Antiferromagnetic interorbital spin-exchange interaction of ${}^{171}\mathrm{Yb}$

We report on the investigation of the scattering properties between the ground state ${}^{1}S_0$ and the metastable state ${}^{3}P_0$ of the fermionic isotope of ${}^{171}\mathrm{Yb}$. We measure the $s$-wave scattering lengths in the two-orbital collision channels as $a_{eg}^{+}=225\left(13\right)a_0$ and $a_{eg}^{-}=355\left(6\right)a_0$, using clock transition spectroscopy in a three-dimensional optical lattice. The results show that the interorbital spin-exchange interaction is antiferromagnetic, indicating that ${}^{171}\mathrm{Yb}$ is a promising isotope for the quantum simulation of the Kondo effect with the two-orbital system.

Figure 1

(a) Singly and doubly occupied sites of ${}^{171}\mathrm{Yb}$($I=1/2$) in an optical lattice at a zero magnetic field. A green (yellow) ball with a black arrow is the ${}^{1}S_0$ (${}^{3}P_0$) atom in a nuclear spin state $m_F=\pm 1/2$. Yellow arrows indicate single-photon excitations to the ${}^{3}P_0$ state using the $\pi$-polarized clock light. A blue (red) ellipse shows a nuclear spin singlet (triplet) state. Note that the interatomic interaction between ${}^{171}\mathrm{Yb}$ atoms in the ground state is much smaller than that between $e$–$g$ atoms [27]. (b),(c) Illustrative plots of the magnetic field dependence of the eigenenergies for the singly occupied sites in the ${}^{3}P_0$ state (dashed lines), the interorbital doubly occupied sites (solid curves with color gradient) and the doubly occupied sites in the ground state (dot-dashed lines) in the case of (b) ferromagnetic and (c) antiferromagnetic spin-exchange interaction. The line color shows the absolute square of the spin-triplet amplitude $|c_{-}^{\alpha }{|}^2$ of the eigenstates $\left|e{g}^{\alpha }\right.〉=c_{+}^{\alpha }\left(B\right)\left|e{g}^{+}\right.〉+c_{-}^{\alpha }\left(B\right)\left|e{g}^{-}\right.〉$, where $\alpha =\text{H}$ and L represent the higher and lower branch, respectively. Yellow arrows indicate the optical coupling with the $\pi$-polarized light for the doubly occupied sites.

Figure 2

Clock transition spectroscopy in a 3D optical lattice for different lattice depths: (a) $s=15$, (b) $s=20$, (c) $s=25$, (d) $s=30$. The horizontal axis shows the detuning of the clock laser from the resonance of singly occupied sites. Two peaks labeled by $n$ correspond to resonances of singly and doubly occupied sites. Error bars show the standard deviations of the mean values obtained by averaging three measurements. (e) Interaction shift as a function of $s$. Error bars are $95\%$ confidence intervals of resonance position fits. The solid line represents fits to the data with Eq. (2).

Figure 3

(a) Magnetic field dependence of clock transition frequencies of ${}^{171}\mathrm{Yb}$ in 3D magic optical lattice. Squares mark transitions to $\left|e\downarrow \right.〉$ and diamonds mark transitions to $\left|e\uparrow \right.〉$. Circle (triangle, only at 145 Gauss) points indicate transitions to $\left|e{g}^{\text{H}}\right.〉$ ($\left|e{g}^{\text{L}}\right.〉$), which asymptotically connect to $\left|e{g}^{-}\right.〉(\left|e{g}^{+}\right.〉$) in a zero magnetic field. The lattice depth is $(s_x,s_y,s_z)=(30,30,30)$, where $s_i$ ($i=x,y,z$) denotes the lattice depth in the units of the recoil energy along the $i$ axis. Error bars are $95\%$ confidence intervals of resonance position fits. Solid lines represent the fits to the data with Eq. (4). (b) Resonance positions as a function of a magnetic field extending to higher magnetic fields.

Figure 4

Relevant energy levels for the clock transition spectroscopy. After the excitation to the ${}^{3}P_0$ state, associated with a yellow arrow, remaining atoms in the ground state are blasted with a strong ${}^1\mathrm{S}\to {}^1\mathrm{P}$ transition, corresponding to a purple arrow. Then the ${}^{3}P_0$ atoms are repumped into the ground state via a ${}^{3}S_1\to {}^3\mathrm{P}\to {}^{1}S_0$ process using simultaneous applications of ${}^3\mathrm{P}\to {}^{3}S_1$ and ${}^3\mathrm{P}\to {}^{3}S_1$ resonant light beams, associated with solid red arrows. Finally, the repumped atoms are captured with a magneto-optical trap using the ${}^1\mathrm{S}\to {}^1\mathrm{P}$ transition. Dashed arrows represent relevant spontaneous decays.

Figure 5

Schematic diagram of the lattice geometry for the Kondo system. (a) Strong transverse potential with the wavelength of 759 nm to achieve a large value of $|V_{\text{ex}}|$. (b) State-dependent lattice with a wavelength of 655 nm to localize the ${}^{3}P_0$ atoms by the Anderson localization. (c) Bichromatic lattice using two beams with the wavelengths of 759 and 650 nm.

Figure 6

Calculation of the dimensionless spin-exchange interaction of ${}^{171}\mathrm{Yb}$ in the state-dependent optical lattice ($\lambda =655$ nm) to localize ${}^{3}P_0$ atoms. A horizontal axis represents the longitudinal lattice depth for the ${}^{1}S_0$ atom in units of the recoil energy $E_{\text{R,655}}=h^2/\left(2m{\lambda }^2\right)$. Circle, diamond, and square points indicate the lattice depths for the transverse confinement of 30, 60, and 90 $E_{\text{R,759}}$, respectively. Note that the expression for the Kondo temperature (C1) is reliable below 0.6 indicated by a dashed line.

Figure 7

Numerical calculation of the IPR in the bichromatic lattice with system size of 500 sites. A vertical axis shows the quasiperiodic disorder strength $\mathrm{\Delta }$ scaled by the hopping $J$. A horizontal axis indicates a label number assigned to an eigenstate. The relative phase $\varphi$ is set to 0.