Coherent Manipulation of
Coupled Electron Spins in
Semiconductor Quantum Dots
J. R. Petta,
1
A. C. Johnson,
1
J. M. Taylor,
1
E. A. Laird,
1
A. Yacoby,
2
M. D. Lukin,
1
C. M. Marcus,
1
M. P. Hanson,
3
A. C. Gossard
3
We demonstrated coherent control of a quantum two-level system based on
two-electron spin states in a double quantum dot, allowing state preparation,
coherent manipulation, and projective readout. These techniques are based on
rapid electrical control of the exchange interaction. Separating and later
recombining a singlet spin state provided a measurement of the spin
dephasing time, T
2
*, of È10 nanoseconds, limited by hyperfine interactions
with the gallium arsenide host nuclei. Rabi oscillations of two-electron spin
states were demonstrated, and spin-echo pulse sequences were used to sup-
press hyperfine-induced dephasing. Using these quantum control techniques, a
coherence time for two-electron spin states exceeding 1 microsecond was
observed.
Quantum coherence and entanglement have
emerged as physical bases for information-
processing schemes that use two-state quantum
systems (quantum bits or qubits) to provide
efficient computation and secure communica-
tion (1, 2). Although quantum control of en-
tanglement has been realized in isolated atomic
systems, its extension to solid-state systems—
motivated by the prospect of scalable device
fabrication—remains a demanding experimen-
tal goal (3, 4), particularly because of the
stronger coupling of solid-state qubits to their
environment. Understanding this coupling and
learning how to control quantum systems in
the solid state is a major challenge of modern
condensed-matter physics (5, 6).
An attractive candidate for a solid-state
qubit is based on semiconductor quantum dots,
which allow controlled coupling of one or
more electrons, using rapidly switchable volt-
ages applied to electrostatic gates (7–9). Re-
cent experiments suggest that spin in quantum
dots may be a particularly promising holder of
quantum information, because the spin relax-
ation time (T
1
) can approach tens of milli-
seconds (10–13). Although gallium arsenide
(GaAs) is a demonstrated exceptional material
for fabricating quantum dots, it has the po-
tential drawback that confined electrons in-
teract with on the order of 10
6
spin-3/2 nuclei
through the hyperfine interaction. Here we
present a quantum two-level system (logical
qubit) based on two-electron spin states (14)
and demonstrate coherent control of this
system through the use of fast electrical control
of the exchange interaction. We first show by
direct time-domain measurements that the
time-ensemble-averaged dephasing time (T
2
*)
of this qubit is È10 ns, limited by hyperfine
interactions. We then demonstrate Rabi oscil-
lations in the two-spin space (including a 180-ps
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SWAP
p
operation between two electron
spins) and implement spin-echo sequences,
showing an extended spin coherence time, T
2
,
beyond 1 ms.
Isolating and measuring two electrons.
Gate-defined double quantum dot devices are
fabricated using a GaAs/AlGaAs heterostruc-
ture grown by molecular beam epitaxy with a
two-dimensional electron gas 100 nm below
the surface, with density È2 10
11
cm
j2
.
When biased with negative voltages, the
patterned gates create a double-well potential
(Fig. 1A). Tunnel barriers [controlled by
voltages V
L
and V
R
(L, left; R, right)] connect
each dot to adjacent reservoirs, allowing
electrons to be transferred into the dots.
Interdot tunneling (at a rate set by voltage
V
T
) allows electrons to be moved between
dots when the detuning parameter e º V
R
–
V
L
is adjusted. Measurements are performed
in a dilution refrigerator with electron tem-
perature T
e
È 135 mK, determined from
Coulomb blockade peak widths. Gates L and
R are connected via low-temperature bias tees
to high-bandwidth coaxial lines, allowing
rapid (È1 ns) pulsing of these gates (15).
High-frequency manipulation of a single elec-
tron, demonstrating the gigahertz bandwidth of
this setup, was reported in (16).
