Introduction

Semiconductor quantum dot (QD) systems facilitate investigations of the interaction between electron spins and nuclear environments, which is known as the central-spin problem1,2. Although the fluctuation of nuclear fields, which is quantified by the effective Overhauser field Bnuc3,4, often acts as a magnetic-noise source for spin qubits3, the hyperfine electron–nuclear spin interaction allows achieving dynamic nuclear polarization (DNP)5,6,7,8. DNP is used for enhancing the signal-to-noise ratio in nuclear magnetic resonance6 and prolonging coherence times in QD-based spin qubits9,10. Gate-defined semiconductor QDs have been used to achieve the fast probing of nuclear environments8,11,12, bidirectional DNP11, and active feedback control of nuclear fields10.

While the DNP achieved by spin-flip mediated transport with an applied bias13,14 allows large DNP13, the QD - reservoir tunnel rate needs to be large enough to allow the finite spin-flip current. On the contrary, the DNP based on the pulsed-gate technique can be demonstrated while maintaining the small tunnel rates ~101 kHz. Because the qubit control typically requires small QD-reservoir tunnel rates transition from the pulsed-gate DNP to qubit experiments is straightforward without additional parameter modulation via the gate voltages. However, spin qubit control combined with DNP has been limited to two-electron singlet–triplet (ST0) spin qubits9,10,11,12,15. Despite the versatility of gate-defined QD systems16,17,18,19, the large singlet–triplet energy splitting EST (~102 h·GHz; h is Planck’s constant) in particular in GaAs limits the access to higher spin states20 in multielectron QDs at moderate external magnetic fields B0 < 1 T or within a typical frequency bandwidth of experimental setups.

Coulomb-correlation-driven Wigner molecules (WMs) in confined systems21,22,23,24,25 may provide new directions for expanding nuclear control to multielectron systems. Recent studies on QDs in various systems have shown clear evidence of WM formation22,23,25,26,27,28,29. It has been demonstrated that the EST can reach down to ~100 h·GHz upon the WM formation27,29 because of strong electron–electron interactions confirmed by full-configuration interaction (FCI)-based theories23,25,28,30. However, most studies have focused on the spectroscopic confirmation of WM formation, and studies on the open system dynamics using correlated states have not been reported to date.

Here, we demonstrate the formation of a WM in semiconductor QDs, which helps achieving efficient spin environment control. We use gate-defined QDs in GaAs and exploit the quenched energy spectrum of the WM (EST ~ 0.9 h·GHz) to enable mixing between different spin subspaces within B0 < 0.3 T. Furthermore, we demonstrate DNP by pulsed-gate control of the electron spin states. Leakage spectroscopy and Landau–Zener–Stuckelberg (LZS) oscillations confirm a sizable bidirectional change in Bnuc ~ 80 mT and the spatial Overhauser field gradient ΔBnuc ~ 35 mT due to the long nuclear spin diffusion time τN ~ 62 s. Further, we demonstrate on-demand control of Bnuc combined with coherent LZS oscillations, providing a new route for realizing controllable DNP using correlated electron states.

Results

Figure 1a shows a gate-defined QD device fabricated on a GaAs/AlGaAs heterostructure, where a 2D electron gas (2DEG) is formed ~70 nm below the surface (see Methods). We focus on the left double QD (DQD) containing three electrons. We designed the V2 gate to form an anisotropic potential, which is predicted to promote WM formation22. An electrostatic simulation of the electric potential at the QD site near V2 shows an oval-shaped confinement potential with anisotropy exceeding 3 (Fig. 1a, right panel). This potential can be tuned by the gate voltage, allowing the controlled electron correlation and localization of the ground state wavefunction within the DQD22,24,26,27. The yellow dot in Fig. 1a. denotes a radio-frequency single-electron transistor (rf-SET) charge sensor utilized for quantum state readout31,32,33. The device was operated in a dilution refrigerator with a base temperature of ~40 mK, an electron temperature Te ~ 150 mK (Supplementary Note 1), and a variable B0 applied to the direction shown in Fig. 1a.

