quantum optics: from basics to advanced topics...3 outline of the talk •introduction...

1

Quantum Optics: From Basics to Advanced topics

量子光学：基礎編及び展開編

Quantum Information Processing Summer School 2011Kyoto Univ, Kyoto, Japan August 13-14, 2011

Takuya Hirano平野琢也

Department of physics, Gakushuin University学習院大学理学部物理学科

2

Self-introduction

１９８７年３月東京大学理学部物理学科卒業前半：塚田研究室，後半：清水忠雄研究室

１９８７年４月～１９９２年３月東京大学物性研究所松岡研究室

Ｍ１：超短パルス，CuCl中ポラリトン，Ｍ２－Ｄ３：パルス光スクイージング（町田さん）

１９９２年４月～１９９３年２月日本学術振興会特別研究員(PD)

アンチバンチング（小芦さん）連続量と離散量

１９９３年～１９９８年東京大学久我研究室助手

ＬＥＤによるサブポアソン光の発生，レーザー冷却（重点領域でＢＥＣ，鳥井さん）

１９９８年～現在学習院大学

連続量の量子暗号，Ｒｂ原子ＢＥＣ，パルス光スクイージング

X1

X2

0o

90o

180o

270o

α

αiα−

αi−

連続量量子鍵配送

TOF15ms

Transmission0 1

mF = +2, +1, 0 -1, -2

Transmission0 1

mF = +2, +1, 0 -1, -2

F = 2 state mF

+2,+1,0,-1,-2

Stern-Gerlachmethod

多成分凝縮体

磁力計ループホールフリー

ＥＰＲ実験、量子通信

3

Outline of the talk

•Introduction

•Quantization of the electromagnetic fields

•Quantum theory of light

Number states (single photon sources),Coherent states, Squeezed states,Homodyne detection

• Nonlinear optics and Quantum state control

• Correlation function

• Representation of density matrix

• Precision measurements using atoms

4

What is light? What is a photon ?

In a letter to Michele Besso in 1951, Albert Einstein wrote:“All the fifty years of conscious brooding have brought me no closer to the answer of the question: What are light quanta? Of course, today every rascal thinks he knows the answer, but he is deluding himself.”

♦Newton’s particle theory

indivisible particle, localized in space

♦Electromagnetic fields described by Maxwell’s equations

vector wave

♦Quantized electromagnetic theory

wave-particle duality

5

Photoelectric effect

hνννν•The ejection rate of the electrons is proportional to the intensity of the incident light.

•There exist a certain minimum frequency of the incident light below which no electrons are ejected.

•The maximum kinetic energy of the ejected electron increases as the frequency of the incident light increases, but it is independent of the intensity of the incident light.

•The time delay between the incidence of the light and the ejection of electron is very small.

φν −= hEK

“Concerning an Heuristic Point of View Toward the Emission and Transformation of Light,”A. Einstein, Translation into English, American Journal of Physics, v. 33, n. 5, May 1965

“According to the concept that the incident light consists of energy quanta of magnitude Rβν /N, however, one can conceive of the ejection of electrons by light in the following way. Energy quanta penetrate into the surface layer of the body, and their energy is transformed, at least in part, into kinetic energy of electrons. The simplest way to imagine this is that a light quantum delivers its entire energy to a single electron: we shall assume that this is what happens.”

6

Photomultiplier tubes = Photoelectric effect + Secondary emission

http://jp.hamamatsu.com/

7

Performance of photomultiplier tubes

http://jp.hamamatsu.com/

8

Semi-classical theory of photoelectric phenomena

“The concept of the photons,” Marlan O. Scully and Murray Sargent III,Physics Today, March 1972, p.38-47.

( )EreH

KytztyErr

r

⋅−=

−=

1

cosˆ),( ω

Fermi’s golden rule

)/)((][2 20

2hh gkkgk tEreP εεωδπ −−=

Wmvgk +=− 2/2εεWmv += 2/2ωh

Energy difference between the ground state and excited state

霜田光一：光子と光波, 数理科学, No. 427, pp. 5-9 (1999).

9

ＴＥＳ（Transition edge sensor）: Photon number resolving detector

「超伝導単一光子検出器」福田大治他、「超伝導単一光子検出器」福田大治他、「超伝導単一光子検出器」福田大治他、「超伝導単一光子検出器」福田大治他、

電子情報通信学会誌電子情報通信学会誌電子情報通信学会誌電子情報通信学会誌, 90, 674 (2007).

http://itaku-kenkyu.nict.go.jp/seika/h20/seika/90/90_aist.pdf

quantum nature of light ?Opt. Express, 16, 3032, (2008): 95%@1556nmOpt. Express, 19, 870 (2011): 98%@850nm

10

Quantum Optics: From Basics to Advanced topics

• Optical technologies continue developing drastically.

• Such development contribute both basic understanding and advanced application.

• In this talk, I would like to review the field of quantum optics emphasizing on fundamental concepts.

11

Outline of the talk

•Introduction








12

Quantization of the electromagnetic field

Maxwell equations

0 div

div

rot

rot

0

000

=

=

∂∂+=

∂∂−=

B

E

t

EjB

t

BE

r

r

rvr

rr

ερ

µεµ

Potential formulation

( )( )0 gradrot grad

0rot div rot

=∂∂−−=

==

φφ Q

rr

rQ

rr

t

AE

AAB

Coulomb gauge0 div =A

r

0

022

2

2 grad

11

ερφ

µφ

=∆−

=∂∂+

∂∂+∆− j

tct

A

cA

vr

r

Free electromagnetic field

0

2)-(1 0 div

1)-(1 01

2

2

2

==

=∂∂+∆−

φA

t

A

cA

r

rr

4)-(1

3)-(1 rot

t

AE

AB

∂∂=

=r

r

rr

13

Quantization of free electromagnetic field (1)

Periodic boundary condition

L

L

L

Plain wave expansion

( ) ( ) ( ) ( ){ }∑ ⋅−+⋅=σ

σσ,

*,,

expexp),(k

kkrkitArkitAtrA

r

rrrrrrrrrr

( )

=

=

L

n

L

n

L

n

kkkk

zyx

zyx

πππ 2,

2,

2

,,r

K,2,1,0,, ±±=zyx nnnSubstituting to eq. (1-2)

( ) ( ) 2,1 ,0*

,,==⋅=⋅ σσσ ktAktA

kk

rrrrrr

Substituting to eq. (1-1)

( ) ( ) , , 222,

2

2,

2

zyxkkkk kkkkcktAt

tA++==−=

∂∂

ωω σσ

r

r rr

The amplitude of each mode satisfies the same equation as a classical harmonic oscillator.

