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Two electron spin qubits in Two
electron spin qubits in
GaAs quantum dots
Hendrik Bluhm
Harvard Universityy
Experimental results presented mostly reflect work in the Yacoby and Marcus groups at Harvard.
Quantum computing – the goal
Principles of quantum mechanics ⇒ Built‐in parallelism ⇒ Exponential speedup p
p
p ((for some problems)
p
)
Classical bits
Classical bits
0 or 1
N bits => 2N
states 0, 1, …, 2N‐1
Quantum bits
Quantum
bits
α|0〉 + β |1〉
N qubits: 2N dimensional Hilbert space
|0〉 |1〉
|0〉, |1〉, …, |2
|2N‐1〉
1〉
2
The case for spin qubits
Quantum computing needs two level systems
p
g
y
⇒Spins natural choice
Compatible with semiconductor technology
ibl i h
i d
h l
⇒ Potential for scalability
Why not charge?
• Charge couples to phonons, photons, Charge couples to phonons, photons,
other charges, cell phones, …
Now: Intel Pentium i7‐980X
Future: Quantum i√2
• Spins are very weakly coupled to other things
ee.g.: Electric vs. magnetic dipole transitions
g : Electric vs magnetic dipole transitions
(Reason: lack of a magnetic monopole)
Reason for weak coupling
• Time reversal symmetry enforces degeneracy at B = 0
(
(Kramer’s doublets) )
=> no dephasing from electric fields
• Matrix elements for decoherence cancel to lowest order
(Van Vleck cancellation)
Decoherence times (bulk)
• P‐ donor electrons in 28Si: T2 = 600 ms
Tyryshkin et al., (unpublished ?)
• 29Si nuclei in purified 28Si: T2 = 25 s at RT Ladd et al., PRB 71, 014401 (2005)
Problem: Single spins difficult to control
Two electron spin qubits
Idea: use two spins for one qubit
p
q
⇒ Electrically controllable exchange interaction
Electrically controllable exchange interaction
• Tunable electric coupling
Fast convenient manipulation
• Fast, convenient manipulation
• Relies on same techniques as single‐spin GaAs qubits in quantum dots (Lars Schreiber)
qubits in quantum dots (Lars Schreiber)
Longest coherence time of all electrically
ongest coherence time of all electrically
controllable solid state qubits.
Outline
Lecture I
• Motivation
• Encoded qubits
• Physical realization in double quantum dots
• Principles of qubit operation
• Single shot readout
Lecture II
• Decoherence
• Hyperfine interaction with nuclear spins
• Recent progress on extending coherence
Outline
Motivation
Encoded qubits
Encoded qubits
Ph i l
Physical realization in double quantum dots
li ti i d bl
t
d t
Principles of qubit operation
Single shot readout
Requirements for qubits
DiVincenzo Criteria for a viable qubit
1. Well‐defined qubit
2. Initialization
3. Universal gates
4. Readout
5. Coherence
Encoded qubits
• Qubit = coherent two level system => single spin ½ most natural qubit
> single spin ½ most natural qubit
1
0
• Any
Any 2D subspace of a quantum system can 2D subspace of a quantum system can
serve as a qubit.
Qubit
subspace
Ad t
Advantages
+ Wider choice of physical qubits
+ Decoherence “free” subspace – choose states that are decoupled from certain perturbations
+ Reduced control requirements – choose subspace with convenient knobs.
Caveats: ‐ Leakage out of logical subspace can cause additional errors.
k
fl i l b
ddi i
l
‐ More complex control sequences. S‐T0 qubit using two spins
Idea: Encode logical qubit in two spins
All i t t
All spin states:
↑↓
↓↑
↑↑
↓↓
(
)
(
)
1
S =
↓↑ − ↑↓
2
1
T0 =
↓↑ + ↑↓
2
T+ = ↑↑ , T− = ↓↓
m = 0 logical
logical subspace
m = ±1
m ±1
Decoherence “free” subspace (DFS)
m = 0 for both logical states ⇒ no coupling to homogeneous magnetic field
⇒ insensitive to fluctuations
i
iti t fl t ti
Simplified operation
p
p
Use exchange coupling between two spins => no need for single spin rotations.
