We have discussed the single-particle potential set up by the lasers, i.e. described by equation (5.7), and the effective interaction potential (5.9). However, many-body physics is commonly expressed in the second quantization framework, therefore we now briefly discuss the second quantized Hamiltonian of the interacting system, which is the Hubbard Hamiltonian, discussed in section 1.1, and also the Bose-Hubbard Hamiltonian, describing interacting fermions or bosons, respectively. Equation (5.7) expresses
5.3. Hubbard Parameters for Optical Lattices 81
the single-particle potential of the system in terms of a trapping potential Vtrap and a periodic lattice potentialVlat, the latter being defined as
Vlat(r) =
3
X
i=1
si cos2(πxi/axi), (5.11)
in units of the recoil energyER. In the regime of sufficiently deep lattices, i.e.s&5the lattice potential is the dominant potential for determining the physics in the center of the trap, i.e. for sufficiently small particle numbers in the system. Because of this, the Hamiltonian of the system is conveniently expressed either in terms of Bloch states|α,ki, which are the eigenstates of the Hamiltonian
Hlat= p2
2m+Vlat (5.12)
and labeled with the band indexα ∈ Nand the quasi-momentumki ∈ [−π/ai, π/ai], or in terms of Wannier states|α, li, which are obtained from the Bloch states via Fourier transformation
|α, li= 1
√ L
X
k
eikRl|αki (5.13)
and for appropriate choice of phase factors are maximally localized around lattice sitel5. The eigenvalue equation for the lattice Hamiltonian then reads
Hlat|α,ki=α(k)|α,ki, (5.14) whereα(k)is a smooth function in terms of the quasi-momentumk, but is well separated in energy for different band indices, i.e. features band gaps
∆α,α0 ≡min|α(k)−α0(k)|>0
forα6=α0. This makes the set of Wannier states a good basis, since although they mix all possible quasi-momenta, they respect the separation of the system into distinct bands. Now, including the spinσas a label for the particles internal degree of freedom, the single-particle Hamiltonian
H = p2
2m+Vlat+Vtrap (5.15)
can be expressed in the basis of Wannier states|α, l, σiaccording to H=− X
α,l,m,σ
|α, l, σitαl,mhα, m, σ|+ X
α,α0,l,m,σ
|α, l, σiVl,mα,α0hα0, m, σ|, (5.16) with the matrix elements of the lattice Hamiltonian
tαl,m=−hα, l, σ|
p2 2m +Vlat
|α, m, σi=−X
k
eik(Rl−Rm)α(k), (5.17) which respect the separation of the system into distinct bands6, and the matrix elements of the trapping potential
Vl,mα,α0 =hα, l, σ|Vtrap|α0, m, σi, (5.18) which are not diagonal with respect to the band indexα. Here, the laser potentials that we are con-sidering do not make a distinction for the different internal states of the particles and therefore there is no additional dependence onσin the matrix elements of the Hamiltonian7. When the band gap∆1,α0 5Maximally localized in this context means that the Wannier states|α, liare the most localized states in the subspace of fixed band indexα.
6The minus sign in (5.17) is convention, leading to the matrix elements being positive.
7There exist experiments which make use of optical potentials, which distinguish between the different hyperfine states of the atoms and therefore would have additional dependence onσin the matrix elements of the Hamiltonian. A corresponding model is discussed in Sec. 5.4.
between the first band (α = 1) and all higher bandsα0 >1is much larger than the typical interaction energy and the temperature of the system and the trapping potential is very flat8, it is sufficient to express the resulting low energy physics in a lowest band approximation, i.e. neglecting all contributions from α > 1. This approximation relies on the fact that the energetically well separated quantum states in the higher bands are only sparsely populated, thereby having no influence on the physics of the system, which is dominated by the hugely populated states in the first band9, therefore we only consider quan-tum states from the lowest band and drop the indexαfurther on. The Wannier states|l, σiare localized around lattice siteland decay exponentially with the distance, which leads to the matrix elementstl,m
andVl,m in equation (5.16) also decaying exponentially in the distance|l−m|. For the case of optical lattice potentials this decay is so strong, that taking only nearest neighbor matrix elements, i.e. those with
|l−m| = 1into account, is a perfectly reasonable approximation [95]. For the matrix elements of the trapping potential (5.18), it is sufficient to only consider the local matrix elements, wherel=m, since the non-local matrix elements decay even faster than for the hopping matrix elements, caused by the lack of a "non-local" operator like the momentum operator in (5.17). Combining all the discussed simplifications and approximations and subsequently switching to the language of second quantization, the Hamiltonian of the system without interactions reads
H =− X
hl,mi,σ
tl,mc†lσcmσ+X
l,σ
Vl,lc†lσclσ, (5.19)
where the creation and annihilation operatorsc†lσ, clσcreate or annihilate a particle with spinσat lattice sitelandhl, mimeans that only the sum over nearest neighborsl, mis evaluated.
The non-interacting Hamiltonian (5.19) is the same for both bosonic and fermionic particles in an optical lattice. The nature of the particles only enters the Hamiltonian in form of the different commutation relations of the creation and annihilation operators, i.e. the fermionic operators anti-commute
{clσ, c†mσ0}=clσc†mσ0+c†mσ0clσ=δlmδσσ0, (5.20) whereas the bosonic operators commute
[clσ, c†mσ0] =clσc†mσ0−c†mσ0clσ=δlmδσσ0. (5.21) As we saw in the previous section, bosonic atoms with the same internal quantum state can interact via s-wave scattering, while for fermions only atoms with different internal quantum states can interact with each other via s-wave scattering. Therefore the simplest interacting bosonic systems are those having no internal degrees of freedom, while for fermions the simplest systems are those which are described by atom with two possible internal quantum states, which can be seen as the spin of the particleσ=↑↓.
For the bosonic system (without internal degrees of freedom), the interacting part of the Hamiltonian is described as
HInt= 1 2
X
l,m,s,t
Ulmstc†lc†mctcs, (5.22) whereas for fermions, it reads
HInt= 1 2
X
l,m,s,t,σσ0
Ulmstc†lσ0c†mσctσcsσ0, (5.23)
with the matrix elements
Ulmst= 2πas m
Z
w∗l(r)w∗m(r)ws(r)wt(r)dr, (5.24) according to (5.9) the same for bosonic and fermionic particles. For sufficiently deep lattices, the Wannier functionswl(r) =hr|liat sitelagain have almost no overlap with their neighboring sites, such that it
8All of which is fulfilled in optical lattice experiments [15].
9For a discussion of the influence of second order processes on the matrix elements in a lowest band approximation see for instance [13, 127]