Coherent Multispin Exchange Coupling in a Quantum-Dot Spin Chain

Heisenberg exchange coupling between neighboring electron spins in semiconductor quantum dots provides a powerful tool for quantum information processing and simulation. Although so far unrealized, extended Heisenberg spin chains can enable long-distance quantum information transfer and the generation of nonequilibrium quantum states. In this work, we implement simultaneous, coherent exchange coupling between all nearest-neighbor pairs of spins in a quadruple quantum dot. The main challenge in implementing simultaneous exchange couplings is the nonlinear and nonlocal dependence of the exchange couplings on gate voltages. Through a combination of electrostatic simulation and theoretical modeling, we show that this challenge arises primarily due to lateral shifts of the quantum dots during gate pulses. Building on this insight, we develop two models that can be used to predict the confinement gate voltages for a desired set of exchange couplings. Although the model parameters depend on the number of exchange couplings desired (suggesting that effects in addition to lateral wave-function shifts are important), the models are sufficient to enable simultaneous and independent control of all three exchange couplings in a quadruple quantum dot. We demonstrate two-, three-, and four-spin exchange oscillations, and our data agree with simulations.

Popular Summary

Electron spins in semiconductors are promising quantum bits (qubits) because of their long coherence times and potential for scalability. A challenge for electron-spin qubits is transferring quantum information between distant qubits, which is required for universal quantum computing. Most current approaches to overcoming this obstacle involve coupling neighboring qubits two at a time. In this work, we devise a means of precisely controlling simultaneous interactions between multiple electron spins, which opens up the possibility of exploring a vast array of new and exciting phenomena that could be harnessed for efficient transfer of quantum information between distant qubits.

Electron spins in semiconductor quantum dots naturally interact with each other via Heisenberg exchange coupling, which results from direct electronic wave-function overlap. This exchange coupling depends sensitively on voltage pulses applied to the confinement gates. We show that a primary reason for this sensitivity is that the position, rather than just the shape, of the electronic wave functions changes in response to these voltage pulses. Using a well-established microscopic theory, we can predict the simultaneous motion of multiple electrons and thus control multiple exchange couplings at the same time. We demonstrate coherent exchange coupling between three and four electrons in our system, and we expect that our approach can scale to even more electrons.

Our results have the potential to unlock a new set of phenomena that could be harnessed for quantum information processing applications that rely on precise control of Heisenberg exchange coupling in extended systems, such as spin-chain-mediated quantum communication.

Figure 1

(a) Scanning electron micrograph of the quadruple quantum-dot device. The scale bar is 200 nm. (b) Measured exchange oscillations with virtual barrier gate $B_3=30\mathrm{mV}$. Inset: Absolute value of the fast Fourier transform (FFT) of data shown in panel (b). As $B_2$ increases, $J_3$ decreases. $P_{S^R}$ is the singlet return probability of the right pair.

Figure 2

Electrostatic simulations. (a) Line cuts of the simulated potential associated with dots 2 and 3 vs the barrier voltage pulse $B_2$. The left dip is the potential of dot 2, and the right dip is the potential of dot 3. The dots move closer together as $B_2$ increases. The dashed lines are guides to the eye. (b) Fitted parameters of the simulated double-dot potential vs $B_2$. Based on the simulated potential of dot 1 (not shown), we also find that dots 1 and 2 move farther apart during the sample pulse.

Figure 3

Dependence of $J_1$ on the barrier-gate voltages. (a) $J_1$ vs $B_1$. (b) $J_1$ vs $B_2$, for $B_1=60\mathrm{mV}$. (c) $J_1$ vs $B_3$, for $B_1=60\mathrm{mV}$. The black data points in each panel are obtained from the fast Fourier transform of a data set similar to Fig. 1. In (a)–(c), the dark blue line is the fit to the exponential model, and the light blue line is the fit to the HL model. Panels (d)–(f) show the difference between the fits and the data for the two models.

Figure 4

Two- and three-spin exchange coupling. (a) Two-spin exchange oscillations obtained by linearly sweeping $j_3$ from 0 to 200 MHz. Inset: FFT of the data. (b) Three-spin exchange oscillations obtained by linearly sweeping $j_3$ from 10 to 150 MHz and fixing $j_2$ at 70 MHz. Inset: FFT of the data. Theoretical predictions of the exchange oscillation frequencies are overlaid in red. (c) Simulated three-spin exchange oscillations corresponding to the data in panel (b). Inset: Simulated FFT. (d) Line cuts from panels (b) and (c) at $j_2=j_3=70\mathrm{MHz}$. In all panels, ${\mathrm{P}}_{\mathrm{S}}^{\mathrm{R}(\mathrm{L})}$ is the singlet return probability of the right (left) pair of dots.

Figure 5

Four-spin exchange oscillations. (a) Experimental data measured on the left side. Inset: FFT of the data. (b) Experimental data measured on the right side. Inset: FFT of the data. (c) Simulated four-qubit exchange oscillation data on the left side. Inset: FFT of the simulation. (d) Simulated four-qubit exchange oscillation data on the right side. Inset: FFT of the simulation.