Toward Testing Quantum Mechanics and Gravity in the Lab

Toward Testing Quantum Mechanics and Gravity in the Lab

Yanbei Chen

California Institute of Technology

Quantum Information: Quo Vadis? Vancouver, Nov 2019

Motivations

2

➤ Quantum Mechanics provides consistent predictions on the probability distribution of measurement results … but not any particular path of outcomes.

➤ Classical General Relativity generically leads to singularities inside black holes and at the beginning of the universe: Penrose-Hawking Singularity Theorems.

Testing QM and Gravity in the Lab …

3

➤ Collapse Models, which “objectively” collapse the quantum state of the universe, including “Gravity Decoherence”.

➤ Testing quantum nature of gravity. Can we formulate gravitational interaction classically?

➤ Alternative formulations of quantum gravity, e.g., the Correlated World Line (CWL) theory. [Partly motivated by gravity decoherence]

➤ Collaborators

➤ Bassam Helou (Caltech) ➤ Sabina Scully, Bram Slagmolen and David McClelland (ANU) ➤ Philip Stamp and Jordan Wilson-Gerow (UBC)

Collapse Models

4

One particular path is somehow chosen; wave function of the universe collapsed by an external agent

x

ψ(x)

x

ψ(x) localize randomly according to |ψ(x)|2

Proposed by Ghirardi, Rimini and Weber, studied extensively by Adler, Bassi, Diosi, et al.

Mathematical Description

5

Zt

0f (t0)x(t0)dt

0 (52)

x(t) = x(0) cos wmt +p(0)mwm

sin wmt +Z

t

0

sin wm(t � t0)

mwm

aa1(t0)dt

0 (53)

b1 = a1 , b2 = a2 + ax(t) . (54)

ZT

0[g1(t)b1(t) + g2(t)b2(t)] dt (55)

det Vadd

h2/4

=14

minW

✓Scl

SSQL

◆2+

e�2q

2

✓WF

Wq

◆2(56)

r(x1, x2; x01, x

02) ! r(x

01, x2; x1, x

02) (57)

x ! x, p ! �p (58)

x1 + x2 ! e�r , p1 � p2 ! e

�r , x1 � x2 ! e+r , p1 + p2 ! e

+r (59)

Vpt +12

K (60)

LCSLr = � lCSL

2p3/2r3CSLamu2

Zd

3s [m(s), [m(s), r]] (61)

m(s) =Z

µ(z)e�(s�z)2/(2r2CSL)d3z (62)

r =i

h[H, r]� l

2[L, [L, r]] (63)

r =i

h[H, r]� Â

j,k

ljk

2⇥Lj, [Lk, r]

⇤(64)

5

Zt

0f (t0)x(t0)dt

0 (52)

x(t) = x(0) cos wmt +p(0)mwm

sin wmt +Z

t

0

sin wm(t � t0)

mwm

aa1(t0)dt

0 (53)

b1 = a1 , b2 = a2 + ax(t) . (54)

ZT

0[g1(t)b1(t) + g2(t)b2(t)] dt (55)

det Vadd

h2/4

=14

minW

✓Scl

SSQL

◆2+

e�2q

2

✓WF

Wq

◆2(56)

r(x1, x2; x01, x

02) ! r(x

01, x2; x1, x

02) (57)

x ! x, p ! �p (58)

x1 + x2 ! e�r , p1 � p2 ! e

�r , x1 � x2 ! e+r , p1 + p2 ! e

+r (59)

Vpt +12

K (60)

LCSLr = � lCSL

2p3/2r3CSLamu2

Zd

3s [m(s), [m(s), r]] (61)

m(s) =Z

µ(z)e�(s�z)2/(2r2CSL)d3z (62)

r =i

h[H, r]� l

2[L, [L, r]] (63)

r =i

h[H, r]� Â

j,k

ljk

2⇥Lj, [Lk, r]

⇤(64)

Htot = H � Âj

f j(t)Lj (65)

5

Htot = H � Âj

f j(t)Lj (65)

h f j(t) fk(t0)i = ljkd(t � t

0) (66)

6

a set of random white forces acting on a set of variables will cause diffusion in linear systems

➤ Lindblad Master Equation: modeling the collapse process.

➤ Can be understood in two steps

➤ Can be constrained in weak force measurement experiments

➤ micro-cantilevers, gravitational-wave detectors, torsional pendulum experiments.

Continuous Stochastic Localization

6

matter distribution in our space-time being measured by external observers

that enter via an extra dimension

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)

Dynamically, this corresponds to adding the following termX

j

F j(t)L j (2)

into the Hamiltonian, with F j stochastic forces that have thefollowing form of correlation functions:

hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)

In the frequency domain, these forces have white cross spec-tra. This is consistent with the fact that Master Equationsmodel Markovian processes. In the single-sided conventionfor spectral density,

S f j fk = 2⇤ jk (4)

In other words, ⇤ jk is 1/2 the single-sided cross spectrum,and is equal to the double-sided cross spectrum. Is this true?Needs checking. Let us emphasize that even though we haveused the master equation, this is simply to describe the e↵ectof the additional forces due to the collapse, which are white.In the proposed experiments, the test masses will be be un-der colored force and sensing noise, in general. We will sim-ply extract the white spectrum of the collapse force from theLindblad operator, and then combine this spectrum with thespectra of other sources of noise.Joint formulation of Collapse Models.– It is possible to for-mulate both the CSL and DP models in terms of a continuousmeasurement of a series of mass-density-induced fields. Thisadds an Lindblad superoperator into the master equation:

L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)

Here the capital A, B, are indices that indicate the fields thatare measured, while ↵AB gives a correlation at which they aremeasured. Furthermore, in both models, �A(s) are determinedby the matter density operator µ(s). Note that the fields at allspatial locations are measured, in an uncorrelated way. Onecan incorporate a correlation structure into the spatial mea-surements, by altering the dependence of �A on µ. Therecould be a temporal correlation — that will in general intro-duce a modification to the system’s dynamics, and we shallnot consider that in this paper.

