Analogous Hawking radiation and quantum entanglement in two-component Bose-Einstein condensates: The gapped excitations

The condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system are studied with the Rabi transition between atomic hyperfine states where the system can be represented by a coupled two-field model of gapless excitations and gapped excitations. We set up the configuration of the supersonic and subsonic regimes with the acoustic horizon between them in the elongated two-component Bose-Einstein condensates, trying to mimic Hawking radiations, in particular due to the gapped excitations. The simplified steplike sound speed change is adopted for the subsonic-supersonic transition so that the model can be analytically treatable. The effective-energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency ${\omega }_{\min }$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the Hawking modes significantly deviates from that of the gapless cases near the threshold frequency due to the modified gray-body factor, which vanishes as the mode frequency is below ${\omega }_{\min }$. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around $\omega \sim {\omega }_{\min }$. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small $\omega$ regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.

Figure 1

The schematic plot depicts two sides of $x=0$. The supersonic region $x<0$ has the sound speed $c_l<v$, while the subsonic region $x>0$ has the sound speed $c_r>v$. This setting is achieved by tuning the intraspecies interaction as $U_r>U_l$ in (7). This works for both gapless and gapped excitations.

Figure 2

The frequency $\omega$ varies with wave number $k$ according to the dispersion relation of gapped excitation (21) (solid line with lighter color) and gapless excitation (19) (dashed line with darker color). In the left panel, a given $\omega$ gives four real-valued roots when $\omega <{\omega }_{\max }$. However, in the right panel two real-valued roots exist only for $\omega \ge {\omega }_{\min }$. We consider the single-metric geometry $c_p=c_d$.

Figure 3

Schematic representation of the scattering modes and the decaying modes. In the downstream (supersonic) region, there are four plane wave modes, while in the upstream (subsonic) region, there are only two plane wave modes and one decaying mode. The amplitudes for each mode can be solved by the matching conditions in (43).

Figure 4

Schematic representation of $ur$ outgoing channel scattering processes when ${\omega }_r(={\omega }_{\min })<\omega <{\omega }_{\max }$ in (a) and when $\omega <{\omega }_r(={\omega }_{\min })$ in (b).

Figure 5

The Hawking spectrum $n_{p\omega }^{ur}$ varies as a function of $\omega$ (in the unit of $m{c}_{p,l}^2$) with different Rabi frequencies; $\mathrm{\Omega }/{\rho }_0{U}_l=0.0$ (red), $3.3\times {10}^{-4}$ (blue), and $6.6\times {10}^{-4}$ (green). The solid lines are based on the numerical computation with details stated in the text, while dashed lines are due to the analytical prediction in (63). We consider $m_{p,l}=7/5,m_{p,r}=3/4$, with $U_r/U_l=8/3,U_{AB}/U_l=1/3$.

Figure 6

Schematic representation of $ul$, out-channel scattering processes when ${\omega }_{\mathrm{r}}<\omega <{\omega }_{\max }$ in (a) and $\omega <{\omega }_r$ in (b).

Figure 7

Schematic representation of $vl$, out-channel scattering processes when ${\omega }_r<\omega <{\omega }_{\max }$ in (a), and $\omega <{\omega }_r$ in (b).

Figure 8

The spectrums of $n_{p-\omega }^{ul}$ in (a) and $n_{p\omega }^{vl}$ in (b) vary as a function of $\omega$ (in units of $m{c}_{p,l}^2$) with different Rabi frequencies; $\mathrm{\Omega }/{\rho }_0{U}_l=0.0$ (red), $3.3\times {10}^{-4}$ (blue), and $6.6\times {10}^{-4}$ (green). The analytical predictions drawn as dashed lines are obtained from (73)–(74) and (79) while the numerical results, whose details are stated in the text, are presented with solid lines. The parameters are the same as that used in Fig. 5.

Figure 9

The density-density correlation function pattern is obtained from numerical computations of the integral (83) with the scattering coefficients, wave numbers and mode amplitudes also numerically obtained with the details stated in the text. We have used the parameters $m_{p,l}=7/5$, $m_{p,r}=3/4$, and $\mathrm{\Omega }/{\rho }_0{U}_l=6.6\times {10}^{-4}$.

Figure 10

The time evolution of ${\mathrm{\Delta }}_{d,\omega }^{ur,ul}$ as a function of the time in unit of period $T_{\omega }=2\pi /\omega$ with the parameters Mach number $m_{p,r}=0.63$, $m_{p,l}=1.63$, the coupling constant $\alpha /U_l=0.13$, the Rabi frequency $\mathrm{\Omega }/{\rho }_0{U}_l=6.6\times {10}^{-4}$. Considering (a) $\omega =0.041T_H<{\omega }_{\min }=0.157T_H$, ${\mathrm{\Delta }}_{d\omega }^{ur,ul}$ behaves as an underdamped oscillation, (b) $\omega =0.124T_H<{\omega }_{\min }=0.157T_H$, ${\mathrm{\Delta }}_{d\omega }^{ur,ul}$ behaves as an overdamped oscillation, and (c) $\omega =0.16T_H>{\omega }_{\min }=0.157T_H$, the behavior of ${\mathrm{\Delta }}_{d\omega }^{ur,ul}$ has mixture of overdamped oscillation for $ul$ mode and underdamped oscillation for $ur$ mode. The dashed lines indicate the values of ${\mathrm{\Delta }}_{d,\omega }^{ur,ul}$ with no corrections from the gapped excitations.

Figure 11

Late-time saturation of ${\mathrm{\Delta }}_{d\omega }^{ur,ul}$ is plotted as function of $\omega$ in unit of effective temperature $T_H$. By varying the Rabi frequency $\mathrm{\Omega }/{\rho }_0{U}_l$: (a) $6.6\times {10}^{-4}$, (b) $1.33\times {10}^{-3}$, and (c) $2\times {10}^{-3}$, we show ${\mathrm{\Delta }}_{d\omega }^{ur,ul}$ with two different values of the mutual coupling constants $\alpha /U_l:0.1$ (red-dotted line) and 0.17 (blue-dashed line). Other parameters are chosen as $m_{p,r}=0.63$, $m_{p,l}=1.63$.