Fig. 1: Wigner molecule formation in a GaAs double quantum dot.
figure 1

a Scanning electron microscope image of a GaAs quantum dot (QD) device similar to the one used in the experiment. Green dots denote the double QD defined for Wigner molecule (WM) formation which is aligned along the [110] crystal axis (black arrow). The inner plunger gate V2 is designed to have anisotropic confinement potential as shown in the right panel to facilitate the localization of the electronic ground state. Yellow circle: a radio-frequency (rf) single-electron transistor (rf-SET) charge sensor for rf-reflectometry. External magnetic field B0 is applied along the direction denoted by the yellow arrow. b Charge stability diagram of the double QD near the three-electron region spanned by V1 and V2 gate voltages. Green-shaded region: the energy-selective tunneling (EST) position for the state readout and initialization. c Landau–Zener–Stückelberg (LZS) oscillation of the WM at B0 = 0 T. The relative phase evolution between the excited doublet (DT) and the ground doublet (DS) results in the oscillation captured by the EST readout. Red-dashed curve in the fast Fourier transformed (FFT) map shows energy dispersion calculated from the toy-model Hamiltonian. The calculation yields quenched orbital energy spacing of the inner dot δR ~ 0.9 h·GHz. d Left (Right) panel: Energy spectrum along the (2,1)–(1,2) charge configuration in the non-interacting (strongly interacting, this work) regime with δL ~ 100 h·GHz (δL ~ 19 h·GHz), and δR ~ 100 h·GHz (δR ~ 0.9 h·GHz). EQ (red-dashed curve) is the energy splitting between the two lowest levels.

Full size image

The three-electron DQD results in two spin doublets and one spin quadruplet state. Without the magnetic field in the (2,1) [(1,2)] charge configuration, doublet-singlet state DS(2,1) [DS(1,2)] with total spin S = 1/2 form the ground state where the two electrons in the left [right] QD form a spin singlet state and fill the ground orbital in the left [right] QD. Here, n [m] denotes the number of electrons in the left [right] QD by (n, m). When the two electrons in the left [right] QD form a spin triplet state and fill the excited orbital in the left [right] QD, the three-electron state result in either the doublet-triplet state DT(2,1) [DT(1,2)] with S = 1/2 or the quadruplet state Q(2,1) [Q(1,2)] with S = 3/2. Because of the orbital splitting, the DT, and Q states usually have higher energy compared to DS states34,35,36. If a finite magnetic field is applied, doublet states with S = 1/2 are split into ms = +1/2 and −1/2 states whereas quadruplet states with S = 3/2 are split into ms = +3/2, +1/2, −1/2, and −3/2 states. Here ms is the spin quantum number related to the z component of the electron spin angular momentum. The explicit spin structures are shown in the Methods. Hereafter, (n, m; ms) notation is used to describe both the charge configuration and spin angular momentum of a state.

First, we show the spectroscopic evidence of the WM at B0 = 0 T by probing EST in the right QD δR. Figure 1b shows a charge stability diagram around (2,1) and (1,2). The green-shaded region near the (2,1)–(1,1) charge transition is exploited for energy-selective tunneling (EST) readout and state initialization27,37,38. We tune the electron tunneling-in (-out) time τin (τout) of the left dot to 14 (7) μs. Starting from the initialized ground doublet state DS in the (2,1) charge configuration, we apply non-adiabatic pulses (Fig. 1b) simultaneously to V1 and V2 with a rise time of ~500 ps and a repetition period of 51 μs ≫ τin to induce coherent LZS oscillation39,40. The oscillation reveals the relative phase evolution between the excited and ground doublet states (DT and DS), the frequency of which is governed by δR.

