Discrete-Time Quantum Walks of the One-Dimensional Dirac Oscillator

Recent experimental studies have demonstrated the simulation of Dirac equation using the discrete-time quantum walks. In this work, we add the oscillator potential and consider the simulation of a one-dimensional Dirac oscillator. Based on the coin operator, we run the quantum walk simulation and consider the effects of oscillator parameters, initial phase shifts, and masses. Compared to the Dirac equation, the probability distribution of a Dirac oscillator becomes more classical. Additionally, we find the probability distribution jumps between two spin states by changing the initial phase shift, and attribute this to the coherent nature of the two spin states. These results are hopefully to be realized in trapped-ion experiments.