optically driven spins in semiconductor quantum dots: toward iii-v based quantum computing
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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 "Nano- Spintronics ” Bad Honnef 2010. Requirements to build a QC (Divincenzo Criteria). Well defined qubits - PowerPoint PPT PresentationTRANSCRIPT
Optically Driven Spins in Semiconductor Quantum Dots:
Toward III-V Based Quantum Computing
DPG Physics School on "Nano-Spintronics”
Bad Honnef 2010
Duncan Steel - Lecture 1
Requirements to build a QC(Divincenzo Criteria)
1. Well defined qubits 2. Universal set of quantum gates (highly
nonlinear) 3. Initializable4. Qubit specific measurements5. 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 nano-
fabrication• High temperature operation – Compared
to a dilution refrigerator• CHALLENGE: spatial placement and
size heterogeneity
InAs
GaAs
GaAs
Cross sectional STMBoishin, Whitman et al.
Coupled QD’s
Coupled QD’s [001]
72 nm x 72 nmAFM Image of Al0.5Ga0.5As QD’s formed on GaAs (311)b substrate. Figure taken from R. Notzel
KEY REQUIREMEMT: CONTROLA 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
Can the Exciton be Controlled in High Dimensional crystals?
Excitons in high dimenisonal crystals do not have a simple atomic like nonlinearity: Quantum gates are hard to imagine
Rabi oscillations in quantum wellsCundiff et al. PRL 1994Schulzgen et al., PRL 1999
Semiconductor with periodic lattice
hole
electron
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?
The exciton in higher dimensional cyrstals is not a well defined qubit.
Bloch Theorem: for a periodic potential of the form The solution to Schrödinger’s equation has the form
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Vr r +
r d ( ) = V
r r +
r d ( )
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ψr r ( ) = e i
r k ⋅
r r u
r r ( ) where u
r r +
r d ( ) = u
r r ( )
hole
electron
Can the Exciton be Controlled in High Dimensional cyrstals: i.e., can you build a universal set of quantum gates?
Rabi Oscillations:Qubit Rotations
Recall the spin paradigm for quantum computing:
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↑
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↓ €
↑
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↓ €
↑
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↓
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.
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↑ or excited
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↓ or ground
x
y
z
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↑ or excited
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↓ or ground
x
y
z
Rabi Precession
Simple Coherent Control in an Atom – Rabi Flops
Laser Pulse
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↑
€
↓
x
y
z
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H =hωo
2−1 00 1 ⎡ ⎣ ⎢
⎤ ⎦ ⎥+
h2
0 ΩRΩR
* 0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥cosωt
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ωo
€
↑
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↓
Controlling t and/or ΩR allows control of the switching between up and down, creating states like:
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ψ = 1 2( ) ↑ + ↓[ ]
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ΩR =r μ ⋅
r E
h
Rabi Oscillations
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ih∂ ψ∂t
= H0 − μE0 sinωt[ ]ψ
H0 un = En un n =1,2; μ = u1 er u2
Pulse Area
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θ =h2
μE0 ′ t ( )d ′ t 0
t∫0
C2 t( )2
1
2
6 7
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
Absorption Saturated absorption
Differential absorption
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.
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α =αo
ω − ω0( )2 + γ 2 1+ I
I sat
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
Pump excitation reduces absorption on excited transition
Pump
ProbeTuning
Differential
Quantum Dot Spectrum
Nearly Degenerate Differential Transmission
CW Nonlinear Spectroscopy Experimental Set-up
ENL ∝ Im[ χ ] Erobe IumIdetected ∝ ENL× Erobe
*
Epump
Eprobe
Eprobe
Esignal
Lock-in amplifier
Acousto-opticModulators ƒ≈100 Mhz
RF electronics
Detector
Frequency stabilizedlasers
Many-Body Effects in High Dimensional Semiconductors
1.508 1.516
Absorbance (a.u.)
Energy (meV)
hh
lh
0
DT/T (a.u.)