Theoretical proposal: J. Levy, PRL 89, 147902 (2002)
Bloch Sphere
1
Ψ =α 0 + β 1
0 −i1
2
0 + 1
0 −1
2
2
0 +i1
Mixed states are statistical mixtures of pure states and
can be inside the Bl h h
Bloch sphere.
1 1
ρ
2
0
• Any pure state of a qubit corresponds to a point on the surface of a sphere.
• They can be identified with the direction of a spin ½.
0 0
ρ = 1/ 4 0 0 + 3 / 4 0 0
Single qubit operations
• Unitary transformations are rotations on the Bloch sphere
• Universal quantum computing requires arbitrary rotations, which can be composed from rotations around two different axis.
1
r
ω
ωz
ωx
0
1 ⎛ ωz
H = ∑ ωiσˆ i = ⎜⎜
2 ⎝ ω x − iω y
i= x, y, z
ω x + iω y ⎞
⎟
ω z ⎟⎠
Standard Rabi control
• Modulate ωx resonant with ωz.
(e g AC magnetic field for spins)
(e.g. AC magnetic field for spins)
• Changing phase of AC signal changes rotation axis in the rotating frame.
g
Gate operations
gμ B ΔBz
1) In field gradient: H =
σz
2
=> and acquire relative phase ↑↓
↑↓
B2
B1
ΔBz = B1 – B2
2) Exchange: H =
J
J
s1 ⋅ s1 = σ x
2
2
↑↓
=> mixing between and
J
↑↓
↑↓
ΔBz
T0
J
↓↑
S
Single spin vs. S‐T0
Single spin qubit
Two‐spin encoded qubit
↑
↑↓
ΔBz
Bz
Bx
↓
• Typically uses resonant modulation of Bx.
• Bx can be an effective field
(e.g. spin‐orbit).
T0
J
S
↓↑
Typically relies on switching of J
Two‐qubit gates
SWAP
• Quantum computing requires (at least) one entangling gate between two (or more)
entangling gate between two (or more) SWAP
qubits (cNOT, cPHASE, ).
↑↓
• Single spins: π/2 exchange provides SWAP
J
↑↓ + i ↑↓
2
• Encoded qubits:
Encoded qubits: construct gates construct gates
from several steps.
• S‐T
S T0: Construction of nAND
C t ti
f AND gate, t
equivalent to cNOT, cPHASE
• In practice, can also use Coulomb interaction In practice can also use Coulomb interaction
to implement cPHASE gate directly.
↓↑
nAND gate for S‐T0 qubit
Evolve in field gradient (π/2)
Evolve in field SWAP inner spins (exchange) gradient (π/2)
Spin 1A
B1
B1
Spin 1B
p
B2
B2
Spin B1
B1
B1
Spin B2
Spin B2
B2
B2
SWAP inner spins
Qubit A
Qubit B
Principle of operation:
0 1 = ↓↑ ↑↓
0 0 = ↓↑ ↓↑
Initial state
Acquire phase
↓↑ ↑↓
Acquires phase
↓↑ ↑↓
↓↓ ↑↑
No phase acquired
p
q
↓↑ ↓↑
Outside logical subspace!
Return to subspace
Exchange‐only with three spins ½
Idea: use m = ½ subspace.
J1(t)
J2(t)
Single qubit:
4 steps
Two qubit:
Two qubit:
27 steps
• No magnetic field required.
• Uses only exchange.
Uses only exchange
DiVincenzo et al. Nature 408, p. 339 (2000)
• Experimental status: Suitable samples developed, E
i
l
S i bl
l d l
d
but no coherent control yet. (Gaudreau et al. arxiv)
J1(t)
J2(t)
Tradeoffs summary
Encoding a qubit in several spins reduces control requirements at the expense of complexity.
