Detecting Entanglement Gravity: Distinguishing Energy-Sourced and Information-Sourced Gravity using High-Q Superconducting Cavities

De ec ing En anglemen G a i y: Dis inguishing
Ene gy-Sou ced and In o ma ion-Sou ced G a i y using High-Q
Supe conduc ing Ca i ies
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
In he s anda d amewo k o Gene al Rela i i y, he g a i a ional eld is comple ely de-
e mined by he ene gy-momen um enso
Tµν
; in he weak-eld limi , as long as he ene gy
densi y and p essu e dis ibu ion emain unchanged, g a i a ional eec s a e independen
o he en anglemen s uc u e o quan um s a es. In con as , a se ies o wo ks based on
en opy, en anglemen , and holog aphic p inciples sugges ha Eins ein's equa ions can be
de i ed om local en opy balance o acuum en anglemen s uc u e, hin ing ha space-
ime geome y may be undamen ally de e mined by quan um in o ma ion. Building on he
p e ious "conse a ion o op ical pa hconse a ion o in o ma ion olume" amewo k, his
pape in oduces a phenomenological "en anglemen g a i y" e m: local on Neumann en-
opy densi y o en anglemen en opy densi y
ρ
en
aec s he Shapi o- ype phase delay o
ligh by modi ying he op ical e ac i e index
n(x)
.
To expe imen ally dis inguish "ene gy g a i y" om "in o ma ion g a i y," we p opose
a class o able- op expe imen s: using high-Q supe conduc ing mic owa e ca i ies, while
keeping he expec ed elec omagne ic eld ene gy
⟨H⟩
inside he ca i y cons an , pe iodically
swi ch be ween low-en anglemen cohe en s a es and highly en angled squeezed s a es o
mul i-mode clus e s a es, and measu e he addi ional op ical pa h leng h and phase delay
ia a high-nesse Fab yPe o p obe beam passing nea he ca i y. This pape cons uc s an
op ical me ic model con aining an in o ma ion-g a i y coupling cons an
λ
en
, and p o es
ha unde weak-eld and pa axial app oxima ions, he phase die ence be ween modula ed
s a es sa ises
|∆Φ ≃λ
en
G∆s
en
,
whe e
∆s
en
is he change in en anglemen en opy su ace densi y inside he ca i y, and
G
is a geome ic ac o de e mined by ca i y geome y and p obe beam pa h. Fu he com-
bining con empo a y supe conduc ing ca i y Q- ac o s and lase in e e ome y echniques,
o ypical pa ame e s (
Q∼109

