TWIN-PHOTONS WITH ORBITAL ANGULAR MOMENTUMEscola de Computação e Informação Quântica

WECIC 2010 / LNCC/ Petrópolis –RJ / BRAZIL - 13 to 15 October 2010

Geraldo A. BarbosaQuantaSEC - Consultoria e Projetos em Criptografia Física

Av. Portugal 1558 Belo Horizonte MG 31550-000 BrazilNorthwestern University

Department of Electrical Engineering and Computer Science2145 N. Sheridan Road, Evanston, Illinois, 60208-3118, USA

GeraldoABarbosa@gmail.com

0∞

∞

⟩ ⟩ ⟩ ⟩∑ ∑m= l

z signal z idlerm=- n=0

|ψ(t) =| + | J = n - m | J = l +m - n

?

CONTENTS

• Laser beam with Orbital Angular Momentum (OAM)• Odd features of OAM• Twin photon generation• Conditions for a photon with OAM• Twin photon generation with OAM• Wave function for twin photons• Some applications of OAM and problems:

o cryptographyo quantum processor: half adder

Creating a laser beam with OAM

≠rbital ngular omentO A umM Spin

( )0V

= dVε × × = +∫J r E B L S

022 ( ) 0l k s n n sπφ πλ

Δ ≡ = Δ = × ≠−

Oemrawsingh et al., Appl. Optics 43, 688 (2004)

OTHERS: lenses, stressed fibers and so on…

Phase mask(quartz for laser or plastic for weak light)

φΔn0 (air)

n (medium)

Static Hologram

one or more singularities(“charges”)

MOST POPULAR METHODS: Static or dynamic hologram or phase mask

from ablazed hologram

Heckenberg et al., Opt. Quant. Elect. 24, S951 (1992)

Dr. Bernhard Kley (Zeilinger’s holograms)(Friedrich Schiller University - Jena)

Laser beam(TEM00)

linearly polarized (spin=0)

Gaussianintensity profile

How to createan OAM mode?

Historical paper on OAM with light:Allen et al, Phys. Rev. A 45, 8185 (1992)

Laser beam with OAM

( ) φω

ρ ρ

ρ ρ φ

× ⟨ × ⟩=

⟨⟨=

× ⟩ ⟩∫∫∫∫r E B

E Bz

z

zJ d

c d dld

W

Orbital angular momentumSpin angular momentum

(circularly polarized light)

L and S areMEASURABLE

Barnett, Corsica 2004

z

phaseUV beam: “donut” shape

E

B

z

pφ

y

x

Sρ

z

( ) ( ) ( )arctan arctanElectric field: ( )

2

2R

2

2 2R

ρ-l -i l (

kρ zl -i22 z +z i 2p+ y / x ) w(z) l+1 z/zlp p

2p l 2R

l

A 2 2ρL e ew(z) w(z)

E ρ,φ,z = ρ e1+(z/

ez )

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦p=Radial index; l=azimuthal index OAM phase

Intensity and Phase signatures of OAM modes

( ) ( ) ( )arctan

2

2 2R

2

R2

ρ-l -il w(z)

kρ zl -i22 z +zlp i 2p+l+1 z/

lzp

l 22R

p

A 2 2ρL e ew(z) w(z)1+(z

E ( , ,z)= ρ e e/z )

φρ φ⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦Electric field:

2| |=pl plI E

l

0,1 0, 1100

Phase of −=

+z cm

E E

X (cm)

X (cm)

y (cm)

y (cm)

0,2 0, 2100

Phase of −=

+z cm

E E

110 ; 282,466 −= =Rz cm k cm

very to interactions:

e air propagationunder turbulences.g.,

Phase sensitive

Gibson et al., Opt. Exp. 12, 5448 (2004)

⇓ BAD for

cryptography(air, water)

l=0 l=1/2

; [ , ] = ( ); i j kijk kSL L i Lε += + a "true" or canonical angular momentumnotJ L S LEnk & Nienhuis, J. Mod. Opt. 41, 963 (1994)

ODD FEATURES

0i lil

j Mt x

∂ ∂+ =∂ ∂gauge invariant

( )0V

= dVε × ×∫ gauge invariantnot

J r E B The angular momentum flux is gauge invariant

S. M. Barnett, J. Opt. B 4, S7 (2002)

Fractionary (any) values of lOemrawsingh,Ma,Aiello,Eliel,Hooft,

and Woerdman, PRL 95, 24051 (2005)

l is really quantum?

l

l=0

Transparent non-linear medium χ(2)

e.g., a crystal(or χ(3) , for example, an optical fiber)