Figure 1c shows the resultant LZS oscillations as a function of evolution time tevol and detuning ε. The EST in GaAs DQDs in the non-interacting regime is typically on the order of 102 h.GHz20 (Fig. 1d). In a charge qubit regime, a steep rise in the LZS oscillation frequency fLZS ~ EQ/h as a function of ε (Fig. 1c, black curve) and short coherence time T2* ~ 10 ps due to strong susceptibility to charge noise is expected41. EQ is the energy splitting between the two lowest levels (Fig. 1d, red-dashed curve). However, we find a significantly smaller fLZS in the (1,2) charge configuration and T2* ~ 10 ns because of the reduced dispersion of fLZS versus ε. This is reminiscent of a QD hybrid qubit27,40,42, but the excited energy is suppressed owing to the electron–electron interaction. WM formation in our previous GaAs device has been recently confirmed by FCI calculation27,28,30. Although such calculation is needed to rigorously determine parameters, we roughly estimate δR ~ 0.9 h.GHz, by fitting the fast Fourier transformed (FFT) spectrum to the calculation result (Fig. 1c, red-dashed curve) derived from a toy-model Hamiltonian37,39,40 (see Supplementary Note 2).

The full energy spectrum calculation of the three-electron states using the parameters obtained experimentally across the (2,1)–(1,2) configuration is illustrated in Fig. 1d (right panel). The suppressed EST of the left dot δL ~ 19 h·GHz is obtained by measuring the width of the EST region in the charge stability diagram with the lever arm of the gate V1 ~ 0.03. The left QD also allows a weak Wigner molecularization as the measured δL is an order of magnitude smaller than the case of non-interacting regime. Because of the small value of δL/(kBTe) ~ 6, where kB is Boltzmann’s constant, thermal tunneling precludes high-fidelity single-shot readout. We obtain data by the time-averaged signal using the correlated-double sampling (CDS) method, which effectively yields the signal proportional to the excited state probability37 (see Supplementary Note 3).

We confirm the WM spin structure via the strongly suppressed energy spectrum in the right QD with varying B0. We focus on five low-lying energy levels among eight possible multiplet states. See Methods for notations used for labeling spin multiplets. As shown in Fig. 2a (left panel), DS(1,2;−1/2) [DS(1,2;1/2)] becomes degenerate with DT(1,2;1/2) or Q(1,2;1/2) [Q(1,2;3/2)] at a certain ε depending on the B0 magnitude. The degeneracies are lifted by the transverse Overhauser field \({B}_{{nuc}}^{\perp }\)8,11. To detect such anticrossings, we first initialize the state to either DS(2,1;−1/2) or DS(2,1;1/2) at the EST position. By pulsing the initialized DS(2,1;−1/2) [DS(2,1;1/2)] towards (1,2) and holding for ~100 ns ≫ T2*, mixing with (or leakage to) states Q(1,2;1/2) or DT(1,2;1/2) [Q(1,2;3/2)] can occur if the pulse amplitude Ap coincides with the anti-crossing position (Fig. 2a, right panel). Upon pulsing back to the (2,1) charge configuration, the resultant excited states Q or the DT probability can be detected via EST27,37,38. Figure 2b shows the leakage spectrum versus AP and B0, mapping out the anti-crossing positions similar to “spin-funnel” measurements in two-electron ST0 qubits reproducing the energy splittings between the ground and excited levels8,16,43,44. The black (red) dashed curves show the calculated splittings (Fig. 1d) between the DS and DT (Q) states at B0 = 0 T, with the Lande g-factor g* ~ −0.4 45,46.

Fig. 2: Leakage spectroscopy and probabilistic nuclear polarization with the Wigner molecule.
figure 2

a Left panel: schematics of the energy levels for different external magnetic fields B0 > 0 T. Crossings between different ms states become anticrossings aided by the transverse nuclear Overhauser field. Right panel: schematic of the pulse sequence for leakage spectroscopy and probabilistic dynamic nuclear polarization (DNP). The pulse diabatically drives the initialized DS(2,1;1/2) [DS(2,1;−1/2)] to (1,2), and hold ε for 100 ns ≫ T2*. Upon the coincidence of the pulse detuning and the anti-crossing, the state probabilistically evolves to Q(1,2;3/2) [Q(1,2;1/2)] and flips the electron spin ΔmS = +1 which accompanies ΔmN = −1. The scale bars on the bottom axis (ε axis) denote 50 μeV, and the scale bars on the left axis (Energy axis) denote 1 h·GHz. b Leakage spectroscopy of the Wigner molecule (WM) state as a function of B0 and the pulse amplitude Ap. Black (Red) dotted curve shows the calculated energy splitting between DT (Q) and DS at B0 = 0 T. Measurement-induced nuclear field shifts the dispersion opposite to the direction of B0. c, d Leakage measurement with an additional probabilistic polarization pulse with amplitude Ap′ applied before each line sweep. The Ap′ is fixed to 370 (450) mV, and the additional distortion in the leakage spectrum is shown as red circles near a pulse amplitude of 370 (450) mV. Black arrows denote the magnetic field sweep direction.

