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Optically Driven Spins in Semiconductor
Quantum Dots:
Toward III-V Based Quantum Computing
Duncan Steel - Lecture 1
DPG Physics School on "NanoSpintronics”
Bad Honnef 2010
Requirements to build a QC
(Divincenzo Criteria)
1. Well defined qubits
2. Universal set of quantum gates (highly
nonlinear)
3. Initializable
4. Qubit specific measurements
5. Long coherence time (in excess of 104
operations in the coherence time)
Quantum Dots:
Atomic Properties But Engineerable
•
•
•
•
•
•
•
•
•
•
Larger oscillator strength (x104)
High Q (narrow resonances)
Faster
Designable
Controllable
Using ultrafast light, we have fast (200
GHz) switching with no ‘wires’.
Integratable with direct solid state photon
sources (no need to up/down convert)
Large existing infrastructure for nanofabrication
High temperature operation – Compared
to a dilution refrigerator
CHALLENGE: spatial placement and
size heterogeneity
AFM Image of Al0.5Ga0.5As QD’s
formed on GaAs (311)b substrate.
Figure taken from R. Notzel
GaAs
InAs
Coupled QD’s
Coupled QD’s
72 nm x 72 nm
GaAs
Cross sectional STM
Boishin, Whitman et al.
KEY REQUIREMEMT: CONTROL
A logic device is highly nonlinear
Requires a two state system: 0 and 1
Semiconductor with periodic lattice
The Principle Physics for Optically Driven
Quantum Computing in semiconductors is
the Exciton
Semiconductor with periodic lattice
Semiconductor with periodic lattice
Can the Exciton be Controlled
in High Dimensional crystals?
hole
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
With coulomb coupling, the e-h
pair forms an exciton:
Extended state of the crystal
Is the Exciton a Well defined qubit in 1, 2, or 3
Dimensional Cystal?
hole
electron
Bloch Theorem: for a periodic potential of the form

 
V r d V rd

The solution to Schrödinger’s equation has the
form
r   eik r ur  where ur  d  ur 

The exciton in higher dimensional cyrstals is not a well defined qubit.

Can the Exciton be Controlled in High Dimensional cyrstals: i.e.,
can you build a universal set of quantum gates?
Recall the spin paradigm for quantum computing:








Rabi Oscillations:
Qubit Rotations




Coherent optical control
•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.
 or excited
 or excited
z
z

y
Rabi
x
 or ground
y
Precession
x
 or ground
Simple Coherent Control in
an Atom – Rabi Flops
h o 1 0 h  0
H

  *
2 0 1 2 R
r r
 E
R 
h
R 
cost
0 
z

o
Laser Pulse


y
x
 Ω allows control
Controlling t and/or
R
of the switching between up and
down, creating states
 like:   1 2     



Rabi Oscillations

ih
 H 0  E 0 sin t 
t
H 0 un  E n un
  u1 er u2
n  1,2;
2
1
C t 
 2
2

0




Pulse Area

6
7
ht
   E 0 tdt
20
Can the Exciton be Controlled
in High Dimensional cyrstals:
i.e., can you build a universal
set of quantum gates?
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
o

   0 
2

2


I
  2 1

I

sat 
Nearly Degenerate Differential Transmission
Quantum Dot Spectrum
Pump
Pump excitation reduces
absorption on excited
transition
Differential
Probe
Tuning
CW Nonlinear Spectroscopy
Experimental Set-up
Frequency stabilized
lasers
Eprobe
Eprobe
Detector
Esignal
Epump
Acousto-optic
Modulators ƒ≈100 Mhz
RF electronics
Lock-in amplifier
ENL  Im[  ] E probe I pump
Idetected  ENL  E*probe
3
Many-Body Effects in High
Dimensional Semiconductors
DT/T (a.u.)
Absorbance (a.u.)
Excitation Wavelength
1.508
hh
lh
1.516
Energy (meV)
0
1.5075
1.5125
1.5175
Energy (eV)
Wang et al. PRL 1993
To Suppress Extended State Wave Function, consider
a zero dimensional system: a Quantum Dot
Still a complex manybody system
Exciton
Electron based qubit
|1>
|0>
|0>
e
300 A
h
Trion
Spin based qubit
|i>
|1>
e
300 A
h
2
4
4
6
Figure of merit ~10 -10 Figure of merit ~10 -10
-9
Dephasing time ~10 sec Dephasing time >>10 -9 sec
(in SAD’s)
Quantum Dot Photoluminescence
as a Function of Laser Excitation Energy
.
Excitation energy (meV)
1630
1628
1626
1624
1622
1621
1622
1623
1624
1625
Detection energy (meV)
1626
Atomic-like spectrum –
Discrete states followed by continuum
1627
Nonlinear Signal Intensity
PL Intensity
Photoluminescence and Nonlinear Spectra Comparison
• The luminescence and
nonlinear spectra have
many lines in common
• The luminescence and
nonlinear techniques do not
measure the same optical
properties
• The nonlinear response is
resonant and highly isolated
Use a Quantum Dot to Build a 2-Qubit Computer?
Empty
Conduction
band

