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Downloaded by guest on September 28, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1807703115 Scully O. hole Marlan black a atoms into from falling to approach optics tmi refl hudei aito ssrrsn omany to an surprising that is notion radiation the emit but should should 10–17, fall results refs. free the is in in field that those atom the to suggests analogous and equivalence be fall of Qualitatively, free field. principle quantized in a the the contains of is and state field) ground atom gravitational Boulware-like a the in supported that (or present accelerated The in -time. differs flat work through accelerated falling was atoms detector) by emission BH. gen- the a the into of to leads In calculation This radiation. detailed of However, modes. field effect zero. of the to eration the and atoms in equal “feel” the is discuss between not held acceleration we does atom 4 mirror an as field Hawk- its A that gravitational the namely 1. us from ; a tells Fig. atoms principle in in infalling equivalence falling shown The Boulware shields as radiation. horizon a ing BH event through a the fall at into atoms (23) when happens what sidering render can we us, around real. all Unruh virtual place view the take to which the state way processes interrupting excited and alternative virtual breaking an an by Namely, to are radiation. 22) jump photon acceleration (21, atoms a shown which emitting was in while it regard processes alternative that virtual and In that interest. new of radiation, are and Unruh important approaches and as Hawking of problems as con- For blend stimulate. startling which a and gravity intrigue yield and to space tinues theory, curved field in quantum effects thermodynamics, quantum fascinating how to showed led horizon has event (20), the entanglement e.g., of physics. quantum optics, analog and and fiber-optical the physics (19), (18), recently (BH) hole More hole black ultraslow black (10–17). a between radiation from connection acceleration emission Bekenstein–Hawking particle and as (3–6), (5–9), such results hole surprising black yield was thermodynamics to and relativity found general of union the later tury G radiation of acceleration Hawking. principle and Einstein Bekenstein the gravity. and into of acceleration insight between entropy provides equivalence BH also analysis the This from dis- to it entropy entropy accelera- (HBAR) this tinguish the call radiation We acceleration of analysis. brightened entropy laser-like Hawking horizon simple the from) a different find via radiation is we tion particular, (but In like acceler- radiation. much emit BH observer looks (BH) distant conditions, hole a initial Duff) black Michael appropriate to and a under Capasso into which, Federico by radiation falling reviewed ation atoms 2018; 4, that May review show for We (sent 2018 24, May Scully, O. Marlan by Contributed Universit Technology, and Quantum Integrated for Center 77843; TX Station, College University, 76798; a Svidzinsky A. Anatoly and nttt o unu cec n niern,TxsAMUiest,CleeSain X77843; TX Station, College University, A&M Texas Engineering, and Science Quantum for Institute ntecascwrs(01)teao o te Unruh–DeWitt other (or atom the (10–17) works classic the In con- by logic that of extension an is paper present The (3–14) others and Unruh, Hawking, works, seminal their In ae nteuino emtyadgaiy() afacen- a Half (2). is gravity (1) and geometry Einstein of by union the developed on originally based as relativity eneral c eateto ehncladArsaeEgneig rneo nvriy rneo,N 08544; NJ Princeton, University, Princeton