Bose 2.2 User Manual download pdf (Page 67)

we show the results of the evaluation of the absorption images from Fig. 4.3. We obtain

independent ﬁtting results of the lattice depth for each diﬀraction order. All of them

match within a small intervall around the mean value

lat

= (144

(disregarding

the weakly populated 6

order).

From the calibration of the lattice depth, taken at the maximal laser power

lat

, we

directly obtain the lattice depths at smaller laser powers by the linear scaling relation

lat

∝ P

lat

. We checked the linearity of the scaling by recording diﬀraction patterns at

diﬀerent laser powers and found indeed good agreement [122, Ch.4.4.2.2].

4.2.3 Dynamics in Shallow and Deep Lattices

Let us now discuss the dynamics, i.e. the movement of a wavepacket in the periodic

potential landscape. As we do not accelerate the condensate in the lattice, we are mainly

interested in the dynamics of a wavepacket close to the zero quasi-momentum,

q ≈

simplifying the discussion below. A thorough analysis of the dynamics of BECs in optical

lattices is given in Ref. [170].

Originally investigated in the context of electrons moving inside a solid body with

crystalline structure [171, 172], travelling wavepackets in periodic potentials are well

understood in physics. The description of the momentum dependent energies

(

) in

terms of energy bands [173] has recently been experimentally demonstrated to be also valid

for superﬂuids in optical lattices [176]. The full solution of the Schrödinger equation

(4.4)

including the movement of the wavepackets (i.e.

q 6

= 0), may be performed numerically.

However, for our purposes, it is suﬃcient to consider two limiting cases:

shallow lattices: In the case of small lattice depths

lat

≈

, the band structure

of the lowest energy band is approximately given by [171],

E(˜q)

= ˜q

−

4˜q

, (4.10)

where

˜q

q/k

lat

−

1. Expanding the expression

(4.10)

into a power series around

= 0 (using

lat

= 1), we obtain

(

)

≈ const. −

015

q/k

lat

98 (

q/k

lat

)

(

). Hence, at low momenta, the wavepackets moving in such

shallow lattice potential show mainly quadratic dispersion, resulting in an almost

free movement along the lattice potential

. As the lattice potential is only a weak

perturbation to the system, the wavepackets are well described by delocalized Bloch

waves extending over the full size of the condensate.

At the edge of the Brillouin-zone, i.e.

lat

, the dispersion curve ﬂattens out, and the particles are

reﬂected by the lattice when accelerated towards this momentum value. This regime is not relevant in

our experiments, as we do not move or accelerate the BEC in the lattice.

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