Download Universal quantum control in two-electron spin quantum bits using

Universal quantum control in two-electron spin quantum
bits using dynamic nuclear polarization.
Sandra Foletti1∗, Hendrik Bluhm1∗ , Diana Mahalu2 , Vladimir Umansky2 & Amir Yacoby1
1
Department of Physics, Harvard University, Cambridge, MA 01238, USA
Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 76100, Israel
*These authors contributed equally to this work.
2
Achieving universal control and prolonging coherence of quantum bits remain some of the
most challenging topics in spin-based quantum computation. Logical quantum bits, constructed out of two or more physical quantum bits, can be designed to prolong coherence
by operating in a subspace that is less susceptible to the effects of the environment1 . However, such decoherence free subspace may come at the expense of increased complexity in the
control of the qubit. Spin based logical qubits composed of two electron spins in a double
quantum dot structure have recently been shown to be dynamically protected against the
fluctuating nuclear spin environment through rapid exchange of two electrons. While such
exchange operation permits rotations around one axis of the qubit, full universal control of
the qubit requires a magnetic field gradient between the two dots to enable rotations around a
second axis. Here we demonstrate full quantum control of the two electron logical spin qubit
with nanosecond operation times. We have used quantum state tomography to characterize
the evolution of the qubit state around a continuously tunable combined axis. The magnetic
filed gradient is internally generated using dynamic nuclear polarization of the underlying
Ga and As nuclear sublattice. Field gradients of several hundred milliTesla can be sustained
for more than 30 minutes and lead to precession frequencies that can exceed 1 GHz.
The potential realization of quantum computers has attracted a lot of attention because of
their promise to perform certain calculations practically intractable for classical computers2. A
variety of possible implementations are currently under investigation, including carbon nanotubes3,
nitrogen vacancies4 and flux qubits5. In the present work, the two-level quantum bit (basic element
of the quantum computer) is encoded in the spin state of two electrons confined in a double-well
potential. This semiconductor-based system has potential for good scalability, manipulations are
all electrical and potentially fast enough to enable 104 gate operations within the coherence time
(essential for quantum error correction). The phase space of the two-level system is on a one-to-one
correspondence with the points on the surface of a three dimensional sphere, the Bloch sphere2 ,
where the basis states (corresponding to the classical 0 and 1) are represented at the north and south
pole (Fig.1g). A generic manipulation of the qubit needed to implement universal gate operations
requires the ability to perform rotations around two axes in the Bloch sphere6 (for example the z
and x-axis). For two-electron spin qubits, rotations around the z axis correspond to the coherent
exchange of two electrons and has been recently demonstrated by Petta et al.7 . Rotations around
the second axis occur in the presence of a non-uniform magnetic field within the double-well
1
potential, making the two spins precess at different rates. To this end, we take advantage of the
interaction of the electrons with the nuclear field of the Ga and As sublattices of the host material.
It has been established though, that the fluctuation of this hyperfine field are also a major source of
decoherence8–11 . In this Letter we demonstrate the possibility of building up a gradient in hyperfine
fields that significantly exceeds the fluctuations and can be sustained for times longer than 30 min.
This is done by employing pumping schemes that transfer spin and thus magnetic moment from
the electronic system to the nuclear sublattice. The internally created gradient of nuclear field, in
excess of 100 mT, together with the coherent exchange of the two electrons allow us to perform
rotations around a variable axis. The coherent manipulation is demonstrated via quantum state
tomography measurement, with which the trajectory of the evolution of a state within the Bloch
sphere can be reconstructed.
