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

Fractionally Quantized Cooper Pair Stair Case in Superconducting Quantum Dots