Twin-photon generation (no OAM)

ωp

or

sωiω

0spontaneous

decay

→Δtpω1

virtuallevel

NO “true” absorption

Possibilities: A continuous rainbow of colors (energy/momentum).Single event: Coherent superposition of all possibilities:Wave function is an ENTANGLED energy-linear momentum state

ω ω ω= += +

p s i

p s ik k k

Type I down-conversion:Signal and idler with same polarization

coincidences

Type I crystals: Ordinary refraction indexfor signal and idler photons:NO azimuthal dependence in propagation

Historical papers:Louisell, Yariv, and Siegman, Phys. Rev. 124, 1646 (1961); Klyshko, Pis’ma Zh. Éksp. Teor. Fiz. 9, 69 (1969) [JETP Lett. 9, 40 (1969)]; Burham and Weinberg, Phys. Rev. Lett. 25, 84 (1970); Ou, Wang, and Mandel, Phys. Rev. A 40, 1428 (1989).

A general condition for a photon state with OAM

ρk

zk

x

y z

k

Phys. Rev. Lett. 85, 286 (2000)Arnaut & Barbosa

( )

( )

†3( ) , , ; ( , ) 0

, , ;

ilz

s

z

t d k g k k s t e a s

g k k s t

φρ

ρ

ψ

φ

⇓⟩ = ⟩

⇒

∑∫

No

k

( ) ( )zL t l tψ ψ⟩ = ⟩

z zi J -i J

= [ , ] 0

: e e i [ , ] ...

z z

z

di J J Hdt

H H H J H Hφ φφ φΔ Δ

=

Δ = + Δ + =⇓

Hamiltonian must have azimuthal symmetry

, assume ( )

Apply rotation o

or angular momentum conse

n

rvation

0

Hamiltonian

Twin-photon wave function with OAM (Type I)

IF the interaction mechanism and the signal and idler propagation have azimuthal symmetry: H does not depend on φ.

Full OAM transfer from pump photons to twins is possible

l

Type I generationSignal and idler with same polarization.

Ordinary refraction indexes for signal and idler photons

|∞

∞

⟩ ⟩ ⟩ ⟩∑ ∑m= l

z signal z idlerm=- n=0

ψ(t) =|0 + | J = n - m | J = l +m - n

Phys. Rev. Lett. 85, 286 (2000)

t

t-ti

i- dτH (τ)h

|ψ(t) = e |0∫

⟩ ⟩

See first theoretical prediction in Phys. Rev. Lett. 85, 286 (2000),Arnaut & Barbosa

|ψ(t)⟩?

signal idler pumpll l+ =“donut” modes in output

First experimental OAM entanglement (Type I)

l1+l2=lp

l1,l2 histograms

Signal and idler entangled in energy, linear momentum, and OAM.

ωs=ωi case

Geometric measurement of OAM (Type I)

4lφΔ ⇒measured

Fit to Δφ

Geometric measurement of OAM

Barbosa and Arnaut, PRA 65, 053801 (2002);Altman, Köprülü,Corndorf,Kumar, and Barbosa, PRL 94, 123601 (2004)

quantization axis

Twin-photon generation Type II (OAM?)

lacks azimuthal symmetry

Signal: ordinary refraction indexIdler: extraordinary index n=n(φ)

ordinary

extraordinary

BBO crystal

Entangled in energy, linear momentum, and polarization.

( )( )11' 1' 1 1' 1

1| | | | |2

H V V H+Ψ ⟩ = ⟩ ⟩ + ⟩ ⟩

1) Are they fully entangled in OAM?2) Just a partial OAM transfer?

OPEN QUESTIONS!

Signal and idler withorthogonal polarizations

Another way for polarization-entanglement (& OAM) : Two crossed Type I crystals (azimuthal symmetry)

Kwiat, Waks, White, Appelbaum, and Eberhard, PRA 60, R773 (1999).

( )( ) 1| | | | |2

H V V H+Ψ ⟩≅ ⟩ ⟩+ ⟩ ⟩from superposition of crystal emissions

Wave function for Types I and II

2 2 ; sin ; ' 'sin ' ( )

2 2; ' '; ( )

' 2 'cos(

or ( )'

'

)R

P

k k

k n k n n n n n

zk

ξ ρ θ ρ θ

π π

ρ ρ ρρ

φ φλ

φ

λ

φ = =

= = = = ¬

⎡ ⎤= + + −⎣ ⎦ variablestransverse

[ ][ ]

