QDs and optical microcavities:
tailoring spontaneous emission
Jean-Michel GERARD
Equipe mixte CEA-CNRS-UJF « Nanophysique et Semiconducteurs »
jmgerard@cea.fr
Grateful thanks to B. Gayral (CEA) as well as to
B. Sermage, A. Lemaître, I. Robert, E. Moreau, I. Abram,
V. Thierry-Mieg, L. Ferlazzo from CNRS/LPN Marcoussis
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Spontaneous emission control : why? (as of ~1982)
Output power
laser diode
Spontaneous
Stimulated
emission
emission
Is
I
250nm
Current I
Only a tiny fraction of
spontaneous emission is useful!
β ~ 10-5
I
100000 times decrease of Is
expected for β ~ 1 !
LABORATOIRE DE
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Two strategies toward improved SE control
Density
of photon modes
kT
quantum
well
emission
Energy
ots
d
m
ntu
a
u
q
Energy
opt
ica
lm
icro
cav
it y
Energy
Optical microcavity = photon confinement on the wavelength scale
discrete photon modes
If the emitter is coupled to a single mode, β=1
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β=1 « Thresholdless »-laser
Ideal conversion of electrical signals into optical signals
. 1 e- -> 1 photon into the (single) mode
. Linear input/output characteristic curve
. No additional noise
. High cut-off frequency
<n>=1
Pout
spontaneous
emission
Proposal :
Kobayashi et al, 1982
Stimulated
emission
Yokoyama, Science 256, 62 (1992)
Pin
Björk et al, IEEE-JQE 27, 2386 (91)
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CQED with QDs : historical background
CQED (80->...)
optical microcavities (90->...)
SE can actually be tailored to a large
extent for atoms in electromagnetic
cavities
it is possible to confine optical
photons on the wavelength scale
in 2D, 1D, 0D
self-assembled QDs (~94->...)
single QD optical spectroscopy
reveals atom-like behavior
CQED with QDs
(96->...)
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Outline
1) Some basic microcavity effects
2) QDs as probes of photonic microstructures
3) CQED effects on QDs
4) Some application prospects :
single-mode photon sources
microlasers
For a recent review see:
Solid state CQED with self-assembled QDs, in
Topics in Applied Physics 30, 269 (2003), P. Michler ed. Springer
LABORATOIRE DE
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SE angular redistribution in planar microcavities
A. Kastler, Appl. Optics 1,17 (1962)
R
R
R
λ/2
R
emitter
in free-space
η~2%
Application to light emitting diodes :
directional emission + high efficiency η~25%
(Blondelle et al, Electron. Lett 31, 1286, 1995)
air
GaAs
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Tailoring the spontaneous emission rate
E
Free space modes
atom
Fermi golden rule: exponential decay
τ ∝ density of modes per unit volume
1
One can tailor spontaneous emission dynamics
. Dimensionality, refractive index...
. Coupling to a single mode (0D cavity, or optical selection rule)
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e.g. : strong coupling for QWs in planar cavities (Weisbuch et al, PRL 69, 3314, 1992)
Emitter in a planar microcavity
L
Ex: Brorson et al,
IEEE JQE 26, 1492 (1990)
ideal metallic mirrors
usual case for
QWs and QDs
Thickness of the cavity layer nL/λ
Both SE rate enhancement and inhibition expected
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InAs QD array in a planar microcavity
Very weak effect on QD spontaneous
emission dynamics !
λ/n
Solomon et al, PRL 86, 3903 (2001)
This is expected since the effective
cavity length is around 4 λ/n
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Strong and weak coupling regimes in 0D cavities
Perfect cavity
=> reversible spontaneous emission
Ω (V)
Cavity damping (Q)
SE into other modes
hΩ = |d . Ε(r)|
Emitter dipole
field per photon
Rabi oscillation
P(t)
Energy
1
|g, 1>
2hΩ
|e, 0>
time
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=>Entangled eigenstates
Strong coupling for a single cesium atom in an optical microcavity
Strong coupling first observed for atoms in the microwave spectral range
(superconductors are IDEAL mirrors!)
