triggered single photon sources and their applications in
TRANSCRIPT
1
Alper Alper KirazKirazDepartmentDepartment of of PhysicsPhysics, K, Kooçç UniversitUniversityyRumelifeneriRumelifeneri YoluYolu,, 34450 34450 SarSarııyeryer, , ĐĐstanbulstanbul
Triggered single photon sources and their applicati onsTriggered single photon sources and their applicati onsin quantum key distribution and quantum information in quantum key distribution and quantum information processingprocessing
http://http://nanonano--opticsoptics..kuku.edu.tr.edu.tr
2
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
References:
• R. Loudon, “The quantum theory of light“, Oxford University Press, 1983 • A. Đmamoğlu and Y. Yamamoto, “Mesoscopic Quantum Optics“, Wiley Inter-Science, 1999• L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics“, Cambridge University Press, 1995
3
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
4
Triggered single photon emission
• A single electron source is not interesting but a single photon source is very interesting:
Photons are bosons Electrons are fermions
• Due to the Pauli exclusion principle fermions tend to be alone, while bosons tend to be together.
• By achieving triggered single photon emission bosons are made to behave like fermions, hence:
Nonclassical Light Emission
A. Đmamoğlu, Y. Yamamoto, Phys. Rev. Lett. 72, 210 (1994)
5
Coulomb blockade of electron/hole tunneling in mesoscopic pn-junction
A. Đmamoğlu, Y. Yamamoto, Phys. Rev. Lett. 72, 210 (1994)J. Kim et al. Nature 397, 500 (1999)
~50 mK temperatues are necessary
6
Triggered Single Photon Emission Based on a Single Two-LevelEmitter
Nonclassical light emission
Photon Antibunching – Proof of a Two-Level Emitter
7
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Young’s Double-Slit Experiment
Degree of First Order Coherence
Degree of Second Order Coherence
Hanbury-Brown and Twiss Experiment
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
8
Young’s Double-Slit Experiment
Đdeal, monockromatic lightsource r1
r2
d
L
d<<λ
L>>λ
zθ r
( )( )tkrir
EtrE ω−= 1
111 exp),(
( ))(exp),( 22
22 tkrir
EtrE ω−=
9
πλπ =zL
d2
πλπ
32 =z
L
d
d
Lz
λ=∆
Young’s Double-Slit Experiment
),(),(Re2),(),(2
1),(),(
2
121
*2
2
2
10
2
210 trEtrEtrEtrEctrEtrEcI ++=+= εε
Intensity recorded on the screen
Interference Term
( )( )rkr
EcI ∆+
≈ cos12
12
0ε)cos(
21 ϑdrr −≈
)cos(22 ϑd
rr +≈ L
dzdr =≈∆ )cos(ϑ
10
Degree of First Order Coherence
∫ +=+T
dttEtET
tEtE )()(1
)()( ** ττ T is large2
*
)1(
)(
)()()(
tE
tEtEg
ττ
+=
∆+
≈ω
ε rkg
r
EcI )1(
2
0 Re12
1
)()()()()()(),(),( **2*12
*1 τ−=∆−=−−= tEtE
c
rtEtE
c
rtE
c
rtEtrEtrE
11
Degree of First Order Coherence
ωττ ieg −=)()1(
Ideal, monochromatic lightsource
Chaotic light, center frequency ω, spectral width 2γ
Intensity pattern on the screen
γτωττ −−= ieg )()1(
ω
ω
Intensity pattern on the screen
12
Degree of First Order Coherence
= ∫∞
0
)1( )exp()(Re1
)( ωττπ
ω igF
Frequency spectrum
220 )(
/)(
γωωπγω
+−=FLorentzian
γττωτ −−= 0)()1( ieg
220 2
1)1( )(
τδτωτ
−−=
ieg
( )
−−=2
20
2 2exp
2
1)(
δωω
πδωFGaussian
13
Coherence Functions
We cannot distinguish a classical light source from a non-classical light source using theYoung’s Double-Slit Experiment.
All the coherence functions should be measured for proper characterization of a lightsource.
( ) 2/12
22
2
11
22*
11*
11)(
)(...)(
)()...()()...()(
nn
nnnnnnn
trEtrE
trEtrEtrEtrEg
++=τ
14
Degree of Second Order Coherence – Classical Fields
222
**
)2(
)(
)()(
)(
)()()()()(
tI
tItI
tE
tEtEtEtEg
ττττ
+=
++=
Light Source
Det.
I(t)
Det.
