Introduction

The excitation energy between the nuclear ground state and the first nuclear excited state (isomeric state) of thorium-229 (Th-229) is approximately 8 eV [1-4]. The optical transition linewidth of the first nuclear excited state is 10^-4 Hz [5, 6]. The low isomeric energy and narrow linewidth provide the potential for a resonator with a quality of a factor of 10¹⁹ and make Th-229 the most suitable choice for the development of nuclear optical clocks [6-8]. Such a Th-229 nuclear clock is expected to be a sensitive probe for the time variation of the fundamental constants of nature and will open opportunities for highly sensitive tests of the fundamental principles of physics, particularly in searches for violations of Einstein’s equivalence principle and new particles [9-11].

The isomeric energy is currently measured in two ways: 1) Measuring the kinetic energy of internal conversion electrons from the Th-229 isomeric state. For example, an electron spectrometer has been used to detect the internal conversion electron energy, and the isomeric energy was obtained as 8.28 ± 0.17 eV [4]. 2) Observing the γ radiation from Th-229 excited nuclei. An isomeric energy of 7.8 ± 0.5 eV was obtained using a magnetic microcalorimeter [2]. Subsequently, a 29.2-keV inter-band excitation of the Th-229 nuclear state was observed using synchrotron radiation [3]. Combined with the in-band 29.2-keV transition observed via nuclear rotational spectroscopy, the Th-229 isomeric state energy was determined to be 8.30 ± 0.92 eV [12]. Recently, a more precise magnetic microcalorimeter with improved energy resolution was developed, and an isomeric energy of 8.10 ± 0.17 eV was obtained [1]. Taking different weights for these measurements, the average value of the isomeric state was determined as 8.12 ± 0.11 eV [13]. Its uncertainty corresponds to 53 THz, which is at least 16 orders of magnitude higher than the narrow linewidth of 10^-4 Hz.

To improve the precision of the isomeric energy, synchrotron radiation has been used to irradiate Th-229 doped in a vacuum ultraviolet (VUV) transparent crystal, and γ photons were detected from the decay of the Th-229 excited nuclei. This measurement excluded approximately half of the favored transition search area [14]. Another scheme was proposed to achieve Th-229 nuclei excitation via electron capture and detect the isomeric state of ions in heavy-ion storage rings [15]. Recently, an optical frequency comb was proposed to irradiate a Th-229 dioxide film, and the isomeric energy was measured based on the internal conversion process [16]. Furthermore, a new approach has been suggested to excite the isomeric Th-229 nuclear state via laser-driven electron recollision. The advantage of this approach is that it does not require precise knowledge of the isomeric energy [17].

In this paper, we propose the excitation of the nuclei of ²²⁹Th³⁺ ions in the ion trap via electronic bridge (EB) excitation and compare it with direct nuclear excitation via a pulsed laser. As shown in the inset of Fig. 1, EB excitation starts from an electric dipole (E1) transition, which is utilized to excite the electrons from an initial state to a virtual intermediate state. Subsequently, the intermediate state decays via the magnetic dipole (M1) transition, and the nuclear ground state 5/2⁺[633] is excited to the isomeric state 3/2⁺[631] simultaneously. To date, EB schemes for nuclear excitation have been theoretically investigated for ^229gTh⁺ [18], ²²⁹Th²⁺ [19], ²²⁹Th³⁺ [20-22], and ²²⁹Th³⁵⁺ [23]. Our proposed experiment is based on EB excitation in triply charged ²²⁹Th³⁺ ions. Owing to their simple electronic energy levels, the trapped ²²⁹Th³⁺ ions can be laser-cooled to form a linear-chain Coulomb crystal using two closed transitions in the nuclear ground state of ²²⁹Th³⁺ (^229gTh³⁺) [24]. When the ^229gTh³⁺ ions are prepared in the 7P_1/2 state, EB excitation is triggered by a 350-nm laser, and isomeric states are expected to be populated. Finally, the isomeric state is detected based on the isomer shift of the cooling lasers [7].

