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IFUAP-BUAP Mini-curso en Nanoestructuras
Parte II. Sergio E. Ulloa – Ohio University
Department of Physics and Astronomy, CMSSCMSS,and Nanoscale and Quantum Phenomena InstituteNanoscale and Quantum Phenomena Institute
Ohio University, Athens, OH
Supported by US DOE & NSF NIRT
¿Preguntas? ¿Más info?
ulloa@ohio.edu
www.phy.ohiou.edu/~ulloa/
nano.gov
Resumen/Outline• Quantum dots – confinement vs interactions
– How to make / study them
• Coulomb blockade & assorted IV characteristics
• Optical effects – excitons: selection rules, field effects
• Transport in complex molecules: the case of DNA
Quantum Dots: L ~ λgood things come in small packages
Confinement: KE~ k2 ~ L-2
Interactions: PE ~ q2/L
For L~100nm, PE ~ 1meVwhile KE ~ 0.5meV (in GaAs)
For L~5nm, PE ~ 20meV ; KE ~ 200meV
Total energy
E = KE + PE
Quantum dot fabrication
Lithographically
Kouwenhoven et al, U Delft Weis et al, MPI Stuttgart
Ensslin et al, ETH Zurich
Quantum dot fabrication
Self-assembly· Stranski – Krastanow islands· MBE· in-plane densities
~ 1010 - 1011 cm-2
· size variations < 10%· sharp photoluminescence
features, frequency ∝ size
AlAs and GaAs “antidots” in InAs
Tenne et al, PRB May 2000
Quantum dot fabrication
Colloidal dots
• Chemical synthesis• CdSe, CdS, InP, etc• Size ~ 5nm diameter• Uses for biotags
Quantum Dot Corp.
excessexcessNaBHNaBH44
solublesolublemetal saltmetal salt(AgNO(AgNO33))
colloidal METAL dots in dendrimers0.6
0.5
0.4
0.3
0.2
0.1
0.0
Abs
orba
nce
800600400Wavelength (nm)
428 nm481 nm
-H2O
• Au or Ag ~3-5nm dots
• pH or hydration changesinterdot separation aggregation changesinteractions & color
G. Van Patten, Ohio University
QD w/tunable size and e- number~ artificial atomsElectronic transport -
Low bias: ground state – Coulomb blockadeHigh bias: excited states – selection rules(!)Rings, phases & resonances
I
ωIncident/outgoing photons -
Visible/optical: excitonFar infrared: internal multi-electron excitationsRaman: excitons/confined phononsω
Capacitance -Ground state vs B-fieldCombination w/optics & in-plane transportB
Resumen/Outline• Quantum dots – confinement vs interactions
– How to make / study them
• Coulomb blockade & assorted IV characteristics
• Optical effects – excitons: selection rules, field effects
• Transport in complex molecules: the case of DNA
Coulomb blockade
I
B field effects
QD molecules
Ramirez et al, PRB 1999
Blick et al Science 2002
future?multidotsfor quantumcomputing
in seriesin parallel
Kondo effectGoldhaber-Gordon et al Nature 1998
Kondo primer
zero-bias feature
Kouwenhoven & Glazman, Phys World 2001
Aharonov-Bohm effect22 1
2 2
ˆ;
p eH p Am m c
B A A Bρϕ
= → −
= ∇× =Φ
B
ρ
Phases, dots, “rings” and resonances
1. Phase in experiments AB interferometer
2. Phase of a resonance (single particle)
3. CB in QD ~ SP resonance?
4. What’s the Fano effect / lineshape?
5. Fano as probe of coherence in QD?
6. QD in Kondo regime phases in expts?
7. Fano+Kondo to probe coherence in QDK?
Phase in experiments AB interferometerSchuster et al, Nature 1997
Yacoby/Heiblum PRL 1995
~ CB peaks
AB oscill’s
|tQD| eiθ
weaker AB signal whenQPC is on
Nature 1998
2 i 2CE QD dir
2 2QD dir QD di Q dirr
0
D
G ~| t | | t e t |
G ~| t | | t | 2 | t | cos(|
/
| t
2
)
ϕ
ϕ
θ θ
π
ϕ
∆= +
∆ = Φ Φ
− − ∆+ +
AB interferometer: “phase measurer”QD
dir
e-
If θdir is indep of Vgate θQD can be measured
QDiQD QD
n
1 nQD C
i / 2t C | t | eE E i / 2
2(E E )tan
θ
θ θ −
Γ= =
− + Γ−
= +Γ
A “pure” resonance has the Breit-Wigner form:
∆θQD/π
BW “SP” resonances OK….coherent propagation through QDCB charging ~ not important
but….
sequence of BW resonanceswould be expected to accumulate while phase “resets” to 0 as tQD ~ 0 !!?
overlapping resonances … BW??
e-e interactions?
