  The electrical control of a spin qubit in a quantum dot relies on spin-orbit
coupling (SOC), which could be either intrinsic to the underlying crystal
lattice or heterostructure, or extrinsic via, for example, a micro-magnet. Here
we show that a key difference between the intrinsic SOC and the synthetic SOC
introduced by a micro-magnet is their symmetry under time reversal.
Specifically, the time-reversal symmetry ($\mathcal{T}$-symmetry) of the
intrinsic SOC leads to not only the traditional van Vleck cancellation known
for spin relaxation, but also vanishing spin dephasing to the lowest order of
SOC, which we term as "longitudinal spin-orbit field cancellation". On the
other 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, the
effective field $\vec\Omega$ experienced by the spin qubit does not depend on
the quantization magnetic field anymore, and a longitudinal component is
allowed for $\vec\Omega$ to the first order of SOC. Consequently, spin
relaxation and dephasing are qualitatively modified compared with the case of
the intrinsic SOC. Furthermore, the fidelity of electric-dipole spin resonance
based on $\vec\Omega$ could be optimized, with potential applications in
spin-based quantum computing.
