Coherent manipulation of spins in quantum dots

Sophia E Economou

sophia.economou@gmail.com

NRL

DPG Physics School 2010, Bad Honnef, September 2010

Naval Research Lab

Washington DC

-2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.81256

1258

1260

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1272

PL

Ene

rgy

(meV

)

Bias (V)

growthspectroscopy

controltheory

Outline• Motivation: quantum information processing

• Quantum dots

• Basics of quantum control

• Coherently controlled spins in QDs– Focus on optical control– Spin rotations– Entangling gates between spins

R. Landauer: “Information is physical”

Why quantum information? (e.g. |0>≡|↑>,|1>≡ |↓>)

• Principles of quantum mechanics determine the properties of information encoded in quantum systems– Superposition

– Entanglement of two spins

– Measurement

] in bit classical [cf. ↓↑↓ +↑ = orβαψ

↑↓ −↓ ↑ = Ψsinglet spin :Example

Not in product form

spin second determinesspin first measuring : Ψfor

0 , 1: two states of a physical system

Some consequences of these principles• N-qubit system described by 2N

parameters [while classical N bits by O(N)]because the different contributions to the superposition evolve ‘in parallel’

Potential computational power: A programmable quantum computer may solve exponentially difficult problems

• Unknown quantum states cannot be clonedOtherwise measurement axiom would be violated

Quantum information cannot be intercepted (cryptography)

Motivation-Why optically controlled spins in QDs?

• Quantum information processing– Spin is naturally a qubit

– Long coherence times ms

– Ultrafast preparation, control & measurement (ps timescale)

– Semiconductor: integrability

– Fixed location of qubits

• New & interesting physics– Interplay of solid state environment & confinement provide new playground to explore quantum effects

– Coherent control

QIP with optically controlled spins in QDs• Quantum dot with single excess electron• e spin carries quantum information

• Operations: optically by Raman transitions via trion (bound state of electron and exciton)

• Each QD emits at a different frequency(Large inhomogeneous broadening allows for ‘zip‐code’assigning and optically selective addressing)

• Inter‐dot coupling: – With common cavity mode

– Optically generated extended state

Alternative use of quantum dots: Emitters of single photons and of entangled photon states for use in Quantum Information tasks

Imamoglu et al, PRL 2001Steel+Gammon, Physics Today 1998

Quantum dots• Semiconductor nanostructures with 3D nanometer confinement for electrons/holes

• Atomic‐like energy levels

D. Gammon et al., PRL 76, 305 (1996)

J. P. Reithmaier et al.,Nature 432, 197 (2004)

J. M. Elzerman et al., Nature 430, 431 (2004)

Energy levels: quantization & HH-LH splittingQuantum dotParticle in a box with effective mass from each band(large enough to have underlying structure; see also shallow impurities)

2/3 h, ±

2/1 h, ±

2/1 e, ±Bands of III-V compounds

21

=J

23

=J

Bulk

2/3 h,±

2/1 h, ±

2/1 e, ±21

=J

23

=J

Confinement—induced H-Lhole splitting

trion

Self assembled quantum dots: growth & spectroscopy

Substrate GaAs: Small lattice constant

Deposited Material InAs: Large lattice constant

WLQDs

Stranski-Krastanov ‘self organized’ QDs

excite quantum dotsthrough submicronapertures

1 mm

0.2-25μm

1.25 1.30 1.35 1.40

norm

. Int

ensi

tät

Energie [eV]

100nm

200nm

300nm

ref

single dot spectroscopy

MBE growth

2/3 h, ±

2/1 h, ±

2/1 e, ±21

=J

23

=J

From band picture to total energy representation

|3/2> |-3/2>

2/1 2/1−

Single particle statesParticle interactions cannot be represented

Energy eigenstates

•The 2 spin states form the qubit•Need spin rotations

Spin control (qubit rotation)• Time-dependent magnetic field (pulsed)

– Direct coupling of spin states

• Time-dependent electric field (gate voltage)– Indirect coupling by

• Spin-orbit coupling• Local change of g factor

• Optical (pulsed laser)– Use of optically excited states

outside qubit subspace

B(t)