( ) ( ) 5)-(1 exp,,

tiAtA kkkωσσ −= rr

rr

14


( ) ( ){ }∑ ⋅−+⋅+−=σ

σσ ωω,

*,,

expexp),(k

kkkkrkitiArkitiAtrA

r

rrrrrrrrrr

Substituting to eq. (1-3), (1-4)

( ) ( ){ }rkitiArkitiAiE

EtrE

kkkkkk

kk

rrrrrrr

rrr

rrr

r

r

⋅−+⋅+−=

=∑

ωωω σσσ

σσ

expexp

),(

*

,,,

,,

( ) ( ){ }rkitiArkitiAkiB

BtrB

kkkkk

kk

rrrrrrrr

rrr

rrr

r

r

⋅−+⋅+−×=

= ∑

ωω σσσ

σσ

expexp

),(

*

,,,

,,

Cycle-averaged energy of the electromagnetic field

( )2

,

22

,

*

,,

20

2

,0

2

,0,

2

1

2

1

2

1

σσ

σσ

σσσ

ω

ωε

µε

kkk

kkk

kkk

qp

AAV

dVBE

rr

rr

rrr

rr

+=

⋅=

+= ∫E

( )

( ) σσσσ

σσσσ

ωωε

ωωε

,,,20

*,

,,,20

,

4

1

4

1

kkkk

k

k

kkkk

k

k

ipqV

A

ipqV

A

rrrr

rrrr

rr

rr

e

e

−=

+=

15


Canonical quantization[ ] σσσσ δδ ′′′′ =

kkkkipq rrrr h

,,ˆ,ˆ

Annihilation and creation operators{ }

{ })(ˆ)(ˆ2

1)(ˆ

)(ˆ)(ˆ2

1)(ˆ

,,

†

,

,,,

tpitqta

tpitqta

kkk

kk

kkk

kk

σσσ

σσσ

ωω

ωω

rrr

rrr

h

h

−=

+=

[ ] σσσσ δδ&

rrrrkkkk

tata ′′′ =)(ˆ),(ˆ †,,

( )

σσ

σσσσ

ωε

ωωε

,,0

,,,20

,

ˆ2

4

1

kkk

kkkk

k

k

aV

iqqV

A

rr

rrrr

rh

rr

e

e

=

+=

( ) ( ){ }

( ) ( ){ }

( ) ( ){ }∑

∑

∑

⋅−−⋅+−×=

⋅−−⋅+−=

⋅−+⋅+−=

σσσ

σσσ

σσσ

ωωωε

ωωεω

ωωωε

σ

σ

σ

,,,

0

,,,

0

,,,

0

expêxpˆ2

),(

expêxpˆ2

),(

expêxpˆ2

),(

†

,

†

,

†

,

kkkkk

k

kkkkk

k

kkkkk

k

rkitiarkitiakV

itrB

rkitiarkitiaV

itrE

rkitiarkitiaV

trA

k

k

k

r

rr

r

rr

r

rr

rrrrrrhrr

rrrrrhrr

rrrrrhrr

r

r

r

e

e

e

16

Outline of the talk

•Introduction








17

Quantum states of light (1)

Hamiltonian and electric field for a single mode

{ } eˆe ˆ2

),(

2

1ˆˆˆ

†

0

†

rkitirkiti aaV

itrE

aaH

rrrrhr)

h

⋅−⋅+− −=

+=

ωω

εω

ω

Fock or number states )(ˆ)(ˆˆ ,ˆ † tatannnnn ==

nnnH

+=2

1ˆ ωh

11 ˆ

1ˆ

† ++=

−=

nnna

nnna

E

0

1

2

3

ωh

ωh

ωh

ωh

discrete energy levels

light quanta in a box

state) (vacuum state ground 0In one-photon state |1>,photon will be detected only at a single location in a box.

18

Fock states (2)

Expectation value of electric field

{ }

+=

=

−= ⋅−⋅+−

2

1),(n

0),(n

eˆe ˆ2

),(

0

2

†

0

nV

ntrE

ntrE

aaV

itrE rkitirkiti

εω

εω ωω

hr)

r)

hr) rrrr

variance

( ) ( ) 2222 ˆ EEEEE)))

−=−=∆

( )2

1

0

+=∆ nV

Eε

ωh)

E

t

Phase is indefinite.There exist vacuum fluctuations.Wave ?

19

Fock states (3)

Description of multimode fields

{ }σσσσσσσ ,,,,,

,2,11,1,2,11,1 knnnnnnn

jjkkkjjkkk

rLLLL rrrrrr ==⊗⊗⊗⊗

σσσσ

σσσσσ

ω

ω

,,,,

,,

†,,,

2

1ˆ

2

1ˆˆˆ , ˆˆ

kkkkk

kkkkkk

nnnH

aaHHH

rrrr

r

rrrr

h

h

+=

+==∑

{ }{ }{ }

σ

σσ

σσσσσσσσσ

ψ

,

,

,

,2,11,1,2,11,1

,2,11,1

,,,,

k

k

k

jjkkkjjkkk

jjkkk

nC

nnnCCC

nn

nnnnnn

r

r

r

rrrrrr

rrr

LLLLLL

∑

∑∑∑

=

=

20

Single photons sources

Single atom

Phys. Rev. Lett. 39, 691 (1977).

"A single-photon server with just one atom", Markus Hijlkema, et al., Nature Physics, 3, 253 (2007)

delivering up to 300,000 photons for up to 30 s.

Single photon sources• Single atoms, molecules• Single “artificial” atoms

Quantum dotsNitrogen vacancies in diamond

• Correlated photons sourcesSpontaneous parametric downconversionFor wave mixing

• Attenuated laser as “quasi” single photon

Two photon correlations will be discussed later.

