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Coherent nonlinear optical
spectroscopy of quantum dots
Cameron Nelson
Steel Group
This work is funded by NSF-CPHOM
Old school atomic spectroscopy
Figure from Annalen der Physik und der Chemie (Poggendorff), Vol. 110 (1860), pp. 161-189 (dated Heidelberg, 1860)
Solar Spectrum
Semiconductor quantum dots: artificial
atoms
Hydrogen atom:
Characteristic length is given by Bohr radius
MBE
Quantum Dot:
Can be grown epitaxially
within a layer of semiconductor
CB
e
Characteristic length is given by
exciton Bohr radius: typically much
larger than atomic Bohr radius.
Exciton: Coulomb-bound
electron-hole pair
(quasiparticle)
h
VB
Quantized Energy Levels
Dipole moment is big: strong
coupling to light!
Nonlinear optical spectroscopy is necessary for a
complete understanding of the resonant interaction of a
laser field and quantum dot.
Quantum Harmonic Oscillator
Quantum Dot Exciton
ON
OFF
The interaction between a laser field and a two-level system is extremely nonlinear.
Compare to a harmonic oscillator, a linear system. Resonant excitation of a linear harmonic
oscillator moves it to a higher energy state. This continues indefinitely.
In a two-level system, you can only excite the system once. If you try to excite it again, you
will drive it back down to the ground state (Rabi oscillations).
Applications of optically controlled
quantum dots
The laser field can be used to coherently control the quantum state of the quantum dot,
resulting in a controllable qubit (on-off switch).
Quantum information processing
Quantum Information Science
Quantum optics in solid state
X. Xu, et al. Science 317, 929-932 (2007).
JQI Website
X. Li, et al. Science 301, 809-11 (2002).
Coherent control of the quantum state
N. H. Bonadeo, et al.
Science 282, 5393 (1998).
All-optical switching
D.A.B. Miller Nat. Photonics.
4, 3 (2010).
The world is demanding more bandwidth, so data distributors
will become more reliant on all-optical switching
Worldwide data usage is increasing every year
http://bradhedlund.com/
Currently, most routing of information requires
optical to electronic conversion to route signals.
Image from Cisco
This is slow, and wasteful (electronics
generate a lot of heat).
Quantum dot-based technology shows some
promise for all-optical switching
A small pulse of light can be used to switch
the quantum state of a quantum dot, thereby
allowing another light pulse to either pass or
reflect.
This offers an extremely low power switch
for controlling an information signal.
(switching power ~attojoules)
Image from NTT photonics
Chip-level interconnects also suffer from
heat-related issues. All-optical switches may be a
good future solution for replacing CMOS transistors.
D.A.B. Miller Nat. Photonics.
4, 3 (2010).
Many quantum applications requiring multiple
qubits will only be realized very far in the future.
1 qubit
Even coupling together two quantum dot
qubits for applications such as quantum
computing is extremely complicated or requires
very long integration times for experimental verification.
Most importantly, the most well-studied materal for
this application, InAs, requires cooling down to 4K
to work properly! .
The search is still on for new computer hardware…
Photoluminescence: useful for characterization of energy
level structure of quantum dots
Quantum Dot:
CB
Camera
Emission
Spectrometer
Pump
laser
(PL)
D
Photons
VB
Photoluminescence: Excite high energy charge carriers that decay nonradiatively
to the lowest energy emitting state. The resulting spectrum is usually similar to the
linear absorption spectrum of the lowest energy states in the system
What I study: InGaN/GaN quantum dots
for room temperature applications
Hanbury Brown-Twiss
Experiment
D
The main advantage of InGaN/GaN systems is a large exciton binding
energy compared to other III-V material. This has allowed for quantum dot
operation up to room temperature.
APD
Time-correlation
Electronics
Quantum Dot
Luminescence
Excitation Pulse
50:50 beamsplitter
D
APD
The InGaN quantum dot is one of the only stable solid state systems that gives
single photon emission behavior up to room temperature.
This has immediate applications in quantum cryptography, for example.
What do you get from nonlinear absorption that you
can’t from linear absorption?
