Light-induced gauge fields for ultracold atoms

Within the ground state manifold (adiabatically eliminating the excited states), the effect of the RWA electric dipole term $\hat{H}_{{\rm dip^{\prime}}}$ is described to second-order by the effective atomic Hamiltonian

$$\begin{equation} \hat{H}_{{\rm eff}} =-\hat{P}_{g}\hat{H}_{{\rm dip^{\prime}}}\hat{H}_{{\rm at}^{\prime}}^{-1}\hat{H}_{{\rm dip^{\prime}}}\hat{P}_{g}, \end{equation} \tag{ 37 }$$

which can be represented as

$$\begin{equation} \hat{H}_{{\rm eff}} =-\case{1}{4}\tilde{E}_{i}^{*}\hat{D}_{i,j}\tilde{E}_{j}. \end{equation} \tag{ 38 }$$

Here we have introduced the rank-2 Cartesian tensor operator

$$\begin{equation} \hat{D}_{i,j} =\hat{P}_{g}\hat{d}_{i}\hat{P}_{e}\hat{H}_{{\rm at}^{\prime}}^{-1}\hat{P}_{e}\hat{d}_{j}\hat{P}_{g}\equiv\hat{P}_{g}\hat{d}_{i}\hat{H}_{{\rm at}^{\prime}}^{-1}\hat{d}_{j}\hat{P}_{g} \end{equation} \tag{ 39 }$$

acting on the ground state manifold. In the second relation of equation (39) we have omitted the excited state projectors $\hat{P}_{e}$, because the atomic ground states do not have a permanent electric dipole moment: $\hat{P}_{g}\hat{d}_{j}\hat{P}_{g}=0$.

Operators of this form can be expressed as the sum

$$\begin{equation} \hat{D}_{i,j} =\hat{D}_{i,j}^{(0)}+\hat{D}_{i,j}^{(1)}+\hat{D}_{i,j}^{(2)} \end{equation} \tag{ 40 }$$

of irreducible tensor operators of rank-0 (transforms as a scalar under rotation)

$$\begin{equation} \hat{D}_{i,j}^{(0)} =\case{1}{3}\delta_{i,j}{\rm Tr}\hat{D}\equiv\case{1}{3}\hat{D}_{l,l}\delta_{i,j}, \end{equation} \tag{ 41 }$$

rank-1 (transforms as a vector under rotation)

$$\begin{equation} \hat{D}_{i,j}^{(1)} =\case{1}{2}\left(\hat{D}_{i,j}-\hat{D}_{j,i}\right) \equiv\case{1}{2}\epsilon_{i,j,k}\epsilon_{i^{\prime}, j^{\prime},k}\hat{D}_{i^{\prime},j^{\prime}}, \end{equation} \tag{ 42 }$$

and rank-2

$$\begin{equation} \hat{D}_{i,j}^{(2)} =\case{1}{2}\left(\hat{D}_{i,j}+\hat{D}_{j,i}\right) -\case{1}{3}\hat{D}_{i,j}^{(0)}, \end{equation} \tag{ 43 }$$

where the summation over the repeated Cartesian indices is again implied. This separation allows one to classify the light–matter interaction in a powerful way.

Let us begin by considering the simple case when A_FS = 0, implying that the excited states are degenerate, and allowing us to completely neglect electronic and nuclear spin. In this case, we replace the atomic Hamiltonian $\hat{H}_{{\rm at}^{\prime}}$ with $\Delta_{e}\hat{P}_{e}$, and the tensor $\hat{D}_{i,j}$ in equation (39) takes the particularly simple form

$$\begin{equation} \hat{D}_{i,j} =\Delta_{e}^{-1}\hat{P}_{g}\hat{d}_{i}\hat{d}_{j}\hat{P}_{g}. \end{equation} \tag{ 44 }$$

In the present case, let us study how equation (44) transforms under rotations as effected by the unitary operator $\hat{R}({\boldsymbol \theta})=\exp(-\rmi\hat{{\bit L}}\cdot{\boldsymbol \theta}/\hbar)$. Because the ground S state has l = 0, it is clear that $\hat{L}_{x,y,z}\hat{P}_{g}=0$, therefore

$$\begin{equation} \hat{R}({\boldsymbol \theta})\hat{D}_{i,j}\hat{R}^{\dagger}({\boldsymbol \theta}) =\hat{D}_{i,j}. \end{equation} \tag{ 45 }$$

Thus $\hat{D}_{i,j}$ transforms like a scalar under rotation and must be proportional to the unit tensor. Consequently only the zero-rank contribution $\hat{D}_{i,j}^{(0)}$ is non-zero

$$\begin{equation} \hat{D}_{i,j}^{(0)} =\hat{P}_{g}\frac{\hat{d}_{l}\hat{d}_{l}}{3\Delta_{e}} \delta_{ij}\hat{P}_{g}=-4u_{s}\delta_{ij}\hat{P}_{g}, \end{equation} \tag{ 46 }$$

where u_s is proportional to the atoms' ac polarizability (without including the fine-structure corrections)

$$\begin{equation} u_{s} =-\frac{\left|\mathinner{\langle{{l=0}}|}|{\bit d}|{\mathinner{|{l^{\prime}=1}\rangle}}\right|^{2}} {12\left(E_{e}-\hbar\omega\right)}. \end{equation} \tag{ 47 }$$

Here $|\mathinner{\langle{{l=0}}|}|{\bit d}| {\mathinner{|{l^{\prime}=1}\rangle}}|^{2} \equiv\sum_{m_{L}^{\prime}=0,\pm1}| \mathinner{\langle{{l=0,m_{L}=0}}|}{\bit d}\times$ ${\mathinner{|{l^{\prime}=1,m_{L}^{\prime}}\rangle}} |^{2}\approx4e^2a_{0}^{2}$ is the commonly used reduced matrix element and a₀ is the Bohr radius (the reduced matrix element is a single number, even though the conventional notation makes it look like it should have three vector components).

Equations (38) and (46) yield the scalar light shift

$$\begin{equation} \hat{H}_{{\rm eff}}^{(0)} =u_{s}|\tilde{{\bit E}}|^{2}\hat{P}_{g}, \end{equation} \tag{ 48 }$$

Alternatively $\hat{H}_{{\rm eff}}^{(0)}$ can be expressed as

$$\begin{equation} \hat{H}_{{\rm eff}}^{(0)} =-\frac{3\pi\hbar^{3}c^{2}I}{2E_{e}^{3}}\frac{\hbar\Gamma_{1\rightarrow0}}{\Delta_{e}}, \end{equation} \tag{ 49 }$$

where we defined the intensity $I=\epsilon_{0}c|\tilde{{\bit E}}|^{2}/2$, used the speed of light c, the transition's natural linewidth Γ_1 → 0, the electric constant ₀, and the expression

$$\begin{equation} \hbar\Gamma_{l^{\prime}\rightarrow l} =\frac{E_{e}^{3}}{3\pi\epsilon_{0}\hbar^{3}c^{3}}\frac{2l+1}{2l^{\prime}+1}\left| \mathinner{\langle{{l}}|}|{\bit d}|{\mathinner{|{l^{\prime}}\rangle}}\right|^{2} \end{equation} \tag{ 50 }$$

for the linewidth [142]. This result is valid whenever it is safe to ignore the excited state fine structure; a good zero-order approximation when the Δ_e = E_e − ℏω detuning from the optical transition is much larger than Δ_FS. This result reminds us of several important facts: the scalar light shift is a potential that is independent of optical polarization, scales like 1/Δ_e, and for red-detuned beams (Δ_e > 0) is attractive.

Let us now turn to the effects of the fine structure by recalling that the excited eigenstates |j, m_J〉 of the Hamiltonian $\hat{H}_{{\rm at}^{\prime}}$ equation (36) corresponding to l = 1 are characterized by the quantum numbers j and m_J of the combined angular momentum $\hat{{\bit J}}=\hat{{\bit L}}+\hat{{\bit S}}$, with j = 1/2 and j = 3/2. Specifically we have $\hat{H}_{{\rm at}^{\prime}}|j,m_{J}\rangle =\Delta_{j}|j,m_{J}\rangle$, where Δ_1/2 = Δ_e − A_FS and Δ_3/2 = Δ_e + A_FS/2 are the detunings for the corresponding fine structure transitions.

Next we shall determine the tensor $\hat{D}_{i,j}=\hat{P}_{g}\hat{d}_{i}\hat{H}_{{\rm at}^{\prime}}^{-1}\hat{d}_{j}\hat{P}_{g}$ describing the light-induced coupling between the atomic ground states, without explicitly involving the excited fine-structure states |j, m_J〉. For this we use equation (36) for $\hat{H}_{{\rm at}^{\prime}}$ and write down a Dyson-type equation for the inverse atomic Hamiltonian $\hat{H}_{{\rm at}^{\prime}}^{-1}$ projected onto the excited-state manifold,

$$\begin{equation} \hat{H}_{{\rm at}^{\prime}}^{-1} =\frac{1}{\Delta_{e}}\hat{P}_{e}-\frac{\alpha}{\Delta_{e}} \hat{{\bit L}}\cdot\hat{{\boldsymbol{\sigma}}}\hat{H}_{{\rm at}^{\prime}}^{-1}, \end{equation} \tag{ 51 }$$

with α = A_FS/2ℏΔ_e and $\hat{{\bit S}}=\hbar\hat{{\boldsymbol \sigma}}/2$.

Calling on the communtation relations $[\hat{L}_{i},\hat{d}_{j}]=\rmi\hbar\epsilon_{ijk}\hat{d}_{k}$ [82] together with $\hat{{\bit L}}\hat{P}_{g}=0$, we replace $\hat{P}_{g}\hat{d}_{i}\hat{{\bit L}}\cdot\hat{{\boldsymbol \sigma}}$ with the commutator $-\hat{P}_{g}[\hat{{\bit L}}\cdot\hat{{\boldsymbol \sigma}}\,,\hat{d}_{i}]=\rmi\hbar\epsilon_{i\ell m}\hat{P}_{g}\hat{d}_{m}\hat{\sigma}_{l}$. Using the relation $\hat{P}_{g}\hat{d}_{i}\hat{d}_{j}\hat{P}_{g}=|\langle|\!|{\bit d}|\!|\rangle|^{2}\hat{P}_{g}\delta_{ij}/3$, equations (39) and (51) yield

$$\begin{equation} \hat{D}_{ij} =\frac{\left|\langle|\!|{\bit d}|\!|\rangle\right|^{2}}{3\Delta_{e}}\delta_{ij} \hat{P}_{g}-\rmi\alpha\hbar\epsilon_{i\ell m}\hat{\sigma}_{\ell}\hat{D}_{mj}, \end{equation} \tag{ 52 }$$

where the reduced matrix element $|\langle|\!|{\bit d}|\!|\rangle|^{2}\equiv | \mathinner{\langle{{l=0}}|}\!|{\bit d}|\!\mathinner|{l^{\prime}}=$ 1〉|² has already been featured in the previous subsection. As equation (52) contains operators that act only within the electronic ground state, we may now omit the projectors $\hat{P}_{g}$. Furthermore, since the only operators acting in the electronic ground state are the Pauli matrices, $\hat{D}_{ij}$ must take the form

$$\begin{equation} \hat{D}_{ij}=D^{(0)}\delta_{ij}+{\rm i}D^{(1)}\epsilon_{ijk}\sigma_{k}, \end{equation} \tag{ 53 }$$

where we also used the fact that $\hat{D}_{ij}$ can be decomposed into a scalar and vector component (no rank-2 component is possible with Pauli matrices alone).

Replacing the proposed solution (53), permuting one Levi–Civita symbol and exploiting the identity $\hat{\sigma}_{i}\hat{\sigma}_{j}=\delta_{ij}+{\rm i}\epsilon_{ijk}\sigma_{k}$, equation (52) gives

$$\begin{eqnarray*}&&\fl \left(D^{(0)}-2\alpha\hbar D^{(1)}\right)\delta_{ij}+{\rm i}\epsilon_{ijk}\left(D^{(1)}-\alpha\hbar D^{(0)}-\alpha\hbar D^{(1)}\right)\hat{\sigma}_{k}\nonumber\\ &&=\frac{\left|\langle|\!|{\bit d}|\!|\rangle\right|^{2}}{3\Delta_{e}}\delta_{ij}. \end{eqnarray*}$$

The resulting pair of linear equations has solutions

$$\begin{eqnarray*}&&\fl D^{(0)} =\frac{1}{3}\left(\frac{2}{\hbar\alpha+1}- \frac{1}{2\hbar\alpha-1}\right)\frac{\left|\langle|\!|{\bit d}|\!|\rangle\right|^{2}}{3\Delta_{e}} \quad {\rm and} \\ &&D^{(1)} =\frac{\alpha\hbar}{1-\hbar\alpha}D^{(0)}. \end{eqnarray*}$$

In the initial notation with ℏα = A_FS/2Δ_e and Δ_e = E_e − ℏω, we associate the denominators appearing in the scalar coefficient D⁽⁰⁾ with the detuning from the excited-state fine-structure split levels E_D1 = E_e − A_FS and E_D2 = E_e + A_FS/2, giving

$$\begin{equation*} D^{(0)} =\frac{\left|\langle|\!|{\bit d}|\!|\rangle\right|^{2}}{9}\left(\frac{2}{E_{D2}- \hbar\omega}+\frac{1}{E_{D1}-\hbar\omega}\right). \end{equation*}$$

Additionally introducing an average $\bar{E}=(2E_{{\rm D1}}+E_{{\rm D2}})/3$ and the fine-structure splitting Δ_FS = 3A_FS/2, the complete tensor operator takes the form

$$\begin{equation} \hat{D}_{ij} =D^{(0)}\left[\delta_{ij}+\rmi\epsilon_{ijk}\frac{2}{3\hbar}\frac{\Delta_{{\rm FS}}}{\bar{E}-\hbar\omega}\hat{J}_{k}\right], \end{equation} \tag{ 54 }$$

where we used $\hat{J}_{i}=\hat{S}_{i}$ as is suitable in the l = 0 electronic ground state. This leads to the light shift

$$\begin{eqnarray} &&\fl \hat{H}_{{\rm eff}} =-\frac{1}{4}\tilde{E}_{i}^{*}\hat{D}_{i,j}\tilde{E}_{j}\nonumber\\ &&=\left[u_{{\rm s}}\left(\tilde{{\bit E}}^{*}\cdot\tilde{{\bit E}}\right)+\frac{\rmi u_{{\rm v}}\left(\tilde{{\bit E}}^{*}\times\tilde{{\bit E}}\right)}{\hbar}\cdot\hat{{\bit J}}\right]\hat{P}_{g} \end{eqnarray} \tag{ 55 }$$

in terms of the scalar and vector polarizabilities

$$\begin{eqnarray} &&\fl u_{{\rm s}} =-\frac{\left|\langle|\!|{\bit d}|\!|\rangle\right|^{2}}{36}\left(\frac{2}{E_{D2}- \hbar\omega}+\frac{1}{E_{D1}-\hbar\omega}\right), \nonumber\\ &&{\rm and} \quad u_{{\rm v}} =\frac{2u_{{\rm s}}\Delta_{{\rm FS}}}{3\left(\bar{E}-\hbar\omega\right)}. \end{eqnarray} \tag{ 56 }$$

When E_D2 = E_D1, we recover our previous result, equations (47)–(48).

As an illustration, consider an atom illuminated with circularly σ⁺ (along e_z) polarized light, $\tilde{{\bit E}}=-|\tilde{{\bit E}}|^{2}({\bit e}_x+\rmi{\bit e}_y)/\sqrt{2}$. For this field $\tilde{{\bit E}}^{*}\times\tilde{{\bit E}}=-\rmi|\tilde{{\bit E}}|^{2}{\bit e}_z$, so the vector light shift is described by the operator $\hat{J}_{z}$ acting on the ground state manifold. Consequently $\hat{H}_{{\rm eff}}^{(1)}$ provides opposite light shifts for the spin-up and spin-down atomic ground states.

In the presence of an external magnetic field, an alkali atom in its electronic ground state manifold is described by the Hamiltonian

$$\begin{equation} \hat{H}_{{{\bit B}}}= A_{{\rm hf}}\hat{{\bit I}}\!\cdot\!\hat{{\bit J}}+\frac{\mu_{\rm B}}{\hbar}{{\bit B}}\!\cdot\!\left(g_{J}\hat{{\bit J}}+g_{I}\hat{{\bit I}}\right), \end{equation} \tag{ 57 }$$

where μ_B is the Bohr magneton (henceforth the projector $\hat{P}_{g}$ is kept implicit in the ground state operators). The first term in equation (57) takes into account the coupling between the electron and nuclear spins of the atom, and is described by the magnetic dipole hyperfine coefficient A_hf. The second (Zeeman) term includes separate contributions from the electronic spin $\hat{{\bit J}}=\hat{{\bit L}}+\hat{{\bit S}}$ and the nuclear angular momentum $\hat{{\bit I}}$, along with their respective Landé factors g_J and g_I. For the alkali atoms |g_I/g_J| ≃ 0.0005, so we will safely neglect the nuclear contribution $\mu_{\rm B}g_{I}{{\bit B}}\cdot\hat{{\bit I}}/\hbar$ to the Zeeman term.

