Neng-Qiang Cai · Guo-Qiang Zhang· Chang-Bo Fu· Yu-Gang Ma,

Abstract The isomer 229mTh 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 7s1/2 electronic shell state of 229mTh3+,with a lifetime of approximately 0.6 s,is chosen as the initial state and the atomic shells (7s-10s) could be achieved as virtual states in a two-photon process. When the virtual states return to the initial state 7s1/2, there is a chance of triggering the nucleus 229Th3+ to its isomeric state 229mTh3+ via EB. Two lasers at moderate intensity((1010-1014）W/m2),with photon energies near the optical range, are expected to populate the isomer at a saturated rate of approximately 109 s-1, which is much higher than that due to other mechanisms. We believe that this twophoton EB scheme can help in the development of nuclear clocks and deserves verification via a series of experiments with ordinary lasers in laboratories.

Keywords Electronic bridge · 229Th · Nuclear clocks ·

1 Introduction

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].In 1967,the International System of Units (SI) second was officially redefined based on the isotope atom133Cs: ‘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] 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], 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],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.,229mTh (8.10 eV) and235mU (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 single229Th3+was first proposed by Peik and Tamm [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 of229Th3+, respectively. In 2012, the single229Th3+ion nuclear clock was further investigated by Campbell et al.[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 isotope229mTh have been suggested, where the key is how to populate the isomeric state. During the last two decades,various theories have been proposed for populating229Th 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. [7] 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 as93Mo and178Hf.In the NEET process, a nucleus is excited and a real electronic shell state is simultaneously deexcited, which is a thirdorder process [7]. Sometimes, it is difficult to distinguish the difference between NEET and EB because they share a similar physics scheme.In Ref.[8],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 on229Th. In the 1970s, the energy was found to be below 100 eV [9] and then below 10 eV in the late 1980s [10]. The energy has shifted from 4.5±1 eV[11]to now 8.10±0.17 eV[12].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 the229Th 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 charged229Th35+ions in an EBIT trap was given by Bilous et al.[13].

In this paper, we propose a new theory for calculating the EB excitation rate with two photons for229Th3+. 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 Γeband electron bridge enhancement R, respectively. Then, we choose specific atomic shells (7s-10s) as the virtual electronic levels to calculate the transition rates for Th3+. We find that the excitation rate Γeband 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

In this section, we deduce a general formalism for twophoton EB excitation. Figure 1a 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 [14], as shown in Fig.1b.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:

Fig. 1 a A two-photon EB process, which absorbs two photons before excitation of the nuclei. b A two-photon bound internal conversion(BIC)process,which emits two photons after deexcitation of the nuclei

One can now obtain the expression of the two-photon EB excitation using the connection between excitation and natural decay rate [14],

Inserting Eq. (4) into Eq. (5) and exchanging (b) and (d),we obtain）

In this study, we follow Ref. [7], 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

Here, ｜g〉 and ｜e〉 are the ground and excited states,respectively.The population density ρexc（t）under resonant laser irradiation can be modeled using Torrey’s solution of the optical Bloch equations [16]. The Rabi frequency Ωegfor the nuclear transition is introduced as in [16]

2.1 Low saturation limit

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[16]

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:

here the (c) index was dropped for easier notation and Δres＝Ω2-Ωres＝Ω2-Ωn-Ωc+Ωb.

Here, ～Γn＝（Γn+Γl）/2. Inserting Eqs. (14) and (16) into Eq. (15), one can obtain the enhancement coefficient for two-photon EB excitation at a low saturation limit:

Here,the(c)index was dropped for easier notation and the subscript ‘ls’ indicates the low saturation limit. At resonance Δs＝0, Δres＝0, and we obtain

2.2 General case

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 [16]

