Limits to the analog Hawking temperature in a Bose-Einstein condensate

Quasi-one-dimensional outflow from a dilute gas Bose-Einstein condensate reservoir is a promising system for the creation of analog Hawking radiation. We use numerical modeling to show that stable sonic horizons exist in such a system under realistic conditions, taking into account the transverse dimensions and three-body loss. We find that loss limits the analog Hawking temperatures achievable in the hydrodynamic regime, with sodium condensates allowing the highest temperatures. A condensate of $30000$ atoms, with transverse confinement frequency ${\omega }_{\perp }=6800\times 2\pi \mathrm{Hz}$, yields horizon temperatures of about $20\mathrm{nK}$ over a period of $50\mathrm{ms}$. This is at least four times higher than for other atoms commonly used for Bose-Einstein condensates.

Figure 1

(Color online) (a) Sketch of an elongated condensate (gray), sliced by multiple blue detuned laser sheets (blue). (b) These sheets (blue) add up to form barriers (black, dotted), which are impenetrable for the condensate. The red, dashed line is the interaction energy $\tilde{U}\tilde{n}$. The left sheets act as a permanent endcap. To initiate supersonic outflow from this reservoir, the optical potentials sketched in light blue in panels (a) and (b) are suddenly removed.

Figure 2

Initial time evolution of the condensate in the reservoir after initiation of the outflow. No three-body loss. (a) Density (solid) and $\tilde{V}\left(x\right)∕{\tilde{U}}_{1\mathrm{D}}$ (dashed) for outflow with no piston. ${\tilde{\mu }}_{1D}=0.61$, ${\tilde{\xi }}_{\mathrm{bg}}=0.64$, ${\tilde{\xi }}_h=3.8$, $\tilde{\sigma }=14\simeq 22{\tilde{\xi }}_{\mathrm{bg}}$, $V_0=0.4$, ${\tilde{x}}_r=80$, ${\tilde{x}}_l=-110$, and $\tilde{d}=0.5\tilde{\sigma }$. The time samples are $\tilde{t}=0$, $60$, $210$, and are shifted up for clarity. The zero of the common axes for $\tilde{n}$ and $\tilde{V}$ for later time samples can be deduced from $\tilde{V}\left(0\right)=0$. ${\tilde{T}}^{*}=0.008$. (b) The same as in (a), but with an optical piston. Here ${\tilde{V}}_0=0.61$, ${\tilde{\xi }}_{\mathrm{bg}}=0.64$, ${\tilde{\xi }}_h=4.66$, and $\tilde{d}=\tilde{\sigma }$; all other parameters are as in panel (a). ${\tilde{T}}^{*}=0.005$. (c) Sample Mach number profile (solid, thick) for the case with piston. The thin black line indicates $M=1$. (d) Comparison of analog Hawking temperatures with $(\circ )$ and without $(\times )$ piston.

Figure 3

(Color online) Time evolution of analog Hawking temperature ${\tilde{T}}_H$ for ${\tilde{\mu }}_{1\mathrm{D}}=0.61$, ${\tilde{\xi }}_{\mathrm{bg}}=0.64$, ${\tilde{\xi }}_h=2.8$, $\tilde{\sigma }=14\simeq 22{\tilde{\xi }}_{\mathrm{bg}}$, and $V_0=0.5$. Initially for these scenarios ${\tilde{T}}^{*}=0.014$. (a) Left end barrier acts as piston with ${\tilde{v}}_p=0.01$. ($\times$, black) ${\tilde{K}}_{3,1\mathrm{D}}=0$. ($엯$, red) ${\tilde{K}}_{3,1\mathrm{D}}=1.4\times {10}^{-7}$. (b) Left end barrier does not move. ($\times$, black) ${\tilde{K}}_{3,1\mathrm{D}}=0$. ($엯$, red) ${\tilde{K}}_{3,1\mathrm{D}}=1.4\times {10}^{-7}$. $\tilde{t}=5000$ corresponds to $t=117\mathrm{ms}$. The three-body-loss constants are for helium with ${\omega }_{\perp }=6800\times 2\pi \mathrm{Hz}$ transverse confinement (Table ).

Figure 4

(Color online) Transverse structure of the BEC near the horizon. The full length of the condensate is about 70 units. The hump potential is centered at ${\tilde{x}}_r=-4$ with width $\tilde{\sigma }=6.8$ [35]. The snapshot is taken at $\tilde{t}=525$ ($12$ ms after initiation). Other parameters: ${\tilde{x}}_l=-76$; $\tilde{d}=0.7\tilde{\sigma }$. (a) Color map of density in $rx$ plane. The density is highest on the left (shaded red) and monotonically decreases to the right (shaded blue). Superimposed are contours of equal Mach number (red) at $M$=$0.5$, $2$, $4$. We marked $M=1$ in violet. (b) Radial structure of Mach number $M$ (dot-dashed), density $n$ (dashed), and approximate Hawking temperature ${\tilde{T}}_p$ (solid; see text for definition) at $\tilde{x}={\tilde{x}}_h=-3.54$. This slice through the condensate is indicated by the white straight line in (a).