Spacetime structure and vacuum entanglement

We derive the structure of the density matrix for two Unruh-DeWitt detectors coupled to a massless scalar field to all orders of perturbation theory, in spacetimes admitting a well-defined Wightman function. Calculating all of its leading terms enables us to fully characterize observable correlations, entanglement, and quantum discord. We apply these results to study detector responses in two locally flat topologically nontrivial spacetimes constructed from identifications of Minkowski space. We demonstrate how local statistics and detector-detector correlations depend on the global spacetime structure. In particular, we show that if the spacetime has a preferred direction, this direction may be inferred from the dependence of correlations between the two detectors on their orientation. While using such measurements to distinguish spacetimes with identical local geometry is apparently impractical, this effect points to fundamental connections between quantum field correlations and the structure of spacetime. This relationship may also be relevant in the phenomenology of the early Universe.

Figure 1

The (a) concurrence $\mathcal{C}({\rho }_{AB})/{\epsilon }_0^2$ and the (b) correlation function ${\text{corr}}_{AB}/{\epsilon }_0^2$ are plotted for two identical detectors in Minkowski space $\mathcal{M}$, as a function of their separation $L$ and their energy gap $\mathrm{\Omega }$ in units of $\sigma$. The red line in (b) corresponds to ${|X|}_M=A_M$, above which the concurrence vanishes exactly. The solid black regions in the plots indicate the plot has been clipped at (a) ${\text{corr}}_{AB}/{\epsilon }_0^2=2$ and (b) ${\mathcal{C}}_{\mathcal{M}}({\rho }_{AB})/{\epsilon }_0^2=0.25$.

Figure 2

The transition probability of a detector in Minkowski space $\mathcal{M}$ is compared to the transition probability of a detector in both (a) the ${\mathcal{M}}_0$ spacetime and (b) the ${\mathcal{M}}_{-}$ spacetime, by plotting it as a function of the energy gap of the detector, $\mathrm{\Omega }$ in units of $\sigma$, for different circumferences $\ell$ of the universe. In both (a) and (b), the dashed blue line is the Minkowski transition probability $A_M$. In the leading order, the probability of the transition $|1⟩\to |0⟩$, $E_1-E_0=\mathrm{\Omega }>0$, is equivalent to the excitation probability $|0⟩\to |1⟩$ with $\mathrm{\Omega }<0$. The inset is a magnification of the excitation probability $\mathrm{\Omega }>0$.

Figure 3

As a measure of the effect the spatial topology has on the observable correlations in ${\rho }_{AB}$, we plot the difference between the correlation function for detectors in Minkowski space and (a) the ${\mathcal{M}}_0$ spacetime, ${\text{corr}}_{AB}^M/{\epsilon }_0^2-{\text{corr}}_{AB}^0/{\epsilon }_0^2$, and (b) the ${\mathcal{M}}_{-}$ spacetime, ${\text{corr}}_{AB}^M/{\epsilon }_0^2-{\text{corr}}_{AB}^{-}/{\epsilon }_0^2$. In addition, we plot the contour lines of the Minkowski correlation function ${\text{corr}}_{AB}^M/{\epsilon }_0^2$ to serve as a benchmark. In both plots, we have chosen both detectors to lie in the $xy$-plane, so $\mathrm{\Delta }z=0$, and choose the length of the identified direction to be $\ell =1$. In plot (b), we have chosen ${\mathbf{d}}_A$ and ${\mathbf{d}}_B$ to be parallel so that $L=d_B-d_A$ and have set $d_A=0.1$. The red region indicates when the correlations between the detectors are greater in Minkowski space than the identified spacetime (${\mathcal{M}}_0$ or ${\mathcal{M}}_{-}$) and vice versa for the blue region. The solid white and black regions indicate where the plots have been clipped, respectively, at 0.10 and $-0.10$.

Figure 4

The concurrence of ${\rho }_{AB}$ for two detectors in the cylindrical universe ${\mathcal{M}}_0$ with circumference $\ell =1$ is plotted as a function of their orientation $\theta$ with respect to the identified $z$-direction. As ${\mathcal{M}}_0$ is translational invariant, we can choose our coordinate system such that detector $A$ is at the spatial origin ${\mathbf{x}}_A=0$ and detector $B$ is located at ${\mathbf{x}}_B=(L\cos \theta ,0,L\sin \theta )$.