Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low energy excitations within vortex cores of type-II superconductors. In the quantum limit, namely $T/T_mathrm{c} ll Delta/E_mathrm{F}$, the energy levels of these states were predicted to be discrete with the basic levels at $E_mu = pm mu Delta^2/E_mathrm{F}$ ($mu = 1/2$, $3/2$, $5/2$, ...). However, due to the small ratio of $Delta/E_mathrm{F}$ in most type-II superconductors, it is very difficult to observe the discrete CdGM states, but rather a symmetric peak appears at zero-bias at the vortex center. Here we report the clear observation of these discrete energy levels of CdGM states in FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus space clearly validates our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi energy in the present system is very small.
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