Fig. 1Full-screen View

Diagram of electronic energy levels and electric-dipole transitions of Th-229 triply charged ions. The insert shows the processes of EB excitation, where the red arrows are the 150-nm M1 transition. The black horizontal lines indicate the nuclear ground state of Th-229, and the red horizontal lines indicate the isomeric state of Th-229. The purple arrow is the 350-nm EB excitation. The lifetime of the electronic excited state is indicated in the parentheses. The optical transition wavelengths are in nm, and the integers near the atomic levels indicate the principal quantum numbers.

Preparation of ^229gTh³⁺ ion Coulomb crystals and the 7P_1/2 state

^229gTh³⁺ ions are typically produced by laser ablation [24, 25]. The linear Paul trap used to trap ^229gTh³⁺ ions are described in detail in our previous papers [26, 27]. The trapped ^229gTh³⁺ ions are laser-cooled to form a Coulomb crystal.

The energy levels of laser cooling and fluorescence detection of the trapped ions are shown in Fig. 2. The ^229gTh³⁺ ions are Doppler-laser-cooled on a 5²F_5/2 ↔ 6²D_3/2 transition at 1088 nm. Alternatively, a closed optical transition of the three-level Λ system, 5²F_5/2 ↔ 6²D_5/2 ↔ 5²F_7/2, can be used to laser-cool the ^229gTh³⁺ ions with two lasers at 690 nm and 984 nm [7]. The population in the three hyperfine levels of $|5F_{5/2}\mathrm{,}F=\mathrm{3,4,5}\rangle$ can be transferred to $|5F_{5/2}\mathrm{,}F=1\rangle$ and $|5F_{5/2}\mathrm{,}F=2\rangle$ via the 6²D_3/2 state, and the population in the $|5F_{7/2}\mathrm{,}F=1\rangle$ state is cumulated as shown in Fig. 2. Therefore, the ions at other hyperfine levels are transferred to the closed system of 5²F_5/2 ↔ 6²D_5/2 ↔ 5²F_7/2, and the second laser cooling cycle is realized.

Fig. 2Full-screen View

State preparation scheme of ^229gTh³⁺ ions. The cooling scheme utilizes the two closed system indicated by the black solid arrows. The wavy arrows indicate spontaneous radiation, and the blue solid arrows indicate the transition to the 7P_1/2 state. The red dashed arrows are the transitions to transfer ions at $|5F_{5/2}\mathrm{,}F=\mathrm{3,4,5}\rangle$ to $|5F_{5/2}\mathrm{,}F=1\rangle$ and $|5F_{5/2}\mathrm{,}F=2\rangle$. The black dashed arrows are the transitions to transfer ions at $|5F_{7/2}\mathrm{,}F=\mathrm{2,3,4,5,6}\rangle$ to $|5F_{7/2}\mathrm{,}F=1\rangle$. The integers near the atomic levels indicate the total angular momentum quantum numbers F.

Once the ^229gTh³⁺ ions are laser-cooled to 50 μK, the Doppler broadening of the transition 6²D_5/2 → 5²F_7/2 at 984 nm is approximately 1 MHz [24]. At this temperature, the corresponding Doppler broadenings of direct nuclear excitation at 150 nm and EB excitation at 350 nm are estimated to be 6.5 MHz and 2.8 MHz, respectively. The shapes of Coulomb crystals can be manipulated into a linear chain by adjusting the radiofrequency (RF) and endcap voltages [28]. Two-dimensional and linear-chain Coulomb crystals can be simulated with single-ion resolution [29, 30]. Using the GPU-accelerated LAMMPS [31] wrapped by the LIon package [32], a Coulomb crystal consisting of 36 ^229gTh³⁺ ions are simulated, as shown in Fig. 3. When an endcap voltage of 0.2 V and an RF voltage (200 V_0-pk) of 2 MHz are applied to the ion trap, a linear-chain Coulomb crystal of 36 ^229gTh³⁺ ions (Fig. 3(c)) is obtained. When an RF voltage (200 V_0-pk) of 2 MHz and endcap voltage of 0.025 V are applied to the ion trap, a linear-chain Coulomb crystal of 100 ^229gTh³⁺ ions is obtained. From the simulation results, the motion amplitude of ions in the x- and y-directions (radial direction) is known to be less than 0.6 μM when the temperature is below 5 mK, which is consistent with the motion amplitude of single ions [33]. Therefore, a strongly focused laser spot size of over 1.2 μM is required to increase the EB and direct nuclear excitation rates.