Quantum point contact (QPC) sensors
fabricated next to each dot serve as local elec-
trometers (17, 18), showing a few-percent
reduction of conductance when a single charge
is added to the adjacent dot. Figure 1B shows
the conductance, g
s
, of the right QPC sensor as
a function of V
L
and V
R
near the two-electron
regime. Each charge state gives a distinct value
of g
s
, decreasing each time an electron is added
to the system or when an electron is transferred
from the left dot to the right dot. Labels (m,n)
in each region indicate the absolute number of
electrons confined on the (left, right) dot in the
ground state. We focus on transitions involving
(0,2) and (1,1) two-electron states, where pre-
vious experiments have demonstrated spin-
selective tunneling (12, 13, 19, 20).
Voltage-controlled exchange. The rel-
ative energy detuning e of the (0,2) and (1,1)
charge states can be rapidly controlled by
RESEARCH ARTICLES
1
Department of Physics, Harvard University, Cam-
bridge, MA 02138, USA.
2
Department of Condensed
Matter Physics, Weizmann Institute of Science,
Rehovot 76100, Israel.
3
Materials Department, Uni-
versity of California at Santa Barbara, Santa Barbara,
CA 93106, USA.
Fig. 1. (A) Scanning electron micrograph of a sample identical to the one measured, consisting of
electrostatic gates on the surface of a two-dimensional electron gas. Voltages on gates L and R
control the number of electrons in the left and right dots. Gate T is used to adjust the interdot
tunnel coupling. The quantum point contact conductance g
s
is sensitive primarily to the number of
electrons in the right dot. (B) g
s
measured as a function of V
L
and V
R
reflects the double-dot charge
stability diagram (a background slope has been subtracted). Charge states are labeled (m,n), where
m is the number of electrons in the left dot and n is the number of electrons in the right dot. Each
charge state gives a distinct reading of g
s
.
30 SEPTEMBER 2005 VOL 309 SCIENCE www.sciencemag.org
2180
applying calibrated voltage pulses to gates L
and R (Fig. 2B). For e 9 0, the ground-state
charge configuration is (0,2). Tight confine-
ment in (0,2) favors a spin-singlet configura-
tion, denoted (0,2)S. The corresponding (0,2)
triplet states are energetically inaccessible,
lying È400 meV above (0,2)S and are ne-
glected in the following discussion. For e G 0,
the ground state configuration is (1,1). In this
case, four spin states are accessible: the singlet
(S 0 0), denoted S [suppressing the (1,1) label];
and three triplets (S 0 1), denoted T
–
,T
0
,and
T
þ
, corresponding to m
s
0 –1, 0, þ1.
In the absence of interdot tunneling, the
two spins in the (1,1) configuration are inde-
pendent; that is, S, T
0
,T
þ
,andT
–
are degenerate.
At finite magnetic fields, S and T
0
are de-
generate. When interdot tunneling is present,
the (0,2) and (1,1) charge states hybridize,
which results in an exchange splitting J(e)
between the S and T
0
spin states of (1,1) that
depends on detuning (Fig. 2B). Near zero
detuning, exchange J (e Y 0) becomes large
(equal to half the splitting of symmetric and
antisymmetric charge states at e 0 0); for large
negative detuning, e ¡ –J(0), exchange van-
ishes, J(e) Y 0, and the spins again become
independent. Except where noted, a perpendic-
ular magnetic field B 0 100mTisusedtosplit
off the T
T
states from T
0
by the Zeeman energy
E
z
0Tg*m
B
B È 2.5 meV (g* 0 –0.44 is the
electron g factor in GaAs; m
B
is the Bohr
magneton). The split-off T
þ
state crosses the
hybridized singlet S when J(e) 0 g*m
B
B
(vertical green line in Fig. 2B), allowing J(e)
to be readily measured, as discussed below.
In all measurements, a cyclical pulse se-
quence is used (see Fig. 2A for a schematic
representation). A pulse transfers the (0,2)S
state into the spatially separated (1,1) singlet
state, S. The singlet state is manipulated with
various control techniques (discussed below).
After manipulation, the resulting (1,1) spin
state is projected back onto (0,2)S for a mea-
surement of the singlet probability P
S
. P
S
is
measured with the QPC: the T states of (1,1)
remain in a spin-blocked configuration, whereas
the S state tunnels directly to (0,2)S. This spin-
to-charge conversion readout is based on the
same mechanism that results in rectification in
dc transport found in similar devices (19, 20).