I(t+τ)
Intensity correlation function
15
Degree of Second Order Coherence – Classical Fields
Some Observations:
22
2121 )()()()(2 tItItItI +≤
N
tItItI
N
tItItI NN22
22
111 )(...)()()(...)()( +++≤
+++
22)()( tItI ≤
)0(1 )2(g≤
[ ] [ ][ ]221
221
211 )(...)()(...)()()(...)()( ττττ ++++++≤++++ NNNN tItItItItItItItI
222)()()( tItItI ≤+τ
)()0( )2()2( τgg ≥
16
Degree of Second Order Coherence – Classical Fields
For chaotic light: Many atoms emitting, monochromatic light while colliding with each other.
2)1()2( )(1)( ττ gg +=
τγτ 2)2( 1)( −+= eg
τ
1
g(2)(τ)
Chaotic light with center frequency ω, spectral width 2γ (thermal light source)
Photon bunching !
17
Degree of Second Order Coherence – Quantized Fields
)(ˆ)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆ)(ˆ)(ˆ)()2(
ττ
τττ
++
++=
+−+−
++−−
tEtEtEtE
tEtEtEtEg
First order photon correlation function
Heisenberg Electric Field Operator
)(ˆ)(ˆ)(ˆ tREtREtRET
rrrrrr−+ +=
Second order photon correlation function
( ) 2/1
22
)1(
)(ˆ)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆ)(
ττ
ττ
++
+=
+−+−
+−
tEtEtEtE
tEtEg
∑ ⋅+−=+
kkkk
k RkitianV
itREr
rrr
r rrrhrr
)exp(ˆ2
)(ˆ0
ωεω
∑ ⋅−= +−
kkkk
k RkitianV
itREr
rrr
r rrrhrr
)exp(ˆ2
)(ˆ0
ωεω
18
Degree of Second Order Coherence – Quantized Fields
n-photon Fock state
Coherent State ( )∑−=n
n
nn 2/1
2
!)
2
1exp(
ααα ααα =a
1ˆˆ
ˆˆˆˆ)( 2
2
2)2( ===
+
++
αα
αααα
τaa
aaaag
11
1)1(
ˆˆ
ˆˆˆˆ)(
22)2( <−=−==
+
++
nn
nn
naan
naaaang τ
n 1ˆ −= nnna 11ˆ ++=+ nnna
Nonclassical light
n=1 0)()2( =τg
19
Hanbury-Brown and Twiss Experiment
Det.
Det.
hνννν hνννν
TAC MCA
Light Source
Det.
I(t)
Det.
I(t+τ)
Ideal detectors ideal intensity correlation function
Coincidence detection
In practice detectors have a dead time following the single photon detection, this is overcomeby using two detectors in the Hanbury-Brown Twiss configuration
20
Single Photon Detectors
Silicon Avalanche PhotoDiode
Maximum spectral range 200-1100 nmHigh quantum efficiency ~60 %Relatively large dead time ~10 ns
Maximum spectral range 200-800 nmHigh quantum efficiency ~30 %
Maximum spectral range 800-1600 nmHigh quantum efficiency ~60 %
InGaAs Avalanche PhotoDiode
21
Hanbury-Brown and Twiss Experiment
-20 -10 0 10 200
1
g(2) (τ
)
τ
-20 -15 -10 -5 0 5 10 150
50
100
150
Coi
ncid
ence
Cou
nts
n(τ)
Delay Time (ns)
0.0
0.5
1.0
1.5
Correlation function g
2(τ)
Triggered Single Photon SourcePhoton Antibunching
22
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Calculation of Second Order Coherence Using Atomic Projection Operators
Photon Antibunching
Cascaded Photon Emission
Triggered Single Photon Emission
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
23
Single Photon Generation Using a Single Dipole
Pulsed Laser Excitation of a Single Two-Level Emitt er(Single quantum dot, single molecule, single N vacancy in diamond, single atom, single ion)
WP
0
1
Γspon
Experimentally difficult to separate the turnstile photons from the pulsed laser
R. Brouri et al., Phys. Rev. A 62, 063817 (2000)
ππππ-pulse is necessary when there is no dephasing
Incoherent excitationlarge dephasing limit
24
Single Photon Generation Using a Single Dipole
Γ−+
−
−∆−ΩΩΓ−Ω−
Ω−−∆
=
−0
0
~
~
2
)(0
)()(
02
)(
~
~
spon
ge
ggee
eg
totP
PsponP
Ptot
ge
ggee
eg
it
i
titi
tii
dt
d
σσσ
σ
γω
γω
σσσ
σ
∆ω = ωeg−ωL, γtot = Γspon/2 + γdeph, ΩP(t) = 2 |µeg| |E(t)| / ħ
Optical Bloch Equations
400 600 800 10000.0
0.5
1.0
400 600 800 10000.0
0.5
1.0
400 600 800 10000.0
0.5
1.0
400 600 800 10000.0
0.5
1.0
400 600 800 10000.0
0.5
1.0
400 600 800 10000.0
0.5
1.0
∫Γ= dttI eesponemission )(σ
10=ΩP 20=ΩP
30=ΩP 40=ΩP
40=ΩP 50=ΩP
π-pulse
ΩP
g
e
Γspon
25
Single Photon Generation Using a Single Dipole – IncoherentExcitation
g
e
hνPulsed laser
J.-M. Gérard, and B. Gayral, IEEE J. Lightwave Tech. 17, 2089 (1999).S. Raymond, K. Hinzer, S. Fafard, and J. L. Merz, Phys. Rev. B 61, 16331(R) (2000).
