Abstract
A basic requirement for quantum information processing is the ability to universally control the state of a single qubit on timescales much shorter than the coherence time. Although ultrafast optical control of a single spin has been achieved in quantum dots, scaling up such methods remains a challenge. Here we demonstrate complete control of the quantumdot charge qubit on the picosecond scale. We observe tunable qubit dynamics in a chargestability diagram, in a time domain, and in a pulse amplitude space of the driven pulse. The observations are well described by Landau–Zener–Stückelberg interference. These results establish the feasibility of a full set of allelectrical singlequbit operations. Although our experiment is carried out in a solidstate architecture, the technique is independent of the physical encoding of the quantum information and has the potential for wider applications.
Introduction
Universal singlequbit gates are key elements in a quantum computer, as they provide the fundamental building blocks for implementing complex operations^{1,2,3}. In the standard circuit model, arbitrary singlequbit rotations^{1}, together with twoqubit controlledNOT gates, provide a universal set of gates. In the alternative measurementbased models, such as the oneway quantum computer, the ability to carry out singlequbit operations from the source of a specific multiparticles state can generate every possible quantum state^{2} and offer practical algorithms. Additionally, with only singlequbit operations and teleportation, one can construct a universal quantum computer^{3}. In the Bloch sphere model of qubit states, a universal singlequbit gate requires arbitrary rotations around at least two axes.
The charge^{4,5} or spin^{6,7,8,9} degrees of freedom of an electron in quantum dots are particularly attractive for the implementations of qubits. Owing to the fast charge or spin decoherence times in semiconductor quantum dots, which are typically less than a few nanoseconds^{4,5,6,7,8,9}, control operating on the picosecond timescale may be necessary. Until now, ultrafast manipulations of a single qubit in quantum dots have been performed using pulsed laser fields^{10,11,12}. Alternatively, electrical pulses can be generated much more easily by simply exciting a local electrode. Logic gate operations and readouts can be carried out allelectrically, much like those in current mainstream semiconductor electronics. Additionally, this allelectric technique provides a simple pathway for greater spatial selectivity to locally address the individual qubits and remove obstacles to scalability. Therefore, universal electrical control on a picosecond scale is highly desirable to allow coherence to be maintained during the completion of a large number of operations.
Over the last few decades, the Landau–Zener–Stückelberg (LZS) interference has served as a textbook model for quantum phenomena^{13}, that occurs when a system sweeps through the anticrossing of two energy levels. LZS has also gained particular interest for quantum control^{8,14} because it is less sensitive to certain types of noise and might enable the implementation of a universal gate with high fidelity^{15,16,17}. Here we experimentally demonstrate such a scheme for a singlecharge qubit in a double quantum dot (DQD), using a single pulse. We may add that very recently LZS interference has been observed under a continuous microwave driving, in both an electrostaticdefined^{18} and a donorbased^{19} semiconductor DQD systems.
Results
Charge qubit in a DQD
Figure 1a provides a scanning electron micrograph of the sample used in the experiments, in which the metal gate pattern electrostatically defines a DQD and a nearby quantum point contact (QPC) detector within a GaAs/AlGaAs heterostructure. All measurements were conducted in a dilution refrigerator equipped with highfrequency lines (details of sample structures and experimental techniques are given in the Methods section). A singleelectron charge in the DQD is used to encode the charge qubit^{4,5,20}. An excess valence electron in the left and right dots defines the basis states L and R, respectively. The schematic diagram in Fig. 1b illustrates that the energies of the qubit states can be continuously tuned by the level detuning ε=E_{R}−E_{L}, in which E_{L} and E_{R} are the energy levels for the electron in the left and right dots for an uncoupled DQD, respectively. In the presence of interdot tunnelling that couples two dots, a characteristic anticrossing occurs between the twoqubit levels near the resonance (ε=0). For our experiments, the anticrossing gap 2Δ is adjusted to 20.7 μeV, which corresponds to a Rabi frequency of 5.0 GHz. We have experimentally determined the Rabi frequency from the coherent oscillations excited by a square, nonadiabatic pulse as shown in Supplementary Fig. S1b. We denote the ground and excited states as 0 and 1, which are roughly the charge eigenstates far from the resonance.
