## Abstract

Periodic band structures are a hallmark phenomenon of condensed matter physics. While often imposed by external potentials, periodicity can also arise through the interplay of couplings that are not necessarily spatially periodic on their own, but this option is generally less explored than the fully-periodic counterpart. Here, we investigate dynamics in a lattice structure that emerges from the simultaneous application of Raman and radio frequency coupling to a dilute-gas Bose-Einstein condensate. We elaborate on the role of Galilean invariance in this system and demonstrate a variety of techniques, including Bloch oscillations and lattice shaking with spin and momentum resolved measurements. This combined coupling scheme allows for tunability and control, enabling future investigations into unconventional band structures such as quasi-flat ground bands and those with semimetal-like band gaps.

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## Introduction

Periodic band structures and spin–orbit coupling play a key role in many modern condensed matter contexts. Implementing these aspects with ultracold quantum gases using optical lattices^{1,2,3,4} and Raman dressing^{5} has opened up avenues for research providing deep insights into topics including synthetic gauge fields^{6,7,8,9}, optical flux lattices^{10}, topological state spaces^{11,12}, supersolids^{13,14,15}, and more^{16,17,18,19,20,21,22}.

Supplementing the Raman-dressing-induced spin–orbit coupling with a suitably chosen radio frequency (RF) drive leads to the emergence of an effective lattice structure even though neither the spin–orbit coupling nor the RF alone produces a periodic structure^{23,24,25}. This synthesized system possesses spin–orbit coupling and a lattice. The lattice is spin-dependent and is called a *Zeeman lattice*. A static version of the spin–orbit coupled Zeeman lattice was first introduced in the pioneering works by^{23} and^{24} for bosonic and fermionic systems, respectively. The dynamical control of a bipartite spin–orbit coupled Zeeman lattice leads to geometrical pumping^{25}. These benchmark investigations witness the powerful amalgamation of the spin–orbit coupling and periodic band structures. Compared to these previous studies, here we extend the treatment of Zeeman lattices to an accelerating case where questions regarding the Galilean invariance of the system become important.

Periodic band structures can be implemented separately for spin–orbit coupled atoms by optical lattices^{7}. Such spin–orbit coupled optical lattice systems have been an important platform to predict novel spin states^{26}, engineer energy band geometry^{27}, and explore spin–orbit coupled dynamics such as novel Bloch oscillation^{22}. However, the lack of Galilean invariance in spin–orbit coupled systems^{7,16,28,29} greatly reduces the controllability of optical lattices and strongly limits applications.

In our system, we find that the spin–orbit coupled Zeeman lattice restores Galilean invariance. Accelerating the atoms by an external force can be substituted by appropriate changes of the Raman drive frequency. We exploit this feature to investigate dynamics and demonstrate experimental techniques to manipulate the lattice. We induce Bloch oscillations via an acceleration of the Zeeman lattice, and spectroscopically probe inter-band transitions by shaking the lattice resonantly. With the restoration of Galilean invariance, this work shows that the Zeeman lattice provides a flexible and robust system for manipulating spin–orbit coupled atoms by equivalently tuning the Raman lasers.

## Results

### Theoretical framework of a Zeeman lattice

A spin–orbit coupled Zeeman lattice is realized via the simultaneous coupling of two hyperfine states (\(\vert \! \uparrow \rangle\) and \(\vert \! \downarrow \rangle\)) using two separate methods: a pair of Raman lasers and an external RF field. The couplings produced by the Raman lasers and by the RF drive are schematically demonstrated in the bare (uncoupled) basis in Fig. 1a. The Raman beams (marked Ω_{R} in Fig. 1b) are arranged such that they induce spin–orbit coupling by imparting momentum on the atoms while also flipping the spins in a two-photon transition. The RF coupling flips the spins without changing the momentum state. Then the coupled states, separated in momentum space by the Raman momentum, can interfere, leading to a periodic structure.

