test of non-locality via non-relativistic quantum...

Test of Non-Locality via Non-RelativisticQuantum Systems

Alessio Belenchia

Optomechanical route to MACROSCOPIC QUANTUMSUPERPOSITIONS, Bratislava

23/03/2018

Alessio Belenchia (IQOQI) Discrete Spacetime Phenomenology 23/03/2018 1 / 19

Introduction

Table of Contents

1 IntroductionThe Quantum Gravity problem

2 Non-Locality in Field Theory

3 Phenomenology

4 Conclusions


Introduction The Quantum Gravity problem

QG tries to reconcile General Relativity with Quantum Theory

Plethora of models/theories:String Theory

LQG

CST

GFT

CDT

...

𝑙𝑝 = ℏ𝐺𝑐3 ≅ 10−35𝑚

𝑡𝑝 = ℏ𝐺𝑐5 ≅ 10−44𝑠

𝐸𝑝 =ℏ𝑐5

𝐺 ≅ 1019𝐺𝑒𝑉

Planck Units

The feature of quantum gravity that challenges its veryright to be considered as a genuine branch of theoreticalphysics is the singular absence of any observed property ofthe world that can be identified unequivocally as the resultof some interplay between general relativity and quantumtheory.


Introduction The Quantum Gravity problem

QG Phenomenology tries to bridge the gap between theory andobservations

Where to look for:Deformed field dynamics

Table-top quantum experiments at thegravity/quantum crossroad

Violations of Lorentz Invariance and of other exactsymmetries

Generalized Uncertainty principle

QG imprints on initial cosmological perturbations


Non-Locality in Field Theory

Table of Contents

1 Introduction

2 Non-Locality in Field TheoryWhy?The example of Causal Set Theory

3 Phenomenology

4 Conclusions


Non-Locality in Field Theory Why?

Non-local field theories:kinetic terms with infinitely many derivatives

Bottom up: nonlocality is a generic feature of QGmodels

field theories with minimal lenght scale

ubiquitous in string theory (SFT, p-adic ST)

field theory on non-commutative spacetimes

spinfoam models

Causal set theory

High-gain/ high-risk:giving up locality — in particular in time — comes with potential problems

classical and quantum instabilities (Ostrogradski, ghosts)

initial value problem and predictability


Non-Locality in Field Theory Why?

Non-local field theories:kinetic terms with infinitely many derivatives

Top down: some form of non-locality have beenconsidered for

cure UV divergences of QFTs

way-out to the info-loss problem?

early Universe and late time cosmology

non-locality as a direct consequence of (local)Lorentz Invariance and discreteness?!?

High-gain/ high-risk:giving up locality — in particular in time — comes with potential problems

classical and quantum instabilities (Ostrogradski, ghosts)

initial value problem and predictability


Non-Locality in Field Theory The example of Causal Set Theory

Discreteness, Lorentz Invariance and Non-Locality

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Discrete partial order which encodes the causalstructure of spacetime

Random lattice which is (Locally) LorentzInvariant [L.Bombelli et al., Mod. Phys. Lett. A(2009)]

Discreteness modifies the dynamics of propagatingfields

One free-parameter to be tested — non-locality scale

Effective Wave Operator

fρ ()φ(x) = −4√

6

[ρ1/2φ(x)− ρ3/2

∫J−(x)

d4y(

1− 9ρV + 8ρ2V2 −43ρ3V3

)φ(y)e−ρV

]= 0

↓ ρ→∞

φ(x) = 0


Phenomenology

Table of Contents

1 Introduction


3 PhenomenologyPhenomenology by way of Non-relativistic systems: the harmonic oscillatorPhenomenology by way of Optomechanics: SummaryForecast of constraints

4 Conclusions


Phenomenology Phenomenology by way of Non-relativistic systems: the harmonic oscillator

Non-relativistic nonlocality and phenomenology I:[AB et al., Phys.Rev.Lett.116 (2016); AB et al. Phys.Rev.D95 (2017)]

Idea: consider the non-relativistic limit of a (generic) non-local Klein-Gordonoperator and its phenomenology

Some physical requirements:

Analyticity

Ghost-free theory

Example: SFT inspired

f () = (−m2)e−−m2

Λ2

Nonlocal Schödinger equation

f (S) = S +∞∑j=2

bj

(−2m~2

)j−1l2j−2n Sj,

f (S)ψ(t, x) = V(x)ψ(t, x)



Non-relativistic nonlocality and phenomenology I:[AB et al., Phys.Rev.Lett.116 (2016); AB et al. Phys.Rev.D95 (2017)]

Idea: consider the non-relativistic limit of a (generic) non-local Klein-Gordonoperator and its phenomenology

