Models  of  spontaneous  wave  func1on  collapse  and  

optomechanics  

Quantum  Interfaces  with  Nano-­‐opto-­‐electro-­‐mechanical  devices:  Applica1ons  

and  Fundamental  Physics  Erice,  1st  –  5th  August  2016  

     

(Angelo  Bassi  –  University  of  Trieste  &  INFN)  

Quantum  Mechanics  

Linearity  è  Superposi1on  Principle  è  Schrödinger's  cat  è  Measurement  Problem    

Modify  the  Schrödinger  equa1on    J.S.Bell  Speakable  and  Unspeakable  in  Quantum  Mechanics  

E.P.  Wigner  in:  Quantum  Op1cs,  Experimental  gravity  and  Measurement  theory,  Plenum,  NY  (1983)  

A.J.  Legge^  Supplement  Progr.  Theor.  Phys.  69,  80  (1980)  

H.P.  Stapp  In:  Quantum  Implica1ons:  Essay  in  Honor  of  David  Bohm,  Routledge  &  Kegan  Paul,  London    (1987)  

S.  Weinberg  Phys.  Rev.  Le6.  62,  486  (1989).  

R.  Penrose  In:  Quantum  Concepts  of  Space  and  Time,  Oxford  U.P.  (1985)  

S.L.  Adler  Quantum  Theory  as  an  emergent  phenomenon,  CUP  (2009)  

 G.C.  Ghirardi,  A.  Rimini,  T.  Weber  Phys.  Rev.  D  34,  470  (1986)  

P.  Pearle  Phys.  Rev.  A  39,  2277  (1989)  

L.  Diosi  L. Diosi, Phys. Rev. A 40, 1165 (1989)

How  to  modify  the  Schrödinger  equa1on?  

The  no-­‐faster-­‐than-­‐light  condi1on  heavily  constraints  the  possible  ways  to  modify  the  Schrödinger  equa1on.    

In  par1cular,  it  requires  that  nonlinear  terms  must  always  be  accompanied  by  appropriate  stochas1c  terms.  

N.  Gisin,  Hel.  Phys.  Acta  62,  363  (1989).  Phys.  Le6.  A    143,  1  (1990)  

N.  Gisin  and  M.  Rigo,  Journ.  Phys.  A  28,  7375  (1995)  

J.  Polcinski,  Phys.  Rev.  Le6.  66,  397  (1991)  

H.M.  Wiseman  and  L.  Diosi,  Chem.  Phys.  268,  91  (2001)  

S.L.  Adler,  “Quantum  Theory  as  an  Emergent  Phenomenon”,  C.U.P.  (2004)  

A.  Bassi,  D.  Dürr  and  G.  Hinrichs,  Phys.  Rev.  Le6.  111,  210401  (2013).  

L.  Diosi,  Phys.  Rev.  Le6.  112,  108901  (2014)    

M.  Caiaffa,  A.  Smirne  and  A.  Bassi,  in  prepara1on  

The  con1nuous  dynamics  (simplified)  

d|⇥⇤t =

� i

~Hdt +⌅�(A� ⇥A⇤t)dWt �

�

2(A� ⇥A⇤t)2dt

�|⇥⇤t

�A⇥t = ��t|A|�t⇥ nonlinear  

quantum   collapse  

The  wave  func1on  is  dynamically  and  stochas1cally  driven  by  the  noise  Wt  towards  one  of  the  eigenstates  of  the  operator  A  

 

This  equa1on  describes  microscopic  physics,  macroscopic  physics,  and  what  happens  in  quantum  experiments  (Born  rule,  collapse  …)  

(Mass-­‐propor1onal)  CSL  model  P.  Pearle,  Phys.  Rev.  A  39,  2277  (1989).  G.C.  Ghirardi,  P.  Pearle  and  A.  Rimini,  Phys.  Rev.  A  42,  78  (1990)  

Two  parameters  

� = collapse strength rC = localization resolution

M(x) = ma†(x)a(x) G(x) =

1

(4⇡rC)3/2exp[�(x)

2/4r2C ]

d

dt| ti =

� i

~H +

p�

m0

Zd3x (M(x)� hM(x)it) dWt(x)

� �

2m20

Z Zd3xd3y G(x� y) (M(x)� hM(x)it) (M(y)� hM(y)it)

�| ti

� = �/(4⇡r2C)3/2

= collapse rate

The  operators  are  func1on  of  the  space  coordinate.  The  collapse  occurs  in  space.  