This coupling scheme is described by the Hamiltonian^{23}

where \({{{{{{{\mathcal{C}}}}}}}}=\frac{\hslash {\Omega }_{{{{{{{{\rm{R}}}}}}}}}}{2}\exp \left[i\left(2{k}_{{{{{{{{\rm{R}}}}}}}}}x+\varphi (t)\right)\right]+\frac{\hslash {\Omega }_{{{{{{{{\rm{RF}}}}}}}}}}{2}\exp \left(i{\omega }_{{{{{{{{\rm{RF}}}}}}}}}t\right)\). Ω_{R} is the two-photon Rabi frequency due to the Raman coupling, Ω_{RF} is the RF Rabi frequency, *ω*_{RF} is the angular frequency of the RF field, and ℏ is the reduced Planck constant. The Raman coupling involves a momentum exchange of 2ℏ*k*_{R} between the atoms and the Raman beams, where *k*_{R} is the effective wave vector of Raman lasers. The energy unit associated with the recoil of the atom due to the absorption of a photon is the recoil energy given by \({E}_{{{{{{{{\rm{R}}}}}}}}}={\hslash }^{2}{k}_{{{{{{{{\rm{R}}}}}}}}}^{2}/2m\). The phase *φ*(*t*) is connected to the angular frequency difference between the Raman lasers Δ*ω*_{R}, which can readily be tuned in the experiment, \(\varphi (t)=\int\nolimits_{0}^{t}d\tau \Delta {\omega }_{{{{{{{{\rm{R}}}}}}}}}(\tau )\). In the Hamiltonian *H*_{0}, *m* is the atomic mass and Δ*ϵ* is the energy difference between the two Zeeman states due to the Zeeman shift of an external 10 G bias field (see Experimental setup for details). Without the RF coupling, the spatial dependence in \({{{{{{{\mathcal{C}}}}}}}}\) generates spin–orbit coupling^{5}, but effectively there is no lattice structure. Due to the presence of RF coupling, the spatial dependence cannot be gauged out, therefore leaving the signature of the spin–orbit coupled Zeeman lattice.

In order to show the spin–orbit coupled Zeeman lattice explicitly, we apply the unitary transformation \(U=\exp \left[i\left(2{k}_{{{{{{{{\rm{R}}}}}}}}}x+\varphi (t)\right){\sigma }_{z}/2\right]\)^{5} to the Hamiltonian *H*_{0} in Eq. (1), leading to *H* = *U*^{†}*H*_{0}*U* − *i*ℏ*U*^{†}∂*U*/∂*t* with

Here, \({{{{{{{\bf{S}}}}}}}}=\left({\sigma }_{x},{\sigma }_{y},{\sigma }_{z}\right)\) are the Pauli matrices. The Zeeman lattice is spin-dependent and is denoted by **B**_{latt}[*x*] = (*b*_{x}, *b*_{y}, *b*_{z}) with \({b}_{x}=\hslash {\Omega }_{{{{{{{{\rm{R}}}}}}}}}+\hslash {\Omega }_{{{{{{{{\rm{RF}}}}}}}}}\cos \left(2{k}_{{{{{{{{\rm{R}}}}}}}}}x\right)\), \({b}_{y}=\hslash {\Omega }_{{{{{{{{\rm{RF}}}}}}}}}\sin \left(2{k}_{{{{{{{{\rm{R}}}}}}}}}x\right)\), and the detuning *b*_{z} = ℏ*d**φ*/*d**t* − Δ*ϵ* = ℏΔ*ω*_{R} − Δ*ϵ*. The time-dependent position offset is \({x}_{0}(t)=\left(\varphi (t)-{\omega }_{{{{{{{{\rm{RF}}}}}}}}}t\right)/2{k}_{{{{{{{{\rm{R}}}}}}}}}\). Therefore, the moving velocity of the Zeeman lattice is

