Harnessing the dynamics of complex quantum systems is an area of much interest and a quantum simulator has emerged as a promising platform to probe exotic topological phases. Since the flexibility offered by various controllable quantum systems has helped gain insight into the quantum simulation of such complicated problems, an analog quantum simulator has recently shown its feasibility to tackle the problems of exploring topological phases. However, digital quantum simulation and the detection of topological phases still remain elusive. Here, we develop and experimentally realize the digital quantum simulation of topological phases with a solid-state quantum simulator at room temperature. Distinct from previous works dealing with static topological phases, the topological phases emulated here are Floquet topological phases. Furthermore, we also illustrate the procedure of digitally simulating a quantum quench and observing the nonequilibrium dynamics of Floquet topological phases. Using a quantum quench, the 0- and π-energy topological invariants are unambiguously detected through measuring time-averaged spin polarizations. We believe our experiment opens up a new avenue to digitally simulate and detect Floquet topological phases with fast-developed programmable quantum simulators.

The nitrogen-vacancy (N-V) center in diamond is a widely used platform for quantum information processing and sensing. The electron-spin state of the N-V center could be initialized, read out optically, and manipulated by resonate microwave fields. In this work, we analyze the dependence of electron-spin initialization on widths of laser pulses. We build a numerical model to simulate this process and to verify the simulation results in experiments. Both simulations and experiments reveal that shorter laser pulses are helpful to the electron-spin polarization. We therefore propose to use extremely short laser pulses for electron-spin initialization. In this new scheme, the spin-state contrast could be improved about 10% in experiments by using laser pulses as short as 4 ns in width. Furthermore, we provide a mechanism to explain this effect, which is due to the occupation time in the meta-stable spin-singlet states of the N-V center. Our new scheme is applicable in a broad range of N-V-based applications in the future.