Quantum information processing with electron spins Florian Meier and David D. Awschalom Funding...

Quantum information processing with electron spins

Florian Meier and David D. Awschalom

Funding from:

Optics & Spin Physics

2F. Meier & D.D. Awschalom

New questions for optics & spin-physics

J.M. Kikkawa and D.D. Awschalom, Nature 397, 139 (1999)

J. A. Gupta et al., Science 292, 2412 (2001)

• optical spin injection• detect spin coherence by time-resolved Faraday Rotation (TRFR)

• spin manipulation with optical pulses (fast)

• spin manipulation in an extended system spin dynamics in QD’s (one qubit)?

• spin dynamics in an external B-field el. spin interactions (CNOT)?

• Laser pulse (many photons) for spin read-out or manipulationsingle photon as qubit?


Outline

1. Towards electron interactions (coupled quantum dots):

M. Ouyang and D. D. Awschalom, Science 301, 1074 (2003) [exp.]

F. Meier et al., PRB 69, 195315 (2004) [thy.]

2. Cavity QED as interface for spin and photon quantum states:

F. Meier and D. D. Awschalom,

cond-mat/0405342. PRB (in press) [thy.]

mode 1

mode 2

QD

σ1+


• pair of coupled QD´s with one exciton• spin dynamics probed by TRFR

Results: • strong delocalization of spin via conjugated molecule• electron exchange interaction relevant for TRFR

EcA

EcB

EvB

EvA

transfer via benzene ring

QD B QD A

Universal quantum computing with electron spins requires electron exchange interaction.

coupled quantum dots

Coupled QD´s


Coupled QD‘s: Absorption

Molecularly coupled QD’s:

Absorption spectroscopy:• Coupled QD’s with different radii

1.7 nm (A) and 3.5 nm (B)

Difference in quantum size levelsallows one to selectively addressboth QD’s of a coupled pair.

• Absorption peaks in the coupled system are red-shifted.

Consistent with a coherentdelocalization of the electronor hole over the coupled system. energy

opt.

abs

orpt

ion


Larmor frequency

Ene

rgy

Faraday rotation:• g-factor size-dependent: distinguish spin in QD A from spin in QD B.

• Pump at low energy: inject exciton into QD B.

• Measure TRFR signal at varying probe energies: Find spin in QD A with probability 10-20% even at T=300 K.

-

EB

B

EA

A

Coupled QD‘s: TRFR


Coupled QD‘s: Theoretical Model

Experimental results: • absorption and TRFR imply delocalization of electrons over both coupled QD’s;

• transfer probability is of order 10-20% even at room T.

Questions for theory:

• simple model which explains the exp. features

• electron exchange interaction?

,ˆˆˆˆ0 TCoul HHHH where

(i)

,;,

,v†,vv,

†,0 ˆˆˆˆˆ

BAccc ccEccEH single-particle energy levels

EcA

EcB

EvB

EvA

UB

tc

,

,,,, ..ˆˆˆˆˆ

chcctcctH BABc

AccT v

†vv

†(iii) transfer of electrons and holes,spin-conserving

el. transfer hole transfer

BA

cccCoul nnnnnnU

H,

vvv ˆˆ2)1ˆ(ˆ)1ˆ(ˆ2

ˆ

Coulomb interaction

e-e repulsion h-h repulsion e-h attraction

(ii)


The only unknown parameters of the model are tc and tv0.

Calculate exciton wave functions and eigenenergy:

Coupled QD‘s: Tunnel matrix elements

0

0

,†,

,†,,

Bv

Ac

BBc

Ac

c

Bv

BcB

ccUEE

tccX

indirect exc.

tc EcA

EcB

EvB

EvA

BBAc

BBv

Bc

BX

UEE

tUEEE

cc

2

en. red-shift

findtc 0.08 eV


TRFR Signal: Theory

)( cn

LF

2

0

220

0

ˆ)(

)()(

P

EEE

EEECE

i

i

i

i

mag. sampleF• FR “macroscopically”: magnetization M

rotates the polarization direction of a linearly polarized Laser beam.

• FR “microscopically”: Because of Pauli blocking, dielectric response

is different for + and -:

EcB

EvB

-: transition blocked

+: transition allowed

+

mathematically:

,v†, ˆˆˆ ccP cwith dipole transition

operator for circularly polarized light.


TRFR in Coupled QD’s: Theory

,)(

21

12

)(

22,

,

22,

,

JEE

JEEpp

EE

EEp

CEE

AX

AXBAAB

AX

AXABF

From transition matrix elements to all bi-exciton states, find:

B

Bc

AcA

Bc

Ac

c UEEUEEtJ

112 2where • bi-exciton exchange splitting;

ABBA pp • probability for el. transfer from QD A to QD B (QD B to QD A);

• EX,A and energy and linewidth of exciton-transition

1. TRFR signal depends on coupling via the transfer probabilities p.2. Electron exchange coupling expected to show up in TRFR signal.


TRFR in Coupled QD’s: Results

1. Probability for electron transfer in coupled QD’s:

%6

%13

AB

BA

p

p • Obtained with tc calculated from absorption data.

• Comparable to exp. spin transfer probability 10%.

2. TRFR signal amplitude as a function of probe energy and Larmor frequ.:

theory experiment

Reentrant behavior is well reproduced by theory.


TRFR: What about Exchange Interaction?

3. Electron exchange interaction is expected to show up in TRFR signal amplitude: Expect several zeroes in F(E).

linecut at fixedLarmor frequency

E[eV]

FR

[a.

u.]

