Quasi-particle interference and vortex ``checkerboard'' in Bi$_2$Sr$_2$CaCu$_2$O$_y$

Relationship between the ``checkerboard'' electronic-state modulation in a vortex core~[1] and the quasi-particle interference effect has been studied using STM/STS in optimally- doped Bi$_2$Sr$_2$CaCu$_2$O$_y$. We found that the vortex- induced signals in Fourier-transform spectroscopic images appear in the close vicinity to some of the ``octet'' scattering vectors for the quasi-particle interference~[2], suggesting that the vortex ``checkerboard'' is associated with the Fermi momentum. Conductance spectrum taken at the center of the vortex core shows a sharp peak at a low energy ($\sim$ meV) in the empty state. We argue the possible relationship between these observations and the quantum-limit nature of the vortex core. [1] J. E. Hoffman {\it et al.}, Science {\bf 295}, 466 (2002). [2] K. McElroy {\it et al.}, Nature {\bf 422}, 592 (2003).