Topological lattice using multi-frequency radiation

We describe a novel technique for creating an artificial magnetic field for ultra-cold atoms using a periodically pulsed pair of counter propagating Raman lasers that drive transitions between 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.


I. INTRODUCTION
Ultracold atoms find wide applications in realising condensed matter phenomena [1][2][3][4]. Since ultracold atom systems are ensembles of electrically neutral atoms, various methods have been used to simulate Lotentz-type forces, with an eye for realizing physics such as the quantum Hall effect (QHE). Lorentz forces are present in spatially rotating systems [5][6][7][8][9][10][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][15][16][17], combining static optical lattices along with laser-assisted spin or pseudo spin coupling [12,13,[18][19][20][21][22][23][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 state-dependent gradient potential that together create a two-dimensional (2D) periodic potential with an intrinsic artificial magnetic field. With no pre-existing lattice potential, there are no a priori 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 time-periodic, 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. 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) = B 0 + B X. This induces a position-dependent splitting g F µ B B between the spin up and down states; g F is the Landé 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 Fig. 1(a)) is at frequency ω + = ω 0 , while the second (down-going in Fig. 1(a)) contains frequency components ω − n = ω 0 + (−1) n (δω + nω); the difference frequency between these beams contains frequency combs centered at ±δω with comb teeth spaced by 2ω, as shown in Fig. 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 B 0 such that g F µ B B 0 = δω 0 , making the frequency difference ω − n=0 − ω + resonant at X = 0, where B = B 0 . Intuitively, each additional frequency component ω − n adds a resonance condition at the regularly spaced points X n = n ω/g F µ B B , a. Schematic b. X = 0 level diagram c. Coupling geometry however, transitions using even-n side bands give a recoil kick opposite from those using odd-n side bands (see Fig. 1(c)). Each of these coupling-locations locally realizes synthetic magnetic field experiment performed at NIST [26], arrayed in a manner to give a rectified artificial magnetic field with a non-zero average that we will show is a novel flux lattice. We formally describe our system by first making the rotating wave approximation (RWA) with respect to the large offset frequency ω 0 . This situation is modeled in terms of a spin-1/2 atom of mass M and wave-vector K with a Hamiltonian
In the RWA only near-resonant terms are retained, giving the Raman coupling described by 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 Eq.(3) we assumed that the coupling amplitude V 0 and the associated recoil wave number K 0 are the same for all frequency components. The coupling Hamiltonian V (t) and therefore the full Hamiltonian H(t) are time-periodic with period 2π/ω, and we accordingly apply Floquet techniques.

III. THEORETICAL ANALYSIS
The outline of this Section is as follows. (1) We begin the analysis of the Hamiltonian given by Eq. (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.

A. Dimensionless units
For the remainder of the manuscript we will use dimensionless units. All energies will be expressed in units of ω, 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 −1 0 , denoted by lowercase letters (x, y) = K 0 (X, Y ). In these units, the Hamiltonian (1) takes the form where E r = 2 K 2 0 /(2M ω) is the dimensionless recoil energy associated with the recoil wavenumber K 0 ; k = K/K 0 is the dimensionless wavenumber. The dimensionless coupling includes a combination of position-dependent detuning and Raman coupling. Here β = ∆ /( ωk 0 ) describes the linearly varying detuning in dimensionless units; the function u(y, The vector Ω(x, y, τ ) is spatially periodic along the y direction with a period 2π. This period can be halved to π by virtue of a gauge transformation U = exp(−iyσ 3 /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 dimensionless Hamiltonian becomesh In the time domain the coupling (5) is with v l (y) = πv 0 e i2y + (−1) l | ↓ ↑ | + H. c. .
In this way we separated the spatial and temporal dependencies in the coupling (6).

