Quantum theory of isomeric excitation of ²²⁹Th in strong laser fields

The Hamiltonian of the laser-nucleus-electron system can be written as

where the Hamiltonians on the right hand side are, respectively, for the atomic electrons, the nucleus, the electron-nucleus coupling, the electron-laser coupling, and the nucleus-laser coupling. They describe the tripartite interaction between the nucleus, the atomic electrons, and the laser field, as illustrated in Fig. 1. The state space of $H_{\mathrm{e}}$ is the atomic eigenstates, including both bound and continuum states, which are denoted by $\left|\varphi_{i}\right\rangle$. The state space of $H_{\mathrm{n}}$ contains the nuclear ground state $\left|I_{g}M_{g}\right\rangle$ and the isomeric excited state $\left|I_{e}M_{e}\right\rangle$. The electron-nucleus coupling $H_{\mathrm{en}}$ is given as a summation over irreducible tensor operators Schwartz_1957

where $\mathcal{M}^{\tau}_{lm}$ is the nuclear multipole moment of type $\tau$ ($E$ for electric, $M$ for magnetic) and rank $l$. $T^{\tau}_{lm}$ is the corresponding multipole operator for the atomic electrons and is given by

where $Y_{lm}(\theta,\phi)$ is spherical harmonics, $\rho_{\mathrm{e}}(\bm{r})$ and $\bm{j}_{\mathrm{e}}(\bm{r})$ are charge and current density operators for the electron.

The nucleus-laser coupling is given by

where $\bm{j}_{n}(\bm{r})$ and $\bm{A}(\bm{r},t)$ are the operators for the nuclear current density and the laser vector potential, respectively. The vector potential $\bm{A}(\bm{r},t)$ satisfies the Coulomb gauge $\bm{\nabla}\cdot\bm{A}(\bm{r},t)=0$. Assume the vector potential has the following form

where $w(t)$ is a temporal function, $\bm{k}$ and $\hat{\bm{z}}$ are the wave vector and the polarization unit vector. By expanding the vector potential in vector spherical harmonics, the nucleus-laser coupling can be rewritten as

where $C_{lm}^{\tau}$ is the multipole expansion coefficient of the vector potential

Here $k_{0}=\omega_{0}/c$ with $\omega_{0}$ the nuclear energy gap, $\hat{\bm{k}}=\bm{k}/k$ is the unit vector along the $\bm{k}$ direction. $\bm{A}_{lm}^{\mathrm{\tau}}(\hat{\bm{k}})$ is the transverse vector spherical harmonics W.J_Aphy_2007

with the orthonormal relation

Similar to the nucleus-laser coupling, the electron-laser coupling can also be expressed as a summation of multipole terms. However, the dipole approximation is usually sufficient, in which only the electric dipole interaction is used to describe the electron dynamics in the laser field. In this approximation, the electron-laser coupling is written as

Here $\bm{D}$ is the dipole moment operator, and $\bm{E}(t)$ is the laser electric field at the position of the atom: $\bm{E}(t)\equiv\bm{E}(\bm{r}=\bm{0},t)=-\partial\bm{A}(\bm{r}=\bm{0},t)/\partial t$.

Without the laser field, the Hamiltonian of the nucleus-electron system is

with eigenstates

where $\mu$ ($\varepsilon$) denotes the nuclear state (electronic state). For example, $\mu=g$ or $e$ for the nuclear ground state or the isomeric excited state, and $\varepsilon=i$ or $f$ for the initial or the final electronic state. The eigenstate $\left|\Psi_{\mu,\varepsilon}\right\rangle$ can be expanded using the uncoupled states $\left|I_{\mu}M_{\mu},\varphi_{\varepsilon}\right\rangle\equiv\left|I_{\mu}M_{\mu}\right\rangle\otimes\left|\varphi_{\varepsilon}\right\rangle$ using perturbation theory

where $E_{\varepsilon}$ and $E_{\mu}$ are the energy eigenvalues corresponding to the atomic state $\left|\varphi_{\varepsilon}\right\rangle$ and the nuclear state $\left|I_{\mu}M_{\mu}\right\rangle$, respectively.

Initially at time $t_{0}$ the nucleus-electron system is assumed to be in its ground state $\left|\Psi_{g,i}\right\rangle$. Then the probability of nuclear isomeric excitation at a later time $t>t_{0}$ is given by

where $U(t,t_{0})$ is the time evolution operator corresponding to the total Hamiltonian $H$, and the summation runs over all final electronic states.

