Controlling dipolar exchange interactions in a dense three-dimensional array of large-spin fermions

Dipolar interactions are ubiquitous in nature and rule the behavior of a broad range of systems spanning from energy transfer in biological systems to quantum magnetism. Here we study magnetization-conserving dipolar induced spin-exchange dynamics in dense arrays of fermionic erbium atoms confined in a deep three-dimensional lattice. Harnessing the special atomic properties of erbium, we demonstrate control over the spin dynamics by tuning the dipole orientation and changing the initial spin state within the large 20-spin hyperfine manifold. Furthermore, we demonstrate the capability to quickly turn on and off the dipolar exchange dynamics via optical control. The experimental observations are in excellent quantitative agreement with numerical calculations based on discrete phase-space methods, which capture entanglement and beyond-mean-field effects. Our experiment sets the stage for future explorations of rich magnetic behaviors in long-range interacting dipoles, including exotic phases of matter and applications for quantum information processing.

Figure 1

(a) Illustration of the total spin space of a single ${}^{167}\mathrm{Er}$ atom in the lowest hyperfine level $|F=19/2\rangle$ with all $20m_F$ states. The angle of the symbols indicates the orientation of the total spin $\left|\mathbf{F}\right|=\sqrt{F(F+1)}$ in relation to the quantization axis. (b) Sketch of the experimental system, an anisotropic three-dimensional lattice structure filled with fermionic ${}^{167}\mathrm{Er}$ with a quantization axis tunable by the angles $\left(\mathrm{\Theta },\varphi \right)$ of the external magnetic field $B$. (c) Illustration of the experimental sequence (from left to right): The system is initialized by preparing all atoms in one starting state, here $|–17/2\rangle$. We activate the spin dynamics by changing the magnetic field to set $\overline{\delta }=0$. Early-time dynamics are happening mainly among nearest-neighbor atoms. Subsequently, interactions between atoms at larger distances are involved in the dynamics.

Figure 2

(a) Measured spin populations in states $m_F=|–17/2\rangle$ (circles) and $m_F\pm 1$ (diamonds and squares) after $50\mathrm{ms}$ hold time as a function of the magnetic field with $\mathrm{\Theta }=0^{\circ }$. A Gaussian fit (not shown) to the data provides a resonant magnetic field value of $\approx 1.67$ G. The top axis shows the mean total detuning $\overline{\delta }$ from the resonance condition. (b) Measured spin population in states $m_F=|–13/2\rangle$ (circles) and $m_F\pm 1$ (diamonds and squares) as a function of the hold time after quenching onto the spin-exchange resonance with $\mathrm{\Theta }=0^{\circ }$. The dashed line shows the simple mean-field result, the dashed-dotted line gives the NNI-GDTWA result, the solid lines represent the full-GDTWA result, and the dotted line shows the full-GDTWA for the ideal case of a lattice with unit filling. The inset shows the total magnetization $M\left(t\right)$. [(c) and (d)] Spin diffusion with initial state $m_F^0=|–13/2\rangle$ plotting the population of states from $m_F-3$ to $m_F+3$ as a function of the hold time, for the experiment (c) and the full-GDTWA model (d), with the same initial conditions as (b). Data points consist of a minimum of four individual realizations and error bars denote the standard error of the mean.

Figure 3

[(a)–(f)] Dynamic evolution of the initial states $|–17/2\rangle$ (a), $|–9/2\rangle$ (b), $|–5/2\rangle$ (c), $|1/2\rangle$ (d), $|9/2\rangle$ (e), and $|13/2\rangle$ (f) and of the corresponding neighboring states $m_F\pm 1$ together with the full-GDTWA results (solid lines) for $\mathrm{\Theta }=0^{\circ }$. (g) Extracted $V_{\text{eff}}$ as a function of the initial state $m_F^0$ from a fit to the experimental data (cyan triangles) and numerically computed from the initial spin distribution (black circles). Error bars denote the $68\%$ confidence interval of the fits. (h) All datasets with $m_F^0<0$ used in (g) together with the corresponding full-GDTWA results (solid lines) plotted in units of the rescaled time $\tau =V_{\text{eff}}/\hslash ·t$. Note that all experiment and theory data are plotted for times $t\le 100\mathrm{ms}$ of (a)–(f). To account for the different preparation fidelity, the populations of the initial states are shifted to 1 by adding a constant offset. For clarity error bars are omitted here.

