Bose–Einstein-Condensate Horizons

A Bose–Einstein-condensate horizon is a controlled transonic background for Bogoliubov quasiparticles. Because the condensate is quantum coherent and its dispersion is known, it can probe both horizon scattering and quantum correlations. The analogue remains a quasiparticle experiment: it does not reproduce gravitational field equations or black-hole thermodynamics.

Condensate horizon and status

Property	Statement
Object class	Analogue system
Medium	Dilute, weakly interacting atomic condensate
Background model	Gross–Pitaevskii field in a specified trap and interaction profile
Perturbations	Bogoliubov–de Gennes quasiparticles with superluminal dispersion
Horizon	Condensate flow crosses the hydrodynamic sound speed
Physical status	Real laboratory quantum fluid; kinematic black-hole analogue only

In the mean-field regime, the condensate order parameter obeys

$$i\partial_t\Psi =\left[-\frac{\nabla^2}{2m_{\rm at}}+V+g_{\rm int}|\Psi|^2\right]\Psi .$$

Writing $\Psi_0=\sqrt{n_0}\,e^{i\theta_0}$ gives $\mathbf v_0=\nabla\theta_0/m_{\rm at}$ and $c_s^2=g_{\rm int}n_0/m_{\rm at}$. Linearization couples $\delta\Psi$ and $\delta\Psi^*$ through the Bogoliubov–de Gennes equations. In a locally homogeneous region their dispersion is

$$(\omega-\mathbf v_0\!\cdot\!\mathbf k)^2 =c_s^2k^2+\frac{k^4}{4m_{\rm at}^2}.$$

The quartic term is not a nuisance to erase: it is the known microscopic completion of the hydrodynamic acoustic metric. Garay et al. 2000, complete short article, pp. 4643–4647 established the condensate black-hole construction, and Macher and Parentani 2009, §§II–V quantified dispersive spectra and correlation signatures in realistic profiles.

Norm, preparation, and observables

Bogoliubov modes carry the indefinite norm

$$(W_1,W_2)=\int dx\, \left(u_1^*u_2-v_1^*v_2\right).$$

A stationary horizon mixes branches of opposite norm. The exact scattering relation is frequency- and profile-dependent; a Planck-shaped ratio appears only in the low-frequency, smooth-horizon regime. Bogoliubov Coefficients and Pair Creation supplies the canonical operator interpretation.

State preparation determines the claim. A classical density wave tests stimulated scattering. A thermal initial quasiparticle population adds stimulated emission. A rapid change in the trap creates quench excitations. To identify spontaneous horizon pairs, one reconstructs the connected equal-time density covariance and looks for the nonlocal Hawking–partner correlation ridge, ideally together with a nonseparability or entanglement witness. Atom-number shot noise, imaging point-spread function, finite sampling, windowing, and background subtraction propagate into that inference.

The hierarchy of observables matters. A correlation ridge establishes paired fluctuations after systematic controls. Nonseparability additionally compares the anomalous correlator with the independently inferred occupations of both modes; entanglement claims require the inequality to survive finite resolution, loss, and mode mixing. Because occupation reconstruction divides by Bogoliubov amplitudes and detector transfer functions, uncertainty in the condensate profile can shift both sides of the same witness. Reporting the covariance, occupations, mode windows, and loss model makes the conclusion reproducible.

Controlled comparisons

First, in the hydrodynamic limit $k/(m_{\rm at}c_s)\to0$, the dispersion reduces to the acoustic light cone and the horizon tends to $|v_\perp|=c_s$. Second, a uniform subsonic condensate has ordinary Bogoliubov pairs but no stationary horizon conversion; applying the same covariance pipeline provides a no-horizon control. Third, propagating the measured incoming covariance through the calibrated Bogoliubov scattering matrix tests whether the output requires a spontaneous component.

Finite length gives an infrared cutoff, while the healing scale controls high-frequency dispersion. A black-hole/white-hole pair can also support cavity amplification. That instability must not be conflated with stationary Hawking emission: Steinhauer 2022, Eqs. (1)–(2), Figs. 1–4, and article pp. 1–4 found stimulated Hawking correlations in a condensate data set but argued that the observed growth was not black-hole lasing.

Strongest evidence and its boundary

Muñoz de Nova et al. 2019, complete short article, pp. 688–691 reported a thermal spectrum and temperature from correlation measurements in an atomic condensate, extending the Hawking–partner and entanglement analysis of Steinhauer 2016, Figs. 1–5 and Methods. This is the strongest controlled experimental regime for spontaneous analogue claims: a quantum fluid, a known dispersive mode theory, and a pair-sensitive observable. It remains one platform with finite lifetime and model-dependent reconstruction; independent variation of temperature, horizon gradient, preparation history, and analysis windows is essential.

At the 20 August 2026 evidence cutoff, the evidence supports Bogoliubov horizon conversion and published spontaneous-correlation observations in condensates. It does not establish Hawking emission by an astrophysical horizon. The analogue temperature $T_{\rm BEC}=\kappa_{\rm ac}/(2\pi)$ is a scattering scale, not a Bekenstein–Hawking thermodynamic temperature; see Surface Gravity and Hawking Temperature.

Principal obstruction and limitations

The dominant obstruction is separating a small spontaneous signal from state-preparation and inference systematics. The condensate is inhomogeneous and time dependent; atom loss changes $n_0$ and $c_s$; transverse modes and nonlinear depletion challenge the one-dimensional Bogoliubov model; repeated images sample an ensemble rather than simultaneous observables. A statistically significant ridge is not automatically entanglement, and thermality over a narrow frequency interval is not universality.

Even an ideal result validates quasiparticle quantum field theory on an emergent metric with known ultraviolet dispersion. The condensate background has no dynamical Einstein equation, and phonon backreaction follows Gross–Pitaevskii many-body physics. Those boundaries are scientific content, not weaknesses of the experiment.

A further limitation is backreaction control. The depletion carried by the measured phonons should remain small enough that the fitted background is self-consistent, yet large occupation can make stimulated signals dominate. Varying the initial temperature and creation ramp separates those regimes more cleanly than fitting a single effective temperature.

References

• Garay, Luis J., J. R. Anglin, J. Ignacio Cirac, and Peter Zoller. “Sonic Analog of Gravitational Black Holes in Bose–Einstein Condensates.” Physical Review Letters 85 (2000), 4643–4647. doi:10.1103/PhysRevLett.85.4643.
• Macher, Jean, and Renaud Parentani. “Black-Hole Radiation in Bose–Einstein Condensates.” Physical Review A 80 (2009), 043601. doi:10.1103/PhysRevA.80.043601.
• Muñoz de Nova, Juan Ramón, Katrine Golubkov, Victor I. Kolobov, and Jeff Steinhauer. “Observation of Thermal Hawking Radiation and Its Temperature in an Analogue Black Hole.” Nature 569 (2019), 688–691. doi:10.1038/s41586-019-1241-0.
• Steinhauer, Jeff. “Observation of Quantum Hawking Radiation and Its Entanglement in an Analogue Black Hole.” Nature Physics 12 (2016), 959–965. doi:10.1038/nphys3863.
• Steinhauer, Jeff. “Confirmation of Stimulated Hawking Radiation, but Not of Black Hole Lasing.” Physical Review D 106 (2022), 102007. doi:10.1103/PhysRevD.106.102007.