Engineering, Aerospace and Mechanical of Department | aiyQED cavity a,b,c,1 hr srltv acceleration relative is there A , Appendix | lc oeentropy hole black tpe Fulling Stephen , a e eateto hsc,Uiest fAbra dotn BTG21 aaa and Canada; 2E1, T6G AB Edmonton, Alberta, of University Physics, of Department | pedxB Appendix qiaec principle equivalence a,d ai .Lee M. David , epoiea provide we tUm -98 l,Germany Ulm, D-89081 Ulm, at ¨ | a o .Page N. Don , rtn tm,w a s h o emn nrp relation entropy Neumann von the use write can to we atoms, erating directly. calculate follows we cavity entropy once and Specifically, the optics (QED), quantum electrodynamics of language quantum the in problem radiation den- the of elements Eq. by diagonal the matrix of sity evolution time the where of radiation emitted the of equa- matrix coarse-grained density form a the the derive for we motion particular, of see In tion references, 28.) and and treatment 27 pedagogical refs. For the 26. of theory ref. the quantum see for the laser, of tool formulation matrix entropy. useful density associated a the the (For and provides in radiation laser, acceleration developed BH the of as analysis of technique, theory find equation quantum We master the 22). quantum (21, 1) the (Fig. that radiation acceleration emitting atoms in given is questions such other and B.) this Appendix from to distance answer the The on BH?” depends emit the acceleration acceleration BH constant atom a falling a into the from falling since comes atom which the radiation can “How Unruh-like reviewers: the of one hsoe cesatcei itiue under distributed is 1 article BY-NC-ND). (CC 4.0 access Hagler License the NonCommercial-NoDerivatives of open for Fellow a Institute This as the University at A&M semester Studies. Texas Advanced a for at spend Institute Engineering will and Duff Science Michael Quantum statement: interest of Conflict College. Imperial M.D., and University; Harvard F.C., Reviewers: paper. the data; wrote analyzed A.A.S. A.A.S. and and W.P.S., M.O.S. D.N.P., and D.M.L., contributed S.F., W.P.S., A.A.S. M.O.S., D.N.P., and tools; W.P.S., S.F., D.N.P., reagents/analytic D.M.L., M.O.S., S.F., new M.O.S., research; research; designed performed A.A.S. D.M.L. and and M.O.S. contributions: Author includ- of for calculations reason another detailed of the into example ing (An taking situations, necessary. two here, been the between has presented differences other quantitative and calculation the this For account detailed 25). and the 24 refs. reasons, in results the (despite people owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To hsaayi lopoie nih noteEnti principle gravity. Einstein and the acceleration into between a equivalence insight of into provides radiation. also falling Hawking from, analysis atoms different is This by but emitted like, looks radiation hole black the relativity, that general show and optics we quantum of combination a Using Significance utemr,w n htoc ehv atteacceleration the cast have we once that find we Furthermore, two-level of consisting cloud atomic an consider we Specifically 7. e ofagP Schleich P. Wolfgang , ρ nn b sgvre ytesuperoperator the by governed is eateto hsc,Byo nvriy ao TX Waco, University, Baylor Physics, of Department S d ˙ eateto ahmtc,TxsA&M Texas Mathematics, of Department p ρ ˙ = nn −k (Mρ) = B ρ ˙ r ˙ Tr( a,f o h edpoue yaccel- by produced field the for pedxB Appendix , ρ f ntttf Institut nn ln , raieCmosAttribution- Commons Creative ρ NSLts Articles Latest PNAS ) rQatnhskand Quantenphysik ur ¨ si h od of words the in is M sgiven as | f6 of 1 [2] [1]

