Topological spin-orbit-coupled fermions beyond rotating wave approximation

The realization of spin-orbit-coupled ultracold gases has driven a wide range of researches and is typically based on the rotating wave approximation (RWA). By neglecting the counter-rotating terms, RWA characterizes a single near-resonant spin-orbit (SO) coupling in a two-level system. Here, we propose and experimentally realize a new scheme for achieving a pair of two-dimensional (2D) SO couplings for ultracold fermions beyond RWA. This work not only realizes the first anomalous Floquet topological Fermi gas beyond RWA, but also significantly improves the lifetime of the 2D-SO-coupled Fermi gas. Based on pump-probe quench measurements, we observe a deterministic phase relation between two sets of SO couplings, which is characteristic for our beyond-RWA scheme and enables the two SO couplings to be simultaneously tuned to the optimum 2D configurations. We observe intriguing band topology by measuring two-ring band-inversion surfaces, quantitatively consistent with a Floquet topological Fermi gas in the regime of high Chern numbers. Our study can open an avenue to explore exotic SO physics and anomalous topological states based on long-lived SO-coupled ultracold fermions.

In this article, we propose and experimentally realize 2D-SO-coupled topological fermions beyond RWA.We prepare the spin-1/2 ultracold Fermi gas of strontium ( 87 Sr) atoms in 2D optical Raman lattice, where two sets of Raman potentials of different resonance conditions are naturally generated without RWA, rendering an intrinsic Floquet SO-coupled system.By developing a highly nonlinear spin-discriminating method, we enhance the lifetime of 2D-SO-coupled fermions by an order of magnitude, achieving a long lifetime on the 100-ms scale.Using the recently developed pump-probe quench measurement method [66], we explore a novel topological phase diagram by indirectly measuring the band topology of 2D-SO-coupled fermions.In particular, various high-Chern-number Floquet topological bands are achieved.This work may open a new avenue in engineering rich long-lived topological systems with SO-coupled fermions.
The scheme.-Our proposed scheme for realizing 2D SO couplings beyond RWA builds on a spin-1/2 Fermi gas in a 2D square optical Raman lattice [52,56], with a temporal modulation naturally emerging from two sets of Raman couplings, which is depicted by a time-dependent Hamiltonian for two spin states |↑⟩ and |↓⟩: Here t is the time, ℏ is the reduced Planck's constant (set to 1 below), k is the atomic momentum, m is the atomic mass, δ 0 denotes two-photon detuning, and σ x,y,z are Pauli matrices.As illustrated in Fig. 1 contains two sets of Raman potentials Ω and Ω ′ : Ω(x, y) = Ω 01 sin k 0 x cos k 0 y +Ω 02 e iδφ cos k 0 x sin k 0 y and Ω ′ (x, y) = Ω ′ 01 sin k 0 x cos k 0 y − Ω ′ 02 e −iδφ cos k 0 x sin k 0 y, with each set including two Raman couplings to drive SO couplings in X and Y directions.Here k 0 = π/a relates to lattice spacing a, ω/2π equals to twice the frequency difference between two Raman coupling beams (say E 1Z and E 2X ), δφ is a tunable relative phase, and Ω 01(02) (Ω ′ 01(02) ) denote the amplitudes of Raman couplings.The two-photon detuning for Raman couplings in Ω ′ is given by δ , where I is the identity matrix, and V 0X↑,↓ (V 0Y ↑,↓ ) denote the optical lattice depths along the X(Y ) direction for the two spin states.An effective Zeeman splitting is further defined as m z ≡ δ 0 /2 + (ϵ ↑ − ϵ ↓ )/2, where ϵ ↑(↓) is the onsite energy of the |↑⟩ (|↓⟩) Wannier function at δ 0 = 0. Note that in the large-ω limit, the Ω ′ term in Eq. ( 2) can be neglected, known as the RWA, and Eq. ( 1) realizes the Qi-Wu-Zhang model [4,52,66].Equations ( 1) and ( 2) describe the new scheme with modulated SO couplings.The main results can be captured by considering tight-binding regime with only the nearest-neighbor hopping terms.We obtain for δφ = π/2 the temporally modulated Bloch Hamiltonian as where q is the Bloch wavevector and U 0 (q) is an overall energy shift.Here, the transverse couplings h x and h y are temporally modulated and defined by

with h
x/y 0 = 2t SO sin(q y/x a) and ω acting as the modulation angular frequency.The time-independent h z is given by h z (q) = m z − 2 t0 [cos(q x a) + cos(q y a)], where t SO and t0 = (t 0↑ +t 0↓ )/2 represent the spin-flip and mean value of spin-conserved (t 0↑,↓ ) hopping coefficients, respectively.
