Many-body bosonic Hamiltonians are a cornerstone of condensed-matter physics. Quantum simulators, that is, quantum-controlled atomic, optical, or solid-state experimental platforms, holdthe promise to explore such models with a degree of precision and flexibility out of reach for real materials. Common Hamiltonians emerging in condensed-matter preserve the number of particles. In contrast, quantum simulators are intrinsically driven-dissipative systems, where processes that break particle-number conservation can be naturally induced either coherently or incoherently. Here we show that such processes, and down-conversion or pair injection in particular, make the phase diagram of paradigmatic examples such as the extended Bose-Hubbard model much richer.

As side-product, we provide an example of a single-mode Hamiltonian that presents a continuous quantum phase transition, a result that challenges the common intuition that only infinite-size systems present such behavior. We prove that the transition satisfies all the standard requirements, specifically the presence of a sensitive thermodynamic limit and well-defined scaling laws, identifying all critical exponents.

We discuss potential implementations of our ideas in modern quantum-controlled platforms such as trapped atoms and ions, nonlinear optics, superconducting circuits, and mechanical devices.