From “strong-field atomic physics” to “strong-field nuclear physics”

Abstract

In the mid-1980s, chirped pulse amplification (Nobel Prize in Physics 2018) overcame previous limits on laser intensity, allowing intensities to exceed the atomic unit threshold (1 atomic unit of laser intensity corresponds to a power density of 3.5×10<sup>16</sup>W/cm<sup>2</sup>). Such intense laser fields induce high-order nonlinear responses in atoms and molecules, resulting in a range of novel phenomena, among which high harmonic generation and attosecond pulse generation (Nobel Prize in Physics 2023) are of particular importance. With advancements in high-power laser technology, laser intensities have now reached the order of 10<sup>23</sup>W/cm<sup>2</sup> and continue to increase. This raises a fundamental question: can such powerful laser fields induce similar high-order nonlinear responses in atomic nuclei, potentially transitioning “strong-field atomic physics” into “strong-field nuclear physics”?<br>To explore this, we investigate a dimensionless parameter that estimates the strength of light-matter interaction: $\eta=\frac{D E_0}{\Delta E}$ where <i>D</i> is the transition moment (between two representative levels of the system), <i>E</i><sub>0</sub> is the laser field amplitude, <i>D</i><i>E</i><sub>0</sub> quantifies the laser-matter interaction energy, and $\Delta E$ is the transition energy. If $\eta \ll 1$, the interaction is within the linear, perturbative regime. However, when <i>η</i>~1, highly nonlinear responses are anticipated. For laser-atom interactions, <i>D</i>~1 a.u. and $\Delta E \sim 1$ a.u., so if <i>E</i><sub>0</sub>~1 a.u., then <i>η</i>~1 and highly nonlinear responses are initiated, leading to the above-mentioned strong-field phenomena.<br>In the case of light-nucleus interaction, it is typical that $\eta \ll 1$. When considering nuclei instead of atoms, <i>D</i> becomes several (~5 to 7) orders of magnitude smaller, while $\Delta E$ becomes several (~5) orders of magnitude larger. Consequently, the laser field amplitude <i>E</i><sub>0</sub> would need to be 10 orders of magnitude higher, or the laser intensity needs to be 20 orders of magnitude higher (~ 10<sup>36</sup>W/cm<sup>2</sup>), which is beyond current technological limit and even surpassing the Schwinger limit, where vacuum breakdown occurs.<br>However, there exists special nuclei with exceptional properties. For instance, the <sup>229</sup>Th nucleus has a uniquely low-lying excited state with an energy of only 8.4 eV, or 0.3 a.u. This unusually low transition energy significantly increases <i>η</i>. This transition has also been proposed for the construction of nuclear clocks, which offer potential advantages over current atomic clocks.<br>Another key factor is nuclear hyperfine mixing (NHM). An electron, particularly one in an inner orbital, can generate a strong electromagnetic field at the position of the nucleus, causing nuclear eigenstates to mix. For <sup>229</sup>Th, this NHM effect is especially pronounced: the lifetime of the 8.4-eV nuclear isomeric state in a bare <sup>229</sup>Th nucleus (<sup>229</sup>Th<sup>90+</sup>) is on the order of 10<sup>3</sup> seconds, while in the hydrogenlike ionic state (<sup>229</sup>Th<sup>89+</sup>) it decreases by five orders of magnitude to 10<sup>-2</sup> s. This 1s electron greatly affects the properties of the <sup>229</sup>Th nucleus, effectively altering the nuclear transition moment from <i>D</i> for the bare nucleus to $D^{\prime}=D+b \mu$ for the hydrogenlike ion, where <i>D</i>~10<sup>-7</sup>a.u.,$b \approx 0.03$ is the mixing coefficient, <i>μ</i><sub>e</sub> is the magnetic moment of the electron, and $D^{\prime} \approx b \mu_e \sim 10^{-4}$ a.u. That is, the existence of the 1s electron increases the light-nucleus coupling matrix element by approximately three orders of magnitude, leading to the five-orders-of-magnitude reduction in the isomeric lifetime.<br>With the minimized transition energy $\Delta E$ and the NHM-enhanced transition moment <i>D</i>', we find that <i>η</i>~1 for currently achievable laser intensities. Highly nonlinear responses are expected in the <sup>229</sup>Th nucleus. This is confirmed by our numerical results. Figure (a) shows the nuclear isomeric excitation probabilities for <sup>229</sup>Th<sup>89+</sup>as a function of laser intensity. Note that the isomeric state has been split into two states with total angular momentum quantum number <i>F</i>=2 and <i>F</i>=1 due to hyperfine interaction, and the excitation probabilities to both of these levels are shown. One can see the nonlinear “bursts” above intensity 10<sup>17</sup> W/cm<sup>2</sup>: a four-order-of-magnitude increase in laser intensity from 10<sup>17</sup> to 10<sup>21</sup> W/cm<sup>2</sup> leads to a 14-orders-of-magnitude increase in excitation probability to the 10% level (per nucleus per laser pulse). In contrast, for the bare nucleus <sup>229</sup>Th<sup>90+</sup>without NHM, the dependency of the excitation probability on the laser intensity remains linear across the whole intensity range up to 10<sup>23</sup> W/cm<sup>2</sup>and the absolute excitation probability remains low (~ 10<sup>-15</sup>).<br>Correspondingly, the intense laser-driven <sup>229</sup>Th<sup>89+</sup> system radiates secondary light in the form of high harmonics, and the spectra are shown in Figure (b) for four different laser intensities. These spectra share similarities with those from laser-driven atoms but also have distinct features.<br>In conclusion, it appears feasible to extend “strong-field atomic physics” to “strong-field nuclear physics”, at least in the case of <sup>229</sup>Th. “Strong-field nuclear physics” is emerging as a new frontier in light-matter interaction and nuclear physics, presenting opportunities for precise excitation and control of atomic nuclei with intense lasers and new avenues for coherent light emission based on nuclear transitions.