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# Quantum optics approach to radiation from atoms falling into a black hole

Contributed by Marlan O. Scully, May 24, 2018 (sent for review May 4, 2018; reviewed by Federico Capasso and Michael Duff)

## Significance

Using a combination of quantum optics and general relativity, we show that the radiation emitted by atoms falling into a black hole looks like, but is different from, Hawking radiation. This analysis also provides insight into the Einstein principle of equivalence between acceleration and gravity.

## Abstract

We show that atoms falling into a black hole (BH) emit acceleration radiation which, under appropriate initial conditions, looks to a distant observer much like (but is different from) Hawking BH radiation. In particular, we find the entropy of the acceleration radiation via a simple laser-like analysis. We call this entropy horizon brightened acceleration radiation (HBAR) entropy to distinguish it from the BH entropy of Bekenstein and Hawking. This analysis also provides insight into the Einstein principle of equivalence between acceleration and gravity.

General relativity as originally developed by Einstein (1) is based on the union of geometry and gravity (2). Half a century later the union of general relativity and thermodynamics was found to yield surprising results such as Bekenstein–Hawking black hole entropy (3⇓⇓–6), particle emission from a black hole (5⇓⇓⇓–9), and acceleration radiation (10⇓⇓⇓⇓⇓⇓–17). More recently the connection between black hole (BH) physics and optics, e.g., ultraslow light (18), fiber-optical analog of the event horizon (19), and quantum entanglement (20), has led to fascinating physics.

In their seminal works, Hawking, Unruh, and others (3⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–14) showed how quantum effects in curved space yield a blend of thermodynamics, quantum field theory, and gravity which continues to intrigue and stimulate. For problems as important and startling as Hawking and Unruh radiation, new and alternative approaches are of interest. In that regard it was shown (21, 22) that virtual processes in which atoms jump to an excited state while emitting a photon are an alternative way to view Unruh acceleration radiation. Namely, by breaking and interrupting the virtual processes which take place all around us, we can render the virtual photons real.

The present paper is an extension of that logic by considering what happens when atoms fall through a Boulware vacuum (23) into a BH as shown in Fig. 1. A mirror held at the event horizon shields infalling atoms from the Hawking radiation. The equivalence principle tells us that an atom falling in a gravitational field does not “feel” the effect of gravity; namely its 4 acceleration is equal to zero. However, as we discuss in *Appendix A*, there is relative acceleration between the atoms and the field modes. This leads to the generation of acceleration radiation. In *Appendix B* we provide a detailed calculation of the photon emission by atoms falling into a BH.

In the classic works (10⇓⇓⇓⇓⇓⇓–17) the atom (or other Unruh–DeWitt detector) was accelerated through flat space-time. The present work differs in that the atom is in free fall and the field is accelerated (or supported in a gravitational field) and contains a Boulware-like ground state of the quantized field. Qualitatively, the principle of equivalence suggests that the results should be analogous to those in refs. 10⇓⇓⇓⇓⇓–17, but the notion that an atom in free fall should emit radiation is surprising to many people (despite the results in refs. 24 and 25). For this and other reasons, the detailed calculation presented here, taking into account the quantitative differences between the two situations, has been necessary. (An example of another reason for including the detailed calculations of *Appendix B* is in the words of one of the reviewers: “How can the atom falling into a BH emit Unruh-like radiation which comes from a constant acceleration since the falling atom acceleration depends on the distance from the BH?” The answer to this and other such questions is given in *Appendix B*.)

Specifically we consider an atomic cloud consisting of two-level atoms emitting acceleration radiation (Fig. 1) (21, 22). We find that the quantum master equation technique, as developed in the quantum theory of the laser, provides a useful tool for the analysis of BH acceleration radiation and the associated entropy. (For the density matrix formulation of the quantum theory of the laser, see ref. 26. For pedagogical treatment and references, see refs. 27 and 28.) In particular, we derive a coarse-grained equation of motion for the density matrix of the emitted radiation of the form**7**.

Furthermore, we find that once we have cast the acceleration radiation problem in the language of quantum optics and cavity quantum electrodynamics (QED), the entropy follows directly. Specifically, once we calculate

Hawking’s pioneering proof that BHs are not black (5, 6) is based on a quantum-field theoretic analysis showing that photon emission from a BH is characterized by a temperature

Hawking showed that the radiation that comes out from the BH is described by the temperature**3** he obtains

In the present paper we analyze the problem of atoms outside the event horizon emitting acceleration radiation as they fall into the BH. The emitted radiation can be essentially, but not inevitably, thermal and has an entropy analogous to the BH result given by Eq. **4**. However, the physics is very different. Here we have radiation coming from the atoms, whereas Hawking radiation requires no extra matter (e.g., atoms).

