## Abstract

Rapid coherent control of electron spin states is required for implementation of a spin-based quantum processor. We demonstrated coherent control of electronic spin states in a double quantum dot by sweeping an initially prepared spin-singlet state through a singlet-triplet anticrossing in the energy-level spectrum. The anticrossing serves as a beam splitter for the incoming spin-singlet state. When performed within the spin-dephasing time, consecutive crossings through the beam splitter result in coherent quantum oscillations between the singlet state and a triplet state. The all-electrical method for quantum control relies on electron–nuclear spin coupling and drives single-electron spin rotations on nanosecond time scales.

Energy-level crossings, in which two quantum states cross in energy as a function of an external parameter, are ubiquitous in quantum mechanics (*1*). Coupling of the quantum states provided by tunnel coupling with strength Δ, for example, leads to hybridization of the states and results in an anticrossing with a minimum energy splitting 2Δ (*2*, *3*). Passing a quantum state through an anticrossing in the level diagram will result in a sweep-rate–dependent nonadiabatic transition probability *P*_{LZ}, commonly known as the Landau-Zener probability (*4*). The theory of Landau-Zener transitions can be applied to a diverse set of problems, ranging from electronic transitions in molecular collisions to chemical reactions to neutrino conversion in the sun (*5*). We apply Landau-Zener transition physics to coherently control electronic spin states in a semiconductor double quantum dot (DQD).

Semiconductor quantum dots have emerged as promising platforms for quantum control of charge and spin degrees of freedom (*6*). Considering future applications of electron spin qubits in quantum information processing, the required elementary building blocks are the exchange gate, which couples two spins, and single-spin rotations (*7*). Extremely fast 200-ps exchange gates have been demonstrated (*6*, *8*). However, coupling to the small magnetic moment of the electron (as required for single-spin rotations) is much more difficult, leading to relatively long, ~100-ns gate-operation times in GaAs quantum dots (*9*). In addition, the ac magnetic fields required for single-spin electron spin resonance (ESR) are difficult to localize on a single quantum dot (~40 nm), hindering extension of the method to a large number of quantum dots operating in close proximity. Several groups have demonstrated fast optical control of single spins, but these methods are also difficult to apply locally (*10*, *11*). In principle, local rotations can be achieved with the use of electrically driven spin resonance, which requires spin-orbit coupling and an ac electric field, but the Rabi frequencies obtained in GaAs quantum dots are approximately a factor of 2 slower than those obtained using conventional ESR (*12*, *13*). We demonstrate an all-electrical method for driving local single-spin rotations on nanosecond time scales.

Our method for coherent quantum control of electron spins is based on two consecutive sweeps through a singlet-triplet anticrossing in a DQD energy-level diagram. Coherent oscillations between the singlet and *m*_{s} = +1 triplet state, T_{+}, occur on a nanosecond time scale and are made possible by the hyperfine interaction between the trapped electron spins and the nuclear-spin bath (*14*–*16*). The oscillations are controlled by tuning the external magnetic field *B*_{E} and the voltage pulse profile that sweeps the quantum dot system through the anticrossing in the energy-level diagram. Similar sweeps through energy-level anticrossings in superconducting qubits have been used to study Landau-Zener interference (*17*–*21*). In addition, deeply bound molecular states have been generated by transferring weakly bound Feshbach molecules through a series of anticrossings in a molecular energy manifold (*22*).

In our device (Fig. 1A), depletion gates are arranged in a triple quantum dot geometry (*23*). A DQD is formed using the middle and right dots of the device. Gate voltages *V*_{L} and *V*_{R} are used to tune the device to the (1,1)-(2,0) charge transition, where *N*_{L} and *N*_{R} indicate the number of electrons in the left and right dots, respectively. High-sensitivity charge sensing is achieved by depleting gates Q_{1} and Q_{2} to form a quantum point contact (QPC) charge sensor with conductance *g*_{Q} (*8*). Energy-level anticrossings (Fig. 1B) in the DQD can be used for quantum control in a manner that is directly analogous to an optical beam splitter (*18*–*20*).