The double-well potential in which the electrons are confined is created by applying a negative voltage on metal gates deposited on top of a two dimensional electron gas embedded in a
GaAs/AlGaAs heterostructure. The negative potential depletes the region underneath the metal
gates creating two isolated puddles of electrons (double quantum dot, Fig.1a). The number of
electrons in the dots can be controlled by tuning the potential on the gates. We restrict the occupation of the double quantum dot to two electrons, and describe their spatial separation by the
parameter ε: for ε ≫ 0 both electrons are in the right quantum dot, the (0,2) configuration; for
ε ≪ 0 one electron occupies each dot, also indicated by (1,1). The dots’ charge configuration
can be continuously swept through the intermediate configurations by varying the voltages on the
metal gates. In the (0,2) charge configuration
√ the only energetically accessible spin configuration
is the singlet state S(0, 2) = (↑↓ − ↓↑)/ 2 (the arrows indicate the direction of the electron
spin in the left and right dot). As we separate the electrons, the two wavefunctions overlap decreases and four spin configurations
become energetically available: the singlet S(1, 1) and three
√
triplets T0 = (↑↓ + ↓↑)/ 2, T− =↓↓ and T+ =↑↑ (Fig.1b). We select the states S(1, 1) and
T0 , both having a vanishing z component of angular momentum, as qubit basis states and lift the
degeneracy with the states T− and T+ by applying an external magnetic field Bext . The Zeeman
energy Ez = gµB B (g = −0.4 the g-factor for GaAs, µB Bohr’s magneton) shifts the T+ state
to lower energies, creating a crossing point with the singlet (marked by a red circle in Fig.1c) at
a value of ε that depends on B = Bext + Bnuc , with Bnuc = (Bnuc,L + Bnuc,R )/2 the average
hyperfine field (Bnuc,L (Bnuc,R ) the nuclear field felt by the electron in the left (right) dot, Fig.1b).
The rate of coherent exchange of the two electrons (rotation around the z-axis) is controlled by
the energy splitting between S(1, 1) and T0 , J(ǫ). The rotation around the x-axis is controlled
by the z-component of the difference in local magnetic field ∆Bnuc = Bnuc,L − Bnuc,R , if we
z
thus p
let a state evolve around a combined axis J/2z + gµB Bnuc
/2x, the precession frequency is
−15
z )2 /h (h = 1 · 10
f = J 2 + (gµB ∆Bnuc
µeV /T is Plank’s constant).
We present two polarization schemes by which the gradient can be increased to values significantly exceeding fluctuations, both making use of the crossing point between S(1, 1) and T+ .
At this point, transitions between the two states are driven by the perpendicular component of
∆Bnuc 12 and accompanied by a spin flip of the nuclei in order to conserve the total angular mo2
Figure 1: Pump and measurement schemes. a, SEM micrograph of a device similar to the
one measured. Gates GL and GR control the charge configuration of the two dots, the central
gates (Nose and Tail) tune the tunnel barrier transparency between the two dots. The average
charge configuration is detected by measuring the conductance (GQP C ) through a capacitively
coupled quantum point contact. b, Bnuc,L and Bnuc,R are the local magnetic fields experienced by
the electrons in the left and right dot through hyperfine coupling with the Ga and As nuclei. c,
Schematic representation of the energy levels at the (0,2)-(1,1) charge transition for finite external
magnetic field. The detuning ε from the degeneracy point is controlled via a change of voltage on
GL and GR. Two pulse cycle are presented: 1) for nuclear pumping the system is moved to point
P where S and T+ are degenerate and can mix 2) for the measurement pulse the system is moved
to a very negative detuning where the states S and T0 can mix. d, The measurement pulse scheme
e, The S-pumping pulse scheme f, The T+ -pumping pulse scheme all shown as a function of GL
and GR. g, Geometrical representation (Bloch sphere) of the two level system (S and T0 ) and the
two rotation axes allowing the implementation of universal single qubit gates.
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mentum. In the newly introduced T+ -pumping scheme, the system is swept slowly into (0, 1) and
subsequently reloaded into the (1, 1) charge state (Fig.1f). First the right and then the left electron
align with the external field due to large Zeeman energy (≈ 12.5 µeV at 500 mT), which preferentially loads a T+ state. Adiabatically sweeping across the S-T+ transition in 100 ns induces a
transition from T+ to S with the transfer of a unit of angular momentum ~ to the nuclei. Alternatively, in the previously used S-pumping scheme13 the system is reset to S(0,2) before the sweep
across the S −T+ -transition, where the adiabatic passage guarantees a spin flip-flop in the direction
opposite to that of the T + cycle (Fig.1e).
