Topological Creutz Ladder in a Resonantly Shaken 1D Optical Lattice
Jin Hyoun Kang Jeong Ho Han Yong-il Shin






yishin@snu.ac.kr
Department of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea 
 and Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea




We report the experimental realization of a topological Creutz ladder for ultracold fermionic atoms in a resonantly driven 1D optical lattice. The two-leg ladder consists of the two lowest orbital states of the optical lattice and the cross inter-leg links are generated via two-photon resonant coupling between the orbitals by periodic lattice shaking. The characteristic pseudo-spin winding in the topologically non-trivial bands of the ladder system is demonstrated using momentum-resolved Ramsey-type interferometric measurements. We discuss a two-tone driving method to extend the inter-leg link control and propose a topological charge pumping scheme for the Creutz ladder system. 





Topological Creutz Ladder in a Resonantly Shaken 1D Optical Lattice
    Y. Shin
    
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Topological phases such as quantum Hall states and topological insulators represent intriguing physics beyond the conventional Landau paradigm of phase transition <cit.>. Motivated further by their novel transport properties, the study of topological phases constitutes one of the frontiers in modern condensed matter physics. Ultracold atoms in optical lattices, featuring tunneling amplitude engineering and tunable interaction strength, provide a unique platform for realizing and exploring such exotic topological states <cit.>. Along with the steady development of experimental techniques, many topological model systems have been recently realized, including the Harper–Hofstadter Hamiltonian in 2D rectangular lattices <cit.>, the Haldane model in a 2D hexagonal lattice <cit.>, and various Hall and topological ladder systems based on additional synthetic dimensions such as internal atomic states <cit.> and lattice orbital states <cit.>.

Periodic lattice shaking is one of the successful tools for exploring exotic phases in optical lattices. Under periodic temporal modulations of the lattice potential, the system parameters such as tunneling magnitude <cit.> and phase <cit.> can be coherently manipulated, giving rise to a hopping configuration that is difficult to realize with static schemes. An outstanding example is the Haldane model realized by circularly shaking a 2D hexagonal optical lattice potential to achieve complex next-nearest-neighbor hopping <cit.>. From the perspective of Floquet band engineering, the lattice shaking method has been extensively discussed even in the resonant regime where the driving frequency is high enough to match the energy gap between two bands <cit.>. Such strong orbital hybridization may enable access to a broader range of effective Hamiltonians <cit.>. In particular, it was anticipated that multi-photon inter-orbital resonant coupling could yield a special route to engineer topological states <cit.>. Thus, it is highly desirable to examine the multifarious scope of Floquet band engineering for the study of topological phases.



In this Letter, we experimentally investigate the effects of two-photon inter-orbital resonant coupling in a periodically driven 1D optical lattice, and demonstrate the realization of a topological Creutz ladder for ultracold fermionic atoms in the shaken lattice system. The Creutz ladder is a cross-linked two-leg ladder system, which has been discussed as a minimal model for 1D topological insulators <cit.>.  In our experiment, the two-leg ladder is formed by the two lowest orbital states, and the cross inter-leg links are generated via the two-photon resonant coupling between orbitals by lattice shaking. Using momentum-resolved Ramsey-type interferometric measurements, we demonstrate the characteristic pseudo-spin winding in the topologically non-trivial bands of the Creutz ladder. We also discuss the extension of the inter-leg link control with two-frequency driving, and propose an experimental scheme for topological charge pumping in the Creutz ladder system.




Our experiment starts by preparing a spin-balanced degenerate Fermi gas of ^173Yb atoms in the F=5/2 hyperfine ground state, as described in Ref. <cit.>. The total atom number is ≈ 1.5× 10^5, and the temperature is ≈ 0.35T_F, where T_F is the Fermi temperature of the trapped sample. The atoms are adiabatically loaded in a 1D optical lattice, which was formed along the x-direction by interfering two laser beams with a wavelength of λ_L = 532 nm. The lattice spacing and depth are a=√(3)λ_L/2 and V_L=8 E_r, respectively, where E_r = h^2/8ma^2 = h× 3.1 kHz. The trapping frequencies of the overall harmonic potential are estimated to be (ω_x,ω_y,ω_z) ≈ 2π× (41,61,130) Hz. After the atom loading, we periodically drive the lattice potential by sinusoidally modulating the frequency difference between the two lattice laser beams as δω_L(t) = A sin(ω t+φ) [Fig. 1a], which results in lattice site vibrations with amplitude d=A/2πωa. In this work, we investigate the resonant driving cases with ħω≈ϵ_sp or 2ħω≈ϵ_sp, where ϵ_sp is the on-site energy difference between the s and p orbitals of the lattice system.










