁ into (30), we obtain

$$\begin{eqnarray} &&\fl \delta f = \frac{(\Gamma'+\delta_m^2/\Gamma')^2\sqrt{1\!-\!\mathrm{e}^{-G\theta}}}{2R\delta_m\sqrt{5N_{\mathrm{eff}}(1-\mathrm{e}^{-\gamma\theta'})}} \frac{\sqrt{\left(1-\frac{\gamma}{G} \right)(1-\mathrm{e}^{-G\theta\!-\!\gamma\theta'})(1+\mathrm{e}^{-G\theta-\gamma\theta'}+\mathrm{e}^{-2G\theta\!-\!2\gamma\theta'})}}{\frac{\gamma}{G^2}(1-\mathrm{e}^{-G\theta})(1-\mathrm{e}^{-G\theta-\gamma\theta'})+\left(1-\frac{\gamma}{G}\right)\theta\mathrm{e}^{-G\theta}(1-\mathrm{e}^{-\gamma\theta'})}, \end{eqnarray} \noindent \tag{ 31 }$$

where G = G(δ_m).

Expression (31) represents a fundamental lower limit on the error of the frequency offset for one interrogation cycle as depicted in figure 4. If the generator has good long-term stability, and the algorithm for frequency stabilization succeeds in the total correction of the measured offset (28), then expression (31) is the frequency error corresponding to an absolute frequency instability of the solid-state nuclear clock after one interrogation cycle.

The frequency error (31) depends on the specific properties of the quantum discriminator (N_Th, γ and Γ'), on the detection system (k Ω), on the intensity of laser radiation determining the excitation rate R and on the regime of interrogation (θ, θ' and δ_m). In practice, the total interrogation time t = 2(θ + θ') will be imposed by the short-term stability of the interrogation laser (i.e. the duration over which frequency drifts and fluctuations are smaller than δf). Also the laser intensity (and therefore the excitation rate R) will have a technical upper limit. On the other hand, the working point δ_m and the relative irradiation time

$$\begin{equation} t_{\mathrm{R}}=\frac{\theta}{\theta+\theta'} \end{equation} \noindent \tag{ 32 }$$

can be adjusted freely. It is therefore of interest to determine the optimal values of t_R and δ_m and study the dependence of the fractional frequency error δf/ω as a function of total interrogation time t and excitation rate R for optimized t_R and δ_m. Because the general expression (31) seems to be too complex for a direct interpretation, we will first focus on the limiting cases of short and long interrogation times without restricting ourselves to explicit experimental parameters. In section 5, we will present numerically optimized results for the specific system of thorium-doped calcium fluoride crystals. A comparison of the analytic expressions with numerical optimizations can be found in appendix B.

Short interrogation time t = 2(θ + θ') refers to γθ' ≪ 1 and Gθ ≪ 1, so the irradiation time and the detection time are shorter than the overall decay time and the inverse excitation rate, respectively. In this case we can expand the exponents in (31) into a Taylor series and keep only the main terms. We obtain

$$\begin{equation} \delta f = \frac{(\Gamma'+\delta_m^2/\Gamma')^2}{2 R \delta_m \sqrt{5 N_{\mathrm{eff}}}} \cdot \frac{\sqrt{3 \left(G-\gamma \right) (G\theta+\gamma\theta')} }{(\theta+\theta')\sqrt{\gamma^3\theta\theta'}}, \end{equation} \noindent \tag{ 33 }$$

and the absolute frequency error scales as t^−3/2 with the total duration of the interrogation cycle. First we fix δ_m and t and find the optimal value of t_R. Therefore we should minimize the function

$$\begin{equation*} \frac{Gt_{\mathrm{R}}+\gamma(1-t_{\mathrm{R}})}{t_{\mathrm{R}}(1-t_{\mathrm{R}})}, \end{equation*}$$

producing the optimal value for t_R:

$$\begin{equation} t_{\mathrm{R}}=\frac{\sqrt{G\gamma}-\gamma}{G-\gamma}. \end{equation} \tag{ 34 }$$

