Strong long-range coupling between distant spins is crucial for spin-based quantum information processing. However, achieving such a strong spin-spin coupling remains challenging. Here we propose to realize a strong coupling between two distant spins via the Kerr effect of magnons in a yttrium-iron-garnet nanosphere. By applying a microwave field on this nanosphere, the Kerr effect of magnons can induce the magnon squeezing, so that the coupling between the spin and the squeezed magnons can be exponentially enhanced. This in turn allows the spin-magnon distance to increase from nano- to micrometer scale. By considering the virtual excitation of the squeezed magnons in the dispersive regime, strong spin-spin coupling mediated by the squeezed magnons can be achieved, and a remote quantum-state transfer, as well as the nonlocal two-qubit iswap gate with high fidelity becomes implementable. Our approach offers a feasible scheme to perform quantum information processing among distant spins.

Cavity optomechanics, a promising platform to investigate macroscopic quantum effects, has been widely used to study nonreciprocal entanglement with the Sagnac effect. Here, we propose an alternative way to realize nonreciprocal entanglement among magnons, photons, and phonons in a hybrid cavity-magnon optomechanics, where the magnon Kerr effect is used. We show that the Kerr effect gives rise to a magnon frequency shift and an additional two-magnon effect. Both of them can be tuned from positive to negative via tuning the magnetic field direction, leading to nonreciprocity. By tuning system parameters such as magnon frequency detuning or the coefficient of the two-magnon effect, bipartite and tripartite entanglements can be nonreciprocally enhanced. By further studying the defined bidirectional contrast ratio, we find that nonreciprocity in our system can be switched on and off, and can be engineered by the bath temperature. Our proposal not only provides a potential path to demonstrate nonreciprocal entanglement with the magnon Kerr effect, but also opens a direction to engineer and design diverse nonreciprocal devices in hybrid cavity-magnon optomechanics with nonlinear effects.

Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect, and we find that the pseudo-Hermitian COM system without $\mathcal{PT}$ symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is $\mathcal{PT}$ symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.

We study the parity-symmetry-breaking quantum phase transition (QPT) in a cavity magnonic system driven by a parametric field, where the magnons in a ferrimagnetic yttrium-iron-garnet sphere strongly couple to a microwave cavity. With appropriate parameters, this cavity magnonic system can exhibit a rich phase diagram, including the parity-symmetric phase, parity-symmetry-broken phase, and bistable phase. When increasing the drive strength beyond a critical threshold, the cavity magnonic system undergoes either a first- or second-order nonequilibrium QPT from the parity-symmetric phase with microscopic excitations to the parity-symmetry-broken phase with macroscopic excitations, depending on the parameters of the system. Our work provides an alternate way to engineer the QPT in a hybrid quantum system containing the spin ensemble in a ferri- or ferromagnetic material with strong exchange interactions.

Cavity magnomechanics has shown great potential in studying macroscopic quantum effects, especially for quantum entanglement, which is a key resource for quantum information science. Here we propose to realize magnon mediated nonreciprocal photon-phonon entanglement, which exhibits asymmetry when opposite magnetic or driving fields are respectively applied to the magnons with the Kerr effect or the photons with the Sagnac effect. We find that the mean magnon number can selectively exhibit nonreciprocal linear or nonlinear (bistable) behavior with the strength of the strong driving field on the cavity. Assisted by this driving field, the magnon-phonon coupling is greatly enhanced, leading to the nonreciprocal photon-phonon entanglement via the swapping interaction between the magnons and photons. This nonreciprocal entanglement can be significantly enhanced with the magnon Kerr and Sagnac effects. Given the available parameters, the nonreciprocal photon-phonon entanglement can be preserved at $\approx 3$ K, showing remarkable resilience against the bath temperature. The result reveals that our paper holds promise in developing various nonreciprocal devices with both the magnon Kerr and Sagnac effects in cavity magnomechanics.

The superradiant phase transition (SPT) has been widely studied in cavity quantum electrodynamics (CQED). However, this SPT is still subject to ongoing debates due to the no-go theorem induced by the so-called ${\mathbf{A}}^2$ term (AT). We propose a hybrid quantum system, consisting of a single-mode cavity simultaneously coupled to both a two-level system and yttrium-iron-garnet sphere supporting magnons with Kerr nonlinearity, to restore the SPT against the AT. The Kerr magnons here can effectively introduce an additional AT tunable and strong enough to counteract the intrinsic AT, via adiabatically eliminating the degrees of freedom of the magnons. We show that the Kerr-magnon-induced SPT can exist in both cases of ignoring and including the intrinsic AT. Without the intrinsic AT, the critical coupling strength can be dramatically reduced by introducing the Kerr magnons, which greatly relaxes the experimental conditions for observing the SPT. With the intrinsic AT, the forbidden SPT can be recovered with the Kerr magnons in a reversed way. Our work paves a potential way to manipulate the SPT against the AT in hybrid systems combining CQED and nonlinear magnonics.