The majority of the duty cycle is spent in the
measurement configuration (e 9 0), so that the
slow (time-averaged) measurement of the QPC
conductance reflects the charge configuration
during the measurement phase (12, 13).
Even though we can coherently control and
measure two-electron spin states electrically,
the local solid-state environment remains crit-
ically important. For our device, each electron
is coupled to roughly 10
6
GaAs nuclei through
the hyperfine interaction. The hyperfine in-
teraction results in an effective random mag-
netic field with magnitude B
nuc
È 1to5mT
(13, 21, 22). These random hyperfine fields
evolve slowly (910 ms) relative to typical pulse
sequence periods and result in spin dephasing,
thereby coupling two-electron spin states
(23–28). At large negative detuning, where
J(e) G g*m
B
B
nuc
, these effective fields mix S
and T states.
The logical qubit. With the T
T
states split
off by an applied field B d B
nuc
, the states S
and T
0
form an effective two-level system (or
qubit) with Hamiltonian
H 0
JðeÞ DB
z
nuc
DB
z
nuc
0
0
@
1
A
where DB
z
nuc
is the difference in random
hyperfine fields along the applied field
direction. To facilitate the following discus-
sion, we define a Bloch sphere for the S-T
0
two-level system that has S and T
0
at the
north and south poles (z axis) and the eigen-
states of the instantaneous nuclear fields
within this subspace, kj,
À
and k,j
À
,asthe
poles along the x axis (Fig. 3A).
Dephasing of the separated singlet.
The pulse sequence described in Fig. 3A is
used to measure the dephasing of the separated
singlet state as a function of the time t
S
that the
system is held at large detuning [with J(e) G
g*m
B
B
nuc
]. This time is a T
2
* time (the asterisk
indicates an average over many experimental
runs), because relative phase evolution of the
separated spins can convert the initial singlet
into a triplet, which will not be able to return
to (0,2)S. The (0,2)S initial state is prepared
each cycle by allowing tunneling to the res-
ervoir with (0,2)S below the Fermi level of
the leads and the (0,2) triplets above. This
energetic configuration is held for 200 ns, and
through a process in which an electron is ex-
changed with the leads, (0,2)S is prepared.
The state is then separated into (1,1) using
rapid adiabatic passage, where e is swept
from a positive value to a large negative value
quickly (È1 ns) relative to the nuclear mixing
time ÈI/(g*m
B
B
nuc
) but slowly as compared
to the tunnel splitting of the hybridized charge
states ÈI/J(0). This yields a separated singlet,
S. After a separation time t
S
, the state is
projected back onto (0,2)S, again using rapid
adiabatic passage, and the system is held at
the measurement point for a time t
M
È 5to
10 ms G T
1
.
The average singlet probability measured
after a separation time of 200 ns, P
S
(e,B, t
S
0
200 ns), is shown in Fig. 2C as a function of
Fig. 2. (A) The control cycle for experiments generally consists of preparation, singlet separation,
evolution of various kinds, and projection onto the (0,2) singlet state (measurement). Projective
measurement is based on the spin-blockaded transition of T states onto (0,2)S, whereas S states proceed
freely, allowing S to be distinguished from T by the charge sensor during the measurement step. (B)
Energy diagram near the (1,1)-to-(0,2) charge transition. A magnetic field splits T states by the Zeeman
energy. At the S-T
0
degeneracy (light blue region) and the S-T
þ
degeneracy (green line), hyperfine fields
drive evolution between S and the respective T states. (C) Singlet probability P
S
after t
s
0 200 ns, as a
function of detuning e and magnetic field B maps out degeneracies of S-T
0
(e G È –1.2 mV) and S-T
þ
(dashed green curve). (D) Dependence of exchange on detuning, extracted from the fit of J(e) 0 g*m
B
B
along the S-T
þ
resonance, assuming g* 0 –0.44 [dashed curve in (C)]. (Inset) For J(e) d g*m
B
B
nuc
,
eigenstates S and T
0
are split by J(e). At large negative detuning, J(e) ¡ g*m
B
B
nuc
, and S and T
0
are
mixed by hyperfine fields but eigenstates kj,
À
and k,j
À
are not.