2.3 2.4 2.5=20 meV

=50 meV=80 meV

Exchange interactionJ 20 meV is too small compared to line-width 50 meV !


Coupled QD’s and Quantum Information

1. Coupled QD’s show strong delocalization of the electron wave function; spin is conserved.

2. Behavior well understood within a simple theoretical model.

3. Perspective: Detect electron exchange interaction spectroscopicallyor by exchange-governed dynamics.


QD’s in Cavities: Interface for Spin & Photon Qubits

Motivation:

Imamoglu, Zoller, Sham, ...:

QD

σ1+-laser

PL

optical selection rules: • spin dependent abs. and PL• optical spin-readout

Haroche, Kimble, Walther, ....:

atom

g

e

Cavity QED: • entanglement of atom and cavity• SWAP atom state onto cavity

Can one swap the spin state of a QD onto the cavity mode?

Using a 2-mode cavity, can implement• spin-photon entanglement;• spin-photon SWAP gate.


2-mode Cavity and QD: The System

propagatingmodes

mode 1

mode 2

QD

σ1+

y2

• QD with excess electron,

• Two cavity modes with circular (mode 1) and linear (mode 2) polarization.

• Strong coupling.

.

Dynamics if a photon is injected into mode 1?

Dynamics depend on • QD level scheme (hh or lh valence band maximum);

• one spin state of QD is always dark! or


Spin-Photon Entanglement: The Hamiltonian

0;; 1 X

QD with hh (|jz|=3/2) val. band maximum. Possible processes ....

2

1

;

;0;

yX

σ1+

sz=±1/2

jz=-3/2

hh

lh

σ1+ ory2

(b) Trion decays by photon emission

into either 1+ or y2; QD returns

to its original spin state.

..;0;;0;ˆ2211 chyXgXgH

(a) For spin state , transition to trion state by photon absorption:

where g1, g2 are coupling constants for modes 1 and 2.(2-mode Jaynes-Cummings model)


Spin-Photon Entanglement: Dynamics

.;0;2

sin2

;2

sin;2

cos)(

1

22

12

Xgti

ygtgt

t

σ1+

y2

?;;)0( 111 Time evolution of

σ1+

X

For g1=g2=g,

12 ;;)(:8/)12( ytghnt nnAt max. entangled states


Entanglement: Master Equation for Cavity Loss

.;0;2

sin2

;2

sin;2

cos)(

1

22

12

Xgti

ygtgt

t

mode 1

mode 2

σ1+

y2

propagating modes1

2

t[/g]

1;

2; y

0;X

Terminate time evolution here!

,ˆˆˆ,ˆˆ LHi

iiii

i aaaaaaLiiiˆˆˆ2ˆˆˆˆˆˆ

2ˆˆ

2,1

Cavity loss is sufficient:

with

Liouville operator for cavity loss.


Entanglement: Von Neumann Entropy

In which direction does the photonleave the cavity for spin state |?

.

)/(4)(

)/(4

;ˆ;

212

21

22

22

0

22

g

g

yydtp

At least one oscillation between

cavity modes.

Cavity lossterminates coh.

evolution exactly after one period.

t[/g]

cav. loss from mode 2

cav. loss from mode 1

Prob. for cavity loss along 2:

For photon loss into mode 2!,/ 21 g

Von Neumann entropy as fctn. of 2:

2[g/]

E

loss frommode 1

inefficienttransfer to 2(linewidth)


Entanglement: Robustness

How sensitive are the above dynamics to experimental fine-tuning?

1. Coupling constants g1g2:

2. QD misalignment by angle :

3. Detuning of cavity modes relative to exciton transition:

2

22

21

22

211F

gg

gg

8

1cos3

cos12F

42

2

2/1F gO

F

g1/g2

F

Resonance condition is crucial!

σ1+


Spin-Photon SWAP: Hamiltonian

0;; 1 X

QD with lh (|jz|=1/2) val. band maximum. Possible processes ....

2

1

;

;0;

zX

σ1+

sz=±1/2

jz=1/2

hh

lh

σ1+

(b) Trion decays by photon emission

into either 1+ or z2; QD spin

can be flipped!

..;0;;0;ˆ2211 chzXgXgH

(a) For spin state , transition to trion state by photon absorption:

z2

Trion couples to two different spin states!


Spin-Photon SWAP: Dynamics

σ1+

z2

?;;)0( 111 Time evolution of

σ1+

X

.;0;2

sin2

;2

sin;2

cos)(

1

22

12

Xgti

zgtgt

t

For g1=g2=g,

.;;)(:8/)12(

12

12

z

ztghnt nnAt QD state swapped onto cavity state!


Experimental Implementation

Main challenge: Scheme requires cavity with

• small mode volume of order 3;

• high Q-factor, Q>104;

• three degenerate modes, which are not all TE or TM;

• QD placed at mode maxima.

Possible (at least in principle) with defectmodes of a photonic crystal.

K. Hennesy et al., APL 83, 3650 (2003)


Summary

1. Spin physics of molecularly coupled QD’s:

• delocalization of electron wave function;

• dynamics driven by electron exchange interaction?

2. QD’s in two-mode cavities:

• create spin-photon entanglement;

• implement spin-photon SWAP gate;

• system robust against experimental imperfections.

y2

mode 1

mode 2

QD

σ1+

Quantum information processing with electron spins Florian Meier and David D. Awschalom Funding...

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