Quantum coherence, radiance, and resistance of gravitational systems
aa r X i v : . [ g r- q c ] S e p Quantum coherence, radiance, and resistance of gravitational systems
Teodora Oniga ∗ and Charles H.-T. Wang † Department of Physics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United Kingdom
We develop a general framework for the open dynamics of an ensemble of quantum particlessubject to spacetime fluctuations about the flat background. An arbitrary number of interactingbosonic and fermionic particles are considered. A systematic approach to the generation of gravi-tational waves in the quantum domain is presented that recovers known classical limits in terms ofthe quadrupole radiation formula and backreaction dissipation. Classical gravitational emission andabsorption relations are quantized into their quantum field theoretical counterparts in terms of thecorresponding operators and quantum ensemble averages. Certain arising consistency issues relatedto factor ordering have been addressed and resolved. Using the theoretical formulation establishedhere with numerical simulations in the quantum regime, we discuss potential new effects includingdecoherence through the spontaneous emission of gravitons and collectively amplified radiation ofgravitational waves by correlated quantum particles.
I. INTRODUCTION
Recent detection of gravitational waves [1] has con-firmed one of the most important predictions of generalrelativity. Their discovery is not only realizing the long-awaited gravitational wave astronomy [2, 3] but also putsthe quest for deeper and wider progress of fundamentalphysics in a new perspective [4]. Despite their prevailingclassical descriptions, the energy density of the observedgravitational waves, close to the source GW150914, isthought to be a small fraction of the Planck density [1].This suggests the effects of quantum gravity and Planckscale physics on gravitational waves are of interest forfurther investigations. Indeed, one asks: What can belearned about quantum gravity from gravitational waves?Gravitons are quantized gravitational waves [5] andcarry the true dynamics of gravitational fields [6–8]. Likephotons, under vacuum fluctuations spontaneous emis-sion of gravitons by energized quantum states undergoingdecay and decoherence has also been postulated. In par-ticular, substantial spontaneous emissions of gravitons inthe early universe following inflation by the matter con-tent subject to quantum-to-classical transitions may beresponsible for entropy production, thermodynamic ar-row of time, structure formation, and the emergence ofthe classical world [9–13]. The precise physical mecha-nisms involved in this chain of processes are however notfully understood at present. The ongoing efforts to ob-serve gravitons of cosmological origin as part of primor-dial and stochastic gravitational waves [14, 15] are ex-pected to provide evidence for the above scenarios havingconsiderable implications on the interplay between cos- ∗ [email protected] † Corresponding author. [email protected] mology, quantum gravity, and potentially the ultimateunified theory at the Planck scale.Driven by the above significant developments with theneed for increased conceptual understanding and techni-cal tools, we report in this paper on a unified frameworkbased on recent theoretical progress of generic gravita-tional decoherence [16–18], and provide an applicationexample using a confined gravitating many-particle sys-tem ready to be generalized. The theory and method-ology are aimed at addressing a wide range of complexand collective quantum dynamical behaviours of realis-tic matter systems that may be isolated in space butopen to spacetime fluctuations (Sec. II). A broad class ofphenomena may be relevant, covering gravitational de-coherence, radiation with reaction and dissipation, andtheir classical reductions. We show that the classical dy-namical structure for gravitational radiation is largelypreserved as the deterministic part of the quantum struc-ture, that also acquires an additional quantum stochas-tic influence from the universal fluctuations of space-time (Sec. III). The generation of gravitational wavesin the quantum domain under our systematic approachbased on the modern formalism of open quantum sys-tems [19] is shown to recover classical limits. In treatingthe quantum mechanisms for gravitational emission andabsorption in terms of quantized operators and quantumensemble averages, we have encountered certain factor or-dering ambiguities, which have fortunately been resolvedthrough consistency considerations (Sec. IV). The estab-lished theoretical formulation, illustrated with numericalsimulations, allows us to demonstrate novel gravitationalradiative phenomena including the collectively amplifiedspontaneous emission of gravitons by a highly coherentstate of identical bosonic particles, in close analogy withthe superradiance of photons [20] (Sec. V). Towards theend, we conclude this work with a summary of its results,implications, and future prospects (Sec. VI).In this work, we will consider the lowest order quantumgravitational effects consistent with the effective quan-tum field theory approach to general relativity [21]. Atlow energy, much less than the Planck scale, this descrip-tion allows one to analyze the propagations of gravitonswith matter interactions using linearized quantum grav-ity to be adopted below, without concerning the non-renormalizability of gravity [22]. Although such a re-stricted framework does not capture higher order quan-tum gravity effects, it is a significant necessary step inmaking progress towards a full quantum gravitationaldescription, which has been useful in probing low-energyquantum gravitational decoherence [16, 23]. A better un-derstanding of the physical effects of linearized quantumgravity may also guide the connections between a fullertheory of e.g. loop quantum gravity [24] with the realworld. It is also sufficient to prove the quantum nature ofgravity using linearized quantum gravity on the more ac-cessible laboratory scales. Therefore, further theoreticaland experimental understandings of linearized quantumgravity effects may bear important implications for fullquantum gravity.Additionally, the linearized quantum gravity frame-work serves as a tradeoff to suspend the problemof time in quantum gravity with full general covari-ance [25], by providing a background Minkowski metric η µν = diag( − , , ,
1) with Lorentz coordinates ( x µ ) =( t, x, y, z ), using Greek indices µ, ν, . . . = 0 , , ,
