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The electrical control of a spin qubit in a quantum dot relies on spin-orbitcoupling (SOC), which could be either intrinsic to the underlying crystallattice or heterostructure, or extrinsic via, for example, a micro-magnet. Herewe show that a key difference between the intrinsic SOC and the synthetic SOCintroduced by a micro-magnet is their symmetry under time reversal.Specifically, the time-reversal symmetry ($\mathcal{T}$-symmetry) of theintrinsic SOC leads to not only the traditional van Vleck cancellation knownfor spin relaxation, but also vanishing spin dephasing to the lowest order ofSOC, which we term as "longitudinal spin-orbit field cancellation". On theother hand, the synthetic SOC from a micro-magnet breaks the$\mathcal{T}$-symmetry, therefore eliminates both the "van Vleck cancellation"and the "longitudinal spin-orbit field cancellation". In other words, theeffective field $\vec\Omega$ experienced by the spin qubit does not depend onthe quantization magnetic field anymore, and a longitudinal component isallowed for $\vec\Omega$ to the first order of SOC. Consequently, spinrelaxation and dephasing are qualitatively modified compared with the case ofthe intrinsic SOC. Furthermore, the fidelity of electric-dipole spin resonancebased on $\vec\Omega$ could be optimized, with potential applications inspin-based quantum computing.

Impact of $\mathcal{T}$-symmetry on spin decoherence and control in a synthetic spin-orbit field