~ 20 ps
~ Γ=1-6 ns
i
Pulsewidth << 1/Γ
Single InAs quantum dots
• Short free carriers lifetime + slow relaxation from level |e>
vanishing probability of re-excitation after first photon emission
• Predominantly radiative recombination
26
Model Using Atomic Projection Operators
−=
ctAta
rr 01ˆ)()(ˆ σSource-Field expression
10ˆ01 =σAtomic transition operator
Wp1 Γ1
0
1
In the Heisenberg representation
)(ˆ)(ˆ
)(ˆ)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆ)(ˆ)(ˆ)(
1111
01011010)2(
τσσστστσσ
ττττ
τ+++
=++
++=
++
++
tt
tttt
tatatata
tatatatag
27
Photon Antibunching from a Dephased Two-Level Emitter
Wp1 Γ1
0
1Dephased two-level emitter can be analyzed in the rate equation limit :
)()()(
)()()(
11100100
11100111
ttWdt
td
ttWdt
td
p
p
σσσ
σσσ
Γ+−=
Γ−=
)().......( 001010 tt σσ
Quantum Regression Theorem
)()()(
)()()(
)2(11
)2(11
)2(
ττττ
τττ
τ
GFWd
dF
GFWd
dG
p
p
Γ+−=
Γ−=
Initial Conditions:
11
1011
)2(
)()0(
0)0(
Γ+==
=
p
p
W
WtF
G
σ
Steady state population of the level |1>
)()()()(
)()()()(
00100010
00111010)2(
tttF
tttG
σσστ
σσστ
=
=
28
Photon Antibunching from a Dephased Two-Level Emitter
0 1 2 3 4 50.0
0.5
1.0
Wp = 0.2Γ
τ (1 / ΓX)
g(2) (τ
)
0 1 2 3 4 50.0
0.5
1.0
Wp = 2Γ
τ (1 / ΓX)
g(2) (τ
)
Wp1 Γ1
0
1
τ
σστσσ
τ )(
2
11
011110)2( 1)(
)()()()( Γ+−−=
+= pWe
t
tttg
)()()(
)()()(
)2(11
)2(11
)2(
ττττ
τττ
τ
GFWd
dF
GFWd
dG
p
p
Γ+−=
Γ−=
11
1011
)2(
)()0(
0)0(
Γ+==
=
p
p
W
WtF
G
σ
( ) ( )ττ σσ
τ )(2
011)(
1
0111)2( 1)(1)(
)( Γ+−Γ+− −=−Γ+
= pp WW
p
p eteW
tWG
29
Dephased Three-Level Cascade Ssystem
ωp1 Γ1
ωp2 Γ2
X0
X1
X2
)()()(
)()()()()(
)()()(
11100100
222112100111
22211222
ttdt
td
tttdt
td
ttdt
td
p
pp
p
σσωσ
σσωσωσ
σσωσ
Γ+−=
Γ++Γ−=
Γ−=
30
0 5 10 15 20 25 300
2x10-3
4x10-3
6x10-3
8x10-3
1x10-2
Unn
orm
. Cor
r. F
unc.
G2 (τ
)
Delay Time (ns)
0.0
0.5
1.0
Norm
. Corr. F
unc. g2(τ)
Case 1 – Single Exciton Emission Self-Correlation
Self-correlation function of 1X state
Initial Conditions:
Quantum Regression Theorem
Low excitation regime
1X Lifetime = 3.6ns
)()0(
0)0(
0)0(
011)2(
00
)2(11
)2(22
tσ=Ψ
=Ψ
=Ψ
)().......( 001010 tt σσ
0tt −=τ
)()()()(
)()()()(
)()()()(
00100010)2(
00
00111010)2(
11
00122010)2(
22
ttt
ttt
ttt
σσστ
σσστ
σσστ
=Ψ
=Ψ
=Ψ
)()()(
)()()()()(
)()()(
)2(111
)2(001
)2(00
)2(222
)2(1121
)2(001
)2(11
)2(222
)2(112
)2(22
ττωτ
ττωτωτ
ττωτ
ΨΓ+Ψ−=Ψ
ΨΓ+Ψ+Γ−Ψ=Ψ
ΨΓ−Ψ=Ψ
p
pp
p
dt
d
GFdt
d
dt
d
31
0 5 10 15 20 25 300
1x10-4
2x10-4
3x10-4
4x10-4
5x10-4
Unn
orm
. Cor
r. F
unc.
