Decrease of the entanglement entropy of the Hawking radiation induced by backreaction in the Bose-Einstein condensate

Abstract

We analytically study the effect of backreaction from analog Hawking radiation on its entanglement entropy in the Bose-Einstein condensate (BEC). The backreaction is expected to play an essential role in the decrease of the entanglement entropy and in realizing the Page curve. Since the BEC theory has microscopic Hamiltonian and thus exhibits unitarity, it is desirable to reproduce the Page curve explicitly by using the Hamiltonian. In order to analyze this in a concrete example, we study the BEC with a step-like configuration that has been extensively studied in the literature. By using the microscopic theory, we derive an explicit form of backreaction from analog Hawking radiation. Combining it with the known results of the Bogoliubov coefficients, we analytically compute the entanglement entropy of the Hawking radiation, and show that it decreases as expected due to the backreaction for sufficiently low energy modes over a wide range of the parameter characterizing the step-like configuration.