At resonance, Δres＝0,Δs1＝0,and the electronic excited state (b) is fixed. Considering I1＝I2≡I, we obtain

where superscript ‘g’ indicates general excitation and

3 Results and discussion

First, we set up some parameters before performing the calculations. Recently, an energy of 8.10±0.17 eV for229mTh was obtained[12],which has the same precision as the value obtained from IC spectroscopy [17]. Generally,the radiative decay rate ΓΓ＝1/τΓof magnetic multipole transitions can be expressed in terms of the energy-independent reduced transition probability B↓（ML）, as in [18]

3.1 Small laser

Fig.2 a Two-photon EB process enhancement factor R as a function of laser intensity I for different atomic shells(b), (c).The dashed line indicates a factor of one. b Three cases for the laser intensity at different atomic shells ((b), (c). The y value in subplot(2) is normalized to 109

Fig. 3 Relationship between two-photon EB excitation rate Γeb and laser intensity I for different atomic shells (b) (c)

Figure 2a shows the relationship between the two-photon EB enhancement R and laser intensity I.The diagram in Fig. 2b is divided into three parts, i.e., the small laser((104-1010) W/m2), moderate laser ((1010-1014) W/m2),and strong laser (≥1018W/m2). Figure 3 shows the twophoton EB excitation rate Γebas a function of the laser intensity I. Here, R1indicates EB enhancement using atomic shell a ＝7s，b ＝7p1/2，c ＝7s. R2stands for a ＝7s，b ＝7p3/2，c ＝7s, R3is a ＝7s，b ＝7p1/2，c ＝8s.R4is a ＝7s，b ＝7p3/2，c ＝8s, R5is a ＝7s，b ＝8p1/2，c ＝8s, and R6is a ＝7s，b ＝8p3/2，c ＝8s.These notations are the same for Γeb. We see that both R and Γebincrease with increasing laser intensity. Surprisingly, from Fig. 3, we note that the excitation rate of the nuclei is only large for a moderate intensity laser, e.g., I ～108W/m2. For example, when we choose a ＝7s，b ＝7p1/2，c ＝7s as our atomic shell, from Table 1, the photon energies of the two laser beams are Ω1＝4.6008 eV and Ω2＝3.4992 eV, respectively. The wavelengths are λ1＝269.02 nm, λ2＝353.57 nm, which are close to the optical range. Assuming equilibrium, the corresponding EB enhancement R1and excitation rate Γeb1are Choosing I ＝107W/m2, from Eqs.(26) and (27), one can obtain R1=3.22×105,Γeb1=92.9 s-1,which is similar to the result 10 s-1for Th+in Ref. [14],where a laser pulses with 10 mJ energy and a spectral width of ΔΩ = 2π× 3 GHz, at a repetition rate of 30 Hz and focusing to a spot size of 0.1 mm;hence,their excitation rate of Γeb=10 s-1.Alternatively, if we let I ＝9×105W/m2, then we obtain Γeb1＝0.75 s-1,R1＝2.89×104,and in[8]the excitation rate Γeb= 0.0281 s-1for Th+. It is interesting to note that the result R1= 3.22× 105is comparable with the result of 10 →106of one-photon EB in the low saturation case[7].

Table 1 Laser intensity I0 when the two-photon EB excitation enhancement factor R reaches its maximum with corresponding excitation Γeb,incident laser energies Ω1,Ω2 at different atomic shells(b), (c) but for the same initial state （a）＝7s

3.2 Moderate laser

In this case, it is quite clear that the threshold will be 1010times 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

4 Summary

In summary, we propose a two-photon EB excitation scheme to populate the isomeric isotope229mTh3+. 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 Γeband its efficiency R were derived under equilibrium conditions. Using the experimentally-known energy levels of229Th3+, we obtained the EB excitation rate of229Th3+and the efficiency R as a function of laser intensity.Three cases of laser intensity were investigated:a small laser ((104-1010) W/m2), moderate laser near the critical ((1010-1014) W/m2), and strong laser (≥1018W/m2). We find that near the critical value ((1010-1014) W/m2), the nuclear excitation rate Γeb, and the electronic bridge efficiency R reach their maximum values under a strong laser (≥ 1018W/m2), 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.