Ω1(t)Ω2(t)

Spin rotationsSome basics• A spin 1/2 (and any quantum two level system) can be

represented as a 3-vector

• ANY unitary operation U can be mapped to a rotation of this vector in 3-space

• Bloch sphere & Bloch vector

• Difference between design of an operationand population transfer: we want to construct U (independence of initial state)

• To construct an arbitrary rotation, rotations about two orthogonal axes suffice

• Rotation operator is R=exp(-isnφ)=

• A rotation about the quantization axis looks like a phase

See Quantum Mechanics textbooks such as e.g. Shankar or Sakurai

1

0

θ

φ

Ψ

12

sin02

cos θθ φie+=Ψ

Exercise: show that any unitary operation in a 2x2 Hilbert space is a (pseudo)spin rotation

⎥⎦

⎤⎢⎣

⎡ −

2/

2/

00φ

φ

i

i

ee

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−

−−−

2sin

2cos

2sin

2sin

2sin

2sin

2sin

2cos

φφφφ

φφφφ

zyx

yxz

innin

ninin

Spin rotations with direct coupling between the two spin states: the Rabi solution

Everything henceforth applies to an arbitrary two‐level system

•Spin states talk to each other through external time dependent magnetic field

•Estimate of field duration: splitting is ωe=g B, with typical QD g-factors |g|=0.2-0.5; we~0.1 meV t~ℏ/ωe ~ 1 ns (useful to remember that ℏ~0.65 meV*ps)

•The Hamiltonian is⎥⎦

⎤⎢⎣

⎡⋅

⋅−2/)()(2/

e

e

tBtB

ωμμω

•We take the field to be product of an envelope function and an oscillation:B(t) = Bo(t)*cos(ωe*t)•cos(ωe*t) = ½ (eiωet + e-iωet): keep only the energy preserving terms: rotating wave approximation (RWA)

After the RWA the Hamiltonian becomes

⎥⎦

⎤⎢⎣

⎡

ΩΩ−

=− 2/)(

)(2/

eti

tie

etet

Hω

ωω

ω

Where we defined ½ μBo(t)=Ω(t)

We can eliminate the oscillating exponents by a transformationU=diageiωt/2, e-iωt/2; this is equivalent to going to the rotating frame of the laser

We then get a new Hamiltonian ⎥⎦

⎤⎢⎣

⎡Δ−Ω

ΩΔ=

2/)( )(2/

tt

Hωω +−=Δ eWhere is the detuning

Now we want to solve the differential equations:

ttittH

∂Ψ∂

=Ψ)()()(

Show how we get this

⎥⎦

⎤⎢⎣

⎡Δ−Ω

ΩΔ=

2/)( )(2/

)(t

ttH t

tittH∂

Ψ∂=Ψ

)()()(

In general, we cannot solve analytically the time-dep Schroedinger eqn, even for the two-level system.

There are some known solutions for specific pulse shapes Ω(t) the simplest being the square pulse (Rabi, Phys Rev 1937)

The Hamiltonian reduces to a time-independent one

Which we know how to solve.

Exercise: use a change of basis to diagonalize this Hamiltonian, find Ud in that basis and transform back to get the evolution operator in the lab frame

t

Ω(t)Ωο

⎥⎦

⎤⎢⎣

⎡Δ−ΩΩΔ

=2/

2/

0

0H

Back in the laser frame the evolution operator is

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω

ΩΔ

+⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω

ΩΩ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω

ΩΩ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω

ΩΔ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω

=

2sin

2cos

2sin2

2sin2

2sin

2cos

0

0

τττ

τττ

eff

eff

effeff

eff

eff

eff

eff

eff

eff

ii

iiU

Since any unitary operator in a 2x2 Hilbert space can be mapped on a rotation, we should be able to do this for U.A rotation is parameterized by 3 real numbers