Fig. 3Full-screen View

Simulation results of a Coulomb crystal consisting of 36 ^229gTh³⁺ ions. With the same RF voltages (200 V_0-pk), the Coulomb crystals’ shapes are changed under different endcap voltages (a) 0.4 V, (b) 0.3 V, and (c) 0.2 V. See text for details.

To induce EB excitation from the 7P_1/2 state of the ^229gTh³⁺ ions, the three-level transition 5²F_5/2 → 6²D_3/2 → 7²P_1/2 can be used to accumulate the population at the 7P_1/2 state. Once the ^229gTh³⁺ ions are laser-cooled to 50 μK, the 690-nm and 984-nm lasers are turned off in sequence to maintain the ions at the 5²F_5/2 state, and the 1088-nm laser is kept on. To transfer a large population to the 7²P_1/2 state with a lifetime of 1 ns, both the 269-nm and 196-nm transitions are excited simultaneously, as shown in Fig. 4. The natural linewidths of the 269-nm and 196-nm transitions are approximately 2π ⋅ 32 MHz and 2π ⋅ 127 MHz, respectively [34]. Continuous wave (CW) lasers at both wavelengths are commercially available [35].

Fig. 4Full-screen View

Expected number of excited nuclei is calculated as a function of time when 100 triply charged Th-229 ions are directly illuminated by a direct excitation laser.

