1. Dynamics of Quantum Gas in Non-Abelian Gauge Field
Mehedi Hasan, Chang Chi Kwong, David Wilkowski
Division of Physics and Applied Physics, NTU, Singapore.

2. Atoms in Vacuum 𝐸 𝑘 𝑦 𝑘 𝑥 Excited State Atomic Cloud
The system we are looking at is cloud of cold Strontium atoms. If the atoms are held in vacuum, the dispersion of each atom is paraboloid. From the energy diagram of Strontium, the ground state has nuclear spin 9/2, this gives us 10 almost degenerate ground states, and we chose three of them to construct the ground state manifoled. The excited state we chose from one of the states of the 3P1 manifold.
Excited State
Atomic Cloud
Ground State Manifold

3. Dressed Atom and Non-Abelian GF𝐸
𝑘 𝑦
Without any laser dressing we have this configuration of bare states with paraboloid dispersion.
𝑘 𝑥
Atomic Cloud

4. Dressed Atom and Non-Abelian GF
𝐻= 𝑝 2 2𝑀 + ℏ𝜅 𝑀 𝑝 𝑦 ℏ𝜅 𝑀 𝑝 𝑥 ℏ𝜅 𝑀 𝑝 𝑥 𝑝 2 2𝑀 − ℏ𝜅 𝑀 𝑝 𝑦 .
Now we dress the atomic cloud with three lasers shown by the black arrows. The horizontally propagating lasers are circular polarized with opposite helicity and the vertical one is linearly polarized. In the dressed picture, two of the states will be degenerate and that is guaranteed by the SU(2) symmetry. This two states quite separated from the two other states and this gives us the freedom to operate in the manifold of this two dark states. One can write down the Hamiltonian in this basis and the dispersion relation looks like this. 𝐸
𝑘 𝑦
𝑘 𝑥
𝜋 Pol.

5. Representation in Reduced Manifold
In Vacuum
Effective Hamiltonian
In 𝑝-space
TDSE:
𝐻=− ℏ 2 2𝑀 𝛻 2 ⊗1+ ℏ𝜅 𝑀 𝜕 𝜕𝑥 ⊗ 𝜎 𝑥 + 𝜕 𝜕𝑦 ⊗ 𝜎 𝑧 .
𝐻 𝑝 = 𝑝 2 2𝑀 + ℏ𝜅 𝑀 𝑝 𝑦 ℏ𝜅 𝑀 𝑝 𝑥 ℏ𝜅 𝑀 𝑝 𝑥 𝑝 2 2𝑀 − ℏ𝜅 𝑀 𝑝 𝑦 .
We start with a cloud of Gaussian shape and in vacuum, the cloud maintains its Gaussian shape as it expands. However, in the presence of non-Abelian gauge field, the up- and down-spins of the cloud expand symmetrically. The reason behind the asymmetry can be traced back to momentum-dependent eigenstate of the Hamiltonian at the left.
𝑖ℏ 𝜕 𝜕𝑡 |𝛹(𝑘) =𝐻(𝑘) |𝛹(𝑘)
|𝛹 = | 𝛹 1 𝑷 | 𝛹 2 𝑷
After 23 ms. in Gauge field

6. Expanding Atomic Cloud
Effective Hamiltonian
𝐻= − ℏ 2 2𝑀 𝛻 2 ⊗1 Kinetic Energy + ℏ𝜅 𝑀 𝜕 𝜕𝑥 ⊗ 𝜎 𝑥 + 𝜕 𝜕𝑦 ⊗ 𝜎 𝑧 Gauge field term .
Now if one would like to be quantitative, we can calculate the size of the each spinor component along x- and y-directions. The plot at the right shows the time-evolution of each. I would like to draw your attention to the green curve that represent the normalized cloud size in the absence of gauge field. Here we see that cloud expands order of magnitude faster and it could be understood via the gauge field-mediated momentum transfer that conspire the cloud to move at a higher rate. This could be one of the signature of gauge field that we are currently trying to observe in our lab.