Creutz ladder in a resonantly shaken 1D optical lattice

We report the experimental realization of a 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 structure in the energy 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.


Introduction
Topological phases such as quantum Hall states and topological insulators represent intriguing physics beyond the conventional Landau paradigm of phase transition [1,2]. 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 [3,4]. Along with the steady development of experimental techniques, many topological model systems have been recently realized, which include the Harper-Hofstadter Hamiltonian in 2D rectangular lattices [5,6], the Haldane model in a 2D hexagonal lattice [7], and various Hall and topological ladder systems based on additional synthetic dimensions such as internal atomic states [8][9][10][11][12][13] and lattice orbital states [14].
Periodic lattice shaking is one of the successful tools for exploring exotic phases in optical lattices. Under periodic modulations of the lattice potential, the system parameters such as tunneling magnitude [15,16] and phase [17] 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 [7,18]. 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 [19]. Such strong orbital hybridization may enable access to a broader range of effective Hamiltonians [20,21]. In particular, it was anticipated that multi-photon inter-orbital resonant coupling could yield a special route to engineer topological states [22,23]. Thus, it is highly desirable to examine the multifarious scope of Floquet band engineering for the study of topological phases.
In this paper, 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 generalized Creutz ladder for ultracold fermionic atoms in the shaken lattice system. The Creutz ladder is a cross-linked two-leg ladder system under a magnetic field, which has been discussed as a minimal model for 1D topological insulators [24][25][26].
Recently, it was extended to an interacting case, referred to as the Creutz-Hubbard model, for the study of correlated topological phases [27][28][29][30][31]. In our experiment, the two-leg ladder is formed by the two lowest orbital states in optical lattice, and the cross inter-leg links are generated via the two-photon resonant coupling between the orbitals by lattice shaking. Using momentum-resolved Ramsey-type interferometric measurements, we demonstrate the characteristic pseudo-spin winding structure in the energy bands of the Creutz ladder. We also discuss the extension of the inter-leg link control with two-frequency driving, where the direct links are additionally controlled by the one-photon resonant coupling between the orbitals. Finally, based on the extented inter-leg link control, we propose an experimental scheme for topological charge pumping in the generalized Creutz ladder system.

Creutz ladder model
The Creutz model describes a spinless fermion system in a two-leg ladder under a magnetic field [24]. The twoleg ladder system is sketched in figure 1, where t denotes the hopping amplitude along the legs, t v along the rungs of the ladder, t d along the diagonal links, and Φ is the gauge flux penetrating each ladder plaquette. The system's Hamiltonian is given by where q is the quasimomentum,  is the identity matrix, and Because the vector h CL is confined in the xz plane, its winding around the origin for the whole Brillouin zone (BZ), p p -<  q , characterizes the topology of the ladder system. A topologically non-trivial state occurs when where h q CL ( ) fully encircles the origin, giving a non-zero integer winding number representing the topological character of the system. The band dispersion of the system is given by It is remarkable that with Φ=π, t v =0 and t d =t, the ladder system has =   E q t 2 ( ) | |, constituting a 1D topological system with two flat bands. The original Creutz model was proposed for a special case of Φ=±π, where the ladder system holds chiral symmetry for σ y H CL, q σ y =−H CL, q [24], and it is equivalent to the Su-Schrieffer-Heeger model, which belongs to the BDI class of the Altland-Zirnbauer classification [30,32]. In an extended case of p F ¹  , the chiral symmetry is broken, but since h CL remains in the xz plane, the topological character of the system is still unambiguously represented by the winding number of h CL . It was discussed that the topological phase of the extended Creutz model is protected by a hidden inversion symmetry [33,34].