3. Whenthe metric is perturbed by a weak compact gravitationalsystem and weak gravitational waves, these coordinatesbehave as mean asymptotic Lorentz coordinates for anobserver distantly exterior to the system. Such time t = x may be measured e.g. by a laboratory which is sta-tionary “relative to a remote star.” This way, while mak-ing no claims to resolve the ambiguity of time measure-ment often encountered in the context of quantum statereduction models [26, 27], prominently by Penrose [28],we circumvent similar discussions with the above choiceof time. Its physical consistency and usefulness withinthe linearized gravity approximation can be justified bythe recovery of the classical limits of the quadrupole ra-diation formula and backreaction dissipation for gravita-tional waves from our quantum derivations, as requiredby the correspondence principle. See Secs. III and IV.In what follows, apart from stated exceptions, wechoose the relativistic units where the speed of lightequals one, c = 1. We retain in particular the reducedPlanck ~ and Newtonian G constants to manifest quan-tum and gravitational couplings. Spatial coordinatesin the Cartesian basis are indexed with Latin letters i, j, . . . = 1 , ,
3. Summation over repeated indices is im-plied should no risk of confusion arise. The time deriva-tive, trace-reversion, Hermitian and complex conjugatesare denoted by an over-dot (˙), over-bar (¯), superscripts( † ) and ( ∗ ) respectively. Symbols H and L are used forthe Hamiltonian and Lagrangian with calligraphic type H and L standing for their densities respectively. II. COVARIANT AND CANONICALVARIABLES OF MATTER-GRAVITY SYSTEMS
We start by considering the quantum dynamics of a(multicomponent) matter field ϕ weakly coupled to grav-ity described by an action functional that can be approx-imated with S M [ ϕ, g αβ ] ≈ S M [ ϕ, η αβ ] + 12 Z h µν T µν d x (1)where the spacetime metric takes the perturbative form g µν = η µν + h µν and T µν = 2 δS M δg µν (cid:12)(cid:12)(cid:12) g = η (2)is the stress-energy tensor of the matter on the Minkowskibackground. Since the matter action S M [ ϕ, g αβ ] abovemay depend on the derivatives of the metric, thereby ac-commodating spin connection for Dirac fields [29, 30], inthis work we can extend the validity of the gravitationalinfluence functional derived in [16] for fermionic as wellas bosonic particles.The expansion (1) gives rise to the matter Lagrangianof the form L M = L (sys)M ( ϕ, ϕ ,α ) + L I ( ϕ, ϕ ,α , h αβ ) (3)where L (sys)M , as the integrand of S M [ ϕ, η αβ ], describesthe dynamics of the unperturbed matter system whengravity is switched off, and L I = 12 h µν T µν (4)describes both the self interaction of matter throughgravity, when switched on, as well as its gravitationalinteraction with the environment. The total Lagrangiandensity L T = L M + L G in terms of L G = (16 πG ) − R yields the linearized Einstein equation G µν = 8 πG T µν (5)using the second order perturbation of the scalar curva-ture R = R (2) [ h αβ ] and the first order perturbation ofthe Einstein tensor G µν = G (1) µν [ h αβ ] whose expressionscan be found in Ref. [31].Note that the Einstein equation (5) based on whichthe time evolution of the system density matrix to be de-veloped is up to first order in metric perturbations. Toobtain such first order field equations, the correspondinggravitational Lagrangian is therefore second order in met-ric perturbations as the fields. Accordingly we have con-sistently used the second order perturbation of the scalarcurvature to enter into the gravitational Lagrangian forlinearized gravity above.The resulting classical theory is invariant under thegauge transformation h µν → h µν + ξ µ,ν + ξ ν,µ inducedfrom the coordinate transformation x µ → x µ − ξ µ forarbitrary displacement functions ξ µ = ( ξ, ξ i ).To establish connection with the standard open systemdescription in Hamiltonian formalism, where the pertur-bative interaction is assumed small, we introduce the con-jugate momentum ̟ of the matter field ϕ with respect to L (sys)M and obtain the corresponding matter Hamiltoniandensity H M = H (sys)M + H I (6)where H (sys)M = ̟ ˙ ϕ − L (sys)M and H I = − h µν T µν . (7)The Hamiltonian density of linearized gravity H G = p ij h ij, − L G takes the ADM form [32] H G = H (env)G + n C G + n i C i G (8)where H (env)G contains kinetic- and potential-like termsquadratic in p ij and h ij respectively counting for thepositive energy of the environmental gravitational waves,and C G = (16 πG ) − ( h ii,jj − h ij,ij ) , C i G = − p ij,j (9)are first class constraints with Lagrangian multipliers n = − h / n i = h i . Therefore by using Eqs. (6), (7)and (8), the total Hamiltonian density H = H M + H G can be expressed as H T = H (sys)M + H G + H I (10)= H (sys)M + H (env)G − h ij T ij + n C + n i C i (11)with the second line (11) above taking an overall ADMform using the constraints C = C G + C M , C i = C i G + C i M (12)including the matter contribution C M = T , C i M = − T i . (13)This Hamiltonian formulation enables the gauge trans-formations of all dynamical variables of the matter-gravity system to be generated by the first class con-straints C and C i through canonical transformations. III. RADIATION, RECEPTION, ANDREACTION OF GRAVITATIONAL WAVES
In the Lorenz gauge ¯ h µν ,ν = 0, the linearized Einsteinequation (5) takes the form h µν,αα = − πG ¯ T µν . (14)with solutions naturally separated into h µν = h (sys) µν + h (env) µν . (15)The first term above is an inhomogeneous solution com-bined from h (sys) µν ( r , t ) = 4 G Z d x ′ ¯ T µν ( r ′ , t − ǫ | r − r ′ | ) | r − r ′ | (16)using the spatial position vector r with norm | r | = r ,for ǫ = 1 as a retarded potential, and ǫ = − ǫ = 1, the amplitude h (sys) µν appears to “leakinto the environment” and becomes observable gravita-tional waves, though technically h (sys) µν is tied to the mat-ter system, and is not part of the environment. Likewise,if the less familiar though physically possible ingoing-wave boundary condition is applied with ǫ = −
1, theamplitude h (sys) µν appears to be “sucked from the environ-ment”, though again h (sys) µν is technically not part of theenvironment.The second term h (env) µν of Eq. (15) above satisfies thehomogeneous part of Eq. (14) and describes the envi-ronmental gravitational waves. As such, the additiontransverse-traceless (TT) condition can be applied to h (env) µν . Since h (env) ij is independent of the mater system,it carries the dynamical degrees of freedom of gravity.The orthogonality of the TT decomposition allows usto split the interacting Hamiltonian density (7) into H I = H (sys)I + H (env)I (17)where H (sys)I = − h (sys) µν T µν (18)describing the self-gravity of the matter system and H (env)I = − h (env) ij τ ij (19)in terms of the TT stress tensor τ ij = T TT ij , describingthe coupling between the matter system and the environ-mental gravitational waves.The interacting matter system H M = H (sys)M + H (sys)I (20)obtained from Eq. (6) by incorporating self-gravityEq. (18) and hence turning off the environmental grav-ity, i.e. h (env) ij = 0, provides a closed dynamics for theclassical radiation (or reception) of gravitational waveswhose wave amplitude is determined by Eq. (16). For anonrelativistic compact matter system of size r (sys) muchless than the wavelength, one obtains the TT part of thiswave amplitude to be h TT ij ( t ) = 2 Gr ¨ I -- TT ij ( t − ǫr ) (21)at a distance r ≫ r (sys) from the matter system havingthe reduced quadrupole moment I -- ij = Z d x (cid:0) x i x j − δ ij r (cid:1) T ( r , t ) . (22)The average radiation (or reception) power can bederived from integrating the total flux associated withEq. (21) using the gravitational wave energy density E = 132 πG h ˙ h TT ij ˙ h TT ij i (23)to be the well-known quadrupole gravitational radiationformula P = G h ... I -- ij ... I -- ij i (24)where h·i denotes classical averaging, which in principleapplies for gravitational reception as well.Since the above gravitationally interacting matter sys-tem is closed, deterministic and conservative, the gravita-tional wave energy escaping to (or feeding from) infinitymust involve balancing (anti-)dissipation. This mecha-nism, at the classical level [33], is indeed provided bythe backreaction from the gravitational wave amplitude h (sys) µν through its time retardation (or advance) inducedeffective (anti-)damping using Eqs. (16) and (18). IV. DECOHERENCE VIA SPONTANEOUSEMISSION AND ABSORPTION OF GRAVITONS