More recently observed in the optical spectral range
Hood et al, PRL 98
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Strong and weak coupling regimes in 0D cavities
Ω >> 1/τdecoherence =>
P(t)
damped
Rabi oscillation
τ ∼ 2 τdecoh
1
Ω (V)
Cavity damping (Q)
time
SE into other modes
Ω < 1/τdecoherence =>
exponential decay
« weak coupling regime »
P(t)
1
τ ∼ f(Q,V,…)
time
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Two important figures of merit
Ability to confine the e.m. field spatially:
Veff :
ε max
hω
=
2ε 0 n 2Veff
d
Actual cavity
d
Equivalent cavity
volume Veff
Ability to store the e.m. energy:
Q:
ω
Q=
∆ω mod e
↔ τ photon =
1
∆ω mod e
Around 1 eV, Q=1000 <-> τphoton =0.6 ps
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The « Purcell effect »
Decoherence mostly due to photon escape (low Q) => ∆ωmode >> ∆ωem
ρ(ω)
~Q
weak
coupling
regime
1
τ
∝ ρ (ω )
rr
i d .ε f
2
~1/Q
free space
Purcell (1946)
3
τ free
3 Q(λ n )
=
τ cav 4π 2 V
Fp
Fp=
magnitude for an ideal emitter
.quasi-monochromatic
.spatial matching
.spectral matching
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Spontaneous emission rate for a « non-ideal » emitter
τ free
∆ωmod e
= Fp
2
τ cav
4(ωmod e − ωem ) 2 + ∆ωmod e
2
Spectral detuning
r r 2
E (r ).d
r 2 r2
E (r ) d
r 2
E (r )
Emax
spatial detuning
If the emitter linewidth is not negligible,
dipole misorientation
« natural » emitter line
ρ (ωem ) → ∫ ρ (ω )L(ω )dω
ω
for a broad emitter Q → Qem and the Purcell effect is washed out
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How to confine light in all 3 dimensions with dielectrics ?
metallic mirrors are too lossy at optical frequencies!
distributed Bragg reflection
(DBR, photonic crystal)
total internal reflection
(e.g. optical fibers)
LABORATOIRE DE
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Some 0D solid-state microcavities
1 µm
2 µm
V=6
3 µm
(λ/n)3
V < (λ/n)3
V=5 (λ/n)3
3D confined modes
but not perfect « photonic dots » (leaky modes)!
Q?
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Probing cavity modes with QDs
diam=4 µm
GaAs/AlAs
micropillar
PL intensity (arb. units)
1 µm
theory:
x10
1.346
1.348
1.350
1.352
1.354
Energy (eV)
Systematic probing of cavity modes (no selection rule)
J.M. Gérard et al, Appl. Phys. Lett. 61, 459 (1996)
QD internal light source widely used to study microdisks or
PC cavities see e.g. D. Labilloy et al, APL 73, 1314 (1998)
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E. Moreau et al,
Probing cavity modes with one (few) QDs
1 µm
PL Intensity (lin. units.)
10K
1 µW
200 µW
1.250
APL 79, 2865 (2001)
1.255
1.260
cavity
mode
1.265
energy (eV)
Very strong homogeneous broadening of the single QD emission under
high excitation conditions
=> broadband light source!
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Probing cavity modes with QDs (2)
: theory
40
30
∆ E (meV)
Appl. Phys. Lett.