G(B)=G(-B) not valid here (open device ∆θ ≠ 0,π)Why? … N theory papers …
nice disc: Aharony et al cond-mat/0205268
Fano effect / lineshapeinterference of discrete “autoionized” statewith a continuum asymmetric peaks in atomic excitation spectra
In QD; expt: Göres PRB 2000th: Clerk PRL 2001 …Ugo Fano, PR 1961
In AB interf + QD;expt: Kobayashi PRL 2002
Fano … ψE
mixVE
E'E E 'E '
a bϕ ψΨ = +∑&ϕ
21 E
2
E 'E
E
'
'
E
'
E
E| V |(E) tanE E F(E
; ')
| V |; F(E
1a s V sin (E
) shift of resonance due to continuumE E '
in ( )b cos (E) (E EV E E
E
)
)V '
ϕ
π
δππ
− = =−
∆ = −−
∆= − ∆ −
−= ∆
− ∑
“excitation” of ΨE via an operator T from an initial state I yields:
E ' E '
E
E E
'
*E
V modified state due to mixE E '
sin| T | I | T | I | T | I cosVψϕ
ψπ
Φ = + →−
∆Ψ = Φ − ∆
∑
Fano lineshape2
E*E E
2 2E
2 2E
| | T | I | (q )| |
E E F|
T | I | 1 transition prob via "autoionize
T | Iq ; cot ; 2 | V |
V | T | I / 2
d" state
ϕε
εψ ε
ππ ψ
− −Φ=
Ψ=
= − ∆ =
+=
+
Γ =Γ
Is G ~ |tQD + tdir|2 a Fano line?
idir d QD
zt e G ; ti
β
ε= =
+
2
d 2
| q |G G1
εε+
=+
withi
d
zq i eG
β−= +
Clerk PRL 2001Notice prob can vanishdue to interference
Fano as probe (proof?) of coherence in QD (CB)
Kobayashi PRL 2002
direct paths “through” QD?
how is QD “intrinsic” width affected by ring?
decoherence?1 vs N passes?
QD in Kondo regime phases in expts?
peaks deform
AB osc’s clear
phase lapses in CBbutplateaus in K valley~ π
Ji, Heiblum, Science 2000
phases in Kondo regime
phase evolves from plateaus ~ πto
lapses to 0 as Γ decreases
th: Gerland PRL 2000
K res phase shift ~ π/2(NRG)
phases in Kondo regime
w/temperature w/DC bias
Fano+Kondo to probe coherence in QDK?
Hofstetter PRL 2001(~Bulka PRL 2001) r 1
dotG ( ) [ ( )]ω ω ε ω −= − − Σ
AB phase/flux
direct ~ W
and for
dot
2R
2
2
b
2
b
L b
2
(e q) sing T "generalized Fano fo
e 2[ (0
rm"
)] /
4 / ; q (1e 1 e
T ) / T cos1
ε
α
ϕα
α ϕ
= +ℜΣ Γ
= Γ
+= +
+Γ Γ = − −
+
references on phases/Kondo/resonances
• Yacoby, PRL 74, 4047 (1995)
• Schuster, Nature 385, 417 (1997)
• Aharony, cond-mat/0205268
• Fano, PR 124, 1866 (1961)
• Clerk, PRL 86, 4636 (2001)
• Kobayashi, PRL 88, 256806 (2002)
• Ji, Science 290, 779 (2000)
• Gerland, PRL 84, 3710 (2000)
• Bulka, PRL 86, 5128 (2001)
• Hofstetter, PRL 87, 156803 (2001)
IFUAP–BUAP – Minicurso en NanoestructurasTarea # 1 — Entrega: 22 Julio 2003
1. [40pts] (a) Calcule el espectro para una partıcula de masa m confinada en una caja “rectan-gular” en 3D y 2D de radio R. Suponga que la caja tiene paredes infinitamente duras.(b) Describa como varıa la energıa de confinamiento con respecto a R.(c) Escriba la forma completa de los eigenstados en 3D/2D.(d) ¿Cual es el valor de la energıa (en eV) para los dos primeros estados en 2/3D si la masa dela partıcula es 0.067m0 (con m0 la masa del electron libre), y el radio es R = 5, 50, 100nm?Hint: funciones de Bessel.
