The kinematics of two-neutron emission following the $beta$-decay of $^{11}$Li was investigated for the first time by detecting the two neutrons in coincidence and by measuring their angle and energy. An array of liquid-scintillator neutron detectors was used to reject cosmic-ray and $gamma$-ray backgrounds by pulse-shape discrimination. Cross-talk events in which two detectors are fired by a single neutron were rejected using a filter tested on the $beta$-1n emitter $^9$Li. A large cross-talk rejection rate is obtained ($> 95 %$) over most of the energy range of interest. Application to $^{11}$Li data leads to a significant number of events interpreted as $beta$-2n decay. A discrete neutron line at $approx$ 2 MeV indicates sequential two-neutron emission, possibly from the unbound state at 10.6 MeV excitation energy in $^{11}$Be.
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