Bose 2.2 User Manual download pdf (Page 127)

is a constant that depends neither on the number of atoms nor on the site index

and is determined by the dimensionality of the system

114

. We now consider the global

properties of the lattice system. As shown in section 4.3.2, we can write the global

chemical potential

as the sum of the local chemical potential

and the local potential

energy

def

= mω

(

lat

)

def

Ω

that arises from the harmonic trapping in the lattice

direction

. With the local chemical potential given by Eq.

(A.13)

, we thus obtain an

expression for the atom number N

on the j-th lattice site,

= max







µ − Ω j

, 0







def

= N

· max







1 −

inv

, 0







, (A.14)

where

def

= (µ/U

)

is the atom number in the central lattice site and

inv

def

µ/Ω

the inversion point above which the lattice sites are not occupied anymore. The global

chemical potential

is then calculated using the condition

115

. We ﬁrst replace

the lattice site index by

j → z

lat

, where the

are the (discrete) positions of the lattice

sites in the

-direction and

lat

is the spacing between the sites. We then express

via

Eq.

(A.14)

and use continuous variables, i.e. we replace

→ z

. By integrating over the

z-direction, we ﬁnally obtain the global chemical potential µ:

µ =



N U

√

Ω



2/5

, (A.15)

which agrees with the results given in Refs. [179, 180]. We can thus express the ground-

state properties of a BEC in a 1D lattice in terms of the trap parameters (harmonic trap

and lattice), the contact coupling strength and the total atom number in the system.

A.4 Variational Calculations with a Gaussian-Shaped Dipolar

BEC

Here, we describe the principle of the Gaussian variational calculations, an eﬃcient

numerical method to calculate the critical scattering length of a trapped dipolar BEC. For

simplicity, we assume a cylindrically symmetric harmonic trap, with the symmetry axis

aligned with the polarization direction

of the dipoles. The according trap frequencies

are given by

and

in the radial and in the axial direction, respectively. Close to the

stability threshold, the interactions are typically suﬃciently weak such that the shape of

the dBEC is well described by a Gaussian (see section 2.4.1). We therefore write the wave

114

In general, the relation is given by

|ψ

, where

= 4

(2 +

) is determined by the

dimensionality D of the system [179]. Therefore, we obtain α = 1 in the quasi-2D system.

115

The condition

ensures that the sum of the local atom numbers

equals the total number

of atoms N in the system.

127

1 2 ... 122 123 124 125 126 127 128 129 130 131 132 ... 154 155

Comments to this Manuals

No comments

Bose 2.2 User Manual Page 127

Comments to this Manuals

Related products and manuals for Soundbar speakers Bose 2.2