21

Coherent states (1)

Eigenstates of annihilation operator

number.complex a is ,ˆ αααα =a

2

*

† ˆˆ

ˆ

α

αααα

αα

αα

=

=

=

=

aa

nn

( ) 22 α=∆n

( )

( ){ }( )

VE

rktV

ntrE

rktV

trE

k

k

ik

0

2

22

0

2

0

2

1sin42

),(n

e , sin2

2),(

εω

θωαεω

ααθωαεωαα θ

h

rrhr)

rrhr)

=∆

+−⋅−=

=−⋅−=

Independent of α, r and t

E

tn

E

2

1

2

1

Aimpitude==∆

α

Negligible for ordinary lasers.

22

Coherent states (2)

Expansion of |α> in terms of |n>

ondistributi Poissonian !

!

2

2

2

0

2/

nen

nn

e

n

n

n

αα

αα

α

α

−

∞

=

−

=

= ∑

22 βαβα −−= e Coherent states are nonorthogonal.

Displacement operator

( )aa

aa

eee

eD

ˆˆ2/

ˆˆ

*†2

*†ˆ

ααα

ααα−−

−

=

=

[ ] BABABA eeeˆˆˆ,ˆˆˆ =++

Q([ ][ ] [ ][ ] 0 ˆ,ˆ,ˆ ˆ,ˆ,ˆ if == BABBAA )

( ) 0ˆ αα D=

( ) ( ) ˆ ,0ˆ

!

ˆ

ˆˆ

0

2/

*†

2

aa

n

n

eDD

nn

e

a

αα

α

ααα

αα

ααα

−

∞

=

−

==•

=•

=•

∑

Three definitions are equivalent except for a phase.

( ) ( )( ) ( ) *†††

†

ˆˆˆˆ

ˆˆˆˆ

αααααα+=

+=

aDaD

aDaD

「基礎からの量子光学」松岡正浩

23

Coherent states (3)

Quadrature phase amplitude

( )

( )†2

†1

ˆˆ2

1ˆ

ˆˆ2

1ˆ

aiaix

aax

+−=

+=

[ ]2

ˆ,ˆ 21

ixx =

( )( )4

1ˆˆ 21 ≥∆∆ xx

{ }

)}2

sin(

)2

cos(ˆ{2

eˆe ˆ2

),(

2

10

†

0

πω

πωε

ω

εω ωω

−⋅−+

−⋅−=

−= ⋅−⋅+−

rktx

rktxV

aaV

itrE rkitirkiti

rr

rrh

hr) rrrr

4

1

2ˆ

2ˆ

2*2

1

*

1

+

+=

+=

αααα

αααα

x

x

αααααα4

ˆˆˆ1ˆˆˆ

4

ˆˆˆˆˆˆˆ

2†††22†††22

1

aaaaaaaaaaaax

++++=+++=

( )4

121 =∆x

( )

( ) ( )2

121 =∆=∆ xx

( )( )4

121 =∆∆ xx

Coherent state is a minimum uncertainty state with equal variance.

24

Squeezed states (1)

Squeeze operator

( ) ( ) ϕζζζζ ireaaS =

−= , ˆˆ

2

1expˆ 2†2*

( ) ( )( ) ( ) rearaSaS

rearaSaSi

i

sinhˆcoshˆˆˆˆ

sinhˆcoshˆˆˆˆ

†††

††

ϕ

ϕ

ζζζζ

−−=

−=

[ ] [ ][ ] LQ +++=− BAABABeBe AA ˆ,ˆ,ˆ!2

1ˆ,ˆˆˆ ˆˆ( )

Squeezed states may be defined as,

( ) ( ) ( ) 0ˆˆˆ, αζαζαζ DSS =≡

rera i sinhcosh,ˆ, * ϕαααζαζ −=

then,

( )212/

21

ˆˆ

ˆˆˆ

yiye

xixai +=

+=ϕ

Quadrature phase amplitude

−=

2

1

2

1

ˆ

ˆ

2/cos2/sin

2/sin2/cos

ˆ

ˆ

y

y

x

x

ϕϕϕϕ

( )212/ ˆˆˆ yiyea i += ϕ

( ) ( )( ) ( ) r

r

eySyS

eySyS

22†

11†

ˆˆˆˆ

ˆˆˆˆ

=

= −

ζζ

ζζ ( )

( ) r

r

ey

ey

2

12

1

2

1

=∆

=∆ −

x1

x2

y1

y2

re−

re

2/ϕ

θ

25

Squeezed states (2)

y1

y2

re−

,0 If == ϕθ

E

t

Figures are inaccurate, sorry.

y1

y2

re

,0,2/ If == ϕπθ

E

t

Amplitude squeezed state Phase squeezed state

26

Squeezed states (3)

Nature 387, 472 (1997)

Quadrature amplitudes were measured by homodyne detection.

27

Quantum states of a single-mode EM field

( )

2/)ˆˆexp()(S

,)(ˆ, states Squeezed

)ˆˆexp()(ˆ

,0)(ˆ statesCoherent

0ˆ!

1 statesFock

2†2*

*†

†

aa

S

aaD

D

an

nn

αςς

αςαςααα

αα

−=

=

−=

=

=

However, in experiments, a quantum state of a plane-wave mode are not observed in most cases.

28

Quantum-enhanced gravitational-wave detector

“Quantum-mechanical noise in an interferometer,” Carlton M. Caves, Phys. Rev. D, 23, 1693 (1981).

m

m

11 ,aa†

22 ,aa †

ｂ：number of bounceτ：observation time

ℏ：planck’s constantω：frequency

ｃ：light velocity

α：laser power

22 ,cc †

11 ccnout†≡

ｚ

Photon counting error

ｒ:squeezing parameter

2

1

4

2

2

2

)sinh

)(2

()(ααω

re

b

cz

r

pc +≅∆−

zc

bnout δφφωαδ

≅2

cos2

sin22

zzz δ+→

out

out

n

n

z

z

δδ∆=∆

11 ,cc†

µπωφ 22 −+=

c

zb

μ：relative phase

laser

Radiation pressure error

))(2

( 1122 bbbbc

b†† −≡℘ ωh

)sinh()2

()( 22222 rec

b r +=℘∆ αωh

( ) ( )

( )2

1222 sinh

2

remc

bm

z

r

rp

+

=

℘∆≅∆

αωτ

τ

h

29

Quantum-enhanced gravitational-wave detector

“Quantum-mechanical noise in an interferometer,” Carlton M. Caves, Phys. Rev. D, 23, 1693 (1981).