2. [60pts] (a) Estime el efecto de la interaccion de Coulomb entre electrones en un punto cuanticode radio R, utilizando U = e2/Rε, donde ε es la constante dielectrica del material (≈ 12 paramaterials tıpicos), si el radio del punto es 5 o 100nm. Compare esta estimacion con la diferenciaentre los dos primeros niveles en un punto como se modelo en 1 arriba, ∆ = E2 − E1. ¿Paraque radio R son estas dos cantidades iguales (U = ∆)? Haga la estimacion tanto en 2D comoen 3D.(b) Use teorıa de perturbaciones para hacer la estimacion mas confiable:
〈Ψn(1)Ψm(2)|V (1 − 2)|Ψk(1)Ψl(2)〉,
en donde V (1 − 2) = e2/ε|r1 − r2| es la interaccion de Coulomb entre dos partıculas, y lasvarias Ψj son las funciones de onda (modeladas/escritas en 1(c) arriba). Use cualquier metodode integracion, numerica incluso, para evaluar (o al menos estimar) la integral en el caso quelos dos electrones estan en el estado base (y diferente spin, por supuesto) en un punto en 3D.Compare con (a) y comente.Hint (aunque no esencial):
1
|r|=
∫ 4π
q2e−iq·rd3q
IFUAP-BUAP Mini-curso en Nanoestructuras
Parte II. – Toma 2Sergio E. Ulloa – Ohio University
Department of Physics and Astronomy, CMSSCMSS,and Nanoscale and Quantum Phenomena InstituteNanoscale and Quantum Phenomena Institute
Ohio University, Athens, OH
Supported by US DOE & NSF NIRT
Resumen/Outline• Quantum dots – confinement vs interactions
– How to make / study them
• Coulomb blockade & assorted IV characteristics
• Optical effects – excitons: selection rules, field effects
• Transport in complex molecules: the case of DNA
Quantum dot fabrication
Self-assembly· Stranski – Krastanow islands· MBE· in-plane densities
~ 1010 - 1011 cm-2
· size variations < 10%· sharp photoluminescence
features, frequency ∝ size
laser power
Findeis et al, Sol. State Comm, 2000
•Excitons•Excited excitons (s- and p-shell)•Biexcitons, X-, X--, etc.
Photoluminescence in SINGLE QD
Excitons and dot shapes• Dot asymmetries reflected in exciton properties:
– Binding energy vs dot size– Oscillator strength / optical response– Influence of magnetic field
• Raman differential cross section and intensity ---experiments and phonon mode confinement:– Selection rules– Carrier masses– Scanning experiments
• Quantum rings: excitonic Aharonov-Bohm effect for neutral/polarizable entity
see part I
Trallero & Ulloa PRB 1999
Simple geometry: Parabolic confinementV(x,y)
Lx
wx
wy
Ly
B
Eg
Vez Vhz
x
y
z
Lz
( )2
2 2 2 2
2
1 1p A ( )2 2
| r r |
e h eh
j j j j x j y j jj
ehe h
H H H H
eH m x y V zm c
eH
ω ω
ε
= + +
= ± + + +
= −−
COM rel cH H H H= + +
Harmonic Oscillator in
2D
Song& SU; Pereyra & SU PRB 2000
effective confinement increases with B field
0
1
/ /
1 (circular limit)
Bx y x y
B
η ω ω η ω ω
η
=
>>
= → =
→
off-diagonal Coulomb interaction
exact expression: Song & SU PRB 1995
off-diagonal one-particle elements
Coulomb interactions in harmonic oscillator basis
exciton size
binding energy
Lx/Ly = 1:1 ◊ 2:1 + 3:1
binding energy = Eexc – E0 = Eb
Eb ~ 1/L
saturates to 2D value for L>>aB & all shapes
exciton “shrinks” as L 0, rS Lbut
rs aB as L>> aB
χ = linear optical susceptibility ~ PLE signal ~ optical response
COM gnd state & replicas
excitedexcited states of internal degrees states of internal degrees of freedomof freedom
area of dots = 5x5 nm2
Magnetic field effects
diamagnetic shift
orbital “Zeeman” splitting
binding energy exciton size
η=1η=1.25 suppression of
Zeeman orbital splitting
asymmetry of structure
SAQD RINGS!!
