Emergent spacetimes from Hermitian and non-Hermitian quantum dynamics*

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 the critical coupling in single/two-mode squeezing, 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.