The collapse rate  

Γ = λn2N

n = number of particles

within rC

+ +

Small superpositions Large superpositions

Collapse NOT effective Collapse effective

+ N = number of such

clusters

⌧ rC � rC

Amplification mechanics

Few particles

no collapse

quantum behavior

Many particles

Fast collapse

classical behavior

CSL  Parameters  

QUANTUM  –  CLASSICAL  TRANSITION  (Adler  -­‐  2007)  

Microscopic  world  (few  par1cles)    

Mesoscopic  world:  Latent  image  forma1on  +  percep1on  in  the  eye  (~  104  -­‐  105  par1cles)  

   Macroscopic  world    (>  1013  par1cles)  

Increasing  size  of  the  system    

 

QUANTUM  –  CLASSICAL    TRANSITION  (GRW  -­‐  1986)  

� ⇠ 10�8±2s�1

� ⇠ 10�17s�1

S.L.  Adler,  JPA  40,  2935  (2007)  

A.  Bassi,  D.A.  Deckert  &  L.  Ferialdi,  EPL  92,  50006  (2010)  

G.C.  Ghirardi,  A.  Rimini  and  T.  Weber,  PRD  34,  470  (1986)  

rC = 1/p↵ ⇠ 10�5cm

Experimental  tests  

The  obvious  way  to  test  collapse  models  is  with  ma^er-­‐wave  interferometry    

Predic1on  of  quantum  mechanics  (no  environmental  noise)  

Predic1on  of  collapse  models  (no  environmental  noise)  

Atom  Interferometry  T.  Kovachy  et  al.,  Nature  528,  530  (2015)    

 M  =  87  amu  d  =  0.54  m  T  =  1  s  

Adler

GRW

10-10 10-8 10-6 10-4 10-2 100

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

rc

CS

L

2 DECEMBER 2011 VOL 334 SCIENCE www.sciencemag.org 1214

PERSPECTIVES

In the movie Dances with Wolves, a lone wolf facilitates Lieutenant John Dunbar’s immersion into the complex culture of

the Sioux Indians. This immersion required overcoming multiple cultural barriers. Ecol-ogists and evolutionary biologists face an equally daunting challenge of understanding how environmental change affects ecological and evolutionary dynamics ( 1). Historically, researchers examined these impacts in isola-tion. However, these dynamics can occur on similar time scales, resulting in a dynamic evolutionary-ecological feedback loop ( 2). Studying these feedbacks directly for long-lived species is often thought to be imprac-tical. On page 1275 of this issue, Coulson et

al. ( 3) overcome this barrier using data from radio-collared gray wolves and state-of-the-art mathematical models.

The 280 radio-collared wolves studied by Coulson et al. are direct descendants of 41 gray wolves reintroduced into Yellowstone National Park between 1995 and 1997 ( 4). This reintroduction was part of a larger effort involving a simultaneous reintroduction in Idaho and a naturally colonized population in Montana. It was extremely successful; by 2010, the Northern Rocky Mountain wolf population had expanded to 1651 individuals ( 5). Individuals within this expanding popula-tion vary substantially in body size, coat color, and other observable (phenotypic) traits. Coat color is particularly enigmatic; gray wolves in North America often have black coats, whereas in Eurasia black coats are rare, but the reason for this difference remains unclear ( 6). These traits were recorded for over a decade (from 1998 to 2009) for each collared wolf and their offspring.

To explore the potential ecological and evolutionary responses of the gray wolves

to environmental change, Coulson et al. fuse integral projection models (IPMs) with clas-sical population genetics. Unlike their matrix model counterparts ( 7), IPMs describe the dynamics of populations with traits that vary continuously, such as body size ( 8), as well as discrete traits, such as coat color ( 9). Tra-ditional IPMs track how the number of indi-viduals with a particular body size changes due to births, deaths, and individual growth. The rules underlying these changes are deter-mined by statistical relationships between the body size of individuals and their vital rates such as fecundity, survivorship, and growth.

