Quantum coherence, radiance, and resistance of gravitational systems

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by  Teodora Oniga

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 particles
subject to spacetime fluctuations about the flat background. An arbitrary number of interacting
bosonicandfermionicparticlesareconsidered. Asystematicapproachtothegeneration ofgravita-
tionalwavesinthequantumdomainispresentedthatrecoversknownclassicallimitsintermsofthe
quadrupole radiation formula and back-reaction dissipation. Classical gravitational emission and
absorption relations are quantized intotheir quantumfield theoretical counterparts in terms of the
correspondingoperatorsandquantumensembleaverages. Certainarisingconsistencyissuesrelated
to factor ordering have been addressed and resolved. Using the theoretical formulation established
herewith numericalsimulations in thequantumregime, we demonstrate newpredictions including
7 decoherence through the spontaneous emission of gravitons and collectively amplified “superradi-
1 ance” of gravitational waves by a highly coherent state of identical particles.
0
2
n I. INTRODUCTION based on recent theoretical progress of generic gravita-
a tional decoherence [16–18], and provide an application
J Recent detection of gravitational waves [1] has con- example using a confined gravitating many-particle sys-

5 firmed one of the most important predictions of general tem ready to be generalized. The theory and method-
1 relativity. Their discovery is not only realizing the long- ology are aimed at addressing a wide range of complex

awaitedgravitationalwaveastronomy[2,3]butalsoputs and collective quantum dynamical behaviours of realis-
]
c the quest for deeper and wider progress of fundamental tic matter systems that may be isolated in space but
q physics in a new perspective [4]. Despite their prevailing opentospacetimefluctuations(Sec.II).Abroadclassof
r- classical descriptions, the energy density of the observed phenomena may be relevant, covering gravitational de-
g gravitational waves close to the source GW150914 are coherence, radiation with reaction and dissipation, and
[ thought to be a small fraction of the Planck density [1]. their classicalreductions. We showthatthe classicaldy-
1 This suggests the effects of quantum gravity and Planck namical structure for gravitational radiation is largely
v scale physics on gravitational waves are of interest for preservedasthedeterministicpartofthequantumstruc-
2 further investigations. Indeed, one asks: What can be ture, that also acquires an additional quantum stochas-
2 learnedaboutquantumgravityfromgravitationalwaves? tic influence from the universal fluctuations of space-
1 Gravitons are quantized gravitational waves [5] and time (Sec. III). The generation of gravitational waves
4
carrythetruedynamicsofgravitationalfields[6–8]. Like in the quantum domain under our systematic approach
0
. photons, under vacuum fluctuations spontaneous emis- based on the modern formalism of open quantum sys-
1 sionofgravitonsbyenergizedquantumstatesundergoing tems [19] is shown to recoverclassical limits. In treating
0
decay anddecoherence has alsobeen postulated. In par- the quantum mechanisms for gravitational emission and
7
1 ticular,substantialspontaneousemissionsofgravitonsin absorptionintermsofquantizedoperatorsandquantum
: the early universe following inflation by the matter con- ensembleaverages,wehaveencounteredcertainfactoror-
v tent subject to quantum-to-classical transitions may be deringambiguities,whichhavefortunatelybeenresolved
i
X responsible for entropy production, thermodynamic ar- throughconsistencyconsiderations(Sec.IV). The estab-
r row of time, structure formation, and the emergence of lishedtheoreticalformulation,illustratedwithnumerical
a the classical world [9–13]. The precise physical mecha- simulations,allowsustodemonstratenovelgravitational
nisms involvedinthis chainofprocessesarehowevernot radiative phenomena including the collectively amplified
fully understood at present. The ongoing efforts to ob- spontaneous emission of gravitons by a highly coherent
serve gravitons of cosmological origin as part of primor- state of identical bosonic particles, in close analogy with
dial and stochastic gravitational waves [14, 15] are ex- the superradianceof photons [20] (Sec. V). Towardsthe
pectedtoprovideevidencefortheabovescenarioshaving end,weconcludethisworkwithasummaryofitsresults,
considerable implications on the interplay between cos- implications, and future prospects (Sec. VI).
mology, quantum gravity, and potentially the ultimate In what follows, apart from stated exceptions, we
unified theory at the Planck scale. choose the relativistic units where the speed of light
Drivenby the abovesignificantdevelopmentswiththe equals one, c=1. We retain in particular the reduced
need for increased conceptual understanding and techni- Planck ~ and Newtonian G constants to manifest quan-
cal tools, we report in this paper on a unified framework tumandgravitationalcouplings. Spatialcomponentsare
indexed with Latin letters with i,j··· = 1,2,3 in the
(x,y,z) Cartesian basis. Spacetime coordinates are la-
belled with Greek alphabets with µ,ν = 0,1,2,3 in the
∗ [email protected] (t,x,y,z)LorentzbasisusingthebackgroundMinkowski
† [email protected] metric η = diag(−1,1,1,1). Summation over repeated
µν
2
indices is implied should no risk of confusion arise. The Toestablishconnectionwiththestandardopensystem
time derivative, trace-reversion, Hermitian and complex description in Hamiltonian formalism, where the pertur-
conjugates are denoted by an over-dot (˙), over-bar (¯), bativeinteractionisassumedsmall,weintroducethecon-
superscripts (†) and (∗) respectively. Symbols H and L jugatemomentum̟ofthematterfieldϕwithrespectto
are used for the Hamiltonian and Lagrangianwith calli- L(sys) and obtain the corresponding matter Hamiltonian
M
graphictypeHandLstandingfortheirdensitiesrespec- density
tively.
