Figure 1

Real ($\eta -1$, solid red line) and imaginary ($\kappa$, dashed blue line) parts of the index of refraction for a uniform gas of ${}^6\mathrm{Li}$ atoms as a function of laser detuning. The atoms are taken to be in the two lowest hyperfine states with densities $n_{\downarrow }=n_{\uparrow }=1/[2(532\hspace{0.5em}\mathrm{nm}){}^3]=3.3\times {10}^{12}$ cm${}^{-3}$ in an external magnetic field of 834 G, yielding a Zeeman splitting of $2{\Delta }_0=2\pi \times 76$ MHz. The laser excites the ${\mathrm{D}}_2$ line with linewidth $\Gamma =2\pi \times 5.9\hspace{0.5em}\mathrm{MHz}\approx 2{\Delta }_0/13$. Thin vertical lines denote the two resonances and their midpoint.Reuse & Permissions

Figure 2

Geometry for scattering from a ($0\hspace{0.5em}0\hspace{0.5em}1$) plane (left) and a $(\frac{1}{2}\hspace{0.5em}\frac{1}{2}\hspace{0.5em}\frac{1}{2})$ plane (right) for a cubic lattice with lattice spacing $a$. The scattering planes are shaded. The thin solid line is normal to the scattering plane. The color of the lattice sites indicates spin; this figure shows AFM ordering. The arrows represent the directions of the incoming and outgoing scattered light. The labeled angles are between the outgoing light and the normal, which is the same value as the angle between the incoming light and the normal. The values of the angles are given by the Bragg condition: $\cos {}^{-1}(|\mathbf{Q}|/2|\mathbf{k}|)$, assuming lattice spacing of $a=532$ nm and probe wavelength of 671 nm.Reuse & Permissions

Figure 3

Staggered spin per atom of the system (red squares) calculated using a quantum Monte Carlo simulation of $6^3$ atoms and maximum scattering cross sections of the magnetic and nonmagnetic Bragg peaks (blue circles and green triangles, respectively) as a function of temperature. The cross sections are calculated using Eq. (4) with the above value of staggered magnetization for a system of ${41}^3$ atoms.Reuse & Permissions

Figure 4

(a)–(c) Differential cross section of the $(001)$ Bragg peak for a fully polarized spherical sample with $N=925$ atoms ($\mathrm{OD}=37$), resonant probe light, and varying mean displacements (disorder) $\Delta x=\sigma /a=0,\hspace{0.5em}0.1,\hspace{0.5em}0.2$ [corresponding to (a), (b), (c)] calculated from Eqs. (10) and (A1) (blue triangles), as a function of outgoing direction $\delta \theta$ relative to the peak center (directions shown in Fig. 2). For comparison, we also show the Born approximation result (red circles). The lines are guides for the eye. (d) Intensity of the $(0\hspace{0.5em}0\hspace{0.5em}1)$ Bragg peak as a function of detuning $\Delta$ in the fully polarized case $\Delta x=0,\hspace{0.5em}0.1,\hspace{0.5em}0.2$ (solid red, dashed green, dotted blue lines, respectively). The resonance is at $\Delta =0$. (e), (f) Intensity of the $(\frac{1}{2}\hspace{0.5em}\frac{1}{2}\hspace{0.5em}\frac{1}{2})$ Bragg peak as a function of $\Delta$ for a Néel AFM state with Zeeman splitting between spin states of $2{\Delta }_0=2\Gamma$ for varying $\Delta x=0,\hspace{0.5em}0.1,\hspace{0.5em}0.2$ (solid red, dashed green, dotted blue lines, respectively) and particle numbers (e) $N=925$ ($\mathrm{OD}=7.4$ at $\Delta =0$) and (f) $N=257$ ($\mathrm{OD}=4.8$ at $\Delta =0$). The data in (d)–(f) have been symmetrized about $\Delta =0$ to remove the effect of residual polarization caused by the small sample size. The vertical dotted lines in (e) and (f) indicate the atomic optical resonance.Reuse & Permissions

Figure 5

Illustration of the Bloch sphere representations of the various AFM configurations. (Left) Isotropic AFM state, where the spins (large red arrows) may lie anywhere on the Bloch sphere. The $z$ component of the staggered spin vector is shown by the small blue arrow. (Center) Canted AFM state, where the spins are restricted to be near the $x$-$y$ plane but lift off of it by some small canting angle (exaggerated here for clarity). In this case, the $z$ projection of the staggered spin vector is zero. (Right) Canted AFM state after applying a $\pi /2$ pulse. Now the staggered spin has a nonzero $z$ component (small blue arrow), which may be detected by Bragg scattering.Reuse & Permissions

Figure 6

Shot-to-shot probability distributions for the intensity of the Bragg scattering signal for two cases. The solid red line shows the case when the staggered magnetization may point in any direction. The dashed blue line shows the case for a canted AFM, after applying a $\pi /2$ pulse to rotate the spins from the $x$-$y$ plane to the $x$-$z$ plane. The vertical lines indicate the mean values of the respective distributions. The vertical scale is normalized such that the areas under both curves are one.Reuse & Permissions