m

m

11 ,aa†

22 ,aa †

ｂ：number of bounceτ：observation time

ℏ：planck’s constantω：frequency

ｃ：light velocity

α：laser power

22 ,cc †

ｚ

ｒ:squeezing parameter

11 ,cc†

µπωφ 22 −+=

c

zb

μ：relative phase

laser

r2sinh|| ≫αr

rp emc

bz ||)(~)( αωτh∆

||)

2(~)(

αω

r

pc

e

b

cz

−

∆

2

122 })(){( rppc zzz ∆+∆=∆rppc zz )()( whenMinimum ∆=∆

222

22 )

1)(

1)((

2

1|| opt

reb

mc αωτω

α == −

h

SQLopt zm

z )()( 2

1

∆≅=∆ τh

30

A Quantum-enhanced prototype gravitational-wave detector

K. Goda, et al. Nature Physics, 4, 472 (2008).

44% improvement in displacement sensitivity of a prototype gravitational-wave detector with suspended quasi-free mirrors

31

“Optical coherence and quantum optics”, L. Mandel and E. Wolf, p.480 and 12.11.5.We have defined photons as quantum excitations of the normal modes of the electromagnetic field, and we have associated them with plane waves of definite wave vector and definite polarization.

Now a plane wave has no localization in space or time, and therefore the excitation in the one-photon state must be regarded as distributed over all space-time.

Pulsed light as a single mode

0ˆ †λa

(IP). )exp()()(),(

,1)( ,0ˆ)(2†

tirutr

a

λλ

λ

λλ

λ

ωλφ

λφφφλφφ

−=Φ

===

∑

∑∑

rrrr

We may define a one-photon state localized in space by a linear superposition,

the corresponding spatial-temporal wave function is

noperator numer photon total theof eigenstatean is φ∑∑ ==⋅=

λλλ

λλφφ aannn ˆˆˆˆ ,1ˆ †

,ˆ)(ˆoperator new a define weIf *∑= λλφ ac

[ ] 1ˆ,ˆ ,0ˆ †† == cccφAny spatial-temporal mode can be regarded as a single mode.

32

Generation of squeezed states

Moritz Mehmetet al. Phys. Rev. A81, 013814 (2010).

33

Two mode squeezed states(1): side-band squeezing

Carlton M. Caves and Bonny L. Schumaker, "New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states," Phys. Rev A31, 3068 (1985).

Bernard Yurke, "Squeezed-coherent-state generation via four-wave mixers and detection via homodyne detectors,“Phys. Rev. A32, 300 (1985).

ε εεω +0εω −0 0ω

−a+a

[ ] [ ][ ] [ ] 0ˆ,ˆˆ,ˆ

1ˆ,ˆˆ,ˆ††

††

==

==

+−−+

−−++

aaaa

aaaa

Quadrature-phase amplitudes can be defined by(in photon-flux stance, Cook’s claim)

( ) ( )−+−+ +−=+= †2

†1 ˆˆ

2

1ˆ , ˆˆ

2

1ˆ aiaiaa χχ

[ ] [ ] [ ] [ ][ ] [ ]

2ˆ,ˆˆ,ˆ

0ˆ,ˆˆ,ˆˆ,ˆˆ,ˆ

†21

†12

†22

†111221

i==

====

χχχχ

χχχχχχχχ

Note that these are not Hermitian

34

Two mode squeezed states (2)

Two-mode squeeze operator

( )[ ]ϕϕϕ ii eaaeaarrS 2††2 ˆˆˆˆexp),(ˆ−+

−−+ −≡

rararSarS i sinhe ˆcoshˆ),(ˆˆ),(ˆ †† ϕϕϕm

−= ±±

r

r

rSrS

rSrS+

−

=

==

eˆ)0,(ˆˆ)0,(ˆ

eˆ)0,(ˆˆ)0,(ˆ ,0when

22†

11†

χχ

χχϕ

Two-mode coherent states

( )aaaDaDaD ˆˆexp),ˆ(ˆ ,0),ˆ(ˆ),ˆ(ˆ *† ααααα −=−−++

Two-mode squeezed-coherent states

0),ˆ(ˆ),ˆ(ˆ),(ˆ−−++= ααϕψ aDaDrS

ψ

35

Two mode squeezed states (3)

Mean-square uncertainty of a non-Hermitian operator

( )2/ˆˆˆˆˆˆ ,ˆˆˆˆ

ˆˆˆˆ

sym†

sym†

sym

*†2

ABBABARRRR

RRRRR

+=−=

−−=∆

R

Mean-square uncertainty of quadrature amplitudes

†222

†2

†22

2

2

†111

†1

†11

2

1

ˆˆˆˆˆˆ2

1

ˆˆˆˆˆˆ2

1

χχχχχχχ

χχχχχχχ

−+=∆

−+=∆

4

1

e 4

1 ,e

4

1

,0 and 0For

2/12

2

2/12

1

22

222

1

=∆∆

=∆=∆

=== −+

χχ

χχ

ϕααr-r-

36

Outline of the talk

•Introduction








37

Homodyne detection (1)

siga

LOa

2a

1a

( )( )

−=

+=

LOsig

LOsig

aaa

aaa

ˆˆ2

1ˆ

ˆˆ2

1ˆ

2

1

††

2†21

†1

ˆˆˆˆ

ˆˆˆˆˆ

LOsigLOsig aaaa

aaaan

+=

−=

When LO is a bright coherent light, θiLOLO na e→

{ }θθ

θθ

sinˆcosˆ2

sin2

ˆˆcos

2

ˆˆ2ˆ

21

††

xxn

i

aaaann

LO

ssssLO

+=

−++=

38


Multimode homodyne detection

( ) ( ))()(2

1)( ,)()(

2

1)( LO2LO1 ωωωωωω aaaaaa ss −=+=

We assume that a photodiode measures photon flux:

( )

( )

PDs. ofbandwidth theis B charge, elementary is e where

e e )()()(

e e )()()(

2

2

2†2

0 0

2

1†1

0 0

1

′−−′−

∞ ∞

′−−′−

∞ ∞

′′=

′′=

∫ ∫

∫ ∫

Bti

Bti

aaddetI

aaddetI

ωωωω

ωωωω

ωωωω

ωωωω

{ } ( )

{ } ( )2

2

e e )()()()(

e e )()()()(

)()()(

LO†SS

†LO

0 0

2†21

†1

0 0

21

′−−′−

∞ ∞

′−−′−

∞ ∞

′+′′=

′−′′=

−=

∫ ∫

∫ ∫

Bti

Bti

aaaadde

aaaadde

tItItI

ωωωω

ωωωω

ωωωωωω

ωωωωωω

39


We assume that the LO is in a monochromatic coherent state.