The preceding paradigm for the radiation, receptionand reaction of gravitational waves changes drasticallywhen the fundamental quantum properties of matter andgravity are taken into account. The field theoreticalnature of linearized gravity means that after quantiza-tion there is a permanent fluctuating gravitational back-ground even at zero temperature. The ambient space-time fluctuations couple universally to all matter systemsthrough the environmental interaction term H (env)I givenby Eq. (19). Like H (sys)I in Eq. (18), this term can drainenergy e.g. at a low environmental temperature, as wellas pump energy e.g. at a high environmental tempera-ture. Therefore, for a quantized gravitating system, thereare now two channels of energy flow from the system: ra-diation reaction with a deterministic character and space-time fluctuations with a stochastic character and hencea capacity to decohere. It may be physically conceivablethat the exchange of gravitational energies, for classical-like macroscopic systems with fluctuations smoothed out,is dominated by radiation reaction, whereas for quan-tumlike microscopic systems with diminishing time re-tardation or advance inside the system, is dominated byspacetime fluctuations.To quantize the total matter-gravity system while pre-serving gauge invariance, we carry out Dirac’s canoni-cal quantization of constrained system [34] based on theHamiltonian density (11) in the Heisenberg picture [16],where the operator forms of the first class constraints C and C i given by Eq. (12) become quantum generatorsof gauge transformation. Accordingly, physical states | ψ i are required to be gauge invariant by satisfying the quan-tum constraints C| ψ i = 0 , C i | ψ i = 0 . (25)In what follows, our perturbative approach would natu-rally admit a “Dirac-Fock” description of quantization,for which it has been shown that only the positive fre-quency modes of the constraints are required to anni-hilate physical states. See e.g. Ref. [35] for relevantdiscussions and further details on the consistent Diracquantization using the Fock representations.The canonical variable operators acting on physicalstates satisfying Eq. (25) then evolve in time accordingto the quantum Heisenberg equations, which are equiv-alent to the quantum linearized Einstein equation (5).In this formalism, supplementary relations can be usedto restrict gauge redundances, as the quantum form ofgauge conditions at no expense of breaking gauge invari-ance as gauge transformations can still be generated by C and C i [16].In this sense, to establish the influence of the quan-tum gravitational environment on the matter system, itis useful to work in the quantum Lorenz gauge so thatthe metric perturbation operator h µν satisfy quantizedEq. (14) with solutions also separated in the same man-ner as Eq. (15). Using quantized Eqs. (8), (10), andEq. (17), and considering only physical states satisfy-ing the quantum constraints (25), we obtain the totalHamiltonian that governs the evolution and coupling ofthe matter-gravity system as follows H T = H (sys)M + H (env)G + H I = H (sys)M + H (env)G + H (sys)I + H (env)I . (26)To investigate the dynamics of matter-gravity couplingand the resulting radiation, decoherence and dissipation,we will from now on employ the interaction picture wherethe interaction Hamiltonian H I , consisting of self ( H (sys)I )and environmental ( H (env)I ) gravity contributions, gener-ates the time evolution of quantum states. We considerthe fluctuating spacetime to resemble an infinite reser-voir in which environmental gravitons with frequencies ω are maintained in an equilibrium Gaussian state with adistribution function N ( ω ) described by a gravitationaldensity matrix ρ G . For thermal equilibrium N ( ω ) is givenby the Planck distribution function and for the zero-pointspacetime fluctuations N ( ω ) vanishes.In terms of the total density matrix ρ T ( t ) of the mattersystem and the gravitational environment, the total timeevolution is determined by the Liouville-von Neumannequation ˙ ρ T = − i ~ [ H I , ρ T ] . (27)The density matrix describing the statistical state of thematter system is reduced from the total system by av-eraging over the ensembles of the gravitational reservoirthrough the partial trace ρ M = Tr G ( ρ T ) . (28)For a matter system initially untangled with the gravi-tational environment at t = 0, when the total state takesthe factored form ρ T (0) = ρ M (0) ⊗ ρ G (29)which may later develop entanglement with the environ-ment, its reduced dynamical evolution is generated bythe non-Markovian master equation˙ ρ = − i ~ [ H (sys)I , ρ ] − πG ~ Z d k π ) k × n Z t d t ′ e − ik ( t − t ′ ) (cid:0) [ τ † ij ( k , t ) , τ ij ( k , t ′ ) ρ ]+ N ( ω k ) [ τ † ij ( k , t ) , [ τ ij ( k , t ′ ) , ρ ]] (cid:1) + H.c. o (30) using Eqs. (27), (28), (29), and the gauge invariant grav-itational influence functional techniques [16]. Above, ρ = ρ M abbreviates the matter system density matrix, ω k = k = | k | denotes the environmental graviton fre-quency associated with wave vector k , and τ ij ( k , t ) = Z τ ij ( r , t ) e − i k · r d x (31)are operators Fourier-transformed from quantized τ ij ( r , t ) introduced in Eq. (19), which have been normal-ordered with particle nonconservation terms neglected inthe low energy domain being considered.Notably, Eq. (30) constitutes an integrodifferentialequation satisfied by the Dyson series solutions ofthe spacetime-ensemble averaged Eq. (27), whose time-nonlocality gives rise to non-Markovianity [16]. In accordwith the perturbation theory of the non-Markovian dy-namics of open quantum systems [36], the order of cou-pling in such a series expansion increases consistently byone, for each Dyson expansion order, with an extra timeintegral.Similarly, in the standard perturbative scattering the-ory with a linear order coupling in the sense of time-local field equations, the transition amplitudes can beobtained from the Dyson expansions containing time-nonlocal integrals with nonlinear coupling orders, phys-ical constraints permitting. For instance, the validity ofsuch expansions for a scattering system may be limitedby whether pair productions or other high-energy effectsare evident. For weak gravitational systems being con-sidered, the size of the dynamical metric perturbationsshould ultimately remain much less than order one forEq. (30) to be valid.It is also worth remarking that, in