PHYSICS we have radiation coming from the atoms, whereas Hawking radiation requires no extra (e.g., atoms). Historically, Bekenstein (3, 4) introduced the BH entropy concept by theory arguments. Hawking (5, 6) then introduced the BH 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 ω which fall into the 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 den- sity matrix of the field due to any one atom, δρi , is small. The (macroscopic) change due to ∆N atoms is then

X i ∆ρ = δρ = ∆N δρ. [5] i Writing ∆N = κ∆t, where κ is the rate of atom injection at random times, we have the coarse-grained equation of motion Fig. 1. A BH is bombarded by a pencil-like cloud of two-level atoms falling radially from infinity. As discussed in Appendix B, a “mirror” is held at the ∆ρ = κδρ. [6] event horizon which shields infalling atoms from the Hawking radiation. ∆t One can imagine that there is a second mirror at large r so that atoms are falling through a cavity. The relative acceleration between the atoms and We thus obtain an evolution equation for the radiation following the field yields generation of acceleration radiation. The physics of the accel- the approach used in the quantum theory of the laser (26). As is eration radiation process correspond to the excitation of the atom together further discussed in Appendices B and C, the coarse-grained time with the emission of the photon (Appendix B). rate of change of the radiation field density matrix for a particular field mode is found to be

to calculate the radiation entropy flux directly. From the present 2 1 dρn,n κg −ξ perspective the acceleration radiation–BH entropy problem is = − e [(n + 1)ρn,n − nρn−1,n−1] close in spirit to the quantum theory of the laser. R dt ω2 Hawking’s pioneering proof that BHs are not black (5, 6) is κg 2 based on a quantum-field theoretic analysis showing that pho- − eξ [nρ − (n + 1)ρ ], [7] ω2 nn n+1,n+1 ton emission from a BH is characterized by a temperature TBH and generalized BH entropy. York (29) gives an analogy between where g is the atom–field coupling constant, ξ = 2πνrg /c, radiation from a BH and total internal reflection in classical ξ optics. He argues that a light beam in a dense medium can R = , [8] undergo total internal reflection at a flat optical surface; but if sinh(ξ) we sprinkle dust particles on the surface, some light will be trans- and ν is the photon frequency far from the BH. Using Eqs. 2 and mitted. Now the flat surface can be likened to the BH event 7, we find that the von Neumann entropy generation rate of the horizon, the dust is replaced by vacuum fluctuations, and light HBAR is (see Appendix D for details) is transmitted through the horizon. Hawking showed that the radiation that comes out from the ˙ 4πkB rg X BH is described by the temperature Sp = n¯˙ ν ν , [9] c ν 3 ~c TBH = . [3] where n¯˙ ν is the flux of photons with frequency ν propagating 8πkB GM away from the BH. Taking into account that the BH change due to pho- Hawking then associates the of the emitted radiation δE 2 P ˙ 2 ton emission is m˙ p c = ~ n¯ν ν, we arrive at the HBAR with the loss in energy of the BH δ(Mc ) and writes the entropy ν 2 entropy/area relation loss as δS = δ(Mc )/TBH. Then using Eq. 3 he obtains 3 kB c 3 S˙ p = A˙ p . [10] 8πG kB c 4 G δS = kB M δM = δA, [4] ~ ~c 4~G Here A˙ p = (2m ˙ p /M )A is the rate of change of the BH area due where the BH area in terms of the gravitational radius rg = to photon emission which we are interested in (Appendix D). 2 2 2 2 4 2GM /c is given by A ≡ 4πrg = 16πG M /c . In the present paper we analyze the problem of atoms out- Discussion and Summary side the event horizon emitting acceleration radiation as they Conversion of virtual photons into directly observable real pho- fall into the BH. The emitted radiation can be essentially, but tons is a subject not without precedent. Moore’s accelerating not inevitably, thermal and has an entropy analogous to the BH mirrors (30), the rapid change of refractive index considered result given by Eq. 4. However, the physics is very different. Here by Yablonovitch (31), and the more recent observation of the