)()()()(ˆ 0LO0LOLOLO ωαωωδαωαω −=a

{ } { } { })()()()()( 0LO†SS0

*LO

0 0

LOLO ωωδαωωωωδαωωαα −′+′−′= ∫ ∫∞ ∞

aaddetI

( )2

e e

′−−′−× Bti

ωωωω

{ }2

0LO0LO e ee)(ee)( )(†S

)(S

0

LO

′−−−−−−

∞

+= ∫Btiitii aadne

ωωωωθωωθ ωωω

. takeand ,: variableof chang 0 B>>∆−= ωωωε

{ }2

LOLO e ee)(ee)( 0†S0SLO

−−−

∆

∆−

+++= ∫Btiitii aadneε

εθεθω

ω

εωεωω

tiitii aadne εθεθω

εωεωω ee)(ee)({ LOLO0

†S0S

0

LO +++= −−∆

∫ 2

LOLO e }ee)(ee)( 0†S0S

−−− −+−+ Btiitii aa

εεθεθ εωεω

0 0ω

ω∆B

40


{ } { } { } e),(e),(e 2)( LO†

LO

0

LOLOLO

2

titiBdnetI εεεω

θεχθεχωαα += −

−∆

∫

Using quadrature-phase amplitude, we obtain

{ }θθ εωεωθεχ ii aa e)(e)(2

1),( where 0

†S0S −++= −

),(),( note † θεχθεχ −=

bygiven is ,)( current, deference theofcomponent frequency - Therfore, εε I

{ } e),(e),(e 2)( LO†

LOLO

2

titiBneI εεε

θεχθεχε += −

−

)analyzer spectrum aby measuredpower Noise εS(bandwidth) (video )( ofcomponent frequency -zero 2 εI⇒

),(),(),(),(e 4)( LOLO†

LO†

LO

2

LO2

2

θεχθεχθεχθεχεε

+=

−BneS

Mean-square uncertainty of the quadrature amplitude

41

Pulsed homodyne detection

Pulsed homodynedetectiondiffers from the usually-used method of using radio-frequency spectral analysis of the current to study noise at certain frequencies. In the pulsed homodyne detection, photoelectrons generated by each pulse are separately amplified by a low-noise amplifier, so a single measurement on the quadrature amplitude of the signal pulse is performed pulse by pulse.

φ φ ′ φ ′′ ,X ,X ′ ,X ′′

A charge sensitive amplifier: repetition freq. < 1MHz ref. D.T.Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1992).

Now, using a low noise op-amp, repetition freq.~100MHz is possible.

continuous-wave

t

n n+1 n+2

Detector’s response~1/B

Pulsed light

tn n+1 n+2

pτT

TBp << /1τ⇒Pulsed readout

42

High speed homodyne detector made of OpAmp

5k

100pF

OPA847, 3.9 GHz, 950 V/μs

OPA847

RBW:1MHz

VBW:100Hz

SweepTime:10s

Useful bandwidth > 200MHzObservation of squeezing over several rep. freq. (f irst)

Okubo et al., Opt. Lett. 33, 1458 (2008).

43・・・・One point is an average of 10 measurements・・・・error bar is a standard deviation of 10 easurements

LO Power 2.0mWPump Power 7.3mW

Squeezing -1.7dBAntisqueezing 8.4dB

Total data 7600 pulses

Effect of amp noise’s low frequency

Correlation coefficient of adjacent squeezed pulse is 0.04

Performance of detector and Measurement Squeezed states in time domainOkubo et al., Opt. Lett. 33, 1458 (2008).

Linearity < 2mW

Independent detectionof 76MHz pulse train

44

Outline of the talk

•Introduction








45

Nonlinear optics and Quantum state control

Electric field and dielectric polarization:

( ) ( ) ( )trPtrEtrD ,,, 0

rrrrrr+= ε

Lrrrrrrr

+++= EEEEEEP E :: )3(0

)2(00 χεχεχε

The polarization may be written as a power series:

(This is not the case for an extremely high electric field.)

Intuitive picture of dielectric polarization:

nucleus

electronEr

-2 -1 1 2

-1

1

2

3

4

3x∝

2x

3x∝

2x

E

P

2 4 6 8 10 12 14

-1

-0.5

0.5

1Green+blue

46

Nonlinear optics and quantum state control (2)

kjijki EEdP )2(0

(2)NL 2)( ε=r

Second order nonlinear polarization:

( )( ) EP

tE

t

EPEEt

it

DH

NL

NLe

rrr

rrrr

rr

r

σε

σχεε

+∂∂+

∂∂=

+++∂∂=

+∂∂=

00

rot

t

EP

tE

tE

Ht

E

t

B

NL ∂∂+

∂∂+

∂∂=∇

=∂∂+∇−

=

∂∂+

rrvr

rr

rr

σµµεµ

µ

02

2

02

2

02

02 0rot

0Erotrot

(magnetic polarization ignored)

x

y

zThree EM waves propagating in z direction with frequencies ω1,ω2,ω3.

( ) ( )( )( ) ( )( )( ) ( )( )c.c.)(2/1,

c.c.)(2/1,

c.c.)(2/1,

333

222

111

3)(

2)(

1)(

+=

+=

+=

−

−

−

zktikk

zktijj

zktiii

ezEtzE

ezEtzE

ezEtzE

ωω

ωω

ωω

Nonlinear polarization at frequency ω1=ω3-ω2

( )( )c.c.)()(

2

1

2,

)()(3

*20

)()(0

)(

23223

321

+=

=

−−− zkkitikjijk

kjijkiNL

ezEzEd

EEdtzP

ωω

ωωω

ε

ε

47


( ) ( )

( )

+

−

∂∂−

∂∂=

+∂∂=

+∂∂=∇

−

−

−

c.c)()(

2)(

2

1

c.c)(2

1

c.c)(2

1,

)(1

21

112

12

)(12

2

)(1

2)(2

11

11

111

zktii

ii

zktii

zktiii

ezEkz

zEik

z

zE

ezEz

ezEz

tzE

ω

ω

ωω

Slowly varying amplitude approximation

λπ2

)()( 1

121

2

=∂

∂<<∂

∂k

z

zEk

z

zE ii

( )

