topological phase and quantum criptography with spin-orbit entanglement of the photon universidade...
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Topological phase and quantum criptography Topological phase and quantum criptography
with spin-orbit entanglement of the photonwith spin-orbit entanglement of the photonTopological phase and quantum criptography Topological phase and quantum criptography
with spin-orbit entanglement of the photonwith spin-orbit entanglement of the photon
Universidade Federal FluminenseUniversidade Federal Fluminense
Instituto de Física - Niterói – RJ - BrasilInstituto de Física - Niterói – RJ - Brasil
Antonio Zelaquett KhouryAntonio Zelaquett Khoury
Financial Support: Financial Support: CNPq - CAPES – FAPERJCNPq - CAPES – FAPERJ
INSTITUTO DO MILÊNIO DE INSTITUTO DO MILÊNIO DE
INFORMAÇÃO QUÂNTICAINFORMAÇÃO QUÂNTICA
OutlineOutline
• Geom. phase for a spin ½ in a magnetic field
• Geometric quantum computation
• The Pancharatnam phase• Beams carrying OAM• Topological phase for entangled states
• BB84 QKD without a shared reference frame
• Conclusions
Geometric phase of a spin 1/2 in a magnetic Geometric phase of a spin 1/2 in a magnetic fieldfield
Spin 1/2 in a time dependent magnetic field Spin 1/2 in a time dependent magnetic field
)(ˆ)())(( tBStBtBH
)(tB
| (0) | , (0)u
0
( / ) ( ( ')) '( )| ( ) | , ( )ˆ
t
n
n
i E B t dti tt e e u t
BERRY PHASE
0( ) ( )ˆB t B u t
Geometric quantum computationGeometric quantum computation
Geometric conditional phase gateGeometric conditional phase gate
]ˆ[]ˆ[ 000 BSIIBSH BBABAA
)(ˆ)(ˆ20 tBStBSSSJHH BBAABzAz
i
i
i
i
e
e
e
e
2
2
2
2
000
000
000
000
| | | |Conditional phase gate
J.A. Jones, V. Vedral, A. Ekert, G. Castagnoll,
NATURE V.403, 869 (2000)
L.-M. Duan, J.I. Cirac, P.Zoller
SCIENCE V.292, 1695 (2001)
The Pancharatnam phaseThe Pancharatnam phase
Pancharatnam phasePancharatnam phase
2/g
S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A, V.44, 247 (1956)
Collected Works of S. Pancharatnam, Oxford Univ. Press, London (1975).
2/ 2/
Beams carrying orbital angular momentumBeams carrying orbital angular momentum
Gauss-Laguerre beams carrying OAMGauss-Laguerre beams carrying OAM
2 , )2 0t
ψ (r zψ + ikz
(Paraxial Wave Equation)(Paraxial Wave Equation)(Paraxial Wave Equation)(Paraxial Wave Equation)
0V
s oL r p dv L L
Angular momentumAngular momentum
Hermite-Gauss (HG)Hermite-Gauss (HG)
RectangularRectangular
Laguerre-Gauss (LG)Laguerre-Gauss (LG)
CylindricalCylindrical
Poincaré representation for beams carrying OAMPoincaré representation for beams carrying OAM
Poincaré representation of first order Gaussian modes
Cylindrical lenses at 45o
Astigmatic mode converter
2
1
i 2
1
Geometric phase from astigmatic mode conversionGeometric phase from astigmatic mode conversion
2/g
E.J. Galvez, P.R. Crawford, H.I. Sztul, M.J. Pysher, P.J. Haglin, R.E. Williams,
Physical Review Letters V.90, 203901 (2003)
Topological phase for entangled statesTopological phase for entangled states
C. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. KhouryC. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. Khoury
IF-UFFIF-UFF
P. Milman P. Milman
LMPQ – Jussieu - FranceLMPQ – Jussieu - France
Geometric representation for two-qubit statesGeometric representation for two-qubit states
TWO QUBITS
| , |1z
| , y | , x
| , | 0z
| , |1z
| , y | , x
| , | 0z
Two Bloch spheres??
Only for product states!!!
Bloch sphere
(or Poincaré sphere)
| , |1z
| , y | , x
| , | 0z
ONE QUBIT
| | 0 |1
Geometric representation for two-qubit PURE statesGeometric representation for two-qubit PURE states
Bloch ball| 0 0 |
|1 1 |
SO(3) sphere
(opposite points identified)
| | 00 | 01 |10 |11
Two-qubit
PURE STATES
21
2
C
C
(Concurrence)
Maximally entangled state 1 0C Bloch ball colapses to a point!!!!