10
,
∆S
en
∼10

102
bi , nesse
F ∼ 105
), we es ima e he
expec ed signal magni ude and noise budge . Resul s show ha unde easonable geome ic
congu a ions and lock-in schemes, i
λ
en
is no lowe han a c i ical alue, he co esponding
phase modula ion ampli ude can each
∆Φ ∼10−9
ad scale, app oaching he phase sensi-
i i y limi o cu en lase in e e ome y and squeezed-ligh eadou chains. Con e sely,
i no die en ial phase signal modula ed a he en anglemen equency is obse ed, di ec
expe imen al uppe bounds can be imposed on
λ
en
, he eby p o iding he  s quan i a i e
cons ain on "whe he en anglemen i sel se es as a g a i a ional sou ce" expe imen ally.
Keywo ds:
En anglemen G a i y; Conse a ion o Op ical Pa h; Shapi o Delay; Supe con-
duc ing Mic owa e Ca i ies; Quan um Op ics; Table- op G a i y Expe imen s
1
1 In oduc ion & His o ical Con ex
Gene al Rela i i y desc ibes g a i y as ou -dimensional space ime geome y wi h Lo en zian
signa u e, whe e he me ic
gµν
is de e mined by Eins ein's equa ions
Gµν =8πG
c4Tµν.
The igh -hand side o his equa ion is uniquely gi en by he ene gy-momen um enso
Tµν
,
and in his amewo k, he g a i a ional sou ce is "ene gy-momen um" a he han independen
in o ma ion quan i ies o en anglemen s uc u es. In he weak-eld limi , he me ic can be
w i en as
gµν =ηµν +hµν
, whe e
hµν
is di ec ly ela ed o he New onian po en ial
Φ
. Classical
Shapi o ime delay expe imen s p ecisely e i y he ligh speed modica ion and op ical pa h
delay p oduced by his ene gy-sou ced g a i y.
On he o he hand, esea ch on "whe he g a i y and space ime o igina e om en opy and en-
anglemen " has de eloped con inuously o e he pas h ee decades. Jacobson de i ed Eins ein's
equa ions om he he modynamic equilib ium ela ion
δQ =T δS
a local Rindle ho izons, in-
e p e ing hem as an "equa ion o s a e." RyuTakayanagi and subsequen wo ks show ha
in AdS/CFT holog aphic co espondence, he en anglemen en opy o con o mal eld heo y
sub egions equals he a ea o co esponding minimal su aces in an i-de Si e space, e eal-
ing a p ecise quan i a i e connec ion be ween quan um en anglemen and geome ic a ea. Van
Raamsdonk u he p oposed ha mac oscopic connec ed space ime can be iewed as he "glue"
o unde lying quan um deg ees o eedom's en anglemen s uc u e, wi h educed en anglemen
leading o geome ic " ea ing" o "b eaking" o space ime egions. Ve linde's en opic g a i y
scheme mo e di ec ly iews g a i y as an en opic o ce ela ed o in o ma ion, en opy, and
holog aphic sc eens.
These wo ks join ly poin o a pic u e: space ime geome y and g a i a ional eld a e, in
some sense, eme gen desc ip ions o quan um in o ma ion and en anglemen s uc u es, and
Eins ein's equa ions can be unde s ood as mac oscopic app oxima ions o some "en anglemen
equilib ium condi ion." Howe e , mos es ablish indi ec connec ions be ween "geome y en-
opy/en anglemen " a he heo e ical le el; expe imen s di ec ly es ing "whe he en anglemen
se es as an independen g a i a ional sou ce" emain absen .
Recen ly, Bose e al. and Ma le oVed al e al. p oposed he amous BMV- ype able- op