Filled 
valence
band


1

2
3

2


1
2
j  12,m j
 3

2
j  32,m j










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
How to Build a Two-Bit
Quantum Computer
Two spin-polarized excitons
Coulomb interaction
Resonant polarizationdependent optical coupling
Need two quantum bits
Need coupling
Need coherent control

|1>
+
|0>

|1>
|0>
|11>
B-Field
Coulomb
Interaction
|01>
|10>
|00>
The Two-Bit System
Optical Field
AlGaAs
GaAs
AlGaAs
|00>
The Two-Bit System
Optical Field
AlGaAs

GaAs
AlGaAs
|01>
The Two-Bit System
Optical Field
AlGaAs

GaAs
AlGaAs
|10>
Formation of the |11> state
Optical Field
AlGaAs
+ -
GaAs
AlGaAs
|11>
Biexciton
Do quantum dots experience pure dephasing?
Detection of coherence is made by measuring an observable

C
proportional to 2C1 where   C1 1  C2 2
The equation of motion for the coherence is
d 
C 2C1  C 2C1  other terms
dt

 arises from either loss of probability amplitude or pure
dephasing due to a randomly fluctuating phase between the
 

two probability amplitudes:
 i 0 t  R t 

C 2C1  c 2c1e 2

1
1
1
Relationship to NMR T1  ; T2  

 2   puredephasing
language

Calculated Coherent Wavelength-Resolved Differential Transmission
from a Two Level System
No pure dephasing Strong pure dephasing
 ph  0
• The coherent contribution
 ph  10 rel
Nonlinear Signal Intensity (a.u.)
leads to an asymmetric
lineshape in the absence of
extra dephasing processes.
• In the presence of strong
extra dephasing processes
the lineshape develops into a
sharp resonance on top of a
broader resonance (Prussian
helmet).
-2
-1
0
1
2
-2
-1
0
Probe detuning (  units)
1
2
Nonlinear Signal Intensity (a.u.)
Measured Coherent Differential Transmission
from a Single Quantum Dot:
No extra dephasing =>quantum coherence is robust
• “Coherent” and “incoherent”
contributions
•Homogeneously broadened
• T1~ 19ps and T2~ 32ps (i.e. T2 ~ 2
T1 , absence of significant extra
dephasing shows dots are robust
against decoherence)
The Two-Bit System
Optical Field
AlGaAs

GaAs
AlGaAs
The Two-Bit System
Optical Field
AlGaAs

GaAs
AlGaAs
First Step Towards Semiconductor Based
Quantum Computing:
Two Exciton-State Quantum Entanglement
- polarized exciton state
c-

1
2
1
2
3
2
3
2
+ polarized exciton state
+ c+ 
1
2
1
2
3
2
3
2
Quantum wave function shows
entanglement of two excitonstates.
e e  c e  3  e  1   c e  1  e  3 




2
2
2
2
+
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
Bloch Spin Vector Basis (Feynman, Vernon, Hellwarth)
Turn off the Coulomb
Correlation
Turn on the Coulomb Correlation
+
Pump


g
+
-
Probe

Pump


g
Probe

Ground
state
depletion
Pump: 
Entanglement
No Signal !!
           1
Total Signal
-2 -1 0
1
2
3 4 5 6
Probe ( )
7
8
9
-2 -1 0
1
2
3 4 5 6 7
Probe ( )
8
9
Experiment : Coulomb Correlation Quantum
Entanglement of two exciton-states
Entanglement of Two Exciton States: Non Factorizable
Wavefunction
  C0 g  C    C    Cb b
Non-interacting Case
Factorizable wavefunction:
b
Cb 
CC
C0
+
g
DE
With Coulomb Correlation
Cb  0
How small Cb is depends on linewidth
of state b and DE
-
b
+
g
The Two (Exciton) Qubit System
Optical Field
AlGaAs
GaAs
AlGaAs
|00>
The Two (Exciton) Qubit System
Optical Field
AlGaAs

