gauge potentials in cold atom experiments [123, 124, 119, 28, 57, 96, 143, 54, 34], reaching from the creation of artificial effective magnetic fields, to optical flux lattices and even time-reversal invariant non-Abelian gauge fields. Here, we will focus on a proposal by Goldmanet al. [57], which focuses on the realization of a combination of Abelian and non-Abelian artificial gauge field for a two-component fermionic system on an atom chip. The model proposed in this letter is the model that the theoretical investigations in the following chapters will be focused on. The proposed setup uses a combination of Raman-assisted tunneling and an alternating Zeeman lattice, which we will explain in the following.
5.4.1 Raman Transitions in the Λ -system
Consider an atomic system consisting of three internal states, two low energy states{|g1i,|g2i}with energiesω1, ω2lying close to each other and a high energy excited state|eiwith a much larger energy ωe, where the two low energy states are coupled to the excited state via two far off-resonant laser beams.
The frequency of the first laser is denoted asω1L = ωe−ω1+δ1, whereδ1is the detuning from the transition frequency, while the frequency of the second laser can be written asωL2 =ωe−ω2+δ2, with the second detuningδ2.
It is more convenient to switch to a rotating frame [25], which simplifies the view on the situation
Figure 5.2: Illustration of theΛ-system in an energy level diagram. The two low energy states|g1i,|g2i are separated by an energy differenceδand the excited state|eiis separated from both states by the energies∆±δ/2.
introduced above. In a rotating frame, the two low energy states are separated by an energy difference δ=δ2−δ1and the excited state is separated from the states|g1i,|g2iby the energies∆±δ/2, where
∆ =δ2+δ1, see Fig. 5.2. After performing a rotating wave approximation (see for instance [151]), an effective, Rabi-type model is obtained, which is described by the Hamiltonian
HRabi= 1 2
−δ 0 Ω∗a 0 δ Ω∗b Ωa Ωb 2∆
, (5.30)
whereΩi,i=a, bis the corresponding Rabi-frequencyΩi=|Ωi|eiφiwith the spatially dependent phase φi = kir. In the following, we consider|∆| >> |δ|,|Ωi|, i.e. very large detuning of the lasers. Then, starting with a state|ψiwithout any population of the excited state, the excited state will stay almost completely unpopulated during the time evolution. Exploiting this fact, an effective model for the low energy states only can be derived. Assuming a state
|ψ(t)i=
α(t) β(t) γ(t)
, (5.31)
5.4. Artificial Gauge Fields for Neutral Atoms in Optical Lattices 85
the Schrödinger time-evolution of this state is determined according to
i∂t
α(t) β(t) γ(t)
=HRabi
α(t) β(t) γ(t)
= 1 2
−δα(t) + Ω∗aγ(t) δβ(t) + Ω∗bγ(t) 2∆γ(t) + Ωaα(t) + Ωbβ(t)
. (5.32)
For stationaryγ(t), i.e.∂tγ= 0, the third row of (5.32) requires γ(t) =−Ωa
2∆α(t)− Ωb
2∆β(t), (5.33)
which allows us to replaceγ(t)byαandβ10. Replacingγby (5.33) in equation (5.32) leads to an effective time-evolution for theαandβcoefficients.
i∂t
α(t) β(t)
= ˜HRabi
α(t) β(t)
, (5.34)
with the simplified Rabi Hamiltonian H˜Rabi=−1
2
δ+|Ω2∆a|2 Ω∗R ΩR −δ+|Ω2∆b|2
!
, (5.35)
where the Rabi frequenciesΩRare defined as ΩR= ΩaΩ∗b
2∆ =|ΩR|ei(ka−kb)r, (5.36)
leading to the Hamiltonian in (5.35) being spatially dependent. Typically, the laser beams which drive the transitions from the low energy states to the excited states have to be chosen to account for the internal structure of the atoms, e.g. for different angular momenta of different interal states, which was not discussed here. A set of laser beams realizing the Hamiltonian (5.35) for a set of two distinct internal states|g1i,|g2iof an atom is called a Raman setup, or a set of Raman lasers. The transitions are called Raman transitions.
5.4.2 Zeeman Lattice plus Raman Beams
The system we consider consists of atoms with two different hyperfine states|g1i,|g2i11, which are con-fined to two dimensions (see section 5.1). In thex-direction, two laser beams are creating a lattice with lattice potential
Vx(x) =sxcos2(πx/ax), (5.37) which acts equally on both hyperfine states. In they-direction, a Zeeman lattice is build by implementing stripes of constant magnetic field along thex-direction but with alternating sign in they-direction [57].