Atom in a resonantly shaken optical lattice
We consider an atom in a 1D optical lattice potential = - where the position x 0 of the lattice potential is periodically driven as Here, V L is the lattice depth and a is the lattice spacing. In the comoving reference frame with the vibrating optical lattice, the Hamiltonian of the system is given by  x L where H stat is the Hamiltonian of the stationary optical lattice system with p x being the atom's momentum and m being the atomic mass, and δH represents the perturbation from the inertial force induced by the lattice shaking, which can make mixing between the energy bands of the stationary lattice system. In this work, we are interested in the situation where the lattice shaking frequency ω is close to the resonance frequency for the coupling between the two lowest, s and p bands of the lattice potential ( figure 2(b)). From a synthetic dimension perspective [35], the 1D shaken lattice system can be regarded as a two-leg ladder system constituted by the s and p orbitals, where the inter-orbital coupling by lattice shaking is depicted as the inter-leg links between the two legs. In a two-band tight-binding approximation, the Hamiltonian of the driven lattice system is expressed as ) , c j, α is the annihilation operator for the atom in the Wannier state añ j, | on lattice site j in orbital αä{s, p}. The matrices K(t) and J(t) are given by are the on-site energy and nearest-neighbor hopping amplitude of the α orbital, respectively, and is the transition element between the α and β orbitals separated by ℓ lattice sites [22].
When the shaking frequency ω is close to w n sp 0 with integer ν, where w =   sp sp 0 and ò sp =ò p −ò s , a resonant inter-orbital coupling would be generated via ν-photon transition process. To explicate the properties of the inter-orbital couplings generated by the one-photon (ν=1) and two-photon (ν=2) resonant processes, we calculate the effective Hamiltonian H ( ν) eff of the system in a rotating frame with frequency νω, using a highfrequency expansion method (appendix A) [36][37][38]. The effective Hamiltonian is obtained as represents an uncoupled two-leg ladder with energy imbalance Δ ν and intra-leg hopping, and n H C ( ) describes the dominant inter-leg coupling generated by the ν-photon resonant process.
The character of the inter-leg coupling given by n H C ( ) is different for the two cases, ν=1 and ν=2. H C ( ) can be understood from parity conservation; since the lattice shaking is an odd-parity operation, the two-photon on-site transition is forbidden between the s and p orbitals, which have opposite parities [23]. In figures 2(c) and (d), we present the schematic diagrams of the resulting ladder systems for ν=1 and 2, respectively. Note that the sign of t p is opposite to that of t s , and under a proper transformation, the ladder systems can be viewed as having a π gauge flux piercing each ladder plaquette.
The Bloch Hamiltonian of the two-leg ladder system is expressed as , where q is expressed in units ofa 1 and the n h q ( ) is given by , the functional form of n H q ( ) on q becomes same as that of H CL, q except the term of D n zˆin n h . Regardless of having the additional term, n h is confined in the plane defined by r n and zˆ, and thus, its winding number determines the topological characteristic of the ladder system. From equation (7), it is obvious that h q 1 ( ) cannot encircle the origin, whereas h q 2 ( ) can give winding number of ±1 when D < t 2 r 2 | | | |. This means that a topologically non-trivial phase would emerge in the shaken lattice system with two-photon resonant coupling (Δ 2 ≈0). The chiral symmetry is generally broken in H q 2 ( ) due to ¹ t 0 r , and it was argued that the topologically non-trivial phase is protected by symmetry composed of time-reversal and mirror symmetries [12]. It is interesting to note that one of the topological bands becomes dispersionless when In figures 2(e) and (f), we display the band structures of the two-leg ladder system with one-photon resonant coupling (Δ 1 =0) and two-photon resonant coupling (Δ 2 =0), respectively. The pseudo-spin distributions of the ground bands over the BZ and their corresponding trajectories on the Bloch sphere are also shown in figures 2(g) and (h). In the ν=2 case, the pseudo-spin trajectory make a great circle on the Bloch sphere, which is the key topological feature of the constructed Creutz ladder system. In the following section, we present our experimental investigation, where the focus is to demonstrate the winding structure of h q 2 ( ) in the resonantly shaken lattice system.

