Passive error correction with a qubit-oscillator system in noisy environment

In this paper, we study an open quantum system consisting of a qubit coupled to a harmonic oscillator subject to two-photon relaxation and demonstrate that such a system can be utilized to construct a cat qubit capable of passive error correction. To this end, we first show that the steady state of the qubit-oscillator system, described by the open quantum Rabi model with two-photon relaxation, undergoes a superradiant phase transition that breaks the strong symmetry of the Lindblad master equation. In the strong symmetry-broken phase, we show that a cat qubit can be stabilized in the steady state by tuning the qubit-oscillator coupling strength and demonstrate that passive error correction can be realized against errors due to fluctuations in the system frequencies. Our study deepens the understanding of dissipative phases in a qubit-oscillator system with strong symmetry and paves the way to utilize them for passive error correction.