Observing the quantum nonlocality in the state of a massive particle Koji Maruyama RIKEN (Institute...

Observing the quantum nonlocality inthe state of a massive particle 1001

Koji Maruyama

RIKEN(Institute of Physical and Chemical Research)

withSahel Ashhab (RIKEN)

Franco Nori (RIKEN / Michigan)

22/3/07 IMS, IC

Phys. Rev. A 75, 022108 (2007)

Is this state entangled?

Trailer

10012

1

Even if it’s in number basis?

Can this single massive particle violate the Bell inequalities?

YES!

A single photon? Or a single massive particle?

with some extra resources/manipulations…

dc 2

1

Backgrounds

a?

beam splittera

b

c

da photon

vacuum

A single photon going through a beam splitter

just a superposition

In (one-photon) mode basis

cdcd

10012

1

Backgrounds

ba01

?

beam splittera

b

c

da photon

vacuum


In the number basis

looks entangled

Is this (mode) entanglement just formal or physical?

10012

1

Backgrounds (Question)


Does this single photon state show quantum nonlocality?

Backgrounds (Question)


Does this single photon state violate the Bell inequalities?

In other words:Can this single photon state be converted to another state(possibly a different physical system) locally so that we canperform the Bell test on it?

N.B. Measurement in the basis is physicallyimpossible!

10 0,0

10012

1

v

History of arguments

Tan et al., 1991; Hardy, 1994:Observing the nonlocality of a single photon with a homodynedetection on each arm.

Vaidman, 1995; GHZ, 1995:No! Hardy’s proposal involves multi-particle states.

Jacobs and Knight, 1996:Eight-port homodyne detection and quantum correlations in quadratures.

Banaszek and Wodkiewicz, 1999:Proposal of Bell-type inequalities with Q- and Wigner functions.

Babichev et al., 2004:Experimental Bell test in the phase space. Violation observed.

van Enk, 2005:Some arguments supporting the existence of ‘real/useful’ entanglement in

.Drezet, 2006:

Entanglement in is just formal (exists only on paper).van Enk, 2006:

Drezet is wrong! Useful entanglement can be withdrawn from .

1001

1001

1001

Mode entanglement of a photon

A simple conversion of mode entanglement into a Bell-testable form

g

g

1

0

gg 10012

1

dcgeaegaH

geeg 2

100

c

d

local interactions only

N.B. Measurement in the basis is possible.eg

Mode entanglement of a massive particle

The same scheme doesn’t work for massive particles.

g

g

1

0

gg 10012

1 gedegc 2

1

c

d

BS for particles

a particle

gegeegeg 2

1Tr atom flying

No entanglement

The conversion scheme (1)

We shouldn’t throw the flying atom away to keep the entanglement.

0

c

d

10012

1

Assume this is given as a ‘resource’.

auxiliary atom

In order to erase the which-path information, we use another mode-entangled single particle.

the same species as the flying atom being tested

Then, shoot a particle to the BS.

g

g0

c

d

flying atom flying flyingaux.target atom

target

Combine two atoms.


x

dcgedegc 2

1

2

1

gegeegeg 2,01,10,22

1

flying target aux.

# of atoms in modes c and d

Measure the number of atoms in each mode.

With probability 1/2, we get . geeg 2

1


(The concurrence of is 1/2.) flyingtarget Tr

However, counting the atoms is a bit tricky.

gecdegdcaux.flyingaux.flying

geeggeeg 1,11,11,1

In the previous slide, we did the following:


But, in principle, and are orthogonal,thus distinguishable. (They aren’t bound at the same location.)

flyingc

aux.c


flyingaux.

Assume that the auxiliary and flying atoms are each bound in a potential well.

merge

flyingaux.aha! distinguishable!

Superposition of the lowest levels with amp. & phases depending on the initial stateand the details of the merging procedure.


flyingc

aux.c

Need to make and indistinguishable.

How?

flyingc

aux.c

The same for the mode d.

But, in principle, and are orthogonal,thus distinguishable. (They aren’t bound at the same location.)

flyingc

aux.c


Recall that there is a perfect correlation between the flyingand target atoms.

dcgedegc 2

1

2

1

flying target aux.


xv

flyingaux.

Recall that there is a perfect correlation between the flyingand target atoms.

flyingaux.

Voila, indistinguishable!

g

e

target

Let lower the potential well for the aux atom. g

equivalent to a CNOT operation.


The conversion scheme (2): more efficient scheme

Yet, this conversion succeeds only with probability 1/2.

Can it be more efficient?

- Mode entanglement of an auxiliary atom

- Indistinguishability between the flying and aux atoms

The keys were:

More auxiliary atoms may be useful in hiding (erasing) thewhich-path information of the flying atom.


Spatially split BEC has large (quantum) fluctuations in thenumber of atoms in each side, thus could be a good eraserof the which-path information.

BEC of N non-interacting atoms

0

BS

02!

1 .,.,

N

auxdauxcBEC

aa

N

.0

, ,aux

N

jjN jNjf

j

Nf

NjN 2

1,where

,

c

d # of atoms in c

# of atoms in d


In the single aux atom case, the whole state before counting the atoms was

dcgedegc 2

1

2

1

flying target aux.

Replacing the auxiliary atom with the BEC, we have

N

jauxjN jNjfgedegc

0., ,

2

1

j

Nf

NjN 2

1,


Merging the flying atom and the BEC gives

N

jjNjN gefegfjNj

0,1,1,

N

jN

geNegN0

1,00,12

1

2

1

j

Nf

NjN 2

1,


Tracing out the BEC gives the density matrix of the target atom

.

0000

010

010

0000

2

1target

ConcurrenceN2

11

xx

An (accidental) coincidence with theone-auxiliary-particle case (N=1).


The state of BEC after detaching the target atoms is

N

jjNc jNjf

0, ,1

,2

1)1(ddccBEC

where

N

jjNd jNjf

0, 1,and .

Whichever we have, the (amplitude) distribution is the sameas that of BEC before the 1st run.

The density matrix depends only on , not the number ofatoms in each mode.

target

jNf ,

jNf ,


Another shot of a flying atom results in the same amount of entanglement in the (new pair of) target atoms.

The resulting BEC will be in a mixed state --- a classical mixtureof states, each of which has again the same amplitude distribution(for different number states).

Shooting flying atoms of the same species as the BEC will generateequally entangled pairs of the target atoms indefinitely.

NC 211


mode entanglement

0catalyst(BEC)

geeg 2

1

‘useful’ entanglementmassiveparticle

Repeatable many times!

Conclusions

Mode entanglement of massive particles carries indeed real entanglement. It can be converted into Bell testable entanglement

with an auxiliary mode-entangled atom pair with probability 1/2.

with an auxiliary mode-entangled BEC with probability .

The BEC can be used as many times as possible (like a catalyst)to generate highly entangled ‘useful’ atom pairs from incident mode-entangled atoms.

N2

11

Possibly useful for some quantum information processing.

Similar scheme for fermions?

References

Beam splitter for massive particles

e.g., see Zhang et al., PRL (2006).

Observing the quantum nonlocality in the state of a massive particle Koji Maruyama RIKEN (Institute...

Documents

Transcript of Observing the quantum nonlocality in the state of a massive particle Koji Maruyama RIKEN (Institute...