H =H(sys)+H (6)
M M I
II. COVARIANT AND CANONICAL where H(sys) =̟ϕ˙ −L(sys) and
M M
VARIABLES OF MATTER-GRAVITY SYSTEMS
1
H =− h Tµν. (7)
I µν
We start by considering the quantum dynamics of a 2
(multicomponent)matterfieldϕweaklycoupledtograv-
The Hamiltonian density of linearized gravity H =
G
itydescribedbyanactionfunctionalthatcanbeapprox-
p h −L takes the ADM form [24]
ij ij,0 G
imated by
S [ϕ,g ]≈S [ϕ,η ]+ 1 h Tµνd4x (1) HG =HG(env)+nCG+niCGi (8)
M αβ M αβ µν
2Z
where H(env) contains kinetic- and potential-like terms
G
where the spacetime metric takes the perturbative form quadratic in pij and h respectively counting for the
ij
g =η +h and positiveenergyoftheenvironmentalgravitationalwaves,
µν µν µν
and
δS
Tµν =2 M (2)
δgµν(cid:12)g=η CG =(16πG)−1(hii,jj −hij,ij), CGi =−2pij,j (9)
(cid:12)
(cid:12)
isthestress-energytensorofthematterontheMinkowski arefirstclassconstraintswithLagrangianmultipliersn=
background. Since the matter action S [ϕ,g ] above −h /2 and n = h . Therefore by using Eqs. (6), (7)
M αβ 00 i 0i
maydepend onthe derivativesofthe metric, therebyac- and (8), the total Hamiltonian density H = H +H
M G
commodating spin connectionfor Dirac fields [21, 22], in can be expressed as
this work we can extend the validity of the gravitational
influence functional derived in [16] for fermionic as well HT =HM(sys)+HG+HI (10)
as bosonic particles.
1
The expansion (1) gives rise to the matter Lagrangian =HM(sys)+HG(env)− 2hijTij +nC+niCi (11)
of the form
with the second line (11) above taking an overall ADM
L =L(sys)(ϕ,ϕ )+L (ϕ,ϕ ,h ) (3)
M M ,α I ,α αβ form using the constraints
where L(Msys), as the integrand of SM[ϕ,ηαβ], describes C =CG+CM, Ci =CGi +CMi (12)
the dynamics of the unperturbed matter system when
gravity is switched off, and including the matter contribution
1 C =T00, Ci =−T0i. (13)
L = h Tµν (4) M M
I µν
2
ThisHamiltonianformulationenablesthegaugetrans-
describes both the self interaction of matter through formations of all dynamical variables of the matter-
gravity, when switched on, as well as its gravitational gravity system to be generated by the first class con-
interaction with the environment. The total Lagrangian straints C and Ci through canonical transformations.
density L = L +L in terms of L = (16πG)−1R
T M G G
yields the linearized Einstein equation
III. RADIATION, RECEPTION, AND
G =8πGT (5)
µν µν REACTION OF GRAVITATIONAL WAVES
using the second order perturbation of the scalar cur-
vature R = R(2)[h ] and the first order perturbation In the Lorenz gauge h¯ ,ν =0, the linearized Einstein
αβ µν
of the Einstein tensor G = G(1)[h ] whose expres- equation (5) takes the form
µν µν αβ
sios can be found in Ref. [23]. The resulting classi- h α =−16πGT¯ . (14)
cal theory is invariant under the gauge transformation µν,α µν
hµν → hµν + ξµ,ν + ξν,µ induced from the coordinate with solutions naturally separated into
transformation xµ →xµ−ξµ for arbitrary displacement
functions ξµ =(ξ,ξi). h =h(sys)+h(env). (15)
µν µν µν
3
The first term above is an inhomogeneous solution com- The average radiation (or reception) power can be
bined from derived from integrating the total flux associated with
T¯ (r′,t−ǫ|r−r′|) Eq. (21) using the gravitationalwave energy density
h(sys)(r,t)=4G d3x′ µν (16)
µν Z |r−r′|
1
using the spatial position vector r with norm |r| = r, E = 32πGhh˙TijTh˙TijTi (23)
for ǫ = 1 as a retarded potential, and ǫ = −1 as an
advancedpotential, describing respectively the radiation to be the well-known quadrupole gravitationalradiation
andreceptionofgravitationalwavesbythematersystem. formula
When the usual outgoing-wave boundary condition is
G ... ...
appliedwithǫ=1,the amplitude h(µsνys) appearsto “leak P = 5 hqijqiji (24)
into the environment” and becomes observable gravita-
tionalwaves,thoughtechnicallyh(sys) istiedtothe mat- where h·i denotes classical averaging, which in principle
µν
tersystem,andisnotpartoftheenvironment. Likewise, applies for gravitationalreception as well.
if the less familiar though physically possible ingoing- Sincethe abovegravitationallyinteractingmattersys-
wave boundary condition is applied with ǫ = −1, the temisclosed,deterministicandconservative,thegravita-
amplitudeh(sys) appearstobe“suckedfromtheenviron- tional wave energy escaping to (or feeding from) infinity
µν
ment”, though again h(sys) is technically not part of the must involve balancing (anti-)dissipation. This mecha-
µν
nism, at the classical level [25], is indeed provided by
environment.