80nm
smallest COHERENTring potential for electronsand/or holes
Lorke et al PRL 1999
J Garcia 1999
Is there AB effect for excitons?
A le edc c
φ∆ = = Φ∫flux lines
v 0→
v exciton charge = ZERO→ no ABE?
AB is the nonadiabatic version of Berry phase in a flux
M. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)
Ground state of 1D excitons (Romer & Raikh 2000): BUT...
Results for microscopic calculationSong & SU PRB 2001
soft/parab walls
hard wall
NO AB oscillations!!binding energy
AB oscillations seen if ring is narrower ~ 1D likewidth = 10nm
Hu et al PRB 2001
heavyhole
lighthole
BUT excitons are polarizable!
flux lines
e-
∆R
80 nm
e-h+
v 0→
v e-
Φe
flux lines
h+∆Φ
( )e h e hec
φ φ∆ − ∆ = Φ − Φ
net flux is nonzero!
*0 0( ) / 2e hR R R a= +strong Coulomb interaction limit:
22
1200
,2excE L EMR
∆Φ= + + Φ
2 2( ) .e hR R Bπ∆Φ = −
correlated motion*
0 0R aweak interaction limit:
2 22 2
2 20 0
,2 2
e hexc e h
e e h hE L L
m R m R Φ Φ
= + + − Φ Φ
2( ) ( ) ,e h e hR BπΦ = e hL L L= +
independent motion
weak interaction limit
optical emission stronglyoptical emission stronglysuppressed in B field suppressed in B field
windows windows
dark window Ltot non zero
bright window Ltot = 0
Govorov et al PRB 2002
strong interaction limit
optical emission stronglyoptical emission stronglysuppressed after critical suppressed after critical
B field B field
effective persistent currentassociated with Berry´s phase
and angular momentum inground state
Optical emission from a charge tunable quantum ring Warburton et al Nature 2000
Optical detection of the AB effect in a charged particleBayer et al PRL 2003
lithographic ring
~90nm
~30nm
PL
no oscillations in X0
only diamagnetic shift
experiments calculations
reasonable agreement w/calculations interactionsdo not change period and only slightly the amplitude
Science 2000
IFUAP–BUAP – Minicurso en Nanoestructuras Tarea # 2
Efecto Aharonov-Bohm en un anillo conductor. (a) En la figura debajo se muestra la conductancia experimental de un anillo
construido en un gas de electrones de dos dimensiones en un semiconductor. Estime de la gráfica el período de éstas oscilaciones de Aharonov-Bohm. El eje horizontal es campo magnético en milésimas de Tesla. ¿Por cierto, porqué se esperaría que la conductancia G fuera simétrica al invertir B, tal y como aparece en la figura?
(b) Suponiendo que éstas oscilaciones son producidas por el efecto AB, la conductancia G se puede aproximar por:
G ≈ G0 |1+ exp(iΦ/Φ0)|2 ,
en done Φ=BA es el flujo a través del anillo de area A, y Φ0=h/e es el cuanto de flujo magnético. Deduzca entonces el diámetro del anillo. (c) Suponga que el anillo fuera “ancho” y que entonces tuviera mas de un canal
transversal. Si el anillo es de 1 micra de radio interno y 2 micras de radio externo, estime el numero de canales permitidos para una longitud de Fermi de 50nm. De este número, estime que rango de períodos de oscilaciones AB se podrían ver en el experimento. ¿Se esperaría que las oscilaciones sobrevivieran? Explique brevemente su respuesta.
IFUAP-BUAP Mini-curso en Nanoestructuras
Parte II. – Toma 3Sergio E. Ulloa – Ohio University
Department of Physics and Astronomy, CMSSCMSS,and Nanoscale and Quantum Phenomena InstituteNanoscale and Quantum Phenomena Institute
Ohio University, Athens, OH
Supported by US DOE & NSF NIRT
Photons, excitons, spins, energy transfer & entaglement – all in QDs
• Gammon/Steel – optical control of quantum dot state
• Imamoglou – single photon source from quantum dots
• Klimov – Förster coupling between quantum dots
• Zrenner – coherent control of quantum dot photodiode
Gammon/Steel – optical control of quantum dot state
Gammon/Steel – optical control of quantum dot stateBonadeo et al PRL 1998
Naturally Formed GaAs QDs
42 Å GaAs layer500 Å Ga0.3 Al0.7As
250 Å Ga0.3 Al0.7As
500 Å GaAs cap layerAl mask with apertures
42 Å STM image showing the island
formation and elongation.
