  We show that entanglement between two solitary qubits in
quasi-one-dimensional Bose-Einstein condensates can be spontaneously generated
due to quantum fluctuations. Recently we have shown that dark solitons are an
appealing platform for qubits thanks to their appreciable long lifetime. We
investigate the spontaneous generation of entanglement between dark-soliton
qubits in the dissipative process of spontaneous emission. By driving the
qubits with the help of oscillating magnetic field gradients, we observe the
formation of long distance steady-state concurrence. Our results suggest that
dark-soliton qubits are good candidates for quantum information protocols based
purely on matter-wave phononics.
