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

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.