The above Bloch Hamiltonian (3) characterizes a novel and robust Floquet topological system resulted from SO couplings beyond RWA, which is different from the counterparts realized using photonic [76][77][78][79][80] and phononic [81] systems and ultracold bosons [82,83].The modulation of near-resonant Raman couplings can drive new types of effective SO couplings from the Floquet high-order effects, which may further contribute to novel topological physics.Fig. 1(b) shows a diagram with rich anomalous topological phases characterized by two winding numbers W 0,π , which count the numbers of chiral edge modes inside the associated quasi-energy gaps and determine the Chern number by Ch 1 = W 0 − W π [84][85][86][87][88][89][90][91][92].Using quasienergy band analysis [90][91][92][93][94], we determine the topological phase boundaries by identifying two types of bandinversion surfaces (BISs), the 0-BIS and π-BIS, which are rings in 2D systems and depict where band crossing occurs, and originate from both the static (Ω) and modulated (Ω ′ ) Raman couplings [91,92].The winding number W 0 (W π ) can be reduced to contributions of all the 0-BISs (π-BISs) [91,92].Thus, via regulating m z and ω, the system will enter a new topological phase whenever one BIS emerges or disappears.Note that under timereversal transformation, both m z and ω reverse sign, and topological invariants change sign accordingly.Thus the ω < 0 part is not shown here but can be easily obtained.
Experimental setup.-Werealize the beyond-RWA scheme in a Fermi gas of 87   previous setups [52,59,61,62,66,96], the two-level energy splitting of the present system is reduced to near zero, such that even moderate SO coupling strength is comparable to the energy splitting, bringing the system into the beyond-RWA regime in a unique manner that is advantageous for reaching a long lifetime.
We make two improvements for the experimental realization.First, we develop a highly nonlinear spindiscriminating (HNSD) method, by which | ↑⟩ and | ↓⟩ are very well isolated from nearby states.As shown in Fig. 2(a), adjacent 3 P 1 spin excited states (with F ′ = 11  2 ) are energetically separated by ∆ e ≈ 13 MHz under a 35-G magnetic field, whereas spin ground states are barely separated (∆ 0 ≪ ∆ e ).Two π-polarized HNSD lasers are applied with frequencies that are near resonance of the m F -conserving π-transitions for the unwanted states (with m F = − 5 2 and − 3 2 ) but detuned from π-transitions for |↑⟩ and |↓⟩.Unlike the far-detuned a.c.Stark shift method where |↑⟩ and |↓⟩ experience large shifts [40,61,66], the HNSD method poses minimal perturbation to |↑⟩ and |↓⟩ (with only kHz-scale shifts), but strongly up-shifts the unwanted states by ∆ H of more than 100 kHz [95] [see Fig. 2(a)].Thus, the HNSD method realizes a well isolated effective spin- 1  2 manifold of energetically nearly degenerate |↑⟩ and |↓⟩ states in a beyond-RWA regime.Second, to minimize detrimental interference effects [66], we implement a large frequency difference ∆ L between similarly polarized beam components (E 1Z,2Z or E 1Y,2X ) [95], such that optical lattices remain essentially static and atomic heating due to moving lattice potentials [66] is vastly suppressed.
Long-lived 2D-SO-coupled fermions.-Underthe new setup, we enhance the lifetime of 2D-SO-coupled fermions by an order of magnitude.Figure 2(b) shows a long lifetime τ 0 that reaches 73 ms, as compared with τ ′ 0 = 7.1 ms based on the conventional far-detuned a.c.Stark shift method [40,66] under similar lattice conditions.With a reference measurement under far-off-resonance SO couplings, we further extract a SO-coupling-restrained characteristic 1/e decay time ∼101 ms, showing the potential of our system for future improvements [95].Figure 2(b) inset shows that the lifetime remains fairly long under various average lattice depths (V 0↑ + V 0↓ )/2 up to about 2E 0 (limited by available optical power), where E 0 ≡ ℏ 2 k 2 0 /(2m) is the recoil energy.Furthermore, we measure the variance of atomic momentum distribution by time-of-flight (TOF) imaging after the fermions are held in the optical Raman lattice for various amount of time [Fig.2(c)], revealing a significant reduction of heating by a factor of 500.These results demonstrate a longlived platform of 2D-SO-coupled fermions.
Verification of SO couplings beyond RWA.-A key feature of our beyond-RWA scheme is the simultaneous generation of two sets of SO couplings naturally related to each other.From Eq. ( 2), the sum of two relative phases for SO couplings driven by Raman potentials Ω and Ω ′ equals an intrinsically fixed value of δφ+(π−δφ) = π [95].Thus, when one SO coupling is tuned to be quasi-1D in the X + Ŷ direction, the other will be quasi-1D in the orthogonal X − Ŷ direction, providing a characteristic signature of SO couplings beyond RWA.
