Topological lattice using multi-frequency radiation

Abstract

We describe anoveltechniqueforcreatinganartificialmagneticfieldforultracoldatomsusinga periodicallypulsedpairofcounterpropagatingRamanlasersthatdrivetransitionsbetween a pair of internal atomic spin states: a multi-frequency coupling term. In conjunction with a magnetic field gradient, this dynamically generates a rectangular lattice with a non-staggered magnetic flux. For a wide range of parameters, the resulting Bloch bands have non-trivial topology, reminiscent of Landau levels, as quantified by their Chern numbers.

1. Introduction

Ultracold atoms find wide applications in realizing condensed matter phenomena[1–4].Since ultracold atom systems are ensembles of electrically neutral atoms, various methods have been used to simulate Lorentz-type forces, with an eye for realizing physics such as the quantum Hall effect(QHE). Lorentzforcesarepresentin spatially rotating systems[5–11]and appear in light-induced geometric potentials[12,13].The magnetic fluxes achieved with these methods are not sufficiently large for realizing the integer or fractional QHE. In optical lattices, larger magnetic fluxes can be created by shaking the lattice potential[14–17],combining static optical lattices along with laser-assisted spin or pseudo spin coupling[12,13,18–24]; current realizations of these techniques are beset with micro motion and interaction induced heating effects. Here we propose a new method that simultaneously creates large artificial magnetic fields and a lattice that may overcome these limitations.

Our technique relies on a pulsed atom-light coupling between internal atomic states along with a statedependent gradient potential that together create a two-dimensional periodic potential with an intrinsic artificial magnetic field. With no pre-existing lattice potential, there are no apriori resonant conditions that would otherwise constrain the modulation frequency to avoid transitions between original Bloch bands [25]. For a wide range of parameters, the ground and excited bands of our lattice are topological, with nonzero Chern number. Moreover, like Landau levels the lowest several bands can all have unit Chern number.

The manuscript is organized as follows. Firstly, we describe a representative experimental implementation of our technique directly suitable for alkali atoms. Secondly, because the pulsed atom-light coupling is timeperiodic, we use Floquet methods to solve this problem. Specifically, we employ a stroboscopic technique to obtain an effective Hamiltonian. Thirdly, using the resulting band structure we obtain a phase diagram which includes a region of Landau level-like bands each with unit Chern number.

2. Pulsed lattice

Figure 1 depicts a representative experimental realization of the proposed method. A system of ultracold atoms is subjected to a magnetic field with a strength B(X) = B₀ + B’X. This induces a position-dependent splitting g_Fμ_BB between the spin up and down states; g_F is the Land g-factor and μ_B is the Bohr magneton. Additionally, the atoms are illuminated by a pair of Raman lasers counter propagating along e_y, i.e., perpendicular to the detuning gradient. The first beam [up-going in figure 1(a)] is at frequency ω⁺ = ω₀, while the second [downgoing in figure 1(a)] contains frequency components ${\omega }_n^{-}={\omega }_0+{(-1)}^n(\delta \omega +n\omega )$; the difference frequency Between these beams contains frequency combs centered at ±δω with comb teeth spaced by 2ω, as shown in figure 1(b). In our proposal, the Raman lasers are tuned to be in nominal two-photon resonance with the Zeeman splitting from the large offset field B₀ such that g_Fμ_BB₀ = ℏδω₀, making the frequency difference ${\omega }_{n=0}^{-}-{\omega }^{+}$ resonant at X = 0, where B = B₀. Intuitively, each additional frequency component ${\omega }_n^{-}$ adds a resonance condition at the regularly spaced points X_n = nℏω/g_Fμ_BB′, however, transitions using even-n sidebands give a recoilkick opposite from those using odd-n sidebands (see figure 1(c)). Each of these coupling locations locally realizes synthetic magnetic field experiment performed at the National Institute of Standards and Technology [26], arrayed in a manner to give a rectified artificial magnetic field with a nonzero average that we will show is a novel fluxlattice.