The spin–orbit coupling is represented by \({{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{soc}}}}}}}}}=\left(0,0,2\hslash {k}_{{{{{{{{\rm{R}}}}}}}}}{p}_{x}/m\right)\). The combination of **B**_{latt}[*x*] and **B**_{soc} in the Hamiltonian *H* represents the spin–orbit coupled Zeeman lattice. If the RF coupling is switched off, then there is no Zeeman lattice. The Zeeman lattice becomes stationary when \(\varphi (t)=\int\nolimits_{0}^{t}d\tau \Delta {\omega }_{{{{{{{{\rm{R}}}}}}}}}(\tau )={\omega }_{{{{{{{{\rm{RF}}}}}}}}}t\) which implies *b*_{z} = ℏ*ω*_{RF} − Δ*ϵ*. The stationary Zeeman lattice can be realized by tuning the angular frequency difference Δ*ω*_{R} between the Raman lasers to exactly equal *ω*_{RF}, which has been explored in^{23,24}.

The lattice structure of the above stationary spin–orbit-RF coupled Hamiltonian [Eq. (2)] can have a meaningful spectrum which has been presented in^{24}. Fig. 2 shows the band structure of the stationary Zeeman lattice in various parameter regimes where the spin polarization 〈*σ*_{z}〉 is indicated by the color of the curve and given by 〈*σ*_{z}〉 = (*N*_{↑} − *N*_{↓})/*N*_{tot}. Here, *N*_{↑} and *N*_{↓} are the occupation of the spin up \(\left\vert \uparrow \right\rangle\) and spin down \(\left\vert \downarrow \right\rangle\) state, respectively, and *N*_{tot} = *N*_{↑} + *N*_{↓}. If one applies Bloch’s theorem to diagonalize the spin–orbit coupling Hamiltonian (without the RF coupling) assuming an infinite set of plane wave solutions, one finds that bands in different Brillouin zones are not coupled and therefore are trivially inaccessible, shown in Fig. 2a. In other words, this shows that there is no lattice structure in the Raman-induced spin–orbit coupling. The inclusion of the RF coupling then opens up gaps and produces a periodic band structure [cf. Fig. 2b]. The Hamiltonian *H* given above can be tuned in highly flexible ways. For example, it can result in a semimetal-like band structure [Fig. 2c] or in a nearly flat band as the lowest Bloch band [Fig. 2d] in different parameter regimes.

### Galilean symmetry of the Zeeman lattice

It is well-known that spin–orbit coupling breaks Galilean invariance^{7,16,28,30}. Our analysis of the synthesized spin–orbit coupled Zeeman lattice reveals that this lattice restores the Galilean invariance in the following sense: For the Zeeman lattice in Eq. (2), the time-dependence relates to both the Raman lasers and the RF field, i.e., *x*_{0}(*t*) = (*φ*(*t*) − *ω*_{RF}*t*)/2*k*_{R}. We can transform into a frame that is co-moving with the lattice by applying the unitary transformation *G*^{31,32},

which can be considered as a generalized Galilean transformation, since *x*_{0}(*t*) may be time-dependent arbitrarily^{33}. With *G*^{†}*x**G* = *x* − *x*_{0} and *G*^{†}*p*_{x}*G* = *p*_{x} − *m**d**x*_{0}/*d**t*, the resultant Hamiltonian in the co-moving frame is *H*_{com} = *G*^{†}*H**G* − *i*ℏ*G*^{†}∂*G*/∂*t* given by

Here \({{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{latt}}}}}}}}}^{{\prime} }[x]=({b}_{x},{b}_{y},{b}_{z}^{{\prime} })\) with \({b}_{z}^{{\prime} }=\hslash {\omega }_{{{{{{{{\rm{RF}}}}}}}}}-\Delta \epsilon\), and *b*_{x} and *b*_{y} are as in Eq. (2). Note that \({{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{latt}}}}}}}}}^{{\prime} }[x]\) is exactly the same as for the stationary Zeeman lattice [*φ*(*t*) = *ω*_{RF}*t*] in Eq. (2). An arbitrary acceleration of the Zeeman lattice leaves its Hamiltonian invariant up to an additional term which represents the apparent force due to the motion of the noninertial reference frame given by \({{{{{{{\mathcal{F}}}}}}}}=-m{d}^{2}{x}_{0}/d{t}^{2}=-(m/2{k}_{{{{{{{{\rm{R}}}}}}}}})\,{d}^{2}\varphi /d{t}^{2}\). Therefore, this analysis reveals that applying an external force to the spin–orbit coupled atoms in a stationary Zeeman lattice is equivalent to accelerating the Zeeman lattice without an external force when viewed from the non-inertial reference frame of the lattice. This equivalence is taken as a signature of the restored Galilean invariance in the system, even though it is in the presence of the spin–orbit coupling.

