wŁasnoŚci i realizacje stanÓw splĄtanych fotonÓwbluebox.ippt.pan.pl/~wnasal/sem wn ztoc ippt...

Seminarium WN ZTOC IPPT PAN 5.02 2010 str. 1

WŁASNOŚCI I REALIZACJE STANÓW SPLĄTANYCH FOTONÓW

Wojciech Nasalski

ZB Nanofotoniki ZTOC IPPT PAN

quantum entanglement, decoherence, erasure and swapping

and their applications in advanced technology


A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. (1935)

“Can quantum-mechanical description of physical reality be considered complete?”

Two particles move from a source S into spatially separated regions A and B, and yet continue to have maximally correlated positions and anticorrelated momenta. This means one may make an instant prediction, with 100% accuracy, of either the position or momentum of particle A by performing a measurement at B.

on the “hidden variable” description: …”we have left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.”

Albert Einstein: I like to think that the moon is there even if I am not looking at it.

N. Bohr, Phys. Rev. (1935)

“Can quantum-mechanical description of physical reality be considered complete?”

E. Schrödinger, Naturwissenschaften (1935)

“The present situation in quantum mechanics”


EINSTEIN-PODOLSKY-ROSEN PARADOKS

Albert Einstein (1935):

But on one supposition we should, in my opinion, absolutely hold fast: The real factual situation of the system S1 is independent of what is done with the system S2, which is spatially separated from the former.

Erwin Schrödinger (1935):

When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. By the interaction the two representatives [quantum states] have become entangled.

A paradox is a seemingly absurd or selfcontradictory statement or proposition that may in fact be true. Compact Oxford English Dictionary, 2006.

S1 S2 S2

S

2

S1

S

2

S1

S

2

S2


Distributing entanglement and single photons through an intra-city, free-space quantum channel

K.J. Resch1, M. Lindenthal1, B. Blauensteiner1, H.R. Böhm1, A. Fedrizzi1, C. Kurtsiefer2,3, A. Poppe1, T. Schmitt-Manderbach2, M. Taraba1, R. Ursin1, P. Walther1, H. Weier2, H. Weinfurter2, and A. Zeilinger1,4

¹Institut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria ²Sektion Physik, Ludwig Maximilians Universität, D-80797, München, Germany 3Quantum Information Technology Lab, National University of Singapore, 2 Science Drive 3, 117542 Singapore 4IQOQI, Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Austria

Abstract: We have distributed entangled photons directly through the atmosphere to a receiver station 7.8 km away over the city of Vienna, Austria at night. Detection of one photon from our entangled pairs constitutes a triggered single photon source from the sender. With no direct time-stable connection, the two stations found coincidence counts in the detection events by calculating the cross-correlation of locally-recorded time stamps shared over a public internet channel. For this experiment, our quantum channel was maintained for a total of 40 minutes during which time a coincidence lock found approximately 60000 coincident detection events. The polarization correlations in those events yielded a Bell parameter, S=2.27±0.019, which violates the CHSH-Bell inequality by 14 standard deviations. This result is promising for entanglement-based free space quantum communication in high-density urban areas. It is also encouraging for optical quantum communication between ground stations and satellites since the length of our free-space link exceeds the atmospheric equivalent.

Our results show that high-fidelity transfer of entangled photons is possible under these real-world conditions.

©2005 Optical Society of America


EINSTEIN-PODOLSKY-ROSEN PARADOKS

the state at B is measured “without one can “predict with certainty” the state disturbing the second system” at A at A, even without any measurement at A

Ψ B = x B xB ; xA Ψ dxB Ψ A = xA xA ; xB Ψ dxA

Φ B = p B pB ; pA Φ dpB Φ A = pA pA ; pB Φ dpA

𝑥 𝐵 Ψ 𝐵 = 𝑥𝐵 ⇒ 𝑥 𝐴 Ψ 𝐴 = 𝒙𝑩 + 𝒙𝟎

𝑝 𝐵 Φ 𝐵 = 𝑝𝐵 ⇒ 𝑝 𝐴 Φ 𝐴 = −𝒑𝑩

𝑥 , 𝑝 = 𝑖ℏ ∆𝑥 ∆𝑝 ≥ ℏ 2

xA ; xB Ψ ≡ Ψ xA , 𝑥𝐵 = 1

2𝜋ℏ 𝑒+𝑖𝑝

ℏ[xA −(𝑥𝐵 +𝑥0)]𝑑𝑝 = 2𝜋ℏ 𝛿(xA − (𝑥𝐵 + 𝑥0))

