Phonon spectrum and correlations in a transonic flow of an atomic Bose gas

Motivated by a recent experiment of J. Steinhauer, we reconsider the spectrum and the correlations of the phonons spontaneously emitted in stationary transonic flows. The latter are described by “waterfall” configurations which form a one-parameter family of stable flows. For parameters close to their experimental values, in spite of high gradients near the sonic horizon, the spectrum is accurately Planckian in the relevant frequency domain, where the temperature differs from the relativistic prediction by less than 10%. We then study the density correlations across the horizon and the nonseparable character of the final state. We show that the relativistic expressions provide accurate approximations when the initial temperature is not too high. We also show that the phases of the scattering coefficients introduce a finite shift of the location of the correlations which has so far been overlooked. This shift is due to the asymmetry of the flow across the horizon, and persists in the dispersionless regime. Finally we show how the formation of the sonic horizon modifies both local and nonlocal density correlations.

Figure 1

Plots of the rescaled atomic density $\rho /{\rho }_{+}$ (top, left) and the Mach number (top, right) for three waterfall solutions similar to that realized in [1]. The flow is from left to right and the subsonic region is on the left side. The asymptotic values of the Mach numbers for the green (dotted), blue (continuous), and orange (dashed) curves are, respectively, $M_{-}\approx 0.52,0.45,0.4$ and $M_{+}=3.75$, 5, 6.25. The unit of the horizontal axis is ${\xi }_{+,5}$; the healing length in the supersonic region for the flow with the central value $M_{+}=5$ (blue curves). The plots are shifted so that the sonic horizon $M=1$ (marked by a horizontal red dashed line the upper right panel) is at $x=0$. The bottom left plot shows the profile of the adimensionalized gradient $\kappa (x){\xi }_{+,5}/c_{+,5}$, where $c_{+,5}$ is the asymptotic downstream sound velocity for the flow with $M_{+}=5$. One clearly sees that $\kappa (x)$ identically vanishes on the right of the potential barrier located near $x/{\xi }_{+,5}=0.4$. When evaluated at $x=0$, $\kappa$ gives the surface gravity ${\kappa }_H$ of Eq. (9). As functions of $M_{+}$, the bottom right plot shows the Hawking temperature $T_H={\kappa }_H/2\pi$ adimensionalized by ${\xi }_H/c_H$ (blue, continuous), ${\xi }_{+}/c_{+}$ (green, dotted), ${\xi }_{-}/c_{-}$ (orange, dashed), and ${\xi }_{-,5}/c_{-,5}$ (red, dot-dashed), for ${\rho }_{+}=1$. The first three values increase with $M_{+}$, unlike the last one which is not monotonic.

Figure 2

Plot of the dispersion relation in homogeneous subsonic (left) and supersonic (right) flows. The horizontal, dashed, black line corresponds to a fixed frequency. The continuous curves correspond to positive values of $\mathrm{\Omega }=\omega -vk$ and the dashed ones to negative values of $\mathrm{\Omega }$. One notices that the two extra roots on the right of the right plot have $\mathrm{\Omega }<0$. As explained in the text, they describe phonons carrying a negative energy in the frame where the potential $V$ is stationary. These two extra roots become complex when $\omega$ reaches the critical frequency ${\omega }_{\max }$ of Eq. (18), which is here indicated by a dotted line.

Figure 3

Space-time diagram of the characteristics associated with the three types of stationary modes propagating in the transonic waterfall flow (from left to right) of Fig. 1 with $M_{+}=5$. The sonic horizon is located at $x=0$, and the characteristics are computed for $\omega =2.5c_{+}/{\xi }_{+}$. As explained in [6, 28], they obey Eq. (15) treated as a Hamilton-Jacobi equation. The initial (final) asymptotic values of their wave vectors are given by the corresponding roots, indicated in the figure at early (late) time. The arrows give the orientation of the group velocity in the rest frame of the potential $V(x)$. The dispersive roots $k_{\omega }^d$, $-k_{-\omega }^d$ and the copropagating one $k_{\omega }^{v,b}$ characterize the three incoming modes.

Figure 4

On the left panel, we show the adimensionalized energy spectrum ${\epsilon }_{\omega }/T_{H,5}$ of the outgoing phonons spontaneously emitted by the scattering on the three flows of Fig. 1, where $T_{H,5}={\kappa }_{H,5}/2\pi$ is the Hawking temperature for the flow with the central value of $M_{+}=5$. The black dashed line shows the adimensionalized Planck energy spectrum evaluated for this flow. On the right panel, we show $T_{\mathrm{eff}}(\omega )$ in units of the corresponding value of $T_H(M_{+})$ for the same three flows. One clearly sees that $T_{\mathrm{eff}}(\omega )$ becomes significantly larger than $T_H$, but this occurs in a domain where ${\epsilon }_{\omega }$ is very small. $T_{\mathrm{eff}}(\omega )$ abruptly drops to 0 when $\omega$ reaches ${\omega }_{\max }$.

