We study field induced quantum phase in weakly-coupled ferromagnetic frustrated chain LiCuVO$_4$ by neutron diffraction technique. A new incommensurate magnetic peak is observed at $H ge 8.5$ T. The field dependent propagation vector is identified with the spin density wave correlation in the theoretically predicted magnetic quadrupole order. Quantum fluctuation, geometrical frustration, and interchain interaction induce the exotic spin density wave long-range order in the insulating magnet.
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