Populating ^229mTh via two-photon electronic bridge mechanism

Populating 229m Th via two-photon electronic bridge mechanism 1 2 first-author 3 2 corresp zhangguoqiang@zjlab.org.cn zhangguoqiang@zjlab.org.cn zhangguoqiang@zjlab.org.cn https://orcid.org/0000-0002-5298-9661 4 corresp cbfu@fudan.edu.cn cbfu@fudan.edu.cn cbfu@fudan.edu.cn https://orcid.org/0000-0002-0031-2077 4 2 1 corresp mayugang@fudan.edu.cn mayugang@fudan.edu.cn mayugang@fudan.edu.cn https://orcid.org/0000-0002-0233-9900 ShanghaiTech University, Shanghai 201210, China ShanghaiTech University, Shanghai 201210, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China The isomer 229m Th is the most promising candidate for clocks based on the nuclear transition because it has the lowest excitation energy of only 8.10±0.17 eV. Various experiments and theories have focused on methods of triggering the transition between the ground state and isomeric state, among which the electronic bridge (EB) is one of the most efficient. In this paper, we propose a new electronic bridge mechanism via two-photon excitation based on quantum optics for a two-level nuclear quantum system. The long-lived 7 s 1/2 electronic shell state of 229m Th 3+ , with a lifetime of approximately 0.6 s, is chosen as the initial state and the atomic shells (7 s –10 s ) could be achieved as virtual states in a two-photon process. When the virtual states return to the initial state 7 s 1/2 , there is a chance of triggering the nucleus 229 Th 3+ to its isomeric state 229m Th 3+ via EB. Two lasers at moderate intensity ((10 10 – 10 14 ) W/m 2 ), with photon energies near the optical range, are expected to populate the isomer at a saturated rate of approximately 10 9 s -1 , which is much higher than that due to other mechanisms. We believe that this two-photon EB scheme can help in the development of nuclear clocks and deserves verification via a series of experiments with ordinary lasers in laboratories. Electronic bridge 229 Th Nuclear clocks Two-photon excitation 1 Introduction 1 Since ancient times, humans have pursued the development of more accurate clocks to arrange social activities and elucidate the secrets of the universe. One of the most important applications of an accurate clock is in global navigation satellite systems, such as the global positioning system (GPS) or BeiDou navigation satellite system (BDS), but they are also used in basic scientific research. Some important units, such as the meter, are defined in relation to a second. Even the time measurement itself would be meaningful, a more precise clock might reveal the intrinsic properties of space and time at the quantum level; e.g., it might be discrete instead of continuous, per the hypothesis of relativity theory. Currently, atomic or optical clocks are the most accurate time and frequency standards [ 1 1 ]. In 1967, the International System of Units (SI) second was officially redefined based on the isotope atom 133 Cs: "The second is the duration of 9192631770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom". In the following decades, the accuracy of this standard was improved from 10 -12 (around 100 ns per day) to 10 -16 [ 2 2 ] because of the significant reduction in the noise-to-signal ratio, with the help of laser cooling. In 2019, scientists from the National Institute of Standards and Technology (NIST) demonstrated an Al + clock with a total uncertainty of 9.4×10 -19 [ 3 3 ], which is the first demonstration of a clock with an uncertainty of less than 10 -18 . Recently, atomic clocks based on optical rather than microwave transitions have achieved higher accuracy (2.5×10 -19 ) and stability performance (within 15 s) [ 4 4 ], which might lead to redefining the current cesium microwave-based SI second in the near future. Despite the great accuracy that atomic and optical clocks have achieved, clocks based on a nuclear transition rather than atomic electron transitions could be more steady and accurate because of their smaller size, with the shielding effects of the surrounding electrons and their higher frequencies. However, nuclei are difficult to control owing to their higher excitation energies (keV to MeV), which have already exceeded those (eV) from modern microwave or laser technologies. Fortunately, two nuclei with excited states lower than 100 eV — i.e., 229m Th (8.10 eV) and 235m U (76 eV) — have been determined thus far. The former has attracted more attention because its transition frequency is closer to the optical range. In 2003, a nuclear optical clock based on a single 229 Th 3+ was first proposed by E. Peik and C. Tamm [ 5 5 ], although a nuclear transition with an energy of 3.5 eV was much lower than the mean experimental value of 8.10 eV. In their pioneering work, a double-resonance method was proposed with two lasers to excite the nuclear shell and the atomic shell of 229 Th 3+ , respectively. In 2012, the single 229 Th 3+ ion nuclear clock was further investigated by Campbell et al. [ 6 6 ], with a total fractional inaccuracy of 1.0×10 -19 , which is approximately an order of magnitude higher than that achieved by the best optical atomic clocks at the time. Instead of exciting an electronic shell state, the nuclear clock proposed by Campbell et al. uses a stretched pair of nuclear hyperfine states in the electronic ground-state configuration, which demonstrates advantages with respect to the achievable quality factor and suppression of the quadratic Zeeman shift. To obtain more precise clocks, an increasing number of proposals for nuclear clocks based on the