Figure 1.

Floquet flux lattice. (a) Experimental schematic depicting a cold cloud of atoms in a gradient magnetic field, illuminated by a pair of counter-propagating laser beams tuned near two-photon Raman resonance. The down-going beam includes sidebands both to the red and blue of the carrier (ω0) in resonance at different spatial positions along ex. (b) Level diagram showing even and odd sidebands linking the ∣↑⟩ and∣↓⟩ stateswith differing detuning fromresonance at X = 0. (c) Spatially dependent coupling. Bottom: different frequency components are in two-photon resonance in different X positions. Top: the recoil kick associated with the Raman transition is along e_y and thus alternates spatially depending on whether the Raman transition is driven from the red or blue sideband of the down-going laser beam.

In practice only a finite number of lattice teeth are needed, owing to the finite spatial extent of a trapped atomic gas. In rough numbers the spatial extent of a quantum degenerate gas is 20 μm, and if we select a very large gradient corresponding to a lattice spacing of 0.5 μm, this gives just 40 comb teeth. Generating the frequency comb is straightforward. In the laboratory, acoustic-optical modulators (AOMs) frequency-shift laser beams by an amount defined by a laboratory radio frequency (rf) source. Therefore creating a comb is a matter of first creating a frequency comb—simple with rf—and then feeding that signal into the AOM. This sort of frequency synthesis is routine in the ultracold atom labs.

We formally describe our system by first making the rotating wave approximation (RWA) with respect to the large offset frequency ω₀.This situation is modeled in terms of a spin-1/2 atom of mass M and wave-vector K = –i∇ with a Hamiltonian

$$H\left(t\right)=H_0+V\left(t\right).$$ (1)

The first term is

$$H_0=\frac{{\hslash }^2{K}^2}{2M}+\frac{\mathrm{\Delta }(X)}{2}{\sigma }_3,$$ (2)

where Δ(X) = Δ′X describes the detuning gradient along e_x, and σ₃ = | ↑ 〉〈 ↑ | − | ↓ 〉〈 ↓ | is a Pauli spin operator. In the RWA only near-resonant terms are retained, giving the Raman coupling described by

$$V(t)=V_0{\displaystyle \sum _n[{\text{e}}^{\text{i}\left(K_{0}Y-2n\omega t\right)}+{\text{e}}^{\text{i}\left(-K_{0}Y-(2n+1)\omega t\right)}]}|\downarrow \rangle \langle \uparrow |+\text{H}.\text{c}.$$ (3)

The first term describes coupling from the sidebands with even frequencies 2nω, whereas the second term describes coupling from the sidebands with odd frequencies (2n + 1)ω. The recoil kick is aligned along ±e_y with opposite sign for the even and odd frequency components. In writing equation (3) we assumed that the coupling amplitude V₀ and the associated recoil wavenumber K₀ are the same for all frequency components. The coupling Hamiltonian V(t) and therefore the full Hamiltonian H(t) are time-periodic withperiod 2π/ω, and we accordingly apply Floquet techniques.

3. Theoretical analysis

The outline of this section is as follows.(1) We begin the analysis of the Hamiltonian given by equation (1) by moving to dimensionless units; (2) subsequently derive an approximate effective Hamiltonian from the single period time evolution operator; (3) provide an intuitive description in terms of adiabatic potentials; and (4) finally solve the band structure, evaluate its topology and discuss possibilities of the experimental implementation.

