Wigner-molecularization-enabled dynamic nuclear polarization

Abstract

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.

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

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 (D_T) and the ground doublet (D_S) 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). E_Q (red-dashed curve) is the energy splitting between the two lowest levels.

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

a Left panel: schematics of the energy levels for different external magnetic fields B₀ > 0 T. Crossings between different m_s 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 D_S(2,1;1/2) [D_S(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 Δm_S = +1 which accompanies Δm_N = −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 A_p. Black (Red) dotted curve shows the calculated energy splitting between D_T (Q) and D_S 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 A_p′ applied before each line sweep. The A_p′ 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.

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

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 t_evol = 0 ns, the sequence corresponds to maximum S-polarization, which brings D_S(1,2;1/2) [D_S(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 Δm_S = +1 and leading to Δm_N = −1 (blue arrow, S-polarization). For t_evol = 600 ns, the sequence corresponds to maximum T-polarization. Herein, the D_T(1,2;1/2) prepared with a (Landau–Zener–Stückelberg) LZS-oscillation-induced π-pulse is adiabatically transferred to D_S(1,2;1/2), resulting in Δm_S = −1 and Δm_N = +1 (red arrow, T-polarization), which has the opposite polarization effect compared to S-polarization. b Change in the nuclear field δB_nuc as a function of t_evol. The gray curve shows the corresponding LZS oscillation measurement reflecting the D_T population. The δB_nuc 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 B_max as a function of ramp time w_R. The B_nuc saturates to B_max when the polarization and the nuclear spin diffusion rate reach an equilibrium. For small w_R, the |B_max| decreases because of the small Landau–Zener transition probability P_LZ for both S- (blue circle) and T-polarizations (red circle). In the case of T-polarization, |B_max| decreases again for long w_R owing to the lattice relaxation of the excited population. d B_max 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 τ_N ~ 62 s, with a polarization magnitude per spin flip of ~2.58 h·kHz·(g*μ_B)⁻¹. g On-demand DNP via t_evol. h Adiabatic ramp amplitude A_R with t_evol = 0 ns realizing self-limiting DNP.

Fig. 4. Field gradient control and measurement.

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 D_T1 – D_S (black arrow, black dotted arrow) and D_T0 – D_S (red arrow) of both the m_s = 1/2 and m_s = −1/2 states. The D_T0 – D_S splitting is constant regardless of the magnetic field gradient ΔB_Z, whereas the D_T1 – D_S energy spacing is modulated by the ΔB_Z depending on the sign of ΔB_Z and m_s. 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 ΔB_Z 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 ΔB_Z in the inset of (d). The simulation in the frequency domain reproduces the branches shown in (b).