Excitation and probing of low-energy nuclear states at high-energy storage rings

${}^{229}\mathrm{Th}$ with a low-lying nuclear isomeric state is an essential candidate for a nuclear clock as well as many other applications. Laser excitation of the isomeric state has been a long-standing goal. With relativistic ${}^{229}\mathrm{Th}$ ions in storage rings, high-power lasers with wavelengths in the visible range or longer can be used to achieve high excitation rates of ${}^{229}\mathrm{Th}$ isomers. This can be realized through direct resonant excitation or excitation via an intermediate nuclear or electronic state, facilitated by the tunability of both the laser-beam and ion-bunch parameters. Unique opportunities are offered by highly charged ${}^{229}\mathrm{Th}$ ions due to the nuclear-state mixing. The significantly reduced isomeric-state lifetime corresponds to a much higher excitation rate for direct resonant excitation. Importantly, we propose electric dipole transitions changing both the electronic and nuclear states that are opened by the nuclear hyperfine mixing. We suggest using them for efficient isomer excitation in Li-like ${}^{229}\mathrm{Th}$ ions, via stimulated Raman adiabatic passage or single-laser excitation. We also propose schemes for probing the isomers, utilizing nuclear radiative decay or laser spectroscopy on electronic transitions, through which the isomeric-state energy can be determined with an orders-of-magnitude higher precision than the current value. The schemes proposed here for ${}^{229}\mathrm{Th}$ could also be adapted to low-energy nuclear states in other nuclei, such as ${}^{229}\mathrm{Pa}$.

Figure 1

Low-lying nuclear levels of ${}^{229}\mathrm{Th}$. These levels belong to two rotational bands (indicated by the horizontal offset) labeled by spin, parity, Nilsson quantum numbers, energy, and (radiative) half-life $T_{1/2}^{\left(\text{rad}\right)}$. $T_{1/2}^{\text{rad}}$ of the isomeric state is derived from Ref. [4]. $T_{1/2}^{\text{rad}}$ of the second excited state is obtained using experimental results in Refs. [2, 15]. 9.3% and 90.7% are the radiative branching ratios. The nuclear magnetic dipole moment $\mu$ and electric quadrupole moment $Q$ are presented for the ground and isomeric states [68, 69, 70].

Figure 2

Laser excitation of ${}^{229m}\mathrm{Th}$ isomers in the storage ring. The laser pulse nearly counter propagates with the relativistic highly charged thorium ions. The intermediate state of the excited ion depends on the specific excitation scheme. The intermediate state shown here corresponds to exciting the 279 eV transition in Li-like ${}^{229}\mathrm{Th}$ ions, see Sec. 3b3. The purple arrow denotes the photon emitted from the ${}^{2}P_{1/2}\to {}^{2}S_{1/2}$ electronic decay, after which the ion remains in the nuclear isomeric state. Produced ${}^{229m}\mathrm{Th}$ will circulate in the storage ring, with the radiative decay indicated by the orange arrow.

Figure 3

Production of ${}^{229m}\mathrm{Th}$ isomers through (a) resonant excitation or (b) excitation via the second nuclear excited state. The second scenario requires a large Lorentz factor of the ion bunch, which could be achieved at the Large Hadron Collider (LHC).

Figure 4

Repeated pumping at (a) SPS and (b) HESR. The blue solid curve is derived from repeatedly calculating the population transfer using Eqs. (5) and (7), which also agrees with Eq. (10) shown as the orange dashed curve. The characteristic time for reaching population saturation is about $60\mathrm{ms}$ in both cases, see the text.

Figure 5

Energy levels of the nuclear ground and isomeric states and related radiative transition rates (${\mathrm{\Gamma }}_{\text{rad}}/\hslash$) in bare thorium nuclei (left) and H-like thorium ions (right). The values are from Ref. [19]. Due to the NHM effect in H-like thorium, the two $F=2$ HFS levels are mixed, leading to significantly higher radiative transition rates from the isomeric state to the ground state compared to bare nuclei. Similar effect also exists in Li-like thorium ions; see the text.

Figure 6

The low-lying energy levels in Li-like ${}^{229}\mathrm{Th}$ ions and transitions between hyperfine components that change both the nuclear and electronic states. The transitions denoted by the solid arrows, although occurring between different nuclear states, are all $E1$ transitions, due to the NHM effect. Since the mixing only occurs between states with the same total angular momentum quantum number $F$, the transitions represented by the blue (orange) arrows are allowed due to mixing of the two $F=2$ levels in the electronic ground (excited) state, ${}^{2}S_{1/2}$ (${}^{2}P_{1/2}$). The transitions denoted by green arrows include contributions from mixing in both ${}^{2}S_{1/2}$ and ${}^{2}P_{1/2}$ manifolds.

