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The number of discrete instabilities in the black hole– white hole configuration increases with the size L of the intermediate supersonic region. In these plots we have used c sub − lhs = 1 . 8, c super = 0 . 7, c sub − rhs = 1 . 9 (with their corresponding v = 1 /c 2 ) and ξ c = 1.
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We examine the linear stability of various configurations in Bose-Einstein condensates with steplike sonic horizons. These configurations are chosen in analogy with gravitational systems with a black hole horizon, a white hole horizon, and a combination of both. We discuss the role of different boundary conditions in this stability analysis, paying...
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... decelerating configurations. When convergence is fulfilled downstream, the configuration is stable, regardless of whether it contains a white hole horizon or not. When this convergence condition is dropped, there is a tendency to destabilization. In the presence of a white hole horizon, the configuration actually becomes dramatically unstable, since there is a huge continuous region of instabilities, and even perturbations with arbitrarily small frequencies destabilize the configuration. In the absence of such a horizon, only a small high-frequency part of this unstable region subsists. Consider flows passing from being subsonic to supersonic and then back to subsonic (Figure 4). The numerical algorithm we have followed to deal with this problem is equivalent to the one presented above, but with a larger set of equations. In this case we have 12 arbitrary constants A j , which have to satisfy 8 + N equations: 4 matching conditions at each discontinuity and N (0 − 8) additional conditions of the form A j = 0, corresponding to modes that do not fulfill the boundary conditions in a particular asymptotic region. When convergence is imposed at the lhs, we do not find any instabilities, regardless of whether the fluid is globally accelerating or decelerating [the final lhs fluid velocity is larger or smaller than the initial rhs one respectively, see Figure 5 cases (a)]. Also when replacing the intermediate supersonic region by a subsonic one, thereby removing the acoustic horizons, the fluid is stable, independently of whether it is globally accelerating or decelerating. When dropping the convergence condition at the lhs the situation changes completely. When the intermediate region is supersonic, i.e. in a black hole–white hole configuration, a discrete set of instabilities appears at low frequencies [Fig. 5 cases (b)]. It is worth mentioning that, when carefully looking at plots of type (a)-cases, we observe some traces of these zeros in the form of local minima which can be understood as particularly soft regions. These regions, although very close to zero in some situations, never give rise to real zeros, as we have carefully checked by zooming in. Notice that these local minima appear in regions with N = 5 where a zero would mean a double degeneracy within the row vectors in the corresponding matrix Λ ̃ ij . When the fluid is globally decelerating, additionally there is a continuous region of instabilities at higher frequencies. Indeed, in this region, as in the case of the white hole configuration, N < 4, and so every frequency in this region automatically represents an instability. When the intermediate region is subsonic, the discrete set of local minima at low frequencies disap- pears, but the continuous strip of instabilities at higher frequencies persists in the case of a globally decelerating fluid. The discrete set of instabilities is therefore a genuine consequence of the existence of horizons. The number of discrete zeros we find in the black hole– white hole configuration increases with the size L of the supersonic region (see Fig. 6), while their Im( ω ) de- creases. This suggests that the region between the horizons acts as a sort of well discretizing some of the instabilities found for the white hole configurations. The larger the well, the larger the amount of instabilities, but the longer-lived these instabilities. To summarize, when requiring convergence in both asymptotic regions, all the types of configurations with two discontinuities that we have discussed are stable. When not requiring convergence at the lhs, discretized instabilities appear associated with the presence of horizons. We have seen in section IV B that configurations with a single black hole horizon do not possess instabilities ...
Citations
... In this context, another remarkable phenomenon is the black-hole laser (BHL) effect [23], i.e., the self-amplification of Hawking radiation due to successive reflections between a pair of horizons, leading to the emergence of dynamical instabilities in the excitation spectrum. In an atomic condensate, the BHL effect can take place because of its superluminal dispersion relation, which allows the radiation reflected at the inner horizon to travel back to the outer one [24][25][26][27][28][29][30]. Other analog setups have been proposed to observe black-hole lasing [31][32][33][34]. ...