GaAs
AlGaAs
|01>
The Two (Exciton) Qubit System
Optical Field
AlGaAs

GaAs
AlGaAs
|10>
The Two (Exciton) Qubit System
Optical Field
AlGaAs
+ -
GaAs
AlGaAs
|11>
Biexciton
NOTE: In semiconductor systems the
“Dipole Blockade” is a naturally
occuring phenomena, but much stronger,
usually, than the dipole term (Coulomb
Blockade).
Photoluminescence and Coherent Nonlinear Optical Spectra
• Superlinear excitation intensity dependence of photoluminescence from the biexciton-to-exciton transition
The Bound Biexciton (Positronium Molecule)
DEbiexciton
binding energy
m=-1/2
m=-3/2
m=1/2
m=3/2
•Higher order Coulomb correlations
lead to 4-particle correlations and
the bound biexciton
•An essential feature of optically
induced entanglement and a
quantum controlled not gate
Quantification of Entanglement: Entropy*
Cg
C+
C-
Cb
DE
-
0.9
0.3
0.3
<<0.005
b
+
g
For two-particle system, the entropy of entanglement goes
between 0 and 1. Zero entropy means product state. Non-zero
entropy indicating entanglement.
From our experiment, using the upper limit for Cb, E  0.08  0.02*
C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A 54, 3824 (1996)
*E~0.2 measured beyond chi-3 limit.
Now up to E~1
Creation of the Bell State
c0
unexcited state
Biexciton state
1
2
1
2
1
2
1
2
3
2
3
2
+ c+-  
+
3
2
3
2
Quantum wave function shows
entanglement of the ground
state and the biexciton.
The Two (Exciton) Qubit System
Rabi Oscillations
Optical Field
AlGaAs
GaAs
AlGaAs
|00>
The Two (Exciton) Qubit System
Rabi Oscillations
Optical Field
AlGaAs

GaAs
AlGaAs
|01>
Rabi Oscillations - qubit rotations

i
 H 0  E0 sin t
t
H0 un  En un

n  1, 2;
  u1 er u2
2
1
C t 
 2
2
0




Pulse Area



ht
E0 tdt


20
One Qubit Rotation in a Single Quantum Dot
The Exciton Rabi Oscillation
Excitonic energy levels
Rabi oscillations


Epump

•Rabi oscillations demonstrate an
arbitrary coherent superposition of
exciton and ground states,
c   c 
-pulse
-pulse
or c   c 
•A pulse area of  gives a complete
single bit rotation,
  
/2-pulse
or
  
“Damping” is due to excitation
induced increase in T1

population:
Time (ps)
final quantum
1
1





state (before
2
2

decoherence):
Time (ps)

  
Time (ps)

  
Physics for Optically Driven Spin
|X>
|0>
Neutral
Exciton
Negative
Exciton
T : trion
Semiconductor
Quantum Coherence
Engineering

Successful coherent
optical manipulation of
the optical Bloch vector
necessary to manipulate
the spin vector



Electronic
Spin Qubit

Optical Excitation of Spin Coherence:
Two-photon stimulated Raman
• Circularly polarized
pump pulse creates
coherent superposition of
spin up and down state.
• Raman coherence
oscillates at frequency of
the Zeeman splitting due
to electron in-plane gfactor and decays with
time.
Single Electron Spin Coherence:
Single Charged Exciton
Raman Quantum Beats
Charged Exciton System
X-
Neutral Exciton System
CNOS (a. u.)


Ensemble Charged Excitons
Single Neutral Exciton
X
Phys. Rev. Lett. - 2005
hs (eV)

0

500
1000
1500
Delay (ps)
2000
2500
T2* >10 nsec at B=0
Anomalous Variation of Beat Amplitude and Phase
Standard
Theory
(a)
• Plot of beat amplitude and phase as a function of the splitting.
(b)
Anomalous Variation of Beat Amplitude and Phase
Standard
Theory
(a)
• Plot of beat amplitude and phase as a function of the splitting.
Spontaneously Generated Coherence (SGC)
Trion


• Coupling to electromagnetic vacuum modes can create coherence* !!
• Modeled in density matrix equations by adding a relaxation term:
Normally forbidden in atomic systems or extremely weak.
Anomalous Variation of Beat Amplitude and Phase:
The result of spontaneously generated Raman coherence
Standard
Theory
(a)
• Plot of beat amplitude and phase as a function of the splitting.