The previous subsection showed that laser fields induce the scalar and vector light shifts featured in the effective ground-state Hamiltonian $\hat{H}_{{\rm eff}}$, equation (55). The two terms can be independently controlled by choosing the laser frequency ω and polarization. Evidently, the vector light shift is a contribution to the total Hamiltonian acting like an effective magnetic field [119, 136, 138, 143, 144]

$$\begin{equation} {{\bit B}}_{{\rm eff}} =\frac{\rmi u_{v}(\tilde{{\bit E}}^{*}\!\times\!\tilde{{\bit E}})}{\mu_{\rm B}{\rm g}_{J}} \end{equation} \tag{ 58 }$$

that affects $\hat{{\bit J}}$ but not the nuclear spin $\hat{{\bit I}}$. Thus the complete effective Hamiltonian for the ground state atoms affected by the magnetic field and light reads

$$\begin{eqnarray} \fl \hat{H}_{{{\bit B}}\&{\rm {{\bit E}}}}=\ u_{s}(\tilde{{\bit E}}^{*}\!\cdot\!\tilde{{\bit E}}) +\frac{\mu_{\rm B}g_{J}}{\hbar}\left({{\bit B}}+{{\bit B}}_{{\rm eff}}\right)\!\cdot\!\hat{{\bit J}}+A_{{\rm hf}}\hat{{\bit I}}\!\cdot\!\hat{{\bit J}}, \end{eqnarray} \tag{ 59 }$$

where B_eff acts as a true magnetic field and adds vectorially with B. Instead of using the full Breit–Rabi equation [145, 146] for the Zeeman energies, we assume that the Zeeman shifts are small in comparison with the hyperfine splitting corresponding to the linear, or anomalous, Zeeman regime. In this case, the effective Hamiltonian for a single manifold of total angular momentum $\hat{{\bit F}}=\hat{{\bit J}}+\hat{{\bit I}}$ states is obtained by replacing $g_{J}\hat{{\bit J}} \rightarrow g_{F}\hat{{\bit F}}$ in the magnetic field interaction term [146], giving

$$\begin{equation} \hat{H}_{{{\bit B}}\&{\rm {{\bit E}}}}^{\left(f\right)}= u_{s}(\tilde{{\bit E}}^{*}\!\cdot\!\tilde{{\bit E}})+\frac{\mu_{\rm B}g_{F}}{\hbar}\left({{\bit B}}+{{\bit B}}_{{\rm eff}}\right)\!\cdot\!\hat{\mathbf{F}}, \end{equation} \tag{ 60 }$$

where we have introduced the hyperfine Landé g-factor

$$\begin{equation*} g_{F} =g_{J}\frac{f(f+1)-i(i+1)+j(j+1)}{2f(f+1)} \end{equation*}$$

and omitted the last term $A_{{\rm hf}}\hat{{\bit I}}\cdot\hat{{\bit J}}=A_{{\rm hf}}(\hat{{\bit F}}^{2}-\hat{{\bit J}}^{2}-\hat{{\bit I}}^{2})/2$ which is constant for the single hyperfine manifold with fixed f. For example, in ⁸⁷Rb's for which i = 3/2 and j = 1/2 in the electronic ground state, the lowest energy hyperfine manifold with f = 1 corresponds to g_F = −g_J/4 ≈ −1/2.

In this way, at second order of perturbation the light shift contains the term $u_{s}(\tilde{{\bit E}}^{*}\cdot\tilde{{\bit E}})$ described as a rank-0 (scalar) tensor operator resulting directly from the electric dipole transition, as well as the term $\mu_{\rm B}g_{F}{{\bit B}}_{{\rm eff}}\cdot\hat{{\bit F}}/\hbar$ described by the rank-1 (vector) tensor operator emerging from adding excited state fine structure. Since the latter operator is linear with respect to the total momentum $\hat{{\bit F}}$, it can induce only transitions which change m_F by ±1.

Adding effects due to the excited state hyperfine coupling leads to an additional rank-2 tensor contribution to the light-induced coupling within the atomic ground-state manifold. Because the scale of this term is set by the MHz to GHz scale excited-state hyperfine couplings, the smallness makes these light shifts unsuitable for most equilibrium-system applications in quantum gases.

We now turn to a situation frequently encountered in the current experiments on artificial gauge potentials [21–23] where an ensemble of ultracold atoms is subjected to a magnetic field B = B₀e_z and is simultaneously illuminated by possibly several laser beams with two frequencies ω and ω + δω. The frequency separation δω = g_Fμ_BB₀/ℏ + δ differs by a small detuning δ (with |δ/δω| ≪ 1) from the linear Zeeman shift between m_F states. The effective magnetic field induced by the complex electric field ${{\bit E}}={{\bit E}}_{\omega_{-}}\exp(-\rmi\omega t)+{{\bit E}}_{\omega_{+}}\exp[-\rmi(\omega+\delta\omega)t]$ can be cast into its frequency components using equation (58)

$$\begin{equation} {{\bit B}}_{{\rm eff}}= {{\bit B}}_{{\rm eff\,0}}+{{\bit B}}_{{\rm eff\,-}}\rme^{-\rmi\delta\omega t}+{{\bit B}}_{{\rm eff\,+}}\rme^{\rmi\delta\omega t}, \end{equation} \tag{ 61 }$$

where

$$\begin{eqnarray} &&\fl {{\bit B}}_{{\rm eff\,0}}= \frac{\rmi u_{v}}{\mu_{{\rm B}}g_{J}}\left({{\bit E}}_{\omega_{-}}^{*}\times {{\bit E}}_{\omega_{-}}+{{\bit E}}_{\omega_{+}}^{*}\times {{\bit E}}_{\omega_{+}}\right),\nonumber\\ &&{{\bit B}}_{{\rm eff\,\mp}}=\frac{\rmi u_{v}}{\mu_{{\rm B}}g_{J}}{{\bit E}}_{\omega_{\mp}}^{*}\times {{\bit E}}_{\omega_{\pm}} \end{eqnarray} \tag{ 62 }$$

are the corresponding amplitudes. The first term B_{eff 0} adds to the static bias field B₀e_z, and the remaining time-dependent terms B_{eff ±}e^±iδωt can drive transitions between different m_F levels.

The real magnetic field is assumed to be much greater than the effective one B₀ ≫ |B_eff|, and the frequency difference δω is taken to be large compared to the kinetic energy scales. In this case, the time-dependence of the Hamiltonian $\hat{H}_{{{\bit B}}\&{\rm {{\bit E}}}}^{(f)}$ in equation (60) can be eliminated via the unitary transformation $\hat{S}=\exp(-\rmi\delta\omega t\hat{F}_{z})$ and the subsequent application of the RWA by removing the terms oscillating at the frequencies δω and 2δω in the transformed Hamiltonian $\hat{S}^{\dagger}\hat{H}_{{{\bit B}}\&{\rm {{\bit E}}}}^{(f)}\hat{S}-\rmi\hbar S^{\dagger}\partial_{t}S$. The transformation does not alter $\hat{F}_{z}$, whereas the raising and lowering operators $\hat{F}_{\pm}=\hat{F}_{x}\pm \rmi\hat{F}_{y}$ change to $\hat{F}_{\pm}\exp(\pm \rmi\delta\omega t)$.

The resulting Hamiltonian can be represented as

$$\begin{equation} \hat{H}_{{\rm RWA}} =V({{\bit r}})\hat{1}+\hat{M}({\bit r}). \end{equation} \tag{ 63 }$$

The state-independent (scalar) potential is given by

$$\begin{equation} V({{\bit r}})=u_{s}\left({{\bit E}}_{\omega_{-}}^{*}\cdot{{\bit E}}_{\omega_{-}}+{{\bit E}}_{\omega_{+}}^{*}\cdot {{\bit E}}_{\omega_{+}}\right) \end{equation} \tag{ 64 }$$

and

$$\begin{equation} \hat{M}({\bit r})={\boldsymbol{ \Omega}}\cdot \hat{{\bit F}}=\Omega_{z}\hat{F}_{z}+\Omega_{-}\hat{F}_{+}+\Omega_{+}\hat{F}_{-}, \end{equation} \tag{ 65 }$$

where

$$\begin{equation} \Omega_{z}=\delta+\frac{\mu_{\rm B}g_{F}}{\hbar}{{\bit B}}_{{\rm eff\,0}}\cdot {\bit e}_z \end{equation} \tag{ 66 }$$

and

$$\begin{equation} \Omega_{\pm}\equiv\frac{\Omega_{x}\pm {\rm i}\Omega_{y}}{2}=\frac{\mu_{\rm B}g_{F}}{2\hbar}\left[{{\bit B}}_{{\rm eff\,\pm}}\cdot\left({\bit e}_x\pm \rmi{\bit e}_y\right)\right] \end{equation} \tag{ 67 }$$

are the circular components of the RWA effective Zeeman magnetic field Ω. The latter Zeeman field is explicitly

$$\begin{eqnarray} {\boldsymbol{ \Omega}}&=& \left[\delta+\rmi\frac{u_{v}g_{F}}{\hbar g_{J}}\left({{\bit E}}_{\omega_{-}}^{*}\times {{\bit E}}_{\omega_{-}}+{{\bit E}}_{\omega_{+}}^{*}\times {{\bit E}}_{\omega_{+}}\right)\cdot {\bit e}_z\right] {\bit e}_z\nonumber\\ && -\frac{u_{v}g_{F}}{\hbar g_{J}}{\rm Im} \left[\left({{\bit E}}_{\omega_{-}}^{*}\times {{\bit E}}_{\omega_{+}}\right)\cdot\left({\bit e}_x -\rmi{\bit e}_y\right)\right]{\bit e}_x\\ && -\frac{u_{v}g_{F}}{\hbar g_{J}}{\rm Re}\left[\left({{\bit E}}_{\omega_{-}}^{*}\times {{\bit E}}_{\omega_{+}}\right)\cdot \left({\bit e}_x-\rmi{\bit e}_y\right)\right]{\bit e}_y.\nonumber \end{eqnarray} \tag{ 68 }$$

The form of the effective coupling operator $\hat{M}({\bit r})$ shows that, while it is related to the initial vector light shifts, the RWA effective Zeeman magnetic field Ω is composed of both static and resonant couplings in a way that goes beyond the restrictive B_eff ∝ iE^* × E form.

In this way, the operator $\hat{F}_{z}$ featured in the vector coupling $\hat{M}({\bit r})$, equation (65), induces the opposites light shifts for the spin-up and spin-down atomic ground states, whereas the operators $\hat{F}_{\pm}$ describe the Raman transitions which change m_F by ±1. If the detuning from atomic resonance Δ_e is large compared to the excited state fine structure splitting Δ_FS, the magnitude of this vector coupling proportional to u_v drops off as $\Delta_{{\rm FS}}/\Delta_{e}^{2}$, not like the scalar shift V(r) which drops off as 1/Δ_e. Since the amount of the off-resonant light scattering is also proportional to $1/\Delta_{e}^{2}$, the balance between the off-resonant scattering and the vector coupling is bounded, and cannot be improved by large detuning. While this is a modest problem for rubidium (15 nm fine structure splitting), it turns to be a serious obstacle for atoms with smaller fine structure splitting such as potassium (≈4 nm) and lithium (≈0.02 nm).

Within this formalism, let us consider a straightforward example of two counter propagating Raman beams depicted in figure 6. Such a setup is often used in current experiments on synthetic gauge fields [21, 62, 147] and SOC [26, 32] for ultracold atoms. In this case

$$\begin{equation} {{\bit E}}_{\omega_{-}} = E\rme^{\rmi k_R x}{\bit e}_y \quad {\rm and} \quad {{\bit E}}_{\omega_{+}} = E\rme^{-\rmi k_R x}{\bit e}_z \end{equation} \tag{ 69 }$$

describe the electric field of two lasers counterpropagating along e_x with equal intensities, crossed linear polarization and wave-number k_R = 2π/λ, giving B_{eff 0} = 0 and ${{\bit B}}_{{\rm eff\,\pm}}=\mp\frac{\rmi u_{v}E^{2}}{\mu_{{\rm B}}g_{J}}\rme^{\pm \rmi 2k_R x}{\bit e}_x$. Consequently Ω_z = δ and

$$\begin{equation} 2\Omega_{\pm}=\Omega_{x}\pm \rmi\Omega_{y}=\Omega_{R}\rme^{\pm \rmi\left(2k_R x-\pi/2\right)}, \end{equation} \tag{ 70 }$$

where Ω_R = (g_F/g_J)u_vE²/ℏ is the Rabi frequency of the Raman coupling. The resulting effective Zeeman field Ω entering the vector light shift is

$$\begin{equation} {\boldsymbol{ \Omega}}=\delta{\bit e}_z+\Omega_{R}\left[\sin\left(2k_R x\right){\bit e}_x-\cos\left(2k_R x\right){\bit e}_y\right]. \end{equation} \tag{ 71 }$$

Together, V(r) = 2u_sE² and Ω(r) describe a constant scalar light shift along with a spatially rotating Zeeman field.

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It is evident from equations (58) and (67) that for this spatially rotating Zeeman field to be non-zero, the effective Zeeman field B_eff must not be parallel to the quantizing axis defined by the applied magnetic field B = B₀e_z. In the present case, this implies that the counter propagating beams cannot be aligned along e_z. Physically, any projection of B_eff along e_z yields a spin-dependent optical lattice [138, 139, 148] rather than a spatially rotating Zeeman field. But since the Raman beams have different frequencies, the spin-dependent lattice is moving fast and vanishes after performing the RWA.

Note also that in order to produce the vector light shifts like the one given by equation (71), there is no need for the two Raman beams to counter-propagate. When the beams intersect with opening angle α, the momentum exchange is changed from 2k_R to 2k_L = 2k_R cos [(π − α)/2]. Since this altered wave-vector simply changes the energy and length scales of the problem, we shall concentrate on the counter propagating setup, equations (69). To obtain an exact correspondence with the notations used in most of the experimental papers on the artificial gauge fields [21–23, 62, 147, 149] and the SOC [25–28, 32], one needs to interchange the e_z and e_y axes in our analysis.

It is instructive to note that the z component of the effective Zeeman field Ω, equation (71), does not depend on the intensity of the Raman beams. The situation is not universal. For example, consider a pair of Raman laser beams with complex electric fields

$$\begin{eqnarray} &&\fl {{\bit E}}_{\omega_{-}} = E_{-}\rme^{\rmi k_R z}\left({\bit e}_x-\rmi{\bit e}_y\right)/\sqrt{2}, \quad {\rm and} \quad {{\bit E}}_{\omega_{+}} = E_{+}\rme^{\rmi k_R x}{\bit e}_z,\nonumber\\ \end{eqnarray} \tag{ 72 }$$

intersecting with angle α = π/2, and where the ω₋ beam is circularly polarized [52]. In this case ${{\bit E}}_{\omega_{-}}^*\times {{\bit E}}_{\omega_{-}} = -\rmi |E_{-}|^2{\bit e}_z$, ${{\bit E}}_{\omega_{+}}^*\times {{\bit E}}_{\omega_{+}} = 0$, and ${{\bit E}}_{\omega_{-}}^*\times {{\bit E}}_{\omega_{+}} = \rmi E_{-}^* E_+ ({\bit e}_x + \rmi{\bit e}_y)/\sqrt{2}$ giving

$$\begin{eqnarray} &&\fl \hbar\Omega_{z}=\hbar\delta+\left(u_{v}\frac{g_{F}} {g_{J}}\right)\left|E_{-}\right|^{2},\nonumber\\ &&\hbar\Omega_{\pm}=-\left(u_{v}\frac{g_{F}}{g_{J}} \right)\frac{E_{-}E_{+}}{\sqrt{2}}\rme^{\pm \rmi k_R\left(z-x\right)}, \end{eqnarray} \tag{ 73 }$$

so Ω_z becomes intensity-dependent, providing an extra control of the light-induced gauge potential.

Finally, more sophisticated spatial dependence of the Zeeman field Ω can be created using additional Raman beams with specially chosen polarizations, for example leading to optical flux lattices [144, 150], to be considered in the section 5.2.3.

Let us now turn to a situation where the atom–light Hamiltonian can be represented as an interaction of a spin $\hat{{\bit F}}$ with an effective Zeeman field

$$\begin{eqnarray} &&\fl \hat{M}({\bit r})={\boldsymbol{ \Omega}}\cdot\hat{{\bit F}}\nonumber\\ &&=\Omega\left(\cos\theta\hat{F}_{z}+\sin\theta\cos\phi \hat{F}_{x}+2\sin\theta\sin\phi\hat{F}_{y}\right) \end{eqnarray} \tag{ 79 }$$

where we have parametrized the Zeeman vector Ω = (Ω_x, Ω_y, Ω_z) in terms of the spherical angles θ and φ shown in figure 9.

$$\begin{equation} \tan\phi =\frac{\Omega_{y}}{\Omega_{x}} \quad {\rm and} \quad \cos\theta =\frac{\Omega_{z}}{\Omega} \end{equation} \tag{ 80 }$$

where Ω is the length of the Zeeman vector. Such a coupling can be produced using bichromatic laser beams considered in the previous section, leading to equation (65) which has the same form as equation (79). For the spin 1 case, the Hamiltonian (79) corresponds to the Λ (ladder) scheme described by the Hamiltonian (74) as long as $\Omega_{1}=\Omega_{2}^{*}$ and Δ = 0. Note that the operator $\hat{{\bit F}}$ does not necessarily represent the true atomic spin operator; it can be other atomic vector operators with Cartesian components obeying the angular momentum algebra (figure 9).

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The coupling Hamiltonian $\hat{M}$ can be diagonalized via a unitary transformation

$$\begin{equation} \hat{R}=\rme^{-\rmi\hat{F}_{z}\phi/\hbar} \rme^{-\rmi\hat{F}_{y}\theta/\hbar}\rme^{\rmi\hat{F}_{z} \phi/\hbar}. \end{equation} \tag{ 81 }$$

The first operator ${\rme^{-\rmi\hat{F}_{z}\phi/\hbar}}$ rotates the spin around the z-axes by the angle φ eliminating the phase φ in $\hat{M}({\bit r})$, equation (79). The subsequent transformation using the second operator $\rme^{-\rmi\hat{F}_{y}\theta/\hbar}$ rotates the spin around the y-axes by the angle θ making the operator $\hat{M}({\bit r})$ proportional to $\hat{F}_{z}$. Although the third operator $e^{i\hat{F}_{z}\phi/\hbar}$ no longer affects the transformed operator $\hat{M}({\bit r})$, its inclusion ensures that the whole transformation $\hat{R}$ reduces to the unit operator for zero polar angle θ = 0 and arbitrary azimuthal angle φ. Thus one has

$$\begin{equation} \hat{R}^{\dagger}\hat{M}\hat{R}=\Omega\hat{F}_{z}. \end{equation} \tag{ 82 }$$

The transformed Hamiltonian $\Omega\hat{F}_{z}$ has eigenstates |f, m〉 ≡ |m〉 characterized by the total angular momentum f and its e_z projection m. In the following, we use both notations m and m_F to denote the projection of the total angular momentum. Therefore the eigenstates |m, Ω〉 ≡ |m, θ, φ〉 of the original atom–light Hamiltonian $\hat{M}$ read

$$\begin{equation} \left|m,{\boldsymbol{ \Omega}}\right\rangle = \hat{R} \left|m \right\rangle =\rme^{\rmi\left(m-\hat{F}_{z}/\hbar\right)\phi}\rme^{-\rmi\hat{F}_{y}\theta/\hbar}\left|m\right\rangle , \end{equation} \tag{ 83 }$$

the corresponding eigenenergies being

$$\begin{equation} \varepsilon_{m} =\hbar m\Omega,\tqs m=-f,\,\ldots\,,f, \end{equation} \tag{ 84 }$$

where both the eigenenergies ε_m(r) and also the local eigenstates |m, Ω(r)〉 are position-dependent through the position dependence of the effective Zeeman vector Ω(r). It is to be noted that similar kinds of eigenstates give rise to artificial gauge potentials describing the rotation of diatomic molecules [112, 172] and the physics of atomic collisions [113, 173].