For the CSL model, � is a scalar field. We have

�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)

For the DP mode, � is a vector field, the gravitational accel-eration induced by the macroscopic object, given by

� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)

We note that for point particles, µ contains � functions, whichleads to divergence of � j(s). This has been treated by addinga cut-o↵ scale in Eq. (8). Both the CSL and the DP formulasneed double checking.Decoupling between CM and internal motions.– For a macro-scopic object, it is necessary to separate its internal motionand its CM motion, in a similar fashion as done by Ref. [1].Let us write the entire object’s Hilbert space H as H =

HCM ⌦ Hint. Let us first suppose the CM motion is decou-pled to the internal motion, so we have

⇢ = ⇢CM ⌦ ⇢int (9)

The fact that our Lindblad term has a translation symmetrydoes preserve this decoupling. Let us evaluate

trintL(⇢CM ⌦ ⇢int) (10)

Using translational invariance, let us note that if we have asingle scalar field, then we can write it down as

�(s) = �int(s � x) (11)

where �int(s) is the field evaluated at s if CM were at 0. Herex is the operator for the CM position. For simplicity, let ussimply consider a 1-D motion first. Let us also compute thegenerating function representation of the Lindblad operator,i.e.,

LJ(µ, ⌫) = trheiµx+i⌫ p

L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p

h�int(s � x), [�int(s � x), ⇢CM ⌦ ⇢int]

io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z

ds trh�int(s � x)�int(s � y)⇢int

i(14)

and we have G(x, y) = G(y, x) and G(x, y) = G(x � z, y � z).We can therefore write

LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)

h�int(s) � �int(s + ⌫)

i⇢inti

(16)

values of a field continuously monitored

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)


j

F j(t)L j (2)


hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)


S f j fk = 2⇤ jk (4)


L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)



�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)


� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)



⇢ = ⇢CM ⌦ ⇢int (9)




�(s) = �int(s � x) (11)



L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p


io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z


i(14)


LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)


i⇢inti

(16)

field generated by matter distribution

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)


j

F j(t)L j (2)


hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)


S f j fk = 2⇤ jk (4)


L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)



�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)


� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)



⇢ = ⇢CM ⌦ ⇢int (9)




�(s) = �int(s � x) (11)



L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p


io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z


i(14)


LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)


i⇢inti

(16)

Each particle sources Gaussian packet (scale rCSL) the total field m(s) gets measured independently at different locations, causing decoherence strength

characterized by λCSL

superpositions separated by less than rCSL does not undergo decoherence

those separated by larger distance will undergo decoherence.

Continuous Stochastic Localization

7

matter distribution in our space-time being measured by external observers

that enter via an extra dimension

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)


j

F j(t)L j (2)


hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)


S f j fk = 2⇤ jk (4)


L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)



�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)


� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)



⇢ = ⇢CM ⌦ ⇢int (9)




�(s) = �int(s � x) (11)



L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p


io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z


i(14)


LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)


i⇢inti

(16)

values of a field continuously monitored

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)


j

F j(t)L j (2)


hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)


S f j fk = 2⇤ jk (4)


L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)



�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)


� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)



⇢ = ⇢CM ⌦ ⇢int (9)




�(s) = �int(s � x) (11)



L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p


io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z


i(14)


LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)


i⇢inti

(16)

field generated by matter distribution

2

equation:

˙⇢ =i~

[H0, ⇢] �1

2~2

X

jk

⇤ jk[L j, [Lk, ⇢]] (1)


j

F j(t)L j (2)


hF j(t)Fk(t0)i = ⇤ jk�(t � t0) . (3)


S f j fk = 2⇤ jk (4)


L⇢ = �CAB

2~2

Zd3

s

h�A(s),

h�B(s), ⇢

ii(5)



�(s) =Zµ(z)e�(s�z)2/(2r2

CSL)d3z (6)

and

CCSL =�CSL

⇡3/2r3CSLamu2

(7)


� j(s) = �Z

Gµ(x)s � x

|s � x|3d3

x , C jkDP =

~

16⇡G� jk (8)



⇢ = ⇢CM ⌦ ⇢int (9)




�(s) = �int(s � x) (11)



L⇢i

= �C

2~2

Zd3

s trneiµx+i⌫ p


io

(12)

This gives

LJ(µ, ⌫) = �C

2~2

D⇥G(x + ⌫, x + ⌫) +G(x, x)

�G(x, x + ⌫) �G(x + ⌫, x)⇤eiµx+i⌫ p

E(13)

Here

G(x, y) =Z


i(14)


LJ = �C

2~2 F(⌫)J(⌫) (15)

where

F(⌫) = 2Z

ds trh�int(s)


i⇢inti

(16)

Φ(s)

time

space

Diosi-Penrose

8

Htot = H � Âj

f j(t)Lj (65)

h f j(t) fk(t0)i = ljkd(t � t

0) (66)

r = rint ⌦ rCM (67)

m(s)| {z }Hint⌦HCM

= m0(s)| {z }Hint

+Âj

∂m(s)∂rj| {z }Hint

rj|{z}HCM

(68)

LCSLrCM = �12 Â

jk

LCSLjk

[rj, [rk, rCM]] (69)

LCSLjk

=lCSL

p3/2r3CSLamu2

Zd

3s

*∂m(s)

∂sj

∂m(s)∂sk

+

int

(70)

LCSLrCM = �lCSL

2p3/2r3CSLamu2

"Zd

3s

*∂m(s)

∂sj

∂m(s)∂sk

+#[rj, [rk, rCM]]

(71)

LDPr = �1

32p2hGÂjk

djk

Zd

3s⇥gj(s), [gk(s), r]

⇤(72)

6

the DP Lindblad term measures gravity acceleration!

gj diverges near a point particle!

regularize?

dx

dt=

[L(x)]Pade

[dE(x)/dx]Pade

(12)

"dE/dx

L

#

Taylor

dx = dt ) x↵(1&x& x

2 & . . .) = t (13)

x = t1/↵(1& t

1/↵ & t2/↵ & . . .) (14)

h(f) = f�7/6

eif�5/3(1& f2/3 & ...) (15)

r↵r↵hµ⌫ + 2Rµ↵⌫�h↵� = 0 . (16)

h+ � ih⇥ =1

R

+1X

l=2

+lX

m=�l

Hl,m(t) �2Ylm(✓,') (17)

: orbital phase

�(s) ! � GM

|s� r|+ �DP(18)

2

L. Diosi proposed that

dx

dt=

[L(x)]Pade

[dE(x)/dx]Pade

(12)

"dE/dx

L

#

Taylor

dx = dt ) x↵(1&x& x

2 & . . .) = t (13)

x = t1/↵(1& t

1/↵ & t2/↵ & . . .) (14)

h(f) = f�7/6

eif�5/3(1& f2/3 & ...) (15)

r↵r↵hµ⌫ + 2Rµ↵⌫�h↵� = 0 . (16)

h+ � ih⇥ =1

R

+1X

l=2

+lX

m=�l

Hl,m(t) �2Ylm(✓,') (17)

: orbital phase

�(s) ! � GM

|s� r|+ �DP(18)

�DP = 10�14 m (19)

2

time

space

g(s)

the larger the σ, the weaker the effect

Bounds on CSL and DP

9

3

spatial states of a macro-molecule, which eventually in-terfere with each other. From the theoretical point ofview, under the rigid-body approximation the moleculecan be treated as a single particle satisfying the collapsedynamics as given by Eq. (1). In this case, the collapserate � for a single nucleon has to be replaced by a rate⇤ associated to the center of mass, which is a function of�, enhanced by a geometric factor depending on the ge-ometry and number of nucleons in the molecule. This isthe mathematical description of the amplification mech-anism [5, 6, 8].