G2 (τ
)
Delay Time (ns)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Norm
. Corr. F
unc. g2(τ)
Case 2 – Biexciton Emission Self-Correlation
Initial Conditions:
Self-correlation function of 2X state
2X Lifetime = 2.6ns and effect of pumping rate
0)0(
)()0(
0)0(
)2(00
022)2(
11
)2(22
=Ψ
=Ψ
=Ψ
tσ
)().......( 012021 tt σσ
)()()()(
)()()()(
)()()()(
01200021)2(
00
01211021)2(
11
01222021)2(
22
ttt
ttt
ttt
σσστ
σσστ
σσστ
=Ψ
=Ψ
=Ψ
32
0 5 10 15 20 25 300
2x10-3
4x10-3
6x10-3
8x10-3
1x10-2
Unn
orm
. Cor
r. F
unc.
G2 (τ
)
Delay Time (ns)
0
1
2
3
4
5
6
7
8
9
10
11
Norm
. Corr. F
unc. g2(τ)
Case 3 – Single Exciton Biexciton Correlation:Single Exciton Emitted after the Biexciton
Correlation function between 1X and 2X states
Initial Conditions:
1X Lifetime = 3.6ns
0)0(
)()0(
0)0(
)2(00
022)2(
11
)2(22
=Ψ
=Ψ
=Ψ
tσ
)().......( 012021 tt σσ
)()()()(
)()()()(
)()()()(
01200021)2(
00
01211021)2(
11
01222021)2(
22
ttt
ttt
ttt
σσστ
σσστ
σσστ
=Ψ
=Ψ
=Ψ
Probability of 1X emission at time t given that a 2X emission has occurred at time t0, for t0<t.
33
0 5 10 15 20 25 300
2x10-4
4x10-4
6x10-4
8x10-4
1x10-3
Unn
orm
. Cor
r. F
unc.
G2 (τ
)
Delay Time (ns)
0.0
0.5
1.0
Norm
. Corr. F
unc. g2(τ)
Case 4 – Single Exciton Biexciton Correlation:Biexciton Emitted after the Single Exciton
Correlation function between 1X and 2X states
Initial Conditions:Probability of 2X emission at time t given that a 1X emission has occurred at time t0, for t0<t.
)()0(
0)0(
0)0(
011)2(
00
)2(11
)2(22
tσ=Ψ
=Ψ
=Ψ
)().......( 001010 tt σσ
)()()()(
)()()()(
)()()()(
00100010)2(
00
00111010)2(
11
00122010)2(
22
ttt
ttt
ttt
σσστ
σσστ
σσστ
=Ψ
=Ψ
=Ψ
34
Demonstration of Cases 3 and 4
Case 4Case 3
Case 3Case 4
-20 -15 -10 -5 0 5 10 150
50
100
Delay Time (ns)
0
100
200
Coi
ncid
ence
Cou
nts
n(τ) 0
100
200
300
400
-20 -15 -10 -5 0 5 10 15
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
Correlation F
unction g2(τ)
0.0
0.5
1.0
1.5
P
5P, 1X sat.
2P
35
Triggered Single Photon Emission – Dephased Two-Level Emitter
Wp1(t) Γ1
0
1
)()()()(
)()()()(
11100100
11100111
tttWdt
td
tttWdt
td
p
p
σσσ
σσσ
Γ+−=
Γ−=
),(),(),(
),(),(),(
)2(11
)2(11
)2(
τττ
τ
τττ
τ
tGtFWd
tdF
tGtFWd
tdG
p
p
Γ+−=
Γ−=
Initial Conditions:
)()0,(
0)0,(
11
)2(
ttF
tG
σ==
)()()(),(
)()()(),(
00100010
00111010)2(
ttttF
ttttG
σσστ
σσστ
=
=
Quantum Regression Theorem
1)0(
0)0(
00
11
=
=
σσ
∫∞→=
T
TdttGG
0
34)2(
exp_34)2( ),(
~lim)(
~ ττ
Pump laser pulse train
36
Triggered Single Photon Emission – Dephased Two-Level Emitter
Wp1(t) Γ1
0
1
G(2
) (τ)
Wp1
(t)
τt
37
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Single Quantum Dots
Single dye molecules
Single N-V centers in diamond
Single Atoms
Single Ions
Electrical Pumping
Practical Issues
Selected Applications
Summary
38
Single Quantum Dot
J.-M. Gérard, and B. Gayral, IEEE J. Lightwave Tech. 17, 2089 (1999).S. Raymond, K. Hinzer, S. Fafard, and J. L. Merz, Phys. Rev. B 61, 16331(R) (2000).