Figure 5

Plots of the squared absolute values of the coefficients of Eq. (19) for the waterfall solution with $M_{+}=5$. The left panel shows the coefficients of the first line as well as $|{\tilde{B}}_{\omega }^2|$ and the Planck distribution with temperature $T_{H,5}$ (dashed line), all multiplied by $\omega /T_{H,5}$. The right panel shows the coefficients involving the incoming copropagating $v$-mode and the difference $(|{\beta }_{-\omega }^2|-|{\beta }_{\omega }^2|)\omega /T_{H,5}$. The smallness of these quantities reveals the weakness of the coupling between the $v$-mode and the two $u$-modes.

Figure 6

Left panel: As functions of the supersonic Mach number $M_{+}$, we show the limit $\omega \to 0$ of $T_{\mathrm{eff}}$ of Eq. (24) (green, continuous), the Hawking temperature $T_H$ (blue, dashed), and $T_{\text{step}}$ obtained with the density profile of Eq. (25) (orange, dotted), all adimensionalized by multiplication by the dispersive time scale ${\xi }_H/c_H$. Right panel: Ratio of $|{\tilde{B}}_{\omega =0}^2/{\beta }_{\omega =0}^2|$ for the waterfall (green, continuous) and for the steplike (orange, dotted) profiles. The very close agreement indicates that, for $\omega \to 0$, the ratio $|{\tilde{B}}_{\omega }^2/{\beta }_{\omega }^2|$ only depends on the asymptotic values of $v$ and $c$.

Figure 7

We represent the absolute value of the spontaneous correlation terms ${\mathbb{A}}_{\omega }$ (blue) and ${\mathbb{B}}_{\omega }$ (orange) defined in Eq. (31), calculated for the waterfall solution with $M_{+}=5$, and adimensionalized by $c_{-}{\rho }_{-}$, so as to get a result invariance under a rescaling of $\lambda$; see footnote . The dotted (red) line shows $|{\mathbb{B}}_{\omega }|$ when the initial state of the copropagating $v$-modes is a thermal state with an initial temperature $T_{\mathrm{in}}=10T_H$. The black, dashed curve shows the relativistic limit for ${\mathbb{B}}_{\omega }$ given in Eq. (32).

Figure 8

Left: We show the profile of $G_2^{\mathrm{rel}}(x,x^{\prime })$ of Eq. (33) evaluated along a line orthogonal to the locus of its minima and located far from the horizon. $x^{\prime \prime }$ is a coordinate along this line, defined by $x^{\prime \prime }=0$ when $G_2$ reaches its minimum and $|d{x}^{\prime \prime }|=\sqrt{d{x}^2+d{x}^{\prime 2}}$. The three curves show the correlation profile for the waterfall solutions with $M_{+}=3.75$ (green, dashed), 5 (blue, continuous), and 6.25 (orange, dotted) considered in Fig. 1. Right: We show $G_2^{\mathrm{rel}}(x,x^{\prime })$ of Eq. (c6) as a function of $x$ and $x^{\prime }$ for the waterfall solution with $M_{+}=5$. The dashed segment indicates the domain used to represent the correlations on the left panel. The broad oblique white band centered along $x=x^{\prime }$ corresponds to values of $G_2$ outside the range represented in colors.

Figure 9

Plot of the quantity ${\mathbb{D}}_{\omega }$ of Eq. (36) computed for the waterfall solution with $M_{+}=5$, and adimensionalized by $c_{-}^2{\rho }_{-}^2$. On the left panel, the blue continuous line gives the numerical value predicted by the Bogoliubov–de Gennes equation, while the black, dashed line shows its relativistic limit given in Eq. (37). The dotted orange line shows the maximum value of ${\mathbb{D}}_{\omega }$ of Eq. (38). On the right panel, we show how ${\mathbb{D}}_{\omega }$ varies when assuming that the copropagating incoming $v$-modes have an initial temperature $T_{\mathrm{in}}$. Curves are shown for $T_{\mathrm{in}}/T_H={10}^{-6}$ (blue), 3 (orange), and 10 (green).

