[Paper Review] Effective low-dimensional Hamiltonian for strongly interacting atoms in a transverse trap

J. P. Kestner, Luming Duan|arXiv (Cornell University)|Jul 30, 2007

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper derives an effective low-dimensional Hamiltonian for strongly interacting ultracold atoms in a transverse trap near a Feshbach resonance, accounting for transverse excitations via dressed molecules. It renormalizes interactions between atoms and dimers, enabling accurate modeling of low-D strongly correlated physics with all parameters fixed for both 1D and 2D systems at any magnetic detuning.

ABSTRACT

We derive an effective low-dimensional Hamiltonian for strongly interacting ultracold atoms in a transverse trapping potential near a wide Feshbach resonance. The Hamiltonian includes crucial information about transverse excitations in an effective model with renormalized interaction between atoms and composite dressed molecules. We fix all the parameters in the Hamiltonian for both one- and two-dimensional cases.

Motivation & Objective

• To develop an effective low-dimensional Hamiltonian that accurately captures strongly correlated physics in ultracold atoms under strong transverse confinement near a Feshbach resonance.
• To account for significant population in transverse excited modes, which invalidates simple projection to the ground state in strongly interacting regimes.
• To introduce and rigorously define the concept of dressed molecules—dimers excluding atomic population in the lowest transverse mode—to decouple transverse excitation effects.
• To derive renormalized interactions between atoms and dressed molecules that are nearly density-independent and fully determined by two-body physics.
• To fix all parameters in the effective Hamiltonian for both 1D and 2D cases at any magnetic field detuning, enabling quantitative many-body modeling.

Proposed method

• Uses a two-channel field theory for Feshbach resonances with dimensionless units (energy in ℏω, length in aₜ = √(ℏ/mω)).
• Expands atomic and molecular field operators in transverse trap eigenmodes and plane waves in untrapped dimensions, transforming the 3D Hamiltonian into a mode-resolved form.
• Derives the effective interaction via T-matrix formalism, incorporating virtual excitation of transverse modes and renormalizing the bare interaction using a self-energy approach.
• Introduces the dressed molecule field to isolate the contribution of transverse excitations, with the molecular field operator defined to exclude atomic population in the lowest transverse mode.
• Solves the T-matrix equations to obtain an effective interaction U_b^eff(E) that depends on the energy and transverse mode structure, leading to a renormalized interaction in the low-D effective theory.
• Fixes all parameters in the effective Hamiltonian using physical scattering data (background scattering length, resonance width, magnetic moment difference), ensuring consistency across detunings.

Experimental results

Research questions

• RQ1 How can a low-dimensional effective Hamiltonian be constructed for strongly interacting ultracold atoms in a transverse trap when transverse excitations significantly contribute to the many-body state?
• RQ2 What is the role of dressed molecules—dimers excluding atomic population in the lowest transverse mode—in renormalizing the effective interaction in low dimensions?
• RQ3 To what extent are the parameters of the effective low-D Hamiltonian independent of atomic density, given the dominance of two-body physics in transverse excitation structure?
• RQ4 How can the effective interaction be systematically derived and parameterized across all magnetic field detunings near a Feshbach resonance?
• RQ5 Can the effective Hamiltonian reproduce the correct two-body bound state and scattering properties in both 1D and 2D systems?

Key findings

• The effective low-dimensional Hamiltonian includes renormalized interactions between atoms and dressed molecules, with the dressed molecule structure fixed by two-body physics and nearly independent of atomic density.
• The effective interaction U_b^eff(E) is derived via T-matrix resummation, incorporating virtual transverse excitations and yielding a self-consistent description of scattering and bound states.
• All parameters in the 1D and 2D effective Hamiltonians are fixed for any magnetic field detuning using physical scattering data, enabling quantitative modeling of low-D strongly correlated systems.
• The dressed molecule field successfully decouples transverse excitation effects, allowing the effective theory to describe both atomic and molecular degrees of freedom on equal footing.
• The effective Hamiltonian correctly reproduces the two-body bound state and scattering length in low dimensions, validating its use for many-body simulations.
• For 6Li and 40K, the effective interaction V_p^eff(2μ) shows distinct dependence on the 3D scattering length a_t/a_s, with curves varying smoothly with detuning and μ_ρ, confirming the model’s predictive power.