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.

• The paper presents a protocol to generate magnonic squeezed states and superpositions via intrinsic longitudinal qubit–magnon coupling under resonant drive.
• It achieves over 8 dB quadrature noise reduction with strong state fidelity despite realistic dissipation and magnon damping.
• The work enables non-Gaussian state engineering for continuous-variable quantum computation and error-corrected logical encoding.

Protocols for Magnonic Squeezing and Superpositions in Hybrid Qubit–Magnon Systems

Introduction

This work presents a detailed protocol for the generation of magnonic squeezed states (MSS) and their symmetric/antisymmetric superpositions (SMSS) in a hybrid quantum system composed of a superconducting flux qubit magnetically coupled to the Kittel mode of a yttrium iron garnet (YIG) sphere (2604.02924). The main innovation lies in utilizing intrinsic longitudinal qubit–magnon coupling, which, under resonant microwave driving, leads to a qubit-state-dependent squeezing Hamiltonian. This protocol enables not only strong magnon quadrature noise reduction but also projective generation of non-Gaussian superpositions with phase-space interference, opening the possibility for bosonic logical encodings with advantageous error-correcting symmetries.

Hybrid System and Conditional Squeezing Hamiltonian

A superconducting flux qubit, comprising three Josephson junctions with tunable asymmetry, is magnetically coupled to the uniform precession (Kittel) magnon mode of a microscale YIG sphere. The qubit's persistent current states yield opposite near fields at the sphere position, resulting in a longitudinal Zeeman interaction with the magnon mode.

The system Hamiltonian in the persistent-current basis incorporates both qubit and magnon dissipation and reads:

$$H = \omega_m m^\dagger m - \frac{\epsilon_z}{2}\sigma_z - \frac{\epsilon_x}{2}\sigma_x + g (m + m^\dagger)\sigma_z,$$

with $\omega_m$ the Kittel mode frequency, $g$ the collective coupling determined by the geometry, and $\sigma_{z,x}$ the qubit Pauli operators. The drive Hamiltonian is

$$H_d = \Omega \cos(\omega_p t + \phi)(\sigma_x - \sigma_z)/\sqrt{2}.$$

Through transformation to the dressed-state basis and successive application of the rotating-wave and large-detuning approximations, an effective time-averaged Hamiltonian is derived that, in the strong-driving regime, reduces to:

$$H_\text{eff}' = \Delta_m m^\dagger m + g_{\rm cs}(m^2 + m^{\dagger 2})\bar{\sigma}_x,$$

where $g_{\rm cs} = -2g_x g_z/\omega_p$ is the qubit- and drive-dependent conditional squeezing rate. When the flux qubit is initialized in an eigenstate of $\bar{\sigma}_x$, the magnon mode undergoes time evolution under a degenerate parametric amplifier, yielding a qubit-state-conditioned squeezed state in the magnon quadratures.

Numerical Validation and Dissipation Analysis

Master equation simulations were performed—both with the full driven system and the approximate effective Hamiltonian—and incorporate all relevant dissipation channels: qubit energy relaxation, dephasing, magnon damping, and finite bath temperature.

Key findings concerning squeezing are as follows:

• Strong squeezing achievable: Quadrature noise reduction exceeding $8$ dB is attained with experimentally feasible parameters, even in the presence of dissipation rates and bath temperatures characteristic of current cQED–magnonics experiments.
• Magnon damping is the primary limitation: In the regime $\kappa < g_{\rm cs}$ and low thermal occupation, the maximum squeezing is near the unitary evolution prediction, while increased magnon linewidth ($\omega_m$0) or higher temperature degrade squeezing performance. Qubit relaxation rates currently achieved have negligible impact relative to magnon dissipation.
• Robust control: The protocol is insensitive to small variations in coupling or drive amplitude, with only minor quantitative changes in attainable squeezing and optimal evolution time.

Superposition of Magnonic Squeezed States and Quantum Codes

By preparing the flux qubit in a superposition of $\omega_m$1 eigenstates and executing a projective measurement in the $\omega_m$2 basis after the conditional squeezing protocol, the magnon is prepared in symmetric ($\omega_m$3 outcome) or antisymmetric ($\omega_m$4 outcome) superpositions of orthogonally squeezed vacuum states. The resulting Wigner functions display clear non-Gaussian features, with phase-space interference fringes and negativity arising from superposition.

Salient implications are:

• Non-Gaussian resource generation: The protocol deterministically prepares macroscopic superpositions of orthogonally squeezed states—an essential resource for continuous-variable quantum computing and bosonic quantum error correction.
• Logical encoding with rotational symmetry: The specific quadrature structure implies fourfold rotational symmetry in phase space, enabling encoding of logical qubits in a manner analogous to multiphoton or “cat” codes. Such symmetry allows for the detection of single-magnon loss and dephasing errors via the approximate Knill–Laflamme conditions.
• Fidelity preservation: Simulations demonstrate that the superposed magnonic squeezed states retain high state fidelity over relevant timescales, with mild fidelity decay in the presence of moderate decoherence.

Implications and Future Directions

The demonstrated protocol establishes the flux-qubit–YIG hybrid system as an effective platform for nonclassical magnon state engineering, particularly for continuous-variable quantum technologies where squeezed and non-Gaussian resource states are pivotal. The leveraging of intrinsic longitudinal qubit–magnon coupling simplifies the required hardware: the squeezing Hamiltonian can be realized solely with resonant driving, obviating the need for complex modulation schemes.

The findings suggest several impactful future research directions:

• Scalable, error-corrected magnonics: Harnessing the logical encoding for bosonic quantum error correction architectures, including active error tracking and stabilization of encoded information.
• Quantum transduction: As magnons readily couple to microwave photons, optical modes, and mechanical systems, engineered squeezed and superposed magnon states can serve as quantum memories or transducers in hybrid networks.
• Multimode and entangled magnon states: Generalization to multimode Kittel and higher-order magnonic modes, enabling distributed and nonlocal encoding and entanglement generation for quantum networks.

Conclusion

The paper presents a detailed theoretical and numerical analysis of magnonic squeezed state generation and its superpositions in a hybrid flux-qubit–YIG platform, relying on intrinsic longitudinal coupling and resonant driving. The protocol achieves strong, robust squeezing and prepares superpositions suitable for logical encoding with high fidelity under realistic dissipation. These results represent an important step toward practical use of magnonic degrees of freedom for fault-tolerant bosonic quantum information processing and pave the way for hybrid architectures exploiting the unique properties of magnonic excitations (2604.02924).