Accelerating many-body entanglement generation by dipolar interactions in the Bose-Hubbard model

The spin-squeezing protocols allow the dynamical generation of massively correlated quantum many-body states, which can be utilized in entanglement-enhanced metrology and technologies. We study a quantum simulator generating twisting dynamics realized in a two-component Bose-Hubbard model with dipolar interactions. We show that the interplay of contact and long-range dipolar interactions between atoms in the superfluid phase activates the anisotropic two-axis countertwisting mechanism, accelerating the spin-squeezing dynamics and allowing the Heisenberg-limited accuracy in spectroscopic measurements.

Figure 1

(a) Time evolution of the spin-squeezing parameter ${\xi }^2$ defined in (1) for different values of anisotropy parameter $\eta =\{0.0,0.5,1.0\}$ (lines from right to left). The two limiting cases, i.e., $\eta =0$ and $\eta =1$ correspond to OAT and TACT dynamics, respectively. Points correspond to the results from the exact many-body numerical simulation of ${\widehat{H}}_{\mathrm{dBH}}$ given by (9), while solid lines to the numerical results from the effective two-mode model (19) when $N=M=10, U/J=0.01$ and $U_{\uparrow \downarrow }/J=0.95U/J$. (b) Color-encoded values of the spin-squeezing parameter ${\xi }^2$ versus $\chi t$ and $\eta$ obtained from the numerical simulations of the two-mode model (19) for $N={10}^3$ atoms. The solid red line indicates the best squeezing time.

Figure 2

Mean-field phase portraits $\epsilon (\varphi ,z)$ indicating geometrical representations of trajectories and directions of evolution for the effective two-mode model (19) for the anisotropy parameter $\eta =\{{10}^{-5},0.04,0.4,1.0\}$ (from left to right). Arrows show the direction set by the vector $\vec{n}=(\dot{\varphi },\dot{z})$. The angles $\theta$ between ingoing and outgoing flow along constant energy lines around the unstable fixed point are marked by red (see main text). The corresponding values of angles $\theta$ are approximately $\pi , 0.87\pi , 0.64\pi , 0.5\pi$ (from left to right). The left-most panel $\eta \approx 0$ corresponds to OAT, where the distribution of a state localized around $z=0$ is stretched during the evolution. While increasing $\eta$ (consecutive panels) we show how the trajectories around $z=0$ change to form the saddle point resulting in stretching of the initial distribution located nearby the saddle in two different directions with relative angle $\theta$, realizing TACT at $\theta =\pi /2$ when $\eta =1.0$.

Figure 3

(a) Examples of the scaling of the best squeezing ${\xi }_{\mathrm{best}}^2$ and the best squeezing time $\chi t_{\text{best}}$ with the number of atoms $N$ for various values of the parameter of anisotropy $\eta =\{1,0.5,{10}^{-3}\}$. (b) Scaling exponents $\alpha$ obtained by fitting the function $\propto N^{-\alpha }$ to the best squeezing ${\xi }_{\mathrm{best}}^2$ is shown by the red points (red solid line is added to guide the eye). The angle $\theta$ between the ingoing and outgoing mean-field trajectories (see main text) is shown by the dashed blue line.

Figure 4

Schematic of the geometry of the system: gray arrows indicate an initial configuration of the elementary dipoles and the green curve of the optical lattice potential.