The magnitude of the induced gradient can be assessed using a measurement scheme known
as T2∗ -pulse7 . This works as follows: after resetting the system into a S(0, 2) state, ε is changed and
z
set to point S in (1,1) for an evolution time τS . Here ∆Bnuc
≫ J(ε)/gµB drives the mixing between
S(1, 1) and T0 and the time evolution of the probability of being in a singlet state oscillates as
z
p(S) ∝ cos2 (gµB ∆Bnuc
·τS /h). When the system is brought back to point M only transitions from
S(1, 1) to S(0, 2) are allowed, while T0 remains blocked in the (1,1) charge configuration. This
spin-blockade effect allows to map the spin configuration of the state onto a charge configuration14 ,
which can be determined with a charge sensor15 . Here the sensor is a quantum point contact (QPC)
positioned near one side of the double quantum dot (Fig1a). The conductance of the QPC is very
sensitive to nearby electrostatic changes: the addition of an electron to one of the dots results in
a detectable change of the conductance. The detector signal, averaged over many T2∗ cycles is
proportional to the probability of being in a singlet state.
We now show that we can reach a steady state polarisation using either cycle and present
evidence that the induced polarisation points in opposite directions for the S and the T+ -pumping
cycles. We achieve nuclear steady states by alternating between one of the pumping cycles for
a time tpump and a measurement pulse T2∗ for 1 s, with a different τS (0 < τS < 30ns) for
each iteration. Each curve in Fig.2a is an average over 30 τS sweeps. An example showing 40
repetitions of τS sweeps for tpump = 60 ms is shown in Fig.2b, where an oscillatory signal with
a frequency fluctuating around a steady mean is clearly visible. This observation demonstrates
a gradient whose mean is larger than its fluctuations and can be kept constant over at least 40
min. The different values of tpump control the steady state value of the gradient in each dataset
z
(Fig.2a). We observe that the oscillations vanish (corresponding to a ∆Bnuc
fluctuating around
0) at moderate S-pumping rather then tpump = 0. This appears to reflect a small polarization
effect from the measurement pulses, that can be compensated with S-pumping (see Supplementary
Material). To compare the two pumping cycles, we have taken a measurement alternating between
them every 40 τS sweeps (see Fig.2c). The data show that upon changing the pump cycle, the
oscillations disappear and then recover after a few minutes in a way that suggests a sign change of
the gradient.
While determining the magnitude of the gradient with measurements of T2∗ , we can probe the
average nuclear field by monitoring the position of the S-T+ transition. This should clarify whether
spins are flipped only in one or both dots. Figs. 3a,b show interleaved measurements of the position
4
+
a
1.10 s T
0.80 s T+
0.42 s T+
0.24 s T+
0.06 s T+
P
S
0.00 s
0.03 s S
0.06 s S
0.09 s S
0.13 s S
0.18 s S
1
0.24 s S
0.42 s S
5
10
τ (ns)
S
1
PS
b
15
time (min)
0
20
25
30
0
21
42
0.4
0
10
20
30 τ (ns)
S
c
0
20
time (min)
0.45 s S
40
60
+
0.40 s T
80
0
5
10
τS (ns)
15
20
25
Figure 2: Build-up of a gradient with two different pumping cycles. a, Singlet return probability
as a function of separation time τS for different tpump . Each line is an average over the last 30 lines
of a measurement as the one shown in b.. b, Singlet return probability as a function of separation
time τS changing between 0 and 30 ns. Before each second of measurement (each pixel on the
line) we run a pump pulse for tpump = 60 ms. Each line is a repetition of the same measurement
z
and the whole sequence shows the stability of ∆Bnuc
over a time scale of 40 min. c, Crossover
between S and T+ -pumping. The intermittance and transient frequency change upon switching the
z
pump pulse suggests that S- and T+ -pumping produce ∆Bnuc
of opposite sign. The plotted data
5
is an average over the last nine of ten repetitions, so that the uppermost lines reflect the transition
from T+ - to S-pumping. Bext = 1.5 T for these data sets.
