electric dipole spin resonance and spin decoherence for heavy holes in quantum dots
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Denis Bulaev and Daniel Loss
Department of PhysicsUniversity of Basel, Switzerland
Electric dipole spin resonance Electric dipole spin resonance and spin decoherenceand spin decoherence
for heavy holes in quantum dotsfor heavy holes in quantum dots
OutlineOutline
Motivation
Spin-Orbit Coupling of Holes
Spin Relaxation and Decoherence of Heavy Holes in QDs
Spin-Manipulation Methods for Heavy Holes
Conclusions
MotivationMotivation
QD based Quantum ComputingLoss & DiVincenzo, PRA 57, 120 (1998)
MotivationMotivation
Koppens et al., Science 309, 1346 (2005)*Amasha et al. cond-mat/0607110
Low T1 (up to 170 ms*)
QD based Quantum ComputingLoss & DiVincenzo, PRA 57, 120 (1998)
MotivationMotivation
Koppens et al., Science 309, 1346 (2005)*Amasha et al. cond-mat/0607110
Low T1 (up to 170 ms*)
Koppens et al., Nature 442, 766 (2006)
ESR
Rabi oscillations
Petta et al., Science 309, 2180 (2005)
QD based Quantum ComputingLoss & DiVincenzo, PRA 57, 120 (1998)
MotivationMotivation
Koppens et al., Science 309, 1346 (2005)*Amasha et al. cond-mat/0607110
Low T1 (up to 170 ms*)
Koppens et al., Nature 442, 766 (2006)
ESR
Rabi oscillations
Petta et al., Science 309, 2180 (2005)
QD based Quantum ComputingLoss & DiVincenzo, PRA 57, 120 (1998)
Fast T2
“T*2 ≈ 10 ns, limited by hyperfine interactions”
Petta et al., Science 309, 2180 (2005)Khaetskii, Loss, Glazman, PRB 67, 195329
(2003)
MotivationMotivation
Heavy-Hole Spin as Qubit:
weak hyperfine interactions with nuclear spins
strong spin-orbit coupling
difficult to manipulate the spin
MotivationMotivation
Heavy-Hole Spin as Qubit:
weak hyperfine interactions with nuclear spins
strong spin-orbit coupling
difficult to manipulate the spin
H LK3D =
Fhh 0 H I
0 Fhh I * - H *
H * I Flh 0
I * - H 0 Flh
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
,
Fj =PP
2
2m jP
+Pz
2
2m j^
,
I = gP-2 ,
H = 2gPzP- .
Spin-Orbit Coupling: 3D vs 2DSpin-Orbit Coupling: 3D vs 2D
Bulk SemiconductorBulk Semiconductor
[001] Quantum Well[001] Quantum Well
E
k
HH
LH
±3/2
±1/2
E
LH
±3/2
±1/2
kⅡ
HH
H LK2 D =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
,
G j =PP
2
2m jP
,
I = gP-2 ,
H = 0.
Spin-Orbit Coupling: 2D vs 0DSpin-Orbit Coupling: 2D vs 0D
[001] Flat Quantum Dot[001] Flat Quantum Dot
E
LH
±3/2
±1/2
HH
H LK0 D =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+ V (rP),
G j =PP
2
2m jP
,
I = gP-2 ,
H = 0.
[001] Quantum Well[001] Quantum Well
E
LH
±3/2
±1/2
kⅡ
HH
H LK2 D =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
,
G j =PP
2
2m jP
,
I = gP-2 ,
H = 0.
Spin-Orbit Coupling: QDSpin-Orbit Coupling: QD
HQD =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+V (rP) + HRSO + HDSO + HZ ,
HRSO = a R JxPy - JyPx( ) =ia R
2
0 0 3P- 0
0 0 0 - 3P+
- 3P+ 0 0 2P-
0 3P- - 2P+ 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HDSO = bD - JxPx + JyPy( ) = -bD
2
0 0 3P+ 0
0 0 0 3P-
3P- 0 0 2P+
0 3P+ 2P- 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HZ = - gmBJ ◊B = -gmB
2
3Bz 0 3B- 0
0 - 3Bz 0 - 3B+
3B+ 0 Bz B-
0 3B- B+ - Bz
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
.
E
LH
±3/2
±1/2
HH
Winkler, PRB 62, 4245 (2000)
DB & Loss, PRL 95, 076805 (2005)
Luttinger, PR 102, 1030 (1956)
HQD =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+V (rP) + HRSO + HDSO + HZ ,
HRSO = a R JxPy - JyPx( ) =ia R
2
0 0 3P- 0
0 0 0 - 3P+
- 3P+ 0 0 2P-
0 3P- - 2P+ 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HDSO = bD - JxPx + JyPy( ) = -bD
2
0 0 3P+ 0
0 0 0 3P-
3P- 0 0 2P+
0 3P+ 2P- 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HZ = - gmBJ ◊B = -gmB
2
3Bz 0 3B- 0
0 - 3Bz 0 - 3B+
3B+ 0 Bz B-
0 3B- B+ - Bz
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
.
