Spin Squeezing with Short-Range Spin-Exchange Interactions

We investigate many-body spin squeezing dynamics in an $XXZ$ model with interactions that fall off with distance $r$ as $1/r^{\alpha }$ in $D=2$ and 3 spatial dimensions. In stark contrast to the Ising model, we find a broad parameter regime where spin squeezing comparable to the infinite-range $\alpha =0$ limit is achievable even when interactions are short ranged, $\alpha >D$. A region of “collective” behavior in which optimal squeezing grows with system size extends all the way to the $\alpha \to \infty$ limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner approximation, are testable in a variety of experimental cold atomic, molecular, and optical platforms.

Figure 1

Representations of the state $|\psi (t)⟩$ of $N=40$ spins initially polarized along the equator and evolved under the OAT Hamiltonian for a time $t$ up to the optimal OAT squeezing time $\chi t_{\mathrm{opt}}^{\mathrm{OAT}}\sim 1/N^{2/3}$. Darker colors at a point $\widehat{\mathit{n}}$ on the sphere correspond to a larger overlap ${|⟨\widehat{\mathit{n}}|\psi (t)⟩|}^2$, where $|\widehat{\mathit{n}}⟩$ is a state in which all spins are polarized along $\widehat{\mathit{n}}$.

Figure 2

The optimal squeezing ${\xi }_{\mathrm{opt}}^2$ (top), minimal squared magnetization $⟨{\mathit{S}}^2{⟩}_{\min }$ (middle), and optimal squeezing time $t_{\mathrm{opt}}$ (bottom) for $N=4096=64^2=16^3$ spins in $D=2$ (left) and $D=3$ (right) spatial dimensions. Spins are initially polarized along the equator and evolved under the $XXZ$ Hamiltonian in Eq. (3). Squeezing ${\xi }_{\mathrm{opt}}^2$ is shown in decibels, and $⟨{\mathit{S}}^2{⟩}_{\min }$ is normalized to its initial value $⟨{\mathit{S}}^2{⟩}_0=(N/2)[(N/2)+1]$. Dashed gray lines mark $\alpha =D$, and dotted gray lines track local minima of $⟨{\mathit{S}}^2{⟩}_{\min }$, marking the boundary between regions of collective and Ising-limited squeezing dynamics, respectively, denoted “S-collective” and “S-Ising.” Other markers in the middle panels indicate vales of $J_z/J_{\perp },\alpha ,D$ that are currently accessible with neutral atoms [42, 43] (cyan line), Rydberg atoms [29, 30, 44] (red dots), polar molecules [31, 32, 45] (green line), magnetic atoms [33, 34] (pink square), and trapped ions [12] (blue line). DTWA results are averaged over 500 trajectories.

Figure 3

Squeezing ${\xi }^2$ and squared magnetization $⟨{\mathit{S}}^2⟩$ over time for the power-law $XXZ$ model with $\alpha =3$ on a 2D lattice of $64\times 64$ spins. Color indicates the value of $J_z/J_{\perp }$, and red lines (at $J_z/J_{\perp }=-2.2$) mark the approximate transition between S-collective and S-Ising phases, when the collective squeezing peak at $\tau \equiv t\times |J_z-J_{\perp }|\sim 6$ drops below the Ising peak at $\tau \sim 1$. For the parameters shown, $⟨{\mathit{S}}^2⟩$ reaches a minimum at $\tau \sim 2$, which means that optimal squeezing at $\tau \sim 1$ is reached before maximal decay of $⟨{\mathit{S}}^2⟩$ in the S-Ising phase.

Figure 4

Optimal squeezing ${\xi }_{\mathrm{opt}}^2$ as a function of system size for the power-law $XXZ$ model with $\alpha =3$ on a 2D lattice of $N=L\times L$ spins. Whereas the amount of squeezing generated in the S-Ising phase is insensitive to system size, squeezing in the S-collective phase grows with system size and as $J_z/J_{\perp }\to 1$ (from below). Dotted gray line tracks minima of $⟨{\mathit{S}}^2{⟩}_{\min }$ as a function of $J_z/J_{\perp }$, as in Fig. 2, marking the approximate dynamical phase boundary.