models’of’spontaneous’wave’ func1on’collapse’and ... · damental tests of quantum...
TRANSCRIPT
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: [email protected]
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: [email protected].
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: [email protected]
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.
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