Parity-symmetry-breaking quantum phase transition via parametric drive in a cavity magnonic system

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.

Figure 1

Schematic of the proposed cavity magnonic system consisting of a small YIG sphere and a microwave cavity. The parametric drive on the cavity, with frequency ${\omega }_d$ and amplitude $G$, is generated by pumping a ${\chi }^{\left(2\right)}$ nonlinear medium inside the cavity. Here, $a$ is the annihilation operator of the cavity mode with frequency ${\omega }_c$ and decay rate $\kappa$, and $b$ is the annihilation operator of the magnon mode with frequency ${\omega }_m$ and damping rate $\gamma$.

Figure 2

(a) Steady-state phase diagram of the cavity magnonic system, where PSP, PSBP, and BP denote the parity-symmetric phase, parity-symmetry-broken phase and bistable phase, respectively. (b)–(d) Time evolution of the scaled magnon number $\langle b^{†}b\rangle /(\gamma /K)$, obtained by numerically solving Eq. (7) at the points indicated by red dots in (a), with (b) $G/\kappa =1.8$, and ${\mathrm{\Delta }}_m/{\mathrm{\Delta }}_c=0.8$, (c) $G/\kappa =2.3$, and ${\mathrm{\Delta }}_m/{\mathrm{\Delta }}_c=1.2$, and (d) $G/\kappa =2.3$, and ${\mathrm{\Delta }}_m/{\mathrm{\Delta }}_c=0.4$. In (b)–(d), ${\langle a\rangle }_{t=0}/\sqrt{\gamma /K}=2.1+1.1i$, and ${\langle b\rangle }_{t=0}/\sqrt{\gamma /K}=2.1-1.1i$ for the (black) solid curves, while ${\langle a\rangle }_{t=0}/\sqrt{\gamma /K}=0.1+0.1i$, and ${\langle b\rangle }_{t=0}/\sqrt{\gamma /K}=0.1-0.1i$ for the (red) dashed curves. Other parameters are ${\mathrm{\Delta }}_c/\kappa =3, g/\kappa =2.4$, and $\gamma /\kappa =1$.

Figure 3

(a) The scaled steady-state magnon number $\langle b^{†}b\rangle /(\gamma /K)$ and (b) the expectation value $\langle \delta b^{†}\delta b\rangle$ versus the reduced drive strength $G/\kappa$ in the case of ${\mathrm{\Delta }}_m/{\mathrm{\Delta }}_c=1$. In (a), the (black) solid curve corresponds to the analytical results in Eq. (9), and the (red) dashed curve corresponds to the numerical results obtained using Eq. (7) with initial conditions ${\langle a\rangle }_{t=0}/\sqrt{\gamma /K}={\langle b\rangle }_{t=0}/\sqrt{\gamma /K}=0.001$. Other parameters are the same as in Fig. 2.

Figure 4

(a) The scaled steady-state magnon number $\langle b^{†}b\rangle /(\gamma /K)$ and (b) the expectation value $\langle \delta b^{†}\delta b\rangle$ versus the reduced drive strength $G/\kappa$ in the case of ${\mathrm{\Delta }}_m/{\mathrm{\Delta }}_c=0.3$, where BP (PSBP) denotes the parity-symmetry-broken phase in the bistable region. In (a), the (black) solid curve corresponds to the analytical results in Eq. (9), and the (red) dashed curve corresponds to the numerical results obtained using Eq. (17) with ${\mathrm{\Omega }}_0/\kappa \sqrt{\gamma /K}=2, \kappa \tau =10$, and the initial conditions ${\langle a\rangle }_{t=0}/\sqrt{\gamma /K}={\langle b\rangle }_{t=0}/\sqrt{\gamma /K}=0$. Other parameters are the same as in Fig. 2.

Figure 5

Time evolution of the scaled magnon number $\langle b^{†}b\rangle /(\gamma /K)$ in the (a) parity-symmetric phase with $G/\kappa =1.8$ and (b) bistable phase with $G/\kappa =2.3$, where the (black) solid and (red) dashed curves correspond to the numerical results obtained using Eqs. (7) and (17), respectively. Other parameters are the same as in Fig. 4.