{ } ( )iNLzkti

i

zktii

ii

Pt

eEi

ezEkz

zEiktzE

111

111

2

2

0)(

11112

110

)(1

21

11

)(2

c.c.2

1)(

c.c)()(

22

1,

ωω

ωω

µσµωωεµ∂∂+

++−=

+

−∂

∂−≈∇

−

−

48


( ) ( )

( )

+

−

∂∂−

∂∂=

+∂∂=

+∂∂=∇

−

−

−

c.c)()(

2)(

2

1

c.c)(2

1

c.c)(2

1,

)(1

21

112

12

)(12

2

)(1

2)(2

11

11

111

zktii

ii

zktii

zktiii

ezEkz

zEik

z

zE

ezEz

ezEz

tzE

ω

ω

ωω

Slowly varying amplitude approximation

λπ2

)()( 1

121

2

=∂

∂<<∂

∂k

z

zEk

z

zE ii

( )

{ } ( )iNLzkti

i

zktii

ii

Pt

eEi

ezEkz

zEiktzE

111

111

2

2

0)(

11112

110

)(1

21

11

)(2

c.c.2

1)(

c.c)()(

22

1,

ωω

ωω

µσµωωεµ∂∂+

++−=

+

−∂

∂−≈∇

−

−

49


kzi

kzi

kzi

eAAiAdz

dA

eAAiAdz

dA

eAAiAdz

dA

∆

∆−

∆−

−−=

−−=

−−=

21333

3*12

22

3*21

11

2

2

2

βα

βα

βα

2130 , , kkkkE

nA

llll

l

ll −−=∆==

εµσα

ω

321

321

321

3212

0

22 nnnc

d

nn

n

k

d ijk

l

llijk ωωωω

ωωωµεβ ==

Second harmonic generation(lossless) 0 , )( 2121 === lAA σωω

kzi

kzi

eAidz

dA

eAAidz

dA

∆

∆−

−=

−=

21

3

3*1

1

β

β

( ) 02

3

2

1 =+ AAdz

d

Energy conservation

0)0( const. ,When 3131 =≈<< AAAA

2

24

1

2

3

213

)2/(

)2/(sin)(

1)(

k

kzAzA

ki

eAizA

kzi

∆∆=

∆−−=

∆

β

β

0 =∆kif24

1

2

3 )( zAzA β=

50

Parametric process

(lossless) 0 , )2 ,( 132121 ==== lAA σωωωω

0)0( const. ,When 3331 =≈<< AAAA

3*1

3*1

1

AigeegA

eAAidz

dA

iikzi

kzi

β

β

ϕϕ =−=

−=

+∆−

∆−

12

21

2

Agdz

Ad =

gzgz eCeCzA −+= 211 )(

gzeAgzAzA i sinh)0(cosh)0()( *111

ϕ−=∴

( ) ( ) rearaSaS i sinhˆcoshˆˆˆˆ note †† ϕζζ −=

51

Outline of the talk

•Introduction








52

Correlation functions

One photon detection probability

( )rkitiaV

itrE

trEtrEtrE

kkk

k

kkk

rrhr

rrr

rr

rrr

⋅+−=

+=

+

−+

ωεω

σσ

σσσ

expˆ2

),(ˆ

),(ˆ),(ˆ),(ˆ

,0

)(,

)(

,

)(

,,

d.annihilate is , modein photon one: σkfir

→

iEEiK

iEffEiKiEfKiEfKP

kk

fkk

fk

fk

)(

,

)(

,

)(,

)(,

2)(

,

2)(

,)1(

ˆˆ

ˆˆˆˆ

+−

+−++

=

=== ∑∑∑

σσ

σσσσ

rr

rrrr

d.annihilate are , modein photon oneanother , , modein photon onewhen σσ ′′kkrr

iEEEEiK

iEEffEEiKiEEfKP

kkkk

fkkkk

fkk

)(

,

)(

,

)(

,

)(

,

)(,

)(,

)(,

)(,

2)(

,)(

,)2(

ˆˆˆˆ

ˆˆˆˆˆˆ

+′′

+−−′′

+′′

+−−′′

+′′

+

=

== ∑∑

σσσσ

σσσσσσ

rrrr

rrrrrr

53

The first order correlation function (1)

2/1)()(

2/1)()(

)()(

)1(

),(ˆ),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),,,(

BBBBAAAA

BBAA

BBAA

trEtrEtrEtrE

trEtrEtrtrg

rrrr

rrrr

+−+−

+−

=

A

B

AA tr ,r

BB tr ,r

Akr

Bkr

tr ,r

( )

( )),(ˆ

expˆ2

expˆ2

),(ˆ

)(

0

0

)(

trE

rkitiaV

i

rkitiaV

itrE

A

AAk

AAAAk

AA

r

rrh

rrhr

+

+

=

⋅+−=

⋅+−=

ωεω

ωεω

crrtt AA /rr −−=

( )

( )),(ˆ

expˆ2

expˆ2

),(ˆ

)(

0

0

)(

trE

rkitiaV

i

rkitiaV

itrE

B

BB

BBBBBB

r

rrh

rrhr

+

+

=

⋅+−=

⋅+−=

ωεω

ωεω

54

The first order correlation function (2)

2/1)()(

2/1)()(

)()(

)1(

),(ˆ),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),,,(

BBBBAAAA

BBAA

BBAA

trEtrEtrEtrE

trEtrEtrtrg

rrrr

rrrr

+−+−

+−

=

A

B

AA tr ,r

BB tr ,r

Akr

Bkr

tr ,r

{ }†)1()1()()()()(

)()()1(

ˆˆˆˆ

),(ˆ),(ˆ),(

gABgABEEEEK

trEtrEKtrP

BBAA +++=

=

+−+−

+− rrr

),(ˆ),(ˆ),(ˆ )()()( trEtrEtrE BA

rrr +++ +=

<=incoherent 0

coherent partial 1

coherentorder -first 1)1(g

55

The second order correlation function (1)