61, 459 (1996)
neff<n
L
20
10
Resonance condition:
0
0.0
0.5
1.0 1.5 2.0
Pillar Radius (µm)
2.5
3.0
planar: L= λ / n
pillar : L= λ/ neff
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Large diameter micropillars (d>4λ/n)
Same profile for guided mode in GaAs and AlAs layers
⇒No mode coupling at interfaces
⇒Fundamental pillar mode built using
L
neff (GaAs) = neff (AlAs)
n (GaAs)
n(AlAs)
=> the structure is still a « λ-cavity » for the novel
resonant wavelength λ = L . neff (GaAs)
=> Cavity Q unchanged
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Probing cavity Q’s
Very weak absorption of the QD array => we probe the « empty » cavity Q
(as long as Q<~10000)
Q
Qplanar
1 000
0
2
4
6
8
diameter (µm)
High Q’s can be achieved (Q=5000, τ~3ps)
Q drops at small sizes : intrinsic or extrinsic effect ?
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Numerical study of the intrinsic Q of GaAs/AlAs micropillars
P. Lalanne et al, APL june 2004
1600
1400
Q
1200
1000
0.4
0.43
800
600
400
200
0.2
0.4
0.6
0.8
1
1.2
1.4
diameter (µm)
Strong oscillations in the small diameter limit !
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Small diameter micropillars (d>4λ/n)
(Slightly) different mode profile in GaAs and
AlAs layers
L
⇒ inter-mode coupling at interfaces
⇒ fundamental pillar mode built using
and
LABORATOIRE DE
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reflectivity
neff<n
Energy
Blue shift of the DBR stopband as a result for the lateral confinement
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reflectivity
Energy of the fundamental
pillar mode
Energy
Resonant cavity mode = α
The
+ β
with |α|>>|β|
component propagates freely through the DBRs
=> interferences and oscillations on Q !
LABORATOIRE DE
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Numerical study of the intrinsic Q of GaAs/AlAs micropillars
P. Lalanne et al, APL june 2004
1600
1400
Q
1200
1000
0.4
0.43
800
600
400
200
0.2
0.4
0.6
0.8
1
1.2
1.4
diameter (µm)
Behavior well understood in a two-mode transfer matrix model
(
+
)
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Probing cavity Q’s
Q
Q2D
see e.g.
T.Rivera et al,
APL 74, 911 (1999)
1 000
0
2
4
6
8
diameter (µm)
High Q’s can be achieved (Q=5000, τ~3ps)
Q drops at small sizes : scattering by sidewall roughness
Fp = 32 achieved for d=1 µm !
LABORATOIRE DE
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Some 0D solid-state microcavities
1 µm
2 µm
V=5 (λ/n)3, Q=2100
Fp=32
V=6 (λ/n)3, Q=12000
Fp =150
3 µm
V=1.2(λ/n)3, Q=13000
Fp=810 (Caltech 04)
High Q, low volume, 3D confined modes
LABORATOIRE DE
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Some solid-state CQED experiments
performed with QDs
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CQED : strong coupling for single QDs?
Energy
|ground, 1 photon>
Ωrabi
|g, 1>
2hΩ
|e, 0>
|excited, 0 photon>
hΩ = |d . εmax|
InAs QD in
solid-state cavity :
huge vacuum field (105 V/cm)
large dipole (0.6 e.nm)
Sizeable vacuum Rabi splitting expected : 2hΩ ~ 50 µeV for V~ 10 (λ/n)3
VRS observable Ù 2hΩ >
∆Eemitter
∆Ecavity
Here, negligible emitter linewidth
⇒the relevant criteria for strong coupling is :
Andreani et al, PRB 60, 13276 (1999)
4 hΩ > ∆Ecavity
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Strong coupling for single QDs ?
Gérard et al, Physica E9, 131 (2001)
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Strong coupling for single QDs ?
Gérard et al, Physica E9, 131 (2001)
Andreani et al, PRB 60, 13276 (1999)
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Fresh news !
The strong coupling regime has been observed by two
research groups :
. micropillars + big InGaAs QDs displaying a « giant »
oscillator strength (Forchel et al, Würzburg)
. InAs QD in high Q photonic crystal microcavity
(Scherer et al, Caltech)
(still unpublished)
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Purcell effect on InAs QDs
1946: Purcell’s paper
1983: Observation for atoms in cavities (x 500! Haroche et al, ENS Paris)
1997: First observation of the Purcell effect in a solid-state microcavity !
