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Controlling Spin Qubits in Quantum Dots C. M. Marcus Harvard University 1

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Page 1: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Controlling Spin Qubits in Quantum Dots

C. M. MarcusHarvard University

1

Page 2: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Theory:Jacob Taylor (MIT)Mikhail LukinMichael Stopa

Material:Art Gossard (UCSB)Loren Pfeiffer (Alcatel-Lucent)

Financial Support:DoDIARPA/ARONSF

GaAs Experiments:David Reilly (Univ. Sydney)Edward LairdChristian BarthelJason Petta (Princeton)Amir Yacoby

Nanotubes:Hugh ChurchillFerdinand KuemmethJennifer Harlow (Boulder)Andrew Bestwick (Stanford)Michael Biercuk (NIST)Nadya Mason (UIUC)

Controlling Spin Qubits in Quantum Dots

C. M. MarcusHarvard University

2

Page 3: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

The modern transistor

3

Page 4: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Quanta

4

Page 5: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

ONOFF

A transistor is a switch controlled by a voltage.

If the transistor can be in more than one state at a time, then it can control another switch that can be in more than

one state at a time, etc.

Mesoscopic Electronics and Quantum Interference

5

Page 6: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Quantum Entanglement: “Spooky Action at a Distance”

|S!

helium

6

Page 7: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

A universal quantum computer

A universal quantum computer can be madefrom 2-bit XOR’s and 1-bit U’s

(from Bennett and DiVincenzo, Nature)

7

Page 8: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

van Meter et al 2006

!"#$%&'(#)%"*#%*+',"#'-*.%-/'#)"0

1%&"23*4,"*52#2$6*72)%*8")92$:)#3

$&9;#2$,<)(:<=2)%<,(<>/

?'"2*@AB@C6*@DDE

F!GH*8")92$:)#3*I(J%%K*%L*!"#2$"2#M)#J*:K)&2:*L$%-*

7<*!#%J*N72)%O6*H<*PQ2*N72)%O6*RRS6*12,0,"*5%%$2*NIGI.O6**,"&*T<*U'>):,M,*NSTTO

FJ,#V:*,*+',"#'-*.%-/'#2$W

! 8:2:*X',"#'-*-2(J,")(,K*2LL2(#:*#%*,((2K2$,#2*(%-/'#,#)%"

! .,"*(,K('K,#2*,*L'"(#)%"*%"*,KK*/%::)QK2*)"/'#*9,K'2:*,#*#J2*:,-2*#)-2

! Y2##)"0*,*':2L'K*,":M2$*%'#*):*#J2*J,$&*/,$#Z

! 5%:#*L,-%':*$2:'K#*):*IJ%$V:*,K0%$)#J-*L%$*L,(#%$)"0*K,$02*"'-Q2$:

U,(#%$)"0*[,$02$*S'-Q2$: U,(#%$)"0*[,$02$*S'-Q2$:time to factor a product of two primes

bits

QC at 1 Hz

QC at 1 kHz

QC at 1 MHz

QC at 1 GHz

Comparing quantum and classical computation

8

Page 9: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

QUBIT REALIZATIONS

!"#$$%&'()*+ ,-.

/01)")2)"3'

3)1%/41%'

5)*6'748(9 /)0%"%*/%

5):'%*;(")*3%*9#1'/)4$1(*6

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+%3(/)*&4/9)" &)9+ +4$%"/)*&4/9(*6'/("/4(9+

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/)4$1(*6

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>1%/9"(/#1'#//%++

making controllable qubits

ion traps Josephson devices

Electron Spins in Dots

9

Page 10: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Quantum computation with quantum dots

Daniel Loss1,2,* and David P. DiVincenzo1,3,†1Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106-4030

2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland3IBM Research Division, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

!Received 9 January 1997; revised manuscript received 22 July 1997"

We propose an implementation of a universal set of one- and two-quantum-bit gates for quantum compu-

tation using the spin states of coupled single-electron quantum dots. Desired operations are effected by the

gating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computed

within a recently derived spin master equation incorporating decoherence caused by a prototypical magnetic

environment. Dot-array experiments that would provide an initial demonstration of the desired nonequilibrium

spin dynamics are proposed. #S1050-2947!98"04501-6$

PACS number!s": 03.67.Lx, 89.70.!c, 75.10.Jm, 89.80.!h

I. INTRODUCTION

The work of the past several years has greatly clarified

both the theoretical potential and the experimental challenges

of quantum computation #1$. In a quantum computer thestate of each bit is permitted to be any quantum-mechanicalstate of a qubit !quantum bit, or two-level quantum system".Computation proceeds by a succession of ‘‘two-qubit quan-tum gates’’ #2$, coherent interactions involving specific pairsof qubits, by analogy to the realization of ordinary digitalcomputation as a succession of Boolean logic gates. It is nowunderstood that the time evolution of an arbitrary quantumstate is intrinsically more powerful computationally than theevolution of a digital logic state !the quantum computationcan be viewed as a coherent superposition of digital compu-tations proceeding in parallel".Shor has shown #3$ how this parallelism may be exploited

to develop polynomial-time quantum algorithms for compu-tational problems, such as prime factoring, which have pre-viously been viewed as intractable. This has sparked inves-tigations into the feasibility of the actual physicalimplementation of quantum computation. Achieving the con-ditions for quantum computation is extremely demanding,requiring precision control of Hamiltonian operations onwell-defined two-level quantum systems and a very high de-gree of quantum coherence #4$. In ion-trap systems #5$ andcavity quantum electrodynamic experiments #6$, quantumcomputation at the level of an individual two-qubit gate hasbeen demonstrated; however, it is unclear whether suchatomic-physics implementations could ever be scaled up todo truly large-scale quantum computation, and some havespeculated that solid-state physics, the scientific mainstay ofdigital computation, would ultimately provide a suitablearena for quantum computation as well. The initial realiza-tion of the model that we introduce here would correspond toonly a modest step towards the realization of quantum com-puting, but it would at the same time be a very ambitiousadvance in the study of controlled nonequilibrium spin dy-

namics of magnetic nanosystems and could point the waytowards more extensive studies to explore the large-scalequantum dynamics envisioned for a quantum computer.