The second term h(env) of Eq. (15) above satisfies the the back-reactionfrom the gravitationalwave amplitude
µν
homogeneous part of Eq. (14) and describes the envi- h(µsνys) through its time retardation (or advance) induced
ronmental gravitational waves. As such, the addition effective (anti-)damping using Eqs. (16) and (18).
transverse-traceless (TT) condition can be applied to
h(env). Since h(env) is independent of the mater system,
µν ij
it carries the dynamical degrees of freedom of gravity. IV. DECOHERENCE VIA SPONTANEOUS
The orthogonality of the TT decomposition allows us EMISSION AND ABSORPTION OF GRAVITONS
to split the interacting Hamiltonian density (7) into
H =H(sys)+H(env) (17) The preceding paradigm for the radiation, reception
I I I and reaction of gravitational waves changes drastically
where whenthefundamentalquantumpropertiesofmatterand
1 gravity are taken into account. The field theoretical
H(sys) =− h(sys)Tµν (18)
I 2 µν nature of linearized gravity means that after quantiza-
tion there is a permanent fluctuating gravitationalback-
describing the self-gravity of the matter system and
ground even at zero temperature. The ambient space-
1
H(env) =− h(env)τ (19) timefluctuationscoupleuniversallytoallmattersystems
I 2 ij ij (env)
throughthe environmentalinteractiontermH given
I
in terms of the TT stress tensor τij = TiTjT, describing by Eq. (19). Like H(sys) in Eq. (18), this term can drain
thecouplingbetweenthemattersystemandtheenviron- I
energy e.g. at a low environmental temperature, as well
mental gravitational waves.
as pump energy e.g. at a high environmental tempera-
The interacting matter system
ture. Therefore,foraquantizedgravitatingsystem,there
H =H(sys)+H(sys) (20) arenowtwochannelsofenergyflowfromthesystem: ra-
M M I
diationreactionwithadeterministiccharacterandspace-
obtained from Eq. (6) by incorporating self-gravity
time fluctuations with a stochastic character and hence
Eq. (18) and hence turning off the environmental grav-
a capacity to decohere. It may be physically conceivable
ity, i.e. h(env) = 0, provides a closed dynamics for the that the exchange of gravitationalenergies, for classical-
ij
classical radiation (or reception) of gravitational waves likemacroscopicsystemswithfluctuationssmoothedout,
whose wave amplitude is determined by Eq. (16). For a isdominatedbyradiationreaction,whereasforquantum-
nonrelativisticcompactmattersystemofsizer(sys) much like microscopic systems with diminishing time retarda-
lessthanthewavelength,oneobtainstheTTpartofthis tionoradvanceinsidethesystem,isdominatedbyspace-
wave amplitude to be time fluctuations.
2G Toquantizethetotalmatter-gravitysystemwhilepre-
hTijT(t)= r q¨iTjT(t−ǫr) (21) serving gauge invariance, we carry out Dirac’s canoni-
cal quantization of constrained system [26] based on the
at a distance r ≫ r(sys) from the matter system having
Hamiltonian density (11) in the Heisenberg picture [16],
the reduced quadrupole moment
where the operator forms of the first class constraints
1 C and Ci given by Eq. (12) become quantum generators
q = d3x xixj − δ r2 T00(r,t). (22)
ij Z 3 ij ofgaugetransformation. Accordingly,physicalstates|ψi
(cid:0) (cid:1)
4
arerequiredtobegaugeinvariantbysatisfyingthequan- which may later developentanglement with the environ-
tum constraints ment, its reduced dynamical evolution is generated by
the non-Markovianmaster equation
C|ψi=0, Ci|ψi=0. (25)
i 8πG d3k
The canonical variable operators acting on physical ρ˙ =− [H(sys),ρ]−
states satisfying Eq. (25) then evolve in time according ~ I ~ Z 2(2π)3k
to the quantum Heisenberg equations, which are equiv- t
aInlenthtistofotrhmeaqliusman,tusumppllienmeaernizteadryErienlastteioinnseqcaunatbioenu(s5e)d. ×nZ0dt′e−ik(t−t′)(cid:0)[τi†j(k,t), τij(k,t′)ρ]
to restrict gauge redundances, as the quantum form of +N(ω )[τ†(k,t), [τ (k,t′), ρ]] +H.c. (30)
gaugeconditions at no expense ofbreakinggaugeinvari- k ij ij o
(cid:1)
ance as gauge transformations can still be generated by usingEqs.(27),(28),(29),andthe gaugeinvariantgrav-
C and Ci [16]. itational influence functional techniques [16]. Above,
In this sense, to establish the influence of the quan- ρ = ρ abbreviates the matter system density matrix,
M
tum gravitational environment on the matter system, it ω = k = |k| denotes the environmental graviton fre-
k
is useful to work in the quantum Lorenz gauge so that quency associated with wave vector k, and
the metric perturbation operator h satisfy quantized
µν
Eq. (14) with solutions also separated in the same man- τ (k,t)= τ (r,t)e−ik·rd3x (31)
ij ij
ner as Eq. (15). Using quantized Eqs. (8), (10), and Z
Eq. (17), and considering only physical states satisfy-
are operators Fourier-transformed from quantized
ing the quantum constraints (25), we obtain the total τ (r,t)introducedinEq.(19),whichhavebeennormal-
ij
Hamiltonian that governs the evolution and coupling of
orderedwith particle proconservationterms neglectedin
the matter-gravity system as follows
the low energy domain being considered. In Ref. [16],
H =H(sys)+H(env)+H bosonic matter has been assumed for simplicity in de-
T M G I riving the master equation (30) and free particles un-
=H(sys)+H(env)+H(sys)+H(env). (26) derspacetimefluctuationswithnegligibleself-gravityare
M G I I
foundinRef.[17]tosustainnodecoherence. Asdiscussed
Toinvestigatethedynamicsofmatter-gravitycoupling
in Sec. II, with our present generalized formulation, the
andthe resultingradiation,decoherenceanddissipation,
validity is indeed extended to fermionic matter as well.
wewillfromnowonemploytheinteractionpicturewhere
Recently, the buildup of decoherence by massless parti-
theinteractionHamiltonianHI,consistingofself(HI(sys)) cles in anoptical-type cavity basedonEq.(30) has been
andenvironmental(H(env))gravitycontributions,gener- briefly discussed in Ref. [18].