1.5075 1.5125 1.5175
Energy (eV)
Wang et al. PRL 1993
Excitation Wavelength
To Suppress Extended State Wave Function, consider a zero dimensional system: a Quantum Dot
ExcitonElectron based qubit
TrionSpin based qubit
|0>
|1>
|0>|1>
|i>
Figure of merit ~10 -10 4 6
e
h300 A
e
h300 A
Figure of merit ~10 -10 2 4
Dephasing time ~10 sec (in SAD’s)
-9 Dephasing time >>10 sec-9
Still a complex manybody system
.
Detection energy (meV)
Exci
tatio
n en
ergy
(meV
)
1622
1624
1626
1628
1630
1621 1622 1623 1624 1625 1626 1627
Quantum Dot Photoluminescence as a Function of Laser Excitation Energy
Atomic-like spectrum – Discrete states followed by continuum
• 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
Photoluminescence and Nonlinear Spectra ComparisonPL
Int
ensi
tyN
onlin
ear S
igna
l In
tens
ity
Use a Quantum Dot to Build a 2-Qubit Computer?
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+12
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+32
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−12
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−32
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j = 32,m j
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j = 12,m j
Filled valence band
Empty Conduction band
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↓
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↑
€
⇑↓↑
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⇓↑↓
First break with atom picture: Lack of spherical symmetry means angular momentum is not a good quantum number
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σ +
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σ −
Ground and first excited states for neutral quantum dot
How to Build a Two-Bit Quantum Computer
Need two quantum bitsNeed couplingNeed coherent control
Two spin-polarized excitonsCoulomb interactionResonant polarization- dependent optical coupling
|0>
|1>+
|0>
|1>
|00>
|10>Coulomb Interaction
B-Field|01>
|11>
σ+ σ-
The Two-Bit System
GaAs
AlGaAs
AlGaAs
Optical Field
|00>
The Two-Bit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ+
|01>
The Two-Bit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ-
|10>
Formation of the |11> state
GaAs
AlGaAs
AlGaAs
Optical Field
Biexciton
σ+ σ-
|11>
Do quantum dots experience pure dephasing?
Detection of coherence is made by measuring an observable proportional to where
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C2∗C1
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ψ =C11 +C2 2The equation of motion for the coherence is
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ddt
C2∗C1 = −γC2
∗C1 + other terms
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γ 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
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C2∗C1 = c2
∗c1e−
Γ2
+iω 0 ⎛ ⎝ ⎜
⎞ ⎠ ⎟t +θ R t( )
T1 = 1
Γ ; T2 =1γ = 1
2Γ +γ puredephasing
Calculated Coherent Wavelength-Resolved Differential Transmission from a Two Level System
• The coherent contribution 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 1 2Probe detuning ( units)
ph =10 rel ph = 0
No pure dephasing Strong pure dephasing
Non
linea
r Sig
nal
Inte
nsity
(a.u
.)
• “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)
Measured Coherent Differential Transmissionfrom a Single Quantum Dot:
No extra dephasing =>quantum coherence is robust
Non
linea
r Sig
nal
Inte
nsity
(a.u
.)
The Two-Bit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ+
The Two-Bit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ-
First Step Towards Semiconductor Based Quantum Computing:
Two Exciton-State Quantum Entanglement
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.
( ) ( ) ( ) ( )23
21
21
23 +−++−−
+−+−−+ΨΨ+ΨΨ=Ψ
eeeeee cc
21-2
1+
23+
23-
21-2
1+
23+
23-
+σ-σ+c- c+
σ- polarized exciton state σ+ polarized exciton state
Quantum wave function shows entanglement of two exciton-states.
+
The Exciton Based Two Qubit SystemBloch Spin Vector Basis (Feynman, Vernon, Hellwarth)
Turn off the CoulombCorrelation
Turn on the Coulomb Correlation
No Signal !!
