Wigner-molecularization-enabled dynamic nuclear polarization

Multielectron semiconductor quantum dots (QDs) provide a novel platform to study the Coulomb interaction-driven, spatially localized electron states of Wigner molecules (WMs). Although Wigner-molecularization has been confirmed by real-space imaging and coherent spectroscopy, the open system dynamics of the strongly correlated states with the environment are not yet well understood. Here, we demonstrate efficient control of spin transfer between an artificial three-electron WM and the nuclear environment in a GaAs double QD. A Landau–Zener sweep-based polarization sequence and low-lying anticrossings of spin multiplet states enabled by Wigner-molecularization are utilized. Combined with coherent control of spin states, we achieve control of magnitude, polarity, and site dependence of the nuclear field. We demonstrate that the same level of control cannot be achieved in the non-interacting regime. Thus, we confirm the spin structure of a WM, paving the way for active control of correlated electron states for application in mesoscopic environment engineering.

on QDs in various systems have shown clear evidence of WM formation 22,23,[25][26][27][28] .It has been demonstrated that the EST can reach down to ~10 0 h•GHz upon the WM formation 25,27 because of strong electron-electron interactions confirmed by full-configuration interaction (FCI)-based theories 23,28,29 .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 a gate-defined QD in GaAs and exploit the quenched energy spectrum of the WM (EST ~ 0.9 h•GHz) to enable mixing between different Sz subspaces within B0 < 0.5 T, where Sz denotes the spin projection to the quantization axis.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 formation 22 .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 DQD 22,24,26,27 .The yellow dot in Fig. 1a.denotes a radiofrequency single-electron transistor (rf-SET) charge sensor utilized for quantum state readout [30][31][32] .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.
First, we show the spectroscopic evidence of the WM at B0 = 0 T by probing EST in the right QD R.Fig. 1b shows a charge stability diagram.The green-shaded region near the (2,1)-(1,1) charge transition is exploited for energy-selective tunneling (EST) readout and state initialization 27,33,34 .We tune the electron tunneling-in (-out) time in (out) of the left dot to 14 (7) s.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s ≫in to induce coherent LZS oscillation 35,36 .The oscillation reveals the relative phase evolution between the excited and ground doublet states (DT and DS), the frequency of which is governed by R.Fig. 1c shows the resultant LZS oscillations as a function of evolution time tevol and detuning .The EST in GaAs DQDs in the non-interacting regime is typically on the order of 10 2 h•GHz 20 (Fig. 1d).In a charge qubit regime, a steep rise in the LZS oscillation frequency fLZS as a function of  (Fig. 1c, black curve) and short coherence time T2 * ~ 10 ps due to strong susceptibility to charge noise is expected 37 .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 .This is a reminiscent of a QD hybrid qubit 27,36,38 , 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 calculation [27][28][29] .Although such calculation is needed to rigorously determine parameters, we roughly estimate R ~ 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 Hamiltonian 33,35,36 (see Methods).
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  Because of the small value of L/(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 probability 33 (see Supplementary Note 2).

Fig. 2.
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 (2d), 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 mS = +1 by the leakage process described above and accompanies flop mN = −1 of the nuclear spin 8,11 .Unlike the electrons in GaAs, nuclei have positive g-factors 8,20 ; therefore, the pulse polarizes Bnuc toward the B0 direction.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, N > 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.
Combining the S-and T-polarizations, we measure the change in Bnuc (Bnuc), 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, Bnuc 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.Fig. 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 PLZ 2,43 .|Bmax| reaches a maximum around wR ~ 0.8 s, after which the maximum efficiency is retained for the S-polarization sequence.In the case of Tpolarization, however, for long wR, |Bmax| decreases because of DT relaxation during the adiabatic passage.
By tuning R via the dc gate voltages and performing similar S-polarization experiments, we find that Bmax decreases with increasing R (Fig. 3d, see Extended Data Fig. 1).As is discussed subsequently, we find that the nuclear diffusion time scale exceeds 60 s regardless of R, but the Overhauser field change per electron flip b0 is strongly suppressed with increasing R.Ultimately, the observation implies that the pulsed-gate-based nuclear control becomes inefficient in the non-interacting regime.
Returning to the condition R ~ 0.9 h•GHz, we demonstrate on-demand nuclear field programming.Fig. 3e  We also demonstrate bidirectional DNP by adjusting tevol in Fig. 3g.Fig. 3h illustrates the programming of Bnuc by adjusting the adiabatic sweep amplitude AR of the S-polarization sequence.Because Bnuc builds in the B0 direction and drives the anti-crossing to deeper  (more to (1,2) charge configuration) under the S-polarization, AR serves as the limiting factor of Bmax.Thus, a self-limiting DNP protocol can be realized.
Using a simple rate equation, we simulate the polarization-probe sequence (red-dashed curve in Fig. 3e, see Methods and Supplementary Note 3) and obtain s and b0 ~ 2.58 h•kHz•(g*B) −1 from the fit.In contrast, the DNP effect is negligible in our device with the twoelectron ST0 qubit 8 under the same repetition rate as in the WM regime (see Supplementary Note 4).Through optimization of the magnitude and direction of B0, b0 ~ 3 h•kHz•(g*B) −1 can be achieved with an ST0 qubit in GaAs 2,8 .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 However, the newly observed phenomena in this study, including the dependence of b0 on the tuning condition, require further investigations 44,45 .
We anticipate several directions for further developments and applications of WMenabled DNP.Similar experiments with a larger L/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 control 10 and tracking 12,47 of nuclear environments in multielectron QDs.
The real-time Hamiltonian estimation also improves frequency resolution for measuring instantaneous Bnuc, which may enable measurements of the degree of spatial localization within WMs.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.

