The local quasi-particle density of states, N s ðE; rÞ, in the isolated vortex of YNi 2 B 2 C has been measured at 0.46 K by low-temperature scanning tunneling microscopy and spectroscopy. The vortex core is found to be fourfold-symmetric star-shaped in real space: Near E ¼ 0 meV, N s ðE; rÞ extends toward h100i (the a-axis) and has a peak at the center of the vortex core. With increasing energy, N s ðE; rÞ comes to extend toward h110i and the peak of N s ðE; rÞ splits into four peaks toward h110i. N s ðE; rÞ in the vortex core exhibits electron-hole asymmetry with the intensity, indicating the first observation of the vortex core in a quantum regime. Low-temperature scanning tunneling microscopy and spectroscopy (LT-STM/STS) is a powerful method for investigating the vortex state in type II superconductors; 1) it is able to measure the local density of states of quasiparticles, N s ðE; rÞ, with atomic spatial resolution and to image the vortex lattice in real space. The quasi-particles in a superconductor are represented by the Bogoliubovde Gennes equation as particles moving in the pair potential ÁðrÞ. Superconductivity is degraded near the vortex core, where the quasi-particles are confined and the bound states are formed. The anisotropy of ÁðrÞ is expected to be determined by imaging the vortex core by STM/STS.We have investigated the vortex core in the borocarbide YNi 2 B 2 C. An anisotropy of a superconducting energy gap has been proposed from measurements of the specific heat, 2-5) thermal conductivity, 6,7) photoemission spectroscopy 8) and ultrasonic attenuation. 9) Our recent LT-STS measurements in zero magnetic field imply a large anisotropy of the gap (Á max % 4.0 meV, Á min 0.5 meV). 10) In an anisotropic superconductor, the anisotropic vortex core reflecting the gap symmetry was predicted by Ichioka et al.11) However, the vortex core observed with STS in YNi 2 B 2 C 12) and the related system LuNi 2 B 2 C 13) at 4.2 K did not exhibit anisotropy. The purpose of this paper is to clarify the detailed structure of the vortex core in YNi 2 B 2 C through measurements of N s ðE; rÞ using LT-STM/STS at lower temperatures. In the present experiment, we have found the fourfold-symmetric vortex core and the bound states in the vortex core in YNi 2 B 2 C by STS for the first time.Single crystals of YNi 2 B 2 C were grown by the floating zone method. The samples for our present STS experiments were obtained from the same batch and the typical shape was 1:0 mm  1:0 mm  3:0 mm 3 . The critical temperature T c was determined to be 15.6 K from dc magnetization measurement. The residual resistivity ratio RRR was 28.8, and the resistivity at T c was 1.2 m cm, while in our previous experiment, 12) the RRR of the sample was about 6. The mean free path ' of the present sample was calculated to be 70 nm from the RRR using the electronic density of states at the Fermi energy.14) The upper critical field H c2 is about 8 T. From the formula H c2 ¼ 0 =2 2 , the coherence length was calculated to be 6.3 nm. Since ' is ten t...