3.1. Dimensionless units

For the remainder of the manuscript we will use dimensionless units. All energies will be expressed in units of ћω, derived from the Floquet frequency ω; time will be expressed in units of inverse driving frequency ω⁻¹, denoted by τ = ωt; spatial coordinates will be expressed in units of inverse recoil momentum $K_0^{-1}$, denoted by lowercase letters (x, y) = K₀(X, Y). In these units, the Hamiltonian (1) takes the form

$$h(\tau )=\frac{H(\tau /\omega )}{\hslash \omega }=E_{\text{r}}{\mathit{k}}^2+\frac{1}{2}\mathbf{\Omega }(\tau )\cdot \sigma ,$$ (4)

where $E_{\text{r}}={\hslash }^2{K}_0^2/(2M\hslash \omega )$ is the dimensionless recoil energy associated with the recoil wavenumber K0; k = K/K₀ is the dimensionless wave number. The dimensionless coupling

$$\mathbf{\Omega }(x,y,\tau )=(2\mathrm{Re}u(y,\tau ),2\mathrm{Im}u(y,\tau ),\beta x)$$ (5)

includes a combination of position-dependent detuning and Raman coupling. Here β = Δ′/(ℏωk₀) describes the linearly varying detuning in dimensionless units; the function $u(y,\tau )=v_0{\sum }_n\{\exp [\text{i}(y-2n\tau )]+\exp [\text{i}(-y-(2n+1)\tau )]\}$ is a dimensionless version of the sum in equation (3)with v0 = V₀/(ћw).

In the time domain the coupling given by equation (5) is

$$\frac{1}{2}\mathbf{\Omega }(\tau )\cdot \sigma =\frac{1}{2}\beta x{\sigma }_3+{\displaystyle \sum _l{v}_l}(y)\delta (\tau -\pi l),$$ (6)

with

$$v_l(y)=\pi v_0\left[{\text{e}}^{\text{i}y}+{(-1)}^l{\text{e}}^{-\text{i}y}\right]|\downarrow \rangle \langle \uparrow |+\text{H}.\text{c}.$$ (7)

In this way we have separated the spatial and temporal dependencies in the coupling(6).

3.2. Effective Hamiltonian

We continue our analysis by deriving an approximate Hamiltonian that describes the complete time evolution over a single period from τ = 0 − ϵ to τ = 2𝜋 - ϵ with ∈ ⟶ 0. This evolution includes a kick v₀ at the beginning of the period τ₊ = 0 and a second kick v1 in themiddle of the period τ₋ = 𝜋; between the kicks the evolution includes the kinetic and gradient energies. In the full time period, the complete evolution operator is a product of four terms:

$$U(2\pi ,0)\equiv \underset{ϵ\to 0}{\lim }U(2\pi -ϵ,0-ϵ)=U_0{U}_{\text{kick}}^{(1)}{U}_0{U}_{\text{kick}}^{(0)}.$$ (8)

Here

$$U_0=\exp \left\{-\text{i}\pi \left[E_{\text{r}}{\mathit{k}}^2+\frac{1}{2}{\sigma }_3\beta x\right]\right\}$$ (9)

is the evolution operator over a half period, generated by kinetic energy and gradient. The operator

$$U_{\text{kick}}^{(l)}=\exp \left[-\text{i}{v}_l(y)\right]$$ (10)

describes a kickat τ = l𝜋.

We obtain an effective Hamiltonian by assuming that the Floquet frequency ω greatly exceeds the recoil frequency, 1 ≫ E_r, allowing us to ignore the commutators between the kinetic energy and functions of coordinates in equation (8).We then rearrange terms in the full time evolution operator (8) and obtain (see appendix)

$$U_{\text{eff}}=\exp [\{-\text{i}2\pi ([E_{\text{r}}{\mathit{k}}^2+v_{\text{eff}})]]\},$$ (11)

where v_eff is an effective coupling defined by

$$\exp \left(-\text{i}2\pi v_{\text{eff}}\right)={\text{e}}^{-\text{i}\pi {\sigma }_3\beta x/2}{U}_{\text{kick}}^{(1)}{\text{e}}^{-\text{i}\pi {\sigma }_3\beta x/2}{U}_{\text{kick}}^{(0)}.$$ (12)