The characterization of the system is shown in the chargestability diagram for the DQD (Fig. 1c), which integrates the fewelectron regime so that ~2–3 electrons occupy each dot (for the full diagram, see Supplementary Fig. S1a). Our experiments are performed for a variety of charge states. As they contain identical physics, for consistency, we present only the data collected from one charge configuration. The system can be conveniently described in a valence electron number configuration that consists of four relevant charge states: (0,0), (1,0), (0,1) and (1,1). A prominent boundary can be observed between (1,0) and (0,1) and marks the interdot transition line corresponding to the ε=0 resonance.
Observations of LZS interference
Our scheme to control a singlecharge qubit using a Gaussianshaped short pulse is shown in Fig. 2a. The system is initially prepared in the R state at a positive detuning ε_{0}, which is far from the resonance. During the rising phase of the pulse, the sweeping pulse takes the system adiabatically to the anticrossing point at t=t_{1} at which a significant probability exists for a nonadiabatic transition to the excited state 1. This probability is the Landau–Zener transition represented by the following formula:
in which υ is the sweep velocity of the driven pulse through the anticrossing point.
As the pulse takes the system further past ε=0, two different trajectories at different energies can coherently interfere. Upon returning to ε=0 at t=t_{2}, the two trajectories, caused by the coherent interference, have also accumulated a phase difference of magnitude
A projective readout is performed at the end of the pulse to measure the L state for a constructive interference and the R state for a destructive interference, known as the LZS interference. Thus, the LZS process consists of both the nonadiabatic level transition and the adiabatic phase accumulation.
The Bloch sphere model provides a convenient picture to understand the quantum control of a charge qubit. Using this model, the charge state is represented as a vector, in which the ground and excited states 0 and 1 are at the north and south poles, respectively. In this model, the dynamics of the qubit can be represented by applying the appropriate sequence of unitary operation matrices to the initial state. The matrices
give rise to a rotation on the Bloch sphere around the x axis by an angle θ and around the z axis by an angle φ.
At a Landau–Zener transition, the initial state becomes a coherent superposition of 1 and 0 with a phase φ_{LZ} related to the Stokes phenomenon^{13}. The relative amplitudes of 1 and 0 depend on P_{LZ}. This behaviour corresponds to the transformation R_{z}(−φ_{LZ})R_{x}(θ_{LZ})R_{z}(−φ_{LZ}), seen as successive x and zrotations (see the Supplementary Discussion for details), and θ_{LZ}=2 sin^{−1} √P_{LZ}. In the phase accumulation stage, the qubit undergoes a single rotation about the z axis, referred to as the phaseshift gate operation R_{z}(φ_{i}). Thus, the combination of R_{x} and R_{z} enables arbitrary onequbit rotations R(θ, φ) on the Bloch sphere and the LZS pulse rotates the input state on the Bloch sphere to the output state
as illustrated in Fig. 2a.
The coherent control of our charge qubit is evident in the stability diagram (Fig. 2b), which is considerably altered as compared with that without an applied pulse. We observe many additional lines parallel to the interdot transition line. These additional lines are a signature of the excitation of the LZS interference by the pulse. In our device, the right barrier to the bath is slightly more open than the left barrier, and one can observe that the chargeaddition lines (0,0)—(1,0) and (0,1)—(1,1) are less visible. As a result, an interesting triangularshaped area exists in which the electron has a high probability of escaping to the reservoir before the interdot transition is completed.
The additional lines in Fig. 2b can be easily understood. For the line labelled 0, the pulse takes the system just past the anticrossing point. At the next line, the pulse can take the qubit further, passing ε=0 and accumulating a total phase of 2π. Therefore, the lines represent the constructive interference fringes between the successive Landau–Zener transitions that correspond to an accumulated total phase of 2πN.
To confirm our identification, we have derived an analytical expression for the locations, ε_{0}^{(N)}, of the constructive interference fingers for a triangular pulse^{21}. The triangular pulse is a simple approximation of the actual Gaussian profile of our pulse and can readily yield an intuitive analytical expression (see the Supplementary Discussion), , in which A is the pulse amplitude with units of energy that can be converted from voltage using the leverarm conversion factor and t_{r} is the pulseraising time. Figure 2c compares the experimental finger positions with a theoretical curve obtained using the above equation, and reasonable agreement can be achieved.
Complete and ultrafast quantum control of the charge states
The ability to fully control the charge qubit is also studied using the LZS interference patterns. Controlling the amplitude of the driven pulse while the time profile of the pulse is fixed sets the speed υ of the passage through the anticrossing point, thus the R_{x}(θ_{LZ}) rotation angle θ_{LZ} and the R_{z}(φ_{LZ}) rotation angle φ_{LZ}. Tuning the pulse time interval in the phase accumulation stage sets the R_{z}(φ_{i}) rotation angle φ_{i}. The parameters θ_{LZ}, φ_{LZ} and φ_{i} are sufficient to rotate the input qubit to any point on the Bloch sphere, or more generally, to implement a universal onequbit operation R(θ, φ) using the LZS pulse profile^{22,23}.
