Motion of an impurity particle in an ultracold quasi-one-dimensional gas of hard-core bosons

The low-lying energy eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaronlike composite object consisting of the impurity clothed by a comoving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio m(i)/m unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact ground state (total linear momentum q=0) and exact boson-impurity distribution function in the thermodynamic limit for arbitrary m(i)/m and arbitrary impurity-boson interaction strength. These results are then extended to states with q > 0.