I
ates the time evolution of quantum states. We consider Here we investigate new nontrivial dynamical conse-
the fluctuating spacetime to resemble an infinite reser- quences of this master equation in a more general phys-
voirinwhichenvironmentalgravitonswithfrequenciesω ical context, covering in particular radiation through
are maintained in an equilibrium Gaussian state with a quantum decoherence and dissipation for particles in
distribution function N(ω) described by a gravitational confined states as opposed to free particles studied in
densitymatrixρ . ForthermalequilibriumN(ω)isgiven Ref. [17]. Kinematically, the finite spatial extension of
G
bythePlanckdistributionfunctionandforthezero-point such a system permits the definitions of outgoing and
spacetime fluctuations N(ω) vanishes. ingoing gravitational waves. Dynamically, the coupling
Intermsofthetotaldensitymatrixρ (t)ofthematter between these waves and the time evolution of the sys-
T
systemandthegravitationalenvironment,thetotaltime tem results in their emissions (or absorptions) through
evolution is determined by the Liouville-von Neumann Eq. (16) in general and Eq. (21) for nonrelativistic sys-
equation tems. On quantization, these equations (16) and (21)
become operator equations.
i
ρ˙ =− [H ,ρ ]. (27) The averagegravitationalwave energy density expres-
T ~ I T
sion(23)thenacquiresquantummeaningbyinterpreting
The density matrix describing the statisticalstate of the hTijT theretobeoperatorsandaveragingtobeoverquan-
matter system is reduced from the total system by av- tum ensembles so that given a variable v we have
eraging over the ensembles of the gravitational reservoir
hvi=Tr(vρ) (32)
through the partial trace
using the matter system density matrix ρ [19]. The
ρ =Tr (ρ ). (28)
M G T quantum radiation formula also takes the same form as
Eq. (24) through quantized Eq. (21). However, factor
Foramattersysteminitiallyuntangledwiththegravi-
ordering requires some care here as the energy density
tationalenvironmentatt=0,whenthe totalstatetakes
relatedtermh˙TTh˙TT in Eq.(23) is normal-ordered. Ac-
the factored form ij ij
cordingly, when the reduced quadrupole moment opera-
ρ (0)=ρ (0)⊗ρ (29) torgivenbyquantizedEq.(22)is expandedinfrequency
T M G
5
modes diation, which we will refer to as “gravitational super-
radiance” that mirrors its original electromagnetic de-
q (t)=a (ω)e−iωt+a† (ω)eiωt (33)
ij ij ij scription [20]. As noted in Sec. IV and by analogy with
standard treatments in quantum optical systems [19],
forsomeoperatorsa (ω)withpositivefrequenciesω,the
ij
we assume the radiation process to be primarily due to
normal-like ordering of these operators
spacetime fluctuations using H(env) by neglecting radia-
aija†kl →a†klaij (34) tion reaction from self-gravity uIsing H(env). As a result,
I
quantumdissipationaloneisresponsiblefortheradiative
should be implemented for consistency.
loss of energy, which we verify explicitly for one particle
Factor ordering for the interaction Hamiltonian H
I
excitedinone dimension. To considerthe nonrelativistic
given by Eq. (17) bears some fundamental significance.
dynamics of the scalar field representing nearly Newto-
For electromagnetic radiative problems,it is known that
nianparticlesweassumethepotentialenergytobemuch
different factor ordering for the analogous interaction
less than the mass energy so that ν ≪1.
Hamiltonian leads to physically distinct mixes and sepa-
In the presence of weak gravity, we have η →η +
rationsofeffectsfromvacuumfluctuationsandradiation µν µν
h with the proper coordinates
reaction[27–31]. Here, the gravitationalcoupling is con- µν
structed from the quantized general action (1) assumed 1
to be Hermitian for any metric perturbation operator xi →xi+ 2h(ijenv)xj +O(h(jskys)xl) (37)
h . Itfollowsthat, the interactionHamiltonianH sep-
µν I
by using (15), with related considerations discussed in
arated from H with arbitrary h factor is necessarily
M µν
Ref. [25]. The resulting fluctuating potential in the TT
Hermitian. Now, from Eq. (17), the interaction Hamil-
tonianH is the sum ofthe environmentalpartH(sys) in gauge for free gravitational waves is given by
I I
aEnqd. (t1h9e),swyshtiecmh ipsaHrter(m18it)i,anwahsichh(ijeinsv)noatndreτaidjilcyomHmerumtei-, ν(xi)→ν(xi)+ 21h(ijenv)xiν,j +O(h(sys)ν). (38)
tian as h(sys) is related to time delayed or advanced Tµν ThesecondandthirdtermsinEq.(38)abovecontribute
µν
through Eq. (16) and so may not commute with Tµν. respectively to H(env) in Eq. (19) and H(sys) in Eq.(18).
Nonetheless, to achieve the Hermiticity of H(sys) and The appearance of gravitational wave induced potential
I fluctuations have also been discussed in Refs. [32, 33].
hence of H , with the correct classical limit, the factor
I
However,iftheconfinementofparticlesislimitedbyfree
ordering for Eq. (17) can be resolved symmetrically as
masses then the corresponding boundaries fluctuate in
follows
thepropercoordinatesinsteadoftheTTcoordinates[18].
1 1
H(sys) =− h(sys)Tµν − Tµνh(sys). (35) The system part of Eq. (36) yields the unperturbed
I 4 µν 4 µν quantum field equation
SimilarlysymmetrizedinteractionHamiltonian[28,29] φ¨=∇2φ−(1+2ν)µ2φ (39)
has been applied in resolving the aforementioned factor
ordering ambiguity in a wide range of problems involv- having solutions of the form
ing electromagnetic fluctuations and radiation reaction.