II. QUANTUM-DOT IMPLEMENTATION

OF TWO-QUBIT GATES

In this paper we develop a detailed scenario for howquantum computation may be achieved in a coupledquantum-dot system #7$. In our model the qubit is realized asthe spin of the excess electron on a single-electron quantumdot; see Fig. 1. We introduce here a mechanism for two-qubit quantum-gate operation that operates by a purely elec-

*Electronic address: [email protected]†Electronic address: [email protected]

FIG. 1. !a" Schematic top view of two coupled quantum dots

labeled 1 and 2, each containing one excess electron (e) with spin

1/2. The tunnel barrier between the dots can be raised or lowered by

setting a gate voltage ‘‘high’’ !solid equipotential contour" or‘‘low’’ !dashed equipotential contour". In the low state virtual tun-neling !dotted line" produces a time-dependent Heisenberg ex-change J(t). Hopping to an auxiliary ferromagnetic dot !FM" pro-vides one method of performing single-qubit operations. Tunneling

(T) to the paramagnetic dot !PM" can be used as a POV read outwith 75% reliability; spin-dependent tunneling !through ‘‘spin

valve’’ SV" into dot 3 can lead to spin measurement via an elec-trometer E. !b" Proposed experimental setup for initial test of swap-gate operation in an array of many noninteracting quantum-dot

pairs. The left column of dots is initially unpolarized, while the

right one is polarized; this state can be reversed by a swap operation

#see Eq. !31"$.

PHYSICAL REVIEW A JANUARY 1998VOLUME 57, NUMBER 1

571050-2947/98/57!1"/120!7"/$15.00 120 © 1998 The American Physical Society

Quantum computation with quantum dots

Daniel Loss1,2,* and David P. DiVincenzo1,3,†1Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106-4030

2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland3IBM Research Division, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

!Received 9 January 1997; revised manuscript received 22 July 1997"

We propose an implementation of a universal set of one- and two-quantum-bit gates for quantum compu-

tation using the spin states of coupled single-electron quantum dots. Desired operations are effected by the

gating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computed

within a recently derived spin master equation incorporating decoherence caused by a prototypical magnetic

environment. Dot-array experiments that would provide an initial demonstration of the desired nonequilibrium

spin dynamics are proposed. #S1050-2947!98"04501-6$

PACS number!s": 03.67.Lx, 89.70.!c, 75.10.Jm, 89.80.!h

I. INTRODUCTION

The work of the past several years has greatly clarified

both the theoretical potential and the experimental challenges

of quantum computation #1$. In a quantum computer thestate of each bit is permitted to be any quantum-mechanicalstate of a qubit !quantum bit, or two-level quantum system".Computation proceeds by a succession of ‘‘two-qubit quan-tum gates’’ #2$, coherent interactions involving specific pairsof qubits, by analogy to the realization of ordinary digitalcomputation as a succession of Boolean logic gates. It is nowunderstood that the time evolution of an arbitrary quantumstate is intrinsically more powerful computationally than theevolution of a digital logic state !the quantum computationcan be viewed as a coherent superposition of digital compu-tations proceeding in parallel".Shor has shown #3$ how this parallelism may be exploited

to develop polynomial-time quantum algorithms for compu-tational problems, such as prime factoring, which have pre-viously been viewed as intractable. This has sparked inves-tigations into the feasibility of the actual physicalimplementation of quantum computation. Achieving the con-ditions for quantum computation is extremely demanding,requiring precision control of Hamiltonian operations onwell-defined two-level quantum systems and a very high de-gree of quantum coherence #4$. In ion-trap systems #5$ andcavity quantum electrodynamic experiments #6$, quantumcomputation at the level of an individual two-qubit gate hasbeen demonstrated; however, it is unclear whether suchatomic-physics implementations could ever be scaled up todo truly large-scale quantum computation, and some havespeculated that solid-state physics, the scientific mainstay ofdigital computation, would ultimately provide a suitablearena for quantum computation as well. The initial realiza-tion of the model that we introduce here would correspond toonly a modest step towards the realization of quantum com-puting, but it would at the same time be a very ambitiousadvance in the study of controlled nonequilibrium spin dy-

namics of magnetic nanosystems and could point the waytowards more extensive studies to explore the large-scalequantum dynamics envisioned for a quantum computer.

