Populating 229mTh via two-photon electronic bridge mechanism

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Populating ^229mTh via two-photon electronic bridge mechanism

Neng-Qiang Cai，

Guo-Qiang Zhang ，

Chang-Bo Fu ，

Yu-Gang Ma

Nuclear Science and Techniques

Vol.32, No.6

Article number 59

Published in print 01 Jun 2021

Available online 12 Jun 2021

DOI：10.1007/s41365-021-00900-3

33200

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 7s_1/2 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 7s_1/2, 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 10⁹s^-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 bridge229ThNuclear clocksTwo-photon excitation

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 atom ¹³³Cs: "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) and ^235mU (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 ²²⁹Th³⁺ was first proposed by E. Peik and C. 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 of ²²⁹Th³⁺, respectively. In 2012, the single ²²⁹Th³⁺ 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 isotope ^229mTh 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 ²²⁹Th 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] 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 ⁹³Mo and ¹⁷⁸Hf. In the NEET process, a nucleus is excited and a real electronic shell state is simultaneously deexcited, which is a third-order process [10]. Sometimes, it is difficult to distinguish the difference between NEET and EB because they share a similar physics scheme. In Ref. [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 ²²⁹Th. In the 1970s, the energy was found to be below 100 eV [7] and then below 10 eV in the late 1980s [8]. The energy has shifted from 4.5 ± 1 eV [9] to now 8.10 ± 0.17 eV [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 ²²⁹Th 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 ²²⁹Th³⁵⁺ ions in an EBIT trap was given by Bilous et al. [16].

In this paper, we propose a new theory for calculating the EB excitation rate with two photons for ²²⁹Th³⁺. 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 (7s-10s) as the virtual electronic levels to calculate the transition rates for Th³⁺. 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

In this section, we deduce a general formalism for two-photon EB excitation. 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], as shown in 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:

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.

\begin{array}{l} Γ_{bic}^{(a \to c)} & = \sum_{d} Γ_{bic}^{(a \to d)} (Δ_{s}) P^{(d \to c)} \\ = \sum_{d} Γ_{bic}^{(a \to d)} (Δ_{s}) \frac{Γ^{(d \to c)}}{Γ_{d}} \end{array}

(1)

Here, Γ^(d→c) denotes the partial natural decay rate from state (d) to (c) and Γd is the total natural decay rate (line width) of state (d). The one-photon BIC is $Γ_{bic}^{(a \to d)} = \sum_{b} Γ_{ic}^{(a \to b)} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(b \to d)}}{Γ_{b}}$ , where the frequency difference between the atomic transition and nuclear transition Δ_s=Ωb-(Ω_n+Ω_a), where Ω_a, Ω_b, and Ω_n are the transition frequencies of (a), (b), and the nucleus. ${\tilde{Γ}}_{s}$ =(Γ_a+Γ_b+Γ_n)/2 and $Γ_{ic}^{(a \to b)}$ , 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]. Inserting $Γ_{bic}^{(a \to d)}$ into Eq. (1), we obtain the expression for the two-photon BIC process:

\begin{array}{l} Γ_{bic}^{(a \to c)} = \sum_{b} \sum_{d} Γ_{i c}^{(a \to b)} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(b \to d)}}{Γ_{b}} \frac{Γ^{(d \to c)}}{Γ_{d}} . \end{array}

(2)

One can now obtain the expression of the two-photon EB excitation using the connection between excitation and natural decay rate [12],

Γ_{exc}^{eb (c)} = \frac{π c^{2} I_{l}}{ℏ {(Ω_{res}^{(c)})}^{3} {\tilde{Γ}}_{res}^{(c)}} Γ_{bic}^{(c \to a)} .

(3)

Here, $I_{l}$ is the intensity of the external laser field. $Ω_{res}^{(c)} = Ω_{n} + Ω_{c} - Ω_{a}$ is the EB resonance frequency for the final state (c). ${\tilde{Γ}}_{res}^{(c)} = (Γ_{res}^{(c)} + Γ_{l}) / 2$ is the decoherence rate of the EB resonance. $Γ_{res}^{(c)}$ is the total linewidth of the two-photon EB resonance, obtained as the convolution of the individual linewidths of the atomic and nuclear transitions, $Γ_{res}^{(c)} = Γ_{n} + Γ_{a} + Γ_{c}$ .

