Derivation of the Hamiltonian of a flux qubit-LC oscillator circuit using the circuit variables

We derive the Hamiltonian of a single-Josephson-junction flux qubit-LC oscillator circuit using the standard quantization procedure [1]. The derived Hamiltonian consists of terms related to the LC oscillator, the flux qubit (and its higher energy levels), and the product of the two flux variables. This Hamiltonian is similar to that of the quantum Rabi model, where a two-level atom and a harmonic oscillator are coupled by a dipole-dipole interaction. Therefore, it is not surprising that the observed transmission spectra of a flux qubit-LC oscillator circuits can be well fitted by the quantum Rabi Hamiltonian [2]. We also find that the simple picture that models the coupling using a common inductor shared by the LC oscillator and the flux qubit needs only a minor modification. We thank Michel Devoret and Motoaki Bamba for stimulating discussions.
[1] U. Vool and M. H. Devoret, arXiv:1610.03438; IJCTA - Special Issue on Quantum Circuits.
[2] F. Yoshihara* and T. Fuse* et al., Nat. Phys. 13, 44 (2017). (*equal contribution)