Correlated spin-flip tunneling in a Fermi lattice gas

We report the realization of correlated, density-dependent tunneling for fermionic ${}^{40}\mathrm{K}$ atoms trapped in an optical lattice. By appropriately tuning the frequency difference between a pair of Raman beams applied to a spin-polarized gas, simultaneous spin transitions and tunneling events are induced that depend on the relative occupations of neighboring lattice sites. This correlated spin-flip tunneling (CSFT) is spectroscopically resolved using gases prepared in opposite spin states, and the inferred Hubbard interaction energy is compared with a tight-binding prediction. We measure the doublons created by the laser-induced correlated tunneling process using loss induced by light-assisted collisions. Furthermore, by controllably introducing vacancies to a spin-polarized gas, we demonstrate that correlated tunneling is suppressed when neighboring lattice sites are unoccupied. We explain how the CSFT quench implemented here prepares and evolves a large number of resonating-valence-bond (RVB) singlets in a Hubbard model, thus allowing exploration of RVB dynamics.

Figure 1

Schematic diagram of Raman transitions. A pair of Raman beams, each detuned approximately 80 GHz from the $4S_{1/2}\to 4P_{1/2}$ transition, with frequencies ${\omega }_{1,2}$ and wave vectors ${\vec{k}}_{1,2}$ are applied to drive transitions between the $|\uparrow \rangle$ (red) and $|\downarrow \rangle$ (blue) states. The Raman wave-vector difference $\overrightarrow{\delta k}={\vec{k}}_1-{\vec{k}}_2$ lies along the $\left(-1,-1,1\right)$ direction of the lattice. Selecting between two distinct processes is achieved by fixing the laser beam frequency ${\omega }_1$ and tuning ${\omega }_2$. (a) If the frequency difference matches the Zeeman energy (i.e., $\mathrm{\Delta }\omega ={\omega }_1-{\omega }_2={\omega }_{\uparrow \downarrow }$), then atoms flip their spin and remain on the same site. (b) When the laser frequency difference accommodates the Hubbard interaction energy $U$ ($\mathrm{\Delta }\omega ={\omega }_{\uparrow \downarrow }-U/\hslash$), then CSFT occurs and atoms tunnel to neighboring occupied sites and flip their spin. For $|\downarrow \rangle$ as an initial state, the condition for resonant CSFT changes to $\mathrm{\Delta }\omega ={\omega }_{\uparrow \downarrow }+U/\hslash$.

Figure 2

Spectroscopy of CSFT. (a) The fraction of atoms transferred between spin states by a 50 m Raman pulse is shown for an initially $|\uparrow \rangle$ (black squares, $f_{\uparrow }$) and $|\downarrow \rangle$ (red circles, $f_{\downarrow }$) spin-polarized state at $s=10$ $E_R$ for varied $\mathrm{\Delta }\omega$. For these measurements, $N=25400\pm 3900$ atoms were cooled to $T/T_F=0.24\pm 0.08$ before turning on the lattice. Each data point is the result from a single experimental run, and the measurement uncertainty is not visible on this scale. Nonzero transfer at large detuning is due to off-resonant spontaneous Raman scattering. (b) The difference $f_{\uparrow }-f_{\downarrow }$ for pairs of points in (a) reveals the CSFT sidebands at approximately $\pm U$. The black line shows a fit to a sum of two Gaussian functions; the individual Gaussians are displayed as shaded regions. The peak at lower (higher) frequency corresponds to CSFT for an initially $|\uparrow \rangle$ ($|\downarrow \rangle$) spin-polarized state. (c) The interaction energy $U$ inferred from fits to data such as those shown in (b) for varied $s$. The error bars are derived from the fit uncertainty. The dashed line is the value of $U$ from a standard tight-binding calculation.

Figure 3

Fraction of doubly occupied sites $D$ measured after a 50 m Raman pulse at various detunings for $s=10$ $E_R$. The inset shows sample LAC for $\left(\mathrm{\Delta }\omega -{\omega }_{\uparrow \downarrow }\right)/2\pi =3.5$ kHz fit to a double-exponential decay with ${\tau }_D=3.1\pm 0.77$ ms and ${\tau }_S=13.7\pm 2.5$ ms. The vertical error bars in $D$ are derived from fits to similar data acquired at different $\mathrm{\Delta }\omega$, while the horizontal error bars show the estimated 0.5 kHz uncertainty in the carrier transition from magnetic-field drift. The solid line in the main panel is the fit from Fig. 2 for $\left|f_{\uparrow }-f_{\downarrow }\right|$ plotted on the same scale as $D$.

