Pseudo-steady-state non-Gaussian Einstein-Podolsky-Rosen steering of massive particles in pumped and damped Bose-Hubbard dimers

We propose and analyze a pumped and damped Bose-Hubbard dimer as a source of continuous-variable Einstein-Podolsky-Rosen (EPR) steering with non-Gaussian statistics. We use and compare the results of the approximate truncated Wigner and the exact positive-$P$ representation to calculate and compare the predictions for intensities, second-order quantum correlations, and third- and fourth-order cumulants. We find agreement for intensities and the products of inferred quadrature variances, which indicate that states demonstrating the EPR paradox are present. We find clear signals of non-Gaussianity in the quantum states of the modes from both the approximate and exact techniques, with quantitative differences in their predictions. Our proposed experimental configuration is extrapolated from current experimental techniques and adds another apparatus to the current toolbox of quantum atom optics.

Figure 1

Averaged positive-$P$ values for ${\kappa }_4$ in the two wells, with $\chi ={10}^{-3}$, $\epsilon =10$, $\gamma =1$, and $J=1$. Damping is at the second well and the horizontal axis is a dimensionless time, $Jt$. The steady-state predictions of the truncated Wigner are shown as dash-dotted lines. All quantities plotted here and in subsequent figures are dimensionless.

Figure 2

The products of the inferred quadrature variances for damping at well 2, with $\chi ={10}^{-3}$, $\gamma =1$, $\epsilon =10$, and $J=1$, optimized for the quadrature angles of greatest violation of the inequalities. The value marked well $i$ refers to the product $\mathrm{\Pi }{V}_{ij}$. The predictions of the positive $P$ and truncated Wigner are indistinguishable for these quantities.

Figure 3

The products of the inferred quadrature variances for damping at well 2 for $\chi ={10}^{-2}$, $\gamma =1$, $\epsilon =10$, and $J=1$, optimized for the quadrature angles of greatest violation of the inequalities. The value marked well $i$ refers to the product $\mathrm{\Pi }{V}_{ij}$. The predictions of the positive $P$ and truncated Wigner are indistinguishable for these quantities.

Figure 4

Averaged values for ${\kappa }_4$ in the two wells, with $\chi ={10}^{-2}$, $\epsilon =10$, $\gamma =1$, $J=1$, and damping at the first well, calculated by using the positive-$P$ representation. The steady-state truncated Wigner predictions are shown as the dash-dotted lines.

Figure 5

The products of the inferred quadrature variances for damping at well 1, with $\chi ={10}^{-2}$, $\gamma =1$, $\epsilon =10$, and $J=1$, optimized for the quadrature angles of greatest violation of the inequalities. The value marked well $i$ refers to the product $\mathrm{\Pi }{V}_{ij}$. The predictions of the positive $P$ and truncated Wigner are indistinguishable for these quantities.