isomeric isotope 229m Th have been suggested, where the key is how to populate the isomeric state. During the last two decades, various theories have been proposed for populating 229 Th to its isomeric state, which can be grouped into laser direct photon excitation, nuclear excitation by electron capture (NEEC), nuclear excitation by electron transition (NEET), and electronic bridges (EB) (see Ref. [ 10 10 ] for a detailed review). Laser direct excitation relies on the precision of the isomeric energy, which has not yet been sufficient. Therefore, indirect excitation schemes — such as NEET, NEEC, and EB — were investigated in alternative ways. NEEC requires a plasma environment to provide free electrons, which seems too harsh to guarantee a low noise level for nuclear clocks. Conversely, NEEC may be a good method for nuclear batteries, such as 93 Mo and 178 Hf. In the NEET process, a nucleus is excited and a real electronic shell state is simultaneously deexcited, which is a third-order process [ 10 10 ]. Sometimes, it is difficult to distinguish the difference between NEET and EB because they share a similar physics scheme. In Ref. [ 11 11 ], Karpeshin claimed that during NEET processes the virtual level is populated after nuclear excitation whereas, in EB processes, a virtual electronic level is populated before nuclear excitation. Considering all of the theories, it seems that the EB is the most promising for nuclear clocks because of its highly efficient transition rate. Thus far, the uncertainty of the energy isomeric state has seriously hindered the development of nuclear clocks based on 229 Th. In the 1970s, the energy was found to be below 100 eV [ 7 7 ] and then below 10 eV in the late 1980s [ 8 8 ]. The energy has shifted from 4.5 ± 1 eV [ 9 9 ] to now 8.10 ± 0.17 eV [ 15 15 ]. Therefore, it was difficult to observe a clear signal from the laser direct excitation experiments. During the EB process, the virtual electronic level tolerates a larger uncertainty of the energy. The EB becomes an important method of populating the 229 Th to its isomeric state. Thus, methods based on EB excitation have been proposed during the last decade. In particular, the EB excitation scheme for highly charged 229 Th 35+ ions in an EBIT trap was given by Bilous et al. [ 16 16 ]. In this paper, we propose a new theory for calculating the EB excitation rate with two photons for 229 Th 3+ . We apply the optical Bloch equation for a two-level nuclear system based on an open quantum system and nuclear quantum optics. Taking electrons and nuclei as an effective two-level system during interaction with laser beams and assuming that the system is at equilibrium, we deduce the general formulae for the excitation rate Γ eb and electron bridge enhancement R , respectively. Then, we choose specific atomic shells (7 s -10 s ) as the virtual electronic levels to calculate the transition rates for Th 3+ . We find that the excitation rate Γ eb and electron bridge enhancement R both reach their maxima when the intensities of the lasers approach the critical value. Moreover, the electron bridge enhancement R should, eventually, be less than one under a relativity intense laser, indicating that populating the isomeric isotope using a two-photon electronic bridge is not an effective method. 2 Theoretical descriptions 1 In this section, we deduce a general formalism for two-photon EB excitation. Figure 1(a) Figure 1(a) shows the Feynman diagram of a two-photon EB excitation process, where the lower case letters a , b , d , and c denote the atomic shells and g and m indicate the ground and excitation(isomeric) states of the nuclei, respectively. To obtain its expression, one can use the connection between the EB excitation process and the corresponding inverse process of the bound internal conversion (BIC) process [ 12 12 ], as shown in Fig. 1(b) Fig. 1(b) . This two-photon BIC process can be regarded as a combination of a subprocess one-photon BIC from ( a ) to ( d ) and the decay from ( d ) to ( c ). Thus, the two-photon BIC rate can be expressed as: <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Δ</mi><mtext>s</mtext></msub><mo stretchy="false">)</mo><msup><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Δ</mi><mtext>s</mtext></msub><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math> Here, Γ ( d → c ) denotes the partial natural decay rate from state ( d ) to ( c ) and Γ d is the total natural decay rate (line width) of state ( d ). The one-photon BIC is <math id="M2"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><msub><mo>∑</mo><mi>b</mi></msub></mstyle><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac></mrow></math> , where the frequency difference between the atomic transition and nuclear transition Δ s = Ω b -( Ω n + Ω a ), where Ω a , Ω b , and Ω n are the transition frequencies of ( a ), ( b ), and the nucleus. <math id="M3"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></math> =( Γ a + Γ b + Γ n )/2 and <math id="M4"><mrow><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> , Γ a , Γ b , and Γ n are the widths of the internal conversion (IC) process from ( a ) to ( b ), ( a ), ( b ), and the isomeric state, respectively [ 10 10 ]. Inserting <math id="M5"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> into Eq. (1), we obtain the expression for the two-photon BIC process: <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msubsup><mi>Γ</mi><mrow><mi>i</mi><mi>c</mi></mrow><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mi>s</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> One can now obtain the expression of the two-photon EB excitation using the connection between excitation and natural decay rate [ 12 12 ], <math display="block" id="M7"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mi mathvariant="script">l</mi></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msubsup><mn>.