Experiment and result 4.1. Experimental setup
Our experiment starts with preparing a spin-balanced degenerate Fermi gas of 173 Yb atoms in the F=5/2 hyperfine ground state, which has all the six spin components equally, in an optical dipole trap [39]. 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 potential, which is formed along the xdirection by interfering two laser beams with a wavelength of λ L =532 nm. (h is the Planck constant). The trapping frequencies of the overall harmonic potential were estimated to be (ω x , ω y , ω z )≈2π×(41, 61, 130)Hz. After loading the atoms in the lattice potential, the fractional population of the p orbital was about 3%. Shaking of the lattice potential is implemented by sinusoidally modulating the frequency difference δω L between the two lattice laser beams as dw figure 2(a)].

Shaking spectroscopy
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 apply the periodic driving over 20 cycles, which is longer than the dephasing time of the system, and we measure n(k) by taking an absorption image of the atoms after a band mapping protocol [40] and a subsequent time of flight. In the band mapping, quasimomentum states are transformed onto the corresponding real momentum states in free space by slowly ramping down the lattice potential, and n(k) shows the quasimomentum distribution of the atoms in the s band for p < k | | with k=q and that in the p band for { } { }kHz for our lattice parameters. The two-photon s-p coupling is observed in the corresponding half-frequency range w p =2 5.5 8kHz( figure 3(b)), where the spectral structure appears consistent with the resonance condition of 2ω=ω sp (q). In comparison with the one-photon resonance case in figure 3(c), it is noticeable that the signal strength is suppressed at w w 2 0 sp ( ) and 2ω∼ω sp (±π), much weaker than that at 2ω∼ω sp (±π/2) ( figure 3(d)). The suppression is consistent with the q-dependence of the transverse field amplitude of h 2 , µ r h q q sin 2, ( ) ( ), which determines the coupling strength between the two orbital-momentum states, ñ q s , | and ñ q p , | . In the frequency range of w p =2 11 14kHz and ω/2π≈8.5kHz, we observe substantial population transfer to the high-momentum region of p > k 2 | | , which result from the two-photon and three-photon couplings between the s and d orbitals, respectively. In the effort to realize an effective two-leg ladder system, coupling to higher orbitals from the s and p orbitals could be detrimental, if its magnitude is not negligible to that of the s-p inter-orbital coupling. In the two-photon resonance condition with Δ 2 =0, which is of our main interest in this work, the atom loss rate out of the s-p ladder system was measured to be ≈0.17t d for A/ 2π=8 kHz (t d /h=0.4 kHz), justifying the effective two-band description of the shaken lattice system. In numerical simulations of an atom in the 1D shaken lattice potential, the atom loss rate due to the higher-band coupling was estimated to be 0.08t d , which is less than a half of the experimentally measured value. Further experimental optimization might help reduce the loss rate, such as phase stabilization of the lattice laser beams and transverse confinement of atoms with additional yz lattices beams. It is worth noting that atom-atom interactions, which are present in our spin-balanced sample, can be a source of heating and atom loss in a periodically driven optical lattice system [41].