expe imen al scheme: using g a i a ional in e ac ion o p oduce obse able spon aneous en an-
glemen be ween mass supe posi ion s a es, he eby p o ing "i wo quan um sys ems become
en angled h ough a eld in e ac ion, hen ha eld mus be a quan um en i y." This app oach
uses "en anglemen as wi ness" o in e he quan um na u e o he g a i a ional eld, bu does
no claim ha en anglemen i sel con ibu es addi ionally o g a i a ional s eng h. Subsequen
analyses u he discuss he easibili y and limi a ions o his scheme unde locali y, quan um
eld heo y modeling, and classical eld e o mula ions.
Die en ly, his pape conside s a mo e adical possibili y: beyond he ene gy-momen um
enso , he e exis s an "in o ma ion g a i y sou ce" ela ed o local en anglemen s uc u e o
in o ma ion p ocessing densi y. In he p e ious "conse a ion o in o ma ion olumeop ical
me ic" amewo k, we p oposed ha in he weak-eld limi , an eec i e e ac i e index eld
n(x)
can encode me ic pe u ba ions, whe e
n(x)
is de e mined by local in o ma ion olume
o in o ma ion p ocessing a e. I his amewo k is co ec , hen unde condi ions o cons an
ene gy densi y, pu ely changing local en anglemen s uc u e should in p inciple cause obse able
op ical pa h changes.
Expe imen ally, o decouple "ene gy" and "en anglemen " as wo ypes o g a i a ional
sou ces unde con olled condi ions equi es sa is ying he ollowing:
1. High-p ecision locking o ene gy expec a ion alue
⟨H⟩
can be achie ed; 2. En anglemen
en opy o on Neumann en opy can be modula ed signican ly while keeping
⟨H⟩
essen ially
2
cons an ; 3. Ex emely weak e ac i e index o op ical pa h changes can be measu ed nea he
a ge olume.
High-Q supe conduc ing mic owa e ca i ies and mode n quan um op ics echniques p ecisely
p o ide such a pla o m. In ecen yea s, wo k based on supe conduc ing RF ca i ies and h ee-
dimensional ca i y QED has achie ed mic owa e ca i y quali y ac o s
Q > 1010
a empe a u es
a ound 10 mK, main aining quan um s a e cohe ence o e millisecond o longe imescales, p o-
iding s able ca ie s o highly en angled op ical elds such as mul i-mode squeezed s a es and
clus e s a es. Meanwhile, in in e e ome y, elying on high-nesse op ical ca i ies and squeezed
ligh injec ion, phase sensi i i y app oaching he quan um limi has been achie ed in g a i a-
ional wa e de ec o s like LIGO.
In his echnical con ex , his pape p oposes and analyzes he ollowing ques ion: Nea a
high-Q supe conduc ing mic owa e ca i y wi h highly s able o al ene gy, i swi ching pe iodi-
cally be ween low-en anglemen and high-en anglemen s a es, will a p obe beam passing h ough
his egion exhibi addi ional Shapi o- ype phase delay, beyond s anda d GR p edic ions, mod-
ula ed a he en anglemen equency? Obse ing such a die en ial signal would suppo he
hypo hesis ha "in o ma ion/en anglemen se es as an independen g a i a ional sou ce"; ail-
u e o obse e i would impose expe imen al uppe limi s on ele an coupling cons an s.
The s uc u e below: Sec ion 2 cons uc s he op ical me ic model o in o ma ion g a i y
and gi es basic assump ions; Sec ion 3 p esen s main esul s, showing in heo em o m ha un-