φ=ψ (r)e−iωnt+H.c. (40)
A recent related discussion and review can be found in n
Ref. [31]. with some orthogonal complex functions ψ (r). Hence
n
Eq. (39) reduces formally to the time-independent
Schr¨odinger equation
V. COLLECTIVE RADIATION BY CONFINED
IDENTICAL PARTICLES ~2
− ∇2ψ +Vψ =E ψ (41)
n n n n
2m
Thetheoreticalframeworkestablishedaboveisapplied where V =mν(r) and
in this section, as an illustrative example, to the quan-
tumgravitationaldecoherenceandradiationofareal,i.e. E = 1 (~2ω2 −m2) (42)
neutral,scalarfieldφwithmassmandtheassociatedin- n 2m n
verse reduced Compton wavelength µ=m/~, subject to
represent the potential and eigen energies respectively.
an external potential ν(r) described by the Lagrangian
Asaconcretephysicalconfiguration,letusconsideran
density
isotropic harmonic potential with frequency ω:
1 1
L=− gαβφ,αφ,β − +ν µ2φ2. (36) V = 1mω2r2. (43)
2 2
(cid:16) (cid:17) 2
We focus on the newly formulated spontaneous emis-
In this case, Eq. (40) becomes
sion of gravitons by nonrelativistic particles through en-
vironmental decoherence at zero temperature and high- ~
light previously undiscovered collective gravitational ra- φ=r2ωn ane−iωnt+a†neiωnt ψn(r) (44)
(cid:0) (cid:1)
6
using the multiple indices n = (n ,n ,n ) for the leading radiative mechanisms by adopting the ro-
1 2 3
n1,n2,n3 =0,1,2..., and functions tatingwaveapproximation,neglectingself-gravityH(sys)
I
(env)
ψn(r)=ψn1(x)ψn2(y)ψn3(z) (45) asunbdstLitaumtibngatnhdeSretasurkltinshgifEtqH. (L5S0) bHacakmiinlttoonEiaqn.s(,3w0)h.en
with the harmonic oscillator wavefunctions ψ (x) and Finally, to focus on the emission of gravitons, we sup-
n
presstheirabsorptionbysettingN(ω)=0,henceretain-
ωn =µ+n1ω1+n2ω2+n3ω3 (46) ing merely zero-point fluctuations in the gravitational
environment. The above considerations lead us to the
arising from the nonrelativistic limit of Eq. (42), where
gravitationalquantum optical master equation
the corresponding ladder operators an and a†n are anni-
hilation and creation operators respectively. ρ˙ = Γ 3δ δ −δ δ A ρA† − 1{A† A ,ρ}
Intermsofthe TT projectorP [16],the TTpartof 2 ik jl ij kl ij kl 2 ij kl
ijkl (cid:0) (cid:1)(cid:0) (cid:1)
the stress-energy tensor follows from Eqs. (2), (36) and (52)
(38) to be
of the Lindblad form, with the transitionrate coefficient
τij(r,t)=Pijkl φ,kφ,l−µ2ω2xkxlφ2 (47) 32G~ω3
Γ= (53)
(cid:0) (cid:1) 15
where the second contribution arises from the second
term in (38). From this, by normal-ordering and ne- and corresponding Lindblad operators
glectingparticlenon-conservationtermsrelevantonlyfor
higher energy scales, we then obtain Aij =Xn qni(nj −δij)a†n−nˆi−nˆjan (54)
τij(k,t)=Fij(n,n′,k)a†n′ane−i(ωn−ωn′)t (48) where nˆ1 = (1,0,0),nˆ2 = (0,1,0),nˆ3 = (0,0,1). It is
important to note that although time t in the original
where
non-Markovian master equation (30) starts with an ini-
~ tiallyfactoredstate(29)untangledwiththeenvironment,
F (n′,n,k)= P (k) d3x
ij µ ijkl Z the Markov assumption used in arriving at Eq. (52) has
effectively pushed that initial time back to the infinite
× ψn′,kψn,l−µ2ω2xkxlψn′ψn (49) past whose memory is lost [19], with time t now reset
(cid:0) (cid:1) to start from any new initial condition for the reduced
using nonrelativistic approximation with n ω ≪ µ as
max matter state ρ=ρ (t).
the kineticenergyismuchlessthantherestmassenergy M
Under the Markovianevolution using Eq. (52) at zero
and related long transmitted gravitational wave length
temperature,anexcitedstateρdecoheresanddecaysto-
condition compared to the spatial extension of occupied
wards the ground state. In the process, gravitons are
harmonic modes.
spontaneous emitted that carry the same amount of en-
To derive the gravitational analogue of the quantum
ergy as being reduced from the matter system. For ex-
optical master equation for the particle system from
ample, let us take an arbitrary one-particle state ρ with
tMhaerkgoevnearpaplrmoxaismteartioenqu[a1t9i]onas(f3o0l)lo,wws.e Fcairrsrty, wouetsuthbe- mveacttroirxfeolremtheentosccρunp,na′ti=onhonf|ρa|nha′irmwoitnhic|nm′iodaesnthebystoantee
stitute Eq. (48) into an integral in Eq. (30) to get
particle. Then we obtain from Eqs. (32) and (52) the
τij(k,t′)e−ik(t−t′) =Fij(n,n′,k)a†n′an dissipation power
dhH(sys)i
×e−i(ωn−ωn′)t tdse−i(k−ωn+ωn′)s. (50) − dt =~ωΓ 2ni(ni−1)ρn,n
Z Xn (cid:8)Xi
0
The nonlocality of this expressionin time represents the + 3ninjρn,n− ninj(ni−1)(nj −1)
non-Markov memory effect, which tends to fade away Xi6=j (cid:2) q
under environmentaldissipation. We “forget”this mem-
ory by taking the limit tds → ∞ds, as it does not ×ρn−2nˆi,n−2nˆj . (55)
0 0 (cid:3)(cid:9)
affectpost-transientdynaRmics,andRapply the Sokhotski- Thisexpressionindeedagreeswiththequadrupoleradia-
Plemelj theorem tionformulaEq.(24)appliedtothepresentconfiguration
and quantized with consistent factor ordering described
∞dse−iǫs =πδ(ǫ)−iP1 (51) inEq.(34)wheretheroleofaij isplayedbytheLindblad
Z ǫ operators A here.