Full size image

Although the calculated curve qualitatively agrees with the experimental curve, the observed spectrum curvature as a function of AP and B0 is smaller because of the DNP induced by the pulse sequence used for leakage spectroscopy. To confirm this, before each line scan of Ap in Fig. 2c, d, a similar step pulse with a fixed amplitude AP′~370 mV (450 mV) is applied for 10 s. Consequently, we observe distortions (red circles) in the spectrum occurring at AP′. This is because, when AP′ matches with the anti-crossing position, the pulse probabilistically flips the electron spin with a change in the angular momentum ΔmS = +1 by the leakage process described above and accompanies flop ΔmN = −1 of the nuclear spin8,11. Unlike the electrons in GaAs, nuclei have positive g-factors8,20; therefore, the ΔmS = +1 electron spin flip, and thereby the ΔmN = −1 nuclear spin flip polarizes Bnuc toward the B0 direction8,11,47. This additionally drags the leakage spectrum opposite to the B0 direction under a specific condition Ap = AP’. These results indicate that leakages induced by hyperfine interaction between the WM and nuclear environment lead to an observable change in Bnuc. Despite the long measurement time per line scan (~7 s) owing to the communication latency between the measurement computer and the instruments, the polarization effect is still visible. Thus, τN > 10 s, as discussed below. Moreover, as the anti-crossing position is a sensitive function of Btot = B0 + Bnuc over 100~300 mT, it can be used to measure Bnuc.

We now show bidirectional DNP combined with coherent control of doublet states at B0 = 230 mT. Figure 3a (top panel) shows the three primary paths through the anticrossings, which can flip the electron spins deterministically by adiabatic passage2,8,11. Paths P1 and P3 describe the S-polarization that flips the electron spin with ΔmS = +1. This is enabled by initializing the state to DS(1,2;−1/2) [DS(1,2;1/2)] at the EST position and then by non-adiabatically pulsing beyond the first anticrossings near the (2,1) charge configuration (Fig. 3a, yellow boxes), followed by adiabatically driving the state through the anti-crossing to Q(1,2;1/2) [Q(1,2;3/2)], which accompanies ΔmN = −1 (Fig. 3a, blue arrows). Because the spin state is initialized to the doublet-singlet state before the adiabatic spin-flip passage, the sequence is named S-polarization. The Q(1,2;1/2) [Q(1,2;3/2)] state is diabatically driven back to the EST position, and one electron quickly tunnels out to the reservoir. Reloading an electron from the reservoir reinitializes one of the Ds states completing the polarization cycle. Both the DS(1,2;−1/2) and DS(1,2;1/2) initial states contribute to the S-polarization. Path P2 denotes the T-polarization (ΔmS = −1, ΔmN = +1), which is possible by driving DT(1,2;1/2) adiabatically to DS(1,2;−1/2) (Fig. 3a, red arrow). To prepare DT(1,2;1/2), we apply a π-pulse to DS(2,1;1/2) before the adiabatic passage (Fig. 3a, bottom panel). Because we prepare the doublet-triplet state before the adiabatic spin-flip passage, the sequence is called T-polarization. The T-polarization is possible only when the state is initialized to DS(2,1;1/2) at the EST position.