If an atom is prepared in a dressed state |m, Ω〉 and if its characteristic kinetic energy is small compared to the energy difference between adjacent spin states ΔE = ℏΩ, the internal state of the atom will adiabatically follow the dressed state |m, Ω〉 as the atom moves, and contributions due to other states with $m_{F}^{\prime}\ne m_{F}$ can be neglected. Projecting the atomic dynamics onto the selected internal eigenstate |m_F, Ω〉 yields a reduced Schrödinger equation for the atomic center of mass motion affected by the geometric vector potential $\pmb{\mathcal A}\equiv\pmb{\mathcal A}_{m}({\bit r})=\rmi\hbar\mathinner{\langle{{m,{\boldsymbol{ \Omega}}}}|}{\boldsymbol \nabla} {\mathinner{|{m,{\boldsymbol{ \Omega}}}\rangle}}$ and the scalar potential W ≡ W_n(r) which are generally defined by equation (19). In the current situation these potentials explicitly read [144]

$$\begin{equation} \pmb{\mathcal A}({\bit r})=\hbar m\left(\cos\theta-1\right){\boldsymbol \nabla}\phi \end{equation} \tag{ 85 }$$

and

$$\begin{eqnarray} &&\fl W_{m}({\bit r}) =\frac{\hbar^{2}}{4m}\left[f(f+1)-m^{2}\right] \left[\sin^{2}\theta\left(\nabla\phi\right)^{2}+ \left(\nabla\theta\right)^{2}\right].\nonumber\\ \end{eqnarray} \tag{ 86 }$$

The geometric vector potential $\pmb{\mathcal A}({\bit r})$ yields the artificial magnetic field

$$\begin{equation} \pmb{\mathcal B}({\bit r})={\boldsymbol \nabla}\times\pmb{\mathcal A}({\bit r})=\hbar m_{F}{\boldsymbol \nabla}\left(\cos\theta\right)\times{\boldsymbol \nabla}\phi , \end{equation} \tag{ 87 }$$

which is non-zero if ∇φ and ∇(cos θ) are not parallel to each other. Both $\pmb{\mathcal A}({\bit r})$ and $\pmb{\mathcal B}({\bit r})$ have the largest magnitude for maximum absolute values of the spin projection m and are zero for m = 0. On the other hand, the geometric scalar potential W(r) is maximum for m = 0 and reduces with increasing |m|.

The atomic motion is affected by three distinct scalar potentials: (a) the state-independent potential V(r) representing the 'scalar light shift' given by equation (64) for bichromatic fields, (b) the 'adiabatic scalar potential 'ε_m(r) ≡ ε_m arising from spatial variations in the magnitude of the Zeeman vector Ω(r), and (c) the 'geometric scalar potential' W(r) described above. All three contribute to the potential energy of atoms in the dressed state basis.

For two counterpropagating Raman beams, the effective Zeeman field Ω, equation (71), defining the gauge fields, is characterized by the azimuthal angle φ = 2k_Rx − π/2 with the gradient

$$\begin{equation} {\boldsymbol \nabla}\phi=2{\bit k_R},\tqs {\bit k_R}=k_R{\bit e}_x, \end{equation} \tag{ 88 }$$

equal to the Raman recoil wave-vector 2k_R. On the other hand, the polar angle θ and the length of the Zeeman vector Ω are determined by the detuning δ and the Raman Rabi frequency Ω_R with

$$\begin{equation} \cos\theta=\frac{\delta}{\Omega},\quad \Omega=\sqrt{\delta^{2}+\Omega_{R}^{2}} . \end{equation} \tag{ 89 }$$

Typically experiments are performed in the lowest energy dressed state, where m = −f assumes its maximum absolute value. Equations (85) and (88)–(89) illustrate two important points: (1) the magnitude of the vector potential $\pmb{\mathcal A}_{m}({\bit r})$ is defined by the recoil momentum of the lasers as well as the spin projection ℏm, and (2) the vector potential is strictly bounded between ±2ℏk_Rm_F, where the maximum possible value is the recoil momentum 2ℏk_R times the spin f.

We note that employing an extra spatially uniform radio-frequency magnetic field adds a constant term to the spatially oscillating x component of effective Zeeman field Ω, equation (71). In that case both the adiabatic energy ε_m and the geometric scalar potential W_m(r) become spatially oscillating functions, thus creating a composite lattice potential in addition to the vector potential $\pmb{\mathcal A}({\bit r})$ [62]. We shall discuss this issue in more detail in the section 8.5.

Returning to equations (87)–(89), we see that in order to have a non-zero $\pmb{\mathcal B}$, either the optical intensity or the detuning must depend on y or z, i.e. their gradient(s) should not be parallel to the wave-vector k_R. The position-dependent detuning can be generated by the vector light shift itself, which is absent for co-propagating light beams. This provides an additional term analogous to δ, but proportional to the Raman Rabi frequency. This is the case if the laser beams propagate at the angle α = π/2, one of them being circularly polarized [52], as one can see in equation (73) of the previous section. Therefore, by simply making δ = 0 and having the intensity-dependent detuning (which adds to δ) depend on position in equation (73), it is possible to create artificial gauge potentials which are truly geometric in nature, as the absolute light intensity then completely vanishes from the azimuthal angle θ.

The first experimental implementation of the synthetic magnetic field [22] was based on a different insight, namely that with constant Ω_R a gradient of the detuning δ due to a magnetic field gradient also generates a non-zero artificial magnetic field $\pmb{\mathcal B}$ providing an artificial Lorentz force. The generated field $\pmb{\mathcal B}$ is not purely geometric, and depends on the relative strength of δ and Ω_R in addition to their geometry.

It is noteworthy that the maximum magnetic flux produced by means of two counterpropagating laser beams amounts to 2fk_RL Dirac flux quanta and is thus proportional to the systems length L [150]. This is a drawback in creating very large magnetic fluxes necessary for reaching the fractional quantum Hall regimes. The use of the optical flux lattices considered in the following section overcomes this drawback making the induced flux proportional to the area rather than the length of the atomic cloud.

Note also that instead of counterpropagating laser fields one can employ co-propagating beams carrying optical vortices with opposite vorticity [154]. For the first-order Laguerre–Gaussian beams the azimuthal angle φ entering the vector and scalar potentials is then twice the real space azimuthal angle. On the other hand, the Raman Rabi frequency Ω_R, entering in equations (88)–(89) for the polar angle, linearly depends on the cylindrical radius in the vicinity of the vortex core. Non-zero artificial magnetic fields can therefore be generated without making the detuning position-dependent. Using vortex beams the number of Dirac flux quanta imparted onto the atomic cloud is determined by the vorticity of the beams [48, 151, 152, 154] and is thus normally much smaller than that induced by counter-propagating laser fields.

Optical flux lattices were introduced by Cooper in [174]. This concept is based on the observation that singularities in the synthetic vector potential could be created and exploited so as to produce periodic structures with non-trivial magnetic fluxes [144, 174]. Such optical flux lattices could be created using a bichromatic laser field [144, 150], as considered in the previous section. Following the proposal by Cooper and Dalibard [150], we consider that the f = 1/2 ground state magnetic sublevels are coupled via Raman transitions (figure 10) involving laser fields with two frequencies. One frequency component ${{\bit E}}_{\omega_{+}}$ shown in green represents a circular σ₋ polarized plane wave propagating along the z axis. Another frequency field ${{\bit E}}_{\omega_{-}}$ shown in red has all three circular polarizations. Dalibard and Cooper considered a situation where the field ${{\bit E}}_{\omega_{-}}$ represents a superposition of three linearly polarized plane waves propagating in the e_x–e_y plane and intersecting at 120°, with polarizations being tilted by some angle with respect to the propagation plane [150]. In that case one can produce triangular or hexagonal state-dependent lattices for the adiabatic motion of laser-dressed atoms affected by a non-staggered magnetic flux.

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A square optical flux lattice is formed if the first frequency component ${{\bit E}}_{\omega_{+}}$ is again a circular σ₋ polarized plane wave propagating along the z axis, yet the second frequency field ${{\bit E}}_{\omega_{-}}$ represents a sum of polarization-dependent standing waves with a π/2 time-phase difference between the standing waves oscillating in the e_x and e_y directions [144, 175]⁶. In that case one arrives at an atom–light coupling described by the state-dependent Hamiltonian $\hat{M}({\bit r})=\boldsymbol{ \Omega}\cdot\hat{{\bit F}}$ with a spatially periodic Zeeman field Ω:

$$\begin{eqnarray} &&\fl \Omega_{x}=b\Omega_{\parallel}\cos(x\pi/a), \quad\Omega_{y}=b\Omega_{\parallel}\cos(y\pi/a), \nonumber\\ &&\Omega_{z}=\Omega_{\parallel}\sin(x\pi/a)\sin(y\pi/a), \end{eqnarray} \tag{ 90 }$$

where a dimensionless ratio b is controlled by changing the polarizations of the standing waves [144, 175].

Atoms with a fixed spin projection m (along the Zeeman field Ω) then undergo an adiabatic motion in a square lattice potential ε_m(r) + W_n(r) with a periodicity in the e_x and e_y directions twice smaller than the periodicity 2a of the atom–light Hamiltonian $\hat{M}({\bit r})$. The adiabatic motion is accompanied by a vector potential $\pmb{\mathcal A}_{m}({\bit r})$ containing two Aharonov–Bohm type singularities (per elementary cell) corresponding to the points where Ω_x = Ω_y = 0 and Ω_z < 0. In the vicinity of each singular point, the vector potential ${\pmb{\mathcal A}}_{m}$ has a Dirac-string piercing the e_x–e_y plane and carrying −2m Dirac flux quanta (with −f ⩽ m ⩽ f) [144]

$$\begin{equation} {\pmb{\mathcal A}}_{m}\rightarrow{\pmb{\mathcal A}}_{m}^{AB}=-2\hbar m\nabla\phi. \end{equation} \tag{ 91 }$$

Note that these singular fluxes are gauge-dependent and non-measurable. One can shift each singular points to another location via a gauge transformation [144], yet the singularities cannot be removed from the e_x–e_y plane.

Due to the periodicity of the system, the total flux over the elementary cell is equal to zero

$$\begin{equation} \frac{1}{\hbar}\oint_{{\rm cell}}\!\pmb{\mathcal A}_{m}\cdot \rmd{\bit r} =\frac{1}{\hbar}\int\hspace{-3pt}\int_{{\rm cell}}\!\pmb{\mathcal B}_{{\rm m}}^{\rm tot}\cdot \rmd{\bit S}=0, \end{equation} \tag{ 92 }$$

where $\pmb{\mathcal B}_{{\rm m}}^{\rm tot}=\pmb{\mathcal B}_{m}({\bit r})+\sum\pmb{\mathcal B}_{{\rm m}}^{AB}({\bit r})$ is the total magnetic flux density composed of the continuous (background) magnetic flux density $\pmb{\mathcal B}_{m}({\bit r})$ and a set of above-mentioned gauge-dependent singular fluxes $\pmb{\mathcal B}_{{\rm m}}^{AB}({\bit r})$ of the Aharonov–Bohm type corresponding to ${\pmb{\mathcal A}}_{m}^{AB}$, equation (91).

Thus it is strictly speaking impossible to produce a non-zero effective magnetic flux over the elementary cell using the periodic atom–light coupling. Yet this does not preclude having a non-staggered continuous magnetic flux density $\pmb{\mathcal B}({\bit r})$ over the elementary cell, because of the existence of non-measurable Dirac strings. Deducting the latter, the continuous physical flux over the elementary cell is

$$\begin{equation} \frac{1}{\hbar}\int\hspace{-3pt}\int_{{\rm cell}}\!\pmb{\mathcal B}_{m}\cdot \rmd{\bit S}=-\frac{1}{\hbar}\sum\oint_{{\rm singul}}\!{\pmb{\mathcal A}}_{m}\cdot \rmd{\bit r}=4m, \end{equation} \tag{ 93 }$$

where the summation is over the singular points of the vector potential (emerging at cos θ → −1) around which the contour integration is carried out.

To summarize, the optical flux lattice contains a background non-staggered magnetic field $\pmb{\mathcal B}$ plus an array of gauge-dependent Dirac-string fluxes of the opposite sign as compared to the background. The two types of fluxes compensate each other, so the total magnetic flux over an elementary cell is zero, as it is required from the periodicity of the Hamiltonian. However, the Dirac-string fluxes are non-measurable and hence must be excluded from any physical consideration. As a result, a non-staggered magnetic flux over the optical flux lattice appears, and topologically non-trivial bands with non-zero Chern numbers can be formed [144, 150]. By choosing the proper laser beams producing the optical flux lattice, one can produce a configuration where the lowest energy band is topologically non-trivial and nearly dispersionless, leading to a possible formation of fractional quantum Hall states [178]. Note also that unlike other lattice schemes involving laser-assisted-tunneling methods, such as proposed in [179, 180] (see section 8), the concept of optical flux lattices is based on the adiabatic motion of the atoms. Finally, it is also convenient to reinterpret optical flux lattices as tight-binding models defined in reciprocal space [181]. This complementary picture offers an efficient method to establish the atom–light coupling configurations leading to non-trivial topological bands: following [181], any tight-binding model exhibiting non-trivial Chern insulating phases, such as the Haldane model [182], could constitute the roots of an interesting optical flux lattice.

We will first consider a coupled two level system with internal states ${\mathinner{|{+}\rangle}}$ and ${\mathinner{|{-}\rangle}}$, where exact solutions can be obtained [183]. Physically, these two states might be two m_F levels in the ground state manifold of an alkali atom. For example, the F = 1 manifold of Rb⁸⁷ at large enough field such that the quadratic Zeeman effect can resolve two out of the three Zeeman sublevels for the Raman or radio frequency transitions. In the frame rotating with angular frequency Δ_R/h, the Raman fields are detuned δ = gμ_BΔB/ℏ from resonance. The atom–light coupling term in the RWA is then

$$\begin{eqnarray} &&\fl \hat H' =\int\!\rmd y \int\!\frac{\rmd k_x}{2 \pi} \left\{\frac{\hbar\Omega_R}{2}\left[\hat \phi_+^\dagger(k_x-2k_R,y)\hat \phi_-(k_x,y) + {\rm h.c.}\right]\right. \nonumber\\ \end{eqnarray} \tag{ 95 }$$

$$\begin{eqnarray}&&\fl \quad+\left.\frac{\hbar\delta}{2}\left[\hat\phi_+^\dagger(k_x,y) \hat \phi_+(k_x,y) - \hat\phi_-^\dagger(k_x,y)\phi_-(k_x,y)\right]\right\}.\nonumber\\ \end{eqnarray} \tag{ 96 }$$

The notation $\hat \phi_\sigma^\dagger(k_x,y)$ denotes the creation of a particle with wave vector k_x along e_x at position y, with σ = ±. Here $\hat H'$ also includes the Raman detuning terms. In the following Ω_R and δ will be treated as spatially varying functions of y, but not x. This is the explicit expansion of the $\boldsymbol{ \Omega} \cdot \hat{F}$ contribution to equation (63) with the effective magnetic field from equation (71) in terms of field operators for the two-level case.

With no coupling, the Hamiltonian is given by $\hat {\mathcal H} =\hat H_x + \hat H_y + \hat{U}+\hat H_{\rm int}$, which represent motion along e_x, motion along e_y, the external potential, and interparticle interactions respectively. When expressed in terms of the real space field operators $\hat \psi_\sigma({\bit r})$ we obtain

$$\begin{eqnarray} \fl \hat H_x &= \int \rmd^2{\bit r}\sum_\sigma\hat\psi^\dagger_\sigma ({\bit r})\left(-\frac{\hbar^2\partial_x^2}{2m}\right) \hat\psi_\sigma({\bit r}) \end{eqnarray} \tag{ 97 }$$

$$\begin{eqnarray} \fl \hat H_y &= \int \rmd^2{\bit r}\sum_\sigma\hat\psi^\dagger_\sigma ({\bit r})\left(-\frac{\hbar^2\partial_y^2}{2m}\right) \hat\psi_\sigma({\bit r}), \end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray} \fl \hat U &= \int \rmd^2{\bit r}\sum_\sigma\hat \psi^\dagger_\sigma({\bit r})U({\bit r})\hat\psi_\sigma ({\bit r}), \end{eqnarray} \tag{ 99 }$$

$$\begin{eqnarray} \fl \hat H_{\rm int} &= \frac{g_{\rm 2D}}{2} \int \rmd^2{\bit r}\sum_{\sigma,\sigma'} \hat\psi^\dagger_\sigma({\bit r}) \hat\psi^\dagger_{\sigma'}({\bit r}) \hat\psi_{\sigma'}({\bit r}) \hat\psi_{\sigma}({\bit r}). \end{eqnarray} \tag{ 100 }$$

The contact interaction for collisions between ultracold atoms in three dimensions is set by the three-dimensional (3D) s-wave scattering length a_s, here assumed to be state independent. Strong confinement in one direction results in an effective 2D coupling constant $g_{\rm 2D}=\sqrt{8\pi}\hbar^2 a_s/m l_{\rm HO}$ where l_HO is the harmonic oscillator length resulting from a strongly confining potential along e_z. Finally, U(r) is an external trapping potential, which we assume also to be state-independent.