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−

−−−== ⋅−

2sin

2cos

2sin

2sin

2sin

2sin

2sin

2cos

2/

φφφφ

φφφφσφ

zyx

yxzni

innin

ninineR

rr

By equating U=R we find eff

zeff

x nnΩΔ

=ΩΩ

=Δ+Ω= 2 τ4 0220φ

where22

04 Δ+Ω=Ωeff

Find the evolution operator and identify the rotation in the lab frame

To summarize so far:Time-dependent magnetic field spin control

– Conceptually straightforward• Time of pulse determines rot. angle

– Rotations in ~ns timescales (compare to ms coherence time)

– No spectral+spatial focusing

– Time dependent magnetic field is accompanied by time-depelectric field (which is stronger, too): effects on orbital part of state can mess things up

– Experimental demonstration by the Delft groupKoppens et al, Nature 442, 766 (2006)

B(t)

Spin control (qubit rotation)• Time-dependent magnetic field (pulsed)

– Direct coupling of spin states

• Time-dependent electric field (gate voltage)– Indirect coupling by

• Spin-orbit coupling• Local change of g factor

• Optical (pulsed laser)– Use of optically excited states

outside qubit subspace

B(t)

Ω1(t)Ω2(t)

A different way of inducing spin rotations

• Optics more focused

• Faster

• But: design not straightforward

The population exits the spin subspace during the operation

– We have to design the pulses such that it returns to the qubit subspace

The problem• Objective: given the 4 states and selection rules

implement the desired rotations– Find laser fields that will yield desired

evol. operator U(tf) from H(t) U=i dU/dt

• During evolution, trion states (decaying) are populated We need:– Population to return to qubit subspace– Operation to be fastWe’d like to find ways to reduce a 3‐ or 4‐level system to an effective 2‐level system (that we have ways to solve)

σ-σ+

|3/2> |-3/2>

|1/2> |-1/2>

B = 0

No Raman transitions |1/2> |-1/2> possible Cannot implement arbitrary spin rotation

Perpendicular field mixes spinstates & enables Raman transitions

Selection rulesz basis x basis

|1/2>-|-1/2> ≡|-x>

σ-σ+

|3/2> |-3/2>

σ-σ+

Bx ≠ 0

|1/2>+-1/2> ≡|+x>

An alternative way to represent the levels+transitions

•Selection rules in z basis are simple•B field in x mixes ground states

T T

z z

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ΩΩ=

T

e

e

ttH

εω

ω

)(0)(0

00

*

Fixing the polarization to σ+ we have in the z basis |-z>,|z>,|T>:

There are 2 timescales in the problem:

•The pulse duration τ•The precession period 1/ωe

When τ >> 1/ωe we can ignore the spin precession during the pulse 3-level systemreduces to 2-level system •All 2 level systems are equivalent

We can use the previous solution but now we need to return the population back to the spin stateIn the |z>, |T> subspace we have:

eeρ

0.2

0.4

0.6

0.8

1

∫Ω= dttA R )(2

2π rotation: returns to -|g> (not |g>)

z

T

Notice what happens for Δ=0 andπτ 2Ω =

The population returns to the ground stateAND the state picks up a minus sign!•In a 2-level system: global phase•In a 3-level system: relativeRotation about z axis by π!

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

ΩΔ

+⎟⎟⎠

⎞⎜⎜⎝

⎛Ω⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

ΩΩ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

ΩΩ

−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

ΩΔ

−⎟⎟⎠

⎞⎜⎜⎝

⎛Ω

=Δ

−−Δ

−−

ΔΔ

2sin

2cos

2sin2

2sin2

2sin

2cos

202

022

τττ

τττ

ττε

ττε

ττ

eff

eff

effiieff

eff

ii

eff

eff

ieff

eff

effi

ieeeie

ieieU

TT

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

100010001

U

We still need: •Rotations by other angles•Rotations about (at least) one more axisNow we consider Δ≠0 but still

πτ 2Ω =eff

We’re only intersted in 2x2 subspace

⎥⎦

⎤⎢⎣

⎡≈⎥

⎦

⎤⎢⎣

⎡=

−

2/

2/

2x2 00

001

φ

φ

φ i

i

i ee

eU where

2τπφ Δ

+=

Issues with the square pulse:•unphysical shape•especially for short pulses, not a good approximation to actual short pulse envelope⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

