Optically controlling quantum dots for
quantum computing
Dan Gammon
Naval Research Laboratory
Washington DC
σ-
Collaborations
Naval Research Lab
Allan Bracker
Alexander Efros
Tom Kennedy
Morgan Ware (NRC postdoc)
Eric Stinaff (NRC postdoc)
Andre Shabaev (NRC postdoc)
Alex Ukhanov (NRC postdoc)
Duncan Steel (U. Michigan)
Vladimir Korenev (Ioffe inst.)
Igor Merkulov
David Gershoni (Technion U.)
Lu Sham (UCSD)
Contents
•
Exciton as qubit
•
Spin as qubit (from the optical point of view)
•
Hyperfine interactions and nuclear spin
Single QD Spectroscopy
Single Qdot
Hole
Luminescence Intensity
0.2 µm
•
•
•
10 Qdots
Apertures in Al mask directly on
sample
Diameters: 200 nm - 25 um
Backscattering or Transmission
1.5 µm
11
105
Qdots
10 mono.
25 µm
1.67
1.68
1.69
Energy (eV)
1.70
AlGaAs
GaAs
AlGaAs
~3 nm
z
x
10-100 nm
Fine Structure Splittings
•
Doublets: ∆ = 10 - 100 µeV
Spin degeneracy
•
Linear Polarization
along [110] axes
STM of GaAs
[110]
[110]
100 nm
[Gammon, et al., PRL 76, 3005 (1996)]
Exciton Spin Structure
|+2> |+1> |−1> |−2>
|−1/2>
|+1/2>
σ−
|−3/2>
1 exciton
states
σ+
Change basis
|+3/2>
Spin interactions lift the degeneracies:
1. Exchange
2. Zeeman
3. Hyperfine
vacuum
δb
δ0
x
y
bright
dark
1 exciton
states
δd
vacuum
Spectral and spatial addressing of QDs
[Qiang Wu, et al., PRB 62, 13022 (2000)]
Quantum Computing
•
•
Quantum bit
Coherence time
•
Probe and control individual qubit
– initialize
– “rotate”
– measure
•
Entangle and operate on two qubits
•
Scalability
|1>
|0>
E1
E0
Crystal
GS
Exciton qubit
1
Ground state
|0>
0
C0
Ψ = c0 0 + c1 1
•
•
Gammon and Steel, Physics Today 55(10), 36 (2002)
Xiaoqin Li, et al., Optic & Photonic News (2004) in press
Exciton
|1>
+
C1
Homogeneous:
– No inhomogeneous broadening
– Fully resolved (fine structure?)
– No line wandering (spectral diffusion)
=> Linewidth is determined by coherence time (T2)
Γhom =
1
T2
=
1
+ γ dep
2T1
Energy relaxation
“inelastic”
E1
-100 0 100
∆E (µeV)
E1
E0
Crystal
GS
Pure dephasing
“elastic”
Γhom~23 µeV <=> T2~60 ps
PL Intensity
Linewidths, Dephasing, and Coherence
•
•
•
Gammon, et al., Science 273, 87 (1996)
Bonadeo, et al., PRL 81, 2759 (1998)
Guest et al PRB 65, 24131 (2002)
Coherent Control
Y
τ
ε (t+ τ )
E1
X
ε (t )
•
Excite QD with pair of coherent
pulses (5 ps pulsewidth) as a
function of delay (τ)
1
2
E0
ε(t)
PL
GS
E
0
τc=40 ps
1
E
1
E
PL Int.
Luminescence intensity (a.u.)