0 ij
ByvirtueofitsinherentLindbladstructure,themaster
to Eq. (50), where P denotes the Cauchy principal value equation(52)is capableofgeneratingnew nonlinearcol-
that gives rise to a non-dissipative Hamiltonian H(env) lective quantum gravity phenomena transferred and in-
LS
fortheenvironmentallyinducedLambandStarkshiftsof spiredfrommoreestablishedquantumoptics areasshar-
energy. Byanalogywithquantumoptics[19],wecapture ing similar dynamical structures.
7
Onesuchnoveleffectisthecollectivelyamplifiedspon- VI. CONCLUSION
taneous emission of gravitons by a matter system in a
highly coherentstate,akintoDicke’s superradiance[20]. Motivated by the need for a better understanding of
To illustrate this, let us consider the present harmonic the fundamental process for quantum matter to deco-
potential containing many particles excited in one direc- hereanddissipatethroughspontaneousemissionandex-
tion,sayalongthex-axis,withamodaloccupationstate changeofgravitonswiththeubiquitousfluctuatinggravi-
vector denoted by |Ni = |{Nn}i = |N0,N1,···i, where tationalenvironment,wehaveextendedarecentlyestab-
n =n1 =0,1,2... labels the harmonic mode in this di- lished theory of quantum gravitationaldecoherence [16],
rection. It follows that the master equation (52) has the nowcompletewiththedynamicaloriginandconsequence
following matrix elements of gravitons mediating spacetime at large and matter,
both bosons and fermions, of interest.
For physically common states subject to a potential,
hN|ρ˙|N′i= Γ An,n′(N,N′)hNn′+|ρ|N′n+i wehaveexplicitlydemonstratedthattheabstractmaster
2n equation describing the general non-Markovian gravita-
−Bn,n′(N)hNn−,n′+|ρ|N′i +(N ↔N′)∗ (56) tiniodneaelddbeecorheedruecnecde,offremeaftrtoemr fUorVm-uculattoeffd,itnoRaemf.o[1re6]ccoann-
o
crete Lindblad form, structurally identical to the fam-
ily of quantum optical master equations widely applied
in terms of nonnegative coefficients in the quantum optics problems. This enables investi-
gations of the theory and phenomenology of quantum
gravity to benefit from a wealth of novel characteris-
An,n′(N,N′)= Nn′Nn′(Nn′+2+1)(Nn′+2+1) tics and solution strategies in the field of quantum op-
(cid:2) tics [19, 20, 34–37]. One such possibility in terms of the
×(n+1)(n′+1)(n+2)(n′+2) 1/2 (57) newly identified superradiance of gravitational waves by
(cid:3) asystemofcoherenceparticleshasbeentheoreticallyde-
Bn,n′(N)= Nn+2Nnn′−(Nn+1)(Nnn′−+2+1) scribed and numerically illustrated in Sec. V.
×(n+(cid:2)1)(n′+1)(n+2)(n′+2) 1/2 (58) Our general framework may serve to clarify various
conceptual issues encountered in the phenomenologi-
(cid:3)
cal approach to quantum gravity [38–44], with first-
principles insights, and to guide further analytical tools,
where Nnn′± = Nn′ ∓δn,n′ ±δn+2,n′. In this case, even mathematical techniques, and modelling methodologies
and odd harmonic modes are disjointly coupled within for possible detections of quantum gravity effects in the
their own parities because of the quadrupole nature of laboratory [45] and observatory [46] on the ground or in
the gravitationalwaves and symmetry of the potential. space [4]. In the context of the cosmological stochastic
gravitationalwaves,since the universe is consideredspa-
Based on the master equation with components (56),
tially flat with a low entropy on exit from inflation [47],
we perform numerical simulations in nondimensional
our theory may describe short-time graviton radiation
time t → Γt initially excited and subsequently relaxed
andreceptionbya distributionofcoherentstateshaving
in the x-direction, with harmonic modes n = n =
1
potentially unexpected but important collective proper-
0,2,4,6, n = n = 0 and a total particle number
2 3
ties including quantumnonlinearity,nonlocality,anden-
N = 1,2,3,4,5 shown in Figs. 1–3. The collective be-
tanglement [17, 18].
haviour of the “superradiant” spontaneous emission of
Another future objective would be to go beyond the
gravitons, due to the quadratic dependence of the par-
perturbative formulation so as to accommodate larger
ticle occupations (of bosonic origin) as well as modal
spacetime fluctuations and curved background or none.
numbers (of quadrupole origin) in Eqs. (57) and (58),
Extensioninthisdirectioncouldallowe.g. thequantum-
is particularly evident in Fig. 1, where the initial state
to-classicaltransitionintheearlyuniverseandnearblack
may be described as maximally coherent. Milder am-
holes with graviton productions to be more accurately
plification of emission power with particle numbers are
analyzed. Anadditionalrationaleforthis finalremarkis
also seen in Figs. 2 and 3, where the initial states may
thatthedevelopmentofopenquantumgravitationalsys-
be described as maximally mixed and maximally en-
temstowardsbackgroundindependence[5–8]mighteven
tangled respectively. The quantum states are enumer-
helpnavigatethesearchforanultimatefullquantumthe-
ated with |pi for p = 1,2,...p with ascending eigen
max
oryof gravitywith compatible andaccessible lowenergy
energies and then the particle number occupations of
effects like gravitationaldecoherence and radiance.
higher harmonic modes. Thus, with N = 5 there are
p = 56 even-mode occupation states |N ,N ...,N i
max 0 1 6
with N = N = N = 0, starting from the ground
1 3 5 ACKNOWLEDGMENTS
state |1i=|5,0,0,0,0,0,0i, then the first excited state
|2i=|4,0,1,0,0,0,0i through |28i=|2,0,1,0,0,0,2i to
This work was supported by the Carnegie Trust for
the highest state |56i=|0,0,0,0,0,0,5i.
the Universities of Scotland (T.O.) and by the EPSRC
GG-Top Project and the Cruickshank Trust (C.W.).
