  Interferometry is a paradigm for most precision measurements. Using $N$
uncorrelated particles, the achievable precision for a two-mode (two-path)
interferometer is bounded by the standard quantum limit (SQL), $1/\sqrt{N}$,
due to the discrete (quanta) nature of individual measurements. Despite being a
challenging benchmark, the two-mode SQL has been approached in a number of
systems, including the LIGO and today's best atomic clocks. Employing
multi-mode interferometry, the SQL becomes $1/[(M-1)\sqrt{N}]$ using M modes.
Higher precision can also be achieved using entangled particles such that
quantum noises from individual particles cancel out. In this work, we
demonstrate an interferometric precision of $2.42^{+1.76}_{-1.29}\,$dB beyond
the three-mode SQL, using balanced spin-1 (three-mode) Dicke states containing
thousands of entangled atoms. The input quantum states are deterministically
generated by controlled quantum phase transition and exhibit close to ideal
quality. Our work shines light on the pursuit of quantum metrology beyond SQL.
