Impurity-Induced Environmental Quantum Phase Transitions in the Quadratic-Coupling Spin-Boson Model

We study the zero temperature properties of the sub-Ohmic spin-boson model with quadratic spin-boson coupling. This model describes experimental set ups at the optimal working point where the linear-coupling between the qubit (spin) and the environmental noise (bosons) is zero and the leading coupling is quadratic. In the strong coupling regime, we find that the existence of spin induces quantum phase transitions (QPTs) between two states of environment: the normal state and a state with local distortions. The phase diagram contains both continuous and the first-order QPTs, with non-trivial critical properties obtained exactly. At the QPTs, the equilibrium state spin dynamics bears power-law $\omega$ dependence in the small frequency limit and a robust coherent Rabi oscillation at high frequency. We discuss the feasibility of observing such environmental QPTs in the qubit-related experiments.