Analog spacetimes from nonrelativistic Goldstone modes in spinor condensates

It is well established that linear dispersive modes in a flowing quantum fluid behave as though they are coupled to an Einstein-Hilbert metric and exhibit a host of phenomena coming from quantum field theory in curved space, including Hawking radiation. We extend this analogy to any nonrelativistic Goldstone mode in a flowing spinor Bose-Einstein condensate. In addition to showing the linear dispersive result for all such modes, we show that the quadratically dispersive modes couple to a special nonrelativistic spacetime called a Newton-Cartan geometry. The kind of spacetime (Einstein-Hilbert or Newton-Cartan) is intimately linked to the mean-field phase of the condensate. To illustrate the general result, we further provide the specific theory in the context of a pseudospin-1/2 condensate where we can tune between relativistic and nonrelativistic geometries. We uncover the fate of Hawking radiation upon such a transition: it vanishes and remains absent in the Newton-Cartan geometry despite the fact that any fluid flow creates a horizon for certain wave numbers. Finally, we use the coupling to different spacetimes to compute and relate various energy and momentum currents in these analog systems. While this result is general, present-day experiments can realize these different spacetimes including the magnon modes for spin-1 condensates such as ${}^{87}\mathrm{Rb}$, ${}^7\mathrm{Li}$, ${}^{41}\mathrm{K}$ (Newton-Cartan), and ${}^{23}\mathrm{Na}$ (Einstein-Hilbert).

Figure 1

Illustration of the different ground-state Bloch-vector manifolds as the parameter $g_3$ is tuned. For $g_3<0$ the ground-state manifold consists of the north and south poles, and thus the system realizes an Ising ferromagnet, spontaneously breaking the ${\mathbb{Z}}_2$ symmetry while maintaining the $\text{U}\left(1\right)$ symmetry. For $g_3=0$ the full $\text{SU}\left(2\right)$ symmetry is realized and the ground-state manifold consists of the entire Bloch sphere. Thus, the system is a Heisenberg ferromagnet which spontaneously breaks the full $\text{SU}\left(2\right)$ down to $\text{U}\left(1\right)\subset \text{SU}\left(2\right)$. Finally, for $g_3>0$ the ground-state manifold consists of the equatorial plane, rendering the system an $XY$ (easy-plane) ferromagnet. Thus, the initial symmetry is $\text{U}\left(1\right)$, which is spontaneously broken to the trivial group.

Figure 2

The total luminosity due to the Hawking radiation for a fixed density profile $\rho \left(x\right)$ and velocity profile $v\left(x\right)$. We see that there is no Hawking radiation when $c_r$ is sufficiently large so that $c_l>v_l$ (recall these are constrained by the continuity equation). When $c_l<v_l$ but $c_r>v_r$ we get a region of subsonic flow that flows into a supersonic region, and we begin seeing traditional Hawking radiation. As we further tune $g_3$, $c_r$ drops below $v_r$ and both regions become supersonic at low frequencies. Evidently, there is still a channel for Hawking radiation emission as seen by the nonzero integrated flux. However, as $c_r$ drops to zero this channel closes, vanishing precisely at the quantum phase transition into the Newton-Cartan geometry ($c_r=0=g_3$). In this plot, $v_l=1.3$, $v_r=0.9$, $m=10$, and $\rho \left(x\right)v\left(x\right)=1$.

Figure 3

The Hawking effect for $g_3$ such that $c_r>v_r$ and $c_l<v_l$ (subsonic to supersonic). In this situation, one side (left) flows faster than the speed of some excitations, and the other side (right) flows slower than the speed of any excitation. The dashed line represents the constant laboratory frame energy $\omega$. The mode that carries away energy from the horizon is the “Hawking mode,” shown by the star marker. Tracing this mode back in time (bottom of figure), we find that it comes from a scattering process that includes positive (red) and negative (blue) norm states. It is the negative norm state to the left of the horizon that is responsible for particle creation in the Hawking channel. Notice that for frequencies larger than those in the labeled “Hawking region,” there is no Hawking effect due to lack of negative energy modes to have scattered from at earlier times.

Figure 4

The Hawking effect for $g_3$ such that $c_r<v_r$ and $c_l<v_l$ (supersonic to supersonic). With both regions flowing faster than the speed of excitations (relative to the horizon), we still have a Hawking region, but now we also have a “super-Hawking” region where the positive and negative normalization modes from both regions can scatter between one another.

Figure 5

Hawking flux $N\left(\omega \right)$ as a function of frequency for the subsonic-to-supersonic case (red) and the supersonic-to-supersonic case (blue). As we approach the Heisenberg symmetric point $g_3=0$, we find the Hawking flux disappears both in its overall magnitude and singular behavior. The black arrow indicates the onset of the “super-Hawking region” responsible for the absence of the singular distribution.

Figure 6

For $g_3=0$ in the Newton-Cartan geometry there is an excitation number conservation that protects negative norm states from scattering into positive norm states and as a result, if we scatter a negative norm state in what used to be the “Hawking region,” we find it fully back scatters into a negative norm state and leaks past the horizon only with an evanescent tail characteristic to a “classically forbidden” region.