  The low-energy excitations of cuprate superconductors exhibit various
characteristics that differ from those of simple Bogoliubov quasiparticles for
pure d_{x^2-y^2}-wave superconductors. Here we report experimental studies of
spatially resolved quasiparticle tunnelling spectra of hole- and electron-type
cuprate superconductors that manifest direct evidences for the presence of
competing orders (COs) in the cuprates. In contrast to conventional type-II
superconductors that exhibit enhanced local density of states (LDOS) peaking at
zero energy near the centre of field-induced vortices, the vortex-state LDOS of
YBa_2Cu_3O_{7-\delta} (Y-123) and La_{0.1}Sr_{0.9}CuO_2 (La-112) remains
suppressed inside the vortex core, with pseudogap (PG)-like features at an
energy larger (smaller) than the superconducting (SC) gap \Delta_{SC} in Y-123
(La-112). Energy histograms of the SC and PG features reveal steady spectral
shifts from SC to PG with increasing magnetic field H. These findings may be
explained by coexisting COs and SC: For hole-type cuprates with PG above T_c,
the primary CO gap (V_{CO}) is larger than \Delta_{SC} and the corresponding
COs are charge/pair-density waves with wave-vectors parallel to
(\pi,0)/(0,\pi). For electron-type cuprates without PG above T_c, V_{CO} is
smaller than \Delta_{SC} and the CO wave-vector is along (\pi,\pi). This CO
scenario may be extended to the ARPES data to consistently account for the
presence (absence) of Fermi arcs in hole- (electron)-type cuprates. Fourier
transformation of the vortex-state LDOS in Y-123 further reveals multiple sets
of energy-independent wave-vectors due to field-enhanced pair- and spin-density
waves. These results imply important interplay of SC with low-energy collective
excitations.