q
p
p
y
Spins/qubit
1
2
Static control requirement
Magnetic field Magnetic field None
difference
AC control AC
control
requirement
(effective) (effective)
Exchange
transverse magnetic field
Exchange
Mechanism M
h i
for 2‐qubit gate
EExchange (or h
(
dipolar)
EExchange
h
(or Coulomb)
E h
Exchange
# of steps in 2‐qubit gate
1
3‐6
19
(experimentally most difficult step in red)
(experimentally most difficult step in red)
3
Outline
Motivation
Encoded qubits
Encoded qubits
Physical realization in double quantum dots
Ph
i l
li ti i d bl
t
d t
Principles of qubit
p
q
operation
p
• Theory of operation
• Experimental procedures
Experimental procedures
Si l h
Single shot readout
d
2D‐electron gas (2DEG)
Wafer surface
GaAs heterostructure
conduction band edge
Dopants induce
electric field
Step at material interface
• Structure grown layer by layer with Molecular Beam Epitaxy
p y ((MBE))
• Atomically smooth transitions
• Ultra‐high purity
Electrons in triangular confining potential occupy lowest subband.
Device fabrication
Fabrication
‐
+
Negative gate voltage pushes electrons away.
2DEG
500 nm
Metal gaate
Goal:
trap two electrons
electrons ‐ +
V
Graphics: Thesis L. Willems van Beveren, TU Delft
Understanding a complex system
Metal gates
90 nm
90 nm
‐
+
+ + + + + + + + + + + +
Dopants, defects and impurities cause disorder
2D electron gas
(Fermi‐sea)
Individual confined electrons
g
Conduction band edge
Electrostatic Electrostatic
potential from gates
First realization and overview of experimental toolbox: Petta et al., Science 309, p. 2180 (2005)
Charge control
E
500 nm
‐ +
‐V
S(0, 2)
Mettal gate
2DEG
‐ +
+V
‐ +
V0
ε<0
0
ε>0
(1, 1)
V(x)
ε
(0, 2)
x
ε = E(1, 1)−E(0, 2) ∝ V
-V
+V
Charge control
E
500 nm
‐ +
V
‐ +
V
ε<0
((1, 1))
((0, 2))
0 ⎞
⎛ε / 2
⎟⎟
H = ⎜⎜
− ε / 2⎠
⎝ 0
0
ε>0
(1, 1)
V(x)
ε
(0, 2)
x
ε = E(1, 1)−E(0, 2) ∝ V
-V
+V
Singlet‐Triplet splitting in (0,2)
First excited state Ψ0 E
S‐T splitt.
Ground state Ψ0
0
(1, 1)
S(0, 2)
T(0, 2)
0
0
⎛ε / 2
⎞
⎜
⎟
H =⎜ 0
−ε / 2
0
⎟
⎜ 0
⎟
0
−
ε
/
2
+
δ
⎝
⎠
ε
(0, 2) states:
Spin singlet: Ψ(x1, x2) = Ψ0(x1) Ψ0(x2)|S>
Spin triplet:
Ψ(x1, x2) = (Ψ0(x1) Ψ1(x2)‐Ψ0(x2) Ψ1(x1))|T>
⇒(0, 2) Triplet has higher energy than (0, 2) Singlet.
Tunnel coupling
E
S(0, 2)
Tunnel coupling
J(ε)
T(1, 1)
S(1, 1)
S(0, 2)
0 ⎞
⎛ε / 2 0
⎜
⎟
H =⎜ 0 ε /2
tc ⎟
⎜ 0
⎟
t
−
ε
/
2
c
⎝
⎠
T
‐>
S
( ) 0⎞
⎛ J (t
⎜⎜
⎟⎟
⎝ 0 0⎠
0
ε
Tunnel coupling
⇒Avoided crossing for singlet
⇒Avoided crossing for singlet
Triplet crossing at larger ε can be ignored. Conveniently described in terms of J(ε)
Zeeman splitting
T0
(
1
↓↑ − ↑↓
2
1
=
↓↑ + ↑↓
2
S =
(
)
)
E
S(0, 2)
m = 0
T+ = ↑↑
m = 1
T− = ↓↓
m = ‐1
Ez = g μB Bext
0
H Z = g * μ B Bz Sˆ z
Bz ~ 10 mT to 1 T
ε
Qubit states
S =
(
)
T0
(
)
1
↓↑ − ↑↓
2
1
=
↓↑ + ↑↓
2
T+ = ↑↑ , T− = ↓↓
E
S(0, 2)
Ez = g μB Bext
↑↓
T0
S
↓↑
0
ε
Qubit dynamics with field gradients E
S(0, 2)
Transitions between S and T+ driven by ΔB⊥.
