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

TL;DR

This paper derives an analytical expression for the momentum distribution of two strongly interacting bosons in a one-dimensional harmonic trap, revealing deviations from fermionic behavior and confirming expected physical properties.

Contribution

The authors provide the first closed-form analytical expression for the momentum distribution of two contact-interacting bosons in a harmonic trap, including hypergeometric functions and high-momentum tail behavior.

Findings

Derived an explicit formula for the momentum distribution $n_B(p)$

Identified deviations from fermionic density matrix due to interactions

Confirmed the $1/p^4$ tail consistent with Tan's relations

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 $\rho_B$ in terms of centre of mass and relative coordinates of the particles. The deviation from $\rho_F$, the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained $\rho_B$ allows us to derive a closed form expression of the corresponding momentum distribution $n_{B}(p)$. We show how the result deviates from the noninteracting fermionic case, the deviation being associated to 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.