For a rigid body, when the wave function of themolecule is delocalized more than its size, as it is thecase for the experiment under consideration, a reason-able expression for ⇤ is [11, 25]:

⇤ =nA

n(rC)

✓mAn(rC)

m0

◆2

�, (4)

where n(rC) is the number of atoms (nuclei) contained ina volume of linear size rC , while mA is the atomic mass,nA is the number of atoms and m0 is the proton referencemass.The interference pattern – Collapse models predict a lossof visibility, with respect to standard quantum mechan-ics. This e↵ect can be used to set an upper bound onthe collapse parameters, and exclude a region of param-eter space, where the parameters take too strong values.Since we are interested in the order of magnitude, a �

2

minimization procedure to compare the theoretical pre-dictions, computed using Eqs. (2), (3) and (4), to theexperimental data will su�ce. The outcome is reportedin Fig. 2.

The plot depicts two exclusion zones. The one atthe bottom comes from the requirement that the modellocalizes macroscopic objects fast enough (shaded grayzone). Specifically, using Eqs. (1), one imposes that theo↵-diagonal elements of the density matrix ⇢(x, x0

, t) aresupressed fast enough. If this does not happen, thenthe model fails to satisfy the fundamental requirementfor which it was first formulated. To be quantitative,we required that a single-layered graphene disk of radius' 0.01 mm (minimum resolution of the human eye) islocalized within ' 10 ms (perception time of the humaneye). The plot shows that according to our classicalitycriterion the original GRW value for � is the lowest possi-ble value (for rC ' 100 nm) for collapse models to explainclassicality. Clearly, this lower bound can be shifted alsoby several orders of magnitude, depending on the chosencriterion for classicality [23].

The exclusion zone at the top comes from compari-son with the KDTL experiment in [3] (shaded red zone).First we have considered the standard CSL model, whichdepends only on � and rC . The exclusion zone is iden-tified by the red line in Fig. 2. The border of the ex-clusion zone highly depends on the shape and size of the

Adler

GRW

Far field

KDTL

Ref.[24]

Ref.[23]

10-10 10-8 10-6 10-4 10-2 100 10210-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

rC (m)

λ(s

-1)

FIG. 2: Parameter diagram for the CSL, dCSL and cCSLmodels. The exclusion zone, given by the gray shaded zoneat the bottom (bordered by the red solid line), arises fromthe requirement that collapse models become e↵ective formacroscopic system. The red shaded zone at the top cor-responds to the upper bounds set by the KDTL [3] experi-ment discussed in the text. We have also reported the boundsfrom the far field experiment [32, 33], given by the the darkgreen exclusion zone, which are roughly 2 orders of magni-tude weaker. For comparison we have included the boundsfrom X-ray experiments [17], valid for the CSL model andthe cCSL model with frequency cuto↵ ⌦ � 1018Hz, givenby the light blue exclusion zone on the left, and the boundsfrom LIGO, LISA Pathfinder and AURIGA [34], analyzed sofar for the CSL model only, given by the exclusion zones onthe right, shaded in light blue, light green and light red, re-spectively. We have also included for reference, the GRW [8]values (� = 10�16s�1

, rC = 10�7m) and the values pro-posed by Adler [11]: (� = 10�8±2s�1

, rC = 10�7m) and(� = 10�6±2s�1

, rC = 10�6m). The dashed blue and purple

lines denote the KDTL bounds estimated using the analysisfrom [23] and [24], respectively. We note that for values of rCsmaller than the size of the macro-molecule (' 10�8

m), thebounds on � become less stringent.

molecule through the amplification mechanics given inEq. (4). In particular, the slope changes significantlyfrom rC = 10�10m (comparable to the atomic radius) torC = 10�8m (comparable to the molecular radius). Theslope of the lower bound instead changes at rC = 10�5

m

(the radius of the disk).

Next, we considered the dCSL model. Besides � andrC , it depends also on the temperature T of the collapsefield and on the average noise field velocity parameteru = (ux, uy, uz). These new parameters can be under-stood by looking at the quantum linear Boltzmann equa-

[Bassi et al., 2016]

[Helou, Slagmolen, McClelland and YC, 2016]

σDP < 4 × 10-14m

similar bounds from LISA pathfinder and Advanced LIGO

Collapse models can be further bounded, but we still need the microphysics underlying these

collapses.

Is Gravity Quantum?

10

3

ÿk + (k2 � a/a)yk = 0 (28)

yk =

8><

>:

⇣1 � i

kh

⌘e�ikh

Ak h < h⇤ ,

ake�ikh + bke

ikh h > h⇤

(29)

ak =�1 � 2ikh⇤ + (kh⇤)2

eikh⇤

2(kh⇤)2(30)

bk =e�2ikh⇤

2

✓e

ikh⇤ +1

(kh⇤)2

◆(31)

1/h (32)

h⇤ (33)

rGW =h

(ah⇤)4

1

w

dw

2p, kh⇤ < 1 (34)

WGW ⌘ 1

rc

drGW

d log f(35)

Sh =3H

2

0

4p2

WGW

f 3(36)

(kh⇤ ⌧ 1) (37)

r ⇠ h∂0hab∂0habi (38)

S12 = gp

S1S2 (39)

f(x) = �Â GMj

|x � xj|(40)

This potential term appears in the Schrödinger Equation

3

ÿk + (k2 � a/a)yk = 0 (28)

yk =

8><

>:

⇣1 � i

kh

⌘e�ikh

Ak h < h⇤ ,

ake�ikh + bke

ikh h > h⇤

(29)

ak =�1 � 2ikh⇤ + (kh⇤)2

eikh⇤

2(kh⇤)2(30)

bk =e�2ikh⇤

2

✓e

ikh⇤ +1

(kh⇤)2

◆(31)

1/h (32)

h⇤ (33)

rGW =h

(ah⇤)4

1

w

dw

2p, kh⇤ < 1 (34)

WGW ⌘ 1

rc

drGW

d log f(35)

Sh =3H

2

0

4p2

WGW

f 3(36)

(kh⇤ ⌧ 1) (37)


S12 = gp

S1S2 (39)

f(x) = �Â GMj

|x � xj|(40)

V = Âj

�1

2Mjf(xj) = � Â

j<k

GMj Mk

|xj � xk|+ (Self Energy) (41)

If quantum information can pass from A to B through , then gravity must be quantum.