Laser
2
0.5
0
0
1622
1624
1626
1628
1630
2
4
τf (fs)
6
Energy (meV)
Bonadeo, et al., Science 282, 1473 (1998)
Decoherence
E
1Y
PL oscillations decay
because of dephasing
Γ
34 µeV
-100
Tc=0 ps
PL Intensity
•
0
∆E (µeV)
100
1
T2=40 ps
Tc=20 ps
0.5
Tc=40 ps
τ
Y
Tc=60 ps
0
0.8
Tc= 80 ps
1.6 2.4 3.2
4
τf (fs)
4.8
5.6
X
ε2(t+τ) ε1(t)
0
6.4
0
20
40
60
80
Delay time (ps)
100
120
Quantum Superposition of Spin States
•
Rotate polarization to create
superposition of x and y states
1
E1X
E1Y
Y
E0
εX(t) εY(t)
X
Tosc= 69 ps
0.8
hω 0
GS
0.6
∆ (60 µeV)
E
-100
1Y
E
0
1X
∆E (µeV)
0.4
0.2
100
0
0
20
40
60
80
Time delay (ps)
100
120
3-level coherence
Ground state
|00>
C00
+
Exciton
|01>
Exciton
|10>
C10
Ψ = c0 0 1 0
2
+
+ c1 1 1 0
2
C01
+ c2 0 1 1
2
0 11
2
110
2
010
2
Biexciton
•
2 excitons
1 exciton
0 excitons
Biexciton state is shifted down by
~4 meV
[Brunner et al., PRL 73, 1138 (1994)]
[Gang Chen et al., PRL 88,117901 (2002)]
Biexciton
Ground state
|00>
C00
+
Exciton
|01>
Exciton
|10>
C10
+
C01
Biexciton
|11>
+
C11
111 2
0 11
2
110
2
010
2
If no biexciton shift, then no entanglement: the excitons factor!
Ψ = c00 0 1 0 2 + c10 1 1 0 2 + c01 0 1 1 2 + c11 1 1 1 2
= (c0 0 1 + c1 1 1 )(c0 0 2 + c1 1 2 )
For a short discussion along these lines, see; Gammon, et al. Physica E 9, 99 (2001).
•For a review; Gang Chen et al., Coherence in Semiconductor Q.Wires and QDs, ed. T. Takagahara (Gordon and Breach
2003). Chapter 9
Qubit Rotations
Ground State Ù Exciton Rabi Oscillations
Approach
τ
• Laser pulse (4 ps) to pump
• Probe population through
differential transmission
• Rabi Oscillations of single
exciton
• 4 ps Π pulse
=> fast operations
ε (t+ τ )
Differential transmission
Results
Y
2
2π
4π
Pulse area
Stievater et al. PRL 87, 133603 (2001).
ε (t )
1
See also:
Kamada et al. PRL (2001)
Htoon et al., PRL (2001)
Zrenner et al., Nature (2002)
X
Rabi Oscillations
Ground State Ù Exciton Rabi Oscillations
2π
4π
Pulse area
Stievater et al., PRL 87, 133603 (2001).
Exciton Ù Biexciton Rabi Oscillations
0π
π (a.u.)
22π
E Field
Pulse area
Elaine Wu et al., Science 301, 809 (2003).
A QNOT gate (exciton)
QNOT Gate
• Coherent control of two interacting
excitons provides an example of a
QNOT gate
Control bit
Target bit
00 → 00
01 → 01
10 → 11
π pulse
|11>
|10>
|01>
|00>
11 → 10
Qubit: exciton
Exciton decoherence / π pulse
|x>
T2 ~ 60 ps
1 ns
τpulse ~ 4 ps
100 fs
T2/τpulse ~ 10 pulses
|0>
104
Use Spin!?
DT (arb units)
Exciton
1.615
Laser
Biexciton
1.616
1.617
1.618
1.619
Energy (eV)
1.620
1.621
Contents
•
Exciton as qubit
•
Spin as qubit (from the optical point of view)
•
Hyperfine interactions and nuclear spin
A spin qubit
qubit: spin
electron
trion
|i>
trion
trion
electron
|0>
|1>
electron
Luminescence from single dot in a diode
AlGaAs
GaAs
QDs
n+-GaAs
N+- I diode with QDs
PL Energy (eV)
X+
X°
1.656
1.654
X-
X+
-
X
1.652
XX°
1.650
XX-
XX+
-1
0
X°
1
Diode Bias (V)
2
The quantum toolbox
Initialization (Optical Pumping)
Rotations (coherent Raman)