Ramsey interferometry
In the shaking spectroscopy, the dispersion curve of ω sp (q) was measured for the two-photon resonant coupling and it confirms the axial component of h 2 is given by = Dh t q 2 cos z r 2, 2 ( ). For verifying the winding stucture of h q 2 ( ), therefore, it would be sufficient to demonstrate the asymmetric property of its transverse component, ( ) ( ). As a means to probe the asymmetric property of h 2, ρ (q), we employ a Ramsey interferometry scheme, where two separate pulses of resonant lattice shaking are applied with a variable time interval T e . The lattice shaking sequence is described in figure 4(a) . Here the global phase factor arising from the term of - t q 2 cos r ( ) in H q is ignored under a proper gauge transformation.
x R n ( ) represents a rotation around the n axis by ξ on the Bloch sphere formed by ñ q p , | and ñ q s , | , and thus, the system's overall evolution is described as a composition of three sequential rotations which correspond to the time evolutions in the first pulse, the intermediate period, and the second pulse, respectively [ figure 4(b)]. As the time T e of the intermediate period increases, the atom population in the ñ q p , | state, = n q c q p p 2 ( ) | ( )| would show oscillations, and its oscillation amplitude and phases can reveal the geometric relation between h q I ( ) and h q II ( ). In our experiment, we examine two cases of (ν I , ν II )=(1, 1) and (2, 1) with j = 0 I II ( ) and t p w = 2 I II I II ( ) ( ) . We measure the momentum distribution n(k) after applying the two pulses of lattice shaking with increasing T e . In figure 5(a) for q=±0.7π. The population oscillations with increasing T e are clearly observed for each q and the oscillation frequency is found to be in good quantitative agreement with ω sp (q) ( figure 7(a)). Next, the measurement result for (ν I , ν II )=(2, 1) is presented in figure 5(b). In the measurement, the driving amplitude for the first pulse (ν I =2) is changed to A I /2π=8 kHz to keep ξ I =π/4 for q=±0.7π, whereas that for the second pulse (ν II =1) is the same as in the previous (ν I , ν II )=(1, 1) measurement. Remarkably, we observe that the Ramsey signals exhibit an asymmetric fringe pattern with respect to k=0, which is in stark contrast to the (ν I , ν II )=(1, 1) case where the Ramsey signals are mirrorsymmetric. Based on the fact that the transverse component of h q II ( ) with ν II =1 is uniform over the BZ  ( ) ( ), thus corroborating the winding structure of h q 2 ( ). In figures 5(c) and (d), we display the population n(k) at k=±1.3π, corresponding to n p (q=m0.7π), as a function of T e for the (ν I , ν II )=(1, 1) and (2, 1) cases, respectively. The population shows damped oscillations with increasing T e and the damping might be attributed to atom-atom interactions and/or the spatial inhomogeneity of the trapped sample. From the damped sinuoidal functions fit to the data, we measured the phase difference between the two oscillation curves to be 0.10π for (ν I , ν II )=(1, 1) and 0.77π for (ν I , ν II )= (2, 1), which are slightly different from the expected values of 0 and π, respectively. We attribute such a small deviation to the off-resonant coupling effect, which is neglected in our model description and will be further discussed in the next section.
As a further comparison of the (ν I , ν II )=(1, 1) and (2, 1) cases, we also examine the dependence of the Ramsey signal on the driving phases j I and j II by measuring n(k=±1.3π) as a function of j I and j II with a fixed evolution time T e =100μ s ( figure 6). For (ν I , ν II )=(1, 1), the two populations at k=±1.3π oscillate in phase with period of 2π in both j I and j II . In contrast, for (ν I , ν II )=(2, 1), they show out-of-phase oscillations with a different period of π in j I and 2π in j II . This observation is consistent with the different j-dependence of r 1 and r 2 in equation (7). The π periodicity of the Ramsey signal with increasing j I for ν I =2 is a direct consequence of that the inter-orbital coupling is generated via two-photon process.