The partial BIC process rate $Γ_{bic}^{(c \to a)}$ can be obtained by changing the initial and final states(a) and (c) in Eq. (2):

Γ_{bic}^{(c \to a)} = \sum_{b} \sum_{d} Γ_{ic}^{(c \to b)} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(b \to d)}}{Γ_{b}} \frac{Γ^{(d \to a)}}{Γ_{d}} .

(4)

Considering two incident laser beams I₁ and I₂ for the two-photon EB scheme, two factors must be considered: $\frac{π c^{2} I_{1}}{Ω_{s}^{3} {\tilde{Γ}}_{s}}$ , $\frac{π c^{2} I_{2}}{Ω_{res}^{3} {\tilde{Γ}}_{res}}$ . Thus, one obtains the proper formula for the two-photon EB excitation rate from Eq. (3),

Γ_{exc}^{eb (c)} = \frac{π c^{2} I_{1}}{ℏ {(Ω_{s}^{(c)})}^{3} {\tilde{Γ}}_{s}^{(c)}} \frac{π c^{2} I_{2}}{ℏ {(Ω_{res}^{(c)})}^{3} {\tilde{Γ}}_{res}^{(c)}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} Γ_{bic}^{(c \to a)} .

(5)

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

Γ_{exc}^{eb (c)} = \sum_{b} \sum_{d} \frac{π c^{2} I_{1}}{ℏ {(Ω_{s}^{(c)})}^{3} {\tilde{Γ}}_{s}^{(c)}} \frac{π c^{2} I_{2}}{ℏ {(Ω_{res}^{(c)})}^{3} {\tilde{Γ}}_{res}^{(c)}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} Γ_{ic}^{(c \to d)} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(d \to b)}}{Γ_{b}} \frac{Γ^{(b \to a)}}{Γ_{d}} .

(6)

Here $Ω_{s} = Ω_{b} - Ω_{a}$ , $Δ_{s} = Ω_{1} - (Ω_{b} - Ω_{a})$ , $Δ_{s_{2}} = Ω_{2} - Ω_{d} + Ω_{b}$ , $Γ_{s_{2}} = Γ_{d} + Γ_{b}$ , $Γ_{s} = Γ_{b} + Γ_{a}$ , ${\tilde{Γ}}_{s} = (Γ_{s} + Γ_{l}) / 2$ , $Ω_{res} = Ω_{n} + Ω_{c} - Ω_{b}$ , $Γ_{ic}^{(c \to d)}$ is the IC process from (c) to (d). The IC process $Γ_{ic}^{(c \to d)}$ takes the form of Eq. [13]

Γ_{ic}^{(c \to d)} = \frac{α_{d}^{(c \to d)} Γ_{Γ}}{Γ_{d}},

(7)

where $α_{d}^{(c \to d)}$ 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

Γ_{exc}^{eb (c)} = \sum_{b} \sum_{d} \frac{π c^{2} I_{1}}{ℏ {(Ω_{s}^{(c)})}^{3} {\tilde{Γ}}_{s}^{(c)}} \frac{π c^{2} I_{2}}{ℏ {(Ω_{res}^{(c)})}^{3} {\tilde{Γ}}_{res}^{(c)}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} \frac{α_{d}^{(c \to d)} Γ_{Γ}}{Γ_{d}} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(d \to b)}}{Γ_{b}} \frac{Γ^{(b \to a)}}{Γ_{d}} .

(8)

In this study, we follow. [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

\begin{array}{l} \hat{ρ} (t) = & ρ_{ee} | e 〉 〈 e | + ρ_{ge} | g 〉 〈 e | + ρ_{eg} | e 〉 〈 g | + ρ_{gg} | g 〉 〈 g | . \end{array}

(9)

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]. The Rabi frequency Ω_eg for the nuclear transition is introduced as in [14]

Ω_{eg}^{2} = 2 Γ_{exc}^{Γ} {\tilde{Γ}}_{n} = \frac{2 π c^{2} I_{l} Γ_{Γ}}{ℏ Ω_{n}^{3}} .

(10)

where $Γ_{exc}^{Γ}$ is the excitation rate. ${\tilde{Γ}}_{n} = Γ_{Γ} + Γ_{nr}$ is the total width of the nuclear transition, including Γ decay (ΓΓ) 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 $Ω_{eb}^{(c) 2} = 2 Γ_{exc}^{eb (c)} {\tilde{Γ}}_{res}^{(c)}$ , 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 $Ω_{eb}^{(c)}$ and ${\tilde{Γ}}_{n}$ for ${\tilde{Γ}}_{res}^{(c)}$ in Eq. (10). In the following, we extend a low saturation limit approximation to a general case.