The function v_l(y) entering the kick operators $U_{\text{kick}}^{(l)}$ is spatially periodic along the y direction with a period 2π. This period can be halved to π by virtue of a gauge transformationU = exp(−iy𝜎3/2). Subsequently,when exploring energy bands and their topological properties, this prevents problems arising from using a twice larger elementary cell. Following this transformation the evolution operator becomes

$$U_{\text{eff}}=\exp \{-\text{i}2\pi [E_{\text{r}}{\left(\mathit{k}+{\sigma }_3{\mathit{e}}_y/2\right)}^2+v_{\text{eff}}]\},$$ (13)

where v_l(y) featured in the kick operators $U_{\text{kick}}^{(l)}$ has now the spatial periodicity π along the y direction, i.e. it should be replace to

$$v_l(y)=\pi v_0[{\text{e}}^{\text{i}2y}+{(-1)}^l]|\downarrow \rangle \langle \uparrow |+\text{H}.\text{c}.$$ (14)

The algebra of Paulimatrices allows us towrite the effective coupling v_eff (r) featured in the evolution equations (12) and (13) as:

$$v_{\text{eff}}(\mathit{r})=\frac{1}{2}{\mathbf{\Omega }}_{\text{eff}}(\mathit{r})\cdot \sigma ,$$ (15)

where Ω_eff = (Ω_eff,1, Ω_eff,2, Ω_eff,3) is a position-dependent effective Zeeman field which takes the analytic form

$$\exp \left(-\text{i}2\pi v_{\text{eff}}\right)=q_0-\text{i}{q}_1{\sigma }_1-\text{i}{q}_2{\sigma }_2-\text{i}{q}_3{\sigma }_3.$$ (16)

Here q₀,q₁,q₂ and q₃ are real functions of the coordinates (x, y), allowing to express the effective Zeeman field as

$${\mathbf{\Omega }}_{\text{eff}}={\pi }^{-1}\frac{\mathit{q}}{\Vert \mathit{q}\Vert }\arccos q_0,$$ (17)

where q is a shorthand of a three dimensional vector (q1, q2, q3). In general the equation (16) gives multiple solutions that correspond for different Floquet bands. Our choice (17) picks only to the two bands that lie in the energy window from −1/2to 1/2 covering a single Floquet period.

Comparing (12) and (16) and multiplying four matrix exponents give explicit expressions

$$q_0=\cos f_1\cos f_2\cos (\pi \beta x),$$ (18)

$$q_1=\sin f_1\cos f_2\cos (y+\pi \beta x)-\cos f_1\sin f_2\sin (y),$$ (19)

$$q_2=\sin f_1\cos f_2\sin (y+\pi \beta x)+\cos f_1\sin f_2\cos (y),$$ (20)

$$q_3=\cos f_1\cos f_2\sin (\pi \beta x)-\sin f_1\sin f_2$$ (21)

with

$$f_1(y)=2\pi v_0\cos (y),$$ (22)

$$f_2(y)=2\pi v_0\sin (y).$$ (23)

These explicit expressions show that the resulting effective Zeeman field (17) and the associated effective coupling (15) are periodic along both e_x and e_y,with spatial periods a_x = 2/β and a_y = 𝜋 respectively. Therefore, although the originalHamiltonian containing the spin-dependent potential slope ∝xσ₃ is not periodic along the x direction, the effective Floquet Hamiltonian is. The spatial dependence of the Zeeman field components Ω_eff,1, Ω_eff,2 and Ω_eff,3 is presented in the figure 2 for β = 0.6 giving an approximately square unit cell. In figure 2 we select v₀ = 0.25 where the absolute value of the Zeeman field Ω_eff is almost uniform, as is apparent from the nearly flat adiabatic bands shown in figure 3 below.

Figure 2.

Coupling components (a) Ω_eff,1(r), (b) Ω_eff,2(r) and (c) Ω_eff,3(r) for v₀ = 0.25 and β = 0.6 calculated using equations (17)–(23). The corresponding eigenvalues of the coupling v_±(r) = ±Ω_eff/2 are presented by the thick red solid lines in the figure 3.