FILMTHICKNESS
MBEIn
As
NU
MB
ER
OF
MO
NO
LAY
ER
SR
S
0
5
10
15
20
25
30
35
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.02 0.03 0.04 0.05 0.06 0.07
COMPOSITION:X
(In)xGa(1-x)As/GaAs
Islands
Layers
MISFIT STRAIN WITH GaAs
14 Å
28 Å
42 Å
56 ÅDislocations
Quantum dotsIslands
FILMTHICKNESS
QDs
W.L.
39
Single Terrylene Molecule
B. Lounis and W.E. Moerner, Nature 407, 491 (2000)
Terrylene
40
Single N-V Center in Diamond
C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, Phys. Rev. Lett. 85, 290 (2000)
Single N (nitrogen) – V (vacancy) center
41
Single Atom
A. Kuhn, M. Hennrich, and G. Rempe, Phys. Rev. Lett. 89, 067901 (2002)J. McKeever et al., Science 303, 5666 (2004)
Single Cesium Atoms
42
Single Trapped Ion
M. Keller et al., Nature 431, 1075 (2004)
Single Calcium Ions
43
Electrical Pumping
Z. Yuan et al., Science 295, 102 (2002)
44
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
45
Practical Issues
Problem 1: Low collection efficiencies can yield single photon sources which do not have a sub-Poissonian statistics
Problem 2: Absence of ideal photodetectors that can distinguish between 0, 1 and 2 photons
46
Photons Emitted by a Pulsed Laser
αα −= en
nPn
!)(
Photons emitted by a pulsed laser obey Poisson’s distribution
Probability of emitting n photons per pulse:
α: average number of photons emitted per pulse
Pulsed laser
47
Photons Emitted by a Pulsed Laser
α = 25 α = 10
α = 0.1α = 1
48
Single Photon Turnstile Device
Single photon turnstile device
Turnstile
49
Pratik Problemler
α = 0.1
Attenuated laser Single photon source
Source of error
0.1*(500 MHz)=50 MHz
80 MHz pump laser repetition frequency
High collection efficiency requiredfor high single photon emissionrate
Typical single photon collectionefficiency < 50 %
50
Solution – Optical Microcavities
Optical microcavities confine light in all 3 dimensionsThey possess high quality, low volume resonancesThey direct light in a specific direction
FabryFabry--PerotPerotMiMiccrorocavitycavity
MiMiccropillarropillar
51
Electrically Pumped Optical Microcavity Structure
Single photon emission rate: 4 MHz !
S. Strauf et al., Nature Photonics 1, 704 (2007)
52
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Quantum physical random number generator
Quantum cryptography
Linear optics quantum teleportation
Linear optics quantum computation
Two photon interference using a single photon source
Two photon interference using a single molecule
Summary
53
Quantum Physical Random Number Generator
Detection: AXXAXAX...
Single photonstream
50/50 beamsplitter
Single Photon Counter:Digital 1
Single Photon Counter: Digital 0
Detection: XAAXAXA...
Random Number:1001010...
54
Quantum Key Distribution
55
Quantum Key Distribution - BB84 Protocol
Fiber-opticcable
Experimental setup for Quantum key distribution. A, attenuator; BS, beam splitter; P-BS, Polarizingbeam splitter; EOM, Electro-optik modulator; D1, D2, D3, D4, single photon counters; FC, fiber coupler
Classical communication line(e.g. internet)
Alice
EOMSingle photonsource
FC
YGA
Bob
λ/4 plate
D1
D2
D3
D4
P-YGA
P-YGA
FC
H,V detection
L,R detection
Synchronizationsignal
HV
L R
56
Quantum Key Distribution - BB84 Protocol
11Key shared by the twoparties
TrueFalseFalseTrueFalseAlice’s answer
LLCLCCLBasis information thatBob sends Alice
HVLHLRHBob’s measurement
LLCCLCCLLRandom basis selectedby Bob
RVHRLHRVLPolarization of thephoton sent by Alice
CLLCCLCLCRandom basis selectedby Alice
110100110Bit sent by Alice
57
Quantum Key Distribution
Quantum cryptography realized under the lake between the town of Nyon, about 23 km north of Geneva,and the center of the city.