0

12

/ 2

arctan( / )( / 2 / 2 ( 2 ) / 2)

sin / 2( )!( ) ( 1) ( )2 ! ! / 2

y xC z C z

l

c zl lPlp lp C

R c zil k ki l l k l z k

l kk l pA lz l p l k

e eπ

ξ ξψ π−

− Δ Δ+ Δ − + Δ

Δ⎡ ⎤ +Δ = − ⎢ ⎥ Δ⎣ ⎦× ×

k plG

longitudinal phase matchconditions

transverse phase matchcondition

OAM phasesignal and idler phase

TWIN-PHOTONS OCCUR AROUND MAX OF LONGITUDINAL AND TRANSVERSE CONDITIONS

Arnaut&Barbosa, PRL 85, 286 (2000),Barbosa,Eur. Phys. J. D 22, 433–440 (2003),Barbosa, PRA 76, 033821 (2007),Barbosa, PRA 80, 063833 (2009)

'( ) ( )ω πδ ω ω ωΔ → + − PT k k3 .( ) ( ) ψ ψ − ΔΔ = ∫

I

ilp lp

V

d r r e k rk 'Δ = + − Pk k k k

3 3, ; ', '

, '0

( , ) ( , ')| ( ) | 0 ' ( ) ( ) ( , ) ' ( , ') | 02 σ σ

σ σω σ ω σψ ω ψ σ σ

ε++

⟩ ⟩ − Δ Δ ⟩∑ ∫ ∫ lp't d k d k A T a a 'k k k k k k k

( )( , , ; ) ( ) ; ( ) Laguerre-Gauss pump field ωρ φ ψ ψ−= =P Pi k z tlp lp lpz t eE r e r

( )1 1. . .2 2

I I

NL

V V

H dV dV= + +∫ ∫D E B H E P

Phase match and coincidence structures in Type II

Barbosa, PRA 80, 063833 (2009)

• Where are the signals of OAM’sfrustrated or partial transfer?• What to look for?

These are OPEN problems!

• Deformations on the coincidence structures?• Poor OAM’s transfer efficiency?• OAM transfer to crystal?

Feng, Chen, Barbosa, and Kumar, arXiv:quant-ph/0703212

coincidence coincidencesingles singles

2| ( ) |0

lpl

ψ Δ=Idler probability, given detection of a signal photon

kBBO crystal

Poor mode identification

Geometrical restrictions in coincidence detection: poor OAM distinguishability

For and from SPDC :θ φΔ Δ

G. A. Barbosa, PRA 79, 055805 (2009)

Polar restriction Azimuthal restriction

Warning:Be careful

Rz

A wave vector set needed to define anl mode in r-space

zero intensityalong z axis

Example: Cryptography with OAM alphabet (size lMax)

G. A. Barbosa, Opt. Letters 33, 2119 (2008)

Basic PROTOCOL

1. A sets a filter for l and lA at every emission2. Signal and idler photons are sent and detected: lA and lB

(Intruse detection and authenticated transmission assumed)3. A uses a public channel to inform B the detected value lA4. Bob obtains lB + lA = l

OK for a wide collecting geometry

lB

A wants to transmit a sequence of secret values l to B. (each l : li ={0,1,2,…lMax})

(2)χ

Al A Bl l l= +

l jt

UV laser

dynamic randomOAM selectors

DM

PBS

detectorA

Bl

Dovesorter

detectors

B

01

lMax

WARNING: IF collecting geometry is too restrictive (e.g., Δθ∼0: collinear wave vectors): indistinguishability

1 2

21 1 1 |2e l lP ψ ψ⎛ ⎞= − − ⟨ ⟩⎜ ⎟⎝ ⎠

Helstrom's bound

5. Scalable (not exponential growth of resources with number of bits).

control - b

INPUT BITS OUTPUT BITS

b plus carry

sum

b1 b - control1

b2

3 3

Toffoli

C-NOT

1 2b b sum carry

1 1 0 1

1 0 1 0

0 1 1 0

0 0 0 0

Truth table

OTHER USES OF OAM? e.g.,QUANTUM HALF-ADDER with twin photons with OAM

2. Optical gates act on this single photon and ancilla photons.

HOW TO IMPLEMENT THIS IDEA?

singlephoton

with OAM

1. A single photon carries two bits to be summed and the carry bit.

DESIRED:

4. Operation at distinct wavelengths can be used for simultaneousmulti-bit operation and for efficiency increase in single operations.

