Analogue Hawking Radiation Overview

• Analogue Hawking radiation is the spontaneous quasi-thermal emission from engineered horizons in laboratory systems, mimicking black hole radiation.
• It relies on Bogoliubov mode mixing in diverse media to produce thermal spectra and measurable quantum entanglement signatures.
• Experimental platforms like ultracold gases, optical fibers, and metamaterials illustrate how dispersion and greybody factors shape the radiation spectrum.

Analogue Hawking radiation refers to the emission of spontaneous quasi-thermal radiation from effective event horizons engineered in laboratory systems, mimicking the predicted quantum effect at the event horizon of an astrophysical black hole. This phenomenon is realized in a diverse set of physical platforms—Bose–Einstein condensates (BECs), water waves, optical fibers, quantum Hall edge states, and electromagnetic metamaterials—where wave excitations in media with spatially varying backgrounds exhibit horizon-like transitions. At such analogue horizons, mixing of positive- and negative-norm (or frequency) modes occurs, resulting in particle creation governed by a Bogoliubov transformation and a spectrum that closely approximates a thermal Bose–Einstein (or Fermi–Dirac, for fermionic systems) distribution, with an effective temperature fixed by the surface gravity—the gradient of the mismatch between background flow and local excitation velocity—at the horizon. Analogue Hawking radiation provides a direct probe of kinematic quantum field theory effects in curved spacetime backgrounds, enabling controlled studies of horizon thermality, entanglement, trans-Planckian regularization, greybody corrections, and dynamical effects beyond reach in astrophysical black holes.

1. Theoretical Foundations and Kinematic Mechanism

The essential requirement for analogue Hawking radiation is the existence of a horizon for wave excitations: a spatial transition in the background of a dispersive medium where the local group velocity $c(x)$ is overtaken by a background flow $v(x)$, i.e., $|v(x)| = c(x)$, termed the analogue (acoustic, optical, etc.) horizon. For example, in a BEC, the one-dimensional flow is engineered so that far on the left ($x\to-\infty$), the flow is supersonic ($|v_l| > c_l$), and far on the right ($x\to+\infty$), it is subsonic ($|v_r| < c_r$). The transition point $x_\mathcal{H}$, where $|v(x_\mathcal{H})| = c(x_\mathcal{H})$, constitutes the analogue horizon (Coutant et al., 2017).

Linearized wave excitations (phonons, surface waves, photons) in these variable backgrounds can always be cast into an effective wave equation,

$$(\partial_t + v(x)\partial_x)^2 \phi - c(x)^2 \partial_x^2 \phi = \text{dispersive terms},$$

which in the low-dispersion limit reduces to the d'Alembertian of a scalar field in a metric of Painlevé–Gullstrand form:

$ds^2 = -[c(x)^2-v(x)^2]dt^2 - 2v(x)dx\,dt + dx^2.$

Solutions in the presence of a horizon demonstrate mode mixing between positive- and negative-norm branches as determined by the Bogoliubov inner product. The resulting scattering matrix relates in-modes to out-modes and gives rise to spontaneous emission: for each frequency,

$$a^{\rm out}_\omega = \alpha_\omega a^{\rm in}_\omega + \beta_\omega (a^{\rm in}_{-\omega})^\dagger,$$

where $|\beta_\omega|^2$ is the mean number of emitted quanta per mode (Steinhauer, 2015, Robertson, 2015, Aguero-Santacruz et al., 2023).

2. Analogue Surface Gravity and Hawking Temperature

The universal prediction of Hawking radiation derives from the exponential redshifting experienced by outgoing modes near the horizon. The corresponding effective surface gravity (in the non-dispersive limit) is given by

$\kappa = \partial_x(v + c)|_{x_\mathcal{H}},$

and the analogue Hawking temperature is

$T_H = \frac{\hbar \kappa}{2\pi k_B},$

where $\hbar$ is the reduced Planck constant and $k_B$ the Boltzmann constant (Steinhauer, 2015, Coutant et al., 2017, Parola et al., 2017, Vieira et al., 2014, Robertson, 2011). This relationship holds for BECs, water waves, metamaterial horizons, optical fibers, and quantum Hall edge horizons (Stone, 2012, Aguero-Santacruz et al., 2020, Bera et al., 2020). The spectrum of created radiation is Bose–Einstein (or Fermi–Dirac for chiral fermions) at $T_H$ modulo possible greybody factors.