This problem can be solved exactly when considering free motion along e_x, i.e., treating only $\hat H_x$ and $\hat H'$, and going to the momentum representation for the atomic motion along the x-axis. The second quantized Hamiltonian for these two contributions can be compactly expressed in terms of the operators $\hat{\varphi}^\dagger_\pm(q_x,y) = \hat\phi^\dagger_\pm(q_x\mp k_R,y)$. Using this set, $\hat H\approx\hat H_x+\hat H'$ reduces to an integral over 2 × 2 blocks

$$\begin{eqnarray} \fl \hat H(q_x,y) &= \left( \begin{array}{@{}cc@{}} \dsty\frac{\hbar^2(q_x - k_R)^2}{2m} + \dsty\frac{\hbar\delta}{2} & \dsty\frac{\hbar\Omega_R}{2} \\\ms \dsty\frac{\hbar\Omega_R}{2} & \dsty\frac{\hbar^2(q_x + k_R)^2}{2m} - \dsty\frac{\hbar\delta}{2} \end{array}\right)\nonumber\\ \end{eqnarray} \tag{ 101 }$$

labeled by q_x and y. The expression in terms of $\hat\varphi$ operators instead of $\hat\phi$ is a gauge transformation which boosts the ${\mathinner{|{+}\rangle}}$ and ${\mathinner{|{-}\rangle}}$ states in opposite directions. The dependence of the two-photon coupling Ω_R and detuning δ on y has been suppressed for notational clarity. The resulting Hamiltonian density for motion along e_x at a fixed y is then

$$\begin{eqnarray} &&\fl \hat H_x + \hat H' = \int\frac{\rmd q_x}{2\pi} \sum_{\sigma,\sigma'} \hat{\varphi}^\dagger_\sigma(q_x,y) \hat H_{\sigma, \sigma'}(q_x,y) \hat {\varphi}_{\sigma'}(q_x,y).\nonumber\\ \end{eqnarray} \tag{ 102 }$$

For each q_x, $\hat H(q_x,y)$ can be diagonalized by the unitary transformation $\hat R(q_x,y) \hat H(q_x,y) \hat R^\dagger(q_x,y)$. Unlike the approximate solutions discussed in the context of adiabatic gauge fields in sections 5.1–5.2, these represent the exact solution for motion along e_x.

The resulting eigenvalues of $\hat H(q_x,y)$,

$$\begin{eqnarray} &&\fl E_\pm(q_x,y)=\nonumber\\ &&\fl E_R\left[\left(\frac{q_x}{k_R}\right)^2 +1\pm\frac{1}{2}\sqrt{\left(\frac{4 q_x}{k_R} - \frac{\hbar\delta}{E_R}\right)^2 + \left(\frac{\hbar\Omega_R}{E_R}\right)^2}\right] \end{eqnarray} \tag{ 103 }$$

give the effective dispersion relations in the dressed basis, $\hat{\tilde{\varphi}}_\sigma(q)=\sum R_{\sigma,\sigma'}(q) \hat {\varphi}_{\sigma'}(q)$. For each q the eigenvectors of H(q_x) form a family of states [184]. In terms of the associated real-space operators $\hat \psi'_\sigma({\bit r})$ these diagonalized terms of the initial Hamiltonian are

$$\begin{eqnarray} &&\fl \hat H_x + \hat H' =\int \rmd^2 {\bit r} \sum_{\sigma=\pm} \hat\psi'^\dagger_\sigma({\bit r}) E_\sigma\left(-\rmi\hbar\frac{\partial}{\partial x},y\right)\hat\psi'_\sigma({\bit r}). \end{eqnarray} \tag{ 104 }$$

In analogy with the term 'crystal-momentum' for particles in a lattice potential, we call the quantum number q_x the 'quasi-momentum'. Here −iℏ∂_x is the real-space representation of the quasi-momentum q_x. The symbol E_±(−iℏ∂_x, y) is a differential operator describing the dispersion of the dressed eigenstates, just as the operator E_x(−iℏ∂_x, y) = (−iℏ∂_x + eBy)²/2m describes quadratic dispersion along e_x of a charged particle moving in a magnetic field Be_z in the Landau gauge.

To lowest order in E_R/ℏΩ_R and second order in q_x/k_R, E_±(q_x, y) can be expanded as

$$\begin{eqnarray} &&\fl E_\pm \approx \frac{E_R}{m^*}\left(\frac{q_x}{k_R} - \frac{\hbar\delta}{4E_R\pm\hbar\Omega_R}\right)^2 + \frac{2E_R\pm\hbar\Omega_R}{2}\nonumber\\ &&+ \frac{\hbar^2\delta^2(4E_R\pm\hbar\Omega_R)}{4(4E_R+\hbar\Omega_R)^2}. \end{eqnarray} \tag{ 105 }$$

Atoms in the dressed potential are significantly changed in three ways: (1) The energies of the dressed state atoms are shifted by a scalar shift analogous to the sum of the adiabatic and geometric potentials V(r) and W(r). (2) Atoms acquire an effective mass m^*/m = ℏΩ_R/(ℏΩ_R ± 4E_R). This change is absent in the adiabatic picture. (3) The center of the dispersion relation is shifted to ${\mathcal A}_x/\hbar k_R = \hbar\delta / (\hbar\Omega_R\pm 4E_R)$. Just as with the adiabatic case considered in the previous subsections 5.1–5.2, ${\mathcal A}_x$ can depend on y by a spatial dependence on Ω_R, or, via δ(y) as described below. In either case, the effective Hamiltonian is that of a charged particle in a magnetic field expressed in the Landau gauge.

Figure 11 shows the dressed state dispersion relations in this model. Panels (a) and (c) show the undressed case (Ω_R = 0) for detuning ℏδ = 0 and 5E_R respectively. Panels (b) and (d) depict the same detunings, for ℏΩ_R = 16E_R, where the exact results (solid line) are displayed along with the approximate dispersion (red dashed line). Figure 11(b) shows the strongly dressed states for large Ω_R, each of which is symmetric about q = 0. When detuned as in panel (c), the dispersion is displaced from q = 0. If space dependent this displacement leads to a non-trivial gauge potential. In the limit of very small Ω_R the dressed curve E₋(q_x, y) forms a double-well potential as a function of k_x. In a related Raman-coupled system, Bose condensation in such double-well potentials were studied theoretically [165, 183, 185, 186], and has been explored in detail in the context of SOC [25–28].

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The Raman coupling leads to a dressed dispersion along e_x, where motion along e_y is largely unaffected. When the detuning is made to vary linearly along e_y, with δ(y) = δ'y, an effective single particle Hamiltonian contains a 2D effective vector potential $\pmb{\mathcal A}/\hbar k_R \approx \hbar\delta' y / (4E_R\pm \hbar\Omega_R) {\bit e}_x$. The synthetic magnetic field is therefore ${\mathcal B}_z/\hbar k_R \approx -\hbar\delta' /(4E_R\pm\hbar\Omega_R)$. Figure 12 shows the vector potential as a function of detuning δ. As expected, the linear approximation discussed above (dashed line) is only valid for small δ. The synthetic field therefore decreases from its peak value as δ increases (top inset).

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This technique also modifies the trapping potential along e_y. In the adiabatic picture, this comprises the sum of the adiabatic and geometric potentials V(r) and W(r) which have no individual identity in this exact solution. When the initial potential U(x, y) is harmonic with trapping frequencies ω_x and ω_y, the combined potential along e_y becomes $U_\pm(y) = m [\omega_y^2+(\omega_\pm^*)^2] y^2/2$, where $m (\omega_\pm^*)^2/2 \approx \hbar^2\delta'^2 (4E_R\pm\hbar\Omega_R)/4\hbar k_R(4E_R+\hbar\Omega_R)^2$.

This contribution to the overall trapping potential is not unlike the centripetal term which appears in a rotating frame of reference, where a synthetic magnetic field $\pmb{\mathcal B}$ arises as well. In the case of a frame rotating with angular frequency Ω, the centripetal term gives rise to a repulsive harmonic term with frequency $\omega_{\rm rot}^2 = (\pmb{\mathcal B} / 2 m)^2$. In the present case the scalar trapping frequency can be rewritten in a similar form $\omega_\pm^{*2} = ({\mathcal B} / 2 m)^2\times|4E_R\pm\hbar\Omega_R|^3/\hbar k_R(4E_R+\hbar\Omega_R)^2$. The scalar potential may be attractive or repulsive, and it increases in relative importance with increasing Ω_R.

We have so far omitted motion along e_y, interactions, and the effect of an external potential. The inclusion of these effects are discussed in detail in [51]. The outcome is that in the limit of large Ω_R these are unchanged by the transformation into the dressed basis. Together these allow the construction of the real space Hamiltonian in the basis of localized spin-superposition states $\hat \psi'({\bit r})$. The density distribution of these localized states is not perfectly localized, but are some fraction of an optical wavelength λ in extent. For interactions, this is important because dressed state atoms separated by this distance will in fact interact. For particles starting in higher bands, transitions to lower bands are energetically allowed and a Fermi's Golden Rule argument thus gives rise to decay from all but the lowest energy dressed state [187].

This technique is not without its limitations. Foremost among them is the range of possible ${\mathcal A}_x/\hbar k_R$ shown in figure 12 where ℏΩ_R = 16E_R. While the linear expansion (dashed) is unbounded, the exact vector potential is bounded by ±k_R. For example, the hybridized combination of ${\mathinner{|{+,q-k_R}\rangle}}$ and ${\mathinner{|{-,q+k_R}\rangle}}$ cannot give rise to dressed states with minima more positive than q = +k_R where the energy of the ${\mathinner{|{+}\rangle}}$ states is minimized without dressing, figure 11(c). The minima can also not be more negative than q = −k_R.

This limitation does not affect the maximum attainable field, only the spatial range over which this field exists. Specifically, a linear gradient in δ(y) gives rise to the synthetic field ${\mathcal B}_z(y)$ which is subject to $\int_{-\infty}^{\infty}{\mathcal B}_z(y)\,\rmd y=2\hbar k_R$. This simply states that the vector potential—bounded by ±ℏk_R—is the integral of the magnetic field. Note however, that along e_x the region of large ${\mathcal B}_z$ has no spatial bounds.

A second limitation of this technique is the assumption of strong Raman coupling between the Zeeman split states. As already pointed out at the end of the section 4.4.1, for alkali atoms, when the detuning from atomic resonance Δ_e is large compared to the excited state fine structure Δ_FS, the two-photon Raman coupling for Δm_F = ±1 transitions drops as $\Omega_R \propto \Delta_{{\rm FS}}/ \Delta_e^{2}$, not $\Delta_e^{-1}$ as for the AC Stark shift. As a result, the ratio between the Raman coupling Ω_R and off-resonant scattering cannot be increased by using larger detuning, and the only solution is to consider atoms with large fine structure splitting Δ_FS, such as rubidium (figure 13).

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The range of possible effective vector potentials can be extended by coupling more states, for example the m_F states of an f > 1/2 manifold in the linear Zeeman regime. We can follow the two-level example above, except there are no compact closed-form solutions. In the adiabatic picture, equations (85) and (88)–(89), additional levels extended the range of the vector potential from ±2ℏk_R (the two level case) to ±2ℏfk_R for arbitrary f.

This can be illustrated by considering an optically trapped system of Rb⁸⁷ atoms in the f = 1 manifold in a small magnetic field which splits the three m_F levels by gμ_rmB |B|. The 3 × 3 blocks $\hat H(q_x)$ describing the three internal states of the f = 1 manifold are

$$\begin{eqnarray} \fl \hat{H}(q_x) = \left( \begin{array}{@{}ccc@{}} \frac{\hbar^2(q_x - 2k_R)^2}{2m} + \hbar\delta & \frac{\hbar\Omega_R}{2} & 0 \\ \frac{\hbar\Omega_R}{2} & {q_x}^2 + \epsilon & \frac{\hbar\Omega_R}{2} \\ 0 & \frac{\hbar\Omega_R}{2} & \frac{\hbar^2(q_x + 2k_R)^2}{2m} - \hbar\delta \end{array}\right).\nonumber\\ \end{eqnarray} \tag{ 106 }$$

In this expression, δ is the detuning of the two photon dressing transition from resonance, accounts for any quadratic Zeeman shift (not included in the section 5.2), Ω_R is the two-photon transition matrix element, and q_x, in units of the recoil momentum k_R, is the atomic momentum displaced by a state-dependent term q_x = k − 2k_R for m_F = −1, q_x = k for m_F = 0, and q_x = k + 2k_R for m_F = +1. Here, the three eigenvalues are denoted by E_± and E₀. As with the two-level case, the states associated with eigenvalues E_± experience an effective vector potential which can be made position-dependent with a spatially varying detuning δ. The E₀ also experiences a synthetic field, but at higher order in $\Omega_R^{-1}$. A magnetic field gradient along e_y gives $\delta\propto y$, and generates a uniform synthetic magnetic field normal to the plane spanned by the dressing lasers and real magnetic field B.

Figure 14 shows the basic principle for creating artificial vector potentials. In these experiments an optically trapped ⁸⁷Rb BEC was adiabatically transferred from the initial ${\mathinner{|{f=-1,m_F = 1}\rangle}}$ state into the ${\mathinner{|{q_x=0, -}\rangle}}$ dressed state, with quasimomentum q = 0, which corresponds to the lowest energy dressed band at δ = 0, as identified by the black arrow in the top-left panel to figure 14(a). Once loaded into this state, the atoms were held for a brief equilibration time, at which point both the Raman lasers and the trapping lasers were suddenly turned off. After the turn off, the atoms were allowed to travel ballistically along e_x, while a magnetic field gradient Stern–Gerlach separated the three m_F components along e_y. This process allowed the direct detection of the spin-momentum superpositions which comprise the Raman dressed states. The top-right panel of figure 14(a) shows that for δ = 0 the ground state BEC is located at q = 0, and that the dressed state is made up of three different m_F states each with momentum given by $k_{m_F} = q - 2 m_F k_R$.

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The lower two panels of figure 14(a) depict an experiment where the BEC is loaded as described above. The detuning is slowly ramped from 0 to ℏδ = −2E_R. During such a ramp, the BEC adiabatically remains at the minimum of E₋(q). The magnitude of the resulting shift is consequently associated with the artificial vector potential. The results of many such experiments is shown in figure 14(b) which compares the measured vector potential. In this case the Raman lasers intersected at 90°. This has the effect of decreasing the effective recoil momentum to $k_L = k_R / \sqrt{2}$ and the recoil momentum to E_L = E_R/2. The prediction of the model is obtained by numerically solving equation (106) with no free parameters.

It is important to note that the BEC was always at rest in these experiments. The uniform and static vector potential would therefore normally not be experimentally detectable. In this case, however, the vector potential's difference from the δ = 0 case is encoded into the spin-momentum superposition making up the dressed state, and can be accessed in experiment. See also [23] for more details of this effect.

With access to an artificial vector potential it is natural to consider what kind of mechanical forces one can induce on the atoms with suitable gradients of this potential. Several possibilities are available. One can for instance change the vector potential from an initial $\pmb{\mathcal A}_i$ to a final $\pmb{\mathcal A}_f$ by suddenly changing δ. If $\pmb{\mathcal A}$ behaves as a real vector potential, then this change should be associated with a force that accelerates the atoms, changing their mechanical momentum by an amount ${\boldsymbol \delta} {\bit p}_{\rm mech} = (\pmb{\mathcal A}_i - \pmb{\mathcal A}_f)$.

There are currently two experiments by Lin et al [23] which illustrate the effect of an artificial electric field. The atoms were first prepared at an initial detuning δ producing a non-zero $\pmb{\mathcal A}_i$, and then suddenly making |δ| ≫ Ω_R. This results in $\pmb{\mathcal A}_f = \pm2\hbar k_L{\bit e}_x$ which depends on the sign of δ, see figure 14(b). Combined with free expansion of the cloud the final momentum can be measured, shown in figure 15(a).

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Alternatively, $\pmb{\mathcal A}$ can be changed only slightly, as illustrated in figure 15(b). By taking the average velocity of all three spin-momentum components the quasimomentum and also the group velocity can be monitored. The subsequent evolution in the harmonic confining potential is then monitored. Figure 15(b) depicts the mechanical momentum p_mech, related to the group velocity, by multiplying the average velocity by the effective mass m^* ≈ 2.5 m. This quantity has the same amplitude as the canonical momentum oscillations, but is centered on zero.

With a synthetic Abelian vector potential present, it is also natural to consider creating an artificial magnetic field $\pmb{\mathcal B}={\boldsymbol \nabla}\!\times\!\pmb{\mathcal A}$. The key ingredient for obtaining a non-zero effective magnetic field is the dressed state, given by a detuning which varies linearly along e_y, with δ(y) = δ'y. This gives $\pmb{\mathcal A}\approx -2\hbar k_R \delta/\Omega_R{\bit e}_x$ when δ ≪ Ω_R.

To create an artificial magnetic field such as in [22], the geometry shown in figure 6 can be used, with the three m_F states of ⁸⁷Rb's hyperfine ground state Zeeman split by a biasing magnetic field along e_y. The states are coupled with a pair of Raman lasers giving a momentum exchange along e_x. Using a pair of quadrupole coils in an anti-Helmholtz geometry aligned along e_z, produces a magnetic field gradient of the form B_quad ≈ β'(xe_x + ye_y − 2ze_z) which adds to the bias field B₀ = B₀e_y. The resulting Zeeman shift is proportional to |B₀ + B_quad| ≈ B₀ + β'y giving the desired detuning gradient along e_y, which in turn results in a synthetic magnetic field $\pmb{\mathcal B} = {\mathcal B}{\bit e}_z$. Figure 16(a) depicts images of atoms after TOF, in which the appearance of an artificial magnetic field is marked in two ways.

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Firstly, the initially symmetric cloud acquires a shear as ${\mathcal B}$ increases. With the gradient present, the vector potential depends on y as ${\mathcal A}_x\propto y$. When the Raman lasers are turned off at the beginning of the time of flight (TOF), ${\mathcal A}_x\rightarrow0$. This process introduces an electric field along e_x proportional to y, and as a result the BEC undergoes a shearing motion during the expansion.

Above a critical gradient vortices spontaneously enter into the non-rotating BEC. In this regime, the BEC is described by a macroscopic wave function ψ(r) = |ψ(r)|e^iφ(r), which obeys the Gross–Pitaevskii (GP) equation. The phase φ winds by 2π around each vortex, with amplitude |ψ| = 0 at the vortex center. The magnetic flux $\Phi_{{\mathcal B}}$ results in N_v vortices and for an infinite, zero temperature system, the vortices are arrayed in a lattice [188] with density ${\mathcal B}/h$. For finite systems vortices are energetically less favorable, and their areal density is below this asymptotic value, decreasing to zero at a critical field ${\mathcal B}_c$.

