The momentum distribution of two bosons in one dimension with infinite contact repulsion in harmonic trap gets analytical

Abstract

For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix rho _mathrm{B} in terms of center of mass and relative coordinates of the particles. The deviation from rho _mathrm{F}, the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained rho _mathrm{B} allows us to derive a closed form expression of the corresponding momentum distribution n_mathrm{B}(p). We show how the result deviates from the noninteracting fermionic case, the deviation being associated with the short-range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum 1/p^4 tail with the appropriate Tan’s coefficient. Numerical results support our findings.