You can measure (among other things):
 Dephasing and population decay rates of exciton
 Inhomogeneous broadening in the system
 Coupling between excited states
Laser Absorption
Quantum Dot:
Laser Absorption
CB
Energy
ωL
1.67σW
2γ
ω0
Lorentzian
VB
ωL – ω0
(Homogeneously Broadened)
γ is the dipole dephasing rate
Gaussian
ωL – ω0
Theoretical description of measurements
• We utilize the density matrix equations of motion for a two-level system to
describe the optical response of a quantum dot exciton.
• This theory is extremely well developed and has been used to successfully
describe the nonlinear optical response of GaAs/InAs quantum dots.
• Most of the following can be found in standard textbooks:
Optical Bloch equations for a homogeneously
broadened two-level system (Lorentzian absorption)
Density matrix: ρ = |ψ><ψ|
Density matrix equations of motion:
From Schrödinger’s equation,
𝑖ℏ𝝆 = [𝐇, ρ]
𝑯 = 𝑯𝟎 + μ ∙ 𝑬
< 𝑃 > = 𝑁(μ12 ρ21 + μ21 ρ12 )
δ
E0
|2> = |X>
Pure dephasing rate
Maxwell-Bloch equation: relation between
absorption and off-diagonal density matrix:
EL
|1> = |0>
Quantum dot systems gain pure dephasing from
environmental fluctuations.
From S. Rand, Lectures on Light
Image by S. Kelly, JQI Univ. of Maryland
Fluctuation-dissipation theorem
The exciton transition energy often undergoes
time-dependent fluctuations due to the solid
state background (i.e. Stark shifts from
surrounding charges).
Optical Bloch equations for a two-level
system with multiple laser fields
There is no analytical steady-state solution for this problem. To understand it,
perturbation theory has to be used. We are particularly interested in third order terms.
ks = kµ - k ν + kσ
δs = δµ - δ ν + δσ
For simplicity, we use two fields, Epump = E1 and Eprobe = E2. This gives a total of 8 terms.
Isolating two nonlinear terms along the
probe beam direction
Sample
C.W. Pump (ω1)
C.W. Probe (ω2)
Pump and probe are crossed at the center of the sample
A detector can be aimed along the probe beam propagation direction. In this way, we can
isolate the detection to nonlinear terms that propagate along the probe beam direction.
Recall that the third order nonlinear density matrix element goes like:
𝐼 (3)
ρ21 ~𝑒 −𝑖(𝑘𝑠 𝑅+ δ𝑠 𝑡)
We therefore select ks = kµ - kν + kσ = k2
This requires μ = ν = 1, σ = 2 and μ = 1, ν = 2, σ = 1.
δs = δ µ - δν + δσ = δ2
1
ρ𝐼21
(3)
= 2𝑖𝑒 −𝑖(𝑘2 𝑅+ δ2 𝑡) |χ1 |2 χ2 {
1
(𝛾 + 𝑖δ2 )2 (𝛾 − 𝑖δ1 )
2
1+
2Γ
γ2 + 𝑖 ω1 − ω2
+[
1
𝛾 2 + 𝑖δ1 2 𝛾 + 𝑖δ2
] 1+
2Γ
}
γ2
Imaginary component of total nonlinear response
No pure dephasing
Some dephasing
Γdeph = 0
1. Population pulsation term:
follows the probe frequency and
has a FWHM ~γ2
Γdeph = 10 γ2
Im(ρ21(3))
2. Saturation term:
does not follow the probe term. Has
a FWHM ~γ
γ2
δ2
δ2
The nonlinear optical spectrum
reveals both the pure dephasing rate
and the exciton decay rate. This
cannot be easily obtained using linear
techniques.
Note: the previous slides only applied to
the case of homogeneous broadening.
• In general, the Bloch equations have to be modified to account for
inhomogeneous broadening (Gaussian lines) properly. The final results are
a lot more complicated, usually.
Inhomogeneously broadened response
Homogeneously broadened response
Nonlinear response of inhomogeneously
broadened transitions: spectral hole burning
• The third order spectra look similar to the homogeneously broadened case,
except the saturation term tracks with the pump laser.
