Understanding the interplay of particle conservation and long-range coherence

Lasers and Bose-Einstein condensates (BECs) exhibit macroscopic quantum coherence in seemingly unrelated ways. Lasers possess a well-defined global phase and are characterized by large fluctuations in the number of photons. In BECs of atoms, instead, the number of particles is conserved and the global phase is undefined. Here, we use gate-based quantum circuits to create a unified framework that connects lasers and BEC states. Our approach relies on a scalable circuit that measures the total number of particles without destroying long-range coherence. We introduce two complementary probes of global and relative phase coherence, and study how they are affected by measurements of the particle number. We find that particle conservation {it enhances} long-range phase coherence, highlighting a mechanism used by superfluids and superconductors to gain phase stiffness.