Class I Photonic sources of entanglement: what is ... ICFO.pdf · from the point of view of common sense. The theory of quantum electrodynamics describes Natures as absurd from the

Class I

Photonic sources of entanglement: what is

entanglement and how to generate it

Juan P. Torres, ICFO-Institut de Ciencies Fotoniques

1. What is entanglement?

• Separability vs entanglement

• What can I do with entanglement?

• How to quantify the degree of entanglement

2. How can I generate entanglement (with photons)?

• Engineering quantum states: an example

entangled isit not, If

:if separable is state quantum A

i

Bi

Aii

AB p

What is entanglement?

AB

E. Schrödinger

1935 AB

photon 1: H, photon 2 : V

photon 1: V, photon 2: H

Correlations…but…is this entanglement?

Paired

photons

H H

V V

PBS PBS

2

1

3

4

drst

rczy

szbx

tyxa

***

**

*

|H>1|H>2 |H>1|V>2 |V>1|H>2 |V>1|V>2

<H|1<H|2

<H|1<V|2

<V|1<H|2

<V|1<V|2

Classical correlations

0000

02/100

002/10

0000

AB

0000

02/12/10

02/12/10

0000

AB

Quantum correlations

0000

02/12/10

02/12/10

0000

AB

2/10

02/1B

2/10

02/1ABB

A Tr

A B

0000

02/100

002/10

0000

AB

Paired

photons

a

b

cosexpsin

expsincos

i

iR

VH

VH

V

H

||

||45

|

|0

0

0

3

1

2

4

Paired

photons

VH bb |sin|cos 3

1

2

4

00a

b

H|

bbP 223 sin

Paired

photons

VH bb |sin|cos 3

1

2

4

045a b

VH ||

4sin 2

23

bbP 5.023 bP

Bell inequalities

E(a,b)=P23(a,b)+P14(a,b)-P24(a,b)-P13(a,b)

S(a,b)=E(a1,b1)+E(a2,b1)+E(a1,b2)-E(a2,b2)

S=-2 or S=+2

J. S. Bell

2

1

3

4

Aspect, Grangier, Roger, PRL 49, 91 (1982)

Star Trek: the transporter

Transporter: Transportation device that converts objects or

persons to energy, sends that energy to the destination, and

reconstitutes the objects/persons back into matter

I can not know the state of a single photon with a single measurement !

......I need to make a thousand experiments !!

I can distinguish

two orthogonal states,

but not

two non-orthogonal states !!

45o

The no-cloning theorem (1982)

|Ya| > + b | > unknown state

|Y |Y|Y amplifier

Photon 1

Photon 2 Photon 3

Photon 3Entanglement

AAAVH Y |||| ba

BABA HVVH Y ||||2

1|

B

}11,10,01,00{

nt)entangleme (maximal 1C nt)entangleme(no0C0C

:ntentangleme much How

Wootters, PRL 80, 2245 (1998)

0000

00

00

0000

* cz

zb

|| zC

Yu and Eberly, PRL 97, 140403 (2006)

Concurrence

Absorción de dos y tres fotonesSpontaneous Parametric Down Conversion (SPDC)

2hf

Pump

virtual state

signal

idler

ground state

hf2

hf1

virtual state

ground state

21 fff

Spontaneous parametric down conversion (SPDC)

Pump photons

(400 nm)

Down converted photons

(signal and idler, 800 nm)

Nonlinear crystal

(L=1 - 50 mm)

)2(c

Efficiency

sphotonsNmWP

Jhc

Enm

p

p

fp

/102100

109.4400

17

19

spairsNefficiencys

/102101 512

2|| iC

iAP

Phase matching conditions

H

VPump, V

Type II

)2(c

screen

pis nnn 2

Spontaneous parametric down conversion

|

Y

BBBAAAAABBBBAA

BBBAAABBAABBAA

qVqHqqfdqddqd

qVqHqqfdqddqd

,,|,,|),,,(

,,|,,|),,,(|

Why bother with space and frequency?

