In this paper, it is shown that a necessary condition for unique identifiability of $K$ chirps from $N$ regularly spaced samples of their mixture is $N\geq 2K$ when $K\geq 2$. A necessary and sufficient condition is that a rank-constrained matrix optimization problem has a unique solution; this is the first result of such kind. An algorithm is proposed to solve the optimization problem and to identify the parameters numerically. The lower bound of $N=2K$ is shown to be tight by providing diverse problem instances for which the proposed algorithm succeeds to identify the parameters. The advantageous performance of the proposed algorithm is also demonstrated compared with the state of the art.