$(2+1)$-dimensional sonic black hole from a spin-orbit-coupled Bose-Einstein condensate and its analog Hawking radiation

We study the properties of a $2+1$-dimensional sonic black hole (SBH) that can be realized in a quasi-two-dimensional two-component spin-orbit-coupled Bose-Einstein condensate (BEC). The corresponding equation for phase fluctuations in the total density mode that describes phonon field in the hydrodynamic approximation is described by a scalar field equation in $2+1$ dimension whose space-time metric is significantly different from that of the SBH realized from a single component BEC that was studied experimentally and, theoretically, meticulously in literature. Given the breakdown of the irrotationality constraint of the velocity field in such spin-orbit-coupled BEC, we study in detail how the time evolution of such a condensate impacts the various properties of the resulting SBH. By time evolving the condensate in a suitably created laser-induced potential, we show that such a sonic black hole is formed in an annular region bounded by an inner and outer event horizon as well as elliptical ergosurfaces. We observe amplifying density modulation due to the formation of such sonic horizons and show how they change the nature of analog Hawking radiation emitted from such a sonic black hole by evaluating the density-density correlation at different times, using the truncated Wigner approximation (TWA) for different values of spin-orbit coupling parameters. We finally investigate the thermal nature of such analog Hawking radiation.

Figure 1

(a) Cross section of the potential $V_{\text{2D}}$ along the $x$ axis, experienced by the SOC-BEC at 20 ms. (b) Schematic illustration of the formation of the supersonic region (area B) and the subsonic region (area A) of the analog black hole formed in this work. Region A represents the outside of the analog black hole and shaded region B represents the inside of the analog black hole. Thus, the emission of the analog Hawking radiation from this SBH takes place in region A (indicated using curved arrows). AEH and IH are also marked. The arrows in region B represents that the flow is from AEH to IH inside the SBH.

Figure 2

Total density evolution for various times corresponding to ${\eta }^{\prime }/\eta =0.4$ (left) and 0.78 (right) is shown. The white dashed lines in each figure represent the location of the circular step.

Figure 3

(a) and (b) The cross-sectional total density at 60 ms, along $\zeta =0^{\circ },{45}^{\circ }$ for ${\eta }^{\prime }/\eta =0.4$, respectively. Inset shows 1D-WFT $n_d(k,r)$ with a window of width $D=6.5 \mu \mathrm{m}$, centered at $r=20.14 \mu \mathrm{m}$. (c) Growth of background density ${\overline{n}}_d$ and the fluctuation ${\tilde{n}}_d$ as a function of time and (d) shows the ratio of fluctuations in total density to background total density as a function of time. (e) and (f) The results corresponding to (c) and (d), respectively, for the case ${\eta }^{\prime }/\eta =0.78$.

Figure 4

Evolution of the SBH: The figure shows the formation of a single horizon (AEH) of the SBH at the initial time and, as time progresses, IH also gets formed, which is followed by the stratified formation of “local” supersonic-subsonic layers in the annular region. The evolution is shown in (a)–(d) for ${\eta }^{\prime }/\eta =0.4$ and in (e)–(h) for ${\eta }^{\prime }/\eta =0.78$, at various times: 0, 30, 70, and 80 ms, respectively. Data beyond IH layer, after its formation, are not shown.

Figure 5

(a)–(c) Comparison of flow in a 2D two-component BEC (without spin-orbit coupling) to the SOC case for the same setup considered in this work: (a) Normalized velocity ($v_x/\left|v\right|,v_y/\left|v\right|$) at 80 ms in the supersonic region, for a two-component BEC without spin-orbit coupling. (b) and (c) To illustrate the azimuthal flow in a SOC-BEC we plot the normalized velocity for ${\eta }^{\prime }/\eta =0.4$, 0.78, respectively, at 80 ms. Length of arrows represents the magnitude of the velocity. The figures are superposed with the boundaries of the event horizon (red) and ergorsurface (black) of the SBH. Inset in (c) shows a slight distinction between the two.

Figure 6

Time evolution of total density for one of the realizations from the ensemble using TWA at various times for (a) ${\eta }^{\prime }/\eta =0.4$ and (b) 0.78.

Figure 7

Density-density correlation $G^{\left(2\right)}(r,r^{\prime };0^{\circ })$ obtained using TWA, for ${\eta }^{\prime }/\eta =0.4$ and 0.78, is shown in (a)–(f) and (h)–(m), respectively, at various times mentioned on the top of each figure. (g) The ensemble-averaged cross-sectional density $\langle n_d(r,0^{\circ })\rangle$. Arrows marked in (b), (c), and (i), along the negative correlation band emanating from the AEH, are indicative of the presence of spontaneous Hawking radiation emitted from AEH as discussed in the text. AEH and IH are marked for all the illustrative times.