The detuning, ε, of the DQD (Fig. 2A) is adjusted using *V*_{L} and *V*_{R} (*24*). For positive detuning, the ground state is the spin-singlet (2,0)S. By decreasing the detuning, a single electron can be transferred from the left dot to the right dot, forming a (1,1) charge state. Here the possible spin state configurations are the spin singlet, S, and the spin triplets T_{0}, T_{–}, and T_{+} with *m*_{S} = 0, –1, and +1 respectively. (2,0)S and S hybridize near ε = 0 due to the interdot tunnel coupling *T*_{c}. The T_{+} and T_{–} states are separated from the T_{0} state by the Zeeman energy, *E*_{Z} = *g*μ_{B}(*B*_{E} + *B*_{N}), where μ_{B} is the Bohr magneton, and *B*_{N} is the Overhauser field (*15*). Throughout this work, we take |*g|* = 0.44, based on previous experiments (*8*, *25*). We focus on the boxed region in Fig. 2A, where hyperfine interactions mix the S and T_{+} states, resulting in an anticrossing in the energy-level diagram. Under appropriate experimental conditions, we show that this anticrossing functions as a beam splitter for incoming quantum states (*18*–*20*).

We first measure the quantum state transition dynamics at the S-T_{+} avoided crossing to verify the mechanism of Landau-Zener tunneling. The analytical expression for the nonadiabatic transition probability is *4*). Here, *ħ* is Planck’s constant divided by 2π, and ν is the energy-level velocity, defined as ν = |*d*(*E*_{1} – *E*_{2})/*dt*|, where *E*_{1} and *E*_{2} are the energies of the states involved in the anticrossing. We determine Δ by measuring *P*_{LZ} as a function of the sweep rate through the S-T_{+} anticrossing. A (2,0)S state is first prepared at positive detuning, then a rapid gate-voltage pulse (~1.1 ns, nonadiabatic with respect to the S-T_{+} mixing rate) shifts the system to negative detuning ε_{S}, which preserves the spin singlet, S. The detuning is then increased during a ramp time *T*_{R}, sweeping the system back through the S-T_{+} avoided crossing. A QPC charge sensor determines the final singlet-state probability *P*_{S} via spin-to-charge conversion (*6*).

*P*_{S} is plotted in Fig. 2B as a function of *T*_{R}. For long ramp times, the initial state should follow the adiabatic branch during the return sweep through the S-T_{+} anticrossing, resulting in a final state T_{+}, as illustrated in the inset of Fig. 2B. We measure *P*_{S} ~ 0.3 at long *T*_{R}, because of the limited measurement contrast set by the spin relaxation time. At short times, *P*_{S} decays exponentially, as expected from the Landau-Zener model, with a characteristic time scale of ~180 ns. Given the detuning pulse amplitude (1.7 mV) and the conversion between gate voltage and energy [|*d*(*E*_{S} – *E*_{T+})/*d*ε| ~ 3.9 μeV/mV], we extract a best fit Δ = 60 neV (*24*). In comparison, time-resolved measurements of the S-T_{+} spin-dephasing time yield *T*_{2}* = 10 ns, corresponding to an energy scale of 66 neV, which is in good agreement with the value of Δ obtained above (*8*). In superconducting flux qubits, this tunnel splitting is set by tunnel junction parameters, whereas in the S-T_{+} qubit, Δ is set by fluctuating transverse hyperfine fields (*15*, *18*).

Quantum control of the S and T_{+} states is achieved by consecutively passing through the S-T_{+} avoided crossing in the coherent limit, where the consecutive crossings take place within the spin-dephasing time (*18*–*21*). The opposite limit, where *T*_{R} >> *T*_{2}*, has been shown to lead to dynamic nuclear polarization (*26*). Our pulse sequence for quantum control is illustrated in Fig. 3A and is analogous in operation to an optical interferometer (Fig. 3C, inset). An initially prepared spin-singlet state is swept through the S-T_{+} avoided crossing. During this detuning sweep, the S-T_{+} avoided crossing “splits” the incoming singlet state into a superposition of states S and T_{+}, with amplitudes *A*_{S} and *A*_{T+}, analogous to an optical beam splitter. In correspondence with the Landau-Zener equation, |*A*_{S}|^{2} = *P*_{LZ}. Spin angular momentum is conserved during this process by coupling to the nuclear-spin bath via the hyperfine interaction, resulting in a small amount of nuclear polarization (*16*, *26*). Detuning is then maintained at a value ε_{S} for the nominal pulse length τ_{S}, which results in a phase accumulation *t* is time) that is equivalent to changing the path length of one leg of an optical interferometer. A second detuning sweep takes the system back through the S-T_{+} anticrossing, resulting in quantum interference of the two paths. The singlet-state return probability *P*_{S} is measured using the QPC charge sensor.

The consecutive sweeps through the S-T_{+} anticrossing and the intermediate phase accumulation ϕ can be treated as unitary operations (Fig. 3B) that act on the initially prepared spin-singlet state (*4*, *27*, *28*). For the ideal case of *P*_{LZ} = 1/2, the S-T_{+} anticrossing functions as a 50:50 beam splitter resulting in the unitary operator _{X} and σ_{Z} are the Pauli matrices). Phase accumulation (ϕ) during the detuning pulse results in a σ_{Z} rotation, _{+} anticrossing in the limit *P*_{LZ} = 1/2 results in a third unitary operation *24*).