A. Shabaev, et al. PRB 68, 201305 (2003).
P. Chen et al., PRB 69, 075320 (2004).
Quantum Toolbox
Entanglement
Measurement (recycling transitions)
J1σ1.σc
Immamoglu et al PRL 83, 4204 (1999)
Piermarcchi et al., PRL 89, 167402 (2002)
Troiani et al., PRL 90, 206802 (2003)
Pazy et al., Europhysics Lett.62, 175 (2003)
Shabaev et al., PRB (2003)
Calarco et al., PRA 68, 12310 (2003)
Selection Rules
-1
+1
+2
-2
σ-
Exchange
Splitting ~100 µeV
σ+
−
3
2
3
2
σ-
σ+
forbidden
0
allowed
−
1
2
1
2
Exciton (X0) vs. Trion (X-) in B-field
E
E
X°
ge - gh
|0⟩
Bz
Voigt
Faraday
BZ
Y
X
σ+
Y
s
σ
ge
s
+
Bz
Bx
Voigt
Bx
Faraday
BZ
Y
X
X
gh
e-
ge
Bx
Bx
X-
σ+
Y
X
s
σ+
[Tischler, et al., PRB 66, 08131 (2002)]
[M. Bayer, et al., PRB 65, 195315 (2002)]
s
Optical Orientation
For GaAs quantum dots:
Nonresonant excitation into QW
~30 meV above QD energy
Fast hole spin flip, Slow electron spin flip
X
σ+
QD
-
σ-
PL polarization of X°, X+, XPL Energy (meV)
1
X°
0
PL polarization
-1
+
-2
I −I
P= +
−
I +I
-
X
-3
X+
-4
Polarization (%)
60
X+
40
X-
X+
Large positive
(long e-spin lifetime)
X°
Zero or small positive at Bz=o
(anisotropic exchange)
X-
Positive or negative,
depends on conditions
20
0
X°
-20
3
4
Diode Bias (V)
−
5
Optical pumping of electron spin
X- polarization:
1. h-spin orientation
2. Ground state e-spin pumping
bright
σ+
X- Polarization (%)
100 W/cm2
10
0
1500 W/cm2
-10
dark
σ-
-20
4.0
Need fast hole flip rate
compared to electron spin
flip rate
4.2
4.4
4.6
4.8
5.0
Diode Bias (V)
Previous work:
Dzhioev et al., Phys. Solid State, 40 1587 (1998)
Cortez et al., PRL 89, 207401 (2002)
The Hanle effect and electron spin lifetimes
geµBBx
E
Z
Polarization ρz (%)
20
BX
10
X+
Y
σ+ S
e
e
Ts−1
X°
Bx
0
Polarization:
-10
ρ z ( B) = ρ z (0)
X-
-20
-0.5
0.0
BX
0.5
(Tesla)
Halfwidth:
1
1 + ( B B1 2 ) 2
(
)
B1 2 = h µ B g e Ts−1
Hanle Halfwidth (G)
ge~0.2
X° : No polarization
X+ : B1/2 = 3.5kG ⇒ TS ≈ 160 psec
X- : B1/2 = 20G ⇒ TS ≈ 30 nsec
X-
100
50
0
0
1
2
3
4
Power (mW)
5
Bracker et al., cond-mat/0408466
Quantum Beats
Xσ+
Charged QD
σ+
Bx
Decay ~ 1.5 nsec
DT (a.u.)
Single Electron!
0
200
Delay(ps)
400
600
Decay ~ 260 psec
X0
Decay ~ 66 psec
Bz
πx
Neutral QD
πx
Ensemble Spin C oherence @ 0.9 T
Ensemble Spin C oherence @ 6.6 T
Single QD X C oherence @ 0.8 T
0
Gurudev Dutt et al. subm.