Gammon et al., Phys. Rev. Lett.76,3005 (1996)40nm
Samples are provided by D.S. Katzer, D. Park, and E. S. Snow, NRL
1623 1623.5 1624 1624.5 1625 1625.5 1626
Photoluminescence of a Single Exciton
Energy (meV)
0.000 T
0.430 T
0.860 T
1.290 T
2.150 T
3.010 T
3.870 T
4.730 T
QD Magneto-excitons * * -- g factorg factorB0
Weak field limit:
* First reported by S. W. Brown et al.,Phys. Rev. B 54, R17 339 (1996)
Hint = g*exµBB0Sz + αB0
2
g*ex = g*
e + g*hh
g*hh = 6κ + 27q/2
0
0.2
0.4
0.6
0.8
0 1 2 3 4 5
splittingshift
Magnetic Field (Tesla)
Zeeman splitting
laser power
Findeis et al, Sol. State Comm, 2000
•Excitons•Excited excitons (s- and p-shell)•Biexcitons, X-, X--, etc.
Photoluminescence in SINGLE QD
• Linear Spectroscopy: PhotoluminescenceIndirect probe of exciton resonancesRequires spectral diffusion of excited carriers
• Coherent Nonlinear Spectroscopy:CW differential transmissionResonant excitationProbe coherent interaction in the system
Experimental Techniques
Lock-in Am
plifierR
F Electronics
Epump
EprobeENL
AOM ƒ~100MHz
Detector
Reference
Imaging
CW FrequencyStabilized Dye
Lasers
5 K
Eprobe
ENL (ε) ~ χ(3) (ε) Epump E*pump Eprobe
Isignal = Re ( ENLE*probe )
D Steel et al U Michigan
NonNon--degenerate Nonlineardegenerate NonlinearSpectroscopy: AdvantagesSpectroscopy: Advantages
• Probe single excitonic state decay dynamics
Measure both T1 and T2 *
• Probe coupling between different excitonic states
Probe inter-dot energy transfer and dot-dot coupling
Study excited states of excitons
Probe multi-exciton correlation effects ;
Study the coherent interaction between exciton doublet .
* Nicolas H. Bonadeo et al., PRL, 81, 2759 (1998)
hhhh ExcitonicExcitonic Level DiagramLevel Diagram
σ+2
1−2
1+
23+
σ−
23−
Non-degenerate experiments can excite both σ+ and σ- excitons by varying the frequency and the polarization of the excitation beams (pump and probe beam).
21−
23+
23−
23+
23−
21+
21−
21+ Two-exciton state
Bound state of multi-excitons
σ+ stateσ- state
Ground state:0-exciton state
∆E
Scattering states
B0 = 0 B0 ≠ 0
c
v
Exciting Two ElectronsExciting Two Electrons
c
Two contributions:1. Incoherent:
Ground state depletion
2. Coherent:
Zeeman coherence
between σ- and σ+ state
21−
23+
23−
23+
23−
21+
σ+ stateσ- state
Ground state
Pumpσ−
Probeσ+
σ+2
1−2
1+
23+
23−
Resonant and coherent
excitation of two electronsσ−
v
3-level diagram in 2-electron basis
Optically Entangling Two Systems: the Importance of Coupling
Optically Entangling Two Systems: the Importance of Coupling
couplingcouplingcoupling
H0 + H´ H0 + H´Htotal=
• Without coupling → Product state of the two subsystems.• A strong coupling allows one system to see the excitation of the other.
Coulomb interaction between charged particles: trapped ionsMagnetic dipole interaction: NMR systemsExciton-exciton Coulomb coupling: excitons
• Mutual coherence between E and E´ is essential .
Coulomb Correlation*:Coulomb Correlation*:Coulomb interaction between electronCoulomb interaction between electron--hole pairs within single QDhole pairs within single QD
Two electrons are involved
within a single QD.Pumpσ−
Probeσ+
21−
23+
23−
23+
23−
21+
21−
21+ Two-exciton state
Bound state of multi-excitons
σ+ stateσ- state
Ground state:0-exciton state
∆E
Scattering states
Coulomb interaction between the two
excitons is important.
Two-exciton state cancels the signal due
to the symmetry in the level diagram.
