Exchange interaction in gate-defined quantum dots beyond the Hubbard model

TL;DR

This work addresses the need for a quantitative description of the exchange coupling $J$ in gate-defined GaAs double quantum dots across a wide detuning range, essential for reliable spin-qubit gate operations. It combines a novel amplitude-frequency (Ramsey+FID) measurement with three theoretical frameworks—3D FC I, 1D FC I with a Coulomb-cutoff length, and an extended Hubbard model that includes excited orbital states—to map $J(\epsilon)$ and extract key parameters. The study finds that 3D FC I captures the full detuning behavior well, while an extended FH model extends the predictive range near the charge transition, and a gauged 1D FC I can reproduce the data with moderate computational effort. These results offer practical guidance for selecting modeling approaches in gate-defined quantum dots and have implications for designing high-fidelity exchange-based qubit operations in GaAs and related materials.

Abstract

A quantitative description of the exchange interaction in quantum dots is relevant for modeling gate operations of spin qubits. By measuring the amplitude and frequency of exchange-driven qubit state oscillations, we measure the detuning dependence of the exchange coupling in a GaAs double quantum dot over three orders of magnitude. Both 1D and 3D full configuration interaction simulations can replicate the observed behavior. Extending a Hubbard model by including excited states increases the range of detuning where it provides a good fit, thus elucidating the underlying physics.

Figures (8)

• Figure 1: (a) Electron micrograph of an identical device. Red circles indicate the dot positions. Gate voltages $V_i$ can be pulsed to change the double dot state on a nanosecond time scale. Reflectance from a nearby sensor dot in an rf tank circuit senses the double dot charge state. We define the x-axis along the inter-dot direction, the z-axis along the out-of-plane direction and y-axis perpendicular to both. (b) Bloch sphere for the $S$-$T_0$ qubit. Exchange $J$ and nuclear magnetic field gradient $\mathrm{\Delta}B_z$ drive rotations around the $z$-/$x$-axis. A finite $J$ tilts the rotation axis, so that the observed frequency is increased and the oscillation amplitude is decreased. (c) Exchange interaction $J$ measured as a function of detuning $\epsilon^\prime$ using a combination of the Ramsey and combined amplitude-frequency technique. $\epsilon^\prime$ is offset to be zero at the $(2,0)$-$(1,1)$ charge transition. The middle of the $(1,1)$ charge region is situated at approximately $\epsilon^\prime = -15m V$. As expected from symmetry, $J(\epsilon\xspace^\prime)$ levels off to a minimum value in the middle of the $(1,1)$ charge region.
• Figure 2: (a) Comparison of the measured $J(\epsilon\xspace)$ with the output of the numerical models. The different curves are labeled with the respective tunnel couplings and offset in $J$ for visual clarity. We use the 3D model to gauge the Coulomb-cutoff length $l_c$ of the 1D model to $l_c = 21.85nm$ using the curve with $t=22µeV$ and employ this 1D model to fit the other two measurements. Using the experimentally determined lever arm we find good qualitative agreement between model and experiment. For $t=22µeV$ the potential parameters are $x_0 = 107.5nm$, $V_0 = 1.575m eV$ and a comparison of extracted parameters with the experiment are shown in Tab. \ref{['tab:1dfit']} (for all parameters of other two measurements see Appendix \ref{['sec:supl:other_exchange_fits']}). The detuning axis is referenced to the $(2,0)$-$(1,1)$ charge transition. (b) Common models with parameters extracted from the exact 1D model. The range over which the extended FH model captures the $J(\epsilon\xspace)$ behavior is much extended, especially in the range of low detuning. Inset: Diagram for the additional terms in the extended FH model.
• Figure A.1: Measured oscillation amplitude as a function of its frequency using the FID pulse for different pulse depths $\epsilon\xspace$ at $t=22µeV$. The two colored traces exemplary show fits of Eq. \ref{['eq:ampfreq']}, which are used to extract $J(\epsilon\xspace)$ shown in figure \ref{['fig:data']}(c) of the main text. The fit range is chosen manually to exclude the noise floor at small $\omega$ and the drop off at large $\omega$, as indicated by the dashed lines
• Figure B.1: (a) Fit $A\text{cos}(\omega)e^{t/T_2^*}$ to the Ramsey measurement results ($J=91MHz$). Using an exponential decay for the oscillations provides a better fit to the data a Gaussian decay, which may be related to non-ergodicity effects of the system. (b) Pulse sequences used in this work (offset in $\epsilon\xspace$). For the Ramsey sequence $\ket{\uparrow \downarrow}$ is adiabatically prepared and read out to observe $z$-rotations. For the FID sequence a singlet is prepared and rapidly separated to observe $S-T_0$ oscillations.
• Figure D.1: (a) Pulse sequence for the excited-state tunneling measurement $t_T$. The $\ket{\uparrow\downarrow}$ state is prepared adiabatically and pulsed back rapidly to some detuning where the charge state is read out for $100ns$ to $200ns$, which is short compared to the lifetime of the prepared state. (b) Charge signal of the singlet ground state (blue) and the $\ket{\uparrow\downarrow}$ state (orange) measurement. The width of the transition labeled ${S}$ (${T_0}$) transition gives $t_S$ ($t_T$) while the distance between both transitions corresponds to the singlet-triplet splitting $\Delta_{ST}$. The linear background reflects the direct effect of the gates on the charge sensor.