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

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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

  • Universal set of quantum gates (highly nonlinear)

  • Initializable

  • Qubit specific measurements

  • Long coherence time (in excess of 104 operations in the coherence time)


InAs

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 ButEngineerable

  • Larger oscillator strength (x104)

  • High Q (narrow resonances)

  • Faster

  • Designable

  • Controllable

  • Using ultrafast light, we have fast (200 GHz) switching with no ‘wires’.

  • Integratablewith 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

AFM 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


Semiconductor with periodic lattice

Can the Exciton be Controlled in High Dimensional crystals?

hole

With coulomb coupling, the e-h 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

Is the Exciton a Well defined qubit in 1, 2, or 3 Dimensional Cystal?

electron

The exciton in higher dimensional cyrstals is not a well defined qubit.


Recall the spin paradigm for quantum computing:

Can the Exciton be Controlled in High Dimensional cyrstals: i.e., can you build a universal set of quantum gates?

Rabi Oscillations:

Qubit Rotations


Coherent optical control

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:


Rabi Oscillations

6







7



0



Pulse Area


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


Nearly Degenerate Differential Transmission

Quantum Dot Spectrum

Pump

Pump excitation reduces absorption on excited transition

Differential

Probe

Tuning


RF electronics

Lock-in amplifier

CW Nonlinear Spectroscopy Experimental Set-up

Frequency stabilized

lasers

E

probe

Detector

E

probe

E

signal

E

pump

Acousto-optic

Modulators ƒ≈100 Mhz


Many-Body Effects in High Dimensional Semiconductors

Excitation Wavelength

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

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

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)

Atomic-like spectrum –

Discrete states followed by continuum


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

PL Intensity

Nonlinear Signal Intensity


Use a Quantum Dot to Build a 2-Qubit Computer?

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>

|1>

B-Field

+

Coulomb

Interaction

|0>

|0>

How to Build a Two-Bit Quantum Computer

Two spin-polarized excitons

Need two quantum bits

Need coupling

Coulomb interaction

Need coherent control

Resonant polarization- dependent optical coupling

|11>

s-

s+

|01>

|10>

|00>


The Two-Bit System

Optical Field

AlGaAs

GaAs

AlGaAs

|00>


The Two-Bit System

Optical Field

AlGaAs

GaAs

s+

AlGaAs

|01>


The Two-Bit System

Optical Field

AlGaAs

GaAs

s-

AlGaAs

|10>


Formation of the |11> state

Optical Field

AlGaAs

GaAs

s-

s+

AlGaAs

|11>

Biexciton


Do quantum dots experience pure dephasing?

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


0

10

ph

ph

rel

-

2

-

1

0

1

2

-

2

-

1

0

1

2

Probe detuning

(

u

n

i

t

s

)

Calculated Coherent Wavelength-Resolved Differential Transmission from a Two Level System

No pure dephasing

Strong pure dephasing

  • 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).

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 robustagainst decoherence)

Nonlinear Signal Intensity (a.u.)


The Two-Bit System

Optical Field

AlGaAs

GaAs

s+

AlGaAs


The Two-Bit System

Optical Field

AlGaAs

GaAs

s-

AlGaAs


s- 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 Exciton-State Quantum Entanglement

Quantum wave function shows entanglement of two exciton-states.

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)


+

-

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 exciton-states


C

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

Non-interacting Case

Factorizable wavefunction:

With Coulomb Correlation

How small Cb is depends on linewidth of state b and DE


The Two (Exciton) Qubit System

Optical Field

AlGaAs

GaAs

AlGaAs

|00>


The Two (Exciton) Qubit System

Optical Field

AlGaAs

GaAs

s+

AlGaAs

|01>


The Two (Exciton) Qubit System

Optical Field

AlGaAs

GaAs

s-

AlGaAs

|10>


The Two (Exciton) Qubit System

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

  • Superlinear excitation intensity dependence of photoluminescence from the biexciton-to-exciton transition


DE=biexciton

binding energy

m=1/2

m=-1/2

m=-3/2

m=3/2

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


C

C

C

C

+

-

b

g

b

DE

s+

<<0.005

s-

0.3

0.9

0.3

g

Quantification of Entanglement: Entropy*

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,

*

C.H. Bennett,D. P. DiVincenzo, J. A. Smolin, W.K. Wootters, Phys. Rev. A54, 3824 (1996)

*E~0.2 measured beyond chi-3 limit.

Now up to E~1


s-

1

1

3

3

-

-

-

-

3

1

3

1

+

+

+

+

2

2

2

2

2

2

2

2

Creation of the Bell State

unexcited state

Biexciton state

Quantum wave function shows entanglement of the ground state and the biexciton.

c+-

c0

+

s+

+


The Two (Exciton) Qubit SystemRabi Oscillations

Optical Field

AlGaAs

GaAs

AlGaAs

|00>


The Two (Exciton) Qubit SystemRabi Oscillations

Optical Field

AlGaAs

GaAs

s+

AlGaAs

|01>


Rabi Oscillations - qubit rotations









0



Pulse Area


Epump

or

One Qubit Rotation in a Single Quantum Dot

The Exciton Rabi Oscillation

Excitonic energy levels

Rabi oscillations

  • Rabi oscillations demonstrate an arbitrary coherent superposition of exciton and ground states,

  • A pulse area of p gives a complete single bit rotation,

p-pulse

p/2-pulse

2p-pulse

or

population:

Time (ps)

Time (ps)

Time (ps)

final quantum

state (before

decoherence):

“Damping” is due to excitation induced increase in T1


Physics for Optically Driven Spin

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


  • Raman coherence oscillates at frequency of the Zeeman splitting due to electron in-plane g-factor and decays with time.

  • Circularly polarized pump pulse creates coherent superposition of spin up and down state.

Optical Excitation of Spin Coherence:Two-photon stimulated Raman


Single Electron Spin Coherence:Raman Quantum Beats

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


Anomalous Variation of Beat Amplitude and Phase

Standard

Theory

(a)

(b)

  • Plot of beat amplitude and phaseas a function of the splitting.


Anomalous Variation of Beat Amplitude and Phase

Standard

Theory

(a)

  • Plot of beat amplitude and phaseas a function of the splitting.


Spontaneously Generated Coherence (SGC)

Trion

G

G

  • 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 phaseas a function of the splitting.


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