Historically, Bekenstein (3, 4) introduced the BH entropy concept by information theory arguments. Hawking (5, 6) then introduced the BH temperature to calculate the entropy. In the present approach we calculate the radiation density matrix and then calculate the entropy directly. To distinguish this from the BH entropy we call it the horizon brightened acceleration radiation (HBAR) entropy.

## The HBAR Entropy via Quantum Statistical Mechanics

As noted earlier, we here consider a BH bombarded by a beam of two-level atoms with transition frequency ω which fall into the event horizon at a rate κ (Fig. 1). The atoms emit and absorb the acceleration radiation.

We seek the density matrix of the field. As in the quantum theory of the laser (26), the (microscopic) change in the density matrix of the field due to any one atom, *Appendices B* and *C*, the coarse-grained time rate of change of the radiation field density matrix for a particular field mode is found to be**2** and **7**, we find that the von Neumann entropy generation rate of the HBAR is (see *Appendix D* for details)

Taking into account that the BH mass change due to photon emission is *Appendix D*).

## Discussion and Summary

Conversion of virtual photons into directly observable real photons is a subject not without precedent. Moore’s accelerating mirrors (30), the rapid change of refractive index considered by Yablonovitch (31), and the more recent observation of the dynamical Casimir effect in a superconducting circuit (32) are a few examples.

The physics behind acceleration radiation are explained in ref. 21 (also ref. 33) where the following is stated:

In conclusion our simple model demonstrates that the ground-state atoms accelerated through a field vacuum-state radiate real photons…. The physical origin of the field energy in the cavity and of the internal energy in the atom is the work done by an external force driving the center-of-mass motion of the atom against the radiation reaction force. Both the present single mode and the many mode effect originate from the transition of the ground-state atom to the excited state with simultaneous emission of photon due to the counterrotating terms in the Hamiltonian.

In other words the virtual processes in which an atom jumps from the ground state to an excited state, together with the emission of a photon, followed by the reabsorption of the photon and return to the ground state, are altered by the acceleration. The atom is accelerated away from the original point of virtual emission, and there is a small probability that the virtual photon will “get away” before it is reabsorbed as is depicted in Fig. 1.

Acceleration radiation involves a combination of two effects: acceleration and nonadiabaticity that produce the emitted light. The energy is supplied by the external force field (e.g., the gravitational field of the star).

Gravitational acceleration of atoms is also a source of confusion. The equivalence principle tells us that the atom essentially falls “force-free” into the BH. How can it then be radiating? Indeed, the atomic evolution in the atom frame is described by the **35**). From the Hamiltonian we clearly see that it is the photon time (and space) evolution which contains effective acceleration. The radiation modes are fixed relative to the distant stars, and the photons (not the atoms) carry the seed of the acceleration effects in

In Fig. 2 we compare the probability of acceleration radiation *A*) the atom is accelerated in Minkowski space-time relative to a fixed mirror; (Fig. 2*B*) the mirror is accelerated in Minkowski space-time relative to a fixed atom, with the field in a Rindler-like ground state (34); and (Fig. 2*C*) the atom freely falls in the gravitational field of a BH (assuming that BH Hawking radiation is shielded). In Fig. 2*A* *B* involves the photon frequency ν. For an atom freely falling in the gravitational field of a BH the Planck factor also contains the photon frequency (measured by an observer at infinity). This provides an insight into Einstein’s equivalence principle. Namely, a fixed atom near an accelerating mirror emits thermal radiation as in Fig. 2*B*; while an atom falling into a BH emits exactly the same thermal spectrum (Fig. 2*C*).

Please note that this is a very different perspective on the equivalence principle (35) than the usual elevator picture. There the elevator observer (the atom) feels the acceleration in his feet to be the same as a uniform gravitational field. Here the atom is stationary (Fig. 2*B*). It is the mirror which is accelerating and this changes the normal modes of the field. In Fig. 2*C* the atom is in free fall and thus it feels no gravity. However, the radiation normal modes are changed by the gravitational field of the BH. Moreover, the atom emits the same way in both cases, Fig. 2*B* and *C*. That is, the acceleration in Fig. 2*B* affects the atom in the same way as the gravitational field in Fig. 2*C*.