The frequency error δf for the optimized value of t_R is

$$\begin{equation} \delta f = \frac{\Gamma'(1+\delta_m^2/\Gamma'^2)^2}{ R (\delta_m/\Gamma') \sqrt{5 N_{\mathrm{eff}}}} \cdot \frac{ \sqrt{6 \left(G(\delta_m)-\gamma\right)^3}\left(\sqrt{G(\delta_m)}-\sqrt{\gamma}\right)} {\sqrt{\gamma^3t^3}}. \end{equation} \tag{ 35 }$$

Taking into account (22) and minimizing (35), we can find optimal values for the working point δ_m and the frequency error δf.

In the limit of a weak field, i.e. for R ≪ γ, we find that

$$\begin{equation} \delta_m=\frac{\Gamma'}{\sqrt{2}} \end{equation} \tag{ 36 }$$

$$\begin{equation} \delta f =\frac{9 \Gamma'}{\gamma \, t^{3/2} \sqrt{R N_{\mathrm{eff}}}}. \end{equation} \tag{ 37 }$$

In the limit of a strong field, i.e. if R ≫ γ, we find that

$$\begin{equation} \delta_m=\Gamma' \end{equation} \tag{ 38 }$$

$$\begin{equation} \delta f =\sqrt{\frac{30}{N_{\mathrm{eff}}}}\,\frac{\Gamma'}{\gamma^{3/2}\, t^{3/2}}\, . \end{equation} \tag{ 39 }$$

We now consider the case when the interrogation time is long compared to the total decay time, i.e. when γt ≫ 1. Here we cannot neglect the exponents in (31) because the optimal value either of θ or of θ' can be comparable to 1/γ or 1/G, respectively. From the structure of (31) we expect a small optimal value of θ because of the presence of $\sqrt {1-\mathrm {e}^{-G\theta }}$ in the numerator. Therefore, we suppose that γθ' ≫ 1 and neglect the terms in e^−γθ'. Under these assumptions the frequency error of the nuclear clock can be expressed as

$$\begin{equation} \delta f = \frac{G(\Gamma'+\delta_m^2/\Gamma')^2\sqrt{(1-\mathrm{e}^{-x})\left(1-\alpha\right)}}{2R\delta_m\sqrt{5N_{\mathrm{eff}}}\left(\alpha(1-\mathrm{e}^{-x})+\left(1-\alpha\right)x\mathrm{\,e}^{-x}\right)}, \end{equation}\noindent \tag{ 40 }$$

where x = Gθ and α = γ/G. To find the optimal value for θ, we minimize (40) with respect to x, which leads to the transcendental equation for x:

$$\begin{equation} \mathrm{e}^x-1 = \frac{(1-\alpha)x} {\alpha+2 (1-\alpha) (1-x)} . \end{equation}\noindent \tag{ 41 }$$

This equation yields the optimal value of θ = x/G for the case of a long total interrogation time. We again consider the cases of weak and strong interrogation fields.

If the laser field is weak, i.e. if R ≪ γ, α is close to 1. Then 1 − α = 5R/(2γ(1 + δ²_m/Γ'²)) and (41) transforms into

$$\begin{equation*} \mathrm{e}^x-1 = \frac{x} {\frac{2 \gamma}{5 R} \left(1+\frac{\delta_m^2}{\Gamma'^2}\right)+1 - 2 x} . \end{equation*}$$

This equation contains a large term proportional to γ/R on the right side of the denominator which has to be compensated for by another large term 2x₀ − 1 to yield a solution x₀ of (41). We obtain

$$\begin{equation} x_0 \simeq \frac{\gamma}{5R}\left(1+\frac{\delta_m^2}{\Gamma'^2}\right)+1 . \end{equation} \tag{ 42 }$$

