and the other four being the bath. The coupling coefficients are again randomly chosen from $[0,1]$ with dimensionless units. The results are analogous to those seen in Fig. 1 and Fig. 2, confirming the general applicability of our UDD control sequence in multi-level quantum systems. Note also that even for the $N=2$ case (middle curve in Fig. 4), decoherence suppression already shows up clearly. The results here may motivate experimental UDD studies using systems analogous to the kicked-top system realized in Ref. [@Jessen].

Discussion and Conclusion
=========================

So far we have assumed that the system-bath coupling, the bath self-Hamiltonian, and the system Hamiltonian in the absence of the control sequence are all time-independent. This assumption can be easily lifted. Indeed, as shown in a recent study by Pasini and Uhrig for single-qubit systems [@Pasini], the UDD result holds even after introducing a smooth time dependence to these terms. The proof in Ref. [@Pasini] is also based on Yang and Liu’s work [@univUDD]. A similar proof can be done for our extension here. Take the two-qubit case with the control operator $Y_1$ as an example. If $H_0$ and $H'$ are time-dependent, then the unitary evolution operator in Eq. (\[Ut\]) is changed 