To use a portion of the full quantum space,

we need to erase all distinguishing information

Frequency

Polarization

Spatial shape

YY

0000

02/12/0

02/2/10

0000

||*,

spacefreq

AB Tr

),,(),,( *

AABBBBAABBAAdqqfdqqfqdddqd

|| C

1),,,(),,,( CqqfqqfIfAABBBBAA

}|||{|||BABAAB

HVVH Y

0000

02/12/0

02/2/10

0000

*

AB

0000

02/12/10

02/12/10

0000

AB

ground

excited

pump Stokesanti

Stokes control

ground

excited

ground

excited

pump Stokes

anti

Stokes control

ground

excited

pump

Stokes

anti

Stokes

control

41Raman transitions in cold atomic ensembles

1. What is entanglement?

Quantum correlations

Degree of entanglement

2. What we can do with entanglement:

Bell inequalities

Teleportation

3. How to generate photonic entanglement:

Type II SPDC

Class II

Some basic thinking: information and

quantum mechanics

Juan P. Torres, ICFO-Institut de Ciencies Fotoniques

1. The role of Information in Quantum Mechanics

• Three experiments

2. What QM tells us (and what does not) about Nature

• Impossible things usually do not happen

3. Much ado about nothing?

A personal view of QM

• Quantum Mechanics is our best

description of the world !!

• A strange theory, but it correct!

classical

11. Incomplete

……..

I have been running so sweaty my

whole life

Urgent for a finish line

And I have been missing the rapture

this whole time

Of being forever incomplete

In our description of Nature the purpose is not

to disclose the real essence of the phenomena

but only to track down, so far as it is possible,

relations between the manifold aspects of our

experience (N. Bohr, Collected papers)

A simple rule:

Information is physical

R. Landauer J. A. Wheeler

It from bit

Detector

Detecto

r

Generator

of single

photons BS 50/50

0

1

2

3

4j

4322 1|exp1|exp| rtirttir jj

j

j

cos12

1

cos12

1

4

3

P

P

Detector 3

Detector 4

phase

phase

1

MinMax

MinMaxV

Generator

of single

photons Beam

splitter 50/50

0

1

2

3

4j

bx

ax

x

||

||

|

:2path

:1path

atinitiallyatom

4322 1|||exp1|||exp| atatatat brtairtbtair jj

4322 1|exp1|exp| rtirttir jj

j

j

cos|||12

1

cos|||12

1

4

3

baP

baP

Detector 3

Detector 4

phase

phase

baV |

Max

Min

ninformatiowayWhich:lityPredictabi 2

|1 baP

VisibilityFringe:Visibility baV |

122 VP

Zou, Wang, Mandel, Phys. Rev. Lett.67, 318 (1991)

Zou, Grayson, Mandel, Phys. Rev. Lett. 69, 3041 (1992)

j

j

sin12

1

|||exp|||

3

223

D

DDD

TP

sittsiritir

D

D

TP

TMinMax

MinMaxV

1

Durr, Nonn, Rempe, Nature 395, 33 (1998)

Durr, Nonn, Rempe, Nature 395, 33 (1998)

Durr, Nonn, Rempe, Nature 395, 33 (1998)

I can not know the state of a single photon with a single measurement !

......I need to make a thousand experiments !!

I can distinguish

two orthogonal states,

but not

two non-orthogonal states !!

45o

C. H. Bennet

| 0>

| 1> 45o

| 0>| 1>

Base 1 Base 2

BB84 Protocol

1 0 1 1 0 0 1 1 0 0 1 1

1 - - 1 0 0 - 1 0 0 - 1

1 0 0 1 0 0 1 1 0 0 0 1

ALICE bit sequence

ALICE base

CODING

BOB detection scheme

BOB measurement

Retained bit sequence

What I am going to tell you about is what we teach our physicsstudents in the third or fourth year of graduate school, and you thinkI’m going to explain it to you so you can understand it? No, you arenot going to be able to understand it. Why, then, am I going to botheryou with all this? Why are you going to sit there all this time, whenyou won’t be able to understand what I am going to say? It is my taskto convince you not to turn away because you don't understand it. Yousee my physics students don't understand it. This is because I don'tunderstand it. Nobody does.

