Electrical Means of Manipulating Electron Spins in Semiconductors
Quantum information processing with electron spins
description
Transcript of Quantum information processing with electron spins
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?
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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+
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• 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
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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
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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
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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)
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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
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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.
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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.
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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.
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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 !
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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.
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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.
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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
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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)
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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
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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.
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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)
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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+
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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!
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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!
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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)
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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+