de xed ene gy condi ions, en anglemen en opy modula ion necessa ily leads o op ical pa h
modula ion; Sec ion 4 combines supe conduc ing mic owa e ca i ies wi h high-nesse op ical
ca i ies o p o ide nume ical es ima es and pa ame e cons ain s; Sec ion 5 discusses expe i-
men al enginee ing implemen a ion and noise budge ; Sec ion 6 discusses isks and ela ionships
wi h exis ing wo k; nally concluding wi h p ospec s, and de ailed de i a ions o op ical me ics,
Shapi o phase, and e o budge a e p o ided in appendices.
2 Model & Assump ions
2.1 Op ical Me ic Model o In o ma ion G a i y
Unde weak-eld s a ic app oxima ion, he s anda d GR line elemen can be w i en as
ds2=−1 + 2Φ(x)
c2c2d 2+1−2Ψ(x)
c2dx2,
whe e
Φ≃Ψ
is he New onian po en ial, sa is ying
∇2Φ(x)=4πGρE(x),
wi h
ρE
being ene gy densi y. Fo pa axial p opaga ing ligh ays, an eec i e e ac i e index
can be in oduced:
n
GR
(x)≃1−Φ(x)
c2,
and Shapi o delay comes om inc eased op ical pa h due o
n
GR
>1
.
In he "conse a ion o op ical pa hconse a ion o in o ma ion olume" amewo k, we
in oduce an addi ional "in o ma ion g a i y po en ial"
Φ
en
, and assume he o al eec i e
po en ial
Φ
e
= ΦE+ Φ
en
,
whe e
ΦE
is he con en ional po en ial de e mined by
Tµν
, and
Φ
en
is di ec ly ela ed o local
en anglemen en opy densi y. Thus he o al e ac i e index
n(x)≃1−Φ
e
(x)
c2= 1 −ΦE(x)
c2−Φ
en
(x)
c2.
3
The phenomenological assump ion adop ed in his pape is: he e exis s a coupling compa ible
wi h he Bekens einHawking a ea-en opy ela ion, linking en anglemen en opy su ace densi y
s
en
(x)
o
Φ
en
:
Φ
en
(x) = −λ
en
c2ℓ2
P
L∗
s
en
(x),
whe e
ℓ
P
is he Planck leng h,
L∗
is he mac oscopic coa se-g aining scale ( aken as ca i y linea
size in his expe imen ),
λ
en
is a dimensionless coupling cons an , and
s
en
has uni s o bi /m
2
.
Thus, he e ac i e index co ec ion con ibu ed by in o ma ion g a i y is
δn
en
(x) = Φ
en
(x)
c2=−λ
en
ℓ2
P
L∗
s
en
(x).
This o m is inspi ed by black hole en opy
S=A/(4ℓ2
P
)
whe e "each
O(ℓ2
P
)
a ea elemen
co esponds o
O(1)
bi ," using
ℓ2
P
s
en
as dimensionless en anglemen "o e densi y," while
L∗
b ings su ace densi y o he app op ia e scale o po en ial. I s physical meaning can be un-
de s ood as: in a gi en mac oscopic olume, i bounda y en anglemen en opy su ace densi y
exceeds a baseline alue, ligh pa h mus be slowed (equi alen o local ime dila ion) o " ee
up compu a ional esou ces," he eby achie ing conse a ion o in o ma ion olume.
2.2 Ca i y Model and En anglemen En opy Densi y
Conside a supe conduc ing mic owa e ca i y wi h leng h
L
ca
and c oss-sec ional a ea
A
ca
,
suppo ing a se o disc e e modes
{ˆak}
, wi h o al Hamil onian
ˆ
H=X
k
ℏωkˆa†
kˆak+1
2.
In his pape , we only ocus on wo mac oscopically unable ca i y quan i ies:
1. Ene gy expec a ion alue
E0=⟨ˆ
H⟩;
2. En anglemen en opy su ace densi y unde a na u al pa i ion (e.g., le - igh pla es,
die en pola iza ion o equency modes)
s
en
=S
en
A
ca
, S
en
=− ˆρAlog ˆρA,
whe e
ˆρA
is he educed densi y ope a o ob ained by acing o e one side's deg ees o eedom.
Expe imen ally, we swi ch be ween wo mac oscopically dis inguishable s a es:

S a e A (baseline s a e): mul i-mode cohe en s a e
|A⟩=O
k|αk⟩,
whose on Neumann en opy and en anglemen en opy a e app oxima ely ze o,
S(A)
en
≃0
;

S a e B ( es s a e): mul i-mode en angled s a e, such as mul iple pai s o wo-mode
squeezed acuum s a es o clus e s a es
|B⟩=
M
O
j=1 |
TMSV
( j)⟩,
by choosing squeezing pa ame e s
j
and mode numbe
M
, adjus o al pho on numbe
⟨ˆ
N⟩=Pk⟨ˆa†
kˆak⟩
o ma ch s a e A, he eby ensu ing
⟨A|ˆ
H|A⟩=⟨B|ˆ
H|B⟩=E0.
4
In hese wo s a es, ene gy densi y
ρE=E0/(A
ca
L
ca
)
emains cons an , while en anglemen
en opy su ace densi y jumps om
s(A)
en
≈0
o
s(B)
en
≫0
. Dene
∆s
en
=s(B)
en
−s(A)
en
,∆S
en
=S(B)
en
−S(A)
en
.
Subs i u ing he abo e in o he in o ma ion g a i y e ac i e index co ec ion gi es he
e ac i e index die ence be ween s a es A and B:
∆n
en
=nB−nA=−λ
en
ℓ2
P
L∗
∆s
en
=−λ
en
ℓ2
P
L∗A
ca
∆S
en
.
In he expe imen al scena io below, we ake
L∗
as
L
ca
, and assume ca i y e ac i e index is
ans e sely uni o m, so
∆n
en
can be iewed as he a e age co ec ion h ough a na ow op ical
pa h passing h ough he ca i y cen e .
2.3 P obe Beam Pa h and Shapi o-Type Phase Delay
Conside a p obe beam wi h wa eleng h
λp
passing h ough he ca i y eec i e egion along an
app oxima ely s aigh op ical pa h, o ming pa o a high-nesse Fab yPe o op ical ca i y.
Le he geome ic leng h o single-pass h ough he ca i y be
L
pass
( o comple e a e sal h ough
ca i y, ake
L
pass
≃L
ca
; o g azing edge passage, ake smalle alue). Unde weak e ac ion
app oxima ion, single-pass addi ional phase delay
δϕ
single
=kpZ
pa h
∆n
en
(l) dl≃kp∆n
en
L
pass
,
whe e
kp= 2π/λp
.
I he p obe beam makes
N
ound ips in he F-P ca i y, he o al addi ional phase delay is
∆Φ = N δϕ
single
≃2πNL
pass
λp−λ
en
ℓ2
P
L
ca
A
ca
∆S
en
.
This gi es he basic o m o Shapi o- ype phase modula ion induced by in o ma ion g a i y
unde xed ene gy condi ions.
3 Main Resul s (Theo ems and Alignmen s)
To acili a e di ec co espondence wi h expe imen al design, his sec ion p esen s h ee co e
esul s based on he p eceding model, o ganized in heo em o m.
3.1 Theo em 1 (Phase Modula ion Induced by En anglemen En opy Mod-
ula ion unde Fixed Ene gy)
Assump ions:
1. When he elec omagne ic eld inside he ca i y swi ches be ween s a e A and s a e B, i
sa ises
⟨A|ˆ
H|A⟩=⟨B|ˆ
H|B⟩=E0,
i.e., he same ene gy expec a ion alue;
2. Ca i y geome y is xed,
L
ca
,
A
ca
unchanged;
3. In o ma ion g a i y is gi en by e ac i e index co ec ion
∆n
en
=−λ
en
ℓ2
P
L
ca
A
ca
∆S
en
;
5

4. P obe beam makes
N≃2F/π
ound ips in he F-P ca i y, whe e
F
is he nesse, wi h
e ac i e index co ec ion exis ing only in he egion o leng h
L
pass
along he op ical pa h.
Then:
The o al phase die ence induced by swi ching be ween s a es A and B is
∆Φ = −λ
en
G∆S
en
,
whe e he geome ic ac o is
G=2πNL
pass
λp
ℓ2
P
L
ca
A
ca
.
In o he wo ds, unde xed ene gy condi ions, all phase modula ion ela ed o in o ma ion
g a i y is linea ly p opo ional o en anglemen en opy change
∆S
en
, independen o ene gy
i sel .
3.2 Theo em 2 (Uppe Bound o GR Con ibu ion unde Ene gy Misma ch)
Assump ions:
1. In ac ual expe imen s, ene gy locking has esidual misma ch
δE =EB−EA
, sa is ying
|δE| ≪ E0
;
2. GR's ene gy-sou ced g a i y con ibu ion modies e ac i e index ia New onian po en ial
∆n
GR
≃ −∆ΦE
c2≃ −G δM
c2R,
whe e
δM =δE/c2
,
R
is ypical dis ance om p obe beam pa h o ca i y cen e ;
3. O he condi ions same as Theo em 1.
Then:
The uppe bound o addi ional phase die ence in oduced by GR is
|∆Φ
GR
|≲2πNL
pass
λp
G|δE|
c4R.
Fo ypical pa ame e s
E0∼10−11
J,
|δE|/E0≲10−9
,
R∼0.1
m, we ob ain
|∆Φ
GR
|≲10−15
ad
,
a below he in o ma ion g a i y a ge signal
∆Φ ∼10−9
ad scale. The e o e, as long as ene gy
locking eaches
10−9
ela i e p ecision, GR con ibu ion can be neglec ed in his expe imen al
scheme.
3.3 Theo em 3 (Expe imen al Cons ain on In o ma ion G a i y Coupling
Cons an )
Le he noise spec al densi y o phase modula ion o he measu emen sys em du ing in eg a ion
ime
T
be
S1/2
Φ
. Then analy ically, unde he condi ion o no obse ing a signican spec al
line a modula ion equency
mod
, he
1σ
uppe bound on
λ
en
is
|λ
en
|≲S1/2
Φ
G|∆S
en
|T−1/2.
Fo ypical pa ame e s
∆S
en
∼10