In gray wolves, a change at a single loca-tion on the genome—the K locus—deter-mines coat color ( 10). To link evolution-ary and ecological dynamics, Coulson et al. extend the IPM to account for this genetic difference between individuals. As a result, the statistical relationships between individ-ual body size and vital rates become geno-

Mathematical Dances with Wolves

ECOLOGY

Sebastian J. Schreiber

Data and modeling of Yellowstone wolf

populations illustrate the complex interrelated

ecological and evolutionary responses to

environmental change.

photon, it could have come from either of the diamond crystals in which one pho-non was excited. The indis-tinguishability of these two possibilities during detec-tion means that the two dia-mond samples coherently shared one phonon, which is the hallmark of a quan-tum-entangled state.

The entanglement be tween the two diamond samples was confi rmed in experiments in which a second laser pulse de-excited the shared phonon and re-emitted a photon that was subsequently detected. By this method, Lee et al. demonstrate that the two diamonds share entanglement at a 98% confidence level. These results provide a striking example that entanglement is not par-ticular to microscopic particles but can mani-fest itself in the macroscopic world, where it could be used in future studies that make fun-damental tests of quantum mechanics.

The demonstration of entanglement in macroscopic systems also has important implications for the ongoing efforts to realize quantum computation and communication. A full-size quantum computer eventually will

need to be a macroscopic device in which entanglement is preserved and used over long times and distances. The lifetime of entangle-ment in the experiment by Lee et al. is still too short for many quantum information applica-tions, in part because of the room-temperature environment and the strong coupling of pho-non modes in solids. However, the experiment emphasizes an important point, that ultrafast optical technology can alleviate the require-ment on quantum coherence time. In future, with improvement of the ultrafast technology, or by using more isolated degrees of freedom in solids—such as as the nuclear spins ( 8) or the dopant rare-earth ions ( 9)—for quantum memory, many more quantum operations

could be done within the coherence time of the solids, even at room temperature.

References and Notes

1. K. C. Lee et al., Science 334, 1253 (2011). 2. L.-M. Duan, Nature 414, 413 (2001). 3. C. W. Chou et al., Nature 438, 828 (2005). 4. T. Chanelière et al., Nature 438, 833 (2005). 5. N. Sangouard, C. Simon, H. de Riedmatten, N. Gisin, Rev.

Mod. Phys. 83, 33 (2011). 6. D. L. Moehring et al., Nature 449, 68 (2007). 7. S. Olmschenk et al., Science 323, 486 (2009). 8. E. Togan et al., Nature 466, 730 (2010). 9. C. Clausen et al., Nature 469, 508 (2011). 10. Supported by the National Basic Research Program of

China (973 Program) 2011CBA00300 (2011CBA00302), Army Research Offi ce, and Air Force Offi ce of Scientifi c Research MURI program.

CR

ED

IT: P. H

UE

Y/S

CIE

NC

E

A B

Mirror

Pump pulse

Pump pulse Scattered photon

Phonon

Beam splitter

Detector

Making quantum connections. The method used by Lee et al. to generate entanglement between two macroscopic diamonds is illus-trated. (A) A pumping laser pulse generates a correlated pair of a phonon inside the diamond as well as a scattered photon. (B) The scattered photons from two diamonds are brought together for interference and detection. When one photon is detected, the two diamonds coherently share a phonon. Thus, the quantum state created has the hallmarks of quantum entanglement.

10.1126/science.1215444

Department of Evolution and Ecology, University of Califor-nia, Davis, CA 95616, USA. E-mail: sschreiber@ucdavis.edu

Published by AAAS

on

May

6, 2

016

http

://sc

ienc

e.sc

ienc

emag

.org

/D

ownl

oade

d fr

om

Interferometric  Experiments  

To  improve  interferometric  tests,  it  will  be  necessary  to  go  to  micro-­‐gravity  environment  in  outer  space.  COST  Ac1on  QTSpace.  h^p://www.cost.eu/COST_Ac1ons/ca/CA15220  

Molecular  Interferometry  S.  Eibenberger  et  al.  PCCP  15,  14696  (2013)  M.  Toros  et  al.,  ArXiv  1601.03672  

 M  =  104  amu  d  =  10-­‐7  m  T  =  10-­‐3  s  

Entangling  Diamonds  K.  C.  Lee  et  al.,  Science.  334,  1253  (2011).  S.  Belli  et  al.,  PRA  94,  012108  (2016)      