R. P. Feynmam,

QED The strange theory of light and matter

Penguin books, page 9, London 1985

The next reason that you might think you do not understand what I am

telling you is, while I am describing to you how Nature works, you won’t

understand why Nature works that way. But you see, nobody understand

that. I can’t explain why Nature behaves in this peculiar way...I am

going to describe to you how Nature is, and if you don’t like it, that’s

going to get in your way of understanding it. It is a problem that

physicist have learned to deal with. They have learned to realize that

whether they like or they don’t like a theory is not the essential

question. Rather, it is whether or not the theory give predictions that

agree with experiment. It is not a question of whether a theory is

philosophical delightful, or easy to understand, or perfectly reasonable

from the point of view of common sense. The theory of quantum

electrodynamics describes Natures as absurd from the point of view of

common sense. And it agrees fully with experiments. So I hope that you

can accept Nature as She is, absurd

R. P. Feynmam,

QED The strange theory of light and matter

Penguin books, page 10, London 1985

A. Peres

A. Peres

I never satisfy myself until I can make a mechanical model of a thing.If I can make a mechanical model I can understand it. As long as Icannot make a mechanical model all the way through I cannotunderstand; and that is why I cannot get the electromagnetictheory. I firmly believe in a electromagnetic theory of light, and thatwhen we understand electricity and magnetism and light we shall seethem all together as parts of a whole. But I want to understand light aswell as I can, without introducing things that we understand evenless of. That is why I take plain dynamics. I can get a model inplain dynamics; I cannot in electromagnetics.

William Thomson, Lord Kelvin,

in Electrodynamics from Ampere to Einstein

(Olivier Darrigol, Oxford University Press)

The goal of understanding electromagnetism was eventuallyabandoned by most physicists, and the search for such “likeness” ofthe ether as the vortex sponge gradually faded as Maxwell’sequations of the electromagnetic field came to be regarded asthemselves fundamentals and self-sufficient. Einstein described theprocess many years later: “One got used to operating with these fieldsas independent substances”, he said, “without finding it necessary togive one’s self an account of their mechanical nature; thus mechanicsas the basis of physics was abandoned, almost unnoticeable,because its adaptability to the facts presented itself as finallyhopeless”. Physicists ceased to feel a need to look for a mechanismbehind electromagnetic laws or believe that their understandingwould be improved by finding one.

Olivier Darrigol,

in Electrodynamics from Ampere to Einstein

(Olivier Darrigol, Oxford University Press)

Most physicist use quantum mechanics every day in their workinglives without needing to worry about the fundamental problems of itsinterpretation. Being sensible people with very little time to follow upall the ideas and data in their own specialties and not having to worryabout it...So irrelevant is the philosophy of quantum mechanics toits use, that one to begins to suspect that all the deep philosophyabout the meaning of measurements is really empty, forced on us byour language, a language that evolved in a world governed verynearly by classical physics. But I admit to some discomfort in workingall my life in a theoretical framework that no one really understand.

S. Weinberg,

Dreams of a final theory

Vintage, p. 66, London 1993

Quantum mechanics has a reputation for being strange,difficult and incomprehensible to ordinary mortals. Yet Ibelieve it is not more difficult to understand quantummechanics than it was to understand Maxwell in 1885.

F. Dyson

in From Eros to Gaia

(Penguin books, 1992)

I have observed in teaching quantum mechanics, and also learning it,that students go through an experience similar to the one Pupindescribes. The student begin by learning the tricks of the trade.He learns how to make calculations and get the right answers….Tolearn the mathematics of the subject and to learn how to use it takesabout six months….

