Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition

We demonstrate a novel way of synthesizing spin-orbit interactions in ultracold quantum gases, based on a single-photon optical clock transition coupling two long-lived electronic states of two-electron ${}^{173}\mathrm{Yb}$ atoms. By mapping the electronic states onto effective sites along a synthetic “electronic” dimension, we have engineered fermionic ladders with synthetic magnetic flux in an experimental configuration that has allowed us to achieve uniform fluxes on a lattice with minimal requirements and unprecedented tunability. We have detected the spin-orbit coupling with fiber-link-enhanced clock spectroscopy and directly measured the emergence of chiral edge currents, probing them as a function of the flux. These results open new directions for the investigation of topological states of matter with ultracold atomic gases.

Figure 1

Sketch of the setup. Ultracold ${}^{173}\mathrm{Yb}$ fermions are trapped in 1D chains by an optical lattice at wavelength ${\lambda }_L$. An ultranarrow clock laser with wavelength ${\lambda }_C$ drives the single-photon transition between the ground state $g={}^1{S}_0$ and the long-lived electronic state $e={}^3{P}_0$. The laser momentum transfer $\delta k=2\pi \cos \theta /{\lambda }_C$ results in a locking between internal state (interpreted as an effective pseudospin) and atomic momentum. The electronic state can also be treated as an effective synthetic dimension made by two sites connected with a coherent tunneling, resulting in a two-leg ladder pierced by a synthetic magnetic flux per plaquette $\varphi =\pi \delta k/k_L$, which can be tuned by adjusting the angle $\theta$.

Figure 2

Spectroscopic observation of SOC. (a) Sketch of the lattice energy bands for $g$ (blue line) and $e$ (green dashed line). Shifting the $e$ band by the SOC momentum $-\delta k$ (green solid line) allows us to represent the clock transition between the two states as a “vertical” process with a momentum-dependent energy. (b) Spectrum of the clock transition in a lattice with $s=3$, $t=2\pi \times 220\mathrm{Hz}$ and $\delta k=1.31k_L$. The plot shows the number of atoms remaining in $g$ after a 800-ms long interrogation time. The horizontal axis shows the detuning with respect to the center of the spectrum. The shape of the spectrum, featuring a broadening and an enhanced response at the edges, is a spectroscopic consequence of SOC. Points are averages over multiple measurements and error bars are standard deviations. (c) Spectra of the clock transition for different $s$. Data sets with $s\ge 4$ have been offset vertically for the sake of presentation. The curves are the result of a single-particle theoretical model (see text). All the spectra have been measured shining the clock light along the lattice direction ($\theta =0$) and using only an additional transverse (vertical) lattice to hold the atoms against gravity.

Figure 3

Chiral currents in the synthetic ladder. (a) Sketch of the dressed (thick, solid line) and bare energy bands (thin, dashed line) for $\varphi =1.31\pi$ and $\mathrm{\Omega }=t$. The color of the lines reflects the state composition of the dressed bands ($e$: green, $g$: blue). In the SD picture, the different composition as a function of $k$ can be interpreted as chiral currents flowing in opposite directions along the two legs of the synthetic ladder. The insets show time-of-flight images of the $g$ atoms for each of the two dressed bands after adiabatic loading ($\varphi =1.31\pi$, $t=2\pi \times 138\mathrm{Hz}$, $\mathrm{\Omega }=2\pi \times 590\mathrm{Hz}$), evidencing complementary momentum distributions along $\widehat{x}$, as expected from the state composition of the bands. (b) Time-of-flight images of the $g$ atoms without the clock laser ($\mathrm{\Omega }=0$, left) and with the clock laser after adiabatic loading of the lowest band ($\mathrm{\Omega }=2\pi \times 590\mathrm{Hz}$, $\varphi =0.58\pi$, right) for $t=2\pi \times 138\mathrm{Hz}$. (c) Integrated lattice momentum distribution $n_g(k)$ of the atoms in the $g$ leg of the ladder for the data reported in panel (b). (d) Asymmetry function $h_g(k)=n_g(k)-n_g(-k)$, evidencing the chiral current induced by the nonzero flux (right).

Figure 4

(a) Dependence of the chiral current $J$ on the flux $\varphi$. Vertical error bars are the standard deviation of a representative set of six measurements taken under the same conditions, while horizontal bars arise from the uncertainty on $\theta$. The dashed line marks the maximum flux that can be obtained ($\theta =0$) with the current setup. The solid curve is the expectation of a theoretical single-particle model (see text). Experimental parameters: $t=2\pi \times 138\mathrm{Hz}$, $\mathrm{\Omega }=2\pi \times 590\mathrm{Hz}$. (b) Chiral currents in the absence of the lattice along the direction of the ladders. In this “continuum” configuration, where a 2D unit cell is not defined, no inversion of $J$ is observed at large SOC momentum transfer $\delta k$. We note that the horizontal scales of the two plots coincide as $\varphi /\pi =\delta k/k_L$.