To describe our 1D shaken lattice system, we take a two-band tight-binding approximation, where the lattice system is regarded as a two-leg ladder system constituted by the s and p orbitals, and the inter-orbital coupling by lattice shaking is depicted as the inter-leg links between the two legs. To explicate the properties of the inter-leg links generated by the one- or two-photon inter-orbital resonant transitions, we obtain the effective Hamiltonian H^(ν)_eff of the system in a rotating frame with frequency νω (ν=1,2), respectively, using a high-frequency expansion method <cit.>. To lowest order, the effective Hamiltonian is expressed as H^(ν)_eff=H_0 + H_ν with  

    H_0 =    ∑_j {Ψ_j^† (Δ_νσ_z)Ψ_j -  [Ψ_j^† (t̅_r𝕀 +t_rσ_z)Ψ_j+1+H.c.]}
    
    H_1 =    ∑_j Ψ_j^†( h_0^sp/2R(φ)σ_x)Ψ_j    
    
    H_2=    ∑_j [ Ψ_j^†(h_0^sph_1/2ħω R(2φ)iσ_y  )Ψ_j+1+H.c.],

where Ψ_j^† = (c_j,p^† , c_j,s^†), c_j,α is the annihilation operator for a spinless fermion in the Wannier state |j,α⟩ on lattice site j in orbital α∈{s,p}, and {𝕀,σ} are the identity and Pauli matrices. H_0 includes the effective orbital energy term with Δ_ν =(ϵ_sp-νħω)/2 and the intra-leg tunneling term with t̅_r = t_p+t_s/2, and t_r =t_p-t_s/2, where t_α is the nearest-neighbor (NN) tunneling amplitude of the α orbital. H_ν represents the dominant inter-orbital coupling generated by the ν-photon processes, where h_ℓ^αβ=ħ A a/2π⟨ j,α|∂/∂ x |j+ℓ,β⟩, h_1=h_1^pp-h_1^ss, and R(φ) = exp(-iφσ_z) <cit.>. 




The two-leg ladder system is characterized by the inter-leg link configuration set by H_ν. H_1 describes the on-site orbital-changing transitions, corresponding to the direct inter-leg links with amplitude t_v=h_0^sp/2. On the other hand, H_2 describes the second-order processes consisting of on-site orbital changing and NN tunneling, which amount to the diagonal inter-leg links with amplitude t_d=h_0^sph_1/(2ħω). Since the lattice shaking is an odd-parity operation, the two-photon on-site transition between the s and p orbitals with opposite parities is forbidden by parity conservation <cit.>. The resulting ladder systems for ν=1 and 2 are schematically described in Figs. 1(c) and (d), respectively. The gray circulating arrow indicates a π gauge flux piercing each ladder plaquette, accounting for the opposite signs of t_p and t_s. Thus, under the two-photon resonance condition, the shaken lattice system realizes the seminal Creutz ladder model <cit.>. The cross-linked two-leg ladder for spinless fermions is equivalent to a spin-1/2 1D chain with spin-flip hopping <cit.>.

 

 
 The Bloch Hamiltonian of the two-leg ladder system is expressed as H^(ν)_q=-2t̅_r cos(q)+ B_ν(q)·σ, where q is the quasimomentum in units of a^-1 and the effective magnetic field B_ν(q) is given by 

    B_1(q)=    	t_vρ̂_1 + [Δ_1 - 2t_r cos(q)]ẑ
    B_2(q)=   	2t_d sin(q) ρ̂_2 + [Δ_2 - 2t_r cos(q)]ẑ

with ρ̂_1=cos(φ) x̂+ sin(φ) ŷ and ρ̂_2=-sin(2φ) x̂+ cos (2φ) ŷ. The band structures of the ladder system with Δ_ν=0 for ν=1 and 2 are displayed in Figs. 1(e) and 1(f), respectively, and the corresponding pseudo-spin distributions of the ground bands and their trajectories on the Bloch sphere as q changes over its space are shown in Figs. 1(g) and (h). The half of the solid angle subtended by the trajectory is the geometric representation of the Zak phase γ_Z of the band <cit.>. We have γ_Z=π for ν=2, whereas γ_Z=0 for ν=1. Having non-zero γ_Z is the key topological character of the cross-linked Creutz ladder system <cit.>, which is preserved once B_2(q) has a winding structure with |Δ_2|<2|t_r|. It is noted that  one of the topological bands becomes dispersionless when |Δ_2|=2|t̅_r| and t_d^2=|t_s t_p|. In this work, our focus is to experimentally demonstrate the winding of B_2(q) in the resonantly shaken lattice system. 