To demonstrate the ability of the LZS method to generate tunable unitary transformations, we use the amplitude of the driven source as a control parameter. The charge state probability P_{L›} as a function of the qubit detuning position ε_{0} and the voltage amplitude A under a 150ps short pulse is provided in Fig. 3a. Given the detuning and driven pulse amplitude, such interference patterns exhibit fringes that rise again from the constructive interference between successive Landau–Zener transitions at φ as a multiple of 2π. A characteristic of the LZS driving occurs when the zrotations result in the total phase φ=2πN; the total xrotation angle θ is generally maximized as 2θ_{LZ}, which increases monotonically with the driven amplitude (see Supplementary Discussion for details). We verify this result by extracting the total rotation angles (θ, φ) of the Bloch vector from Fig. 3b (a horizontal cut at ε_{0}=400 μeV of Fig. 3a) as it undergoes LZS interference. These rotation angles are also parametrically plotted in Fig. 3c, demonstrating the expected behaviour as a function of the pulse amplitude. In addition, we note that the fit of φ is better than that of θ, indicating that the phaseshift gates have higher intrinsic resistance to certain decoherence^{15,16}. These findings suggest that the LZS pulse amplitude should be an important tuning parameter for optimizing the quantum control of twolevel systems.
The LZS interferences are further studied in the timedomain. We use a lowpass filter to shape the timevarying, squaredriven pulse into an approximately Gaussian profile for the investigations. Figure 4 shows the charge state occupation P_{L›} as a function of both the detuning energy position ε_{0} and the pulse width t_{p}. Up to 10 LZS interference fringes can be clearly observed. The brightest line in the figure can be understood as the detuning pulse precisely at the anticrossing point, and the subsequent finer lines corresponds to full phase accumulations of 2π, 4π and so on. We also simulate the evolution of the charge qubit by numerically solving the master equations as described in the Supplementary Discussion. This simulation (as shown in Supplementary Fig. S2) is in reasonable agreement with the experimental data. In particular, fast timeevolutions are observed in the insert of Fig. 4. For example, at a detuning energy of ε_{0}=400 μeV, only ~10 ps is required to accumulate a phase of 2π that corresponds to a full cycle of R_{z} operations of the qubit. For gate defined GaAs qubits, the phase rotation time can be electrically manipulated at this time scale.
To further highlight the importance of the LZS interference for a general singlequbit gate, we consider the following example: Tuning the speed of passage to yield P_{LZ}=1/2, the total rotation is a Hadamard gate with an arbitrary phase^{22}. As the Rabi frequency can be reliably tuned to 10 GHz, the total rotation can be completed in ~50 ps.
Decoherence information in the amplitude spectroscopy
Decoherence of the qubit due to its environment can be readily extracted from the amplitude spectroscopy^{24,25} (for details, see the Supplementary Discussion). It is useful to evaluate the Fourier transform of the occupation probability
in which k_{A} and k_{ε} represent the reciprocalspace variables corresponding to the realspace variables A and ε, respectively. The twodimensional Fourier transform of the data in Fig. 3a is provided in Fig. 5. As k_{ε} is practically a timescale, decoherence leads to an attenuation in the k_{ε} direction as follows^{25}:
Therefore, both the intrinsic dephasing time T_{2} and the inhomogeneous broadening T_{2}* can be extracted from the overall amplitude decay. One must emphasize that this amplitude spectroscopy has an advantage over the two types of dephasing times because T_{2} and T_{2}* exhibit different k_{ε} dependences.
In our data, the Fourier intensity, threedimensionally plotted in a log scale, is apparently dominated by a linear k_{ε} dependence. Therefore, T_{2} is extracted without requiring spinecho experiments. A typical trace form Fig. 5 yields an estimation of T_{2}=4±0.6 ns, while T_{2}* is difficult to extract as the quadratic term is relatively small and is masked by noise. Nevertheless, the T_{2} decoherence time is much longer than the 10 ps required for a 2π phase rotation. The decoherence and relaxation times can greatly effect the singlequbit operation fidelity as shown in Supplementary Fig. S3.