The both hyperfine states, due to their distinct internal spin, feel a lattice potential in they-direction.
In contrast to the lattice inx-direction, the lattice in they-direction is state dependent, i.e. |g1ifeels a maximum in the potential at the place where|g2ifeels a minimum and vice versa. Additionally, there is a total energy shiftB between the two hyperfine states. In the proposal of [57], the Zeeman lattice is created on an atom chip by current carrying wires, oriented in thex-direction, such that the current travels in the opposite direction in two neighboring wires. Approximating the Zeeman lattice by acos -function, which is sufficient for the tight-binding approximation, then leads to the lattice potential
V(x, y) =Vx(x)1+Vy(y)σz+Bσz, (5.38)
10It is evident that the choice ofγin (5.33) is itself time dependent and therefore not self-consistent with the assumption [25, 94].
However, the approximation is well established since it leads to results in good agreement with exact solutions. For a detailed analysis, see [20]
11There must be more than two internal states for these atoms, however, we consider all others to be essentially unpopulated.
whereσzis the Pauliz-matrix, acting on the subspace of the two hyperfine states and
Vy(y) =sycos2(πy/ay) (5.39) is the potential for the first hyperfine state. To create a square lattice in the end, we choseay= 2ax= 2a.
The Hamiltonian for the Zeeman lattice configuration then reads H0= p2
2m1+Vx(x)1+Vy(y)σz+Bσz. (5.40) In thex-direction, a particle can hop from one lattice site to the next lattice site without changing the internal state and therefore the system is described by a tight-binding model in thex-direction. On the other hand, hopping from one lattice site to a neighboring site in they-direction requires the change of the particle’s hyperfine state because of the alternating potential in this direction. Without changing the hyperfine state, the particle can only hop to the next nearest neighbor, which is strongly suppressed due to a small spatial overlap and therefore, without additional potentials that change the hyperfine states of the particles, no hopping iny-direction takes place.
To allow for tunneling in they-direction, additional operators are required, which change the internal state of the particles. Equation (5.35) exactly describes such an operator, which is the operator describing two Raman beams acting on the system. Adding these two Raman beams, leads to the total Hamiltonian of the system
H = p2
2m1+Vx(x)1+Vy(y)σz+Bσz+ ˜HRabi (5.41) which contains diagonal and off-diagonal terms in the hyperfine states. The corresponding tight-binding Hamiltonian in second quantization can now be expressed as
H = X
hi,ji,σ,σ0
tσσij0c†iσcjσ0, (5.42)
wherei, jlabel the lattice sites of the two-dimensional system and the sum is running only over pairs of nearest neighbors. The indexσ=g1, g2labels the two hyperfine states. For hopping in thex-direction, only the hopping amplitudes between equal hyperfine states are non-zeros, while for they-direction only hopping amplitudes between distinct hyperfine states are non-zero, due to the Raman coupling.
As shown in (5.36), the off-diagonal terms in the Hamiltonian (5.42) carry a position dependent phase factorei(ka−kb)·r, which can be adjusted by changing the wave-vectors of the Raman beams ka,b. To realize phase factors which break time-reversal symmetry, i.e. realize quantum Hall physics, one can realize the Landau gauge by adjusting the phase factor such that they depend on thex-coordinate only.
Then the hopping matrix elements in thex-direction would obtained by the common formula txij =
Z
w(r)H11w(r+axex)dr=tx, (5.43) withH11being the first matrix element of the Hamiltonian (5.41) andexthe unit vector inx-direction.
Therefore, the hopping elements in thex-direction can be chosen purely real and are uniform.
On the other hand, the hopping matrix elements in they-direction are defined according to tyij =
Z
w(r)H12w(r+ay/2ey)dr= |ΩR| 2
Z
w(r)ei(ka−kb)xw(r+ay/2ey), (5.44) where we have already exploited that the Raman beams are chosen such that the phase factor only de-pends on thex-coordinate. Solving the equations above leads to the hopping matrix elements for the y-direction being
tyij =tyei2παxi (5.45)
with uniform magnitude and spatial dependent phaseei2παxi, where α = (ka −kb)ay/2π = (ka − kb)a/4π. The absolute value of the hopping amplitude inx- andy- direction can be tuned independently, allowing the realization of the desired case ofty = tx ≡ tby adjusting the experimental parameters.
On the other, the ”flux” parameterαcan be adjusted the same way, allowing for the special cases, where