Quantum description: The Bose-Hubbard Hamiltonian

h·idenote a cycle average. If the detuning of the laser is negativeω_L <ω₀(“red-detuned”) then the induced dipole D= α(ω_L)E is in phase with the electric field. Therefore, the potential energy is minimal when the intensity of the laser is maximal and the atoms are attracted to the bright spots. In the opposite case of a blue-detuned laser (ω_L>ω₀) the atoms are attracted by the dark spots in the field. This is the experimentally favorable situation since the scattering rateΓscatteffectively decreases with the decreasing light intensity. The depthV₀of the optical trap is [153]

V₀∝ I_p

ω₀−ω_L . (2.3)

Accordingly, in order to have a conservative potential one has to work at the largest detuning possible since then the dissipative part given by (2.1) can be neglected.

At this point the atoms are radially confined to the waist size of the laser beam but there is not lattice yet. The simplest way to create a periodic optical potential is to make two laser beams counter-propagate and let them interfere. For a wavelengthλ_L this results in a potential of the form

V(x) =V₀cos²(πx/d), (2.4) where d =λ_L/2 is the distance between two minima in the direction of the laser beam.

In practice, such a lattice can be created, for example, by retro-reflecting a laser beam and inserting an opto-acoustical modulator. This device allows for fast (less than a microsecond) and precise control of the laser intensity and also introduces a shift of the laser frequency of tens of MHz [153]. Alternatively, one can use two phase-coherent beams and introduce a frequency shiftbetweenthem. As a consequence the lattice is not stationary but moving and by increasing the shift one can even create accelerated optical lattices.

After the condensate has evolved on the lattice, the next step is to measure the atomic cloud. A popular technique is time of flight measurements. The trapping potential is switched off and a resonant laser is shone on the atoms from above while a CCD cam-era takes an image of the light distribution from below the condensate. Since this proce-dure destroys the condensate, a time-resolved experiment requires repetitive measurements, starting each time with the same initial conditions. This is not an obstacle since the of cool-ing of the atoms can be achieved within few seconds while the condensates can exist in the traps up to the order of several minutes [141, 12].

2.2. Quantum description of cold bosons on a lattice:

The Bose-Hubbard Hamiltonian

We now turn to the mathematical description of (ultra-)cold bosons loaded on a lattice.

Here we derive the Bose-Hubbard Hamiltonian (BHH), the simplest non-trivial quantum

model that takes into account the competition between the interaction energy and the kinetic energy of the system. Then we discuss the BHH’s validity.

2.2.1. Derivation from second quantization

As mentioned above, the particle density in the atomic cloud of a BEC is extremely low, hence three-body collisions are rare events. Accordingly, the many-body Hamiltonian de-scribingNinteracting bosons confined by an external potential is given in second quantiza-tion by [125, 67]

Hˆ = Z

drΨˆ^†(r)

−h¯²

2m∇²+V_lat(r) +V_ext(r)

Ψ(r)ˆ +1

2 Z Z

drdr⁰Ψˆ^†(r)Ψˆ^†(r⁰)V(r−r⁰)Ψ(rˆ ⁰)Ψ(r)ˆ (2.5) where ˆΨ(r)and ˆΨ^†(r)are the bosonic field operators that annihilate and create a particle at the position r respectively andV(r−r⁰) is the two-body inter-atomic potential. The termV_lat(r) describes the (optical) lattice potential while V_ext(r) accounts for a possibly present additional potential which is slowly varying along the lattice like the magnetic trap used for the evaporative cooling. Due to the extremely low temperatures (typically sev-eral nanokelvin) the predominant inter-atomic interaction results from s-wave scattering.

Since also the particle density (and hence the mean inter-atomic distance) is very low we can approximate the otherwise complicated two-body interaction potentialV(r−r⁰)with a delta-like contact-potential [141]

V(r−r⁰)≈4πa_sh¯²

m ×δ(r−r⁰), (2.6)

wherea_sis thes-wave scattering length andmis the atomic mass. Even with this simplified potentialV(r−r⁰) solving (2.5) is impractical if not impossible but we can use the fact that the underlying potential is periodicV_lat(r) =V_lat(r+d)withdbeing the lattice vector:

The eigenstates of a single atom moving in a potentialV_lat(r), would be the well-known Bloch functions φq,n(r) =e^iqru_q.n(r), where u_q,n(r+d) =u_q,n(r) and q is the so-called quasimomentum. The presence of the periodic potential leads to the formation of the so-called Bloch bands in the energy spectrum which are labeled by the sub-indexn. As in a substantial part of the experimental studies [47, 202, 159, 102, 48] we consider deep lattices in this work. Therefore, it is useful to work in a basis where the eigenfunctions are localized at the sitesi. Such a basis is given by the Wannier-functions

w_n(r−r_i) = 1

√f

∑

q

e^−iqriφ_q,n(r), (2.7)

which are obtained via a uniform transformation from the Bloch basis. Above, the summa-tion is done over the quasi-momentum in the first Brillouin zone and f denotes the number

