  We demonstrate experimentally the existence of magic wavelengths and
determine the ratio of the oscillator strengths for a single trapped ion. For
the first time, two magic wavelengths for the 40Ca+ clock-transition are
measured simultaneously with high precision, giving Lamda|mj|=1/2 = 395.7992(7)
nm and Lamda|mj|=3/2 = 395.7990(7) nm. By tuning a laser to an intermediate
wavelength between two transitions (4s1/2-4p1/2 and 4s1/2-4p3/2) of 40Ca+, the
sensitivity of the clock transition Stark shift to the oscillator strengths for
the resonance transition has been greatly enhanced. With the measured magic
wavelengths, we determine the ratio of the oscillator strengths to sub-0.5%
accuracy. Our experimental method may be applied to measure magic wavelengths
for other ion clock-transitions, and, promisingly, the measurement of these
magic wavelengths paves the way to building lattice ion clocks.