</mn></mrow></math> Here, <math id="M8"><mrow><msub><mi>I</mi><mi mathvariant="script">l</mi></msub></mrow></math> is the intensity of the external laser field. <math id="M9"><mrow><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>Ω</mi><mtext>n</mtext></msub><mo>+</mo><msub><mi>Ω</mi><mi>c</mi></msub><mo>−</mo><msub><mi>Ω</mi><mi>a</mi></msub></mrow></math> is the EB resonance frequency for the final state (c). <math id="M10"><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>Γ</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><msub><mi>Γ</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></mrow></math> is the decoherence rate of the EB resonance. <math id="M11"><mrow><msubsup><mi>Γ</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> is the total linewidth of the two-photon EB resonance, obtained as the convolution of the individual linewidths of the atomic and nuclear transitions, <math id="M12"><mrow><msubsup><mi>Γ</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>Γ</mi><mtext>n</mtext></msub><mo>+</mo><msub><mi>Γ</mi><mi>a</mi></msub><mo>+</mo><msub><mi>Γ</mi><mi>c</mi></msub></mrow></math> . The partial BIC process rate <math id="M13"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> can be obtained by changing the initial and final states(a) and (c) in Eq. (2): <math display="block" id="M14"><mrow><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> Considering two incident laser beams I 1 and I 2 for the two-photon EB scheme, two factors must be considered: <math id="M15"><mrow><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><msubsup><mi>Ω</mi><mtext>s</mtext><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow></math> , <math id="M16"><mrow><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac></mrow></math> . Thus, one obtains the proper formula for the two-photon EB excitation rate from Eq. (3), <math display="block" id="M17"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><msubsup><mi>Γ</mi><mrow><mtext>bic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msubsup><mn>.</mn></mrow></math> Inserting Eq. (4) into Eq. (5) and exchanging ( b ) and ( d ), we obtain <math display="block" id="M18"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> Here <math id="M19"><mrow><msub><mi>Ω</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>Ω</mi><mi>b</mi></msub><mo>−</mo><msub><mi>Ω</mi><mi>a</mi></msub></mrow></math> , <math id="M20"><mrow><msub><mi>Δ</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>Ω</mi><mn>1</mn></msub><mo>−</mo><mo stretchy="false">(</mo><msub><mi>Ω</mi><mi>b</mi></msub><mo>−</mo><msub><mi>Ω</mi><mi>a</mi></msub><mo stretchy="false">)</mo></mrow></math> , <math id="M21"><mrow><msub><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow></msub><mo>=</mo><msub><mi>Ω</mi><mn>2</mn></msub><mo>−</mo><msub><mi>Ω</mi><mi>d</mi></msub><mo>+</mo><msub><mi>Ω</mi><mi>b</mi></msub></mrow></math> , <math id="M22"><mrow><msub><mi>Γ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow></msub><mo>=</mo><msub><mi>Γ</mi><mi>d</mi></msub><mo>+</mo><msub><mi>Γ</mi><mi>b</mi></msub></mrow></math> , <math id="M23"><mrow><msub><mi>Γ</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>Γ</mi><mi>b</mi></msub><mo>+</mo><msub><mi>Γ</mi><mi>a</mi></msub></mrow></math> , <math id="M24"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Γ</mi><mtext>s</mtext></msub><mo>+</mo><msub><mi>Γ</mi><mtext>l</mtext></msub><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></mrow></math> , <math id="M25"><mrow><msub><mi>Ω</mi><mrow><mtext>res</mtext></mrow></msub><mo>=</mo><msub><mi>Ω</mi><mtext>n</mtext></msub><mo>+</mo><msub><mi>Ω</mi><mi>c</mi></msub><mo>−</mo><msub><mi>Ω</mi><mi>b</mi></msub></mrow></math> , <math id="M26"><mrow><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> is the IC process from ( c ) to ( d ). The IC process <math id="M27"><mrow><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> takes the form of Eq. [ 13 13 ] <math display="block" id="M28"><mrow><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M29"><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> is the internal conversion coefficient. If only the bound states for both the initial and final electron states, a and c , are considered, Eq. (6) becomes <math display="block" id="M30"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mi>s</mi><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> In this study, we follow. [ 10 10 ], which takes the nuclear ground and excited states as a two-level quantum system in an external laser field. The corresponding evolution density matrix for this system is <math display="block" id="M31"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>ρ</mi><mrow><mtext>ee</mtext></mrow></msub><mo>|</mo><mi>e</mi><mo>〉</mo><mo>〈</mo><mi>e</mi><mo>|</mo><mo>+</mo><msub><mi>ρ</mi><mrow><mtext>ge</mtext></mrow></msub><mo>|</mo><mi>g</mi><mo>〉</mo><mo>〈</mo><mi>e</mi><mo>|</mo><mo>+</mo><msub><mi>ρ</mi><mrow><mtext>eg</mtext></mrow></msub><mo>|</mo><mi>e</mi><mo>〉</mo><mo>〈</mo><mi>g</mi><mo>|</mo><mo>+</mo><msub><mi>ρ</mi><mrow><mtext>gg</mtext></mrow></msub><mo>|</mo><mi>g</mi><mo>〉</mo><mo>〈</mo><mi>g</mi><mo>|</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Here, | g ⟩ and | e ⟩ are the ground and excited states, respectively. The population density ρ exc ( t ) under resonant laser irradiation can be modeled using Torrey’s solution of the optical Bloch equations [ 14 14 ]. The Rabi frequency Ω eg for the nuclear transition is introduced as in [ 14 14 ] <math display="block" id="M32"><mrow><msubsup><mi>Ω</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mo>=</mo><mn>2</mn><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mi mathvariant="script">l</mi></msub><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><mi>ℏ</mi><msubsup><mi>Ω</mi><mtext>n</mtext><mn>3</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></math> where <math id="M33"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup></mrow></math> is the excitation rate. <math id="M34"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><mo>=</mo><msub><mi>Γ</mi><mi>Γ</mi></msub><mo>+</mo><msub><mi>Γ</mi><mrow><mtext>nr</mtext></mrow></msub></mrow></math> is the total width of the nuclear transition, including Γ decay ( Γ Γ ) and other non-radiative decay channels ( Γ nr ). Similarly, to calculate the two-photon EB process using this nuclear quantum optics, we define the Rabi frequency for the EB process as <math id="M35"><mrow><msubsup><mi>Ω</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mtext> </mtext><mn>2</mn></mrow></msubsup><mo>=</mo><mn>2</mn><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> , which can be used to model the on-resonance nuclear population density as a function of time for EB excitation when substituting Ω eg for <math id="M36"><mrow><msubsup><mi>Ω</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> and <math id="M37"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow></math> for <math id="M38"><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> in Eq. (10). In the following, we extend a low saturation limit approximation to a general case. 2.1 low saturation limit 2 Assuming that the intensity of the laser is sufficiently low (so that the excited state is far less populated than the ground state), the solution for the optical Bloch equation is[ 14 14 ] <math display="block" id="M39"><mrow><msubsup><mi>ρ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo>≡</mo><msub><mi>ρ</mi><mrow><mtext>ee</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mi>Ω</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>Γ</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mi>Ω</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>Γ</mi><mtext>n</mtext></msub><mi>t</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> Given sufficient time, t , the system evolves; when the population of the excited state is in equilibrium, i.e., the excitation rate is equal to the total decay rate, the total decay rate can be expressed as a product of the population density and the natural decay rate: <math display="block" id="M40"><mrow><msubsup><mi>ρ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo>≡</mo><msub><mi>ρ</mi><mrow><mtext>ee</mtext></mrow></msub><mo>→</mo><mfrac><mrow><msubsup><mi>Ω</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>Γ</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msubsup><mi>Δ</mi><mtext>n</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext><mn>2</mn></msubsup></mrow></mfrac><mo>=</mo><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext><mn>2</mn></msubsup><mo>/</mo><msub><mi>Γ</mi><mtext>n</mtext></msub></mrow><mrow><msubsup><mi>Δ</mi><mtext>n</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext><mn>2</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></math> With nonzero detuning <math id="M41"><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> of the laser light with respect to the EB resonance, we can obtain <math display="block" id="M42"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Δ</mi><mrow><mtext>res</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msubsup><mi>ρ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Δ</mi><mrow><mtext>res</mtext></mrow></msub><mo stretchy="false">)</mo><msub><mi>Γ</mi><mtext>n</mtext></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> Inserting Eq. (8) into Eq. (13), an additional factor <math id="M43"><mrow><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> for the EB excitation rate at a low saturation limit. <math display="block" id="M44"><mrow><msubsup><mi>Γ</mi><mrow><mtext>ls</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo>≡</mo><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Δ</mi><mrow><mtext>res</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>1</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mi>Ω</mi><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mn>2</mn></msub></mrow><mrow><mi>ℏ</mi><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mi>s</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mi>s</mi><mn>2</mn></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mi>s</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mi>s</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> here the ( c ) index was dropped for easier notation and Δ res = Ω 2 - Ω res = Ω 2 - Ω n - Ω c + Ω b . Note that this equation is similar to that in [ 10 10 ] and others [ 11 11 , 12 12 ] but our formula does not require a certain excited electronic state ( b ). An interesting feature of the low-limit saturation two-photon EB excitation is its double resonance effect. When Δ s =0 is satisfied in the system, one will obtain Δ res =0 and vice versa. At resonance Δ s =0, Δ res =0, when a =7 s , c =7 s , b =7 p 1/2 , d =8 s , <math id="M45"><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><msub><mi>I</mi><mn>2</mn></msub><mo>≡</mo><mi>I</mi><msup><mrow><mo>=</mo><mn>3</mn><mo>×</mo><mn>10</mn></mrow><mn>5</mn></msup></mrow></math> W/m 2 , Γ l =10 -5 eV are satisfied, we obtain <math id="M46"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo>=</mo><mn>0.082</mn></mrow></math> s -1 , which is comparable to the results reported in [ 11 11 , 12 12 ], where the values of 10 s -1 and 0.0281 s -1 for Th + were obtained, respectively. For convenience, considering two incident lasers with the same laser intensity <math id="M47"><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><msub><mi>I</mi><mn>2</mn></msub><mo>≡</mo><mi>I</mi></mrow></math> , the EB enhancement coefficient R can be defined as the ratio between the EB excitation cross-section and the direct photon excitation cross-section. <math display="block" id="M48"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>σ</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>σ</mi><mi>Γ</mi></msub></mrow></mfrac><mo>≃</mo><mfrac><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup></mrow></mfrac></mrow></math> Here, <math id="M49"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> denotes the EB-excitation rate at resonance, and <math id="M50"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup></mrow></math> is the excitation rate for direct photon excitation [ 10 10 ]: <math display="block" id="M51"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup><mo>=</mo><mfrac><mrow><msub><mi>σ</mi><mi>Γ</mi></msub><msub><mi>I</mi><mi mathvariant="script">l</mi></msub></mrow><mrow><mi>ℏ</mi><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mi mathvariant="script">l</mi></msub></mrow><mrow><mi>ℏ</mi><msubsup><mi>Ω</mi><mtext>n</mtext><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow></mfrac><msub><mi>Γ</mi><mi>Γ</mi></msub><mn>,</mn></mrow></math> Here, <math id="M52"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mi>n</mi></msub></mrow></math> =( Γ n + Γ l )/2. Inserting Eq. (14) and Eq. (16) into Eq. (15), one can obtain the enhancement coefficient for two-photon EB excitation at a low saturation limit: <math display="block" id="M53"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mrow><mtext>res</mtext></mrow><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></math> Here the ( c ) index was dropped for easier notation and the subscript ‘ls’ indicates the low saturation limit. At resonance Δ s =0, Δ res =0, and we obtain <math display="block" id="M54"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>2</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math> 2.2 general case 2 When the laser beam is sufficiently large or there is a double resonance effect, the excitation rate is large. In this case, the low saturation limit can no longer provide a good prediction. Then, the general steady-state solution of the optical Bloch equation is adopted [ 14 14 ] <math display="block" id="M55"><mrow><msub><mi>ρ</mi><mrow><mtext>ee</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mi>Ω</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><mfrac><mrow><mn>2</mn><mi>Γ</mi></mrow><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover></mfrac><mrow><mo>(</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mi>Ω</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><msubsup><mi>Ω</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></math> Using the same procedure from quantum opticals as in the low-saturation limit, <math id="M56"><mrow><msub><mi>Ω</mi><mrow><mtext>eg</mtext></mrow></msub><mo>→</mo><msub><mi>Ω</mi><mrow><mtext>eb</mtext></mrow></msub></mrow></math> , <math id="M57"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub><mo>→</mo><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></math> <math display="block" id="M58"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>Γ</mi><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><msubsup><mi>ρ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mi>Γ</mi><mtext>n</mtext></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mfrac><mrow><msubsup><mi>Γ</mi><mrow><mtext>ls</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow><mrow><mfrac><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msubsup><mi>Γ</mi><mrow><mtext>ls</mtext></mrow><mrow><mtext>eb</mtext></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> At resonance, Δ res =0, <math id="M59"><mrow><msub><mi>Δ</mi><mrow><msub><mi>s</mi><mn>1</mn></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> , and the electronic excited state (b) is fixed. Considering <math id="M60"><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>=</mo><msub><mi>I</mi><mn>2</mn></msub><mo>≡</mo><mi>I</mi></mrow></math> , we obtain <math display="block" id="M61"><mrow><msup><mi>Γ</mi><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfrac><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mtext>ls</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mtext>ls</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mn>1</mn><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M62"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mfrac><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>1</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><msup><mi>R</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>'</mo></mrow></msup></mrow></mfrac><mn>,</mn></mrow></math> where superscript ‘ g ’ indicates general excitation and <math