Figure 7

(a) A generic STIRAP scheme. The pump and Stokes and pulses interact with a three-level $\mathrm{\Lambda }$ system. Under STIRAP conditions, the population from state 1 can be fully transferred to state 3, while the population of state 2 remains nominally zero throughout the process. (b) Excitation of ${}^{229m}\mathrm{Th}$ via the electronic state $1s{}^{2}2p_{1/2}\left({}^{2}P_{1/2}\right)$ in Li-like thorium. Red solid lines denote energy levels of thorium ions in the nuclear isomeric state $I^P=3/2^{+}$. Other energy levels are for thorium ions in the nuclear ground state $I^P=5/2^{+}$. Here, as an example, we consider a pump laser driving the transition between $|5/2^{+},{}^{2}S_{1/2},F=3\rangle$ and $|5/2^{+},{}^{2}P_{1/2},F=3\rangle$, and a Stokes laser driving the transition between $\overline{|3/2^{+},{}^{2}S_{1/2},F=2\rangle }$ and $|5/2^{+},{}^{2}P_{1/2},F=3\rangle$. The latter transition is allowed due to the NHM effect; see the text.

Figure 8

Population transfer into the isomeric state $|3\rangle$ via the STIRAP set-up. (a) Population transfer under different laser-pulse intensities and pulse delays $\mathrm{\Delta }{\tau }^{\prime }$ (given in the unit of $t_{\text{p}}^{\prime }$). (b) Population evolution under $I_{\text{S}0}^{\prime }=6\times {10}^{22}\mathrm{W}/{\mathrm{m}}^2$ with delay $\mathrm{\Delta }{\tau }^{\prime }=0.07t_{\text{p}}^{\prime }$ (the minimum $I_{\text{S}0}^{\prime }$ for 100% population transfer). ${\rho }_{ii}$ denotes the relative population of the state $|i\rangle$. (c) Similar to (b) but for $I_{\text{S}0}^{\prime }=6\times {10}^{23}\mathrm{W}/{\mathrm{m}}^2$ and $\mathrm{\Delta }{\tau }^{\prime }=0.76t_{\text{p}}^{\prime }$. The laser intensities are given for $I_{\text{S}0}^{\prime }$. $I_{\text{P}0}^{\prime }$ is related to $I_{\text{S}0}^{\prime }$ by letting ${\mathrm{\Omega }}_{\text{P}}$ and ${\mathrm{\Omega }}_{\text{S}}$ have the same peak value.

Figure 9

Repeated pumping at the LHC. The blue solid curve is derived from the theoretical calculation described in the text, which also agrees with Eq. (10) shown as the orange dashed curve. Nearly full population transfer is reached after $n_{\text{col}}\approx 6300$ collisions ($t_{\text{ex}}\approx 0.56\mathrm{s}$).

Figure 10

(a) Population transfer into the isomeric state $|3\rangle$ via the STIRAP set-up under different laser-pulse intensities and pulse delays. (b) Population evolution under $0.6\widetilde{I_{\text{P/S,0}}^{\prime }}$ and $\mathrm{\Delta }{\tau }^{\prime }=0.78t_{\text{p}}^{\prime }$.

Figure 11

Detection of ${}^{229m}\mathrm{Th}$ via laser spectroscopy on the ${}^{2}S_{1/2}\to {}^{2}P_{1/2}$ electronic transition ($\approx 271\mathrm{eV}$) of Li-like thorium ions in (a) the nuclear ground state and (b) the isomeric state. The ${}^{2}S_{1/2}$ and ${}^{2}P_{1/2}$ energy levels without considering the HFS are denoted by dashed lines on the left. The HFS levels are shown as solid lines on the right and corresponding energy level splittings are indicated. All energies are given in eV. See the text for details of the transition energies, $E^{229g}$ and $E^{229m}$, and the hyperfine splittings.

Figure 12

Relative excitation rates of (a) transitions from $|{}^{2}S_{1/2},I,F_g\rangle$ to $|{}^{2}P_{1/2},I,F_e\rangle$ in unpolarized Li-like ${}^{229}\mathrm{Th}$ ions; see Eq. (28), and (b) transitions from $|{}^{2}S_{1/2},I,F_g{M}_{F_g}\rangle$ to $|{}^{2}P_{1/2},I,F_e{M}_{F_e}\rangle$ in fully polarized Li-like ${}^{229}\mathrm{Th}$ ions; see Eq. (30). The horizontal axis denotes photon energies in the ion frame. Solid lines and dashed lines are for ions with nuclei in the ground and isomeric state, respectively.