Figure 8

(a)–(c) and (d)–(f) Density-density correlation for ${\eta }^{\prime }/\eta =0.4$, along $\zeta ={45}^{\circ }$ and ${90}^{\circ }$, respectively, at various times mentioned on the top. (g)–(i) $G^{\left(2\right)}(r,r^{\prime };{45}^{\circ })$ at these illustrative times for ${\eta }^{\prime }/\eta =0.78$. Arrows marked in $G^{\left(2\right)}$ at the initial times in (a), (d), and (g) are just symbolic of the presence of spontaneous Hawking radiation emitted from the horizon. AEH and IH are also marked.

Figure 9

$G^{\left(2\right)}(r,r^{\prime };0^{\circ })$ for ${\eta }^{\prime }/\eta =0.4$ at various times, evaluated using consideration of atom number fluctuations in the simulation. AEH and IH are marked.

Figure 10

(a) and (b) Correlation function (with rescaled spatial coordinate) for ${\eta }^{\prime }/\eta =0.4$ along $\zeta =0^{\circ }$ at 20 and 50 ms, respectively. Here the red (thick) line corresponds to the correlation between the distant points located on opposite sides of the horizon with equal propagation times. The corresponding region consisting of these points are also marked in the total density (blue and red) in (c) and (d), respectively. (e) and (f) The profile of $G^{\left(2\right)}$ along the black line in (a) and (b), respectively. (g) and (h) The spectrum of the radiation corresponding to (a) and (b) at the respective times.

Figure 11

(a)–(c) Correlation function (with rescaled spatial coordinate) for ${\eta }^{\prime }/\eta =0.78$ at various times mentioned and angle $\zeta$ in each of the figures. (d)–(f) The corresponding density of outside (red) and inside (blue) regions of SBH, where the correlations are observed in (a)–(c), respectively. (g)–(i) The profile of $G^{\left(2\right)}$ along the black line in (a)–(c), respectively. (j)–(l) The spectrum of the radiation emitted corresponding to (a)–(c), respectively, at the mentioned times.

Figure 12

(a) and (b) Correlation function (with rescaled spatial coordinate) for a two-component BEC (without spin-orbit coupling), along $\zeta =0^{\circ }$, at 20 and 50 ms, respectively. The red (thick) line corresponds to the correlation between the pair of distant points ($r,r^{\prime }$) located on the opposite sides of the horizon, with equal propagation times. The corresponding region consisting of these points are also marked in the total density (blue and red) in (c) and (d), respectively. (e) and (f) The profile of $G^{\left(2\right)}$ along the black line in (a) and (b), respectively. (g) and (h) The spectrum of the radiation corresponding to (a) and (b) at the respective times. Inset shows the agreement between the spectrum evaluated using Eq. (21) (solid curve) with that of the predicted thermal spectrum (dashed curve).

Figure 13

Comparison of the predicted spectrum at 20 ms is shown, for the case of two-component BEC (without SOC) (solid curve blue) with two sets of SOC strengths: ${\eta }^{\prime }/\eta =0.4$ (dashed black) and 0.78 (dotted red). The results along $\zeta =0^{\circ }$ are labeled, with a marker “o” and results along $\zeta ={45}^{\circ }$ are without any marker.

Figure 14

(a) The basic tripod system and (b) the four states obtained from the atom-laser interaction Hamiltonian and a schematic that we are going from dark states to lower energy subspace.

Figure 15

Dispersion relation in subsonic region and supersonic regions for ${\eta }^{\prime }/\eta =0.4$ at 20 ms along $\zeta =0^{\circ }$.

Figure 16

We plot the one-dimensional cross-sectional view of the subsonic and supersonic regimes at 50 ms for ${\eta }^{\prime }/\eta =0.78$ superposed with the space-time diagram.

Figure 17

Comparison of quantum pressure term with the interaction energy is shown for the cross section of density component $n_{+}$ along $0^{\circ },{45}^{\circ }$, and ${90}^{\circ }$ for (a) ${\eta }^{\prime }/\eta =0.4$ (on the left), (b) 0.78 (on the right) at $t=50$ ms.

Figure 18

Total density (left) and its corresponding phase variation (right) for ${\eta }^{\prime }/\eta =0.78$ at 80 ms. The marker “$+$” in the phase shows the presence of vortices at few of the marked minima location of the left figure, where the interference fringes are formed.

Figure 19

Time evolved density for a two-component BEC.

Figure 20

Density-density correlation for a two-component BEC $G^{\left(2\right)}(r,r^{\prime };0^{\circ })$ without SOC, obtained using TWA, is shown at various times.

Figure 21

A cross-sectional profile of time-dependent GPE (TDGPE) density evolution superimposed with those obtained from TWA along $\zeta =0^{\circ }$ using Figs. 2 and 6. In both cases, amplification of density in the supersonic region (where density modulations appear), is observed. However, in this region, slight shifts in the amplitude were observed in the density profiles obtained using TWA.