ωxx2+ωyy2+ωzz2, (2.29) we can determine the so-called Thomas-Fermi radius, which in the i-th dimension is given by
Rtfi = 1 ωi
r2µ
m. (2.30)
The normalization conditionN =R d~r|φ(~r)|of the wave function leads to an expres-sion for the chemical potentialµ
µ= ¯hωho 2
15N a
rmωho
¯ h
2/5
, (2.31)
with geometric mean of trapping frequenciesωho = (ωxωyωz)1/3, total number of atoms N, and scattering length a[116]. By simply combining Eq. 2.30 and 2.31, we finally obtain
Rtfi = 15N¯h2a m2ωi2
!1/5
. (2.32)
Unfortunately, the TF approximation itself is not valid for analyzing the free ex-pansion of a condensate after release from the trapping potential. Once in free fall, the internal energy is converted into kinetic energy and thus not negligible anymore.
First calculations towards the expansion of a condensate by changing the trapping parameters or the external potential have been carried out by solving the GP equation numerically accompanied by comparisons with real experimental data [128].
Another approach is based on classical scaling laws. Here, the dynamic of the macro-scopic wave function is described in the evolution of three scaling parameters, which are obtained by a suitable coordinate transformation [129, 130]. This method will be used to predict the free expansion of condensates in extended free fall experiments at the drop tower (Sec. 5.2).
But before we are able to analyze the temporal evolution of freely expanding con-densates, we obviously have to trap atoms first. This will be discussed in the following section.
2.3 Forming a magnetic trap for neutral atoms
The magnetic dipole interaction energyU of a paramagnetic atom in a given magnetic fieldB~ is
U =−~µ·B~ =−µBcosθ, (2.33)
with~µbeing the magnetic moment andθthe angle between the magnetic moment and the magnetic field orientation. This angle is stabilized by means of the rapid Larmor precession of ~µ around the magnetic field direction with the frequency ωL = µB/¯h.
Quantum mechanically speaking, this becomes
U =gµBmF|B~|, (2.34)
whereg is the Landé factor,µB the Bohr magneton andmF the magnetic quantum number associated with the projection of the total angular momentum L~ =~µ/µBgF onto the direction of B. In Eq. 2.34, we assume the external field to be sufficiently~ small, so the energy of each unperturbed magnetic sublevel is linearly shifted with respect to its hyperfine state (linear Zeeman effect).
Paramagnetic atoms therefore can be trapped making use of the interaction of the magnetic moment with a spatially varying magnetic field, causing a magnetic dipole force (Stern-Gerlach force) as
F~SG=−∇U =−gµBmF∇|B~|. (2.35) Depending on the magnetic polarization of the atoms, we need to create a field minimum or maximum to trap atoms and therefore the product of the magnetic quan-tum number mF and the Landé factor gF allows to classify atoms in three distinct categories:
• mFgF >0: weak field seeker, whose energy increase with increasing magnetic field strength. Therefore, the F~SG is pointing toward a local minimum of the external field,
• mFgF = 0: to first order insensitive to magnetic fields,
• mFgF <0: strong field seeker, whose energy levels decreases with increasing magnetic field strength.
Since there are no magnetic monopoles as Maxwell’s equations prohibit a magnetic field maximum for static current configurations in free space (Earnshaw theorem), one only can trap week field seekers in local magnetic minima. For a sufficient trapping efficiency, these minima should maintain a non-zero value in all dimension, to avoid one of the major loss channels.
Majorana losses
Atoms must maintain a given magnetic field orientation with respect to the local field, otherwise losses occur if an atom changes its state from weak field seeking to strong field seeking or even to a state with mF = 0. This process in known as Majorana spin flips [131]. The trap is only stable, if the precessing atomic spin can follow the changing magnetic field direction adiabatically. In order to maintain the projection of the magnetic moment mF, only slow changes in the projection angle θ w.r.t. to the Larmor frequency are allowed, leading to the condition
dθ
dt < ωL. (2.36)
The presence of spin flips would strongly limit the lifetime of the atoms. Static magnetic traps can be classified by the magnitude of the magnetic field in the center of the trap. Very simple magnetic traps have a zero field in the minimum, B0 = 0, letting non-adiabatic Majorana transitions play a significant role, since losses occur due to unavoidable spin flips. More advanced magnetic traps show a non-zero field component in the minimum, B0 6= 0, and are usually used for cold atom and BEC experiments. Two common magnetic field configurations, representing both cases, will be introduced in the following.
2.3.1 Quadrupole trap
The Quadrupole trap provides a simple magnetic field configuration in which a local field minimum can be realized. The magnetic field scales linearly in all dimensions and may be created by a pair of coils implemented in anti-Helmholtz configuration.
With an axial symmetry along the z-direction, the magnetic field can be approximated around the minimum by
Obviously, the main disadvantage of this trap configuration is a zero field in the minimum. To circumvent this, one may add a homogeneous magnetic field which is rotating in the area perpendicular to the symmetry axis of the quadrupole coils, thus forming a time-orbiting-average-potential (TOP) trap [67, 132]. Another method may be to add a repulsive, optical dipole potential (blue-detuned to the atomic resonance) which prohibits atoms from reaching the zero field [69].
2.3.2 Ioffe-Pritchard trap
A well known and extensively studied example of a magnetic trap with a local minimum is the Ioffe-Pritchard trap (IPT), first discussed by Ioffe [133] and adapted to neutral atoms by Pritchard [134, 135]. The IPT provides a quadratic confinement and has a non-zero magnetic field in the trap center. Assuming an axially symmetric case, the trapping field is given by [115, 136]
B~(r) =B0 It can be created macroscopically with a complex coil assembly [69, 67] or simply by overlapping the magnetic field of an appropriately bent, current carrying wire (e.g., atom chip) with an additional, external homogenous field (Sec. 2.4.2). Independent of the technology, the center of the trapping potential for paramagnetic atoms can be approximated as harmonic,
U(x, y, z)≈ m 2
ωxx2+ωyy2+ωzz2, (2.39)
with trapping frequencies in the axial (ωz) and radial (ωx=ωy ≡ωrad) direction of
ωz =
rµBgFmF m ·√
B′′, (2.40)
ωrad =
rµBgFmF
m ·
s B′2
B0 −B′′
2 . (2.41)
The axial trap steepness thus scales with the curvature of the magnetic fieldB′′only, whereas in the radial direction additionally the gradient B′ and the trap bottom B0
need to be considered. Strongly confined atoms are favorable for high collision rates, necessary for fast re-thermalization during the evaporation process (Sec. 2.5.2).
After introducing the basics of Bose-Einstein condensation and standard principles of magnetic trapping of neutral atoms, the QUANTUS-I apparatus and the experimental sequence used to create degenerate gases with an atom chip will be presented.