J(ε)
0
ε << 0: Free precession
Bext±ΔB
Δ z/2
/
↑↓
ε
ΔBz
ε ~< 0: Coherent exchange
T0
J
↓↑
S
Effective Hamiltonians
T0
S
⎛ J (t ) ΔBz / 2 ⎞
⎟⎟
J , ΔBz << Bext : H = ⎜⎜
0 ⎠
⎝ ΔBz / 2
In logical subspace:
S
T‐
All spin states:
H=
H =
Coish and Loss, PRB 72, 125337
T0
T+
Outline
Motivation
Encoded qubits
Encoded qubits
Physical realization in double quantum dots
Ph
i l
li ti i d bl
t
d t
Principles of qubit
p
q
operation
p
• Theory of operation
• Experimental procedures
Experimental procedures
Si l h
Single shot readout
d
Isolating two electrons
# electrons in each dot
G
(1, 0)
(nL, n
nR)=(1, 1)
)=(1 1)
VL
Gqpc
Gqpc
‐ +
VL
‐ +
VR
(0 0)
(0, 0)
VR
(1, 1)
V(x)
(0, 1)
10 mV
Conductance depends on electric field from electrons
l
i fi ld f
l
(0, 2)
(0, 2)
2 mV
2 mV
(1, 2)
VL
(1, 1)
x
VL
VR
VL
VR G
qpc
(0, 2)
(0, 1)
VR
Tuning the tunnel coupling
Measure current through double dot
10
VL
VSD =0.4 mV 0
I
20
2 mV
VGateR
Isd (pA)
VL
Gqpc
VL
‐ +
VL
(1, 2)
(1, 1)
(0, 2)
(0, 1)
VR
‐ +
VR
2 mV
VGateR
Gqpc
Magnitude and variation of current and charge signal reveal tunnel couplings.
T
Target: t
t tc ~ 20 μeV
20 V
Tunneling rate to leads ~ 100 MHz
Pulsed Measurements
S
(1, 1)
M
R
(0 2)
(0, 2)
Gqpc
1 ns gate control
Typical pulse cycle for qubit operation
1) Initialize S at reload point R.
(1, 1)
V(x)
(0, 2)
2) M
Manipulate (nearly) separated i l t (
l )
t d
electrons (S)
x
Q
3) Return to M for measurement. Return to M for measurement
Readout
E
S(0, 2)
Goal: distinguish S and T state of separated electrons
state of separated electrons.
Mechanism:
•Increase ε.
•(1, 1)S adiabatically ( , )
y
transitions to (0, 2).
ε
0
Life time long enough to time long enough to
•Life
detect charge signal.
X
S
•TT stays in (1, 1) stays in (1, 1)
(metastable).
T0
Q
Q
Johnson et al., Nature 435, p. 925 (2005)
Readout region and Initialization
S
((1, 1))
Region in which (1, 1)T is long lived (S i Bl k d )
(Spin Blockade)
ε
(1, 1)
M
(0 2)
(0, 2)
R
Outside blocked region, (1, 1) can decay to lead.
can decay to lead. E
(0 2)
(0, 2)
Gqpc
Initialization of S at reload point R after
a measurement:
S(0, 2)
If in (0, 2)S, nothing happens.
(1, 1)T ‐> (0, 1) ‐> (0, 2)S via exchange with leads.
‐
‐
0
ε
Duration ~ 100 ns.
D
ti
100
High fidelity due to large S‐T splitting
Outline
Motivation
Encoded qubits
Encoded qubits
Ph i l
Physical realization in double quantum dots
li ti i d bl
t
d t
Principles of qubit operation
Single shot readout
Single shot readout
For many experiments, can average signal over many pulses.
• No high readout bandwidth required.
g
q
• Reduce noise by long averaging. => Can use standard low‐freq lock‐in measurement with room‐
temperature amplification to measure G
lf
QPC.
