Bose 2.2 User Manual download pdf (Page 73)

parameter (

a, U

lat

, ω

ρ,z

), we also change the local values of

such that it diﬀers from

one lattice site to another. As we can calculate both



and

(see section 4.3.2), we

obtain analytical expressions for the phases

(

) =

(

j d

lat

)

t/~

of the independent

condensates in the lattice. The resulting deterministic evolution of the relative

phases can be probed for example by the time-of-ﬂight (TOF) technique, when the

dephased matter waves interfere after switching oﬀ the trapping potential

In conclusion of this section, we have determined the ground-state properties of a

contact interacting BEC in a 1D lattice potential. The results obtained here for e.g. the

populations of the lattice sites provide a good estimate for the case of a dipolar BEC at

a weak relative dipole strength, i.e. for





1. Since the time-of-ﬂight technique is a

fundamental tool in our experiments to probe the ground-state of the BEC in the lattice,

we next investigate the expansion of a condensate from a 1D lattice potential.

4.4 Expansion of a BEC from the Lattice

Due to the limited imaging resolution of around 6

in our experiment, we cannot resolve

the in-trap density distribution of the BEC in the lattice. Therefore, we use the TOF

technique to expand the atomic cloud before taking an absorption image. Since the

spatially separated on-site condensates overlap during the expansion, the TOF technique

is essentially an interference measurement in lattice experiments. It thus becomes a

valuable tool to probe the relative phases between the on-site condensates.

We ﬁrst discuss the expansion of a coherent array of BECs, i.e. with all the on-site

condensates in phase with each other. Then, we examine the expansion from a lattice of

increasing lattice depths. By comparing experimental data with simulations, we are able

to identify the lattice depth above which we can neglect the tunneling of atoms between

the lattice sites on experimental time scales.

4.4.1 Expansion of a Coherent Array of Condensates

We consider the case of a BEC in a lattice of intermediate depth, where tunneling is

present and the chemical potential

is constant over all the lattice sites. The system is

then a coherent array of condensates, with the expansion ﬁrst investigated in Ref. [180].

To describe the expansion process by analytical expressions we neglect all inter-atomic

interactions. This is typically a valid assumption since the density of the cloud rapidly

decreases after release from the trap and interactions do not play a role anymore.

Experiments with a well controlled dephasing of independent BECs have been carried out in Refs. [186,

187]. There, the TOF technique has been extended to the so-called band-mapping technique, allowing

for instance for the observation of the Talbot-eﬀect with matter waves.

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