QD array in micropillar
QD array in microdisk
QD array in PC cavity
x5
x 18
x 12
x 12
single QD in microdisk
x6
single QD in micropillar > x 3
x4
x5
J.M. Gérard et al, PRL 81, 1110 (1998)
B. Gayral et al, Physica E7, 641(2000)
B. Gayral et al, APL 78, 2828 (2001)
T. Happ et al, PRB 66, R041303 (2002)
A. Kiraz et al, APL 78, 3932 (2001)
E. Moreau et al, APL 79, 2865 (2001)
G. Solomon et al, PRL 86, 3903 (2001)
J. Vuckovic et al, APL 82, 3596 (2003)
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Purcell effect for a collection of QDs (d=1 µm, Fp=32)
PL intensity (arb. units)
PL intensity (lin. units)
QBs out of resonance
QBs in resonance
τ = 1.0 5 ns
τ ~ 0.2 5 ns
1.346
0
1000
2000
time (ps)
PL decay faster only for on-resonance QDs : intrinsic effect
SE rate enhancement (x5) much smaller than Fp
LABORATOIRE DE
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PRL 81, 1110 (1998)
1.348
How to understand the magnitude of the Purcell effect
M o d e d e n s ity
d
E n e rg y
Random distribution of QDs
=> spatial + spectral averaging
LABORATOIRE DE
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Fp : the relevant figure of merit
6
experiment
model for 2D dipole
+random distribution
5
τ0 / τd
on
4
3
Q= 1600 Q= 2250
d=0.9 µ m d=1 µ m
2
Q= 2000
d=1. 3 µ m
1
0
Enhancement factor
depends as expected
on Fp , not d or Q
Q= 5200
d=4.2 µ m
0
5
Good agreement
with theory
Q= 960
d=1 µ m
10
15
20
25
30
35
Purcell factor FP
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PRL 81, 1110 (1998)
Dependence of SE rate enhancement on the spectral detuning
from
Solomon et al, PRL 86, 3903 (2001)
see also
Graham et al, APL 74, 3408 (1999)
Experiment in good agreement with
the expected spectral dependence
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SE inhibition in metal-coated micropillars
No inhibition in standrad micropillars, due to leaky modes => gold-coated pillars
cavity mode energy
M. Bayer et al,
PRL 86, 3168 (2001)
Efficient reduction of the density
of non-resonant modes by the
metallic coating
=> strong SE inhibition (/10)
for out-of-resonance QDs
LABORATOIRE DE
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Intensity (lin. scale)
Purcell effect for QDs in microdisks
Off-resonance
2 µm
B. Gayral et al
APL 78, 2828 (2001)
Q=7000
Q=10000
200
400
600
800
1000
1200
1400
1600
1800
Time (ps)
Up to x 12 SE rate enhancement
LABORATOIRE DE
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Purcell effect in microdisks
B. Gayral et al, APL 78, 2828 (2001)
Intensity PL (lin. scale)
experiment
models:
Fp+spectral/spatial averaging
+ "jitter" due to relaxation time
(30 ps)
0
200
400
600
800
Time (ps)
LABORATOIRE DE
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Purcell effect in a c.w. experiment
8K
1E 7
in-resonance QDs
PL intensity (log)
PL
10 0000 0
1 0000 0
B. Gayral et al
Physica E7, 641 (2001)
1000 0
out-of-resonance QDs
100 0
10 0
1E -4
1E -3
0.01
0.1
1
excitation power (log)
Saturation due to filling of fundamental QD states
Purcell effect (x18 here!)=> saturation occurs for higher pumping power
Same behavior observed for QDs in PC cavities (Happ et al, PRB 2002)
LABORATOIRE DE
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Few QDs in a micropillar
1 µm
PL Intensity (lin. units.)
