Geometric Phase of the Two-Particle Bethe Wavefunction

TL;DR

The paper analyzes how repulsive interactions affect geometric phase generation for two bosons on a ring with a localized defect within the Lieb-Liniger framework. It formulates the problem via a Bethe ansatz with four momentum configurations due to reflection, deriving spectral relations whose solutions lie on intersecting curves in the $(k_1,k_2)$ plane, yielding energies $E^{(ij)} = (k_1^{(i,j)2} + k_2^{(i,j)2})/2$. The main finding is that the geometric phase $\theta_g$ increases with the interaction strength $c$ for a fixed defect variation, with stronger defect reflection (larger $\eta$) amplifying the effect; in the zero-interaction limit the spectrum is solvable analytically and the phase is dominated by the global phase, approaching $-\arccos\left(1/\cosh\eta\right) + \pi/2$ as $\eta$ grows. The work points to potential metrological applications in BEC-based inertial sensors and outlines extensions to many-particle and non-inertial settings.

Abstract

We consider a problem of geometric phase generation in a system of two interacting bosons confined in a narrow ring potential with a localized defect. Geometric phase emerges from variation of parameters of the defect. Particle interaction is taken into account within a framework of the Lieb-Liniger model. The energy spectrum is evaluated and its dependence on the parameters of the problem is described. It is shown that the interaction leads to increase of the geometric phase for a given contour of variations. The work is motivated by earlier proposed ideas of quantum gyroscope and quantum accelerometer based on atomic Bose-Einstein condensates.

Figures (3)

• Figure 1: The schematics of the ring-shaped potential with a defect. Transmitted and reflected waves are indicated.
• Figure 2: Two sets of curves yielding the solutions of spectral relations for $\eta=2,\alpha=0,c=0.1,L=5$. The roots of the spectral solutions are marked by circles. Each root appears as an intersection vertical and horizontal curve numerated by integers. The dashed line connects two roots with numbers $(1,3)$ and $(3,5)$ transforming into each other upon variation of $\alpha$ by $2\pi$ (see the text).
• Figure 3: Geometric phase for $(1,3)$ steady state as a function of interaction parameter $c$ for different value of $\eta$. For higher values of $\eta$ and $c$ the geometric phase approaches its maximal asymptotical value of $\pi$. For small values of $\eta$ and $c$ its asymptotic is $\pi/2$.