3

ÿk + (k2 � a/a)yk = 0 (28)

yk =

8><

>:

⇣1 � i

kh

⌘e�ikh

Ak h < h⇤ ,

ake�ikh + bke

ikh h > h⇤

(29)

ak =�1 � 2ikh⇤ + (kh⇤)2

eikh⇤

2(kh⇤)2(30)

bk =e�2ikh⇤

2

✓e

ikh⇤ +1

(kh⇤)2

◆(31)

1/h (32)

h⇤ (33)

rGW =h

(ah⇤)4

1

w

dw

2p, kh⇤ < 1 (34)

WGW ⌘ 1

rc

drGW

d log f(35)

Sh =3H

2

0

4p2

WGW

f 3(36)

(kh⇤ ⌧ 1) (37)


S12 = gp

S1S2 (39)

f(x) = �Â GMj

|x � xj|(40)

V = Âj

�1

2Mjf(xj) = � Â

j<k

GMj Mk


M1M2

M3

M4

x1

x2

x3

x4

However, directly confirming quantum information transfer via gravity is very hard. [Kafri & Taylor, 2014]

3

ÿk + (k2 � a/a)yk = 0 (28)

yk =

8><

>:

⇣1 � i

kh

⌘e�ikh

Ak h < h⇤ ,

ake�ikh + bke

ikh h > h⇤

(29)

ak =�1 � 2ikh⇤ + (kh⇤)2

eikh⇤

2(kh⇤)2(30)

bk =e�2ikh⇤

2

✓e

ikh⇤ +1

(kh⇤)2

◆(31)

1/h (32)

h⇤ (33)

rGW =h

(ah⇤)4

1

w

dw

2p, kh⇤ < 1 (34)

WGW ⌘ 1

rc

drGW

d log f(35)

Sh =3H

2

0

4p2

WGW

f 3(36)

(kh⇤ ⌧ 1) (37)


S12 = gp

S1S2 (39)

f(x) = �Â GMj

|x � xj|(40)

V = Âj

�1

2Mjf(xj) = � Â

j<k

GMj Mk


w� =q

w2

0+ w2

g (42)

D ⇡w2

g

2w0

(43)

4

V = Âj

�1

2Mjf(xj) = � Â

j<k

GMj Mk


w� =q

w2

0+ w2

g (42)

D ⇡w2

g

2w0

(43)

wSi

g<⇠p

Gr ⇠ 4 ⇥ 10�4

s�1

(44)

Gravity

6

w� =q

w2

0+ w2

g (73)

wg<⇠p

Gr ⇠ 4 ⇥ 10�4

s�1

(74)

wq =q

w2c + w2

SN(75)

wSN =

sGm

12p

pDx2zp

(76)

⇠ 4 ⇥ 10�2

s�1

(77)

Dxzp (78)

w0 (79)

6

w� =q

w2

0+ w2

g (73)

wg<⇠p

Gr ⇠ 4 ⇥ 10�4

s�1

(74)

wq =q

w2c + w2

SN(75)

wSN =

sGm

12p

pDx2zp

(76)

⇠ 4 ⇥ 10�2

s�1

(77)

Dxzp (78)

w0 (79)

Alternative point of view: If Gravity is classical, self-gravitating objects will not be completely quantum.

[e.g., Feynman, Lectures on Gravitation, 1957]

Demonstrating Quantum Nature of Gravity

11

Spin Entanglement Witness for Quantum Gravity

Sougato Bose,1 Anupam Mazumdar,2 Gavin W. Morley,3 Hendrik Ulbricht,4 Marko Toroš,4

Mauro Paternostro,5 Andrew A. Geraci,6 Peter F. Barker,1 M. S. Kim,7 and Gerard Milburn7,81Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom

2Van Swinderen Institute University of Groningen, 9747 AG Groningen, The Netherlands3Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom

4Department of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom5CTAMOP, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN Belfast, United Kingdom

6Department of Physics, University of Nevada, Reno, 89557 Nevada, USA7QOLS, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom

8Centre for Engineered Quantum Systems, School of Mathematics and Physics,The University of Queensland, QLD 4072, Australia

(Received 6 September 2017; revised manuscript received 6 November 2017; published 13 December 2017)

Understanding gravity in the framework of quantum mechanics is one of the great challenges in modernphysics. However, the lack of empirical evidence has lead to a debate on whether gravity is a quantumentity. Despite varied proposed probes for quantum gravity, it is fair to say that there are no feasible ideasyet to test its quantum coherent behavior directly in a laboratory experiment. Here, we introduce an idea forsuch a test based on the principle that two objects cannot be entangled without a quantum mediator. Weshow that despite the weakness of gravity, the phase evolution induced by the gravitational interaction oftwo micron size test masses in adjacent matter-wave interferometers can detectably entangle them evenwhen they are placed far apart enough to keep Casimir-Polder forces at bay. We provide a prescription forwitnessing this entanglement, which certifies gravity as a quantum coherent mediator, through simple spincorrelation measurements.

DOI: 10.1103/PhysRevLett.119.240401

Quantizing gravity is one of the most intensively pursuedareas of physics [1,2]. However, the lack of empiricalevidence for quantum aspects of gravity has lead to a debateon whether gravity is a quantum entity. This debateincludes a significant community who subscribe to thebreakdown of quantum mechanics itself at scales macro-scopic enough to produce prominent gravitational effects[3–7], so that gravity need not be a quantized field in theusual sense. Indeed it is quite possible to treat gravity as aclassical agent at the cost of including additional stochasticnoise [8–11]. Moreover, oft-cited necessities for quantumgravity (e.g., the big bang singularity) can be averted bymodifying the Einstein action such that gravity becomesweaker at short distances and small time scales [12]. Thus itis crucial to test whether fundamentally gravity is aquantum entity. Proposed tests of this question havetraditionally focused on specific models, phenomenology,and cosmological observations (e.g., [2,13–16]) but are yetto provide conclusive evidence. More recently, the idea oflaboratory probes (proposed originally by Bronstein[17,18] and Feynman [19]) that emphasize the interactionof a probe mass with the gravitational field created byanother mass [20–25], has started to take hold. However,this approach does not yet clarify how the possible quantumcoherent nature of gravity can be unambiguously certifiedin an experiment. In this Letter, we present the scheme foran experiment that not only would certify the potential

quantum coherent behavior of gravity, but would also offera much more prominent witness of quantum gravity thanexisting laboratory-based proposals.We show that the growth of entanglement between two

mesoscopic test masses in adjacent matter-wave interfer-ometers [Fig. 1(b)] can be used to certify the quantumcharacter of the mediator (gravitons) of the gravitationalinteraction—in the same spirit as a Bell inequality certifiesthe “nonlocal” character of quantum mechanics. We maketwo striking observations that make the test for quantumgravity accessible with feasible advances in interferometry:(i) For mesoscopic test masses ∼10−14 kg (with whichinterference experiments might soon be possible [26])separated by ∼100 μm, the quantum mechanical phaseE τ=ℏ induced by their gravitational interaction (with Ebeing their gravitational interaction energy, and τ ∼ 1 stheir interaction time) is significant enough to generate anobservable entanglement between the masses; (ii) if we usetest masses with embedded spins and a Stern-Gerlachscheme [27,28] to implement our interferometry, then, atthe end of the interferometry, the gravitational interaction ofthe test masses actually entangles their spins which arereadily measured in complementary bases (necessary inorder to witness entanglement). Additionally, although ourapproach is independent of the specifics of any quantumtheory of gravity (in the same spirit as using entanglementto study the nature of unknown processes [29,30]), we

