## Abstract

We observed mixing between two-electron singlet and triplet states in a double quantum dot, caused by interactions with nuclear spins in the host semiconductor. This mixing was suppressed when we applied a small magnetic field or increased the interdot tunnel coupling and thereby the singlet-triplet splitting. Electron transport involving transitions between triplets and singlets in turn polarized the nuclei, resulting in marked bistabilities. We extract from the fluctuating nuclear field a limitation on the time-averaged spin coherence time *T* _{} of 25 nanoseconds. Control of the electron-nuclear interaction will therefore be crucial for the coherent manipulation of individual electron spins.

A single electron confined in a GaAs quantum dot is often referred to as artificial hydrogen. One important difference between natural and artificial hydrogen, however, is that in the first, the hyperfine interaction couples the electron to a single nucleus, whereas in artificial hydrogen, the electron is coupled to about one million Ga and As nuclei. This creates a subtle interplay between electron spin eigenstates affected by the ensemble of nuclear spins (the Overhauser shift), nuclear spin states affected by time-averaged electron polarization (the Knight shift), and the flip-flop mechanism that trades electron and nuclear spins (*1*, *2*).

The electron-nuclear interaction has important consequences for quantum information processing with confined electron spins (*3*). Any randomness in the Overhauser shift introduces errors in a qubit state, if no correcting measures are taken (*4*–*6*). Even worse, multiple qubit states, like the entangled states of two coupled electron spins, are redefined by different Overhauser fields. Characterization and control of this mechanism will be critical both for identifying the problems and finding potential solutions.

We studied the implications of the hyperfine interaction on entangled spin states in two coupled quantum dots—an artificial hydrogen molecule—in which the molecular states could be controlled electrically. A random polarization of nuclear spins creates an inhomogeneous effective field that couples molecular singlet and triplet states and leads to new eigenstates that are admixtures of these two. We used transport measurements to determine the degree of mixing over a wide range of tunnel coupling and observed a subtle dependence of this mixing on magnetic field. We found that we could controllably suppress the mixing by increasing the singlet-triplet splitting. This ability is crucial for reliable two-qubit operations such as the SWAP gate, which interchanges the spin states of the two dots (*3*).

Furthermore, we found that electron transport itself acts back on the nuclear spins through the hyperfine interaction, and time-domain measurements revealed complex, often bistable, behavior of the nuclear polarization. Understanding the current-induced nuclear polarization is an important step toward electrical control of nuclear spins. Such control will be critical for electrical generation and detection of entangled nuclear spin states (*7*) and for transfer of quantum information between electron and nuclear spin systems (*8*, *9*). It may also be possible to control the nuclear field fluctuations themselves in order to achieve longer electron spin coherence times (*10*–*12*).

We investigated the coupled electron-nuclear system using electrical transport measurements through two dots in series (*13*), in a regime where the Pauli exclusion principle blocks current flow (*14*, *15*). The dots were defined with electrostatic gates on a GaAs/AlGaAs heterostructure (Fig. 1E) (*16*). The gate voltages were tuned such that one electron always resides in the right dot, and a second electron could tunnel from the left reservoir, through the left and right dots, to the right reservoir (Fig. 1D). This current-carrying cycle can be described with the occupations (*m, n*) of the left and right dots: (0,1) → (1,1) → (0,2) → (0,1). When an electron enters from the left dot, the two-electron system forms either a molecular singlet, S(1,1), or a molecular triplet, T(1,1). From S(1,1), the electron in the left dot can move to the right dot to form S(0,2). From T(1,1), however, the transition to (0,2) is forbidden by spin conservation [T(0,2) is much higher in energy than S(0,2)]. Thus, as soon as T(1,1) is occupied, further current flow is blocked (we refer to this effect as Pauli blockade).

