Black-hole powered quantum coherent amplifier

Atoms falling into a black hole (BH) through a cavity are shown to enable coherent amplification of light quanta powered by the BH gravitational vacuum energy. This process can harness the BH energy towards useful purposes, such as propelling a spaceship trapped by the BH. The process can occur via transient amplification of a signal field by falling atoms that are partly excited by Hawking radiation reflected by an orbiting mirror. In the steady-state regime of thermally equilibrated atoms that weakly couple to the field, this amplifier constitutes a BH-powered quantum heat engine. The envisaged effects substantiate the thermodynamic approach to BH acceleration radiation.

Introduction: Imagine a scene that can play out in a science fiction movie (Fig. 1): a spaceship is helplessly falling into a black hole (BH) because its fuel supply is dwindling and does not suffice for a breakaway maneuver.Luckily, its SOS message has been received by a faraway spaceship, which is equipped with a powerful laser that can transfer coherent energy to its distressed sister ship.Unlike heat, coherent energy transfer is associated with ergotropy [1][2][3][4][5][6][7][8][9][10] that can perform mechanical work [11] to propel the ship.Unfortunately, coherent energy transfer would have poor efficiency due to diffraction and BH gravitational lensing over large distances between the ships.Yet a revolutionary technique may still rescue the ill-fated spaceship: the laser signal can be coherently amplified in a novel fashion by atoms in free fall through a cavity.Namely, the amplification can only occur through excitation of the free-falling atoms by BH Hawking radiation redirected by an orbiting mirror.The envisioned amplification can strongly enhance the coherent power transfer to the falling spaceship, providing it with enough thrust to free itself from the grip of the BH.
What is the theoretical basis for this fantastic story?It is the mind-boggling idea that the Unruh vacuum [12][13][14][15] yields thermal Hawking radiation near the BH horizon, but cannot directly excite atoms falling into the BH, as opposed to a bright star that can directly heat up falling atoms in its vicinity.By contrast, near a BH the free-falling atoms feel the heat only if the Hawking radiation is redirected by a mirror placed on a stable orbit around the BH (Fig. 1).Then, counter-intuitively, BH gravity can act on atoms as a heat bath, although the process is purely unitary [12][13][14][15][16][17].
For atoms falling into a BH during their passage through a cavity, a perturbative (master-equation) approach maps this BHgravitational problem onto that of a quantum heat engine that acts as a two-level maser/laser without population inversion coupled to two baths at different temperatures [18].Here the piston of the heat engine is the signal laser field whereas the BH scalar field modes redirected by a mirror replace the hot bath as the energy source and the cold bath as the entropy dump of the engine.This uniquely quantum mechanical manifestation of anomalous, gravitational vacuum effect unequivocally demonstrates the validity of the thermodynamic approach to acceleration radiation near a BH.Another intriguing limit is the strong-coupling field-atom regime mediated by the BH vacuum state, a novel manifestation of gravity-induced quantum electrodynamics.
Analysis: A cloud of two-level atoms (TLA) initially in their ground state, is freely falling towards the BH through a cavity.The TLA are coupled to the gravitational field of the BH by a quantized scalar field [12][13][14][15] where H.c. stands for the Hermitian conjugate, index i labels the field modes, r = (r, Θ) denotes the radial and angular coordinates, and âi is the i−th mode annihilation operator.The scalar field is coupled with the TLA as depicted in the space-time diagram (Fig. 1b).An atom freely falling into a nonrotating BH while still above the horizon can (see App. A) be resonant with the following scalar field modes (in the Kruskal-Szekeres coordinates) φ where θ is the step function and Ω > 0. From the perspective of the free-falling atom the modes (2)-( 3) harmonically oscillate as a function of the atom's proper time with positive frequency.The form of the outgoing mode (2) and the ingoing mode (3) derived here (App.A) is, as shown below, key to our ability to employ the BH as a source of useful quanta.