3. Dispersion, Robustness, and Greybody Effects

All experimental platforms exhibit nontrivial dispersion at high frequencies, modifying the simple relativistic theory. The most general wave equation includes higher-order derivatives or a nonlinear $k$–dependence—e.g., for a BEC:

$(\omega - v k)^2 = c^2 k^2 + \frac{k^4}{4m^2},$

where $m$ is the atomic mass. Weak dispersion justifies a WKB treatment away from the horizon and a matched asymptotic analysis near the turning point (Belgiorno et al., 2020, Porro et al., 20 Jun 2024).

Dispersion regularizes the infamous trans-Planckian problem: instead of infinite blueshift, high-$k$ excitations are either cut off (subluminal) or pass through the horizon (superluminal), but for small $\omega/\Lambda$ (with $\Lambda$ the dispersive scale), thermal Hawking emission persists to high accuracy (Coutant et al., 2017, Porro et al., 20 Jun 2024, Robertson, 2011).

Greybody factors arise from partial reflection or mode mixing outside the near-horizon region. Analytically, they appear as prefactors $R(\omega)$, $B(\omega)$, or transmission coefficients modifying the Planck spectrum. In the BEC BdG model, these are controlled by background matching: vanishing in the "conformal coupling" limit where $v(x) c(x) = \text{const}$, maximizing the "purity" of the outgoing Hawking spectrum (Coutant et al., 2017). In dynamical Casimir setups, cavity geometry, acceleration profile, and finite lifetime create oscillatory greybody features (Martín-Caro et al., 18 Jul 2025, Martín-Caro et al., 2023).

4. Experimental Realizations and Measurement Techniques

Laboratory detection of analogue Hawking radiation has been achieved in ultracold atomic BECs and classical water wave flumes (Steinhauer, 2015, Weinfurtner et al., 2010, Weinfurtner et al., 2013).

• BEC platforms: Steinhauer's experiments (Steinhauer, 2015) detected spontaneous Hawking emission and measured two-point density–density correlations via high-resolution in situ imaging, extracting the characteristic negative off-diagonal bands representing entangled phonon pairs. The nonseparability criterion $|C(k)|^2 > n(k)[n(k)+1]$ was used to verify quantum entanglement at high energies. Greybody suppression enhanced the signature at optimal matching parameters (Coutant et al., 2017, Steinhauer, 2015).
• Hydrodynamic analogues: Water wave experiments use surface gravity waves encountering depth transitions: the blocking region acts as an effective horizon, with mode conversion measured by analyzing the amplitude spectrum of outgoing positive- and negative-norm components, which fit the Boltzmann distribution with temperature set by the local surface gravity (Weinfurtner et al., 2010, Weinfurtner et al., 2013).
• Optical analogues: Stimulated and spontaneous Hawking-like emission is observed in optical fibers by tracking the frequency conversion of probe light interacting with intense pump pulses generating moving refractive-index perturbations; the emission spectrum is analyzed for thermal components and greybody oscillations resulting from the effective "acceleration" profile (Aguero-Santacruz et al., 2020, Bera et al., 2020).
• Metamaterials and Quantum Hall analogues: In dielectric–metamaterial composites with vanishing refractive index, complex-path (tunneling) methods reveal thermal photon flux signatures at the effective electromagnetic horizon, and the spectrum is Planckian up to losses and bandwidth limits (Bera et al., 2020). Chiral edge states in quantum Hall systems at $\nu=1$ have also been shown to exhibit Fermi–Dirac distributed Hawking radiation, with straightforward identification of partner correlations (Stone, 2012).
• Cavity QED and Dynamical Casimir analogues: Moving-boundary cavities implement effective geometries with time-dependent horizons, producing Hawking-like emission with fit spectra including robust greybody modulations (Martín-Caro et al., 18 Jul 2025, Martín-Caro et al., 2023).

