Synthetic Spin-Orbit Coupling in an Optical Lattice Clock

We propose the use of optical lattice clocks operated with fermionic alkaline-earth atoms to study spin-orbit coupling (SOC) in interacting many-body systems. The SOC emerges naturally during the clock interrogation, when atoms are allowed to tunnel and accumulate a phase set by the ratio of the “magic” lattice wavelength to the clock transition wavelength. We demonstrate how standard protocols such as Rabi and Ramsey spectroscopy that take advantage of the sub-Hertz resolution of state-of-the-art clock lasers can perform momentum-resolved band tomography and determine SOC-induced $s$-wave collisions in nuclear-spin-polarized fermions. With the use of a second counterpropagating clock beam, we propose a method for engineering controlled atomic transport and study how it is modified by $p$- and $s$-wave interactions. The proposed spectroscopic probes provide clean and well-resolved signatures at current clock operating temperatures.

Figure 1

(a) A clock laser along the $Z$ direction of wavelength $\lambda$ and Rabi frequency $\mathrm{\Omega }$ interrogates the ${}^1{S}_0(g)\text{−}{}^3{P}_0(e)$ transition in fermionic alkaline-earth atoms trapped in an optical lattice with magic wavelength ${\lambda }_m$. The transverse confinement is provided by Gaussian curvature of the lattice beams with harmonic frequency $h{\nu }_{\perp }$. Many transverse modes $\mathbf{n}$ are populated at current operating temperatures. (b) The phase difference $\varphi$ between adjacent sites $j$ and $(j+1)$ induces SOC when atoms can tunnel with mode-dependent tunnel coupling $J_{\mathbf{n}}$, realizing a synthetic two-leg ladder with flux $\varphi$ per plaquette.

Figure 2

(a) SOC band structure for $\delta =-2J_{\mathbf{0}}$, ${\mathrm{\Omega }}_{\mathbf{0}}=J_{\mathbf{0}}$ (the solid lines) and ${\mathrm{\Omega }}_{\mathbf{0}}^p=0.05J_{\mathbf{0}}$ (the dashed lines). The axial depth is $V=12$ recoils, $J_{\mathbf{0}}/h=42\mathrm{Hz}$, and ${\nu }_{\perp }\approx 900\mathrm{Hz}$. Colors correspond to state character, with $g$ ($e$) being more blue (red). (b) Chiral Bloch vector angle, ${\theta }_{\mathbf{0}q}$, in the $xz$ plane extracted from Rabi spectroscopy using the protocol explained in the text [39]. The figure shows three temperatures for the parameters of (a).

Figure 3

A clock laser pulse with pulse area ${\theta }_1$ imprints a phase difference $\varphi$ between atoms in neighboring sites. Atom tunneling, $J_{\mathbf{n}}$, allows for $s$-wave interactions, $\propto U_{eg}^{-}$, which are signaled as a density shift in Ramsey spectroscopy after a second pulse of area ${\theta }_2$ is applied.

Figure 4

(a) An additional counterpropagating probing beam with a differential phase $\mathrm{\Upsilon }$ generates a sliding superlattice potential, shown for $\varphi =7\pi /6$, corresponding to a ${}^{87}\mathrm{Sr}$ OLC. For weak tunneling $J\ll \mathrm{\Omega }$, transport is energetically suppressed except at resonant defect points (circled). (b) Two-particle interaction sectors classified by the total polarization $M_x$ and spatial symmetry of the dressed states. An oval (figure eight) denotes a symmetric (antisymmetric) spatial wave function. (c) Dynamics of a single particle at the tunneling resonance for two temperatures (the red solid and blue dashed lines) and an off-resonant site (the black dotted line) for $J_{\mathbf{0}}/h=8\mathrm{Hz}$, ${\mathrm{\Omega }}_{\mathbf{0}}/h=1\mathrm{kHz}$, $\varphi =7\pi /6$. (d) Normalized differential excitation extracted as explained in the text with the interaction parameters of Ref. [22]. The $M_x=0$ components (the solid lines) involve the $s$-wave sector, and thus display a strong dependence of contrast on temperature, while the $M_{\pm 1}$ sectors (the dashed lines) experience only weaker single-particle thermal dephasing.