𝑝𝐴 , 𝑝𝐵 Φ ≡ Φ 𝑝𝐴 , 𝑝𝐵 =1

2𝜋ℏ 𝑒−𝑖𝑥

ℏ[𝑝𝐴 +𝑝𝐵 )]𝑑𝑥 = 2𝜋ℏ 𝛿(𝑝𝐴 + 𝑝𝐵)


EINSTEIN-PODOLSKY-ROSEN-BOHM PARADOKS

D. Bohm, Prentice-Hall (1951)

“Quantum theory”

The Bohm gedanken EPR experiment. Two spin -1/2 particles prepared in a singlet state move from the source into spatially separated regions A and B, and give anticorrelated outcomes for JA and JB.

J. S. Bell, Physics (1964)

“On the Einstein-Podolsky-Rosen paradox”

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. (1969)

“Proposed experiment to test local hidden-variable theories”

The failure of local hidden variables is then indicative of a failure of local realism.


stany czyste cząstki o spinie ½

wzdłuż osi z: ↑z ↓z

𝝍 = 𝜶 ↑𝒛 + 𝜷 ↓𝒛

wzdłuż osi x: ↑x =1

2 ↑z + ↓z ↓x =

1

2 ↑z − ↓z

wzdłuż osi y: ↓y = 1

2 ↑z + i ↓z ↓y = 1

2 ↑z − i ↓z

S z =ℏ

2 ↑z

↑z − ↓z ↓z S z ↑z

, ↓z = ±ℏ

2 ↑z

, ↓z

S x =ℏ

2 ↑z

↓z + ↓z ↑z S x ↑x

, ↓x = ±ℏ

2 ↑x

, ↓x

S y =ℏ

2i ↑z

↓z − ↓z ↑z S y ↑y

, ↓y = ±ℏ

2 ↑y

, ↓y


superpozycje stanów cząstki spinie ½

𝝍 = 𝜶 ↑𝒛 + 𝜷 ↓𝒛 𝝋 = 𝜶′ ↑𝒛 + 𝜷′ ↓𝒛

para cząstek o spinie ½ ½

sA, sB ⇒ +1

2, +1

2 −1

2, −1

2 +1

2, −1

2 −1

2, +1

2

A B ⇒ ↑ ↑ ↓ ↓ ↑ ↓ ↓ ↑

A B ⇒ 0 0 1 1 0 1 1 0

stany Bell’a - maksymalnie splątane stany EPR

B00 = ↑ ↓ + ↓ ↑ / 2 B01 = ↑ ↓ − ↓ ↑ / 2

B10 = ↑ ↑ + ↓ ↓ / 2 B11 = ↑ ↑ − ↓ ↓ / 2

„EPR spin singlet” - całkowity spin równy zero

𝐵01 = 𝑠 = 0, 𝑠𝑧 = 0 = ↑ ↓ − ↓ ↑ / 2


bity i qubity

Bit: 0 , 1 ↑ , ↓

Dwa bity: 0 0 , 0 1 , 1 0 , 1 1

Qubit: Ψ1 = 𝛼 0 + 𝛽 1 , 𝛼 2 + 𝛽 2 = 1

Ψ′1 = 𝛼′ 0 + 𝛽′ 1 , 𝛼′ 2 + 𝛽′ 2 = 1

Dwa qubity: Ψ2 = c00 00 + c01 01 + c10 10 + c11 11

stany nie splątane – faktoryzują się:

Ψ2 = Ψ1 ⨂ Ψ′

1 = (𝛼 0 + 𝛽 1 )⨂(𝛼′ 0 + 𝛽′ 1 )

𝑐00 = 𝛼𝛼′, 𝑐01 = 𝛼𝛽′, 𝑐10 = 𝛽𝛼′, 𝑐11 = 𝛽𝛽′

stany splątane - bez faktoryzacji:

Ψ2 = Ψ1 Ψ′

1 = = c00 00 + c01 01 + c10 10 + c11 11

𝑐00 ≠ 𝛼𝛼′, 𝑐01 ≠ 𝛼𝛽′, 𝑐10 ≠ 𝛽𝛼′, 𝑐11 ≠ 𝛽𝛽′


Operacje na jednym qubicie

Ψ1 = 𝛼 0 + 𝛽 1 = 𝛼𝛽 ⇒ 𝑈 Ψ1 = Ψ′1 = 𝛼′ 0 + 𝛽′ 1 =

𝛼′𝛽′

Unity: 1 00 1

𝛼𝛽 =

𝛼𝛽

Phase: 1 00 𝑖

𝛼𝛽 =

𝛼𝑖𝛽

Hadamard: 1 11 −1

𝛼𝛽 =

𝛼 + 𝛽𝛼 − 𝛽

Pauli X: 0 11 0

𝛼𝛽 =

𝛽𝛼

Pauli Y: 0 −𝑖𝑖 0

𝛼𝛽 = 𝑖

−𝛽+𝛼

Pauli Z: 1 00 −1

𝛼𝛽 =

+𝛼−𝛽


operacje przelączania na stanach splątanych

Ψ2 = c00 00 + c01 01 + c10 10 + c11 11

transformacje na obydwu (pierwszej i drugiej) cząstkach: 00 → _ _

0 → 0 0 → 0 00 → 00

0 → 0 0 → 1 00 → 01

0 → 1 0 → 0 00 → 10

0 → 1 0 → 1 00 → 11

stany maksymalnie splątane

B00 = 00 + 11 / 2 B01 = 01 + 10 / 2

B10 = 00 − 11 / 2 B11 = 01 − 10 / 2

transformacje tylko na jednej (np. drugiej) cząstce: B00 = 00 + 11 → _ _ + _ _

0 → 0 1 → + 1 B00 → B00

0 → 0 1 → − 1 B00 → B10

0 → 1 1 → + 0 B00 → B01

0 → 1 1 → − 0 B00 → B01


COHERENCE

coherent qubit

ΨA = 𝛼 0 + 𝛽 1

𝜌 𝐴 = ΨA ΨA = 𝛼 2 0 0 + 𝛽 2 1 1 + 𝛼𝛽∗ 0 1 + 𝛽𝛼∗ 1 0

DECOHERENCE

entangled twin qubits

ΦAB = 𝛼 0𝐴 0𝐵 + 𝛽 1𝐴 1𝐵

𝜌 𝐴 = 𝑇𝑟𝐵 ΦAB ΦAB = 𝛼 2 0𝐴 0𝐵 + 𝛽 2 1𝐴 1𝐵

QUANTUM ERASURE

measurement on one part of the entangled twin AB qubit state

ΦAB ⇒ Φ′AB

0𝐴 0𝐴 ΦAB = 𝛼 0𝐴 0𝐵 = 𝛼 Φ′

AB

𝜌 ′𝐴 = Φ′AB Φ′

AB = 0𝐴 0𝐵


stany spinu elektronów w kierunkach Z, X, Y

↑ , ↓

↑𝑥 = ↑ + ↓ / 2

↓𝑥 = ↑ − ↓ / 2

↑𝑦 = ↑ + 𝑖 ↓ / 2

↓𝑦 = ↑ − 𝑖 ↓ / 2

stany polaryzacji fotonów w płaszczyźnie X - Y

↕ , ↔

↗ = ↕ + ↔ / 2

↖ = ↕ − ↔ / 2

↻ = ↕ + 𝑖 ↔ / 2

↺ = ↕ − 𝑖 ↔ / 2

↕ ⟶ 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒕𝒐𝒓 𝑿 ⟶ ↕ ⟶ 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒕𝒐𝒓 𝒀 ⟶ 0

↕ ⟶ 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒕𝒐𝒓 𝑿 ⟶ ↕ ⟶ 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒕𝒐𝒓 𝑿𝒀 (𝟒𝟓𝒐) ⟶ ↗ ⟶ 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒕𝒐𝒓 𝒀 ⟶ ↔