Figure 3.

Adiabatic Floquet potentials given by equation (29) for β = 0.6 ms. (a) Thin black dotted lines denote the spin-dependent gradient slopes without including the Raman coupling (v0 = 0); (b) thin blue solid lines denote effective adiabatic potentials for weak Raman coupling (v₀= 0.05) (c) red solid lines denote nearly flat adiabatic potentials that are achieved for stronger Raman coupling (v₀ = 0.25). All the curves are projected into x plane for various y values. A weak y dependence of the adiabatic potentials is seen to appear in the strong coupling case (c) making the superimposed red lines thicker.

3.3. Adiabatic evolution and magnetic flux

Before moving further to an explicit numerical analysis of the band structure, we develop an intuitive understanding by performing an adiabatic analysis of motion governed by effective Hamiltonian

$$h_{\text{eff}}(\mathit{r})=E_{\text{r}}{\left(\mathit{k}+{\sigma }_3{\mathit{e}}_y/2\right)}^2+\frac{1}{2}{\mathbf{\Omega }}_{\text{eff}}\cdot \sigma$$ (24)

featured in the evolution operator U_eff, equation (13). The coupling field Ω_eff (r) is parametrized by the spherical angles 𝜃(r) and ø(r) defined by

$$\cos \theta =\frac{{\mathrm{\Omega }}_{\text{eff},3}}{{\mathrm{\Omega }}_{\text{eff}}},$$ (25)

$$\tan \varphi =\frac{{\mathrm{\Omega }}_{\text{eff},2}}{{\mathrm{\Omega }}_{\text{eff},1}}.$$ (26)

This gives the effective coupling [12]

$$\frac{1}{2}{\mathbf{\Omega }}_{\text{eff}}\cdot \sigma =\frac{1}{2}{\mathrm{\Omega }}_{\text{eff}}\left[\begin{array}{cc}\cos \theta & {\text{e}}^{-\text{i}\varphi }\sin \theta \\ {\text{e}}^{\text{i}\varphi }\sin \theta & -\cos \theta \end{array}\right],$$ (27)

characterized by the position-dependent eigenstates

$$|+\rangle =\left(\begin{array}{c}\cos (\theta /2)\\ e^{\mathrm{i}\varphi }\sin (\theta /2)\end{array}\right),|-\rangle =\left(\begin{array}{c}-{\text{e}}^{-\text{i}\varphi }\sin (\theta /2)\\ \cos (\theta /2)\end{array}\right).$$ (28)

The corresponding eigenvalues

$$v_{\pm }(\mathit{r})=\pm \frac{1}{2}{\mathrm{\Omega }}_{\text{eff}},$$ (29)

are shown in figure 3 for various value of the Raman coupling v0. As one can see in figure 3, for v₀ = 0.25 the resulting bands v±(r) (adiabatic potentials) are flat and have a considerable gap ≈𝜔/2, a regime suitable for a description in terms of an adiabatic motion in selected bands [27].

As in [28], we consider the adiabatic motion of the atom in one of these flat adiabatic bands with the projection Schrödinger equation that includes a geometric vector potential

$${\mathit{A}}_{\pm }(\mathit{r})=\pm \frac{1}{2}(\cos \theta -1)\nabla \varphi .$$ (30)

This provides a syntheticmagnetic flux density B±(r) = ∇ × A±(r). The geometric vector potential A±(r) may contain Aharonov–Bohm type singularities, that give rise to a synthetic magnetic flux over an elementary cell

$${\alpha }_{\pm }=-{\displaystyle \sum {\displaystyle {\oint }_{\text{singul}}\text{d}}}\mathit{r}\cdot {\mathit{A}}_{\pm }(\mathit{r}).$$ (31)

