  The transverse field Ising chain (TFIC) model is ideally suited for testing
the fundamental ideas of quantum phase transitions, because its well-known
$T=0$ ground state can be extrapolated to finite temperatures. Nonetheless, the
lack of appropriate model materials hindered the past effort to test the
theoretical predictions. Here we map the evolution of quantum fluctuations in
the TFIC based on Nuclear Magnetic Resonance (NMR) measurements of
CoNb$_2$O$_6$, and demonstrate the finite temperature effects on quantum
criticality for the first time. From the temperature dependence of the
$^{93}$Nb longitudinal relaxation rate $1/T_1$, we identify the renormalized
classical, quantum critical, and quantum disordered scaling regimes in the
temperature ($T$) vs. transverse magnetic field ($h_{\perp}$) phase diagram.
Precisely at the critical field $h_{\perp}^{c}=5.25 \pm 0.15$ T, we observe a
power-law behavior, $1/T_{1} \sim T^{-3/4}$, as predicted by quantum critical
scaling. Our parameter-free comparison between the data and theory reveals that
quantum fluctuations persist up to as high as $T \sim 0.4 J$, where the
intra-chain exchange interaction $J$ is the only energy scale of the problem.