of the S–T+ transition16 and the oscillatory S–T0 mixing as a function of tpump using the T+ -pump
cycle. A shift of the S–T+ transition to the left corresponds to the build-up of an average field Bnuc
oriented opposite to the external magnetic field, consistent with spin flips from down to up in the
z
nuclear system. Fig.3e shows that at Bext = 500 mT, ∆Bnuc
reaches 230 mT while Bnuc is about 130
mT. The ratio of nearly a factor 2 indicates that the nuclei are polarized predominantly in one of the
two dots. However, measurements at different Bext show a less pronounced difference, implying
some degree of polarization in both dots. Data obtained using the S-pumping cycle (Fig.3e) show a
z
∆Bnuc
that tends to be slightly smaller than the average field. Note that this analysis may be subject
to various systematic errors. An imperfect alignment of the sample with the external magnetic field
may lead to orbital effects contributing to the calibration of the position of the S-T+ transition via
its dependence on Bext . This calibration may also be affected by a small measurement-induced
polarization, either in the calibration data or at the beginning of each measurement. Finally, the
z
S-T+ transition might also shift in response to ∆Bnuc
rather than just Bext + Bnuc , although we
z
estimate this effect to be negligible at least for small values of ∆Bnuc
. For the measurements as
in Fig. 3b, we introduced an overshoot towards larger ε before returning to M in the measurement
z
pulse in order to mitigate a loss of contrast for large ∆Bnuc
(see Supplementary Material).
z
In conjunction with the electrically controllable exchange operation, a controllable ∆Bnuc
allows single qubit rotations around an axis that can be tilted to any desired angle between 0 and
π/2 away from the x-axes (angle θ in Fig. 4d). We demonstrate and characterize such a rotation
using state tomography, consisting of three independent measurements of the probability of being
in a |Si ≡ |Zi, in an | ↑↓i ≡ |Xi and in a |Si+i|T0i ≡ |Y i state17 , with pulses shown in Fig. 4a.
This allows us to fully reconstruct the state time evolution. For each of the measurements, we first
prepare an | ↑↓i by loading a (0,2)S and adiabatically switching off J(ε) in (1, 1). The desired
rotation is performed by instantaneously setting J to a finite value for a time τrot . Rapidly returning
to S(0, 2) allows to measure p(|Zi) ≡ |hZ|ψi|2, whereas slowly increasing J brings | ↑↓i onto
|Si and | ↓↑i onto |T0 i, thus allowing the readout of p(|Xi) ≡ |hX|ψi|2. To obtain p(|Y i) ≡
|(hY |)|ψi|2, J is turned off for a time corresponding to a π/2 rotation around the x-axes before
z
rapidly returning to M. Results of this procedure for a particular choice of J and ∆Bnuc
are shown
in Fig. 4c as a function of τrot . For ideal pulses, one would expect p(|Xi) (p(|Zi)) to oscillate
sinusoidally between 1 (1/2) and some other value determined by the rotation axis, whereas p(|Y i)
should vary symmetrically around 1/2. Deviations from this behavior can be attributed to a finite
pulse rise time and high pass filtering of the pulses. The first causes an approximately adiabatic
drift of the rotation axis which prevents p(|Si) to return to the starting point, whereas the second
leads to slightly different ε-offsets and thus different J(ε) for different pulses, causing a 25 %
frequency change. Fits to a model (see Supplementary Material) incorporating these effects and
z
inhomogeneous broadening due to fluctuations in ∆Bnuc
give a good match with the data. In Fig.
4d, the data and fits are displayed in the Bloch sphere representation. We estimate that the errors
due to measurement noise, pulse imperfections other than those included in the model, incomplete
ensemble averaging over nuclear fluctuations and uncertainties in the QPC conductance calibration
(Supplementary Material) are on the order of 0.15 for all three probability measurements. They
could be substantially reduced by improving the characteristics of our high frequency setup, such
6
0
a
b
tpump (s)
0.3
0.6
PS 1
0.7
0.9
0.4
1
ε (mV)
c
1
d
0
5
τS (ns)
10
0
5
τS (ns)
10
P
S
0.9
0.8
0.7
0.7
−0.2
0
0.2 0.4
ε (mV)
0.6
300
e
S
200
(Bznuc,L+Bznuc,R)/2
|∆B |
z
(Bznuc,L+Bznuc,R)/2
−|∆Bz|
0
B
nuc
(mT)
100
−100
+
T
−200
0
0.2
0.4
0.6
0.8
1
tpump (s)
Figure 3: Comparison of ∆Bznuc and Bnuc a, Line scans of the position of the S–T+ transition as
a function of pump time per 2 s measurement interval for T+ -pumping. b, Singlet return probability
versus separation time τS under the same pumping conditions. c, Single line scan from the data in a
at tpump = 0.6 s. d, corresponding τS sweep from b. The continuous red line is a sinusoidal fit, the
dashed line reconstructs the actual, unaliased time dependence of the singlet probability. e, Bnuc
z
and ∆Bnuc
extracted from fits as shown in c, d for increasing and decreasing tpump for both T + and
S-pumping. The shift of the S-T + transition was converted to Bnuc using its measured dependence
z
on Bext (see supplementary material) 16 , ∆Bnuc
was obtained from the fitted oscillation frequency
after correcting for the aliasing. The dotted line shows the same data before this correction. Bext =
500 mT for these data sets.