58
Two Photon Interference
0 probability for identical photons
59
Two-Photon Interference
−=
+
+
φ
φ−
+
+
)t(a
)t(a
1e
e1
2
1
)t(a
)t(a
4
3i
i
2
1
( )43i
43i 20e02e
2
1 φ−φ −=( ) 0aaaaaaeaae2
10aa11 344344
i33
i2121
++++++φ−++φ++ −+−==
Two-photon entangled state
Two otherwise indistinguishable photons which enter a 50/50 beam splitter throughdifferent input channels will leave through the same output channel
1
1
1
23
4
=0 BOSON
++++ = ijji aaaa ++++ −= ijji aaaa
Bosons Fermions
•First experimental demonstration using twin-photons generated by parametric down conversionC. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
60
Two Photon Interference
011 21++=⊗ aa
1
1
BS
( )( ) ( ) ( )2,00,22
02
02
122112121 +−=+−=+−− ++++++++ i
bbbbi
bibibb
−−
=
+
+
+
+
2
1
2
1
1
1
2
1
b
b
i
i
a
a
2
1
2x2 unitary matrix
a2+
a1+
b2+
b1+
61
Linear Optics Quantum Teleportation
01 ⊗⊗ϕ
ϕ
1
0
10 βαϕ +=
M1
M2BS
BS
)0110(2
1i−⊗ϕ
c
100,1 21 βα +=⇒== cMM
101,0 21 βα −=⇒== cMM
00,2 21 β=⇒== cMM
02,0 21 β=⇒== cMMM1, M2 should be able todistinguish 2 and 1 photonFock states
3
2
1
+⊗− 1002
αi
( )++⊗− 10102
βαi
( )+−⊗ 100121 βα
+⊗− 02022
βi
00222
β⊗− i
−−
=
+
+
+
+
2
1
2
1
1
1
2
1
b
b
i
i
a
a
62
A Brief Introduction to Linear Optics Quantum Computation
Two observations:
• Non-deterministic linear optics quantum computation is possible• Probability of success can be increased arbitrarily close to 1
E. Knill, R. Laflamme, and G. J. Milburn, Nature (London) 409, 46 (2001).
Ideal photodetectors (n-photon detection)Stringent requirements on single photon sources:
large total collection efficiency and 100% indistinguishability
63
Nonlinear Phase Shift
210 210 αααψ ++= 210 210 ααα −+
ππππ
0
n
n 0
1
θθθθ1
θθθθ2θθθθ3
1
θ1=22.5o , φ1=0o θ2=65.5302o , φ1=0o θ3=-22.5o , φ3=0o
ψ
64
Conditional Sign Flip
ba ( ) baab1−
11100100 3321 αααα +++ 11100100 3321 αααα −++
Universal quantum gate – Proves the ability of quantum computation
101 = 010 =Qubits coded with two optical modes
65
Conditional Sign Flip
NS-1
NS-1
45o -45o
13130220 +
1
2
3
4
Success rate 1/16
10
n 1n 0
10
n 1n 0
Q1
Q2
Proves nondeterministic quantum computation
66
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Quantum physical random number generator
Quantum cryptography
Nonlinear optics quantum teleportation
Linear optics quantum computation
Two photon interference using a single photon source
Two photon interference using a single molecule
Summary
67
Two-Photon Interference Using a Single Photon Source
1hν
hν 1
BS
Single photonsource
Single photonsource
)(ˆ)(ˆ)(ˆ)(ˆ),( 344334)2( tatatatatG τττ ++= ++
3
4
1
2
68
Two-Photon Interference Using a Single Photon Source
)(ˆ)(ˆ)(ˆ)(ˆ),( 344334)2( tatatatatG τττ ++= ++
g
e
hν
−=c
tAta ge
rr σ)()(ˆ
Source-Field relationship
egge =σ
( )234
)2( )(ˆ)(ˆ)(ˆ)(ˆ21
),(~
tttttG geegeeee στστσστ +−+=
For a balanced beam splitter, θ=π/4:
( )2)1( ),(
~)(ˆ)(ˆ
2
1 ττσσ tGtt eeee −+=
Atomic projection operator
Solve the problem of two-photon interference using the microscopic properties of the emitter7
69
Incoherently Pumped Single Photon Source
( ) )(ˆ)(ˆ)(ˆ2
)()(ˆttt
ti
dt
tdpgrelaxgppg
Lpp σσσσ
Γ−−Ω−=
)(ˆ)(ˆ)(ˆ
ttdt
tdeesponpprelax
ee σσσ
Γ−Γ=
[ ] ( )eepppeeprelaxH
idt
d σρρσσρσρρ ˆˆˆˆˆˆˆ22
ˆ,ˆ1ˆ
int −−Γ+=h
( )eeeeeggespon σρρσσρσ ˆˆˆˆˆˆˆ22
−−Γ
+
dephspon γγ +
Γ=
2
( ))(ˆ)(ˆ2
)()(ˆ
)(ˆtt
tit
dt
tdggpp
Lpgrelax
pg σσσγσ
−Ω−=
( )gppgL tiH σσ ˆˆ)(ˆint −Ω= h
)(ˆ)(ˆ τσσ +tt eeeeSolve for using Optical Bloch Equations ( )ATrA ˆˆˆ ρ=
( )212
)1(34
)2( ),(~
)(ˆ)(ˆ21
),(~ ττσστ tGtttG eeee −+=
70
Incoherently Pumped Single Photon Source
Quantum Regression Theorem),(~),(
~)1(
)1(
τγτ
τtG
d
tGd −=
)(ˆ)(ˆ
tdt
tdeg
eg σγσ
−=
)(ˆ)0,(~ )1( ttG eeσ=
)(ˆ)(ˆ),(~ )1( tttG geeg στστ +=
( )212
)1(34
)2( ),(~
)(ˆ)(ˆ21
),(~ ττσστ tGtttG eeee −+=
71
71Normalization
∫ τ=τ∞→
T
0
34)2(
Texp_34
)2( dt),t(G~
lim)(G~
Det.