For verification, we probe fermions with quasi-1D SO couplings using the recently developed pump-probe quench measurement (PPQM) method [66].Atoms initially prepared in the |↑⟩ state experience SO-couplinginduced, quasi-momentum-dependent spin flip into | ↓⟩ under a short pulse of optical Raman lattice, and are subsequently probed by spin-resolved TOF measurements.Fig. 3(a) shows TOF images measured for various ω.As ω increases, we observe two groups of | ↓⟩ atoms: one group (black dashed lines) moves toward the center of the first Brillouin zone (FBZ) along the X − Ŷ direction; the other (magenta dashed lines) moves away from the center along X + Ŷ .The two-photon detunings δ 0 ∼ ω/2 and δ ′ 0 = δ 0 − ω ∼ −ω/2 [see Fig. 1(a)] indicate that the effective Zeeman splittings for the two sets of SO couplings depend on ω in reverse ways.We identify the two groups as pumped respectively by the two quasi-1D SO couplings in the orthogonal directions of X ∓ Ŷ , corre- Verification of the beyond-RWA scheme using fermions with quasi-1D SO couplings.(a) Pump-probe quench measurement (PPQM) of momentum distribution of |↓⟩ atoms at various ω.As ω increases, two groups of atoms (marked by black and magenta dashed lines) move toward or away from the center of the FBZ (white square) in opposite manners, revealing two quasi-1D SO couplings in orthogonal directions of X ∓ Ŷ .Accordingly, the two relative phases for SO couplings are π and 0, showing the key feature of a phase sum of π for beyond-RWA SO couplings.
Here we set Ω01 = Ω ′ 02 = 0.36E0, Ω02 = Ω ′ 01 = 0.14E0, sponding to relative phases of π and 0 for the two SO couplings (with a sum of π).The measured highly symmetric movements (with respect to ω) of the two atomic groups with opposite effective Zeeman splittings show unambiguously the realization of SO couplings beyond RWA, as further supported by agreement between measurements and numerical results by exact diagonalization in the non-tight-binding regime [66,95] (Fig. 3(b)).
The PPQM resolution can be sharpened by optimizing the pulse length t p under a fixed pulse area [95].Figure 3(c) shows a minimum atomic ring width at t p ≈ 700 µs that is used in subsequent measurements.
We tune the SO coupling beyond RWA to the optimum 2D configuration by setting δφ = π/2, and then perform a systematic PPQM study by varying ω.In Fig. 4(a), we observe two BISs that emerge from M (±k 0 , ±k 0 ) and Γ (0, 0) points in the FBZ, evolving toward each other, switching positions at ω = 0, and finally shrink at Γ and M, respectively [109].This observation agrees with numerical simulation in Fig. 4(b) [95].In Fig. 4(c), we quantify the observation by measuring sizes of the two BISs versus ω, which also shows agreement with numerical results.The systematic measurements of BIS configurations, which match well with the numerical study, unveil the underlying nontrivial topological regimes realized in the experiment, as we elaborate below.
In Fig. 4(d), we compute a simplified phase diagram with experimental parameters by considering only leading-order SO couplings [110], and compare it with the systematic measurements in Fig. 4(a), which scan through the high-Chern-number regime with Ch 1 = ±2 [95].This regime is characterized by that the two BISs observed in Fig. 4(a) correspond to a 0-BIS (black) and a π-BIS (magenta), and are induced by two sets of SO couplings, respectively [95].Such two BISs correspond to opposite effective Zeeman splittings and carry opposite winding numbers according to previous stud-ies [91,92], yielding W 0 = −W π = ±1 and a high Chern number Ch 1 = ±2.At ω = 0, the measurement reduces to a single-ring BIS with |Ch 1 | ≤ 1 for a static Hamiltonian.These results indirectly show that the present experiment has achieved a fermionic anomalous Floquet topological system.A direct observation of topological invariants can be achieved by measuring winding numbers on BISs [91,92,105] and will be presented in next studies.
Conclusion and discussion.-Insummary, we have realized 2D SO coupling beyond RWA for ultracold fermions with a long lifetime and observed various BIS configurations that provide an indirect measurement of the nontrivial topology engineered in the current system.The SO coupling beyond RWA renders an intrinsic temporal engineering, giving rise to a rich Floquet topological phase diagram with high-Chern-number states.Together with the HNSD method developed in our experiment, the 2D-SO-coupled Floquet fermion system has a long lifetime, which can be further improved [95] and fulfills a key prerequisite for future studies of novel correlated topological phases including the topological superfluid, dynamical gauge fields, and quantum magnetism in the interacting regimes [52,[111][112][113][114].The present scheme can be naturally generalized to engineer an isolated manifold of an arbitrary number of spin states for SO-coupled Fermi gases [65,115].Applying the beyond-RWA scheme to such large-spin systems can lead to fundamentally new type of SO couplings and can potentially bring about profound spin-orbit and topological physics, which deserves efforts in future studies.