deriving Eq. (30),the averaging over the Gaussian environment with zero-mean fluctuating gravitational fields assimilates theDyson expansion into a cumulant expansion that termi-nates at the second order, making the non-Markovianmaster equation (30) truncation-free [16].The second coupling order with fluctuating linearizedgravity has also emerged previously in calculating transi-tion amplitudes under a gravitational bath [37, 38] usingFeynman’s path integral approach [39]. Indeed, second-order master equations have been a prevalent featurefor models of stochastic quantum evolutions under weakgravitational fluctuations [26, 27].Here we investigate new nontrivial dynamical conse-quences of this master equation in a more general phys-ical context, covering in particular radiation throughquantum decoherence and dissipation for particles inconfined states as opposed to free particles studied inRef. [17]. Kinematically, the finite spatial extension ofsuch a system permits the definitions of outgoing andingoing gravitational waves. Dynamically, the couplingbetween these waves and the time evolution of the sys-tem results in their emissions (or absorptions) throughEq. (16) in general and Eq. (21) for nonrelativistic sys-tems. On quantization, these equations (16) and (21)become operator equations.The average gravitational wave energy density expres-sion (23) then acquires quantum meaning by interpreting h TT ij there to be operators and averaging to be over quan-tum ensembles so that given a variable v we have h v i = Tr( vρ ) (32)using the matter system density matrix ρ [19]. Thequantum radiation formula also takes the same form asEq. (24) through quantized Eq. (21). However, factorordering requires some care here as the energy densityrelated term ˙ h TT ij ˙ h TT ij in Eq. (23) is normal-ordered. Ac-cordingly, when the reduced quadrupole moment opera-tor given by quantized Eq. (22) is expanded in frequencymodes I -- ij ( t ) = a ij ( ω ) e − iωt + a † ij ( ω ) e iωt (33)for some operators a ij ( ω ) with positive frequencies ω , thenormal-like ordering of these operators a ij a † kl → a † kl a ij (34)should be implemented for consistency.Factor ordering for the interaction Hamiltonian H I given by Eq. (17) bears some fundamental significance.For electromagnetic radiative problems, it is known thatdifferent factor ordering for the analogous interactionHamiltonian leads to physically distinct mixes and sepa-rations of effects from vacuum fluctuations and radiationreaction [41–45]. Here, the gravitational coupling is con-structed from the quantized general action (1) assumedto be Hermitian for any metric perturbation operator h µν . It follows that, the interaction Hamiltonian H I sep-arated from H M with arbitrary h µν factor is necessarilyHermitian. Now, from Eq. (17), the interaction Hamil-tonian H I is the sum of the environmental part H (sys)I inEq. (19), which is Hermitian as h (env) ij and τ ij commute,and the system part (18), which is not readily Hermi-tian as h (sys) µν is related to time delayed or advanced T µν through Eq. (16) and so may not commute with T µν .Nonetheless, to achieve the Hermiticity of H (sys)I andhence of H I , with the correct classical limit, the factorordering for Eq. (17) can be resolved symmetrically asfollows H (sys)I = − h (sys) µν T µν − T µν h (sys) µν . (35)Similarly symmetrized interaction Hamiltonian [42, 43]has been applied in resolving the aforementioned factor ordering ambiguity in a wide range of problems involv-ing electromagnetic fluctuations and radiation reaction.A recent related discussion and review can be found inRef. [45]. V. COLLECTIVE RADIATION BY CONFINEDIDENTICAL PARTICLES
The theoretical framework established above is appliedin this section, as an illustrative example, to the quantumgravitational decoherence and radiation of a real, i.e.,neutral, scalar field φ with mass m and the associatedinverse reduced Compton wavelength µ = m/ ~ , subjectto an external nongravitational potential ν ( r ) describedby the Lagrangian density L = − g αβ φ ,α φ ,β − (cid:16)
12 + ν (cid:17) µ φ . (36)We focus on the newly formulated spontaneous emis-sion of gravitons by nonrelativistic particles through en-vironmental decoherence at zero temperature and high-light previously undiscovered collective gravitational ra-diation, which we will refer to as “superradiance of grav-itational waves” that mirrors its original electromagneticdescription [20]. As noted in Sec. IV and by analogy withstandard treatments in quantum optical systems [19],we assume the radiation process to be primarily due tospacetime fluctuations using H (env)I by neglecting radia-tion reaction from self-gravity using H (env)I . As a result,quantum dissipation alone is responsible for the radiativeloss of energy, which we verify explicitly for one particleexcited in one dimension. To consider the nonrelativisticdynamics of the scalar field representing nearly Newto-nian particles we assume the potential energy to be muchless than the mass energy so that ν ≪ η µν → η µν + h µν with the proper coordinates x i → x i + 12 h (env) ij x j + O ( h (sys) jk x l ) (37)by using (15), with related considerations discussed inRef. [33]. The resulting fluctuating potential in the TTgauge for free gravitational waves is given by ν ( x i ) → ν ( x i ) + 12 h (env) ij x i ν ,j + O ( h (sys) ν ) . (38)The second and third terms in Eq. (38) above contributerespectively to H (env) in Eq. (19) and H (sys) in Eq. (18).The appearance of gravitational wave induced potentialfluctuations have also been discussed in Refs. [37, 38].However, if the confinement of particles is limited by freemasses then the corresponding boundaries fluctuate inthe proper coordinates instead of the TT coordinates [18].The system part of Eq. (36) yields the unperturbedquantum field equation¨ φ = ∇ φ − (1 + 2 ν ) µ φ (39)having solutions of the form φ = Ψ n ( r ) e − iω n t + H.c. (40)with some orthogonal operators Ψ n ( r ). Hence Eq. (39)reduces formally to the time-independent Schr¨odingerequation − ~ m ∇ Ψ n + V Ψ n = E n Ψ n (41)where V = m ν ( r ) and E n = 12 m ( ~ ω n − m ) (42)represent the potential and eigen energies respectively.As a concrete physical configuration, let us consider anisotropic harmonic potential with frequency ω : V = 12 mω r . (43)In this case, Eq. (40) becomes φ = r ~ ω n (cid:0) a n e − iω n t + a † n e iω n t (cid:1) ψ n ( r ) (44)using the multiple indices n = ( n , n , n ) for n , n , n = 0 , , . . . , and functions ψ n ( r ) = ψ n ( x ) ψ n ( y ) ψ n ( z ) (45)with the harmonic oscillator wave functions ψ n ( x ) and ω n = µ + ( n + n + n ) ω (46)arising from the nonrelativistic limit of Eq. (42), wherethe corresponding ladder operators a n and a † n are anni-hilation and creation operators respectively.In terms of the TT projector P ijkl [16], the TT part ofthe stress-energy tensor follows from Eqs. (2), (36) and(38) to be τ ij ( r , t ) = P ijkl (cid:0) φ ,k φ ,l − µ ω x k x l φ (cid:1) (47)where the second contribution proportional to φ arisesfrom the second term in Eq. (38), which is induced frommetric fluctuations having no electromagnetic analogueas discussed