The spontaneous radiation rate of the hyperfine structure level transitions between two electronic energy levels is [36] ${\text{Γ}}_{\text{f}\to i}={\left|\langle {\xi }_{\text{f}}{J}_{\text{f}}{I}^{\text{g}}{F}_{\text{f}}\Vert d\Vert {\xi }_{\text{i}}{J}_{\text{i}}{I}^{\text{g}}{F}_{\text{i}}\rangle \right|}^2\frac{4{\omega }^3}{3\hslash c^3}\mathrm{,}$ (1) where ξ_i (ξ_f) incorporates all other electronic quantum numbers, J_i (J_f) is the total electronic quantum number of the initial (final) state, F_i (F_f) is the total quantum number of the initial (final) state, I^g is the nuclear spin of the nuclear ground state, d is the electron dipole, ω is the angular frequency of the electronic transition, $\hslash$ is the reduced Planck constant, and c is the speed of light in a vacuum. ${\left|\langle J_{\text{f}}{I}^{\text{g}}{F}_{\text{f}}\Vert d\Vert J_{\text{i}}{I}^{\text{g}}{F}_{\text{i}}\rangle \right|}^2$ is the transition strength of the hyperfine structure level, which can be expressed as $\begin{array}{c}{\left|\langle {\xi }_{\text{f}}{J}_{\text{f}}{I}^{\text{g}}{F}_{\text{f}}\Vert d\Vert {\xi }_{\text{i}}{J}_{\text{i}}{I}^{\text{g}}{F}_{\text{i}}\rangle \right|}^2=(2F_{\text{i}}+1)(2F_{\text{f}}+1)\\ {\left\{\begin{array}{ccc}{J}_{\text{f}}& F_{\text{f}}& I^{\text{g}}\\ F_{\text{i}}& J_{\text{i}}& 1\end{array}\right\}}^2{\left|\langle J_{\text{f}}\Vert d\Vert J_{\text{i}}\rangle \right|}^2\mathrm{,}\end{array}$ (2) where ${\left|\langle J_{\text{f}}\Vert d\Vert J_{\text{i}}\rangle \right|}^2$ denotes the transition strength of the electronic energy levels. The spontaneous radiation rate of the electronic energy levels is ${\text{Γ}}_{\text{spont}}={\left|\langle {\xi }_{\text{f}}{J}_{\text{f}}\Vert d\Vert {\xi }_{\text{i}}{J}_{\text{i}}\rangle \right|}^2\frac{4{\omega }^3}{3\hslash c^3}\mathrm{.}$ (3) The excitation rate of the hyperfine structure levels between the two electronic energy levels is [37] ${\text{Γ}}_{\text{exc}}={\text{Γ}}_{\text{f}\to \text{i}}\frac{{\pi }^2{c}^2}{\hslash {\omega }^3}{P}_{\omega }\frac{2F_{\text{f}}+1}{2F_{\text{i}}+1}\mathrm{,}$ (4) where Pω denotes the laser spectral intensity. The stimulated emission rate of the hyperfine structure levels between the two electronic energy levels is ${\text{Γ}}_{\text{sti}}={\text{Γ}}_{\text{exc}}\frac{2F_{\text{i}}+1}{2F_{\text{f}}+1}\mathrm{.}$ (5) With a laser power of 1 mW and 100-μM laser spot sizes for 1088 nm, 269 nm, and 196 nm lasers, the steady-state condition can be fulfilled for all 5²F_5/2, 7²S_1/2, 6²D_3/2, and 7²P_1/2 states under continuous illumination. For multiple hyperfine structure transitions between electronic transitions 5²F_5/2 ↔ 6²D_3/2 and 7²S_1/2 ↔ 7²P_1/2, the power of the lasers is divided equally. Thus, after state preparation, the population of the $|7P_{1/2}\mathrm{,}F=2\rangle$ state accounts for approximately 12% of the total number of trapped ions.

Calculation of the population in the nuclear excited state

The time-dependent nuclear excitation probability ρ_exc(t) for a single nucleus under resonant irradiation is given by Torrey’s solution to the optical Bloch equations as follows [38-40]: $\begin{array}{c}①{\text{Ω}}_{\text{eg}}>\frac{|\text{Γ}-\widetilde{\text{Γ}}|}{2}\\ {\rho }_{\text{exc}}(t)=\frac{{\text{Ω}}_{\text{eg}}^2}{2(\text{Γ}\widetilde{\text{Γ}}+{\text{Ω}}_{\text{eg}}^2)}\\ \left[1-e^{-\frac{1}{2}(\text{Γ}+\widetilde{\text{Γ}})t}\times \left(\cos (\lambda t)+\frac{\text{Γ}+\widetilde{\text{Γ}}}{2\lambda }\sin (\lambda t)\right)\right]\mathrm{,}\end{array}$ (6) $\begin{array}{c}②{\text{Ω}}_{\text{eg}}<\frac{|\text{Γ}-\widetilde{\text{Γ}}|}{2}\\ {\rho }_{\text{exc}}(t)=\frac{{\text{Ω}}_{\text{eg}}^2}{2(\text{Γ}\widetilde{\text{Γ}}+{\text{Ω}}_{\text{eg}}^2)}\\ \left[1-e^{-\frac{1}{2}(\text{Γ}+\widetilde{\text{Γ}})t}\times \left(\cosh (\lambda t)+\frac{\text{Γ}+\widetilde{\text{Γ}}}{2\lambda }\sinh (\lambda t)\right)\right]\mathrm{.}\end{array}$ (7) Here, Ω_eg denotes the Rabi frequency between the isomeric and nuclear ground states. Γ=Γ_γ+Γ_nr is the total linewidth of the isomeric state, and Γ_γ denotes the γ decay rate of the isomeric state. Because the ionization energy of ^229gTh³⁺ (28.6 eV) is considerably greater than the energy of the isomeric state, the internal conversion process is strongly forbidden [41]. The possible non-radiative transition linewidth Γ_nr is dominated by the EB transition for ^229gTh³⁺. $\widetilde{\text{Γ}}=\frac{\text{Γ}+{\text{Γ}}_{\text{L}}}{2}+{\text{Γ}}_{\text{add}}$ denotes the total transition linewidth [40], Γ_L denotes the laser linewidth used for nuclear excitation, and Γ_add is the additional incoherent linewidth, such as phonon coupling of ions in a Coulomb crystal. Because Γ_add is significantly smaller than the laser linewidth Γ_L, the influence of the incoherent linewidth is negligible. t denotes the interaction time between the laser and ^229gTh³⁺ ions. Here, $\lambda =|{\text{Ω}}_{\text{eg}}^2-{(\text{Γ}-\widetilde{\text{Γ}})}^2/4|$.