A characteristic measurement of this blockade is shown in Fig. 1A. The suppression of current (<80 fA) in the region defined by dashed lines is a signature of Pauli blockade (*14*, *15*) (fig. S1 and supporting text). Fig. 1B shows a similar measurement, but with a much weaker interdot tunnel coupling *t*. Strikingly, a large leakage current appears in the Pauli blockaded region, even though the barrier between the two dots is more opaque. Furthermore, this leakage current was substantially reduced by an external magnetic field of only 100 mT (Fig. 1C). Such a strong field dependence is unexpected at first glance, because the in-plane magnetic field, *B*_{ext}, couples primarily to spin but the Zeeman energies (*E*_{Z}) involved are very small (*E*_{Z} ∼2.5 μeV at *B*_{ext} = 100 mT, as compared with a thermal energy of ∼15 μeV at 150 mK, for example).

Leakage in the Pauli blockade regime occurs when singlet and triplet states are coupled. The T(1,1) that would block current can then transition to the S(1,1) state and the blockade is lifted (Fig. 1D). As we will show, coupling of singlets and triplets (Fig. 1, B and C) in our measurements is caused by the hyperfine interaction between the electron spins and the Ga and As nuclear spins [other leakage mechanisms can be ruled out (supporting text)].

The hyperfine interaction between an electron with spin and a nucleus with spin has the form (), where *A* characterizes the coupling strength. An electron coupled to an ensemble of *n* nuclear spins experiences an effective magnetic field , with *g* the electron *g* factor and μ_{B} the Bohr magneton (*1*). For fully polarized nuclear spins in GaAs, *B*_{N} ∼5 T (*17*). For unpolarized nuclear spins, statistical fluctuations give rise to an effective field pointing in a random direction with an average magnitude of 5 T/√*n* (*4*, *5*, *18*). Quantum dots like those measured here contain *n* ∼10^{6} nuclei, so .

Nuclei in two different dots give rise to effective nuclear fields, and , that are uncorrelated. Although the difference in field is small, corresponding to an energy , it nevertheless plays a critical role in Pauli blockade. The (1,1) triplet state that blocks current flow consists of one electron on each of the two dots. When these two electrons are subject to different fields, the triplet is mixed with the singlet and Pauli blockade is lifted. For instance, an inhomogeneous field along *ẑ* causes the triplet to evolve into the singlet . Similarly, the evolution of the other two triplet states, |*T*_{+} 〉 = |↑↑ 〉 and |*T*_{–}〉 = |↓↓〉, into the singlet is caused by *x̂* and *ŷ* components of .

The degree of mixing by the inhomogeneous field depends on the singlet-triplet energy splitting, *E*_{ST}. Singlet and triplet states that are close together in energy (*E*_{ST} « *E*_{N}) are strongly mixed, whereas the perturbation caused by the nuclei on states far apart in energy (*E*_{ST} » *E*_{N}) is small.

The singlet-triplet splitting depends on the interdot tunnel coupling *t* and on the detuning of left and right dot potentials Δ_{LR}. Δ_{LR} and *t* were controlled experimentally with gate voltages (Fig. 1E). Gate voltage *V*_{t} controlled the interdot tunnel coupling. *V*_{L} and *V*_{R} set the detuning, and thereby determined whether transport was inelastic (detuned levels), resonant (aligned levels), or blocked by Coulomb blockade (Fig. 1F). The coupling of the dots to the leads was held constant with *V*_{lead}.

The effect of the two tunable parameters *t* and Δ_{LR} on the singlet and triplet energies is illustrated in Fig. 2, A and B. For weak tunnel coupling (*t* ∼0), and in the absence of a hyperfine interaction (*E*_{N} ∼0), the (1,1) singlet and (1,1) triplet states are nearly degenerate (Fig. 2A). A finite interdot tunnel coupling *t* leads to an anticrossing of S(1,1) and S(0,2). The level repulsion results in an increased singlet-triplet splitting that is strongly dependent on detuning (Fig. 2B). At the resonant condition (Δ_{LR} = 0, aligned levels), the two new singlet eigenstates are equidistant from the triplet state, both with *E*_{ST} = √2*t*. For finite detuning (finite but still smaller than the single dot S-T splitting), one singlet state comes closer to the triplet state (*E*_{ST} ∼*t*^{2}/Δ_{LR}), whereas the other moves away. In Fig. 2, A and B, singlet and triplet states are pure eigenstates (not mixed), and therefore Pauli blockade would be complete.