Therefore, for the case of a weak laser field, Gθ ∼ x₀ ≫ 1, and we can neglect e^−Gθ in (40). Optimization for δ_m yields

$$\begin{equation} \delta_m=\frac{\Gamma'}{\sqrt{2}} \end{equation} \tag{ 43 }$$

$$\begin{equation} \delta f =\frac{1}{2}\left(\frac{3}{2}\right)^{3/2}\Gamma' \sqrt{\frac{\gamma}{R\, N_{\mathrm{eff}}}}\, . \end{equation}\noindent \tag{ 44 }$$

We now consider the case when the laser field is strong, i.e. when R ≫ γ. We then can set α = 0 in (41). We obtain

$$\begin{equation}\theta = \frac{x_0}{G} = \frac{0.643\,798}{G}. \end{equation}\noindent \tag{ 45 }$$

In this case, the optimal working point δ_m and the frequency fluctuation δf are

$$\begin{equation} \delta_m=\Gamma' \end{equation} \tag{ 46 }$$

$$\begin{equation} \delta f =\frac{2.2778 \, \Gamma'}{\sqrt{N_{\mathrm{eff}}}}\, . \end{equation}\noindent \tag{ 47 }$$

The last expression determines the ultimate stability limit that can, in principle, be attained by the fluorescence spectroscopy method in a single interrogation cycle for a given quantum discriminator. It is valid for a large range of parameters as discussed in appendix B. The optimized value for δf (47) is exclusively determined by the decoherence rate Γ' (which is estimated in appendix A) and the number of interrogated nuclei; it does not depend on the exact values of the isomer transition energy or lifetime. To reach this performance level, the conditions R ≪ Γ' and R ≪ Γ_gg' have to be fulfilled; otherwise an accumulation of populations in quantum states not interacting with the laser field will decrease N_eff over time. However, the very large ratio of Γ'/γ ∼ 10⁶ gives a broad range of parameters where R ≫ γ and R ≪ Γ',Γ_gg' can be satisfied simultaneously.

Note that in the limit of long interrogation time, the equilibrium value of the excited state population is attained in a short fraction of the interrogation cycle and the optimal excitation time takes only a small fraction of the interrogation cycle. In contrast, in the limit of short interrogation time, the equilibrium value cannot be attained in one interrogation cycle and the optimal excitation time θ should be equal to the photon collection time θ'.

Here we have followed a method developed in [48] for the calculation of the effect of a stationary random perturbation on the elements of the density matrix. The evolution of the density matrix is described by the Liouville–von Neumann equation

$$\begin{equation} \frac{\mathrm{d}\skew3\hat{\rho}}{\mathrm{d}t}=-\frac{\mathrm{i}}{\hbar}\left[\skew3\hat{H},\skew3\hat{\rho}\right], \quad \mathrm{where}~{\skew3\hat{H}=\skew3\hat{H}_0+\skew3\hat{H}_1^C+\skew3\hat{H}_1}. \end{equation} \tag{ A.1 }$$

Here $[\skew3\hat {A},\skew3\hat {B}]\equiv \skew3\hat {A}\skew3\hat {B}-\skew3\hat {B}\skew3\hat {A}$ is a commutator, $\skew3\hat {H}_0=\hbar \sum _\alpha |\alpha \rangle \,\omega _{\alpha }\langle \alpha |$ is the Hamiltonian of the unperturbed system, $\skew3\hat {H}_1^C$ is the Hamiltonian of interaction with the coherent external field and $\skew3\hat {H}_1$ is the Hamiltonian of interaction with a random field responsible for the relaxation, in our case with the random magnetic field $\mathcal {\skew5\vec {H}}$. In order to eliminate the strong contribution $\skew3\hat {H}_0$ from (A.1), one applies the unitary transformation $\skew3\hat {L} \rightarrow \skew3\hat {L}'= \mathrm {e}^{\mathrm {i}\skew3\hat {H}_0t/\hbar }\skew3\hat {L}(\mathrm {e})^{-\mathrm {i}\skew3\hat {H}_0t/\hbar }$ to all the operators in (A.1) (interaction representation). We obtain the equation for the density matrix,

$$\begin{equation} \frac{\mathrm{d}\skew3\hat{\rho}'}{\mathrm{d}t}=-\frac{\mathrm{i}}{\hbar}\left[\skew3\hat{H}_1^{C\prime},\skew3\hat{\rho}'\right]- \frac{\mathrm{i}}{\hbar}\left[\skew3\hat{H}_1',\skew3\hat{\rho}'\right]. \end{equation} \tag{ A.2 }$$

