Floquet systems that are generally defined by periodically driven or time-dependent Hamiltonians with H(t+T)=H(t) for a fixed period T offer new opportunities to observe quantum Floquet matter [1]. In the area of topology, one appealing aspect of Floquet engineering is the ability to generate Floquet topological phases of matter [2–4] that are inaccessible in static equilibrium systems, thus providing a way to better understand the associated influence such as Floquet Majorana modes [5–7], anomalous topological phases with a zero Chern number [2,3], and a chiral topological phase with 0- or π-energy topological edge states [8–10]. Searching a topological phase of matter has raised considerable attention in condensed-matter physics; however, realizing Floquet-engineered topological phases in materials remains theoretical.

Remarkable advances in quantum simulation [11,12] have revolutionized our understanding of complex systems. A high degree of controllability enables ultracold atoms [13–15], trapped ions [16,17], superconducting circuits [18,19], and photonic systems [20] to offer the feasibility to tackle problems that are intractable on classical computers. Quantum simulators would not only unveil new results that cannot be otherwise predicted or classically simulated, but they would also allow us to test various models. For instance, analog quantum simulators (AQSs) mimic the time evolution of one specific model Hamiltonian; as a result, they are used to investigate the topological phases and effects with great experimental progresses recently [21–39]. As the counterpart of AQS, digital quantum simulation (DQS) [11,12,40–43] encodes the state of the quantum system onto qubits and emulates the time evolution through repeated cycles of qubit rotations (quantum gates) by means of a quantum algorithm. Such a circuit-based simulator can, in principle, efficiently simulate any finite-dimensional local Hamiltonian, hence owning the advantage of universality. Although the DQS of many-body physics has been intensively studied in various programmable quantum simulators [44–50], digital simulation and detection of topological phases are still less explored [51].

In this paper, we report the realization of the DQS of Floquet topological phases in a solid-state digital quantum simulator at room temperature [52,53]. In contrast to the AQS approach simulating static topological phases and anomalous π modes in a photonic system [54], it is illustrated here that digital quantum simulator constitutes a natural platform to simulate Floquet topological phases. We further exhibit the DQS of a quantum quench and observe the nonequilibrium dynamics of Floquet topological phases. Most strikingly, after such quench, we show that the signature 0- and π-energy topological invariants associated with Floquet topological phases could be detected through measuring the time-averaged spin polarizations, where the experimental observation is in good agreement with theoretical results. We also report the first experimental observation of the topological winding number ν=5, which is much higher than one. Further applications of this protocol could enable studies of high-dimensional and complex Floquet topological phases that go beyond the conventional topological systems [55,56].

Floquet topological phases. In this work, we consider simulating the two-band Floquet topological insulator phases described by this Floquet Hamiltonian, HF=dx(kx)σx+dy(kx)σy,(1)where dx and dy are the spin-orbit fields. Specifically, we simulate a one-dimensional (1D) periodically driven system formed by two units in one driven period [57,58]. Suppose U is the Floquet operator that describes such a periodically driven system evolving over one period T; i.e., U^=e−iH2T2e−iH1T2,(2)where H^1=tysin(kx)σ^y and H^2=txcos(kx)σ^x. Then, the Floquet Hamiltonian HF describing the emerged Floquet topological phases is defined as U^=e−iH^FT.(3)As we will show, H1 and H2 both can be simulated by applying two microwave pulses on the solid-state NV-center qubit, which further allows us to digitally realize the Floquet operator U.

Regarding a Floquet topological system, there are two quasi-energy gaps centered around E=0 and E=π. The topological features of Floquet topological phases are rooted in these gaps, which are characterized by the topological invariants ν0 and νπ, respectively. According to bulk-edge correspondence associated with topological phases, the values of topological invariants defined in the momentum space count the number of the edge modes defined in the real space. The edge modes correspond to the eigenmodes of the real-space lattice Hamiltonian with their densities maximally localized at the edges. In our Floquet topological systems, the value of ν0 (νπ) determines the number of the edge modes with eigenenergy E=0 (E=π).