From an experimental point of view, this is a significant feature. The motion of spin–orbit coupled atoms can be manipulated by performing operations on the Zeeman lattice. Moving, accelerating, or shaking a lattice by changing the laser frequency in one of the beams can typically be performed with much higher precision than directly moving the atoms^{1,31}. The restoration of Galilean invariance is, hence, an important aspect. The restoration is specific for the spin–orbit coupled Zeeman lattice. A standard spin–orbit coupled lattice lacks the Galilean invariance. A typical Hamiltonian for a standard spin–orbit coupled optical lattice is^{7,26}\({H}_{{{{{{{{\rm{soc}}}}}}}}}={p}_{x}^{2}/2m+\gamma {p}_{x}{\sigma }_{z}+{V}_{0}{\cos }^{2}[x+{x}_{0}(t)]\), where *γ* is the spin–orbit coupling strength, \({V}_{0}{\cos }^{2}[x+{x}_{0}(t)]\) is the moving optical lattice, and *V*_{0} is the lattice depth. Transforming into the co-moving frame by applying the generalized Galilean transformation results in \({G}^{{{{\dagger}}} }{H}_{{{{{{{{\rm{soc}}}}}}}}}G-i\hslash {G}^{{{{\dagger}}} }\partial G/\partial t={p}_{x}^{2}/2m+\gamma {p}_{x}{\sigma }_{z}-\gamma m{\sigma }_{z}d{x}_{0}/dt+{V}_{0}{\cos }^{2}(x)+{{{{{{{\mathcal{F}}}}}}}}x\). The lack of the Galilean invariance is demonstrated by the appearance of the term *γ**m**σ*_{z}*d**x*_{0}/*d**t,* which depends on the frame velocity *d**x*_{0}/*d**t*. Physically, the optical lattice generated by optical lattice lasers and the spin–orbit coupling generated by Raman lasers have two independent frames. Working in the co-moving frame with the optical lattice must affect the spin–orbit coupling frame in the way that Raman laser frequencies are Doppler shifted. Therefore it leads to the appearance of an effective detuning *γ**m**σ*_{z}*d**x*_{0}/*d**t*. The lack of Galilean invariance in this system means that the manipulation of motions of the spin–orbit coupled atoms via the optical lattice is inevitably accompanied by the changing of the spin degree of freedom. For the spin–orbit coupled Zeeman lattice, the frames for the Zeeman lattice and the spin–orbit coupling are not independent. The RF coupling does not produce an additional reference frame for the atoms due to the negligible Doppler shift associated with RF frequencies (see Supplementary Note 1 for details). The Zeeman lattice presented here produces a spin-dependent lattice with a reference frame that only depends on one parameter—the angular frequency difference between Raman lasers Δ*ω*_{R}(*t*). Therefore, in our experiments described below we only change Δ*ω*_{R} and keep *ω*_{RF} constant.

### Bloch oscillation in the Zeeman lattice

To investigate dynamics and exploit the Galilean invariance of the Zeeman lattice, we first consider Bloch oscillations, i.e., the oscillatory motion of particles confined in a periodic potential under the influence of a constant force. Here we explore the band structure shown in Fig. 2(b) where the lowest band is well isolated from the first excited band, suppressing Landau–Zener tunneling^{31}.

