Understanding Hawking Radiation from Simple Models of Atomic Bose-Einstein Condensates

Abstract

This chapter is an introduction to the Bogoliubov theory of dilute Bose condensates as applied to the study of the spontaneous emission of phonons in a stationary condensate flowing at supersonic speeds. This emission process is a condensed-matter analog of Hawking radiation from astrophysical black holes but is derived here from a microscopic quantum theory of the condensate without any use of the analogy with gravitational systems. To facilitate physical understanding of the basic concepts, a simple one-dimensional geometry with a stepwise homogenous flow is considered which allows for a fully analytical treatment.

Appendix

In this appendix, we give the explicit expressions for S Matrix coefficients, to first order in the limit of small ω, for the supersonic-supersonic condition treated in Sect. 9.6. Note in particular how those involved in the vacuum emission (S _vl,ur , S _ul,vr , S _3r,4l , S _4r,3l ) grow as $\sqrt{\omega}$ at low ω, while (S _4r,vr , S _3r,ur , S _ul,3l , S _vl,4l ) tend to constants.

$$\begin{aligned} S_{vl,vr} = & \sqrt{\frac{v_{0}+c_{l}}{v_{0}-c_{l}}}\frac {(c_{r}^{2}-c_{l}^{2})\sqrt{\omega\xi_{l}}}{2\sqrt {2}(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})}, \\ S_{ul,vr} = & \sqrt{\frac{v_{0}-c_{l}}{v_{0}+c_{l}}}\frac {(c_{r}^{2}-c_{l}^{2})\sqrt{\omega\xi_{l}}}{2\sqrt {2}(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})}, \\ S_{3r,vr} = & \frac{\sqrt{v_{0}^{2}-c_{l}^{2}}+\sqrt {v_{0}^{2}-c_{r}^{2}}}{2(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})^{1/4}}, \\ S_{4r,vr} = & \frac{\sqrt{v_{0}^{2}-c_{l}^{2}}-\sqrt {v_{0}^{2}-c_{r}^{2}}}{2(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})^{1/4}}, \\ S_{vl,ur} = & \sqrt{\frac{v_{0}+c_{l}}{v_{0}-c_{l}}}\frac {(c_{l}^{2}-c_{r}^{2})\sqrt{\omega\xi_{l}}}{2\sqrt {2}(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})}, \\ S_{ul,ur} = & \sqrt{\frac{v_{0}-c_{l}}{v_{0}+c_{l}}}\frac {(c_{r}^{2}-c_{l}^{2})\sqrt{\omega\xi_{l}}}{2\sqrt {2}(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})}, \\ S_{3r,ur} = & \frac{\sqrt{v_{0}^{2}-c_{r}^{2}}-\sqrt {v_{0}^{2}-c_{l}^{2}}}{2(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})^{1/4}}, \\ S_{4r,ur} = & \frac{\sqrt{v_{0}^{2}-c_{l}^{2}}+\sqrt {v_{0}^{2}-c_{r}^{2}}}{2(v_{0}^{2}-c_{l}^{2})^{1/4}(v_{0}^{2}-c_{r}^{2})^{1/4}}, \\ S_{vl,3l} = & \frac{c_{l}+c_{r}}{2\sqrt{c_{l}c_{r}}}, \\ S_{ul,3l} = & i\frac{c_{l}-c_{r}}{2\sqrt{c_{l}c_{r}}}, \\ S_{3r,3l} = & \frac {(v_{0}^{2}-c_{r}^{2})^{1/4}(c_{l}^{2}-c_{r}^{2})\sqrt{c_{l}\xi _{l}\omega}}{2\sqrt {2c_{r}}(c_{r}-v_{0})(v_{0}^{2}-c_{l}^{2})}, \\ S_{4r,3l} = & \frac {(v_{0}^{2}-c_{r}^{2})^{1/4}(c_{l}^{2}-c_{r}^{2})\sqrt{c_{l}\xi _{l}\omega}}{2\sqrt {2c_{r}}(c_{r}-v_{0})(v_{0}^{2}-c_{l}^{2})}, \\ S_{vl,4l} = & \frac{c_{l}-c_{r}}{2\sqrt{c_{l}c_{r}}}, \\ S_{ul,4l} = & i\frac{c_{l}+c_{r}}{2\sqrt{c_{l}c_{r}}}, \\ S_{3r,4l} = & \frac {(v_{0}^{2}-c_{r}^{2})^{1/4}(c_{l}^{2}-c_{r}^{2})\sqrt{c_{l}\xi _{l}\omega}}{2\sqrt {2c_{r}}(c_{r}+v_{0})(c_{l}^{2}-v_{0}^{2})}, \\ S_{4r,4l} = & i\frac {(v_{0}^{2}-c_{r}^{2})^{1/4}(c_{r}^{2}-c_{l}^{2})\sqrt{c_{l}\xi _{l}\omega}}{2\sqrt{2c_{r}}(c_{r}+v_{0})(c_{l}^{2}-v_{0}^{2})} . \end{aligned}$$

(9.126)