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In open quantum systems, we study the geometric phases acquired for atwo-level atom coupled to a bath of fluctuating vacuum massless scalar fieldsdue to linear acceleration and circular motion without and with a boundary. Infree space, as we amplify acceleration, the geometric phase acquired purely dueto linear acceleration case firstly is smaller than the circular accelerationcase in the ultrarelativistic limit for the initial atomic state$\theta\in(0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi)$, then equals to thecircular acceleration case in a certain acceleration, and finally, is largerthan the circular acceleration case. The spontaneous transition rates show asimilar feature. This result is different from the case of a bath offluctuating vacuum electromagnetic fields that has been studied. Consideringthe initial atomic state $\theta \in(0,\pi)$, we find that the geometric phaseacquired purely due to linear acceleration always equals to the circularacceleration case for the certain acceleration. The feature implies that, in acertain condition, one can simulate the case of the uniformly acceleratedtwo-level atom by studying the properties of the two-level atom in circularmotion. Adding a reflecting boundary, we observe that a larger value of ageometric phase can be obtained compared to the absence of a boundary. Besides,the geometric phase fluctuates along $z$, and the maximum of geometric phase iscloser to the boundary for a larger acceleration. We also find that geometricphases can be acquired purely due to the linear acceleration case and circularacceleration case with $\theta \in(0,\pi)$ for a smaller $z$.

Geometric phases acquired for a two-level atom coupled to fluctuating vacuum scalar fields due to linear acceleration and circular motion