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Two-axis control of a singlet–triplet qubit with an integrated micromagnet
Contributed by S. N. Coppersmith, June 30, 2014 (sent for review March 19, 2014)

Significance
Qubits, the quantum-mechanical analog of classical bits, are the fundamental building blocks of quantum computers, which have the potential to solve some problems that are intractable using classical computation. This paper reports the fabrication and operation of a qubit in a double-quantum dot in a silicon/silicon–germanium (Si/SiGe) heterostructure in which the qubit states are singlet and triplet states of two electrons. The significant advance over previous work is that a proximal micromagnet is used to create a large local magnetic field difference between the two sides of the quantum dot, which increases the manipulability significantly without introducing measurable noise.
Abstract
The qubit is the fundamental building block of a quantum computer. We fabricate a qubit in a silicon double-quantum dot with an integrated micromagnet in which the qubit basis states are the singlet state and the spin-zero triplet state of two electrons. Because of the micromagnet, the magnetic field difference ΔB between the two sides of the double dot is large enough to enable the achievement of coherent rotation of the qubit’s Bloch vector around two different axes of the Bloch sphere. By measuring the decay of the quantum oscillations, the inhomogeneous spin coherence time
Fabricating qubits composed of electrons in semiconductor quantum dots is a promising approach for the development of a large-scale quantum computer because of the approach’s potential for scalability and for integrability with classical electronics. Much recent progress has been made, and spin manipulation has been demonstrated in systems of two (1⇓⇓⇓–5), three (6, 7), and four (8) quantum dots. A great deal of attention has focused on the singlet–triplet qubit in quantum dots (1, 2, 9⇓⇓⇓⇓⇓⇓⇓⇓–18), which consists of the Sz = 0 subspace of two electrons, for which the basis can be chosen to be a singlet and a triplet state. Full two-axis control on the Bloch sphere is achieved by electrical gating in the presence of a magnetic field difference ΔB between the two dots. In previous experiments (2, 9⇓⇓⇓⇓–14), ΔB arises from coupling to nuclear spins in the material, and slow fluctuations in these nuclear fields lead to inhomogeneous decoherence times that, without special nuclear state preparation, typically are shorter than the period of the quantum oscillations. In III–V materials, ΔB is large, so fast oscillation periods of order 10 ns are achievable, but the inhomogeneous dephasing time is also ∼10 ns, so that oscillations from ΔB are overdamped, ending before a complete cycle is observed (2). The fluctuations of the nuclear spin bath can be mitigated to some extent (10), but inhomogeneous dephasing times in III–V materials are short enough that high-fidelity control is still very challenging. Coupling to nuclear spins in silicon is substantially weaker, leading to longer coherence times, but also smaller field differences and hence slower quantum oscillations (14, 19).
Here, we report the operation of a singlet–triplet qubit in which the magnetic field difference ΔB between the dots is imposed by an external micromagnet (20, 21). Because the field from the micromagnet is stable in time, a large ΔB can be imposed without creating inhomogeneous dephasing. We present data demonstrating underdamped quantum oscillations, and, by investigating a variety of voltage configurations and two ΔB configurations, we show that the micromagnet indeed increases ΔB without significantly increasing inhomogeneous dephasing rates induced by coupling to nuclear spins.