Det.
hννννhνννν
Coincidence Detection
A0 A1A-1
∫ ∫
∫ ∫
+=
pulse
pulse
T
A
eeee
T
A
dtdtt
dtdtG
A
A
0
0
34)2(
1
0
1
0
)(ˆ)(ˆ
),(~
ττσσ
ττ
Normalized coincidence rateA. Kiraz et al., Phys. Rev. A. 69, 032305 (2004).
72
Effect of Dephasing
0.01 0.1 1 10
0.0
0.5
1.0
Indistinguishability
γe_deph
(Γe)
0,01 0,1 1 10
0,0
0,5
1,0
Indistinguishability T
2/2T
1
γe_deph
(Γe)
Τ1=1/Γspon : lifetimeΤ2: coherence time 1/Τ2= 1/2T1 + 1/ T2
*
dephspon γγ +
Γ=
2
T2/2T1
73
Effect of Dephasing
21 ψ≠ψ
0)t(adte0
)t(i2/tspon +∞ φ+Γ−∫∝ψ
Single photon wavefunction
φ(t): random function describing the pure dephasingΓspon: spontaneous emission rate
x(t)
1
22*2
21 T2T
)t(y)t(dtx ==ψψ ∫Mean overlap integral
Single photons should possesslarge coherence lengths !
74
Two-Photon Interference Using a Single Emitter
Experimentally difficult to find two single emitters emitting at the same exact photonfrequency.
Instead use two photons subsequently emitted from a single emitter
∆t
hνhν1
1
BS BS
∆t = Pulse separation
75
Two-Photon Interference Using a Single Emitter
C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, Nature 419, 594 (2002) → Quantum dots
T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 93, 070503 (2004) → Atoms
76
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Quantum physical random number generator
Quantum cryptography
Nonlinear optics quantum teleportation
Linear optics quantum computation
Two photon interference using a single photon source
Two photon interference using a single molecule
Summary
77
Vibronic Excitation
Nuclear Coordinate
Zero-Phonon-Line
|S0 >
|S1 >
Room Temperature
absorption emission
Vibronic excitation was not much explored at the single molecule level at low temp eratures!
A. Kiraz et al., J. Chem. Phys. 118, 10821 (2003).
Zero-phonon-line emission from a single molecule is highly coherent
78
Terrylenediimide
N N
O
OO
O
O
Terrylenediimide (TDI)Dimensions: 3.18nm x 0.92nm x 1.14nm
n-Hexadecane C16H34Shpol’skii matrixSemi-Crystalline
79
Experimental Setup
Cryostat, 1.4 K
Counter APD
λexc
Pinhole
Scanning Fabry-PerotSpectrum Analyzer
λ/2 plate
galvano optic scanner
Autoscan single modecw dye laser 565-615nm
Monochromator
CCD
APD
Spectral resolution:1 MHz (excitation) 35 GHz (monochromator)15 MHz (spectrum analyzer)
Spatial resolution:<1µm
Aspheric lensNA = 0.55
80
Emission Spectrum
0.0
0.5
1.0
Nor
mal
ized
Inte
nsity
14700 14800 14900 15000
0.00
0.05
ZP
L
Wavenumber (cm-1)
~40% emission through the ZPL!
20 40 60 80 100
14950
15000
15050
Intensity (a.u.)
