Hamiltonian of a flux qubit-LC oscillator circuit in the deep-strong-coupling regime*

We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian [1]. The energy level structure of the circuit Hamiltonian can be fitted well by the quantum Rabi Hamiltonian even when the qubit-oscillator circuit is in the deep-strong-coupling regime [2]. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the capacitive-coupling variant of the quantum Rabi Hamiltonian even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.

We are grateful to M. Devoret for valuable discussions.

[1] F. Yoshihara et al., arXiv: 2008.07708.

[2] F. Yoshihara* and T. Fuse* et al., Nat. Phys. 13, 44 (2017) *equal contribution.