The topological invariants ν0 and νπ are defined in terms of a symmetry time framework [8], where the starting time point of the Floquet operator U is shifted to two symmetry time points, leading to the following two Floquet operators: $$\begin{gathered} U^1=e−iH^1T4e−iH^2T2e−iH^1T4, \\ U^2=e−iH^2T4e−iH^1T2e−iH^2T4. \end{gathered}$$(4)Both can be further rewritten in terms of Floquet Hamiltonian as U^1=e−iH^F1T,U^2=e−iH^F2T,(5)where H^F1,F2=Ens·σ^, with ns=(nsx,nsy,0) (s=1,2) [59], E=±arccos{cos[txcos(kx)]cos[tysin(kx)]}, and σ^=(σ^x,σ^y,σ^z) are Pauli spin operators.

Of great interest is the fact that both H^F1 and H^F2 are protected by a chiral symmetry with the chiral operator Γ^=σ^z, supporting chiral topological phases characterized by these topological winding numbers, νs=12π∫dkx(nsx∂kxnsy−nsy∂kxnsx),s=1,2.(6)

The topological invariants ν0 and νπ are defined as ν0=ν1+ν22,νπ=ν1−ν22.(7)Figure 1 presents the numerical results of the values of the topological invariants ν0 and νπ as a function of tx and ty. Surprisingly, such a simple periodically driven system has a rich topological phase diagram, and even can support topological phases with topological invariants larger than one [57,58]. On the other hand, a topological phase transition is known to occur accompanied by a gap closing. In Fig. 1, we also plot the quasi-energy gap closing points at E=0 and E=π. It is shown that the values of the topological invariant ν0 (νπ) would change when crossing the gap closing at E=0 (E=π).

Digital simulation of Floquet topological phases. To digitally simulate the Floquet operator U^(T) (the time evolution of the Floquet topological Hamiltonian H^F), we use a negatively charged nitrogen-vacancy (NV) center in type-IIa, single-crystal synthetic diamond sample (Element Six). ms=−1 and ms=0 in A23 are encoded as spin down |g⟩ and up |e⟩ of the electron spin qubit (Fig. 2). The state of the qubit can be manipulated with microwave pulses (ωMW=2π×1404.3MHz), while the spin level ms=+1 remains idle due to large detuning. By applying a laser pulse of 532 nm wavelength with the assistance of intersystem crossing (ISC) transitions, the spin state can be polarized into ms=0 in the ground state. This process can be used to initialize and read out the spin state of the NV center. The fluorescence photons are detected using the single photon counting module (SPCM). By using a permanent magnet, a magnetic field (about 520 G) is applied along the NV axis, and the nearby nuclear spins are polarized by optical pumping, improving the coherence time of the electron spin.

The key ingredient in our experiment is to individually engineer the time evolution of the Hamiltonian H^1=tysin(kx)σ^y and H^2=txcos(kx)σ^x, offered by such a well-controlled solid state quantum simulator [60]. Both of the Hamiltonians can be emulated by manipulating the electron spin qubit via a microwave pulse with H^1=Ωyσ^y,H^2=Ωxσ^x and the associated Rabi frequencies Ωx and Ωy. Therefore, by tuning Ωx=txcos(kx), Ωy=tysin(kx), the Floquet operators depicted in Eq. (4) can be naturally simulated in a digital way [Figs. 2(c) and 2(d)]. It is worth mentioning that this method is genetic and not limited to the 1D topological phases studied here. This approach can be also mapped into the high-dimensional Brillouin zone and generalized to digital simulation of the Floquet operators associated with high-dimensional Floquet topological phases.

Digital simulation of quantum quenches and detection of topological invariants. The topological invariants ν0 and νπ featuring the topological properties of the quasi-energy gaps centered around E=0 and E=π are seminal hallmarks of Floquet topological phases [8]. Recent theoretical study has shown that, after a quantum quench, the topological invariants associated with static topological phase can be directly measured through the time-averaged spin polarizations on the band inversion surfaces (BISs) [61]. We thus proceed to show that such a method also can be used to measure the topological invariants associated with Floquet topological phases. Specifically, we digitally perform a quantum quench of Floquet topological phases and employ such a quench to measure the topological winding numbers ν1 and ν2, allowing us to detect the topological invariants ν0 and νπ according to Eq. (7).

