Scheme for the excitation of thorium-229 nuclei based on electronic bridge excitation

Thorium-229 possesses the lowest first nuclear excited state, with an energy of approximately 8 eV. The extremely narrow linewidth of the first nuclear excited state, with an uncertainty of 53 THz, prevents direct laser excitation and realization of the nuclear clock. We present a proposal using the Coulomb crystal of a linear chain formed by 229\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{229}$$\end{document}Th3+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{3+}$$\end{document} ions, where the nuclei of 229\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{229}$$\end{document}Th3+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{3+}$$\end{document} ions in the ion trap are excited by the electronic bridge (EB) process. The 7P1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1/2}$$\end{document} state of the thorium-229 nuclear ground state is chosen for EB excitation. Using the two-level optical Bloch equation under experimental conditions, we calculate that 2 out of 36 prepared thorium ions in the Coulomb crystal can be excited to the first nuclear excited state, and it takes approximately 2 h to scan over an uncertainty of 0.22 eV. Taking advantage of the transition enhancement of EB and the long stability of the Coulomb crystal, the energy uncertainty of the first excited state can be limited to the order of 1 GHz.


I. INTRODUCTION
The excitation energy between the nuclear ground state and the nuclear first excited state (isomeric state) of thorium-229 (Th-229) is about 8 eV [1][2][3][4].The optical transition linewidth of the nuclear first excited state is 10 −4 Hz [5,6].The low isomeric energy and narrow linewidth provide potential for a resonator quality with the factor of 10 19 and make Th-229 the most suitable choice for the development of nuclear optical clocks [6][7][8].Such a Th-229 nuclear clock is expected to be a sensitive probe for time variation of the fundamental constants of nature; besides, it will open opportunities for highly sensitive tests of fundamental principles of physics, particularly in searches for violations of Einstein's equivalence principle and new particles [9][10][11].
The isomeric energy is currently measured in two ways: 1) Measuring the kinetic energy of the internal conversion electrons from the Th-229 isomeric state.For example, an electron spectrometer was used to detect the internal conversion electron energy and the isomeric energy obtained was 8.28 ± 0.17 eV [4].2) Observing the γ radiation from the Th-229 excited nuclei.An isomeric energy of 7.8 ± 0.5 eV was obtained by a magnetic microcalorimeter [2].Later, a 29.2 keV inter-band excitation of the Th-229 nuclear state was observed using synchrotron radiation [3].Combining 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, the isomeric energy of 8.10 ± 0.17 eV was obtained [1].Taking different weights for these measurements, the average value of the isomeric state is determined to be 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, the synchrotron radiation is used to irradiate the Th-229 doped in a vacuum ultraviolet (VUV) transparent crystal, and then the γ photons are detected from the decay of the Th-229 excited nuclei, this measurement excludes roughly half of the favored transition search area [14].Another scheme is proposed to achieve the Th-229 nuclei excitation by electron capture, and detect the isomeric state of ions in heavy-ion storage rings [15].Recently, an optical frequency comb is proposed to irradiate the Th-229 dioxide film, the isomeric energy can be measured based on the internal conversion process [16].Besides, a new approach is suggested to excite the isomeric Th-229 nuclear state via a laser-driven electron recollision.The advantage of the approach is that it does not require knowl-edge of the isomeric energy precisely [17].