If atoms are ejected randomly, the photon statistics will be thermal (21, 22). For Fig. 2*A* the average photon occupation number in the mode with frequency ν reads (21, 22)

The present model is simple enough to allow a direct calculation of the HBAR entropy. It is a much more tractable problem then the daunting BH entropy issue. It is interesting that the answer for the HBAR entropy we found is essentially the same as the formula for the Bekenstein–Hawking BH entropy.

## Appendix A. Motion of Particle in Rindler and Schwarzschild Space-Time

When atoms are in free fall, their operator time dependence in the interaction picture goes as *i*) special relativity, (*ii*) Rindler metric, and (*iii*) Schwarzschild metric.

#### Special Relativity.

First of all we note that finding **16** we obtain

#### Rindler.

The Rindler metric for a particle undergoing uniformly accelerated motion is obtained from the Minkowski line element **12** if we make a coordinate transformation**19** and **20** with Eqs. **16** and **18** shows that a particle moving along a trajectory with constant

#### Schwarzschild.

Finally we make an observation that the **22**, curves of constant

## Appendix B. Acceleration Radiation from Atoms Falling into a Black Hole

Here we consider a two-level (a is the excited level and b is the ground state) atom with transition angular frequency ω freely falling into a nonrotating BH of mass M along a radial trajectory from infinity with zero initial velocity. We choose the gravitational radius **28** yields**32** dominate and one can approximately write**9**. However, a proper cavity arrangement as alluded to in the caption of Fig. 1 could be envisioned as yielding effectively a single mode behavior. Furthermore, a properly configured dense atomic cloud could in itself be used to select the desired mode structure. Finally we note that the “mirror” of Fig. 1 could be thought of as completely surrounding the BH. For the purpose of this appendix we assume that the Boulware vacuum has been arranged.

The interaction Hamiltonian between the atom and the field mode **34** is

The probability of excitation of the atom (frequency ω) with simultaneous emission of a photon with frequency ν is due to a counterrotating term **38** at **38** becomes**40** reads**40** comes from the detailed calculation, i.e., is not put in by hand. The result is equivalent to that for a constant acceleration because the main contribution comes from the event horizon. We note also that the mirror “edge effects” are not a problem.

## Appendix C. Density Matrix for the Field Mode

The (microscopic) change in the density matrix of a field mode **35**. The time τ is the atomic proper time, i.e., the time measured by an observer riding along with the atom.

In the case of random injection times, the equation of motion for the density matrix of the field is*Appendix B*,**44**. Steady-state solution of Eq. **44** is given by the thermal distribution (26):*Appendix B*, but we can then take the limit as

## Appendix D. Entropy Flux

The time rate of change of entropy due to photon generation,**45**. Inserting it into **[****47****]** gives

Recalling the BH area

## Acknowledgments

We thank M. Becker, S. Braunstein, C. Caves, G. Cleaver, S. Deser, E. Martin-Martinez, G. Moore, W. Unruh, R. Wald, and A. Wang for helpful discussions. We note that both referees, quantum optics–Casimir effect expert Federico Capasso (NAS member) and general relativity expert Michael Duff (FRS), have provided insightful criticism which have improved the presentation and physics of the paper. This work was supported by the Air Force Office of Scientific Research (Award FA9550-18-1-0141), the Office of Naval Research (Awards N00014-16-1-3054 and N00014-16-1-2578), the National Science Foundation (Award DMR 1707565), the Robert A. Welch Foundation (Award A-1261), and the Natural Sciences and Engineering Research Council of Canada.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: scully{at}tamu.edu.

Author contributions: M.O.S. and D.M.L. designed research; M.O.S., S.F., D.N.P., W.P.S., and A.A.S. performed research; M.O.S., S.F., D.M.L., D.N.P., W.P.S., and A.A.S. contributed new reagents/analytic tools; M.O.S., S.F., D.M.L., D.N.P., W.P.S., and A.A.S. analyzed data; and M.O.S. and A.A.S. wrote the paper.

Reviewers: F.C., Harvard University; and M.D., Imperial College.

Conflict of interest statement: Michael Duff will spend a semester at the Institute for Quantum Science and Engineering at Texas A&M University as a Fellow of the Hagler Institute for Advanced Studies.

- Copyright © 2018 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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- The HBAR Entropy via Quantum Statistical Mechanics
- Discussion and Summary
- Appendix A. Motion of Particle in Rindler and Schwarzschild Space-Time
- Appendix B. Acceleration Radiation from Atoms Falling into a Black Hole
- Appendix C. Density Matrix for the Field Mode
- Appendix D. Entropy Flux
- Acknowledgments
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