Optically Driven Spins in Semiconductor Quantum Dots: Toward IIIV Based Quantum Computing. Duncan Steel  Lecture 1. DPG Physics School on "Nano Spintronics ” Bad Honnef 2010. Requirements to build a QC (Divincenzo Criteria). Well defined qubits
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Duncan Steel  Lecture 1
DPG Physics School on "NanoSpintronics”
Bad Honnef 2010
GaAs
Coupled QD’s
[001]
Coupled QD’s
GaAs
72 nm x 72 nm
Cross sectional STM
Boishin, Whitman et al.
Quantum Dots:Atomic Properties ButEngineerableAFM Image of Al0.5Ga0.5As QD’s formed on GaAs (311)b substrate. Figure taken from R. Notzel
Semiconductor with periodic lattice
Semiconductor with periodic lattice
Semiconductor with periodic lattice
hole
With coulomb coupling, the eh pair forms an exciton:
Extended state of the crystal
electron
Excitons in high dimenisonal crystals do not have a simple atomic like nonlinearity: Quantum gates are hard to imagine
Rabi oscillations in quantum wells
Cundiff et al. PRL 1994
Schulzgen et al., PRL 1999
Bloch Theorem: for a periodic potential of the form
The solution to Schrödinger’s equation has the form
hole
electron
The exciton in higher dimensional cyrstals is not a well defined qubit.
Recall the spin paradigm for quantum computing:
Rabi Oscillations:
Qubit Rotations
z
z
Coherent optical control of an electronic state means controlling the state of the spin or pseudo spin Bloch vector on the Bloch sphere.
It is a highly nonlinear optical process and is achieved with a combination of Rabi oscillations and precession.
y
y
Precession
Rabi
x
x
Simple Coherent Control in an Atom – Rabi Flops
z
Laser Pulse
y
x
Controlling t and/or ΩR allows control of the switching between up and down, creating states like:
Excitons in high dimenisonal crystals do not have a simple atomic like nonlinearity: Quantum gates are hard to imagine
What does an atomic like nonlinearity look like in the laboratory: Saturation (Spectral Hole Burning) Spectroscopy
Quantum computing is a highly nonlinear system (intrinsic feature of a two level system in contrast to a harmonic oscillator. Nonlinear spectroscopy quantifies the behavior.
Absorption
Saturated absorption
Differential absorption
Nearly Degenerate Differential Transmission laboratory: Saturation (Spectral Hole Burning) Spectroscopy
Quantum Dot Spectrum
Pump
Pump excitation reduces absorption on excited transition
Differential
Probe
Tuning
RF electronics laboratory: Saturation (Spectral Hole Burning) Spectroscopy
Lockin amplifier
CW Nonlinear Spectroscopy Experimental SetupFrequency stabilized
lasers
E
probe
Detector
E
probe
E
signal
E
pump
Acoustooptic
Modulators ƒ≈100 Mhz
Excitation Wavelength
Wang et al. PRL 1993
Still a complex manybody system
Exciton
Electron based qubit
Trion
Spin based qubit
1>
i>
1>
0>
0>
e
e
300 A
300 A
h
h
4
6
2
4
Figure of merit ~10 10
Figure of merit ~10 10
9
9
Dephasing time >>10 sec
Dephasing time ~10 sec
(in SAD’s)
1630 dimensional system: a Quantum Dot
1628
1626
1624
1622
1621
1622
1623
1624
1625
1626
1627
Quantum Dot Photoluminescence
as a Function of Laser Excitation Energy
.
Excitation energy (meV)
Detection energy (meV)
Atomiclike spectrum –
Discrete states followed by continuum
Photoluminescence and Nonlinear Spectra Comparison dimensional system: a Quantum Dot
PL Intensity
Nonlinear Signal Intensity
Empty Conduction band
Filled valence band
Ground and first excited states for neutral quantum dot
First break with atom picture: Lack of spherical symmetry means angular momentum is not a good quantum number
1> dimensional system: a Quantum Dot
1>
BField
+
Coulomb
Interaction
0>
0>
How to Build a TwoBit Quantum ComputerTwo spinpolarized excitons
Need two quantum bits
Need coupling
Coulomb interaction
Need coherent control
Resonant polarization dependent optical coupling
11>
s
s+
01>
10>
00>
Optical Field
AlGaAs
GaAs
s
s+
AlGaAs
11>
Biexciton
Detection of coherence is made by measuring an observable proportional to where
The equation of motion for the coherence is
arises from either loss of probability amplitude or pure dephasing due to a randomly fluctuating phase between the two probability amplitudes:
Relationship to NMR language
dimensional system: a Quantum Dot
0
10
ph
ph
rel