8
[1] B.P.Abbottetal,(LIGOScientificandVirgoCollabora- [22] R.W.TuckerandC.H.-T.Wang,BlackholeswithWeyl
tions),ObservationofGravitationalWavesfromaBinary charge and non-Riemannian waves, Classical Quantum
BlackHoleMerger,Phys.Rev.Lett.116,061102(2016). Gravity 12, 2587 (1995).
[2] B. F. Schutz, Gravitational wave astronomy, Classical [23] C.W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation
QuantumGravity 16, A131 (1999). (Freeman, New York,1973).
[3] A. Sesana, Prospects for Multiband Gravitational-Wave [24] R.Arnowitt,R.DeserandC.W.Misner,inGravitation:
Astronomy after GW150914, Phys. Rev. Lett. 116, AnIntroductiontoCurrentResearch,editedbyL.Witten
231102 (2016). (Wiley, New York,1962)
[4] G.Amelino-Cameliaetal,GAUGE:TheGrAndUnifica- [25] M. Maggiore, Gravitational Waves, Vol. 1: Theory and
tionGravityExplorer,ExperimentalAstronomy23,549 Experiments (Oxford University Press, Oxford, 2008)
(2009), and references therein. [26] P.A.M.Dirac,LecturesonQuantumMechanics(Yeshiva
[5] A. Ashtekar, C. Rovelli, and L. Smolin, Gravitons and University,New York,1964)
loops, Phys.Rev.D 44, 1740 (1991). [27] J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, Radia-
[6] C. H.-T. Wang, Conformal geometrodynamics: True de- tionReactionandRadiativeFrequencyShifts,Phys.Rev.
grees of freedom in a truly canonical structure, Phys. Lett. 30, 456 (1973)
Rev.D 71, 124026 (2005). [28] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji,
[7] C. H.-T. Wang, Unambiguous spin-gauge formulation of Vacuum fluctuations and radiation reaction: identifica-
canonical general relativity with conformorphism invari- tion of tlieir respective contributions, J. Physique 43,
ance, Phys.Rev. D 72, 087501 (2005). 1617 (1982)
[8] C. H.-T. Wang, New “phase” of quantum gravity, Phil. [29] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji,
Trans. R. Soc. A 364, 3375 (2006). Dynamicsofasmallsystemcoupledtoareservoir: reser-
[9] C. Kiefer, D. Polarski, and A. A. Starobinsky, Entropy voir fluctuations and self-reaction, J. Physique 45, 637
of gravitons produced in the early universe, Phys. Rev. (1984)
D 62, 043518 (2000). [30] J. Gea-Banacloche, M. 0. Scully, and M. S. Zubairyz,
[10] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, Vacuum Fluctuations and Spontaneous Emission in
and I.-O. Stamatescu, Decoherence and the appearance Quantum Optics, Physica Scripta T21,81 (1988)
of a classical world in quantum theory (Springer, Berlin, [31] G. Menezes and N. F. Svaiter, Vacuum fluctuations and
2003). radiation reaction in radiative processes of entangled
[11] M. Schlosshauer, Decoherence and the quantum-to- states, Phys. Rev. A 92, 062131 (2015), and references
classical transition (Springer,Berlin, 2008). therein.
[12] C. Kiefer, Emergence of a classical Universe from quan- [32] G. Schafer and H. Dehnen, On the gravitational radia-
tum gravity and cosmology, Phil. Trans. R. Soc. A 370, tion formula, J. Phys.A 13, 2703 (1980), and references
4566 (2012). therein.
[13] E. A. Lim, Quantum information of cosmological corre- [33] G. Schafer, Gravitational radiation resistance, radiation
lations, Phys.Rev. D 91, 083522 (2015). dampingandfieldfluctuations,J.Phys.A14,677(1981).
[14] B. P. Abbott et al, (LIGO Scientific and Virgo Collab- [34] A. Peres, Separability Criterion for Density Matrices,
orations), GW150914: Implications for the Stochastic Phys. Rev.Lett. 77, 1413 (1996)
Gravitational-WaveBackgroundfromBinaryBlackHole, [35] M. Horodecki, P. Horodecki, and R. Horodecki, Separa-
Phys.Rev.Lett. 116, 131102 (2016) bility of Mixed States: Necessary and Sufficient Condi-
[15] LIGO Scientific and Virgo Collaborations, An upper tions, Phys.Lett. A 223, 1 (1996).
limit on thestochastic gravitational-wave backgroundof [36] R.HanburyBrownandR.Q.Twiss,Correlationbetween
cosmological origin, Nature(London) 460, 990 (2009) photonsintwocoherentbeamsoflight,Nature(London)
[16] T.OnigaandC.H.-T.Wang,Quantumgravitationalde- 177, 27 (1956).
coherence of light and matter, Phys. Rev. D 93, 044027 [37] H. J. Metcalf, Laser Cooling and Trapping (Springer,
(2016). New York,2013), and references therein.