7
that pulse compensation schemes would be simpler and pulse dependant variations of J could be
eliminated.
The mechanism responsible for the large gradient due to pumping is currently unknown.
While the spin from an electron is not necessarily deposited equally in the two dots18, 19 , the known
z
for both S and T+ -pumping. A possible cause might
mechanism cannot explain the large ∆Bnuc
be a different relaxation rate in the two dots, for example due to a small difference in size20 . The
z
relation between our results and those in Ref.16 , where a strong suppression of ∆Bnuc
was reported,
is currently under investigation. The apparent contradiction with the observation of Reilly et al.
that S-pumping becomes ineffective at fields exceeding a few tens of mT might be due to a different
coupling between the dots and the electron reservoirs (see Supplementary Material).
In the present experiment the inhomogenous dephasing of the precession, associated with
z
slow fluctuations of ∆Bnuc
is in the order of 20 ns. With a faster waveform generator (our is limited
to 1 ns steps) more than 10 coherent oscillations could be exploited within this time. Furthermore,
z
the fluctuations in ∆Bnuc
could be eliminated with the implementation of feedback loops that could
rapidly adjust the pumping time or the pumping efficiency. Compared to previous reports of gate
driven spin manipulation21, 22, our approach has the advantage of much shorter pulse times. This
means that if the coherence time at high fields is ≈ 10-100 µs as theoretically predicted23–25 , 104
coherent gate operations are within reach, allowing quantum error correction, essential to perform
arbitrary long computation.
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a
c 1
p(|↑↓〉)
b
0.3
p(|Y〉)
0.8
p(|S〉)
0.2
1
0.8
0.6
0.4
0
c Preparation
|S〉
dd
|S〉
10
τS (ns)
20
30
e
|S〉
X-readout |S〉
|↑↓〉
|↑↓〉
|↓↑〉
θ
|−Y〉
|↑↓〉
Y-readout |S〉
|↑↓〉
|T 〉
0
|Y〉
|T 〉
0
Figure 4: State tomography and universal gate. a, Pulse schemes to measure the singlet probability p(|Si) ≡ p(|Zi) , the | ↑↓i probability p(| ↑↓i) ≡ p(|Xi) and the |Si + i|T0 i probability
p(|Si + i|T0 i) ≡ p(|Y i) after rotation around a tilted, tunable axis. b, Measurements taken with
the X,Z and Y pulses (dots) and fits (line) to a numerical solution of the Schrödinger equation of
the S–T0 Hamiltonian incorporating the finite pulse rise time and inhomogeneous broadening due
z
to fluctuations in ∆Bnuc
. c, The preparation of an | ↑↓i state is done by adiabatically turning off
J. d, Representation of the measured and fitted trajectory in the Bloch sphere. In order to eliminate phase shifts due to the slightly different frequencies (see text), the time scales for the X-data
has been rescaled using spline interpolation so that the expected phase relations are maintained.
The blue line is a spline interpolation of the data points in panel b.. e, Visualization of the X and
Y-readout schemes on the Bloch sphere.
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Acknowledgements We are grateful for useful discussions with M.D.Lukin, M.Gullans, M.Stopa, J.J.Krich,
B.I.Halperin and C.Barthel. We acknowledge support from (GRANTS...)
Competing Interests
Correspondence
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to A. Y. (email: [email protected]
11