↕ = ↗ + ↖ / 2 ↔ = ↗ − ↖ / 2

Uniform field polarization and phase in the beam transverse plane


stany czyste polaryzacji (jednego) fotonu

w płaszczyźnie X - Y

𝒃𝒂𝒛𝒂: 𝒔𝒕𝒂𝒏𝒚 𝒐 𝒑𝒐𝒍𝒂𝒓𝒚𝒛𝒂𝒄𝒋𝒊 𝒍𝒊𝒏𝒊𝒐𝒘𝒆𝒋 ↕ 𝒊 ↔ , 𝒄𝒛𝒚𝒍𝒊 𝑻𝑴 𝒊 𝑻𝑬

Ψ𝛼𝛽 = 𝛼 ↕ + 𝛽 ↔ =

𝛼𝛽 𝜌 𝛼𝛽 = 𝛼 ↕ ↕ + 𝛽 ↔ ↔ =

𝛼 00 𝛽

stany czyste i separowalne polaryzacji dwóch fotonów mierzone w A i B

Ψ𝐴𝐵 = Ψ𝐴

⊗ Ψ𝐵 =

Ψ𝛼𝛽

Ψ𝛾𝛿 𝜌 𝐴𝐵 = Ψ𝐴

Ψ𝐴 + ΨB ΨB = 𝜌 𝛼𝛽 0

0 𝜌 𝛾𝛿

𝑃𝐴 = Ψ𝐴 𝜌 𝐴 Ψ𝐴 𝑃𝐵 = Ψ𝐵 𝜌 𝐵 Ψ𝐵 𝑷𝑨𝑩 = 𝚿𝑨𝑩 𝝆 𝑨𝑩 𝚿𝑨𝑩 = 𝑷𝑨𝑷𝑩

stany splątane (nieseparowalne) polaryzacji dwóch fotonów mierzone w A i B

Ψ𝐴𝐵 = Ψ𝐴

Ψ𝐵

𝑃𝐴 = Ψ𝐴 𝜌 𝐴 Ψ𝐴 𝑃𝐵 = Ψ𝐵 𝜌 𝐵 Ψ𝐵 𝑷𝑨𝑩 = 𝚿𝑨𝑩 𝝆 𝑨𝑩 𝚿𝑨𝑩 ≠ 𝑷𝑨𝑷𝑩


przestrzenne rozkłady modów wiązek falowych

0 1

Intensity transverse distribution of the EHG beams: (b) a EHG3,3 beam pattern, (c) a EHG4,4 beam pattern.

0 1

Phase transverse distribution of the ELG beam: (b) a ELG2,4 beam pattern, (c) a ELG3,2 beam pattern.


stany Bell’a fotonów

z ortogonalnymi modami przestrzennymi

𝐵00 = + 0 𝐴 1 𝐵 + 1 𝐴 0 𝐵 2 𝐁𝟎𝟏 = + 0 𝐴 1 𝐵 − 1 𝐴 0 𝐵 2

𝐵10 = + 0 𝐴 0 𝐵 + 1 𝐴 1 𝐵 2 𝐵11 = + 0 𝐴 0 𝐵 − 1 𝐴 1 𝐵 2

symetrie ze względu na przestawienia cząstek 𝐴 ⇔ 𝐵

B00, B10, B11 – symetryczne, B01 – antysymetryczne

Fotony – bozony – stany symetryczne względem przestawień 𝐴 ⇔ 𝐵

Polaryzacja liniowa wiązek fotonów: TE ↕ , TM ↔ ,

stany Bell’a fotonów

z ortogonalnymi stanami polaryzacji

𝐵00 = + ↕ 𝐴 ↔ 𝐵 + ↔ 𝐴 ↕ 𝐵 2 𝐁𝟎𝟏 = + ↕ 𝐴 ↔ 𝐵 − ↔ 𝐴 ↕ 𝐵 2

𝐵10 = + ↕ 𝐴 ↕ 𝐵 + ↔ 𝐴 ↔ 𝐵 2 𝐵11 = + ↕ 𝐴 ↕ 𝐵 − ↔ 𝐴 ↔ 𝐵 2


stany splątane fotonów

Mieszanie trzech fal na nieliowości drugiego rzędu z dopasowaniem fazowym typu II

𝜔𝑝 = 𝜔↕ + 𝜔↔ 𝒌𝑝 = 𝒌↕ + 𝒌↔


generacja koherentnej superpozycji stanów fotonów

(from above) (↘) in 𝑟 = + 1

2, 𝑡 = + 1

2 out (↗) (to above)