3. Each single photon detected gives the sum operation plus carry.

QUANTUM CIRCUITS: Barenco, Bennett, Cleve, DiVicenzo, Margolus, Shor, Sleator, Smolin, Weinfurter, Phys. Rev. A 52, 3457 (1995)

1 1 1( )signal in

p s m| psn psm

p s mψ

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⟩ ⇒ = ⊗ ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

V VV

polarization

space

l-momentum

INPUT BITS

p-control

l plus carry

sum

L L L L

C-NOTToffoli

( )o o o out

p s m��

��

�

( )in

psm

1 1 1 1( )p s m2 2 2 2( )p s m

3 3 3 3( )p s m

1 2 3 4

PHYSICAL INPUTS

How to implement the adder algorithm with OAM?A possibility:

GA Barbosa, PRA 73, 052321 (2006)

polarization space OAM

in

out PBS

PBS

M

M

M

Dove

BS

BS

phaseshifter

phaseshifter

j

H V

T R L

B L R

FIRST LOOP

1 1 1 1 1 1( ) ( ) 2inCNOT .V psm p s m p p,s p s psψ ψ= ⇒ = = + −

( )inpsmψ

L miniature:1

Parallel operation (l):1. Simultaneous bit processing2. Redundancy for efficiency increase

T

B

LR

[ ]1 1 1(1 )in

p s mV | sV s I

p s mψ

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⟩ ⇒ ⊗ ⊗ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Effective “interaction”: OAM superposition

C-NOT: Fiorentino and Wong,

Phys. Rev. Lett. 93, 070502 (2004).

ancilla photon (OAM)

V acts on the orbital

angular momentum subspace

L1 IMPLEMENTATION: V+CNOT

1 1

2 2

3 3

4 4

pilot photons

ancilla photons

a b

a b

a b

a

l l ll l ll l ll l l

+ = ⇒

+ = ⎫⎪+ = ⇒⎬⎪+ = ⎭

Assume source with simultaneous multi-pairs:

L1

L2L3

L4

M

M M

M

M M

M

M

M

MM

BS

PBS

PBS

PBS

PBS

PBS

PBS

PBSBS

BS

BS

BSBS

Dove

Dove

Dove

D D

D

D

D

D

l-mask l-mask

l-mask

l-mask

INPUT

PDC (bright oron-demand source)

PDC

PDCPDC

a

b

input

auxiliary

in-loop beam

1

2

t ir

ir t

t r= =

4i

e

�

4i

e

�

4i

e

�

4i

e

�4

i

e

�

N

to polarizersand detectors

to polarizers,OAM masks

and detectors

grating

gratingV VV

polarization

space

l-momentum

INPUT BITS

p-control

l plus carry

sum

L L L L

C-NOTToffoli

( )o o o out

p s m��

��

�

( )in

psm

1 1 1 1( )p s m2 2 2 2( )p s m

3 3 3 3( )p s m

1 2 3 4

1

1

1

1

output:

2 superposition

Lp p,s p s ps,m

== + −=

2

2

2

2

output:

superposition

Lp p,s s,m

===

3

3

3

3

output:

2

Lp p,s ps,m ps m psm

=== + −

4 output:

2 2

o

o

o

Lp p,s p s ps,m ps m psm

== + −= + −

ADDER CIRCUITS: a modular system

SCALABLE? YES: Resources proportional to number of bits

2

1 2pn Mμ μ= ⇒ = 2NHilbert space dimension for N qubits :

1

1

2 ( )

T

T

T

n

n i iN i i i

Tni

p qp q n=

=

Δ ΔΔ Δ =

∏∏∼ = number of processes

3 2 M q= ×Total number of modes : (signal and idlers,p,s,m)

2pn q=Number of photons :

q signals q idlers

( )( -1) 12 1

( 1)

pnpN

p

M n !M ! n ! μ

+ ⎛ ⎞→ +⎜ ⎟− ⎝ ⎠

2 2

and 1 1log 1 log 1μμ μ

= = ⇒⎡ ⎤ ⎡ ⎤+ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

SCALABLEpN Nn M

⇓BASIC PROBLEM:

Source for multiple entangled photons?

GA Barbosa, PRA 73, 052321 (2006)

Hilbert space dimension for npphotons in M modes:

Blume-Kohout,Caves,Deutsch, Found. Phys. 32, 1641 (2002)

REMARK on

Algorithms and implementations

Quantum computation algorithms are necessaryfor physical implementations, but not sufficient.

Quantum processes occur in the physical space andrelevant physical implementations may be

very HARD to achieve!

Strong cooperation needed between people creating algorithms and physicists, engineers, chemists…

0 z idle

m= l

z signalm=- 0

rn=

| J|ψ(t) =| + | = l +m -J n= n - m∞

∞

⟩ ⟩ ⟩⟩ ∑ ∑

The End

Thank You