The singularities appear at pointswhere 𝜃 = 𝜋, where the angle ø and its gradient ∇ø are undefined and cos 𝜃 = −1. The term cos 𝜃 - 1 in (30) is nonzero and does not remove the undefined phase ∇ø. Our unit cell contains two such singularities located at r = (ax, 3ay)/4 and r = (3ax, ay)/4, containing the same flux, so that they do not compensate each other, giving the synthetic magnetic flux ±2𝜋 in each unit cell. Note that usually the optical lattices are sufficiently deep, and the ±2𝜋 flux per elementary is topologically trivial. In that case the tight binding model can be applied, with the tunneling taking place only between the nearest-neighboring sites of the square plaquette. The ±2𝜋 flux over the square plaquette can then be eliminated by a gauge transformation. Yet if the lattice is shallow enough, the tight binding model is not applicable and the above arguments do not work. In the present situation, the most interesting topological lattice appears for a flat adiabatic trapping potential shown by a solid red curve in figure 3. In such a situation there are no well defined lattice sites, and the ±2𝜋 flux per elementary cell results in topologically non-trivial bands explored in the next subsection.

For a weak coupling (such as v = 0.05) the geometric flux density B(r) ≡ B±(r) is concentrated around the intersection points of the gradient slopes shown in in figure 3 and has a very weak y dependence. With increasing the coupling v, the flux extends beyond the intersection areas and acquires a y dependence. Figure 4 shows the geometric flux density B(r) ≡ B₊(r) for the strong coupling (v₀ = 0.25) corresponding to the most flat adiabatic bands. In this regime the flux develops stripes in the x direction and has a strong y dependence. For the whole range of coupling strengths 0 ≤ v₀ ≤ 1/2 the total synthetic magnetic flux per unit cell is 2π and is independent of the Floquet frequency ω and the gradient β.

Figure 4.

Geometric flux density B±(r) = ∇ × A±(r) computed for v₀ = 0.25 and β = 0.6 using equation (30). The overall spatial structure of this flux density does not depend on the gradient β; rather it scaleswith the corresponding lattice constant a_x = 2/β.

Now let us discuss the effect of an extra spin-independent trapping potential. The present scheme requires a large spin-dependent energy gradient which would have a huge influence on the relative trapping for the two spin states without the Raman coupling or for a weak Raman coupling. In that case one would expect that the stable positions for any trapped sample of the two spin states would live at entirely distinct locations, possibly with no overlap. Yet we are interested mostly in a sufficiently strong Raman coupling where the two spin states get mixed, and the atomic motion takes place in almost flat adiabatic potentials shown in red in figure 3. Therefore the atoms are no longer affected by the steep spin-dependent potential slopes, and the spinindependent trapping potential would not cause separation of different spin states. Instead, the extra spinindependent parabolic trapping potential would simply make the flat adiabatic trapping potentials parabolic. Of course, one needs to be all the time in the regime where the Raman coupling is strong compared to the characteristic energy of the spin-dependent potential slope. That is why we propose to introduce the spindependent potential gradient only at the final stage of the adiabatic protocol discussed in section 3.5.

3.4. Band structure and Chern numbers

Weanalyze the topological properties of this Floquet flux lattice by explicitly numerically computing the band structure and associated Chern number using the effective Hamiltonian (24) without making the adiabatic approximation introduced in section 3.3. Again the gradient of the original magnetic field is such that we approximately get a square lattice, β = 0.6. Furthermore, we choose the Floquet frequency to be ten times larger than the recoil energy, E_r = 0.1. Note that one can alter the length of the plaquette along the x direction (and thus the flux density) by changing β representing the potential gradient along the x axis.