0 → 0 + 1 / 2

1 → 0 − 1 / 2

(from below) (↗) in 𝑟′ = − 1

2, 𝑡′ = + 1

2 out (↘) (to below)

Transformacja Hadamarda: 𝐻 0

1 =

1

2 0 + 1

0 − 1


interferencja koherentnych superpozycji stanów fotonów

0 → 0 + 1 / 2 → 0

1 → 0 − 1 / 2 → 1

𝐻 𝐻 0

1 =

0

1


generacja splątanych stanów Bell’a fotonów (wersja I)

outA symmetric inA, inB symmetric outB

+ 0 𝐴 1 𝐵 + 1 𝐴 0 𝐵 2 0 𝐴 1 𝐵 1 𝐴 0 𝐵 + 0 𝐴 1 𝐵 + 1 𝐴 0 𝐵 2

0 𝐴 0 𝐵

1 𝐴 1 𝐵

+ 0 𝐴 1 𝐵 − 1 𝐴 0 𝐵 2 1 𝐴 0 𝐵 0 𝐴 1 𝐵 − 0 𝐴 1 𝐵 + 1 𝐴 0 𝐵 2

outA antisymmetric inA, inB antisymmetric outB

Ψ± = 0 𝐴 1 𝐵 ± 𝑒𝑖𝜑 1 𝐴 0 𝐵 / 2


generacja splątanych stanów fotonów wersja II


+ 0 𝐴 0 𝐵 + 1 𝐴 1 𝐵 2 0 𝐴 0 𝐵 1 𝐴 1 𝐵 + 0 𝐴 0 𝐵 + 1 𝐴 1 𝐵 2

0 𝐴 1 𝐵

1 𝐴 0 𝐵

+ 0 𝐴 0 𝐵 − 1 𝐴 1 𝐵 2 1 𝐴 1 𝐵 0 𝐴 0 𝐵 − 0 𝐴 0 𝐵 + 1 𝐴 1 𝐵 2


Ψ± = 0 𝐴 0 𝐵 ± 𝑒𝑖𝜑 1 𝐴 1 𝐵 / 2


pomiar A skorelowany z pomiarem B

na splątanej parze bliźniaczych cząstek – spin singlet: 0 1 − 1 0 ⇔ ↑ ↓ − ↓ ↑

Pomiar 1: Bob makes no measurements ⇒ Alice’s measurements 𝑺 z or 𝑺 x show random results

Pomiar 2: Bob measures 𝑺 z (↑ 𝒐𝒓 ↓) ⇒ Alice measures 𝑺 z (↓ 𝒐𝒓 ↑) with 100% correlation

Pomiar 3: Bob measures 𝑺 z (↑ 𝒐𝒓 ↓) ⇒ Alice measures 𝑺 x (↗ 𝒐𝒓 ↖) with random corellation

↑𝑥 = ↗ = ↑ + ↓ / 2 ↓𝑥

= ↖ = ↑ − ↓ / 2


Pomiar 2 typu A(↑, ↓) 𝑖 𝐵( ↑, ↓ ) stanu splątanego dwóch cząstek ↑ ↓ − ↓ ↑

Alice Bob

↑ ↓ − ↓ ↑

| ↓ ↑

| ↓ ↑ ↓ ↑

↑ ↓ − ↓ ↑

| ↑ ↓

| ↑ ↓ | ↑ ↓

S1 S2 S2

S

2

S1

S

2

S1

S

2

S2

S

2

S2

S

2

S1

S

2

S2

S

2

S1

S

2

S1

S

2

S1

S

2

S2

S

2

S2

S

2


Pomiar 3 typu A(↗, ↖) 𝑖 𝐵( ↑, ↓ ) stanu splątanego dwóch cząstek ↑ ↓ − ↓ ↑

↑ ↓ − ↓ ↑ = ( ↗ + ↖ ) ↓ − ( ↗ − ↖ ) ↑ = − ↗ ↖ + ↖ ↗

Alice Bob

↑ ↓ − ↓ ↑

↓ = ↗ − ↖ ↑

↑ ↗ or ↑ ↖ ↑ ↗ or ↑ ↖

↑ ↓ − ↓ ↑

↑ = ↗ + ↖ ↓

↓ ↗ or ↓ ↖ ↓ ↗ or ↓ ↖

S1 S2 S2

S

2

S1

S

2

S1

S

2

S2

S

2

S2

S

2

S1

S

2

S2

S

2

S1

S

2

S1

S

2

S1

S

2

S2

S

2

S2

S

2


wymiana informacji poprzez operacje na jednej cząsteczce

C. H. Bennett and S. J. Viesner, Phys. Rev. Lett. (1992)

Alice ↑ ↓ − ↓ ↑ Bob

1

2 ↑ − ↓ 1

2 ↓ − ↑

1

2 ↑ 𝑈(↓) − ↓ 𝑈(↑)