The mean-field phase diagram of the homogenous, V(r) = 0, spin–orbit-coupled BEC is already altered by a simple interaction term of the form [248]

$$\begin{equation} \hat{H}_{\rm int} = \int \rmd{\bit r} \, g \hat n_1 ^2 + g \hat n_2 ^2 + g_{12} \hat n_1 \hat n_2, \end{equation} \tag{ 119 }$$

where the two scattering parameters g and g₁₂ account for spin-dependent collisions in a minimal manner, and where $\hat n_{\mu} = \hat \Psi^{\dagger}_{\mu} \hat \Psi_{\mu}$ denote the density operators. The mean-field phase diagram described below has been obtained by Wang et al [248], through a minimization of the GP free energy for the two-component order parameter $\Psi ({\bit r})=\langle \hat \Psi ({\bit r}) \rangle$, derived from the SOC Hamiltonian (118)–(119). When g₁₂ < 2g, the densities associated with the two spin components ρ_1,2(r) = |Ψ_1,2|² are uniform in space, but the phase of the condensate $\Psi ({\bit r})=\langle \hat \Psi ({\bit r}) \rangle$ varies periodically along one direction, spontaneously breaking rotational symmetry. This regime characterizes the 'plane wave phase', where the macroscopic wave function of the condensate takes the simple form [216, 248]

$$\begin{equation} \Psi ({\bit r})= \sqrt{(\rho_1+\rho_2)/2} \, \rme^{\rmi \kappa x} \, (1 , 1)^{T}. \end{equation} \tag{ 120 }$$

Without loss of generality, we have taken the plane wave to be along e_x, reflecting the spontaneous symmetry breaking. When g₁₂ > 2g, the condensate Ψ(r) is described by a superposition of two such plane waves,

$$\begin{eqnarray} &&\fl \Psi ({\bit r})= \frac{1}{2} \sqrt{\rho_1+\rho_2} \, \biggl [ \rme^{\rmi \kappa x} \, \left(\begin{array}{@{}c@{}}1\\ 1\end{array}\right) + \rme^{-\rmi \kappa x} \, \left(\begin{array}{@{}c@{}}1 \\ -1\end{array}\right) \biggr ] \nonumber\\ &&= \sqrt{\rho_1+\rho_2} \left(\begin{array}{@{}c@{}}\cos(\kappa x)\\ \rmi\sin(\kappa x)\end{array}\right), \end{eqnarray} \tag{ 121 }$$

yielding a periodic spatial modulation of the spin-density

$$\begin{equation*} \rho_{s}({\bit r})=\vert \Psi_{1} ({\bit r}) \vert^2 -\vert \Psi_{2} ({\bit r}) \vert^2 = (\rho_1+\rho_2) \cos (2 \kappa x). \end{equation*}$$

This phase is known as the 'stripe superfluid' or 'standing wave phase' [248]. The mean-field phase diagram of the homogeneous system, obtained from the GP equation derived from equations (118)–(119) is robust against deformation of the SOC term $\hat{\pmb{\mathcal A}}$ (without losing the cylindrical symmetry), and generally holds in the presence of additional fields, e.g. Zeeman terms [216].

The phase diagram is enriched by the harmonic confinement V(r) = mω²r²/2 generally present in cold-atom experiments [34]. Remarkably, novel phases emerge in trapped systems [232], even under the strong (and generally unphysical) assumption that all the interaction processes are described by a single scattering parameter, i.e. when the interaction term reduces to the spin-independent form $\Psi^{\dagger} \hat G\Psi=g_{{\rm eff}} (\rho_1(r)+\rho_2(r))^2$, where g_eff is an effective scattering length describing all collisions [232]. This strong simplification aims to isolate the main effects introduced by the trap in spin–orbit coupled gases⁸. The external trap introduces an additional energy scale ℏω into the homogenous problem discussed above, hence it is natural to describe the resulting phase diagram in terms of the dimensionless parameters $\mathfrak{k}=\kappa l_0$ and $\mathfrak{g}=g_{{\rm eff}} m / \hbar^2$, where $l_0=\sqrt{\hbar/m\omega}$.

In the non-interacting regime where SOC dominates ( $\mathfrak{g}=0$, $\mathfrak{k} \gg 1$), the upper branch E₊(k) can be neglected, and, going to the momentum representation, one obtains the single particle wavefunction Ψ(k) = ψ(k) u₋(k), where

$$\begin{equation} {\bit u}_{-}({\bit k})=\frac{1}{\sqrt{2}}\left(\begin{array}{@{}c@{}} 1 \\ \frac{k_x+\rmi k_y}{k}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{@{}c@{}} 1 \\ \rme^{\rmi \varphi}\end{array}\right) \end{equation} \tag{ 122 }$$

is an eigenstate of the homogeneous system in the lowest branch E₋(k); and φ denotes the polar angle of k. Adding the confining potential to the single-particle Hamiltonian, $H ({\bit k}) \propto (\hbar {\bit k} - \hat{\pmb{\mathcal A}})^2 - \nabla_k^2$, and solving the corresponding Schrödinger equation using the ansatz ψ(k) = ∑_lk^−1/2 f_l(k) e^ilφ, yields the eigenenergies [232, 259]

$$\begin{equation} E_{nl}=\frac{(l+\frac{1}{2})^2}{2\mathfrak{k}^2}+n+\frac{1}{2}, \end{equation} \tag{ 123 }$$

where n labels the radial excitations around the minima of the Mexican-hat potential k ≈ κ (see figure 19) and $l \in \mathbb{Z}$. The degenerate ground states

$$\begin{equation} \Psi_{l=\{0,-1\}}(k)\propto \rme^{-(k-\kappa)^2 l_0^2/2+\rmi l \varphi}{\bit u}_- (\varphi) \end{equation} \tag{ 124 }$$

have quantum numbers n = 0 and l = {0, −1}. In real space, and using polar coordinates (r, θ), the two degenerate ground states become

$$\begin{eqnarray} &&\fl \Psi_0(r,\theta)\propto \left(\begin{array}{@{}c@{}} J_0(\mathfrak{k} r) \\ \rme^{\rmi\theta}J_1(\mathfrak{k} r) \end{array}\right),\nonumber\\ &&{\rm and} \quad \Psi_{-1}(r,\theta)\propto \left(\begin{array}{@{}c@{}} \rme^{-\rmi\theta}J_1(\mathfrak{k} r) \\ J_0(\mathfrak{k} r) \end{array}\right). \end{eqnarray} \tag{ 125 }$$

These half-vortex states [165, 216, 232, 242, 248, 262] appear in various contexts, including topological quantum computing [263], superfluid ³He [264] and triplet superconductors. The degenerate states in equation (125) and any linear combination thereof, yield the rotationally symmetric density distribution illustrated in figure 20(a). As pointed out in [232], this half-vortex regime (HV 1/2) survives for finite interaction strength $\mathfrak{g} < \mathfrak{g}_1$, where $\mathfrak{g}_1 \sim \mathfrak{k}^{-2}$ is related to the energy difference between the ground (|l + 1/2| = 1/2) and higher energy (|l + 1/2| = 3/2) states. Indeed, for large SOC strength k² ≫ 1, the energy difference between successive angular excitations (equation (123)) is small compared to radial excitations. Thus, the population of higher angular momentum states (l ⩾ 1) should be favored at greater, but still reasonably small, interaction strength $\mathfrak{g} \ge \mathfrak{g}_1$ [232]. A condensate formed with higher angular momentum states satisfying |l + 1/2| = 3/2, called the (HV 3/2)–phase, occurs in a certain range of the interaction strength $\mathfrak{g}_1 <\mathfrak{g} < \mathfrak{g}_2$, and has the ring-shaped density distribution depicted in figure 20(b).

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For larger interaction strength, $\mathfrak{g} > \mathfrak{g}_2$, the gas enters a 'lattice phase', where its density is modulated in the form of a hexagonal lattice, see figure 20(c). Interestingly, rotational symmetry is broken and the lattice period is independent of the interaction strength $\mathfrak{g}$. In fact, the distance between the lattice minima is solely dictated by the SOC strength k. Consequently, the resulting momentum distribution has a robust ring-like structure with six peaks, see figure 20(d).

The three phases discussed above all have in common that their densities satisfy (continuous or discrete) rotational symmetry. This is no longer the case for large interaction strength $(\mathfrak{g} \gg \mathfrak{g}_2)$. In this 'highly' interacting regime, the spin–orbit coupled cloud enters a 'striped phase', where the density is sinusoidally modulated in a specific direction, see figures 20(e)–(f). This anisotropic state shares similarities with the 'stripe phase' in the homogeneous system with spin-dependent interactions g₁₂ > g (see equation (121) and section 7.1.1). Mapping out the full phase diagram for the trapped spin–orbit coupled gas, by scanning the dimensionless parameters $\mathfrak{g}$ and $\mathfrak{k}$, reveals a surprisingly intricate picture, see figure 21. For readers further interested in the effects of finite temperature, anisotropic Rashba coupling $\hat{\pmb{\mathcal A}} = \hbar (\kappa_x \hat \sigma_x, \kappa_y \hat \sigma_y)$ and beyond-mean-field descriptions, we refer to [216, 242], and references therein. Finally, the interplay between SOC and dipolar interactions, leading to diverse spin and crystalline structures, have been explored in [265, 266], and the effects due to the position-dependent SOC have been studied in [241].

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Consider a particle subject to a gauge potential $\pmb{\mathcal A}\equiv q{\bit A}$ (where the coupling constant q would be the electric charge in conventional electromagnetism) confined to the e_x − e_y plane $\mathbb{R}^2$, described by the single-particle Hamiltonian

$$\begin{equation} H=\frac{1}{2m} \left[ {\bit p} - \pmb{\mathcal A} ({\bit r}) \right] ^2. \end{equation} \tag{ 129 }$$

When the particle follows a path from a reference point j to a point k, whose coordinates are denoted r_j and r_k, respectively, it acquires a 'magnetic phase factor' [40, 41, 306, 307]

$$\begin{equation} \psi ({\bit r}_k)= \exp\left(\frac{\rmi}{\hbar} \int_j^k \pmb{\mathcal A} \cdot {\rm d} {\bit l}\right) \psi_0 ({\bit r}_k)=U_{jk} \psi_0 ({\bit r}_k), \end{equation} \tag{ 130 }$$

where ψ₀(r_k) denotes the wavefunction in the absence of the gauge potential, as in figure 22(a). Here, we introduced the link variable [288]

$$\begin{equation} U_{jk}=\exp\left(\frac{\rmi}{\hbar} \int_j^k \pmb{\mathcal A} \cdot {\rm d} {\bit l}\right) = \rme^{\rmi \phi_{jk}} = \bigl ( U_{kj}\bigr)^{*} \in {\rm U}(1), \end{equation} \tag{ 131 }$$

embodying the effect of the gauge potential on the fictitious link connecting the points j and k. From the perspective of fiber-bundle theory, the link variables define the parallel transport on a principle fiber bundle $P ({\rm U(1)}, \mathbb{R}^2)$, where the connection $\mathbb{A}=\rmi \mathcal{A}_{\mu} \,\rmd x^{\mu}$ is determined by the gauge potential ${\mathcal A}$ (see [76]). The link variables U_jk = exp(iφ_jk) are generally called 'Peierls phases' in condensed-matter physics (see section 8.3), where they naturally appear in the description of solids subjected to magnetic fields [305, 308]. Under gauge transformations, the link variables (131) change according to

$$\begin{equation} \pmb{\mathcal A} \rightarrow \pmb{\mathcal A}'= \pmb{\mathcal A} + {\boldsymbol \nabla} \chi \end{equation} \tag{ 132 }$$

$$\begin{equation} U_{jk} \rightarrow U_{jk}'=U_{jk}\exp\left[\rmi \frac{\chi ({\bit r}_k) - \chi ({\bit r}_j)}{\hbar} \right]. \end{equation} \tag{ 133 }$$

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To develop lattice models, we introduce the notion of a plaquette, a closed region in space delimited by a set of points {r₁, r₂, r₃, ..., r_L}, connected by links. When the particle performs a loop □ around such a plaquette, it acquires an Aharonov–Bohm phase [41, 307]

$$\begin{eqnarray}&&\fl \psi({\bit r}_1) \mathop{\longrightarrow}\limits^{\square} \exp\left(\frac{\rmi}{\hbar} \oint_{\square} \pmb{\mathcal A} \cdot {\rm d} {\bit l}\right) \psi({\bit r}_1)= \exp\left(2 \pi \rmi\Phi_{\square}\right) \psi({\bit r}_1),\nonumber\\ \end{eqnarray} \tag{ 134 }$$

where $\Phi_{\square}=h^{-1} \int \pmb{\mathcal B} \cdot {\rm d} {\bit S}$ is the number of magnetic flux quanta Φ₀ = h penetrating the plaquette □ and $\pmb{\mathcal B} = {\boldsymbol \nabla} \times \pmb{\mathcal A}$ is the magnetic field associated with the gauge potential A, see figure 22(bc). This is the well-known Aharonov–Bohm effect [307, 309], which can also be expressed in terms of the link variables U_jk = exp(iφ_jk) defined in equation (131) as

$$\begin{eqnarray} &&\fl \rme^{\rmi 2 \pi \Phi_{\square}}=U_{12} \, U_{23} \, U_{34} \dots U_{L-1 \, L} \, U_{L1}=\prod_{\square}U_{jk}\nonumber\\ &&=\exp\left(\rmi \sum_{\square} \phi_{jk}\right). \end{eqnarray} \tag{ 135 }$$

Using equation (133), we see that the magnetic flux obtained through the 'loop' product (135) is a gauge-invariant quantity associated with the link variables

$$\begin{eqnarray} &&\fl \rme^{\rmi 2 \pi \Phi'_{\square}}=\prod_{\square}U_{jk}'= \underbrace{\exp\left\{\frac{i}{\hbar} \sum_{\square} \left[ \chi ({\bit r}_k) - \chi ({\bit r}_j)\right]\right\}}_{=1} \, \prod_{\square}U_{jk}\nonumber\\ &&=\rme^{\rmi 2 \pi \Phi_{\square}}. \end{eqnarray} \tag{ 136 }$$

Here, $\tilde \Phi= \Phi + N$ is physically equivalent to Φ for $N \in \mathbb{Z}$, since $\exp(\rmi 2 \pi \tilde \Phi)=\exp(\rmi 2 \pi \Phi)$.

These general considerations suggest that the lattice description of a quantum system subject to a gauge potential $\pmb{\mathcal A}$ should include

• (a)  
  a set of lattice sites {r_j},
• (b)  
  a set of links {j − k} connecting the sites and defining plaquettes □,
• (c)  
  a set of link variables {U_jk} associated with the links {j − k}, see figure 22(c).

Here, the information in $\pmb{\mathcal A}({\bit r})$ is fully contained in the link variables {U_jk}. The physical gauge-invariant quantity is the magnetic flux penetrating each plaquette Φ_□, which can be derived from the link variables (see equation (135), and figure 22(c)). The gauge-invariance relation (136) is interesting from a quantum-simulation point of view, as it defines the simplest set of link variables {U_jk} for any magnetic flux configuration. The generalization of these concepts to higher dimensions is straightforward.

B ∼ 50 T is the largest magnetic field that can routinely be applied to materials. Because the typical lattice spacing is a ≈ 10⁻¹⁰ m, the flux per elementary plaquette Φ_□ ∼ 10⁻⁴ is tiny. At higher fields, with Φ ∼ 0.1–1 (requiring a fantastically large magnetic field B ∼ 10⁴ T), the interplay between magnetic and lattice structures is captured by the Hofstadter model discussed in section 8.3.1. In contrast, synthetic magnetic fields for cold atoms trapped in optical lattices (section 8.4) can be engineered in the high-flux regime Φ ∼ 0.1–1.

The proceeding discussion focused on Abelian gauge potentials and magnetic fields, where the link variables U_jk ∈ U(1) are phase factors that do not couple to internal degrees of freedom. These concepts can be generalized to a wide family of multi-component systems with electronic spin, color, flavor or atomic spin degrees of freedom. In the following, we will generically refer to these as 'spin' degrees of freedom. In this context, gauge potentials and link variables may act differently on the different spin states, and they are therefore matrix-valued objects (see [42, 288]). The link variables belong to a Lie group, e.g. $\hat U_{jk} \in {\rm U(N)}$ or $\hat U_{jk} \in {\rm SU(N)}$, while the non-Abelian gauge potential is an element of the corresponding Lie algebra, e.g. $\hat{\mathcal{A}}_{\mu} \in \mathfrak{u}(N)$ or $\hat{\mathcal{A}}_{\mu} \in \mathfrak{s}\mathfrak{u}(N)$. A formal generalization of equation (131) defines the link variable

$$\begin{equation} \hat U_{jk}=\mathcal{P} \biggl \{\exp\left[\frac{\rmi}{\hbar} \int_j^k \hat{\pmb{\mathcal A}}({\bit r}) \cdot {\rm d} {\bit l}\right] \biggr \}, \end{equation} \tag{ 137 }$$

in terms of the non-Abelian gauge potential $\hat{\pmb{\mathcal A}}({\bit r})$ (the coupling constant is again subsumed into the definition of $\hat{\pmb{\mathcal A}})$. The path-ordered integral $\mathcal{P}(\cdot)$ is required because the matrices $\hat{\mathcal A}_{x,y}({\bit r})$ at different points of the path do not necessarily commute [288, 290, 291]. The link variable $\hat U_{jk}$ and the non-Abelian gauge structures are clearly connected in the continuum limit, where the sites r_{j, k} are sufficiently close to each other, r_j = r and r_k = r + e_μ, where  ≪ 1, e_μ is the unit vector along the μ direction and μ = x, y. In this case, the link variables are [288–291]

$$\begin{eqnarray} &&\hat U_{{\bit r} , {\bit r} + \epsilon {\bit e}_{\mu}}= \hat U_{\mu} ({\bit r})=\exp\left[\rmi \frac{\epsilon}{\hbar} \hat{\mathcal A}_{\mu} ({\bit r} ) \right], \\ &&\hat U_{{\bit r} + \epsilon {\bit e}_{\mu}, {\bit r}}= \hat U_{-\mu} ({\bit r} + {\bit e}_{\mu})=\exp\left[-\rmi \frac{\epsilon}{\hbar} \hat{\mathcal A}_{\mu} ({\bit r} ) \right], \nonumber \end{eqnarray} \tag{ 138 }$$

which lead to the 'loop' product around a unit square plaquette,

$$\begin{eqnarray} &&\fl \hat U_{x}({\bit r}) \hat U_{y}({\bit r} + \epsilon {\bit e}_{x}) \hat U_{-x}({\bit r} + \epsilon ({\bit e}_{x}+{\bit e}_{y}) ) \hat U_{- y}({\bit r} + {\bit e}_{y}) \nonumber\\ &&=\prod_{\square} \hat U_{jk}=\exp\left[{\frac{\rmi \epsilon ^2}{\hbar} \hat{\mathcal{F}}_{x y}} ({\bit r})\right]. \end{eqnarray} \tag{ 139 }$$

Equations (138)–(139) define the Yang–Mills field strength [310] (analogous to the antisymmetric tensor $\hat{{\mathcal{F}}}_{kl}$ defined by equation (23) for the geometric gauge potentials)

$$\begin{equation} \hat{\mathcal{F}}_{\mu \nu} ({\bit r})= \partial_{\mu} \hat{\mathcal{A}}_{\nu} ({\bit r}) - \partial_{\nu} \hat{\mathcal{A}}_{\mu} ({\bit r}) + \frac{\rmi}{\hbar} [\hat{\mathcal{A}}_{\mu} ({\bit r}), \hat{\mathcal{A}}_{\nu} ({\bit r})], \end{equation} \tag{ 140 }$$

to leading order in (see [288] and [291] for derivations). In general, the commutator, $[\hat{\mathcal{A}}_x,\hat{\mathcal{A}}_y]\ne0$, which is the hallmark of non-Abelian gauge theories [288]. The loop product (139) thus generalizes equation (135) to the non-Abelian case. This result connects the link variables, defined in the lattice context, and the field strength which plays a fundamental role in continuum theories.

