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

Spin qubits in semiconductor quantum dots are one of the promising 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 paper, 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 us 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 frequencies. 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 ${\ensuremath{\omega}}_{th}$, when the two-level system reacts to the statistical distribution of the fluctuator states. We show that the dephasing time ${T}_{2}({\ensuremath{\omega}}_{th})$ 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, ${T}_{2}({\ensuremath{\omega}}_{th})$ remains bounded due to dephasing induced by the nondiagonal terms of the stochastic perturbation Hamiltonian. On the other hand, our simulations reveal that the spin relaxation, usually characterized by the time ${T}_{1}$, 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 multilevel simulations including the coupling to phonons should be necessary to describe the relaxation phenomenon in this type of qubit.