Study and characterisation of polarisation entanglement

JABIR M V

Photonic sciences laboratory, PRL

Plan of talk• What is entanglement ?

• Why we need entangled system ?

• Question on completeness of QM

• Bell-CHSH inequality for discrete variables

• Source of entanglement

• Measuring entangled state

• Experimental setup and results

What is entanglement ? Suppose we have a composite system which composes of two subsystems and , if we can write

then it is a product state.

then we call it is an entangled state.

Pairs or groups of particles in such that the quantum state of each particle cannot be described independently – instead, a

quantum state may be given for the system as a whole.

| | a | b

| | |a b

| | |a b

Why we need entangled system ?

• Quantum teleportation

• Quantum cryptography

• Quantum computation

Question on completeness of QM

• EPR questioned element of reality and locality of Quantum theory

Assertion

1Quantum mechanics is

complete if incompatible quantities can not have

simultaneous reality

2Quantum mechanics is

incomplete if incompatible quantities can have simultaneous reality

Measurement by Alice does not change Bob’s system- locality Bob’s spin component is predetermined - realism

Measurement of position on electron and momentum on positron give simultaneous reality

Assertion one is failed- Quantum description of physical reality is incomplete

Introduce local hidden variable to explain this contrary

non-seperable system

Bell-CHSH inequality for discrete variables

• Hidden variable theory: Hidden variables must exist which determine EPR results being necessary to extend quantum mechanics to a complete local and realistic theory.

• Bell’s role: One can find bounds between a local and nonlocal prediction of quantum mechanics.

• Clauser, Horne, Shimony and Holt (CHSH) inequality : Experimental adaptation of Bell’s inequality

• λ - hidden variable ; p(λ) - probability distribution which determines measurement results.

Bell-CHSH inequality for discrete variables

• A(a,λ), A(a’,λ) and B(b,λ), B(b’,λ) – two measurement outcomes for particle A and B. a, a’, b and b’ projection angle

• Possible outcomes are ±1

• Principle of locality - A(a,λ), A(a’,λ) independent of B(b,λ), B(b’,λ) and vice versa

• Thus the correlation value E of the measurement on particle A and B

( , ) p( ) ( , ) ( , ) ( ). ( )E a b d A a A b E a E b ...(1)

• Using the constraints of A and B, a parameter S can be defined as

Lets calculate possible outcome..

S | ( , ) ( , ) ( , ) ( ', ) | | ( ', ) ( , ) ( ', ) ( ', ) |A a B b A a B b A a B b A a B b

2 |)),'(),(( ),'(| |)),'(),(( ),(| bBbBaAbBbBaA .... (2)

±1 ±10±2

0±2

1)

2)

1 0 1 2 2

1 2 1 0 2

3) 1 0 1 2 2

4) 1 2 1 0 2

• Using the constraints of A and B, a parameter S can be defined as

S | ( , ) ( , ) ( , ) ( ', ) | | ( ', ) ( , ) ( ', ) ( ', ) |A a B b A a B b A a B b A a B b

2 |)),'(),(( ),'(| |)),'(),(( ),(| bBbBaAbBbBaA .... (2)

±1 ±10±2

0±2

• Using inequality we can derive the CHSH inequality as

dxxfdxxf )()(

2 |)','(),'(||)',(),(| baEbaEbaEbaES

CHSH inequality

• For local realistic system, and for entangled system

Here,

Where,

is the coincidence counts at angle a and b of corresponding analysers

2 2S

(a, b) (a', b) (a,b') (a', b')S E E E E

(a, b) N(a,b ) N(a ,b) N(a ,b )(a, b)

(a, b) N(a,b ) N(a ,b) N(a ,b )

NE

N

(a, b)N

2 2 2S

EPR-Bell statesThere are mainly four maximally entangled bipartite system called EPR-Bell states.

1| (| HV | VH )

2

1| (| HV | VH )

2

1| (| HH | VV )

2

1| (| HH | VV )

2

Sources of entanglementOne of the popular source of entanglement is spontaneous

parametric down converted photons.

Type-Ι phase matching

for positive uniaxial ( )

for negative uniaxial crystal ( )

3 1 2o e e

p i sn n n

03 1 2e o

p i sn n n

e on n

e on n

1| (| | )

2iHH e VV

Type- ΙΙ phase matching

for negative uniaxial crystal ( )

3 1 2o o e

p i sn n n

3 1 2e e o

p i sn n n

for positive uniaxial crystal ( ) e on n

e on n

A

B

1| (| | )

2iHV e VH

Measuring entangled states

• Characterisation is done by projective measurements.• Here we use polarizer for project to desired state

single state of photons which we can pass through PA

So the coincidence count,

• For diagonal projection-

For linear projection-

| ip

2| | |C N

1 2| | |p p

1| (| | V )

2p H

| | |p H or V

• For linear projection ,

| H 2 21

| cos2

C

Experimental Setup

1. Blue diode laser2. Half wave plate3. Cascaded BIBO crystal4. Polarising beam splitter5. Interference filter @810nm6. Collimator7. Single photon counting module(SPCM)8. Time to digital converter10. Half wave plate

Experimental Setup

1

1. Blue diode laser2. Half wave plate3. Cascaded BIBO crystal4. Polarising beam splitter5. Interference filter @810nm6. Collimator7. Single photon counting module(SPCM)8. Time to digital converter10. Half wave plate

Analyser -I Analyser -II

Result and discussion

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Coincedence(counts /10 s) Singles (counts /10 s) Theoretical fit

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r counts

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CHSH inequality

Where,

is the coincidence counts at angle a and b of corresponding analysers

(a, b) (a', b) (a,b') (a', b')S E E E E

(a, b) N(a,b ) N(a ,b) N(a ,b )(a, b)

(a, b) N(a,b ) N(a ,b) N(a ,b )

NE

N

(a, b)N

• Visibility, V=91.5±0.2%

• Bell’s parameter S = 2.518835± 0.126864

• The state which we have produced is,

1| (| | )

2iHH e VV

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Coincidence Theoretical fit

Coin

cidence

Analyser