Fig. 3: Bidirectional and controllable dynamic nuclear polarization enabled by Wigner molecularization.
figure 3

a Top panel: Schematic of the anticrossings used for deterministic dynamic nuclear polarization (DNP). Bottom panel: pulse sequence used for S- and T-polarizations. For tevol = 0 ns, the sequence corresponds to maximum S-polarization, which brings DS(1,2;1/2) [DS(1,2;−1/2)] adiabatically across the anti-crossing to Q(1,2;3/2) [Q(1,2;1/2)] flipping the electron spin with ΔmS = +1 and leading to ΔmN = −1 (blue arrow, S-polarization). For tevol = 600 ns, the sequence corresponds to maximum T-polarization. Herein, the DT(1,2;1/2) prepared with a (Landau–Zener–Stückelberg) LZS-oscillation-induced π-pulse is adiabatically transferred to DS(1,2;1/2), resulting in ΔmS = −1 and ΔmN = +1 (red arrow, T-polarization), which has the opposite polarization effect compared to S-polarization. b Change in the nuclear field δBnuc as a function of tevol. The gray curve shows the corresponding LZS oscillation measurement reflecting the DT population. The δBnuc oscillates out of phase to the LZS oscillation owing to the oscillation of the S- and T-polarization ratio. c The magnitude of the maximum polarization Bmax as a function of ramp time wR. The Bnuc saturates to Bmax when the polarization and the nuclear spin diffusion rate reach an equilibrium. For small wR, the |Bmax| decreases because of the small Landau–Zener transition probability PLZ for both S- (blue circle) and T-polarizations (red circle). In the case of T-polarization, |Bmax| decreases again for long wR owing to the lattice relaxation of the excited population. d Bmax as a function of δR. The polarization gets more efficient for smaller δR indicating a strong dependence of the nuclear polarization efficiency on the Wigner parameter. e, f Dynamic nuclear control with the S (T)-polarization sequence. The red dotted line is the numerical fit derived from the simple rate equation-based model. The fit yields the nuclear spin diffusion time τN ~ 62 s, with a polarization magnitude per spin flip of ~2.58 h·kHz·(g*μB)−1. g On-demand DNP via tevol. h Adiabatic ramp amplitude AR with tevol = 0 ns realizing self-limiting DNP.

Full size image

Combining the S- and T-polarizations, we measure the change in Bnuc (δBnuc), where the repeated polarization pulse sequence (Fig. 3a, bottom panel) with variable tevol and a repetition rate of ~ 20 kHz is applied for 10 s before each line scan. For Fig. 3b, a waiting time ~10 min was added after each sweep to allow the polarized nuclei to diffuse and minimize the polarization effect in the next sweep. As shown in Fig. 3b, δBnuc oscillates with tevol, which is anti-correlated with the LZS oscillation that represents the population of DT(1,2;1/2). This confirms that the net polarization rates can be controlled by adjusting tevol. Accordingly, we calibrate tevol = 0 (0.62 ns) for S (T)-polarization. We also calibrate the duration of the adiabatic spin transfer wR. Figure 3c shows the maximum nuclear field change Bmax reachable as a function of wR, where both S- and T- polarizations are ineffective for short wR because of negligible adiabatic transfer probability PLZ2,48. |Bmax| reaches a maximum around wR ~ 0.8 μs, after which the maximum efficiency is retained for the S-polarization sequence. In the case of T-polarization, however, for long wR, |Bmax| decreases because of DT relaxation during the adiabatic passage.

By tuning δR via the dc gate voltages and performing similar S-polarization experiments, we find that Bmax decreases with increasing δR (Fig. 3d, see Supplementary Note 4). As is discussed subsequently, we find that the nuclear diffusion time scale exceeds 60 s regardless of δR, but the Overhauser field change per electron flip b0 is strongly suppressed with increasing δR. Ultimately, the observation implies that the pulsed-gate-based nuclear control becomes inefficient in the non-interacting regime. We suspect the degree of electronic wavefunction localization which depends on the Wigner parameter may be affecting the contact hyperfine interaction between the electron and the nuclear spins and altering the DNP efficiency as a result.

Returning to the condition δR ~ 0.9 h·GHz, we demonstrate on-demand DNP. Figure 3e, f shows the result of optimized S (T)-polarization with tevol = 0 ns, wR = 1000 ns (tevol = 0.62 ns, wR = 600 ns). Although the local fluctuations of the nuclear spins lead to random drift of the anti-crossing positions without the polarization pulse, Bnuc builds toward (opposite to) the B0 direction faster than the nuclear spin diffusion timescale when the polarization pulse is applied before each line scan. δBnuc rises to Bmax 80 mT (−40 mT) until a dynamic equilibrium is reached. Because only the ms = 1/2 states contribute to the T-polarization, |Bmax| for the T-polarization is about half of that for the S-polarization, implying that the state initialize to both ms states with nearly equal probability at the EST position.

We also demonstrate bidirectional DNP by adjusting tevol in Fig. 3g. Figure 3h illustrates the control of Bnuc by adjusting the adiabatic sweep amplitude AR of the S-polarization sequence. Under the S-polarization, Bnuc builds in the B0 direction and drives the anti-crossing to deeper ε (more to (1,2) charge configuration). Because the pulse cannot have a finite polarization effect if the anti-crossing position is driven beyond ε reachable with AR, AR serves as the limiting factor of Bmax. Thus, a self-limiting DNP protocol, where the DNP field is limited by experimental parameters used in the pulse shape rather than the interplay between pumping rate and nuclear diffusion, can be realized. This self-limiting property can be useful in future DNP experiments as the steady state DNP field can be simply controlled by adjusting the pulse amplitude.

Using a simple rate equation, we simulate the polarization-probe sequence (red-dashed curve in Fig. 3e, see Methods and Supplementary Note 5) and obtain τΝ ∼ 62 s and b0 ~ 2.58 h·kHz·(g*μB)−1 from the fit. In contrast, the DNP effect is negligible in our device with the two-electron ST0 qubit8 under the same repetition rate as in the WM regime (see Supplementary Note 6). We further find our DNP is still effective when the repetition rate is as low as 5 kHz (see Supplementary Note 7) showing that the Wigner molecule allows sizable DNP that cannot be achieved with conventional QDs. Through optimization of the magnitude and direction of B0, b0 ~ 3 h kHz·(g*μB)−1 can be achieved with an ST0 qubit in GaAs2,8. However, the obtained result shows that robust nuclear control can be achieved with WMs even in the regime where the same level of control cannot be achieved with an ST0 qubit. In addition, residual polarization ~21.5 mT exists after turning off the polarization sequence (Fig. 3e), which diffuses within ~30 min. The large Knight shift gradient originating from the non-uniformly broadened WM wavefunction may be a possible cause of the long τΝ. However, the newly observed phenomena in this study, including the dependence of b0 on the tuning condition, require further investigations47,49.

Furthermore, the WM’s coherent LZS dynamics provide a novel approach to measure the spatial Overhauser field gradient ΔBZ between QDs. When ΔBZ is larger than the exchange splitting between DT(1,2;1/2) [DT(1,2;–1/2)] and Q(1,2;1/2) [Q(1,2; − 1/2)], the eigenstates are expected to become DT1(1,2;1/2) = \(|\downarrow \rangle|{T}_{+}\rangle\) [DT1(1,2; − 1/2) = \(|\uparrow \rangle|{T}_{-}\rangle\)] and DT0(1,2;1/2) = \(|\uparrow \rangle|{T}_{0}\rangle\) [DT0(1,2; − 1/2) = \(|\downarrow \rangle|{T}_{0}\rangle\)]35. Because both states can tunnel-couple to DS(1,2;1/2) [DS(1,2;−1/2)], the LZS oscillation reveals the DT1 – DS and DT0 – DS energy splittings. As can be inferred from the Hamiltonian (see Supplementary Note 8), although the DT0 – DS splitting is independent of the ΔΒZ and BZ, the DT1 – DS splitting is modulated by ΔΒZ depending on the sign of ΔΒZ and ms, providing the direct measure of ΔΒZ. Because the states can initialize to both DS(1,2;1/2) and DS(1,2;−1/2) at the EST position, the LZS oscillation captures the dynamics of both ms = 1/2 and ms = −1/2 subspaces.