 M  =  1016  amu  d  =  10-­‐11  m  T  =  10-­‐12  s  

Lower  bound:  Collapse  effec1ve  at  the  macroscopic  level  Graphene  disk:  N  =  1011  amu,  d  =  10-­‐5  m,  T  =  10-­‐2  s    

Non  interferometric  tests  

The  collapse  induces  a  Brownian  mo1on  on  the  system  

Charged  free  par1cle    1.  Quantum  mechanics  

 

2.  Collapse  models  

Spontaneous  photon  emission  

Nano-­‐par1cle  in  a  op1cal  cavity  

 

Extra  shi\  and  broadening    

M.  Bahrami,  M.  Paternostro,  A.  Bassi,  H.  Ulbricht,  Phys.  Rev.  Le6.  112,  210404  (2014).  S.  Nimmrichter,  K.  Hornberger,  K.  Hammerer,  Phys.  Rev.  Le6.  113,  020405  (2014).  L.  Diósi,  Phys.  Rev.  Le6.  114,  050403  (2015)    

Non-­‐Interferometric  Experiments  

Gravita1onal  wave  detectors  M.  Carlesso  et  al.,  ArXiv  1606.04581    

M  =  1030  amu  T  =  ∞  

X-­‐Ray  C.  Curceanu  et  al.,  J.  Adv.  Phys.  4,  263  (2015).      

M  =  1  Kg  T  =  days  /  months  

Can1lever  A.  Vinante  et  al.,  Phys.  Rev.  Le^.  116,  090402  (2016)    

M  =  1014  amu  T  =  ∞  

Cold  Atom  Gas  M.  Bilardello  et  al.,  ArXiv  1605.01891      

M  =  87  amu  T  =  1  s  

Cold  Atom  Gas    F.  Laloë  et  al.  Phys.  Rev.  A  90,  052119  (2014)    

M  =  87  amu  T  =  30  s  

A superconducting quantum interference device based read-out of asubattonewton force sensor operating at millikelvin temperatures

O. Usenko, A. Vinante, G. Wijts, and T. H. Oosterkampa!

Leiden Institute of Physics, Leiden University, The Netherlands

!Received 4 January 2011; accepted 15 February 2011; published online 30 March 2011"

We present a scheme to measure the displacement of a nanomechanical resonator at cryogenictemperature. The technique is based on the use of a superconducting quantum interference deviceto detect the magnetic flux change induced by a magnetized particle attached on the end of theresonator. Unlike conventional interferometric techniques, our detection scheme does not involvedirect power dissipation in the resonator, and therefore, is particularly suitable for ultralowtemperature applications. We demonstrate its potential by cooling an ultrasoft silicon cantilever toa noise temperature of 25 mK, corresponding to a subattonewton thermal force noise of0.5 aN /#Hz. © 2011 American Institute of Physics. $doi:10.1063/1.3570628%

Due to its excellent sensitivity, optical interferometry isthe most widely used technique to detect the motion of ul-trasensitive mechanical resonators, for applications whichrange from magnetic resonance force microscopy !MRFM",1investigation of quantum effects in mechanical systems,2 andfundamental physics experiments.3 Unfortunately, optical de-tection becomes hard to implement when the size of the reso-nator is pushed to the nanoscale, because of the diffractionlimit, and when low or ultralow temperatures are required toreduce the thermal force noise, as for single spin MRFM. Inthe latter case, resonator heating due to light absorption isfound to limit the effective cooling of the resonator.4 Thisproblem can be partially circumvented only by substantiallyreducing the input light power, at the price of reducing thedisplacement sensitivity. Other techniques have been re-cently demonstrated to be more compatible with ultralowtemperatures. In particular, both single electron transistors5

and microwave cavities6–8 have demonstrated outstandingdisplacement sensitivity for the detection of nanomechanicalresonators at temperatures below 100 mK. So far, however,their implementation has been limited to systems where de-tector and resonator are tightly integrated, which is not prac-tical for scanning probe applications. Moreover, for micro-wave techniques the direct photon absorption still remains anissue at millikelvin temperatures, which again can only bemitigated by reducing the input power. Displacement sensorsbased on quantum point contacts have also been demon-strated in an off-board setup9 but so far their use has beenlimited to liquid helium temperature.

