  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.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons