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Control and Detection of Singlet-Triplet Mixing in a Random Nuclear Field

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Science  26 Aug 2005:
Vol. 309, Issue 5739, pp. 1346-1350
DOI: 10.1126/science.1113719

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 Math 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 (46). 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 (1012).

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).

Fig. 1.

Pauli blockade and leakage current. (A) Color-scale plot of the current through two coupled dots as a function of the left and right dot potentials (voltage bias, 800 μeV; VT = –108 mV). The experimental signature of Pauli blockade is low current (<80 fA) in the area denoted by dashed gray lines. (B) Analogous data for smaller interdot tunnel coupling (Vt = –181 mV), with the same color scale as in (A). A marked increase of leakage current is seen in the lower part of the Pauli blockaded area (the green and yellow band). Inset: One-dimensional trace along the solid gray line, with Coulomb blockaded, resonant, and inelastic transport regimes marked as defined in (F). (C) Analogous data for the same tunnel coupling as in (B), but for Bext = 100 mT. The leakage current from (B) is strongly suppressed. (D) Two level diagrams that illustrate Pauli blockade in coupled quantum dots. When the (1,1) triplet evolves to a (1,1) singlet (red arrow), Pauli blockade is lifted. (E) Scanning electron micrograph showing the device geometry. White arrows indicate current flow through the two coupled dots (dotted line). (F) Level diagrams illustrating three transport regimes. △: Coulomb blockade; transport would require absorption of energy. ▢: Resonant transport; the dot levels are aligned. +: Inelastic transport; energy must be transferred to the environment, for instance, by emitting a phonon.

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, Bext, couples primarily to spin but the Zeeman energies (EZ) involved are very small (EZ ∼2.5 μeV at Bext = 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 Math and a nucleus with spin Math has the form (Math), where A characterizes the coupling strength. An electron coupled to an ensemble of n nuclear spins experiences an effective magnetic field Math, with g the electron g factor and μB the Bohr magneton (1). For fully polarized nuclear spins in GaAs, BN ∼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 ∼106 nuclei, so Math.

Nuclei in two different dots give rise to effective nuclear fields, Math and Math, that are uncorrelated. Although the difference in field Math is small, corresponding to an energy Math, 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 causes the triplet Math to evolve into the singlet Math. Similarly, the evolution of the other two triplet states, |T+ 〉 = |↑↑ 〉 and |T〉 = |↓↓〉, into the singlet is caused by and ŷ components of Math.

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

The singlet-triplet splitting depends on the interdot tunnel coupling t and on the detuning of left and right dot potentials ΔLR. ΔLR and t were controlled experimentally with gate voltages (Fig. 1E). Gate voltage Vt controlled the interdot tunnel coupling. VL and VR 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 Vlead.

The effect of the two tunable parameters t and ΔLR 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 (EN ∼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 (ΔLR = 0, aligned levels), the two new singlet eigenstates are equidistant from the triplet state, both with EST = √2t. For finite detuning (finite but still smaller than the single dot S-T splitting), one singlet state comes closer to the triplet state (ESTt2LR), 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.

Fig. 2.

Two-electron level diagrams showing energy as a function of detuning ΔLR. Detuning is defined so that the energy of T(1,1) remains constant as ΔLR varies (fig. S1B and supporting text). T(0,2) is not shown as it occurs far above the energies shown here. The panels on the left illustrate the effect of t; the panels on the right include the additional effect of an inhomogeneous magnetic field. Pure singlet and triplet states are drawn in blue and red, respectively; strong admixtures are in purple. The blue (△), white (▢), and yellow (+) background corresponds to the Coulomb blockade, resonant, and inelastic transport regimes, respectively. (A) For small tunnel coupling, T(1,1) and S(1,1) are nearly degenerate. (B) For finite t, level repulsion between the singlet states results in a larger singlet-triplet splitting than shown in (A), which depends on detuning. The tunnel coupling does not mix singlet and triplet states. For large ΔLR (that are still smaller than the single dot S-T splitting), ESTt2LR. (C and D) An inhomogeneous field mixes triplet and singlet states that are close in energy (purple lines). For clarity, only one triplet state is shown in the main panels. (C) For small t, T(1,1) and S(1,1) mix strongly over the full range of detuning. (D) For large t, T(1,1) mixes strongly with the singlet only for large detuning. The insets to (C) and (D) show the effect of an external magnetic field on the two-electron energy levels. All three triplets are shown in the insets; the triplets |T+〉 and |T〉 split off from |T0〉 because of Bext. The leakage current is highest in the regions indicated by black dotted ellipses.

The additional effect of the inhomogeneous nuclear field is shown in Fig. 2, C and D. For small t (√2t, t2LR < EN), 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 t2LR < EN < √2t, triplet and singlet states mix strongly only for finite detuning. This is because EST is larger than EN for aligned levels but smaller than EN at finite detuning. For still larger t (√2t, t2LR > EN, 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 EST and EN 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 ΔLR = 0 and inelastic leakage as the shoulder at ΔLR > 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 EST 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 EST being much smaller for finite detuning than for aligned levels (t2LR « √2t) (Fig. 2, B and D).

Fig. 3.

The measured leakage current results from a competition between EN, EST, and EZ. (A) One-dimensional traces of the leakage current as a function of detuning at Bext = 0, for a wide range of tunnel couplings (analogous to the inset of Fig. 1B). Coulomb blockade, resonant transport, and inelastic transport are indicated as in Fig. 2. Inset: Leakage current along the dotted gray and orange lines is shown as a function of Vt. Resonant and inelastic leakage (gray and orange markers) reach a maximum at different tunnel couplings (Vt = –190 mV and –150 mV, respectively). (B) For small tunnel coupling (<EN), both the resonant and inelastic leakage currents drop monotonically with Bext. Inset: Magnetic field dependence of the inelastic current along the dotted line (ΔLR = 40 μeV). (C) For larger t (>EN), the resonant leakage current is maximum at Bext = 10 mT. Inset: Field dependence of the resonant peak height (dotted line). (D) For still larger t, the resonant current is strongly reduced at low field (main panel), then becomes unstable for higher field (inset).

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 EN. For this reason, monitoring the field dependence allowed a more detailed examination of the competing energy scales EST, EZ, and EN.

The competition between EZ and EN 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 EZ exceeds EN, 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 √〈ΔB N2〉 = 1.73 ± 0.02 mT (EN = 0.04 μeV), largely independent of ΔLR over the parameter range studied (21). The value for the effective nuclear field fluctuations in a single dot was obtained from the relation Math, giving √〈B N2〉 = 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 EST, EZ, and EN is most clearly visible in the resonant current. At an intermediate value of tunnel coupling, Math (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 Bext = 0, the resonant current in Fig. 3C was suppressed compared to the current in Fig. 3B, because EST was greater than EN at that point. Increasing Bext enhanced the mixing as the |T+〉 and |T〉 states approached the singlet states. The maximum leakage occurred when the states crossed, at EST (= √2t) = EZ. Here, EZ was 0.25 ± 0.03 μeV at the current maximum, from which we extract t = 0.18 ± 0.02 μeV for this setting of Vt. At still larger Bext, |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).

Fig. 4.

Time dependence of the leakage current reveals the dynamics of the electron-nuclear system. This time dependence occurs in the regime corresponding to Fig. 2D. (A) Surface plot of electrical transport for Vt = –151 mV. Instability on the resonant peak is visible as sharp current spikes. The sweep direction is from positive to negative ΔLR, for fields stepped from negative to positive Bext. (B) Explicit time dependence of the resonant current exhibits bistability (Vt = –141 mV, Bext = 100 mT). (C) Lower axis: Dynamic nuclear polarization due to electron transport through the device (Vt = –141 mV, ΔLR = 0, Bext = 200 mT), after initialization to zero polarization by waiting for 5 min with no voltage applied. Top axis: In order to measure the nuclear spin relaxation time, we waited for the current to saturate, switched off the bias voltage for a time trel, and then remeasured the leakage current. An exponential fit gives a time constant of 80 ± 40 s (measurements of these long time scales result in large error bars, ±20 fA, because of 1/f noise). (D) The field dependence of the resonant current is hysteretic in the sweep direction (Vt = –149 mV). Each trace takes ∼7 min.

We found that time-dependent behavior was a consistent feature of resonant transport for (EST, EZ) » EN. 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 ΔLR, t, and Bext. 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 ΔLR 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 gμB √〈BN2 = 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 Math [as defined by Merkulov et al. (4)], comparable to the Math found in recent optical spectroscopy measurements (28). This value is four orders of magnitude shorter than the theoretical prediction for the electron spin T2 in the absence of nuclei, which is limited only by spin-orbit interactions (2931).

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

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