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(also 21 a are (32) circuit superconducting examples. a few in effect Casimir dynamical ubri h oewt frequency 2A with mode Fig. the in For number 22). (21, thermal 2 C. Fig. 2B in field Fig. gravitational cases, the both as way in same way BH. same the the of and emits radiation field the atom gravitational However, the the gravity. by Moreover, no changed feels 2C are it Fig. modes thus In normal and and field. fall accelerating the free atom is of in which the modes is mirror Here normal the the field. is changes gravitational It this uniform 2B). a (Fig. stationary as feet is same his in the acceleration be the feels to atom) There (the picture. observer elevator elevator usual the the than (35) an principle equivalence while 2B; thermal Fig. same the in exactly acceler- emits 2C). as (Fig. BH an radiation a into near thermal falling atom atom emits the Einstein’s fixed mirror BH into a insight ating a Namely, an of provides principle. by This field equivalence (measured infinity). frequency gravitational at photon observer the the an contains in also falling factor Planck freely atom 2B an Fig. for factor 2A Planck frequency Fig. atom In the containing shielded). ω factor is Planck the radiation to (assum- Hawking proportional BH a BH of field (Fig. that gravitational and the ing (34); in falls state freely ground atom Rindler-like the fixed a 2C) in a field to the relative the with space-time 2B) atom, Minkowski (Fig. in mirror; accelerated fixed is a mirror to relative space-time Minkowski in in P effects acceleration space) the of (and (not seed photons time the the carry and photon atoms) stars, radiation the distant the the The to relative is acceleration. fixed are effective it modes contains that which see evolution clearly we tonian clye al. et Scully ex n h nu temperature Unruh the and rvttoa ceeaino tm sas oreo confu- of source a also is atoms of acceleration Gravitational effects: two of combination a involves radiation Acceleration jumps atom an which in processes virtual the words other In ref. in explained are radiation acceleration behind physics The faosaeeetdrnol,tepoo ttsiswl be will statistics photon the randomly, ejected are atoms If the on perspective different very a is this that note Please radiation acceleration of probability the compare we 2 Fig. In oteectdsaewt iutnoseiso fpoo u othe to due Hamiltonian. photon the atom of many in emission ground-state terms the simultaneous the counterrotating with and of state external excited mode transition the the single an to from present by originate the done effect Both radi- mode work the of force. against the and atom reaction cavity is the of ation the atom motion in center-of-mass the the energy driving in field pho- force the energy real of internal radiate origin the physical vacuum-state The field . . a . ground-state tons. the through that accelerated demonstrates model atoms simple our conclusion In e hti,teaclrto nFg 2B Fig. in acceleration the is, That C. h tmi accelerated is atom the 2A) (Fig. configurations: three for i ωτ emi h aitna (Eq. Hamiltonian the in term n ¯ ν = novstepoo frequency photon the involves exp T h vrg htnoccupation photon average the 2π U 1 a c = ω ν ~a  ed 2,22) (21, reads − /2π 1 .Fo h Hamil- the From 35). fet h tmi the in atom the affects . k B c ncnrs,the contrast, In . V h atom the ˆ (τ ν P For . ). ex [11] is h htnsetu sflt htis, that flat; is spectrum photon The in discussed as radiation Hawking from (C shielded and is B. (34); and state Appendix BH, ground Rindler-like a relative a into space-time in falls field Minkowski a atom the in to with accelerated atom, relative fixed is space-time a mirror to Minkowski the in (B) accelerated mirror; is fixed atom the (A) wherein 2. Fig. hnaosaei refl,teroeao iedpnec in dependence time operator as their goes fall, picture interaction free the in are atoms When Schwarzschild and Rindler in Particle of Space-Time essen- Motion is A. found Appendix BH we Bekenstein–Hawking the entropy for formula HBAR tractable interesting the entropy. the more is as same It much for the issue. a tially answer entropy is the BH It daunting that entropy. the then HBAR problem the of culation field gravitational distribution. the in falling freely frequency ton rprtm fteao.Tecrepnigtm vlto of evolution time corresponding The is operator atom. radiation-field the the of time proper .. h oriaetm n oiino h tmi em of relativity. special terms in in problem atom a the really the from is of Namely, time, position proper and atom’s time the coordinate the i.e., relativity, metric. . Schwarzschild (i) Special (iii) steps: and three metric, in Rindler results (ii) the obtain we follows field the of ization ψ B A C (t h rsn oe ssml nuht lo ietcal- direct a allow to enough simple is model present The , z ) he ofiuain n h orsodn xiainprobabilities excitation corresponding the and configurations Three stemd ucinadtesaeadtm parameter- time and space the and function mode the is ν ncnrs,fra ceeae irro atoms or mirror accelerated an for contrast, In . 2D is falw oeta finding that note we all of First t (τ ikwk ieelement line Minkowski ) ds and 2 = z c a ˆ (τ σ ˆ 2 k + + dt ) (t (τ 2 r ob eemnd nwhat In determined. be to are ˆ = ) n − ˆ = ) ¯ ν dz a sidpneto h pho- the of independent is n ¯ σ k + ν 2 NSLts Articles Latest PNAS + (0)ψ (0)e sgvnb h Planck the by given is [t i ωτ (τ where , ), t (τ z (τ ) and where )], τ | sthe is z f6 of 3 [12] the ) (τ ),