A. Preparation and detection of Fermi gases
The preparation of strontium ( 87 Sr) ultracold Fermi gases and the detection of two spin states are performed in a way similar to Ref. 66.The two relevant spin ground states are |↑⟩ = 5s 2 1 S 0 |F = 9  2 , m F = − 9 2 ⟩ and |↓⟩ = |9/2, −7/2⟩.Fermions of 87 Sr are first laser-cooled and then evaporatively cooled to a temperature below 200 nK.The corresponding ultracold Fermi gas has about 4 × 10 4 atoms in a far-detuned crossed dipole trap, with more than 85% of these atoms initially populated in the |↑⟩ state.
After the Fermi gas is loaded into the optical Raman lattice and the experiment is performed, we shut off all lasers within 1 µs and perform spin-resolved time-of-flight measurements to extract the distribution of the |↑⟩ and |↓⟩ atoms [66].

B. Two-dimensional optical Raman lattice
The experimental setup for the two-dimensional optical Raman lattice is illustrated in Fig. S1(a).Two Raman coupling beams propagate along the X and − Ŷ horizontal directions, intersect at the atoms, and are each phaseshifted and retro-reflected to form two-dimensional (2D) optical lattices for the |↑⟩ and |↓⟩ states.Here, X, Ŷ , and Ẑ denote a set of orthogonal spatial axes, with − Ẑ being the direction of the gravity.
The Raman coupling beams of wavelength λ 0 ≈ 689.4 nm are set to a frequency of about -0.8 GHz with respect to the 1 S 0 (F = 9  2 ) → 3 P 1 (F ′ = 11 2 ) intercombination transition for the 87  tials that heat up the atoms [66].Such detrimental effect can be suppressed by implementing a large frequency difference between two pairs of electric fields, namely the vertically polarized pair (E 1Z , E 2Z ) and the horizontally polarized pair (E 1Y , E 2X ).
As shown in Fig. S1(b), we introduce large frequency differences (close to ∆ L = 4 MHz) between these pairs of similarly polarized electric fields; see also Fig. 2(a) and Fig. 1(a) in the main text.Here, ∆ L is on the megahertz scale and is three orders of magnitude larger than two physical quantities: (a) the kilohertz-scale level splitting ∆ 0 between the |↑⟩ and |↓⟩ states and (b) typical two-photon detuning values that are also on the kilohertz scale.In the experimental implementation, for each Raman coupling beam, the vertical and horizontal linear polarization components are two single-frequency laser beams of distinct frequencies, with a large frequency difference ∆ L between the two.We note that the large ∆ L and the two-photon detuning (δ 0 , δ ′ 0 ) for Raman couplings are fully independent parameters that are precisely controlled in the experiment.The detailed structure of this optical implementation is not required for the understanding of this work, and will be presented elsewhere in a more specialized journal.
The choice of ∆ L is based on two considerations.First, in our experiment, we have verified that as long as ∆ L reaches a few MHz, the heating effect due to moving lattice potentials can be sufficiently suppressed.Second, ∆ L can not be too large, either.This is because, under the same physical distance between atoms and the retro-reflection mirror, the frequency difference between two polarization components will cause a mismatch between the lattice potential minima for these two polarization components.Under ∆ L = 4 MHz and a distance of L a,m ≈ 0.5 m between atoms and the retroreflection mirror, the mismatch is about 1.3 × 10 −2 a, where a = λ 0 /2 ≈ 344.7 nm is the lattice spacing for both polarization components.This mismatch is sufficiently small and thus neglected in our analysis.

C. The highly nonlinear spin-discriminating method
We design and implement a highly nonlinear spindiscriminating (HNSD) method to isolate an effective two-spin manifold out of the ten nuclear spin ground states of 87 Sr.In our case of two-photon Raman transitions, we need to well separate the two nearby spin ground states ( 1 S 0 |F = 9  2 , m F = − 5 2 ⟩ and | 9 2 , − 3 2 ⟩) from the two-spin manifold of the |↑⟩ and |↓⟩ states.To achieve this goal, we first magnetically separate the energy levels of 3 P 1 excited states, and then apply HNSD beams that are (1) near-resonance for the π-transitions of the m F = − 5 2 or − 3 2 state and (2) relatively farther-detuned from transitions of the m F = − 9 2 and − 7 2 states.In our experiment, a relatively large magnetic field of 35 G is applied along the + Ẑ direction, which both determines the quantization axis for the atomic states and induces a large level splitting of ∆ e ≈ 13 MHz between adjacent spin excited states in the 5s5p 3 P 1 |F ′ = 11  2 , m F ′ ⟩ excited manifold.By contrast, adjacent spin ground states are separated by a much smaller amount of about 6 kHz under the same field.Under such magnetic Zeeman splittings, we apply to atoms a π-polarized HNSD beam that propagate in the horizontal plane along a direction 20 degree with respect to X.This HNSD beam, with 80 µW power and a 1/e 2 beam radius of 200 µm, contains two frequency modes in its spectrum.Each frequency mode has a narrow linewidth below 1 kHz and is about 100 kHz blue-detuned relative to the resonance of the π-transition As a result, the HNSD beam induces large upward level shifts (more than 100 kHz) for the | 9 2 , − 5 2 ⟩ and | 9 2 , − 3 2 ⟩ spin ground states.By contrast, this beam only induces kilohertz-scale small level shift for the |↑⟩ and |↓⟩ states, which has a high stability on the 10-hertz scale based on laser intensity stabilization tech-nique.Thus, compared to the Zeeman shift that is linear to the lowest order (and at small magnetic field), the HNSD method engineers highly nonlinear energy level shifts among the spin states, which both enhances the lifetime of SO-coupled fermions and holds promise for creating an isolated manifold with an arbitrary number of spin states.

