Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results

TL;DR

This paper evaluates the accuracy of the local density approximation for computing local exchange coefficients in strongly interacting one-dimensional multi-component systems, finding it effective for harmonic-like potentials but less so for complex barriers, impacting quantum state transfer fidelity.

Contribution

The study extends the assessment of the local density approximation to non-harmonic potentials relevant to experiments, providing insights into its limitations and effects on quantum information protocols.

Findings

LDA performs well for harmonic-like potentials.

LDA breaks down with larger barriers.

Small deviations in exchange coefficients significantly affect quantum state transfer fidelity.

Abstract

One-dimensional multi-component Fermi or Bose systems with strong zero-range interactions can be described in terms of local exchange coefficients and mapping the problem into a spin model is thus possible. For arbitrary external confining potentials the local exchanges are given by highly non-trivial geometric factors that depend solely on the geometry of the confinement through the single-particle eigenstates of the external potential. To obtain accurate effective Hamiltonians to describe such systems one needs to be able to compute these geometric factors with high precision which is difficult due to the computational complexity of the high-dimensional integrals involved. An approach using the local density approximation would therefore be a most welcome approximation due to its simplicity. Here we assess the accuracy of the local density approximation by going beyond the simple harmonic oscillator that has been the focus of previous studies and consider some double-wells of current experimental interest. We find that the local density approximation works quite well as long as the potentials resemble harmonic wells but break down for larger barriers. In order to explore the consequences of applying the local density approximation in a concrete setup we consider quantum state transfer in the effective spin models that one obtains. Here we find that even minute deviations in the local exchange coefficients between the exact and the local density approximation can induce large deviations in the fidelity of state transfer for four, five, and six particles.