We first investigate the resonance condition for inter-obital coupling by measuring the momentum distribution n(k) of the atoms as a function of the driving frequency ω. After preparing the atoms in the s band of the static lattice, we suddenly applied the periodic driving over 20 cycles, and measure n(k) using a band mapping technique <cit.>. In Fig. 2, the change in the momentum distribution, Δ n(k), from that of the non-driven sample is displayed as a function of ω, where n(k) is normalized as ∫ n(k)dk=2π and k is expressed in units of a^-1. A strong spectral signal was observed in the range ω/2π=11∼17 kHz [Fig. 2(a)]. The spectral peak position follows the one-photon resonance condition ω=ω_sp(q) ≈[ϵ_sp-4t_r cos (q)]/ħ, where {ϵ_sp,t_r}/h={13.4,-0.54} kHz for our lattice parameters. The two-photon s-p coupling was observed in the corresponding half-frequency range ω/2π=5.5∼8 kHz [Fig. 2(b)]. The spectral structure appears consistent with the resonance condition of 2ω=ω_sp(q), but compared with the one-photon resonance case, the signal strength is significantly suppressed at k=0 and ±π [Figs. 2(c) and (d)]. This is attributed to the q-dependence of the transverse field amplitude, B_2,ρ(q)∝sin(q), which determines the coupling strength between the two orbital-momentum states, |q,s⟩ and |q,p⟩. The two-photon and three-photon couplings between the s and d orbitals were also observed for ω/2π=11∼14 kHz and ω/2π≈ 8.5 kHz, respectively, with substantial population transfer to the high-momentum region of |k|>2π.











To further examine the structure of B_ν(q) in the q space, we employed an interferometric measurement technique using two pulses of resonant lattice shaking separated by a time interval T_e [Fig. 3(a)]. The frequency modulations were set as δω_L(t)=A_Isin (ω_I t +φ_I) for the first pulse in 0<t<τ_I and δω_L(t)=A_IIsin (ω_II t'+φ_II) with t'=t-(τ_I+T_e) for the second pulse in 0<t'<τ_II. Here, ħω_I(II)=ϵ_sp/ν_I(II) with ν_I(II)=1 or 2, and τ_I(II) is the first (second) pulse duration. The measurement scheme can be described as Ramsey interferometry, where two pulses of lattice shaking play the roles of rotating magnetic fields [Fig. 3(b)]. In the rotating frame with frequency ϵ_sp/ħ, an atom was initially prepared in the |q,s⟩ state in the presence of the axial magnetic field of B_z(q)=2t_rcos(q) and it experienced the transverse fields of B_ν_I(II),ρ(q) along the ρ̂_ν_I(II) directions over the pulse durations. Then, the interferometric signal was obtained by measuring the atom population of the |q,p⟩ state.




Because the Ramsey fringe signal is determined by the directions of the pulsed transverse fields and the accumulated precession angle, its q-dependence faithfully reflects the structure of the effective magnetic field. For example, in the case of (ν_I,ν_II)=(1,1), the transverse field is B_1,ρ=t_v, uniform over the whole q space, and the Ramsey signal will appear as a conventional form of S(q)∼1/2[1+cos{ω_sp(q)T_e+φ_I-φ_II}] <cit.>. Here, the evolution rate with T_e directly reveals B_z(q) via ω_sp(q). Another interesting case is the one with (ν_I,ν_II)=(2,1), where the first transverse field has opposite directions for q>0 and q<0, i.e., B_2,ρ(-q)=-B_2,ρ(q), so the Ramsey signal would be asymmetric in q as S(q)∼1/2[1-sgn(q)×sin{ω_sp(q)T_e+2φ_I-φ_II}] <cit.>. As B_2,z=2t_r cos(q) was confirmed by the two-photon resonance condition of 2ω=ω_sp(q), the establishment of the asymmetric relation of B_2,ρ(-q)=-B_2,ρ(q) would be sufficient for verifying the winding structure of B_2.




In Figs. 3(c) and 3(d), we display the measurement results of the Ramsey interferometry for (ν_I,ν_II)=(1,1) and (ν_I,ν_II)=(2,1), respectively, as a function of T_e. Here (φ_I,φ_II)=(0,0) and τ_I(II)= 2π/ω_I(II). We set A/2π = 5 or 8 kHz for ν_I(II)=1 or 2, respectively, to obtain a π/2-pulse for q=± 0.7π. The population oscillations are clearly observed for each q and the oscillation frequency with T_e is found to be in good quantitative agreement with ω_sp(q) <cit.>. Also, it was observed that the Ramsey signals are mirror-symmetric with respect to k=0 for ν_I=1 and asymmetric for ν_I=2 as expected. In Figs. 3(e) and 3(f), we display the population evolutions at k=±1.3π with increasing T_e for ν_I=1 and 2, respectively. From the damped sinusoidal functions fit to the data, we measure the phase difference between the two oscillation curves to be 0.10 π for ν_I=1 and 0.77 π for ν_I=2, which are slightly different from the expected values of 0 and π, respectively.  Such a deviation might be attributed to the off-resonant coupling effect <cit.>, which was neglected in our description. The dependence of the Ramsey signal on the driving phases φ_I and φ_II was also examined and confirmed to be consistent with our description <cit.>. In particular, the double periodicity of the Ramsey signal with increasing φ_I was observed  in the ν_I=2 case [Fig. 3(f) inset], which is a direct consequence of the fact that two photons are involved in the inter-orbital coupling.










