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 ≥ 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 longrange order in the insulating magnet. KEYWORDS: ferromagnetic frustrated chain, nematic state, spin density waveOne of hot topics in condensed matter science is to search a spin liquid 1) that exhibits order not in conventional two spin correlation but in others such as magnetic multipole 2) or spin chirality. 3) 1D frustrated spin chain with ferromagnetic nearest neighbor interaction (J 1 ) and antiferromagnetic next nearest neighbor interaction (J 2 ) is a zoo of such novel states.While in case of the classical spin the spontaneous rotational symmetry breaking induces spiral long-range order (LRO) for |J 1 |/J 2 < 4, those of quantum spin does not exhibit the LRO because of Mermin-Wagner theorem. 4) Instead solely the spin chirality, κ = S i × S j does order with Z 2 symmetry broken and vector chiral (VC) phase appears. 5,6) Applying magnetic field makes the system more complex and interesting. At the field close to the ferromagnetic polarized phase, a pair of magnons form bound state 7) and its Bose condensation at around q = π induces quasi-LRO of the magnetic quadrupole. 8) The quadrupole phase is characterized by the following longitudinal spin and transverse nematic correlations, 9)The former exhibits magnitude modulation with the characteristic wave vector 2k F = 2πρ, where ρ is the density of bound two-magnon, ρ = 1/2(1/2− S z ). A z , A m , andà m are positive