Figure 4

(a) Exemplary measurements of the time evolution for the starting spin state $|–17/2\rangle$ for $\mathrm{\Theta }={40}^{\circ },{80}^{\circ }$. Solid lines show the full-GDTWA results. (b) Extracted $V_{\text{eff}}$ as a function of $\mathrm{\Theta }$ from a fit to the experimental data (orange circles) and numerically computed from the initial spin distribution (black circles). Error bars denote the $68\%$ confidence interval of the fits. (c) Time evolution of the initial state $|–9/2\rangle$ at $\overline{\delta }=0$ and $\mathrm{\Theta }=0^{\circ }$ without (filled circles) and with (open circles) switching on an additional light field after $20\mathrm{ms}$ of evolution. Solid (dashed) lines are the corresponding full-GDTWA calculations. The inset shows the population of the initial spin state after $50\mathrm{ms}$ evolution time as a function of the quadratic Zeeman shift without (filled circles) and with (open circles) the additional light field. Determining the centers of the resonances via a fit yields an absolute shift of the resonance condition by $h\times 27\left(1\right)\mathrm{Hz}$ between both conditions.

Figure 5

(a) Energy levels of the ground-state hyperfine manifold in the dressed-state picture in dependence of the detuning between the applied just RF and the atomic resonance condition for the $|–1/2\rangle$ to the $|1/2\rangle$ hyperfine levels. The solid red arrow exemplary shows the RAP for preparation of atoms into the $m_F^0=|–7/2\rangle$ state. The insets show a zoom of one avoided crossing. [(b) and (c)] Spin-preparation and atom-counting efficiency measured for $|–17/2\rangle ,|–15/2\rangle ,|–9/2\rangle$, and $|9/2\rangle$. The obtained values are interpolated linearly assuming a linear dependence on the $m_F^0$ state.

Figure 6

Absorption images for a nonadjusted RAP and for the preparation of $|–9/2\rangle ,|3/2\rangle$, and $|5/2\rangle$. Whereas for the $|–9/2\rangle$ and $|5/2\rangle$ case no residual atoms in other spin states are visible, for the $|3/2\rangle$ case we observe a small amount of residual atoms in other spin states due to a nonperfect preparation.

Figure 7

Spin dynamics for $m_F^0=|–9/2\rangle$ as shown in Fig. 3 of the main text. The solid line represents the spin population and the shaded area denotes the standard deviation of individual trajectories which shows that the spread of the GDTWA trajectories does not grow large with time. The inset visualizes 10 typical trajectories obtained from the GDTWA simulation. Note that the different trajectories include both the quantum noise, which is essential to account for beyond-mean-field effects, and statistical noise, which averages out when sampling over enough trajectories.

Figure 8

Histogram showing the average number of atoms in distances normalized to the lattice direction along $y$ for random removal of atoms and for removal depending on the number of nearest neighbors (NN-dependent removal).

Figure 9

The dotted-dashed line exemplary shows the fit of Eq. (I2) to the experimental data to extract $V_{\mathrm{eff}}$ for $m_F^0=|–9/2\rangle$. The solid green line indicates the time $t_{\mathrm{fit}}$ up to which the fit is performed.

Figure 10

Expansion of Fig. 3 showing all measured spin states and the corresponding theory predictions.

Figure 11

Comparison between spin dynamics obtained with the reduced spin model Eq. (F1) (solid lines) and the Hamiltonian Eq. (J12) accounting for superexchange interactions (circles) for initial state $m_F^0=|-17/2\rangle$ and typical experimental conditions. Results are shown for the three most populated spin states, $m_F^0,m_F^0\pm 1$. For the on-site interaction strengths, we adopt for $a_{18}$ the value measured in Ref. [29] and assume the rest $a_S$'s are randomly distributed within $\pm 40\%$ with respect to $a_{18}$. For identical $a_S$ and absent of $U_{dd}$ the superexchange Hamiltonian has a $\mathrm{SU}(2F+1)$ symmetry that does not change spin population. We expect that the reasonable variation between $a_S$'s of ${}^{167}\mathrm{Er}$ atoms does not exceed $\pm 40\%$.

Figure 12

Spin dynamics from the Fermi-Hubbard model with $F=3/2$ (symbols). A one-dimensional system with eight sites is used, with uniform nearest-neighbor tunneling rate $J_{\mathbf{i},\mathbf{j}}=J$. To reduce finite-size effect, a periodic boundary condition is adopted. We assume an on-site contact interaction $U_s$ with $\mathrm{SU}(2F+1)$ symmetry and strength $U_s/J=200$, approximately corresponding to the value measured in Ref. [29] and $J/\hslash =10$ Hz. Since $U_s$ is much stronger than the on-site dipolar interaction for the erbium experiment [29], we do not include $U_{dd}$ in these calculations. We have also neglected external inhomogeneous fields [Eq. (J3)] in the model. Initially five sites are occupied with atoms all prepared in the $m_F^0=1/2$ state. The spin dynamics is calculated using exact diagonalization and averaged over all possible initial distribution of empty sites. To provide a reference for the $F=19/2$ erbium case, we have focused on a timescale similar to the one probed in experiment, with $V_{\mathrm{eff}}$ defined as in the main text and setting $F=3/2$ in the expression for $\gamma \left(m_F^0\right)$. As a comparison, the spin dynamics obtained with the frozen spin model [Eq. (F1)] with the same lattice configuration is also plotted with solid lines.