As for the Abelian case, a lattice system with a non-Abelian gauge structure can be entirely described by (a) the lattice topology: the sites coordinates, and the links, delimiting the unit plaquettes, and (b) the link variables $\{\hat U_{jk} \}$ associated with the links. Under local gauge transformations, the link variables $\hat U_{jk} \in U(N)$ transform according to [288, 291]

$$\begin{equation} \hat U_{jk} \longrightarrow \hat U_{jk}'= \hat T_j \hat U_{jk} \hat T_k^{\dagger}, \end{equation} \tag{ 141 }$$

where $\hat T_k \in U(N)$ is a local transformation (a rotation in spin space at the lattice site r_k). Therefore, the loop product around a unit plaquette, delimited by the lattice sites {r₁, ..., r_L},

$$\begin{equation} \hat U_{\square}= \prod_{\square} \hat U_{jk}=\hat U_{12} \, \hat U_{23} \, \hat U_{34} \dots \hat U_{L-1 \, L} \, \hat U_{L1}, \end{equation} \tag{ 142 }$$

is not in general gauge invariant [42, 288], since

$$\begin{equation} \hat U_{\square}\longrightarrow \hat U_{\square}'= \hat T_1 \hat U_{\square}\hat T_1^{\dagger}. \end{equation} \tag{ 143 }$$

This is related to the fact that the field strength $\hat{\mathcal{F}}_{\mu \nu}$, or curvature [76], of the continuum theory is not gauge-invariant. We therefore introduce the Wilson loop [288, 289, 311], a gauge-invariant quantity associated to each plaquette □,

$$\begin{equation} W(\square)={\rm Tr} \bigl (\hat U_{\square} \bigr ), \end{equation} \tag{ 144 }$$

where Tr(·) indicates the trace of the N × N matrix. The gauge invariant Wilson loop (144) generalizes the notion of flux per plaquette Φ_□ to non-Abelian gauge fields.

Genuine non-Abelian properties result from the non-commutativity of the gauge structure. When a particle performs two successive loops γ₁ and γ₂, that both start and end at the same point r₀, it will acquire a geometric phase factor U ∈ U(N). If, for every pair of loops γ_1,2, this geometric phase factor does not depend on the order of the operations U = U₁₂ = U₂₁ (where U₁₂ (respectively, U₂₁) corresponds to the situation where the loop γ₁ (respectively, γ₂) has been performed first), then the system is Abelian in nature. The observable effects related to these phase factors are those of a commutative gauge theory. On the other hand, if the phase factors differ U₁₂ ≠ U₂₁ for some loops γ₁ and γ₂, then the gauge theory is genuinely non-Abelian. The manifestations of the underlying non-Abelian gauge structure do not have any Abelian counterpart [42, 126, 288–291].

In continuum gauge theories, the non-Abelian property U₁₂ ≠ U₂₁ can be traced back to the non-commutativity of the field strength, $[\hat{\mathcal{F}}_{\mu \nu} ({\bit r}), \hat{\mathcal{F}}_{\mu^{\prime} \nu^{\prime}} ({\bit r}^{\prime})] \ne 0$, which naturally stems from the non-commutativity of the gauge potential's components $[\hat{\mathcal{A}}_{\mu}, \hat{\mathcal{A}}_{\nu}] \ne 0$, see [42].

As already pointed out in section 3.3, the criterion $[\hat{\mathcal{A}}_{\mu}, \hat{\mathcal{A}}_{\nu}] \ne 0$ is generally used in the literature to specify 'non-Abelian gauge potentials', but it is not sufficient to attest that the system hosts genuine non-Abelian properties, which are captured by the field strength $\hat{\mathcal{F}}_{\mu \nu}$ or the loop operators U, as described above [42, 126]. To avoid any ambiguity, we stress that the term non-Abelian gauge fields should be entirely based on the non-commutativity of the field strength or loop operators (see also equation (145) below), as described in this section.

An analogous criterion can be introduced in the lattice framework, in terms of the loop operator $\hat U_{\square}$ defined in equation (142). The lattice gauge structure associated with the link variables $\{\hat U_{jk} \}$ is said to be genuinely non-Abelian if there exists a pair of loops $\hat U_{\square_1}$ and $\hat U_{\square_2}$, both starting and ending at some site j = 1, such that

$$\begin{equation} \hat U_{\square_1} \hat U_{\square_2} \ne \hat U_{\square_2} \hat U_{\square_1}. \end{equation} \tag{ 145 }$$

Note that the non-Abelian criterion, based on the non-commutativity property (145), is gauge invariant, as can be verified using equation (143). Therefore, the link variables $\{\hat U_{jk} \}$ which lead to non-commutating loop operators, provide genuine non-Abelian effects [42, 126, 261] on the lattice.

Besides, one can introduce a simple criterion to detect lattice configurations in which all the loop matrices $\hat U_{\square}$ can be simultaneously gauge-transformed into a simple phase factor

$$\begin{equation} \hat U_{\square}=\exp (\rmi 2 \pi \Phi) \hat{1}_{N \times N} , \quad \forall \, {\rm loops} \, \square. \end{equation} \tag{ 146 }$$

If such a reduction (146) was possible over the whole lattice, then the multi-component system would behave as a collection of uncoupled Abelian subsystems subjected to the same flux Φ [311]. On the square lattice, denoting $\hat U_{x,y}$ the link variables along e_x and e_y, the condition $[\hat U_x, \hat U_y]\ne 0$ would constitute a natural, but incomplete criterion to detect non-Abelian structures. For example, the link variables $\hat U_{x,y}=\exp ({\rm i} \pi \hat \sigma_{x,y} /2)$ have $[\hat U_x, \hat U_y]= - 2\rmi \hat\sigma_z\ne 0$, while the loop matrix around a unit cell is $\hat U_{\square}=\exp (\rmi \pi ) \hat{1}_{2 \times 2}$: this configuration therefore corresponds to a two-component lattice subjected to a uniform (Abelian) flux per plaquette Φ = 1/2⁹. As shown in [311], the Wilson loop (144) provides an unambiguous criterion to determine whether the set $\{\hat U_{jk} \}$ leads to uncoupled Abelian subsystems captured by equation (146). Indeed, the loop operator $\hat U_{\square} \in U(N)$ reduces to a simple phase factor $\hat U_{\square}=\exp (\rmi 2 \pi \Phi) \hat{1}_{N \times N}$ if and only if the Wilson loop satisfies |W(□)| = N.

The criterion based on non-trivial Wilson loops |W(□)| ≠ N is a necessary condition for attesting that the link variables $\{\hat U_{jk} \}$ produce a non-Abelian gauge field (equation (145)). However, we stress that this condition is not sufficient. For instance, a spin-1/2 lattice satisfying $\hat U_{\square}=\exp (\rmi 2 \pi \Phi \hat \sigma_z)$ in all its plaquettes is clearly Abelian in the sense of equation (145), while it is generally associated with a non-trivial Wilson loop |W(□)| ≠ 2, since the different spin components feel an opposite magnetic flux $\Phi \hat \sigma_z$, i.e. the trivialization (146) is not satisfied in this case. We note that in the context of the non-Abelian Aharonov–Bohm effect [313], such a gauge field, although Abelian, is loosely referred to as a 'non-Abelian flux'. Again, to avoid any ambiguity, we stress that non-Abelian gauge fields defined on a lattice are those that satisfy the strict criterion based on non-commutating loop operations (equation (145)).

A square lattice with a uniform magnetic field $\pmb{\mathcal B}= {\boldsymbol \nabla} \times \pmb{\mathcal A}=B {\bit e}_z$ is an iconic model—called the Hofstadter model [305]—of a two-dimensional electron gas (2DEG) in a strong magnetic field, and is the most simple case of a non-trivial gauge field in a lattice. Here, particles undergo the traditional (i.e. Abelian) Aharonov–Bohm effect when they circulate around the unit square plaquettes, ψ(r) → ψ(r)exp(i2πΦ). This model is described by equation (147), with the specific form

$$\begin{equation} \small{\hat{H}= -J \sum_{m,n} \rme^{\rmi \phi_{x} (m,n)} \hat{c}_{m+1,n}^{\dagger} \hat{c}_{m,n} + \rme^{\rmi \phi_{y}(m,n)} \hat{c}_{m,n+1}^{\dagger} \hat{c}_{m,n} + {\rm h.c.}}, \end{equation} \tag{ 148 }$$

where the lattice sites are located at (x, y) = (ma, na), $m,n \in \mathbb{Z}$, and a is the lattice period. Here, the link variables $U_{jk}=\exp \bigl [\rmi \phi_{x,y} (m,n) \bigr] \in {\rm U(1)}$, commonly known as 'Peierls phases', satisfy (135),

$$\begin{eqnarray} \prod_{\square} U_{jk}&= \rme^{\rmi \left[(\phi_{x} (m,n) + \phi_{y} (m+1,n) - \phi_{x} (m+1,n+1) - \phi_{y} (m,n+1) \right]}, \nonumber\\ &=\rme^{\rmi 2 \pi \Phi}. \end{eqnarray} \tag{ 149 }$$

In a tight-binding treatment [292, 305, 308], equation (148) can be obtained through the Peierls substitution $E_0(\hbar {\bit k} \rightarrow \hat{{\bit p}} -\pmb{\mathcal A})$, where E₀(k) is the tight-binding model's field-free dispersion relation¹⁰. The Hofstadter Hamiltonian (148) can also be obtained from the continuum Hamiltonian (129), by discretizing the spatial coordinates and the derivative operators [314]. This continuum approach is only rigorously valid in the limit a → 0, namely for small flux Φ = Ba²/Φ₀, where the energy structure only slightly deviates from the Landau levels. Away from this continuum limit, Φ ∼ 1, the tight-binding Hamiltonian (148) displays new features originating from the underlying lattice structure. The hopping phases in the Landau gauge are

$$\begin{equation} \phi_x=0 {\rm ,} \, \phi_y (m)= 2 \pi \Phi m, \mbox{see figure 23 (a).} \end{equation} \tag{ 150 }$$

For rational flux Φ = p/q, with $p,q \in \mathbb{Z}$, the Hamiltonian commutes with the magnetic translation operators $T_x^q \psi (m,n)= \psi (m+q,n)$ and T_yψ(m, n) = ψ(m, n + 1). In this gauge, the system is described by q × 1 magnetic unit cells (gauge dependent) and its energy spectrum splits into q (gauge independent) subbands, see figure 23(b). The spectrum has a fractal self-similar pattern set of eigenenergies, called the Hofstadter butterfly [305], see figure 23(c), and it has quantum Hall phases with topological order, to be discussed in section 9 (see also figure 23(b)). At the special 'π-flux' case Φ = 1/2, the Hofstadter lattice reduces to a two-band model displaying conical intersections at zero energy [315–317], similar to graphene's pseudo-relativistic spectrum [295, 318]. In this singular case, the system satisfies time-reversal symmetry (TRS) H(Φ = 1/2) ≡ H(−Φ). The Hofstadter model can easily be extended to other 2D lattices [319–323], and in particular to the honeycomb lattice [316, 324], making it an efficient tool to investigate the electronic properties of graphene or other exotic materials in strong magnetic fields [325]. Finally, the effects of interactions in the Hofstadter model have been analyzed in [326–329].

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In the early 1980s, the quantum Hall effect was discovered in 2D electronic systems at modest magnetic fields [77], and until a seminal work by Haldane [182], it was believed that the presence of a uniform magnetic field was necessary to produce this effect. In [182], Haldane introduced a lattice model demonstrating that the essential ingredient was not the magnetic field, but rather the associated breaking of TRS. In Haldane's model, while TRS is broken locally, the magnetic flux per unit cell is zero. Figure 23(d) shows this model's honeycomb lattice topology, featuring both nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping. While the NN hopping is trivial (positive and real valued), the NNN hopping terms are accompanied by non-trivial Peierls phases, indicating the presence of a TRS-breaking gauge potential. The corresponding Hamiltonian (147) is

$$\begin{equation} \hat H = - J \sum_{\langle k,l \rangle} \hat c_{k}^{\dagger} \hat c_{l}+ \rmi \lambda \sum_{\langle\!\langle k,l \rangle\!\rangle} \nu_{k,l} \hat c_{k}^{\dagger} \hat c_{l}, \end{equation} \tag{ 151 }$$

where 〈k, l〉 and 〈〈k, l〉〉 denotes the NN and NNN sites respectively of a honeycomb lattice, and ν_{k, l} = ±1 depending on the orientation of the path connecting the NNN sites 〈〈k, l〉〉. The Peierls phases U_kl = exp(i πν_{k, l}/2) are complex for NNN sites, leading to non-zero local magnetic flux Φ_α within triangular 'subplaquettes' α, even though the total flux Φ = ∑_α Φ_α = 0 inside each honeycomb unit cell. For λ ≠ 0, an energy gap opens in the spectrum, see figure 23(e), leading to non-trivial topological orders and quantum 'anomalous' Hall phases (see also section 9). The Haldane model played a surprising role in the prediction of the quantum spin Hall effect, an early precursor to TR preserving topological insulators (see below). It has recently been implemented with cold atoms using time-modulated optical lattices [490].

The Kane–Mele model was introduced as a tight binding framework to explore the effects of SOCs in graphene [330, 331]. Based on this model, Kane and Mele predicted a quantum spin Hall effect, leading to the current interest in TR-invariant topological insulators (see also [73, 74, 332] and [296]). Interestingly, the intrinsic SOC term can be modeled by a NNN Haldane hopping term, acting oppositely on the two spin components σ = ↑, ↓. In the presence of SOC, each spin component feels a local 'magnetic' flux, yet TRS is preserved because the fluxes are opposite for ↑ and ↓. The corresponding link variables can be represented as 2 × 2 matrices, with $\hat U_{kl}^{{\rm SO}}=\rmi\nu_{k,l} \hat \sigma_z$. In solids, the Rashba SOC produced by an external electric field [333, 334], is modeled by a NN hopping term that mixes the spin components non-uniformly. It involves SU(2) link variables such as $\hat U_{kl}^{{\rm R}} = (\hat{{\boldsymbol \sigma}} \times \hat{{\bit d}}_{kl})_z \propto \hat \sigma_{x,y}$, where $\hat{{\bit d}}_{kl}$ denotes the unit vector along the link connecting the NN sites k, l. The total Kane–Mele Hamiltonian is

$$\begin{eqnarray} \hat H =& - J \sum_{\langle k,l \rangle} \hat c_{k}^{\dagger} \hat c_{l}+ \rmi \lambda_{{\rm SO}} \sum_{\langle\!\langle k,l \rangle\!\rangle}\nu_{k,l} \hat c_{k}^{\dagger} \, \hat \sigma_z \, \hat c_{l} \nonumber\\ &+ \rmi \lambda_{{\rm R}} \sum_{\langle k,l \rangle} \hat c_{k}^{\dagger} \, (\hat{{\boldsymbol \sigma}} \times \hat{{\bit d}}_{kl})_z \, \hat c_{l}, \end{eqnarray} \tag{ 152 }$$

where the three terms correspond to: direct hopping between NN sites, intrinsic SOC, and Rashba SOC [330, 331]. The Kane–Mele model for graphene can therefore be viewed as the direct SU(2) analog of the Haldane model (151) with an extra Rashba term characterized by the strength λ_R featured in equation (152). Just as the Haldane model produces quantum Hall states, the Kane–Mele model produces quantum spin Hall states, which can be viewed as two superimposed, and opposite, spin-filtered QH phases. A 3D generalization of the Kane–Mele model (152), where the link variables $\hat U_{kl}^{{\rm SO}}$ are defined along the links of a diamond lattice [335] reveals the exciting physics of 3D topological insulators [73, 74, 296].

In the same spirit, Goldman et al [336] introduced a spinful SU(2) analog of the Hofstadter model on a square lattice. In this model, particles with spin σ = ↑, ↓ experience a uniform magnetic flux per plaquette Φ_↑ = −Φ_↓ opposite in sign for the two spin components. The corresponding Hamiltonian is

$$\begin{eqnarray} &&\fl \mathcal{H}\!=\!-J\sum_{m,n} \hat c_{m+1,n}^{\dagger} \rme^{\rmi 2\pi \gamma \hat \sigma_x} \hat c_{m,n} +\hat c_{m,n+1}^{\dagger} \rme^{\rmi 2\pi m \Phi \hat \sigma_z} \hat c_{m,n}+{\rm h.c.}, \nonumber\\ \end{eqnarray} \tag{ 153 }$$

where the SU(2) link variables $\hat U_{x},\hat U_y(m)$ act on the two-component field operator $\hat c_{m,n}$, defined at lattice site (x, y) = (ma, na). For γ = 0, this models corresponds to two decoupled copies of the spinless Hofstadter model (148)–(150). The effect of the link variable $\hat U_y (m) \propto \hat \sigma_z$ is therefore analogous to the intrinsic SOC in equation (152). For γ ≠ 0, the two spin components are mixed as they tunnel from one site to its NN. The link variable $\hat U_x \propto \sin (2 \pi \gamma) \hat \sigma_x$ plays a role similar to the Rashba coupling in equation (152). This model therefore captures the essential effects of the Kane–Mele TRS-invariant model in a multi-band framework, but offers the practical advantage of only involving NN hopping on a square lattice. The optical-lattice implementation of this model has been reported by Aidelsburger et al [64] and investigated by Miyake et al [65]. The effects of interactions in the SU(2) Hofstadter model have been analyzed by Cocks, Orth and co-workers in [338, 339].

