We report the results of muon-spin relaxation measurements on the low-dimensional antiferromagnet Rb4Cu(MoO4)3. No long-range magnetic order is observed down to 50 mK implying a ratio T_N/J<0.005 (where J is the principal exchange strength along the spin chains) and an effective ratio of interchain to intrachain exchange of |J_perp/J|<2 x 10^-3, making the material an excellent realization of a one-dimensional quantum Heisenberg antiferromagnet. We probe the persistent spin excitations at low temperatures and find that ballistic spin transport dominates the excitations detected below 0.3 K.
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