in Sec. IV. From this, by normal-ordering and neglecting particle nonconservation terms relevantonly for higher energy scales, we then obtain τ ij ( k , t ) = F ij ( n , n ′ , k ) a † n ′ a n e − i ( ω n − ω n ′ ) t (48)where F ij ( n ′ , n , k ) = ~ µ P ijkl ( k ) × Z d x (cid:0) ψ n ′ ,k ψ n ,l − µ ω x k x l ψ n ′ ψ n (cid:1) (49)using nonrelativistic approximation with n max ω ≪ µ asthe kinetic energy is much less than the rest mass energyand related long transmitted gravitational wave lengthcondition compared to the spatial extension of occupiedharmonic modes.To derive the gravitational analogue of the quantumoptical master equation for the particle system fromthe general master equation (30), we carry out theMarkov approximation [19] as follows. First, we sub-stitute Eq. (48) into an integral in Eq. (30) to get Z t d t ′ τ ij ( k , t ′ ) e − ik ( t − t ′ ) = F ij ( n , n ′ , k ) a † n ′ a n × e − i ( ω n − ω n ′ ) t Z t d s e − i ( k − ω n + ω n ′ ) s . (50)The nonlocality of this expression in time represents thenon-Markov memory effect, which tends to fade awayunder environmental dissipation. We “forget” this mem-ory by taking the limit R t d s → R ∞ d s , as it does notaffect post-transient dynamics, and apply the Sokhotski-Plemelj theorem Z ∞ d s e − iǫs = πδ ( ǫ ) − i P 1 ǫ (51)to Eq. (50), where P denotes the Cauchy principal valuethat gives rise to a nondissipative Hamiltonian H (env)LS forthe environmentally induced Lamb and Stark shifts of en-ergy. By analogy with quantum optics [19], we capturethe leading radiative mechanisms by adopting the ro-tating wave approximation, neglecting self-gravity H (sys)I and Lamb and Stark shift H (env)LS Hamiltonians, whensubstituting the resulting Eq. (50) back into Eq. (30).Although our general description covers both emissionand absorption of gravitons, for a typical environmentwith a very low level of gravitational wave background,let us focus on the emission of gravitons in the following,leaving the absorption to a separate discussion [46]. Thuswe suppress the absorption of gravitons by setting theirenvironmental distribution function N ( ω ) = 0, hence re-taining merely zero-point fluctuations in the gravitationalenvironment. The above considerations lead us to thegravitational quantum optical master equation˙ ρ = Γ2 (cid:0) δ ik δ jl − δ ij δ kl (cid:1)(cid:0) A ij ρA † kl − { A † ij A kl , ρ } (cid:1) (52)of the Lindblad form, with the transition rate coefficientΓ = 32 G ~ ω c (53)where the speed of light c has been reinstated, and theassociated Lindblad operators A ij = X n q n i ( n j − δ ij ) a † n − ˆ n i − ˆ n j a n (54)where ˆ n = (1 , , , ˆ n = (0 , , , ˆ n = (0 , , t in the originalnon-Markovian master equation (30) starts with an ini-tially factored state (29) untangled with the environment,the Markov assumption used in arriving at Eq. (52) haseffectively pushed that initial time back to the infinitepast whose memory is lost [19], with time t now reset tostart from any new initial condition for the reduced mat-ter state ρ = ρ M ( t ). The detailed derivation of Eq. (52)is given in Appendix A.The justification of the above Markov assumption nec-essarily requires the evolution time scale ∆ t for masterequation (52) to be much greater than the system timescale τ = 2 π/ω , i.e. over many circles of the system oscil-lations, for Eq. (51) to provide a good approximation tothe last integral of Eq. (50), where k is fixed to be 2 ω byEq. (A14) as shown in Appendix A. Likewise, the rotatingwave approximation requires ∆ t ≫ τ for the evolutiontime scale ∆ t to be long enough to average out oscilla-tions on a faster time scale of 2 π/ω . Therefore, for thesystem transition time scale using Eq. (53) to be validitywe must require 1 / Γ ≫ τ for a single particle system.This can be practically satisfied, thanks to the smallnessof Γ for conceivable oscillators. In a broad context ofquantum Brownian motion [40], non-Markovianity canarise even without an integrodifferential structure andthe justification of the Markov assumption may requiremore than time-scale comparisons. Nonetheless, for alarge class of open quantum oscillator models, Marko-vian master equations are shown to often provide goodapproximations at sufficiently high temperature and forsufficiently weak system-environment coupling at low orzero temperature [19, 40]. The latter condition amountsto 1 / Γ ≫ τ stated above in our case. However, for col-lectively amplified transitions with a particle number N to be discussed below, the condition beyond which non-Markovian effects could start to occur may become morestringent, as the transition rate scales with N .Under the Markovian evolution using Eq. (52) at zerotemperature, an excited state ρ decoheres and decays to-wards the ground state. In the process, gravitons are spontaneous emitted that carry the same amount of en-ergy as being reduced from the matter system. For ex-ample, let us take an arbitrary one-particle state ρ withmatrix elements ρ n , n ′ = h n | ρ | n ′ i with | n ′ i as the statevector for the occupation of a harmonic mode n by oneparticle. Then we obtain from Eqs. (32) and (52) thedissipation power − d h H (sys) i d t = ~ ω Γ X n (cid:8) X i n i ( n i − ρ n , n + X i = j (cid:2) n i n j ρ n , n − q n i n j ( n i − n j − × ρ n − n i , n − n j (cid:3)(cid:9) . (55)This expression indeed agrees with the quadrupole radia-tion formula Eq. (24) applied to the present configurationand quantized with consistent factor ordering describedin Eq. (34) where the role of a ij is played by the Lind-blad operators A ij here. See Appendix B for an explicitproof.By virtue of its inherent Lindblad structure, the masterequation (52) is capable of generating new nonlinear col-lective quantum gravity phenomena transferred and in-spired from more established quantum optics areas shar-ing similar dynamical structures.One such novel effect is the collectively amplified spon-taneous emission of gravitons by a matter system in ahighly coherent state, akin to Dicke’s superradiance [20].To illustrate this, let us consider the present harmonicpotential containing many particles excited in one direc-tion, say along the x -axis, with a modal occupation statevector denoted by | N i = |{ N n }i = | N , N , · · · i , where n = n = 0 , , . . . labels the harmonic mode in this di-rection. It follows that the master equation (52) has thefollowing matrix elements h N | ˙ ρ | N ′ i = Γ2 n A n,n ′ ( N , N ′ ) h N n ′ + | ρ | N ′ n + i−B n,n ′ ( N ) h N n − ,n ′ + | ρ | N ′ i o + ( N ↔ N ′ ) ∗ (56)in terms of nonnegative coefficients A n,n ′ ( N , N ′ ) = (cid:2) N ′ n N n ′ ( N ′ n +2 + 1)( N n ′ +2 + 1) × ( n + 1)( n ′ + 1)( n + 2)( n ′ + 2) (cid:3) / (57) B n,n ′ ( N ) = (cid:2) N n +2 N n − n ′ ( N n + 1)( N n − n ′ +2 + 1) × ( n + 1)( n ′ + 1)( n + 2)( n ′ + 2) (cid:3) / (58)where N n ± n ′ = N n ′ ∓ δ n,n ′ ± δ n +2 ,n ′ . In this case, evenand odd harmonic modes are disjointly coupled withintheir own parities because of the quadrupole nature ofthe gravitational waves and symmetry of the potential. (d) (c) t = 0.5 (b) t = 0.05 (a) t = 0.005 ––––– N = 5 ––––– N = 4 ––––– N = 3 ––––– N = 2 ––––– N = 1 t (1,1) (56,1) (1,1) (56,1) (1,1) (56,1) (1,56) (56,56) (1,56) (56,56) (1,56) (56,56) FIG. 1. Plots (a)–(c) show the simulation of the symmetric modulus of the density matrix | ρ p,p ′ | ( t ) for p, p ′ = 1 , , . . . n = 6. While releasing a short burst of gravitational wave, they spontaneouslydecay towards the ground state in the bottom left corner, where all 5 particles occupy the n = 0 mode. Plot (d) shows theaverage radiation power per particle as a function of time for a similar initial state as in plots (a)–(c) but with different particlenumbers. (d) (c) t = 0.5 (b) t = 0.05 (a) t = 0.005 ––––– N = 5 ––––– N = 4 ––––– N = 3 ––––– N = 2 ––––– N = 1 t (1,1) (56,1) (1,1) (56,1) (1,1) (56,1) (1,56) (56,56) (1,56) (56,56) (1,56) (56,56) FIG. 2. Plots (a)–(c) show the simulation of | ρ p,p ′ | ( t ). Here 5 scalar bosons in a harmonic trap are initially equally distributedalong the diagonal of the density matrix for harmonic modes n = 0 , , , n = 0 mode. Plot (d) shows the average radiation power per particle for a similar initial statewith different particle numbers. Based on the master equation with components (56),we perform numerical simulations in nondimensionaltime t → Γ t initially excited and subsequently relaxed inthe x -direction, with harmonic modes n = n = 0 , , , n = n = 0 and a total particle number N = 1 , , , , | p i for p = 1 , , . . . p max with ascending eigenenergies and then the particle number occupations ofhigher harmonic modes. Thus, with N = 5 there are p max = 56 even-mode occupation states | N , N . . . , N i with N = N = N = 0, starting from the groundstate | i = | , , , , , , i , then the first excited state | i = | , , , , , , i through | i = | , , , , , , i tothe highest state | i = | , , , , , , i .While classical sources of gravitational waves areof astronomical scales, the mechanism of collectivelyenhanced quantum gravitational radiation consideredabove may open up a future prospect of a lab-sizedgravitational wave transmitter. Based on the ongoingrapid development of high- Q nanomechanical resonatorsdemonstrated in the quantum regime [47–50], one could0envisage a high-density cluster of nanoresonators in sucha correlated state that they behave like a system of N identical harmonic oscillators with frequency ω . Suppos-ing these oscillators occupy around the n -th harmonicmode, then following discussions of Eq. (56), the max-imum spontaneous decay rate due to collective grav-itational radiation is approximately given by Γ max = N n Γ. For example, a future such microfabricated clus-ter consisting of up to N = one mole of nanoresonators at ω/ π = 10 GHz excited with n = 1000 could in principlehave an observable peak decay rate of up to Γ max = 1Hz via the superradiant spontaneous emissions of gravi-tons. Furthermore, such gravitons could also be detectedusing a similar cluster of nanoresonators instead of anensemble of atoms as a gravitational radiation receiverdescribed in Ref. [46]. The quantum nature of gravitycould then be probed through the quantum properties ofthe nanoresonators imparted by the absorbed gravitons. ––––– N = 5 ––––– N = 4 ––––– N = 3 ––––– N = 2 ––––– N = 1 t (d) (c) t = 0.5 (b) t = 0.05 (a) t = 0.005 (1,1) (56,1) (1,1) (56,1) (1,1) (56,1) (1,56) (56,56) (1,56) (56,56) (1,56) (56,56) FIG. 3. Plots (a)–(c) show the simulation of | ρ p,p ′ | ( t ). Here, 5 scalar bosons in a harmonic trap are initially equally andfully distributed in the density matrix for harmonic modes n = 0 , , , n = 0 mode. Plot (d) shows the average radiation power per particle for a similar initial state withdifferent particle numbers. VI. CONCLUSION
Motivated by the need for a better understanding ofthe fundamental process for quantum matter to deco-here and dissipate through spontaneous emission and ex-change of gravitons with the ubiquitous fluctuating gravi-tational environment, we have extended a recently estab-lished theory of quantum gravitational decoherence [16],now complete with the dynamical origin and consequenceof gravitons mediating spacetime at large and matter,both bosons and fermions, of interest.For physically common states subject to a potential,we have explicitly demonstrated that the abstract masterequation describing the general non-Markovian gravita-tional decoherence of matter formulated in Ref. [16] canindeed be reduced, free from UV-cutoff, to a more con-crete Lindblad form, structurally identical to the fam-ily of quantum optical master equations widely appliedin the quantum optics problems. This enables investi-gations of the theory and phenomenology of quantumgravity to benefit from a wealth of novel characteris-tics and solution strategies in the field of quantum op-tics [19, 20, 51–54]. One such possibility in terms of thenewly identified superradiance of gravitational waves bya system of coherence particles has been theoretically de- scribed and numerically illustrated in Sec. V.Our general framework may serve to clarify variousconceptual issues encountered in the phenomenologicalapproach to quantum gravity [26, 55–60], with first-principles insights, and to guide further analytical tools,mathematical techniques, and modelling methodologiesfor possible detections of quantum gravity effects in thelaboratory [61] and observatory [62] on the ground or inspace [4]. In the context of the cosmological stochasticgravitational waves, since the universe is considered spa-tially flat with a low entropy on exit from inflation [63],our theory may describe short-time graviton radiationand reception by a distribution of coherent states havingpotentially unexpected but important collective proper-ties including quantum nonlinearity, nonlocality, and en-tanglement [17, 18]. In this regard, the theoretical frame-work reported here has recently been applied and furtherextended to address the possible detection of stochas-tic gravitational waves using correlated atoms [46] andpotential observation of spacetime fluctuations throughgravitational lensing [64].Another future objective would be to go beyond theperturbative formulation so as to accommodate largerspacetime fluctuations and curved background or none.Extension in this direction could allow the quantum-to-1classical transition in the early universe with gravitonproductions to be more accurately analyzed. This maybe initiated by generalizing our non-Markovian masterequation (30) to accommodate cosmological perturba-tions [65], in addition to its existing gravitational fluc-tuations in vacuum. A qualitative study of quantum-to-classical transition may follow from the resulting decoher- ence of the content of the early universe in the presence ofcosmological perturbations. An additional rationale forthis final remark is that the development of open quan-tum gravitational systems towards background indepen-dence [5–8, 66] might even help navigate the search foran ultimate full quantum theory of gravity with compat-ible and accessible low energy effects like gravitationaldecoherence and radiance.
ACKNOWLEDGMENTS
This work was supported by the Carnegie Trust for the Universities of Scotland (T. O.) and by the EPSRC GG-TopProject and the Cruickshank Trust (C. W.).