Time (sec)
Wav
enum
ber
(cm
-1)
0
20.00
40.00
60.00
80.00
100.0
Discrete spectral jumps: Proof for single molecule detection
A. Kiraz et al., J. Chem. Phys. (2003).
81
Emission Linewidth Measurements
A. Kiraz et al., Appl. Phys. Lett. 85, 920 (2004).
T2 = 4.9ns coherence length“Almost transform limited ZPL emission !”
1 scan with500 ms integration time at each bin
0
50
100
150
200
250
-+FWHM=65 10 MHz
Num
ber
of C
ount
s
0 100 200 300 400 500 600
-500
Res
idua
ls
Bins
Compared with ~ 40 MHz transform limit
2T1/T2≈1.6
Counter APD
interference filter2 nm FWHM
82
Experimental Setup
Cryostat
λexc
Pinhole
λ/2 plateAspheric lens
NA = 0.55xyz piezo scanner
Autoscan single modecw dye laser 565-615nm
λ/2 plate
t0t0+∆t
2APD
AP
D
Michelson InterferometerTAC MCA
start
stop
Hanbury Brown and Twiss
∆t = 4.6 ns >> coherence length / 2
83
Experimental Setup
λ/2plate
84
Two-Photon Interference Using cw Excitation
)t(a)t(a)t(a)t(a
)t(a)t(a)t(a)t(a)(g
4433
3443)2(34 τ+τ+
τ+τ+=τ
++
++
( )( )2)1()2()2(||34 )(g1)(g
21
)(g τ−+τ=τ
( )1)(g21
)(g )2()2(34 +τ=τ⊥
hν1
2 3
4
First-order photon correlationfunction
-0,5 0,0 0,50,0
0,2
0,4
0,6
0,8
1,0
|| Polarization Polarization
g(2) 34
(τ)
τ (1 / Γspon
)
Signature of two-photoninterference
BS
hνcw excitation
λ/2 plate
85
Experimental Results
-30 -20 -10 0 10 20 300.0
0.5
1.0
g(2) (τ
)
(e)
(d)
(a)
Delay Time (ns)
-30 -20 -10 0 10 20 300.0
0.5
1.0
g(2) (τ
)
(b)
Delay Time (ns)
-30 -20 -10 0 10 20 30
-0.1
0.0
0.1
0.2
0.3
(g(2
) -g(2
) )/g(2
)
(c)
Delay Time (ns)
-30 -20 -10 0 10 20 30
0.0
0.2
(g(2
) -g(2
) )/g(2
)
Delay Time (ns)
0.0
0.2
(g(2
) -g(2
) )/g(2
)
⊥
⊥⊥
||||
(⊥-||) / ⊥
⊥||
(||1-||2) / ||1
(⊥1- ⊥2) / ⊥1
|| 1|| 2
|| 1⊥
1⊥
2⊥
1
A. Kiraz et al., Phys. Rev. Lett. (2005)
Coincidence reduction factor: 24.0)0(
)0()0()0(
)2(||
)2(||
)2(
=−
= ⊥
g
ggV
Poor mode-matching in the beam splitter!
86
Experimental Results
-30 -20 -10 0 10 20 30
0,0
0,5
1,0
g(2) (τ
)
Delay Time (ns)
-30 -20 -10 0 10 20 30
0,0
0,5
1,0
g(2) (τ
)
Delay Time (ns)
-30 -20 -10 0 10 20 30
-0,1
0,0
0,1
0,2
(g(2
) -g(2
) )/g(2
)
Delay Time (ns)
⊥
⊥⊥
||||
(⊥-||) / ⊥
⊥||(a) (b)
(c)
-30 -20 -10 0 10 20 30
-0,1
0,0
0,1
0,2
-0,1
0,0
0,1
0,2
(g(2
) -g(2
) )/g(2
)
Delay Time (ns)
(g(2
) -g(2
) )/g(2
)
(||1-||2) / ||1
(⊥1- ⊥2) / ⊥1
(d)
(e)|| 1
|| 2|| 1
⊥1
⊥1
⊥2
87
Outline
Triggered Single Photon Sources
Fluctuations Properties of Light
Single Photon Generation Using a Single Dipole
Review of Available Experimental Systems
Practical Issues
Selected Applications
Summary
88
Summary
At the end of this course I hope that you became familiar with:
The current status in single photon source research
Single photon sources are now very much real!
The calculation of the second order correlation function of light emitted by a single dipole
Quantum physical random number generator
Quantum key distribution
Linear optics quantum teleportation
Linear optics quantum computation
Two photon interference phenomenon
References:
• R. Loudon, “The quantum theory of light“, Oxford University Press, 1983 • A. Đmamoğlu and Y. Yamamoto, “Mesoscopic Quantum Optics“, Wiley Inter-Science, 1999• L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics“, Cambridge University Press, 1995