Taming Multi-Headed Schrödinger Cats with Nonlinear Reservoir Engineering

Quantum harmonic oscillators play an important role in quantum technologies, enabling precision sensing, quantum computation, and bosonic error-correcting codes that exploit the infinite-dimensional Hilbert space for robust encoding with low hardware overhead. Two major classes of bosonic codes are translation-symmetric GKP codes and rotation-symmetric cat codes; the latter are especially suited to protecting against phase and motional dephasing errors in oscillator and trapped-ion platforms. However, standard trapped-ion control methods operate within the linear Lamb–Dicke regime, limiting access to the non-Gaussian state manifolds required for bosonic codes.

We demonstrate the nonlinear preparation, stabilization, and measurement of multi-headed Schrödinger cat manifolds in the motional degree of a single Beryllium ion confined in a microfabricated Penning trap outside the Lamb–Dicke regime. In this strongly nonlinear atom–light interaction regime, higher-order motional sidebands, combined with dissipation, implement nonlinear reservoir engineering (NLRE), driving the oscillator toward stable multi-headed, rotation-symmetric cat states. By measuring characteristic functions, we reconstruct Fock distributions and Wigner functions that confirm the generation of these states. By tailoring chosen high-order Hamiltonians, we implement symmetry-selective measurements that enable purification of specific manifold components. To our knowledge, this is the first experimental demonstration of high-order nonlinear processes for direct preparation and control of non-Gaussian oscillator states in a trapped-ion system. The resulting rotation-symmetric cat manifolds form a resource for bosonic codes, offering a platform for logical qubits, quantum computation, and quantum sensing.