In this paper, we propose to excite the nuclei of 229 Th 3+ ions in the ion trap by the EB excitation and compare with the direct nuclear excitation by a pulsed laser.As shown in the insets of Fig. 1, the EB excitation starts from an electric dipole (E1) transition that is utilized to excite the electron from an initial state to a virtual intermediate state.Then the intermediate state decays via the magnetic dipole (M 1) transition, and the nuclear ground state 5/2 + [633] is excited to the isomeric state 3/2 + [631] simultaneously.So far, the EB schemes for the nuclear excitation have been theoretically investigated for 229g Th + [18], 229 Th 2+ [19], 229 Th 3+ [20][21][22] and 229 Th 35+ [23].Our proposed experiment is based on the EB excitation in the triply charged 229 Th 3+ ions.Due to the simple electronic energy levels, the trapped 229 Th 3+ ions can be laser-cooled to form a linear-chain Coulomb crystal using two closed transitions in the nuclear ground state of 229 Th 3+ ( 229g Th 3+ ) [24].When the 229g Th 3+ ions are prepared in the state 7P 1/2 , the 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 by the cooling lasers [7].
The 229g Th 3+ ions are typically produced by the laser ablation [24,25].The linear Paul trap for the trapping 229g Th 3+ ions has been described in detail in our previous works [26,27].The trapped 229g Th 3+ ions are laser-cooled to form a Coulomb crystal.
Once the 229g Th 3+ ions are laser-cooled down to 50 µK, the Doppler broadening of the transition 6 2 D 5/2 → 5 2 F 7/2 at 984 nm is about 1 MHz [24].At this temperature, the corresponding Doppler broadenings of the direct nuclear excitation at 150 nm and the 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].The two-dimensional and linear-chain Coulomb crystals can be simulated with a single ion resolution [29,30].Using the GPU-accelerated LAMMPS [31] wrapped by the LION package [32], a Coulomb crystal consisting of 36 229g Th 3+ ions are simulated as shown in Fig. 3.When the endcap voltage of  ions is obtained.From the simulation results, the motion amplitude of ions in x-and y-direction (the 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 for increasing the EB excitation rate and the direct nuclear excitation rate.
To excite EB excitation from the state 7P 1/2 of the 229g Th 3+ ions, the three-level transition 5 2 F 5/2 → 6 2 D 3/2 → 7 2 P 1/2 can be used to accumulate the population at the 7P 1/2 state.Once the 229g Th 3+ ions are laser-cooled down to 50 µK, the 690 nm and 984 nm laser are turned off in sequence to maintain the ions all at the 5 2 F 5/2 state, the 1088 nm laser is kept on.To transfer a large population to the 7 2 P 1/2 state with its lifetime of 1 ns, both 269 nm and 196 nm transitions are excited simultaneously as shown in Fig. 1.The natural linewidths of 269 nm transition and 196 nm transition are about 2π • 32 MHz and 2π • 127 MHz, respectively [34].The continuous wave (CW) lasers at both the wavelengths are commercially available [35].
The spontaneous radiation rate of the hyperfine structure level transitions between two electronic energy levels is [36] where ξ i (ξ f ) incorporates all other electronic quantum numbers, J i (J f ) is the total electronic quantum number of initial (final) state, F i (F f ) is the total quantum number of initial (final) state, I g is the nuclear spin of nuclear ground state, d is electron dipole, ω is the angular frequency of the electronic transition, is the reduced Planck constant, c is the speed of light in vacuum.
2 is the transition strength of the hyperfine structure level, which can be expressed as where is the transition strength of the electronic energy levels.The spontaneous radiation rate of the electronic energy levels is The excitation rate of the hyperfine structure levels between two electronic energy levels is [37] where P ω is the laser spectral intensity.The stimulated emission rate of the hyperfine structure levels between two electronic energy levels is With the laser power of 1 mW and 100 µm laser spot size for 1088 nm, 269 nm and 196 nm lasers, the steady-state condition can be fulfilled for all 5 2 F 5/2 , 7 2 S 1/2 , 6 2 D 3/2 and 7 2 P 1/2 states under continuous illumination.For the multiple hyperfine structure transitions between electronic transitions 5 2 F 5/2 ↔ 6 2 D 3/2 and 7 2 S 1/2 ↔ 7 2 P 1/2 , the power of the lasers is divided equally.Thus, after state preparation, the population of the | 7P 1/2 , F = 2 state accounts for about 12% of the total number of trapped ions.