But
21−
23−2
3+
21+
σ+σ-
* Kner et al Phys. Rev. Lett. 78, 1319 (1997)Ostreich et al Phys. Rev. Lett. 74, 4698 (1995)
Turn Turn offoff the Coulombthe CoulombCorrelation
Turn Turn onon the Coulomb Correlationthe Coulomb CorrelationCorrelation
Pumpσ−
+-Probe
σ+g Pumpσ−
- +Probe
σ+g
4 5 6 7 8 9-2 -1 0 1 2 3
σ− σ+
Pump: σ−
1==== −+−+ γγγγ No Signal !!
4 5 6 7 8 9-2 -1 0 1 2 3Probe ( γ )
Ground-state
Depletion
ZeemanCoherence
Total Signal
Probe ( γ )
1632.3 1632.4 1632.5 1632.6Energy (meV)
Degenerate and Non-degenerate Coherent Nonliear Response
σ− σ+
pump
probe
Degenerate
Non-degenerate
1.3 Tesla5K
σ−
σ+
Evidence for Zeeman Coherence in Single QDs
Experiment : Coulomb Correlation and Experiment : Coulomb Correlation and ZeemanZeeman CoherenceCoherence
4 5 6 7 8 9
Ground state
depletion
ZeemanCoherence
Total Signal
-2 -1 0 1 2 3
σ+ σ−
Pump: σ−
1==== −+−+ γγγγ
Probe ( γ )
Creation of two-electron entanglement
Ultimate Goal
• Combine:Optical control of individual QDsLong spin lifetimesQD nanostructure engineering
• To produce:Qubit register of QD spins
Coherence of a single QD can be controlled – Gammon et al have demonstrated this for excitons [Stievator et al., PRL (2001)]
Next: QNOT and other quantum gates
Spins have long coherence times: dephasing times T2* = 5ns-300ns[Dzhioev et al., PRL (2002)]
Next: explore in single QDs; optical read-out and initialization schemes
Science 2001
Imamoglou – single photon source from QDs
Imamoglou – single photon source from QDsMichler et al., Science 290, 2282 (2000)
• It is difficult to isolate a single photon, or fix the number of photons in a pulse
• Fluctuations in photon number mask the quantum features of light
• A stable train of laser pulses have Poissonian (photon number) statistics
It is desirable to have single photon sources:
single photon turnstile
Complete regulation of photon generation
Single Photon SourcesQuantum Cryptography: secure key distribution by single photon pulses
Quantum Computation: single photons + linear optical elementsE. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46 (2001)
Available sources:
• Highly attenuated laser pulse ⇒ Poisson fluctuations
• Parametric down conversion ⇒ Random generation of single photons
Possible solution: Deterministic (triggered) single photon emission:Single Photon Turnstile DeviceA. Imamoglu, Y. Yamamoto, Phys. Rev. Lett. 72, 210 (1994)
Experiments:
Coulomb blockade of electron/hole tunneling in a mesoscopic pn-junction:
J. Kim et al. Nature 397, 500 (1999)
Single Molecule at room temperature:
B. Lounis and W.E. Moerner, Nature 407, 491 (2000)
Single InAs Quantum Dot in a microcavity:
P. Michler et al., Science 290, 2282 (2000)
Signature of a triggered single-photon source
2)2(
)(:)()(:
)(tItItI
gτ
τ+
=• Intensity (photon) correlation function:
1
0 τ
g(2)(τ)• Single quantum emitter (I.e. an atom) driven by a cw laser field exhibits photon antibunching.
g(2)(τ)
τ0
• Triggered single photon source: absence of a peak at τ=0 indicates that none of the pulses contain more than 1 photon.
Photon antibunching• Intensity correlation (g(2)(τ)) of light generated by a
single two-level (anharmonic) emitter.
• Assume that at τ=0 a photon is detected:- We know that the system is necessarily in theground state |g>
- Emission of another photon at τ=0+ε is impossible.⇒ Photon antibunching: g(2)(0) = 0.
• g(2)(τ) recovers its steady-state value in a timescale given by the spontaneous emission time.
pump
• If three are two or more 2-level emitters, detection of a photon at τ=0 can not ensure that the system is in the ground state (g(2)(0) >0.5).
(nonclassical light)
photon
at ωp
Single InAs Quantum Dots
Energy (eV)1.25 1.30 1.35
Inte
nsity
(a.u
.)