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detuning at the separation point and applied
field (29). Evident in the data are a funnel-
shaped feature where S and T
þ
cross (vertical
green line in Fig. 2B) and are rapidly mixed
by hyperfine fields. The degeneracy occurs at
J(e) 0 g*m
B
B, allowing J(e) to be measured
(Fig. 2D) by mapping the location of this
feature in P
S
(e,B). At larger detuning, where
J(e) G g*m
B
B
nuc
,theSandT
0
states approach
degeneracy and are susceptible to hyperfine
mixing, which reduces P
S
(light blue area of
Fig. 2B). When B G B
nuc
, all three degenerate
triplet states can mix with S at large detuning,
which further reduces P
S
as compared to the
finite-field case.
For applications involving the manipulation
of entangled pairs of electrons, a relevant
question is how long the electrons can be
spatially separated before losing phase coher-
ence. We measure this time by varying the
singlet separation time t
S
. The time evolution
of the average singlet return probability,
P
S
(t
S
), measured using the pulse sequence in
Fig. 3A with e 0 –6 mV, is shown in Fig. 3B.
As t
S
increases, P
S
decreases from È1ona
10-ns time scale, saturating after 20 ns to
P
S
È 0.5 (0.7) for B 0 0 (100) mT.
A semiclassical model of dephasing of the
separated singlet was investigated in (23). It
assumes independent quasistatic nuclear fields
acting on the two spins (26, 30) and ideal mea-
surement contrast, and yields Gaussian-like
decay on a time scale T
2
*fromP
S
(t
S
0 0) 0 1
to long-time saturating values P
S
(t
S
d T
2
*) 0
1/3 for B ¡ B
nuc
and P
S
(t
S
d T
2
*) 0 1/2 at
B d B
nuc
. The field dependence is caused
by the lifting of the triplet degeneracy with
the external field, although the naı¨ve expecta-
tion based on incoherent mixing would be
P
S
(t
S
d T
2
*) 0 1/4, not 1/3, at B 0 0. Fits to
the measured P
S
(t
S
)yieldT
2
* 0 10 T 1ns,
corresponding to B
nuc
0 2.3 mT, consistent
with previous measurements (13, 22, 31). An
observed È40% reduction of contrast is treated
as a fit parameter. The predicted weak over-
shoot of P
S
for B 0 0, a remnant of Rabi
oscillations (23), is not seen in these data.
Spin SWAP and Rabi oscillations in the
kj,
À
, k,j
À
basis. By initializing from (0,2)S
using slow ramping of detuning, the (1,1)
system can be initialized into the ground state
of the nuclear field [defined as kj,
À
(Fig. 2D,
inset)] instead of the singlet state S. This
initialization scheme is illustrated in Fig. 4A:
after preparing (0,2)S (as described above),
detuning is swept to e G 0 slowly relative to
tunnel splitting but quickly relative to the
nuclear mixing time through the S-T
þ
degen-
eracy. The system is then ramped slowly as
compared to the nuclear mixing time (t
A
È
1 ms d T
2
*) to large negative detuning. This
slow lowering of J(e) leads to adiabatic
following of the initial state S into the state
kj,
À
, the ground state of the Hamiltonian with
J Y 0(30, 32). Readout follows the same
steps in reverse: ramping slowly out of the
large detuning region unloads kj,
À
to S and k,j
À
to T
0
. Then, moving quickly though S-T
þ
de-
generacy and finally projecting onto (0,2)S
measures the fraction that was in the state kj,
À
before readout.
Once initialized in kj,
À
, the application of a
finite exchange J(e) for a time t
E
rotates the
spin state about the z axis of the Bloch sphere,
in the plane containing kj,
À
and k,j
À
, through
an angle f 0 J(e)t
E
/I. The case J(e)t
E
/I 0 p
constitutes a SWAP operation, rotating the
state kj,
À
into the state k,j
À
.
Figure 4B shows P
S
(e,t
E
) oscillating as a
function of both t
E
and e, with minima of the
singlet probability corresponding to J(e)t
E
/I 0
p,3p,5p,I. The inset shows theoretical
predictions P
S
0 {1 þ cos[J(e)t
E
/I]}/2, using
values for J(e) obtained independently from
the S-T
þ
resonance measurement as in Fig.