II. QUANTUM-DOT IMPLEMENTATION

OF TWO-QUBIT GATES

In this paper we develop a detailed scenario for howquantum computation may be achieved in a coupledquantum-dot system #7$. In our model the qubit is realized asthe spin of the excess electron on a single-electron quantumdot; see Fig. 1. We introduce here a mechanism for two-qubit quantum-gate operation that operates by a purely elec-

*Electronic address: [email protected]†Electronic address: [email protected]

FIG. 1. !a" Schematic top view of two coupled quantum dots

labeled 1 and 2, each containing one excess electron (e) with spin

1/2. The tunnel barrier between the dots can be raised or lowered by

setting a gate voltage ‘‘high’’ !solid equipotential contour" or‘‘low’’ !dashed equipotential contour". In the low state virtual tun-neling !dotted line" produces a time-dependent Heisenberg ex-change J(t). Hopping to an auxiliary ferromagnetic dot !FM" pro-vides one method of performing single-qubit operations. Tunneling

(T) to the paramagnetic dot !PM" can be used as a POV read outwith 75% reliability; spin-dependent tunneling !through ‘‘spin

valve’’ SV" into dot 3 can lead to spin measurement via an elec-trometer E. !b" Proposed experimental setup for initial test of swap-gate operation in an array of many noninteracting quantum-dot

pairs. The left column of dots is initially unpolarized, while the

right one is polarized; this state can be reversed by a swap operation

#see Eq. !31"$.

PHYSICAL REVIEW A JANUARY 1998VOLUME 57, NUMBER 1

571050-2947/98/57!1"/120!7"/$15.00 120 © 1998 The American Physical Society

10

Page 11: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

11

Page 12: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

State Preparation|S!

|T0!

| !"#| !"#S

(0,2)S

(0,2)S

εEn

ergy

ST0

T-

T+

T-

T+

Voltage-controlled tilt(0,2) is GS

(1,1) is GS

12

Page 13: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

|S!

|T0!

| !"#| !"#

State Preparation

S

(0,2)S

(0,2)S

εEn

ergy

ST0

T-

T+

T-

T+

Voltage-controlled tilt(0,2) is GS

(1,1) is GS

13

Page 14: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

|S!

|T0!

| !"#| !"#

Eigenstates of magnetic field difference

Eigenstates of exchange

ΔBZJ

The fully controlled singlet-triplet qubit

14

Page 15: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Hyperfine-Mediated Gate-Driven Electron Spin Resonance

E. A. Laird,1 C. Barthel,1 E. I. Rashba,1,2 C. M. Marcus,1 M. P. Hanson,3 and A. C. Gossard3

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2Center for Nanoscale Systems, Harvard University, Cambridge, Massachusetts 02138, USA

3Materials Department, University of California at Santa Barbara, Santa Barbara, California 93106, USA(Received 3 July 2007; published 13 December 2007)

An all-electrical spin resonance effect in a GaAs few-electron double quantum dot is investigatedexperimentally and theoretically. The magnetic field dependence and absence of associated Rabioscillations are consistent with a novel hyperfine mechanism. The resonant frequency is sensitive tothe instantaneous hyperfine effective field, and the effect can be used to detect and create sizable nuclearpolarizations. A device incorporating a micromagnet exhibits a magnetic field difference between dots,allowing electrons in either dot to be addressed selectively.

DOI: 10.1103/PhysRevLett.99.246601 PACS numbers: 76.20.+q, 76.30.!v, 76.70.Fz, 78.67.Hc

The proposed use of confined electron spins as solid-state qubits [1] has stimulated progress in their manipula-tion and detection [2–8]. In such a proposal, the mostgeneral single-qubit operation is a spin rotation. One tech-nique for performing arbitrary spin rotations is electronspin resonance (ESR) [9], in which a pair of magneticfields is applied, one static (denoted B) and one resonantwith the electron precession (Larmor) frequency (denoted~B). Observing single-spin ESR is challenging because ofthe difficulty of combining sufficient ~B with single-spindetection [2,3,5]. In GaAs quantum dots, where a highdegree of spin control has been achieved [4,6,7], ESRwas recently demonstrated using a microstripline to gen-erate ~B [8].

An alternative to ESR is electric dipole spin resonance(EDSR) [10–12], in which an oscillating electric field ~Ereplaces ~B. EDSR has the advantage that high-frequencyelectric fields are often easier to apply and localize thanmagnetic fields, but requires an interaction between ~E andthe electron spin. Mechanisms of EDSR include spin-orbitcoupling and inhomogeneous Zeeman coupling [12–15].

In this Letter, we present the first experimental study of anovel EDSR effect mediated by the random inhomogeneityof the nuclear spin orientation. The effect is observed viaspin-blocked transitions in a few-electron GaAs doublequantum dot. For B " jBj< 1 T the resonance strengthis independent of B and shows no Rabi oscillations as afunction of time, consistent with a theoretical model wedevelop but in contrast to other EDSR mechanisms. Wemake use of the resonance to create nuclear polarization,which we interpret as the backaction of EDSR on thenuclei [8,16–18]. Finally, we demonstrate that spins maybe individually addressed in each dot by creating a localfield gradient.

The device for which most data are presented [Fig. 1(a)]was fabricated on a GaAs=Al0:3Ga0:7As heterostructurewith two-dimensional electron gas (density 2#1015 m!2, mobility 20 m2=Vs) 110 nm below the surface.Ti=Au top gates define a few-electron double quantum dot.

A charge sensing quantum point contact (QPC), tuned toconductance gs $ 0:2e2=h, is sensitive to the electron oc-cupation %NL; NR& of the left and right dots [19,20]. Thevoltages VL and VR on gates L and R, which control theequilibrium occupation, are pulsed using a TektronixAWG520; in addition, L is coupled to a Wiltron 6779Bmicrowave source gated by the AWG520 marker. A staticin-plane field B was applied parallel to '110(. Measure-ments were performed in a dilution refrigerator at 150 mKelectron temperature, known from Coulomb blockadewidth.