P (W/cm2)
650
400
210
105
55
17 x5 in intensity
x2 in intensity
s-Shell
p-ShellInAs/GaAs Single QD
1X2X
T=4K
Two main wavelengths emitted from each shell
(For s-Shell)•2X recombination while there is already an e-h
pair in the s-shell (biexciton)•1X recombination while there is no other e-h pair
in the s-shell (exciton)
Due to carrier-carrier interactionTypically hν1X = hν2X + 2-3meV
Unique wavelengths for 1X and 2X transitions
Exciton linewidth measured by a scanning Fabry-Perot
Under non-resonant pulsed excitation
500 1000 15000
50
100
150
200
250
Inte
nsity
(a.u
.)
Bins
Free Spectral Range: 62 µeV
linewidth: 5.6 µeV
1.325 1.330 1.335
X2
X1
Inte
nsity
(a.u
.)
Energy (eV)
Photon antibunching from a Single Quantum Dot
(a)
(b)
(c)
-10 0 10 20 30 40 500.0
0.5
1.0
1.5
2.0
0
20
40
60
Delay Time τ (ns)
Cor
rela
tion
Func
tion
g(2) (τ
) Coincidence C
ounts n( τ)
0.0
0.5
1.0
1.5
2.0
0
20
40
60
-10 0 10 20 30 40 50
0.0
0.5
1.0
1.5
2.0
0
20
40
60
80
0.5
0.0
-10 50
Energy (eV)1.335 1.340 1.345
Inte
nsity
(a.u
.) 1X
2X
105 W/cm2
g2(0)=0.1τ = 750 ps
55 W/cm2
g2(0)=0.0τ = 1.4 ns
15 W/cm2
g2(0)=0.0τ = 3.6 ns
proof of atom-like behavior
PL spectrumP=55W/cm2
T=4K
A single quantum dot excited with a short-pulse laser can provide single-photon pulses on demand
BUT
How about embedding quantum dots in a microcavityto increase collection efficiency and fast emission?
Microdisk Cavities
No roughness on the sidewall up to 1nm !Q>18000 for 4.5µm diameter microdiskQ=9000 for 2µm diameter microdisk
Fundamental whispering gallery modes cover a ring with width ~ λ/2n on the microdisk
Q>17000
Q=13500
Michler et al., Appl. Phys. Lett. 77, 184 (2000)
A single quantum dot in a microdisk
Energy (eV)1.315 1.320 1.325 1.330
Inte
nsity
(a.u
.)
1X
2X
WGM
Q = 6500
Larger width of the peaks due to larger lifetime of the quantum dot
P=20W/cm2
T=4K
Pump power well above saturation level
Tuning the quantum dot into resonance with a Cavity Mode
EWGM-E1X (meV)-1.5 -1.0 -0.5 0.0 0.5 1.0
WG
M In
tens
ity (a
.u.)
0
200
400
600
800
Temperature (K)0 10 20 30 40 50 60
Ener
gy (m
eV)
1318
1319
1320
1321
1322
1323
WGM1X
T=44K
• Small peak appears at τ=0:• Purcell effect: reduction of emission time
Cavity coupling can provide better collection
Klimov – Förster coupling between QDs
Klimov – Förster coupling between QDsCrooker et al PRL 2002
D* + A D + A*
Non-Radiative Energy Transfer Mechanism
Coulomb-driven interaction
Dipole-dipole interaction (Förster 1946) Higher multipoles interaction (Förster – Dexter)
Exchange-driven interaction (Dexter)
trioctylphosphine oxide (TOPO) ~11A
A acceptor: larger dot
D donor: smaller dot
NQDs have bettercharacteristics than
biological light harvesting compounds, eg LH2
(a) PL decays from a dense film of monodisperse R=12.4A/9A CdSe/ZnS NQDs at the energies specified in the inset. Inset: cw PL spectra from film (solid) and original solution (dashed).
(b) Dynamic redshift of the peak emission.Inset: PL spectra at the specified times.
Crooker et al PRL 2002
(a) Schematic of NQD energy-gradient bilayer for light harvesting—13 A dots on 20.5 A dots.
(b) ‘‘Instantaneous’’ PL spectra at 500 ps intervals (from 0 to 5 ns), showing rapid collapse of emission from 13 A dots.