First, let us consider the case where v₀ = 0.25 corresponding to the most flat adiabatic potential. In this situation the Chern numbers of the first five bands appear to be equal to the unity, as one can see in the left part of figure 5. Thus the Hall current should monotonically increase when filling these bands. This resembles the QHE involving the Landau levels. Second, we check what happens when we leave the regime v₀ = 0.25 where the adiabatic potential is flat, and consider lower and higher values of the coupling strength v₀. Near v₀ = 0.175 we find a topological phase transition where the lowest two energy bands touch and their Chern numbers change to c₁ = 0 and c₂ = 2, while the Chern numbers of the higher bands remain unchanged, illustrated in figure 6. In a vicinity of v₀ = 0.3 there is another phase transition, where the second and third bands touch, leading to a new distribution of Chern numbers: c₁ = 1, c₂ = – 1, c₃ = 3,c₄ = 1. Interestingly the Chern numbers of the second and the third bands jump by two units during such a transition.

Figure 5.

Left: band structure calculated using the effective Hamiltonian (24) for v₀ = 0.25, β = 0.6 and E_r = 0.1. Right: the band gap Δ₁₂ between the first and second bands for E_r = 0.1 and various values of v₀ and β. The letters A, B andCindicate the regions where the Chern numbers of the first three bands correspond to the right, middle and left parts of the figure 6.

Figure 6.

Dependence of Chern number for the bands calculated using the effective Hamiltonian (24) on the coupling strength v₀ for β = 0.6 and E_r = 0.1. Here we present the Chern numbers c₁, c₂ and c₃ of the three lowest bands.

Finally, we explore the robustness of the topological bands. The right part of figure 5 shows the dependence of the band gapΔ₁₂ between the first and second bands on the coupling strength v₀ and the potential gradient β. One can see that the band gap is maximum for v₀ = 0.25 when the adiabatic potential is the most flat. The gap increases by increasing the gradient β, simultaneously extending the range of the v0 values where the band gap is nonzero. Therefore to observe the topological bands, one needs to take a proper value of the Raman coupling v₀ ≈ 0.25 and a sufficiently large gradient β, such as β = 0.6.

Wenow make some numerical estimates to confirm that this scheme is reasonable. Weconsider an ensemble of ⁸⁷Rb atoms, with ∣↑⟩ = ∣ f = 2, m_F = 2⟩ and ∣↓⟩ = ∣ f = 1, m_F = 1⟩; the relative magnetic moment of these hyperfine states is ≈2.1 MHz G⁻¹, where 1G = 10⁻⁴T. For a reasonable magnetic field gradient of 300G/cm, this leads to the Δ′/ћ≈ 2𝜋 × 600 MHz cm⁻¹ = 2π × 60 kHz μm⁻¹ detuning gradient. For ⁸⁷Rb with λ = 790 nmlaser fields the recoil frequency is ω_r/2π = 3.5 kHz. Along with the driving frequency ω = 10ω_r, this provides the dimensionless energy gradient β = Δ′/(ћωk₀) ≈ 1.3, allowing easy access to the topological bands displayed in figure 5.

3.5. Loading into dressed states

Adiabatic loading into this lattice can be achieved by extending the techniques already applied to loading into Raman dressed states [29]. The loading technique begins with a Bose–Einstein condensate (BEC) in the lower energy ∣ ↓ ⟩ state in a uniform magnetic field B0. Subsequently one slowly ramps on a single off resonance RF coupling field and the adiabatically ramp the RF field to resonance (at frequency δω). This RF dressed state can be transformed into a resonant Raman dressed by ramping on the Raman lasers (with only the ω₀ + δω frequency on the k^– laser beam) while ramping off the RF field. The loading procedure then continues by slowly ramping on the remaining frequency components on the k^– beam, and finally by ramping on the magnetic field gradient (essentially according in the lattice sites from infinity). This procedure leaves the BEC in the q = 0 crystal momentum state in a single Floquet band.

4. Conclusions

Initial proposals [30–32] and experiments [26] with geometric gauge potentials were limited by the small spatial regions over which these existed. Here we described a proposal that overcomes these limitations using laser coupling reminiscent of a frequency comb: temporally pulsed Raman coupling. Typically, techniques relying on temporal modulation of Hamiltonian parameters to engineer lattice parameters suffer from micro-motion driven heating. Because our method is applied to atoms initially in free space, with no optical lattice present, there are no a priori resonant conditions that would otherwise constrains the modulation frequency to avoid transitions between original Bloch bands [25].