The black-hole laser (BHL) effect is the self-amplification of Hawking radiation in the presence of a pair of horizons which act as a resonant cavity. In a flowing atomic condensate, the BHL effect arises in a finite supersonic region, where Bogoliubov-Cherenkov-Landau (BCL) radiation is coherently excited by any static perturbation. Thus, experimental attempts to produce a black-hole laser unavoidably deal with the presence of a strong BCL background, making the observation of the BHL effect still a major challenge in the analog gravity field. Here, we perform a theoretical study of the BHL-BCL crossover using an idealized model where both phenomena can be unambiguously isolated. By drawing an analogy with an unstable pendulum, we distinguish three main regimes according to the interplay between quantum fluctuations and classical stimulation: quantum BHL, classical BHL, and BCL. Based on quite general scaling arguments, the nonlinear amplification of the initial amplitude of the quantum fluctuations up to saturation is identified as the most robust trait of a quantum BHL. A classical BHL behaves instead as a linear quantum amplifier, where the output is proportional to the input. The BCL regime also acts as a linear quantum amplifier, but its gain is exponentially smaller as compared to a classical BHL. In addition, we find that the decrease in the amplification for increasing BCL amplitude or the nonmonotonic dependence of the growth rate with respect to the background parameters are complementary signatures of black-hole lasing. We also identify interesting analog phenomena such as Hawking-stimulated white-hole radiation or quantum BCL-stimulated Hawking radiation. The results of this work not only are of interest for analog gravity, where they help to distinguish each phenomenon and to design experimental setups leading to a clear observation of the BHL effect, but they also open the prospect of finding applications of analog concepts in quantum technologies.
... In this context, another remarkable phenomenon is the black-hole laser (BHL) effect [22], i.e., the selfamplification of Hawking radiation due to successive reflections between a pair of horizons, leading to the emergence of dynamical instabilities in the excitation spectrum. A BHL requires a superluminal dispersion relation, as that of an atomic condensate [23][24][25][26][27][28][29], so the radiation reflected at the inner horizon can travel back to the outer one. Other analogue setups have been also proposed to observe the BHL effect [30][31][32][33]. ...
The black-hole laser (BHL) effect is the self-amplification of Hawking radiation in the presence of a pair of horizons which act as a resonant cavity. Its clear observation still remains a major challenge in the analogue gravity field. In a flowing atomic condensate, the BHL effect arises in a finite supersonic region, where Bogoliubov-Cherenkov-Landau (BCL) radiation is resonantly excited by any static perturbation. Thus, any experimental attempt to produce a BHL will deal with the presence of a BCL background, as already observed in experiments. Here, we perform a theoretical study of the BHL-BCL crossover using an idealized model where both phenomena can be unambiguously isolated. By drawing an analogy with an unstable pendulum, we distinguish three main regimes according to the interplay between quantum fluctuations and classical stimulation: quantum BHL, classical BHL, and BCL. Based on quite general scaling arguments, the nonlinear amplification of quantum fluctuations until saturation is identified as the most robust trait of a quantum BHL. A classical BHL behaves instead as a linear quantum amplifier, where the output is proportional to the input. Finally, the BCL regime also acts as a linear quantum amplifier, but its gain is exponentially smaller as compared to a classical BHL. The results of this work not only are of interest for analogue gravity, where they help to distinguish unambiguously each phenomenon and to design experimental schemes for a clear observation of the BHL effect, but they also open the prospect of finding applications of analogue concepts in quantum technologies.
... To date, QNMs have only been observed in two-dimensional rotating flows [45]. They have also been studied in the context of large amplitude perturbations destabilising the horizon and the so-called black-hole laser in one-dimensional conservative quantum fluids [46][47][48][49], but it is the first time they are observed in a driven-dissipative fluid. ...
Analogue gravity enables the laboratory study of the Hawking effect, correlated emission at the horizon. Here, we use a quantum fluid of polaritons as a setup to study the statistics of correlated emission. Dissipation in the system may quench quasi-normal modes of the horizon, thus modifying the horizon structure. We numerically compute the spectrum of spatial correlations and find a regime in which the emission is strongly enhanced while being modulated by the quasi-normal modes. The high signal-to-noise ratio we obtain makes the experimental observation of these effects possible, thus enabling the quantitative study of the influence of dissipation and of higher order corrections to the curvature on quantum emission.
... Another interesting analog feature that can be observed in a Bose-Einstein condensate due to its superluminal dispersion relation is the so-called black-hole laser (BHL) effect [26], in which a configuration displaying a pair of horizons can give rise to the self-amplification of Hawking radiation due to successive reflections between them, like in a laser cavity, translated into the appearance of dynamical instabilities in the spectrum of excitations [27][28][29][30][31][32][33]. A complete experimental characterization of the BHL effect is still one of the major present challenges in the gravitational analog field, since its first reported observation [34] has sparked intense discussions within the community [35][36][37][38][39]. ...
For flowing quantum gases, it has been found that at long times an initial black-hole laser (BHL) configuration exhibits only two possible states: the ground state or a periodic self-oscillating state of continuous emission of solitons. So far, all the works on this subject are based on a highly idealized model, quite difficult to implement experimentally. Here we study the instability spectrum and the time evolution of a recently proposed realistic model of a BHL, thus providing a useful theoretical tool for the clear identification of black-hole lasing in future experiments. We further confirm the existence of a well-defined phase diagram at long times, which bespeaks universality in the long-time behavior of a BHL. Additionally, we develop a complementary model in which the same potential profile is applied to a subsonic homogeneous flowing condensate that, despite not forming a BHL, evolves toward the same phase diagram as the associated BHL model. This result reveals an even stronger form of robustness in the long-time behavior with respect to the transient, which goes beyond what has been described in the previous literature.
