Entanglement dynamics through electromagnetic interactions in single-electron traps

Technical Summary: Entanglement Dynamics through Electromagnetic Interactions in Single-Electron Traps

Problem Statement
This work investigates the dynamics of quantum entanglement generated between two harmonically trapped electrons interacting via electromagnetic forces. While the trapping and control of charged particles (e.g., in Paul traps) are well-established for precision measurements and quantum computing, the specific role of electromagnetic interactions in generating or degrading entanglement between single-electron motional states remains a subject of theoretical and experimental interest. The authors aim to characterize how different initial quantum states—specifically thermal single-mode squeezed states and thermal two-mode squeezed states—evolve under electromagnetic coupling, and how physical parameters such as temperature and squeezing coefficients influence this evolution.

Methodology
The study employs a theoretical framework based on Gaussian state theory and the covariance matrix formalism.

• System Model: The system consists of two electrons of mass $m$ and charge $-e$, trapped in adjacent harmonic potentials along the $x$-axis with frequency $\omega$. The interaction Hamiltonian includes the Coulomb potential ($\hat{H}_C$) and the Darwin Hamiltonian ($\hat{H}_D$), representing the leading-order relativistic correction. The distance $d$ between equilibrium positions is assumed to be much larger than the quantum fluctuations.
• Dynamics: The time evolution is solved by deriving the Heisenberg equations of motion for the ladder operators. This yields a linear system solvable via a time-dependent Bogoliubov transformation, represented by a symplectic matrix $U(t) = \exp[-iHt]$.
• Entanglement Quantification: The authors utilize the covariance matrix formalism to describe the system's state. Since the system is considered to be in thermal equilibrium (a mixed state) rather than a pure state, entanglement is quantified using the logarithmic negativity ($E_N$). This requires calculating the smallest symplectic eigenvalue ($\tilde{d}_-$) of the partially transposed covariance matrix.
• Initial Configurations: Two scenarios are analyzed:
  1. Single-Mode Squeezing: Two initially separable thermal states, each locally squeezed.
  2. Two-Mode Squeezing: An initially entangled thermal two-mode squeezed state.

Key Contributions and Results
The paper provides analytical solutions for the time evolution of the system and numerical results for the logarithmic negativity under various conditions.

1. Dominance of Coulomb Interaction: In the parameter regime analyzed (distances $d=3\,\mu\text{m}$, frequency $\omega=10\,\text{GHz}$), the Darwin term ($\tilde{g}_D$) is found to be negligible compared to the Coulomb term ($\tilde{g}_C$). Consequently, the analysis focuses primarily on the Coulomb interaction.
2. Single-Mode Squeezing Scenario:
   • Starting from separable states, the electromagnetic interaction generates entanglement over time.
   • Thermal Noise: Higher thermal occupation numbers ($N$) delay the onset of detectable entanglement. For $N=0.5$, no entanglement is generated within the observed cycle, whereas for $N=0$ and $N=0.01$, entanglement appears, exhibiting phenomena such as "sudden death and revival" around $\omega t = \pi$.
   • Squeezing: Increasing the squeezing parameter ($|\xi|$) reduces the depth of the entanglement dip and accelerates the growth of entanglement. The evolution also shows a dependence on the squeezing angle $\theta$ (specifically comparing $\theta=0$ and $\theta=\pi$).
3. Two-Mode Squeezing Scenario:
   • Starting from an already entangled state, the interaction leads to a degradation of quantum correlations over time.
   • The rate of degradation depends on the sign of the squeezing coefficient and the thermal number. For instance, with negative squeezing and $N=0.5$, entanglement decays significantly earlier than in other configurations.
4. Experimental Relevance: The authors identify parameter regimes accessible to current and near-future experiments. Using a frequency of $10\,\text{GHz}$ and temperatures achievable in recent experiments ($50\text{--}100\,\text{mK}$), the thermal number is estimated at $N \sim 0.1$. The results suggest that entanglement dynamics are observable within these constraints, particularly for lower thermal numbers.

Significance and Claims
The paper claims to provide a comprehensive analytical and numerical description of entanglement dynamics in a specific, experimentally relevant setup: two single-electron traps interacting via electromagnetic forces.

• It demonstrates that electromagnetic interactions can act as a source of entanglement generation for initially separable states, provided the thermal noise is sufficiently low.
• Conversely, it shows that for pre-entangled states, the same interaction acts as a source of decoherence, degrading correlations.
• The work identifies the specific parameter regimes (squeezing magnitude, thermal occupation, and interaction time) where these effects are most pronounced.
• The authors modestly frame the work as a foundation for future studies, suggesting that extending this framework to three-electron traps could enable the study of tripartite entanglement, and that developing specific entanglement witnesses for electron trap systems is a potential future direction. The paper does not propose new experimental setups but rather analyzes the theoretical feasibility of observing these dynamics in existing or near-future single-electron trap experiments.