≡⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=Δ

−−Δ

−−

Δ

22

2

00

00001

00

00

001

ττε

φ

ττε

τ

ii

i

ii

i

ee

e

ee

eUTT

|1/2>-|-1/2> ≡|-x>

σ-σ+

|3/2> |-3/2>

σ-σ+

Bx ≠ 0

|1/2>+-1/2> ≡|+x>

The one ‘special’ direction was z; we already made use of thatThe other one is the B-field axis (x)

Obvious way to generalize to the x axis: focus to one transition only by narrow-width (i.e. long) pulses

Not practical for ωe small (we shall revisit it later for large values of ωe)

There is a way to do this without using (too) long pulses

But first some background...

Some more tools in our toolbox•Analytically solvable pulses: the sech pulse•Approximate methods•Coherent population trapping

Vge=Ω(t) eiΔt =Ω0sech(βt) eiΔt|e>

|g>

t−1β

Ω(t)

History of hyperbolic secant pulse

1932: Rosen & Zener find the solution first in the context of a spin ½ in a magnetic field

1969: McCall & Hahn discover Self Induced Transparency

1975: Allen & Eberly find a solution even when there is frequency modulation

1980s: More general (but complicated) solutions of which the sech is a special case

|e>

|g>

sech pulses in two level systemsRosen & Zener Phys. Rev. ’32

( )

( )[ ]

equation tricHypergeome

1tanh21

variableChangedetuning theis and ,sech

0)/( 2

+=

ΔΩ==+−Δ+

t

tfcfcffic eee

βζ

β&&&&

Schrödinger equation

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−++−+−

++−+−−=

),,,(),1,,(

),1,,(),,,(

***

*

*

* ζζζ

ζζζ

caaFccacaFcia

ccacaFciacaaF

Uc

c

Then the evolution operator is

Phase induced by sech pulse in 2-level system

|e>

|g>

When Ω = β (notice that we get a 2π pulse independent of detuning) population returns to |g> with an acquired phase:

Economou et al. PRB 74 (2006)

State: |g> eiφ |g> ≡ |g> (Global for 2-level system)In the presence of third level |g’>, it alters the quantum state:

eiφ |g>+|g’>≠ |g>+|g’>

⎟⎠⎞

⎜⎝⎛

Δ=

βφ arctan2

Fast pulse (β >> ωe, ωh)Spin precession ~ ‘frozen’ during pulse2-level system + 2 uncoupled levels

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

≈−

⇓↑↓⇑↑↓↑↓

10000000000001

φ

φ

i

i

ee

U

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡≈

−

2/

2/2/

00

001

φ

φφ

φ i

ii

ispin ee

ee

U

z rotations with sech pulse

σ+

|3/2> |-3/2>

|1/2>

z basis

B

|-1/2>

Experimental relevance

No need to change power only detuning

Short pulses can be well approximated by a sechenvelope

⎟⎠⎞

⎜⎝⎛

Δ=

βφ arctan2

z basis

T T

z z zzx −=

zzx +=

TTTx +=TT

Tx

−

=T T

zzx −=

zzx +=

x basis mixed basis

Rotations about other axes: useful to think in various bases

Use of linearly polarized light decouples the 4-level system to two 2-level systems:

•Use a single 2π sech pulse: couples to both transitions•We take advantage of property of sech that 2π pulse is detuning independent to avoid long pulses •A different phase is induced to each of the ground states |x>, |-x>:

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡≈

−

−−+−

−

−

2/)(

2/)(2/)(

21

2121

2

1

00

00

φφ

φφφφ

φ

φ

i

ii

i

i

spin ee

ee

eU

221

2 )(arctan2x

xx β

βφφφ+ΔΔ

Δ−Δ=≡− 1

21

Economou & Reinecke, Phys. Rev. Lett. 99, 217401 (2007)

H

|3/2> - |-3/2>

|+x>

|3/2> + |-3/2>

H

|-x>

x rotations

General rotations: combine rotations about z and x to make arbitrary rotations

Composite rotations

• Rn(φ)=Rz(θ)Rx(φ)Rz(-θ) where θ=angle (n,z)