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[14]

ρ_{exc}^{eb} \equiv ρ_{ee} = \frac{Ω_{eg}^{2} {\tilde{Γ}}_{n} / (2 Γ_{n})}{{(Δ Ω)}^{2} + {\tilde{Γ}}_{n}^{2}} (1 - e^{- Γ_{n} t}) .

(11)

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:

ρ_{exc}^{eb} \equiv ρ_{ee} \to \frac{Ω_{eg}^{2} {\tilde{Γ}}_{n} / (2 Γ_{n})}{Δ_{n}^{2} + {\tilde{Γ}}_{n}^{2}} = Γ_{exc}^{Γ} \frac{{\tilde{Γ}}_{n}^{2} / Γ_{n}}{Δ_{n}^{2} + {\tilde{Γ}}_{n}^{2}} .

(12)

With nonzero detuning $Δ_{res}^{(c)}$ of the laser light with respect to the EB resonance, we can obtain

\begin{array}{l} Γ_{exc}^{eb} (Δ_{res}) & = ρ_{exc}^{eb} (Δ_{res}) Γ_{n} \\ = Γ_{exc}^{eb} (0) \frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}} . \end{array}

(13)

Inserting Eq. (8) into Eq. (13), an additional factor $\frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}}$ for the EB excitation rate at a low saturation limit.

Γ_{ls}^{eb} \equiv Γ_{exc}^{eb} (Δ_{res}) = \sum_{b} \sum_{d} \frac{π c^{2} I_{1}}{ℏ {(Ω_{s}^{(c)})}^{3} {\tilde{Γ}}_{s}^{(c)}} \frac{π c^{2} I_{2}}{ℏ {(Ω_{res}^{(c)})}^{3} {\tilde{Γ}}_{res}^{(c)}} \frac{{\tilde{Γ}}_{s 2}^{2}}{Δ_{s 2}^{2} + {\tilde{Γ}}_{s 2}^{2}} \frac{α_{d}^{(c \to d)} Γ_{Γ}}{Γ_{d}} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{Γ^{(d \to b)}}{Γ_{b}} \frac{Γ^{(b \to a)}}{Γ_{d}} \frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}}

(14)

here the (c) index was dropped for easier notation and Δ_res=Ω₂-Ω_res=Ω₂-Ω_n-Ω_c+Ω_b.

Note that this equation is similar to that in [10] and others [11, 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=7s, c=7s, b=7p_1/2, d=8s, $I_{1} = I_{2} \equiv I {= 3 \times 10}^{5}$ W/m², Γ_l=10^-5 eV are satisfied, we obtain $Γ_{exc}^{eb} = 0.082$ s^-1, which is comparable to the results reported in [11, 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 $I_{1} = I_{2} \equiv I$ , the EB enhancement coefficient R can be defined as the ratio between the EB excitation cross-section and the direct photon excitation cross-section.

R_{eb}^{(c)} = \frac{σ_{eb}^{(c)}}{σ_{Γ}} ≃ \frac{Γ_{exc}^{eb (c)}}{Γ_{exc}^{Γ}}

(15)

Here, $Γ_{exc}^{eb (c)}$ denotes the EB-excitation rate at resonance, and $Γ_{exc}^{Γ}$ is the excitation rate for direct photon excitation [10]:

Γ_{exc}^{Γ} = \frac{σ_{Γ} I_{l}}{ℏ Ω_{n}} = \frac{π c^{2} I_{l}}{ℏ Ω_{n}^{3} {\tilde{Γ}}_{n}} Γ_{Γ},

(16)

Here, ${\tilde{Γ}}_{n}$ =(Γ_n+Γl)/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:

R_{ls}^{(b)} = \sum_{d} {(\frac{Ω_{n}}{Ω_{s}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \frac{π c^{2} I}{Ω_{res}^{3} {\tilde{Γ}}_{res}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} \frac{α_{d}^{(c \to d)}}{Γ_{d}} \frac{Γ^{(d \to b)}}{Γ_{d}} \frac{Γ^{(b \to a)}}{Γ_{b}} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}} .