[17] T. Oniga and C. H.-T. Wang, Spacetime foam induced [38] G. Amelino-Camelia, Gravity-wave interferometers as
collective bundling of intense fields, Phys. Rev. D 94, probes of a low-energy effective quantum gravity, Phys.
061501(R) (2016). Rev. D 62, 024015 (2000).
[18] T. Oniga and C. H.-T. Wang, Quantum dynamics [39] S. Schiller, C. Lammerzahl, H. Muller, C. Braxmaier, S.
of bound states under spacetime fluctuations, IARD Herrmann, and A. Peters, Experimental limits for low-
2016 Proceedings, J. Phys. Conf. Ser. (Accepted), frequency space-time fluctuations from ultrastable opti-
arXiv:1612.01097. cal resonators, Phys. Rev.D 69, 027504 (2004).
[19] H.-P. Breuer and F. Petruccione, The Theory of Open [40] C.H.-T.Wang,R.BinghamandJ.T.Mendonca,Quan-
Quantum Systems (Oxford University Press, New York, tumgravitationaldecoherenceofmatterwaves,Classical
2002). Quantum Gravity 23, L59 (2006).
[20] R. H. Dicke, Coherence in Spontaneous Radiation Pro- [41] B. Lamine, R. Herve, A. Lambrecht and S. Reynaud,
cesses, Phys.Rev. 93, 99 (1954). Ultimate Decoherence Border for Matter-Wave Interfer-
[21] I. M. Benn and R. W. Tucker, An Introduction to ometry, Phys. Rev.Lett.96, 050405 (2006).
Spinors and Geometry with Applications in Physics [42] C. Anastopoulos and B. L. Hu, A master equation for
(Adam Hilger, Bristol, 1987). gravitationaldecoherence: probingthetexturesofspace-
time, Classical Quantum Gravity 30, 165007 (2013).
9
[43] G. Amelino-Camelia, Quantum-Spacetime Phenomenol- [45] C. Pfister et al, A universal test for gravitational deco-
ogy, Living Rev. Relativ. 16, 5 (2013), and references herence, Nat. Comms. 7 13022 (2016).
therein. [46] V. Vasileiou, J. Granot, T. Piran, and G. Amelino-
[44] V. A. De Lorenci and L. H. Ford, Decoherence induced Camelia, A Planck-scale limit on spacetime fuzziness,
by long wavelength gravitons, Phys. Rev. D 91, 044038 Nat. Phys. 11, 344 (2015).
(2015). [47] C. H.-T. Wang et al, A consistent scalar-tensor cosmol-
ogyforinflation,darkenergyandtheHubbleparameter,
Phys. Lett. A 380, 3761 (2016), and references therein.
10
(1,56) (56,56) (1,56) (56,56) (1,56) (56,56)
0.47 (a) 0.14 (b) 0.16 (c) (d) ––––– N = 5
0.24 t = 0.005 0.07 t = 0.05 0.08 t = 0.5 ––––– N = 4
––––– N = 3
0 0 0 ––––– N = 2
––––– N = 1

(1,1) (56,1) (1,1) (56,1) (1,1) (56,1) t
FIG. 1. Plots (a)–(c) show the simulation of the symmetric modulus of the density matrix |ρp,p′|(t) for p,p′ = 1,2,...56
consisting of even harmonic modes for the quantum transitions through the superradiance of gravitational waves. Five scalar
bosons in a harmonic trap are initially in the same highest energy state at the top right corner of the density matrix, where
all 5 particles occupy the harmonic mode n=6 as a maximally coherent state. While releasing a short burst of gravitational
wave, they spontaneously decay towards the ground state in the bottom left corner, where all 5 particles occupy the n = 0
mode. Plot(d)showstheaverageradiationpowerperparticleasafunctionoftimeforasimilarinitialstateasinplots(a)–(c)
but with different particle numbers.
(1,56) (56,56) (1,56) (56,56) (1,56) (56,56)
0.026 (a) 0.097 (b) 0.18 (c) (d) ––––– N = 5
0.013 t = 0.005 0.049 t = 0.05 0.09 t = 0.5 ––––– N = 4
––––– N = 3
0 0 0 ––––– N = 2
––––– N = 1

(1,1) (56,1) (1,1) (56,1) (1,1) (56,1) t
FIG.2. Plots(a)–(c)showthesimulation of|ρp,p′|(t). Here5scalarbosonsinaharmonictrapareinitially equallydistributed
along the diagonal of the density matrix for harmonic modes n = 0,2,4,6 as a maximally mixed state. While releasing a
continuously decreasing gravitational wave, they spontaneously decay towards the ground state in the bottom left corner,
whereall 5 particles occupythen=0mode. Plot (d) shows theaverage radiation power perparticlefor a similar initial state
with different particle numbers.
(1,56) (56,56) (1,56) (56,56) (1,56) (56,56)
0.048 ( a ) 0.19 (b) 0.48 (c) (d) ––––– N = 5
0.024 t = 0.005 0.097 t = 0.05 0.24 t = 0.5 ––––– N = 4
––––– N = 3
0 0 0 ––––– N = 2
––––– N = 1

(1,1) (56,1) (1,1) (56,1) (1,1) (56,1) t
FIG. 3. Plots (a)–(c) show the simulation of |ρp,p′|(t). Here 5 scalar bosons in a harmonic trap are initially equally and
fully distributed in the density matrix for harmonic modes n = 0,2,4,6 as a maximally entangled state. While releasing a
continuouslydecreasinggravitationalwave,theyspontaneouslydecaytowardsthegroundstateinthebottomleftcorner,where
all 5 particles occupy the n=0 mode. Plot (d) shows the average radiation power per particle for a similar initial state with
different particle numbers.