φ(x) = e−i mc2~ tψ(t, x)

Lrel.c→∞→ LNR = ψ∗(t, x)f (S′)ψ(t, x)+c.c.+V(x)ψ∗ψ

S′ = −2m~2

S = −2m~2

(i~∂

∂t+

~2

2m∇2)

Nonlocal Schödinger equation

f (S) = S +∞∑j=2

bj

(−2m~2

)j−1l2j−2n Sj,

f (S)ψ(t, x) = V(x)ψ(t, x)



Non-relativistic non-locality and phenomenology II

Perturbatively:

ψ = ψ0 +∑n=1

εnψn, ε =mω~Λ2

=l2n

2x2zpm

ψ0 → coherent states solution of the h.o. Schödinger equation

O(1) : (S− V)ψ0 = 0O(ε) : (S− V)ψ1 = J1

O(ε2) : (S− V)ψ2 = J2

etc.,

J1 =2a2

~ΩS2ψ0,

J2 = J2 (ψ0, ψ1)

〈x〉 =√

2α cos(t)(

1 +14εα

2a2 [cos(2t)− 1]

)+ O(ε

2),

〈p〉 =√

2α sin(t)(

1 +14ε a2

[α

2(7 + 3 cos(2t))− 2

])+ O(ε

2)

--

-

-

-



Var(x) =12

(1− εa2

[(6α2 − 1

)sin2(t)

])+ O(ε2),

Var(p) =12

(1 + εa2

[(6α2 − 1

)sin2(t))

])+ O(ε2),

-4 -2 0 2 4

-4

-2

0

2

4

x

p


Phenomenology Phenomenology by way of Optomechanics: Summary

Where to Look for: [AB et al. Phys.Rev.D95 (2017)]

Average variance for the ground state

ln < 2xzpm√

∆meas

Average variance for coherent states

ln <xzpm

α

√2∆meas

3

Third harmonic distortion in the mean position for coherent states

H3 =εα2

8⇒ ln <

4α

xzpm√

H3

Spontaneous, time periodic squeezing:Reconstruction of the time depend positionvariance or (even better) of the Wigner function(via QNM?)


Phenomenology Forecast of constraints

HEP constraints

ln ≤ 10−19m[T. Biswas and N. Okada, Nucl. Phys. B 898 (2015)]

Achievable constraint withdetectors

ln ≤ 10−24m

“Reasonable forecast” withoptomechanics

ln ≤ 10−26m


Conclusions

Table of Contents

1 Introduction


3 Phenomenology

4 Conclusions


Starting from general ideas of QG testable phenomenological models can be obtained Discreteness and LI combined lead to non-locality, new opportunity for phenomenology Low-energy, quantum, massive systems provide a new avenue for tests of QG effects

General lessons:

Outlooks:

Hamiltonian formulation (work in progress!!)

Introduction of the environment and role of decoherence

Account of the coupling with light: more realistic description of an experimental situation

AB, Benincasa, Liberati, Nonlocal Scalar Quantum Field Theory from Causal Sets, JHEP 03, p. 036

AB, Benincasa, F. Dowker, The continuum limit of a 4-dimensional causal set scalar d’Alembertian, Class.Quant.Grav. (2016)

AB, Benincasa, Liberati, Marin, Marino, and Ortolan, Tests of Quantum Gravity induced non-locality via opto-mechanical quantum oscillators, Phys.Rev.Lett.116 (2016)

AB, Benincasa, Liberati, Marin, Marino, and Ortolan, Tests of Quantum Gravity-Induced Non-Locality via Opto-mechanical Experiments Phys.Rev.D95 (2017)

Selected References:

Conclusions

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Relativistic Quantum Information North 2018 Vienna, 24-27 September

Order online at https://www.postersession.com/order/

ORGANIZING COMMITTEE:

Markus Aspelmeyer Alessio Belenchia

Časlav Brukner Esteban Castro-Ruiz Flaminia Giacomini

Philipp Höhn

The Relativistic Quantum Information – North 2018 conference at the University of Vienna is held under the auspices of the International Society for Relativistic Quantum Information and is the ninth in the series of such meetings taking place in the Northern Hemisphere. This workshop series aims at bringing together researchers working across quantum information science, quantum field theory in curved spacetime, and quantum gravity.