D. Lifetime measurement and analysis
The lifetime τ 0 of SO-coupled fermions is determined as follows.We hold the atoms in the optical Raman lattice for various length of time, measure the total number of atoms in the |↑⟩ and |↓⟩ states as a function of holding time, and fit the measure with a first-order exponential decay function, extracting a 1/e decay lifetime.
As described in the main text, we enhance the lifetime of 2D-SO-coupled fermions by an order of magnitude.Such enhancement builds on two technical improvements that are also described in the main text, namely the HNSD method and the upgraded method for generating the optical Raman lattice.These two improvements for the experimental realization are both indispensable to the lifetime enhancement.In fact, during our experimental period, we first implemented the optical Raman lattice upgrade, and observed the lifetime doubled, which was encouraging but still insufficient.Later we developed and implemented the HNSD method, and achieved another enhancement factor of about 5 for the lifetime.These two improvements together enable the one-orderof-magnitude lifetime enhancement.
In addition, we perform a similar measurement when the SO couplings are far-off-resonance, and extract a reference lifetime τ Ref ≈ 265 ms under the same lattice depths and Raman coupling strengths as those in Fig. 2(b) in the main text.Comparing τ 0 ≈ 73 ms and τ Ref , we see that the near-resonant SO-couplings still play an important role in affecting the lifetime.We further extract a SO-coupling-constrained characteristic 1/e decay lifetime, t SOC , based on an empirical form: 1/τ 0 ≡ 1/τ SOC + 1/τ Ref .This analysis yields τ SOC ≈ 101 ms, which is shorter than τ Ref .Therefore, compared to other factors (revealed by τ Ref ), the SO couplings currently do impose a primary constraint on the lifetime of SO-coupled fermions, which can be further improved in future experiments; see next paragraph.At the same time, such a SO-coupling constrained lifetime already reaches the 100 ms scale and can enable a number of precise measurements for various topological phases.
The lifetime of 2D-SO-coupled fermions in our system is currently limited by several factors that can be further improved in future experiments.First, from the fundamental aspect, when the SO coupling is close to the two-photon resonance, the fluctuation of the two-photon detuning can induce heating effects, which is similar to the magnetic-field-induced heating effect in the 2D-SOcoupled bosonic systems [59].Thus, the lifetime can be further enhanced by improving the stability of the relative energy splitting between the | ↑⟩ and | ↓⟩ states.Second, we can further reduce the photon scattering in our setup.For example, applying a larger magnetic field can reduce the residual scattering rate due to the HNSD beams for the |↑⟩ and |↓⟩ states.The photon scattering due to the Raman beams can also be reduced by implementing a Ti:Sapphire laser that provides a cleaner spectrum, higher optical power and larger single-photon detunings for the optical Raman lattices.Third, the heating due to residual moving lattice potentials can be further suppressed by improving the laser polarization purity with better polarizing optics.The systematic technical improvements can further remove the major technical impediments to enhancing the lifetime and allow us to focus on improving the fundamental, SO-couplingrelated factors for the Fermi gases.Overall, our system of 2D-SO-coupled Fermi gas holds the promise to reach an even longer lifetime that approaches the performance in Ref. 38 or even that in the bosonic systems [59,96].
We also note that, when the lattice depth is increased to even higher values, the aforementioned several improvements will similarly alleviate the issues of increased scattering rate and residual moving lattice potentials.This is beneficial for reaching a sufficiently long lifetime under the corresponding experimental condition and for future studies of novel correlated topological physics in the interacting regimes.

E. Optimization of PPQM pulse length
The long lifetime enables us to sharpen the detection resolution of the PPQM measurement.Under a given PPQM pulse area, the Fourier-limited width of atomic distribution for the observed |↓⟩-state atoms decreases as the pulse becomes longer.The key in this optimization is to search for a balance between the Fourier broadening and other factors.