Example: π rotation about y axis

z ro

tati

on

x r

ota

tion

z ro

tati

on

Experimental setup (theorist’s view)

Pump‐probe experiment

Sampleτ1

τ2

Detector

Differential transmission as fn of delay time τ2 − τ1

σ+

+z

Bloch sphere diff

. tra

nsm

issi

on

Dutt et al., PRL 2005

Experimental results

Greilich, Economou, Spatzek, Yakovlev, Reuter, Wieck, Reinecke, Bayer,

Nature Physics 5, 262 (2009)

⎟⎠⎞

⎜⎝⎛

Δ=

σφ arctan2

z

control pulse Spin rotations also demonstrated optically by Yamamoto’s group (Nature 2008) and the Steel group (PRL 2010)

Brief summary of yesterday’s lecture

• Spin rotations by – time dependent B field– Optical control

• Analytically solvable pulses: square pulse (Rabi), sechpulse (Rozen & Zener)

• Obtaining effective 2‐level system by fast pulse approximation

• Resonant 2π pulse induces a minus sign to state

Other ways to get an effective 2‐level system

• Adiabatic elimination or Optical Stark Effect

• Coherent population trapping (CPT)‐based methods

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΩΩΩΩ−

=**

2/002/

βα

β

α

δδ

H

•This method is appropriate for large detunings Δ>>|δ|,|Ωa||,|Ωb|•Since the detuning is very large the trion population is assumed negligible

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

γβα

ψ )(t

The Hamiltonian in the rot. frame is

The state vector is

γβαγ

γβδβ

γαδα

βα

β

α

Δ+Ω+Ω=

Ω+−=

Ω+−=

∗∗)(2

)(

2)(

ti

ti

ti

&

&

&

Eqns of motion areSetting dγ/dt=0 we get

ΔΩ+Ω−= ∗∗ /)( βαγ βα

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΔΩ−ΔΩΩ−ΔΩΩ−ΔΩ−−

= ∗

∗

/||2////||2/2

2

βαβ

βαα

δδ

effH

for Δ 0 this blows up

Other ways to get an effective 2‐level system

• Adiabatic elimination or Optical Stark Effect

• Coherent population trapping (CPT)‐based methods

Define new basis |D> = Ω1 |g2> - Ω2 | g1 > and |B> = Ω1 | g1 >+ Ω2 | g2 > Take matrix element of laser between |D> and |E> :

<E|V (Ω1 |g2> - Ω2 |g1>) = Ω1<E|V|g2> - Ω2<E|V|g1> =Ω1Ω2-Ω2Ω1=0

State Ω1 |g2> − Ω2 |g1> is decoupled, even though it contains states that couple to the excited state (destructive quantum interference)

The orthogonal state |B> = Ω1 | g1 >+ Ω2 | g2 > can be thought of as the only state coupling to |E>

1g 2g

E

Ω1Ω2

B D

E

Ω

• Above scheme requires large bandwidths for z rotations

• For QDs with large Zeeman splittings such lasers may not be available and/or other levels may couple

Use narrowband pulses to select a Λ systemTotal laser field

Choosing equal detuning and same f(t) creates a coherently trapped state

Alternative spin rotation method based onCoherent population trappingfor large Zeeman splittings

|1/2>-|-1/2>≡ |-x>

σ+

|3/2> |-3/2>

σ+

|1/2>+-1/2> ≡ |+x>

where

Bright state coupling to trion is

where

Coherent population trapping in electron/trion system

For bright/dark states to be time-independent, they are related to the energy eigenstates by the transformation:

xT

B

xT

D

, xB TV

xT|xT|

B| D|

xTBV ,|

Want the total pulse acting on bright state to have area 2π :

Coherent population trapping + 2π sechpulses: analytic sln to Λ system

xT

B

xT

D

, xB TV

Then we have a rotation

Δ is the (common) detuning

Axis of rot. determined by phase and relative strength of two lasers

σ+σ+

Parameters for CdSe QDs used

Example: π/2 rotation about z axis

Other theoretical works using CPT for rotation:Kis and Renzoni, PRA 2002Chen et al, PRB 2004Experimental CPT with CW pulses: Xu et al, Nature 2008