The interplay between Abelian and non-Abelian gauge fields can be studied in a simple two-component model defined on a square lattice [311, 340], where the link variables along e_x and e_y are

$$\begin{eqnarray} &&\hat U_x= \rme^{\rmi \alpha \hat \sigma_y},\quad {\rm and} \quad \hat U_y (m)= \rme^{\rmi \beta \hat \sigma_x} \rme^{\rmi 2 \pi \Phi m}. \end{eqnarray} \tag{ 154 }$$

The Abelian part of the gauge structure given by the U(1) phase φ_y(m) = exp(i2 πΦm) in $\hat U_y$, corresponds to a uniform magnetic flux per plaquette Φ (see section 8.3.1). The SU(2) part is controlled by the parameters α and β. In two limiting cases this system is purely Abelian, i.e. |W(□)| = 2, when α = integer × π or β = integer × π, and when α = integer × π/2 and β = integer × π/2. In both cases, the system reduces to two uncoupled (Abelian) models. For arbitrary values of α, β the system features non-Abelian fluxes (see section 8.2.3) where the Wilson loop is uniform and non-trivial |W(□)| ≠ 2. This model was investigated in the contexts of quantum Hall physics [340–345], transition to the Mott-insulating phase [346] and non-Abelian anyonic excitations [129]. Alternative non-Abelian lattice models have been proposed and studied [261, 347–353], showing the rich properties stemming from non-Abelian structures in non-relativistic quantum systems.

This method is based on the possibility to couple atoms living on neighboring sites of an optical lattice, hence controlling their tunneling over the lattice, as illustrated in figure 24(a). Peierls phases are then engineered by controlling the phase of the coupling, which allows to simulate lattice (tight-binding) Hamiltonians with non-trivial link variables U_jk, for example leading to the Hofstadter Hamiltonian (148) in 2D. This method was further developed by several authors, such as Mueller [356], Gerbier–Dalibard [180], Anisimovas et al [362], Goldman et al [336, 363] and Mazza et al [20, 365]. The method, which can be extended to non-Abelian structures [261, 363, 365] and more exotic lattice geometries [366], can be summarized as follows:

• (a)  
  Gauge fixing: what link variables U_jk should be generated?
• (b)  
  Prevent spontaneous hopping along the links for which U_jk ≠ 1.
• (c)  
  Atom–light coupling: induce the hopping externally and engineer the desired link variables U_jk by tuning the coupling lasers, see figure 24(a).

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Any physical scheme accomplishing the crucial steps (b)–(c), relies on the specific properties of the atoms used in the experiment. For instance, Gerbier and Dalibard proposed an elegant method exploiting the unique properties of alkaline-earth or ytterbium atoms [180] (see also Anisimovas et al [362]). In their proposal, the induced-hopping (c) involves a coupling between the ground state and a long-lived metastable excited electronic state. For these atomic species, metastable states have remarkably long lifetimes, and thus, the single-photon transitions to the excited state have greatly reduced spontaneous emission rates (as compared, e.g., to schemes based on two-photon transitions). For the more common alkali atoms, e.g. Li, K, Rb, alternative schemes based on Raman couplings, i.e. two-photon transitions, are required [61, 64, 65, 179, 364].

We point out that the concept of laser-induced tunneling, which is based on the coupling of atoms living on neighboring lattice sites (figure 24(a)), is intimately related to methods exploiting time-periodic modulations ('shaking') of the optical lattice [59, 63, 96, 304, 357, 367]. This analogy is illustrated in figure 24(b). Here, atoms are trapped in an optical superlattice potential, where neighboring sites are shifted in energy by the offset Δ so as to inhibit the natural hopping. A resonant modulation of the lattice then allows to re-establish the hopping in a controllable manner. In this framework, and in contrast with the standard laser-assisted method described in section 8.4.1, the atoms can be prepared in a single internal state. Designing a modulation that also transfers momentum to the atoms, e.g. using two Raman beams, also allows to generate Peierls phases in the effective hopping matrix elements (see section 8.4.3 and works by Kolovsky [357], Creffield and Sols [358, 359], Bermudez et al [360, 361], Lim et al [412] and Baur–Schleier–Smith–Cooper [103]). This 'shaking' scheme was considered in the Munich [61, 64, 364] and MIT [65] experiments to imprint space-dependent Peierls phase in a 2D optical lattice, with a view to creating the Hofstadter model (148) with cold atoms. We point out that off-resonant potential modulations can also be considered to engineer Peierls phases in optical lattices, as recently demonstrated in Hamburg [95, 97] (see section 8.5 for a more detailed discussion on periodically driven cold-atom systems).

Below, we present the main ingredients for engineering Abelian gauge structures in optical lattices, based on the experiment performed in Munich in 2011 [61, 364]. The generalization to non-Abelian (matrix-valued) link variables $\hat U_{jk} \in {\rm U}(N)$ is briefly discussed in section 8.4.6. Other interesting schemes are further discussed in section 8.5.

Choosing the gauge.

Preventing natural hopping.

Atom–light coupling.

The tunneling between neighboring sites j → k must be induced and controlled externally, leading to an effective hopping matrix element $J^{{\rm eff}}_{j\rightarrow k}$. In this section, we consider the superlattice configuration used in the Munich experiment [61, 364], which is similar to the tilted lattice configuration implemented in the subsequent Munich and MIT experiments [64, 65]. We refer the reader to Refs. [80, 179, 180, 362, 363] for a description of the laser-induced tunneling methods using different internal states together with state-dependent lattices and inter-species coupling. In the superlattice illustrated in figure 25(a), atoms in neighboring sites occupy Wannier functions w_l(r − r_j) and w_h(r − r_k), respectively (here the indices l and h refer to the low energy and high energy sites in the superlattice). For a given energy offset Δ, tunneling can be induced by an external time-dependent perturbation

$$\begin{equation} V_{{\rm coupl}}({\bit r},t)= \hbar \Omega \cos ({\bit q} \cdot {\bit r} - \omega_L t), \end{equation} \tag{ 155 }$$

from a pair of Raman lasers with wave vectors k_1,2 and frequencies ω_1,2. The ℏω_L = ℏ(ω₁ − ω₂) ≈ Δ energy difference allows resonant coupling between the staggered lattice sites, see figure 25(b). Here, Ω denotes the Rabi frequency characterizing the strength of the atom–light coupling. The key ingredient of the scheme is that the momentum transfer ℏq = ℏ(k₂ − k₁) can be adjusted by tuning the angle between the Raman beams. The resulting hopping amplitude, from a low energy site r_j to a high energy site r_k, is then given by the integral [61, 64, 65] (see also [179, 180])

$$\begin{eqnarray} J^{{\rm eff}}_{j\rightarrow k}&= \frac{\hbar \Omega}{2} \int w^{*}_h ({\bit r} - {\bit r}_k) w_l ({\bit r} - {\bit r}_j) \rme^{\rmi {\bit q} \cdot {\bit r}} {\rm d}{\bit r} , \nonumber\\ & = J^{{\rm eff}}_{0} \rme^{\rmi {\bit q} \cdot {\bit r}_j} = J^{{\rm eff}}_{0} \, U_{jk}, \end{eqnarray} \tag{ 156 }$$

where δ_kl = r_k − r_j = a is the vector connecting neighboring sites j and k. The effective hopping is therefore characterized by the amplitude $J^{{\rm eff}}_{0}$, along with a complex Peierls phase factor determined by the momentum transfer ℏq, see figure 25(b). In other words, the time-dependent perturbation (155) generates a link variable given by $U_{jk}=\exp \bigl [\rmi \phi ({\bit r}_j) \bigr]$, where φ(r_j) = (k₂ − k₁) · r_j. The effective hopping in equation (156) can also be written in the 'symmetric' notation

$$\begin{eqnarray} &&\fl J^{{\rm eff}}_{j\rightarrow k} ({\bit r}_j,{\bit r}_k)= \frac{\hbar \Omega}{2} \rme^{\rmi{\bit q} \cdot ({\bit r}_j + {\bit r}_k)/2}\nonumber\\ &&\quad \times \int w^{*}_h ({\bit r} - {\boldsymbol \delta}_{kl}/2) w_l ({\bit r} + {\boldsymbol \delta}_{kl}/2) \rme^{\rmi {\bit q} \cdot {\bit r}} {\rm d}{\bit r}\nonumber\\ &&= \tilde{J}^{{\rm eff}}_{0} \rme^{\rmi{\bit q} \cdot ({\bit r}_j + {\bit r}_k)/2}, \end{eqnarray} \tag{ 157 }$$

which is appropriate for certain geometries, especially for the non-square lattices [363, 368–370]. Importantly, the effective hopping from a high energy site r_k to a low energy site r_j

$$\begin{eqnarray} J^{{\rm eff}}_{k\rightarrow j} ({\bit r}_k) &= \bigl (J^{{\rm eff}}_{j\rightarrow k} ({\bit r}_j) \bigr ) ^* \nonumber\\ &= J^{{\rm eff}}_{0} \rme^{- \rmi {\bit q} \cdot {\bit r}_j}= \tilde{J}^{{\rm eff}}_{0} \rme^{-\rmi{\bit q} \cdot ({\bit r}_j + {\bit r}_k)/2}, \end{eqnarray} \tag{ 158 }$$

results from a momentum transfer −ℏq reversed in sign as compared to the low-to-high case¹¹. Finally, we point out that the effective tunneling amplitude can be expressed in terms of a Bessel function of the first kind $J^{{\rm eff}}_{0} = J \mathcal{J} (\kappa)$, with κ ∝ Ω/Δ, as expected for shaken lattice systems (see [64, 65, 357, 364], and also section 8.5.1 where off-resonant driving is discussed in a related context).

Staggered flux configuration.

Equations (156)–(158) and figure 25 show that the induced tunneling is necessarily accompanied with alternating Peierls phases: $\phi_y (m,n)= - \phi_y (m,n+1)=\phi_y (m,n+2)=\dots$. Generating such Peierls phases on a square lattice,

$$\begin{equation} \phi_x=0 {\rm ,} \, \phi_y (m,n)= (-1)^{n} \, 2 \pi \Phi m, \end{equation} \tag{ 159 }$$

leads to a staggered flux configuration, where successive plaquettes along e_y are penetrated by fluxes ±Φ. Here, the synthetic magnetic flux Φ is governed by the Raman coupling lasers Φ ∼ a|q| = a|k₂ − k₁|, and it can thus be set in the high-flux regime Φ ∼ 0.1 − 1, equivalent to the effects of a gigantic magnetic field B ∼ 10⁴T for electrons in a crystalline lattice. However, this staggered configuration does not break TRS, and is only equivalent to the Hofstadter model (150) in the 'π-flux' limit Φ → 1/2. The experimental realization of the staggered flux model in 2011 [61] constituted a first important step towards the realization of the Hofstadter model (i.e. uniform magnetic flux over the whole lattice). Staggered magnetic fluxes have also been realized in off-resonant shaken triangular optical lattices in 2013 [97].

The uniform flux configuration.

Additional coupling lasers (see above) are needed for special lattice geometries, such as the honeycomb lattice. As was shown in [368, 369] and [362], it is possible to generate local Haldane-like fluxes using two intertwined triangular sublattices, denoted A and B, coupled together by a single Raman coupling A ↔ B. In this setup, the natural hopping between the NNN sites of the honeycomb lattice (i.e. the NN sites of the triangular sublattices A and B) remains, while the laser-induced hopping with Peierls phases acts between NN sites. For arbitrary values of the Raman wave vectors k₁ − k₂ = q, this configuration has non-trivial fluxes inside the subplaquettes of the unit hexagonal cells, and therefore reproduces the Haldane model's quantum Hall phases. A complete description of the phase diagrams and flux configurations stemming from this cold-atom setup can be found in [369]. This scheme could also be easily implemented with Yb atoms, in which case it is convenient to set the lattice potential at a so-called 'anti-magic' wave-length [180]. In this case, the two internal states are automatically trapped in two intertwined triangular and honeycomb lattices, and the coupling can be directly addressed with a single-photon transition, see Anisimovas et al [362].

This honeycomb system could be also extended to the spinful Kane–Mele model for Z₂ topological insulators [330]. In this case, each sublattice must trap atoms in two internal atomic states (see section 8.4.6) coupled independently by lasers such that the tunneling operators (2 × 2 matrices acting between NN sites j and k) are $\hat U (j, k)= \exp ({\rm i} \hat\sigma_z \delta {\bit p} \cdot ({\bit r}_{j} + {\bit r}_{k})/2)$. In this spinful honeycomb lattice configuration, non-trivial Z₂ topological phases, featuring helical edge states exist [369]. This system can include TR breaking perturbations, testing the robustness of the Z₂ topological phases and exploring phase transitions between helical and chiral edge textures [348, 352].

The general atom-coupling method presented in section 8.4.3 for creating Peierls phases $U_{jk}=\exp \bigl (\rmi \phi ({\bit r}) \bigr)$ can be extended to produce matrix-valued link variables (e.g. $\hat U_{jk} \in {\rm U} (N))$. For this, each lattice site must host N nearly degenerate spin states with index τ = 1, ..., N. Using external (real) magnetic fields to lift any degeneracies, each state τ can then be individually trapped and coupled to other sublevel states τ ≠ τ'. In particular, the effective tunneling of an atom in a specific internal state τ, from site j to k, could then be induced and controlled individually by external couplings [261, 311, 340, 363]. In principle, the coupling can be chosen to flip the atomic spin τ → τ' ≠ τ during the hopping process, resulting in a non-diagonal hopping matrix $\hat J^{\tau \tau'}_{j \rightarrow k} \not\propto \hat{1}_{N \times N}$. Considering the simplest case N = 2, this scheme can produce synthetic SOC terms with $\hat U_{jk} \sim \hat{\sigma}_{\mu}$. Such non-Abelian gauge potentials, based on spin-dependent hopping, lead to the spin-1/2 models presented in section 8.3.3, making possible topological insulators in 2D and 3D optical lattices [20, 64, 336, 337, 365]. Non-trivial hopping operators along both spatial directions, using laser-induced tunneling techniques, generally require checkerboard lattices [363].

A quite different strategy consists in using off-resonant periodically driven optical lattices [96, 98]. One can generate artificial magnetic flux in such lattices by combining lattices and time-dependent quadrupolar potentials [101, 371], by modulating the lattice depth (i.e. the tunneling amplitude) in a directional manner [105] or by shaking optical lattices [59, 63, 96, 304, 357, 367]. Similarly, SOCs could be generated by subjecting an optical lattice to time-dependent magnetic fields [101, 203, 204]. A general formalism describing periodically driven quantum systems and effective gauge structures can be found in [101, 105].

Motivated by the recent experiments in Hamburg [63, 95, 97], let us consider a scheme based on shaken optical lattices, as already outlined in section 2.2. The method relies on off-resonant modulations of the lattice potential V_OL(r'), where r' ≡ r'(t) = r − r₀(t) is the position vector in the oscillating frame of reference (see equation (4)). The displacement vector r₀(t) = r₀(t + T) is time-periodic, with period T = 2π/ω, and is given by equation (5) for linear or circular harmonic shaking. As presented in section 2.2, the transformation to a non-inertial frame of reference modifies the Hamiltonian by adding a spatially homogeneous vector potential $\pmb{\mathcal A}(t)$ to the momentum, see equation (7):

$$\begin{eqnarray} &&\fl H^{\prime}=\frac{\left({\bit p}-\pmb{\mathcal A} (t)\right)^{2}}{2m}+V_{\mbox{OL}}\,\left({\bit r}\right),\quad{\rm with}\quad\pmb{\mathcal A}=m\dot{{\bit r}}_{0}\left(t\right).\nonumber\\ \end{eqnarray} \tag{ 160 }$$

The full Hamiltonian H' shares the same spatial periodicity as the lattice potential V_OL (r), hence, the quasimomenta are still good quantum numbers for the instantaneous eigenstates and eigenenergies of the Hamiltonian H'. Since the Hamiltonian is also time-periodic, it is appropriate to deal with the quasi-energies and the corresponding eigenstates of the effective (Floquet) Hamiltonian $\hat{H}_{{\rm F}}$ (defined below in equation (162)) when analyzing the topological properties and effective gauge structures induced by the driving [101, 105, 106].

Let us consider a situation where the potential V_OL (r) traps atoms in a deep 1D optical lattice, directed along x, which is subjected to a harmonic modulation x₀(t) = κ sin (ωt). Using the tight-binding approximation, the second-quantized lattice Hamiltonian reads

$$\begin{eqnarray} &&\fl \hat{H}(t)=-J\left(\hat{T}\rme^{\rmi{\mathcal{A}}\left(t\right) a/\hbar}+\hat{T}^{\dagger}\rme^{-\rmi{\mathcal{A}}\left(t\right) a/\hbar}\right),\quad\hat{T}=\sum_{j}\hat{a}_{j+1}^{\dagger}\hat{a}_{j},\nonumber\\ \end{eqnarray} \tag{ 161 }$$

with ${\mathcal{A}}(t)=m\dot{x}_{0}(t)=m\kappa\omega\cos(\omega t)$, where the operator $\hat{a}_{j}^{\dagger}$ creates a particle at lattice site x = ja, a is the lattice spacing and J is the tunneling matrix element between the neighboring sites. The time-dependent phase factors $\exp(\pm {\rm i}{\mathcal{A}}(t) a/\hbar)$ accompanying the hopping terms appear by simple application of the Peierls substitution [305, 372]. It is to be emphasized that these 'time-dependent Peierls phases' oscillate fast in time, and in particular, they should not be mistaken with the standard Peierls phases associated with (synthetic) magnetic flux discussed in section 8.3.1.

