Bose 2.2 User Manual download pdf (Page 63)

with the parameter

def

= E/E

−U

lat

) and the lattice parameter

def

= U

lat

) =

4. The solutions of Eq.

(4.5)

are the so-called Mathieu functions

(

A, Q, ˜z

), which can

be computed numerically for known parameters

and

. Recalling the periodicity of

the lattice potential and according to Bloch’s theorem, any eigenfunction of the system

may be written in the form [170]

(

) =

iqz

(

). Here, the function

(

) is periodic in

with a period

lat

, and

is the quasi-momentum of the wavepacket. For the moment, we

consider negligible center of mass motion (

= 0) such that any eigenfunction

(

) of the

system exhibits the characteristic lattice periodicity.

As a consequence of Floquet’s theorem, any Mathieu function can be written in the form

(

A, Q, ˜z

) =

ir˜z

(

A, Q, ˜z

), with

(

A, Q, ˜z

) being a 2

-periodic function in

˜z

. While in

general complex and non-periodic, the Mathieu functions become real and periodic

for

integer values of

, as requested by Bloch’s theorem. Then, for a given

, the parameter

can only take a ﬁnite number of values and is called the characteristic parameter,

with

being the characteristic Mathieu exponent. Note that, at given

and

, there

exists only one value

solving the Mathieu equation: for even values of

the Mathieu

functions are symmetric, while they are anti-symmetric if r is odd.

4.2.2 Calibration of the Lattice Depth by BEC Diﬀraction

Since the lattice spacing

lat

is given via the laser wavelength and the geometry of the

experimental setup (see section 4.1.1), the lattice depth

lat

is the only parameter that

we need to determine from the experiment to have a full characterization of our system

Among the existing methods [170], we choose the BEC diﬀraction method for a daily

calibration of the lattice depth as it provides a precise result, is applicable for deep lattices,

and consumes only little measurement time.

The underlying principle of this calibration scheme is closely related to the diﬀraction

of laser light from a phase grating (see e.g. [175, Ch.12.6]). The diﬀerence is mainly the

inverted role of matter and light: We ﬁrst prepare a coherent matter wave (BEC) in

its trap. Then, we suddenly turn on the periodic lattice potential and after a variable

evolution time

pulse

we switch oﬀ all optical trapping potentials to observe the diﬀracted

BEC in the “far ﬁeld”, i.e. after a suﬃciently long time-of-ﬂight

At the time

= 0, i.e. directly before switching on the lattice potential, we can assume

the wave function of the BEC

(

) to be constant over the extent of two lattice sites.

When we suddenly turn on the lattice, the system does not adiabatically evolve from the

The periodic Mathieu functions are normalized via the relation

−π

M(A

, Q, ˜z) M

∗

, Q, ˜z)d˜z = π.

In our system, we do not control the absolute position of the lattice sites, i.e. there is no active phase

stabilization of the lattice laser. This does not play a role in most of our experiments, as the lattice

spacing is typically much smaller than the overall size of the BEC. We will comment on the inﬂuence of

the random lattice phase shift when it is relevant for the measurements.

Not only the role of matter and light are exchanged, but also the role of space and time: while in optics

the components are located at diﬀerent positions in space, we apply the elements (lattice potential,

imaging) at diﬀerent instants in time.

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