  We succeeded in measuring phase shift spectra of a microsphere cavity coupled
with a tapered fiber using a weak coherent probe light at the single photon
level. We utilized a tapered fiber with almost no depolarization and
constructed a very stable phase shift measurement scheme based on polarization
analysis using photon counting. Using a very weak probe light (\bar{n} = 0:41),
we succeeded in observing the transition in the phase shift spectrum between
undercoupling and overcoupling (at gap distances of 500 and 100 nm,
respectively).We also used quantum state tomography to obtain a 'purity
spectrum'. Even in the overcoupling regime, the average purity was 0.982 \pm
0.024 (minimum purity: 0.892), suggesting that the coherence of the
fiber-microsphere system was well preserved. Based on these results, we believe
this system is applicable to quantum phase gates using single light emitters
such as diamond nitrogen vacancy centers.
