In Chap. 17 , we derived the Hamiltonian of a Cooper pair box coupled to an external voltage and found that it depended on the offset charge capacitively induced by the external voltage. In this chapter, we will diagonalize the Hamiltonian analytically to understand the nature of its eigenstates. Since obtaining an analytical solution is difficult, we will only consider a two-level system which in principle does not give the correct solution. However, it will give us a lot of insight and it is instructive. We will discuss how to reduce its charge sensitivity by operating at the sweet spot. To further reduce the charge sensitivity, we discuss the creation of a transmon qubit by increasing the capacitance of the shunt capacitor.