Spin-Orbit Coupling: QDSpin-Orbit Coupling: QD
E
LH
±3/2
±1/2
HH
No LH-HH Coupling
Winkler, PRB 62, 4245 (2000)
DB & Loss, PRL 95, 076805 (2005)
Luttinger, PR 102, 1030 (1956)
HQD =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+V (rP) + HRSO + HDSO + HZ ,
HRSO = a R JxPy - JyPx( ) =ia R
2
0 0 3P- 0
0 0 0 - 3P+
- 3P+ 0 0 2P-
0 3P- - 2P+ 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HDSO = bD - JxPx + JyPy( ) = -bD
2
0 0 3P+ 0
0 0 0 3P-
3P- 0 0 2P+
0 3P+ 2P- 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HZ = - gmBJ ◊B = -gmB
2
3Bz 0 3B- 0
0 - 3Bz 0 - 3B+
3B+ 0 Bz B-
0 3B- B+ - Bz
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
.
Spin-Orbit Coupling: QDSpin-Orbit Coupling: QD
E
LH
±3/2
±1/2
HH
No LH-HH Coupling
No SO Coupling of HHs
Winkler, PRB 62, 4245 (2000)
Luttinger, PR 102, 1030 (1956)
Spin-Orbit Coupling: QDSpin-Orbit Coupling: QD
HQD =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+V (rP) + HRSO + HDSO + HZ ,
HRSO =ia R
2
0 0 3P- 0
0 0 0 - 3P+
- 3P+ 0 0 2P-
0 3P- - 2P+ 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HDSO = -bD
2
0 0 3P+ 0
0 0 0 3P-
3P- 0 0 2P+
0 3P+ 2P- 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HZ = -gmB
2
3Bz 0 3B- 0
0 - 3Bz 0 - 3B+
3B+ 0 Bz B-
0 3B- B+ - Bz
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
.
E
LH
±3/2
±1/2
HH
Spin-Orbit Coupling: QDSpin-Orbit Coupling: QD
HQD =
Ghh 0 0 I
0 Ghh I * 0
0 I Glh - D 0
I * 0 0 Glh - D
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
+V (rP) + HRSO + HDSO + HZ ,
HRSO =ia R
2
0 0 3P- 0
0 0 0 - 3P+
- 3P+ 0 0 2P-
0 3P- - 2P+ 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HDSO = -bD
2
0 0 3P+ 0
0 0 0 3P-
3P- 0 0 2P+
0 3P+ 2P- 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
,
HZ = -gmB
2
3Bz 0 3B- 0
0 - 3Bz 0 - 3B+
3B+ 0 Bz B-
0 3B- B+ - Bz
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜
.
E
LH
±3/2
±1/2
HH
Hhh
eff ª Hhh +V +V
D
SO due to BII-field
Dresselhaus SO coupling
Rashba SO coupling
H = H0 + HSO ,
H0 =P2
2m+
mw02
2x2 + y2( ) -
1
2g^ mBBzs z ,
HSO = ia s + P-3 - s - P+
3( )
- b s + P- P+ P- + s - P+ P- P+( )
+3ggmB
Ds + B- P-
2 + s - B+ P+2( ).
R. Winkler, PRB 62, 4245 (2000)
Effective Hamiltonian of Heavy HolesEffective Hamiltonian of Heavy Holes
DB & D. Loss, PRL 95, 076805 (2006)
DB & D. Loss, cond-mat/0608410
Energy Level StructureEnergy Level Structure
0 Ø
0 ≠
1 Ø
2 Ø
3 Ø
DSO
SO(BII)
RSO
Spin Evolution. No DampingSpin Evolution. No Damping
QuickTime™ and aJPEG 2000 decompressor
are needed to see this picture.
B0
≠
Ø
S zS x
zt
zt
Spin Evolution. DampingSpin Evolution. Damping
B0
≠
Ø
exp[- t / T1]
exp[- t / T2 ]
S zS x
zt
zt
Spin Relaxation and DecoherenceSpin Relaxation and Decoherence
DSO SO(BII) RSO
h = 5 nm; l = 40 nm; Low T (kBT = DE), T2 = 2T1
T1 ª ms
DB & D. Loss, cond-mat/0608410
Spin Relaxation and DecoherenceSpin Relaxation and Decoherence
DSO SO(BII) RSO
h = 5 nm; l = 40 nm; Low T (kBT = DE), T2 = 2T1
S.Sasaki et al., PRL 95, 056803 (2005)
DB & D. Loss, cond-mat/0608410
GaAs Quantum Dot (g = 2.5) InAs Quantum Dot (g = -2.2)
l = 30 nm, h = 5 nm, T = 0.1 K, BP = 0.