P. Milman and R. Mosseri, Phys. Rev. Lett. 90, 230403 (2003).
P. Milman, Phys. Rev. A 73, 062118 (2006).
Topological phase for maximally entangled statesTopological phase for maximally entangled states
* *| | 00 | 01 |10 |11
Cyclic evolutions preserveing maximal entanglement (“Closed” trajectories)
Two homotopy classes:
0top
top
0-type trajectories
π-type trajectories
| ( ) | (0)T
| ( ) | (0)T
SO(3) sphere
0
Separable polarization-OAM modesSeparable polarization-OAM modes
0( ) ( ) ( ) ˆ ˆH VE r E r r e e
( )r
( )r
ˆHe
Ve
Nonseparable polarization-OAM modesNonseparable polarization-OAM modes
* *( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆH V H VE r r e r e r e r e
Geometric representation on the SO(3) sphere 1
23
4
0 01 2
0 03 4
( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ2 2
( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ2 2
H V V H
V H H V
E EE r r e r e E r r e r e
iE iEE r r e r e E r r e r e
Nonseparable mode preparationNonseparable mode preparation
Holographic preparation of the LG modesHolographic preparation of the LG modesHolographic preparation of the LG modesHolographic preparation of the LG modes
LG LG LG0-1 00 01
(a)
(b)Laser
2/
PBS
1
E
Interferometric measurementInterferometric measurement
1
23
4
4’
1 2 3 4
4 1 (θ = 00) / 4’ 1 (θ = 900)41
/ 2 / 2 / 2
/ 4
/ 4
CCDθ = 45 0 θ = - 45 0θ = 0 0
θ = 0 0
θ = 0 0, 22.5 0, 45 0, 67.5 0, 90 0
Experimental resultsExperimental results
Unseparable mode
Separable mode
θ=00 θ=22.50 θ=450 θ=67.50 θ=900
θ=00 θ=22.50 θ=450 θ=67.50 θ=900
Theoretical expressionsTheoretical expressions
Unseparable mode
2 2( ) ( ) 2 ( ) 1 cos2 cos sin 2 sin 2 siniqyE r e E r r qy qy
2 2( ) ( ) 2 ( ) 1 cos2 cosiqyE r e E r r qy
Separable mode
Calculated imagesCalculated images
Unseparable mode
Separable mode
Partial separability and concurrencePartial separability and concurrence
Partially separable mode
0( ) ( ) 1 ( )ˆ ˆ
H VE r E r e r e
Interference pattern (θ=450)
2( ) 2 ( ) 1 2 (1 ) sin 2 sin
I r r qy
CONCURRENCE
BB84 Quantum key distribution without a BB84 Quantum key distribution without a shared reference frameshared reference frame
C. E. Rodrigues de Souza, C. V. S. Borges, C. E. Rodrigues de Souza, C. V. S. Borges,
J. A. O. Huguenin and A. Z. KhouryJ. A. O. Huguenin and A. Z. Khoury
IF-UFFIF-UFF
L. Aolita and S. P. Walborn L. Aolita and S. P. Walborn
IF-UFRJIF-UFRJ
The BB84 protocolThe BB84 protocol
0
0
1
1
0
0
1
1
ALICE
Bennett and Brassard
1984Polarizers
HV
+/-
HV
+/-
Polarizers
BOB
H - 45o45oV
Photons
010111100Result
HVHVHV+/-HV+/-+/-+/-HVBasis
000111101Result
HV+/-HVHVHV+/-HV+/-+/-Basis
0 1 1 0 0
Alice and Bob check their basis, but not their results !
ALICE
BOB
Spin-orbit entanglementSpin-orbit entanglement
1
2[ ]
L 0L 1L
Logic basis +/-
Logic basis 0/1
1
2[ ]1L
1
2[ ]
1
2[ ]0L 1
2[ ]
Invariant under rotations ! ! ! !
L. Aolita and S. P. Walborn
PRL 98, 100501 (2007)
BB84 without frame alignmentBB84 without frame alignment
BASIS BASIS
{ 0L 1L },
{ L L },
{ 0L 1L },
{ L L },Photons
0L 1L, L L,,
Robust against alignment noise ! ! ! !
ALICE BOB
Procedure sketchProcedure sketch
??
0L 1LL L
0L 0L
0 1
+ -
BOB
CNOTXX
R(φ)
ALICE
R(θ)
Experimental setupExperimental setup
Experimental resultsExperimental results
Bob’s detector 1
State sent by Alice
Bob’s detector 0
Rotation of Alice’s setup
Bob’s detector 1Alice sends 1
Bob’s detector 0
{ 0L 1L },
Bob`s detection basis:
ConclusionsConclusions
ConclusionsConclusions
• Spin-orbit entanglement
• Topological phase for spin-orbit transformations
• Potential applications to conditional gates
• Quantum criptography without frame alignment