10K
1 µW
200 µW
cavity
mode
1.250
1.255
1.260
1.265
energy (eV)
LABORATOIRE DE
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ET DE NANOSTRUCTURES
Purcell effect on isolated QDs in a micropillar
(Q2D=1000, Q=500, Fp=7)
10 µW Cavity Mode
100 nW
Relative Intensity
0,1
QD1
τ ~ 400 ps
0,01
QD2
τ ~ 1.1 ns
τ ~ 1.3 ns
970
QD3
980
λ (nm)
990
1E-3
0
2
4
6
Delay (ns)
8
SE enhancement > x 3
10
Moreau et al,
APL 79, 2865 (2001)
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Beware !
Unlike experiments on large collections of QDs, quantitative modelling not possible
since the QD location (and τfree) are not precisely known !
(see e.g. Gayral et al, PRL 90, 229701, 2003)
Coupling the QD in/out of resonance is a good way to highlight the Purcell effect and
to estimate its magnitude
LABORATOIRE DE
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Purcell effect for a single QD in a microdisk
Kiraz et al, APL 78, 3932 (2001)
UCSB
Intensity (a.u.)
1
T=4K
1.75nsec
0.1
T=48K
1.75nsec
T=31K
850psec (resonance)
0
2
4
6
8
10
12
14
16
Time (ns)
F>2
LABORATOIRE DE
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18
20
Applications of the Purcell effect
LABORATOIRE DE
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ET DE NANOSTRUCTURES
(Nearly) single-mode spontaneous emission
ρ(ω)
Purcell effect
=
selective enhancement of
SE into the resonant mode
F/τfree
QD
ω
γ/τfree
(γ∼1)
F
β=
≈1
F +γ
J.M. Gérard et al
PRL 81, 1110 (1998)
Very interesting for single photon sources
and microlasers !
LABORATOIRE DE
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ET DE NANOSTRUCTURES
What is a single photon source ?
Source able to emit single photons pulses on demand
clic
N.B. : Non-classical state of light
⇒Impossible to generate it using a thermal source or a laser
Applications : quantum cryptography, metrology...
LABORATOIRE DE
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ET DE NANOSTRUCTURES
A single-mode single-photon source
Isolated InAs QD
GaAs/AlAs micropillar
20 nm
=> Single-photon emission
⇒ Efficient collection
+
single-mode behavior
proposal: J.M. Gérard et B. Gayral,
J Lightwave Technol. 17, 2089 (1999)
exp :
E. Moreau et al, APL 79, 2865 (2001)
LABORATOIRE DE
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Some properties of this single photon source
<2% !
Single photon collection efficiency
44%
38%
air
GaAs
Moreau et al, Physica E 13, 418 (2002)
Pelton et al, PRL 89, 233602 (2002)
All photons prepared in the same mode
(spatial+ polarization)
~ No two-photon pulses
Key device for the practical implementation
of quantum cryptography
LABORATOIRE DE
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ET DE NANOSTRUCTURES
SPS efficiency for GaAs/AlAs micropillars
scattering by
sidewall roughness
Q2D
ε<β
Q
Q spoiling
1 000
1
1
1
=
+
Q Qint Qscat
0
2
4
6
diameter (µm)
scattering by sidewall roughness !
8
ε =β
Q
Qint
ε<β
a careful optimisation of the intrinsic
losses is necessary!
LABORATOIRE DE
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ET DE NANOSTRUCTURES
J.M. Gérard et al, quant-ph/0207115
W. Barnes et al, Eur. Phys. J. D.18, 197 (2002)
Polarization control in single-mode micropillars
Elliptical cross section
1.4 µm
x-polarized
PL Intensity (lin.)
