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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
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
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…