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arXiv:2008.01433 · Physical review. D/Physical review. D.

Correlations between a Hawking particle and its partner in a 1 + 1 D Bose-Einstein condensate analog black hole

Richard A. Dudley, Alessandro Fabbri, Paul R. Anderson, Roberto Balbinot

Published Nov 4, 2020• 6 citations
arXivHTMLPDF

Abstract

The Fourier transform of the density-density correlation function in a Bose-Einstein condensate (BEC) analog black hole is a useful tool to investigate correlations between the Hawking particles and their partners. It can be expressed in terms of $⟨{^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{ext}}\text{ }\text{ }{^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{int}}⟩$, where ${^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{ext}}$ is the annihilation operator for the Hawking particle and ${^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{int}}$ is the corresponding one for the partner. This basic quantity is calculated for three different models for the BEC flow. It is shown that in each model the inclusion of the effective potential in the mode equations makes a significant difference. Furthermore, particle production induced by this effective potential in the interior of the black hole is studied for each model and shown to be nonthermal. An interesting peak that is related to the particle production and is present in some models is discussed.

DOI
10.1103/physrevd.102.105005

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The Fourier transform of the density-density correlation function in a Bose-Einstein condensate (BEC) analog black hole is a useful tool to investigate correlations between the Hawking particles and their partners. It can be expressed in terms of ⟨{^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{ext}}\text{ }\text{ }{^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{int}}⟩, where {^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{ext}} is the annihilation operator for the Hawking particle and {^{\mathrm{out}}\stackrel{^}{a}}_{\mathrm{up}}^{\mathrm{int}} is the corresponding one for the partner. This basic quantity is calculated for three different models for the BEC flow. It is shown that in each model the inclusion of the effective potential in the mode equations makes a significant difference. Furthermore, particle production induced by this effective potential in the interior of the black hole is studied for each model and shown to be nonthermal. An interesting peak that is related to the particle production and is present in some models is discussed.
Observing quantum particle creation by black holes (Hawking radiation) in the astrophysical context is, in ordinary situations, hopeless. Nevertheless the Hawking effect, which depends only on kinematical properties of wave propagation in the presence of horizons, is present also in nongravitational contexts, for instance in stationary fluids undergoing supersonic flow. We present results on how to observe the analog Hawking radiation in Bose-Einstein condensates by a direct measurement of the density correlations due to the phonon pairs (Hawking quanta-partner) created by the acoustic horizon.
We extend the Gross-Pitaevskii equation to incorporate the effect of quantum fluctuations onto the flow of a weakly interacting Bose-Einstein condensate. Applying this framework to an analog black hole in a quasi-one-dimensional, transonic flow, we investigate how acoustic Hawking radiation back-reacts on the background condensate. Our results point to the emergence of stationary density and velocity undulations in the supersonic region (analogous to the black hole interior) and enable to evaluate the change in upstream and downstream Mach numbers caused by Hawking radiation. These findings provide new insight into the interplay between quantum fluctuations and analog gravity in Bose-Einstein condensates.
Inspired by a recent experiment by Steinhauer and co-workers, we present a simple model which describes the formation of an acoustic black hole in a Bose-Einstein condensate, allowing an analytical computation of the evolution in time of the corresponding density-density correlator. We show the emergence of analog Hawking radiation out of a "quantum atmosphere" region significantly displaced from the horizon. This is quantitatively studied both at T=0 and even in the presence of an initial temperature T, as is always the case experimentally.
This thesis covers three applications of Bose-Einstein condensates and related phenomena, in the theme of pair creation and pair annihilation. First, Bose-Einstein condensates (BEC) are viewed as one of the candidates to implement a sonic black hole. This can lead to the observation of analog Hawking radiation, resulting from a phonon pair creation at a black-hole horizon (BH). Such implementation has been achieved in a resent experiment by J. Steinhauer, in which a black-hole/white-hole pair has been produced. He also reported the observations of self-amplifying Hawking radiation, via a lasing mechanism operating between the black and white-hole horizons. Through our simulations, we find that the observations should be attributed not to the black hole laser effect, but rather to a growing zero-frequency bow wave, generated at the white-hole horizon. The relative motion of the two horizons produces a Doppler shift of the bow wave at the BH, where it stimulates a monochromatic Hawking radiation. We also find that shot-to-shot atom number variations and quantum fluctuations give density-density correlations consistent with those reported in the experiments. In particular, atom number variations can produce a spurious correlation signal. Secondly, a sonic black hole/white hole pair and phonon pair creation can also be realized using a ring-shaped condensate. Here we focus on the phonon spectroscopy of a ring condensate. We probe the phonon excitation spectrum by applying a harmonically driven barrier to a 23Na Bose-Einstein condensate in a ring-shaped trap. When excited resonantly, these wavepackets display a regular periodic structure. The resonant frequencies depend upon the particular configuration of the barrier, but are commensurate with the orbital frequency of a sound wave traveling around the ring. Energy transfer to the condensate over many cycles of the periodic wavepacket motion causes enhanced atom loss from the trap at resonant frequencies. Solutions of the Gross-Pitaevskii (GP) equation exhibit quantitative agreement with the experimental data. Thirdly, positronium (Ps) BECs have been of experimental and theoretical interest due to their potential application as the gain medium of a gamma-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho- (o-Ps) and para-positronium (p-Ps), whose annihilation lifetimes differ by three orders of magnitude. We study the spinor dynamics and annihilation processes in the p-Ps/o-Ps system using both solutions of the GP equations and a rate-equation approach. For an initially unpolarized condensate, there is a threshold density at which spin mixing between o-Ps and p-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.