102,F ∼ 105, L
pass
∼L
ca
∼0.1
m
, A
ca
∼10−4
m
2,
and lase in e e ome y chain phase noise
S1/2
Φ∼10−10
ad
/√
Hz, wi h eec i e in eg a ion ime
T∼103
s,
|λ
en
|
can be cons ained wi hin a ni e ange. I a phase signal modula ed a
∆S
en
equency is obse ed, he eec i e alue o
λ
en
can be  ed om da a and compa ed wi h o he
expe imen al and as ophysical cons ain s.
6
4 P oo s
This sec ion de i es he abo e h ee heo ems. Since in o ma ion g a i y i sel is a new hypo h-
esis, he s a ing poin o de i a ion is he s anda d ela ion be ween op ical me ic and Shapi o
delay, and he phenomenological coupling in Sec ion 2.
4.1 De i a ion o Theo em 1: Linea Rela ion be ween Op ical Pa h and
En anglemen En opy
F om he deni ion in Sec ion 2.3, he single-pass phase o p obe ligh is
ϕ=kpZ
pa h
n(l) dl≃kpL
geom
+kpZ
pa h
δn(l) dl,
whe e
L
geom
is geome ic op ical pa h,
δn
is small de ia ion om g a i a ional equi alen e ac-
i e index co ec ion. When compa ing s a es A and B, geome ic op ical pa h and GR ene gy
sou ce componen a e he same (comple ely iden ical unde ideal xed ene gy assump ion), wi h
die ence only om in o ma ion g a i y co ec ion
∆n
en
. Thus single-pass phase die ence
δϕ
single
=kpZ
pa h
∆n
en
(l) dl.
Assuming e ac i e index in ca i y egion can be iewed as cons an
∆n
en
, pa h leng h is
L
pass
, hen
δϕ
single
≃kp∆n
en
L
pass
=2π
λp−λ
en
ℓ2
P
L
ca
A
ca
∆S
en
L
pass
.
Conside ing
N
ound ips in F-P ca i y, eec i e phase amplica ion is
∆Φ = N δϕ
single
=−λ
en
2πNL
pass
λp
ℓ2
P
L
ca
A
ca
∆S
en
=−λ
en
G∆S
en
,
consis en wi h he o m s a ed in Theo em 1.
4.2 De i a ion o Theo em 2: Uppe Bound o GR Phase om Ene gy Mis-
ma ch
Fo a local ene gy co ec ion
δE
, app oxima ely iewing i as a mass pe u ba ion
δM =δE/c2
,
i s New onian po en ial co ec ion a dis ance
R
is
∆ΦE≃ −GδM
R=−GδE
c2R.
F om e ac i e index-po en ial ela ion
n≃1−Φ/c2
, we ge
∆n
GR
=−∆ΦE
c2≃GδE
c4R.
Thus uppe bound o single-pass phase die ence is
δϕ
GR,single
≃kp∆n
GR
L
pass
≲2π
λp
G|δE|
c4RL
pass
,
To al phase die ence
|∆Φ
GR
|≲N|δϕ
GR,single
|≲2πNL
pass
λp
G|δE|
c4R.
Subs i u ing ypical nume ical alues gi es he o de o magni ude s a ed in Theo em 2.
Since
|δE|
can be locked o ex emely low le els h ough ac i e eedback and SQUID eadou ,
his e m's con ibu ion can be sa ely iewed as seconda y sys ema ic e o .
7
4.3 De i a ion o Theo em 3: S a is ical Cons ain on Coupling Cons an
Le he single-sided powe spec al densi y o phase in he de ec ion chain be
SΦ( )
, which can
be iewed as cons an
SΦ
nea modula ion equency
mod
. Wi hin in eg a ion ime
T
, s a is ical
e o o na ow-band spec al line  ing ollows
σΦ≃S1/2
ΦT−1/2.
I no signican spec al line is obse ed, he
1σ
uppe bound o ac ual phase modula ion
ampli ude
|∆Φ|
can be aken as
σΦ
. F om he linea ela ion in Theo em 1
|∆Φ|=|λ
en
||G∆S
en
|,
hus
|λ
en
|≲σΦ
|G∆S
en
|≃S1/2
Φ
G|∆S
en
|T−1/2.
This gi es he s a is ical uppe bound o m o Theo em 3.
5 Model Apply
This sec ion, unde he abo e heo e ical amewo k, combines ealis ic easible expe imen al
pa ame e s o es ima e signal magni ude and discuss po en ial cons ain s eng h on
λ
en
.
5.1 Typical Geome y and Ca i y Pa ame e s
Conside he ollowing ep esen a i e design:

Supe conduc ing mic owa e ca i y: leng h
L
ca
= 0.1
m, c oss-sec ional a ea
A
ca
=
π(1
cm
)2≃3.1×10−4
m
2
;

Ca i y quali y ac o :
Q∼109

10
, equency
ωc/2π∼10
GHz, co esponding single-pho on
ene gy
hν ∼6.6×10−24
J;

A e age pho on numbe
⟨ˆ
N⟩ ∼ 1012
, hen o al ene gy
E0∼6.6×10−12
J, ene gy densi y
ρE∼2×10−9
J
/
m
3
, con en ional GR g a i a ional eec ex emely weak;

P obe ligh : wa eleng h
λp= 1064
nm, compa ible wi h LIGO-class echnology;

P obe beam pa h: passing h ough ca i y cen e , single-pass eec i e leng h
L
pass
≃0.1
m;

F-P ca i y nesse:
F= 105
, hen numbe o ound ips
N≃2F
π≃6.4×104,
eec i e op ical pa h
L
e
=NL
pass
≃6.4×103
m.
Unde such geome y, he geome ic ac o
G=2πNL
pass
λp
ℓ2
P
L
ca
A
ca
.
Subs i u ing nume ical alues
ℓ2
P
≃2.6×10−70
m
2
,
λp≃10−6
m,
L
ca
=L
pass
= 0.1
m,
A
ca
≃3×10−4
m
2
,
2πNL
pass
λp∼2π×6.4×104×0.1
10−6∼4×1010,
8
ℓ2
P
L
ca
A
ca
∼2.6×10−70
0.1×3×10−4∼10−66,
hus
G ∼ 4×1010 ×10−66 ∼4×10−56
ad/bi
.
A his scale, e en wi h
∆S
en
∼102
, i
λ
en
∼1
, he esul ing phase modula ion
∆Φ ∼λ
en
G∆S
en
∼4×10−54
ad
,
a below any easible expe imen al sensi i i y. Thus i he coupling coecien di ec ly adop s
ℓ2
P
scale, expe imen s canno measu e i a labo a o y scales.
This eec s he huge hie a chical gap be ween Planck scale and labo a o y scale, consis en
wi h he gene al dicul y o obse ing adi ional quan um g a i y eec s in able- op expe i-
men s.
Howe e , in he eme gen pic u e o in o ma ion g a i y,
λ
en
need no necessa ily be o he
same o de as Planck scale. I unde lying QCA o o he disc e e on ology exhibi s amplied
eec i e in o ma ion g a i y coupling mac oscopically (e.g., due o collec i e eec s o many-
body en anglemen ), hen
λ
en
could be much la ge han 1, making
∆Φ
all wi hin measu able
ange.
To quan i y his, we can w i e
λ
en
as
λ
en
= 10Γ,
hen
∆Φ ∼4×10−54 ×10Γ×∆S
en
102
ad
.
I expec ing
∆Φ ∼10−9
ad, hen need
10−9∼4×10−54 ×10Γ⇒Γ∼45.
In o he wo ds, his expe imen unde he abo e pa ame e s can de ec o exclude whe he
λ
en
is a ound
1045
magni ude. This seems huge, bu since
λ
en
is a comple ely new eec i e
pa ame e , cu en ly lacking any expe imen al e idence cons ain , i s heo e ical na u al alue
is unclea .
5.2 Compa ison wi h Phase Sensi i i y
Taking LIGO and o he in e e ome ic de ices as e e ence, unde condi ions o in oducing
equency-dependen squeezed ligh , single- equency phase noise spec al densi y can each
S1/2
Φ∼10−10
ad
/√
Hz o e en lowe . Unde in eg a ion ime
T∼103