(17)

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

\begin{array}{l} R_{ls}^{(b)} & = \sum_{d} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \frac{π c^{2} I}{Ω_{2}^{3} {\tilde{Γ}}_{s}} \\ \times \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} \frac{α_{d}^{(c \to d)}}{Γ_{d}} \frac{Γ^{(d \to b)}}{Γ_{d}} \frac{Γ^{(b \to a)}}{Γ_{b}} \end{array}

(18)

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 [14]

ρ_{ee} = \frac{Ω_{eg}^{2}}{\frac{2 Γ}{\tilde{Γ}} ({(Δ Ω)}^{2} + {\tilde{Γ}}^{2}) + 2 Ω_{eg}^{2}} .

(19)

Using the same procedure from quantum opticals as in the low-saturation limit, $Ω_{eg} \to Ω_{eb}$ , ${\tilde{Γ}}_{n} \to {\tilde{Γ}}_{res}$

\begin{array}{l} Γ^{eb (g)} & = \sum_{b} ρ_{exc}^{eb (g) (b)} Γ_{n} \\ = \sum_{b} \frac{Γ_{ls}^{eb} (0) {\tilde{Γ}}_{res} Γ_{res}}{\frac{Γ_{res}}{{\tilde{Γ}}_{res}} (Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}) + 2 Γ_{ls}^{eb} (0) {\tilde{Γ}}_{res}} . \end{array}

(20)

At resonance, Δ_res=0, $Δ_{s_{1}} = 0$ , and the electronic excited state (b) is fixed. Considering $I_{1} = I_{2} \equiv I$ , we obtain

Γ^{eb (g) (b)} = \frac{Γ_{exc}^{eb (ls) (b)}}{1 + 2 Γ_{exc}^{eb (ls) (b)} \frac{1}{Γ_{res}}},

(21)

R_{eb}^{(b)} = \frac{R_{ls}^{(b)}}{1 + 2 \frac{Γ_{Γ}}{Γ_{res}} \sum_{d} \frac{π c^{2} I}{Ω_{1}^{3} {\tilde{Γ}}_{s}} R^{(d)'}},

(22)

where superscript ‘g’ indicates general excitation and

\begin{array}{l} R_{ls}^{(b)} & = \sum_{d} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \frac{π c^{2} I}{Ω_{2}^{3} {\tilde{Γ}}_{s}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} \\ \times \frac{Γ_{ic}^{(c \to d)}}{Γ_{Γ}} \frac{Γ^{(d \to b)}}{Γ_{d}} \frac{Γ^{(b \to a)}}{Γ_{b}}, \end{array}

(23)

R^{(d)'} = \frac{π c^{2} I}{Ω_{2}^{3} {\tilde{Γ}}_{res}} \frac{{\tilde{Γ}}_{s_{2}}^{2}}{Δ_{s_{2}}^{2} + {\tilde{Γ}}_{s_{2}}^{2}} \frac{Γ_{i c}^{(c \to d)}}{Γ_{Γ}} \frac{Γ^{(d \to b)}}{Γ_{d}} \frac{Γ^{(b \to a)}}{Γ_{b}} .

(24)

It seems that the general formalism for Γ^eb(g)(b) is simply multiplied by a decay factor, such as $\frac{1}{1 + 2 Γ_{exc}^{eb (ls) (b)} \frac{1}{Γ_{res}}}$ 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 Th³⁺.

3 Results and discussion

First, we set up some parameters before performing the calculations. Recently, an energy of 8.10±0.17 eV for ^229mTh was obtained [15], which has the same precision as the value obtained from IC spectroscopy [17]. Generally, the radiative decay rate $Γ_{Γ} = 1 / τ_{Γ}$ of magnetic multipole transitions can be expressed in terms of the energy-independent reduced transition probability $B_{↓} # x 2193; (M L)$ , as in [18]

Γ_{Γ} = \frac{2 μ_{0}}{ℏ} \frac{L + 1}{L {[(2 L + 1)!!]}^{2}} {(\frac{Ω_{n}}{c})}^{2 L + 1} B_{↓} (M L) .

(25)