In Fig. 3(c) of the main text, we employ a fixed pulse area of Ω 2 01 + Ω 2 02 t p /2ℏ ≈ 0.6π, the measured momentum-width of pumped |↓⟩-state atoms indeed decreases as the pulse length t p increases from a small value (about 100 µs), which is consistent with a reduction of the Fourier limit of a finite pulse (∼ 1/t p ); see the blue dashed line in Fig. 3(c).For t p exceeding 800 µs, the atomic width stops improving, which is likely due to other factors such as the finite imaging resolution.An optimum value of t p ≈ 700 µs is found, providing substantial better resolving power than the 200 µs used in Ref. 66.

II. KEY CHARACTERISTIC FEATURE OF SO COUPLINGS BEYOND RWA
The two Raman potentials, Ω and Ω ′ , are generated from the same electric fields in two different but intrinsi-cally related ways.Below we shall show that the sum of the two relative phases for SO couplings driven by Raman potentials Ω and Ω ′ equals to a deterministic value of π, which is dictated by the phase relation between dipole oscillations with respect to different spatial axes.
For the Raman potential Ω, the relative phase between the two Raman processes [Ω 01 and Ω 02 in Figs.1(a) and 2(a) in the main text] is determined by where δψ is the relative phase shift induced by the laser and optics in the setup [66], and θ ij ê denotes the phase of the dipole matrix element ⟨m , where the transition is driven by the dipole operator ⃗ d and an unit vector ê along the relevant electric field polarization, with ê ∈ { X, Ŷ , Ẑ} in our experimental implementation.In Eq. (S1), the minus sign in −θ 79 X marks the "emission" of a photon.Here for simplicity of expression, we only show the terms for σ transitions driven by horizontally polarized electric fields.The phase terms of the π transitions driven by the vertically polarized electric fields are neglected because they do not influence the main conclusion in this section [Eq.( S3)].
Likewise, in the Raman potential Ω ′ , the relative phase between Raman processes Ω ′ 01 and Ω ′ 02 is given by where the minus sign in −θ 79 Ŷ again marks the "emission" of a photon.Here in Eq. (S2), the contribution from lasers and optics (−δψ) reverses sign as compared to that in Eq. (S1), which is because each pair of electric fields in two-photon Raman transitions reverse their roles of absorption and emission.Finally, it is straightforward to verify the relation that θ 79 Ŷ − θ 79 X = θ 97 X − θ 97 Ŷ , and that these two phase differences both equal to π 2 [56].We thus derive the key characteristic feature of SO couplings beyond RWAthat the two relative phases δϕ 1 and δϕ 2 must have a deterministic relation: which corresponds to the expression δφ + (π − δφ) = π in the "Verification of SO couplings beyond RWA" section of the main text, with the correspondence δϕ 1 = δφ.
where the unshown matrix elements are zero.Under finite lattice depths in the experiment, the form of the Hamiltonian matrix requires more elaboration and can be accurately determined based on expansions under a planewave basis [66].
We note that under proper conditions, in the derivation of a simplified time-independent Floquet Hamiltonian under the |↑⟩ and |↓⟩ basis, one can retain the form of the Bloch Hamiltonian by introducing renormalized SO coupling strength and other parameters [31,91,92,116].If the SO coupling strength is relatively small compared to the modulation frequency, the corresponding 0th-order Bessel function approximately takes the value of unity, and the effective Hamiltonian can be further simplified.
The Hamiltonian matrix can be numerically diagonalized [66], revealing the properties of this Floquet topological system, such as quasi-energies as functions of quasi-momentum.We have numerically confirmed that this exact diagonalization method yields results in agreement with results based on an evolution operator scheme.In the latter method, an effective Floquet Hamiltonian can be deduced by ĤeF = i log Û (t F )/t of the main text, the topology of our model system is characterized by two winding numbers W 0 and W π , with the Chern number of the Floquet bands given by Ch 1 = W 0 − W π [85,86].Here, the Chern number can be also numerically computed as the integral of Berry curvature F over the first Brillouin zone, namely, where the Berry curvature is given by F = i( ∂ ∂qx ⟨ψ(q)| ∂ ∂qy |ψ(q)⟩ − ∂ ∂qy ⟨ψ(q)| ∂ ∂qx |ψ(q)⟩), with ψ(q) denoting a state of the ground quasi-energy band [10][11][12].