As in previous measurements [8], we detect spin tran-sitions with the device configured in the spin blockaderegime [21,22]. In this regime, accessed by tuning VLand VR, a source-drain bias Vsd across the device induces

FIG. 1 (color online). (a) Micrograph of a device lithograph-ically identical to the one measured, with schematic of themeasurement circuit. The direction of B and the crystal axesare indicated. (b) QPC conductivity gs measured at Vsd $600 !eV near the %1; 1&-%0; 2& transition. The spin blockaderegion is outlined. Equilibrium occupations for different gatevoltages are shown, as are gate configurations during the mea-surement or reinitialization (M) and manipulation (C) pulses. Aplane background has been subtracted. (c) Energy levels of thedouble dot during the pulse cycle (see text).

PRL 99, 246601 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 2007

0031-9007=07=99(24)=246601(4) 246601-1 ! 2007 The American Physical Society

a)

c)

b)2 !m

Magnet

Magnet

1.4

1.2

1.0

Fre

quency (

GH

z)

220200180

Magnetic Field (mT)

30

20

10

0

!VQPC

(nV)

15

Page 16: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

S

T0

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#"

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Electrostatic Two-Qubit Gate

! store storetest ! store storesense

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16

Page 17: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Data

Ancillablock

Datablock

Ancillablock

Encodingunit A

Contr

ol cir

cuit

Encodingunit B

Contr

ol cir

cuit

Encoding B

AncillaAncillablock

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block

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cuit

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cuit

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(a)

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MW

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MW

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tic g

ate

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tic g

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ple 2-qubit couple

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Classical control circuits

Qubit A

SET/QPC couple

Qubit B

FIG. 1: for Taylor et al.

1

17

Page 18: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

! !

Electrostatic Two-Qubit Gate

18

Page 19: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

|S!

|T0!

|!"

#|!"

#

ΔBZS

T0

scope~

3

sum of two gaussian peaks because part of the popu-lation initially in the (1,1) state relaxes into the (0,2)state during the integration time !M, with the relaxationtime constant T1 [8]. The normalized number of eventsn(Vrf) = N(Vrf)/

!!"!N(Vrf)dVrf is modeled as

n(Vrf) = nS(Vrf) + nT (Vrf) (1)

with the events originating from singlet states nS(Vrf)and from triplet states nT (Vrf). These are given by

nS(Vrf) = (1! "PT #) e"(Vrf!V

(S)rf )2

2!21$2"#

(2)

nT (Vrf) = e"!M/T1"PT # e"(Vrf"V (T )rf )2/(2"2) +

" !

"!

!M

T1

"PT #!Vrf

e"V!V

(S)rf

!Vrf

"MT1 e"

(Vrf!V )2

2!2dV$2"#

(3)

with the triplet probability "PT #, averaged over all !S.The plot of Eq. (1) in Fig. 2(b) uses "PT # = 0.5, T1 =34 µs, and peak positions V (S)

rf , V (T )rf , determined as de-

scribed below [Fig. 2(c,e)]. The width # [23] is obtainedfrom the control experiment.

The parameters T1 and "PT # are extracted from theraw data Vrf(!), which is plotted as a function of thetime ! spent at point M in Fig. 2(c). Each data pointfor ! = 0.5! 15 µs, is averaged over all 7000 cycles withvarying !S. The signal is fitted with [8]

Vrf(!) = V (S)rf + "PT #!Vrf e"!/T1 (4)

using fit parameters T1 and "PT #. The singlet positionV (S)

rf and the peak spacing !Vrf [23] are fixed, as ob-tained from a fit of the model Eq. (1) to the histogram for!M = 15 µs [Fig. 2(e)]. For the fit of Eq. (1) the parame-ters T1 and "PT # are self-consistently fixed to the valuesextracted from the raw data Vrf(!) [Eq. (4), Fig. 2(c)].

Maximizing the fidelity by optimization of the inte-gration time !M is a tradeo" between increasing the sig-nal to noise ratio % $

!M and limiting relaxation during!M. The histograms of single-shot outcomes "Vrf#!M inFig. 2(d), for integration times !M = 0.25 ! 15 µs showthat the two peaks can no longer be clearly resolved for!M < 1 µs while the relative height of the triplet peakreduces with increasing !M. Common benchmarks forsingle-shot readout [24] are the fidelities FS , FT of sin-glet, triplet measurement:

FS = 1!" !

VT

nS(V )dV, FT = 1!" VT

"!nT (V )dV. (5)

The integral in the expression for FS (FT ) is the proba-bility to assign a singlet as a triplet (a triplet as a sin-glet). Both quantities are combined to define the visibil-ity V = FS + FT ! 1. The fidelities and the visibility fora single-shot measurement with !M = 7 µs are calculatedfrom the data in Fig. 2(b) and plotted in Fig. 2(f) as

(a) (b)

(c)

VT

FIG. 3: (a) Single-shot outcomes !Vrf"!M for 6000 cycles, puls-ing to ! = !S [Fig. 1(b)] for "S, stepped by # 17 ns every 200cycles. Points in the green (blue) region are above (below)the threshold VT and assigned as triplet (singlet). (b) Single-shot outcomes (gray markers) and triplet probabilities (blackcircles) over "S with three di!erent periods. (c) Rapid acqui-sition of 108 PT traces at times t. PT is determined from 400measurements per "S.

