Long Time Energy Oscillation Between Electron Shell and Nucleus in $^{229}$Th Ions and Coherent Electron Bridge for Nuclear Quantum Battery

Technical Summary: Long Time Energy Oscillation Between Electron Shell and Nucleus in 229Th Ions and Coherent Electron Bridge for Nuclear Quantum Battery

Problem Statement
The paper addresses the interaction between the electronic shell and the nucleus in Thorium-229 ($^{229}\text{Th}$) ions. Specifically, it investigates the scenario where a magnetic dipole (M1) transition in the electron shell has an energy ($\omega_A$) close to the energy of the low-lying isomeric state in the $^{229}\text{Th}$ nucleus ($\omega_N \approx 8.4$ eV). While previous studies on Nuclear Excitation by Electron Transition (NEET) in systems like $^{197}\text{Au}$ and $^{193}\text{Ir}$ have shown that energy transfer between atomic and nuclear shells is possible, these processes typically occur with very low probability. This is because the interaction energy between electron and nuclear currents is several orders of magnitude smaller than the radiative widths of the atomic levels involved, limiting transfer to a single, rare event. The paper explores whether a different regime exists in $^{229}\text{Th}$ ions where the interaction energy is sufficiently strong to sustain coherent, multiple energy oscillations between the two subsystems, even without strict resonant coincidence of transition energies.

Methodology
The authors model the system as two nested qubits: an atomic qubit (electron shell) and a nuclear qubit (ground and isomeric states), with the nuclear qubit spatially embedded within the atomic one. The dynamics are described using the Lindblad equation for the density matrix $\rho$ of the two-qubit system.

• Hamiltonian: The coherent interaction is governed by a Hamiltonian $H$ containing terms for the individual qubit energies and an interaction term $g_{ij}S^+_i S^-_j$, representing the coherent coupling of the nuclear and atomic magnetic dipoles.
• Interaction Energy: The coupling strength $g_{ij}$ is derived from the real part of the interaction energy $E_{\text{int}}$, calculated via second-order perturbation theory involving a virtual photon propagator. The imaginary part of $E_{\text{int}}$ accounts for incoherent decay channels.
• Numerical Estimation: The study performs numerical estimates for two specific cases: a highly charged ion ($\text{Th}^{39+}$) and a low-charge ion ($\text{Th}^{+}$). For $\text{Th}^{39+}$, the authors utilize existing calculations for the NEET probability. For $\text{Th}^{+}$, they calculate the matrix elements for the M1 transition between $7S_{1/2}$ and $8S_{1/2}$ states using the relativistic Hartree-Fock-Slater approximation, accounting for electron core polarization.
• Dynamics Analysis: The system is solved analytically under various conditions, including resonance ($\Delta E \ll \Gamma_A$) and large detuning ($\Gamma_A \ll \Delta E \ll g$). The paper also extends the model to include coherent laser radiation to simulate the "charging" of the system.

Key Contributions and Results

1. Strong Coupling Regime: The authors demonstrate that in $^{229}\text{Th}$ ions, the interaction energy $g$ (the real part of $E_{\text{int}}$) significantly exceeds the radiative width of the atomic transition $\Gamma_A$. For $\text{Th}^{39+}$, $g \approx 6 \times 10^{-6}$ eV, and for $\text{Th}^{+}$, $g \approx 10^{-6}$ eV. This is orders of magnitude larger than the incoherent damping terms ($\gamma_{AN}$), which are negligible ($10^{-15}$ to $10^{-18}$ eV).
2. Sustained Coherent Oscillations: Unlike the standard NEET process which is a single, low-probability event, the strong coupling in $^{229}\text{Th}$ allows for long-time, coherent energy oscillations between the electron shell and the nucleus.
   • Resonance: When transition energies are close, the probability of nuclear excitation oscillates with a frequency $\nu \approx 10^9 \text{ s}^{-1}$ and persists for a duration $t \approx 1/\Gamma_A$ ($10^{-5} - 10^{-4}$ s).
   • Large Detuning: Remarkably, these oscillations persist even when the energy detuning $\Delta E$ is much larger than the atomic width ($\Gamma_A \ll \Delta E \ll g$). In this regime, the nuclear excitation probability remains close to unity, and the system oscillates rapidly without requiring resonant energy matching.
3. "Breathing" Electron Shell: The energy exchange manifests physically as the electron shell periodically expanding and contracting ("breathing"). In low-charge ions, this involves a size change from $(3-4)a_B$ to $(7-8)a_B$. The authors suggest this effect could be observed in ion traps via the intensity of scattered light, as the scattering cross-section differs significantly between the ground and excited states.
4. Entanglement: The coherent interaction spontaneously generates entanglement between the atomic and nuclear qubits, quantified by the concurrence $C(t)$, which oscillates in phase with the energy transfer.
5. Nuclear Quantum Battery: The system is proposed as a "nuclear quantum battery." The electron shell acts as a donor, and the nucleus as an acceptor. The paper demonstrates that by applying coherent laser radiation tuned to the atomic transition, one can drive synchronous coherent oscillations between the ground state ($|A_g N_g\rangle$) and the doubly excited state ($|A_e N_e\rangle$). This allows for the efficient "charging" (excitation) of the nuclear isomer $^{229m}\text{Th}$ via a coherent electron bridge mechanism.
   • For $\text{Th}^{39+}$, the authors estimate that with a laser power of $\approx 10^{-5}$ W focused on a $10^{-2}$ cm spot, the system can achieve effective Rabi frequencies sufficient to excite the nucleus within seconds.

Significance and Claims
The paper claims that the unique energy structure of $^{229}\text{Th}$ ions creates a model system where the interaction between atomic and nuclear degrees of freedom is strong enough to support macroscopic quantum phenomena (coherent oscillations) over observable timescales.

• Extended Excitation Range: The findings extend the energy range for exciting the $^{229m}\text{Th}$ isomer via an electron bridge, removing the strict requirement for resonant coincidence of transition energies.
• Quantum Battery Realization: The authors assert that this system represents a viable candidate for a nuclear quantum battery at the current level of technological development. The "charging" mechanism utilizes established methods for quantum batteries (coherent excitation followed by coherent transfer), offering a pathway to excite the nuclear isomer without direct gamma-ray irradiation.
• Observability: The paper suggests that the "breathing" effect and the associated light scattering modulation provide a direct experimental signature for verifying these long-time energy oscillations in ion traps.

The work is framed as a theoretical proposal based on existing data and standard quantum mechanical models, highlighting the potential for $^{229}\text{Th}$ ions to serve as a platform for studying strong atom-nucleus coupling and developing nuclear quantum technologies.