2C. In Fig. 4C, we plot exchange oscillations at
the four values of detuning marked by the
dashed lines in Fig. 4B. Data are fit using an
exponentially damped cosine with offset, am-
plitude, decay time, and phase as fit param-
eters. To achieve faster p-pulse times, J(e)can
be increased by setting V
T
to increase interdot
tunnel coupling and by moving to less
negative (or even positive) detunings during
the exchange pulse (Fig. 4D). The fastest
p-pulse time obtained using these methods
is È350 ps (33).
We note that the observed decay time of
Rabi oscillations is proportional to the Rabi
period, suggesting that dephasing scales with
the value of J(e) during the exchange pulse and
may reflect gate noise during the t
E
interval.
The contrast (È45%) seen in Fig. 4, B and C,
is consistent with the contrast obtained in the
singlet separation measurement of T
2
*.
Singlet-triplet spin-echo. Voltage-
controlled exchange provides a means of
refocusing the separated singlet to undo de-
phasing due to the local hyperfine fields. The
pulse sequence is shown in Fig. 5A and is sim-
ilar to refocusing sequences used in nuclear
magnetic resonance (34, 35). The separated
singlet S will dephase at large negative de-
tuning [J(e) È 0] due to local hyperfine fields
after a separation time t
S
. In the Bloch sphere
representation, hyperfine dephasing results in a
rotation by a random nuclear-field–dependent
angle about the x axis. Thus, in each run the
Bloch vector rotates by a random amount about
the x axis. The dephased (1,1) state can be
refocused to S by applying a pulse of finite
exchange J(e)foratimet
E
,whereJ(e)t
E
/I 0
p,3p,5p,I, which rotates the Bloch vector
around the z axis by an angle p,3p,5p,I,and
waiting for a time t
S
¶ 0 t
S
.
The singlet probability P
S
(e,t
E
) measured
using the spin-echo sequence (Fig. 5A) is
shown as a function of detuning and t
E
in
Fig. 5B. Singlet recoveries (black regions) are
observed for p,3p,and5p exchange pulses. A
plot of the theoretical prediction P
S
0 {3 –
cos[J(e)t
E
/I]}/4 (Fig. 5B, inset) using values
Fig. 3. (A) Pulse sequence used to measure T
2
*. The system is initialized into
(0,2)S and transferred by rapid adiabatic passage to the spatially separated S
state. With T
T
separated by a Zeeman field, S and T
0
mix at large detuning
(light blue region), where hyperfine fields drive rotations about the x axis in
the Bloch sphere. After a separation time t
S
, the state is projected onto (0,2)S.
(B) Singlet probability P
S
measured using the calibrated QPC charge sensor, as
afunctionoft
S
at100mT(blackcurve)and0mT(redcurve).Fort
S
¡ T
2
*,
thesingletstatedoesnothaveampletimetodephase,andP
S
È 1. For t
S
d
T
2
*, P
S
È 0.7 at 100 mT and P
S
È 0.5 at 0 mT. A semiclassical model of
dephasing due to hyperfine coupling (23)predictsP
S
È 1/2 at high field and
P
S
È 1/3 at zero field. Fits to the model (solid curves), including a parameter
adjusting measurement contrast, give T
2
* 0 10 ns and B
nuc
0 2.3 mT.
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for J(e) measured independently from the
S-T
þ
resonance condition compares well with
experiment. We note greater noise in these data
than in Fig. 4. We speculate that this noise,
which is È100 times noisier than the QPC
sensor readout instrument noise, is likely due to
slow fluctuations in the nuclear system. Noise
from a possibly similar origin was recently
observed in dc transport through a double
quantum dot system (36). Figure 5C shows P
S
(red) as a function of the difference in
dephasing and rephasing times, t
S
– t
S
¶,forin-
creasing values of the total time spent at large
detuning, t
S
þ t
S
¶, averaged over 10 data sets.
Differences in t
S
and t
S
¶ result in imperfect
refocusing and decrease the recovery amplitude
onacharacteristictimescalet
S
– t
S
¶ 0 T
2
*.