SPEAKERS:

Hartmut Abele, Technische Universität Wien Paul Alsing, Air Force Research Laboratory Giovanni Amelino-Camelia, Università Federico II Miles Blencowe, University of Dartmouth Sougato Bose, University College London Fabio Costa, University of Queensland Astrid Eichhorn, University Heidelberg Ivette Fuentes, University of Nottingham Domenico Giulini, Leibniz University Hannover Lucien Hardy, Perimeter Institute Bei Lok-Hu, University of Maryland Mark Kasevich, Stanford University Achim Kempf, University of Waterloo Adrian Kent, University of Cambridge Juan Leon, CSIC Madrid Stefano Liberati, SISSA Trieste Robert Mann, University of Waterloo Mercedes Martín-Benito, Universidad Complutense de Madrid Eduardo Martin-Martinez, University of Waterloo Tobias Osborne, Leibniz Universität Hannover Timothy Ralph, University of Queensland Rafael Sorkin, Perimeter Institute Rupert Ursin, IQOQI Vienna Magdalena Zych, University of Queensland Silke Weinfurtner, University of Nottingham William Unruh (tbc), University of British Columbia Mark Van Raamsdonk (tbc), University of British Columbia

QI

2018

https://rqi.iqoqi.univie.ac.at/


www.postersession.com

Relativistic Quantum Information North 2018 Vienna, 24-27 September

Order online at https://www.postersession.com/order/

ORGANIZING COMMITTEE:

Markus Aspelmeyer Alessio Belenchia

Časlav Brukner Esteban Castro-Ruiz Flaminia Giacomini

Philipp Höhn

The Relativistic Quantum Information – North 2018 conference at the University of Vienna is held under the auspices of the International Society for Relativistic Quantum Information and is the ninth in the series of such meetings taking place in the Northern Hemisphere. This workshop series aims at bringing together researchers working across quantum information science, quantum field theory in curved spacetime, and quantum gravity.

SPEAKERS:

Hartmut Abele, Technische Universität Wien Paul Alsing, Air Force Research Laboratory Giovanni Amelino-Camelia, Università Federico II Miles Blencowe, University of Dartmouth Sougato Bose, University College London Fabio Costa, University of Queensland Astrid Eichhorn, University Heidelberg Ivette Fuentes, University of Nottingham Domenico Giulini, Leibniz University Hannover Lucien Hardy, Perimeter Institute Bei Lok-Hu, University of Maryland Mark Kasevich, Stanford University Achim Kempf, University of Waterloo Adrian Kent, University of Cambridge Juan Leon, CSIC Madrid Stefano Liberati, SISSA Trieste Robert Mann, University of Waterloo Mercedes Martín-Benito, Universidad Complutense de Madrid Eduardo Martin-Martinez, University of Waterloo Tobias Osborne, Leibniz Universität Hannover Timothy Ralph, University of Queensland Rafael Sorkin, Perimeter Institute Rupert Ursin, IQOQI Vienna Magdalena Zych, University of Queensland Silke Weinfurtner, University of Nottingham William Unruh (tbc), University of British Columbia Mark Van Raamsdonk (tbc), University of British Columbia

QI

2018

https://rqi.iqoqi.univie.ac.at/

Back-up slides

Table of Contents

5 Back-up slidesLIVCausal Set theoryCurved spacetime and EEP


Back-up slides

Back-up slides


p2 −m2 + Σ = 𝑝0 2 − 𝒑2 −𝑚2 − 𝑔2𝑎0 𝑝0 2 − 𝑔2𝑎1𝒑

2

~ 𝑝0 2 − 1 + 𝑔2 𝑎1 − 𝑎0 −𝑚2

Δ𝑐 = 𝑔2(𝑎1 − 𝑎0)

Perturbatively in g

Physical Significance of the Violating Coefficient

Back-up slides Causal Set theory

Hierarchy of causality conditions


GREEN functions in QFT

𝑊 𝑥, 𝑦 = ⟨0 𝜙 𝑥 𝜙 𝑦 0⟩

i Δ 𝑥, 𝑦 = ⟨0 ,𝜙 𝑥 𝜙 𝑦 - 0⟩

H 𝑥, 𝑦 = ⟨0 *𝜙 𝑥 𝜙 𝑦 + 0⟩

GR 𝑥, 𝑥′ = −𝜃 𝑡 − 𝑡′ 𝑊(𝑥, 𝑥′)

GA 𝑥, 𝑥′ = 𝜃 𝑡′ − 𝑡 𝑊(𝑥, 𝑥′)

i GF 𝑥, 𝑥′ = ⟨0 𝑇(𝜙 𝑥 𝜙 𝑦 ) 0⟩

G 𝑥, 𝑦 = 2𝜋 −𝑛 𝑑𝑛𝑘 𝑒𝑖𝑘⋅𝑥

−𝑘2−𝑚2

Integral Kernel representation of Non-local wave operators

𝑛𝑙𝜙 𝑥 = 𝑑𝐷𝑦 𝐾𝐷 𝑥, 𝑦 𝜙 𝑦𝐽−(𝑥)

= 𝑑𝐷𝑤 𝐾𝐷 −𝑤 𝑒𝑤⋅𝜕𝑥𝜙 𝑥 = 𝑓𝐷 𝜙 𝑥𝐽−(0)

From Discrete to Continum

• Divide the spacetime into cells of volume Δ𝑉 • Let 𝑚𝐼 be the number of elements sprinkled between x and the I-th cell • Define

• The mean of the sum over the first layer 𝐿1 is

Back-up slides Curved spacetime and EEP

What about curved spacetime?