a function of the threshold voltage VT. The maximumvisibility & 90% is achieved for VT slightly less than themean of V (T )

rf and V (S)rf so that a triplet decaying towards

the end of !M still gets counted correctly.To determine the optimal values of !M and VT the max-

imum visibility V max is calculated as a function of !M

from Eq. (1) using the parameters T1, "PT #, determinedfrom Fig. 2(c), V (T )

rf and V (S)rf , from Fig. 2(e) and #(!M),

determined from the control experiment [23]. The thresh-old VT for which the visibility is maximized is plotted to-gether with V max in Fig. 2(g). The maximum visibility,obtained for !M & 6 µs, is V max ! 90%.The single-shot readout is applied to observe the evo-

lution of the singlet triplet qubit at point S [4], drivenby the di"erence in the hyperfine induced e"ective mag-netic (Overhauser) field !Bnuc

z between the left and rightquantum dot. Single-shot outcomes "Vrf# are shown as afunction of !S in Fig. 3(a) for a pulse sequence [Fig. 1(d)]with !S = 1! 500 ns stepped by 17 ns every 200 cycles,for a total of 6000 consecutive cycles. Points that arein the green (blue) region are above (below) the thresh-old VT and are assigned as triplet (singlet) states. Foreach !S the triplet probability PT is the percentage ofsingle-shot outcomes above threshold. Probabilities PT

for the single-shot data in Fig. 3(a) are shown in thetop graph of Fig. 3(b) as a function of !S. The twographs below show probability traces with identical pa-rameters. Single-shot outcomes from which the proba-

SINGLET

TRIPLET

Single-shot S-T detection

19

Page 20: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Dynamic Nuclear Polarization with Single Electron Spins

J. R. Petta,1, 2, ! J. M. Taylor,1, 3, ! A. C. Johnson,1 A. Yacoby,1

M. D. Lukin,1 C. M. Marcus,1 M. P. Hanson,4 and A. C. Gossard4

1Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 021382Department of Physics, Princeton University, Princeton, New Jersey 08544

3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 021394Materials Department, University of California, Santa Barbara, California 93106

We polarize nuclear spins in a GaAs double quantum dot by controlling two-electron spin statesnear the anti-crossing of the singlet (S) and mS=+1 triplet (T+) using pulsed gates. An initializedS state is cyclically brought into resonance with the T+ state, where hyperfine fields drive rapidrotations between S and T+, ‘flipping’ an electron spin and ‘flopping’ a nuclear spin. The resultingOverhauser field approaches 80 mT, in agreement with a simple rate-equation model. A self-limitingpulse sequence is developed that allows the steady-state nuclear polarization to be set using a gatevoltage.

PACS numbers: 03.67.Bg, 73.21.La, 76.70.-r

Semiconductor quantum dots share many features withreal atoms: they possess discrete electronic energy states,obey Hund’s rules as the states are filled, and can be cou-pled to one another, creating artificial molecules [1, 2]. Incontrast to atoms, where electrons are coupled to a sin-gle nucleus, electrons within quantum dots interact withmany (typically !106) lattice nuclei [2]. The dynamicsof spin systems comprising few electrons interacting withmany nuclei is an interesting and complex many-bodyproblem in condensed matter physics [3, 4, 5, 6].

Electrically controlled nuclear polarization in semicon-ductor microstructures has been investigated in gate-defined quantum point contacts (QPCs) in GaAs, wherenuclear polarization driven by scattering between spin-polarized edge states induced hysteresis in conductanceas a function of magnetic field [7, 8]. In few-electronquantum dots, transport in the so-called Pauli blockaderegime, which requires a spin flip for conduction, canexhibit long-time-scale oscillations and bi-stability, alsounderstood to result from a build-up and relaxation ofnuclear polarization during spin-flip-mediated transport[6, 9, 10, 11]. Depending on conditions, sizable polariza-tions can result from transport in this regime [12]. Spinrelaxation at low magnetic fields, (time scale T1) andspin dephasing (time scale T !

2 ) in GaAs double quantumdots are limited by hyperfine interactions with host nuclei[13, 14, 15, 16, 17]. It has been suggested that electronspin dephasing times may be extended by cooling nu-clear spins through nuclear polarization and projectivemeasurement [6, 18, 19]. In addition, nuclear spins maybe used as a quantum memory due to long nuclear spincoherence times [20]. For these reasons, it is desirable tohave the ability to control interactions between a singleconfined electron spin and a bath of quantum dot nuclearspins [21].

In this Letter, we use fast pulsed-gate control of two-electron spin states in a double quantum dot to createand detect nuclear polarization. Rather than using spin-

100

50

0

-50

-100

B0

(mT

)

!S

(mV) 0-1.0

1 "ma)

d)

b)S(0,2)S

(0,2)S

Energ

y

0 !

0.6

1.0

0.8

PS

VR (mV)

VL

(mV

)

-345 -341

-470

-474

gS

(10

-3e

2/h

)

0

20

40

60(1,2)

(0,2)(0,1)

(1,1)

c)

P

P', M

S

RL

T

gS

!#

!$%

T0

T+

T-

EZ=g*"BBtot

S

FIG. 1: (a) SEM image of a device similar to the one usedin this experiment. Gates L, R set the occupancy of the dou-ble dot while gate T tunes the interdot tunnel coupling. AQPC with conductance, gS, senses charge on the double dot.(b) Charge sensor conductance, gS, measured as a functionof VL and VR. A pulse sequence used to polarize nuclei issuperimposed on the charge stability diagram (see text). Thebright signal at point M that runs parallel to the !=0 line isa result of Pauli-blocked (1,1)!(0,2) charge transitions andreflects S-T+ mixing at point S. A background plane has beensubtracted from the data. (d) Singlet return probability PS

for separation time "S=200 ns, as a function of separationposition !S and applied field B0. The field dependent “spinfunnel” corresponds to the S-T+ anti-crossing position, !S=!!.

flip-mediated transport with an applied bias [11, 12], weuse a cycle of gate voltages to prepare a spin singlet (S)state, induce nuclear flip-flop at the singlet-triplet (S-T+)anti-crossing, and measure the resulting nuclear polariza-tion from the position of that anti-crossing. In this way,dynamic nuclear polarization (DNP) is achieved (andmeasured) by the controlled flipping of single electronspins. Additionally, we demonstrate a self-limiting pulse

arX

iv:0

709.0

920v2 [c

ond-m

at.m

es-h

all]

7 S

ep 2

007

2

cycle with fixed period that allows voltage-controlled“programming” of steady-state nuclear polarization. Ex-perimental results are found to be in qualitative agree-ment with a simple rate-equation model.