Our Goal
Study the excitation energy transfer in quantum-dotarrays using an appropriate model Hamiltonian
† † † † † †( ) † †, ,
N N NH T c c T d d Uc c d d U c c d de NNi j i j i i i i i i j jh
N
i j i i jV c
NNd d cs j ji ii j
= + + + +∑ ∑ ∑=
∑≠
Period of small oscillation ~ 1.3 ~1
2T Ve cProbability of each basis state as a function of time
0 10 20 30 400.0
0.2
0.4
0.6
0.8
1.0 |1100> |1001> |0110> |0011>
Vct/(2π)
0 30 600.5
1.0
1.5
2.0
10 12 14 16 18 20 22 240.5
0.6
0.7
0.8
0.9
1.0
tota
l pro
babi
lity
of e
xcito
n st
ate
Vct/(2π )
3.6
3.4
-13.7
-14.28
Period of large oscillation ~ 15 ~ 12 fV Vc
0.2
Probability of each basis as a functionof time without tunneling
Probability of each basis as a functionof time without Förster
Calculations for two dots
0 10 20 30 40 50
0.0
0.5
1.0
1.5
2.0
Vct/(2π)
|1100> |1001> |0110> |0011>
Te/Vc=0Vf/Vc=-0.04Vcnn/Vc=-0.1
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
Vc/(2π)
|1100> |1001> |0110> |0011>
Probability of each basis as a function of time
Te/Vc=-0.8Vf=0Vcnn/Vc=-0.1
Time evolution of the oscillator strength of an exciton initially localized at dot 12 in a 24 dot chain
0 10 20 30 40 50 60 70 800.0
0.5
1.0
1.5
2.0
2.5
3.0
Osc
illat
or s
tren
gth
Vct/2π
1 Psec
Vc=10 mev Vc=100 mev
4Te/Vc=0.30.8 1.6
Vc,nn/Vc=0.1Vf/Vc=0.04Th=0
G. W. Bryant, Physica B 314, 15 (2002).
30 20 10 0 10 20 300.0
0.2
0.4
0.6
0.8
1.0
Vct/2π
(12) (11) (14) (10) (15) (9) (16) (8) (17) (7) (18) (6) (19) (5) (20) (4) (21) (3) (22) (2) (23) (1) (24)
The “movie” of the 24 dots
exciton probability at each dot
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
Vc/(2π)
12
13
1415 16 17 18 19
2021 22
24
23
probability of exciton state in each dot
Förster rate for polymer
2 23.11 8 8Vh ctn n U U V Vc c
π π= = ≈→ +
From graph/calculation
22.2tVc
π≈
exciton probability at each dot
Vct/2π
efficient interdot transfer rate
Zrenner – coherent control of quantum dot photodiodes
Zrenner – coherent control of quantum dot photodiodesZrenner et al Nature 2002
Shadow mask
Top contact: 5nm Ti (semitransparent)
QDs InGaAs
Back contact
Laser focus
Possibles Possibles measurementmeasurementss
Photolumenescence (PL) spectrum (for τrad< τtunnel)
Photocurrent (PC) spectrum (for τrad> τtunnel)
Rabi Oscillation in a two level systemRabi Oscillation in a two level system
0ω
0 ( ) cos( )2 2z x
tH tω σ ω σΩ= +
( )( ) tt µ ε⋅Ω =
For For ω = ωω = ω0 0 and using RWAand using RWA
20 sin
2XP →Θ =
( )t dtτ
−∞
Θ = Ω∫0.0
0.5
1.0
πPulsed Area (Θ)
3π 4π2π0
Exciton Population
Time scales for the deviceTime scales for the device
rad 1 nsτ tunnel 3 ps τ → ∞ Tuned by VbTuned by Vb
PhotoluminescencePhotoluminescence τrad< τtunnel τrad< τtunnel τrad> τtunnel
NeHe laser
PhotocurrentPhotocurrent τrad> τtunnel
Ti:sapphire laser
pulse 1 psτ
pulse 82 MHzfF. Findeis et al. Appl. Phys. Lett. 78, 2958 (2001)
Artificial ion (charged exciton)Artificial ion (charged exciton)τrad< τtunnel
F. Findeis et al. Phys. Rev. B 63, 121309 (2001)
How does the Zrenner device work?How does the Zrenner device work?
τrad> τtunnel
A. Zrenner et al. Nature 418, 612 (2002)
How does the Zrenner device work?How does the Zrenner device work?
pulse 1 psτ
pulse 82 MHzf
pulseXXI e fρ=
2sin2XXρ Θ =
Mesoscopic optical spectrum analyserMesoscopic optical spectrum analyser
QCSEQCSE
Rabi Oscillation in the photocurrentRabi Oscillation in the photocurrent
pulseXXI e fρ=
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