Off-resonance coupling effect
In the model description of the resonantly shaken lattice system, we consider only the effects of the resonant coupling between the s and p orbitals. Although the model is efficient in capturing the essential topological features of the system, its improvement, in particular, by including the off-resonant inter-orbital coupling effects would be necessarily desirable for further development of the ladder system. To evaluate the limitations of the current model, we make a quantitative comparison between the experimental results in the Ramsey interferometry measurement and the prediction from the model. In figure 7, we characterize the Ramsey interferometry data presented in figures 5(a) and (b) with the oscillating frequency ω f (k) and the phase f(k), where {ω f (k), f(k)} are determined from a damped sinusoidal function fit to the fringe signal n(T e ; k) at each k. In the (ν I , ν II )=(1, 1) case, we find that the measured {ω f (k), f(k)} are well described by the model [figures 7(a) and (c)]. Here, the model prediction was numerically obtained by calculating equation (8) for the experimental conditions. On the other hand, in the (ν I , ν II )=(2, 1) case, we observe that the measured oscillation phase f(k) shows a quantitatively non-negligible deviation from the model prediction [ figure 7(d)]. The deviation is pronounced near k=0 and ±π. These momentum regions are where the two-photon resonant coupling is weak for µ r h q q sin 2, ( ) ( ) so that it is not only technically difficult to precisely determine f but also likely to be sensitive to any off-resonant coupling effect.
In the two-photon resonance condition, the dominant off-resonant coupling must arise from the onephoton inter-orbital transition process. We speculate that its effect might be approximated to direct leakage links between the two legs, and then represented by a uniform transverse component in h as observed in the case of one-photon resonant coupling. To test the possibility, we recalculate equation (8) for the (ν I , ν II )=(2, 1)  The presence of such leakage links has an important implication that depending on the complex amplitude of the leakage links, h might not be confined in a plane so that its winding number would be ill-defined. The off-resonant coupling effects, including those from the coupling to higher orbitals, warrant further theoretical and experimental investigations in future study.

Shaking with two resonant frequencies
The observation that one-and two-photon resonant couplings bring about the distinctive effects of direct and cross inter-leg links, respectively, prompts us to discuss an extension of the inter-leg link control in the ladder by using both types of resonant couplings. Specifically, we consider a situation in which the lattice shaking is applied as dw Here, together with the ω driving generating two-photon resonant coupling, the 2ω driving provides one-photon resonant coupling. We find that the interorbital coupling effects of the two drivings appears additive in the effective Hamiltonian at the same level of approximation used in equation (7) (see appendix B), giving . This suggests that the direct inter-leg links can be generated independently in the Creutz ladder by simply adding a 2ω driving; furthermore, its complex amplitude can be controlled with the driving parameters Ã and j. Even with the off-resonant coupling effects discussed in the previous subsection, this twotone driving method would provide a flexible way to control the inter-leg links of the Creutz ladder system.
The extended control of the inter-leg links will enable to explore many interesting aspects of the topological ladder system. First of all, the topological phase transition of the Creutz ladder can be studied in a controlled manner, which occurs at accompanied by the band gap closing at q=±π/2 and a sudden change of the winding number of h q ( ). Anomalous scaling of defect formation was predicted for the phase transition dynamics due to the topological properties of the system [25]. Equipped with the dynamic control of the inter-leg links, we can also envisage a realization of a topological charge pump in the Creutz ladder [42]. For example, when the direct link parameters j t , v {˜} are controlled to adiabatically encircle one of the critical points in the parameter space, the Zak phase γ Z [47] continuously changes by 2π per cycle (figure 8), which would result in one lattice site shift of atoms in the ladder.

Conclusion
We have demonstrated the realization of the Creutz ladder in a periodically shaken 1D optical lattice via twophoton resonant coupling. The topological structure of the Creutz ladder was characterized with the winding structure of h q 2 ( ) in the BZ. By the shaking spectroscopy, the longitudinal component of h 2 was directly measured as =h t q 2 cos z r ( ), and using the Ramsey interferometry, the asymmetric relation of its transverse component, ( ) ( ), was demonstrated. From the observation that one-and two-photon resonant couplings generate direct and diagonal inter-leg links, respectively, in the ladder system, we suggested the two-tone driving method of simultaneously exploiting both resonant couplings, which would allow the controlled study of the topological phase transition dynamics and the realization of a topological charge pump. As another extension of this work, topological flat band engineering may be pursued with the Creutz ladder system, which would provide interesting opportunities for observing possible emergence of correlated topological phases [43,44]. In interacting two-leg ladder systems, many correlated topological phases and associated edge states were theoretically discussed [27][28][29][30][31]. Finally, we also 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 [45,46]. , whereas ω/2π=13.7 or 6.8kHz, and t r /h=0.54 kHz, justifying the approximation taken in (A.9) and (A.13).