Fractionally Quantized Cooper Pair Stair Case in Superconducting Quantum Dots

We study clean superconducting quantum dots (SQD) and also site dependent Josephson couplings, on site charging energies and the intersite interactions in presence of gate voltage. We predict the existence of different fractionally quantized Cooper pair stair case with many interesting physical properties. The appearance of stair case is not only due to the Coulomb blocked phenomena but also for the site dependent Josephson couplings. We also explain physically the absence of other fractionally quantized Cooper pair stair case. The physics of fractionally quantized Cooper pair stair case has close resemblance with the fractionally quantized magnetization plateau of a spin chain system under a magnetic field.