Far-from-Equilibrium Spin Transport in Heisenberg Quantum Magnets

We study experimentally the far-from-equilibrium dynamics in ferromagnetic Heisenberg quantum magnets realized with ultracold atoms in an optical lattice. After controlled imprinting of a spin spiral pattern with an adjustable wave vector, we measure the decay of the initial spin correlations through single-site resolved detection. On the experimentally accessible time scale of several exchange times, we find a profound dependence of the decay rate on the wave vector. In one-dimensional systems, we observe diffusionlike spin transport with a dimensionless diffusion coefficient of 0.22(1). We show how this behavior emerges from the microscopic properties of the closed quantum system. In contrast to the one-dimensional case, our transport measurements for two-dimensional Heisenberg systems indicate anomalous superdiffusion.

Figure 1

Experimental sequence for the measurement of the spiral evolution. The illustrations on top show the spin distribution in the transverse plane at different stages of the experiment as indicated by the gray shading [before (i) and after (ii) spiral imprinting and after evolution (iii)]. The pictures below are single shot measurements at the respective times after an additional $\pi /2$ pulse (not shown) and removal of the $|\uparrow ⟩$ state. The experimental sequence is depicted at the bottom.

Figure 2

Decay of a 1D spin spiral. Measured decay of an exemplary 1D spin spiral with wavelength $\lambda =5.7a_{\text{lat}}$ at $10E_r$ lattice depth. The solid blue line is an exponential fit used to extract the lifetime. We also show theoretical predictions of the Heisenberg model (gray, dashed line) and the $t$-$J$ model for 0.08 hole probability (gray, solid line). The inset shows three measured correlations $g_2(d)$ at $t_1=0$, $t_2=0.7\hslash /J_{\text{ex}}$, and $t_3=2.8\hslash /J_{\text{ex}}$ (dark to bright blue), from which the visibility is extracted via sinusoidal fits.

Figure 3

Wave vector dependence of the spin spiral lifetimes. We plot the data for 1D (a) and 2D (b) spirals double logarithmically and extract the exponents via power law fits (black lines). The lifetime is scaled with the superexchange rate $\hslash /J_{\mathrm{ex}}$, which results in a collapse of the measurements at different lattice heights [in 1D: $8E_r$ (blue), $10E_r$ (green), $12E_r$ (yellow), and $13E_r$ (red) and in 2D: $12E_r$ (blue), $14E_r$ (green), and $16E_r$ (yellow)]. Additionally, predictions of the Heisenberg model (numerically solved in 1D, spin wave calculations in 2D) are shown as gray solid lines. The gray band in (a) is obtained numerically from the $t$-$J$ model with hole probabilities between 0.04 and 0.12. The insets show the experimental data without scaling of the lifetimes.

Figure 4

Microscopic view of the diffusionlike behavior in 1D. The energy spread of eigenstates contributing to the spin spiral decay grows quadratically with wave vector $Q$. This leads to the observed quadratic $Q$ dependence of the decay rate, as expected for classical spin diffusion. The data shown for two system sizes of 12 (blue) and 16 (red) sites are obtained from full diagonalization of 1D Heisenberg chains. The insets show energy histograms weighted with the overlap of the initial spiral and the eigenstates for $Q=\pi /4a_{\text{lat}}$ and $Q=\pi /2a_{\text{lat}}$ for systems with 16 spins. A spin spiral with wave vector $Q$ is a superposition of many-body eigenstates with wave vectors $k$ that are integer multiples of $Q$.

Figure 5

Dependence of the diffusion constant in 1D on the hole density. The diffusion constant $D$ increases approximately linearly with hole probability. The gray area is the numerical result of the $t$-$J$ model with its 95% confidence interval. The inset shows the decay rate $1/\tau$ of the spin spiral versus the squared wave vector $Q^2$ for the lowest (blue) and highest (red) hole probability.