${|{b}\rangle}$ are linearly independent but not orthogonal to each other, then $\hat A$ and $\hat B$ act on different but not independent oscillators. We can define an orthogonal basis via Gram-Schmidt orthogonalization and we can thus define $\hat X_A = \hat X_1$ and $$\begin{gathered}
  \label{eq:Xb-general}
  \hat P_B =
  \big( \hat X_1 \cos\varphi + \hat P_1 \sin\varphi \big)
  \cos\vartheta + \hat P_2 \sin\vartheta,\end{gathered}$$ where $\varphi = \arg {\langle{a}|{b}\rangle}$ as previously and $\cos\vartheta = |{\langle{a}|{b}\rangle}| / ( \|a\| \|b\|)$ describes how parallel the two vectors are. The quadrature operators in this case have a non-canonical commutation relation, $[\hat X_A, \hat P_B] /i < 1$. See Appendix \[app:two-oscillators\] for a derivation of the formulae.

Motion in rotating frame {#sec:rot-frame}
------------------------

In a typical experiment, the atomic spins are placed in a homogeneous magnetic field where they precess coherently and independently of each other. The reference state then also precesses, and we will describe the collective excitations as deviations from this time dependent state.

In general, we can split the Hamiltonian into two parts, $\hat H = 
\hat H_0 + \hat H_1$, one that acts coherently on each particle and another that describes, e.g., interaction with an external quantum field. Using the interaction free 