In this letter, we demonstrate a rather simple alternativedetection technique, based on the use of a dc superconduct-ing quantum interference device !SQUID", which in prin-ciple does not require any power to be directly dissipated inthe mechanical resonator. Our method involves attaching aferromagnetic particle to the end of the resonator $Fig. 1!a"%which, whenever the resonator moves, causes a change inmagnetic flux in a superconducting detection coil, positionedclose to the resonator $Fig. 1!b"%. A cantilever displacement xis thus converted into a coil flux !="x, where the constant "is proportional to the magnetic moment # of the ferromag-

netic particle and depends in a complex way on the coilgeometry and the relative position and orientation of mag-netic moment and coil. The flux change in the detection coilis measured by the dc SQUID amplifier via a superconduct-ing flux transformer of total inductance Lt, which includes acalibration transformer and the SQUID input coil.

In our experiment, we use a silicon resonator consistingof a 100 nm thick single crystal beam, 5 #m wide and

a"Electronic mail: usenko@physics.leidenuniv.nl.

FIG. 1. !a" An electron microscopy image of the silicon resonator with amagnetic sphere attached to its end. The single crystal beam is 100 nm thick,5 #m wide, and 100 nm long. The 4.5 #m diameter magnetic sphere ismade of a neodymium based alloy with remanence Br=0.75 T. The fre-quency of the lowest flexural mode of the resonator is 3084 Hz, with aquality factor of 3.8$104. !b" Circuit diagram illustrating the detectionscheme. The motion x of the resonator induces a flux !="x in the detectioncoil and a current I=−! /Lt in the superconducting detection loop, which ismeasured by the dc SQUID.

APPLIED PHYSICS LETTERS 98, 133105 !2011"

0003-6951/2011/98"13!/133105/3/$30.00 © 2011 American Institute of Physics98, 133105-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:140.105.16.64 On: Thu, 03 Dec 2015 09:34:06

Updated  version  of:  S.L.  Adler  and  A.  Bassi:,  Science  325,  275  (2009)    

Lower  bound:  Collapse  effec1ve  at  the  macroscopic  level  Graphene  disk:  N  =  1011  amu,  d  =  10-­‐5  m,  T  =  10-­‐2  s    

Beyond  CSL  

The  collapse  is  driven  by  a  random  noise.  In  the  CSL  model,  the  noise  is  white  and  no  dissipa1ve    effects  are  included.      This  makes  the  model  rela1vely  easy  to  work  with,  but  not  physically  realis1c.      Progress  has  been  made  in  generalizing  CSL,  both  with  a  colored  spectrum,  as  well  as  with  dissipa1on.          

Collapse  models  in  space  REVIEW: A. Bassi and G.C. Ghirardi, Phys. Rept. 379, 257 (2003)

REVIEW: A. Bassi, K. Lochan, S. Satin, T.P. Singh and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013)

White noise models

All frequencies appear with the same weight

Colored noise models

The noise can have an

arbitrary spectrum

Infinite temperature

models

No dissipative effects

GRW / CSL G.C. Ghirardi, A. Rimini, T. Weber , Phys.

Rev. D 34, 470 (1986) G.C. Ghirardi, P. Pearle, A. Rimini, Phis.

Rev. A 42, 78 (1990) QMUPL

L. Diosi, Phys. Rev. A 40, 1165 (1989)

DP L. Diosi, Phys. Rev. A 40, 1165 (1989)

Non-Markovian CSL

P. Pearle, in Perspective in Quantum Reality (1996)

S.L. Adler & A. Bassi, Journ. Phys. A 41, 395308 (2008). arXiv: 0807.2846

Non-Markovian QMUPL A. Bassi & L. Ferialdi, Phys. Rev. Lett. 103,

050403 (2009)

Finite temperature

models

Dissipation and thermalization

Dissipative QMUPL A.  Bassi, E. Ippoliti and B. Vacchini,

J. Phys. A 38, 8017 (2005).

Dissipative GRW & CSL A. Smirne, B. Vacchini & A. Bassi Phys. Rev. A 90, 062135 (2014)