Appendix A: Derivation of the gravitational quantum optical master equation
To derive Eq. (52), we first introduce˜ τ ij ( k , t ) = Z ∞ d s τ ij ( k , t ′ ) e − iks = π X n , n ′ F ij ( n , n ′ , k ) a † n ′ a n e − i ( ω n − ω n ′ ) t δ ( k − ω n + ω n ′ ) (A1)using Eqs. (50) and (51). Note that since k ≥
0, we have nonzero δ ( k − ω n + ω n ′ ) = 0 only if ω n ≥ ω n ′ . The followingrelations then hold τ † ij ( k , t )˜ τ ij ( k , t ) ρ = X δ ( k − ω (∆ n )) A † ij ( n , ∆ n , k ) A ij ( m , ∆ m , k ) ρ (A2)˜ τ ij ( k , t ) ρ τ † ij ( k , t ) = X δ ( k − ω (∆ n )) A ij ( m , ∆ m , k ) ρ A † ij ( n , ∆ n , k ) (A3) ρ ˜ τ ij ( k , t ) τ † ij ( k , t ) = X δ ( k − ω (∆ n )) ρ A ij ( m , ∆ m , k ) A † ij ( n , ∆ n , k ) (A4) τ † ij ( k , t ) ρ ˜ τ ij ( k , t ) = X δ ( k − ω (∆ n )) A † ij ( n , ∆ n , k ) ρ A ij ( m , ∆ m , k ) (A5)summing over n , m , ∆ n , ∆ m subject to ω (∆ n ) = ω (∆ m ) in terms of the operators A ij ( n , ∆ n , k ) = √ πF ij ( n + ∆ n , n , k ) a † n a n +∆ n . (A6)Using Eqs. (A2)–(A5), we have[ τ † ij ( k , t ) , ˜ τ ij ( k , t ) ρ ] = τ † ij ( k , t )˜ τ ij ( k , t ) ρ − ˜ τ ij ( k , t ) ρτ † ij ( k , t )= X δ ( k − ω (∆ n )) (cid:2) A † ij ( n , ∆ n , k ) A ij ( m , ∆ m , k ) ρ − A ij ( m , ∆ m , k ) ρ A † ij ( n , ∆ n , k ) (cid:3) (A7)and [ τ † ij ( k , t ) , [˜ τ ij ( k , t ) , ρ ]] = τ † ij ( k , t )˜ τ ij ( k , t ) ρ − ˜ τ ij ( k , t ) ρτ † ij ( k , t ) + ρ ˜ τ ij ( k , t ) τ † ij ( k , t ) − τ † ij ( k , t ) ρ ˜ τ ij ( k , t )= X δ ( k − ω (∆ n )) (cid:2) A † ij ( n , ∆ n , k ) A ij ( m , ∆ m , k ) ρ − A ij ( m , ∆ m , k ) ρ A † ij ( n , ∆ n , k )+ ρ A ij ( m , ∆ m , k ) A † ij ( n , ∆ n , k ) − A † ij ( n , ∆ n , k ) ρ A ij ( m , ∆ m , k ) (cid:3) . (A8)From Eqs. (A7) and (A8), we see that (cid:2) τ † ij ( k , t ) , ˜ τ ij ( k , t ) ρ ( t ) (cid:3) + N ( k ) (cid:2) τ † ij ( k , t ) , (cid:2) ˜ τ ij ( k , t ) , ρ ( t ) (cid:3)(cid:3) + H.c.= − X δ ( k − ω (∆ n )) h (1 + N ( k )) (cid:0) A ij ( m , ∆ m , k ) ρA † ij ( n , ∆ n , k ) − { A † ij ( n , ∆ n , k ) A ij ( m , ∆ m , k ) , ρ } (cid:1) + N ( k ) (cid:0) A † ij ( n , ∆ n , k ) ρA ij ( m , ∆ m , k ) − { A ij ( m , ∆ m , k ) A † ij ( n , ∆ n , k ) , ρ } (cid:1)i . (A9)2Substituting Eq. (A9) into Eq. (30) with negligible H (sys)I , we have˙ ρ = G π ~ X ω (∆ n ) Z dΩ( k (∆ n )) h (1 + N ( ω (∆ n ))) (cid:0) A ij ( n ′ , ∆ n ′ , k (∆ n )) ρA † ij ( n , ∆ n , k (∆ n )) − { A † ij ( n , ∆ n , k (∆ n )) A ij ( n ′ , ∆ n ′ , k (∆ n )) , ρ } (cid:1) + N ( ω (∆ n )) (cid:0) A † ij ( n , ∆ n , k (∆ n )) ρA ij ( n ′ , ∆ n ′ , k (∆ n )) − { A ij ( n ′ , ∆ n ′ , k (∆ n )) A † ij ( n , ∆ n , k (∆ n )) , ρ } (cid:1)i (A10)where k (∆ n ) denotes k with k = ω (∆ n ). Furthermore, from Eq. (A6), we have A ij ( n ′ , ∆ n ′ , k (∆ n )) = √ πF ij ( n ′ + ∆ n ′ , n ′ , k (∆ n )) A ( n ′ , ∆ n ′ ) A † ij ( n , ∆ n , k (∆ n )) = √ πF ∗ ij ( n + ∆ n , n , k (∆ n )) A † ( n , ∆ n )in terms of the operator A ( n , ∆ n ) = a † n a n +∆ n . (A11)We then substitute the above into Eq. (A10) to isolate the solid angle integral as follows:˙ ρ = X F ( n , n ′ , ∆ n , ∆ n ′ ) h (1 + N ( ω (∆ n ))) (cid:0) A ( n ′ , ∆ n ′ ) ρA † ( n , ∆ n ) − { A † ( n , ∆ n ) A ( n ′ , ∆ n ′ ) , ρ } (cid:1) + N ( ω (∆ n )) (cid:0) A † ( n , ∆ n ) ρA ( n ′ , ∆ n ′ ) − { A ( n ′ , ∆ n ′ ) A † ( n , ∆ n )) , ρ } (cid:1)i (A12)where F ( n , n ′ , ∆ n , ∆ n ′ ) = Gπ ~ ω (∆ n ) Z dΩ( k (∆ n )) F ∗ ij ( n + ∆ n , n , k (∆ n )) F ij ( n ′ + ∆ n ′ , n ′ , k (∆ n )) (A13)subject to ω (∆ n ) = ω (∆ n ′ ). For ω (∆ n ) >
0, we obtain from Eq. (49) that ω (∆ n ) = 2 ω (A14)which in turn requires ∆ n = 2 ˆ n , n , n , ˆ n + ˆ n , ˆ n + ˆ n , ˆ n + ˆ n (A15)and, furthermore, the expression F ij ( n + ∆ n , n , k ) = − ~ ω P ijkl ( k ) f kl ( n + ∆ n , n ) (A16)in terms of f ( n + ∆ n , n ) = p ( n + 1)( n + 2) δ ∆ n , δ ∆ n , δ ∆ n , f ( n + ∆ n , n ) = p ( n + 1)( n + 2) δ ∆ n , δ ∆ n , δ ∆ n , f ( n + ∆ n , n ) = p ( n + 1)( n + 2) δ ∆ n , δ ∆ n , δ ∆ n , f ( n + ∆ n , n ) = p ( n + 1)( n + 1) δ ∆ n , δ ∆ n , δ ∆ n , f ( n + ∆ n , n )) = p ( n + 1)( n + 1) δ ∆ n , δ ∆ n , δ ∆ n , f ( n + ∆ n , n ) = p ( n + 1)( n + 1) δ ∆ n , δ ∆ n , δ ∆ n , . Using the above relations, Eq. (A13) then becomes F ( n , n ′ , ∆ n , ∆ n ′ ) = 2 G ~ ω π Z dΩ( k (∆ n )) P ijkl ( k (∆ n )) f ij ( n + ∆ n , n ) f kl ( n ′ + ∆ n ′ , n ′ ) . (A17)Through the identities P ijkl ( k ) = 12 h δ ik δ jl + δ il δ jk − δ ij δ kl i + 12 k h δ ij k k k l + δ kl k i k j − δ jk k i k l − δ ik k j k l − δ il k j k k − δ jl k i k k i + k i k j k k k l k Z dΩ( k ) k i k j = 4 πk δ ij Z dΩ( k ) k i k j k k k l = 4 πk h δ ij δ kl + δ ik δ jl + δ il δ jk i we obtain that Z dΩ( k ) P ijkl ( k ) = 4 π h δ ik δ jl + 3 δ il δ jk − δ ij δ kl i . (A18)Substituting (A18) into (A17), we have F ( n , n ′ , ∆ n , ∆ n ′ ) = Γ4 X i,j,k,l h δ ik δ jl + 3 δ il δ jk − δ ij δ kl i f ij ( n + ∆ n , n ) f kl ( n ′ + ∆ n ′ , n ′ )= Γ2 X i,j h f ij ( n + ∆ n , n ) f ij ( n ′ + ∆ n ′ , n ′ ) − f ii ( n + ∆ n , n ) f jj ( n ′ + ∆ n ′ , n ′ ) i (A19)where Γ is given by Eq. (53). From Eq. (A19) we have F ( n , n ′ , n i , n i ) = Γ f ii ( n + 2 ˆ n i , n ) f ii ( n ′ + 2 ˆ n i , n ′ ) , ( i = 1 , ,
