Dynamics of an itinerant spin-3 atomic dipolar gas in an optical lattice

Arrays of ultracold dipolar gases loaded in optical lattices are emerging as powerful quantum simulators of the many-body physics associated with the rich interplay between long-range dipolar interactions, contact interactions, motion, and quantum statistics. In this work we report on our investigation of the quantum many-body dynamics of a large ensemble of bosonic magnetic chromium atoms with spin $S=3$ in a three-dimensional lattice as a function of lattice depth. Using extensive theory and experimental comparisons, we study the dynamics of the population of the different Zeeman levels and the total magnetization of the gas across the superfluid to the Mott insulator transition. We are able to identify two distinct regimes. At low lattice depths, where atoms are in the superfluid regime, we observe that the spin dynamics is strongly determined by the competition between particle motion, on-site interactions, and external magnetic-field gradients. Contact spin-dependent interactions help to stabilize the collective spin length, which sets the total magnetization of the gas. On the contrary, at high lattice depths, transport is largely frozen out. In this regime, while the spin populations are mainly driven by long-range dipolar interactions, magnetic-field gradients also play a major role in the total spin demagnetization. We find that the dynamics at low lattice depth is qualitatively reproduced by mean-field calculations based on the Gutzwiller ansatz; on the contrary, only a beyond-mean-field theory can account for the dynamics at large lattice depths. While the crossover between these two regimes does not display sharp features in the observed dynamical evolution of the spin components, our simulations indicate that it would be better revealed by measurements of the collective spin length.

Figure 1

Schematic of the system diagram indicating the 3D geometry of the optical lattice, the interactions, and tunneling processes. The magnetic field (with direction $\widehat{\mathbf{B}}$ in the ${\mathbf{u}}_X\text{−}{\mathbf{u}}_Y$ plane) has a gradient lying along the ${\mathbf{u}}_Z$ direction (lower inset). Each spin-3 chromium atom also has its sublevels shifted by a quadratic Zeeman term (upper inset).

Figure 2

Comparison of predictions for $N_0$ population dynamics in the double-well system ${\widehat{H}}_2$. (a) Doublon fraction of the two-particle initial state. (b)–(f) The population of the $m=0$ substate as a function of time for a double-well system allows us to compare the short-time perturbative expansions at second-order to the Gutzwiller dynamics, as well as the exact dynamics for five different lattice depths. In the deep lattice the second-order formula is a good approximation for longer times. The simulations parameters are given in Table 1, with the tunneling and nearest-neighbor DDIs adjusted so the lattice depth at which the transition occurs is similar to the 3D system and $q/h=2\mathrm{Hz}$. The inset in (b) shows the perturbative result (12) extended to include doublons, demonstrating that they give rise to rapid short-time dynamics captured in the exact solution, but limiting the applicability of the perturbative result to short times. (g) Difference of Gutzwiller and second-order populations compared to the exact population at 4 ms, as a function of lattice depth.

Figure 3

Comparison of simulation (lines) to experiment (markers). Blue (solid line with circles) indicates $m=0$, red (dashed line with squares) indicates $m=1$, yellow (dot-dashed line with crosses) indicates $m=2$, and purple (dotted line with diamonds) indicates $m=3$. Pale markers indicate the negative $m$ experimental populations. For $V_0<V_c$ the Gutzwiller model is shown, while for $V_0\ge V_c$ the GDTWA model is shown, with the corresponding $q/h=2\mathrm{Hz}$ Gutzwiller solution shown in gray. Experimental error bars are from statistical standard errors and from $10\%$ uncertainty in the estimated lattice depths. (o) Quadratic Zeeman field $q$ fitted at each lattice depth in the simulations based on the Gutzwiller ansatz and the $q/h=2\mathrm{Hz}$ value used in the GDTWA. The light gray region indicates the range of acceptable $q$ values. The light blue circles correspond to optimal values for the Gutzwiller approximation which however are outside the acceptable range. (p) Distribution of the whole cloud across the lattice at $t=0$ (black line) and distribution of individual Zeeman levels across the lattice at $t=5\mathrm{ms}$.

Figure 4

(a)–(c) Gutzwiller (solid line) and GDTWA (dashed line) $m=0$ populations compared to experimental results (symbols) as a function of lattice depth. (d) Time $\tau$ for $m=0$ population to decay to $N_0/N_{\mathrm{tot}}=0.25$.

Figure 5

Total spin as defined by Eq. (13) for a range of lattice depths. The blue solid line indicates the total spin using the Gutzwiller model, with the results beyond the lattice depth $V_c$ that the Gutzwiller model is considered reliable shown in gray. The red dot-dashed line indicates the GDTWA result and the green dashed line gives the short-time second-order expansion, found by summing Eq. (14) over all pairs of atoms. (o) Total spin in a double-well system (see Fig. 2) using the exact method with different lattice depths.

Figure 6

Comparison of DDI cutoff approximations in the Gutzwiller model for $V_0=15E_r$ and $q/h=15\mathrm{Hz}$. When $D_{\mathrm{cut}}=0$ only on-site DDIs are included, when $D_{\mathrm{cut}}=1$ nearest neighbors (including diagonal nearest neighbors) are included, and when $D_{\mathrm{cut}}=2$ next-nearest neighbors are also included. We see that the contribution from next-nearest neighbors is small compared to the nearest neighbors. As intersite DDIs are most important in the deep lattice limit, this conclusion will hold for all lattice depths.

Figure 7

Experimental values for the total number of atoms in the lattice, along side approximate exponential decay with $\gamma =100$ and $170{\text{s}}^{-1}$.