𝐵11 = ↑ ↓ − ↓ ↑ / 2,

𝐵01 = ↑ ↓ + ↓ ↑ / 2,

𝐵10 = ↑ ↑ − ↓ ↓ / 2,

𝐵00 = ↑ ↑ + ↓ ↓ / 2

.

↑ 𝑈(↓) − ↓ 𝑈(↑) 𝑈(↓) + 𝑈(↑)

S2

S

2

S1

S

2

S1

S

2

S2

S

2


High-fidelity transmission of polarization encoded qubits

from an entangled source over 100 km of fiber

Hannes Höbel1, Michael R. Vanner

1, Thomas Lederer

1, Bibiane Blauensteiner

1,

Thomas Lorünser2, Andreas Poppe

1, Anton Zeilinger

1,3

1Quantum Optics, Quantum Nanophysics and Quantum Information, Faculty of Physics,

University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria 2Austrian Research Centers GmbH - ARC, Donau-City-Str. 1, 1220 Vienna, Austria

3Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

Abstract: We demonstrate non-degenerate down-conversion at 810 and 1550 nm for long-distance fiber based quantum communication using polarization entangled photon pairs. Measurements of the two-photon visibility, without dark count subtraction, have shown that the quantum correlations (raw visibility 89%) allow secure quantum cryptography after 100 km of non-zero dispersion shifted fiber using commercially available single photon detectors. In addition, quantum state tomography has revealed little degradation of state negativity, decreasing from 0.99 at the source to 0.93 after 100 km, indicating minimal loss in fidelity during the transmission.

© 2007 Optical Society of America

810 nm

1550 nm

Alice Bob

Pol. Cont.

Pol 1550

Pol 810

0 . . . 100 km

Si-APD

PolarizationEntangled

Source

Delay

GatedInGaAs-

APD

Counter

Trigger-

signal

Fig. 1. (color online) Distribution and Measurement Scheme: A source of polarization en-tangled photon pairs produces photons at 810 and 1550 nm. The 810 nm pair-photons arepolarization analyzed locally by Alice with a rotatable polarizer (Pol 810) and subsequentlydetected by her silicon APD (Si-APD). The partner photons at 1550 nm are transmitted toBob via telecom fibers on spools, analyzed (Pol 1550) and detected by an InGaAs-APD.The random rotation in polarization is compensated by an electronic polarization controller(Pol.Cont.). The trigger signals are carefully matched by an electronic delay generator (De-lay) to the transmission time through the fiber spools and the measured coincidence rate isdisplayed on a counter.

larization state due to depolarization, mainly polarization mode dispersion (PMD). With theseeffects in mind, long-distance fiber transmission has been performed using time-bin entangledphotons, reaching distances of 50 km [5] and 60 km [6].

Recent experiments have demonstrated polarization entanglement distribution up to100 km [7] in fibers. However, in these experiments quantum correlation could only be shownafter subtraction of background counts. In addition, the use of standard wavelength-division-multiplexing components in the quantum channel was demonstrated for polarization entangledphoton pairs [7, 8] combining classical and quantum signals in a single fiber.

These results are encouraging as polarization encoded qubits can be easily produced, manipu-lated and detected with simple optical components. Hence the use of polarization entanglementin optical fibers brings several advantages; for instance, the construction of entanglement basedQKD setups with only passive components reduces the risk of side channel attacks. Also, itallows more convenient control for interaction with atoms, thereby giving greater opportunityfor the development of quantum memory and repeater devices (e.g. conversion of polarizationencoded photons into atomic qubits [9]).