A. Smirne & A. Bassi Nat. Sci. Rept. 5, 12518 (2015)

Non-Markovian & dissipative QMUPL

L. Ferialdi, A. Bassi Phys. Rev. Lett. 108, 170404 (2012)

Atom  Interferometry  T.  Kovachy  et  al.,  Nature  528,  530  (2015)    

 M  =  87  amu  d  =  0.54  m  T  =  1  s  

Adler

GRW

10-10 10-8 10-6 10-4 10-2 100

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

rc

CS

L

2 DECEMBER 2011 VOL 334 SCIENCE www.sciencemag.org 1214

PERSPECTIVES

In the movie Dances with Wolves, a lone wolf facilitates Lieutenant John Dunbar’s immersion into the complex culture of

the Sioux Indians. This immersion required overcoming multiple cultural barriers. Ecol-ogists and evolutionary biologists face an equally daunting challenge of understanding how environmental change affects ecological and evolutionary dynamics ( 1). Historically, researchers examined these impacts in isola-tion. However, these dynamics can occur on similar time scales, resulting in a dynamic evolutionary-ecological feedback loop ( 2). Studying these feedbacks directly for long-lived species is often thought to be imprac-tical. On page 1275 of this issue, Coulson et

al. ( 3) overcome this barrier using data from radio-collared gray wolves and state-of-the-art mathematical models.

The 280 radio-collared wolves studied by Coulson et al. are direct descendants of 41 gray wolves reintroduced into Yellowstone National Park between 1995 and 1997 ( 4). This reintroduction was part of a larger effort involving a simultaneous reintroduction in Idaho and a naturally colonized population in Montana. It was extremely successful; by 2010, the Northern Rocky Mountain wolf population had expanded to 1651 individuals ( 5). Individuals within this expanding popula-tion vary substantially in body size, coat color, and other observable (phenotypic) traits. Coat color is particularly enigmatic; gray wolves in North America often have black coats, whereas in Eurasia black coats are rare, but the reason for this difference remains unclear ( 6). These traits were recorded for over a decade (from 1998 to 2009) for each collared wolf and their offspring.

To explore the potential ecological and evolutionary responses of the gray wolves

to environmental change, Coulson et al. fuse integral projection models (IPMs) with clas-sical population genetics. Unlike their matrix model counterparts ( 7), IPMs describe the dynamics of populations with traits that vary continuously, such as body size ( 8), as well as discrete traits, such as coat color ( 9). Tra-ditional IPMs track how the number of indi-viduals with a particular body size changes due to births, deaths, and individual growth. The rules underlying these changes are deter-mined by statistical relationships between the body size of individuals and their vital rates such as fecundity, survivorship, and growth.

In gray wolves, a change at a single loca-tion on the genome—the K locus—deter-mines coat color ( 10). To link evolution-ary and ecological dynamics, Coulson et al. extend the IPM to account for this genetic difference between individuals. As a result, the statistical relationships between individ-ual body size and vital rates become geno-

Mathematical Dances with Wolves

ECOLOGY

Sebastian J. Schreiber

Data and modeling of Yellowstone wolf

populations illustrate the complex interrelated

ecological and evolutionary responses to

environmental change.

photon, it could have come from either of the diamond crystals in which one pho-non was excited. The indis-tinguishability of these two possibilities during detec-tion means that the two dia-mond samples coherently shared one phonon, which is the hallmark of a quan-tum-entangled state.

The entanglement be tween the two diamond samples was confi rmed in experiments in which a second laser pulse de-excited the shared phonon and re-emitted a photon that was subsequently detected. By this method, Lee et al. demonstrate that the two diamonds share entanglement at a 98% confidence level. These results provide a striking example that entanglement is not par-ticular to microscopic particles but can mani-fest itself in the macroscopic world, where it could be used in future studies that make fun-damental tests of quantum mechanics.

The demonstration of entanglement in macroscopic systems also has important implications for the ongoing efforts to realize quantum computation and communication. A full-size quantum computer eventually will

need to be a macroscopic device in which entanglement is preserved and used over long times and distances. The lifetime of entangle-ment in the experiment by Lee et al. is still too short for many quantum information applica-tions, in part because of the room-temperature environment and the strong coupling of pho-non modes in solids. However, the experiment emphasizes an important point, that ultrafast optical technology can alleviate the require-ment on quantum coherence time. In future, with improvement of the ultrafast technology, or by using more isolated degrees of freedom in solids—such as as the nuclear spins ( 8) or the dopant rare-earth ions ( 9)—for quantum memory, many more quantum operations

could be done within the coherence time of the solids, even at room temperature.