Here is the quantum quench procedure. First, the initial state of the system is initialized in the ground state of a trivial Floquet topological Hamiltonian H^Fi. Then, N series of the Floquet operators U^1,2 are digitally performed, as shown in Fig. 2(c) or 2(d), which digitally simulates the time evolution of a nontrivial Floquet topological Hamiltonian over N periods; i.e., U^1N=e−iNH^F1T,U^2N=e−iNH^F2T.(8)As a consequence, a sudden change from a trivial Floquet topological phase to a nontrivial Floquet topological phase is effectively performed, realizing a quantum quench. The initial quantum state will thereby evolve under the final Hamiltonian H^F1,F2.

In our work, two cases in detail are exhibited: (1) tx=0.5π, ty=0.5π, with (ν1=1, ν2=1) and (ν0=1, νπ=0); (2) tx=2.5π, ty=0.5π, with (ν1=1, ν2=5) and (ν0=3, νπ=−2). In particular, we implement a y-direction quench for the Floquet topological Hamiltonian H^F1 and an x-direction quench for the Floquet topological Hamiltonian H^F2.

To measure the topological winding number ν1 in case (1), the initial state of the system is prepared into |ψ(t=0)⟩=(|g⟩−i|e⟩)/2, the ground state of a trivial topological Hamiltonian H^F1i=n1xσ^x+(my+n1y)σ^y with my≫1. After that, we repeat U^1 operation N times. Consequently, the dynamics of the initial state |ψ(t=0)⟩ is governed by a nontrivial topological Hamiltonian H^F1, fulfilling a quantum quench from my≫1 to my=0. In this y-direction quench process, the BIS appears when n1y=0 [61], which yields kxBIS=0,π. After the quantum quench, we measure the time evolution of the spin polarization ⟨σ^x,y⟩ for each kx in the Brillouin zone, from which we extract the time-averaged spin polarization ⟨σ^x,y⟩¯=1N∑t=1N⟨σ^x,y⟩t, with the time-resolved spin polarization ⟨σ^x,y⟩t=⟨ψ(t=0)|(U^1−1)t·σ^x,y(U^1)t|ψ(t=0)⟩.

When N is very large, the BISs kxBIS and the topological winding number ν1 both can be directly measured through ⟨σ^x,y⟩¯ [61]; i.e., $$\begin{gathered} ⟨σ^y(kxBIS)⟩¯=0, \\ ν1=12[gx(kxBIS=π)−gx(kxBIS=0)], \end{gathered}$$(9)where gx(kxBIS)=−sgn(∂k⊥⟨σ^x⟩¯) is related to the sign of the slope of the time-averaged spin polarization ⟨σ^x⟩‾ at BISs, with k⊥ denoting the momentum perpendicular to BIS and points from n1y<0 to n1y>0. The theoretical results for N=60 are presented in Fig. 3(b), showing that ⟨σ^y(kxBIS=0,π)⟩¯=0 and gx(kxBIS=π)=−gx(kxBIS=0)=1.

The achievable repetition number N is limited by the coherence time T2* of the electron spin [T2*≈2.6μs measured with a Ramsey free precession sequence, as shown in Fig. 1(d)], which is also the main error source in the operation. We take the optimal number of N=10, experimentally. Figure 3(a) shows the experimentally measured results of ⟨σ^x,y⟩‾ are in good agreement with the theoretical results. Based on these results, it is found that the time-averaged spin polarization ⟨σ^y⟩‾ is zero when kx=0,π, which allows us to clearly identify the BISs. Note that n1y>0 when kx∈(0,π); otherwise, n1y<0. Therefore, the slope of the time-averaged spin polarization ⟨σ^x⟩‾ at kxBIS=0 (kxBIS=π) is recognized as 1 (−1), yielding gx(kxBIS=π)=−gx(kxBIS=0)=1. According to Eq. (9), we can determine the topological winding number value as ν1=1.