104
s, s a is ical e o
σΦ∼S1/2
ΦT−1/2∼10−11

12
ad
.
The e o e, once
∆Φ ≳10−10
ad, i can be de ec ed wi h
>5σ
signicance. Con e sely, i no
spec al line is obse ed below his h eshold, Theo em 3 gi es
|λ
en
|≲1044

45,
co esponding o di ec expe imen al uppe bound on in o ma ion g a i y coupling.
6 Enginee ing P oposals
This sec ion discusses enginee ing elemen s and op ional ou es needed o implemen he abo e
scheme.
9

Typical ca i y scale
R∼0.1
m.
New onian po en ial
|ΦE| ∼ GM
EM
R∼6.7×10−11 ×7.3×10−29
0.1∼5×10−39
J/kg
.
Rela i e o
c2∼9×1016
m
2/
s
2
, dimensionless po en ial

ΦE
c2∼5×10−56,
co esponding e ac i e index co ec ion
∆n
GR
∼10−56.
E en accumula ing
L
e
∼104
m op ical pa h in F-P ca i y, o al phase
∆Φ
GR
∼2πL
e
λp
∆n
GR
≲10−42
ad
,
comple ely unde ec able.
The e o e, any phase modula ion obse able in his expe imen , synch onized wi h s a e
swi ching equency, canno be explained by con en ional GR ene gy g a i y, and mus o igina e
om non-s anda d mechanisms (o expe imen al sys ema ic e o s).
D Example Cons uc ion o En anglemen En opy
To gi e conc e e cons uc ion example o
∆S
en
∼10

102
, conside ealizing
M
pai s o wo-mode
squeezed acuum s a es in ca i y
|
TMSV
( )⟩= exp h (ˆaˆ
b−ˆa†ˆ
b†)i|0,0⟩,
each pai 's educed en opy is
S
pai
( ) = cosh2 log(cosh2 )−sinh2 log(sinh2 ),
a e age pho on numbe
⟨ˆna+ ˆnb⟩= 2 sinh2 .
Taking
∼1
,
sinh2 ∼1.4
, each pai con ibu es pho on numbe
∼3
, co esponding en-
anglemen en opy
S
pai
∼2
bi scale (in app op ia e basis). By aking
M∼5

50
mode pai s,
can ealize
∆S
en
∼10

102
ange, while o al pho on numbe
∼3M
ma ches wi h same-scale
cohe en s a e, he eby signican ly changing en anglemen s uc u e while ene gy locking.
Mo e complex clus e s a es o g aph s a es can in oduce highe many-body en anglemen
deg ee unde same o al pho on numbe , bu since en anglemen en opy deni ion depends
on chosen deg ees o eedom pa i ion, i s specic con ibu ion o in o ma ion g a i y needs o
co espond wi h "na u al pa i ion" o unde lying QCA o eld heo y on ology. This pape only
uses wo-mode squeezed s a e as example, showing ha unde exis ing ci cui QED echnology,
∆S
en
∼10

102
ange is ully achie able.
16