Minimum averaging: 30 ms, 3000 pulses.
Single shot readout
Determine qubit state after each single pulse with high fidelity.
Benefits and applications: • Quantum error correction.
• Verify entanglement through correlations and Bell inequalities.
Verify entanglement through correlations and Bell inequalities
• Fundamental studies (e.g. projective measurement)
Fast and accurate data acquisition.
• Fast and accurate data acquisition.
RF‐reflectometry
Demodulation
Goal: increase bandwidth and sensitivity of
charge readout with RF lock‐in technique.
g
q
Reilly et al., APL 91, 162101 (2007)
Reeflected signal
RF components 50 RF
components 50
Ω, sensor 50 kΩ
=> Impedance matching with LC with LC
resonator.
Excitatio
on
Low noise cryogenic y g
amplifier
Single shot readout
Sensor signal
Histogram of cycle‐averages
Reinitialization and manipulation of qubit
=> random new state
Averaging window
Averaging
window
(μs scale)
Need to distinguish state before the metastable triplet can decay (μs scale).
Barthel et al., PRL 103 160503 (2009)
• Each
Each peak corresponds to peak corresponds to
one qubit state.
• Broadening due to (
(amplifier) readout noise.
lifi )
d t i
Improvement with quantum dot sensor
Quantum point contact
Quantum dot
(single electron transistor)
Qubit state modulates single tunnel barrier.
Modulation of ability to add electron to island
Quantum dot
Quantum dot
Factor 3 increase in sensitivity
=> factor 10 reduction in averaging time.
Peaks need to be well separated to distinguish
separated to distinguish states.
Barthel et al., PRB 81 161308(R), 2010
QPC
Readout summary
• Qubit is read out by spin‐to‐charge conversion
utilizing spin blockade.
• State is read using a charge sensor before the metastable (1, 1)T decays.
X
S
T0
Q
• RF reflectometry allows single shot readout
• Fidelity > 90 %
Q
Measuring coherent exchange
(
(gate volt
age)
ε
Exchange pulse
initialize
evolve
readout
(0, 2)
(1, 1)
τ
t
Petta et al., Science 2005
E
↑↓
S
T0
J(ε)
J
S(0, 2)
ε
↓↑
Decay reflects dephasing
Decay
reflects dephasing due due
to electric noise.
Exchange echo
(gate vo
oltage)
ε
↑↓
initialize
evolve
τ/2
readout
(0, 2)
(1, 1)
τ/2 + Δτ
t
π
π
ΔBz
ΔBz ‐ rotation
t ti
Echo signal
T0
S
J
↓↑
T2 = 1.6 μs
τ = 2 μs
Coherence times
x CPMG
• Hahn‐echo
Hahn echo
All data fitted with ~1 nV/Hz1/2 white noise with 3 MHz cutoff. Consistent with expected Johnson noise in DC wires => improvement with filtering.
Outline
Lecture I
• Conceptual and theoretical background
• Physical realization and principles of qubit operation
• Single shot readout
Lecture II
• Decoherence
• Hyperfine interaction with nuclear spins
• Recent progress on extending coherence
Main results
• Used qubit as quantum p
pp
feedback loop to suppress nuclear fluctuations and enhance T2*.
• Detailed picture of bath d
dynamics and decoherence
i
dd h
from echo experiments.
• T2 ≈ 200 μs achieved with q
quantum decoupling.
p g
• Universal control.
Outline
Background
• Error correction
• Decoherence
• Hyperfine interaction
Measuring and manipulating the nuclear hyperfine field
Measuring and manipulating the nuclear hyperfine field
Universal control
Reduction of nuclear fluctuations via 1‐qubit feedback loop
Coherence with echo and dynamic decoupling
Decoherence vs. control – the challenge
• Qubits are analog => small errors matter
small errors matter
• Using phase => Uncertainty relation forbids any leakage of information
However:
• Need to manipulate qubit
Need to manipulate qubit
• Qubits have to interact
• Eventually want to measure qubit
ventually want to measure qubit
⇒ need extremely tight control over interactions.
Impossible? – not quite.
Only need need ~10
102 ‐ 106 coherent operations per error coherent operations per error
“Only”
with quantum error correction.