Spin Relaxation: Electrons vs. HolesSpin Relaxation: Electrons vs. HolesDB & D. Loss, PRL 95, 076805 (2006)
Thh
Tel
ª16
9
gel
ghh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4mel
mhh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4l
h
Ê
ËÁÁÁ
ˆ
¯˜̃˜
4D so
2
(Eg + D so )2 (low B, hwZ < < kBT).
GaAs Quantum Dot (g = 2.5) InAs Quantum Dot (g = -2.2)
l = 30 nm, h = 5 nm, T = 0.1 K, BP = 0.
Spin Relaxation: Electrons vs. HolesSpin Relaxation: Electrons vs. HolesDB & D. Loss, PRL 95, 076805 (2006)
Thh
Tel
ª16
9
gel
ghh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4mel
mhh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4l
h
Ê
ËÁÁÁ
ˆ
¯˜̃˜
4D so
2
(Eg + D so )2 (low B, hwZ < < kBT).
Abstreiter Group, ICPS Conference (July 2006)
T1 : 0.3 ms
Spin Relaxation due LH-HH SOISpin Relaxation due LH-HH SOI
Other TheoriesWoods, Reinecke, & Kotlyar, PRB 69, 125330 (2004)Lü, Cheng, & Wu, PRB 71, 076308 (2005)
T1(B = 1 T) : 10 ns
T1 µ B- 9 l
h
Ê
ËÁÁÁ
ˆ
¯˜̃˜
8
5 orders of magnitude longer than that due to RSO, DSO, and SO(BII)
Spin ManipulationSpin Manipulation
Heavy-Hole Spin as Qubit:
weak hyperfine interactions with nuclear spins
strong spin-orbit coupling Long T1, T2
difficult to manipulate the spin
weak
rf
ESR. Rabi OscillationsESR. Rabi Oscillations
B0
≠
Ø B1
rf = z — Spin ResonanceS z
z
Rt
ESR. Rabi OscillationsESR. Rabi Oscillations
B0
≠
Ø B1
Koppens et al., Nature 442, 766 (2006)Engel & Loss, PRL 86, 4648 (2001)
S z
rf
z
Rt
Electric Dipole Spin ResonanceElectric Dipole Spin Resonance
+ S ◊B(t) - = 0 No magnetic-dipoletransitions!!!
DB & D. Loss, cond-mat/0608410
Electric Dipole Spin ResonanceElectric Dipole Spin Resonance
0 Ø
0 ≠
1Ø 2 Ø 3 Ø
DSO
SO(BII)
RSO
+ = 0 ≠ + ib + 1 Ø + g+ B+ 2 Ø + a + 3 Ø ,
- = 0 Ø + ib - 1≠ + g - B- 2 ≠ + a - 3 ≠
DSO SO(BII) RSO
+ S ◊B(t) - = 0 No magnetic-dipoletransitions!!!
Electric-dipoletransitions!!!
DB & D. Loss, cond-mat/0608410
Electric Dipole Spin ResonanceElectric Dipole Spin Resonance
E(t) = E(sin wt,- coswt,0),
dSO =b | e | mhw0
2
w(w02 + wc
2 / 4)
w-2
w- - wZ
+w+
2
w+ + wZ
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃,
H E (t) = - dSO ◊E(t),
w± = w02 + wc
2 / 4 ± wc / 4
&r z = 2(dSOE / h )r - - (r z - r zT ) / T1,
&r + = (wZ - w)r - - r + / T2 ,
&r - = - (wZ - w)r - - 2(dSOE / h )r + - r - / T2 ,
Bloch equations:
0 dSO=0 (min)
0 dSO=|e| l Z / 2(max)
Electric Dipole Spin ResonanceElectric Dipole Spin Resonance
P = - d H E (t) / dt =
2w(dSOE)2T2r zT / h
1+ (wZ - w)2T22 + (2dSOE / h )2T1T2
Power:
Electric Dipole Spin ResonanceElectric Dipole Spin Resonance
P = - d H E (t) / dt =
2w(dSOE)2T2r zT / h
1+ (wZ - w)2T22 + (2dSOE / h )2T1T2
Power:
B^r ,1 = hw / g^ mB
B^
r ,2 =hw0
g^ mB 1+ 2m0 / g^ m
B^
d =hw0
2g^ mB 2m0 / g^ m
B^
r ,3 =4hw0
g^ mB 1+ 4m0 / g^ m
Rabi OscillationsRabi Oscillations
wR = (dSOE / h )2 - (T1- 1 - T2
- 1) / 4 (Rabi frequency)
B = 0.5 TB =0.8 TB = 0.865 T
SummarySummary
Spin-orbit effect suppressed for flat QDs
T2 = 2T1 at low temperatures
Spin relaxation time T1 can be milliseconds
Coherent spin manipulation by RF electric fields
Strong control of Rabi oscillations
Heavy holes in quantum dots:
Quantum Error CorrectionsQuantum Error Corrections
wR = (dSOE / h )2 - (T1- 1 - T2
- 1) / 4 (Rabi frequency)
QEC threshold wRT2 ≥ 104
scalable QC
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