The solution to this equation can be written as

$$\begin{equation} \skew3\hat{\rho}'(t)=\skew3\hat{\rho}'(0)-\frac{\mathrm{i}}{\hbar}\int_0^{t}\left[(\skew3\hat{H}_1^{C\prime}+\skew3\hat{H}_1'),\skew3\hat{\rho}'(t')\right]\mathrm{d}t'. \end{equation} \tag{ A.3 }$$

Substituting (A.3) into the second term on the right-hand side of (A.2) yields

$$\begin{eqnarray} &&\fl \frac{\mathrm{d}\skew3\hat{\rho}'}{\mathrm{d}t}=-\!\frac{\mathrm{i}}{\hbar}\left[\!\skew3\hat{H}_1^{C\prime}(t),\skew3\hat{\rho}'(t)\!\right]\!-\! \frac{\mathrm{i}}{\hbar}\left[\!\skew3\hat{H}_1^{\prime}(t),\skew3\hat{\rho}'(0)\!\right]\!-\!\frac{1}{\hbar^2}\int_0^t \left[ \skew3\hat{H}_1^{\prime}(t), \left[ \bigg(\skew3\hat{H}_1^{C\prime}(t')+\skew3\hat{H}_1^{\prime}(t')\bigg),\skew3\hat{\rho}'(t')\right] \right] \mathrm{d}t'.\nonumber\\ \end{eqnarray} \tag{ A.4 }$$

We assume that the characteristic time scale of the correlations of fluctuation of the random field $\skew3\hat {H}_1^{\prime }(t)$ is much smaller than all other characteristic time scales, including the characteristic time scale of variation of the density matrix⁷. Then one can choose the origin in such a way that the density matrix would not change noticeably during this time interval (one may substitute ρ'(t') by ρ'(t)) but all the correlations of the random field will disappear (one may take −∞ as a lower limit of integration). Also suppose that all the matrix elements of $\skew3\hat {H}_1^{\prime }(t)$ have zero mean values (otherwise we would add them to $\skew3\hat {H}_1^{C\prime }$). Performing the averaging over the ensemble (indicated by overline), we obtain

$$\begin{equation} \frac{\mathrm{d}\skew3\hat{\rho}'}{\mathrm{d}t}=-\frac{\mathrm{i}}{\hbar}\left[\skew3\hat{H}_1^{C\prime}(t),\skew3\hat{\rho}'(t)\right] -\frac{1}{\hbar^2}\int_{-\infty}^t \left[\, \overline{\skew3\hat{H}_1^{\prime}(t), \bigg[ \skew3\hat{H}_1^{\prime}(t')},\skew3\hat{\rho}'(t)\bigg] \right] \mathrm{d}t'. \end{equation} \tag{ A.5 }$$

The first term in (A.5) corresponds to the evolution of the density matrix under the action of a coherent field. The second term describes the relaxation. If the perturbations are stationary, it can be written as

$$\begin{equation} \mathbb{R}\skew3\hat{\rho}'(t)\equiv -\frac{1}{\hbar^2}\int_0^{\infty} \left[\, \overline{\skew3\hat{H}_1^{\prime}(t), \bigg[ \skew3\hat{H}_1^{\prime}(t-\theta)},\skew3\hat{\rho}'(t)\bigg] \right] \mathrm{d}\theta. \end{equation} \tag{ A.6 }$$