The time evolution is computationally very demanding due to the large state space (which comes mostly from the electronic states). However, one notices that the couplings of the nucleus to the laser field and to the electrons are weak. This allows the usage of perturbative treatments for the laser-nucleus and electron-nucleus couplings, reducing the computation load substantially. The laser-electron coupling, in contrast, is very strong and must be treated non-perturbatively. Using the time-dependent perturbation theory, the time evolution operator $U(t,t_{0})$ becomes

where $V_{I}(t)$ is defined by

In the above expressions, $U_{0}(t,t_{0})$ is the time evolution operator corresponding to $(H_{\mathrm{e}}+H_{\mathrm{el}})$.

Substituting Eqs. (13) and (15) into Eq. (14), $P_{\mathrm{exc}}(t)$ can be derived into the following form after averaging over initial nuclear states and summing over final states:

The favorable feature of the above formula is that the nuclear transitions are packed in $B(\tau l,g\rightarrow e)$ and the electronic transitions are packed in $N^{\tau;fi}_{lm}$. The former is the reduced nuclear transition probability

and $N^{\tau;fi}_{lm}$, depending only on the electronic initial and final states, is given by

In the above formula $\left|\varphi_{i/f}(t)\right\rangle$ is the electronic state evolved by $U_{0}(t,t_{0})$ from the state $\left|\varphi_{i/f}\right\rangle$. Note that both LDEE and OE channels emerge. The first three lines of Eq. (18) all contain the (time-dependent) electronic states and they describe the LDEE channel. As the atomic state space includes both bound and free states, the contributions from bound-bound, bound-free, and free-free electronic transitions to the nuclear excitation are taken into account intrinsically. The last line, which does not contain electronic states, describes the OE channel, with the $C_{lm}^{\tau}$ given in Eq. (7). The OE channel does not change the electronic state, hence the $\delta_{fi}$.

For the ²²⁹Th nucleus, the leading nuclear transitions from the ground state to the isomeric state are magnetic dipole ($M1$) and electric quadrupole ($E2$). In our calculation we use the reduced transition probability values $B(M1,e\rightarrow g)=0.005$ W.u. and $B(E2,e\rightarrow g)=30$ W.u., as predicted recently by Minkov and Pálffy Minko_PRC_2021 .

The dynamics of the atomic electrons driven by a strong laser pulse has been extensively studied in strong-field atomic physics. Theoretically, most strong-field phenomena can be well understood by solving the time-dependent Schrödinger equation under a single-active-electron (SAE) approximation ADK ; Kulander-87 ; Schafer-93 ; Corkum-93 ; Awasthi-08 ; Le-16 , which assumes that only the outermost electron actively responds to the external laser field, with the remaining electrons contributing a mean-field potential.

The difference between the current work and traditional strong-field atomic physics is the addition of the nuclear degree of freedom. We find that this difference makes the Schrödinger equation insufficient. The main contribution to nuclear excitation comes from electron wave functions very close (around 10^-2 a.u.) to the nucleus due to the factor $r^{-l-1}$ in the electronic operator $T_{lm}^{\tau}$ of Eq. (3). Yet the amplitudes of the Schrödinger wave functions, even for low electron energies, can be very different from those of the Dirac wave functions in this region due to the high nuclear charge ($Z=90$). Therefore relativistic effects are important for the isomeric excitation. A similar conclusion has also been given in nuclear excitation by inelastic electron scattering Zhang-22 .

The straight way to calculate the nuclear excitation probability is to solve the time-dependent Dirac equation, which, however, is very time consuming. An alternative and much more economical approach is used in this paper. We adopt for $H_{\mathrm{e}}$ a so-called zero-order-regular-approximation (ZORA) Hamiltonian Chang_1986 ; Lenthe_1993 ; Lenthe_1994 , instead of the Dirac Hamiltonian. The ZORA Hamiltonian is an effective two-component Hamiltonian giving an accurate approximation of relativistic effects:

where $\bm{\sigma}$ is the Pauli matrix, $\alpha$ is the fine structure constant, and $V(r)$ is a central potential felt by the electron.

Figure 2 shows the radial wave function of the 7$s$ orbital of the ²²⁹Th atom, for the Schrödinger case, the (large component of the) Dirac case, and the ZORA case. One can see that the ZORA wave function is almost identical to the Dirac one, whereas the Schrödinger wave function has smaller amplitudes close to the nucleus, leading to an underestimation of the nuclear excitation probability up to an order of magnitude. The potential energy $V(r)$ used in Fig. 2 and in the results below is calculated by the RADIAL package radial_2019 based on a self-consistent Dirac-Hartree-Fock-Slater method.