Spin‐spin interactions for 2‐qubit gates & entanglement

• Two‐qubit logic gates are necessary for QIP

• Need for (switchable) interaction between spins

– exchange interaction based (Coulomb nature)

– cavity mediated interactions

• Growth– Coupled quantum dots (quantum dot molecules)

– QD photonic crystal samples

• Design– Switchable interactions

– Given the interactions design gates

Quantum dot ‘molecules’

SAQD

Growth:

TruncatePartial cap Repeat

Strain-inducednucleation

Spectroscopy:

⇓↑

⇑↓

⇓↑

⇑↓

Applied Bias

Photoluminescence

Example: exciton in a quantum dot molecule with hole state tunneling

Quantum dot ‘molecules’For a switchable interaction we want:•Ground state not coupled•Excited state coupled

Need the 2‐qubit gate to be compatible with the spin rotations:•Additional states to realize two‐qubit gate (i.e., we do not want the trion states used before to tunnel‐couple)

•Subband states can be used

2/3 h, ±

2/1 h, ±

2/1 e, ±21

=J

23

=J

Quantum dot ‘molecules’ for 2‐qubit gates

Find Coulomb interactions: direct interactions strongest

Exchange interactions: e‐e strongest, diagonalize first

Electron‐hole interactions are small but nonzero and they’ll prove to be very important!

3 electrons and 1 hole

3 distinct orbital electron states available: all spin configurations (no Pauli excl.)

3 spin ½ particles: total e‐spin can be 3/2, ½, ½

See also earlier proposal by Lu Sham group using RKKY type interaction, Piermarocchi et al, PRL 2002

Exchange interactions—general• Exchange interactions are between spins

• Arise from Coulomb interaction + Pauli principle

• Electric in nature, so stronger than magnetic dipole

• Example: 2 spins

• Possible spin states are singlet and 3 triplets (multiplied by the symm. + asymm. orbital states respectively for antisymmetrization)

First find energy eigenstates of 3e and 1h without EHEI

• External B field along x (in-plane direction)

• Diagonalize using 3 electronic orbital states & tensor product with hole

2Jee

Jee

Se=1/2

Se=1/2

Se=3/2

L RE

electron‐hole exchange interactions

• In our picture, electrons and holes are different quasiparticles (in conduction and valence band respectively)

• But, there is a mixing term (k dot p) that couples different bands (see effective mass approximation)

• 3rd order process:

v

c

pk⋅ pk⋅

VCoulomb

Pikus & Bir,

Electron-hole exchange• EHEI of electron i can be written using hole

pseudospin j as

• Total spin Hamiltonian of 3 e’s and hole can be written as

• Parity good quantum number Hspin is block diagonal

λλλ

λα jsrH iizyx

iexch )(

,,∑=

=

.. )(3

)()(

3

)()()(

2121

,,

321

pcjssrr

jSrrrjSHzyx

xhxespin

+−−

+

++++= ∑

=

λλλλλ

λλλ

λλλ

αα

αααωω

jSJ

He spinJi x

+=

= ,0],[ π

Diagonalizing Hspin gives the excited (trion) states

• Color corresponds to odd and even parity

• States mostly mix within Se

2Jee

Jee

Se=1/2

Se=1/2

Se=3/2

EHEI

We focus on the 4x4 subspace in the dashed box

Describe entangling gates, CNOT, CZ• For quantum information processing we need single qubit gates & 1 entangling two‐qubit gate

• The most well‐known is probably the CNOT:– If spin 1 is up, flip the other spin; if it is down, do nothing

• In the basis

• An alternative is the CZ gate

• Entangling gates between spins also useful for entangled photon generation!

↓↓↓↑↑↓↑↑ ,,,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000010000010010

cnot

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=

1000010000100001

czExercise: show that the CZ gate is equivalent to the CNOT up to single qubit operations

ideal

ikikki

kkiii

ii

UUI

IIIIIf

+

∗

≠

∗

=

++= ∑∑ )(201||

101 2

Single transition isolated by EHEI• Left: without EHEI the transitions

are doubly degenerate• Cannot address only one

transition• EHEI enables a new transition

with non degenerate frequency ω

• Now CZ gate can be implementedUideal= diag-1,1,1,1

• Fidelity calculation

ω0

ω0

ω+Δω

EHEI

↑↑

↓↑↑↓ ,

↓↓

ω

~90%

What about experimental demonstration?