The free-falling atoms may resonantly interact with the outgoing plane-wave field φ 1Ω and with the ingoing Rindler field φ 2Ω .However, in the Unruh vacuum, which by consensus represents the state of the evaporating BH field [15], there are no photons in the modes ( 2) and (3).Consequently, free-falling atoms cannot become excited in the Unruh vacuum (see App. A).Instead, we might consider exciting these atoms by the outgoing Rindler photons, which fill the Unruh vacuum and constitute the Hawking radiation [19,20].They thermally populate the modes Yet, it can be shown (App.A) that these outgoing Rindler photons cannot excite free-falling atoms.Is there another way to excite these atoms by BH radiation?Indeed, there is: we show that free-falling atoms can be excited by redirecting the outgoing Rindler photons (Hawking radiation) towards the BH via a mirror.The mirror should orbit the BH at a fixed radius r = r 0 .To be stable, the mirror orbit should lie at r > 3r g , r g being the gravitational radius, but otherwise the value of r does not affect the result (see below).In the presence of such a mirror, the mode function satisfying the boundary condition φ(t, r 0 ) = 0 at the mirror surface acquires a new, advantageous form This hitherto unexplored scalar field mode has two parts: the outgoing Rindler photon mode (the first term on the rhs) and a part reflected from the mirror into the ingoing Rindler mode (the second term on the rhs).This ingoing Rindler mode acts as a hot bath mode, denoted as φ h (r, t) with frequency Ω = Ω h , that can excite the free-falling atom.The outgoing Rindler modes act as a cold-bath (vacuum state) mode denoted as φ c (r, t).
We wish to show that the redirected Hawking radiation can enable coherent amplification of a signal mode.The complete field-atom interaction Hamiltonian has then the form Here b stands for the signal-mode annihilation operator, âhi is the i-th mode annihilation operator of the hot bath mode φ hi of the redirected Hawking radiation, and ĉj for that of the j-th cold bath mode φ cj of the redirected Hawking radiation (Eq.( 5)).
The atom-scalar field interaction (first term on the rhs of Eq. ( 6)) represents an anti-resonant Raman process whereby a scalarfield quantum in the i−th redirected Hawking-radiation mode φ hi is converted into a signal photon by the atomic transition between the ground (g) and excited (e) states, with coupling strength g hi .The interaction Hamiltonian of the atom with the cold bath φ cj involves the same atomic transition operator |e g| with coupling strength g cj .Our goal is to maximize the energy gain of the signal mode in a non-passive (ergotropy-carrying) form, capable of delivering work [11].
Strong TLA-BH coupling: Here we assume that while traversing the cavity, the atom is strongly coupled to one redirected Hawking radiation mode φ h with a coupling strength g h that overwhelms the coupling strengths g cj to all cold bath modes.This scenario corresponds to a high-Q cavity which allows for strong coupling of a single Hawking radiation mode to the atom.To render the problem single-mode, we choose the TLA resonant frequency ω 0 , the cavity frequency ω c , the signal ν and the Ω h frequency of the redirected mode φ h in ( 5) such that ν ≈ Ω h − ω 0 .Then the interaction Hamiltonian in Eq. ( 6) simplifies to The basis for the combined atom-field energy states can then be where |n s and |n h are Fock states of the signal mode and the BH φ h mode respectively.At short times, where first-order transitions between the atom and the field modes predominate, the subspace in Eq. ( 8) is decoupled from other subspaces, whilst keeping the total number of excitations constant.Let us assume that the atom and the signal mode are initially in the ground and Fock state |n s respectively.Thus, the initial state of the combined system is ρ i = |g g| ⊗ |n s n s | ⊗ ρ Tc ⊗ ρ T h , where ρ Tc and ρ T h are the thermal field states at temperature T c and T h , respectively.In this problem, T c = 0. Then the initial state is a mixture of the pure states |g |n s |n h with probabilities p n h = e −β h Ω h n h /Z β h , where is the effective BH (Hawking) temperature [19,20].
The final-states of the atom and the signal mode after their unitary evolution over time t are then (App.B) where The work capacity (ergotropy) change following the interaction in the cavity is which is maximized for |v| = 1, |u| = 0.
