## Abstract

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 ∼10 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 suppress 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 communication (*1*, *2*). Although quantum control of entanglement has been realized in isolated atomic systems, its extension to solid-state systems—motivated by the prospect of scalable device fabrication—remains a demanding experimental 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 voltages applied to electrostatic gates (*7*–*9*). Recent experiments suggest that spin in quantum dots may be a particularly promising holder of quantum information, because the spin relaxation time (*T*_{1}) can approach tens of milliseconds (*10*–*13*). Although gallium arsenide (GaAs) is a demonstrated exceptional material for fabricating quantum dots, it has the potential drawback that confined electrons interact 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 ∼10 ns, limited by hyperfine interactions. We then demonstrate Rabi oscillations in the two-spin space (including a 180-ps operation between two electron spins) and implement spin-echo sequences, showing an extended spin coherence time, *T*_{2}, beyond 1 μs.

**Isolating and measuring two electrons.** Gate-defined double quantum dot devices are fabricated using a GaAs/AlGaAs heterostructure grown by molecular beam epitaxy with a two-dimensional electron gas 100 nm below the surface, with density ∼2 × 10^{11} cm^{–2}. 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 ϵ ∝ *V*_{R} – *V*_{L} is adjusted. Measurements are performed in a dilution refrigerator with electron temperature *T*_{e} ∼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 (∼1 ns) pulsing of these gates (*15*). High-frequency manipulation of a single electron, demonstrating the gigahertz bandwidth of this setup, was reported in (*16*).

Quantum point contact (QPC) sensors fabricated next to each dot serve as local electrometers (*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 previous experiments have demonstrated spin-selective tunneling (*12*, *13*, *19*, *20*).

**Voltage-controlled exchange.** The relative energy detuning ϵ of the (0,2) and (1,1) charge states can be rapidly controlled by applying calibrated voltage pulses to gates L and R (Fig. 2B). For ϵ > 0, the ground-state charge configuration is (0,2). Tight confinement in (0,2) favors a spin-singlet configuration, denoted (0,2)S. The corresponding (0,2) triplet states are energetically inaccessible, lying ∼400 μeV above (0,2)S and are neglected in the following discussion. For ϵ < 0, the ground state configuration is (1,1). In this case, four spin states are accessible: the singlet (S = 0), denoted S [suppressing the (1,1) label]; and three triplets (S = 1), denoted T_{–}, T_{0}, and T_{+}, corresponding to *m*_{s} = –1, 0, +1.

In the absence of interdot tunneling, the two spins in the (1,1) configuration are independent; that is, S, T_{0},T_{+}, and T_{–} are degenerate. At finite magnetic fields, S and T_{0} are degenerate. When interdot tunneling is present, the (0,2) and (1,1) charge states hybridize, which results in an exchange splitting *J*(ϵ) between the S and T_{0} spin states of (1,1) that depends on detuning (Fig. 2B). Near zero detuning, exchange *J* (ϵ → 0) becomes large (equal to half the splitting of symmetric and antisymmetric charge states at ϵ = 0); for large negative detuning, ϵ « –*J*(0), exchange vanishes, *J*(ϵ) → 0, and the spins again become independent. Except where noted, a perpendicular magnetic field *B* = 100 mT is used to split off the T_{±} states from T_{0} by the Zeeman energy *E*_{z} = ±*g**μ_{B}*B* ∼2.5 μeV (*g** = –0.44 is the electron *g* factor in GaAs; μ_{B} is the Bohr magneton). The split-off T_{+} state crosses the hybridized singlet S when *J*(ϵ) = *g**μ_{B}*B* (vertical green line in Fig. 2B), allowing *J*(ϵ) to be readily measured, as discussed below.

In all measurements, a cyclical pulse sequence 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 measurement 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 (ϵ > 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 critically important. For our device, each electron is coupled to roughly 10^{6} GaAs nuclei through the hyperfine interaction. The hyperfine interaction results in an effective random magnetic field with magnitude *B*_{nuc} ∼1 to 5 mT (*13*, *21*, *22*). These random hyperfine fields evolve slowly (>10 μs) 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*(ϵ) < *g**μ_{B}*B*_{nuc}, these effective fields mix S and T states.

**The logical qubit.** With the T_{±} states split off by an applied field *B* » *B*_{nuc}, the states S and T_{0} form an effective two-level system (or qubit) with Hamiltonian where is the difference in random hyperfine fields along the applied field direction. To facilitate the following discussion, 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 eigenstates of the instantaneous nuclear fields within this subspace, |↑↓〉 and |↓↑〉, as the 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 τ_{S} that the system is held at large detuning [with *J*(ϵ) < *g**μ_{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 reservoir 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 exchanged with the leads, (0,2)S is prepared. The state is then separated into (1,1) using rapid adiabatic passage, where ϵ is swept from a positive value to a large negative value quickly (∼1 ns) relative to the nuclear mixing time but slowly as compared to the tunnel splitting of the hybridized charge states . This yields a separated singlet, S. After a separation time τ_{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 τ_{M} ∼5 to 10 μs < *T*_{1}.

