量子漫步作為一種簡單而有趣的量子現象，在許多領域都有廣泛的應用。這裡我們考慮一個修改位移算符的一維量子漫步系統。常見量子漫步的實驗中量子位元（qubit）的實踐採用的是光子，而量子漫步的位移算符於光子系統中是利用分光鏡（beam splitter）實踐，其特色為只有兩個輸出端，因此在不同量子漫步系統中位移算符的數學描述多半採用相對簡單的矩陣形式。

然而在利用原子作為量子位元的時候，對於原子的動量變化會有更多可能的操作。在此我們考慮在諸多操作中相對簡單的形式，即量子位元動量的變化不再是兩個值，而是三個可能的輸出量。

我們分析了這種條件下量子漫步的行為，展示了解析解與數值解，通過改變不同行為間的相對相位，可以顯示更豐富的特徵。我們也可以擴展上述的情形，藉由引進更多種類的位移算符對應至不同情況的量子漫步，並探討對應的特色。

Quantum walks is a simple and interesting quantum phenomenon, and quantum walks have a wide range of applications in many fields. Here, we consider a one-dimensional quantum walk system with a modified displacement operator

In common quantum walk experiments, quantum bits are practiced with photons. The displacement operator of a quantum walks when being practiced with a beam splitter in a photonic system is characterized by the two outputs of the beam splitter, so the mathematical descriptions of the displacement operators are straightforward. However, when using atoms as quantum bits, there are more probable operations on the momentum space of atoms. Here we consider a relatively simple form of one of the many operations, i.e., the momentum changes of a quantum bit no longer has two values, but three possible outcomes.

We analyze the behavior of quantum walks under such conditions, showing analytical solutions that can demonstrate rich features by varying the relative phases and transition probabilities between different states. We can also extend the above scheme by introducing more complicated displacement operators to have different models and explore the corresponding physics.

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