display="block" id="M63"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>2</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mfrac><mrow><msubsup><mi>Γ</mi><mrow><mtext>ic</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M64"><mrow><msup><mi>R</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>'</mo></mrow></msup><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>2</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>Γ</mi><mrow><mi>i</mi><mi>c</mi></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> It seems that the general formalism for Γ eb( g )( b ) is simply multiplied by a decay factor, such as <math id="M65"><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mn>2</mn><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mrow><mtext>eb</mtext><mo stretchy="false">(</mo><mtext>ls</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mn>1</mn><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac></mrow></mfrac></mrow></math> and is similar to R . Therefore, we may have a maximum for both Γ eb( g )( b ) and R , as presented in the next section. Note that this is a general result and, in this work, we use it to calculate Th 3+ . 3 Results and discussion 1 First, we set up some parameters before performing the calculations. Recently, an energy of 8.10±0.17 eV for 229m Th was obtained [ 15 15 ], which has the same precision as the value obtained from IC spectroscopy [ 17 17 ]. Generally, the radiative decay rate <math id="M66"><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub><mo>=</mo><mn>1</mn><mo>/</mo><msub><mi>τ</mi><mi>Γ</mi></msub></mrow></math> of magnetic multipole transitions can be expressed in terms of the energy-independent reduced transition probability <math id="M67"><mrow><msub><mi>B</mi><mo>↓</mo></msub><mo>#</mo><mi>x</mi><mn>2193</mn><mo>;</mo><mo stretchy="false">(</mo><mi>M</mi><mi>L</mi><mo stretchy="false">)</mo></mrow></math> , as in [ 18 18 ] <math display="block" id="M68"><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow><mi>ℏ</mi></mfrac><mfrac><mrow><mi>L</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>L</mi><msup><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>2</mn><mi>L</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo>!</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mi>c</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>L</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>B</mi><mo>↓</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>L</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, L denotes multipolarity and Ω n is the angular frequency of the nuclear transition. However, in this study, we choose <math id="M69"><mrow><msub><mi>B</mi><mo>↓</mo></msub><mo>#</mo><mi>x</mi><mn>2193</mn><mo>;</mo><mo stretchy="false">(</mo><mi>M</mi><mi>L</mi><mo stretchy="false">)</mo></mrow></math> =5.0×19 -3 W . u . and τ Γ =1.2×10 4 s , as in Ref.[ 19 19 ]. The width of the laser is approximately Γ l =10 -5 eV. The energy spectrum of the valence electron is obtained from experiments by J. Blaise and J. Wyart [ 20 20 ]. Transition rates, such as Γ ( c → d ) , and lifetimes for different intermediate states are from [ 21 21 ] [ 22 22 ], which considers 229 Th 3+ to be a Fr-like atom. Because of the metastability and long lifetime of 7 s , τ 7 s =0.6 s in Ref. [ 22 22 ], we chose 7 s as the initial state ( a ). The electronic shell can be effectively transferred from the ground state 5 F 5/2 to the 7 s state by making use of the Stimulated Raman Adiabatic Passage (STIRAP) method [ 23 23 ] via the intermediate state 6 d 3/2 and more details about STIRAP are in ref.[ 24 24 ]. In addition, we make use of the fact (also considered in [ 25 25 ]) that the coupling of the nuclear transition is maximal to an M 1 electronic transition between s orbitals; thus, ns ( n =7,8,9,10) for the final state ( c ) is considered. To obtain <math id="M70"><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> an M 1 electronic transition, the ns ( n =7,8,9,10) atomic shells are only considered for ( d ). Furthermore, from Eq. (13), two factors Γ ( b → a ) and Γ ( d → b ) indicate that state ( b ) should be np j ( n =7,8,9,10), ( j =1/2, 3/2) to make E 1 transitions between them. For a detailed analysis of the results, we divided this section into three parts: a small laser ((10 4 –10 10 ) W/m 2 ), moderate laser((10 10 – 10 14 ) W/m 2 ), and strong laser ( ≥ 10 18 W/m 2 ). 3.1 Small laser 2 Figure 2(a) Figure 2(a) shows the relationship between the two-photon EB enhancement R and laser intensity I . The diagram in Fig. 2 (b) Fig. 2 (b) is divided into three parts, i.e., the small laser ((10 4 –10 10 ) W/m 2 ), moderate laser((10 10 - 10 14 ) W/m 2 ), and strong laser ( ≥ 10 18 W/m 2 ). Figure 3 Figure 3 shows the two-photon EB excitation rate Γ eb as a function of the laser intensity I . Here, R 1 indicates EB enhancement using atomic shell a =7 s , b =7 p 1/2 , c =7 s . R 2 stands for a =7 s , b =7 p 3/2 , c =7 s , R 3 is a =7 s , b =7 p 1/2 , c =8 s . R 4 is a =7 s , b =7 p 3/2 , c =8 s , R 5 is a =7 s , b =8 p 1/2 , c =8 s , and R 6 is a =7 s , b =8 p 3/2 , c =8 s . These notations are the same for Γ eb . We see that both R and Γ eb increase with increasing laser intensity. Surprisingly, from Fig. 3 Fig. 3 , we note that the excitation rate of the nuclei is only large for a moderate intensity laser, e.g., I 10 8 W/m 2 . For example, when we choose a =7 s , b =7 p 1/2 , c =7 s as our atomic shell, from Table 1 Table 1 , the photon energies of the two laser beams are Ω 1 =4.6008 eV and Ω 2 =3.4992 eV, respectively. The wavelengths are λ 1 =269.02 nm, λ 2 =353.57 nm, which are close to the optical range. Assuming equilibrium, the corresponding EB enhancement R 1 and excitation rate <math id="M71"><mrow><msubsup><mi>Γ</mi><mn>1</mn><mrow><mtext>eb</mtext></mrow></msubsup></mrow></math> are a Here, the unit for laser intensity is 10 10 W/m 2 . 