Fast initialization of Bell states with Schrödinger cats in multi-mode systems

TL;DR

This work presents a robust and scalable protocol for fast initialization and entanglement of Schrödinger-cat qubits in multi-mode Kerr parametric oscillators. By combining a tunable cross-Kerr interaction with a two-mode drive, the authors show how to adiabatically connect Fock states to all four Bell-cat states and how to extend the approach to multi-mode entanglement, including the use of geometric (Berry) phases for state manipulation. The protocol preserves the protected cat-qubit subspace during initialization, exhibits strong robustness to finite-time ramps and switching imperfections, and is compatible with superconducting circuit architectures using SQUID-based couplers. The results provide a practical path toward scalable, bias-preserving quantum information processing with bosonic codes, with potential applicability to other platforms beyond superconducting circuits.

Abstract

Schrödinger cat states play an important role for applications in continuous variable quantum information technologies. As macroscopic superpositions they are inherently protected against certain types of noise making cat qubits a promising candidate for quantum computing. It has been shown recently that cat states occur naturally in driven Kerr parametric oscillators (KPOs) as degenerate ground states with even and odd parity that are adiabatically connected to the respective lowest two Fock states by switching off the drive. To perform operations with several cat qubits one crucial task is to create entanglement between them. Here, we demonstrate efficient transformations of multi-mode cat states through adiabatic and diabatic switching between Kerr-type Hamiltonians with degenerate ground state manifolds. These transformations can be used to directly initialize the cats as entangled Bell states in contrast to initializing them from entangled Fock states.

Figures (8)

• Figure 1: Degenerate eigenstates of the two-mode Kerr parametric oscillator depending on the drives and coupling between the modes. The undriven system (top row) is described by the Hamiltonian in Eq. \ref{['eq:H_full']} for $\epsilon_1=\epsilon_2=\epsilon_{12}=0$ and $K_{12}=0$. The driven system (last row) is uncoupled, $K_{12}=\epsilon_{12}=0$.
• Figure 2: Drives and cross-Kerr coupling as a function of time that achieve the adiabatic connection between Fock states $\ket{0,0}$ and $\ket{1,0}$ and corresponding Bell cat states. The two-mode drive $\epsilon_{12}$ must be tuned according to the relations in the corresponding gray boxes at all times.
• Figure 3: Difference between Berry phases accumulated by states $\ket{C^+_{\alpha}}$ and $\ket{C^-_{\alpha}}$ after two rotations of the drive amplitude $\epsilon$. The dashed lines indicate a phase difference of $\pi$.
• Figure 4: Layout of a possible physical implementation of the coupling via two transmons that are connected by a tunable coupler. The transmons as well as the coupler circuit are realized using SQUIDs.
• Figure 5: Simulation of the transformation to Bell state $\ket{\Phi^+_{\alpha_f,\alpha_f}}$ from Fock state $\ket{0,0}$. Plotted is the logarithmic infidelity of the final state as a function of displacements $|\alpha_1| = |\alpha_2| \equiv \alpha_f$ and of the scaled ramp-up time $\tau K_{12}$, where the ramping of the drive is $\epsilon_1(t)= \epsilon_2(t)\propto \tanh{(4 (2 t/\tau-1))}+1$.