Dumb Hole Spectroscopy: Probing Analog Horizons

• Dumb Hole Spectroscopy is the study of frequency-dependent acoustic emissions from analogue black holes in fluids or BECs.
• It employs quantum noise analysis and quasi-normal mode spectroscopy to measure parameters like circulation, drain rate, and surface gravity.
• Experimental setups using draining vortex flows and BECs validate theoretical models, offering non-invasive insights into horizon physics and quantum field effects in curved spacetime.

Dumb hole spectroscopy is the study of frequency-dependent responses (spectra) emitted by sonic analogues of black holes (“dumb holes”), typically engineered in fluid systems or Bose–Einstein condensates (BECs), to extract detailed information about the effective spacetime geometry and flow parameters that define the horizon and ergoregion. Central to this discipline is the analysis of the quantum and classical acoustic radiation—phonons or surface waves—that is spontaneously emitted by the dumb-hole horizon, with a spectrum that is, under broad conditions, exactly thermal. This synthesis of amplifier quantum noise, quasi-normal mode (QNM) spectroscopy, and fluid–gravity correspondence enables the direct experimental investigation of horizon physics, non-invasively characterizing analogue systems and providing a precision window into the foundations of quantum field theory in curved spacetime (Unruh, 2011, Torres et al., 2019, Das et al., 2010).

1. Fluid–Gravity Analogy and Acoustic Horizons

The fluid–gravity analogy, first formalized by Unruh, establishes that small perturbations (sound or surface waves) in an irrotational, barotropic, inviscid fluid propagate according to a wave equation identical to that of a massless scalar field in an effective metric. For two-dimensional flows, such as the draining bathtub (DBT) vortex, the background velocity field

$$\mathbf v_0(r) = -\frac{D}{r}\hat e_r + \frac{C}{r}\hat e_\theta,$$

with drain rate $D$ and circulation $C$, defines the analogues of black hole and ergoregion horizons. The “acoustic horizon” is located at $r_{\rm hor} = D/c$, where $c$ is the sound speed, corresponding to the locus where inward fluid speed equals the local wave speed. More generally, in conformal charged fluids, the emergent acoustic metric for phonon perturbations is deduced from the energy-momentum tensor structure and exhibits horizons at $|v| = c_s$ (Das et al., 2010).

2. Quantum Noise, Thermality, and the Sonic Hawking Effect

The quantum theory of linear amplifiers, when applied to dumb-hole horizons, demonstrates that the minimum unavoidable quantum noise is of exactly thermal (Planckian) character (Unruh, 2011). Modeling the horizon as a two-mode, phase-insensitive quantum amplifier, the transformation of annihilation and creation operators for input (“in”) and output (“out”) channels is

$$C_{\rm out}(\omega) = A(\omega) a_{\rm in}(\omega) + B(\omega) b_{\rm in}^\dagger(\omega)$$

with $A(\omega) = \cosh\mu(\omega)$ and $B(\omega) = \sinh\mu(\omega)$ determined by requirements of unitarity and preservation of canonical commutators.

The mean occupation number in an outgoing mode is

$$n_C(\omega) = \langle 0| C_{\rm out}^\dagger C_{\rm out} |0\rangle = |B(\omega)|^2 = \sinh^2\mu(\omega),$$

and the outgoing channel acquires a thermal density matrix

$$\rho_C \propto \exp[-\Lambda(\omega) C_{\rm out}^\dagger C_{\rm out}],$$

with $$e^{-\Lambda(\omega)} = |B(\omega)|^2 / [ |A(\omega)|^2 - |B(\omega)|^2 ] = |B(\omega)|^2$$ for $|A|^2 - |B|^2 = 1$. The output spectrum is therefore thermal by construction, enforced by the commutation relations.

The Hawking temperature for the dumb hole is set by the fluid’s surface gravity analogue,

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

where $$\kappa = \left.\partial_x\left[v(x)-c_s\right]\right|_{x_H}$$ is the gradient at the horizon $x_H$.

3. Quasi-Normal Modes and Ringdown Spectroscopy

Perturbations initiated near a dumb hole induce ringdown, a process governed by superpositions of quasi-normal modes—solutions that are ingoing at the acoustic horizon and outgoing at infinity. For azimuthal number $m$ and overtone $n$, the modes are of the form

$\phi_m(r,t) \sim e^{-i\omega_{mn} t} u_{mn}(r),$

with complex frequency $\omega_{mn} = \Re\omega_{mn} - i \Gamma_{mn}$, where $\Gamma_{mn}$ is the decay rate. Techniques for extracting the QNM spectrum include both analytic (hydron/light-ring) approximations, which rely on geodesics of the effective acoustic metric, and direct numerical evolution of the PDEs arising from linearized fluid dynamics (Torres et al., 2019).