To experimentally demonstrate Bloch oscillations in the system, we prepare the ground state of the stationary Zeeman lattice (see Materials and methods for details) and then ramp up the frequency difference between the Raman beams in such a way that the relative phase between the two Raman beams evolves as *φ*(*t*) = *ω*_{RF}*t* + *α**t*^{2} with *α* = 15*π* MHz s^{−1}, or, in different experimental runs, *α* = 15*π* MHz s^{−1}. Therefore the Zeeman lattice is accelerated as *x*_{0}(*t*) = [*φ*(*t*) − *ω*_{RF}*t*]/2*k*_{R} = *α**t*^{2}/2*k*_{R} with the constant acceleration *α*/*k*_{R} = 8.24 m s^{−2}, or *α*/*k*_{R} = − 8.24 m s^{−2}, respectively. The velocity of the lattice evolves as *v*_{lat} = *α**t*/*k*_{R}. Equivalently, in the frame of the moving lattice atoms feel a constant force \({{{{{{{\mathcal{F}}}}}}}}=-m\alpha /{k}_{{{{{{{{\rm{R}}}}}}}}}\) from the last term in Eq. (5). It linearly changes the quasimomentum of the atoms, \(k(t)=-0.97{k}_{{{{{{{{\rm{R}}}}}}}}}+{{{{{{{\mathcal{F}}}}}}}}t/\hslash =-0.97{k}_{{{{{{{{\rm{R}}}}}}}}}-m{v}_{{{{{{{{\rm{lat}}}}}}}}}/\hslash\). The acceleration is applied for up to 2 ms, which corresponds to moving through more than two Brillouin zones. The ramp can be performed in either direction (i.e., ± *α*), yielding four Brillouin zones worth of data. At various stages along the ramp, and thus for various lattice velocities, we perform expansion imaging of the BEC and from the observed velocity components determine the average lab-frame velocity *v*_{lab} of the cloud. The obtained *v*_{lab} as a function of *v*_{lat} are presented in Fig. 3a. The staircase structure is the characteristics of Bloch oscillations. In the co-moving frame with the lattice, the lattice is at rest and the BEC has the group velocity, *v*_{g} = *v*_{lab} − *v*_{lat}, which forms the sinusoidal pattern as shown in Fig. 3b. In theory, the group velocity can be predicted by calculating the gradient of the lowest band *ϵ*_{0}(*k*) shown in Fig. 2b, *v*_{g} = ∂*ϵ*_{0}(*k*)/ℏ∂*k*. The theoretically calculated group velocity is shown by the orange curves in Fig. 3. The good agreement between the observations and theoretical predictions confirms that we can effectively accelerate atoms in the spin–orbit coupled lattice by accelerating the lattice instead.

The condensate atoms experience modulational instability during their Bloch oscillations, resulting in a fringe-like substructure within each momentum class. This instability, influenced by parameters like duration within the unstable region, the system’s nonlinearity related to the condensate atom numbers, and external perturbations, leads to random variations in the width of momentum peaks, suggesting no inherent significance to the differences observed in the insets of Fig. 3a.

### Band spectroscopy of the Zeeman lattice

Next, we show that a periodic modulation of the frequency difference between the Raman beams leads to a shaken Zeeman lattice, which can induce inter-band transitions. This provides a practical means for performing band spectroscopy in the Zeeman lattice.

We begin with a BEC prepared in the ground state of the stationary Zeeman lattice with ℏΩ_{R} = (2.6 ± 0.13) *E*_{R}, ℏΩ_{RF} = (1.15 ± 0.03) *E*_{R}, and *b*_{z} = *h* × (500 ± 100) Hz (see Materials and methods for details and Supplementary Fig 2). To modulate the phase of the lattice^{1}, we apply a sinusoidal modulation to the Raman frequency difference such that \({x}_{0}(t)=[\varphi (t)-{\omega }_{{{{{{{{\rm{RF}}}}}}}}}t]/2{k}_{{{{{{{{\rm{R}}}}}}}}}={\phi }_{0}\sin (2\pi ft)/2{k}_{{{{{{{{\rm{R}}}}}}}}}\), where *ϕ*_{0}/2*k*_{R} and *f* are the shaking amplitude and frequency, respectively. In the co-moving frame with the shaken lattice, atoms experience the oscillating force as \({{{{{{{\mathcal{F}}}}}}}}=2{\pi }^{2}m{\phi }_{0}{f}^{2}\sin (2\pi ft)/{k}_{{{{{{{{\rm{R}}}}}}}}}\). To induce inter-band transition, the amplitude of \({{{{{{{\mathcal{F}}}}}}}}\) should be a perturbation to the system. We shake the lattice with an amplitude of *ϕ*_{0}/2*k*_{R} = *π*/20*k*_{R} for 6 ms using various frequencies *f*. The measured spin polarization as a function of *f* is shown in Fig. 4a. The observed response of the spin polarization centered at *f* = (2.02 ± 0.05) kHz is the signature of an inter-band transition.