Threshold theorem
Small enough error probability per gate operation
=> error correction can make QC fault tolerant without Q
exponential overhead.
Basic idea:
Basic
idea:
• Encode logical qubits redundantly in several physical qubits, 〉 | L〉 = ||000〉〉.
e.g. ||1L〉 = ||111〉, |0
• Can detect errors that leave the logical subspace => encoded information is not extracted.
• Correct errors if detected.
Correct errors if detected
Hurdle: Error correction operations will be subject to errors themselves.
Solution: Solution:
• (Error probability) x #(physical gate operations per logical gate) < 1
=> reduce error by hierarchically concatenating error correction codes (i.e. using th l i l bit f l l th h i l bit f th
the logical qubits of on level as the physical qubits of the next higher level).
t hi h l l)
Steane Code
1
( 0000000 + 1010101 + 0110011 + 1100110 + 0001111 + 1011010 + 0111100 + 1101001 )
8
1
( 1111111 + 0101010 + 1001100 + 0011001 + 1110000 + 0100101 + 1000011 + 0010110 )
1L =
8
0L =
Ancilla
qubits
7 physical qubits encoding a logical qubits
(from Nielsen and Chuang)
Measurement Measurement
indicating if and what error occurred.
Decoherence
Decoherence = loss of information stored in a qubit. Classical picture of environment: Fluctuation of Hamiltonian
Quantum mechanical picture: Entanglement with environment
Quantum mechanical picture: Entanglement with environment.
1
Decoherence turns pure states into mixed states
=> Ψ goes into Bloch sphere.
0
Energy relaxation
1
1
E01
• Corresponds to classical bit flip error
• Due to noise at f = E01/h
• Timescale T1
0
0
•Practically not important for spins in GaAs
•Measured T1 in GaAs
•Measured T
in GaAs up to 1 s
up to 1 s (Amasha et al., PRL 100, 046803 (2008)) et al PRL 100 046803 (2008))
Dephasing
= Loss of phase information due to variation of E01.
T2: true decoherence from fast, uncorrelated noise. Needs to be weak enough to enable error correction.
1
T2* : broadening from slow fluctuations b d i f
l fl t ti
(or ensemble measurements). Long temporal correlations
Long temporal correlations help to remove it.
0
Rough measure of error probablility:
Duration of operation/Coherence time.
(
(exact only for exponential decay from Markovian
t l f
ti l d
f
M k i (unstructured) ( t t d)
bath, otherwise misleading.)
Noise sources
Noise limits measurements and causes decoherence
and gate errors.
d t
Local environment
Local
environment
Fluctuating spins (electron, nuclear)
Phonons
Charge traps
Superconducting vortices.
Relevance for GaAs
Relevance
for GaAs spin qubits
spin qubits
Dominant source of decoherence
?
Wafer dependent
None
Electrical noise
Pulse generator voltage sources
Pulse generator, voltage sources
Interference
Johnson noise from resistors
Generally avoidable
Generally
avoidable
(but devil in the details).
Some work to be done. Hyperfine basics
Confined s‐band electron in GaAs
ψ ( x)
50 nm
N ~ 106 nuclei
B
r
r
m ≈ μN I
m=nIA
B = n I / L = m/V
≈ m δ(xj)
2
r
r
2
H = ∫ B ( x) ⋅ s ψ ( x)
r r
2
= A∑ I j ⋅ s ψ ( x j )
j
=∑
r r
Aj I j ⋅ s
j
Electron feels an effective magnetic field. Typical magnitude = A / N1/2 ~ 2 mT.
Typical magnitude = A
2 mT
Fluctuations of this field cause decoherence.
Nuclear dynamics
Flip‐flops: 100 μs
(Dipolar interaction)
Bext
Spin diffusion: 1 s – 1 min
Slow enough for real time probing, manipulation
=> Slow enough for real time probing, manipulation
Larmor precession: 0.1 – 1 μs.