For each value of t
S
þ t
S
¶,thedataarefit
to a Gaussian form giving T
2
* 0 9 T 2ns,
consistent with measurements of the singlet
decay discussed above. The best-fit heights
for each t
S
þ t
S
¶ time are plotted as the black
data points in Fig. 5C. A fit to an exponential
decay with an adjustable offset to correct for
the finite measurement contrast gives a char-
acteristic coherence time of 1.2 ms, which sets a
lower bound on T
2
. Comparing measured val-
ues of T
2
* and this bound on T
2
, we note that
a simple spin-echo sequence extends the
coherence time of a spatially separated singlet
by more than a factor of 100. We find that
two spin-echo pulse sequences applied in
series (Carr-Purcell) extends the bound on T
2
by at least another factor of 2. The coherence
time of our qubit using the simple spin-echo
sequence exceeds the
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SWAP
p
operation time
by a factor of È7000. Because the echo se-
quence relies on gate-voltage control of J(e),
it is susceptible to charge dephasing during
the exchange pulse. The interplay between
charge dephasing during the exchange pulse
and dephasing due to nuclear processes war-
rants further investigation (30, 37).
Summary and outlook. We have dem-
onstrated coherent quantum control of a logical
qubit based on two-electron spin states. Spin
states are prepared, manipulated, and measured
using fast control of the exchange interaction.
Rapid electrical control of the exchange
interaction is used to measure T
2
*, to demon-
strate Rabi oscillations and a 180-ps
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SWAP
p
operation, and to greatly reduce dephasing of
a spatially separated spin-singlet state with
spin-echo techniques. Moreover, the echo se-
quence implements a dynamical decoherence-
free subspace (38, 6), which allows arbitrary
two-electron spin states in S-T
0
subspace to be
protected from noise. Furthermore, our results
show that even in the presence of dephasing,
such an encoded logical qubit can be manip-
ulated efficiently with effectively long coher-
ence times. This two-electron spin qubit may
provide a starting point for implementation of
quantum computation schemes with consider-
able practical advantages: All operations for
preparing, protecting, and measuring entangled
electron spins can be implemented by local
electrostatic gate control. We anticipate that
the techniques developed in this work will lead
to intriguing prospects for experimental real-
izations of ideas from quantum information
science in semiconductor nanostructures.
Fig. 4. (A) Pulse sequence demonstrating exchange control. After initializing
into (0,2)S, detuning e is swept adiabatically with respect to tunnel
coupling through the S-T
þ
resonance (quickly relative to S-T
þ
mixing),
followed by a slow ramp (t
A
È 1 ms) to large detuning, loading the system in
the ground state of the nuclear fields kj,
À
. An exchange pulse of duration t
E
rotates the system about the z axis in the Bloch sphere from kj,
À
to k,j
À
.
Reversing the slow adiabatic passage allows the projection onto (0,2)S to
distinguish states kj,
À
and k,j
À
after time t
E
. Typically, t
S
0 t
S
¶ 0 50 ns. (B) P
S
as a function of detuning and t
E
.Thez-axis rotation angle f 0 J(e)t
E
/I results
in oscillations in P
S
as a function of both e and t
E
.(Inset)ModelofP
S
using
J(e)extractedfromS-T
þ
resonance condition, assuming g* 0 –0.44 and ideal
measurement contrast (from 0 to 1). (C) Rabi oscillations measured in P
S
at
four values of detuning indicated by the dashed lines in (B). Fits to an
exponentially damped cosine function, with amplitude, phase, and decay
time as free parameters (solid curves), are shown. Curves are offset by 0.3 for
clarity. (D) Faster Rabi oscillations are obtained by increasing tunnel coupling
and by increasing detuning to positive values, resulting in a p-pulse time of
È350 ps.
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References and Notes
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15. Except where noted, a Tektronix AWG520 pulse
generator with a 1-ns minimum pulse width was
used for fast gate control. The sample response to
fast pulses was checked by measuring g
s
with pulses
applied individually to gates L and R. A doubling of
the charge stability diagram was observed for pulse
widths down to 1 ns (12).
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29. The separation detuning can be changed by keeping pulse
displacements fixed and sweeping the measurement
point detuning using dc gates, or by keeping the
measurement point fixed and changing pulse parameters.