Further transitions can be observed by driving the system at higher frequencies. Here, we modulate the phase by *ϕ*_{0}/2*k*_{R} = *π*/40*k*_{R} for 2 ms at each frequency. In this case, the clearest signature of the transition is seen in the fractional population of the bare momentum states, which can be understood by inspecting the spin and bare-momentum composition of the underlying bands. The availability of different but correlated observables for the spectroscopy is a particular strength of our Zeeman lattice. Figure 4b shows that the fractional population of the zero-momentum state exhibits a single broad feature, indicating transitions out of the ground band. Plotting the fractional populations in the ± 2*k*_{R} states resolves this broad peak into two peaks centered at (8.24 ± 0.04) kHz and (8.65 ± 0.02) kHz, respectively, where the uncertainty in the line center is given by the standard error of the fit to the experimental data.

These observed features are in reasonable agreement with the theory calculations for the excitations to the *n* = 1, 2, 3 states. For a BEC placed at a minimum of the lowest band occurring at the quasimomentum *k* = (0.97 ± 0.02) *k*_{R}, the diagonalization of Eq. (2) for the stationary lattice predicts the *n* = 0 → 1, 2, 3 transitions to occur at frequencies 2.05, 8.12, and 8.79 kHz for our experimental parameters, as marked with dashed lines in Fig. 4. These values are calculated based on a two-state model for an infinite, noninteracting, two-state system. We note that the resonance frequencies are quite sensitive to the atomic momentum—displacing the momentum along the noninteracting dispersion by just 0.02 *k*_{R} produces a near perfect agreement between experiment and theory for all three resonances observed here. The detuning *b*_{z} breaks the parity symmetry \({{{{{{{\mathcal{P}}}}}}}}{\sigma }_{x}\) of the stationary spin–orbit coupled Zeeman lattice, where \({{{{{{{\mathcal{P}}}}}}}}\) is the parity operator with \({{{{{{{{\mathcal{P}}}}}}}}}^{{{{\dagger}}} }x{{{{{{{\mathcal{P}}}}}}}}=-x\). So the observation of above three band transitions is possible without the constraint of the selection rule due to the parity symmetry (see Supplementary Note 3 for details).

## Conclusions

We have studied the rich dynamics of atoms in a spin–orbit coupled Zeeman lattice that emerges from the confluence of spin–orbit coupling and an external radio frequency field. We have discussed the role of Galilean invariance and find that this invariance is restored in the Zeeman lattice, while spin–orbit coupling alone breaks Galilean invariance. This has important consequences for experiments: controlling the motion of the lattice structure is typically much more precise and flexible than controlling the motion of the atoms. We have applied this insight to characterize the Zeeman lattice using Bloch oscillations and resonant band spectroscopy. In our band spectroscopy, we have shown that both spin and momentum composition are useful observables, demonstrating the multifaceted nature of the Zeeman lattice. The availability of a variety of correlated observables, such as spin and momentum populations, affords powerful experimental tools for performing detailed studies. This work shows that the spin–orbit coupled Zeeman lattice introduced in ^{23,24} provides powerful methods for further investigations of spin–orbit coupling including spin-dependent lattices, the characterization of a semimetal-like band structure (Fig. 2c), dynamics of a BEC in a tunable flat ground band (Fig. 2d), the realization of Wannier-Stark ladders^{34}, and exploration of the spin Hall effect^{35}, to name a few.