Linear Absorption,
Inhomogeneously Broadened Transition
σW = 20 γ2
Pump Wavelength
20
10
0
10
20
γ2
Nonlinear Absorption
Nonlinear Absorption
Γ= 0 γ2
20
10
Γ= 3γ2
10
20 20
10
10
20
γ2
• The nonlinear spectrum can be used to distinguish inhomogeneous and
homogeneous broadening.
Measuring nonlinear signals with
differential transmission
Optical Chopper
Sample
C.W. Pump (ω1)
C.W. Probe (ω2)
dT Tprobe with pump on  Tprobe with pump off

T
Tprobe with pump off
Lock-in
Amplifier
Function
Gen
The lowest order contribution comes from
the third order nonlinear signal ~χ(3)|Epu|2|Epr|2
Positive dT/T: Pump beam modifies sample absorption so that the probe beam absorbs
less → saturation
Negative dT/T: Pump beam modifies samples absorption so that the probe beam
absorbs more
Our study: Coherent nonlinear optical
spectrum of InGaN disks in GaN nanowires
Sample grown by Zetian Mi’s
group at McGill University
Current studies show evidence
of a compact quantum dot in
the center of the disk.
Disk-in-nanowires show single photon emission
behavior, which allows for unique quantum
applications.
Quantum cryptography requires quantum key generation. For
this, can use polarization of photons from a single quantum dot,
for example.
A novel quantum key distributor can be made using
single quantum dot
a single electrically injected quantum disk LED
1 μm
Polarized Emission
Background states can cause a significant spectral
diffusion (dephasing) for the InGaN exciton
Charges captured by traps may cause a
Stark shift of the exciton resonance
M. Holmes et al. Phys Rev. B 115447 (2015)
Stark Shifts from background states in
InGaN dots-in-nanowires can cause extremely
large optical linewidths.
-Very small nonlinear signal
The photoluminescence spectrum of a single
InGaN quantum dot shows fluctuations in the
transition energy over time.
J.H. Rice et al. Appl. Phys. Lett. 84, 4110 (2004).
InGaN materials are known to be
very messy (lots of background
disorder states)
Optical Setup
Preliminary sample data: bright PL
from quantum confined excitons
Photoluminescence
The PL data shows evidence
for radiative recombination
from quantum confined exciton
states: there is no continuous
blue shift as a function of
excitation intensity, a signature
of quantum confinement.
The PL originates from disksin-nanowires with different
diameters, and therefore
different emission
wavelengths. The broad
emission comes from
inhomogeneous broadening.
Ongoing work: Analyzing the full
coherent nonlinear spectrum
Exciton excited states
Emissive Exciton States
10 K
Nondegenerate nonlinear optical spectrum reveals evidence of saturation resonances with
~100 fs dephasing rates, this is at least 10,000x faster dephasing than similar QD systems.
Coherent population pulsations of excitons reveal that
excitons are stable up to room temperature
Total nonlinear spectrum
Pump-probe detuning (GHz)
Pump-probe detuning (GHz)
We observe strong coherent population pulsation resonances in
the sample that reveal the population decay time (T1) of the exciton
states all the way up to room temperature.
We find that the nonlinear signal is relatively robust against
temperature, but fast dephasing is still present, likely due to
very fast Stark shifting from background states.
400
200
0
200
400
400
200
0
200
400
400
200
0
200
400
400
200
0
200
400
Δpu,pr (γ2 units)
Conclusions
•
We observe coherent population pulsation resonances that allow us to
extract fundamental decay parameters of the exciton states and to verify a
low density of metastable trap states.
•
For the first time, we show that the quantum dot exciton is likely
homogeneously broadened, and the dipole dephasing rate is ~100 fs,
probably due to background disorder states.
•
Our results show that excitons can be stably excited up to room
temperature,
InGaN is typically grown along polar axes,
which leads to internal electric fields
InGaN has a wurtzite crystal structure
The internal electric fields changes the
band diagram and pushes the
electron and hole apart along the
growth direction