The additional effect of the inhomogeneous nuclear field is shown in Fig. 2, C and D. For small *t* (√2*t, t*^{2}/Δ_{LR} < *E*_{N}), the (1,1) singlet and (1,1) triplet are close together in energy and therefore strongly mixed (purple lines) over the entire range of detuning. For *t* such that *t*^{2}/Δ_{LR} < *E*_{N} < √2*t*, triplet and singlet states mix strongly only for finite detuning. This is because *E*_{ST} is larger than *E*_{N} for aligned levels but smaller than *E*_{N} at finite detuning. For still larger *t* (√2*t, t*^{2}/Δ_{LR} > *E*_{N}, not shown in Fig. 2), mixing is weak over the entire range of detuning. In the cases where mixing between S and T is strong, as in Fig. 2, C and D (for large detuning), Pauli blockade is lifted and a leakage current results.

The competition between *E*_{ST} and *E*_{N} can be seen experimentally by comparing one-dimensional traces of leakage current as a function of detuning over a wide range of *t* (Fig. 3A). Resonant current appears as a peak at Δ_{LR} = 0 and inelastic leakage as the shoulder at Δ_{LR} > 0 (*19*). When the interdot tunnel coupling was small, both resonant and inelastic transport were allowed because of singlet-triplet mixing, and both rose as the middle barrier became more transparent. As the tunnel coupling was raised further, a point was reached where *E*_{ST} became larger than the nuclear field and Pauli blockade suppressed the current (Fig. 1A). The maximum resonant current occurred at a smaller value of *t* compared to the maximum inelastic current (Fig. 3A, inset). This is consistent with *E*_{ST} being much smaller for finite detuning than for aligned levels (*t*^{2}/Δ_{LR} « √2*t*) (Fig. 2, B and D).

The experimental knob provided by electrostatic gates is very coarse on the energy scales relevant to the hyperfine interaction. However, the external magnetic field can easily be controlled with a precision of 0.1 mT, corresponding to a Zeeman splitting (2 neV) that is 50 times smaller than *E*_{N}. For this reason, monitoring the field dependence allowed a more detailed examination of the competing energy scales *E*_{ST}, *E*_{Z}, and *E*_{N}.

The competition between *E*_{Z} and *E*_{N} is clear for small interdot tunnel coupling (Fig. 3B). Leakage current was suppressed monotonically with the magnetic field, on a scale of ∼5 mT and ∼10 mT for inelastic and resonant transport, respectively. The qualitative features of this field dependence can be understood from the insets to Fig. 2C. At zero field, all states are mixed strongly by the inhomogeneous nuclear field, but when *E*_{Z} exceeds *E*_{N}, the mixing between the singlet and two of the triplet states (|*T*_{+}〉 and |*T*_{–}〉) is suppressed. An electron loaded into either of these blocks further current flow, explaining the disappearance of leakage at high field in the measurement.

The magnitude of the fluctuating Overhauser field can be extracted from the inelastic peak shape in the limit of small *t* (Fig. 3B, inset). We fit the data to a model that describes the transport cycle with the density matrix approach (*20*) (supporting text). From this fit, we found the magnitude of the inhomogeneous field √〈Δ*B* _{N}^{2}〉 = 1.73 ± 0.02 mT (*E*_{N} = 0.04 μeV), largely independent of Δ_{LR} over the parameter range studied (*21*). The value for the effective nuclear field fluctuations in a single dot was obtained from the relation , giving √〈*B* _{N}^{2}〉 = 1.22 mT. This is consistent with the strength of the hyperfine interaction in GaAs and the number of nuclei that are expected in each dot (*4*, *22*).