Expanding the commutators in (A.6), one obtains

$$\begin{equation} \left(\mathbb{R}\skew3\hat{\rho}'\right)_{\alpha \alpha'}=\sum_{\beta \beta'} R_{\alpha \alpha' \beta \beta'} \, \rho'_{\beta \beta'}\, \exp[\mathrm{i}(\omega_{\alpha \beta}-\omega_{\alpha' \beta'})t], \end{equation} \tag{ A.7 }$$

where ω_ij = ω_i − ω_j,

$$\begin{eqnarray} &&\fl R_{\alpha \alpha' \beta \beta'}=\frac{1}{\hbar^2}\Bigg(j_{\alpha \beta \alpha' \beta'}(\omega_{\alpha \beta})+j_{\beta' \alpha' \beta \alpha}(\omega_{\beta' \alpha'}) -\delta_{\alpha \beta} \sum_{\gamma}j_{ \beta' \gamma \alpha' \gamma} (\omega_{\beta' \gamma})-\delta_{\alpha' \beta'} \sum_{\gamma}j_{\gamma \beta \gamma \alpha} (\omega_{\gamma \beta})\Bigg) \nonumber\\ \end{eqnarray} \tag{ A.8 }$$

and

$$\begin{equation} j_{\alpha \beta \alpha' \beta'}(\omega)= \int_0^{\infty} \overline{\skew3\hat{H}_{1\alpha \beta}(t) \skew3\hat{H}_{1\beta' \alpha'}(t-\theta)}\, \mathrm{e}^{-\mathrm{i}\omega\theta}\, \mathrm{d}\theta \end{equation} \tag{ A.9 }$$

(note that on the right-hand side of the last expressions, the matrix elements of $\skew3\hat {H}_1$ are taken in the 'initial' Schrödinger representation). Expression (A.7) for the relaxation operator $\mathbb {R}$ contains oscillating terms that can be neglected. As a result we keep only the terms where ω_αβ = ω_α'β'. Except for some special cases, this condition is fulfilled only if α = α' and β = β' (mixing of populations) or if α = β and α' = β' (relaxation of coherences).

Using (A.6)–(A.9) one can calculate the relaxation and population mixing rates.

Let us consider a single fluorine nucleus. The spin of this nucleus can take one of two projections m = ± 1/2 with respect to the arbitrary quantization axis z. Let us denote |1〉 = |m = −1/2〉 and |2〉 = |m = 1/2〉. The magnetic moment operator $\skew3\hat {\skew5\vec {\mu }}$ for the spin of this nucleus can be expressed as $\skew3\hat {\skew5\vec {\mu }}=\skew3\hat {\skew5\vec {s}}\, \mu _{\mathrm {F}}/s$. Therefore, we can express the operator $\skew3\hat {H}_1=-\skew3\hat {\skew5\vec {\mu }} \cdot \skew5\vec {\mathcal {H}}$ as

$$\begin{eqnarray} &&\fl \skew3\hat{H}_1=\mu_{\mathrm{F}} \bigg(\mathcal{H}_z\left(|1\rangle \langle 1|-|2\rangle \langle 2| \right) -\mathcal{H}_x\left(|1\rangle \langle 2|+|2\rangle \langle 1| \right)- \mathrm{i} \mathcal{H}_y\left(|1\rangle \langle 2|-|2\rangle \langle 1| \right)\bigg). \end{eqnarray} \tag{ A.10 }$$

In this work, we use the 'diffusion' model for the correlations of the random magnetic field:

$$\begin{equation} \left\langle\mathcal{\skew5\vec{H}}(t) \right\rangle=0, \quad \left\langle\mathcal{\skew5\vec{H}}^2(t) \right\rangle=\mathcal{H}_0^2, \quad \left\langle\mathcal{H}_i(t)\mathcal{H}_j(t-\theta) \right\rangle=\delta_{ij}\frac{\mathcal{H}_0^2}{3}\, \mathrm{e}^{-|\theta|/t_{\mathrm{c}}}. \end{equation} \tag{ A.11 }$$

The average square of the random magnetic field $\mathcal {H}_0^2$ and the correlation time t_c will be determined later.

