Abstract
We show that quantum dynamics of any systems with SU(1,1) symmetry give rise to emergent Anti-de Sitter spacetimes in 2+1 dimensions (AdS₂₊₁). Using the continuous circuit depth, a quantum evolution is mapped to a trajectory in AdS₂₊₁. Whereas the time measured in laboratories becomes either the proper time or the proper distance, quench dynamics follow geodesics of AdS₂₊₁. Such a geometric approach provides a unified interpretation of a wide range of prototypical phenomena that appear disconnected. For instance, the light cone of AdS₂₊₁ underlies expansions of unitary fermions released from harmonic traps, the onsite of parametric amplifications, and the exceptional points that represent the PT symmetry breaking in non-Hermitian systems. Our work provides a transparent means to optimize quantum controls by exploiting shortest paths in the emergent spacetimes. It also allows experimentalists to engineer emergent spacetimes and induce tunnelings between different AdS₂₊₁.