PHYSICS we can write yields the Rindler metric (36) r Z τ Z t V 2 2 τ = dτ = 1 − dt, [13] 2 z¯ 2 ¯2 2 2 ds = 2 c dt − dz¯ . [26] 0 0 c 4rg

where V = dz/dt. One can show that for a particle moving with According to Eq. 22, curves of constant z¯ (or r¯) correspond to constant proper acceleration a uniformly accelerated motions with

at 2 2 V = q [14] c c 1 a2t2 a = = q . [27] 1 + c2 z¯ 2rg rg 1 − ¯r and, therefore, Z t dt c  at  Appendix B. Acceleration Radiation from Atoms Falling into τ = = sinh−1 , [15] q 2 2 a c a 0 1 + a t c2 Here we consider a two-level (a is the excited level and b is the or ) atom with transition angular frequency ω freely c  aτ  t(τ) = sinh . [16] falling into a nonrotating BH of mass M along a radial trajectory a c from infinity with zero initial . We choose the gravita- 2 Likewise, integration of V (t) yields tional radius rg = 2GM /c as a unit of distance and rg /c as a unit of time and introduce the dimensionless distance, time, and r ! Z t c2 a2t 2 frequency as z(t) − z(0) = V (t)dt = 1 + 2 − 1 . [17] 0 a c r → rg r, t → (rg /c)t, ω → (c/rg )ω. 2 Setting z(0) = c /a and using Eq. 16 we obtain In dimensionless Schwarzschild coordinates the atom trajectory is described by the equations c2  aτ  z(τ) = cosh . [18] a c dr 1 dt r = − √ , = , [28] dτ r dτ r − 1 Rindler. The Rindler metric for a particle undergoing uniformly accelerated motion is obtained from the Minkowski where t is the dimensionless time in Schwarzschild coordinates 12 if we make a coordinate transformation and τ is the dimensionless for the atom. Integration of Eq. 28 yields z¯  a¯¯t  2 t = sinh , [19] τ = − r 3/2 + const, [29] c c 3 √ 2 √  r − 1   a¯¯t  t = − r 3/2 − 2 r − ln √ + const. [30] z =z ¯ cosh , [20] 3 r + 1 c For a scalar photon in the Regge–Wheeler coordinate where a¯ is a constant. This leads to the line element  a¯z¯  r = r + ln(r − 1) [31] ds2 = 2c2d¯t 2 − dz¯2, [21] ∗ c2 the field propagation equation reads which is the Rindler line element describing uniformly acceler- ated motion. Comparison of Eqs. 19 and 20 with Eqs. 16 and 18  ∂2 ∂2  1  1 ∆  − + 1 − − ψ = 0, [32] z¯ 2 2 3 2 shows that a particle moving along a trajectory with constant ∂t ∂r∗ r r r in Rindler space has τ =a ¯¯t/a and is uniformly accelerating in with acceleration where ∆ is the angular part of the Laplacian. We are interested in solutions of this equation outside of the event horizon, that is, c2 a = . [22] for r > 1. If the dimensionless photon angular frequency ν  1 z¯ and is neglected, then the first two terms in Eq. 32 dominate and one can approximately write Schwarzschild. Finally we make an observation that the t − r  ∂2 ∂2  part of the , − ψ = 0. [33] ∂t 2 ∂r 2   ∗ 2 rg 2 ¯2 1 2 ds = 1 − c dt − rg dr¯ , [23] r¯ 1 − ¯r We consider a solution of this equation describing an outgoing wave 2 iν(t−r∗) iν[t−r−ln(r−1)] where rg = 2GM /c is the gravitational radius, can be approxi- ψ = e = e , [34] mated around rg by Rindler space by using the coordinate 0 < where ν is the wave frequency measured by a distant observer. z¯  rg defined by 2 In general we will have many modes of the field ( z¯ ν 9 r¯ = rg + . [24] ) which we will sum over as in Eq. . However, a proper 4rg cavity arrangement as alluded to in the caption of Fig. 1 could be Expanding around rg , envisioned as yielding effectively a single mode behavior. Fur- thermore, a properly configured dense atomic cloud could in 2 rg z¯ itself be used to select the desired mode structure. Finally we 1 − ≈ 2 , [25] r¯ 4rg note that the “mirror” of Fig. 1 could be thought of as completely