PRL 119, 240401 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

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0031-9007=17=119(24)=240401(6) 240401-1 © 2017 American Physical Society

Gravitationally Induced Entanglement between Two Massive Particles is SufficientEvidence of Quantum Effects in Gravity

C. Marletto1 and V. Vedral1,21Clarendon Laboratory, Department of Physics, University of Oxford, England

2Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore(Received 6 September 2017; published 13 December 2017)

All existing quantum-gravity proposals are extremely hard to test in practice. Quantum effects in thegravitational field are exceptionally small, unlike those in the electromagnetic field. The fundamentalreason is that the gravitational coupling constant is about 43 orders of magnitude smaller than the finestructure constant, which governs light-matter interactions. For example, detecting gravitons—thehypothetical quanta of the gravitational field predicted by certain quantum-gravity proposals—is deemedto be practically impossible. Here we adopt a radically different, quantum-information-theoretic approachto testing quantum gravity. We propose witnessing quantumlike features in the gravitational field, byprobing it with two masses each in a superposition of two locations. First, we prove that any system (e.g., afield) mediating entanglement between two quantum systems must be quantum. This argument is generaland does not rely on any specific dynamics. Then, we propose an experiment to detect the entanglementgenerated between two masses via gravitational interaction. By our argument, the degree of entanglementbetween the masses is a witness of the field quantization. This experiment does not require any quantumcontrol over gravity. It is also closer to realization than detecting gravitons or detecting quantumgravitational vacuum fluctuations.

DOI: 10.1103/PhysRevLett.119.240402

Contemporary physics is in a peculiar state. The mostfundamental physical theories, quantum theory and generalrelativity, claim to be universally applicable and have beenconfirmed to a high accuracy in their respective domains.Yet, it is hard to merge them into a unique corpus of laws.We still do not have an uncontroversial proposal forquantum gravity. Some approaches are based on applyinga quantization procedure to the gravitational field [1], inanalogy with the electromagnetic field; some others arebased on “geometrizing” quantum physics [2], while othersmodify both into a more general theory (e.g., string theory[3]) containing both quantum physics and general relativityas special cases. All of them are affected by acute technicaland conceptual difficulties [4–6].There is, however, an even more serious problem.

Current proposals for quantum gravity lead to seeminglyuntestable predictions [7,8]. On this ground, some haveeven argued that quantizing gravity is not needed after all[9] or that gravity may not even be a fundamental force[10,11]. Ronsenfeld summarized the problem as follows:“the incorporation of gravitation into a general quantumtheory of fields is an open problem, because the necessaryempirical clues for deciding the question of the quantiza-tion of the gravitational fields are missing. It is not so mucha matter here of the mathematical problem of how oneshould develop a quantum formalism for gravitation, butrather of the purely empirical question, whether thegravitational field—and thus also the metric—evidencequantumlike features” [12].

How would one confirm experimentally that the gravi-tational field has “quantumlike features”? A good startingpoint, though not sufficient, is a thought experimentFeynman proposed during the Chapel Hill conference ongravity [13]. A test mass is prepared in a superpositionof two different locations and then interacts with thegravitational field.Then, the gravitational field and the mass would pre-

sumably become entangled (Feynman used different ter-minology, but that is what a fully quantum treatment wouldimply). To conclude that the field must be quantized,Feynman proposed to perform a full interference of themass. If the mass did interfere, Feynman’s reasoning goes,gravity would be quantum since remerging the two spatialbranches would then reverse the coupling to gravity,confirming the unitary dynamics in quantum theory. Ofcourse, Feynman also acknowledged that quantum theorycould stop applying at a certain scale. This would thenpresumably constitute a new law of nature—for instance,see the existing “gravitational collapse” literature [9,14,15].Even if successful in showing the full interference of a

single macroscopic mass, Feynman’s thought experiment isnot enough to conclude that the gravitational field isquantum. This is because his proposed interference onlyrequires that the two spatial states of the mass acquiredifferent phases during the experiment. These phases couldsimply be induced by interaction with an entirely classicalgravitational field, without ever requiring entanglementbetween the mass and the field. There is indeed a long


15 DECEMBER 2017

0031-9007=17=119(24)=240402(5) 240402-1 © 2017 American Physical Society

Quantum correlation of light mediated by gravity

Haixing Miao,1, ⇤ Denis Martynov,1, † and Huan Yang2, 3, ‡

1School of Physics and Astronomy, and Institute for Gravitational Wave Astronomy,University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

2Perimeter Institute for Theoretical Physics, Waterloo, ON N2L2Y5, Canada3University of Guelph, Guelph, ON N2L3G1, Canada

We consider using the quantum correlation of light in two optomechanical cavities, which are coupled to eachother through the gravitational interaction of their end mirrors, to probe the quantum nature of gravity. Theoptomechanical interaction coherently amplifies the correlation signal, and a unity signal-to-noise ratio can beachieved within one-year integration time by using high-quality-factor, low-frequency mechanical oscillators.Measuring the correlation can test classical models of gravity, and is an intermediate step before demonstratingthe gravity-mediated entanglement which has a more stringent requirement on the thermal decoherence rate.

Introduction.—Constructing a consistent and verifiablequantum theory of gravity is a challenging task of mod-ern physics [1–3], which is partially due to the di�culty inobserving quantum e↵ects of gravity. This, to certain ex-tents, motivates some theoretical models that treat gravity asa fundamental classical entity [4–11] or being emerged fromsome yet-to-known underlying microphysics [12–15]. Prob-ing the quantum nature of gravity experimentally is there-fore essential for providing hints towards constructing the cor-rect model [16, 17]. Recently, there are two experimentalproposals about demonstrating gravity-induced quantum en-tanglement between two mesoscopic test masses in matter-wave interferometers [18, 19], motivated by an early sugges-tion of Feynman [20]. The setup involves two interferom-eters located close to each other and their test masses areentangled through the gravitational interaction. There aresome discussions regarding whether the gravity-mediated en-tanglement in the Newtonian limit proves the quantumnessof gravity or not [21–25], because the radiative degrees offreedom—graviton, are not directly probed in these exper-iments. Given the lack of experimental evidence, such ex-periments are important steps towards understanding gravityin the quantum regime. Interestingly, they are also sensi-tive to gravity-induced decoherence models for explaining thequantum-to-classical transition [26–31].