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 = 2.6×10 6 cm 2 (V•s) −1 with electron density n = 4.0×10 11 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 interaction 2 , 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 (10 1 -10 10 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 detection 27,[30][31][32][33] .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 lownoise 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 s were generated by a fieldprogrammable-gate array (FPGA, Digilent, Zedboard) to synchronize the timing of the AWG and the boxcar integrator for the CDS 33 .

Eigenstates of three-electron spin states
Three-electron spin-multiplet structure consists of eight different eigenstates, which are four quadruplet states

    
When B0 = 0 T, the DS states, DT states, and Q states are degenerate respectively, resulting in three different branches in the energy dispersion.We use a simple toy-model Hamiltonian adopted from the double QD hybrid qubit 35,36 , which leads to a 6 × 6 Hamiltonian with the charge states considered as below.The ordered basis for the Hamiltonian is [DS(2,1), DT(2,1), Q (2,1), DS (1,2), DT (1,2), Q (1,2)], where n (m) denotes the number of electrons in the left (right) QD by (n, m).
Here,  is the energy detuning between the double QD, ti is the tunnel coupling strength between different orbitals (i = 1, 2, 3, 4), and L (R) is the orbital energy splitting in the left (right) dot.
Further,  is a factor to account for the different lever-arms of the ground and excited states in the (1,2) WM states 50 , recently shown to be the consequence of many-body effects 28,29 .The Hamiltonian is utilized to obtain the energy spectra shown in Fig. 1.As we discuss in detail in Supplementary Note 6, the LZS oscillation at non-zero B0 is simulated by adding the hyperfine interaction terms 46,48 to the aforementioned Hamiltonian and by solving the time-dependent Schrodinger equation with the experimentally obtained parameters.

Rate equation
Nuclear spin polarization and the diffusion process were phenomenologically modeled using a rate equation: where N is the nuclear spin diffusion time, b0 is the Overhauser field change per electron spinflip, Pflip is the nuclear spin flop probability obtained from the Landau-Zener transition probability PLZ and the false initialization probability (see Supplementary Note 3), and Trep is the pulse repetition period.Using Eq. ( 2), 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.

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 Bnuc ~ 35 mT due to the long nuclear spin diffusion time N ~ 62 s.Further, we demonstrate on-demand control of Bnuc combined with coherent LZS oscillations, providing a new route for realizing programmable DNP using correlated electron states.
(right panel).The suppressed EST of the left dot L ~ 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.
(3f) 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.Bnuc rises to Bmax 80 mT (−40 mT) until a dynamic equilibrium is reached.Because only the SZ = 1/2 states contribute to the Tpolarization, |Bmax| for the T-polarization is about half of that for the S-polarization, implying that the state initialize to both SZ states with nearly equal probability at the EST position.

Figure captions Figure 1 .
Figure captions

Figure 2 .
Figure 2. Leakage spectroscopy and probabilistic nuclear polarization with the Wigner

Figure 3 .
Figure 3. Bidirectional and programmable dynamic nuclear polarization enabled by

Figure 4 .
Figure 4. Field gradient control and measurement.Landau-Zener-Stückelberg (LZS) , where r = z, +, -, Bd denotes the magnetic field on the d th electron, and ̄ = -n.The transverse magnetic field B + and B -couple different SZ subspaces with |mS| = 1, where SZ is the spin projection to the quantization axis.Note that the spin-flip terms corresponding to |mS| = 2 are not present.For the LZS oscillation simulation shown in Fig.4c, 4d, we assume that 1) the transverse Overhauser field B + and B -are negligibly small compared to B Z , and 2) the spatial magnetic field gradient within a single QD is insignificant compared to that between the left and right QDs.In the (1,2) charge configuration in a DQD, we use BL = Bd=1 to denote the magnetic field on the electron in the left QD and BR = Bd=2 = Bd=3 to denote the magnetic field experienced by the two electrons inside the right QD.Based on the notation and the two assumptions above, Hhf can be simplified as (SE3).