Modelling of spin decoherence in a Si hole qubit perturbed by a single charge fluctuator

Spin qubits in semiconductor quantum dots are one of the promizing devices to realize a quantum processor. A better knowledge of the noise sources affecting the coherence of such a qubit is therefore of prime importance. In this work, we study the effect of telegraphic noise induced by the fluctuation of a single electric charge. We simulate as realistically as possible a hole spin qubit in a quantum dot defined electrostatically by a set of gates along a silicon nanowire channel. Calculations combining Poisson and time-dependent Schr\"odinger equations allow to simulate the relaxation and the dephasing of the hole spin as a function of time for a classical random telegraph signal. We show that dephasing time $T_2$ is well given by a two-level model in a wide range of frequency. Remarkably, in the most realistic configuration of a low frequency fluctuator, the system has a non-Gaussian behavior in which the phase coherence is lost as soon as the fluctuator has changed state. The Gaussian description becomes valid only beyond a threshold frequency $\omega_{th}$, when the two-level system reacts to the statistical distribution of the fluctuator states. We show that the dephasing time $T_{2}(\omega_{th})$ at this threshold frequency can be considerably increased by playing on the orientation of the magnetic field and the gate potentials, by running the qubit along "sweet" lines. However, $T_{2}(\omega_{th})$ remains bounded due to dephasing induced by the non-diagonal terms of the stochastic perturbation Hamiltonian. Our simulations reveal that the spin relaxation cannot be described cleanly in the two-level model because the coupling to higher energy hole levels impacts very strongly the spin decoherence. This result suggests that multi-level simulations including the coupling to phonons should be necessary to describe the relaxation phenomenon in this type of qubit.