4 5 6 7 8 9
Ground state
depletion
Entanglement
Total Signal
-2 -1 0 1 2 3
σ- σ+
Pump: σ-
1==== −+−+
Pumpσ-
+-Probeσ+g Pump
σ-
- +Probeσ+g
Probe ( )4 5 6 7 8 9-2 -1 0 1 2 3Probe ( )
Experiment : Coulomb Correlation Quantum Entanglement of two exciton-states
Entanglement of Two Exciton States: Non Factorizable Wavefunction
ψ =C0 g +C+ σ + +C - σ - +Cb b
Non-interacting CaseFactorizable wavefunction:
With Coulomb CorrelationHow small Cb is depends on linewidth of state b and DE
b
σ+σ-
g
Cb =C+C−C0
b
σ+σ-
g
DE
Cb ≈0
The Two (Exciton) Qubit System
GaAs
AlGaAs
AlGaAs
Optical Field
|00>
The Two (Exciton) Qubit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ+
|01>
The Two (Exciton) Qubit System
GaAs
AlGaAs
AlGaAs
Optical Field
σ-
|10>
The Two (Exciton) Qubit System
GaAs
AlGaAs
AlGaAs
Optical Field
Biexciton
σ+ σ-
|11>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)
• 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
m=-3/2 m=3/2
m=-1/2 m=1/2
DE=biexciton binding energy
Cg C+ C- Cb
0.9 0.3 0.3 <<0.005
Quantification of Entanglement: Entropy*
b
σ+σ-
g
DE
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*
*E~0.2 measured beyond chi-3 limit. Now up to E~1
C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A 54, 3824 (1996)
Creation of the Bell State
21-2
1+
23+
23-
21-2
1+
23+
23-
+ σ-σ+c0 c+-
unexcited state Biexciton state
Quantum wave function shows entanglement of the ground state and the biexciton.
+
The Two (Exciton) Qubit SystemRabi Oscillations
GaAs
AlGaAs
AlGaAs
Optical Field
|00>
The Two (Exciton) Qubit SystemRabi Oscillations
GaAs
AlGaAs
AlGaAs
Optical Field
σ+
|01>
Rabi Oscillations - qubit rotations
ih∂ ψ∂t = H0 −μE0sinωt[ ]ψ
H0 un =En un n=1,2; μ = u1 er u2
Pulse Area
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θ =h2
μE0 ′ t ( )∫ d ′ t 0
t∫
0
C2 t( )2
1
2
One Qubit Rotation in a Single Quantum DotThe Exciton Rabi Oscillation
• Rabi oscillations demonstrate an arbitrary coherent superposition of exciton and ground states,
• A pulse area of gives a complete single bit rotation,
/2-pulse -pulse -pulse
↑↓population:
Time (ps) Time (ps) Time (ps)
final quantumstate (beforedecoherence):
c↓↓↓↓ +c↑↓↑↓ or c↓↓↓↓ +c↓↑↓↑
↓↓→ ↑↓ or ↓↓ → ↓↑ ψ = 1
2 ↓↓ + 12 ↑↓ ψ = ↑↓ ψ = ↓↓
Excitonic energy levels Rabi oscillations
↓↓
Epump
↓↑↑↓
“Damping” is due to excitation induced increase in T1
Physics for Optically Driven Spin
Semiconductor Quantum Coherence
Engineering
|0>
|X>
Neutral Exciton
Electronic Spin Qubit
Successful coherent optical manipulation of the optical Bloch vector necessary to manipulate
the spin vector
Negative Exciton
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↑
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↓€
T : trion
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 g-factor and decays with time.
CN
OS
(a. u
.)
Single Electron Spin Coherence:Raman Quantum Beats
X -
X
Charged Exciton System
Neutral Exciton System
0 500 1000 1500 2000 2500Delay (ps)
Single Charged Exciton
Ensemble Charged Excitons
Single Neutral Exciton
T2* >10 nsec at B=0
hs (m
eV)
Phys. Rev. Lett. - 2005
Anomalous Variation of Beat Amplitude and Phase
(a) (b)
StandardTheory
• Plot of beat amplitude and phase as a function of the splitting.
(a)
StandardTheory
Anomalous Variation of Beat Amplitude and Phase
• 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
(a)
StandardTheory
• Plot of beat amplitude and phase as a function of the splitting.