A top view of the double-quantum dot device, which is fabricated in a Si/SiGe heterostructure, is shown in Fig. 1A; fabrication techniques are discussed in Materials and Methods, and an optical image of the micromagnet can be found in SI Appendix. The charge occupation of the two sides of the double dot is determined by measuring the current through a quantum point contact (QPC) next to one of the dots, as shown in Fig. 1A. Fig. 1B shows a charge stability diagram, obtained by measuring the current through the QPC as a function of gate voltages on the left plunger (LP) and right plunger (RP); the number of electrons on each side of the dot is labeled. The qubit manipulations are performed in the (1, 1) region (detuning ε > 0), and initialization and readout are carried out in the (0, 2) region (ε < 0). Fig. 1C shows the energy-level diagram at small but nonzero magnetic field. The three triplet states T− = |↓↓〉,
(A) Scanning electron micrograph of a device identical to the one used in the experiment before deposition of the gate dielectric and accumulation gates. An optical image of a complete device showing the micromagnet is included in SI Appendix. Gates labeled left side (LS) and right side (RS) are used for fast pulsing. The curved arrow shows the current path through the QPC used as a charge sensor. (B) IQPC measured as a function of VLP and VRP yields the double-dot charge stability diagram. Electron numbers in the left and right dot are indicated on the diagram. The red arrow denotes the direction in gate voltage space
By increasing the rise time of the pulse, so that it is slower than that used to observe the spin funnel, the voltage pulse can be used to cause S to evolve into a superposition of the S and T− states. In this case, the pulse remains adiabatic with respect to the S(0, 2)–S(1, 1) anticrossing; it is, however, only quasi-adiabatic with respect to the S-T− anticrossing, enabling use of the Landau–Zener mechanism to initialize a superposition between states S and T− (Fig. 1C, Inset) (22, 26⇓–28). Because the voltage pulse takes these states to larger detuning, an energy difference arises between the pair of states, and there is a relative phase accumulation between them. The return pulse leads to quantum interference between these two states and to oscillations in the charge occupation as a function of the acquired phase. Fig. 1D illustrates the ideal case, in which the rising edge of the pulse transforms S into an equal superposition of S and T−, followed by accumulation of a relative phase difference of π after pulse duration τS. Fig. 1F shows S-T− oscillations at Bext = −4 mT, obtained by applying a pulse with a rise time of 45 ns. Fig. 1H reports a line scan of the singlet probability for S-T− oscillations measured at Bext = −4 mT; for this measurement, the tip of the voltage pulse reaches large enough detuning that
We investigate the S-T0 oscillations, which correspond to a gate rotation of the S-T0 qubit, in more detail by changing the applied magnetic field Bext to −30 mT, and by working with faster pulse rise times. Here the S-T− anticrossing occurs at negative ε, as shown in Fig. 2A, making it easier to pulse through that anticrossing quickly enough so that the state remains S. In this situation, the relevant Hamiltonian H for ε > 0, in the S-T0 basis, is
(A) Schematic energy level diagram near the (0, 2)–(1, 1) charge transition at external field Bext = −30 mT. (B) Pulse sequence used to observe S-T0 oscillations. Starting at point M in the S(0, 2) ground state, a fast adiabatic pulse into (1, 1) is applied [it is adiabatic for the S(0, 2)–S(1, 1) anticrossing and sudden for the S(1, 1)-T0 anticrossing], to point P, where the exchange coupling J is comparable to or less than h, the energy from the magnetic field difference. The speed and axis of the rotation on the Bloch sphere during the pulse of duration τS depend on both J and h. Readout is performed by reversing the fast adiabatic pulse, which converts S(1, 1) to S(0, 2) but does not change the charge configuration of T0. (C) Probability PS of being in state S as a function of the detuning voltage at the pulse tip,
Rotations around the x axis of the Bloch sphere (the “ΔB gate”) are implemented using the simple one stage pulse shown in Fig. 2B, starting from point M in the (0, 2) charge state. The pulse rise time of a few nanoseconds is slow enough that the pulse is adiabatic through the S(0, 2) to S(1, 1) anticrossing. As ε increases, the eigenstates transition from S(1, 1) and T0 to other combinations of |↑↓〉 and |↓↑〉, and in the limit of ε → ∞, the eigenstates become |↑↓〉 and |↓↑〉. The voltage pulse applied is sudden with respect to this transition in the energy eigenstates, so that immediately following the rising edge of the pulse the system remains in S(1, 1). At large detuning, J ≤ h, and S-T0 oscillations are observed following the returning edge of the pulse. These oscillations arise from the x component of the rotation axis and have a rotation rate that is largely determined by the magnitude of h. Fig. 2C shows the singlet probability PS plotted as a function of the detuning voltage at the pulse tip,
Fig. 3 shows oscillations around the z axis of the Bloch sphere, obtained by applying the exchange pulse sequence pioneered in (2). Starting from point M in S(0, 2), we first ramp from M to N at a rate that ensures fast passage through the S(0, 2)-T− anticrossing, converting the state to S(1, 1), and then ramp adiabatically from N to P, which initializes to the ground state in the J < h region. The pulse from P to E increases J suddenly so that it is comparable to or bigger than h, so that the rotation axis is close to the z axis of the Bloch sphere. Readout is performed by reversing the ramps, which projects |↓↑〉 into the S(2, 0) state, enabling readout. Fig. 3C shows the singlet probability PS as a function of τS and the detuning of the exchange pulse
(A and B) Pulse sequence used to observe S-T0 oscillations when J > h. We initialize into the S(1, 1) state by preparing the S(0, 2) ground state at point M and ramping adiabatically through the (0, 2)–(1, 1)S anticrossing to an intermediate point N and then to P, where the singlet and triplet states are no longer energy eigenstates. Decreasing ε suddenly brings the state nonadiabatically to a value of the detuning where J is comparable or greater than h, inducing coherent rotations. The Bloch vector rotates around the new axis for a time τs. Reversing the sequence of ramps projects the state into S(0, 2) for readout. (C) Probability PS of observing the singlet as a function of the detuning voltage of the exchange pulse
Dependence of the inhomogeneous dephasing time
We also implemented both the ΔB and exchange gate sequences after performing a different cycling of the external magnetic field, which resulted in a different value of ΔB, corresponding to h ≃ 32 neV. The results obtained are qualitatively consistent with those shown in Figs. 2 and 3 (data shown in the SI Appendix, Fig. S3).
