Experimental realization of a SU(3) color-orbit coupling in an ultracold gas

Abstract

Spin-orbit interaction couples the spin of a particle to its motion and leads to spin-induced transport phenomena such as spin-Hall effects and Chern insulators. In this work, we extend the concept of internal-external state coupling to higher internal symmetry, exploring features beyond the established spin-orbit regime. We couple suitable resonant laser beams to a gas of ultracold atoms, thereby inducing artificial SU(3) non-Abelian gauge fields that act on a degenerate ground state manifold comprised of three dark states. We demonstrate the inherent all-state connectivity of SU(3) systems by performing targeted geometric transformations. Then, we investigate color-orbit coupling, an extension of SU(2) spin-orbit coupling to SU(3) systems. We reveal a rich dynamical interplay between three distinct oscillation frequencies, which possesses interesting analogies with neutrino oscillations and quark mixing mechanisms. In the future, the system should provide a testbed for exploring topological properties of SU(3) systems.

Figures (3)

• Figure 1: Experimental setup. a. A laser beam (red lines), lying in the ($x,y$)-plane, is split into three parts and redirected on a cold atomic cloud of strontium atoms (grey disk). The polarization of each beam is indicated on the graph. Electro- and acousto-optic modulators (EOM, AOM) and quarter-wave plates (QWP) control each beam frequency, Rabi frequency, and polarization state. A magnetic field bias $\textit{B}=67\,$G is applied along the $y$-axis. The in-plane atomic mean momentum p and its polar angle $\theta$ are defined in the laser beams reference frame (see Methods). b. The red arrows show the quasi-resonant atomic transitions of the intercombination line driven by the laser beams. These transitions form a 2-tripod scheme made of a left and a right tripod configuration sharing the common ground state $\left| {5/2} \right \rangle_g$. The 2-tripod laser Rabi frequencies and single photon detunings are $\Omega^{l, r}_{1,2,3}$ and $\delta^{l, r}_{1,2,3}$, respectively. Here, all the Rabi frequencies are equal, namely $\Omega^{l,r}_{1,2,3} = \Omega$. c. The bright ($\left| {\textrm{B}} \right \rangle_j$, $j=1,2,3,4$) and color ($\left| {R} \right \rangle$, $\left| {B} \right \rangle$, $\left| {G} \right \rangle$) dressed states diagonalize the atom-laser coupling Hamiltonian in the rotating wave approximation. At resonance ($\delta^{l, r}_{1,2,3} = 0$), the color states are degenerated with a zero eigenenergy and separated from the bright states by a frequency shift of the scale $\Omega$. For moving atoms and quasi-resonant beams, the color states are quasi-degenerate as long as the Doppler effect and detunings are much smaller than the Rabi frequency. Then, the atomic dynamics constrained to the color space are described by an SU(3) gauge field. d. Left panel: Time-of-flight (TOF) image showing the measured velocity distribution (in units of the recoil velocity $v_r$) of the atoms in the different ground states after the initial color state preparation. Right panel: From the TOF image, we extract the populations in the different ground states (blue vertical bars) and compare them to the populations expected from the targeted initial state (green vertical bars). The black error bars are the experimental standard deviation. The fidelity of our state preparation protocol is $\mathcal{F}= 0.97(2)$.
• Figure 2: Cartan-Weyl transformation. (a) Schematic of the Cartan-Weyl SU(3) ladder operators structure leading to the all-state connectivity. (b) Time evolution of the populations of states $\left| {R} \right \rangle$ (red disks), $\left| {G} \right \rangle$ (green disks) and $\left| {B} \right \rangle$ (blue disks) for the Hamiltonian $\hat{H}_{RG}$ defined in Eq. \ref{['eq:CW_H']}. With an initial state as $\left| {G} \right \rangle$, we observe a direct $\left| {G} \right \rangle\rightarrow\left| {R} \right \rangle$ population transfer. The colored solid lines are the theoretical predictions for an atom at rest and the colored-dashed curves correspond to theoretical predictions taking into account the momentum dispersion. (c) Same as (b) but for Hamiltonian $\hat{H}_{RGB}$. We observe an indirect $\left| {G} \right \rangle\rightarrow\left| {R} \right \rangle$ population transfer mediated by $\left| {B} \right \rangle$. The practical implementation of the transformations is detailed in Methods.
• Figure 3: Color-orbit dynamics. (a) Temporal evolution of the atomic velocity components $v_x$ (green circles) and $v_y$ (red squares) around their mean values $\langle v_x\rangle= -0.14(2)v_r$ and $\langle v_y\rangle= 1.91(11)v_r$. The time origin corresponds to the end of the adiabatic ramp sequence transferring bare state $\left| {9/2} \right \rangle_g$ to a color state and the start of the evolution in the SU(3) gauge field. The plain curves are the theory prediction considering the finite momentum spread of gas. The (conserved) mean initial momentum of the gas is ${\bf p}=8 \hbar k \,\hat{{\bf x}}$. (b) Fourier Transformation (FT) amplitude of $v_y-\langle v_y\rangle$ obtained by Fourier transform of the time signal in (a). (c) Bare state populations spectral density (PSD) as indicated on each panel. In all panels, the plain curves are numerical predictions using Eq. \ref{['eq:CW_H']} and the finite momentum dispersion of the gas, see methods. The dashed black vertical lines indicate the predicted three Bohr oscillation frequencies $4\omega_r$, $8\omega_r$ and $12\omega_r$ for an atom at rest, see methods.