III. THE 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 of the optical Bloch equations as the following [38][39][40]: 1 2 Here Ω eg denotes the Rabi frequency between the isomeric state and the nuclear ground state.Γ = Γ γ + Γ nr is the total linewidth of the isomeric state, Γ γ denotes the γ decay rate of the isomeric state.Because the ionization energy of 229g Th 3+ is 28.6 eV is much greater than the energy of the isomeric state, the internal conversion process is strongly forbidden [41].The possible non-radiation transition linewidth Γ nr is then dominated by the electronic bridge transition for 229g Th 3+ .Γ = Γ+ΓL 2 + Γ add denotes the total transition linewidth [40], Γ L denotes the laser linewidth used for the nuclear excitation, Γ add is the additional incoherent linewidth such as phonon coupling of ions in a Coulomb crystal.Since Γ add is much smaller than the laser linewidth Γ L , the influence of incoherent linewidth is negligible.t denotes the interaction time between the laser and the 229g Th 3+ ions.Here λ = |Ω 2 eg − (Γ − Γ) 2 /4|.Under the resonant condition, the Rabi frequency between the isomeric state and the nuclear ground state is [39] Here I is the intensity of the excitation laser, ω m is the angular frequency of the nuclear transition.The Zeeman splitting cuased by the geomagnetic field between the two outermost lines is below 1 kHz and can be negligible.So the Clebsch-Gordan coefficient of the sub-states transition between the isomeric state and the nuclear ground state C eg is 1 [16].
Assuming the intensities of all the lasers have Gaussian distribution, the Rabi frequency can be modified as the following two situations: i) If the laser linewidth (Γ L ) is greater than the Doppler broadening linewidth (Γ D ), only a part of the laser with the matched frequency range can interact with the triply charged thorium ions effectively.The equivalent Doppler broadening linewidth is The effective laser intensity is The modified Rabi frequency is ii) If the laser linewidth (Γ L ) is smaller than the Doppler broadening linewidth (Γ D ), only a part of ions interact with the laser light is on the frequency resonant at a time, and the equivalent laser linewidth is The effective Doppler broadening linewidth is The modified Rabi frequency is IV. DIRECT NUCLEAR EXCITATION Th-229 isomeric energy is 8.12 ± 0.11 eV, corresponding to a wavelength of 150.7−154.8nm.Currently, the available light sources such as the VUV pulsed laser [42], the synchrotron radiation light source [14], and the 7th harmonic of a Yb-doped fiber laser [16,[43][44][45] are suitable candidates for the direct nuclear excitation.Here, based on the resonanceenhanced four wave mixing, the parameter 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, so the corresponding laser intensity I = ELRL πr 2 = 4.67 • 10 6 W/m 2 can be reached.The temperature of Th 3+ ions is 50 µK, the corresponding Doppler broadening linewidth Γ D = 6.5 MHz is narrower than the laser linewidth.So the modified Rabi frequency on resonance Ω ′ eg (δ = 0) = 107 Hz can be obtained based on Eq. ( 9)−( 11), which is much smaller than the total transition linewidth Γ as shown in Table 1.In this case, Ω ′ eg < |Γ− Γ| 2 is satisfied.The number of excited nuclei is calculated as a function of time based on Eq. ( 7).As shown in Fig. 4, the number of excited Th-229 nuclei is only 0.12 out of 100 Th-229 ions and reaches saturated at the irradiation time of 30000 s.Such required laser irradiation time is beyond the stable trapping time, therefore, no Th-229 isomeric state can be detected under the current experimental condition.