Figure 4a (4b) illustrates the LZS oscillation measurement of the WM multiplet states at B0 = 230 mT in the time (frequency) domain with the S-polarization turned on and off at specific laboratory times. The FFT spectrum exhibits three different branches corresponding to the DT0 – DS (red arrow) and DT1 – DS (black and black-dashed arrows) where the beating patterns vary as the S-polarization induces changes in ΔBz. Two different DT1 – DS branches correspond to different ms subspaces, where the sign of ΔBz should be known to distinguish the ms for each branch. The DT0 – DS splitting is the same for both ms subspaces and is displayed as a single branch (red arrow). Figure 4c, d shows the simulated time (frequency) domain signal of the same LZS oscillation, which agrees well with the experimental result (see Supplementary Note 9). As expected, the DT0 – DS splitting is constant regardless of ΔBz, whereas the DT1 – DS splitting is modulated along the polarization sequence.

Fig. 4: Field gradient control and measurement.
figure 4

Landau–Zener–Stückelberg (LZS) oscillation of the Wigner molecule (WM) states at B0 = 230 mT in a the time domain and b the frequency domain with the S-polarization sequence. The oscillation reveals the relative phase oscillation of the DT1 – DS (black arrow, black dotted arrow) and DT0 – DS (red arrow) of both the ms = 1/2 and ms = −1/2 states. The DT0 – DS splitting is constant regardless of the magnetic field gradient ΔBZ, whereas the DT1 – DS energy spacing is modulated by the ΔBZ depending on the sign of ΔBZ and ms. The resultant beating is visible in e, f the time (frequency) domain line-cut when the polarization is on (green arrow in a) and off (blue arrow in a). The line cuts in the time domain are numerically fitted to the sum of three sine functions (solid lines in e) with different amplitudes. Three separate peaks are visible in the frequency domain (f) when the ΔBZ is largely polarized in the bottom panel (blue line) in (f) Simulated LZS oscillation in c the time domain and d the frequency domain with the ΔBZ in the inset of (d). The simulation in the frequency domain reproduces the branches shown in (b).

Full size image

The DT0 – DT1 splitting without the polarization sequence implies the built-in ΔBz ~ 200 h·MHz·(g*μB)−1 (35 mT), which is also confirmed by the ST0 oscillation (see Supplementary Note 6). ΔBz increases to 400 h·MHz·(g*μB)−1 (70 mT) with the S-polarization and decreases to 200 h MHz·(g*μB)−1 after turning the polarization off. Thus, we conclude that the S-polarization yields the asymmetric pumping effect (ΔBnuc ~ 200 h MHz·(g*μB)−1) about the QD sites, whereas the ΔBnuc direction can be experimentally checked, for example, via single-spin electric-dipole spin resonances46. Furthermore, the DT0 – DS splitting comprises the decoherence-free subspace for the qubit operations resilient to magnetic noises, where the coherent microwave control combined with the large polarization may enable leakage-free and state-selective transitions.

Discussion

The present work uncovers the spin and energy structure of the WM states and explores the central-spin problem with strongly correlated WM states in semiconductor QDs. With the energy splitting of the WM ~ 0.9 h·GHz, we confirm the controllable DNP of Bnuc (ΔBnuc) reaching (but not limited to) 80 mT (35 mT) via leakage spectroscopy and LZS oscillations. The τN exceeds 60 s, which, together with bidirectional polarizability, is beneficial for stabilizing the nuclear bath fluctuation and realizing long-lived nuclear polarization10,15.

We anticipate several directions for further developments and applications of WM-enabled DNP. Similar experiments with a larger δL/Te ratio can enable high-fidelity single-shot readout for a faster measurement of the dynamics of nuclear polarization. This would further enable feedback loop control10 and tracking12,50,51 of nuclear environments in multielectron QDs which can be utilized to narrow the nuclear field distribution for electron coherence enhancement. The real-time Hamiltonian estimation also improves frequency resolution for measuring instantaneous ΔBnuc, which may enable measurements of the degree of spatial localization within WMs. We expect more asymmetric QD geometry would allow a smaller energy gap and thereby the broader electronic wavefunction. Along with the tunability of the energy gap via the gate voltage, as shown here, this may facilitate the investigation of the spatial noise characteristics within a single QD which has been impractical with the typical QD geometries. Furthermore, DNP becomes inefficient with increasing EST of the WM, as discovered herein. This implies that the pulsed-gated electron–nuclear flip-flop probability is a strong function of the Wigner parameter, the microscopic origin of which requires more rigorous investigations.

