Hawking Radiation As Stimulated Emission

We regard the Parikh-Wilczek’s tunnelling model of Hawking radiation as a quantum mechanical process of stimulated emission. The hypothesized microstates are found at the horizon with double degeneracy. A Jaynes-Cummings toy model for a black hole in the cavity is proposed to demonstrate how to write a qubit via the angular-dependent transition coupling, which might be related to the soft Goldstone hairs at analytic continuation. At last, we show how information is retained in the black hole by computing the time evolution of mutual entanglement entropy in the cavity-black holes system.


I. INTRODUCTION AND SUMMARY
Parikh and Wilczek earlier gave a semiclassical derivation of Hawking radiation as a tunneling process, similar to pair creation in a constant electric field [1]. If one considers a particle with energy ω is emitted from a black hole with mass M. The emission rate reads 1 Γ ∼ e −8πω(M −ω/2) = e −ω/T H Boltzmann factor e +4πω 2 . (1) It is impressive a nonthermal spectrum can be obtained by assuming conservation of energy during the tunneling process. In addition, the exponent happens to be the very change of entropy, conservation of information is therefore achieved in terms of mutual information [2]. Nevertheless, it is unclear whether the infamous paradox of lost information can be resolved in this macroscopic picture since it still cannot reveal those hidden microstates responsible for the black hole entropy. In this letter, we model the black hole as an atom with many degenerate states and regard the Hawking radiation as a quantum mechanical process of stimulated emission [3]. Our quantum mechanical model of black holes has following features: • We argue that transition from different degenerate excited states maybe responsible for the featured radiation as depicted in the Fig. 1. In the section II, we explicitly show that at large black hole limit, the nonthermal spectrum in the Parikh-Wilczek tunneling model can be reproduce. This agreement implies the hypothesized microstates with double degeneracy were seated at the horizon. It is also tempted to guess they are closed related to the soft gravitons which claim the black hole information.
• To justify our statement, in the section III we propose a Jaynes-Cummings (JC) toy model for a black hole in the cavity and demonstrate how to write a qubit via the angular-dependent transition coupling. In the section IV, we further argue that after analytic continuation, the coupling strength is related to the soft hairs, which is represented by the Goldstone boson modes. to the quantum state of black hole with mass M a(b) after(before) radiation. According to the common notation, let A be the spontaneous emission rate and B ba ρ(ω) be the transition rate for stimulated emission. We have following additional assumptions: • Each state has g a(b) degeneracy, which should be exponentially large enough to accommodate degrees of freedom in a black hole.
• Those degrees of freedom are located somewhere at or outside the event horizon.
Applying the detailed balancing to the cavity-black holes system, the radiation spectrum reads In thermal equilibrium at Hawking temperature T H , the black hole population in the state To have simple Maxwell-Boltzmann distribution, one would have demanded the relation g a B ab = g b B ba . In our model, however, we have assumed g a(b) ∼ e α 4 A(βM a(b) ) in order to respect the area law such that it can host black hole's Bekenstein-Hawking entropy. To be precise, we assume the hypothesized microstates are located at radius r = βM a(b) and A(βM a(b) ) is the area of sphere with that radius in the isotropic coordinate. Here the isotropic metric is adopted to honestly reflect spatial distance in sphere: The proportionality coefficients α and β will be determined shortly. We are looking for large black hole limit where 1/M ≪ ω ≪ M, then equation (2) can be cast into where We remark the choices for coefficients α and β as follows: • To recover the Boltzmann factor, we choose β = 1/2 such that the leading term in function C(M, ω) vanishes. This suggests those degrees of freedom are seated at the horizon 2 .
This implies that the degeneracy at each energy level is twice amount of the black hole entropy, for S BH = A/4.
It has been conjectured that black hole microstates are given by those soft hairs which enjoy the Bondi-Metzner-Sachs (BMS) transformation of supertrasnlation and superrotation [4][5][6][7][8][9][10][11][12]. This infinite dimensional global symmetry is expected to be broken by the horizon geometry to form a finite number of gapless Bogoliubov-Goldstone modes [13,14]. Our Einstein model of stimulated emission happens to agree with that picture that double at 2 We note that in the isotropic coordinate, the horizon locates at r = M/2. the horizon could be interpreted as N massless spin-2 soft gravitons at criticality, where N ∼ e M 2 /l 2 p respecting the area law. Those microstates in coordinate representation can be modeled as Boolean qubit at the lattice of Planckian spacing [15,16] or in momentum representation as eigenstates labeled by angular quantum numbers, say |l, m, s , where s = ±2, |m| < l and l ≤ l max ∼ √ N [17].