For the choice δ = 0, g h t|φ h | = (2m + 1)π/2, where m is an integer, the atom is transferred to the excited state and the signal adds a photon to its mode, ρ f s = |n s + 1 n s + 1|.The highest amplification per atom is achieved for n s = 1.The efficiency of work extraction by the signal from the BH is then This efficiency can closely approach the Scovil-Schulz-Dubois (SSD) bound of quantum heat engine/amplifiers [21] ν/(ω 0 + ν).
In turn, the SSD efficiency η SSD can approach the Carnot efficiency η C if T h Tc Ω h ωc .However, as T c → 0, the atom resonant frequency must approach zero in order to attain the Carnot efficiency, which is unfeasible.
The maximal average power of work extraction in this regime is given by where the maximal power corresponds to m = 0. Spectacular power boost can be obtained in the Dicke regime of N atoms that are collectively coupled to the hot bath mode.Following [22], we can have Weak TLA-BH coupling: Let us now consider the opposite limiting regime of a cavity with insufficiently high Q, such that its leakage to cold bath modes φ c outside the cavity is stronger than the coupling of the atom to the Hawking radiation mode φ h .In this regime, the atom that is energized by the redirected Hawking radiation reaches a steady state (equilibrates) under the action of the cold bath while in the cavity.Hence, the process is analogous to our continuously operating heat-engine maser based on a TLA [18].Here, the atom together with the signal at frequency ν are coupled to a hot field mode near resonantly, but the coupling strength g h is assumed to be weaker than the coupling to the cold modes g cj .The atom then reaches a steady state under the action of the cold bath (App.C).
The atom-scalar field interaction obeys the Raman Hamiltonian that in the interaction picture reads (cf.Ref [18] for derivation) Under this interaction, we then get a master equation for the state of the hot scalar field.By tracing out the atom, which has reached a steady population under the influence of the cold bath, we then find the time evolution of the signal mode (see SI) The ergotropy (work capacity) of the signal state in this regime that corresponds to coherent amplification grows as where |α 0 | is the mean initial signal amplitude and G is the gain (see SI).The power of the gained work is therefore given by As in the strong-coupling regime, N -fold collective (Dicke) power boost [22] is attainable by N atoms.The efficiency can be computed as the ratio of power generated by the signal to the heat flux from the BH, Qh .This efficiency evaluates to (see App. C) where |α 0 | is the mean initial signal amplitude.It approaches the Scovil-Schulz-Dubois (SSD) bound ν/(ω 0 + ν) as |α 0 | >> 1 (Fig. 2).In Fig. 3 we show that the division of the gained signal energy between ergotropy and heat tends in favor of ergotropy (coherent work production) as the gain increases.

Conclusions:
We have put forth the possibility of black hole (BH) gravity to act as the energizing source of coherent light amplification.The amplification is mediated by the Hawking radiation of the BH in the presence of an orbiting mirror that transforms outgoing Hawking radiation into ingoing Rindler quanta.It can be viewed as a BH-fueled heat engine that converts Hawking radiation into work in a coherent signal mode.
The main energy source in our model is Hawking radiation, and not the kinetic or potential energy of the atoms.In principle, one can also use the kinetic energy of ground-state atoms passing through the cavity to amplify light [23].Our results corroborate the view [12][13][14][15] that, despite the unitarity of such processes, a BH can act as a heat source on falling matter (cf.[24]).
Concepts of quantum information theory and optics have been gaining prominence in the context of quantum effects of gravity [25][26][27][28].We here venture in yet another direction, demonstrating that such effects may find practical use, such as propelling a spaceship by atoms falling into a BH.These results open a new avenue that bridges quantum optics, quantum thermodynamics and BH gravity.

FIG. 1 :
FIG.1: a) Coherently amplified energy transfer between spaceships is enabled by a cavity filled with atoms mounted on a spaceship that is freely falling into a BH provided the BH radiation is redirected by an orbiting mirror.b) Space-time diagram of the relevant scalar modes in the Kruskal coordinates.c) Schematic description of the amplification process in the cavity.