30. J. M. Taylor et al., in preparation.
31. The semiclassical model developed in (23, 30) yields
P
S
ðt
S
Þ 0 1 j
C
1
2
1 j e
jðt
S
=T
2
Þ
2
for B d B
nuc
and
P
S
ðt
S
Þ0 1j
3
4
C
2
1j
1
9
1 j 2e
j
1
2
t
S
=T
2
ðÞ
2
ðt
S
=T
2
Þ
2
j
nn
1
o
2
for B ¡ B
nuc
. A fit to the 100-mT data gives
C
1
0 0.62 T 0.01 and T
2
* 0 10 T 1 ns. A fit to the B 0 0
data with T
2
* 0 10 ns fixed gives C
2
0 0.74 T 0.01.
32. We have verified that the system is in the ground
state of the nuclear fields by varying the pulse
parameters t
S
and t
S
¶. No dependence on these
parameters was observed when varied as in the
singlet-triplet spin-echo measurements.
33. Two synchronized Tektronix AWG710B pulse gen-
erators were used for the fast Rabi measurements.
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39. We acknowledge useful discussions with S. Das
Sarma, H.-A. Engel, X. Hu, D. Loss, E. Rashba, and P.
Zoller. Funding was provided through the Army
Research Office under grants DAAD55-98-1-0270
and DAAD19-02-1-0070; the Defense Advanced
Research Projects Agency–Quantum Information
Science and Technology program; NSF under grant
DMR-0072777; the NSF Career Program; the Harvard
Center for Nanoscale Systems; and the Sloan and
Packard Foundations.
5 July 2005; accepted 22 August 2005
Published online 1 September 2005;
10.1126/science.1116955
Include this information when citing this paper.
Fig. 5. (A) Spin-echo pulse sequence. The system is initialized in (0,2)S and
transferred to S by rapid adiabatic passage. After a time t
S
at large negative
detuning, S has dephased into a mixture of S and T
0
due to hyperfine
interactions. A z-axis p pulse is performed by making detuning less negative,
moving to a region with sizable J(e)foratimet
E
. Pulsing back to negative
detunings for a time t
S
¶ 0 t
S
refocuses the spin singlet. (B) P
S
as a function
of detuning and t
E
.Thez-axis rotation angle f 0 J(e)t
E
/I results in
oscillations in P
S
as a function of both e and t
E
.(Inset)ModelofP
S
using
J(e)extractedfromtheS-T
þ
resonance condition, assuming g* 0 –0.44 and
ideal measurement contrast (from 0.5 to 1). (C) Echo recovery amplitude P
S
plotted as a function of t
S
– t
S
¶ for increasing t
S
þ t
S
¶ (red points), along
with fits to a Gaussian with adjustable height and width. The best-fit
width gives T
2
* 0 9 ns, which is consistent with the value T
2
* 0 10 ns
obtained from singlet decay measurements (Fig. 3B). Best-fit heights (black
points) along with the exponential fit to the peak height decay (black curve)
give a lower bound on the coherence time T
2
of 1.2 ms.
R ESEARCH A RTICLES
30 SEPTEMBER 2005 VOL 309 SCIENCE www.sciencemag.org
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![Fig. 2. (A) The control cycle for experiments generally consists of preparation, singlet separation, evolution of various kinds, and projection onto the (0,2) singlet state (measurement). Projective measurement is based on the spin-blockaded transition of T states onto (0,2)S, whereas S states proceed freely, allowing S to be distinguished from T by the charge sensor during the measurement step. (B) Energy diagram near the (1,1)-to-(0,2) charge transition. A magnetic field splits T states by the Zeeman energy. At the S-T0 degeneracy (light blue region) and the S-Tþ degeneracy (green line), hyperfine fields drive evolution between S and the respective T states. (C) Singlet probability PS after ts 0 200 ns, as a function of detuning e and magnetic field B maps out degeneracies of S-T0 (e G È –1.2 mV) and S-Tþ (dashed green curve). (D) Dependence of exchange on detuning, extracted from the fit of J(e) 0 g*mBB along the S-Tþ resonance, assuming g* 0 –0.44 [dashed curve in (C)]. (Inset) For J(e) d g*mBBnuc, eigenstates S and T0 are split by J(e). At large negative detuning, J(e) ¡ g*mBBnuc, and S and T0 are mixed by hyperfine fields but eigenstates kj,À and k,jÀ are not.](/figures/fig-2-a-the-control-cycle-for-experiments-generally-consists-vixwqlmu.webp)