Figure 10

Left: As a function of $\omega /{\omega }_{\max }$, we show the phase of ${\alpha }_{\omega }{\beta }_{-\omega }^{*}$ for the flow of Fig. 1 with $M_{+}=5$. The dashed line shows the relativistic phase which is obtained from Eq. (d6). The dotted line shows $\arg ({\alpha }_{-\omega }{\beta }_{\omega }^{*})-\arg ({\alpha }_{\omega }{\beta }_{-\omega }^{*})$. Its smallness is another test of the validity of the relativistic approximation, which predicts that this difference should vanish. Right: As a function of $\gamma$ of Eq. (42), we show the shift $\mathrm{\Delta }x=x_M(0)-x_H$, see Eq. (41), evaluated in the small-frequency limit and adimensionalized by the surface gravity length $c_H/{\kappa }_H$. The four dots are obtained from the numerical integration of Eq. (42), whereas the horizontal line shows the relativistic result of Eq. (d3).

Figure 11

Plots of the local density adimensionalized by $a_s$ (left) and Mach number (right) for the waterfall solution of the NPS Eq. (a2) in a steplike potential. The asymptotic densities ${\rho }_{\pm }$ and the value of $M_{+}$ are chosen to model the flow of [1]; see the text. The horizontal dashed line shows the density at the horizon (left) and $M=1$ (right). The unit of the coordinate $x$ is $a_{\perp }$.

Figure 12

Effective temperature (left, in units of $T_H$) and relative phase of the coefficients ${\alpha }_{\omega }$ and ${\beta }_{-\omega }$ (right) as functions of $\omega /{\omega }_{\max }$ for the waterfall solution shown in Fig. 11. There are no significant differences with respect to the results obtained in the main text; see Fig. 4 right panel and Fig. 10.

Figure 13

Integrated squared density perturbation over a waterfall solution as a function of time. The background solution is the waterfall with $M_{+}=5$. The initial perturbation has the form Eq. (b3) with $A=1$, $\chi =4$, and $x_p=-20$ (left) and 20 (right). For these simulations, $x_{\max }\approx 249.5$ and the squared density perturbation is integrated between $-x_{\max }/8$ and $x_{\max }/8$. The spatial grid is made of 12800 uniformly spaced points; the time step is 0.0025. The oblique dashed lines show $10^4{t}^{-3}$ (left) and $10^3{t}^{-3}$ (right). [In this plot, the scale $\lambda$ of Eq. (7) is fixed so that ${\rho }_{+}=1$.]

Figure 14

Plots of equal-time correlation functions evaluated in a background flow given by Eqs. (c15) and (c16). In the upper left panel, we show the stationary contribution of $v$-modes given by Eq. (c7). The contribution of $u$-modes given by Eq. (c13) evaluated at $t=0$ is shown on the upper right panel, whereas that evaluated at ${\kappa }_{H}t\approx 5.9$ is shown on the lower left panel. The growth of the correlations on opposite sides is the main signal. One should also notice that on both sides, there is a narrowing of the autocorrelations which encodes the thermal radiation emitted from the horizon. On the lower right panel, we show the sum of the $u$ and $v$ contributions evaluated again at ${\kappa }_{H}t\approx 5.9$.

Figure 15

As a function of the downstream Mach number $M_{+}$, we show the relativistic shift $\mathrm{\Delta }x$ of Eq. (d3) adimensionalized by the length associated with the surface gravity $c_H/{\kappa }_H$.

Figure 16

Profile of $c-v$ for the perturbed tanh flow of Eq. (d16) with $\gamma =\eta =0.5$. The horizon is located at $x\approx 0.48$.

Figure 17

Two-point correlation function Eq. (c5) (left) and Eq. (d20) (right) (rescaled so that the maximum represented value is 1) for a perturbed tanh profile given by Eq. (d17), with $\gamma =2\eta =1$. The black, dashed line shows the locus $u_R(x)=u_L(x^{\prime })$ and $u_L(x)=u_R(x^{\prime })$. The white, straight, dashed half-lines are their asymptotes. In these plots, the origin of $x$ has been put where they meet, i.e., $x_M=0$. Hence the sonic horizon is located at $x_H=-\mathrm{\Delta }x$. On the right plot, we clearly see that the maxima of $G_2^{(tt)}$, which give $x(x^{\prime })$, start from $x=x_H$. On the left plot, we see that the maxima of $G_2^{(xx)}$ emerge from a broad region which is approximately centered on $x_M=0$, and not on $x_H$.

Figure 18

Phase of the scattering coefficients ${\alpha }_{\omega }$ (left) and ${\beta }_{-\omega }$ (right) for the waterfall solution with $M_{+}=5$. Continuous curves show the phases of ${\alpha }_{\omega }$ and ${\beta }_{-\omega }$ obtained by numerical integration and working with the definition of the asymptotic modes used in the main text. Dots show the analytical prediction of [47] reexpressed with the same modes. The dashed line on the left plot shows the result of the naive WKB approximation; see the text for explanation.