Generalized Causal Set d’Alebertians

1 Linearity2 Retardedness3 Label invariance4 Neighbourly

democracy

nlφ(x) = ρ2/daφ(x) + ρ(2+d)/dO∫

J−(x)

√−g e−ρV(x,y)φ(y)ddy

O =

Lmax∑n=0

bn

n!(−1)nρn ∂

n

∂ρn

limρ→+∞

nlφ = gφ−R2φ

EEP in 4 dimensions requires a conformalcoupling to hold [Sonego, Faraoni, Class. and Quant.Grav. 10.6 (1993): 1185], if not

Possible sub-luminal tails for masslesswaves

Possible luminal propagation for massiveone

Further investigation of the massive case

Explore the quantum version of the theory



EEP and couplings I

(ηab∂a∂b −m2)φ = 0 → (gab∇a∇b −m2 − ξR)φ = 0

ξ = 0 minimal coupling, requiring the same formal structure

ξ = 1/6 conformal coupling, conformal invariance and EEP

EP:WEP: Universality of free fall

EEP: WEP+LLI+LPISEP, GWEP,.....

Local physics should be described by SR according to EEP⇒ Physical features ofthe solution should be locally the same

GR(x′, x) = Σ(x′, x)δR(Γ(x′, x)) + V(x′, x)ΘR(−Γ(x′, x))

limx′→xΣ(x′, x) = limx′→xΣM(x′, x)

limx′→xV(x′, x) = limx′→xVM(x′, x)



EEP and couplings II

Σ(x′, x) =1

4π+ O1(x′, x)

ΣM(x′, x) =1

4π

V(x′, x) = − 18π

[m2 +

(ξ − 1

6R(x)

)]+ O2(x′, x)

VM(x′, x) = −m2

8π

(gab∇a∇b −m2 − ξR + αR2 + βRµνRµν + γRabcdRabcd + ....)φ+ λF(φ) = 0

EP require:

ξ =16, α = β = γ = ..... = 0



EEP and couplings III

A non vanishing V(x′, x) implies wave tails inside the light cone.

Flat spacetime:Type a: due to m 6= 0, limx′→xVM(x′, x) 6= 0

Type b: due to backscattering off a potential, limx′→xVM(x′, x) = 0

In curved spacetimes we have the same classification.

If ξ = 1/6:

Tails of type a iff m 6= 0

If ξ 6= 1/6:

If m = 0 there could be type a tails (HP violations)

When m 6= 0 if m2 + (ξ − 1/6)R(x) = 0 then no tails of type a⇒ Propagation ofmassive particles on the light cone, unacceptable if local physics has to bedescribed by SR according to EEP



Spontaneous emission in the T →∞ limit:

∆(ln,Ω,T) ≈23

c−2|Ω|2l2n

The result can be obtain computing the rate when the window functions are removed:

Rate for a massive local field−iµ4π2

∫ ∞−∞

d∆τK1(iµ(∆τ − iε))

∆τ − iεe−iΩ∆τ

,

K1(z) has two branch points at z = 0,∞, and a branch-cut between them. In particular, the branch-cutcan be placed on the negative real axis Re(z) < 0, corresponding to Im(∆τ) > ε.

Note that for large z, K1(z) ≈ e−z/√

z: when Ω > −µ the contour integral vanishes

When Ω < −µ, the contour integral gives

−iµ4π2

[2πi(−Ω

µ

)+ 2πi

∫ ∞0

drI1(−r)

re−|Ω|r/µ

],

The rate of the detector is given by

Fµ =1

2πΘ(−Ω− µ)

√Ω2 − µ2.

and the relative rate,

Fnonlocal − Flocal

Flocal=

∫ |Ω|20 dµ2ρ(µ2) 1

2π

√Ω2 − µ2

12π |Ω|Θ(−Ω)

. (1)


• Red Detuning 𝚫 ≈ −𝜴: relevant for cooling and quantum state transfer between the radiation and mechanical mode

• Blue Detuning 𝚫 ≈ 𝜴 : relevant for creating correlations between the two modes • On Resonance Regime 𝚫 ≈ 𝟎 : cavity is operated as an interferometer and can be

used to read out the mechanical motion

Cavity Optomechanics

test of non-locality via non-relativistic quantum...

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