Still, no technique is without its limitations, and this proposal does not resolve the second standing problem of Raman coupling techniques: spontaneous emission process from the Raman lasers. Our new scheme extends the spatial zone where gauge fields are present by adding sidebands to Raman lasers, ultimately leading to a $\propto \sqrt{N}$ increase in the required laser power (where N is the number of frequency tones), and therefore the spontaneous emission rate. As a practical consequence it is likely that this technique would not be able reach the low entropies required for many-body topological matter in alkali systems [13], but straightforward implementations with single-lasers on alkaline-earth clock transitions [33, 34] are expected to be practical.

Appendix. Stroboscopic evolution operator

The stroboscopic evolution operator (8) reads explicitly

$$U(2\pi ,0)=U_0{U}_{\text{kick}}^{(1)}{U}_0{U}_{\text{kick}}^{(0)}={\text{e}}^{-\text{i}\pi \left[E_{\text{r}}{\mathit{k}}^2+\frac{1}{2}{\sigma }_3\beta x\right]}{\text{e}}^{-\text{i}{v}_1(y)}{\text{e}}^{-\text{i}\pi [E_{\text{r}}{\mathit{k}}^2+\frac{1}{2}{\sigma }_3\beta x]}{\text{e}}^{-\text{i}{v}_0(y)}.$$ (32)

In the main we have approximated the evolution operator by equation (11). To estimate the validity of the latter equation, let us make use of the Baker–Campbell–Hausdorff formula

$$e^X{e}^Y=e^Z\text{with}Z=X+Y+\frac{1}{2}[X,Y]+\dots$$ (33)

and consider this expansion up to the leading term $\frac{1}{2}[X,Y]$, essentially the second term in the Magnus expansion.

Neglecting the commutation between E_rk² and σ3βx, one can write

$$U_0={\text{e}}^{-\text{i}\pi \left[E_{\text{r}}{\mathit{k}}^2+\frac{1}{2}{\sigma }_3\beta x\right]}\approx {\text{e}}^{-\text{i}\pi E_{\text{r}}{\mathit{k}}^2}{\text{e}}^{-\text{i}\frac{\pi }{2}{\sigma }_3\beta x}.$$ (34)

The error in doing so is approximately $-\text{i}\frac{\pi }{4}{E}_{\text{r}}\beta {\sigma }_3[x,{\mathit{k}}^2]=\frac{\pi }{2}{E}_{\text{r}}\beta k_x{\sigma }_3.$ Since E_rβ ≪ 1, this provides a small momentum shift along the x direction. Furthermore, we shall neglect the commutation between E_rk² and v_l(y). The error in doing so is approximately $-\mathrm{i}\frac{\pi }{2}{E}_{\text{r}}{\sigma }_{x,y}[y,{\mathit{k}}^2]=\pi E_{\text{r}}{k}_y{\sigma }_3.$ Since the Floquet frequency ω greatly exceeds the recoil frequency E_r ≪ 1 and β < 1, this also provides a small momentum shift along the y direction. With these assumptions, one has

$$U(2\pi ,0)=U_0{U}_{\text{kick}}^{(1)}{U}_0{U}_{\text{kick}}^{(0)}\approx {\text{e}}^{-\text{i}2\pi E_{\text{r}}{\mathit{k}}^2}{\text{e}}^{-\text{i}2\pi v_{\text{eff}}},$$

where

$${\text{e}}^{-\text{i}2\pi v_{\text{eff}}}={\text{e}}^{-\text{i}\pi {\sigma }_3\beta x/2}{\text{e}}^{-\text{i}{v}_1(y)}{\text{e}}^{-\text{i}\pi {\sigma }_3\beta x/2}{\text{e}}^{-\text{i}{v}_0(y)}.$$

Finally under the above assumptions one canmerge the exponents inU (2p, 0), giving equation (11).