5. Entanglement, Decoherence, and Nonseparability

A defining quantum signature of Hawking radiation is the creation of correlated particle–partner pairs straddling the horizon. In BEC experiments, this is measured via two-point density correlation matrices $G^{(2)}(x,x')$, whose off-diagonal (ingoing→outgoing) "tongues" correspond to spontaneous emission of Hawking pairs (Steinhauer, 2015, Steinhauer, 2015). The degree of quantum entanglement is tested through violation of classical Cauchy–Schwarz inequalities and the Peres–Horodecki criterion. Experiments reveal that entanglement persists in the high-frequency wing of the spectrum, while the low-frequency sector is only classically correlated, both features quantitatively explained by Bogoliubov theory (Steinhauer, 2015, Coutant et al., 2017).

Greybody-induced partner mixing and background temperature reduce the strength and visibility of entanglement, especially at low frequencies. Optimization strategies (e.g., conformal coupling, noise minimization, smooth horizon engineering) directly increase the observable quantum signal (Coutant et al., 2017, Palan et al., 2022). Decoherence due to losses, atom heating, or stimulated emission enhances the overall signal but can mask the spontaneous quantum component, mandating careful background subtraction and dynamic modeling (Palan et al., 2022).

6. Exact Solutions, Dynamical Protocols, and Generalizations

Beyond semiclassical methods, exactly solvable models (e.g., the 1D Tonks–Girardeau gas) establish strict conditions for true thermal analogue Hawking emission. Only for appropriately smooth and subsonic–supersonic transitions does the upstream phonon occupation become genuinely Planckian—sharp steps or barriers yield nonthermal or featureless spectra (Parola et al., 2017). Confluent Heun function solutions in analytic black-hole metrics yield direct calculation of the outgoing spectra and greybody corrections for acoustic and optical configurations (Vieira et al., 2014).

Hamilton–Jacobi (tunneling) approaches, both in continuous fluid and discrete or $\mathcal{PT}$-symmetric systems, provide unified derivations of emission rates and clarify the robustness of the thermal spectrum against ultraviolet/Planck-scale modifications, with energy-dependent surface gravity corrections parameterizing deviation from thermality in the presence of strong dispersion or in horizonless regimes (Porro et al., 20 Jun 2024, Bagchi et al., 2023).

In reversible "bouncing" geometries (e.g., dynamically modulated cavities), even in the absence of a strict event horizon, quasi-thermal emission occurs if null rays undergo exponential peeling—a hallmark originally identified in black-hole collapse calculations—while the detailed greybody factors reveal imprints of transient acceleration and cavity boundaries (Martín-Caro et al., 2023, Martín-Caro et al., 18 Jul 2025).

7. Outlook: Universality, Limits, and Experimental Frontiers

The universality of analogue Hawking radiation is now established across diverse systems, unified by the kinematic conditions of a mode-mixing horizon and the existence of positive/negative-norm branches linked by a conserved pseudo-norm (Aguero-Santacruz et al., 2023). The general criterion for a system to exhibit Hawking radiation comprises: (i) existence of an apparent horizon for some excitation sector, (ii) finite surface gravity, (iii) mode-mixing producing nonzero Bogoliubov $\beta$-coefficients, and (iv) sufficient adiabaticity for in/out bases (Aguero-Santacruz et al., 2020).

Experimental limitations arise predominantly from low effective temperatures (e.g., sub-nK in BECs), thermal/stimulated backgrounds, atom/photon losses, and detection sensitivity. Nevertheless, ongoing advances in ultracold quantum gases, photonic microstructures, superconducting circuits, and synthetic topological platforms are extending reach towards direct observation of not only spontaneous but also entangled and dynamical aspects of analogue Hawking quanta. The ability to tune dispersion, geometry, noise, and acceleration in these platforms provides an arena for probing fundamental questions in quantum field theory in curved spacetimes, including information recovery, trans-Planckian regularization, and the stability of evaporating (analogue) horizons (Steinhauer, 2015, Coutant et al., 2017, Aguero-Santacruz et al., 2020).

Analogue Hawking radiation thus provides a bridge connecting mathematical structures in gravitation, condensed matter, optics, and quantum information, with ongoing research aimed at refining detection, controlling decoherence, and engineering tailored horizon geometries for precision measurement of quantum field phenomena in laboratory analogues.