The key to demonstrate the entanglement is a low thermaldecoherence rate, so the quantum coherence from the gravi-tational interaction can build up significantly. As shown byEq. (25) and also Appendix A, there is an universal require-ment on the thermal decoherence rate that is independent ofthe size of the two test masses:

�mkBT ~G⇢ . (1)

Here �m is the damping rate and also quantifies the strengthof the thermal Langevin force according to the fluctuation-dissipation theorem [32, 33]; kB is the Boltzmann constant; Tis the environmental temperature; G is the gravitational con-stant; ⇢ is the density of the test mass. For test masses that are

⇤ [email protected]† [email protected]‡ [email protected]

mechanical oscillators with resonant frequency !m, it implies

TQm 1.5 ⇥ 10�18K

1 Hz!m/2⇡

! ⇢

19 g/cm3

!, (2)

where Qm ⌘ !m/�m is the mechanical quality factor and adensity close to Tungsten or Gold is assumed. This require-ment is beyond the state-of-the-art, and needs further experi-mental e↵orts.

In this paper, we propose an intermediate step beforedemonstrating the entanglement by using optomechanical de-vices [34, 35] to realise gravity-mediated quantum correlationof light, which is not constrained by Eq. (1). The setup isshown schematically in Fig. 1. Two optomechanical cavitiesare placed close to each other with their end mirrors (as thetest masses) interacting through gravity. Di↵erent from thesingle-photon nonlinear regime studied by Balushi et al. [36],we are considering the linear regime with the cavity drivenby a coherent laser, and having the light (optical field) andthe mirrors (mechanical oscillators) in Gaussian states. Thequantum correlation of light is measured by cross-correlatingthe homodyne readouts of two photodetectors. With the sys-tem being in a steady state, the signal-to-noise ratio (SNR) forthe correlation measurement grows in time. As shown later inEq. (18), the integration time for achieving a unity SNR is

⌧ ⇡ 1.0 year

nth/C0.4

! !m/2⇡1 Hz

!3 106

Qm

! 19 g/cm3

⇢

!2

, (3)

where nth is the thermal occupation number, and C is the op-tomechanical cooperativity. To constrain the integration time

FIG. 1. Schematics showing the setup of two optomechanical cav-ities with their end mirrors coupled to each other through gravity.The quantum correlation of light is inferred by cross-correlating thereadouts of two photodiodes.

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Information Content of the Gravitational Field of a Quantum Superposition

Alessio Belenchia,1, ⇤ Robert M. Wald,2, † Flaminia Giacomini,3, ‡

Esteban Castro-Ruiz,3, § Caslav Brukner,3, ¶ and Markus Aspelmeyer3, ⇤⇤

1Centre for Theoretical Atomic, Molecular, and Optical Physics,School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom.

2Enrico Fermi Institute and Department of Physics, The University of Chicago,5640 South Ellis Avenue, Chicago, Illinois 60637, USA

3Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria.(Dated: May 14, 2019)

When a massive quantum body is put into a spatial superposition, it is of interest to considerthe quantum aspects of the gravitational field sourced by the body. We argue that in order tounderstand how the body may become entangled with other massive bodies via gravitationalinteractions, it must be thought of as being entangled with its own Newtonian-like gravitationalfield. Thus, a Newtonian-like gravitational field must be capable of carrying quantum information.Our analysis supports the view that table-top experiments testing entanglement of systemsinteracting via gravity do probe the quantum nature of gravity, even if no “gravitons” are emittedduring the experiment.

*Corresponding author

First prize essay written for the Gravity Research Foundation 2019 Essays on Gravitation

⇤Corresponding author: [email protected]

†[email protected]

‡[email protected]

§[email protected]

¶[email protected]

⇤⇤[email protected]

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�mkBT ~G⇢ . (1)




TQm 1.5 ⇥ 10�18K

1 Hz!m/2⇡

! ⇢

19 g/cm3

!, (2)



⌧ ⇡ 1.0 year

nth/C0.4

! !m/2⇡1 Hz

!3 106

Qm

! 19 g/cm3

⇢

!2

, (3)



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show, in Supplemental Material [31], that off-diagonalterms between coherent states (a signature of the quantumsuperposition principle) of the Newtonian gravitationalfield are necessary for the development of the entanglementbetween the test masses.Our proposal relies on two simple assumptions: (a) the

gravitational interaction between two masses is mediatedby a gravitational field (in other words, it is not a directinteraction at a distance) and (b) the validity of a centralprinciple of quantum information theory: entanglementbetween two systems cannot be created by local operationsand classical communication (LOCC) [38]. It can readilybe proved that, in the absence of closed timelike loops [39](i.e., under the assumption of validity of the chronologyprotection conjecture [40]) and as long as the notion ofclassicality itself is not extended significantly [41], LOCCkeeps any initially unentangled state separable. Translatingto our setting of two test masses in adjacent interferometersany external fields (including the gravitational fields fromother masses around them) can only make LOs on theirstates, while a classical gravitational field propagatingbetween the test masses can only give a CC channelbetween them. These LOCC processes cannot entanglethe states of the masses. Thus it immediately follows thatif the mutual gravitational interaction entangles the state oftwo masses, then the mediating gravitational field isnecessarily quantum mechanical in nature.