• Recent progress by NRL experimental group– Kim, Carter, Greilich, Bracker and Gammon, arxiv 2010

• Control of entanglement demonstrated

• Coupling is in the ground state, not excited state

Energy Level Diagram

Applied Bias (V)

2e

Applied Bias (V)

2e

X2-

ST

ST

T

X

S

kineticexchange

Δee

S S

S S

T

Short vs. Long optical pulses

Short pulses Long pulses

Time domain: • Acts on single spin state because faster than exchange interaction

Time domain: • Acts on joint spin state because slower than exchange interaction

Frequency domain: Frequency domain:

T

X

S

T

X

S

Acts on S + T Acts just on S

– Cavity‐waveguide systems with a quantum dot coupled with the cavity at each node

– Need switchable interactions

– Need coherent control design such

that only the target qubit(s) is (are) affected

– Measurement schemes

Current/future work: Multiqubit and long range control: Cavity/waveguide systems

initialize measure

cavity 2cavity 1Entanglement

initialize measure

Photonic coupling

Other related uses of optically active quantum dots

• In many quantum information processing tasks, single and entangle photon sources are needed

• QDs are good single photon emitters (large dipole moments, simple level structure, polarization memory even at B=0)

• Entangled photon emitters– Biexciton

– Periodically pulsed spins

Coupled quantum dots as emitters of entangled photons

Main idea:Entangled emitters emit entangled photons

Measurement‐based or one‐way quantum computing: an alternative way of doing quantum computationRaussendorf & Briegel, Phys. Rev. Lett. 86, 5188 (2001)

• Creation of a highly entangled multi‐qubit state upfront

• Only single qubit measurements needed

• State collapses

• Measurement determines the ‘answer’

• Envisioned as ‘the way to go’ for photonic QC

• Problem: creation of the ‘cluster state’

Single QD based linear cluster state• Broken symmetry of quantum dot along growth axis gives rise to unusual selection rules

• Spontaneous emission rate γvery fast

• Can consider 2 indep. 2‐level systems when ωZeeman<γ

σ+

Bσ-

23,

23

23,

23

−

21,

21

21,

21

−e spin states

Trion states

Spontaneous emission:

23,

23

21,

21

23,

23

−21,

21

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−++−++⎟⎟

⎠

⎞⎜⎜⎝

⎛−−++

⎯⎯ →⎯−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−++⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

⎯⎯ →⎯−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−++⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

⎯⎯⎯ →⎯−−++→−+

σσσσσσ

σσ

σσ

σσ

21

21

21

21

23

23

23

23

21

21

21

21

21

21

23

23

decay

excite

precession

Lindner+Rudolph PRL 2009

entangled photon states:

cluster state generation

lines denote entanglement

1st set of photons

time

QD 1

QD 2...

2nd set of

photons

Protocol:• Initialize spins

• Precession under B field

• Apply CZ gate

• Excite each spin to trion with linearly polarized light

• Fast decay+photon emission

• Precession

• …Economou, Lindner, RudolphPRL 2010

yR

yR

yR

yR

top dot

bottom dot

photonphoton

photonphoton

Outlook• Multiqubit systems

– Design cavity‐waveguide configurations– Pulse design in many‐qubit context

• Nuclei– Understand physics of simultaneous hyperfine interaction and pulses at the quantum level & compare results from different experiments

– Possible use for quantum memory?

• QDs as emitters of entangled photon states– Design of many particle state production

Summary• Single qubit rotations can be done directly by time‐dep B fields, or by lasers through auxiliary optically excited states

• Square pulse and sech pulse 2 of the few analytically solvable pulses of Schroedingereqn. for 2 level syst.

• Resonant 2π rotation induces a minus sign to a two‐level system

• Exchange interactions (Pauli+Coulomb) and cavity mediated interactions can be used to couple spins in different dots