Measurements are carried out using a gated-definedGaAs double quantum dot [14]. The device is configurednear the (1,1)-(0,2) charge transition (see Fig. 1(a)). AQPC charge sensor is used to read out the time-averagedcharge configuration of the dot, from which spin statescan be inferred. Relevant energies of spin states as afunction of detuning, !, are shown in Fig. 1(b). The(1,1) singlet, S, and (0,2) singlet, denoted (0, 2)S, hy-bridize near !=0 due to interdot tunneling. Triplets of(0,2) are not considered, due to the large singlet-tripletsplitting, EST!400 µeV. The triplet states of (1,1), T+,T!, and T 0, do not hybridize with (0, 2)S due to spinselection rules. An external magnetic field, "B0, appliedperpendicular to the plane of the two-dimensional elec-tron gas (2DEG), adds to the hyperfine field, "Bnuc, result-ing in a total e!ective Zeeman field "Btot= "B0+ "Bnuc. TheZeeman energy, EZ=g"µBBtot, lifts the triplet-state de-generacy, where Btot=| "Btot| and g"!-0.4 is the e!ectiveelectronic g-factor in GaAs. The nuclear Zeeman field isproportional to nuclear polarization and has a magnitude| "Bnuc|=5.2 T for fully polarized nuclei in GaAs [22].

Hyperfine fields rapidly drive transitions between Sand T0 on a 10 ns time scale at large negative detuning,where these states become degenerate [10, 14]. However,transitions between S and T0 do not polarize nuclei sincethere is no change in the total spin component along thefield, "mS=0. In contrast, the transition from S to T+

involves a change in electron spin, "mS=1. When drivenby the hyperfine interaction, this electron spin ‘flip’ is ac-companied by a nuclear spin ‘flop’ with "mI=-1. Thechange in Bnuc associated with this flip-flop transitionis along the direction of external field, taking into ac-count the positive nuclear g factors for Ga and As [2].The cyclic repetition of the transition from S to T+ canthereby lead to a finite time-averaged Bnuc oriented alongthe external field.

The value of detuning where the S-T+ anti-crossingoccurs, denoted !" (see Fig. 1(b)), is a sensitive functionof Btot for |Btot|"80 mT, providing a straightforwardmeans of measuring Bnuc within that range. To calibratethe measurement, the dependence of !" on external fieldamplitude B0 is measured using the pulse sequence shownin Fig. 1(c): The (0, 2)S state is first prepared at pointP. A delocalized singlet in (1,1) is created by moving topoint S (detuning !S) via point P#. The system is heldat point S for a time #S#T "

2 then moved to point M andheld there for the longest part of the cycle. The sequenceis then repeated. When !S=!", rapid mixing of S and T+

states occurs. When the system is moved to point M, the(1,1) charge state will return to (0,2) only if the separatedspins are in a singlet configuration. The probability of

Adiabatic nuclear polarizationProbabilistic nuclear polarization

Rapid Adiabatic Passage

Slow Adiabatic Passage

Prepare Singlet

Rapid Adiabatic Passage, S-T+ evolution

PS

Prepare Singlet

t=!S

t=0

Rapid Adiabatic Passage, Projection

a) b)

t=!S

t=0

1-PS

Energ

y

"(0,2)S

0

(0,2)S T0

T+

T-

Energ

y

"(0,2)S

0

(0,2)S T0

T+

T-

Energ

y

"(0,2)S

0

(0,2)S T0

T+

T-E

nerg

y

"(0,2)S

0

(0,2)S T0

T+

T-

Energ

y

"(0,2)S

0

(0,2)S T0

T+

T-

"S

Energ

y

"(0,2)S

0

(0,2)S T0

T+

T-

"F

"*

FIG. 2: (a) Probabilistic nuclear polarization sequence. (b)Adiabatic nuclear polarization sequence (see text).

being in the singlet state after time #S thus appears ascharge signal—the di!erence between the (1,1) and (0,2)charge states—detected by measuring the time averagedQPC conductance, gS (see Fig. 1(a)). Figure 1(c) showsgS as a function of VL and VR with this pulse sequence ap-plied. The field dependence of this signal is shown in Fig.1(d), which plots the calibrated singlet state probability,PS , as a function of B0 and !S. In Fig. 1(d), PS!0.7at the S-T+ degeneracy, corresponding to a probabilityof electron-nuclear flip-flop per cycle (1$ PS)!0.3. Theposition of the anti-crossing, !S=!", becomes more nega-tive as B0 decreases toward zero (Fig. 1(d)), as expectedfrom the level diagram, Fig. 1(b).