Here, L denotes multipolarity and Ω_n is the angular frequency of the nuclear transition. However, in this study, we choose $B_{↓} # x 2193; (M L)$ =5.0×19^-3W.u. and τΓ=1.2×10⁴ s, as in Ref.[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]. Transition rates, such as Γ^(c→d), and lifetimes for different intermediate states are from [21] [22], which considers ²²⁹Th³⁺ to be a Fr-like atom. Because of the metastability and long lifetime of 7s, τ_7s=0.6s in Ref. [22], we chose 7s as the initial state (a). The electronic shell can be effectively transferred from the ground state 5F_5/2 to the 7s state by making use of the Stimulated Raman Adiabatic Passage (STIRAP) method [23] via the intermediate state 6d_3/2 and more details about STIRAP are in ref.[24]. In addition, we make use of the fact (also considered in [25]) that the coupling of the nuclear transition is maximal to an M1 electronic transition between s orbitals; thus, ns (n=7,8,9,10) for the final state (c) is considered. To obtain $α_{d}^{(c \to d)}$ an M1 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 npj (n=7,8,9,10), (j=1/2, 3/2) to make E1 transitions between them. For a detailed analysis of the results, we divided this section into three parts: a small laser ((10⁴–10¹⁰) W/m²), moderate laser((10¹⁰ – 10¹⁴) W/m²), and strong laser (≥ 10¹⁸ W/m²).

3.1 Small laser

Figure 2(a) shows the relationship between the two-photon EB enhancement R and laser intensity I. The diagram in Fig. 2 (b) is divided into three parts, i.e., the small laser ((10⁴–10¹⁰) W/m²), moderate laser((10¹⁰ - 10¹⁴) W/m²), and strong laser (≥ 10¹⁸ W/m²). Figure 3 shows the two-photon EB excitation rate Γ^eb as a function of the laser intensity I. Here, R₁ indicates EB enhancement using atomic shell a=7s, b=7p_1/2, c=7s. R₂ stands for a=7s, b=7p_3/2, c=7s, R₃ is a=7s, b=7p_1/2, c=8s. R₄ is a=7s, b=7p_3/2, c=8s, R₅ is a=7s, b=8p_1/2, c=8s, and R₆ is a=7s, b=8p_3/2, c=8s. 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, we note that the excitation rate of the nuclei is only large for a moderate intensity laser, e.g., I 10⁸ W/m². For example, when we choose a=7s, b=7p_1/2, 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 R₁ and excitation rate $Γ_{1}^{eb}$ are

Fig. 2.

(a) The 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) and (c). The y value in subplot(2) is normalized to 10⁹.

Fig. 3.

The relationship between two-photon EB excitation rate Γ^eb and laser intensity I for different atomic shells (b) (c).

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

(b)	(c)	Ω₁ (eV)	Ω₂ (eV)	Γeb(10⁸ s^-1)	R(10⁸)	I₀[1]
7p_1/2	7s	4.6008	3.4992	2.17	3.45	1.20
7p_3/2	7s	6.1899	1.9101	3.96	11.5	1.19
7p_1/2	8s	4.6008	15.459	2.27	1.68	4.68
7p_3/2	8s	6.1899	13.870	3.97	2.40	5.72
8p_1/2	8s	13.810	6.2500	0.78	0.50	5.50
8p_3/2	8s	14.474	5.5860	1.32	0.18	23.5

^aHere, the unit for laser intensity is 10¹⁰ W/m².

R_{1} = \frac{0.0322 I}{(2.18 \times 10^{- 21} I^{2} + 1)}

(26)

Γ_{1}^{eb} = \frac{9.29 \times 10^{- 13} I^{2}}{(1 + 2.04 \times 10^{- 21} I^{2})}

(27)

Choosing I=10⁷ W/m², from Eqs.(26) and (27) one can obtain R₁=3.22 × 10⁵, $Γ_{1}^{eb}$ =92.9 s^-1, which is similar to the result 10 s^-1 for Th⁺ in Ref.[12], 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 Γ^eb=10 s^-1. Alternatively, if we let I=9×10⁵ W/m², then we obtain $Γ_{1}^{eb} = 0.75$ s^-1, R₁=2.89×10⁴, and in [11] the excitation rate Γ^eb=0.0281 s^-1 for Th⁺. It is interesting to note that the result R₁=3.22× 10⁵ is comparable with the result of 10 → 10⁶ of one-photon EB in the low saturation case [10].

3.2 Moderate laser

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=I₀ is satisfied. From Fig. 1b and Table (1), we see that R₂ has the largest value, i.e., R₂=1.15×10⁹ when I₀=1.19×10¹⁰ W/m². At the same time, the excitation rate Γ^eb also has a maximum value, as shown in 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 $I_{0}^{'}$ . It is quite amazing that when I 10¹⁰ W/m², we obtain R 10⁸; however, note that we only consider the case of Δ_s=0 and Δ_res=0. If the conditions( $Δ_{s_{2}} = 0$ but Δ_s ≠ 0 and Δ_res ≠ 0) are satisfied, from Eq.(17), there are two extra resonant factors, namely, $\frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}}$ , $\frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}}$ . Thus one can obtain