1 1 ( b ) 1 1 ( c ) 1 1 Ω 1 (eV) 1 1 Ω 2 (eV) 1 1 Γ eb (10 8 s -1 ) 1 1 R (10 8 ) 1 1 I 0 [ 1 1 ] 1 1 7 p 1/2 1 1 7s 1 1 4.6008 1 1 3.4992 1 1 2.17 1 1 3.45 1 1 1.20 1 1 7 p 3/2 1 1 7s 1 1 6.1899 1 1 1.9101 1 1 3.96 1 1 11.5 1 1 1.19 1 1 7 p 1/2 1 1 8s 1 1 4.6008 1 1 15.459 1 1 2.27 1 1 1.68 1 1 4.68 1 1 7 p 3/2 1 1 8s 1 1 6.1899 1 1 13.870 1 1 3.97 1 1 2.40 1 1 5.72 1 1 8 p 1/2 1 1 8s 1 1 13.810 1 1 6.2500 1 1 0.78 1 1 0.50 1 1 5.50 1 1 8 p 3/2 1 1 8s 1 1 14.474 1 1 5.5860 1 1 1.32 1 1 0.18 1 1 23.5 <math display="block" id="M72"><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>0.0322</mn><mi>I</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2.18</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>21</mn></mrow></msup><msup><mi>I</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> <math display="block" id="M73"><mrow><msubsup><mi>Γ</mi><mn>1</mn><mrow><mtext>eb</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><mn>9.29</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>13</mn></mrow></msup><msup><mi>I</mi><mn>2</mn></msup></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>2.04</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>21</mn></mrow></msup><msup><mi>I</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> Choosing I =10 7 W/m 2 , from Eqs.(26) and (27) one can obtain R 1 =3.22 × 10 5 , <math id="M74"><mrow><msubsup><mi>Γ</mi><mn>1</mn><mrow><mtext>eb</mtext></mrow></msubsup></mrow></math> =92.9 s -1 , which is similar to the result 10 s -1 for Th + in Ref.[ 12 12 ], where a laser pulses with 10 mJ energy and a spectral width of Δ Ω = 2 π × 3 GHz, at a repetition rate of 30 Hz and focusing to a spot size of 0.1 mm; hence, their excitation rate of Γ eb =10 s -1 . Alternatively, if we let I =9×10 5 W/m 2 , then we obtain <math id="M75"><mrow><msubsup><mi>Γ</mi><mn>1</mn><mrow><mtext>eb</mtext></mrow></msubsup><mo>=</mo><mn>0.75</mn></mrow></math> s -1 , R 1 =2.89×10 4 , and in [ 11 11 ] the excitation rate Γ eb =0.0281 s -1 for Th + . It is interesting to note that the result R 1 =3.22× 10 5 is comparable with the result of 10 → 10 6 of one-photon EB in the low saturation case [ 10 10 ]. 3.2 Moderate laser 2 Usually, owing to the low excitation cross-section of the nuclei, a low saturation limit for excitation is assumed and satisfied in most cases. However, in this work, cross-sections from the two-photon EB excitation under resonance conditions are quite large so that the low saturation limit will no longer be satisfied. We extend the low saturation limit to a general situation, including the large excitation case for the first time, as shown in the general case of the theoretical descriptions. We find that the two-photon EB enhancement R reaches its maximum when I = I 0 is satisfied. From Fig. 1b Fig. 1b and Table (1) Table (1) , we see that R 2 has the largest value, i.e., R 2 =1.15×10 9 when I 0 =1.19×10 10 W/m 2 . At the same time, the excitation rate Γ eb also has a maximum value, as shown in Fig. 3 Fig. 3 . Surprisingly, at a certain point, the excitation rate Γ eb increases very slowly as the intensity I increases; we refer to this point as <math id="M76"><mrow><msubsup><mi>I</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> . It is quite amazing that when I 10 10 W/m 2 , we obtain R 10 8 ; however, note that we only consider the case of Δ s =0 and Δ res =0. If the conditions( <math id="M77"><mrow><msub><mi>Δ</mi><mrow><msub><mtext>s</mtext><mn>2</mn></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> but Δ s ≠ 0 and Δ res ≠ 0) are satisfied, from Eq.(17), there are two extra resonant factors, namely, <math id="M78"><mrow><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> , <math id="M79"><mrow><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac></mrow></math> . Thus one can obtain <math display="block" id="M80"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>2</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Assuming Γ l =10 -5 eV, and Δ s , Δ res ∼ eV, <math id="M81"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub><mo>~</mo><msub><mi>Γ</mi><mtext>l</mtext></msub></mrow></math> , <math id="M82"><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mrow><mtext>res</mtext></mrow></msub><mo>∼</mo><msub><mi>Γ</mi><mtext>l</mtext></msub></mrow></math> , one obtains the two resonant factors <math id="M83"><mrow><mfrac><mrow><msubsup><mi>Γ</mi><mtext>l</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac></mrow></math> , and <math id="M84"><mrow><mfrac><mrow><msubsup><mi>Γ</mi><mtext>l</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> , respectively. Thus Eq.