We now present evidence that the inhomogeneous dephasing is dominated by detuning noise and by fluctuating nuclear fields, and that it does not depend on the field from the micromagnet. Following ref. 18, we write
In summary, we have demonstrated coherent rotations of the quantum state of a singlet–triplet qubit around two different directions of the Bloch sphere. Measurements of the inhomogeneous dephasing time at a variety of exchange couplings and two different field differences demonstrate that using an external micromagnet yields a large increase in the rotation rate around one axis on the Bloch sphere without inducing significant decoherence. Because the materials fabrication techniques are similar for both quantum dot-based qubits and donor-based qubits in semiconductors (35), it is reasonable to expect micromagnets also should be applicable to donor-based spin qubits (36⇓–38). Micromagnets allow a difference in magnetic field to be generated between pairs of dots that does not depend on nuclear spins, and thus offer a promising path toward fast manipulation in materials with small concentrations of nuclear spins, including both natural Si and isotopically enriched 28Si.
Materials and Methods
All measurements reported in this manuscript were made on a double-quantum dot device fabricated in an undoped Si/Si0.72Ge0.28 heterostructure with a 12-nm-thick Si quantum well located 32 nm below the heterostructure surface. The double-quantum dot is defined using two layers of electrostatic gates (39⇓⇓⇓⇓–44). The lower layer of depletion gates is shown in Fig. 1A. The upper and lower layer of gates are separated by 80 nm of Al2O3 deposited via atomic layer deposition. The upper layer of gates is positively biased to accumulate a 2D electron gas in the Si well. The micromagnet, a rectangular thin film of cobalt, is deposited via electron-beam evaporation on top of the gate structure, 1.78 μm from the double-dot region (SI Appendix). A uniform in-plane magnetic field Bext is applied, and cycling Bext to relatively large values is used to change the magnetization of the micromagnet. All measurements were made in a dilution refrigerator with an electron temperature of ∼120 mK, as determined using the method of ref. 45.
Acknowledgments
We thank M. Rzchowski for help in characterizing cobalt films, and acknowledge useful correspondence with W. Coish and F. Beaudoin. This work was supported in part by Army Research Office Grant W911NF-12-0607; National Science Foundation (NSF) Grants DMR-1206915 and PHY-1104660; and the Department of Defense. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the US Government. Development and maintenance of growth facilities used for fabricating samples is supported by Department of Energy Grant DE-FG02-03ER46028, and nanopatterning made use of NSF-supported shared facilities (DMR-1121288).
Footnotes
- ↵1To whom correspondence may be addressed. Email: snc{at}physics.wisc.edu or maeriksson{at}wisc.edu.
Author contributions: X.W., D.R.W., J.R.P., D.K., J.K.G., R.T.M., Z.S., M.F., S.N.C., and M.A.E. designed research; X.W., D.R.W., J.R.P., D.K., J.K.G., R.T.M., Z.S., D.E.S., M.G.L., M.F., S.N.C., and M.A.E. performed research; D.R.W. contributed new reagents/analytic tools; D.R.W., D.E.S., and M.G.L. fabricated the sample; X.W., M.F., S.N.C., and M.A.E. analyzed data; and X.W., M.F., S.N.C., and M.A.E. wrote the paper, with input from all the authors.
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1412230111/-/DCSupplemental.
Freely available online through the PNAS open access option.
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