A two-mode model for black hole evaporation and information flow

TL;DR

This work develops a minimal two-mode model for black hole evaporation by pairing a negative-energy geometric oscillator with a positive-energy radiation oscillator in a bilinear Hamiltonian $H=\tfrac{1}{2}(p_x^2+\omega_x^2 x^2)-\tfrac{1}{2}(p_y^2+\omega_y^2 y^2)+g\,x y$. It derives exact normal-mode solutions, introduces smooth envelope functions $A(x)$ and $B(x)$ to bridge discrete modal coefficients with a continuous geometric coordinate, and analyzes energies, occupation numbers, and reduced entropies within Gaussian-state formalism. Numerical simulations in a truncated Fock space reveal roughly out-of-phase energy exchange with near-equal mean occupations $\langle n_x\rangle \approx \langle n_y\rangle$ and periodic growth of the reduced entropy $S_x(t)$, illustrating entanglement dynamics consistent with information flow in evaporation. Overall, the model provides a tractable, analytically solvable platform capturing key qualitative features of black-hole evaporation and information transfer, while suggesting concrete extensions and potential quantum-simulation implementations.

Abstract

We develop and analyze a two-oscillator model for black hole evaporation in which an effective geometric degree of freedom and a representative Hawking radiation mode are described by coupled harmonic oscillators with opposite signs in their free Hamiltonians. The normal-mode structure is obtained analytically and the corresponding modal amplitudes determine the pattern of energy exchange between the two sectors. To bridge the discrete and semiclassical pictures, we introduce smooth envelope functions that provide a continuous effective description along the geometric variable. Numerical simulations in a truncated Fock space show that the two oscillators exchange quanta in an approximately out-of-phase manner, consistent with an effective conservation of $\langle n_x\rangle - \langle n_y\rangle$. The reduced entropy $S_x(t)$ exhibits periodic growth, indicating entanglement generation. These results demonstrate that even a minimal two-mode framework can capture key qualitative features of energy transfer and information flow during evaporation.

Figures (8)

• Figure 1: Continuous envelope functions $A(x)$ (blue) and $B(x)$ (red) constructed from the discrete modal coefficients $A_j,B_j$. The dots and squares correspond to the numerically obtained values of the amplitudes for two representative modes, while the smooth curves illustrate effective interpolations that describe how the black hole and radiation sectors vary along the geometric variable $x$. These continuous profiles are used in the subsequent analytical and numerical analysis to connect the discrete two-mode model to a semiclassical, quasi-continuous description of the evaporation process.
• Figure 2: Behavior of the metric functions $A(x)$ and $B(x)$ defined by $A(x)=1-\frac{2M_0}{x}$ and $B(x)=\frac{e^{-kx}}{x}$ for an evaporating black hole with initial mass $M_0=1$ and evaporation parameter $k=0.1$. The function $A(x)$ vanishes at $x=2M_0$, indicating the classical event horizon, while $B(x)$ exhibits an exponential suppression that models the effect of Hawking evaporation.
• Figure 3: Probability distribution $P_{n_x}(t)$ of the geometric oscillator as a function of time, obtained from the full spectral simulation with cutoff $N_{\mathrm{cut}}=8$. Parameters: $\omega_x=1.0$, $\omega_y=0.8$, $g=0.1$, and $\alpha=1.0$. The color scale indicates the occupation probability of each Fock level $|n_x\rangle$ at time $t$. The initial state is a coherent excitation centered near $n_x\simeq|\alpha|^2\approx1$, while higher excitations become transiently populated due to coupling with the radiation mode. The recurrent rise and decay of probability in successive $n_x$ bands illustrate a quasi-periodic energy exchange that parallels the black hole mass loss and partial recovery phases. This pattern reproduces qualitatively the population–transfer diagrams shown in kiefer2009blackhole, confirming that the simplified two-mode Hamiltonian already encodes the essential evaporation dynamics.
• Figure 4: Spatial dependence of the entanglement entropy $S(x)$ obtained from the coupled oscillator model of black hole evaporation. The parameters are chosen as $\omega_x=\omega_y=1$ and $g=0.05$. The smooth variation of $S(x)$ reflects how the local degree of correlation between the "black hole" and "radiation" modes changes along the effective geometric variable $x$.
• Figure 5: Time evolution of the coupled oscillator system representing black hole evaporation. The upper panel shows the expectation values of the occupation numbers $\langle n_x(t)\rangle$ and $\langle n_y(t)\rangle$ for the "black hole" and "radiation" modes, respectively, while the lower panel displays the corresponding von Neumann entropy $S_x(t)$ of the reduced density matrix of the black hole mode. The parameters are $\omega_x=1$, $\omega_y=0.8$, coupling constant $g=0.1$, and initial coherent amplitude $\alpha=1$. The entropy oscillations follow the energy exchange between the two subsystems, reflecting the alternating dominance of the geometric and radiative degrees of freedom.