Detecting a topological transition of quantum braiding in a threefold-degenerate eigensubspace

The braiding operations of quantum states have attracted substantial attention due to their great potential for realizing topological quantum computations. In this paper, we show that a threefold-degenerate eigensubspace can be obtained in a four-level Hamiltonian which is the minimal physical system. Braiding operations are proposed to apply to dressed states in the subspace. The topology of the braiding diagram can be characterized through physical methods once the sequential braiding pulses are adopted. We establish an equivalent relationship function between the permutation group and the output states where different output states correspond to different values of the function. The topological transition of the braiding happens when two operations overlap, which is detectable through the measurement of the function. Combined with the phase variation method, we can analyze the wringing pattern of the braiding. Therefore, the experimentally feasible system provides a platform to investigate braiding dynamics, the SU(3) physics, and the qutrit gates.