),(ˆ),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),(ˆ),(ˆ),,,(

)()()()(

)()()()(

)2(

BBBBAAAA

AABBBBAA

BBAAtrEtrEtrEtrE

trEtrEtrEtrEtrtrg rrrr

rrrrrr

+−+−

++−−

=

second order autocorrelation functionBeam splitter

cτ

DA

DB

)(tI

)( τ+tI

)(ˆ)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ)(ˆ)(

)()()()(

)()()()(

)2(

ττ

τττ

++

++=

+−+−

++−−

tEtEtEtE

tEtEtEtEg

Classical theory

)()(

)()()()2(

ττ

τ+

+=

tItI

tItIg C

( )1

)(

)(

)(

)()(

)(

)()0( 2

2

2

2

2

2

)2( ≥+−

==tI

tI

tI

tItI

tI

tIg C

) )()()()( ( 1)()2(lim τττττ

++= →∞→∞→

tItItItIg C Q

56

The second order correlation function (2)

Quantum theory

2†

22†

)2(

ˆˆ

ˆˆ)0(

aa

aag =

Fock states

)1(ˆˆ 22† −= nnnaan

≥−=

=)1( /11

)0( 0)0()2(

nn

ng

ngantibunchi 1)0()2( <g

Nature Physics, 3, 253 (2007)

Coherent states

1)0()2( =g 1ˆˆ

ˆˆ)0( 2†

†

)( ==aa

aag

nn

n

Squeezed states( ) ( ) ( ) ( ){ }

( ) ( ){ }{ }2222222

422242242

)2(

sinh2/cos2/sin

sinh2sinh2/cos2/sin1)0(

ree

rreeeeg

rr

rrrr

+−+−

++−−+−−+=

−

−−

θϕθϕα

θϕθϕα

Koashi, et al. PRL 71, 1164 (1993).

Thermal light

1)1( ++=

n

n

n n

nP2)0()2( =g

57

Representation of density matrix (1)

Representation in Fock state basis

mnmn

mmnn

nmnmmn

mn

ρρρ

ρρ

ˆ ,

ˆˆ

==

=

∑∑

∑∑

Example :coherent state

ααρ =ˆ!!

*

2

mne

mn

nm

ααρ α−=

Representation in coherent state basis ? Yes.22 βαβα −−= e

ππ

αααααα α

==

=

∑

∫∑∑∫∞

=

−∞

=

∞

=

!!

!!

0

2*

00

22

nn

mn

demn

mnd

n

mn

mn

coherent states are overcomplete.

58


Glauber-Sudarshan P-representation

∫= ααααρ )( ˆ 2 Pd

Expectation value of normally ordered operators

( ) ( ) ( )mn

mnmnmnmn

Pd

aaPdaaPdaaaa

αααα

ααααααααρ

*2

†2†2††

)(

ˆˆ)( Trˆˆ)( TrˆˆˆTrˆˆ

∫

∫∫=

===

( )

{ }[ ]{ }2**2

2*2**22

2†2††22

12

12

1

)()()( 14

1

)(12)( 4

1

2

ˆˆ

4

1ˆˆˆ2ˆ

αααααα

αααααααα

+−++=

+−+++=

+−+++=−=∆

∫

∫

Pd

Pd

aaaaaaxxx

αα ⇒⇒ aa ˆ,ˆ *†

( )

−−−+=∆ ∫

2**

222 )()()( 1

4

1

iiPdx

αααααα

∫

∫=

=*2*

2

)(

)(

αααα

αααα

Pd

Pd

( ) ( )4

1 and

4

1

,0)( if

22

21 ≥≥

≥

∆x∆x

P α

59


( )

[ ]2222

222

)(

αααα −+=

−=∆

∫ Pdn

nnn

( ) n∆nP ≥≥ 2 ,0)( if α

classical light : 0)( ≥αP

Normally ordered characteristic function:

( )αηηα

ηη

αα

ρηχ**

*†

)(

ˆTr)( 2

ˆˆN

−

−

∫=

=

eePd

ee aa

ααηη αη iyxiyx +=+= ,αηαηααηχ yixxiy eePd 222

N )( )( −∫=

{ }{ })()2/(

)2/()(

N

N

ηχααηχ

1-F

F

=

=

P

P

Wigner distribution function:

( ) 2/ˆˆ2*†

)(ˆTr)( ηηη ηχρηχ −− == ee Naa

∫

∫

∫

−−

−−

−

=

=

≡

2

**2

**

22

2/2

2

)( 2

)(

)( ) W(

αβ

ηααηη

ηααη

ββπ

ηχη

ηχηα

ePd

eed

ed

N

Example :coherent state

( )2

2

2

20

(2)2

2

2

) W(

αβ

αβ

π

αβδβπ

α

−−

−−

=

−= ∫

e

ed

60


Wigner distribution function:

Example (2) :squeezed vacuum state

( )[ ]rr eyeyyy 222

22121 2exp

2), W( +−= −

πExample (3) :fock state |n>

rn

n erLxx 2221 )4()1(

2), W( −−=

πLn:Laguerre polynomial

can be negative

Relation to homodyne measurement

qUUqq )(ˆˆ)(ˆ),(pr † θρθθ =θθθ ini eaUaUeU −− == ˆˆˆˆ ,)(ˆ †ˆ

∫∞

∞−

−−= dppqpqWq )cossin,sincos(),(pr θθθθθ

ｘφ

Ｐ（ｘ　）φ

Ｗ（ｘφ、ｐφ）

ｘｘｘｘφφφφ

ｐφφφφ

It is possible to reconstruct Wigner function from homodyne data.

61


Husimi Q distribution:

( )†* ˆˆˆTr)( aaA ee ηηρηχ −=

**

)( 1

)Q( 22

ηααηηχηπ

α −∫= ed A

∫−−=

=

2

)( 1

ˆ1

)Q(

2 αβββπ

αραπ

α

ePd

mnmn Qdaa *2† )( ˆˆ αααα∫=

62

Outline of the talk

•Introduction








63

Atomic Magnetometer

review article: D. Budker and M. Romalis, Nature Phys. 3, 227 (2007).

pump light

B

probe light

64

"A subfemtotesla multichannel atomic magnetometer," I. K. Kominis, T. W. Kornack, J. C. Allred & M. V. Romalis, Nature, Vol. 422, pp. 596-599 (2003).

VtnTB

2

1

γδ = )12(/ += Ig B hµγ

65

"A subfemtotesla multichannel atomic magnetometer," I. K. Kominis, T. W. Kornack, J. C. Allred & M. V. Romalis, Nature, Vol. 422, pp. 596-599 (2003).