The effective (Floquet) Hamiltonian $\hat{H}_{{\rm F}}$ ruling the time-averaged dynamics can be defined through the evolution operator over a cycle [101, 105, 106]

$$\begin{eqnarray} &&\fl \hat{U}(T)=\mathcal{T}\exp\left(-\rmi\int_{0}^{T}\hat{H}(\tau) {\rm d}\tau\right)=\exp\left(-\rmi T\hat{H}_{{\rm F}}\right), \end{eqnarray} \tag{ 162 }$$

where $\mathcal{T}$ denotes time ordering. In the absence of additional confining trap (e.g. a harmonic trap V_conf ∼ x²), and imposing the periodic boundary conditions, one verifies that the 1D tight-binding Hamiltonian in equation (161) commutes with itself at different times. In this case, the time ordering can be omitted in equation (162), and the effective Hamiltonian $\hat{H}_{{\rm F}}$ is then simply obtained through the time average

$$\begin{eqnarray} \fl \hat{H}_{{\rm F}}&=\left(1/T\right)\int_{0}^{T}\hat{H}(\tau) \,{\rmd}\tau , \end{eqnarray} \tag{ 163 }$$

$$\begin{eqnarray} \fl &=-J\mathcal{J}_{0}(\xi_{0})\left[\sum_{j}\hat{a}_{j+1}^{\dagger} \hat{a}_{j}+a_{j-1}^{\dagger}\hat{a}_{j}\right],\quad \xi_{0}=m\kappa a\omega/\hbar, \nonumber\\\fl & \end{eqnarray} \tag{ 164 }$$

where $\mathcal{J}_{0}(\xi_{0})$ denotes the Bessel function of the first kind [59, 101, 357].

Thus, modulating an optical lattice enables one to control the sign and amplitude of the tunneling between the neighboring sites: $J_{j\rightarrow k}^{{\rm eff}}=-J\mathcal{J}_{0}(\xi_{0})$. This can have non-trivial consequences for triangular optical lattices, where a change of sign $\mathcal{J}_{0}(\xi_{0})<0$ leads to staggered synthetic magnetic fluxes and frustrated magnetism [95]. The effects of additional potentials, which would require time-ordering in equation (162), can be evaluated by considering a perturbative treatment in (1/ω) [101, 104, 373, 374]. In particular, we note that the cycle-averaged effective Hamiltonian in equation (163) corresponds to the lowest-order term of the Magnus expansion [373], which is generally relevant for sufficiently short periods T.

To induce complex effective Peierls phase factors, $J_{j\rightarrow k}^{{\rm eff}}\rightarrow J_{j\rightarrow k}^{{\rm eff}}\rme^{\rmi\theta_{jk}}$, non-sinusoidal driving is necessarily required, so that certain temporal symmetries should be broken by the forcing [63]. The specific protocol implemented by Struck et al [63], consists in a sinusoidal forcing over a period T₁ interrupted by a short periods of rest T₂, so the lattice perturbation has a total period of τ = T₁ + T₂ and a zero mean value. Under such conditions, the effective tunneling operator is [63]

$$\begin{equation} \frac{J_{j\rightarrow k}^{{\rm eff}}(\xi_{0},\omega)}{J}=\frac{T_{2}}{\tau}\rme^{\rmi K(T_{1}/\tau)} +\mathcal{J}_{0}(\xi_{0})\frac{T_{1}}{\tau}\rme^{-\rmi K(T_{2}/\tau)}, \end{equation} \tag{ 165 }$$

leading to a constant but complex valued effective Peierls phase. This scheme was realized experimentally in Hamburg, for bosons in a 1D optical lattice, where the phase θ affected the lowest Bloch band dispersion through $E(k)=-2\vert J_{j\rightarrow k}^{{\rm eff}}(\xi_{0},\omega)\vert\cos(ka-\theta)$. In this setup, the effective Peierls phase θ can therefore be evaluated by measuring the quasimomentum distribution of the BEC, whose superfluid ground state is reached at a finite value dictated by θ [63].

The extension of this method to generate synthetic (staggered) magnetic fluxes in 2D optical lattices has been implemented for a triangular optical lattice [95, 97]. The Haldane model has been realized experimentally using such time-modulated honeycomb optical lattices [490] as already mentioned in section 8.4.2. Similar schemes can be envisaged to produce synthetic SOCs, uniform magnetic fields and topological insulating states with shaken optical lattices [100]. Recently, and in direct analogy with solid-state-physics Floquet topological states [104, 105, 107, 109, 375], it has been suggested that (circularly) shaken hexagonal optical lattices could reproduce the physics of graphene subjected to circularly polarized light (see section 2.2 and [102, 103]), with a view to realizing the Haldane model [182] and topological bands with cold atoms.

In general, the great versatility of periodically driven systems can be exploited to generate a wide family of gauge structures in cold-matter systems. Let us illustrate this fact by considering a general static Hamiltonian $\hat H_0$ subjected to a single-harmonic modulation $\hat V(t)= \hat A \cos (\omega t) + \hat B \sin (\omega t)$, where $\hat A$ and $\hat B$ are arbitrary operators. Following [101], the effective Hamiltonian reads

$$\begin{eqnarray} &&\fl \hat H_{{\rm eff}}= \hat H_0 + \frac{\rm i}{2 \omega} [\hat A, \hat B] + \frac{1}{4 \omega ^2} \left ( [[ \hat A, \hat H_0], \hat A] + [[ \hat B, \hat H_0], \hat B] \right )\nonumber\\ &&+ \mathcal{O} (1/ \omega ^3). \end{eqnarray} \tag{ 166 }$$

Hence, convenient choices for the static Hamiltonian $\hat H_0$ and 'pulsed' operators $\hat A$ and $\hat B$ can generate a large variety of gauge structures, such as SOC (e.g. $[\hat A, \hat B] \sim p_x \sigma_x + p_y \sigma_y)$ and orbital magnetism (e.g. $[\hat A, \hat B] \sim x p_y - y p_x)$.

The main advantage of the shaking method is that it can be applied to different lattice geometries and various atomic species. In this sense, it is particularly interesting when considering the physics of fermionic species, where state-dependent lattices and Raman coupling (see section 8.5.3) are generally associated with higher spontaneous emission rates. However, periodically driven systems might also suffer from heating issues, which could naturally emanate from the external forcing. For instance, inter-particle collisions due to the micro-motion—which is inherent to the fast modulations captured by the time-dependent Hamiltonian, e.g. equation (161)—or transitions to higher energy bands could potentially lead to uncontrollable heating processes. Moreover, the manner by which a driven quantum system absorbs energy strongly depends on its characteristics (e.g. energy spectrum), which indicates that this question should generally be addressed on a case-by-case basis. The thermodynamics of periodically driven systems is an important subject, which has been recently studied in [376–379].

Recently, Jimenez-Garcia et al proposed and implemented an original setup producing periodic potential for cold atoms [62], with naturally complex tunneling operators $J^{{\rm eff}}_{j\rightarrow k} \rme^{\rmi \theta}$. In other words, this scheme produces the lattice and the Peierls phases θ simultaneously. In this method, ⁸⁷Rb atoms in the f = 1 ground level are concurrently subjected to a radio-frequency (RF) magnetic field and to two counter-propagating Raman laser beams with wave number k_R. The Raman and RF fields resonantly coupled the three spin states |m_F = 0, ±1〉, yielding the coupling Hamiltonian, see sections 4.4.1 and 5.2.2,

$$\begin{equation} \hat H_{{\rm RF + R}} ({\bit r}) = \boldsymbol{ \Omega} ({\bit r}) \cdot \hat{{\bit F}} + {\rm constant}, \end{equation} \tag{ 167 }$$

where $\hat{{\bit F}}$ is the angular momentum operator and

$$\begin{eqnarray*}&&\fl \boldsymbol{ \Omega} ({\bit r})\equiv\boldsymbol{ \Omega} (x)\nonumber\\ &&= \frac{1}{\sqrt{2}}(\Omega_{{\rm RF}} + \Omega_{{\rm R}} \cos ( 2 k_{{\rm R}} x) , - \Omega_{{\rm R}} \sin ( 2 k_{{\rm R}} x) , \sqrt{2} \delta). \end{eqnarray*}$$

Here Ω_RF and Ω_R are the coupling strength, or Rabi frequencies, associated with the RF and Raman fields, and δ is the detuning from Raman resonance [62]. The two non-trivial eigenvalues obtained of (167) are

$$\begin{equation} \Omega (x) = \pm \sqrt{\Omega_{{\rm RF}} \Omega_{{\rm R}} \cos ( 2 k_{{\rm R}} x) + {\rm constant}}, \end{equation} \tag{ 168 }$$

producing a periodic potential along e_x when Ω_RF, Ω_R ≠ 0. Importantly, equation (167) describes a spin-1 particle subjected to an effective space-dependent magnetic field B_eff(r) = ℏΩ (r)/g_Fμ_B, where g_F and μ_B are the Landé factor and Bohr magneton, respectively. Now, according to Berry [41], an eigenstate of Hamiltonian (167) acquires a geometric (Berry) phase θ when a loop γ is performed in the parameter space spanned by the 'magnetic' field B_eff(r). Here, B_eff(r) is space-dependent, and therefore, the spin will precess and make a loop each time an atom hops from one lattice site to its neighboring sites (the lattice structure being dictated by equation (168)). The hopping process is therefore naturally accompanied with a non-trivial 'Peierls' phase θ, proportional to the solid angle subtended B_eff(r). This strategy relies on the fine tuning of both the Raman and RF fields, and simultaneously produces a periodic structure intrinsically featuring Peierls phases, as demonstrated experimentally at NIST [62].

Extensions of this setup could realize non-trivial magnetic flux configurations in 2D lattices, through space-dependent Peierls phases. Again, this requires more complicated but realistic arrangements, for example, combining a state-independent optical lattice along one direction with a state-dependent 'vector' lattice along the other [62]. Alternatively, one could make use of the optical flux lattices [144, 150, 174, 175, 178, 181] considered in section 5.2.3.

Most proposals based on laser-coupling methods for the alkali atoms, involve two-photon Raman transitions coupling different internal atomic states. In this context, a crucial experimental issue concerns the minimization of undesired heating stemming from spontaneous emission, which cannot be reduced even using large detuning of the Raman lasers, as pointed out at the end of the section 4.4.1. This drawback is particularly severe for alkali fermions, e.g. ⁴⁰ K and ⁶ Li, where the possible detunings are so small that they necessarily imply large spontaneous emission rates. Therefore alternative methods are required to investigate the physics of fermionic atoms subjected to synthetic gauge potentials. Several solutions have already been invoked above, such as exploiting the coupling to long-lived electronically excited states (e.g. using Ytterbium) or engineering the gauge potential through lattice-shaking methods.

Another solution consists in trapping and coupling different hyperfine levels using the magnetic (Zeeman) potentials produced by an array of current-carrying wires, therefore avoiding the use of optical state-dependent lattices and Raman coupling [336]. Here we describe how such a setup could be implemented to reproduce the Abelian gauge structure (150) of the Hofstadter model (a generalization of this atom-chip method has been described in [336] to produce the SU(2) Hamiltonian (153)). First, consider two atomic states ${\mathinner{|{g}\rangle}} = {\mathinner{|{f=1/2,m_F=1/2}\rangle}}$, ${\mathinner{|{e}\rangle}} = {\mathinner{|{{3/2,1/2}}\rangle}}$ of ⁶Li, trapped in a primary (state-independent) optical lattice potential V₁(x) = V_x sin ²(kx) along e_x. Importantly, the corresponding pair of laser beams are incident on an atom chip's reflective surface, trapping the atoms about 5 µm above the device. The atom chip consists of an array of conducting wires, with alternating currents ±I [380], which traps the atoms in a deep 1D Zeeman lattice along e_y. Since the chosen states | e, g〉 have opposite magnetic moments, the Zeeman shifts produce a state-dependent lattice along this direction. Then, additional moving Zeeman lattices [380], with space-dependent currents I_m = I sin (2πΦm − ωt), act as a time-dependent (RF) perturbation directly coupling the internal states (in direct analogy with (155)), and thereby producing the desired space-dependent Peierls φ_y(m) = 2 πΦm. Such an elegant and versatile method, suitable both for bosonic and fermionic species, is currently in development at NIST, Gaithersburg.

Jaksch and Klein proposed immersing the optical lattice and the atoms of interest (A-atoms) into a BEC formed by B-atoms [381]. Then, the interaction between the A and B atoms result in phonon excitations, which in turn induce effective interactions between A-atoms located at NN sites [381]. When B-BEC is made to rotate, the resulting effective interactions imprint space-dependent Peierls phases to the tunneling matrix elements of the A-atoms. Such a system could potentially lead to the realization of a synthetic magnetic field in the dynamics of the A-system.

Finally, interesting properties of 2D systems can be captured by 1D systems, through the concept of dimensional reduction [296]. This fact, which is particularly relevant for topological systems, offers an interesting route for the exploration of topological order using 1D optical lattices [382, 383]. Also, this strategy suggests an alternative way to reproduce the effects of magnetic fields in a simple optical-lattice environment.

Let us illustrate this concept for 2D topological systems, namely, systems exhibiting robust edge states at their 1D boundaries with energies E_edge located within bulk gaps [73]. First, suppose that the system is defined on a 2D square lattice and described by a general Hamiltonian

$$\begin{equation} \hat H = \sum_{j,k} \hat c^{\dagger}_{j} \, h_{j k} \, \hat c_{k}, \end{equation} \tag{ 169 }$$

where $c^{\dagger}_{j}$ creates a particle at lattice site r_j = (m, n) and m, n = 1, ..., L. In this open 2D geometry, topological edge states are localized along the single 1D boundary delimiting the large square L × L [296]. Now, consider the cylindrical geometry obtained by identifying the opposite edges at n = 1 and n = L, namely, by imposing periodic boundary conditions along the y direction only. In this geometry, and considering Bloch's theorem, the system is well described by the spatial coordinate m = 1, ..., L and the quasi-momentum k_y = (2π/L) l, where l = 1, ..., L. In particular, the system Hamiltonian (169) can be decomposed as a sum

$$\begin{equation} \hat H = \sum_{m=1}^L \sum_{k_y} \hat H (m; k_y) = \sum_{k_y} \hat H_{{\rm 1D}} (k_y), \end{equation} \tag{ 170 }$$

which indicates that the 2D system can be partitioned into independent 1D chains [296]. The edge states are now located at the two opposite edges of the cylinder, defined at m = 1 and m = L, and their dispersion relations are expressed as E_edge = E_edge(k_y). The general dimensional-reduction strategy can be formulated as follows: the energy spectrum and edge-state structures emanating from the 2D Hamiltonian (169)–(170) could be captured by a family of 1D models with Hamiltonians $\hat H_{{\rm 1D}} (\theta)$, where the controllable parameter θ ∈ [0, 2 π] should be identified with the quasi-momentum k_y. In other words, engineering a 1D system with tunable Hamiltonian $\hat H_{{\rm 1D}} (\theta)$ would provide useful informations related to the full 2D system of interest (169).

To be specific, let us consider the Hofstadter model introduced in section 8.3.1 and described by the Hamiltonian (148)–(150). Using the Landau gauge, the single-particle wave function can be written as ψ(m, n) = exp(ik_yn) u(m), where the function u(m) satisfies a 1D Schrödinger equation and where k_y is the quasi-momentum along e_y. The Schrödinger equation associated with this 'Hofstadter cylinder' takes the form of the Harper–Aubry–André equation [78, 305, 384]

$$\begin{eqnarray} &&\fl E \psi (m)= - J \big [ \psi (m+1) + \psi (m-1)\nonumber\\ &&+ 2 \cos (2 \pi \Phi m - k_y) \psi (m) \big ], \end{eqnarray} \tag{ 171 }$$

where m = 1, ..., L. Solving this equation yields the projected bulk bands of the Hofstadter model E(k_x, k_y) → E(k_y), which display q − 1 bulk gaps for $\Phi=p/q \in \mathbb{Q}$. Moreover, since the cylinder is a partially opened geometry (with edges), new states with energies E_edge(k_y) are located within the bulk gaps. These states are topological edge states, which play an important role in the quantum Hall effect [78]. A typical spectrum, showing the bulk and edge states dispersions, is shown in figure 26. Originally, the Aubry–André model (171) was studied as a simple model for Anderson localization [384]. In this 1D model, k_y is an adjustable parameter (not related to quasi-momentum), and it is therefore treated on the same level as the parameter Φ. Recently, this model has been realized with cold atoms [385], in a quasi-periodic 1D lattice created by interfering two optical lattices with incommensurate wave numbers k_1,2. In this context, the parameters in equation (171), Φ = k₂/k₁ and k_y, are tuned by the lasers creating the two lattices [385]. This reproduces the dimensional reduction of the Hofstadter optical lattice, where the synthetic magnetic flux Φ can be easily adjusted and where the 'quasi-momentum' k_y is fixed by the laser phases. Considering fermionic atoms in such a 1D setup, and setting the Fermi energy inside a bulk gap, it is possible to populate topological edge states for a certain range of the tunable phase k_y [382, 386].

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Extending this scheme to two-component 1D optical lattices, and considering well-designed state-dependent bichromatic optical lattices [383], Mei et al obtained the dimensional reduction of the Z₂ topological insulator in equation (153). By sweeping the tunable phase k_y, one is then able to transfer topological (helical [330]) edge states with opposite spin, from one edge to the other [383]. The interacting bosonic version of the 1D bichromatic optical lattice described above has been explored by Deng and Santos [387], where the topological phase diagram has been obtained in terms of the interaction strength, the atomic filling factor and the strength of the auxiliary lattice.

The dimensional reduction strategy therefore allows to access gauge structures and topologically ordered phases in a rather simple manner. These methods could be extended to access unobserved phenomena, such as the 4D quantum Hall effect [388].

Finally, 2D atomic lattice systems could be realized through the concept of synthetic dimensions [389], where the dimensionality of the physical optical lattice D_phys is augmented by a synthetic dimension spanned by the internal states of the atoms. For instance, a 2D (semi-)synthetic lattice could be obtained from a D_phys = 1 optical lattice with L lattice sites (with spatial coordinates x = ma, where a is the physical lattice spacing and m = 1, 2, ..., L), and filled with a N-component atomic gas: in such a configuration, the 2D synthetic lattice is characterized by L × N lattice sites located at the coordinates r_site = (m, n), where m = 1, ..., L and n = 1, ..., N. In synthetic lattices, the hopping is natural along the physical direction x, while it is assisted along the synthetic (spin) direction through atom–light coupling (e.g. Raman transitions). The realization of synthetic magnetic fluxes in synthetic lattices has been described by Celi et al in [390], where it was shown that such setups could be exploited to observe the Hofstadter butterfly spectrum and the evolution of chiral (topological) edge states, using atoms with N > 2 internal states. The propagation of chiral edge states in ladder systems (N = 2) penetrated by synthetic magnetic fields has been studied by Hügel and Paredes [391]. An experimental implementation of such a ladder scheme (without involvement of the synthetic dimension) has been recently reported by Atala et al [392].