Electric Aharonov-Bohm effect without a loop in a Cooper pair box

We predict the force-free scalar Aharonov-Bohm effect of a Cooper pair box in an electric field at a distance without forming a closed path of the interfering charges. The superposition of different charge states plays a major role in eliminating the closed loop, which is distinct from the original topological Aharonov-Bohm effect. The phase shift is determined by the charge-state-dependent local field interaction energy. In addition, our proposed setup does not require a pulse experiment for fast switching of a potential, which eliminates the major experimental obstacle for observing the ideal electric Aharonov-Bohm effect.