The second stage comes when the student begins to worry because hedoes not understand what he has been doing. He worries because hehas no clear physical picture in his head. He gets confused intrying to arrive at a physical explanation for each of the mathematicaltricks he has been taught…

Then, unexpectedly, the third stage begins. The student suddenlysays to himself, “I understand quantum mechanics,”, or rather says,“I understand now that there isn’t anything to be understood…”.What has happened is that he has learned to think directly andunconciously in quantum mechanical language

Class III

The Orbital Angular Momentum of light

Juan P. Torres, ICFO-Institut de Ciencies Fotoniques

1. What is the Orbital Angular Momentum (OAM) of light?

2. What can we do with OAM?

3. The quantum OAM

• Correlations

• Spiral spectrum

singular beams

light with OAM

Twisted light

optical tornadoes

vortex beams

phase singularity

tiikzimw

tzyxE

beamGaussLaguerre

mm

expexpexp),,,(

:

2

2

0

tiikzimw

tzyxE

beamGaussLaguerre

mm

expexpexp),,,(

:

2

2

0

ˆˆ2

12

20

2

0

EcmzEcnS

),(),(exp),(

),(),(),(

yxEyxiHyxE

yxEyxTyxE

inT

inT

:HologramsPhase

:HologramsAmplitude

m=0

m=0

m=1

m=-1

M=+1

A computed generated hologram

Phase plates

l

Spatial Light Modulators (SLM)

Angular momentum = (paraxial regime)

“spin” angular momentum

(polarization)

+

Orbital angular momentum

(transverse spatial shape)

Light has energy

NmLz

Light has momentum

hNP

NhfU

NLz RL

Osorio, Molina-Terriza, and Torres, J. Opt. A 11, 094013 (2009)

8. Spiral Phase Contrast Microscopy

Christian Maurer, Stefan Bernet, and Monika Ritsch-Marte

of%25.0 Human cheek cell (50 mm)

4. Trapping and Rotation of Particles in Light Fields

with Embedded Optical Vortices

Michael Mazilu and Kishan Dholakia

12. Rotating Atoms with Light

Kristian Helmerson and William D. Phillips

12. Rotating Atoms with Light

Kristian Helmerson and William D. Phillips

3. Helically Phased Beams, and analogies with Polarisation

Miles Padgett

imEE mm 2exp,00

Frequency

Polarization

Spatial shape

The quantum orbital angular momentum of light

Molina-Terriza, Torres, Torner, PRL 88, 013601 (2002)

Torner, Torres, Carrasco, Opt. Express 13, 813 (2005)

Spiral spectrum

25

Molina-Terriza, Torres and Torner

Nat. Physics 3, 305 (2007).

Spatial entanglement = OAM entanglement

0,0|,| qiapsaqpdqdp

2

sinsinc o sc o sta nta nta n 2121210 yyisxxxxp qpkkqpqpkk

What is the quantum state?

CorrelationsSpiral bandwidth

(entanglement)

2,2

2221112,1,2,11,1

,|,||pm

ppmmpm

pmpmC

Spatial entanglement (OAM)

a new resource for multidimensional

Quantum Information

Protocols in higher dimensional spaces

New ideas, new paradigms

A new tool in quantum information!

m=-1

m=-1

m=0

m=-2

M=+1

&

Beam-

preparation

Crystal

Hologram

Monomode-

Fiber

Coincidence

Detection

&

Beam-

preparation

Crystal

Hologram

Monomode-

Fiber

Coincidence

Detection

m=1

m=1

m=2

m=0

M=+1

&

Beam-

preparation

Crystal

Hologram

Monomode-

Fiber

Coincidence

Detection

&

Beam-

preparation

Crystal

Hologram

Monomode-

Fiber

Coincidence

Detection

&

Beam-preparation

Crystal

Hologram

Monomode-Fiber

CoincidenceDetection

&

Beam-preparation

Crystal

Hologram

Monomode-Fiber

CoincidenceDetection

&

Beam-preparation

Crystal

Hologram

Monomode-Fiber

CoincidenceDetection

Mair, Vaziri, Weihs, Zeilinger, Nature, 412, 313 (2001)

How to measure quantum OAM

m=0

m=0

m=1

m=-1

M=+1

2

sinsinc o sc o sta nta nta n 2121210 yyisxxxxp qpkkqpqpkk

OAM correlations

mp=m1+m2?

