[Paper Review] Bose-Einstein Condensates in Non-abelian Gauge Fields

Tin-Lun Ho, Shizhong Zhang|arXiv (Cornell University)|Jul 5, 2010

Cold Atom Physics and Bose-Einstein Condensates15 references3 citations

TL;DR

This paper proposes that in a non-abelian gauge field generated via laser-driven Raman transitions in a spin-1 Bose-Einstein condensate, the ground state develops a spontaneous momentum-carrying stripe order due to two non-orthogonal dressed states with distinct momenta. The key result is the emergence of macroscopic spin-orbit coupling in bosons, realized as spatially modulated density patterns in each spin component, which can be detected via time-of-flight expansion imaging.

ABSTRACT

The recent success of the NIST group in generating abelian gauge field in cold atoms has created opportunities to simulate electronic transports in solids using atomic gases. Very recently, the NIST group has also announced in a DARPA Meeting the creation of non-abelian gauge fields in a pseudo spin-1/2 Bose gas. While there have been considerable theoretical activities in synthetic gauge fields, non-abelian fields have not been generated until now. Here, we show that in a non-abelian gauge field, a spinor condensate will develop a spontaneous stripe structure in each spin component, reflecting a ground state made up of two non-orthogonal dressed states with different momenta. Depending on interactions, this ground state can reduce back to a single dressed state. These momentum carrying stripes are the {\em macroscopic} bosonic counterpart of the spin-orbit phenomena in fermions that are being actively studied in electron physics today.

Motivation & Objective

• To investigate the ground state structure of a spin-1 Bose-Einstein condensate under non-abelian gauge fields generated via Raman coupling and magnetic field gradients.
• To demonstrate that non-abelian gauge fields induce spontaneous formation of momentum-carrying stripe order in each spin component, distinct from abelian cases.
• To show that the ground state consists of two non-orthogonal dressed states with different momenta, leading to macroscopic spin-orbit coupling in bosons.
• To propose a generalized adiabatic scheme for generating increasingly complex gauge fields by successively including low-energy spin states.
• To provide a theoretical framework for detecting the stripe phase via time-of-flight expansion, where the cloud splits into components with distinct momenta.

Proposed method

• The system is modeled using a single-particle Hamiltonian with a spatially periodic spin-dependent potential $ W({f r}) $, derived from counter-propagating lasers and a magnetic field gradient.
• A unitary transformation to a rotating frame in spin space eliminates time dependence, yielding a static effective Hamiltonian with emergent gauge fields.
• The gauge field is non-abelian when multiple low-lying spin states (e.g., $ m=1,0 $) are degenerate and coupled via the Raman term $ ilde{f A} eq 0 $.
• The Gross-Pitaevskii equation is solved for the condensate wavefunction in the Thomas-Fermi approximation, with chemical potential renormalized by the trap potential.
• The condensate wavefunction is expressed as a superposition of two dressed states with momenta $ p_+ $ and $ p_- $, leading to spatially modulated density profiles.
• The phase structure is analyzed using perturbations such as field gradients or quantum fluctuations to resolve the relative phase ambiguity in the GP approach.

Experimental results

Research questions

• RQ1 How does a non-abelian gauge field affect the ground state of a spin-1 Bose-Einstein condensate?
• RQ2 What is the nature of the spontaneous order that emerges in the presence of non-abelian gauge fields, and how does it differ from abelian cases?
• RQ3 Can the macroscopic spin-orbit coupling observed in fermions be realized in bosonic systems through synthetic gauge fields?
• RQ4 How can the stripe-ordered phase be detected experimentally in a trapped atomic gas?
• RQ5 What is the role of the generalized adiabatic scheme in generating increasingly complex gauge fields from low-energy spin subspaces?

Key findings

• The ground state of the spinor condensate develops a spontaneous stripe order in each spin component due to the coexistence of two non-orthogonal dressed states with different momenta.
• The stripe wavelength is $ \pi/k_0 $, with a contrast of 70% at the center in the simulated 87 Rb system with $ N=2.5\times10^5 $ atoms.
• For $ \beta=0 $, the amplitudes of the two dressed states are equal when $ \alpha < \alpha_c $, but degeneracy occurs at $ \alpha > \alpha_c $, leading to a phase transition.
• In the presence of a harmonic trap, the condensate wavefunction exhibits spatially modulated density profiles with $ \tilde{n}_1(x,0) $ showing periodic oscillations with period $ \pi/k_0 $.
• Time-of-flight expansion reveals two distinct atomic clouds moving with momenta $ p_+ $ and $ p_- $, confirming the two-component momentum structure.
• The non-abelian gauge field cannot be gauged away due to non-commutativity, leading to physical effects such as spin rotation during particle motion, unlike in abelian cases.