Momentum correlations of the Hawking effect in a quantum fluid

The Hawking effect – the spontaneous emission of correlated quanta from horizons – can be observed in laboratory systems where an acoustic horizon forms when a fluid transitions from subcritical to supercritical flow. Although most theoretical and experimental studies have relied on real-space observables, the frequency-dependent nature of the Hawking process motivates a momentum-space analysis to access its spectral structure and entanglement features. Here, we numerically compute the momentum-space two-point correlation function in a quantum fluid using the truncated Wigner approximation, a general method applicable to both conservative and driven-dissipative systems. We consider a polaritonic fluid of light in a realistic configuration known to yield strong real-space correlations between Hawking, partner, and witness modes. We find signatures that are directly accessible in state-of-the-art experiments and offer a robust diagnostic of spontaneous emission. Our results form the basis for a new theoretical framework to assess a variety of effects, such as quasi-normal mode emission or modifications of the horizon structure on the Hawking spectrum.

💡 Research Summary

The paper investigates the Hawking effect—spontaneous emission of correlated quanta from an acoustic horizon—in a quantum fluid of polaritons, focusing on momentum‑space observables rather than the more common real‑space density correlations. The authors employ the truncated‑Wigner approximation (TWA) to simulate the driven‑dissipative Gross‑Pitaevskii dynamics of a one‑dimensional polariton wire that contains a localized repulsive potential. By shaping the pump intensity and wave‑vector profile, they create a step‑like flow: subcritical upstream (flow speed v < c_B) and supercritical downstream (v > c_B), thereby establishing a Killing horizon for collective excitations.

In each asymptotic region the Bogoliubov‑de Gennes dispersion relation ω±(k) (Eq. 3) yields a positive‑norm branch (ω₊) and a negative‑norm branch (ω₋). In the supercritical region the negative‑norm branch is pulled up to positive laboratory frequencies, allowing a negative‑energy partner mode (denoted dn) to exist alongside the Hawking radiation mode (HR) that propagates upstream. Scattering at the horizon is described by a 3 × 3 S‑matrix (Eq. 4) that mixes incoming modes (in, p, d) into outgoing modes (HR, down, dn). Because dn carries negative norm, the relevant operator in observables is b_dn (not b_dn†), leading to a dominant anomalous two‑mode correlation ⟨b_HR b_dn⟩. This correlation violates the Cauchy–Schwarz inequality, signalling non‑separability (quantum entanglement) of the emitted pair.

The numerical study proceeds by generating many stochastic realizations of the polariton field ψ(x,t) within TWA, Fourier‑transforming to ψ(k), and evaluating the momentum‑space second‑order correlation function

 δg₂(k,k′) = ⟨ψ†(k′)ψ†(k)ψ(k)ψ(k′)⟩ / ⟨ψ†(k′)ψ(k′)⟩⟨ψ†(k)ψ(k)⟩ − 1.

Two window configurations are examined. In the first (Fig. 2a) both windows are identical and centered on the horizon, producing a symmetric pattern in (k,k′) that largely reflects self‑correlations on the same side of the horizon. In the second (Fig. 2b) the windows are placed on opposite sides of the horizon; this suppresses the large autocorrelation background and reveals off‑diagonal features associated with cross‑horizon mode pairs. The resulting map (Fig. 2d) displays clear “Hawking traces”: (i) an anomalous HR–dn* line (the true Hawking‑partner correlation) and (ii) an HR–down line (the grey‑body channel). These traces are strongest near the low‑frequency cutoff ω_min, peak around that frequency, and decay toward the upper cutoff ω_max, mirroring the frequency dependence of the Bogoliubov coefficients. Additional zero‑correlation bands appear along k = k_down and k′ = k_up, originating from spectral leakage of the amplified pump modes.

The analysis shows that momentum‑space correlations directly encode the dispersion relations k_i(ω) of each outgoing mode, providing mode‑resolved spectral information that is inaccessible in integrated real‑space measurements. Moreover, the TWA proves capable of capturing quantum statistics even in a driven‑dissipative setting, suggesting the method’s applicability to a broad class of analog gravity platforms (photonic, atomic, superconducting). The study also highlights practical considerations: increasing the distance between the windows and the horizon reduces correlation strength and raises noise due to finite polariton lifetime, emphasizing the need to place detection regions close to the horizon in experiments.

In conclusion, the work delivers the first comprehensive numerical calculation of momentum‑space Hawking correlations in a transcritical quantum fluid, identifies experimentally feasible signatures (mode‑specific off‑diagonal lines, Cauchy–Schwarz violation), and establishes a versatile framework for probing not only the basic Hawking process but also more intricate phenomena such as quasi‑normal‑mode emission, horizon‑structure modifications, and superradiant scattering. The results pave the way for future experiments that can directly verify the spectral and entanglement structure of analog Hawking radiation.