B. Effective Hamiltonian
We continue our analysis by deriving an approximate Hamiltonian that describes the complete time evolution over a single period from τ = 0 − to τ = 2π − with → 0. This evolution includes a kick v 0 at the beginning of the period τ + = 0 and a second kick v 1 in the middle 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: Here is the evolution operator over the half period, generated by kinetic energy and gradient. The operator describes a kick at 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 eq.(7). We then rearrange terms in the full time evolution operator (7) and obtain where v eff is an effective coupling defined by The algebra of Pauli matrices allows us to write the effective coupling in a form where Ω eff = (Ω eff,1 , Ω eff,2 , Ω eff,3 ) is a position-dependent effective Zeeman field which takes the analytic form Here q 0 , q 1 , q 2 and q 3 are real functions of the coordinates (x, y), allowing to express the effective Zeeman field as where q is a shorthand of a three dimensional vector (q 1 , q 2, q 3 ). In general the equation (13) gives multiple solutions that correspond for different Floquet bands. Our choice (14) picks only to the two bands that lie in the energy window from −1/2 to 1/2 covering a single Floquet period.
These explicit expressions show that the resulting effective Zeeman field (14) and the associated effective coupling (12) are periodic along both e x and e y , with spatial periods a x = 2/β and a y = π respectively. Therefore, although the original Hamiltonian containing the spin-dependent potential slope ∝ xσ 3 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 fig. 2 for β = 0.6 giving an approximately square unit cell. In fig. 2 we select v 0 = 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 fig. 3 below.

C. 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 featured in the evolution operator U eff , Eq. (10). The coupling field Ω eff (r) is parametrized by the spherical angles θ(r) and φ(r) defined by tan φ = Ω eff,2 Ω eff,1 . This gives the effective coupling [12] characterized by the position-dependent eigenstates The corresponding eigenvalues are shown in Fig. 3 for various value of the Raman coupling v 0 . As one can see in Fig. 3, for v 0 = 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 Ref. [28], we consider the adiabatic motion of the atom in one of these flat adiabatic bands with the projection Shrodinger equation that includes a geometric vector potential This provides a synthetic magnetic 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 The singularities appear at points where θ = π, where the angle φ and its gradient ∇φ are undefined and cos θ = −1.
The term cos θ − 1 in (27) is non zero and does not remove the undefined phase ∇φ. Our unit cell contains two such singularities located at r = (a x , 3a y )/4 and r = (3a x , a y )/4, containing the same flux, so that they do not compensate each other, giving the synthetic magnetic flux ±2π in each unit cell. 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 Fig. 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. Fig. 4 shows the geometric flux density B(r) ≡ B + (r) for the strong coupling (v 0 = 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 ≤ v 0 ≤ 1/2 the total synthetic magnetic flux per unit cell is 2π and is independent of the Floquet frequency ω and the gradient β.

D. Band structure and Chern numbers
We analyze the topological properties of this Floquet flux lattice by explicitly numerically computing the band structure and associated Chern number using the effective Hamiltonian (21) without making the adiabatic approximation introduced in Sec. III C. 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. Finally, we explore the robustness of the topological bands. The right part of Fig. 5 shows the dependence of the band gap ∆ 12 between the first and second bands on the coupling strength v 0 and the potential gradient β. One can see that the band gap is maximum for v 0 = 0.25 when the adiabatic potential is the most flat. The gap increases by increasing the gradient β, simultaneously extending the range of the v 0 values where the band gap is nonzero. Therefore to observe the topological bands, one needs to take a proper value of the Raman coupling v 0 ≈ 0.25 and a sufficiently large gradient β, such as β = 0.6.
We now make some numerical estimates to confirm that this scheme is reasonable. We consider an ensemble of 87 Rb 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, where 1 G = 10 −4 T. For a reasonable magnetic field gradient of 300 G/cm, this leads to the ∆ / ≈ 2π × 600 MHz/cm = 2π × 60 kHz/µm detuning gradient. For 87 Rb with λ = 790 nm laser 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 = ∆ /( ωk 0 ) ≈ 1.3, allowing easy access to the topological bands displayed in Fig. 5.

E. Loading into dressed states
Adiabatic loading into this lattice can be achieved by extending the techniques already applied to loading in to Raman dressed states [29]. The loading technique begins with a BEC in the lower energy ↓ state in a uniform magnetic field B 0 . 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 ω 0 + 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.

IV. CONCLUSIONS
Initial proposals [30][31][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 side-bands to Raman lasers, ultimately leading to a ∝ √ 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.