Bose 2.2 User Manual download pdf (Page 77)

In the regime 0

< U

lat

< U

ﬁx

, we calculate the equilibrium atom number distribution in

the lattice, given by Eq.

(4.20)

, for the diﬀerent lattice depths used in the experiment. We

then perform the TOF simulations, independently expanding the on-site condensates as

explained in Ref. [89, Ch.13]. In this procedure, we approximate the on-site wave functions

by 1D Gaussians of width σ = a

lat

and with equal phases ϕ

= 0.

When reaching the lattice depth

ﬁx

, we model the suppression of the tunneling by ﬁxing

the on-site populations

such that

(

lat

> U

ﬁx

) =

(

ﬁx

). We now perform real-time

simulations of the evolution of the independent phases

(

) of the separated on-site

condensates: we split the linear lattice ramp into small time intervals ∆

t 

(

)

−1

, with

the local chemical potential on the central lattice site. Then, we let the phases

(

)

evolve linearly in time with

(

+ ∆

) =

(

) +

(

lat

) ∆

t/~

, where

(

lat

) is

the position-dependent chemical potential introduced in section 4.3.2. Finally, we simulate

the TOF procedure at regular time intervals of the loading time, as explained above.

The results of the simulations are shown in Fig. 4.6(b), next to the experimental data.

Choosing the threshold

ﬁx

= 13

for the suppression of the tunneling, we can reproduce

the main features observed in the experiment. We observe initially the discrete momentum

peaks that we expect from the interference of a coherent array of condensates. Interestingly,

the on-site condensates show a signiﬁcant dephasing (indicated by the multi-peak structure

after TOF) only at a lattice depth

lat

∼

, much larger than the value

ﬁx

. This is

an eﬀect of both, the weak dependence

µ ∝ U

1/8

lat

of the chemical potential on the lattice

depth and the timing of the experiment, i.e. the ramping speed of the lattice potential.

We see that we can model the measured phase dynamics of the dipolar

Cr condensate

in the lattice by considering only contact interactions and a deterministic phase evolution

between the on-site condensates. Indeed, it was shown in Ref. [51] that, in a 1D optical

lattice, the eﬀect of the on-site dipolar interactions on the phases of the condensates

can be accounted for by introducing an eﬀective

-wave scattering length. This eﬀective

scattering length replaces the true scattering length

, but the basic description of the

system remains the same. The eﬀect of the inter-site dipolar interactions on the phase

dynamics of the system is typically small, as investigated also in Ref. [51].

Understanding the in-trap phase dynamics and the resulting density patterns after the

TOF is required to interpret correctly the absorption images collected in our experiments.

In particular, the dephasing of the on-site condensates in the deep lattice regime has

consequences for our measurements: we have to adapt our evaluation routine of the

absorption images to cope with the multi-peak patterns, as we explain in detail in

section 5.3.1. Furthermore, our investigations of the collapse dynamics of a dipolar BEC

in chapter 6 greatly rely on the discussion of the expansion of a BEC from the lattice.

In this chapter, we could explain all experimental observations even without taking into

account the dipolar interactions. The situation is diﬀerent, however, when entering the

strongly dipolar regime for suﬃciently low

-wave scattering lengths. Then, the dipolar

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