Einstein-Podolsky-Rosen correlations between two uniformly accelerated oscillators

We consider the quantum correlations, i.e. the entanglement, between two systems uniformly accelerated with identical acceleration $a$ in opposite Rindler quadrants which have reached thermal equilibrium with the Unruh heat bath. To this end we study an exactly soluble model consisting of two oscillators coupled to a massless scalar field in $1+1$ dimensions. We find that for some values of the parameters the oscillators get entangled shortly after the moment of closest approach. Because of boost invariance there are an infinite set of pairs of positions where the oscillators are entangled. The maximal entanglement between the oscillators is found to be approximately 1.4 entanglement bits.

Figure 1

Minkowski and Rindler coordinates. The Minkowski time and space coordinates are $t$ and $x$; $V$, $U=t\pm x$ are the Minkowski light like coordinates; ${\tau }_R$, ${\tau }_L$ and ${\rho }_R$, ${\rho }_L$ denote the Rindler time and space coordinates in the right and left Rindler quadrants, respectively. Note that ${\tau }_R$ increases toward the future, whereas ${\tau }_L$ increases toward the past. The uniformly accelerated oscillators follow the trajectories ${\rho }_R=a^{-1}$, and ${\rho }_L=a^{-1}$ which are indicated in the figure by bold curves. Because of Lorentz invariance, there is an infinite set of pairs of points for which the squared invariant distance $\Delta s^2$ between the oscillators is minimal, such as the pair $(P_L,P_R)$ or the pair $(P_L^{\prime },P_R^{\prime })$ indicated on the figure. The two oscillators are only entangled shortly after having reached positions where $\Delta s^2$ is minimal. For instance if the left oscillator is at $P_L$ (or $P_L^{\prime }$), the right oscillator should be along its world line, slightly after $P_R$ (or $P_L^{\prime }$).

Figure 2

Maximum entanglement between the oscillators, as a function of the dimensionless parameters $\gamma$ and $\omega$. The outer line is the frontier of the region where the oscillators get entangled. The contour lines correspond to increases in the logarithmic negativity $E_N$ in steps of 0.1. The maximum entanglement occurs for $\gamma \simeq 0.703$, $\omega \simeq 0.0845$ when $E_N=1.406$.

Figure 3

Evolution of the logarithmic negativity $E_N$ between the two oscillators as a function of $T={\tau }_R-{\tau }_L$ for the values $\gamma =0.703$, $\omega =0.0845$ for which it reaches its maximal value of 1.406. In these units the period of the oscillator is $2\pi /\omega \simeq 74$, and the ”lifetime” of the first excited state of the oscillator is ${\Gamma }^{-1}=(\gamma \omega {)}^{-1}=17$. (We put quotes because the concept of lifetime is not well defined when it is smaller than the oscillation period, although it serves as a useful guiding concept for understanding what are the different time scales involved). The oscillator is thus entangled for less than one oscillation period, and during this time it exchanges several quanta with the Unruh heat bath. Note also that in this case the Boltzman factor $e^{-2\pi \omega }\simeq 0.6$ is of order 1 which implies that several of the oscillator levels have significant probability of being populated -this is necessary since $E_N>1$ requires a system with dimension greater than 2-. The shapes of the curves for other values of $\gamma$ and $\omega$ are similar, except that the maximum entanglements reached are smaller, and that the times at which these maxima are reached are different.