Under the resonant condition, the Rabi frequency between the isomeric state and nuclear ground state is [39] ${\text{Ω}}_{\text{eg}}=\sqrt{\frac{2\pi c^{2}I{C}_{\text{eg}}^2\text{Γ}}{\hslash {\omega }_{\text{m}}^3}}\mathrm{.}$ (8) Here, I is the intensity of the excitation laser, and ω_m is the angular frequency of the nuclear transition. The Zeeman splitting caused by the geomagnetic field between the two outermost lines is below 1 kHz and is negligible. Therefore, the Clebsch–Gordan coefficient of the sub-state transition between the isomeric state and the nuclear ground state C_eg is 1 [16].

Assuming that the intensities of all lasers have a Gaussian distribution, the Rabi frequency can be modified via the following two situations:

(i) If the laser linewidth (Γ_L) is greater than the Doppler-broadening linewidth (Γ_D), only part of the laser with a matching frequency range can effectively interact with the triply charged thorium ions. The equivalent Doppler-broadening linewidth is ${\text{Γ}}_{\text{D}}^{\text{'}}=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}{\text{Γ}}_{\text{D}}\mathrm{.}$ (9) The effective laser intensity is $I_{\text{eff}}=I{\displaystyle {\int }_{-\frac{{\text{Γ}}_{\text{D}}^{\text{'}}}{2}}^{\frac{{\text{Γ}}_{\text{D}}^{\text{'}}}{2}}}{g}_{\text{L}}(\omega )\text{d}\omega \mathrm{.}$ (10) The modified Rabi frequency is ${\text{Ω}}_{\text{eg}}^{\text{'}}=\sqrt{\frac{2\pi c^2{I}_{\text{eff}}{C}_{\text{eg}}^2\text{Γ}}{\hslash {\omega }_{\text{m}}^3}}\mathrm{.}$ (11)

(ii) If the laser linewidth (Γ_L) is smaller than the Doppler-broadening linewidth (Γ_D), only part of the ions interact with the laser light at the frequency resonance at a certain time, and the equivalent laser linewidth is ${\text{Γ}}_{\text{L}}^{\text{'}}=\frac{1}{2}\sqrt{\frac{\pi }{\ln 2}}{\text{Γ}}_{\text{L}}\mathrm{.}$ (12) The effective Doppler-broadening linewidth is ${\text{Γ}}_{\text{eff}}=\text{Γ}{\displaystyle {\int }_{-\frac{{\text{Γ}}_{\text{L}}^{\text{'}}}{2}}^{\frac{{\text{Γ}}_{\text{L}}^{\text{'}}}{2}}}{g}_{\text{D}}(\omega )\text{d}\omega \mathrm{.}$ (13) The modified Rabi frequency is ${\text{Ω}}_{\text{eg}}^{\text{'}}=\sqrt{\frac{2\pi c^{2}I{C}_{\text{eg}}^2{\text{Γ}}_{\text{eff}}}{\hslash {\omega }_{\text{m}}^3}}\mathrm{.}$ (14)