To measure the topological winding number ν2 in case (1), the initial state of the system is prepared into |ψ(t=0)⟩=(|g⟩−|e⟩)/2, the ground state of a trivial topological Hamiltonian H^F2i=(mx+n2x)σ^x+n2yσ^y with mx≫1. Subsequently, an x-direction quantum quench from mx≫1 to mx=0 is implemented by repeating U^2N times, thus causing the time evolution of the initial state |ψ(t=0)⟩ to be governed by a nontrivial topological Hamiltonian H^F2. The BIS appears when n2x=0 [61], which gives kxBIS=±0.5π. The BISs and the topological winding number ν2 are measured through [61] $$\begin{gathered} ⟨σ^x(kxBIS)⟩¯=0, \\ ν2=12[gy(kxBIS=0.5π)−gy(kxBIS=−0.5π)], \end{gathered}$$(10)where gy(kxBIS)=−sgn(∂k⊥⟨σ^y⟩¯) is related to the sign of the slope of the time-averaged spin polarization ⟨σ^y⟩‾ at the BISs, with k⊥ denoting the momentum perpendicular to BIS and pointing from n2x<0 to n2x>0. As theoretically presented in Fig. 3(d) for N=60, ⟨σ^x(kxBIS=0,π)⟩¯=0 and gy(kxBIS=0.5π)=−gy(kxBIS=−0.5π)=1. The corresponding experimental results for N=10 are shown in Fig. 3(c) and agree well with theoretical results. From these experimental results, we can unambiguously conclude ⟨σ^x(kx=±0.5π)⟩‾=0 and gy(kxBIS=0.5π)=−gy(kxBIS=−0.5π)=1, identifying the location of the BISs and the value of the topological winding number ν2=1. Hence, according to Eq. (7), the 0- and π-energy topological invariants are measured as ν0=1 and νπ=0.

Regarding case (2), the same procedure is applied to extract the topological winding number ν1 and ν2. The experimental results on the time-averaged spin polarizations after N=10 times U^1 and U^2 are shown in Figs. 4(a) and 4(c), respectively, which agree well with the theoretical results. Figure 4(a) illustrates that the BIS appears at kxBIS=0,π in which ⟨σy⟩‾=0 and gx(kxBIS=π)=−gx(kxBIS=0)=1. The value of the topological winding number is measured through ν1=12[gx(kxBIS=π)−gx(kxBIS=0)]=1. Figure 4(c) shows that the BIS appears at kxBIS=±0.5π,±arccos(±0.4),±arccos(±0.8). At such points, ⟨σ^x⟩‾ is not strictly zero but still maximal and approaching zero. As suggested by the theoretical results shown in Fig. 4(d) for N=60, the slight difference from zero is a result of the fact that the number N we chose in the experiment is not large enough. This does not affect the measurement of the slopes of the time-averaged spin polarization ⟨σy⟩‾ at the BISs; i.e., gy(kxBIS∈k+)=−gy(kxBIS∈k−)=1, where k+=0.5π,arccos(±0.8),−arccos(±0.4) and k−=−0.5π,−arccos(±0.8),arccos(±0.4). The topological winding number ν2 is measured through ν2=12[∑kxBIS∈k+gy(kxBIS)−∑kxBIS∈k−gy(kxBIS)], giving the value of the topological winding number as ν2=5. Substituting the above values into Eq. (7), the 0- and π-energy topological invariants are determined as ν0=3 and νπ=−2.

We emphasize that gx,y(kxBIS) is related to the slopes of the time-averaged spin polarizations ⟨σx,y⟩‾ at the BISs and quite robust to the experimental imperfections. This feature manifests topological protection and enables accurate measurements of the topological invariants of the Floquet topological phases.

In summary, we have reported the digital simulation and detection of Floquet topological phases with a solid-state quantum simulator. To measure Floquet topological invariants, quantum quenches of Floquet topological phases are digitally simulated. The method developed here can be directly applied to other well-developed platforms of quantum simulators, such as superconducting circuits and trapped ions. We believe our work opens the door for the DQS of topological phases with programmable quantum simulators, including high-dimensional Floquet topological insulators [2–4,62], Floquet Z2 topological phases [63], and Floquet Hopf insulators [64] that are hard to engineer in other topological systems. This also paves the way for DQS of nonequilibrium topological phases [65–67], where quantum quenches could be digitally simulated.