III. JAYNES-CUMMINGS MODEL OF BLACK ATOM
We now construct a Jaynes-Cummings (JC) toy model for the black hole atom fit to our thought experiment. The JC model was to study the interaction of a two-level atom with a single electromagnetic field [18]. Now we would like to replace the atom by a black hole. It could be dangerous to model a black hole without its spacetime context. However, it has been argued that Hawking radiation, including the nonthermal feature, could be well formulated without spacetime [19]. Therefore, our quantum mechanical model may be just enough to serve our purpose to explain this featured radiation. After rotating-wave approximation, the JC Hamiltonian, which is composed of energy of black holes, radiation energy and interaction, reads where indices i, j label each degeneracy state.α andα † are annihilation and creation operators of photons. The couplings g ij are responsible for emission and absorption. Without loss of generality, we only focus on transition among just few states, say |n, a 1 for the ground state, and |n − 1, b 1 , |n − 1, b 2 for two degenerate excited states. We define the product state, for instance, |n, a 1 = |n ⊗ |a 1 , where n is the photon occupation number in the cavity and spin index s is suppressed in atom states. The eigenvalues and eigenstates are calculated in the Appledix. In particular, at the resonance ∆ = 0 and even coupling strength g 11 = g 12 = g, we have following remarks: • After analytic continuation of coupling g → ig, the transition (A4) can be regarded as the Bogoliubov transformation between in state · 0 and out state · t similar to that in the parametric amplifier [20] or a harmonic oscillator with upside-down potential.
The Rabi oscillation frequency Ω ∼ O(g) after analytic continuation can be related to the Unruh temperature and therefore the coupling g ∼ 1/M, which also agrees with the gravitational self-coupling strength at criticality in [17].
• The hopping probability among those states of same energy level, say |n − 1, b 1 and |n − 1, b 2 , may represent the scramble rate, that is t n − 1, b 2 |n − 1, b 1 0 2 = − 1 2 √ 2 + 1 2 cos Ωt 2 . This scramble is somehow triggered by interaction with resonant photons through coupling g. In general, one may also introduce additional hopping coupling Hamiltonian (6). This is about to shift the eigenvalue by some constant proportional to J as well as Ω → 2g 2 n + J 2 .
• To account for the large degeneracy in a black hole, the amount of various couplings g ij is of order ∼ O(e M 2 ). Therefore the Rabi frequency will be enhanced by the same power, which leads to fast scramble.
• In the case of uneven coupling strength g 11 = g 12 , the transition amplitude to n−1, b 1 and n − 1, b 2 are different. This could be responsible for featured emission and therefore encoding information. In terms of the BMS transformation of soft hairs, the variation in photon coupling g ij and hopping coupling J ij can be realized in some arbitrary angle-dependent function F (ν, θ, φ) [13,21].
• It was argued in [17] that the degeneracy of above-mentioned Goldstone modes will be lifted by quantum fluctuation. That is, one expects nontrivial tidal force due to gravitational perturbation, like the gravity version of Zeeman effect, contributes to a small energy gap 3 . We may estimate its order by considering fluctuation in Newton's gravitational potential δU ∼ ωδg 00 ∼ Mωδr/r 2 | r=2M . Given the quantized radiation ω ∼ /4M [22] and δr ∼ ω, one obtains the energy gap ∆E ∼ δU ∼ /M 3 in agreement with [17]. However, its contribution to Ω is minor due to its small magnitude.

IV. SOFT-HAIR DRESSED COUPLING STRENGTH
We have argued that uneven coupling strength could be responsible for featured emission, from which one might be able to decode the information. While processing information in quantum mechanics is reversible thanks to unitarity property, we now show how to create a desired superposition state in the JC model by tuning the ratio of uneven coupling strength. First a qubit is prepared at ground state ψ(0) = |n, a 1 . Given time, it will evolve accordingly, where we regard the coupling strength ratio δ = g 11 /g 12 as a controllable parameter. We remark that at late time t r = π/2Ω, a superposition of excited states ψ(t r ) is created with coefficients tuned by δ. This suggests that the coupling strength should be dressed with angular dependence, which might be closely related to the BMS transformation of soft hairs. With that being said, the analytic continuation Ω → iΩ ′ will bring equation (7) to following form in comparison to the Goldstone boson mode 4 where both Ω ′ andΩ are of order 1/M. Assuming there exists a black hole state |BH such thatÂ ± |BH ∼ |E ± . This similarity indeed implies that δ is a function of angular dependence.
At last, we may label those degenerate black hole states |m i ,θ i ,φ i by mutually commuting quantum numbers. Letm i denotes the principle quantum number, which is given in two-level Hamiltonian (6) are refined as for angular dependent photon coupling g ij (among different mass/energy levels) and hopping coupling J ij (among same mass/energy levels).

V. DEGREE OF ENTANGLEMENT
On the other hand, given a mixed excited state such as that in the previous section, one is interested in the time evolution of its degree of entanglement. In this letter, we adopt the mutual entropy method (DEM) proposed in [23] and its application to JC model [24]. We prepare the initial state and photons are in a coherent state The time evolution for the JC model between black atom and coherent field is generated byĤ, namely, given the unitary operator U t = exp(−itĤ), we have Following [24], one can obtain the reduced density operator ρ(t) A and ρ(t) F for black atom and radiation field respectively. Then the DEM is computed as  concern. We will leave them for future study.