Direct nuclear excitation

The Th-229 isomeric energy is 8.12 ± 0.11 eV, corresponding to a wavelength of 150.7-154.8 nm. Currently, available light sources such as the VUV pulsed laser [42], synchrotron radiation light source [14], and the 7th harmonic of a Yb-doped fiber laser [16, 43-45] are suitable candidates for direct nuclear excitation. Here, based on resonance-enhanced four-wave mixing, the parameters of a tunable and pulsed VUV laser source are a pulse energy of E_L = 13.2  μJ around 150 nm, a bandwidth of δν_L = 15 GHz, and a repetition rate of R_L = 10 Hz, which are used to calculate the direct nuclear excitation rate [42].

Assuming a strongly focused laser with a 3-μm spot radius to irradiate the linear chain of 100 ions, the corresponding laser intensity $I=\frac{E_{\text{L}}{R}_{\text{L}}}{\pi r^2}=4.67\times {10}^6$ W/m² can be obtained. The temperature of Th³⁺ ions is 50 μK, and the corresponding Doppler-broadening linewidth Γ_D = 6.5 MHz is narrower than the laser linewidth. Therefore, the modified Rabi frequency at resonance ${\text{Ω}}_{\text{eg}}^{\text{'}}(\delta =0)$ = 107 Hz can be obtained based on Eq. (9)-(11), which is significantly smaller than the total transition linewidth $\widetilde{\text{Γ}}$ shown in Table 1. In this case, ${\text{Ω}}_{\text{eg}}^{\text{'}}<\frac{|\text{Γ}-\widetilde{\text{Γ}}|}{2}$ is satisfied. The number of excited nuclei is calculated as a function of time using Eq. (7). As shown in Fig. 1, the number of excited Th-229 nuclei is only 0.12 out of 100 Th-229 ions and reaches saturation at an irradiation time of 30000 s. The required laser irradiation time is beyond the stable trapping time; therefore, no Th-229 isomeric state can be detected under the current experimental conditions.

Electronic bridge excitation

Direct nuclear excitation is limited by the natural linewidth of the isomeric state and the power of the excitation laser. If EB excitation occurs, the transition linewidth can be increased by approximately 40 times compared with direct nuclear excitation [46]. EB excitation can be utilized by exciting the quantum state of the electronic state 7P_1/2 of ^229gTh³⁺ (7P_1/2 5/2⁺[633]) to the electronic state 7S_1/2 of ^229mTh³⁺ (7S_1/2 3/2⁺[631]), as shown in Fig. 1. The prepared ^229gTh³⁺ Coulomb crystal comprising a linear chain of 36 ions is shown in Fig. 3(c), among which there are four ions (12% of the total number of trapped ions) in the 7P_1/2 state under steady-state conditions (see Sect. 2).

The wavelength of the laser used for EB excitation is 350 nm. A 350-nm laser with a power of 250 mW can be obtained from Toptica DLC TA-SHG pro and focused to a spot size of 10  μmeter. Thus, an intensity of 7.95 × 10⁸ Wm^-2 can be achieved. Therefore, the Rabi frequency is estimated as 334 kHz.

The EB laser linewidth, typically 500 kHz at 350 nm, is narrower than the Doppler broadening of the trapped thorium ions. The modified Rabi frequency ${\text{Ω}}_{\text{eg}}^{\text{'}}$ is 140 kHz at resonance and 98 kHz upon the detuning Γ_D/2 (shown in Table 2). Then, the condition ${\text{Ω}}_{\text{eg}}^{\text{'}}<\frac{|\text{Γ}-\widetilde{\text{Γ}}|}{2}$ is satisfied, and thus the number of excited nuclei as a function of time can be obtained based on Eq. (7), as shown in Fig. 5. If the excitation energy from the 7P_1/2 state of ^229gTh³⁺ to the 7S_1/2 state of ^229mTh³⁺ is at resonance, approximately 2 ^229gTh³⁺ ions can be excited to the isomeric state after 300-μs irradiation by the 350-nm laser. If the excitation laser is detuned by Γ_D/2, approximately 1.8 ^229gTh³⁺ ions can be excited to the isomeric state within the same irradiation time.

