Nonclassicality and entanglement for wave packets

Mode-entanglement-based criteria and measures become insufficient for broadband emission, e.g., from spasers (plasmonic nanolasers). We introduce criteria and measures for the (i) total entanglement of two wave packets, (ii) entanglement of a wave packet with an ensemble, and (iii) total nonclassicality of a wave packet. We discuss these criteria in the context of (i) entanglement of two wave packets emitted from two initially entangled cavities (or two initially entangled atoms) and (ii) entanglement of an emitted wave packet with an ensemble or atom for spontaneous emission and single-photon superradiance. We also show that, (iii) when two constituent modes of a wave packet are entangled, this creates nonclassicality in the wave packet as a noise reduction below the standard quantum limit. The criteria we introduce are all compatible with near-field detectors.

Figure 1

Two cavities are initially in an entangled state and they decay into two different reservoirs. We examine the time evolution of the onset of the entanglement between the two reservoirs or, in other words, correlations in the electric field measurements of the wave packets (WPs) emitted into the two reservoirs. We also calculate the total entanglement of the two wave packets in Sec. 4.

Figure 2

Hillery-Zubairy and Simon-Peres-Horodecki criteria, which become ${\lambda }_{\mathrm{HZ}}\left(t\right)={\lambda }_{\mathrm{SPH}}\left(t\right)$ for the two wave packets emitted from two initially entangled cavities depicted in Fig. 1. In contrast to pointwise, $E_1({\mathbf{r}}_1,t)\leftrightarrow E_2({\mathbf{r}}_2,t)$, electric-field correlations studied in Sec. 2, ${\lambda }_{\mathrm{HZ},\mathrm{SPH}}\left(t\right)<0$ witnesses a kind of total entanglement between the two wave packets emitted into two different reservoirs.

Figure 3

Spontaneous emission from a single atom into the vacuum. Evolution of the entanglement, ${\mu }_{\mathrm{HZ}}\left(t\right)<0$, between the atom and the emitted wave packet. Superradiant single-photon emission from an ensemble shows a similar behavior except for the emission time determined by collective decay ${\gamma }_N$ in place of single-atom decay $\gamma$.