The authors model a one-dimensional, dilute bosonic gas in a ring of length 20, with periodic boundaries and a circulating flow whose wave number is quantized as k_n = 2nπ/l. Their mean-field description is the Gross–Pitaevskii equation, which evolves the condensate wave function in one space and one time dimension. Before the interaction is switched on, the condensate is uniform, with a constant density ρ₀. At positive time, a spatially varying Gaussian coupling is introduced, causing the local sound speed to vary around the ring while the flow velocity remains tied to the circulation.

The variation creates regions where the local Mach number, defined as flow velocity divided by sound speed, crosses unity. In the n = 3 configuration, the flow becomes supersonic between two points: the first crossing is an analogue black-hole horizon, and the second, where the flow returns to subsonic speed, is an analogue white-hole horizon. Phonons generated in the supersonic region cannot propagate upstream against the flow, providing the acoustic analogue of a one-way event horizon. By comparison, the n = 2 case remains subsonic everywhere and contains no such horizon pair. Numerical evolution of the mean field indicates broadly similar density behavior in the two configurations over an initial interval from time zero to 3.9.

To measure what happens outside the coherent condensate, the study decomposes the bosonic field into a mean-field wave function and a fluctuation operator. The linearized fluctuation equation includes both the usual density perturbations and anomalous terms that pair creation and annihilation of excitations. A Bogoliubov transformation mixes particle annihilation and creation operators, allowing the quadratic fluctuation Hamiltonian to be expressed in terms of quasiparticles. In that transformed vacuum, a nonzero single-particle occupation equal to the squared magnitude of each Bogoliubov coefficient persists even at zero temperature. This is the origin of quantum depletion: interactions generate a population of particles outside the condensate mode even when there is no thermal excitation.

The same mode-mixing mathematics links the condensate calculation to the description of Hawking particle production, where outgoing modes are built from incoming positive- and negative-frequency modes. In both cases, a vacuum defined in one basis contains particles when measured in another. The authors emphasize that quantum depletion and Hawking radiation are not the same physical process. In the analogue system, the acoustic horizons supply a mechanism for Hawking-like phonon production, and the objective is to determine how that production contributes to depletion of the condensate as the analogue Hawking temperature changes.

For the evolving, spatially inhomogeneous background, the researchers solve the time-dependent Bogoliubov–de Gennes equations numerically. The resulting local depletion density is the sum of the squared Bogoliubov coefficients across modes, while the global depletion D is its spatial average over the ring. Early in the evolution, configurations with and without horizons deplete at similar rates because mean-field interaction dynamics act across the condensate immediately. A horizon-related difference emerges only after phonons have had time to propagate through the system, on a characteristic timescale of L/c_s, where L is the system size and c_s is the sound speed. The separation between these two effects makes the comparison clearest at later times.

To distinguish steady particle production from short-lived transients, the authors fit the depletion evolution with a constant offset, a linear growth term and a damped oscillatory component. The constant accounts for depletion already present immediately after the quench; the linear coefficient, Γ, is interpreted as the asymptotic depletion rate; and the damped term captures transient excitations and interference among finite-size modes. The numerical data were reproduced with R² greater than 0.995 for all investigated Hawking temperatures. The resulting rate rises monotonically with analogue Hawking temperature, indicating that hotter analogue horizons are associated with faster accumulation of particles outside the condensate mode.

The baseline analogue Hawking temperature is approximately 0.027 in the paper’s dimensionless energy units, expressed there as mc₀² = 1/ξ₀². To examine a broader temperature range while retaining the black-hole–white-hole pair, the authors vary the background density by as much as a factor of ten. At the upper end of the explored range, around T_H ≈ 0.1 mc₀², global depletion reaches approximately 25%. At that level, about 75% of particles remain in the condensate mode, so the state is still predominantly condensed. Nevertheless, the authors argue that the linearized treatment is becoming less reliable because higher-order correlations, beyond-mean-field physics and backreaction are no longer negligible.

The temperature increase creates a trade-off that is central to the proposed connection between theory and tabletop analogue gravity. A higher Hawking temperature produces a faster depletion signal, which may be easier to distinguish from noise or finite-temperature effects during an experiment. At the same time, it pushes the condensate toward a regime in which quantum excitations can influence the mean-field background and the Bogoliubov–de Gennes approximation eventually breaks down. The study therefore does not claim a sharp universal transition at 25%; rather, it uses that level as a practical benchmark within its modeled parameter range. Tuning the background density is also more controllable in the calculations than changing the spatial interaction profile, which brings the two horizons close together and makes local depletion harder to resolve.

Overall, the simulations suggest that the long-time depletion rate, rather than the accumulated depletion at an arbitrary observation time, is the clearest indicator of horizon-induced production in this model. The approach offers a way to study microscopic effects associated with analogue horizons while condensation remains robust, but it also identifies where the present framework must give way. The authors propose future work beyond Bogoliubov theory that explicitly incorporates strong backreaction, as well as a closer examination of the microscopic structure of depletion near the two horizons. Their results frame condensate depletion as a possible experimental probe of Hawking-related mode mixing, without equating the laboratory system with an astrophysical black hole or claiming that gravitational Hawking radiation has been observed.

Subject of Research: Numerical study of Hawking temperature and quantum depletion in a ring-shaped Bose–Einstein condensate with analogue horizons

Article Title: Probing Hawking temperature threshold via quantum depletion in Bose–Einstein condensate

Article References: Rana, A. (2026). Probing Hawking temperature threshold via quantum depletion in Bose–Einstein condensate. The European Physical Journal C, 86(10), Article 1193. https://doi.org/10.1140/epjc/s10052-026-16404-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16404-6

Keywords: Bose–Einstein condensate, Hawking temperature, quantum depletion, analogue gravity, Bogoliubov theory, black hole horizon, backreaction, condensate coherence, phonon production, Probing, Hawking, temperature