Entanglement engineering in magnomechanical system via cross-Kerr interaction and mechanical parametric amplification

Technical Summary: Entanglement Engineering in Magnomechanical Systems via Cross-Kerr Interaction and Mechanical Parametric Amplification

Problem Statement
Quantum entanglement is a critical resource for modern quantum technologies, including communication, sensing, and computing. While hybrid systems involving magnonic modes (collective spin excitations) offer tunability and rich mode spectra, generating robust quantum correlations often requires strong coupling strengths or specific interaction mechanisms. Previous work established that generating entanglement in magnomechanical systems typically relies on strong effective cross-Kerr coupling when mechanical coupling and parametric driving are absent. The challenge addressed in this work is to determine how quantum entanglement can be engineered and enhanced in a hybrid magnomechanical system under more feasible conditions, specifically by leveraging mechanical coupling and parametric amplification to reduce the reliance on strong nonlinearities.

Methodology
The authors propose a theoretical model based on a yttrium iron garnet (YIG) microsphere (radius 100 nm) mechanically coupled to the free-moving end of a cantilever beam. The system comprises three interacting modes:

1. A magnonic mode ($m$).
2. An acoustic phononic mode ($a$) associated with the cantilever.
3. A center-of-mass motion (CMM) phononic mode ($b$) associated with the microsphere.

The dynamics are governed by a Hamiltonian that includes:

• Couplings: Magnon-acoustic ($g_{ma}$), magnon-phonon ($g_{mb}$), and a weak mechanical phonon hopping rate ($J_m$) between the acoustic and CMM modes.
• Nonlinearity: A cross-Kerr interaction ($g_{ck}$) between the magnonic and phononic modes.
• Driving: A mechanical parametric amplification (MPA) acting on the phononic mode ($b$) via modulation of the spring constant, and an external driving field for the magnon.

The study employs the Quantum Langevin Equations (QLEs) to describe the system's dynamics. By linearizing these equations around steady-state mean values and applying the rotating wave approximation (RWA), the authors derive a drift matrix. The stability of the system is verified using the Routh-Hurwitz criterion. The steady-state properties are characterized by a $6 \times 6$ covariance matrix (CM), obtained by solving the Lyapunov equation.

To quantify entanglement, the authors utilize:

• Logarithmic Negativity ($E_N$): To measure bipartite entanglement between pairs of modes (magnon-acoustic, magnon-phonon, acoustic-phonon).
• Minimum Residual Contangle ($R_{min}^{\tau}$): To quantify genuine tripartite entanglement.
• Purity ($\mu$): Calculated via the trace of the squared reduced density matrix to assess the quality of the generated states.

Key Contributions and Results
The study systematically investigates the interplay between cross-Kerr nonlinearity, phonon hopping ($J_m$), and mechanical parametric amplification ($\chi$).

1. Threshold Reduction via Combined Interactions:

   • In the absence of mechanical coupling ($J_m=0$) and parametric amplification ($\chi=0$), the generation of entanglement requires a relatively strong cross-Kerr coupling strength ($\tilde{G}_{ck} \gtrsim 0.04\omega_b$).
   • When phonon hopping and MPA are included, the threshold for the cross-Kerr coupling is effectively eliminated. Quantum entanglement can be generated even in the weak-coupling limit of the cross-Kerr nonlinearity ($\tilde{G}_{ck} \approx 0$).

2. Role of Phonon Hopping ($J_m$):

   • The phonon hopping rate is identified as a key parameter for redistributing quantum correlations.
   • A threshold in $J_m$ is required for entanglement onset in the absence of cross-Kerr nonlinearity. Specifically, $J_m \lesssim 0.04\omega_b$ is sufficient for magnon-acoustic ($E_{ma}$), magnon-phonon ($E_{mb}$), and tripartite entanglement, while a slightly higher value ($J_m \approx 0.08\omega_b$) is needed for acoustic-phonon entanglement ($E_{ab}$).
   • Increasing $J_m$ monotonically improves the strength of entanglement, with an optimal value observed at $J_m = 0.1\omega_b$.

3. Enhancement by Parametric Amplification ($\chi$):

   • The combined effect of phonon hopping and parametric driving extends the regime of entanglement generation.
   • The presence of cross-Kerr interaction further lowers the required mechanical coupling threshold for specific entanglement types, particularly for the magnon-phonon pair ($E_{mb}$), which benefits from the direct nature of the cross-Kerr coupling.

4. Robustness and Purity:

   • Thermal Robustness: The generated entanglement is robust against thermal noise. While entanglement vanishes as temperature increases, higher phonon hopping rates enhance the system's resilience to thermal fluctuations.
   • State Purity: The purity of the generated steady-state entangled states remains high ($\mu \approx 0.92$) across the parameter regimes studied. This indicates that the engineering of entanglement does not significantly increase the mixing of the involved states, preserving the system near an ideal pure state.

Significance
The paper claims that the joint action of mechanical coupling, parametric amplification, and cross-Kerr nonlinearity provides a versatile platform for engineering robust and large-scale quantum entanglement in hybrid magnomechanical systems. The primary significance lies in demonstrating that strong cross-Kerr nonlinearities are not strictly necessary for entanglement generation if mechanical coupling and parametric driving are optimized. This finding suggests a pathway to creating stable quantum correlations in magnomechanical structures using feasible experimental parameters. The authors state these results are useful for modern quantum applications, specifically citing quantum information processing, quantum communication, and quantum computational tasks, without proposing specific new experimental setups or unverified future implications beyond these established domains.