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1807703115 Scully et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 netn eeEs 93 eobtain we 29–31 Eqs. here Inserting from trajectory atomic the over integral ∞ an as written be can event, this of probability frequency with term photon counterrotating a of emission simultaneous (23). vacuum Boulware frequency and the with in state modes is ground the field the for photons in no is are atom there the Initially “photons.” (-0) that assume We and constant. operator, lowering atomic the operator the where ntelimit the In have in we exponential terms the leading the under function the Eq. expanding of behavior asymptotic The where variable integration the of change another make we Next into variable integration the of change a Making mode arranged. been has assume vacuum we appendix Boulware this the of that purpose the For BH. the surrounding clye al. et Scully P 2ω P 3 V exc ˆ exc h rbblt fectto fteao (frequency atom the of excitation of probability The h neato aitna ewe h tmadtefield the and atom the between Hamiltonian interaction The (y oteeethorizon event the to (τ = = − P = ) 34 g exc 1) 4g 2 is 9 ~g φ(x

2 n find and = Z ω P

1 h g Z ∞ P exc = )  a ˆ 2 1 P ν exc ∞

dr e Z exc 1 = ω −i x ∞ dye φ(x √ = Eq. ln 2 + =g 1 ω g = ν + a ˆ re dr 2 2 t ~ ν 1 a −i 2 ( ˆ )  2 ω

g 38 τ ν −i + Z

 ≈ stepoo niiainoperator, annihilation photon the is )+i 2 r 2 ν + 1 Z

0 σ  ˆ Z [ ν d becomes 1 = ∞ dr ln 2 + 3

3 2 + [ Z ν g d τ y 2 3 r d 0 + 1 dxe 2ω +y 3x τ nteitrcinHmloin The Hamiltonian. interaction the in ∗ r ≈  ∞ τ 3/2 ( as e τ e 2/3 h1 i os hc stecs o scalar for case the is which const  ) dxe i −2i ν 2ω 3x +r ν H.c + 38 t ν 2/3 t (  +2y ( , τ +2 ν r −i  2ω a ) g at ) x −i ln 2 + −i | 1/3 1/3 √ ( νφ steao–edcoupling atom–field the is V x ˆ ν ω  r ν i e r 2ln +2 + 2ln +2 r (τ ∗ x  −  −ix ∗ ) (τ σ ( ˆ e + 1 ) 2x r 1 ω ) e 1 −ix ) |0, (  ( e ( e −i 1+ √ a eotie by obtained be can . y i . i ωτ 1/3 ωτ 2ω r 3x b

ωτ −1 2ν 2 ω i

, (r

y −1 2  H.c + 2 ) )] 1/ω

) = , e 2 ν 1/3

ν )] 2 − ota the that so , e r . sdet a to due is − 3 2 3/2 Keeping . i 3 2 ω  ω i r , ω yields 3/2 with ) y [36] [35] [38] [37]

σ x ˆ r 2

2 . = = is . h mcocpc hnei h est arxo edmode field a of matrix density the δρ in Mode Field change the (microscopic) for The Matrix Density C. Appendix mirror the problem. contribu- that a also main not equivalent note are the We is effects” horizon. because “edge result event acceleration the The from constant hand. comes a tion by for in that put to not is Eq. i.e., in calculation, factor Planck the that note We changing by obtained is absorption photon ν of probability The where integrals frequency the of by given photon a to pling where Eq. that units assume large can very one frequencies following with atomic emitted typical be hand, not will radiation P [x π/ where Using ent htteasrto n msinmti lmnso the in of as elements are matrix Hamiltonian emission interaction and absorption the that note We o h est arxo h edis field the of matrix density the for atom. the with along riding Eq. observer by an given by mode field time the The and 35. atom and the states, between atom Hamiltonian over trace the denotes where hra itiuin(26): distribution thermal Eq. obtain and exc −ν → ntecs frno neto ie,teeuto fmotion of equation the times, injection random of case the In i u oa tmijce ttime at injected atom an to due sinh(π eoe xoetal ml for small exponentially becomes Tr ξ Γ(z Γ T V hc for which , 2 = e atom int a 40 and ) = d πν stegmafnto,adteproperty the and function, gamma the is x Z stepoe tmfil neato time, interaction atom–field proper the is reads ρ h0, dt τ )] 0 h δρ n ge r Γ ∞ V steaoi rprtm,ie,tetm measured time the i.e., time, proper atomic the is ,n ˆ taysaeslto fEq. of solution Steady-state 44. g efind we a a i −i /c = P (τ dxx = | = r msinadasrto ae u ocou- to due rates absorption and emission are P exc ξ/π V ω ω 0 ˆ and exc − ), −Γ − 2 2i  (τ = h I ~ ν 1 P = e Γ V ) 2 + 1 e ˆ ν e ν ,a ω abs a g | [(n ix Z 1, (τ 4π stemd rqec a rmteBH. the from far frequency mode the is yields 2 2 = [n τ = i = b 4π τ 00 2ν c ω g ρ − + 1 i 1)ρ + i ), ω +T 2 − n e  , g ~ r 2 i ,n ρ 2 ω 4π 2 g ih(2πν sinh int atom ν Z ν ν 2ν

