Entangled phonons in atomic Bose-Einstein condensates

We theoretically propose the measurement of phonon entanglement as a tool to study the quantum dynamics of atomic Bose-Einstein condensates. In particular, we show that nonseparability of the phonon modes offers a unambiguous signature of the quantum origin of the phonon emission by analog Hawking and dynamical Casimir processes. The method is numerically validated by applying a generalized Peres-Horodecki criterion to a truncated Wigner description of the condensate. Viable strategies to implement the proposed scheme in state-of-the-art experiments are discussed.

Figure 1

Entanglement between phonons generated by the analog DCE. The shading indicates the regions of the $k_1\text{−}{k}_2$ momentum plane where the GPH function $\mathcal{P}<0$ (darker gray, larger $\left|\mathcal{P}\right|$) for wave packets centered at $\Delta x=|x_1-x_2|=120{\xi }_{-}$ (left) and $240{\xi }_{-}$ (right) with $\sigma =35{\xi }_{-}$, at $k_{B}T/{\mu }_{-}=0$ (top) and $0.2$ (bottom). Dashed lines indicate $k_2=-k_1$. In all plots $t_{\mathrm{obs}}=120\hslash /{\mu }_{-}$.

Figure 2

Entanglement between phonons generated by the analog DCE. The shading indicates the region of the $\Delta x\text{−}{t}_{\mathrm{obs}}$ plane where the GPH function $\mathcal{P}<0$ (darker gray, larger $\left|\mathcal{P}\right|$) for wave packets with $k_2=-k_1=0.25/{\xi }_{-}$ (top) and $1/{\xi }_{-}$ (bottom) with $\sigma =15{\xi }_{-}$, at $T=0$. Dashed lines indicate $t_{\mathrm{obs}}=\Delta x/2\left|v^{\mathrm{gr}}\left(k_{1,2}\right)\right|$. In all plots $t_{\mathrm{obs}}=120\hslash /{\mu }_{-}$.

Figure 3

Cut of the left panel of Fig. 2 along the horizontal dotted line: The GPH $\mathcal{P}$ function is plotted as a function of the distance $\Delta x$ between the centers of two wave packets with opposite wave vectors $k_2=-k_1=0.25/{\xi }_{-}$ and spatial width $\sigma =15{\xi }_{-}$, at $t_{\mathrm{obs}}=110\hslash /{\mu }_{-}$ after the sudden temporal modulation originating the analog DCE. Solid curve: Best fit of the Gaussian function of Eq. (10); values of the parameters in Eq. (11).

Figure 4

Flowing condensate showing a sonic horizon at $x=0$. Upper panel: Flow velocity (dashed line) and speed of sound (solid line). Lower panels: Bogoliubov dispersion in the subsonic (left) and supersonic (right) regions. Open and solid dots indicate the modes where the dominant Hawking emission occurs.

Figure 5

Entanglement between Hawking phonons. The shading indicates the region of the $k_1\text{−}{k}_2$ momentum plane where the GPH function $\mathcal{P}<0$ (darker gray, larger $\left|\mathcal{P}\right|$) for initial temperatures $k_{B}T/{\mu }_{\mathrm{sub}}=0$ (left) and $0.1$ (right) and a late observation time $t_{\mathrm{obs}}=480\hslash /{\mu }_{\mathrm{sub}}$, where ${\mu }_{\mathrm{sub}}=m{c}_{\mathrm{sub}}^2$ in the subsonic region. To optimize the signal the wave packets are located at $x_1=-61{\xi }_{\mathrm{sub}}$ and $x_2=59{\xi }_{\mathrm{sub}}$ with $\sigma =35{\xi }_{\mathrm{sub}}$. Dashed lines indicate the energy conservation condition (see text). The velocity profile is the one shown in Fig. 4.

Figure 6

Entanglement between Hawking phonons. The shading indicates the region of the $x_1\text{−}{x}_2$ plane where the GPH function $\mathcal{P}<0$ (darker gray, larger $\left|\mathcal{P}\right|$) for different evolution times $t_{\mathrm{obs}}{\mu }_{\mathrm{sub}}/\hslash =240$ (left) and 480 (right). Wave packets are centered at $k_2=k_1=-0.25/{\xi }_{\mathrm{sub}}$ with $\sigma =15{\xi }_{\mathrm{sub}}$. Initial temperature $T=0$. Dashed lines indicate the ballistic condition (see text). The velocity profile is the one shown in Fig. 4.

Figure 7

Sketch of the proposed experimental cavity configuration. The cavity mode has a ring structure with two intersecting running waves generating a longitudinal standing-wave pattern in the region of interest.