The observation that one- and two-photon resonant couplings bring about the distinctive effects of direct and cross inter-leg links, respectively, prompted us to discuss the extension of our Creutz ladder scheme using both types of resonant couplings. Specifically, we considered a situation in which the lattice shaking was applied as δω_L(t) =A sin(ω t)+ Ãsin (2ω t+φ) with Δ_2=0. Here, the additional 2ω driving provides  one-photon resonant coupling. In our effective Hamiltonian description, we found that the effects of the two drivings were additive <cit.>, resulting in the effective magnetic field 

    B(q)=t̃_v ρ̂_1(φ) +2t_d sin(q) ŷ + -2t_r cos(q) ẑ,

with t̃_v=(Ã/A)t_v. This shows that the direct inter-leg link of the Creutz ladder is generated by the additional 2ω driving; furthermore, its complex amplitude can be directly controlled with the driving parameters Ã and φ.

The flexible control of the inter-leg links with two-tone driving would enable to study the topological phase transitions of the Creutz ladder system, which occurs at {t̃_v,φ}_c={2t_d,±π/2}, accompanied by the band gap closing for B(q)=0 at q=∓π/2 and a sudden change of the Zak phase γ_Z by π. Interestingly, a topological charge pump can be realized by dynamically controlling the system  parameters of {t̃_v,φ} to encircle one of the critical points (Fig. 4) <cit.>. As the system adiabatically moves along the closed pumping path, γ_Z continuously changes by 2π per cycle, which would result in one lattice site shift of atoms in the ladder. In our experiment with Δ_2=0 and A/2π=8 kHz (t_d/h=0.4 kHz), the atom loss rate out of the s-p ladder system was observed to be ≈ 0.17 t_d. For robust implementation of the pump scheme, further experimental optimization seems to be necessary such as phase stabilization of the lattice laser beams and transverse confinement of atoms with additional yz lattices beams.   

 
    
In conclusion, we demonstrated the realization of the topological Creutz ladder in a periodically shaken 1D optical lattice via two-photon resonant coupling. Besides the implementation of the topological charge pump using the two-tone driving, we may pursue topological flat band engineering with our ladder system, which would be interesting for possible emergence of correlated topological phases such as fractional quantum Hall states <cit.>. Also, we expect that the two-photon resonant coupling method can be readily applied to 2D optical lattice systems, providing an alternative route to investigate anomalous quantum Hall states <cit.>. 


We thank Wei Zheng and Hui Zhai for helpful discussions. This work was supported by the Institute for Basic Science in Korea (Grant No. IBS-R009-D1) and the National Research Foundation of Korea (Grants No. NRF-2018R1A2B3003373, and No. 2014-H1A8A1021987).






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SI See Supplemental Material for the derivation of the effective Hamiltonian, numerical simulations of the Ramsey interferometry, additional interferometric measurement data, and discussion of the two-tone driving method. 

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footnote1 The long-range tunneling in the p band is included in our calculation of ω_sp(q).


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§ SUPPLEMENTAL MATERIAL




 §.§ Atom in a driven optical lattice

In the experiment, a 1D optical lattice was formed by spatially interfering two laser beams. The lattice potential is expressed as 

    V(x) = V_L/2cos[2π/a(x-x_0)],

where a is the lattice spacing and x_0=a/2πϕ represents the lattice position determined by the relative phase ϕ between the two laser beams. Periodic shaking of the lattice was achieved by sinusoidally modulating the frequency difference between the two lattice laser beams as δω_L(t) = Asin(ω t+φ) with the modulation frequency ω, the amplitude A, and the phase φ. From the relation,  ϕ(t)-ϕ(0) =∫_0^t δω_L(τ)dτ, the lattice position is given by x_0(t)=-dcos(ω t + φ) with d =A/2πωa. 

The Hamiltonian of an atom in the periodically driven optical lattice is given by

    H_Lab = p^2/2m + V_L/2cos[ 2π/a(x + dcos(ω t+φ)) ].

Taking the unitary transformation of  

    U_1(t) =exp[ i/ħ d cos(ω t+φ) p]

for  x → x-dcos(ω t+φ), we obtain the system's Hamiltonian H in the reference frame comoving with the optical lattice, and H is expressed as H=H_stat+δ H with 

    H_stat   = p^2/2m+V_L/2cos(2π/a x) 
    δ H     = -dωsin(ω t+φ)p,

where H_stat is the Hamiltonian of the stationary lattice system and δ H represents the perturbation from the inertial force induced by the lattice shaking.



