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
•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
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
Quantization of free electromagnetic field (2)
( ) ( ){ }∑ ⋅−+⋅+−=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
Quantization of free electromagnetic field (3)
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
expˆexpˆ2
),(
expˆexpˆ2
),(
expˆexpˆ2
),(
†
,
†
,
†
,
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
•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
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
•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
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
Homodyne detection (2)
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
Homodyne detection (3)
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
Homodyne detection (4)
{ } { } { } 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
•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
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
Nonlinear optics and quantum state control (3)
( ) ( )
( )
+
−
∂∂−
∂∂=
+∂∂=
+∂∂=∇
−
−
−
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
Nonlinear optics and quantum state control (3)
( ) ( )
( )
+
−
∂∂−
∂∂=
+∂∂=
+∂∂=∇
−
−
−
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
Nonlinear optics and quantum state control (4)
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
•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
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
Representation of density matrix (2)
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
Representation of density matrix (3)
( )
[ ]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
Representation of density matrix (4)
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
Representation of density matrix (5)
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
•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
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
•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
• 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). )
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