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Project supported by the National Natural Science Foundation of China (Grant Nos. 11547024, 11791240178, and 11674338.
We study the phenomena of the sonic horizon in an ultracold atomic Fermi system in an elongated harmonic trap. Based on the one-dimensional Gross–Pitaevskii equation model and variational method combined with exact derivation approach, we derive an analytical formula which describes the occurrence of the sonic horizon and the associated Hawking radiation temperature. Using a pictorial demonstration of the key physical quantities we identify the features reported in prior numerical studies of a three-dimensional (3D) ultracold atomic system, proving the applicability of the theoretical model presented here.
For a long time, black hole related problems have been a fascinating subject in macro-world physics fundamental problem scrutiny. With their close analog in ultracold atomic systems, black hole related physics has recently drawn special attention. As a key concept, a black hole has its origin in astrophysics. General relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole: a region of spacetime exhibiting such a strong gravitational effect that even light cannot escape from inside it. The event horizon is a boundary in spacetime beyond which events cannot affect an outside observer, and for a black hole, it is the boundary of the region from which escape is impossible. One of the particular features of the black holes that are predicted by Hawking is the blackbody radiation due to quantum effects near the event horizon. However, Hawking radiation from normal black holes is extremely weak so the experimental investigation of the phenomenon would seem to be virtually impossible, or would depend on the highly unlikely discovery of a small black hole near the Earth.
The quantum fluid supplies the necessary scenario where black hole related physics can be investigated in a practical and controllable way.[1–7] As pointed out by Unruh,[8] a quantum fluid could be used to form an analogue black hole: sonic black hole,[9,10] with the formation of the boundary which marked transition from subsonic flow to supersonic flow. This is analogous to a black-hole event horizon and is called sonic horizon in such setting. As a quantum fluid, a Bose–Einstein condensate (BEC) is a promising candidate for realizing a sonic black hole. Recent experiments of accelerating an elongated BEC under a step like potential also demonstrated the formation of a sonic black hole, and self-amplifying Hawking radiation and spontaneous Hawking radiation were detected.[11,12] Besides, an alternative approach to creating a sonic black hole in a BEC is to trigger condensate expansion by changing the interaction which can be controlled by the Feshbach resonance. Additionally, other intriguing methodologies for the study of sonic black hole require specific attention, for example, a similar technique was employed in observing “Bose–Novae” jets and bursts in a collapsing condensate.[13] Recently, a scheme of the quantum simulation of traversable wormhole spacetimes in a BEC by using the Feshbach resonance has been proposed.[14] In this paper, our investigation is based on the method of tuning the inter-particle nonlinear interaction.
In the previous work,[15] the sonic horizon phenomenon was studied for oscillating Bose–Einstein condensates in an isotropic harmonic potential based on the Gross–Pitaevskii equation model[16,17] and the variational method, where the analytical formula for the criteria and lifetime of the formation of the sonic horizon were derived, and the derived analytical results matched very well with the results obtained by a numerical simulation. Taking the advantage of the flexible modulation of nonlinear interaction strength, it is interesting to investigate similar phenomena in a degenerate Fermion system and inspect sonic black hole-related problems[18] in the Bardeen–Cooper–Schrieffer (BCS) to BEC crossover.[19,20] In this paper, we use the variational method combined with an exact derivation approach to study the evolution of the sonic horizon in a quasi-one-dimensional ultracold degenerate Fermion system. Across the BCS–BEC crossover regime, we derive the criterion formula for the occurrence of the sonic horizon and give the analytical expression for the associated Hawking radiation temperature. The results exhibit identical features compared with those obtained by the study based on a numerical method.[21]
This paper is organized as follows: the next section presents the theoretical model, together with the methodology for the calculation of the results for the sonic horizon formation. This is followed by a discussion. The last section presents conclusive remarks.
To study the formation of a sonic horizon in a one-dimensional ultracold degenerate Fermi gas in the BCS–BEC crossover regime, we use the one-dimensional GPE incorporating polytropic approximation.[22] We firstly consider a quasi-one-dimensional Fermi gas in an elongated harmonic potential V(x,y,z) = (1/2)m(ωxx2 + ωyy2 + ωzz2) (ωx ≪ ωy, ωz). The system is effectively one-dimensional along the x direction. The formation of the sonic horizon can be initiated by expanding the system along the x direction. One way to do this is to enforce an abrupt change in the system’s inter-particle scattering length via the Feshbach resonance experimental technique. Under certain experimental settings, the sonic horizon can be formed during the system’s dynamical evolution process. This is done by solving the one-dimensional time-dependent GPE for the system in an elongated harmonic trap. The one-dimensional GPE used here can be expressed as[23]
We use a variational method to obtain a quantitative description of the dynamical evolution of the system. According to the analytical results from prior work,[23–25] the bright soliton solution can be obtained from Eq. (
The Euler–Lagrangian equation for the action S for the Lagrangian Eq. (
![](cpb_27_10_100312/cpb_27_10_100312_ieqn3.gif)
![](cpb_27_10_100312/cpb_27_10_100312_ieqn4.gif)
We now try to derive the oscillation mode based on the exact solution of Eq. (
![](cpb_27_10_100312/cpb_27_10_100312_ieqn5.gif)
Using the dynamical evolution parameters derived in the previous subsections, we proceed by calculating the key physical quantities related to the occurrence of the sonic horizon. From Eq. (
![](cpb_27_10_100312/cpb_27_10_100312_ieqn6.gif)
Figure
The variation of both the fluid velocity v0 (ascending) and the sound velocity cs (descending) with the location x (shown in Fig.
In this study, based on the one-dimensional Gross–Pitaevskii equation and the variational method combined with exact derivation approach, we calculated the evolution of the a quasi-one-dimensional, harmonically trapped ultracold Fermi system under the assumption of an abrupt change of scattering wavelength using Feshbach resonance technique. We derived a formula which describes the criteria for the occurrence of a sonic black hole and the associated Hawking radiation temperature. Our theoretical results show qualitative agreement with results obtained via the numerical method (for 3D isotropic BEC system with harmonic trapping potential) in prior work. The theoretical results presented here are useful to guide future experimental investigations into sonic black holes in quasi-one-dimensional ultracold systems.
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