V. ELECTRONIC BRIDGE EXCITATION
The direct nuclear excitation is limited by the natural linewidth of the isomeric state and the power of the excitation laser.If the EB excitation takes place, the transition linewidth can be increased by about 40 times compared with the direct nuclear excitation [46].The EB excitation can be utilized by exciting the quantum state of the electronic state 7P 1/2 of 229g Th 3+ (7P  The wavelength of the laser for the EB excitation is 350 nm.A 350 nm laser with the power of 250 mW can be obtained from Toptica DLC TA-SHG pro and the laser is focused to a spot size of 10 µm.Then an intensity of 7.95 • 10 8 Wm −2 can be achieved.Therefore, the Rabi frequency is estimated to be 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 Ω ′ eg is 140 kHz on reso-nance and 98 kHz on the detuning Γ D /2, respectively (shown in Table 2).Then the condition Ω ′ eg < |Γ− Γ| 2 is satisfied, so the number of excited nuclei as a function of time can be obtained based on Eq. ( 7) and shown in Fig 5 .If the excitation energy from the state 7P 1/2 of 229g Th 3+ to the state 7S 1/2 of 229m Th 3+ is on resonance, about 2 229g Th 3+ 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, about 1.8 229g Th 3+ ions can be excited to the isomeric state with the same irradiation time.
After a successful EB excitation, the Th-229 ions at the isomeric state can be identified based on the different hyperfine structures of the isomeric state and the nuclear ground state.The detailed hyperfine structures of the isomeric state are shown in the Appendix B. After EB nuclear excitation, the EB laser at 350 nm and the state-preparation lasers at 196 nm and 269 nm are both turned off, the cooling lasers at 690 nm and 984 nm are switched back on immediately after and combined with the 1088 nm laser to cool the ions.The nuclear-excited ions stay at the isomeric states and accumulate at the electronic state of 5 2 F 5/2 as illustrated in Fig. 1.The rest nuclear-unexcited ions remain in the nuclear ground state of the Doppler cooling cycle of 5 2 F 5/2 , 6 2 D 5/2 , 5 2 F 7/2 and 6 2 D 3/2 .Because the isomer shift of 6D 5/2 and 6D 3/2 electronic state are both about 400 MHz [24,47].The 690 nm and 984 nm lasers are off-resonance for the Doppler cooling of ions at the isomeric state.As a result, the nuclear-excited 229m Th 3+ ions leave the 690 nm and 984 nm lasers' cooling cycle and appear as the dark ions in the Coulomb crystal [7].In the linear-chain 229 Th 3+ ions Coulomb crystal, single ions are distinguishable and any dark 229m Th 3+ ions in the dark generated by a successful EB excitation event can be detected.The current reported uncertainty of the isomeric state energy is 0.22 eV [13].To cover this uncertainty range, it requires about 1.9 • 10 7 scans for each scanning interval of the Doppler broadening linewidth Γ D .The time for a single scan step takes about 300 µs, therefore the total scan time takes about 5670 s as shown in Table 3.The final isomeric energy is determined by the sum of the energy interval between the electronic state 7P 1/2 of 229g Th 3+ and the electronic state 7S 1/2 of 229m Th 3+ and the photon energy of the 350 nm laser.Since the uncertainty of the natural linewidth of the state 7P 1/2 of 229g Th 3+ is 1 GHz which is much larger than the laser linewidth and Doppler broadening linewidth, the uncertainty of isomeric energy is on the order of 1 GHz (corresponding to 4 • 10 −6 eV) dominated by the natural linewidth of the state 7P 1/2 .

VI. CONCLUSION
In this paper, taking account of the Doppler broadening of thorium ions in the ion trap, we show that 2 229g Th 3+ ions out of 36 trapped ions can be excited to the isomeric state under the resonant condition via the EB excitation.We present a feasible proposal that a total measurement time of about 2 hours can be achieved for the current uncertainty of the isomeric state energy 0.22 eV.If a cryogenic linear Paul trap is used, the ions can be stably trapped for a longer period of time and the radiative lifetime of the isomeric state can be measured.The utilization of the EB excitation can reduce the uncertainty of Th-229 isomeric energy to about 1 GHz.The calculation is based on the theoretical prediction that the transition linewidth of the EB excitation is 40 times larger than the direct nuclear excitation.If the 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 expectedly strong or the explored nuclear transition energy range is not accurate, and the laser energy used for the EB excitation needs to be increased.On the other hand, successfully observing the 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.