An alternative sequence that (in principle) determinis-tically flips one nuclear spin per cycle and allows greatercontrol of the steady-state nuclear polarization is shownin Fig. 2(b): In this case, the initial (0, 2)S is separatedquickly (!1 ns), to a value of detuning beyond the S-T+ resonance, !S<!". Since the time spent at the S-T+

resonance during the pulse is short, the singlet is pre-served with high probability. Next, detuning is broughtback toward zero, to a value !F, on a time scale slowcompared to T "

2 (!100 ns). This converts S to T+ andflips a nuclear spin each cycle. Detuning is then rapidlymoved to the point M (again, over a time !1 ns). Whileslightly di!erent, both pulse sequences rely on bringingthe S and T+ states into resonance; the otherwise largedi!erence between nuclear and electron Zeeman energieswould prevent flip-flop processes [23]. We now explorecharacteristics of the resulting nuclear polarization foreach of the processes illustrated in Fig. 2.

We begin by examining the statistical polarization se-quence shown in Fig. 2(a). We first measure PS asa function of B0 and !S, with #S=100 ns, #P=300 ns,#P!=100 ns, #M=32 µs. These data, shown in Fig. 3(a),map out the S-T+ anti-crossing in the limit of minimal

Bex

ΔBnu~

Dynamic Nuclear Polarization with Single Electron Spins

J. R. Petta,1,2 J. M. Taylor,1,3 A. C. Johnson,1 A. Yacoby,1 M. D. Lukin,1 C. M. Marcus,1 M. P. Hanson,4 and A. C. Gossard4

1Department of Physics, Harvard University, 17 Oxford St., Cambridge, Massachusetts 02138, USA2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA4Materials Department, University of California, Santa Barbara, California 93106, USA

(Received 6 September 2007; published 11 February 2008)

We polarize nuclear spins in a GaAs double quantum dot by controlling two-electron spin states nearthe anticrossing of the singlet (S) and mS ! "1 triplet (T") using pulsed gates. An initialized S state iscyclically brought into resonance with the T" state, where hyperfine fields drive rapid rotations between Sand T", ‘‘flipping’’ an electron spin and ‘‘flopping’’ a nuclear spin. The resulting Overhauser fieldapproaches 80 mT, in agreement with a simple rate-equation model. A self-limiting pulse sequence isdeveloped that allows the steady-state nuclear polarization to be set using a gate voltage.

DOI: 10.1103/PhysRevLett.100.067601 PACS numbers: 76.70.Fz, 73.21.La

Semiconductor quantum dots share many features withreal atoms: they possess discrete electronic energy states,obey Hund’s rules as the states are filled, and can becoupled to one another, creating artificial molecules[1,2]. In contrast to atoms, where electrons are coupledto a single nucleus, electrons within quantum dots interactwith many (typically #106) lattice nuclei [2]. The dynam-ics of spin systems comprising few electrons interactingwith many nuclei is an interesting and complex many-bodyproblem in condensed matter physics [3–6].

Electrically controlled nuclear polarization has beeninvestigated in gate-defined quantum point contacts(QPCs) in GaAs, where nuclear polarization driven byscattering between spin-polarized edge states induced hys-teresis in conductance as a function of magnetic field [7,8].In few-electron quantum dots, transport in the so-calledPauli blockade regime, which requires a spin flip for con-duction, can exhibit long time scale oscillations and bi-stability, also understood to result from a buildup andrelaxation of nuclear polarization during spin-flip-mediated transport [6,9–11]. Sizable polarizations can re-sult from transport in this regime [12]. Spin relaxation atlow magnetic fields (time scale T1) and spin dephasing(time scale T$

2) in GaAs double quantum dots are limitedby hyperfine interactions with host nuclei [13–17]. Theorysuggests that electron spin dephasing times may be ex-tended by cooling nuclear spins through nuclear polariza-tion and projective measurement [6,18,19]. In addition,nuclear spins may be used as a quantum memory due tolong nuclear spin coherence times [20]. For these reasons,it is desirable to have the ability to control interactionsbetween a single confined electron spin and a bath ofquantum dot nuclear spins [21].

In this Letter, we use fast pulsed-gate control of two-electron spin states in a double quantum dot to create anddetect nuclear polarization. Rather than using spin-flip-mediated transport with an applied bias [11,12], we use acycle of gate voltages to prepare a spin singlet (S) state,

induce nuclear flip-flop at the singlet-triplet (S-T") anti-crossing, and measure the resulting nuclear polarizationfrom the position of that anticrossing. In this way, dynamicnuclear polarization (DNP) is achieved (and measured) bythe controlled flipping of single electron spins. Addition-ally, we demonstrate a self-limiting pulse cycle with fixedperiod that allows voltage-controlled ‘‘programming’’ ofsteady-state nuclear polarization. Experimental results arefound to be in qualitative agreement with a simple rate-equation model.

Measurements are carried out using a gated-definedGaAs double quantum dot configured near the (1,1)-(0,2)charge transition [see Fig. 1(a)] [14]. A QPC charge sensoris used to read out the time-averaged charge configurationof the dot, from which spin states can be inferred. Relevantenergies of spin states as a function of detuning, ! / VR %VL, are shown in Fig. 1(b). The (1,1) singlet, S, and (0,2)singlet, denoted &0; 2'S, hybridize near ! ! 0 due to inter-dot tunneling. Triplets of (0,2) are not considered, due tothe large singlet-triplet splitting, EST # 400 "eV. Thetriplet states of (1,1), T", T%, and T0 do not hybridizewith &0; 2'S due to spin selection rules. An external mag-netic field, ~B0, applied perpendicular to the plane of thetwo-dimensional electron gas (2DEG), adds to the hyper-fine field, ~Bnuc, resulting in a total effective Zeeman field~Btot ! ~B0 " ~Bnuc. The Zeeman energy, EZ ! g$"BBtot,lifts the triplet-state degeneracy, where Btot ! j ~Btotj andg$ # %0:4 are the effective electronic g-factor in GaAs.The nuclear Zeeman field is proportional to nuclear polar-ization and has a magnitude j ~Bnucj ! 5:2 T for fully po-larized nuclei in GaAs [22].