Dephasing : ~100 μs
: 100 μs
Bext
Outline
Background
Measuring and manipulating the nuclear hyperfine field
Universal control
Reduction of nuclear fluctuations via 1‐qubit feedback loop
Coherence with echo and dynamic decoupling
Coherence with echo and dynamic decoupling
Probing ΔBz
↑↓
Bext+Bnuc,
ΔBz
z
T0
⎛ ωτ ⎞
Sensor signal ∝ cos 2 ⎜ S ⎟
⎝ 2 ⎠
S
ω = g * μBΔBz / h
N ~106 nuclei
↓↑
r r
r
ΔB = BL − BR
Q
10 mT
Q (e)
∝ 1/ΔBz
Typical time trace of hyperfine gradient
Typical time trace of hyperfine gradient
ΔBz
Data
0.55 s of data:
Fit
Manipulating Bnuc
E
T+ ‐> S
S(0, 2)
T = ↑↑
ε
T+‐loading
loading
Δmz = ‐1
→ S =
SS‐loading
loading
Δmz = +1
Quantities of interest
• Average polarization of both dots (Petta et al., Reilly et al.)
• Bi‐directional real time control of gradient.
(
1
↓↑ − ↑↓
2
)
Effect of pumping on ΔBz
Apply pump pulses between measurements (typically ~106 cycles)
Real time control of ΔB
Real time control
of ΔBz
S‐loading
pump
T+‐loading
pump
0
500
Time (s)
Steady state when relaxation
when relaxation compensates pumping.
1000
Outline
Background
Summary of device operation
• Measure nuclear field gradient reflected in S‐T0 mixing frequency every second.
• Manipulate gradient by nuclear polarization between measurements.
t
Use of gradient control
Use
of gradient control
• Universal qubit control
• Reduction of nuclear fluctuations by operating Reduction of nuclear fluctuations by operating
qubit as a feedback loop
Coherence with echo and dynamic decoupling
Universal single qubit gates
↑↓
Foletti et al., Nature Physics 5, p. 903 (2009)
E
• Fully electrical
•• Nuclei turned into resource
ΔBz
Nanosecond gate time
tc
• Fast (ns gate times)T
S
J
ΔBz ⎞
⎛ J
J(ε)
• Fully electrical
⎜
⎟
H =⎜
ΔBz
0 ⎟⎠
⎝
• Extrapolated fidelity of 99.99 % at QEC threshold
0
in S , T0 basis.
Adiabatic preparation
↓↑
ε
Evolution
S
S
↓↑
↑↓
↑↓
T0
S(0, 2)
Data
Data
Model
Model
T0
Dephasing due to nuclear fluctuations
Fluctuation of ΔBz over time
Q (ee)
Precession in “instantaneous” ΔBz
(0 55 s acquisition time)
(0.55 s acquisition time)
Q (e)
Time ‐ average
Preparing the bath via feedback
Control and measurement faster than bath dynamics => Software feedback – adjust pump rate to keep ΔB
=> Software feedback adjust pump rate to keep ΔBz stable.
stable
gΔ B z //h (MHz)
250
Fixed pumping
Feedback
200
150
100
0
500
1000
1500
t (s)
2000
• Qubit measures the nuclear bath
• Qubit manipulates bath
=> let it do all the feedback!
l t it d ll th f db k!
2500
3000
Pulses with built‐in feedback
smaller ΔBz => more pumping => ΔBz increases
E
larger ΔBz => less pumping => ΔB
ΔBz decreases
intermediate ΔBz
=> stable fixpoint
=> stable fixpoint
S(0, 2)
Ez
ε
↑↓
ΔBz
T0
S
Singlet prob.
Fixed precession time
1
0
↓↑
τ
T2* enhancement and narrowing
p(ΔBz)
Q
Q (e)
No feedback
Q (e)
Qubit feedback
p(ΔBz)
Operated qubit
O
t d bit as a complete feedback loop stabilizing l t f db k l
t bili i
its own environment and enhancing coherence.
HB et al., arxiv:1003.4031
Outline
Background
Measuring and manipulating the nuclear hyperfine field
Universal control
Universal control
Reduction of fluctuations via feedback
So far: Averaging over slow fluctuations (T2*)
Coherence time and short time dynamics (T2)
• Hahn echo
• Nuclear dynamics and model
• 200 μs coherence time with Carr Purcell Meiboom Gill (CPMG) decoupling
Carr‐Purcell‐Meiboom‐Gill (CPMG) decoupling
Hahn echo
↑↓
ΔBz
T0
J
S
• Perfect refocussing for static ΔBz
• Decoherence reveals bath dynamics.