## Methods

### Experimental setup

In our experiment, we prepare a BEC of approximately 2.5 × 10^{5}^{87}Rb atoms in the \(\left\vert F,{m}_{F}\right\rangle =\left\vert 1,-1\right\rangle\) Zeeman state. The BEC is confined in a crossed optical dipole trap characterized by the harmonic trap frequencies ** ω** = (

*ω*

_{x},

*ω*

_{y},

*ω*

_{z}) = 2

*π*(20 ± 2, 160 ± 10, 190 ± 10) Hz. A 10 G ± 0.1 mG bias field applied along the

*x*-axis lifts the degeneracy among the three states in the

*F*= 1 hyperfine manifold. We generate spin–orbit coupling in the

*x*-direction using two Raman beams with a wavelength of 789 nm intersecting at approximately 45° angles with the

*x*-axis, as shown in Fig. 1b. This leads to an effective recoil energy of

*E*

_{R}=

*h*× 1.9 kHz. The frequencies of the Raman beams are tuned such that the \(\left\vert 1,-1\right\rangle\) and \(\left\vert 1,0\right\rangle\) states are nearly resonantly coupled via the two-photon Raman transition. These two states are additionally coupled by an RF drive (Fig. 1a). The quadratic Zeeman shift places the \(\left\vert 1,+1\right\rangle\) state far out of resonance so that effectively a spin-1/2 system composed of \(\left\vert \uparrow \right\rangle \equiv \left\vert 1,-1\right\rangle\) and \(\left\vert \downarrow \right\rangle \equiv \left\vert 1,0\right\rangle\) as the two pseudo-spin states is realized. After performing the experiments described in the previous sections, imaging is performed by suddenly turning off the trap and all driving fields, and allowing time of flight in the presence of a Stern–Gerlach field. An absorption image is then taken which resolves the BEC into the bare-state spin and momentum components. The existence of lattice structure has been experimentally detected by Kapitza–Dirac scattering

^{23}(see Supplementary Note 2 for details and Supplementary Fig. 1).

### Zeeman lattice ground state preparation

We begin an experiment by preparing the BEC in a minimum of the lowest band of the stationary Zeeman lattice: we adiabatically dress the condensate with a Raman coupling field of strength ℏΩ_{R} = (2.6 ± 0.13) *E*_{R} and a detuning *b*_{z} = *h* × (500 ± 100) Hz, and then adiabatically ramp on the strength of the radio frequency drive to a coupling strength of ℏΩ_{RF} = (2.3 ± 0.07) *E*_{R} over 100 ms while satisfying the stationary lattice condition *φ* − *ω*_{RF}*t* = 0.

## Data availability

All data in the main text or the supplementary materials are available upon reasonable request.

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## Acknowledgements

M.K.H.O., A.M., E.C., S.M., T.B., and P.E. acknowledge funding from NSF through Grant No. PHY-1912540. P. E. acknowledges support through the Ralph G. Yount Distinguished Professorship at WSU. H.H. and Y.Z. are supported by National Natural Science Foundation of China with Grants Nos. 12374247 and 11974235.

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P.E. and Y.Z. conceived and supervised the research project. Y.Z., H.H., S.M., E.C., and M.K.H.O. performed the theoretical analysis and numerics. P.E., S.M., T.B., E.C., M.K.H.O., and A.M. did the experiment, data analysis, and interpretation. All authors discussed the results and contributed to the paper preparation, review, and editing.

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*Communications Physics* thanks Erich Mueller and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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### Cite this article

Ome, M.K.H., He, H., Mukhopadhyay, A. *et al.* Galilean invariant dynamics in an emergent spin-orbit coupled Zeeman lattice.
*Commun Phys* **7**, 9 (2024). https://doi.org/10.1038/s42005-023-01506-4

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DOI: https://doi.org/10.1038/s42005-023-01506-4

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