The three-way interplay between *E*_{ST}, *E*_{Z}, and *E*_{N} is most clearly visible in the resonant current. At an intermediate value of tunnel coupling, (Fig. 3C), the resonant peak was split in magnetic field, with maxima at ±10 mT (Fig. 3C, inset). The lower inset to Fig. 2D illustrates this behavior. At *B*_{ext} = 0, the resonant current in Fig. 3C was suppressed compared to the current in Fig. 3B, because *E*_{ST} was greater than *E*_{N} at that point. Increasing *B*_{ext} enhanced the mixing as the |*T*_{+}〉 and |*T*_{–}〉 states approached the singlet states. The maximum leakage occurred when the states crossed, at *E*_{ST} (= √2*t*) = *E*_{Z}. Here, *E*_{Z} was 0.25 ± 0.03 μeV at the current maximum, from which we extract *t* = 0.18 ± 0.02 μeV for this setting of *V*_{t}. At still larger *B*_{ext}, |*T*_{+}〉 and |*T*_{–}〉 moved away from the singlet states again, and the leakage current was suppressed.

The system entered into a new regime for still higher tunnel coupling (Figs. 3D and 4), where it became clear that the electron-nuclear system is dynamic. The zero field resonant leakage was further suppressed, and above 10 mT, prominent current spikes appeared (Fig. 3D, inset). The spikes are markedly visible in a three-dimensional surface plot of leakage over a broader range of field (Fig. 4A). For fixed experimental parameters, the current still fluctuated in time (Fig. 4B).

We found that time-dependent behavior was a consistent feature of resonant transport for (*E*_{ST}, *E*_{Z}) » *E*_{N}. For some device settings, the time dependence was fast (for example, the fluctuations in Fig. 4, A and B), but for others, the leakage changed much more slowly (Fig. 4C). Starting from an equilibrium situation (bias voltage switched off for 5 min), the current was initially very small after the bias was turned on. It built up and then saturated after a time that ranged from less than a second to several minutes. This time scale depended on Δ_{LR}, *t*, and *B*_{ext}. When no voltage bias was applied, the system returned to equilibrium after ∼80 s at 200 mT. Similar long time scales of the nuclear spin-lattice relaxation times have been reported before in GaAs systems (*23*) and quantum dots (*24*). We thus associate the slow time dependence observed in our system with current-induced dynamic nuclear polarization and relaxation.

Evidence that the fast fluctuations too are related to current-induced nuclear polarization (and cannot be explained by fluctuating background charges alone) is found in their dependence on sweep direction and sweep rate (*23*, *25*). When the magnetic field was swept while fixed Δ_{LR} was maintained, the current showed fluctuations at low field but suddenly became stable at high field (Fig. 4D). The crossover from unstable to stable behavior occurred at a field that was hysteretic in sweep direction (Fig. 4D), and this hysteresis became more pronounced at higher sweep rates (faster than ∼1 mT/s). The connection between the fluctuations and nuclear polarization is also evident from time traces, in which instability developed only after the nuclear polarization was allowed to build for some time (fig. S3).

Unlike the regular oscillations that have been observed in other GaAs structures (*1*, *26*), the fluctuations in our measurements were random in time and, in many cases, suggested bistability with leakage current moving between two stable values. We discuss the origin of such fast bistable fluctuations in the supporting text.

The ensemble of random nuclear spins that gives rise to the mixing of two-electron states as observed in this experiment also gives rise to an uncertainty of *g*μ_{B} √〈*B*_{N}^{2} = 0.03 μeV in the Zeeman splitting of one electron. When averaged over a time longer than the correlation time of the nuclear spin bath (∼100 μs) (*27*), this implies an upper limit on the time-averaged spin coherence time of [as defined by Merkulov *et al.* (*4*)], comparable to the found in recent optical spectroscopy measurements (*28*). This value is four orders of magnitude shorter than the theoretical prediction for the electron spin *T*_{2} in the absence of nuclei, which is limited only by spin-orbit interactions (*29*–*31*).

One way to eliminate the uncertainty in Zeeman splitting that leads to effective dephasing is to maintain a well-defined nuclear spin polarization (*12*). Many of the regimes explored in this paper show leakage current that is stable when current-induced polarization is allowed to settle for some time. These may in fact be examples of specific nuclear polarizations that are maintained electrically.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1113719/DC1

SOM Text

Figs. S1 to S3

References and Notes