3) (A20) F ( n , n ′ , n i , n j ) = − Γ2 f ii ( n + 2 ˆ n i , n ) f jj ( n ′ + 2 ˆ n j , n ′ ) , ( i = j ) (A21) F ( n , n ′ , ˆ n i + ˆ n j , ˆ n i + ˆ n j ) = 3 Γ f ij ( n + ˆ n i + ˆ n j , n ) f ij ( n ′ + ˆ n i + ˆ n j , n ′ ) , ( i = j ) (A22) F ( n , n ′ , ∆ n , ∆ n ′ ) = 0 , (for other ∆ n , ∆ n ′ ) . (A23)For N ( ω ) = 0, the master equation (A12) then becomes˙ ρ = X F ( n , n ′ , ∆ n , ∆ n ′ ) h A ( n ′ , ∆ n ′ ) ρA † ( n , ∆ n ) − { A † ( n , ∆ n ) A ( n ′ , ∆ n ′ ) , ρ } i (A24)which can be expanded as˙ ρ = Γ X n , n ′ X i f i ( n + 2 ˆ n i , n ) f i ( n ′ + 2 ˆ n i , n ′ ) h A ( n ′ , n i ) ρA † ( n , n i ) − { A † ( n , n i ) A ( n ′ , n i ) , ρ } i − Γ2 X n , n ′ X i = j f i ( n + 2 ˆ n i , n ) f j ( n ′ + 2 ˆ n j , n ′ ) h A ( n ′ , n j ) ρA † ( n , n i ) − { A † ( n , n i ) A ( n ′ , n j ) , ρ } i +3 Γ X n , n ′ X i By construction of the Lindblad operators (54), we have the following relations A † ii A ii | m i = m i ( m i − | m i (B1) A ii A † ii | m i = ( m i + 1)( m i + 2) | m i (B2) A † ii A jj | m i = q m j ( m j − m i + 1)( m i + 2) | m + 2 ˆ n i − n j i (B3) A ii A † jj | m i = q m i ( m i − m j + 1)( m j + 2) | m − n i + 2 ˆ n j i (B4) A † ij A ij | m i = m i m j | m i (B5) A ij A † ij | m i = ( m i + 1)( m j + 1) | m i (B6)where i = j and no sums are implied.Applying the master equation (52) or equivalently (A25) to the one-particle density matrix ρ , we have h n ′ | ˙ ρ | n i = Γ2 X i h n ′ | h A ii ρA † i − A † i A ii ρ − ρA † ii A ii i | n i − Γ4 X i = j h n ′ | h A jj ρA † ii − A † ii A jj ρ − ρA † ii A jj i | n i + 3Γ4 X i = j h n ′ | h A ij ρA † ij − A † ij A ij ρ − ρA † ij A ij i | n i . Using Eqs. (B1)–(B6), the above yields the one-particle master equation h n ′ | ˙ ρ | n i = Γ X i q ( n ′ i + 1)( n ′ i + 2)( n i + 1)( n i + 2) ρ n ′ +2 n i , n +2 n i − Γ2 X i (cid:2) n ′ i ( n ′ i − 1) + n i ( n i − (cid:3) ρ n ′ , n − Γ2 X i = j q ( n ′ j + 1)( n ′ j + 2)( n i + 1)( n i + 2) ρ n ′ +2 n j , n +2 n i + Γ4 X i = j q n ′ i ( n ′ i − n ′ j + 1)( n ′ j + 2) ρ n ′ − n i +2 n j , n + Γ4 X i = j q n i ( n i − n j + 1)( n j + 2) ρ n ′ , n − n i +2 n j + 3Γ2 X i = j q ( n ′ i + 1)( n ′ j + 1)( n i + 1)( n j + 1) ρ n ′ + n i + n j , n + n i + n j − X i = j (cid:2) n ′ i n ′ j + n i n j (cid:3) ρ n ′ , n . (B7)To obtain the power dissipation for one-particle states, we first use the system Hamiltonian H S = ~ ω X i X n n i a † n a n (B8)derived from T of the scalar field φ in the nonrelativistic limit and master equation (B7). This givesdd t h H S i = X n ′ h n ′ | ( H S ˙ ρ ) | n ′ i = ~ ω Γ X i,k X n ( n k − δ ik ) n i ( n i − ρ n , n − ~ ω Γ X i,k X n n k n i ( n i − ρ n , n + 3 ~ ω Γ2 X i = j,k X n ( n k − δ ik − δ jk ) n i n j ρ n , n − ~ ω Γ2 X i = j,k X n n k n i n j ρ n , n − ~ ω Γ4 X i = j,k X n ( n k − δ ik − δ jk ) q n i ( n i − n j + 1)( n j + 2) ρ n , n − n i +2 n j + ~ ω Γ4 X i = j,k X n n k q n i ( n i − n j + 1)( n j + 2) ρ n , n − n i +2 n j (B9)5yielding the quantum dissipation power through spontaneous emission of gravitons P (se) = − dd t h H S i (B10)given by Eq. (55). Note that the last cross term in Eq. (B9) involving ρ n − n i , n − n j represents a quantum correctionof the gravitational wave emission process. 2. Quantum quadrupole radiation formula for one-particle states In the nonrelativistic limit, using Eq. (44) we have I ij = Z d x x i x j T = m X n , n ′ h a † n ′ a n e − i ( ω n − ω n ′ ) t + a † n a n ′ e i ( ω n − ω n ′ ) t i Z d x x i x j ψ n ( r ) ψ n ′ ( r )yielding ... I ij = 4 i ~ ω h A ij e − iωt − A † ij e iωt i . In terms of the traceless part ... I -- ij = ... I ij − δ ij ... I kk of the above we obtain and the time averaged product h ... I -- ij ... I -- ij i time av = 16 ~ ω X i = j h A † ij A ij + A ij A † ij i − ~ ω X i = j h A † i A j + A i A † j i + 323 ~ ω X i h A † i A i + A i A † i i . (B11)The one-particle matrix elements of Eq. (B11) can be evaluated using Eqs. (B1)–(B6) and applying consistent factorordering described in Eq. (34) with respect to the Lindblad operators A ij and A † ij to be h m ′ | ... I -- ij ... I -- ij | m i = 323 ~ ω X i h m i ( m i − 1) + ( m i + 1)( m i + 2) i δ m ′ , m +16 ~ ω X i = j h m i m j + ( m i + 1)( m j + 1) i δ m ′ , m − ~ ω X i = j q m j ( m j − m i + 1)( m i + 2) δ m ′ , m +2ˆ n i − n j − ~ ω X i = j q m i ( m i − m j + 1)( m j + 2) δ m ′ , m − n i +2ˆ n j . (B12)Finally, by using Eqs. (24), (B11), (B1)–(B6) and (B12) we obtain the quantum quadrupole radiation formula P (qr) = G h ... I -- ij ... 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