Methods

Device fabrication

A quadruple QD device was fabricated on a GaAs/AlGaAs heterostructure with a 2DEG formed ~70 nm below the surface. The transport property of the 2DEG showed mobility μ = 2.6 × 106 cm2(V s)−1 with electron density n = 4.0 × 1011 cm−2 at temperature T = 4 K. Electronic mesa around the QD site was defined by the wet etching technique, and thermal diffusion of a metallic stack of Ni/Ge/Au was used to form the ohmic contacts. The depletion gates were deposited on the surface using standard e-beam lithography and metal evaporation of 5 nm Ti/30 nm Au. The lithographical width of the inner QD along the QD axis direction was designed to be ~10% wider than the outer dot to facilitate WM formation. The QD array was aligned to the [110] crystal axis, as shown in Fig. 1a. Although the magnetic field B0 was intended to be applied perpendicular to the [110] axis to minimize the effect of spin-orbit interaction2, the angular deviation was not strictly calibrated.

Measurement

The device was placed on a ~40 mK plate in a commercial dilution refrigerator (Oxford instruments, Triton-500). Ultra-stable dc-voltages were generated by battery-powered dc-sources (Stanford Research Systems, SIM928). They were then combined with rapid voltage pulses from an arbitrary waveform generator (AWG, Keysight M8195A with a sample rate up to 65 GSa/s) via homemade wideband (101–1010 Hz) bias tees to be applied to the metallic gate electrodes. An LC-tank circuit with a resonant radio frequency (rf) of ~120 MHz was attached to the ohmic contact near the SET charge sensor to enable high-bandwidth (fBW > 1 MHz) charge detection27,31,32,33,37. The reflected rf-signal was first amplified by 50 dB using two-stage low-noise cryo-amplifiers (Caltech Microwave Research, CITLF2 ×2 in series) at a 4 K plate. Next, it was further amplified by 25 dB at room temperature using a homemade low-noise rf-amplifier. The signal was then demodulated by an ultra-high-frequency lock-in amplifier (Zurich Instruments, UHFLI), which was routed to the boxcar integrator built in the UHFLI. Trigger signals with a repetition period of 51 μs were generated by a field-programmable-gate array (FPGA, Digilent, Zedboard) to synchronize the timing of the AWG and the boxcar integrator for the CDS37.

Eigenstates of three-electron spin states

Three-electron spin-multiplet structure consists of eight different eigenstates, which are two DS states, two DT states, and four quadruplet states For simplicity, we show only the spin states with (1,2) charge configuration in Table 134,35,36, where the spin state in the first (second) bracket indicates the single- (two-) electron spin state in the left (right) QD.

Table 1 Three-electron spin states
Full size table

Rate equation

Nuclear spin polarization and the diffusion process were phenomenologically modeled using a rate equation:

$$\frac{d{{B}}_{{{{{\rm{nuc}}}}}}}{d{t}}=-{{B}}_{{{{{\rm{nuc}}}}}}/{\tau }_{N}+{{b}}_{0}{{P}}_{{{{{\rm{flip}}}}}}/{{T}}_{{{{{\rm{rep}}}}}},$$
(1)

where τN is the nuclear spin diffusion time, b0 is the Overhauser field change per electron spin-flip, Pflip is the nuclear spin flop probability obtained from the Landau–Zener transition probability PLZ and the false initialization probability (see Supplementary Note 5), and Trep is the pulse repetition period. Using Eq. (1), we simulated the polarization-probe sequence shown in Fig. 3 with the experimental parameters including the time required for the amplitude sweep in the leakage probe step.