Entanglement due to gravitational interaction.—We firstconsider a schematic version that clarifies how the states oftwo neutral test masses 1 and 2 (masses m1 and m2), eachheld steadily in a superposition of two spatially separatedstates jLi and jRi as shown in Fig. 1(a) for a time τ, getentangled. Imagine the centers of jLi and jRi to beseparated by a distance Δx, while each of the states jLiand jRi is a localized Gaussian wave packet with widths≪ Δx so that we can assume hLjRi ¼ 0. There is aseparation d between the centers of the superpositions asshown in Fig. 1(a) so that even for the closest approach ofthe masses (d−Δx), the short-range Casimir-Polder forceis negligible. Distinct components of the superpositionhave distinct gravitational interaction energies as themasses are separated by different distances and therebyhave different rates of phase evolution. Under thesecircumstances, the time evolution of the joint state of thetwo masses is purely due to their mutual gravitationalinteraction, and given by

jΨðt ¼ 0Þi12 ¼1ffiffiffi2

p ðjLi1 þ jRi1Þ1ffiffiffi2

p ðjLi2 þ jRi2Þ ð1Þ

→ jΨðt ¼ τÞi12 ¼eiϕffiffiffi2

p"jLi1

1ffiffiffi2

p ðjLi2 þ eiΔϕLRjRi2Þ

þ jRi11ffiffiffi2

p ðeiΔϕRLjLi2 þ jRi2Þ#; ð2Þ

where ΔϕRL ¼ ϕRL −ϕ, ΔϕLR ¼ ϕLR −ϕ, and

ϕRL ∼Gm1m2τℏðd−ΔxÞ

; ϕLR ∼Gm1m2τℏðdþ ΔxÞ

;

ϕ∼Gm1m2τ

ℏd:

One can now think of each mass as an effective “orbitalqubit” with its two states being the spatial states jLi andjRi, which we can call orbital states. As long as1=

ffiffiffi2

pðjLi2 þ eiΔϕLRjRi2Þ and 1=

ffiffiffi2

pðeiΔϕRLjLi2 þ jRi2Þ

are not the same state (which is very generic, happeningfor any ΔϕLR þ ΔϕRL ≠2nπ, with integral n), it is clearthat the state jΨðt ¼ τÞi12 cannot be factorized and isthereby an entangled state of the two orbital qubits.Witnessing this entanglement then suffices to prove thata quantum field must have mediated the gravitationalinteraction between them.It makes sense to start with particles of the largest

possible masses, namely, m1 ∼m2 ∼10−14 kg for whichthere have already been realistic proposals for creatingsuperpositions of spatially separated states such as jLi andjRi [26]. Note that we are constrained to design anexperiment in which only the gravitational interaction isactive. This means that the allowed distance of closestapproach is d−Δx≈200 μm, which is the distance at

x1

d

m1 m1

x2

m2m2

h00

t Spin Correlation Measurements Certifying Entanglement

0

x1

d

m1 m1

x2

m2m2

h00

t S i C l ti M t C tif i E t l t

0

(a)

(b)L

1R

1L

2R

2

L,1 R,

1L,

2R,

2

C1

C2

C1

C2

FIG. 1. Adjacent interferometers to test the quantum nature ofgravity: (a) Two test masses held adjacently in superposition ofspatially localized states jLi and jRi. (b) Adjacent Stern-Gerlach(SG) interferometers in which initial motional states jCij ofmasses are split in a spin dependent manner to prepare statesjL;↑ij þ jR;↓ij (j ¼ 1, 2). Evolution under mutual gravitationalinteraction for a time τ entangles the test masses by impartingappropriate phases to the components of the superposition. Thisentanglement can only result from the exchange of quantummediators—if all interactions aside gravity are absent, then thismust be the gravitational field (labeled h00 where hμν are weakperturbations on the flat space-time metric ημν). This entangle-ment between test masses evidencing quantized gravity can beverified by completing each interferometer and measuring spincorrelations.


15 DECEMBER 2017

240401-2

Using Newtonian Gravity Field to Transfer Quantum Information

12

Can gravity be “classical”?

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)

Gµn = 8phTµni (60)

r2f = 4pGhri ) f(x) = �Z

d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)[Møller 1962, Rosenfeld 1963; Kibble 1976; … ; Guilini 2012; H. Yang et al., 2013]

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12

m ⌧ alattice ⇠ 10�10

m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN=

Gm

12p

px3

ZPF

� w2

g (65)

wSi

SN= 4 ⇥ 10

�2s�1 ⇡ 100 wSi

g (66)

˙x = p/M (67)

˙p = �Mw2

CMx � Mw2

SN(x � hxi) (68)

w2

Q= w2

CM+ w2

SN(69)

Quantum “Self Gravity”

𝜓

particle carries own gravity field gravity field entangled with particle

back action negligible

𝜙

Classical “Self Gravity”

𝜓

unique classical field wave packet attracted by its own potential

𝜙

Schroedinger Newton Phenomenology

13

a macroscopic crystal made up from atoms

center of mass xZPF alattice

xCM xCM xCM

xCM ≪xZPF

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN= (65)

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

MwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN=

Gm

12p

px3

ZPF

� w2

g (65)

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN=

Gm

12p

px3

ZPF

� w2

g (65)

wSi

SN= 4 ⇥ 10

�2s�1 ⇡ 100 wSi

g (66)

dx

dt=

[L(x)]Padé

[dE(x)/dx]Padé

(12)

dE/dx

L

�

Taylordx = dt ) x

a(1 & x & x2 & . . .) = t (13)

x = t1/a(1 & t

1/a & t2/a & . . .) (14)

h( f ) = f�7/6

ei f

�5/3(1 & f2/3 & ...) (15)

rarahµn + 2Rµa

nb

hab = 0 . (16)

h+ � ih⇥ =1R

+•

Âl=2

+l

Âm=�l

Hl,m(t) �2Ylm(q, j) (17)

y: orbital phase

r = � i

h[H0, r] + Â

m,namn

⇢LnrL

†m � 1

2

⇣rL

†mLn + L

†mLnr

⌘�(18)

H = H0 �12 Â

n

⇣fnLn + fnL

†n

⌘, h fm(t) f

⇤n (t

0)i = amn (19)

wOsSN = 0.4 s�1 ⇡ 2p ⇥ 64 mHz . (20)

2

Schroedinger Newton Phenomenology

14

ωCMωQ

phase space

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN=

Gm

12p

px3

ZPF

� w2

g (65)

wSi

SN= 4 ⇥ 10

�2s�1 ⇡ 100 wSi

g (66)

˙x = p/M (67)

˙p = �Mw2

CMx � Mw2

SN(x � hxi) (68)

5

kh⇤ < 1 (54)

n ⇠ n1 + n2E cos 2w0t (55)

SGW

12= g( f )SGW (56)

Sth

12⇠ (WT)�1/2

Sn (57)

E ⇠...Q

2

⇠ m2L

2x

2w6(58)

gGW

w=

1

w

Emw2x2

⇠ mL2w3 ⇠ mw

✓wL

c

◆2

<⇠Gmw

c3

⇡ 10�24

m

100 kg

w

2p ⇥ 1 GHz(59)



d3y

Ghr(y)i|x � y| (61)

ih∂ty(x1, . . . , xn) = H0y(x1, . . . , xn)�1

2Â

j

Mjf(xj)y(x1, . . . , xn) (62)

xZPF ⇠s

h

mwDebye

⇠ 10�12


m (63)

ih∂YCM

∂t=

"� h

2r2

2M+

1

2Mw2

CMx

2 +1

2Mw2

SN(x � hxi)2

#YCM (64)

w2

SN=

Gm

12p

px3

ZPF

� w2

g (65)

wSi

SN= 4 ⇥ 10

�2s�1 ⇡ 100 wSi

g (66)

˙x = p/M (67)

˙p = �Mw2

CMx � Mw2

SN(x � hxi) (68)

w2

Q= w2

CM+ w2

SN(69)

Quantum noise ellipse rotate at a different frequency:

mass

light

ωQωC

classical peak quantum

peak

WS x

H. Yang et al., 2013

Nonlinear QM and Measurement

15

Nonlinear QM in two steps:

H(t, λ), λ = λ[ |ψ⟩]Hamiltonian depends on quantum state

time

A B

instantaneous quantum state

reduction

measurementB will feel the effect right away!