2.2. Quantum description: The Bose-Hubbard Hamiltonian 9 of lattice sites.⁴ If the lattice is deep enough such that the chemical potential is too small to excite states outside the first Bloch band [151, 125] we can expand the field operators ˆΨ of the above Hamiltonian (2.5) in the local modesw_n(r−r_i)of the wells keeping only the states belonging to the lowest band

Ψ(r) =ˆ

f i=1

∑

bˆ_iw₀(r−r_i), (2.8)

where ˆb_i annihilates a boson at siteiand f is the size of the lattice. The Hamiltonian (2.5) then reduces to the Bose-Hubbard Hamiltonian [125, 151]

Hˆ = wherehi,jiindicates summation over adjacent sites j=i±1. In the second step we used the canonical commutation rules for the bosonic annihilation (creation) operators ˆb_i, (ˆb^†_i)

[bˆ_i,bˆ^†_j] =δ_i,j, (2.11) and the definition of the number operators

ˆ

n_i=bˆ^†_ibˆ_i. (2.12)

The parameters in Eq. (2.10) are

v_i = Herev_iis the on-site potential at each lattice site,U_iis the on-site interaction strength,⁵and k_{i j} parameterizes the coupling strength which accounts for the tunneling of particles be-tween neighboring sites.⁶ Therefore,k_{i j} is the proportionality factor for the kinetic energy.

4Here we are interested in one-dimensional lattices, but the derivation also applies to higher dimensions.

5The inverse of the integral in (2.14) is also referred to as the effective mode volume V_eff⁻¹ = Rd³r|w₀(r−r_i)|⁴.

6Apart from Chapter 6 we will consider a setup whereU_i=Uandk_{i j}=k.

The BHH was originally conceived [153] to describe superfluid He in restricted geome-tries like porous media (see e.g. [88]) and later suggested by Jaksch et. al [125] as a description of BEC in optical lattices. Both works were focused on quantum phase transi-tions which is not the topic of this thesis but an important result that we mention here. In contrast to their classical analog, quantum phase transitions occur atT =0. In other words they are not driven by the temperatureT but result only from quantum (not thermal) fluc-tuations [181]. The phase transition in the Bose-Hubbard Hamiltonian (2.10) is a result of the competition between the interaction energy and the kinetic energy. Roughly speaking, the nonlinearity tries to localize the bosons, while the coupling tries to delocalize them.

Consider the limitUkfor an optical lattice with exactly one atom per well, i.e., afilling factorn¯=N/f =1. The energy cost to move one atom in this case isU and determines the energy gap to the first excited (i.e. conducting) state. If this energy is not provided by, say, an external potential that sufficiently tilts the lattice, this configuration is insulating (no atom current) and the system is in the so-called Mott-insulator state [88, 125]. In the other limit ofkU one can neglect the interaction term. The resulting Hamiltonian is then diagonal in the Bloch basis [71] and all particles will be completely delocalized over the entire lattice. For vanishing on-site potentialsv_i=0 the BEC ground state corresponds to a quasimomentumq=0. In contrast to the previous case the atoms form a superfluid. Thus, tilting the lattice even slightly will cause the bosons to move. As the parameterU/k is changed from∞→0 the BEC exhibits the so-called Mott-insulator to superfluid transition which was experimentally confirmed in a seminal paper of Greiner et al. [102]. We note that this transition is appreciable only forsmall integerfilling factors ¯n. If the filling factor is non-integer then there is always one atom that can move. This is sufficient to ensure phase coherence between wells and hence the system is in the superfluid regime.

One of the advantages of realizing the BHH with optical lattices is that all parameters are accessible in the experiment. In other words, while the on-site potentialv_i is given by V_ext(r_i), U_i and k_i are determined by the wavefunction w₀(r−r_i) which depends on the lattice depthV₀, i.e. on the intensity of the interfering laser beams. For typical lattice con-figurations the tunneling strength decreases exponentially with the lattice depthk∼e^−V0, while the on-site interaction grows algebraically [177]U ∼V₀^D/4 (here D is the dimen-sionality of the lattice). Additionally, the scattering lengtha_s can be tuned using Feshbach resonances [141] – by applying an additional magnetic field the hyperfine levels that deter-mine thes-wave scattering are shifted. With this technique, the parametera_scan be changed over several orders of magnitude [119] including a change in sign. For negative scattering lengths a_s the condensate becomes unstable above a certain boson number N due to the attractive interaction. Unless stated otherwise, we consider only repulsive interactions, i.e.

a_s>0.