Quantum simulation with electron spins in quantum dots

More is more applies in particular to systems with interacting parts. These interactions enable the emergence of collective behaviour. Examples can be found among the behaviour of animals, such as the V-shaped formation of migrating geese and the flight of a flock of starlings. More examples are found among the electromagnetic properties of materials. For properties that rely on quantum-mechanical correlations it quickly becomes infeasible for classical numerical simulations to provide accurate results. An appealing alternative is to study these properties with quantum simulators, which mimic the material properties themselves. Besides being of scientific interest for the field of condensed matter physics, insights obtained from quantum simulations could in the future serve as input for the synthesis of novel materials. Developing quantum simulators requires the engineering of quantum systems. One such quantum system is that of electrons in gate-defined quantum dots, which are formed by three-dimensional confinement at the nano-scale. Experiments with quantum dots have already demonstrated measurement and coherent control of both individual charges and spins, and their operation as quantum bits. The first quantum simulation experiments with quantum dots have been performed in the last couple of years. Further development of quantum dots as platform for quantum simulations forms the overarching motivation for this thesis. The first experiment in this thesis describes the automated tuning of the tunnel coupling between quantum dots. This automation builds on previously developed automated tuning of double quantum dots. The automated tuning relies on image processing to extract parameters from measurement results. This step is part of a feedback loop in which the voltages on the gates are iteratively adjusted. This loop repeats until the target tunnel coupling is achieved. The second experiment further studies the tuning of tunnel couplings. For operation of gate-defined quantum dots it is common practice to independently control chemical potentials with so-called virtual gates. These virtual gates compensate for crosstalk effects due to cross-capacitances of the physical gates. The control of multiple tunnel couplings similarly suffers from crosstalk, but efficient compensation techniques were lacking. This chapter reports an efficient calibration scheme for such crosstalk, and demonstrates independent control of tunnel couplings with enhanced virtual gates. The third experiment demonstrates a method to measure charge and spin in large quantum dot arrays. The charge configuration of a quantum dot array is typically measured with a charge sensor, which is usually another quantum dot. To measure the spin configuration it is first mapped onto a charge configuration, which for singlet-triplet measurements is based on the Pauli exclusion principle. The charge measurement relies on Coulomb repulsion, which decays with distance, thus only charge and spin close to the sensor can be reliably measured. This chapter presents how, inspired by the effect of toppling dominoes, a cascade of hopping electrons induced by Coulomb repulsion can effectively convert the information about motion of a distant charge to the motion of a charge close to the sensor. The benefit of cascade-based readout is demonstrated by comparing singlet-triplet measurements with or without the cascade activated. The most involved experiment described in this thesis is a proof-of-principle quantum simulation of Heisenberg magnetism, which is one of the most famous models in condensed-matter physics. Specifically, this experiment demonstrates how a linear array of quantum dots can be operated as a Heisenberg spin chain. The first part of the experiment shows the characterization of the energy spectrum, which is based on degeneracies between spin states with different magnetization. From the energy spectroscopy the conditions are identified for which the exchange couplings are homogeneous. Next, the coherence is studied by inducing global exchange oscillations, and evolution in different subspaces of the Heisenberg Hamiltonian is demonstrated. The final step of the experiment consists of the adiabatic preparation of the low-energy global singlet state for a homogeneous chain, and its characterization with pairwise singlet-triplet measurements for each of the nearest-neighbours and correlations therein. These techniques and results form the basis for the operation of quantum dots to simulate larger spin systems and different lattice structures. The final experiment, shifts the focus from spin-spin interactions to electron-electron interactions. For gate-defined quantum dots, the Coulomb repulsion results in both on-site and inter-site interactions between electrons. The interaction is experimentally characterized with a linear array of six dots in which the tunnel couplings are tuned to be homogeneous. The decay of the interaction as a function of distance is modelled with both the method of image charges, where the gate metal acts as screening layer, and with a Yukawa type potential as a heuristic model. The latter provides an intuitive interpretation for the decay of the interaction in terms of a screening length. The characterization of the long-range electron-electron interaction is relevant for the operation of quantum dot arrays as hosts of spin qubits, but also for quantum simulations in which the charge degree of freedom and electron-electron interactions play an important role. Some examples of many-body physics for which long-range interactions are essential, are quantum chemistry, Wigner crystallization, and high-temperature superconductivity. Summarizing, this thesis reports novel techniques for the control and measurement of larger quantum dot arrays, the operation of such an array as quantum simulator of Heisenberg magnetism with control over the spin-spin interactions, and characterization of the electron-electron interactions. These results pave the way for future quantum simulations with quantum dots.