Extreme spin squeezing from deep reinforcement learning

Spin squeezing (SS) is a recognized resource for realizing measurement precision beyond the standard quantum limit $\propto 1/\sqrt{N}$. The rudimentary one-axis twisting (OAT) interaction can facilitate SS and has been realized in diverse experiments, but it cannot achieve extreme SS for precision at the Heisenberg limit $\propto 1/N$. Aided by deep reinforcement learning (DRL), we discover size-independent universal rules for realizing nearly extreme SS with a OAT interaction using merely a handful of rotation pulses. More specifically, only six pairs of pulses are required for up to ${10}^4$ particles, while the time taken to reach extreme SS remains on the same order of the optimal OAT squeezing time, which makes our scheme viable for experiments that reported OAT squeezing. This Rapid Communication highlights the potential of DRL for controlled quantum dynamics.

Figure 1

(a) Evolution of SS coefficients are compared among the max-probability protocol from DRL (black solid circles with solid guiding line) under Hamiltonian (1), OAT (blue dashed line) with Hamiltonian $H_{\mathrm{OAT}}=\chi J_z^2$, and the effective TACT (red dashed-dotted line) with Hamiltonian $H_{\mathrm{TACT}}=\chi (J_z^2-J_y^2)/3$ for $N=100$. The initial state is a CSS polarized along the $x$ axis. The lower panel illustrates the corresponding DRL pulse sequence. A zoomed-in region for one of the pairs (shaded) is shown in (b), with the distributions on the Bloch spheres denoting the squeezed states before and after the $\pm \pi /2$ pulse pair, representing a nonlinear anticlockwise rotation around the $x$ axis.

Figure 2

The $N$ dependence of the optimal SS coefficients obtained by applying the optimal DRL policy trained with $N=100$ to other $N$ (black solid circles with solid guiding line and black dotted line). For comparison, the SS coefficients for OAT (blue dashed line) and the effective TACT (red dashed-dotted line) are also shown. Inset: Zoomed-in view of the marked region with $N\in [50,150]$.

Figure 3

The $N$ dependence of the optimal SS coefficients ${\xi }^2$ (a) and the corresponding evolution time (b) for the IS (black solid circles with solid guiding line), OAT (blue dashed line), and the effective TACT (red dashed-dotted line). Inset of (b): Ratios of the optimal squeezing time of IS with respect to OAT ($t_{\mathrm{IS}}/t_{\mathrm{OAT}}$, blue dashed line), and to the effective TACT ($t_{\mathrm{IS}}/t_{\mathrm{TACT}}$, red dashed-dotted line). (c) Same as in (a) except that the vertical axis denotes the SS coefficients ${\xi }_R^2$. Inset of (c): Same as in (b) except that the vertical axis denotes the corresponding evolution time when ${\xi }_R^2$ is optimal. The shaded region in (a) and (c) displays the corresponding total number of pulse pairs used in IS (right vertical axes).

Figure 4

Histograms from 1000 simulations with Gaussian stochastic distributions for $\delta t$ (left panel) and temporal instants of pulse pairs (right panel) for $N=2000$. The ideal IS without noise gives an optimal SS coefficient ${\xi }^2=1.02\times {10}^{-3}$ (black dashed line), while OAT gives ${\xi }^2=6.56\times {10}^{-3}$ and the effective TACT gives ${\xi }^2=0.91\times {10}^{-3}$.