The measured *P*_{S} shows clear Stückelberg oscillations between S and T_{+} as a function of τ_{S} and ε_{S} (*4*, *5*). At negative detunings, far from the avoided crossing, the oscillation period is set by *E*_{S} – *E*_{T+} = *E*_{Z}. For *B*_{E} = 100 mT, the Zeeman energy corresponds to a period of 1.6 ns, assuming |*g*| = 0.44, in good agreement with the ~1.5-ns period observed in the data for ε_{S} = –1.7 mV. The curvature of the interference pattern is partially due to the voltage pulse profile, which is smoothed to maintain some degree of adiabaticity during the sweep through the S-T_{+} anticrossing. In these data (Fig. 3C), the first bright interference fringe corresponds to the condition where the detuning pulse exactly reaches the S-T_{+} anticrossing. The second bright interference fringe corresponds to a configuration in which U_{2} gives a 2π pulse about the *z* axis of the Bloch sphere.

Singlet-state probability as a function of pulse length *P*_{S}(τ_{S}) is plotted in Fig. 3D for two different values of detuning. The oscillation visibility ranges from 15 to 30% for these data and is a function of detuning, as the spin relaxation time and *P*_{LZ} are detuning-dependent (*6*). Higher-visibility oscillations are obtained when the level velocity ν is small at the S-T_{+} anticrossing (Fig. 3D, insets). Maximum visibility would be obtained for *P*_{LZ} = 1/2 (the limit of a perfect 50:50 beam splitter). To achieve this, detuning ramp times on the order of 160 ns >> T_{2}* are required, which is no longer in the coherent limit. These data suggest that active pulse shaping with subnanosecond resolution could be used to increase the fidelity of the gate operations by lowering the level velocity only in the vicinity of the S-T_{+} anticrossing.

We confirm that the interference fringes are caused by consecutive sweeps through the S-T_{+} beam splitter by varying *B*_{E}. Landau-Zener interference patterns are plotted in Fig. 4, B to D, for *B*_{E} = 90, 70, and 50 mT, respectively. A reduction in field results in two major differences: (i) The first oscillation shifts to more negative ε_{S}, and (ii) the oscillation frequency decreases. Both observations are consistent with the level diagram shown in Fig. 4A.

To quantitatively model the data, we calculate the probability to return to the spin-singlet state *P*_{S} by considering the action of the unitary operations (Fig. 3B) on the initially prepared spin-singlet state. Neglecting relaxation and dephasing, we find *19*, *24*). We calculate ϕ by combining our knowledge of the voltage pulse profile with the measured *E*_{S}(ε) – *E*_{T+}(ε), as determined by energy-level spectroscopy (*24*). The visibility of the calculated oscillations (Fig. 4, B to D, insets) is 15% and is set by *P*_{LZ} = 0.96, as determined for these sweep conditions using the data in Fig. 2B. Overall, the observed and calculated Landau-Zener interference patterns are in very good agreement. The decay of the oscillations as a function of τ_{S} is most likely due to fluctuations in the Overhauser field (*8*).

Whereas commonly used single-spin rotation mechanisms rely on gigahertz frequency magnetic fields, the coherent rotations between S and T_{+} demonstrated here occur on a nanosecond time scale set by the Zeeman energy and are solely driven with local gate-voltage pulses. As a result, it will be feasible to scale this quantum control method to a large number of spin qubits operating in close proximity. In addition, it is possible that the spin-flip mechanism employed here, which relies on coupling to the nuclear-spin bath, could be harnessed under the appropriate conditions to create a nuclear-spin memory (*29*).

## Supporting Online Material

www.sciencemag.org/cgi/content/full/327/5966/669/DC1

Materials and Methods

SOM Text

Figs. S1 to S3

References

## References and Notes

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- We thank G. Burkard, B. Coish, D. Haldane, D. Huse, D. Loss, and H. Ribeiro for useful discussions and C. Laumann for technical contributions. Research at Princeton Univ. was supported by the Sloan Foundation, the Packard Foundation, and the NSF through the Princeton Center for Complex Materials (grant DMR-0819860) and CAREER award (grant DMR-0846341). Work at UCSB was supported by the Defense Advanced Research Projects Agency grant N66001-09-1-2020 and the UCSB National Science Foundation DMR Materials Research Science and Engineering Center.