While γ BO (Γ) = 0 or π is a topological phase, and is thus an intrinsic property of the Hamiltonian of the system, γ(Γ,t) depends on the dynamics that the system undergoes and is a geometric quantity. In addition, since the TDVP is related to the nuclear momentum field of the molecular system as shown in Eq. ( ), its circulation is gauge-independent and representation-independent. Therefore, Eq. ( ) is a robust physical observable. Note that, in addition, the curl of the TDVP is a physical observable as well, since it is the curl of the nuclear momentum field. Preliminary studies on geometric phases in the exact factorization were conducted by Gross and coworkers employing its static formulation and drew different conclusions depending on the systems that were considered . In Ref. [ 124], for instance, it was shown that, in the studied model for proton coupled-electron transfer, the concept of topological phase was merely a consequence of the BO approximation. Thus, it completely disappeared in the framework of the exact factorization, only to reappear as a topological quantity in the limit of infinite nuclear mass, i.e., the BO limit. Instead, in Ref. [125], a model of a pseudorotating triatomic molecule 236 was used to show that the topological phase reverts to a geometric quantity if the system manifests a degenerate ground state, even within the exact factorization. These studies suggest, then, that the topological phase of the Born-Huang representation becomes geometric in particular cases related to degeneracies of the full Hamiltonian, while neither topological nor geometric phases appear otherwise. Later on, to shed some light on the time-dependent picture of geometric phases in the exact factorization and to relate theoretical observations to experiments, our group investigated the dynamics generated by the E ⊗ e Jahn-Teller Hamiltonian recently engineered in a trapped-ion quantum simulator 220 . The experiment allowed to directly image the evolution of the nuclear density at the conical intersection in order to detect directly the signature of the conical intersection. The working hypothesis was that, since the real-valued electronic wavefunction changes sign when transported around the conical intersection, in order to yield a single-valued molecular wavefunction, the nuclear wavefunction in Eq. ( ) needs to change sign as well. Therefore, the nuclear wavefunction has to develop a node or a zero-density line, which are indications of destructive interferences caused by the geometric phase. However, this hypothesis relies on the BO approximation and, as shown in Ref.