Generation of magnonic squeezed state and its superposition in a hybrid qubit-magnon system

Abstract

We propose a protocol for generating magnonic squeezed states (MSS) and their superpositions (SMSS) in a hybrid system comprising a superconducting flux qubit magnetically coupled to the Kittel mode of a yttrium iron garnet (YIG) sphere. The flux qubit provides an intrinsic longitudinal interaction with the magnon mode, which, under resonant microwave driving, gives rise to an effective qubit-state-dependent squeezing Hamiltonian. Numerical simulations incorporating realistic dissipation demonstrate that magnon quadrature noise reduction exceeding $8~\mathrm{dB}$ is achievable with experimentally accessible parameters.~By preparing the qubit in a superposition state followed by projective measurement, we further obtain symmetric and antisymmetric superpositions of orthogonally squeezed magnon states exhibiting clear phase-space interference fringes.~We discuss how the fourfold rotational symmetry of these states supports a bosonic logical encoding with potential for protecting against dominant error channels in magnonic platforms.

Figures (7)

• Figure 1: Schematic of the magnetically coupled hybrid system. A YIG sphere is positioned at the center of a three–Josephson-junction superconducting flux qubit (junctions indicated by yellow markers). A static bias field $B_z$ is applied along the $z$ axis to tune the Kittel-mode frequency. The two persistent-current states, carrying clockwise and counterclockwise currents $I_P$ and $I'_P$, generate opposite near fields at the sphere and thereby mediate a magnetic dipole (Zeeman) coupling between the qubit and the YIG.
• Figure 2: (a) Magnon--qubit coupling strength $g$ as a function of the YIG radius $R$ and the flux-qubit persistent current $I_p$, with the loop side length fixed at $L=10~\mu{\rm m}$ and the sphere centered at the loop center $\mathbf{r}_0=\mathbf{0}$. The coupling is evaluated from Eq. \ref{['eq:g_beff_compact1']} using the Biot--Savart field of the square loop; in panel (a) we employ the point-sphere approximation $B_{\rm eff}^x\simeq B_{\rm loop}^x(\mathbf{r}_0)$ with $N=\rho(4\pi R^3/3)$. Dashed curves show isovalue-line $g$ contours from $0.05$ to $0.29$ in steps of $0.04$. (b) Effective coupling $g(\mathbf{r}_0,R)$ versus the radius $R$ and the out-of-plane offset $x_0$ for a fixed $I_p=0.4~\mu{\rm A}$. The square loop lies in the $y$--$z$ plane (normal along $\hat{x}$), and the sphere center is chosen as $\mathbf{r}_0=(x_0,0,0)$ along the symmetry axis. In panel (b), $B_{\rm eff}^x$ is obtained from the volume average over the sphere according to Eq. \ref{['eq:g_beff_compact2']}. Dashed curves show sovalue-line $g$ contours from $0.04$ to $0.20$ in steps of $0.04$.
• Figure 3: Time evolution of (a) the minimum quadrature variance $\zeta_B^2$, (b) the corresponding squeezing degree $S$, and (c) the mean magnon number $\langle m^{\dagger}m\rangle$. The red solid curves correspond to the dynamics governed by the total Hamiltonian in Eq. \ref{['eq:Htot']} and obtained by solving the master equation \ref{['eq:me']}, whereas the black dashed curves are calculated from the effective Hamiltonian in Eq. \ref{['eq:conditional_squeezing']} by solving the corresponding effective master equation \ref{['eq:me2']}. The system is initialized in the magnon vacuum state, while the qubit prepared in the $\bar{\sigma}x$ eigenstate $(\ket{g} + \ket{e})/\sqrt{2}$. The parameters are chosen as $\omega/2\pi = 1.513~{\rm GHz}, \omega_p/2\pi = 3.002~{\rm GHz}, \Omega/2\pi = 0.5~{\rm GHz}, \phi=\pi$, $g=2\pi\times0.15~\mathrm{GHz}$, $g_x=g_z=g/\sqrt{2}$ with $\theta=\pi/4$, and $\nu=2\pi\times3~\mathrm{GHz}$. The dissipation rates are set to $\kappa/2\pi = 0.5~\mathrm{MHz}$, $\gamma/2\pi = 3~\mathrm{kHz}$, and $\gamma_\phi = \gamma$, with a bath temperature $T=10~\text{mK}$.
• Figure 4: Time evolution of the squeezing degree $S$ (dB) for (a) several magnon damping rates $\kappa$ and (b) various bath temperatures $T$. In panel (a) we fix $T=10~\mathrm{mK}$, while in panel (b) we set $\kappa/2\pi=0.5~\mathrm{MHz}$. All other parameters are identical to those used in Fig. \ref{['fig:fig2']}.
• Figure 5: (a) Maximum squeezing $S$ as a function of the magnon and qubit dissipation rates $\kappa$ and $\gamma$. For each pair $(\kappa,\gamma)$, the reported value corresponds to the largest squeezing obtained by optimizing the evolution time and the squeezing angle. The blue dot marks the parameter set used in Fig. \ref{['fig:fig2']}. All other parameters are the same as those in Fig. \ref{['fig:fig2']}.