Protecting Spin Coherence in a Tunable Heisenberg Model

Using an ensemble of atoms in an optical cavity, we engineer a family of nonlocal Heisenberg Hamiltonians with continuously tunable anisotropy of the spin-spin couplings. We thus gain access to a rich phase diagram, including a paramagnetic-to-ferromagnetic Ising phase transition that manifests as a diverging magnetic susceptibility at the critical point. The susceptibility displays a symmetry between Ising interactions and $XY$ (spin-exchange) interactions of the opposite sign, which is indicative of the spatially extended atomic system behaving as a single collective spin. Images of the magnetization dynamics show that spin-exchange interactions protect the coherence of the collective spin, even against inhomogeneous fields that completely dephase the noninteracting and Ising systems. Our results underscore prospects for harnessing spin-exchange interactions to enhance the robustness of spin squeezing protocols.

Figure 1

Experimental scheme for engineering Heisenberg interactions. (a) Atomic spins in optical cavity precess about magnetic field $\mathbf{B}$ at angle $\mathrm{\Theta }$ from cavity axis ${\widehat{\mathbf{z}}}_c$. The enlargement shows spin texture with three distinct regions $A$, $B$, $C$ as starting point for Hamiltonian tomography. (b) Atom-induced birefringent splitting ${\omega }_{c+}-{\omega }_{c-}=2\mathrm{\Omega }\mathcal{F}·{\widehat{\mathbf{z}}}_c$ of ${\sigma }_{\pm }$ cavity modes. (c) Effect of precessing spins on intracavity intensities of ${\sigma }_{\pm }$ light: projection of static $z$ component (green) shifts dc level to generate Ising interactions, while oscillating projections of transverse components (red) modulate the intensities to generate spin-exchange interactions.

Figure 2

Hamiltonian tomography for determining (a) Ising and (b) $XY$ couplings. (i) Schematic of initial state and its evolution, showing probe spins precessing about mean field along (a) $-\widehat{\mathbf{z}}$ or (b) $-\widehat{\mathbf{y}}$. (ii) Representative measurements at $\mathrm{\Theta }\approx 53°$ with $|\mathbf{B}|=3\mathrm{G}$. The initial state is indicated above each plot, showing the direction of the spin vector in regions A, B, C. These measurements show (a) phase $\varphi$, with opacity indicating transverse spin length or (b) magnetization $⟨f_z⟩$. We measure with both signs of drive detuning $\delta$ to compare antiferromagnetic (AFM) [(a) $\delta =2\pi \times 7.5\mathrm{MHz}$, (b) $2\pi \times 5.5\mathrm{MHz}$] and ferromagnetic (FM) [(a) $\delta =-2\pi \times 5.5\mathrm{MHz}$, (b) $-2\pi \times 5.5\mathrm{MHz}$] couplings. (iii) Cuts through the regions initially polarized along $\widehat{\mathbf{x}}$ (crimson) and $-\widehat{\mathbf{x}}$ (pink) showing (a) $\varphi (t)$ with linear fits and (b) $⟨f_z(t)⟩$ with sinusoidal fits. Crosses (triangles) are for AFM (FM) couplings. (c) $J^z$ (blue circles) and $J^{xy}$ (green squares) versus $\mathrm{\Theta }$. Dark (light) markers are for blue (red) drive detuning $\delta =\pm 2\pi \times 5.3(4)\mathrm{MHz}$.

Figure 3

Magnetic susceptibility of Ising and $XY$ models. (a) Phase diagram of $H_{XXZ}+h_x{F}_x$ for collective spin model in large-$\mathcal{F}$ limit. Color shows prediction for $\log [\chi ]$, which diverges at transition between paramagnetic (PM) and ferromagnetic (FM) phases. Data for (b)–(d) and corresponding theory curves were taken along blue, green, and orange cuts. (b) Representative measurements of $⟨f_z⟩$ versus $h_z$ at points (i)–(iii) in phase diagram; $\chi (J^z)$ is extracted from spatially averaged data (left subplots with dark blue fit curves) [55]. (c) Susceptibility $\chi$ versus $J^z$ (blue circles) and versus $J^{xy}$ (green squares). Dashed gray line indicates maximum detectable slope. (d) Susceptibility versus $\mathrm{\Theta }$ at fixed drive power. For noninteracting spins, $\chi =1$ [black lines in (c) and (d)].

Figure 4

Protection against dephasing. (a),(b) Phase $\varphi$ and global contrast $C_g$ at $\mathrm{\Theta }=\pi /2$ for (a) no interactions and (b) $XY$ interactions [${\mathrm{\Lambda }}^{xy}/\mu \text{L}=-0.43(4)$], in magnetic field gradient $|\mu |=2\pi \times 2.1(1)\mathrm{kHz}/z_R$. Opacity indicates length of transverse spin component. (c) Phase winding ${\varphi }_L(t)$ for ferromagnetic $XY$ interactions (squares, $\mathrm{\Theta }=\pi /2$) or antiferromagnetic Ising interactions (circles, $\mathrm{\Theta }=0$) of equal strength. Dashed lines show ${\varphi }_L$ in noninteracting systems at $\mathrm{\Theta }=0$ (blue) and $\mathrm{\Theta }=\pi /2$ (green). (d) Global contrast $C_g$ versus collective interaction strength $|{\mathrm{\Lambda }}^{xy,z}|$ for $XY$ (green squares) and Ising (blue circles) interactions, at $t=0.5\mathrm{ms}$. Solid (dotted) curves show mean-field model with (without) free-space scattering. Dashed gray line indicates initial contrast $C_0$.