Hyperfine fields rapidly drive transitions between S andT0 on a 10 ns time scale at large negative detuning, wherethese states become degenerate [10,14]. However, thesetransitions do not polarize nuclei since !mS ! 0. In con-trast, a transition from S to T" flips an electron spin,!mS ! 1. When driven by the hyperfine interaction, this

PRL 100, 067601 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending15 FEBRUARY 2008

0031-9007=08=100(6)=067601(4) 067601-1 ! 2008 The American Physical Society

nuclear state preparation

20

Page 21: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Suppressing Spin Qubit Dephasing byNuclear State PreparationD. J. Reilly,1 J. M. Taylor,2 J. R. Petta,3 C. M. Marcus,1* M. P. Hanson,4 A. C. Gossard4

Coherent spin states in semiconductor quantum dots offer promise as electrically controllablequantum bits (qubits) with scalable fabrication. For few-electron quantum dots made from galliumarsenide (GaAs), fluctuating nuclear spins in the host lattice are the dominant source of spindecoherence. We report a method of preparing the nuclear spin environment that suppressesthe relevant component of nuclear spin fluctuations below its equilibrium value by a factor of ~70,extending the inhomogeneous dephasing time for the two-electron spin state beyond1 microsecond. The nuclear state can be readily prepared by electrical gate manipulationand persists for more than 10 seconds.

Q

www.sciencemag.org SCIENCE VOL 321 8 AUGUST 2008

nuclear state preparation

21

Page 22: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Nuclear Zamboni

22

Page 23: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

some mostly-zero-nuclear-spin materials

23

Page 24: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

-2.89

-2.88

-2.87

-2.86

-3.04 -3.03 -3.02 -3.01 -3.04 -3.03 -3.02 -3.01 -3.04 -3.03 -3.02 -3.01

VL

P

(V)

VRP (V)

0 – 0.004 e2/h 0 – 0.05 e2/h 0 – 0.12 e2/h

-2.96

-2.95

-2.9440

20

0

-2.96

-2.95

-2.89 -2.88 -2.87 -2.86 -2.85 -2.84

-2.94

gd

d (1

0-3

e2/h

)d

gs/d

VL

P (a

.u.)

VL

P

(V)

VRP (V)

15

0

-10

a

b

Si/Ge Nanowire with Integrated Charge Sensor

Si

Ge

Yongjie Hu, Hugh O. H. Churchill, David J. Reilly, Jie Xiang, Charles M. Lieber, Charles M. Marcus, Nature Nanotechnology 2, 622 (2007).

24

Page 25: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Si/Ge Nanowire with Integrated Charge Sensor

-2.94

-2.92

-2.90

-2.85 -2.84 -2.83

VRP (V)

-2.96

-2.94

-2.92

VLP

(V

)

-2.89 -2.88

VRP (V)

dgs/dVLP (a.u.)-2 2 -20 20

dgs/dVLP (a.u.)

a b

-2876

-2873VLP

(mV

)-3038 -3035

VRP (mV)

1.0

0.5

0.0

m

- M

1.0

0.5

0.0

-1.0 -0.5 0.0 0.5 1.0! (mV)

gs (10-3 e2/h)

m

- M

(M+1, N+1)

(M,N+1) (M,N)

(M+1,N)

2510

! a

b

0.15 K (fit) 0.5 K 1.0 K

Temperature

-863 58

-855 31

-851 18

-850 ~ 0

V3 (mV) t (µeV)

2t

(M+1,N)

(M+1,N)(M,N+1)

(M,N+1)

Yongjie Hu, Hugh O. H. Churchill, David J. Reilly, Jie Xiang, Charles M. Lieber, Charles M. Marcus, Nature Nanotechnology 2, 622 (2007).

25

Page 26: Controlling Spin Qubits in Quantum DotsFinancial Support: DoD IARPA/ARO NSF GaAs Experiments: David Reilly (Univ. Sydney) Edward Laird Christian Barthel Jason Petta (Princeton) Amir

Summary

coupling to nuclei: resource or headache?-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

Scope V

oltage (

V)

6000500040003000200010000

Measurement Cycle

2.5

2.0

1.5

1.0

0.5

0.0

Trip

let

pro

ba

bili

ty

500ns4003002001000

Separation time ts

ens_b_singleshot_766

SINGLET

TRIPLET

quantum bits on a chip

nuclear spin free materials for quantum spin

Data

Ancillablock

Datablock

Ancillablock

Encodingunit A

Contr

ol cir

cuit

Encodingunit B

Contr

ol cir

cuit

Encoding B

AncillaAncillablock

EncodingEncodingEncodingunit

block

Contr

ol cir

cuit

Contr

ol cir

cuit

Classical control circuity

(a)

(b)

(c)

Ancillaunit

Dataunit

Ancillaunit

EncodingEncodingunit Bunit A unitunit

Ancilla

Contr

ol cir

cuit

Inject/eject electron

Spin shuttle gates

MW

gate

MW

gate

MW

gate

MW

gate

MW

gate

MW

gate

Sta

tic g

ate

Sta

tic g

ate

Sta

tic g

ate

Sta

tic g

ate

SET/

QPC

cou

ple 2-qubit couple

Qubit (in transit)

Classical control circuits

Qubit A

SET/QPC couple

Qubit B

FIG. 1: for Taylor et al.

1

how to build a quantum processor with error correction

26