 §.§ Two-band model description


In the two-band tight binding approximation, the Hamiltonian of the lattice system is expressed as

    H = ∑_j Ψ_j^† K(t) Ψ_j -∑_j [Ψ_j^† J(t) Ψ_j+1+H.c.],

where Ψ_j^† = ( c_j,p^†,c_j,s^†), c_j,α is the annihilation operator for the atom in the Wannier state |j,α⟩ on lattice site j in orbital α∈{s,p}. The matrices K(t) and J(t) are given by

    K(t)     = 
    	[                ϵ_p -ih_0^spsin(ω t+φ);  ih_0^spsin(ω t+φ)                ϵ_s ]
    
    J(t)    =
    	[ t_p-ih_1^ppsin(ω t+φ)     ih_1^spsin(ω t+φ);    -ih_1^spsin(ω t+φ) t_s-ih_1^sssin(ω t+φ) ]

with

    ϵ_α =   ⟨ j, α|H_stat |j,α⟩
    
    t_α =    -⟨ j, α| H_stat|j+1,α⟩
    
    h_ℓ^αβ =    ħ A a/2π⟨ j, α|(∂/∂ x)|j+ℓ,β⟩.

 ϵ_α and t_α are the on-site energy and nearest-neighbor hopping amplitude of the α orbital, respectively, and h_ℓ^αβ is the transition element between the α and β orbitals separated by ℓ lattice sites. In terms of the identity and Pauli matrices {𝕀,σ_x,σ_y,σ_z}, 

    K(t) =    ϵ̅𝕀+ ϵ_sp/2σ_z +h_0^spsin(ω t+φ)σ_y 
    
    J(t) =    [t̅_r-ih̅_1 sin(ω t+φ)] 𝕀
       + [t_r-ih_1 sin(ω t+φ)] σ_z 
       -h_1^spsin(ω t+φ) σ_y

with ϵ̅=ϵ_p+ϵ_s/2, ϵ_sp=ϵ_p-ϵ_s, t̅_r =t_p+ t_s/2, t_r =t_p-t_s/2, h̅_1 = h_1^pp+h_1^ss/2, and h_1 = h_1^pp-h_1^ss/2.



To investigate the effects of the ν-photon inter-orbital resonant coupling, we obtain the Hamiltonian H' in a rotating reference frame with frequency νω, taking a unitary transformation of

    U_2(t) = ∑_j[ e^-iνω/2 tc_j,p^† c_j,p +e^iνω/2 tc_j,s^† c_j,s].

 The two matrices K(t) and J(t) are transformed to, respectively,

    K'(t) =    Δ_νσ_z +h_0^spsin(ω t+φ)R(-νω t)σ_y 
    
    J'(t) =    [t̅_r-ih̅_1 sin(ω t+φ)] 𝕀
       + [t_r-ih_1 sin(ω t+φ)] σ_z 
       -h_1^spsin(ω t+φ)R(-νω t) σ_y,

where Δ_ν = (ϵ_sp-νħω)/2 and R(θ) =exp(-iθσ_z). Here we ignore the energy offset of ϵ̅𝕀 in K(t). 





  §.§.§ Effective Hamiltonian for ν=1

Using a high frequency expansion method up to second order processes <cit.>, the effective Floquet Hamiltonian of the periodically driven system is obtained as

    H_eff^(ν) = H_0 + ∑_n=1^∞[H_n,H_-n]/nħω,
  
where H_n is the nω Fourier component of the original time-periodic Hamiltonian H'(t), such that

    H'(t) = H_0 +∑_n≠ 0 H_n e^i n ω t.


In the case of ν=1, from Eq. (S8),

    H_0    = ∑_j Ψ_j^†[Δ_1σ_z+h_0^sp/2R(φ)σ_x]Ψ_j    
       +∑_j {Ψ_j^†[-t̅_r 𝕀 -t_r σ_z+h_1^sp/2R(φ)σ_x]Ψ_j+1+H.c.}
    
    H_1    =∑_j{Ψ_j^†[e^iφ/2(h̅_1𝕀+h_1σ_z) ]Ψ_j+1
                 +Ψ_j+1^†[-e^iφ/2(h̅_1𝕀+h_1σ_z) ]Ψ_j}
    
    H_2    = ∑_j Ψ_j^†[-h_0^sp/2e^iφσ^+ ]Ψ_j 
       +∑_j{Ψ_j^†[-h_1^sp/2e^iφσ^+ ]Ψ_j+1+Ψ_j+1^†[-h_1^sp/2e^iφσ^+ ]Ψ_j}

and H_-n = H_n^†, where σ^± = (σ_x ± iσ_y)/2. Then,

    H_eff^(1)   = 
    ∑_j    Ψ_j^†[(Δ_1+(h_0^sp)^2+(h_1^sp)^2/8ħω)σ_z+h_0^sp/2R(φ)σ_x]Ψ_j    
    
    	   +∑_j   {Ψ_j^†[-t̅_r 𝕀+(h_0^sph_1^sp/4ħω-t_r) σ_z+h_1^sp/2R(φ)σ_x]Ψ_j+1
         +Ψ_j^†[ (h_1^sp)^2/8ħωσ_z ]Ψ_j+2+ H.c.}.