Figure 4

Density dependence of CSFT. The CSFT spectroscopy signal taken with fixed $\left(\mathrm{\Delta }\omega -\mathrm{\Delta }{\omega }_{\uparrow \downarrow }\right)\approx \pm U/\hslash$ is shown for varied fraction $\delta N$ of atoms randomly removed from an $s=8$ $E_R$ lattice gas. For these data, $N=47000$–81 000, and the gas was cooled to $T/T_F\approx 0.35$ before turning on the lattice. Data obtained with the $+U$ sideband are shown as red circles and those for $-U$ as black squares. The sideband frequencies were determined using a double-Gaussian fit to CSFT spectroscopy data, as in Fig. 2. The dashed line is a prediction for the probability to find adjacent sites occupied based on a calculation of the density profile after the removal procedure. The inset shows the procedure for controllably introducing vacancies. Atoms (shown as transparent) that are not shelved in the $F=7/2$ state via microwave transitions are removed using resonant light.

Figure 5

Energy levels of three-level atoms with two lasers of frequencies ${\omega }_1$ and ${\omega }_2$.

Figure 6

Evolution of doublon fraction $\langle n_{i\uparrow }{n}_{i\downarrow }\rangle /\langle n_{i\uparrow }+n_{i\downarrow }\rangle$ from a numerical simulation with constant $\mathrm{\Omega }$. The solid lines shows the simulation with the full Hamiltonian $H_0$, while the dashed line shows that with the effective Hamiltonian $H_{\mathrm{eff}}$ derived from second-order perturbation theory. The states are initialized with one spin-up fermion on every site, and the parameters are determined by experiment: $t/h=0.25\mathrm{kHz}$, $U/h=3.22\mathrm{kHz}$, $\mathrm{\Omega }=0.1U$, and $\overrightarrow{\delta k}·\vec{d}=\pi /2\sqrt{3}$, where $\vec{d}$ is a lattice vector.

Figure 7

CSFT signal for varied Raman pulse time. The Raman detuning for these measurements is fixed to the $+U$ CSFT sideband. The measurements are shown using black circles and a theoretical simulation is displayed as a red line. The simulation is performed with $\mathrm{\Omega }$ in $H_0$ replaced with $\mathrm{\Omega }{e}^{i\varphi \left(t\right)}$, where $\langle \varphi {\left(t\right)}^2\rangle =8$, and the characteristic time scale of the fluctuations in $\varphi \left(t\right)$ is 2 ms. The dynamics has been averaged (indicated by the notation $\langle \rangle$) over five realizations of $\varphi \left(t\right)$. For these parameters, the carrier Rabi oscillations are not strongly perturbed.

Figure 8

Schematic diagram showing CSFT for a two-site two-fermion system. CSFT happens as a two-step process via a virtual state. Two possible channels between the initial state ${|\uparrow ,\uparrow \rangle }_W$ and the final state ${|\uparrow \downarrow ,0\rangle }_W+{|0,\uparrow \downarrow \rangle }_W$ happen simultaneously but with amplitudes carrying opposite signs. The probability to observe a doublon-hole pair is affected by interference between these channels.

Figure 9

Fraction of atoms with nearest neighbors at various removal fractions $\delta N$ for $N=61000$ and $S/N=2.89k_B$, which corresponds to $k_B\tilde{T}=9.7t$ and chemical potential $\mu =6.4t$ in the lattice. The insets at the right show sample occupation profiles (with one black dot per atom) through a central slice of the gas. The inset at bottom left schematically illustrates the procedure for counting pairs.

Figure 10

Measurements of $f_{\uparrow }$ (black squares) and $f_{\downarrow }$ (red circles) for varied Raman detuning $\mathrm{\Delta }\omega$ taken using the same procedure as for Fig. 2 in the main text. For these data, $\delta N\approx 0.57$.