Encoding multiple quantum coherences in non-commuting bases

Multiple quantum coherences are typically characterised by their coherence number and the number of spins that make up the state, though only the coherence number is normally measured. We present a simple set of measurements that extend our knowledge of the multiple quantum state by recording the coherences in both the $x$ basis and the usual $z$ basis. The coherences in the two bases are related by a similarity transformation. We characterize the growth of the multiple quantum coherences via measurements in the two bases, and show that the rate varies with the coefficient of the driving term in the Hamiltonian. Such measurements in non-commuting bases provides additional information over the 1D method about the state of the spin system. In particular the measurement of coherences in a basis other than the usual $z$ basis allows us to study the dynamics of the spin system under Hamiltonians, such as the secular dipolar Hamiltonian, that conserve $z$ basis coherence number.