[Paper Review] Non-universal bound states of two identical heavy fermions and one light particle

Arghavan Safavi-Naini, Seth T. Rittenhouse+2 more|Queensland's institutional digital repository (The University of Queensland)|Jan 8, 2013

Cold Atom Physics and Bose-Einstein Condensates1 references3 citations

TL;DR

This paper investigates non-universal three-body bound states in a system of two identical heavy fermions and one light particle with short-range interactions. By introducing a short-range hyperradial boundary condition via a logarithmic derivative phase $\delta(R_0)$, the authors show that a non-universal state emerges and couples to universal trimers, altering their energies—particularly for $\kappa > \kappa_1 = 8.173$—and providing a quantum defect theory framework that accurately predicts these deviations from universality.

ABSTRACT

We study the behavior of the bound state energy of a system consisting of two identical heavy fermions of mass M and a light particle of mass m. The heavy fermions interact with the light particle through a short-range two-body potential with positive s-wave scattering length a_s. We impose a short-range boundary condition on the logarithmic derivative of the hyperradial wavefunction and show that, in the regime where Efimov states are absent, a non-universal three-body state "cuts through" the universal three-body states previously described by Kartavtsev and Malykh [O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007)]. The presence of the non-universal state alters the behavior of the universal states in certain regions of the parameter space. We show that the existence of the non-universal state is predicted accurately by a simple quantum defect theory model that utilizes hyperspherical coordinates. An empirical two-state model is employed to quantify the coupling of the non-universal state to the universal states.

Motivation & Objective

• To investigate deviations from universal three-body behavior in a system of two identical heavy fermions and one light particle with positive s-wave scattering length.
• To explore the role of short-range three-body boundary conditions, parameterized by hyperradial phase $\delta(R_0)$, in modifying the energy spectrum.
• To determine whether non-universal three-body states exist outside the universal regime predicted by Kartavtsev and Malykh, especially for $\kappa < \kappa_1$.
• To develop a quantum defect theory (QDT) model that accurately predicts the onset of non-universal states and their coupling to universal trimers.
• To quantify the coupling strength between non-universal and universal states using a two-state model and relate it to experimental parameters like $R_0$ and $\kappa$.

Proposed method

• Formulates the three-body Hamiltonian using zero-range two-body pseudopotentials with s-wave scattering length $a_s$, employing hyperspherical coordinates to separate the hyperradial and hyperangular degrees of freedom.
• Imposes a short-range boundary condition on the hyperradial wavefunction via a logarithmic derivative phase $\delta(R_0)$ at hyperradius $R_0$, allowing exploration of the full range of physically allowed boundary conditions.
• Solves the hyperradial Schrödinger equation numerically for varying $\delta(R_0)$, $R_0$, and mass ratio $\kappa$ to map the three-body energy spectrum.
• Applies quantum defect theory (QDT) to analytically describe the non-universal state and its energy dependence on $\delta(R_0)$ and $R_0$, showing excellent agreement with numerical results.
• Constructs a two-state model coupling the non-universal state to universal trimers, with coupling strength $\beta$ quantified relative to the two-body energy $E_{\text{2b}}$.
• Compares results with prior work (e.g., Endo et al.) using momentum cutoffs, suggesting a correspondence between $\delta(R_0)$ and $\Lambda_c$ in momentum-space formulations.

Experimental results

Research questions

• RQ1 How do short-range three-body boundary conditions, parameterized by $\delta(R_0)$, affect the energy spectrum of a two-heavy-fermion, one-light-particle system?
• RQ2 Does a non-universal three-body bound state exist outside the universal regime predicted by Kartavtsev and Malykh, particularly for $\kappa < \kappa_1 = 8.173$?
• RQ3 Can quantum defect theory accurately predict the onset and energy of the non-universal state as a function of $\delta(R_0)$ and $R_0$?
• RQ4 How does the coupling strength $\beta$ between the non-universal state and universal trimers depend on $R_0$ and $\kappa$?
• RQ5 What is the physical interpretation of the transition from universal to Efimov-like trimers in terms of boundary condition variation?

Key findings

• A non-universal three-body bound state exists and 'cuts through' the universal trimers predicted by Kartavtsev and Malykh, particularly for $\kappa > \kappa_1 = 8.173$, even when the universal states are not supported.
• The non-universal state emerges at a critical phase $\delta_c(R_0)$, where it becomes bound, and its energy deviates significantly from the universal spectrum near this point.
• Quantum defect theory (QDT) accurately predicts the critical phase $\delta_c(R_0)$ for the onset of the non-universal state across different $R_0$ and $\kappa$ values.
• The coupling strength $\beta/E_{\text{2b}}$ increases with both $\kappa$ and $R_0$, indicating stronger mixing between the non-universal and universal states at larger mass ratios and longer length scales.
• The non-universal state persists even for $\kappa < \kappa_1 = 8.173$, where universal trimers do not exist, demonstrating that non-universality is not confined to the universal regime.
• The deviations from universality observed here are interpreted as a smooth interpolation between the Kartavtsev-Malykh universal trimers and Efimov trimers for $\kappa \gtrsim 13.606$, with the boundary condition $\delta(R_0)$ acting as a continuous tuning parameter.