When ħω≫ h_0^sp≫ |h_1^sp|, the effective Hamiltonian is approximated as

    H_eff^(1)   ≈∑_j Ψ_j^†[Δ_1σ_z+t_v R(φ)σ_x]Ψ_j    
       -∑_j {Ψ_j^† (t̅_r 𝕀+t_r σ_z)Ψ_j+1+H.c.}.

with t_v=h^sp_0/2. The Bloch Hamiltonian is given by 

    H_q^(1) =-2t̅_rcos(q)𝕀+B_1(q) ·σ

with

    B_1(q) =     t_v ρ̂_1+ [Δ_1-2t_rcos(q)]ẑ
    ρ̂_1=   cos(φ)x̂+sin(φ)ŷ,

where q is the quasimomentum normalized in units of a^-1. 




  §.§.§ Effective Hamiltonian for ν=2

In the case of ν =2,

    H_0 = ∑_j    Ψ_j^† (Δ_2σ_z)Ψ_j -∑_j [ Ψ_j^†(t̅_r𝕀 +t_rσ_z)Ψ_j+1+H.c.] 
    
    H_1 =∑_j   Ψ_j^†[ h_0^sp/2e^-iφσ^+ ]Ψ_j
    
    	   +∑_j   {Ψ_j^†[h_1^sp/2e^-iφσ^+ +e^iφ/2(h̅_1𝕀+h_1σ_z) ]Ψ_j+1
       +Ψ_j+1^†[h_1^sp/2e^-iφσ^+ -e^iφ/2(h̅_1𝕀+h_1σ_z) ] Ψ_j}
    
    H_3 = ∑_j    Ψ_j^†[-h_0^sp/2e^iφσ^+ ]Ψ_j 
    
    +∑_j   {Ψ_j^†[-h_1^sp/2e^iφσ^+ ]Ψ_j+1+Ψ_j+1^†[-h_1^sp/2e^iφσ^+ ]Ψ_j},

and H_-n = H_n^†, giving 

    H_eff^(2)   = 
    ∑_j   Ψ_j^†[(Δ_2+(h_0^sp)^2+(h_1^sp)^2/3ħω)σ_z -h_1^sph_1/2ħωR(2φ) σ_x]Ψ_j 
    
    +∑_j    {Ψ_j^†[-t̅_r𝕀 +(2h_0^sph_1^sp/3ħω-t_r )σ_z  +h_0^sph_1/2ħωR(2φ)iσ_y]Ψ_j+1
        +Ψ_j^†[(h_1^sp)^2/3ħωσ_z +h_1^sph_1/2ħωR(2φ) i σ_y ]Ψ_j+2 +H.c.}.

With ħω≫ h^sp_0 > |h_1| > |h^sp_1|, H^(2)_eff is approximated as

    H_eff^(2)   ≈∑_j  Ψ_j^† (Δ_2σ_z)Ψ_j 
       -∑_j  {Ψ_j^†[ t̅_r𝕀 +t_rσ_z - t_d R(2φ)iσ_y]Ψ_j+1+H.c.}

with t_d=h_0^sp h_1 / (2ħω). Here, we ignore the ac Stark shift, δ_2^ac≈(h_0^sp)^2/3ħω, in the orbital energy, which is comparable to t_d in magnitude but much smaller than t_r in the experiment. The Bloch Hamiltonian is given by

    H_q^(2) =-2t̅_r cos(q)𝕀+B_2(q) ·σ

with

    B_2(q) =     2t_d sin(q) ρ̂_2+[Δ_2- 2t_rcos(q)] ẑ
    ρ̂_2=   -sin(2φ)x̂+cos(2φ)ŷ.
 

For our experimental condition, h_0^sp:|h_1|:|h_1^sp|=6.4:1.7:0.3 and h_0^sp/h<1.3 kHz, whereas ω/2π=13.7 or 6.8 kHz, and |t_r|/h=0.54 kHz, justifying the approximation taken in Eqs. (S13) and (S17).





 §.§ Ramsey interferometry using two pulses of lattice shaking


We performed the Ramsey interferometry by modulating the frequency difference δω_L as

    δω_L (t)= 
    	
    		A_Isin (ω_I t +φ_I)         for 0<t<τ_I, 
          0     for τ_I<t<τ_I+T_e, 
    
    		A_IIsin (ω_II t'+φ_II)     for 0<t'<τ_II

with t'=t-(τ_I+T_e). Here, ω_I(II)=ω^0_sp/ν_I(II) with ν_I(II)=1 or 2, and ħω^0_sp=ϵ_sp, so the ν_I(II)-photon resonant coupling was generated during the first (second) pulse duration.  τ_I(II) was set to the shaking period of 2π/ω_I(II), satisfying the Floquet picture as well as ensuring the continuous changing of δω_L. 

