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\[ \begin{equation} \large e^{i\omega t} = \cos\omega t + i \sin\omega t \end{equation} \]

The red ball represents the position of the complex number $e^{i\omega t}$ as it moves through the complex plane with real numbers plotted horizontally and imaginary number plotted vertically. The black circle is the unit circle. The blue ball represents the position of $\cos \omega t$ and the green ball represents the position of $i\sin \omega t$. The simulation on the left is a graphical representation of the formula on the right.

Oscillations that can be described by $\sin \omega t$ or $\cos \omega t$ are called harmonic oscillations. From the simulation it is clear that there is a relationship between circular motion and harmonic oscillations. If you look at the motion of the red ball from above, it moves in a circle but if you look at the motion of the red ball from the side, it executes harmonic motion.

The relationship between circular motion and harmonic oscillations is described easily using complex numbers. Sometimes when we observe a harmonic oscillation it is convenient to imagine that we are looking at circular motion from the side. We can't measure the component of the motion in the imaginary direction, we just imagine it.

For phonons the solutions are typically written as a complex exponential,

\[ \begin{equation} \vec{u}_{lmn}=\vec{A}_k \exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right). \end{equation} \]

Here $lmn$ are integers that label the unit cell. For $p$ atoms in the basis, the displacement vector $\vec{u}_{lmn}$ has $3p$ components corresponding to the $x-$, $y-$, and $z-$displacements of all the atoms in the basis. Consider one of these components. It has a time dependence described by,

\[ \begin{equation} \exp\left(i\left(l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3-\omega t\right)\right) = e^{il\vec{k}\cdot\vec{a}_1}e^{im\vec{k}\cdot\vec{a}_2}e^{in\vec{k}\cdot\vec{a}_3}e^{-i\omega t}. \end{equation} \]

This is a phasor rotating in the complex plane. All of the atoms oscillate with the same frequency but the oscillations of the different atoms are shifted in phase with respect to each other. The motion that we will observe is the real part of the phasor. In this case the real part is,

\[ \begin{equation} \cos (-\omega t + \delta_{lmn}), \end{equation} \]

where

\[ \begin{equation} \delta_{lmn} = l\vec{k}\cdot\vec{a}_1+m\vec{k}\cdot\vec{a}_2+n\vec{k}\cdot\vec{a}_3, \end{equation} \]

is the phase at atom $lmn$.