Entanglement properties of photon–magnon crystal from nonlinear perspective

Quantifying the entanglement between two continuous bosonic modes, such as magnons and photons, is not trivial. The state-of-the-art for today is the logarithmic negativity, calculated through the quantum Langevin equations subjected to thermal noise. However, due to its complexity, this method requires further approximation. Namely, after the linearization procedure, quantum operators are replaced by their semiclassical expectation values calculated near the steady state . However, the phase space of a generic nonlinear system contains topologically different regions, and the steady state may correspond to the different types of fixed points, such as Saddle Points, Stable or unstable Spirals, and Nodes. Through the conventional linearization procedure, one obtains equations for the photon and magnon number operators, but the character of the fixed point is unexplored. In the present work, we propose a new procedure. Namely, we derived the complete set of nonlinear equations, which includes equations for the magnon and photon number operators and phases. We show that not only number operators but also phases are important for exploring the character of the fixed point, and the character of the fixed point influences the magnon–photon entanglement. We showed that methods of the qualitative theory of nonlinear differential equations are also relevant for photon–magnon entanglement problems. Our main finding is that entanglement is not defined in the Saddle Point region. On the other hand, the maximum of the entanglement corresponds to the region near the border between the Stable node and Stable spiral regions. Our approach is quite general. However, we did calculations for a particular system: photon–magnon crystal based on the yttrium iron garnet (YIG) film with the periodic air holes drilled in the film. Our interest focuses on magnons with a particular wavelength and frequency corresponding to the magnon condensate. Those magnons couple strongly with the photons of similar frequency. We discuss in detail the interaction between magnons and photons originating from the magneto-electric coupling and the effective Dzyaloshinskii–Moriya interaction. We show that this interaction is responsible for the robust photon–magnon entanglement in the system.