\begin{array}{l} R_{ls}^{(b)} & = \sum_{b} \sum_{d} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \frac{π c^{2} I}{Ω_{2}^{3} {\tilde{Γ}}_{s}} \frac{α_{d}^{(c \to d)}}{Γ_{d}} \frac{Γ^{(d \to b)}}{Γ_{d}} \\ \times \frac{Γ^{(b \to a)}}{Γ_{b}} \frac{{\tilde{Γ}}_{s}^{2}}{Δ_{s}^{2} + {\tilde{Γ}}_{s}^{2}} \frac{{\tilde{Γ}}_{res}^{2}}{Δ_{res}^{2} + {\tilde{Γ}}_{res}^{2}} . \end{array}

(28)

Assuming Γ_l=10^-5 eV, and Δ_s, Δ_res∼ eV, ${\tilde{Γ}}_{s} ~ Γ_{l}$ , ${\tilde{Γ}}_{res} \sim Γ_{l}$ , one obtains the two resonant factors $\frac{Γ_{l}^{2}}{Δ_{s}^{2}}$ , and $\frac{Γ_{l}^{2}}{Δ_{res}^{2}}$ , respectively. Thus Eq.(28) becomes

R_{ls}^{(b)} = \sum_{b} \sum_{d} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \frac{π c^{2} I}{Ω_{2}^{3} {\tilde{Γ}}_{s}} \frac{α_{d}^{(c \to d)}}{Γ_{d}} \frac{Γ^{(d \to b)}}{Γ_{d}} \frac{Γ^{(b \to a)}}{Γ_{b}} \frac{Γ_{l}^{2}}{Δ_{s}^{2}} \frac{Γ_{l}^{2}}{Δ_{res}^{2}}

(29)

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

We notice that when the intensity of the laser reaches $I_{0}^{'}$ , 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 $Γ_{exc}^{Γ}$ . Figure 2(a) demonstrates this tendency. When the intensity of the laser reaches I (10¹⁹-10²⁰) W/m², one obtains R 1. The case of R<1 is common in the literature. For example, in Ref.[26], they choose an initial electron state i=5f_5/2 and a final electron state f=6d_3/2,6d_5/2,7s_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

\begin{array}{l} R_{eb}^{(g) (b)} & = \frac{R_{ls}^{(b)}}{1 + 2 \frac{Γ_{Γ}}{Γ_{res}} \sum_{d} \frac{π c^{2} I}{Ω_{1}^{3} {\tilde{Γ}}_{s}} R^{(d)'}} \\ = \frac{R_{ls}^{(b)}}{1 + 2 \frac{Γ_{Γ}}{Γ_{res}} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} \sum_{d} \frac{π c^{2} I}{Ω_{1}^{3} {\tilde{Γ}}_{s}} R^{(d)'}} \end{array}

(30)

from Eq. (23) and (24) one can obtain

R_{ls}^{(b)} = \sum_{d} {(\frac{Ω_{n}}{Ω_{1}})}^{3} {(\frac{{\tilde{Γ}}_{n}}{{\tilde{Γ}}_{s}})}^{3} R^{(d)'}

(31)

Thus Eq. (32) becomes

R_{eb}^{(g) (b)} = \frac{R_{ls}^{(b)}}{1 + 2 \frac{Γ_{Γ}}{Γ_{res}} \frac{π c^{2} I}{Ω_{1}^{3} {\tilde{Γ}}_{s}} R_{ls}^{(b)}}

(32)

From Eq. (23), we obtain a linear dependence on the laser intensity in the low saturation case, $R_{ls}^{(b)} \sim I$ while, at a strong laser intensity from Eq.(32), we obtain $R_{eb}^{(g) (b)}$ ∼ a₁I/(1+b₁I²), which is no longer linear. When the condition I→ ∞ is satisfied, then one obtains $R_{eb}^{(g) (b)}$ ∼ 0. At an intensity (10¹⁹-10²⁰) W/m², 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

In summary, we propose a two-photon EB excitation scheme to populate the isomeric isotope ^229mTh³⁺. 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 ²²⁹Th³⁺, we obtained the EB excitation rate of ²²⁹Th³⁺ 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(≥ 10¹⁸ W/m²). We find that near the critical value ((10¹⁰ – 10¹⁴) W/m²), the nuclear excitation rate Γ^eb, and the electronic bridge efficiency R reach their maximum values under a strong laser (≥ 10¹⁸ W/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.

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