Coupling two order parameters in a quantum gas

Controlling matter to simultaneously support coupled properties is of fundamental and technological importance¹ (for example, in multiferroics^2-5 or high-temperature superconductors^6-9). However, determining the microscopic mechanisms responsible for the simultaneous presence of different orders is difficult, making it hard to predict material phenomenology^10,11 or modify properties^12-16. Here, using a quantum gas to engineer an adjustable interaction at the microscopic level, we demonstrate scenarios of competition, coexistence and mutual enhancement of two orders. For the enhancement scenario, the presence of one order lowers the critical point of the other. Our system is realized by a Bose-Einstein condensate that can undergo self-organization phase transitions in two optical resonators¹⁷, resulting in two distinct crystalline density orders. We characterize the coupling between these orders by measuring the composite order parameter and the elementary excitations and explain our results with a mean-field free-energy model derived from a microscopic Hamiltonian. Our system is ideally suited to explore quantum tricritical points¹⁸ and can be extended to study the interplay of spin and density orders¹⁹ as a function of temperature²⁰.