Uniqueness of Squeezed States for One and Two Modes, and a No-Go Beyond

Abstract: We investigate the structure and uniqueness of squeezed vacuum states defined by annihilation conditions of the form $(a - \alpha a^\dagger)|\psi\rangle = 0$ and their multimode generalizations. For $N=1$ and $N=2$, we rigorously show that these conditions uniquely define the standard single- and two-mode squeezed states in the Fock basis. We then analyze a cyclically coupled $N$-mode system governed by $(a_i - \alpha_i a_{i+1}^\dagger)|\psi\rangle = 0$ with $a_{N+1} \equiv a_1$. Although the recurrence structure restricts solutions to equal-photon-number states, we prove that for $N>2$ no such state satisfies the full set of conditions. This establishes a sharp no-go result for multipartite squeezed states under cyclic annihilation constraints, underscoring a fundamental structural limitation beyond pairwise squeezing.