Bose 2.2 User Manual download pdf (Page 71)

where we have neglected any contributions from the tunneling. The global chemical poten-

tial

is thus expressed as the sum of the on-site potential energy

def

= mω

(

lat

)

def

Ω

arising from the underlying harmonic trapping in the z-direction, and the local chemical

potential

, as illustrated in Fig. 4.5. Note that Eq.

(4.18)

is valid only if the tunneling is

not fully suppressed: then, the atoms distribute such among the lattice sites that, in the

combined potential of ODT and lattice, the global chemical potential is constant across

the full system. From the calculations performed explicitly in appendix A.3, we obtain

BEC

Fig. 4.5, BEC in the combined potential V

ext

of ODT and lattice:

Left: For the

tight-binding approximation to be valid, we consider a trapping potential with

the modulation

lat

of the lattice much larger than the chemical potential

of the condensate. Right: Zoom into the region

ext

≈ µ

. The global chemical

potential

is the sum of the on-site potential energy

and the local chemical

potential µ

the local chemical potential

= U

· N

1/2

, with U

def

m˜g ω

, (4.19)

where we use

˜g

def

= g/

(

√

2πa

lat

), with

the contact coupling strength deﬁned in Eq.

(2.4b)

and

lat

def

~/(mω

lat

)

the harmonic oscillator length in the lattice direction. The radial

TF radius R

(j)

⊥

of the j-th on-site condensate is then given by R

(j)

⊥

2µ

/(mω

The most interesting quantity regarding our experiments is the atom number distribution

in the lattice. Inserting the local chemical potential from Eq.

(4.19)

, into Eq.

(4.18)

we obtain

= N

· max







1 −

inv

, 0







, (4.20)

1 2 ... 66 67 68 69 70 71 72 73 74 75 76 ... 154 155

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