Cold Atoms in Non-Abelian Gauge Potentials: From the Hofstadter "Moth" to Lattice Gauge Theory

We demonstrate how to create artificial external non-Abelian gauge potentials acting on cold atoms in optical lattices. The method employs atoms with $k$ internal states, and laser assisted state sensitive tunneling, described by unitary $k\times k$ matrices. The single-particle dynamics in the case of intense $U(2)$ vector potentials lead to a generalized Hofstadter butterfly spectrum which shows a complex mothlike structure. We discuss the possibility to realize non-Abelian interferometry (Aharonov-Bohm effect) and to study many-body dynamics of ultracold matter in external lattice gauge fields.

Figure 1

Optical lattice setup for U(2) gauge fields: Red and blue open semicircles (closed semicircles) denote atoms in states $|g_1⟩$ and $|g_2⟩$, respectively ($|e_1⟩$ and $|e_2⟩$). Top: Hopping in the $x$ direction is laser assisted and allows for unitary exchange of colors; it is described by the same unitary hopping matrix $U_x$ for both $|g_i⟩$ and $|e_i⟩$ states. Hopping along the $y$ direction is also laser assisted and attains “color dependent” phase factors. Bottom: Trapping potential in the $y$ direction. Adjacent sites are set off by an energy $\Delta$ due to the lattice acceleration, or a static inhomogeneous electric field. The lasers ${\Omega }_{\pm i}$ are resonant for transitions $|g_i⟩\to |e_i⟩$ for $n\to n\pm 1$. Because of the spatial dependence of ${\Omega }_{\pm i}$ (running waves in the $\pm x$ direction), the atoms hopping around the plaquette get the unitary transformation $\widehat{U}=U_y^{†}(m)U_x{U}_y(m+1)U_x^{†}$, where $U_y(m)=\exp (2\pi im\mathrm{diag}[{\alpha }_1,{\alpha }_2])$, as indicated in the upper figure.

Figure 2

The Hofstadter moth spectrum. Forbidden eigenenergies $\epsilon$ are plotted versus ${\alpha }_i=p_i/q_i,\in [0,=0.5]$ ($i=1,2$), where $q_i\le 41$ and ${\alpha }_1\ne {\alpha }_2$.