The average singlet probability measured after a separation time of 200 ns, *P*_{S}(ϵ,*B*, τ_{S} = 200 ns), is shown in Fig. 2C as a function of detuning at the separation point and applied field (*29*). Evident in the data are a funnel-shaped feature where S and T_{+} cross (vertical green line in Fig. 2B) and are rapidly mixed by hyperfine fields. The degeneracy occurs at *J*(ϵ) = *g**μ_{B}*B*, allowing *J*(ϵ) to be measured (Fig. 2D) by mapping the location of this feature in *P*_{S}(ϵ,*B*). At larger detuning, where *J*(ϵ) < *g**μ_{B}*B*_{nuc}, the S and T_{0} states approach degeneracy and are susceptible to hyperfine mixing, which reduces *P*_{S} (light blue area of Fig. 2B). When *B* < *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 coherence. We measure this time by varying the singlet separation time τ_{S}. The time evolution of the average singlet return probability, *P*_{S}(τ_{S}), measured using the pulse sequence in Fig. 3A with ϵ = –6 mV, is shown in Fig. 3B. As τ_{S} increases, *P*_{S} decreases from ∼1 on a 10-ns time scale, saturating after 20 ns to *P*_{S} ∼0.5 (0.7) for *B* = 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 measurement contrast, and yields Gaussian-like decay on a time scale *T*_{2}* from *P*_{S}(τ_{S} = 0) = 1 to long-time saturating values *P*_{S}(τ_{S} » *T*_{2}*) = 1/3 for *B* « *B*_{nuc} and *P*_{S}(τ_{S} » *T*_{2}*) = 1/2 at *B* » *B*_{nuc}. The field dependence is caused by the lifting of the triplet degeneracy with the external field, although the naïve expectation based on incoherent mixing would be *P*_{S}(τ_{S} » *T*_{2}*) = 1/4, not 1/3, at *B* = 0. Fits to the measured *P*_{S}(τ_{S}) yield *T*_{2}* = 10 ± 1 ns, corresponding to *B*_{nuc} = 2.3 mT, consistent with previous measurements (*13*, *22*, *31*). An observed ∼40% reduction of contrast is treated as a fit parameter. The predicted weak overshoot of *P*_{S} for *B* = 0, a remnant of Rabi oscillations (*23*), is not seen in these data.

**Spin SWAP and Rabi oscillations in the** |↑↓〉**,** |↓↑〉 **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 |↑↓〉 (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 ϵ < 0 slowly relative to tunnel splitting but quickly relative to the nuclear mixing time through the S-T_{+} degeneracy. The system is then ramped slowly as compared to the nuclear mixing time (τ_{A} ∼1 μs » *T*_{2}*) to large negative detuning. This slow lowering of *J*(ϵ) leads to adiabatic following of the initial state S into the state |↑↓〉, the ground state of the Hamiltonian with *J* → 0 (*30*, *32*). Readout follows the same steps in reverse: ramping slowly out of the large detuning region unloads |↑↓〉 to S and |↓↑〉 to T_{0}. Then, moving quickly though S-T_{+} degeneracy and finally projecting onto (0,2)S measures the fraction that was in the state |↑↓〉 before readout.

Once initialized in |↑↓〉, the application of a finite exchange *J*(ϵ) for a time τ_{E} rotates the spin state about the *z* axis of the Bloch sphere, in the plane containing |↑↓〉 and |↓↑〉, through an angle . The case constitutes a SWAP operation, rotating the state |↑↓〉 into the state |↓↑〉.

Figure 4B shows *P*_{S}(ϵ,τ_{E}) oscillating as a function of both τ_{E} and ϵ, with minima of the singlet probability corresponding to . The inset shows theoretical predictions , using values for *J*(ϵ) obtained independently from the S-T_{+} 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, amplitude, decay time, and phase as fit parameters. To achieve faster π-pulse times, *J*(ϵ) 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 π-pulse time obtained using these methods is ∼350 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*(ϵ) during the exchange pulse and may reflect gate noise during the τ_{E} interval. The contrast (∼45%) 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 dephasing due to the local hyperfine fields. The pulse sequence is shown in Fig. 5A and is similar to refocusing sequences used in nuclear magnetic resonance (*34*, *35*). The separated singlet S will dephase at large negative detuning [*J*(ϵ) ∼0] due to local hyperfine fields after a separation time τ_{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*(ϵ) for a time τ_{E}, where , which rotates the Bloch vector around the *z* axis by an angle π, 3π, 5π,..., and waiting for a time τ_{S}′ = τ_{S}.

The singlet probability *P*_{S}(ϵ,τ_{E}) measured using the spin-echo sequence (Fig. 5A) is shown as a function of detuning and τ_{E} in Fig. 5B. Singlet recoveries (black regions) are observed for π,3π, and 5π exchange pulses. A plot of the theoretical prediction (Fig. 5B, inset) using values for *J*(ϵ) measured independently from the S-T_{+} resonance condition compares well with experiment. We note greater noise in these data than in Fig. 4. We speculate that this noise, which is ∼100 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, τ_{S} – τ_{S}′, for increasing values of the total time spent at large detuning, τ_{S} + τ_{S}′, averaged over 10 data sets. Differences in τ_{S} and τ_{S}′ result in imperfect refocusing and decrease the recovery amplitude on a characteristic time scale τ_{S} – τ_{S}′ = *T*_{2}^{*}.

For each value of τ_{S} + τ_{S}′, the data are data are fit a Gaussian form giving *T*_{2}* = 9 ± 2 ns, consistent with measurements of the singlet decay discussed above. The best-fit heights for each τ_{S} + τ_{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 characteristic coherence time of 1.2 μs, which sets a lower bound on *T*_{2}. Comparing measured values 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 operation time by a factor of ∼7000. Because the echo sequence relies on gate-voltage control of *J*(ϵ), 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 warrants further investigation (*30*, *37*).

**Summary and outlook.** We have demonstrated 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 demonstrate Rabi oscillations and a 180-ps operation, and to greatly reduce dephasing of a spatially separated spin-singlet state with spin-echo techniques. Moreover, the echo sequence 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 manipulated efficiently with effectively long coherence times. This two-electron spin qubit may provide a starting point for implementation of quantum computation schemes with considerable 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 realizations of ideas from quantum information science in semiconductor nanostructures.