Bose 2.2 User Manual download pdf (Page 68)

deeper lattices: At increasing lattice depths, the movement of the wavepackets

becomes gradually inhibited by the strong conﬁnement at the positions of the

individual lattice sites. In this case, the delocalized Bloch waves are not well suited

for an intuitive description of the system anymore. In contrast, the so-called Wannier

functions provide an orthogonal set of wave functions that are localized on the

individual lattice sites

. The Wannier function

(

z − z

), describing a wavepacket

localized at z = z

in the lowest energy band, is constructed via a superposition of

the Bloch functions [170]:

(z − z

) =

lat

dq e

−iz

1,q

(z), (4.11)

where

1,q

(

) =

iqz

imz2π/d

lat

with

being integer numbers

and the coeﬃ-

cients c

deﬁne the weight of the diﬀerent plane wave states.

Using the localized wave functions, given by Eq.

(4.11)

, we can understand the

movement of the wavepackets in terms of particles tunneling from one lattice site to

the next one. The corresponding tunneling matrix element

, which measures the

inter-site kinetic energy in the system, is then calculated by [177]

J =

dz w(z − z

)

−

∂

∂z

+ V

lat

(z)

w(z − z

j+1

). (4.12)

In the approximation

s 

1, the energy in the lowest band may be calculated

analytically by solving the 1D Mathieu equation

(4.5)

, with the resulting energy

spectrum [178]

E(q)

√

s − 2

cos (qd

lat

) (4.13a)

with J =

√

3/4

−2

√

. (4.13b)

We see from Eq.

(4.13a)

that the tunneling matrix element

is directly related to the

bandwidth of the lowest energy band via

= (

max

(

)

− min

(

))

4. Furthermore,

using Eq.

(4.13b)

, we may estimate the relevance of the tunneling in the system:

when the tunneling time

h/J

is large on experimental time scales

(1-10 ms), the

particle exchange between the lattice sites becomes negligible. While there is no

clear threshold, we may neglect the tunneling in our experiments for lattice depths

lat

. In this regime of very deep lattices, the calculated tunneling time is

larger than 40 ms.

Note that the Wannier functions are not eigenfunctions of the lattice system.

For usual lattice parameters it is suﬃcient to sum over m

max

∼ 5 plane wave states [177].

The experimental time scale are estimated by the inverse trapping frequencies which typically range

from 100 Hz to 1000 Hz.

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