Fig. 5Full-screen View

Expected number of excited nuclei is calculated as a function of time when 36 triply charged Th-229 ions are trapped and four ions are at the 7P_1/2 state.

After successful EB excitation, Th-229 ions in the isomeric state can be identified based on the different hyperfine structures of the isomeric and nuclear ground states. Detailed hyperfine structures of the isomeric state are shown in Appendix B. After EB nuclear excitation, the EB laser at 350 nm and the state-preparation lasers at 196 nm and 269 nm are all turned off, and the cooling lasers at 690 nm and 984 nm are immediately switched back on and combined with the 1088-nm laser to cool the ions. The nuclear-excited ions remain in the isomeric states and accumulate in the electronic state of 5²F_5/2, as illustrated in Fig. 6. The remaining nuclear-unexcited ions remain in the nuclear ground state of the Doppler cooling cycle of 5²F_5/2, 6²D_5/2, 5²F_7/2 and 6²D_3/2. The isomer shifts of the 6D_5/2 and 6D_3/2 electronic states are both approximately 400 MHz [24, 47]. The 690-nm and 984-nm lasers are off-resonance for the Doppler cooling of ions in the isomeric state. As a result, the nuclear-excited ^229mTh³⁺ ions leave the 690-nm and 984-nm lasers’ cooling cycle and appear as dark ions in the Coulomb crystal [7]. In the linear-chain ²²⁹Th³⁺ ion Coulomb crystal, single ions are distinguishable, and any dark ^229mTh³⁺ ions generated by a successful EB excitation event can be detected.

Fig. 6Full-screen View

Experimental steps and the time of each laser switch.

The current reported uncertainty on the isomeric state energy is 0.22 eV [13]. To cover this uncertainty range, approximately 1.9 × 10⁷ scans are required for each scanning interval of the Doppler-broadening linewidth Γ_D. The time required for a single scan step is approximately 300 μs; therefore, the total scan time is approximately 5670 s, as shown in Table 3. The final isomeric energy is determined by the sum of the energy intervals between the electronic state 7P_1/2 of ^229gTh³⁺ and the electronic state 7S_1/2 of ^229mTh³⁺ and the photon energy of the 350-nm laser. Because the uncertainty on the natural linewidth of the 7P_1/2 state of ^229gTh³⁺ is 1 GHz, which is significantly larger than the laser and Doppler-broadening linewidths, the uncertainty on the isomeric energy is of the order of 1 GHz (corresponding to 4 × 10^-6 eV), dominated by the natural linewidth of the 7P_1/2 state.

Conclusion

In this paper, considering the Doppler broadening of thorium ions in the ion trap, we show that 2 ^229gTh³⁺ ions out of 36 trapped ions can be excited to the isomeric state under resonant conditions via EB excitation. We propose that a total measurement time of approximately 2 h can be achieved for the current uncertainty on the isomeric state energy of 0.22 eV. If a cryogenic linear Paul trap is used, the ions can be stably trapped for a longer period and the radiative lifetime of the isomeric state can be measured. The utilization of EB excitation can reduce the uncertainty on the Th-229 isomeric energy to approximately 1 GHz. The calculation is based on the theoretical prediction that the transition linewidth of EB excitation is 40 times larger than that of direct nuclear excitation. If EB excitation is not observed, as proposed in this paper, it may indicate that the coupling between the electrons and the nuclear core is not as strong as expected or that the explored nuclear transition energy range is not accurate, and the laser energy used for the EB excitation needs to be increased. However, the successful observation of EB excitation will further improve the accuracy of isomeric energy, pave the way for the development of a nuclear optical clock, test the temporal variation of fundamental constants, and provide new methods for studying nuclear physics.