Entanglement from thermal blackbody radiation

Two noninteracting quantum systems which couple to a common environment with many degrees of freedom initially in thermal equilibrium can become entangled due to the indirect interaction mediated through this heat bath. I examine here the dynamics of reservoir-induced entanglement for a heat bath consisting of a thermal electromagnetic radiation field, such as blackbody radiation or the cosmic microwave background, and show how the effect can be understood as result of an effective induced interaction.

Figure 1

Entanglement of formation $E$ for two initially not entangled DQDs with $d=10\mathrm{nm}$ coupled to the CMB at $T=2.73\mathrm{K}$ as a function of ${\log }_{10}(t_0∕\tau )$ and ${\log }_{10}(t∕\tau )$, $\tau =\beta \hslash$. Black means perfect entanglement, $E=1$, white no entanglement $E=0$. Entanglement is created only for $t∕\tau \gtrsim {10}^{12}{(t_0∕\tau )}^3$.

Figure 2

(Color online) Entanglement of formation for $t⪢t_0⪢\tau$ as function of $v={\omega }_{\mathit{max}}d∕c_0$ and the phase ${\phi }_{-}$. The latter increases linearly with time $t$ leading to entanglement oscillations, and decays as $t_0^{-3}={(R∕c_0)}^{-3}$, which makes the entanglement creation very slow for large distances $R$.

Figure 3

Entanglement of formation $E$ for a hypothetical coupling to the cos waves only. Entanglement production is basically independent of $R$ in the interval shown, and thus possible quasiinstantaneously $(t<t_0)$. Same parameters and gray-scale code as in Fig. 1.

Figure 4

Entanglement of formation for $t_0={10}^6\tau$ as function of $\gamma$ and ${\log }_{10}(t∕\tau )$. The parameter $\gamma$ measures the coupling strength to the sin waves (0 no coupling, 1 full coupling). Already values of $\gamma$ only slightly smaller than 1 would lead to the entanglement creation before the light cone $(t<t_0)$. Same parameters and gray-scale code as in Fig. 1.