Our two-photon distribution and measurement scheme is depicted in Fig. 1. A source ofpolarization entanglement distributes photon pairs to Alice and Bob. Alice performs local po-larization analysis on the 810 nm photon of the pair. The 1550 nm photon is transmitted to Bobwho performs his own polarization analysis. The measured coincidences obtained by Alice andBob can be used to characterize the shared quantum state or obtain a secret key according to theBBM92 protocol for entangled photons [2]. Such a QKD scheme has been used to demonstratethe successful distribution of secret keys in an urban environment [10].

Even though one photon of the entangled pair is measured directly, projecting the state of theother photon, the correlation carried by this photon is of quantum nature. For this reason, quan-tum communication protocols are made possible even if the measurements are not performedsimultaneously. For example, our setup could in principle be used in a teleportation experiment,where a Bell state analysis is performed on the 810 nm photon and the to-be-teleported state.Our 1550 nm photon would then carry the teleported polarization state through the fiber. Theresulting state of such a teleportation to a far away receiver is determined by the quantum corre-lation. The quality of these correlations, measured by quantum state tomography, are presentedin this paper. In addition we investigate how the quantum correlation evolves in long distancefiber transmission and what are the principle causes.

#81491 - $15.00 USD Received 26 Mar 2007; revised 1 Jun 2007; accepted 5 Jun 2007; published 8 Jun 2007

(C) 2007 OSA 11 June 2007 / Vol. 15, No. 12 / OPTICS EXPRESS 7855

400 GHz

1550nmppKTP×2

HV

L1HWP

BP 810

1 nm

L3

L2

SMF

Dichroic

Alice

BP1550Pumplaser

532nm

SMF

Bob

810nm

Birefringentwedges

Fig. 2. (color online) Optical Setup: The solid state diode pumped laser (Pump laser 532nm) is focused (L1) at the interface of the two periodically-poled KTP nonlinear crys-tals (ppKTP) for highly non-degenerate collinear down-conversion. The half-wave-plate(HWP) rotates the pump to excite the horizontal (H) and vertical (V) crystals equally. The810 and 1550 nm photons are spatially separated (Dichroic), recollimated by lens L2 andL3 and coupled into single-mode fibers (SMF). The wavepackets are spectrally confinedand matched with filters BP 1550 and BP 810. A betraying timing offset between the pho-tons of one pair is compensated by the birefringent wedges in the 810 nm arm.

2. Source of polarization entangled photons

For long-distance fiber communication systems it is essential to have a high flux of pair gener-ation, which we realized by producing entangled photons using spontaneous parametric down-conversion (SPDC) [11, 12, 13] in the orthogonally oriented two crystal geometry [14].

Our compact source (40×40 cm), Fig. 2, is pumped by a 532-nm-laser. For equal crystalexcitation, the pump polarization is rotated to 45◦ with respect to the crystal axes and the beamis focussed at the boundary of two crystals. It is believed that optimal focussing is obtainedwhen the Rayleigh range z0, is comparable to the individual crystal length L (L = 4 mm forboth crystals) [15]. The beam size at the focus was measured to be 55 μm, giving a Rayleighrange of ∼ 4.5 mm and hence a L/z0 ratio of 0.9.

The two nonlinear crystals used in the source are quasi-phase matched periodically-poledKTiOPO4 (ppKTP), with a grating spacing of 9.7 μm, which has been tailored for type-Icollinear generation of an asymmetric photon pair at 810 and 1550 nm from a 532 nm pump.These wavelengths were selected because of the efficient detection at 810 nm, low fiber ab-sorption at 1550 nm and readily available stable radiation sources at 532 nm. The crystals arehoused in a temperature controlled copper mount, heated to approximately 65 ◦C. Varying thetemperature allows wavelength tuning for collinear emission. Each ppKTP crystal produces apair with an intrinsic bandwidth of 800 GHz, which was reduced to 400 GHz with filters BP810 and BP 1550 to minimize chromatic dispersion.

If the down-conversion processes in the two crystals with orthogonal polarizations are in-distinguishable in terms of spectral, spatial and temporal degrees of freedom, the presenceof a photon pair does not reveal in which crystal it was produced. The superposition of thetwo possible creation events gives rise to the polarization entangled state of the photon pair:|φ〉 = 1√

2

(|H810 H1550〉+ eiφ |V810 V1550〉)

However, chromatic dispersion between the 810 and 1550 nm photon inside the crystal leadsto a temporal distinguishability between pairs generated in the two different crystals and henceloss of entanglement [16]. This was compensated by transmission through birefringent quartzwedges which also allow control of the phase in the entangled state to obtain either of the twotype-I Bell states |Φ±〉.




