Relaxation of the Collective Magnetization of a Dense 3D Array of Interacting Dipolar $S=3$ Atoms

We report on measurements of the dynamics of the total magnetization and spin populations in an almost unit-filled lattice system comprising about ${10}^4$ spin $S=3$ chromium atoms, under the effect of dipolar interactions. The observed spin population dynamics is unaffected by the use of a spin echo and fully consistent with numerical simulations of the $S=3$ $XXZ$ spin model. On the contrary, the observed magnetization decays slower than in simulations and, surprisingly, reaches a small but nonzero asymptotic value within the longest timescale. Our findings show that spin coherences are sensitive probes to systematic effects affecting quantum many-body behavior that cannot be diagnosed by merely measuring spin populations.

Figure 1

(a) The experimental system consists of a 3D array of dipolar Cr atoms, with a spatially varying magnetic field $B(\mathbf{r})$. (b) Experimental rf sequence: Cr atoms are initialized in the $m_s=-3$ spin state and rotated by the first $\pi /2$ pulse to align with the $x$ direction; the second $\pi /2$ pulse is used to measure the magnetization. Because of magnetic noise, the collective spin makes an angle $\varphi$ with respect to $x$ when this second pulse is imparted. We impart or not an echo pulse using an additional $\pi$ pulse at half the evolution time $t/2$. (c) Time evolution of the fractional populations $p_{m_s}$ of four $m_s$ spin components, with and without spin echo (full circles and empty triangles, respectively). Each data point corresponds to the average of ten realizations. Error bars correspond to statistical uncertainties. The solid (dashed) lines show the numerical results obtained with GDTWA with (without) spin echo, for a lattice with unit filling and $B_Q/h=-4\mathrm{Hz}$. From bottom to top: $m_s=-3$ (black), $m_s=-2$ (green), $m_s=-1$ (orange), $m_s=0$ (blue). The colored bands in the GDTWA calculations account for a $\pm 2\%$ experimental error in the first pulse area.

Figure 2

Normalized spin component $M_z$ measured after a Ramsey sequence, with 60–100 realizations for each spin dynamics duration. Top: absolute values of $M_z$; blue triangles and green circles correspond to experiments without and with spin echo, respectively. Inset: the parameter $\eta$ (see text) is evaluated from the corresponding $M_z$ distributions. The horizontal lines show the expected value for respectively the PD of a classical spin (dashed, red) and the Gaussian PD (dotted, purple). Bottom: histograms of the $M_z$ distributions are shown together with the PD used to fit them (solid line); see text.

Figure 3

Values of the normalized magnetization $\ell$ derived by fitting the distributions of $M_z$ after the Ramsey sequence. Error bars represent the 68% confidence interval and are detailed in the Supplemental Material [23]. Filled circles and empty triangles are measurements with and without the spin-echo pulse, respectively. The black dashed line shows the numerical results with the spin-echo pulse applied, obtained with GDTWA for the same lattice configuration as in Fig. 1. The blue dashed line shows the dynamics without the spin echo, obtained from the Gaussian TWA simulations [23]. The blue solid line corresponds to GDTWA simulations effectively accounting for atomic motion in the lattice, with $B_Q/h=-2\mathrm{Hz}$. Inset: difference between the two experimental determinations of $\ell$, comparing the results of the fit of the experimental probability distributions to Eq. (5). Error bars correspond to the quadratic average of the standard deviations associated with either methods.