0.7 µm
y-polarized
Non degenerate fundamental mode
Gayral et al, APL 72, 1421, 1998
~ 90% linear polarisation degree
of single QD emission
Moreau et al, APL 79, 2865, 2001
1.265
1.270
1.275
Energy (eV)
1.280
LABORATOIRE DE
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ET DE NANOSTRUCTURES
Indistinguishable single photons and the issue of decoherence
t
Decoherence events
t
Indistinguishable single photons Ù T2 = 2 T1
LABORATOIRE DE
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ET DE NANOSTRUCTURES
InAs QD SPS : photon coalescence experiment
Santori et al, Nature
419, 594 (2002)
Probability of detection on D1+D2
D1
D2
InAs QD in bulk GaAs : T2 (~ 600ps) << 2 T1 (~2 ns)
Here, emission of nearly-Fourier-transform limited single photons by
the InAs QD, thanks to the Purcell effect
LABORATOIRE DE
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ET DE NANOSTRUCTURES
Toward « nanolasers »
β ~ 10-5
Ith = 5 mA
β ~0.04
β ~ 0.1
Ith = 36 µA
Huffaker et al,
APL 71, 1449 (1997)
Ith = 40 µA
Fujita et al,
Electron. Lett. 36, 790 (2000)
Sub µA threshold current expected in the β~1 limit !
LABORATOIRE DE
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ET DE NANOSTRUCTURES
Purcell effect at 300K ???
QDs
The smallest PC cavities exhibit
V’s as low as 0.2 (λ/n)3
Best QD arrays ~ 15 meV linewidth
at 300K
Assuming Qcav > Q QDs , the SE rate
enhancement factor is :
3
(
)
τ free
λ
Q
n
3 QDs
=
τ cav 4π 2
V
and β > 0.95 !
Probably the best strategy to date for approaching the regime of
« thresholdless » lasing at 300K
LABORATOIRE DE
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~ 30
Conclusion
Using QDs has been the key for observing CQED
effects
Solid-state CQED is a very lively domain
Still many basic effects to be observed !
Important potential applications in optoelectronics and
quantum information science
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES
What is beyond?
moderate β, huge Q
200 µW at 300K
for InAs QDs
(ENS/LKB2003)
moderate Q, high β
using the Purcell effect
β∼1 in 3D PCs
ex: QDs in 2D PCs
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES
Single quantum dot laser?
Low temperature required !!!
β
>γ
τ
ut t
ti ho effec
w ll
rce
Pu
Ith~10 pA predicted
Pelton et Yamamoto,
PRA 59, 2418 (1999)
⇒ ∆ ω em > ∆ ω cav
Pur with
cel
l ef
fe
ct
∆ω cav < ∆ω em < hΩ
Strong coupling
Τ
∆ ω cav <
h Ω < ∆ ω em
Weak coupling and lasing
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES
Semiconductor microdisks
2 µm
GaAs disk
GaAlAs pedestal
3D photon confinement using waveguiding + total internal reflexion
(whispering gallery modes)
V~ 5 (λ/n)3 !
LABORATOIRE DE
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PL Intensity (lin. units)
Probing WGMs with QDs
unprocessed QB array
(weak excitation)
B. Gayral et al
APL (1999)
2.5 mW
100 µW
50 µW
1.2
1.3
Energy (eV)
Observation of high Q WGMs
Great sensitivity of Q on sidewall roughness
Q>10000 for wet-etched microdisks
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES
QD internal light source also commonly used to probe PC cavities
see e.g. D. Labilloy et al, APL 73, 1314 (1998)
An ideal dielectric microcavity
E. Yablonovitch, PRL, 2238 (1991)
photonic crystal + defect
β=1
All photons are collected/prepared in a single mode !
Rem:
spontaneous emission control
= main initial motivation for developping photonic crystals
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES
3D PCs and SPSs
To be done !!
Photonic defect
+
SC or diamond nanoX, molecule
LABORATOIRE DE
DE PHOTONIQUE
ET DE NANOSTRUCTURES