The temporal evolution of the electronic state $\left|\varphi_{i}(t)\right\rangle$ obeys the time-dependent ZORA equation

The laser electric field is assumed to be linearly polarized along the $z$ axis with amplitude $F_{0}$, envelope function $f(t)$, and angular frequency $\omega$. Thus, the dipole coupling between the laser and the active electron is written as $H_{\mathrm{el}}(t)=zF_{0}f(t)\sin\omega t$. Equation (20) is numerically solved using a generalized pseudospectral method Yao_1993_CPL ; Tong_1997_CP ; Chu_2004_PR . The time propagation of the ZORA equation can be realized using a split-operator method

where $dt$ is the time step. From the above equation, the time evolution of the wave function from $t$ to $t+dt$ is completed by three steps: (i) evolution for a half-time step $dt/2$ in the energy space spanned by $H_{\mathrm{ZORA}}$; (ii) evolution for one time step $dt$ in the coordinate space under the influence of the electron-laser coupling $H_{\mathrm{el}}$; (iii) evolution for another half-time step $dt/2$ in the energy space spanned by $H_{\mathrm{ZORA}}$. In contrast to Refs. Yao_1993_CPL ; Tong_1997_CP ; Chu_2004_PR , here the wave function at time $t$ is expanded in spherical spinors rather than spherical harmonics for adapting the ZORA Hamiltonian

where $\left|R_{\kappa}(r,t)\right\rangle$ is the (time-dependent) radial wave function, $\left|\Omega_{\kappa m}(\theta,\phi)\right\rangle$ is spherical spinors W.J_Aphy_2007 with quantum number $\kappa$ and magnetic quantum number $m$ , and $K_{\mathrm{max}}$ is an integer to truncate the orbital angular momentum. Here, $m$ is fixed due to $\Delta m=0$ in the linearly polarized laser field. Spherical spinors $\left|\Omega_{\kappa m}(\theta,\phi)\right\rangle$ are orthonormal

and they satisfy the eigenvalue equation

The calculation is performed in a spherical box with radius 150 a.u. and 300 spatial grid points (nonuniform grid, denser near the origin). The time step is $dt=0.1$ a.u. The orbital angular momentum is truncated at $K_{\mathrm{max}}=80$ in the partial-wave expansion of Eq. (22). A boundary absorbing function $1/[1+\mathrm{exp}(br-r_{0})]$ with $b=1.25$ and $r_{0}=120$ a.u. is used to avoid boundary reflection. The convergency has been carefully checked by varying the calculation parameters. The initial state $\left|\varphi_{i}\right\rangle$ for the first, second, third, and fourth electrons sequentially pulled out by the laser field is the $7s_{1/2}$ orbital of the Th atom, the $7s_{1/2}$ orbital of the Th⁺ ion, the $6d_{5/2}$ orbital of the Th²⁺ ion, and the $5f_{5/2}$ orbital of the Th³⁺ ion, respectively. The magnetic quantum number $m$ is fixed at $1/2$.

Figure 3 shows the nuclear excitation probability $P_{\mathrm{exc}}(t)$ during a laser pulse for two different laser wavelengths, namely, 800 nm and 400 nm. For both cases, the laser pulse has a duration of 5 optical cycles with a temporal envelop function $f(t)=\sin^{2}(\pi t/NT)$, where $T=2\pi/\omega$ is the period and $N=5$ is the number of optical cycles. The peak intensity of the laser pulse is $10^{14}$ $\mathrm{W}/\mathrm{cm}^{2}$. This intensity has the ability to drive the outermost three electrons of the Th atom, with negligible effects on the fourth electron, which lies too deeply (A higher intensity is needed to drive this electron, as shown in an example later). Based on the SAE approximation, we calculate separately the first electron, the second electron, and the third electron. For each case, the total excitation probability as well as separated contributions from the $M1$ or $E2$ channels are presented.

With these laser parameters, contributions from the OE channel are found to be 3 to 4 orders of magnitude lower than those from the LDEE channel. This is due to the fact that both 800 nm (photon energy 1.55 eV) and 400 nm (photon energy 3.10 eV) are far off resonance to the nuclear energy gap of 8.28 eV, albeit with a high intensity. Therefore the excitation probabilities presented in Fig. 3 are almost solely from LDEE. For 800 nm, the end-of-pulse nuclear excitation probability is about $8.2\times 10^{-12}$, $8.3\times 10^{-13}$, and $6.7\times 10^{-14}$ for the first electron, the second electron, and the third electron, respectively. The total excitation probability is about $9.1\times 10^{-12}$. For 400 nm, the end-of-pulse nuclear excitation probability is about $1.5\times 10^{-11}$, $8.4\times 10^{-12}$, and $3.1\times 10^{-12}$ for the first electron, the second electron, and the third electron, respectively. The total excitation probability is about $2.7\times 10^{-11}$. This is about three times higher than the 800-nm case.