In the rotating frame with frequency ω^0_sp=ν_I(II)ω_I(II), as shown in Eqs. (S14) and (S18), the effective Bloch Hamiltonian of the system is expressed as H_q(t)=-2t̅_rcos(q)𝕀+B(q,t)·σ with

    B(q,t)   =B_ρ(q,t)ρ̂(θ(t)) + B_z(q)ẑ,

where ρ̂(θ)=cos(θ)x̂+sin(θ)ŷ and B_z(q)=-2t_r cos (q). The magnitude B_ρ(>0) and the azimuthal angle θ of the transverse component of B(q,t) are determined by the lattice shaking parameters, {A_I(II),ν_I(II),φ_I(II)}, in each pulse duration. Then, the system's evolution over a pulse duration τ is described by R_b̂(ξ)=exp(-i ξb̂·σ) with b̂=B/|B| and ξ=|B|τ/ħ. This operation corresponds to a rotation by ξ around the b̂ axis on the Bloch sphere formed by the |q,p⟩ and |q,s⟩ states. Here the effect of the spin-independent term, -2t̅_r cos(q), in H_q is nullified by a gauge transformation. 

For the initial state of (0,1)^T, i.e., |q,s⟩, the atom's final state after the Ramsey interferometry sequence is given by   

    [ c_p(q); c_s(q) ]   =
    R_b̂_II(ξ_II)R_ẑ(B_z T_e/ħ)R_b̂_I(ξ_I)
    [ 0; 1 ],

where b̂_I(II) and ξ_I(II) are the direction of B and the rotation angle for the first (second) lattice-shaking pulse, respectively. When |B_z|≪ B_ρ and ξ_I=ξ_II =π/4, b̂_I(II)≈ρ̂_I(II), and the evolution matrix can be approximated as

    R_ρ̂_II   (π/4)R_ẑ(B_z T_e/ħ)R_ρ̂_I(π/4) 
    ≈    1/2[ e^-iB_z T_e/ħσ_z-e^i(B_z T_e/ħ+θ_I-θ_II) σ_z
          -iσ_x(e^-i(B_z T_e/ħ +θ_I)σ_z+e^i(B_z T_e/ħ -θ_II)σ_z)].

Then, the interferometric signal n_p(q)=|c_p(q)|^2 is given by 

    n_p(q) = 1/2{ 1+cos[2B_z(q) T_e/ħ+θ_I-θ_II]}.







In the case of {ν_I,ν_II}={1,1} with A_I=A_II=A and τ_I=τ_II=τ, we have B_ρ(q)=t_v, θ_I=φ_I, and θ_II=φ_II-ω_sp^0(τ+T_e)=φ_II-ω_sp^0 T_e (mod 2π)  [Eq. (S14)]. Then Eq. (S20) gives

    n_p(q) = 1/2{ 1+cos[ω_sp(q) T_e+φ_I-φ_II]}

with ħω_sp(q)=ϵ_sp-4t_r cos (q). 

In the case of {ν_I,ν_II}={2,1},  B_ρ(q)=2t_d |sin (q)|  and θ_I=2φ_I+sgn (q) (π/2) for the first pulse [Eq. (S18)], and B_ρ(q)=t_v and θ_II=φ_II-ω_sp^0 T_e  for the second pulse. Although the assumption of |B_z|≪ B_ρ is not valid near q=0 and π, we see that Eq. (S20) results in

    n_p(q) = 1/2{ 1-sgn(q)sin[ω_sp(q)T_e+2φ_I-φ_II]}.

Because of the sgn(q) factor, the Ramsey fringe signal is asymmetric for opposite momenta ± q. Another noticeable feature is that the dependence of n_p(q) on φ_I is different from that in the ν_I=1 case, so the interference signal has π periodicity rather than 2π with increasing φ_I [Fig. 3(f) inset and Fig. S1].




















 §.§ Numerical simulation of the Ramsey interferometry


Using Eq. (S19), we numerically calculated the Ramsey interferometry signals for our experimental conditions. The results for {ν_I,ν_II}={1,1} and {2,1} are shown in Fig. S2(a) and S2(b), respectively. The oscillations at each k is characterized by a sinusoidal curve, n(k,T_e)∝sin[ω_f(k)T_e+ϕ(k)] with ω_f(k) = ω_sp(k) [Fig. S2(c)]. We observe ϕ(k)-ϕ(-k) =0 for {ν_I,ν_II}={1,1} and |ϕ(k)-ϕ(-k)|=π for {ν_I,ν_II}={2,1}  [Fig. S2(d)], which are consistent with the expectations from Eqs. (S21) and (S22). 


In Fig. S3, we compare the experimental data in Fig. 3(c) and 3(d) with the numerical results. Here the measured spectra are characterized by fitting a sinusoidal curve with frequency ω_f(k) and phase ϕ(k) to the oscillating signal at each k. The obtained ω_f(k) is found to be in good agreement with the  calculated ω_sp(k) [Fig. S3(a) and S3(b)]. In  the {ν_I,ν_II} ={1,1} case, the oscillation phase ϕ(k) is also found to be well described by the numerical results [Fig. S3(c)]. The small deviations near k=±π might be attributed to the imperfection of the band mapping technique or some systematic effects in our experiment. 

