  Whether a quantum bath can be approximated as classical noise is a
fundamental issue in central spin decoherence and also of practical importance
in designing noise-resilient quantum control. Spin qubits based on bismuth
donors in silicon have tunable interactions with nuclear spin baths and are
first-order insensitive to magnetic noise at so-called clock-transitions (CTs).
This system is therefore ideal for studying the quantum/classical nature of
nuclear spin baths since the qubit-bath interaction strength determines the
back-action on the baths and hence the adequacy of a classical noise model. We
develop a Gaussian noise model with noise correlations determined by quantum
calculations and compare the classical noise approximation to the full quantum
bath theory. We experimentally test our model through dynamical decoupling
sequence of up to 128 pulses, finding good agreement with simulations and
measuring electron spin coherence times approaching one second - notably using
natural silicon. Our theoretical and experimental study demonstrates that the
noise from a nuclear spin bath is analogous to classical Gaussian noise if the
back-action of the qubit on the bath is small compared to the internal bath
dynamics, as is the case close to CTs. However, far from the CTs, the
back-action of the central spin on the bath is such that the quantum model is
required to accurately model spin decoherence.