2,2

2221112,1,2,11,1

,|,||pm

ppmmpm

pmpmC

C. I. Osorio, G. Molina-Terriza, and J. P. Torres. Phys. Rev. A 77, 015810 (2008)

pump beam

waist

walk-off

mpump=m1+m2

25

PBS

Dichroic

mirrorCollinear PP Type II SPDC

|V>pump |H>signal |V>idler

|V>idler

|H>signal

|V> pump

Coincidence

counter

Laser

DF

DF

mpump=msignal+midler

Effective finite dimensional Hilbert space

Frequency entanglement,

Law et al., Phys. Rev. Lett., 84, 5304 (2000)

Spatial entanglement,

Torres et al., Phys. Rev. A, 68, 050301 (2003)

Spiral bandwidth

Pires et al., Phys. Rev. Lett. 104, 020505 (2010)

|=1/2 [-|00>+ exp (iq1) |11> + exp(iq2) |22> + |33>]

Controlling the spatial shape of the pump beamTorres et al., Phys. Rev. A, 67, 052313 (2003)

1. What is OAM?

• Twisted light

• What can we do with twisted light?

2. OAM in the quantum domain:

• OAM correlations

• Spiral bandwidth

Class IV

Quantum engineering of light:

How to avoid entanglement and how to control the

bandwidth of paired photons

Juan P. Torres, ICFO-Institut de Ciencies Fotoniques

1. The Hong-Ou-Mandel interferometer (HOM)

2. How can I generate separability (or avoid entanglement)?

• Why?

• How

3. Control of the bandwidth of paired photons

• Why?

• How?

Entanglement

Information

Superposition

Purity

1

2

3

4

214

213

2

1

2

1

2

1

2

1

aaia

aiaa

1

2

3

4

1

2

3

4

1

2

3

41

2

3

4

1

2

3

4

3443

44

33

aaaa

aa

aa

R. Glauber

Hong, Ou, Mandel, Phys. Rev. Lett. 59, 2044 (1987).

Hendrych, Micuda, Torres, Opt. Lett. 32, 2339 (2007).

222

211

21212 1:

TrPTrP

TrPPurity

1

2

3

4

cc

The HOM interferometer

11 222

211

2112

TrPTrP

I hate entanglement!!

Frequency

Polarization

Spatial shape

ABspacefreq

ABpol

AB |||

:ntentanglemeonPolarizati

Frequency

Polarization

Spatial shape

AFrequency

Polarization

Spatial shape

B

BAAB |||

HV

VH

||

||

tctc

tctc

tctc

tctc

HVVV

VVHV

VHVH

HHHH

||||

||||

||||

||||

One - photon gate

Two - photon gate

signal

pump

crystal

idler

isiiisssiissisAB HHqqqqdd ||,|,|,,,|

Polarization

Frequency A

Polarization

B

Spatial shape

Polarization

Frequency A

Polarization

B

Spatial shape Spatial shape

isoiissisisAB UHUHdd 0,|,|||,|

2

11exp

2

11exp

2

1111)(,

L

vvi

L

vvi

L

vvvvE

ip

i

sp

s

i

sp

s

sp

ispis sinc

sp vv

pis vvv

111

2

1

sp vv

2 mm BBO pumped at 400 nm 5 mm KDP pumped at 415 nm

Mosley et al., Phys. Rev. Lett. 100, 133601 (2008).

1

2

3

4

cc

The HOM interferometer

cc

3

1

2

4

Making paired photons with a thousand colors

2|| iC

iAP

Phase matching conditions

Green Laser Pointer

ps

isp

nn

kkk

0

zizizz

2exp

2

2exp

2

2cos)(

)2()2()2()2(

N. Bloembergen

02

isp kkk

)(2 sp nn

m

pn

sn

nmps

35.9

8873.18304.1

532064.1

m

Quasi-phase matching

Generation of light with enhanced bandwidth

Carrasco et al., Opt. Lett. 29, 2429 (2004)

ICFO

+

BOSTON UNIVERSITY

+

STANFORD

Generation of quantum light with enhanced bandwidth

Nasr et al., Phys. Rev. Lett. 100, 183601 (2008)

Hendrych, Micuda, Torres, Opt. Lett. 32, 2339 (2007).

Bandwidth: 300 nm, HOM dip: 7.1 fs

1. The HOM

2. How to generate separable quantum

states?

• How to do it and measure it (HOM)

2. How to generate and measure paired

photons with a thousand colors:

• Chirped QPM and HOM, again