− ω c  r V , Z pedxB, Appendix τ g n e ω i Z τ 0  ν Γ (n e ,n τ i ∞ 2 τ ν 4π (τ P +T e  i = i τ 40 ,a hn ntedimensional the in Then, . − e π 1)ρ + exc i c is r i ν dxx +τ g 4πν h1, 1 int ) e 1 = ν n oe rmtedetailed the from comes  NSLts Articles Latest PNAS ⊗ −πν V . Γ(−2i )Γ ˆ 0 ρ 1 n,teeoe nthe in therefore, and, − κ|gI V 2i a d n − ρ(t (τ hs acceleration Thus, 1. | n −1,n ν e 1 τ V ,a +1,n e 1 ˆ ) 0 . e (τ d ix d . steinteraction the is ,a (τ 44 ν τ −1 τ i

| nteother the On . 00 2 )) +1 ν ) , 2 sgvnb the by given is . ) and , ]− |0, ii , ], | , Γ(−ix b i , I e Tr | ,a )| f6 of 5 [42] [41] [40] [39] [44] [43] atom 2 are =

PHYSICS S.S. ρn,n = exp (−2ξn)[1 − exp (−2ξ)]. [45] where n¯˙ ν is the photon flux from the infalling atoms. 2 Recalling the BH area A ≡ 4πrg , where the gravitational To approach this steady-state solution, we need a cavity to 2 2 P ˙ radius rg = 2MG/c and m˙ p c = ~ ν n¯ν ν is the car- restrict the modes to a finite range of the Regge–Wheeler coor- ried away by the emitted photons, we arrive at the HBAR dinate r∗, so the bottom mirror must be at rb > rg , and the top entropy/area relation must be at rt < ∞. This will modify the analysis of Appendix B, r → r r → ∞ 3 but we can then take the limit as b g and t . kB c S˙ p = A˙ p . [49] Appendix D. Entropy Flux 4~G 2 4 The time rate of change of entropy due to photon generation, Here A˙ p = 32πG M m˙ p /c is the rate of change of the BH area X due to photon emission. The BH rest mass changes as M˙ = S˙ p = −kB ρ˙n,n ln ρn,n , [46] m˙ atom +m ˙ p due to the atomic cloud adding to and the emit- n,ν ted photons taking from the mass of the BH. The BH area A is 2 ˙ ˙ ˙ ˙ to a good approximation can be written as proportional to M and, hence, A = (2M /M )A = Aatom + Ap .

˙ X S.S. ACKNOWLEDGMENTS. We thank M. Becker, S. Braunstein, C. Caves, Sp ≈ −kB ρ˙n,n ln ρn,n [47] G. Cleaver, S. Deser, E. Martin-Martinez, G. Moore, W. Unruh, R. Wald, and n,ν A. Wang for helpful discussions. We note that both referees, quantum optics– expert Federico Capasso (NAS member) and general once one has approached the steady-state solution. The steady- relativity expert Michael Duff (FRS), have provided insightful criticism which S.S. have improved the presentation and physics of the paper. This work was state density matrix ρn,n is given by Eq. 45. Inserting it into supported by the Air Force Office of Scientific Research (Award FA9550- [47] gives 18-1-0141), the Office of Naval Research (Awards N00014-16-1-3054 and 4πkB rg X N00014-16-1-2578), the National Science Foundation (Award DMR 1707565), S˙ ≈ n¯˙ ν, [48] p c ν the Robert A. Welch Foundation (Award A-1261), and the Natural Sciences ν and Engineering Research Council of Canada.

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