  We study a weakly-interacting one-dimensional Bose gas with two impurities
coupled locally to the boson density. We derive analytical results for the
induced interaction between the impurities at arbitrary coupling and separation
$r$. At $r\lesssim \xi$, where $\xi$ denotes the healing length of the Bose
gas, the interaction is well described by the mean-field contribution. Its form
changes as the coupling is increased, approaching a linear function of $r$ at
short distances in the regime of strong coupling. The mean-field contribution
decays exponentially at arbitrary coupling for $r\gg\xi$. At such long
distances, however, the effect of quantum fluctuations becomes important,
giving rise to a long-ranged quantum contribution to the induced interaction.
At longest distances it behaves as $1/r^3$, while at strong coupling we find an
intermediate distance regime with a slower decay, $1/r$. The quantum
contribution in the crossover regime is also calculated. The induced
interaction between impurities (i.e., polarons) is attractive and leads to the
formation of their bound state, known as bipolaron. We discuss its binding
energy.