We use the topological characterization theory based on band-inversion surfaces (BISs) [91,92,97] to determine the invariants W 0,π .Here a BIS denotes the momentum subspace where the spin bands are inverted, and is determined by h z (q) − nω/2 = 0 (n = 0, ±1, • • • ).We denote the gap around the quasienergy 0 ( π T ) as the 0gap (π-gap) and the BIS living in this gap as the 0-BIS (π-BIS) [91,92].According to the BIS characterization, the winding number W 0 (W π ) is contributed by all the 0-BISs (π-BISs) [91,92]: where j ) denotes the topological invariant associated with the jth 0-BIS (π-BIS), characterizing the winding of spin-orbit coupling on this BIS.Under the experimental conditions with finite lattice depths, the positions of the BISs can be numerically computed under a plane-wave basis [66].
Analytical forms.-UnderSO coupling strengths that are weaker compared to ℏω, the topological phase boundaries can be computed by performing BIS analysis and considering the coupling between pairs of quasi-energy bands for ĤsF [91,92,97].We provide the characteristic phase boundary equations f 1 [n] ≈ 0 and f 2 [n] ≈ 0 (marked by solid lines in Fig. 1(b) of the main text), where in the tight-binding regime, f 1 and f 2 take the following analytical forms: Here the sign function sgn[x] takes values of 1, 0, -1 under the conditions x > 0, x = 0, x < 0, respectively.Here, n takes odd integer values (n = ±1, ±3, • • • ) and |n| denotes the order number of Raman couplings that induces a certain band inversion.By considering the contribu-tions from all BISs, we find an analytical form for the Chern number: where l and m are odd integers given by l = The analytical form in Eq. (S9) agrees with our numerical computation based on Eq. (S6).

B. Negligibility of higher-order SO couplings and the simplified phase diagram
In our present setup, Raman couplings and optical lattices are relatively weak, such that the relevant topology is dominated by leading-order (i.e., |n| = 1) SO couplings.Specifically, the observed double-ring BISs reveal the lowest-order Raman transition processes allowed by our model.Here we further note that the higher-order processes correspond to energy gaps on the scale of tens of hertz or smaller (namely 10 −2 E 0 or smaller) , which are much weaker than Ω 01,02 or Ω ′ 01,02 .For example, under the experimental parameters listed in the Fig. 4 caption of the main text, the leading-order SO coupling strength Ω 2 01 + Ω 2 02 is about 0.18E 0 ≈ h × 0.9kHz.The largest higher-order coupling is a third-order coupling, and can be estimated as ( Ω 2 01 + Ω 2 02 ) 3 /(ℏω) 2 .We note that the value of ω cannot be exactly zero, otherwise all BISs will overlap, which prevents the identification of a certain BIS.Thus, using a characteristic value of ω/4π ∼ 2kHz (namely ℏω ∼ h × 4kHz), we derive a higher-order strength on the scale of h × 40Hz.Such strength for the higher-order coupling is weak and rather marginal for the detection of the corresponding higher- order SO effect based on the present setup.
Thus, even with our current long lifetime of SOcoupled fermions, the corresponding higher-order couplings can hardly cause any band inversion or detectable spin flipping in a realistic experimental sequence.Thus, when only the leading-order SO couplings are considered, a full topological phase diagram (Fig. 1(b) in the main text) can be simplified into Fig.4(d), as further illustrated and explained by Fig. S2.The topological invariant can be described by Eq. (S9) with summation over n = 1 and −1, which leads to the following expression Ch 1 = sgn[f 1 [1]] + 1 2 × sgn[f 2 [1]] for the Chern number as a function of m z and ω (see Eq. (S8) for example).Equation (S10) provides a useful expression for depicting and understanding the simplified phase diagram that is numerically computed based on leading-order SO couplings only (Fig. 4(d) in the main text), where n = 1 and n = −1 correspond to the rotating and counter-rotating leading-order terms for SO coupling beyond RWA.
In the future, if we improve the setup (including implementing a laser with larger optical power) and increase the leading-order SO coupling strength by a factor of two, the third-order coupling strength can in principle increase by almost an order of magnitude to h× several hundred hertz, which will provide a much more favorable condition for the detection of higher-order SO physics.This subsection provides supporting numerical evidence for the main-text note that "in Fig. 4(a), the observed weak connections between two BISs are contributed from the |↑⟩ atoms initially populated outside the first Brillouin zone".As shown in Fig. 4(a) in the main text, in addition to the two-ring band-inversion surface, we notice some weak connections between the two rings.Here, in Fig. S3, we simulate the contributions to the PPQM result based on (a) |↑⟩ atoms initially populated in the first Brillouin zone, (b) |↑⟩ atoms initially populated in the second Brillouin zone, and (c) |↑⟩ atoms initially populated in the first and second Brillouin zones.Indeed, in Fig. S3(a) we observe the two-ring BIS configuration.In Fig. S3(b) and (c), we see that the atoms initially populated in the second Brillouin zone contributes to the "connections" (in green) between the two BIS rings (in orange) that reflect the ground band topology.In future experiments, our system holds promise for exploring more phase regions in the 2D Floquet phase diagram.This can be realized via independent control of ω and the energy splitting between the |↑⟩ and |↓⟩ states.For example, by tuning the magnetic field independent of ω, the magnetic-field-induced energy splitting between |↑⟩ and |↓⟩ will be tunable independent of ω.Thus m z will be similarly tunable, too.Based on independent control of two parameters like ω and the magnetic field, our system will be capable of exploring the 2D Floquet topological phase diagram in the future.