APPENDIX A. THE EXPERIMENTAL STEPS
As shown in Fig. 6, the experiments are divided into three steps: 1 Laser cooling: the 1088 nm, 984 nm and 690 nm lasers are turned on to laser-cool the ions into a Coulomb crystal of a linear chain.Later, the 690 nm laser and 984 nm laser is turned off in sequence to keep all the population at the state 5F 5/2 .
2 State preparation and nuclear excitation: once the 690 nm laser is turned off, the 269 nm and 196 nm lasers are turned on to prepare the 229g Th 3+ ions at the state of 7P 1/2 and the 350 nm laser is turned on to excite the Th-229 nuclei from the nuclear ground state to the isomeric state.
3 Detection of the isomeric state: the 350 nm laser is turned off to prohibit the electronic bridge transition from 7S 1/2 of the isomeric state and maintain the population of isomeric state.Simultaneously, the 196 nm and 269 nm lasers are turned off in sequence to cumulate the ions at the four lowest electronic states.Then, 690 nm and 984 nm lasers are turned on to detect the isomeric state based on the isomer shift.where K = F (F + 1) − J(J + 1) − I m (I m + 1).
Based on Eq.B17, the hyperfine structures of the 229m Th 3+ four lowest levels are calculated as shown in Table.5 The isomer shift is determined by where the difference in the mean-square radii of the isomeric and ground states in 229 Th is δ < r 2 > = 0.012(2)f m 2 [49] and the field-shift constant F ′ can be obtained in ref [42].So the centers of hyperfine structures of 229m Th 3+ can be obtained as shown in Table .6.Both the hyperfine structures of 229m Th 3+ and 229g Th 3+ are displayed in Fig. 7 and the ∆ν m i indicates the isomer shift.The transition frequency of | 5F 5/2 , F = 1 → | 6D 5/2 , F = 0 in 229g Th 3+ is 434282480 (32) MHz.The hyperfine transitions between the state 5F 5/2 and the state 6D 5/2 of 229m Th 3+ is listed in Table .7.   8.As a result, the 690 nm and 984 nm lasers used for cooling of the 229g Th 3+ ions can not form the closed optical cycle for laser-cooling the 229m Th 3+ ions.

Fig. 1 .
Fig. 1.Diagram of electronic energy levels and electric-dipole transitions of Th-229 triply charged ions.Inserts show the processes of the EB excitation, the red arrows are the 150 nm M 1 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, respectively.The purple arrow is the 350 nm EB excitation.The lifetime of electronic excited state is indicated in the parentheses.Optical transition wavelengths are in nm and the integers near atomic levels indicate the principal quantum numbers.

Fig. 3 .
Fig. 3.The simulation results of Coulomb crystal consisting of 36 229g Th 3+ 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.

Fig. 4 .
Fig. 4. The expected number of the excited nuclei is calculated as a function of time when 100 triply charged Th-229 ions are directly illuminated by a direct excitation laser.

Fig. 5 .
Fig. 5.The expected number of the excited nuclei is calculated as a function of time when 36 triply charged Th-229 ions are trapped and 4 ions are at the state 7P 1/2 .

Fig. 6 .
Fig. 6.The experimental steps and the time of each laser switch.

Fig. 7 .
Fig. 7.The four lowest-lying fine structure levels of 229 Th 3+ , including hyperfine structures.The integers near atomic levels indicate total angular momentum quantum numbers F .

Table 2 .
Values of variables used for the calculation of the number of excited nuclei based Eq. (7).

Table 3 .
Main parameters for 350 nm irradiation on 36 trapped ions.

Table 5 .
The hyperfine structures of the 229m Th 3+ four lowest levels (MHz).

Table 6 .
The centers of the hyperfine structures of 229 Th 3+ four lowest levels (MHz).