According to (A.8) the relaxation rate R₁₂₁₂ of the coherence ρ₁₂ is

$$\begin{eqnarray} &&\fl R_{1212}=\frac{1}{\hbar^2} \left[ j_{1122}(0)+j_{2211}(0)-j_{2121}(0) - j_{2222}(0)-j_{1111}(0)-j_{2121}(0) \right]. \end{eqnarray} \tag{ A.12 }$$

From (A.9) and (A.11), one finds that

$$\begin{eqnarray} &&R_{1212}=\frac{1}{\hbar^2} \left[ j_{1122}(0)+j_{2211}(0) - j_{2222}(0)-j_{1111}(0)-2 j_{2121}(0) \right], \end{eqnarray} \tag{ A.13 }$$

where

$$\begin{eqnarray} j_{1111}(0)&=&j_{2222}(0)=-j_{1122}(0)=-j_{2211}(0)=\frac{\mu_{\mathrm{F}}^2 \mathcal{H}_0^2}{3}t_{\mathrm{c}}, \nonumber \\ j_{2121}(0)&=&2 \frac{\mu_{\mathrm{F}}^2 \mathcal{H}_0^2}{3}t_{\mathrm{c}}, \end{eqnarray} \tag{ A.14 }$$

which yields the coherence relaxation rate

$$\begin{equation} R_{1212}=-\frac{8}{3}\frac{\mu_{\mathrm{F}}^2 \mathcal{H}_0^2 t_{\mathrm{c}}}{\hbar^2}. \end{equation} \tag{ A.15 }$$

Similarly, one finds that the population mixing rate is

$$\begin{eqnarray} &&R_{1111}=-R_{1122} =-\frac{j_{2121}(0)+j_{1212}(0)}{\hbar^2} =-\frac{4}{3}\frac{\mu_{\mathrm{F}}^2 \mathcal{H}_0^2 t_{\mathrm{c}}}{\hbar^2}. \end{eqnarray} \tag{ A.16 }$$

Now we can estimate the correlation time t_c which is determined by the population relaxation rate:

$$\begin{equation} t_{\mathrm{c}}=\sqrt{\frac{3 \hbar^2}{4\mu_{\mathrm{F}}^2 \mathcal{H}_0^2}}. \end{equation} \tag{ A.17 }$$

To find $\mathcal {H}_0^2$ we recall that in CaF₂ the fluorine nuclei form a simple cubic lattice with a distance of a/2 = 2.73 Å between two neighbors. Spins of different fluorine nuclei are not correlated; therefore the average square $\mathcal {H}_0^2$ of the random magnetic field $\skew5\vec {\mathcal {H}}$ is the sum of squares of magnetic fields produced by different fluorine nuclei:

$$\begin{equation} \mathcal{H}_0^2=\sum_a\mathcal{H}_{a}^2. \end{equation} \tag{ A.18 }$$

$\skew5\vec {\mathcal {H}}_a$ is the magnetic field created by the ath fluorine nucleus whose radius-vector is $\skew5\vec {r}_a$ with respect to the nucleus of interest. Using $\skew3\hat {\skew5\vec {\mu }}_a=2 \skew3\hat {\skew5\vec {s}}_a\, \mu _{\mathrm {F}}$ and $\mathcal {\skew5\vec {H}}_a=(3\skew5\vec {r} (\skew3\hat {\skew5\vec {\mu }}_a\skew5\vec {r}_a) - \skew3\hat {\skew5\vec {\mu }}_a r_a^2)/r_a^5$, we find that

$$\begin{equation} \mathcal{H}_{a}^2=\frac{3 (\skew3\hat{\skew5\vec{\mu}}_a\skew5\vec{r}_a)^2+\skew3\hat{\skew5\vec{\mu}}_a^2r_a^2}{r_a^8} = \frac{6\mu_{\mathrm{F}}^2}{r_a^6}. \end{equation} \tag{ A.19 }$$