References and Notes

1. K. C. Lee et al., Science 334, 1253 (2011). 2. L.-M. Duan, Nature 414, 413 (2001). 3. C. W. Chou et al., Nature 438, 828 (2005). 4. T. Chanelière et al., Nature 438, 833 (2005). 5. N. Sangouard, C. Simon, H. de Riedmatten, N. Gisin, Rev.

Mod. Phys. 83, 33 (2011). 6. D. L. Moehring et al., Nature 449, 68 (2007). 7. S. Olmschenk et al., Science 323, 486 (2009). 8. E. Togan et al., Nature 466, 730 (2010). 9. C. Clausen et al., Nature 469, 508 (2011). 10. Supported by the National Basic Research Program of

China (973 Program) 2011CBA00300 (2011CBA00302), Army Research Offi ce, and Air Force Offi ce of Scientifi c Research MURI program.

CR

ED

IT: P. H

UE

Y/S

CIE

NC

E

A B

Mirror

Pump pulse

Pump pulse Scattered photon

Phonon

Beam splitter

Detector

Making quantum connections. The method used by Lee et al. to generate entanglement between two macroscopic diamonds is illus-trated. (A) A pumping laser pulse generates a correlated pair of a phonon inside the diamond as well as a scattered photon. (B) The scattered photons from two diamonds are brought together for interference and detection. When one photon is detected, the two diamonds coherently share a phonon. Thus, the quantum state created has the hallmarks of quantum entanglement.

10.1126/science.1215444

Department of Evolution and Ecology, University of Califor-nia, Davis, CA 95616, USA. E-mail: sschreiber@ucdavis.edu

Published by AAAS

on

May

6, 2

016

http

://sc

ienc

e.sc

ienc

emag

.org

/D

ownl

oade

d fr

om

Interferometric  experiments  and  the  new  models  

The  new  experimental  bounds  are  robust  against  changes  in  the  noise.  It  comes  not  as  a  surprise,  as  these  are  direct  tests  of  the  superposi1on  principle      

Molecular  Interferometry  S.  Eibenberger  et  al.  PCCP  15,  14696  (2013)  

 M  =  104  amu  d  =  10-­‐7  m  T  =  10-­‐3  s  

Entangling  Diamonds  K.  C.  Lee  et  al.,  Science.  334,  1253  (2011).      

 M  =  1016  amu  d  =  10-­‐11  m  T  =  10-­‐12  s  

Non-­‐Interferometric  Experiments  and  the  new  models  

Gravita1onal  wave  detectors  Perhaps  weakened  if  dissipa1on  is  taken  into  account  

X-­‐Ray  Disappears  for  any  reasonable  cutoff  in  the  spectrum  

Can1lever  Perhaps  weakened  if  dissipa1on  is  taken  into  account  

Cold  Atom  Gas  Weakened  only  for  strong  dissipa1on  

Cold  Atom  Gas    

Collapse  and  gravity  

Fundamental  proper1es  of  the  collapse  

It  occurs  in  space.  

It  scales  with  the  mass  of  the  system.    

 

The  possible  role  of  gravity  

The  “natural”  way  to  describe  it  mathema1cally,  is  to  couple  the  noise  field  to  the  energy  density  (the  stress-­‐energy  tensor,  in  a  rela1vis1c  framework).    

Gravity  provides  such  a  coupling.    

 

Problem    

The  coupling  is  not  the  standard  one  prescribed  by  quantum  theory  (which  would  be  linear).  No  one  knows  why  gravity  should  couple  as  prescribed  by  collapse  models  

Diosi  –  Penrose  model  L.  Diosi,  Phys.  Rev.  A  40,  1165  (1989)  

G(x) =G

~1

|x| Gravity.  And  no  other  free  parameter.  

d| ti =

� i

~Hdt+

Zd3x (M̂(x)� hM̂(x)it)dWt(x)

� 1

2

Zd3xd3yG(x� y)(M̂(x)� hM̂(x)it)(M̂(y)� hM̂(y)it)dt

�| ti

Same  equa1on  as  that  of  the  CSL  model.  The  only  difference  is  in  the  noise:  