Polchinski 1991

space

Nonlinear QM + Instantaneous State Reduction lead to superluminal communication

light co

ne

H(t, λA, λB)λA = λA[ |ψA⟩cond], λB = λB[ |ψB⟩cond]

time

A Bspace

Force at each location only depends on results within past light cone [Helou, 2018; Scully, in prep]

Gravity as classical feedback! (Kafri & Taylor)

λA λB

Experimental Signatures

16








�mkBT ~G⇢ . (1)




TQm 1.5 ⇥ 10�18K

1 Hz!m/2⇡

! ⇢

19 g/cm3

!, (2)



⌧ ⇡ 1.0 year

nth/C0.4

! !m/2⇡1 Hz

!3 106

Qm

! 19 g/cm3

⇢

!2

, (3)



arX

iv:1

901.

0582

7v2

[qua

nt-p

h] 7

Mar

201

9

Quantum radiation-pressure-induced motion of A will not cause motion in B via gravity [Miao, Martynov & Yang]

A B

Out-going field spectrum has a peak near ωQ [S. Scully, in prep.]

Notes on Schrodinger-Newton

Yanbei Chen

Here we have the Heisenberg equations of motion:

�M⌦2x = �M!2c x+ iM�m⌦x�M!2

SN(x� hxi) + ↵a1 (1)

b⇣ = a⇣ +↵

~ sin ⇣x (2)

As we had discussed before, we have

x� hxi = R(0) (3)

where R(0) is residual of the following system:

�M⌦2x(0) = �M!2qx

(0) + iM�m⌦x(0) + ↵a1 (4)

b(0)⇣ = a⇣ +↵

~ sin ⇣x(0) (5)

where

x(0) = K(0)⇣ b(0)⇣ + R(0) (6)

We can approximately write

x(0) = � ↵

M

a1(⌦� !q + i�m/2)(⌦+ !q + i�m/2)

, (7)

b(0)⇣ = a1

cos ⇣ � ↵2 sin ⇣

~M1

(⌦� !q + i�m/2)(⌦+ !q + i�m/2)

�+ a2 sin ⇣ (8)

We will need to causal whiten b(0⇣ . For this, we �nd

Sbb =(⌦� ⇠)(⌦+ ⇠⇤)(⌦+ ⇠)(⌦� ⇠⇤)

(⌦� !q + i�m/2)(⌦+ !q + i�m/2)(⌦� !q � i�m/2)(⌦+ !q � i�m/2)(9)

where ⇠ has positive real part and negative imaginary part:

⇠ =

r!2q +

↵2 sin ⇣

~M e�i⇣ . (10)

In this way, the whitened version of b(0)⇣ is given by

z =(⌦� !q + i�m/2)(⌦+ !q + i�m/2)

(⌦� ⇠)(⌦+ ⇠⇤)b(0)⇣ (11)

We will have

K⇣b⇣ = [Sxz(⌦)]+z(⌦) (12)

where [. . .]+ takes only poles from lower half of complex plane. We get

K⇣ =~

↵ sin ⇣

!2q � ⇠⇠⇤ + ⇠⇤⌦� ⇠⌦

(⌦� ⇠)(⌦+ ⇠⇤)(13)

In this way, we get

R(0) = (1� ↵

~ sin ⇣K)x(0 �K(a1 cos ⇣ + a2 sin ⇣)

= � ↵a1M(⌦� ⇠)(⌦+ ⇠⇤)

�K(a1 cos ⇣ + a2 sin ⇣) (14)

self gravity back action

mass

light

a1,2b1,2

0 1 2 3 41

2

5

10

f

Sζ

2.90 2.95 3.00 3.05 3.101.0

1.2

1.4

1.6

1.8

f

Sζ

Correlated World Line Theory

17

EXPECTATIONS/HOPES (1) Full non-linearity of General Relativity would make quantized theory non-linear → breakdown of superposition principle (2) Anthropocentric nature of Quantum Measurement be eliminated

1

2

1

2Path A Path A

Path BPath B

Incorporates key ideas from General Relativity and Quantum Mechanics

Equivalence Principle General Covariance

Indistinguishability

adapted from Philip Stamp’s slides

Inter-path Communication and Correlation via gravity

CWL vs Conventional Quantum Gravity

18

Propagator in Conventional QG

Diagrams

Propagator in CWL

Diagrams


Low speed & Weak Gravity

19


+1 2

Crude Approximation

Can be viewed as propagator for two-particle Schroedinger Equation

CWL Phenomenology, 1-Dimension

20

i∂tΨ(Q, y, t) = [−∂2

Q

2M−

∂2y

2μ+ V(Q + y/2) + V(Q − y/2) + Vg(y)] Ψ(Q, y, t), Q =

q + q′�2

, y = q − q′�

ωSN = 2ωc

Quo vadis?• We discussed • collapse models, quantum/classical nature of gravity, and CWL

• Signatures that deviate from QM may be found. • What about anthropogenic nature of quantum measurement?

21

device B

device A

Measurement Process

EPR pair

Collapse of Quantum State: Random

system B

system A

EPR pair

Dynamical Process

Unitary Evolution (Deterministic)

???

No-Go Theorems!!

Quo vadis?

22

device B

device A

EPR pair

Contradiction required by proving No-Go Theorems: no superluminal communication

ϕA ϕB

Results of devices [evolutions of systems] depends on the other polarizer: evolution allows B to know A’s setting before 2L/c, then build a time machine to tell A what her setting would be.

time

For this to happen, both systems must be closed, and at pre-determined states. B cannot get out of the closed system to build his time machine.

Quo vadis?

23

System Device 1

System Device 1 Device 2

System Device 1 Device 2 Device 3

Test Mass Photon Photoelectron Electrons in amplifier

System Device 1 Device 2 Device 3 Device N…

Only closed “measuring systems” can be mapped to a unitary evolution?

Documents

Toward Testing Quantum Mechanics and Gravity in the Lab