... Another interesting analog feature that can be observed in a Bose-Einstein condensate due to its superluminal dispersion relation is the so-called black-hole laser (BHL) effect [26], in which a configuration displaying a pair of horizons can give rise to the self-amplification of Hawking radiation due to successive reflections between them, like in a laser cavity, translated into the appearance of dynamical instabilities in the spectrum of excitations [27][28][29][30][31]. A complete experimental characterization of the black-hole laser effect is still one of the major present challenges in the gravitational analog field, since its first reported observation [32] has sparked intense discussions within the community [33][34][35][36]. ...
For flowing quantum gases, it has been found that at long times an initial black-hole laser (BHL) configuration exhibits only two possible states: the ground state or a periodic self-oscillating state of continuous emission of solitons. So far, all the works on this subject are based on a highly idealized model, quite difficult to implement experimentally. Here we study the instability spectrum and the time evolution of a recently proposed realistic model of a BHL, thus providing a useful theoretical tool for the clear identification of black-hole lasing in future experiments. We further confirm the existence of a well-defined phase diagram at long times, which bespeaks universality in the long-time behavior of a BHL. Additionally, we develop a complementary model in which the same potential profile is applied to a subsonic homogeneous flowing condensate which, despite not forming a BHL, evolves towards the same phase diagram as the associated BHL model. This result reveals an even stronger form of robustness in the long-time behavior with respect to the initial condition which goes beyond what has been described in the previous literature.
... Another important issue concerning our system with two horizons is its stability, which is sensitive to boundary conditions and the presence of the different horizons [75]. For the initial noise that does not strongly violate the frequency condition, we have found the steep horizon to be subject to eventual instability, with the respective lifetime t decay 600 (≈0.438 s, in physical units) for L = 320. ...
We propose emulation of Hawking radiation (HR) by means of acoustic excitations propagating on top of persistent current in an atomic Bose-Einstein condensate (BEC) loaded in an annular confining potential. The setting is initially created as a spatially uniform one, and then switches into a nonuniform configuration, while maintaining uniform BEC density. The eventual setting admits the realization of sonic black and white event horizons with different slopes of the local sound speed. A smooth slope near the white-hole horizon suppresses instabilities in the supersonic region. It is found that tongue-shaped patterns of the density-density correlation function, which represent the acoustic analog of HR, are strongly affected by the radius of the ring-shaped configuration and number of discrete acoustic modes admitted by it. There is a minimum radius that enables the emulation of HR. We also briefly discuss a possible similarity of properties of the matter-wave sonic black holes to the known puzzle of the stability of Planck-scale primordial black holes in quantum gravity.
... Another important issue concerning our system with two horizons is its stability, which is sensitive to boundary conditions and the presence of different horizons [70]. For the initial noise that does not strongly violate the frequency condition, we have found the steep horizon to be subject to eventual instability, with lifetime t decay ≃ 600 (≈ 0.438 s, in physical units) for L = 320. ...
We propose emulation of Hawking radiation by means of acoustic excitations propagating on top of a persistent current in an atomic Bose-Einstein condensate loaded in an annular confining potential. The setting admits realization of sonic black and white event horizons. It is found that density-density correlations, representing the acoustic analogue of the Hawking radiation, are strongly affected by the perimeter of the ring-shaped configuration and number of discrete acoustic modes admitted by it. Remarkably, there is a minimum radius of the ring which admits the emulation of the Hawking radiation. We also discuss a possible similarity of properties of the matter-wave sonic black holes to the known puzzle of the stability of Planck-scale primordial black holes in quantum gravity.
... The analogue of the Hawking radiation in this system is the spontaneous emission of entangled phonons by the acoustic horizon into the subsonic and supersonic regions [10][11][12][13][14][15][16][17][18][19][20][21][22][23], similar to the particle-antiparticle creation at the event horizon of a BH. In addition, a flowing condensate presenting a finite-size supersonic region (giving rise to a pair of acoustic horizons) provides the analog of a black-hole laser [16,[24][25][26][27][28][29]. ...
... Due to the structure of the equations, the conjugatez n is also a mode with energy − * n . An interesting property of the eigenvalue problem of Equation (24) is that it is non-Hermitian, and thus, it can yield complex eigenvalues. In particular, eigenvalues with a positive imaginary part correspond to dynamical instabilities, i.e., exponentially-growing modes: the presence of such dynamical instabilities in a finite region of a condensate flow are the origin of the black-hole laser effect, discussed in the next section. ...