Source of polarization entangled photons

For long-distance fiber communication systems it is essential to have a high flux of pair generation, which we realized by producing entangled photons using spontaneous parametric downconversion in the orthogonally oriented two crystal geometry. The two nonlinear crystals used in the source are quasi-phase matched periodically-poled KTiOPO4, with a grating spacing of 9.7 μm, which has been tailored for type-I collinear generation of an asymmetric photon pair at 810 and 1550 nm from a 532 nm pump. These wavelengths were selected because of

the efficient detection at 810 nm,

low fiber absorption at 1550 nm and

readily available stable radiation sources at 532 nm. If the down-conversion processes in the two crystals with orthogonal polarizations are indistinguishable in terms of spectral, spatial and temporal degrees of freedom, the presence of a photon pair does not reveal in which crystal it was produced. The superposition of the two possible creation events gives rise to the polarization entangled state of the photon pair:

𝝋 =𝟏

𝟐 𝐇𝟖𝟏𝟎𝐇𝟏𝟓𝟓𝟎 +𝒆𝒊𝝋 𝐕𝟖𝟏𝟎𝐕𝟏𝟓𝟓𝟎 =

𝟏

𝟐 ↔𝟖𝟏𝟎↔𝟏𝟓𝟓𝟎 +𝒆𝒊𝝋 ↕𝟖𝟏𝟎↕𝟏𝟓𝟓𝟎

HHHVVH

VVHH HV VH VV

0

0.25

0.5

HHVVH

VV

0

5

HHHVVH

VVHH HV VH VV

0

0.25

0.5

HHVVH

VV

0

5

HHHVVH

VVHH HV VH VV

0

0.25

0.5

HHVVH

VV

0

5

HHHVVH

VVHH HV VH VV

0

0.25

0.5

HHVVH

VV

0

5

0 km

101 km75 km

50 km(a)

Fig. 3. (color online) (a) Real part of measured density matrices at fiber lengths of 0, 50,75 and 101 km (raw data). The imaginary part is close to zero with no element higher than0.09. (b) Uncorrected (�) and corrected (�) logarithmic negativity.

The logarithmic negativity lies between zero (separable states) and one (maximal entangledstates). It also bounds the maximal amount of entanglement that can be extracted (distilled)from the given state [21].

Figure 3(b) displays the values of EN for increasing fiber lengths. EN is calculated from theraw data and also from corrected data, where the background has been removed. The actualbackground was measured for each individual data point by adding an additional 10 ns delaythereby temporally displacing the gate window from the coincidence peak. The recorded co-incidences are then composed of detector dark counts and multi pair contributions only andwere subtracted from the measured coincidences to calculate a corrected density matrix. Theuncorrected values decrease from 0.94 at 0 km to 0.83 at 75 km. At 101 km, the negativity ishigher at 0.88 due to lower losses in set II. When background counts are removed E N decreasesfrom 0.99 to 0.93 in the first 50 km then remaining constant up to 101 km.

4.2. Two-photon visibility

Although the tomography portrays the full information about the quantum state, in quantumcommunication applications like QKD only a subspace of measurement bases is used in prac-tice. We therefore measured the visibility in the HV and the +− basis for fiber lengths up to101 km. The two-photon visibility (V ) was calculated using the definition V = Max−Min

Max+Min , whereMax is the maximal coincidence rate as obtained for parallel polarizer settings at Alice and Boband Min is the coincidence rate at orthogonal settings.

The measured data points in Fig. 4(a) show the uncorrected visibilities (no background sub-tracted) of the HV and the +− basis as a function of fiber length. We display the average of thetwo visibilities since any negative effect on the polarization induced by the fiber should affectboth polarizations basis equally. Indeed, we found the difference of the two visibilities (HV,+−) to be less than 3%.

After the first spool, a decrease in visibility from 96.6% at the source to 93% is observed.The visibility remains approximately this value for transmission distances up to 60 km beforedecreasing steadily to 82.7% at 88 km. For the 101 km fiber we see yet a higher value of 88.6%,




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