１チャンネルの雑音

（磁気シールドの熱雑音）

隣接チャンネルの差から

もとめた固有雑音

0 140 Hz

66

Quantum magnetometer

“Squeezed spin states,” Masahiro Kitagawa and Masahito Ueda, Phys. Rev. A 47, 5138–5143 (1993)

Angular momentum system

),,( zyx SSSS =r

Cyclic commutation relation

kijkji SiSS ε=],[222

4

1))(( kji SSS ≥∆∆

For N atoms in mF=F along the quantization axis z,

FNS z =A magnetic field along y axis causes a rotation of S in x-z plane. Polarization of light propagating along x will be rotated propotional to Sx. This measumentis limited by the projection noise of the atom,

2/2

)( 2 FNS

S zx ==∆

and light shot noise of polarization measurement.

67

"Sub-Projection-Noise Sensitivity in Broadband Atomic Magnetometry," M. Koschorreck, M. Napolitano, B. Dubost, and M. W. Mitchell, Phys. Rev. Lett., Vol. 104, 093602 (2010).

Laser cooled Rb atoms, 106, 25µK, dipole trap20 times QND measurement w/o dipole trap

“Spin Squeezing of a Cold Atomic Ensemble with the Nuclear Spin of One-Half”,T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, Phys. Rev. Lett. 102, 033601 (2009).

68

"Squeezed-Light Optical Magnetometry," Florian Wolfgramm, Alessandro Cerè, Federica A. Beduini, Ana

Predojević, Marco Koschorreck, and Morgan W. Mitchell , Phys. Rev. Lett., Vol. 105, 053601 (2010).

Hot Rb atoms, 794.7 nm, D1 line

-3.2 dB, 3.2 x 10-8 T/√√√√Hz

69

Outline of the talk

•Introduction








• Continuous-variable quantum key distribution

70

A brief review of CV QKD

• Home-(or hetero-)dyne detection is utilized to detect weak light.

→Quadrature-phase amplitudes (continuous variables) are measured.

♦merit: room temperature operation with conventional devices

♦drawback: needs local oscillator phase-locked to the signal

• Quantum states sent by Alice

coherent states (easy to generate), or squeezed states, or non-Gaussian

• Coherent-states modulation schemes

♦Continuous modulation : e.g. Gaussian modulation

Gaussian distribution of quadrature-phase amplitudes

♦Discrete modulation (i.e. QPSK)

Similarities with QAM optical transmission sheme

Unconditional security proofs have been presented.

• Mode of electromagnetic field

pulsed light, frequency sidebands

71

QAM system by Prof. Nakazawa group, Tohoku Univ.

72

Similarities♦ Coherent states modulated in phase-space are sent♦ Homodyne detection is utilized to readout quadrature-phase amplitudes

A coherent optical communication system operating in quantum noise limit is ready for CV QKD.

Similarities between the QAM optical communication and the CV-QKD may allow us in future to perform both

high speed data transmission and unconditionally secure QKD with same equipments by switching its operation mode.

CV QKD and QAM optical transmission

speed

security=small excess noise

CV-QKD QAM system(QNRC)

73

CV QKD and 3-dB loss limit

3dB loss limit“Continuous Variable Quantum Cryptography Using Coherent States”,

F. Grosshans and P. Grangier, PRL 88 ,057902 (2002).

Eve

Bob

Recipe for beating 3dB-loss-limit

Post-selection

“Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit”, Ch. Silberhorn, T. C. Ralph, N. Lütkenhaus, and G. Leuchs, Phys. Rev. Lett. 89, 167901 (2002).

“Quantum cryptography using balanced homodyne detection,”T. Hirano, T. Konishi, R. Namiki, quant-ph/0008037; Extened abstract for EQIS 2001.

"Security of quantum cryptography using balanced homodyne detection", R. Namiki, T. Hirano, Phys. Rev. A, vol. 67, no.2, 022308-1-7 (2003).

Reverse reconciliation

“Quantum key distribution using gaussian-modulated coherent states” , F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, Nature 421, 238 (2003).

Bit 0Bit 1

-5 -4 -3 -2 -1 0 1 2 3 4 5

10

100

1000

ＣＯ

ＵＮ

Ｔ

直交振幅

Quadrature amplitude

Num

ber

of c

ount

s

Postselection

ＸＸＸＸ００００

－－－－ＸＸＸＸ００００

74

Probability distribution of quadrature amplitude

Bob sets up a threshold value X0.If the measured value Xφ< -X0, Bob judges that φ=180o.If Xφ> X0, Bob judges that φ=0o.If -X0 <Xφ< X0, Bob abandons the judgement.

Increasing X0, the error rate, eB, decreases to an arbitrary small value.The “effective” detection efficiency, P, also decreases.

-X0 X0

<n>＝1

75

Security of CV QKD: loss limit caused by excess noise (1)

1. Eve performs a simultaneous measurement on both quadratureamplitudes (x,p) using a beam splitter.

2. Eve resends a coherent state to Bob.>+ )(2| ipx

No secret key because Eve knows the state that Bob measures.(entanglement breaking)

Applicable to any CV QKD protocol using coherent states and homodyne detection

R. Namiki and TH, Phys. Rev. Lett., vol. 92, 117901 (2004).

76

0.01 0.1 10.1

1

10

100

1000

BS followed by amplification

NO SECURE KEY

dis

tan

ce (

km)

excess noise δ

δ＜２η

R. Namiki and TH, Phys. Rev. Lett., vol. 92, 117901 (2004).

Security of CV QKD: loss limit caused by excess noise (2)

Define excess noise δ :

Intercept & resend attack increases excess noise:

Excess noise decreases by loss: δ =2η, η:transmissivity.

δ <2η is a necessary condition.

δ =2.

77

M. Koashi, quant-ph/0609180

0.2dB/km

ξ=0.01, 50km, 10-3

ξ=0.006, 100km, 10-4

ξ=0.004, 150km, 10-5

ξ=0.002, 200km, 10-6

Excess noise should be small for unconditional security.

Anthony Leverrier and Philippe Grangier, Phys. Rev. Lett., 102, 180504 (2009).( Erratum: PRL 106, 259902 (2011). )

quantum optics: from basics to advanced topics...3 outline of the talk •introduction...

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