... First, we analyze 1D homogeneous stationary flows, characterized by GP plane waves of the form ψ 0 (x) = √ ne iqx+φ 0 , with n the density of the condensate, q its momentum and φ 0 some phase. After removing the phase of the condensate from the field operator,φ(x, t) → e iqx+φ 0φ (x, t), it is straightforward to show that the eigenmodes of the BdG Equation (24) are plane waves with wave vector k and frequency ω, giving rise to the following dispersion relation: ...
From both a theoretical and an experimental point of view, Bose–Einstein condensates are good candidates for studying gravitational analogues of black holes and black-hole lasers. In particular, a recent experiment has shown that a black-hole laser configuration can be created in the laboratory. However, the most considered theoretical models for analog black-hole lasers are quite difficult to implement experimentally. In order to fill this gap, we devote this work to present more realistic models for black-hole lasers. For that purpose, we first prove that, by symmetrically extending every black-hole configuration, one can obtain a black-hole laser configuration with an arbitrarily large supersonic region. Based on this result, we propose the use of an attractive square well and a double delta-barrier, which can be implemented using standard experimental tools, for studying black-hole lasers. We also compute the different stationary states of these setups, identifying the true ground state of the system and discussing the relation between the obtained solutions and the appearance of dynamical instabilities.
... This approximation is not valid for high overtone QNMs and for that reason is not relevant to this paper. In addition, Barcelo et al [10,11] calculate the QNMs of an analog black hole in the background of a BEC with one-dimensional flow and step-like discontinuities at the sound horizon. They keep the quantum potential in their linearized Gross-Pitaevskii equation and consider the full Bogoliubov dispersion relation (not the hydrodynamic approximation as in [9]). ...
... They keep the quantum potential in their linearized Gross-Pitaevskii equation and consider the full Bogoliubov dispersion relation (not the hydrodynamic approximation as in [9]). Therefore, in this paper we focus on the work of [10,11]. ...
... We follow the footsteps of Barcelo et al [10,11] where they use the full Bogoliubov dispersion relation to perform stability analysis and calculate the QNMs for the 1+1 dimensional (one-dimensional flow) black hole in a BEC with step-like discontinuities at the sound horizon. After substituting the Madelung representation in the Gross-Pitaevskii equation, one can linearize the equations. ...
The goal of this paper is to build a foundation for, and explore the
possibility of, using high overtone quasinormal modes of analog black holes to
probe the small scale (microscopic) structure of a background fluid in which an
analog black hole is formed. This may provide a tool to study the small scale
structure of some interesting quantum systems such as Bose-Einstein
condensates. In order to build this foundation, we first look into the
hydrodynamic case where we calculate the high overtone quasinormal mode
frequencies of a 3+1 dimensional canonical non-rotating acoustic black hole.
The leading order calculations have been done earlier in the literature. Here,
we obtain the first order correction. We then analyze the high overtone
quasinormal modes of acoustic black holes in a Bose-Einstein condensate using
the linearized Gross-Pitaevskii equation. We point out that at the high
overtone quasinormal mode limit, the only term that is important in the
linearized Gross-Pitaevskii equation is the quantum potential term, which is a
small scale effect.
... One point of particular interest and controversy is the stability or otherwise of spacetimes containing white holes. While black holes are always found to be stable [51,52], white holes were found in [51] to be intrinsically unstable, and, conversely, to be stable in [53]. A systematic numerical study in [52] found that the stability or otherwise of white holes depends crucially on the boundary conditions imposed, though the nature of the physically appropriate boundary conditions remains for now a moot point (see also section 5.2.4 of [18] for a concise description of this problem and the issues involved). ...
... While black holes are always found to be stable [51,52], white holes were found in [51] to be intrinsically unstable, and, conversely, to be stable in [53]. A systematic numerical study in [52] found that the stability or otherwise of white holes depends crucially on the boundary conditions imposed, though the nature of the physically appropriate boundary conditions remains for now a moot point (see also section 5.2.4 of [18] for a concise description of this problem and the issues involved). Another spacetime showing signs of instability is the black hole-white hole pair. ...
Hawking radiation, despite being known to theoretical physics for nearly 40 years, remains elusive and undetected. It also suffers, in its original context of gravitational black holes, from practical and conceptual difficulties. Of particular note is the trans-Planckian problem, which is concerned with the apparent origin of the radiation in absurdly high frequencies. In order to gain better theoretical understanding and, it is hoped, experimental verification of Hawking radiation, much study is being devoted to laboratory systems which use moving media to model the spacetime geometry of black holes, and which, by analogy, are also thought to emit Hawking radiation. These analogue systems typically exhibit dispersion, which regularizes the wave behaviour at the horizon at the cost of a more complicated theoretical framework. This tutorial serves as an introduction to Hawking radiation and its analogues, developing the moving medium analogy for black holes and demonstrating how dispersion can be incorporated into this generalized framework.