On the other hand, in the {ν_I,ν_II} ={2,1} case, we observe that the measured ϕ(k) shows a non-negligible deviation from the numerical result [Fig. S3(d)], although it reasonbly demonstrates the asymmetric property of |ϕ(k)-ϕ(-k)|=π. The deviation is most pronounced near k=0, where it should be technically difficult to determine ϕ because of small oscillation amplitude. In order to understand the measurement result, it might be necessary to take into account the one-photon off-resonant effect that is neglected in our description. As an attempt, we examined a situation where an additional transverse field B_x is added along the x̂ direction for the first pulse period to mimic the off-resonant effect. We observed that with B_x∼ B_ρ/3, the numerically recalculated fringe signal exhibits similar deviating behavior to the experimental result [Fig. S3(d)].




 §.§ Two-tone driving scheme


In this section, we describe the two-tone driving scheme where the optical lattice is driven as 

    δω(t) = A sin(ω t)+Ãsin(2ω t+φ).

Following the derivation in Sec. A, we obtain the Hamiltonian in the comoving frame as 

    H =   p^2/2m+V_L/2cos(2k_L x)-d ωsin(ω t)p 
       -2d̃ωsin(2ω t+φ)p.

In two-band tight binding approximation, the Hamiltonian H' in a rotating reference frame with frequency 2ω is given by

    H' = ∑_j {Ψ_j^† K'(t) Ψ_j -(Ψ_j^† J'(t) Ψ_j+1+H.c.)},

with

    K'(t) =   Δ_2 σ_z +h_0^spsin(ω t)R(-2ω t)σ_y 
       +h̃_0^spsin(2ω t+φ)R(-2ω t)σ_y, 
    
    J'(t) =     t̅_r 𝕀+ t_r σ_z 
       -i[h̅_1 sin(ω t)+h̅̃̅_1 sin(2ω t+φ)] 𝕀
       -i[h_1 sin(ω t)+h̃_1 sin(2ω t+φ)] σ_z 
       -h_1^spsin(ω t)R(-2ω t) σ_y 
       -h̃_1^spsin(2ω t+φ)R(-2ω t) σ_y.

Here the tilde symbol indicates that the elements are derived from the 2ω driving with the amplitude Ã. 





The effective Hamiltonian, H_eff, of this peridocially driven system can be obtained in a same manner shown in Sec. B using the high frequency expansion method. Here, it is important to note that the Fourier components, H_± 1 and H_± 3 are generated only by the ω driving to have the same forms expressed in Eq. (S15), and H_± 2 and H_±4 are generated only by the 2ω driving, resulting in the same forms of H_±1 and H_±2 in Eq. (S11). Then, from Eq. (S9), we can see that the effects of the two drivings arise additively in the system, thus leading to 

    H_eff≈   ∑_j  Ψ_j^†[Δ_2σ_z+t̃_v R(φ)σ_x]Ψ_j     
        -∑_j [ Ψ_j^†(t̅_r𝕀 +t_rσ_z-t_d iσ_y)Ψ_j+1+H.c.]

with t̃_v = h̃_0^sp/2 and t_d =h_0^sph_1/2ħω. Here the same approximations used in Eqs. (S13) and (S17) are applied. Accordingly, in the quasimomentum space, the Bloch Hamiltonian is given by 

    H_q = -2t̅_rcos(q)𝕀+B(q) ·σ

with

    B(q)    = t̃_v ρ̂+ 2 t_dsin(q) ŷ+ [Δ_2-2t_rcos(q)]ẑ
    ρ̂   =cos(φ)x̂+sin(φ)ŷ.




In this two-tone driving scheme, the topological phase transitions of the Creutz ladder system can be studied. 
For example, when φ = π/2 and 

    B(q) =     [t̃_v + 2t_dsin(q)] ŷ+ [Δ_2-2t_rcos(q)]ẑ,

the system undergoes a topological phase transition as t̃_v/t_d changes across the critical point of t̃_v/t_d=2. At the critical point, the band gap closes at q_c such that Δ_2-2t_rcos(q_c)=0 giving |B(q_c)|=0 [Fig. S4(b)], and  the Zak phase γ_Z suddenly changes by π [Fig. S4(f)]. The Zak phase is numerically calculated as γ_Z = i ∫_BZ dq ⟨ u_q |∂_q|u_q⟩, where u_q is the Bloch function of the band <cit.>. 

On the other hand, when φ=0, i.e., the two transverse fields from the ω and 2ω drivigns are orthogonal to each other, 

    B(q) =    t̃_v x̂ + 2t_dsin(q) ŷ+ [Δ_2-2t_rcos(q)]ẑ.

In this case, γ_Z can change continuously with varying t̃_v/t_d [Fig. S4(f)], and the system can make a topological phase transition without having band gap closing [Figs. S4(a,d,e)]. 