2

1
0
1
2

2

1
0
1
2
Probe detuning
(
u
n
i
t
s
)
Calculated Coherent WavelengthResolved Differential Transmission from a Two Level SystemNo pure dephasing
Strong pure dephasing
Nonlinear Signal Intensity (a.u.)
Measured Coherent Differential Transmission dimensional system: a Quantum Dot
from a Single Quantum Dot:
No extra dephasing =>quantum coherence is robust
Nonlinear Signal Intensity (a.u.)
s dimensional system: a Quantum Dot polarized exciton state
s+ polarized exciton state
s
c+
c
+
s+
+
1
3
1
3




1
3
3
1
+
+
+
+
2
2
2
2
2
2
2
2
First Step Towards Semiconductor Based Quantum Computing:Two ExcitonState Quantum EntanglementQuantum wave function shows entanglement of two excitonstates.
Quantum entanglement in the wave function is a key feature in quantum computers. This is the property which allows them to surpass classical computers in computational ability.
The Exciton Based Two Qubit System dimensional system: a Quantum Dot
Bloch Spin Vector Basis (Feynman, Vernon, Hellwarth)
+ dimensional system: a Quantum Dot

Probe
s+
Pump
s
g
Turn off the Coulomb
Correlation
Turn on the Coulomb Correlation
+

Probe
s+
Pump
s
g
s
s+
Ground
state
depletion
Pump: s
Entanglement
No Signal !!
Total Signal
3
2
1
0
1
2
4
5
6
7
8
9
3
2
1
0
1
2
4
5
6
7
8
9
Probe ( g )
Probe ( g )
Experiment : Coulomb Correlation Quantum Entanglement of two excitonstates
C excitonstates
g
C
C
C
b
y
=
+
s
+
s
+
0
+
+


b
b
b
DE
s+
s
s+
s
g
g
Entanglement of Two Exciton States: Non Factorizable Wavefunction
Noninteracting Case
Factorizable wavefunction:
With Coulomb Correlation
How small Cb is depends on linewidth of state b and DE
Optical Field
AlGaAs
GaAs
s
s+
AlGaAs
NOTE: In semiconductor systems the “Dipole Blockade” is a naturally occuring phenomena, but much stronger, usually, than the dipole term (Coulomb Blockade).
11>
Biexciton
Photoluminescence and Coherent Nonlinear Optical Spectra excitonstates
DE= excitonstatesbiexciton
binding energy
m=1/2
m=1/2
m=3/2
m=3/2
The Bound Biexciton (Positronium Molecule)
C excitonstates
C
C
C
+

b
g
b
DE
s+
<<0.005
s
0.3
0.9
0.3
g
Quantification of Entanglement: Entropy*
For twoparticle system, the entropy of entanglement goes between 0 and 1. Zero entropy means product state. Nonzero entropy indicating entanglement.
From our experiment, using the upper limit for Cb,
*
C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A54, 3824 (1996)
*E~0.2 measured beyond chi3 limit.
Now up to E~1
s excitonstates
1
1
3
3




3
1
3
1
+
+
+
+
2
2
2
2
2
2
2
2
Creation of the Bell Stateunexcited state
Biexciton state
Quantum wave function shows entanglement of the ground state and the biexciton.
c+
c0
+
s+
+
Optical Field
AlGaAs
GaAs
AlGaAs
00>
Optical Field
AlGaAs
GaAs
s+
AlGaAs
01>
E excitonstatespump
or
One Qubit Rotation in a Single Quantum Dot
The Exciton Rabi Oscillation
Excitonic energy levels
Rabi oscillations
ppulse
p/2pulse
2ppulse
or
population:
Time (ps)
Time (ps)
Time (ps)
final quantum
state (before
decoherence):
“Damping” is due to excitation induced increase in T1
Neutral Exciton
Negative Exciton
X>
0>
Electronic Spin Qubit
Semiconductor Quantum Coherence Engineering
Successful coherent optical manipulation of the optical Bloch vector necessary to manipulate the spin vector
Charged Exciton System
X 
CNOS (a. u.)
G
G
Neutral Exciton System
X
G
G
hgs (meV)
T2* >10 nsec at B=0
Phys. Rev. Lett.  2005
Standard
Theory
(a)
(b)
Standard
Theory
(a)
Trion
G
G
Standard
Theory
(a)