Measuring entanglement entropy through the interference of quantum many-body twins

Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is rapidly gaining prominence in diverse fields ranging from condensed matter to quantum gravity. Despite this generality, measuring entanglement remains challenging. This is especially true in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Leveraging our single-site resolved control of ultra-cold bosonic atoms in optical lattices, we prepare and interfere two identical copies of a many-body state. This enables us to directly measure quantum purity, Renyi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly-correlated many-body systems.Comment: 14 pages, 12 figures (6 in the main text, 6 in supplementary material

Observing the emergence of a quantum phase transition -- shell by shell

Many-body physics describes phenomena which cannot be understood looking at a systems' constituents alone. Striking manifestations are broken symmetry, phase transitions, and collective excitations. Understanding how such collective behaviour emerges when assembling a system from individual particles has been a vision in atomic, nuclear, and solid-state physics for decades. Here, we observe the few-body precursor of a quantum phase transition from a normal to a superfluid phase. The transition is signalled by the softening of the mode associated with amplitude vibrations of the order parameter, commonly referred to as a Higgs mode. We achieve exquisite control over ultracold fermions confined to two-dimensional harmonic potentials and prepare closed-shell configurations of 2, 6 and 12 fermionic atoms in the ground state with high fidelity. Spectroscopy is then performed on our mesoscopic system while tuning the pair energy from zero to being larger than the shell spacing. Using full atom counting statistics, we find the lowest resonance to consist of coherently excited pairs only. The distinct non-monotonic interaction dependence of this many-body excitation as well as comparison with numerical calculations allows us to identify it as the precursor of the Higgs mode. Our atomic simulator opens new pathways to systematically unravel the emergence of collective phenomena and the thermodynamic limit particle by particle.Comment: 14 pages, 10 figure