Boundary conditions for the modal analysis are purely ingoing at the horizon, and purely outgoing at spatial infinity. The QNM frequencies encode geometry: their real parts reflect oscillation frequencies while the imaginary components capture dissipative timescales. For well-resolved spectra, these appear as Lorentzian peaks in the frequency domain, with positions and linewidths determined by system parameters.

4. Spectroscopic Methodology and Parameter Extraction

The methodology of analogue black hole spectroscopy (ABHS) comprises:

1. Measurement: Field observables (surface elevation, phonon density) are recorded at fixed radii in a laboratory dumb hole (surface-wave tank, BEC).
2. Spectrum Extraction: One forms the analytic signal, performs Fourier transforms (azimuthal and temporal), and computes the power spectral density $P_m(\omega, r) \propto |\tilde\phi_m(\omega, r)|^2$. The least-damped QNMs appear as horizontal ridges in $(r,\omega)$ plots.
3. Parameter Determination: For trial parameters $(C, D)$ (circulation, drain), theoretical QNM spectra $\omega_\star(C, D; m)$ are computed; the measured frequencies are fit by minimizing the mean-square error. Co- and counter-rotating branches (different sign $m$) allow full parameter recovery. This approach achieves order-percent accuracy on inferred parameters (Torres et al., 2019).

Spectroscopic analysis can also recover the surface gravity $\kappa$ (from $T_H$) and test the Planckian nature of the outgoing spectrum by fitting the observed noise power spectral density

$$S(\omega) = \frac{\hbar\omega}{\exp(\hbar\omega/(k_BT_H)) - 1}$$

over low frequency ranges.

5. Gravity Duals, AdS/CFT, and Fluid-Gravity Correspondence

In systems with charged conformal fluid flows, the fluid–gravity correspondence enables a dual description in terms of deformations of anti-de Sitter (AdS) black brane geometries (Das et al., 2010). Dumb-hole quasinormal modes in the fluid correspond to gravitational perturbations in the bulk geometry, whose spectra encode both discrete QNM frequencies and the thermal envelope set by $T_H$. The poles of the retarded Green's function $G_R(\omega)$, located at QNM frequencies $\omega_n$, yield the Lorentzian spectral function

$$\rho(\omega) \approx \sum_n Z_n \frac{\Gamma_n}{(\omega - \Re\,\omega_n)^2 + \Gamma_n^2}$$

with Planckian envelope. The thermal nature and linewidths of the emission, both in dumb-hole fluids and their AdS duals, are governed by the horizon gradient via $T_H = (\hbar/2\pi k_B)|\partial_x v|_{horizon}$.

6. Experimental Realizations and Precision Metrology

Dumb-hole spectroscopy is implemented in platforms such as draining vortex flows in water tanks and atomic BECs. The experimental workflow consists of:

1. Generation of a steady, near-ideal flow forming an acoustic horizon,
2. Controlled perturbation of the flow (impulse or pulse excitation),
3. Multichannel acquisition of the propagating field,
4. Extraction of the QNM spectrum and Planckian noise parameters,
5. Inference of system properties such as circulation, drain, and surface gravity with competitive accuracy versus invasive methods (Torres et al., 2019).

Correlation measurements between outgoing channels (entangled Hawking partner modes) provide further verification of the two-mode squeezing structure imposed by Bogoliubov transformations.

7. Spectral Signatures and Theoretical Implications

The spectral signatures in dumb-hole systems consist of a discrete set of Lorentzian peaks (the QNM spectrum), superimposed on a thermal Bose–Einstein statistical envelope at the Hawking temperature, with explicit Planckian form for the mean phonon occupation

$$n(\omega) = \frac{1}{e^{\hbar\omega/(k_BT_H)} - 1}.$$

For $\hbar\omega \ll k_BT_H$, the Rayleigh–Jeans limit applies; for $\hbar\omega \gg k_BT_H$ the population is exponentially suppressed. The peak of $S(\omega)$ is located at $\omega \approx 0.83\,k_BT_H/\hbar$, with spectral width $\sim k_BT_H/\hbar$.

This combination of thermal noise enforced by unitarity, horizon-induced QNM spectra, and direct extractability from experimental data makes dumb hole spectroscopy a rigorous, non-invasive probe of horizon physics and the fluid–gravity analogy (Unruh, 2011, Torres et al., 2019, Das et al., 2010).