(28) becomes <math display="block" id="M85"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>b</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mtext>1</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>2</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><mfrac><mrow><msubsup><mi>α</mi><mi>d</mi><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>d</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>Γ</mi><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>→</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msub><mi>Γ</mi><mi>b</mi></msub></mrow></mfrac><mfrac><mrow><msubsup><mi>Γ</mi><mtext>l</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mtext>s</mtext><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><msubsup><mi>Γ</mi><mtext>l</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>Δ</mi><mrow><mtext>res</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></math> In this case, it is quite clear that the threshold will be 10 10 times larger than the low saturation limit, as shown in Eq. (18), indicating that a stronger laser field is required to reach a larger EB excitation rate. 3.3 Strong laser 2 We notice that when the intensity of the laser reaches <math id="M86"><mrow><msubsup><mi>I</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> , the excitation rate Γ eb saturates as the intensity I increases. As the intensity increases, the EB enhancement factor R will eventually be less than one because of the rapid increase in <math id="M87"><mrow><msubsup><mi>Γ</mi><mrow><mtext>exc</mtext></mrow><mi>Γ</mi></msubsup></mrow></math> . Figure 2(a) Figure 2(a) demonstrates this tendency. When the intensity of the laser reaches I (10 19 -10 20 ) W/m 2 , one obtains R 1. The case of R < 1 is common in the literature. For example, in Ref.[ 26 26 ], they choose an initial electron state i =5 f 5/2 and a final electron state f =6 d 3/2 ,6 d 5/2 ,7 s 1/2 . In these cases, the EB enhancement factors β are 0.015, 0.0015, and 2×10 -9 , respectively. Note that the case of R < 1 is shell-independent in our situation, it will eventually be less than one. To see this, from Eq. (22), we obtain <math display="block" id="M88"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mfrac><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>1</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><msup><mi>R</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>'</mo></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mfrac><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>1</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><msup><mi>R</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>'</mo></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> from Eq. (23) and (24) one can obtain <math display="block" id="M89"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>d</mi></munder></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Ω</mi><mtext>n</mtext></msub></mrow><mrow><msub><mi>Ω</mi><mn>1</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>n</mtext></msub></mrow><mrow><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msup><mi>R</mi><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>'</mo></mrow></msup></mrow></math> Thus Eq. (32) becomes <math display="block" id="M90"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mfrac><mrow><msub><mi>Γ</mi><mi>Γ</mi></msub></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>res</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi></mrow><mrow><msubsup><mi>Ω</mi><mn>1</mn><mn>3</mn></msubsup><msub><mover accent="true"><mi>Γ</mi><mo>˜</mo></mover><mtext>s</mtext></msub></mrow></mfrac><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac></mrow></math> From Eq. (23), we obtain a linear dependence on the laser intensity in the low saturation case, <math id="M91"><mrow><msubsup><mi>R</mi><mrow><mtext>ls</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>∼</mo><mi>I</mi></mrow></math> while, at a strong laser intensity from Eq.(32), we obtain <math id="M92"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> ∼ a 1 I /(1+ b 1 I 2 ), which is no longer linear. When the condition I → ∞ is satisfied, then one obtains <math id="M93"><mrow><msubsup><mi>R</mi><mrow><mtext>eb</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> ∼ 0. At an intensity (10 19 -10 20 ) W/m 2 , the two-photon EB process is not an effective method of exciting nuclei. Generally, it is difficult to determine whether an EB process is effective but we see that, at high laser intensity, the two-photon EB process is not an effective method. In this case, the direct photonuclear excitation rate dominates. Actually, we do not include any field or nonlinear effects in this calculation, which should be an interesting task in the future. 4 Summary 1 In summary, we propose a two-photon EB excitation scheme to populate the isomeric isotope 229m Th 3+ . Based on the nuclear quantum optics for two-level open quantum systems, we deduce an expression for the two-photon EB excitation rate in an electron-nucleus system. The nuclear excitation rate Γ eb and its efficiency R were derived under equilibrium conditions. Using the experimentally-known energy levels of 229 Th 3+ , we obtained the EB excitation rate of 229 Th 3+ and the efficiency R as a function of laser intensity. Three cases of laser intensity were investigated: a small laser((10 4 –10 10 ) W/m 2 ), moderate laser near the critical((10 10 – 10 14 ) W/m 2 ), and strong laser( ≥ 10 18 W/m 2 ). We find that near the critical value ((10 10 – 10 14 ) W/m 2 ), the nuclear excitation rate Γ eb , and the electronic bridge efficiency R reach their maximum values under a strong laser ( ≥ 10 18 W/m 2 ), the two-photon electronic bridge efficiency R will eventually be less than one. In this calculation, we do not consider the hyperfine structure due to electromagnetic splitting, which will be conducted in future work. We believe that this two-photon EB scheme can help to realize nuclear clocks and suggest verifying the scheme through a series of experiments with ordinary lasers in laboratories. References A. D. Ludlow , M. M. Boyd , J. 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