Discussion
In summary, we have used a shaped electrical pulse to create LZS interference in a semiconductor DQD charge qubit. The LZS interferenceinduced xrotation operations along with the dynamic phasegate operations can form a basis for rapid, universal, allelectric onequbit gate operations in a few tens of picoseconds. These results represent progress towards the implementation of semiconductor quantum dotbased qubits. Our results are an important addition to the rapidly growing toolbox of quantum information processing because they are generally applicable to systems that avoid crossings, including both artificial and natural twolevel systems. This wellcontrolled, solidstate system could also be seen as an analogue quantum simulator of a real atom undergoing LZS interferometry^{26}. A similar timescale for electrical control was also observed in ref. 27 in a quantumdot charge qubit using nonadiabatic voltage pulses.
Methods
Devices
The DQD device is defined by electron beam lithography on a molecular beam epitaxially grown GaAs/AlGaAs heterostructure. The twodimensional electron gas is located 95 nm below the surface. The twodimensional electron gas has a density of 3.2 × 10^{11} cm^{−2} and a mobility of 1.5 × 10^{5} cm^{2} V^{−1} s^{−1}. Figure 1a provides the scanning electron micrograph of the surface gates. Six gates—A1, A2, A3, B1, B2 and B3—shape the DQD. Gates B4 and B2 form a QPC chargesensing channel for counting the DQD electron occupations via capacitive coupling.
Control and measurements
The experiments were performed in an Oxford Triton dilution refrigerator with a base temperature of 30 mK. An Agilent 81134A pulse generator with a time resolution of 1 ps was used to deliver fast pulse trains through semirigid coaxial transmission lines to the A3 side gate of the device. The conductance through the QPC, G_{QPC}, depends on the change in local charge configuration and provides a sensitive metre for the number of electrons in the left and right dots. The charge state probability is determined by normalizing the charge sensor conductance to the adjacent plateaus in the chargestability diagram. This measurement technique has been reportedly used in singlecharge qubits and offers the experimental convenience of integrating initialization, manipulation and measurement in the same pulse^{7}. In our experiment, a pulse repetition rate of 30 MHz was chosen to ensure that the qubit is relaxed to the initial state and to carry out a sufficient number of projective measurements (~10^{7} times) for an adequate signaltonoise ratio. The ensemble averaging of these measurements, in terms of the average charge detector conductance, allows us to directly obtain the probability of the qubit states.
Additional information
How to cite this article: Cao, G. et al. Ultrafast universal quantum control of a quantumdot charge qubit using Landau–Zener–Stückelberg interference. Nat. Commun. 4:1401 doi: 10.1038/ncomms2412 (2013).
Change history
13 November 2013
In the original version of this Article, the authors claimed electrical control of a quantumdot charge qubit on a timescale orders of magnitude faster than previous measurements on electrically controlled charge or spinbased qubits. After publication, they became aware of a related work by Dovzhenko et al. on charge qubits that showed electrical control on a comparable timescale. Therefore, the Abstract of this Article has now been corrected to 'Here we demonstrate complete control of the quantumdot charge qubit on the picosecond scale'. Furthermore, the following statement has been added to the Discussion section to recognise the work of Dovzhenko et al.: 'A similar timescale for electrical control was also observed in ref. 27 in a quantumdot charge qubit using nonadiabatic voltage pulses.
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Acknowledgements
This work was supported by the National Fundamental Research Programme (Grant No. 2011CBA00200), and National Natural Science Foundation (Grant Nos. 11222438, 10934006, 11274294, 11074243, 11174267 and 91121014).
Author information
Author notes
 Gang Cao
 & HaiOu Li
These authors contributed equally to this work.
Affiliations
Key Laboratory of Quantum Information, Department of Optics and Optical Engineering, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, China
 Gang Cao
 , HaiOu Li
 , Tao Tu
 , Li Wang
 , Cheng Zhou
 , Ming Xiao
 , GuangCan Guo
 & GuoPing Guo
Department of Physics and Astronomy, University of California at Los Angeles, California 90095, USA
 HongWen Jiang
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Contributions
G.C., H.O.L., C.Z., M.X. and H.W.J. fabricated the samples and performed the measurements. T.T., L.W. and G.C.G. designed the experiment, provided theoretical support and analysed the data. G.P.G. and H.W.J. supervised the project. All authors contributed to write the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to GuoPing Guo.
Supplementary information
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Supplementary information
Supplementary Figures S1S3, Supplementary Discussion and Supplementary References
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