500
1000
1500
D elay (psec)
2000
2500
Measurement
trion
trion
|i>
trion
electron
electron
electron
|0>
|1>
Contents
•
Exciton as qubit
•
Spin as qubit (from the optical point of view)
•
Hyperfine interactions and nuclear spin
s++
σ
Nuclear spin
Nuclear spin => effective magnetic field
•
Average => Overhauser shifts
•
Fluctuations => spin dephasing
h
e
•Brown et al. PRB 54, R17339 (1996)
•Gammon et al., Science 277, 85 (1997)
•Gammon et al., PRL 86, 5176 (2001)
•For a review see; Gammon et al., Coherence in
Semiconductor Q.Wires and QDs, ed. T. Takagahara
(Gordon and Breach 2003) Chap. 8.
•Bracker et al., cond-mat/0408466
Nuclear spin fluctuations
Electrons dephase in the effective magnetic field
produced by a net fluctuation of the nuclear spin
Merkulov et al., PRB 66 153409 (2002)
Khaetskii et al., PRL 88, 186802 (2002)
Dzhioev et al., PRB 65 205309 (2002)
δBeff
Inhomogeneous dot-to-dot variation OR
Long-time variation in a single dot
(spectral diffusion)
Ts ≈ h N / A
N ≈ 105 atoms
A=90 µeV (hyperfine constant in GaAs)
expect: Ts ≈ 6 ns
measure: 4-30 ns
Pumping of nuclei; Overhauser effect
Oriented
e-h pair
+
Hyperfine Interaction
(electron-nuclear spins)
mutiple
cycles
Overhauser effect
(hyperfine)
BN
Bext
Effective
B-field
σ+
Electron spin splitting
depends on laser pol.
Key Point:
BN affects only electron spin
• Not hole spin (p-like)!
• Not orbital (i.e. no diamagnetic shift)!
Observed:
Max. Overhauser shift
∆EN= (∆Eσ +-∆Eσ -)/2 = 81 µeV
Nuclear Polarization
PN = ∆EN / (3/2)*Ahf = 60%
Effective B-Field
BN = ∆EN / 2geµB = 14 Tesla
X°, X+, X- in longitudinal magnetic field
σσ+ PL polarization:
™ Spin splittings vary with laser polarization:
PL Intensity (a.u.)
X°
→ signature of optical pumping of nuclear spins
Bext = 5T
σ-
σ
Laser: +
σ
1.6560
1.6565
X
PL Intensity (a.u.)
40
BZ
-
Y
Praw (%)
1.6555
60
1.6530
PL Intensity (a.u.)
X+
σ+
s
Spin splitting (meV)
1.6525
πx σ
-
B
X° 3.9 V
X- 4.1 V
X+ 2.5 V
X+
20
X°
0
XC
0.6
X
0.5
Amplitude:
Polarization
“memory”
Offset:
Thermal
polarization
-20
-40
X
1.6520
π y σ+
-
X°
Amplitude:
Overhauser shift
X+
Offset:
Intrinsic splitting
0.4
0.3
0.25 0.50 0.75 1.00 1.25
1.6530
1.6535
1.6540
PL Energy (eV)
Retardance (λ)
Bias dependence of polarization, Bz=5T
-1
-2
-3
-4
XX+
By increasing the bias and injecting
unpol. e’s:
• X- polarization increases
• Nuclear polarization decreases
Electron polarization
is the link !
X°
40
X+
20
X-
0
-20
0
C
0
X°
20
40
X-
X+
3
-40
4
Diode Bias (V)
Allan Bracker et al., cond-mat/0408466
-20
5
Nuclear pol. (%)
X°
0
Polarization (%)
60
Overhauser shift (µeV)
PL Energy (meV)
1
Future Directions
•
Beyond the perturbative regime
– π pulses, etc.
•
Ground state (spin) coherences
– Λ vs. V transitions
•
Decoherence physics
•
QD molecules and more complex structures
– entanglement
•
Quantum computing?