The relative importance between $M1$ and $E2$ varies from case to case. One sees from the first electron that $E2$ is more important than $M1$ almost for the entire pulse, for both 800 nm and 400 nm. However, the situation reverses for the second electron, where $M1$ dominates during the entire pulse. The situation for the third electron is different for 800 nm and 400 nm. For 800 nm, $M1$ dominates most of the time during the pulse, but near the end of the pulse, the two have almost equal contributions to nuclear excitation. For 400 nm, in contrast, $E2$ dominates for most of the pulse duration except for the initial stage. These results have no simple interpretations but they can be attributed to the dependency of the matrix element of $T_{lm}^{\tau}$ on the time-dependent electronic states.

In Fig. 4, we show the nuclear excitation probability $P_{\mathrm{exc}}(t)$ with laser pulses of different durations ($N=10$ and 15 optical cycles). The wavelength and intensity of the laser pulses are fixed at 400 nm and $10^{14}$ W/cm², so this figure is to be compared with the lower row of Fig. 3 which is for a shorter pulse of $N=5$. The outermost three electrons contribute to the nuclear excitation and they are calculated separately.

For the case of $N=10$, the end-of-pulse nuclear excitation probability is about $3.5\times 10^{-12}$, $7.0\times 10^{-12}$, and $1.5\times 10^{-12}$ for the first electron, the second electron, and the third electron, respectively. The total excitation probability is about $1.2\times 10^{-11}$.

For the case of $N=15$, the end-of-pulse nuclear excitation probability is about $2.7\times 10^{-12}$, $7.4\times 10^{-12}$, and $1.7\times 10^{-12}$ for the first electron, the second electron, and the third electron, respectively. The total excitation probability is about $1.2\times 10^{-11}$, almost identical to the $N=10$ case. This value is about two times lower than the $N=5$ case shown above (Fig. 3).

Another noticeable difference is that for $N=5$, the first electron contributes the most to the nuclear excitation, whereas for $N=10$ or 15 the second electron contributes the most. The major difference from the longer pulses is to reduce the contribution from the first electron (from $1.5\times 10^{-11}$ for $N=5$, to $3.5\times 10^{-12}$ for $N=10$ and $2.7\times 10^{-12}$ for $N=15$). Without presenting analyses involving too many details, this pulse-duration effect can be briefly understood as follows: The shorter pulse provides a larger bandwidth such that it drives the first electron to favorable states for the nuclear excitation. Longer pulses reduce the bandwidth and diminish electronic transitions to these favorable states. Nevertheless, the pulse-duration effect is not severe: Under the same peak intensity, pulses with different durations lead to nuclear excitation probabilities within a factor of two or three.

Figure 5 shows the nuclear excitation probability during two pulses of different peak intensities, namely, $2\times 10^{14}$ and $4\times 10^{14}$ W/cm². Both laser pulses have wavelength 400 nm and duration 10 optical cycles. With intensity $2\times 10^{14}$ W/cm², the first three electrons contribute to the nuclear excitation, whereas with the higher intensity $4\times 10^{14}$ W/cm², the fourth electron starts to contribute to the nuclear excitation.

For the lower intensity, the end-of-pulse nuclear excitation probability is about $6.5\times 10^{-12}$, $6.4\times 10^{-12}$, and $2.5\times 10^{-12}$ for the first electron, the second electron, and the third electron, respectively. The total nuclear excitation probability is about $1.5\times 10^{-11}$.

For the higher intensity, the end-of-pulse nuclear excitation probability is about $3.5\times 10^{-12}$, $3.0\times 10^{-12}$, $8.0\times 10^{-12}$, and $3.5\times 10^{-12}$ for the first electron, the second electron, the third electron, and the fourth electron, respectively. The total nuclear excitation probability is about $1.8\times 10^{-11}$. From Fig. 4(a) and Fig. 5, one can see that the total nuclear excitation probability increases with the laser intensity.

In all the above results, the LDEE channel dominates the nuclear excitation, and the OE channel is weaker by 3 or 4 orders of magnitude. As explained, this is because both 800 nm and 400 nm are far off resonance with the nuclear isomeric energy. The OE channel is important when the laser frequency is close to the isomeric resonance (8.28 eV = 0.304 a.u.). Fig. 6(a) shows a few near-resonant examples for peak intensity 10¹⁴ W/cm² and pulse duration 20 optical cycles. The nuclear excitation probability can reach about 10^-12 for exact resonance, but drops in the presence of a detuning.

However, this does not mean that the LDEE channel is negligible. Fig. 6(b) shows the comparison between OE and LDEE channels for $\omega=0.300$ a.u., which is very close to the resonant frequency. The calculation is performed with the Th²⁺ ion. One can see that the two channels are comparable in this example, and the LDEE channel is even higher. Although there is no easy way of obtaining intense laser pulses with photon energies around 8.28 eV, we want to use this example to emphasize that both OE and LDEE processes exist when the ²²⁹Th atom is radiated by a laser field, and that it would be dangerous to neglect one of them without doing the calculations. This is why a theory including both channels in a single framework is important.