↓↑
Dephasing during free precession
during free precession
Bext+Bnuc,z
π – pulse via coherent exchange
pulse via coherent exchange
Experiment
Data
Fits
Bext ≥ 400 mT:
400 mT
(
Echo ∝ exp − (τ / 30 μ s ) 4
)
Mostly dipolar spin diffusion
Normalization:
1: complete refocussing, no decoherence
0: fully dephased, mixed state
Experiment
Data
Fits
Bext ≥ 400 mT:
400 mT
(
Echo ∝ exp − (τ / 30 μ s ) 4
)
Mostly dipolar spin diffusion
Lower fields:
Periodic collapses and revivals due to Larmor precession.
due to Larmor
precession
Decoherence model
Predicted by Cywiński, Das Sarma et al., (PRL,PRB 2009) based on quantum treatment.
Intuitive picture: Yao et al., PRB 2006, PRL 2007
Classical model ⊥
B nuc
B
⊥
2
B
(
t
)
z
Hˆ (t ) = γBext Sˆ z + γBnuc
(t ) Sˆ z + γ nuc
Sˆ z
2 Bext
z
nuc
Btot ≈
B ext
2
B
z
nuc
+ Bext
⊥
B nuc
+
2 Bext
Spin diffusion : p
field independent decay
(
expp − (τ / 35 μ s ) 4
)
(e.g. Witzel et al. PRB 2006)
z
B nuc
Origin of revivals
⊥
B nuc
2
⊥
BIsotope
Abundance Gyromag. ratio
nuc oscillates due to relative Larmor
75As 50 % 7 MHz/T
precession.
71Ga Ga
20 %
20 %
13 MHz/T
13 MHz/T
69Ga 30 %
10 MHz/T
71Ga
69Ga
G
Bext
Total phase = 0 when evolving over whole period
whole period
⇒ Revivals
75As
⊥
B nuc
2
Random phase otherwise
⇒ Collapses
τ/2
Dephasing of Larmor precession (dipolar, quadrupolar shifts)
=>> faster low
faster low‐field
field envelope decay
envelope decay
t
Echo revivals
Fit model: average over initial conditions. Exactly y
reproduces quantum results.
Field independent fit parameters:
#nuclei = 4 4 x 106
#nuclei = 4.4 x 10
Spread of Larmor fields = 3 G
Spin diffusion decay time = 37 μs
Data
Fits
Carr‐Purcell‐Meiboom‐Gill (CPMG)
Hahn echo
CPMG
Init
τ/2
τ/2n
π
Read
τ/2
τ/n
…
= concatenation of Hahn echo sequences.
Prediction: Witzel et al., PRL 2007 τ/n
τ/2n
CPMG ‐ data
no
ormalizeed echo
o amplittude
B = 0.4 T
Subtracted mixed‐state reference (no π‐pulses), normalize by τ = 0 data
data.
Initial linear decay may reflect single‐spin relaxation.
Linear fit extrapolates to
Linear fit extrapolates to
τ = 276 μs
HB et al., arxiv:1005.2995
Summary
• Semiclassical model provides detailed understanding of Hahn echo decay. • Dynamic decoupling highly effective.
y
p g g y
Figures of merit for qubit
Figures of merit for qubit
• Memory time T2 ≥ 200 μs, sub‐ns gates .
=> Exceeding 105 operations within T
=> Exceeding 10
operations within T2. • Extrapolated gate error from nuclear fluctuations ~10‐4.
Future directions
Quantum computing •
•
•
•
•
Two‐qubit gates.
gates
High fidelity gates.
Decoupled gates.
p g
Multi‐qubit devices.
Materials improvement.
N clear bath ph sics
Nuclear bath physics
• Interplay with spin orbit coupling
• Short time polarization dynamics
Short time polarization dynamics
• Ultimate limit of (nuclear) decoherence?