The  localiza1on  1me  is:  

⌧(x,x0) =~

U(x� x

0)� U(0)U(x) = �G

Zd3rd3r0

M(r)M(r0)

|x+ r� r

0|

Penrose’s  idea:  quantum  superposi1on  è  space1me  superposi1on  è  energy  uncertainty    è  decay  in  1me      (R.  Penrose,  Gen.  Rel.  Grav.  28,  581  -­‐  1996)  

Cri1cisms  

1.  The  model  is  not  derived  following  some  guideline.  It  does  not  explain  why  gravity  enters  the  game  (expect  for  G).  

 

2.  G  and  1/r  do  not  appear  in  the  coupling  between  ma^er  and  gravity,  but  in  the  correla1on  func1on  of  the  noise.  There  is  no  reason  for  that  to  be  the  case.  (Gravity  induced  vs.  gravity  related  collapse  model.)  

 

3.  The  model  diverges  for  point-­‐like  par1cles.  One  needs  to  introduce  a  cut  off.  Then  the  model  depends  on  a  parameter,  the  cut-­‐off  R0.    

 Diosi’s  original  proposal:  R0  =  10-­‐15  m  =  Compton  wavelength  of  a  nucleon.  This  is  jus1fied  by  the  requirement  that  the  model  is  non-­‐rela1vis1c.  

 However,  this  model  pumps  energy  at  a  very  high  rate,  contradic1ng  experimental  data.  To  avoid  this,  one  has  to  introduce  a  large  cut  off,  which  at  present  has  no  jus1fica1on.    

The  Schrödinger-­‐Newton  equa1on  

L.  Diósi.  Phys.  Le^.  A  105,  199  (1984).  R.  Penrose,  Gen.  Relat.  Gravit.  28,  581  (1996).  D.  Giulini  and  A.  Grossardt,  Class.  Quantum  Grav.  29,  215010  (2012)    

i~ d

dt (x, t) =

✓� ~22m

r2 �Gm2

Zd3y

| (y, t)|2

|x� y|

◆ (x, t)

 quantum  spread   gravita1onal  collapse  

It  comes  from  semi-­‐classical  gravity  if  taken  as  a  fundamental  theory  =  ma^er  is  fundamentally  quantum  and  gravity  is  fundamentally  classical,  and  they  couple  as  follows  

Gµ⌫ =8⇡G

c4h |T̂µ⌫ | i The  term  on  the  right  is  nonlinear  in  the  

wave  func1on  

Wrong  collapse  

It  collapses  the  wave  func1on,  but  not  as  prescribed  by  the  Born  rule  

Double  slit  experiment  according  to  standard  QM  

Double  slit  experiment  according  to  the  Schrödinger-­‐Newton  equa1on  

But  there  are  smarter  ways  of  tes1ng  the  equa1on    H.  Yang,  H.  Miao,  D.-­‐S.  Lee,  B.  Helou,  Y.  Chen,  Phis.  Rev.  Le6.  110,  170401  (2013)  A.  Großardt,  J.  Bateman,  H.  Ulbricht,  A.  Bassi,    Phys.  Rev.  D  93,  096003  (2016)    

+  

It  does  faster-­‐than-­‐light  Consider  the  usual  “Alice  &  Bob  sharing  an  entangled  spin  state”  scenario.  

Alice  first  measure  along  the  z  direc1on:  

Then  Alice  measures  along  the  x  direc1on  

M.  Bahrami,  A.  Grossardt,  S.  Donadi  and  A.  Bassi,  New  J.  Phys.  16,  115007  (2014)    

Acknowledgments  The  Group  (www.qmts.it)    •  Postdocs:  S.  Donadi,  F.  Fassioli,  A.  Grossardt    •  Ph.D.  students:  G.  Gasbarri,  M.  Toros,  M.  Bilardello,  M.  Carlesso,  S.  Bacchi,  L.  

Curcuraci  •  Graduate  students:  A.  Rampichini  

Collabora1ons  with:  S.L.  Adler,  M.  Paternostro,  A.  Smirne,  H.  Ulbricht,  A.  Vinante,  C.  Curceanu.      

www.infn.it  

www.units.it  

www.cost.eu