In addition, the energy splitting between |↑⟩ and |↓⟩ can also be changed by choosing a different pair of spin states to act as |↑⟩ and |↓⟩.For example, instead of the current choice of |m F = − 9 2 ⟩ and | − 7 2 ⟩, if we choose |↑ ′ ⟩ = | + 9  2 ⟩ and |↓ ′ ⟩ = | + 7 2 ⟩, the magnetic-field-induced energy splitting between the two states will reverse its sign, which again makes m z change independent of ω.This method further enlarges the phase region that can be explored in the 2D Floquet phase diagram.

FIG. 1 .
FIG. 1. Diagram of SO-coupled fermions beyond RWA.(a) Raman processes beyond RWA.Left: processes coupling |↑⟩ and |↓⟩ via absorbing a photon from Raman coupling beam 1 (red/light red) and emitting one into beam 2 (blue/cyan).Right: processes via absorption from beam 2 and emission into beam 1. Incorporating all four Raman couplings yields two sets of SO couplings in Eq. (2).(b) Topological phase diagram regarding the effective Zeeman splitting mz and modulation angular frequency ω.The anomalous topological phases are characterized by two winding numbers W0,π.Right panels show examples of quasi-energy bands in two phase regions.
FIG. 3.Verification of the beyond-RWA scheme using fermions with quasi-1D SO couplings.(a) Pump-probe quench measurement (PPQM) of momentum distribution of |↓⟩ atoms at various ω.As ω increases, two groups of atoms (marked by black and magenta dashed lines) move toward or away from the center of the FBZ (white square) in opposite manners, revealing two quasi-1D SO couplings in orthogonal directions of X ∓ Ŷ .Accordingly, the two relative phases for SO couplings are π and 0, showing the key feature of a phase sum of π for beyond-RWA SO couplings.Here we set Ω01 = Ω ′ 02 = 0.36E0, Ω02 = Ω ′ 01 = 0.14E0,V 0X↑ = V 0Y ↑ = 0.26E0, V 0X↓ = V 0Y ↓ = 0.03E0.(b) Measured movement of peak positions L1,2 of |↓⟩ atoms (circles) with ω, in comparison to numerical results (dashed lines).(c) Width of pumped atomic ring, measured as a function of the PPQM pulse length tp under a fixed pulse area.Blue dashed line denotes the Fourier limit.Insets: sample atomic rings measured under tp ≈ 200 µs (left) and 700 µs (right).Error bars represent 1σ statistical uncertainties.
FIG. S1.Setup and diagram for two-dimensional optical Raman lattice.(a) Schematic of the experimental setup.The two incident Raman coupling beams propagate along the X and − Ŷ directions.In the optical path along the − Ŷ direction, λ ph denotes a composite wave plate used for controlling the relative phase δφ [66].(b) Energy level diagram and frequencies of the four polarization components.Among the four linearly-polarized electric fields (E1Z , E1Y , E2Z , and E2X ), the field E1Z (E2X ) is additionally frequency-shifted with respect to E1Y (E2Z ) by a large amount of ∆L, as also illustrated in Fig. 2(a) of the main text.
F , where Û (t) = T exp[−i t 0 Ĥ(τ )dτ ] with T denoting the timeordering operator.Topological invariants and Floquet topological phase diagram.-Asshown in the phase diagram in Fig. 1(b) FIG. S2.Simplification of the topological phase diagram when only the leading-order SO couplings are considered.(a) A full topological phase diagram is plotted for both the ω > 0 part (Fig. 1(b) of the main text) and ω < 0 part.Furthermore, those phase boundary lines marked by green color are due to higher-order SO couplings and are neglected when only the leading-order SO couplings are considered.Thus, in a simplified phase diagram, the only remaining phase boundary lines consist of the three negative-slope lines and the horizontal and vertical axes.(b) For convenience of comparison, Fig. 4(d) of the main text is provided here on the right side with the same aspect ratio, with E0 ≈ 4 t0 under the experimental condition of Fig. 4.

C
. Effect of |↑⟩ atoms initially populated outside the first Brillouin zone D. On the future experimental exploration of the 2D Floquet topological phase diagram At present, Fig. 4 in the main text explores a line in the 2D Floquet topological phase diagram because the modulation angular frequency ω and the effective Zeeman term m z are linearly related.