Therefore

$$\begin{equation} \mathcal{H}_{0}^2=\frac{6\mu_{\mathrm{F}}^2}{r_a^6} \sum\limits_{\scriptstyle{l,m,n=-\infty}}^{\infty} \left[l^2+m^2+n^2\right]^{-3}\simeq \frac{6\mu_{\mathrm{F}}^2}{r_a^6} \cdot 8.4 \end{equation} \tag{ A.20 }$$

(here the sum is taken over l,m,n excluding l = m = n = 0) or $\mathcal {H}_{0}^2\simeq 21.5$ G², which leads to a relaxation time t_c ≃ 15 μs according to (A.17). Note that this value is quite close to experimentally observed spin–spin relaxation times ∼20 μs [49].

We now consider an individual thorium nucleus interacting with the random magnetic fields created by the surrounding fluorine nuclei. The quantization axis is fixed by the direction of the main axis of the electric field gradient. Let us denote the different states of the thorium nucleus by |i〉, where i is a number ranging from 1 to 10 corresponding to the spin state shown in figure A.1.

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The average square $\mathcal {H}_{\mathrm {Th}}^2$ of the magnetic field at the position of the thorium nucleus is given by

$$\begin{equation} \mathcal{H}_{\mathrm{Th}}^2=\frac{6\mu_{\mathrm{F}}^2}{r_a^6} \sum \limits_{l,m,n=-\infty}^{\infty}\left[\sum\limits_{k=l,m,n}\left(k-\frac{1}{2}\right)^2\right]^{-3}\simeq \frac{6\mu_{\mathrm{F}}^2}{r_a^6} \cdot 19\simeq 48.5 \, \mathrm{G}^2. \end{equation}\noindent \tag{ A.21 }$$

The relaxation rates Γ_ij = −R_ijij for optical coherences ρ_ij due to interaction with the random magnetic field created by the surrounding fluorine spins can be calculated in the same manner as for a single fluorine spin above. These rates (for allowed magnetic dipole transitions) are equal to

$$\begin{eqnarray} \Gamma_{4,8}&=&\Gamma_{3,9}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2+\frac{2}{15}\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\frac{19}{25}\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 142 \,\mathrm{Hz}, \nonumber \\ \Gamma_{3,8}&=&\Gamma_{4,9}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2-\frac{2}{15}\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\frac{19}{25}\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 150 \,\mathrm{Hz}, \nonumber \\ \Gamma_{2,8}&=&\Gamma_{5,9}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2-\frac{2}{5}\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\frac{9}{25}\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 84 \,\mathrm{Hz}, \nonumber \\ \Gamma_{3,7}&=&\Gamma_{4,10}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2-\frac{2}{5}\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\frac{19}{25}\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 158 \,\mathrm{Hz}, \nonumber \\ \Gamma_{2,7}&=&\Gamma_{5,10}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2-\frac{6}{5}\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\frac{9}{25}\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 108 \,\mathrm{Hz}, \nonumber \\ \Gamma_{1,7}&=&\Gamma_{6,10}=\frac{\mathcal{H}_{\mathrm{Th}}^2 t_{\mathrm{c}}}{3 \hbar^2}\left[\mu_{\mathrm{e}}^2-2\mu_{\mathrm{e}} \mu_{\mathrm{g}}+\mu_{\mathrm{g}}^2 \right]\simeq 2\pi\cdot 251 \,\mathrm{Hz.} \end{eqnarray}\noindent \tag{ A.22 }$$

Here we used the values of the thorium nuclear magnetic moment in the ground state μ_g = 0.45μ_N and in the isomer state μ_e = −0.076μ_N, respectively [50]. The decoherence rates Γ_3,8 = Γ_4,9 have been used in the evaluations of the nuclear clock performance in section 5.