Wildermuth PhD thesis 1105

Wildermuth PhD thesis 1105
Dissertation
submitted to the
Combined Faculties for the Natural Sciences
and for Mathematics
of the Ruperto-Carola University of Heidelberg,
Germany
for the degree of
Doctor of Natural Science
presented by
Dipl.-Phys. Stephan Wildermuth
born in: Köln, Germany
Oral examination: October 19th , 2005
One-dimensional
Bose-Einstein condensates
in micro-traps
Referees: Prof. Dr. Jörg Schmiedmayer
Prof. Dr. Markus Oberthaler
Zusammenfassung
Eindimensionale Bose-Einstein Kondensate in
Mikrofallen
Eine neuartige auf einem stromführenden Draht basierende magneto-optische
Falle wurde entwickelt und mit ihr bis zu 3 · 108 kalte Atome in der Nähe einer reflektierenden Oberfläche eines Atomchips gefangen. Diese Atome wurden schrittweise in Mikrofallen umgeladen, die von Drähten auf dem Atomchip erzeugt
werden. In diesen Fallen wurden Bose-Einstein Kondensate (BEK) hergestellt,
die sich im eindimensionalen Thomas-Fermi Regime befinden. Der Übergang von
dreidimensionalen zu eindimensionalen BEK wurde studiert, indem die transversale Größe der BEK nach ballistischer Expansion vermessen wurde. Die Ergebnisse dieser Messung zeigen gute Übereinstimmung mit der Theorie. Als Anwendung der eindimensionalen BEK wurde ein mikroskopischer Magnetfeldsensor
entwickelt. Dieser Sensor ermöglicht Magnetfeldmessungen in einem Bereich, der
für heute gebräuchliche Magnetfeldsensoren nicht zugänglich ist. Eine Feldsensitivität von 4nT wurde bei einer räumlichen Auflösung von 3µm erreicht. Zur
Vermessung der Phaseneigenschaften eines eindimensionalen BEK wurde dieses
kohärent aufgespalten und darauf aufbauend ein Interferometer auf dem Atomchip entwickelt.
Abstract
One-dimensional Bose-Einstein condensates in
micro-traps
A novel wire-based magneto-optical trap has been demonstrated which enables
to collect up to 3 · 108 cold atoms close to the reflecting surface of an atom
chip. These atoms are subsequently transferred to micro-traps generated by wires
mounted on the atom chip and Bose-Einstein condensation has been achieved.
The Bose-Einstein condensates (BECs) created in the micro-traps form in the
one-dimensional Thomas-Fermi regime. The cross-over between three-dimensional
and one-dimensional BECs has been investigated by monitoring the transverse
size of the BEC after ballistic expansion. Good agreement to theory has been
found. As an application, one-dimensional BECs have been used to implement
a microscopic magnetic field sensor. This sensor enables field measurements in
a region which is not accessible for today’s state-of-the-art sensors. A field sensitivity of 4nT at a spatial resolution of 3µm has been demonstrated. To investigate the phase-properties of a one-dimensional BEC, coherent splitting of a
one-dimensional BEC has been achieved and interferometry on an atom chip has
been demonstrated.
Contents
1 Introduction
1
2 Optimized U-MOT setup for BEC production
2.1 Experimental setup . . . . . . . . . . . . . . .
2.1.1 Laser system . . . . . . . . . . . . . .
2.1.2 Vacuum chamber . . . . . . . . . . . .
2.1.3 Imaging system . . . . . . . . . . . . .
2.1.4 Chip mounting . . . . . . . . . . . . .
2.1.5 Computer control and data acquisition
2.2 Magnetic wire traps . . . . . . . . . . . . . . .
2.2.1 Magnetic trapping of atoms . . . . . .
2.2.2 Basic wire traps . . . . . . . . . . . . .
2.2.3 Finite size effects . . . . . . . . . . . .
2.3 Designing magnetic potentials: the U-MOT .
2.3.1 Mirror MOT . . . . . . . . . . . . . . .
2.3.2 Optimization of magnetic field . . . . .
2.3.3 Measurements on U-MOT . . . . . . .
2.4 BEC production close to surfaces . . . . . . .
2.4.1 Experimental cycle . . . . . . . . . . .
2.4.2 BEC in a copper-Z trap . . . . . . . .
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3 Micromanipulation of BECs on atom chips
3.1 BEC in atom chip traps . . . . . . . . .
3.1.1 Chip wire design . . . . . . . . .
3.1.2 Loading of pure chip traps . . . .
3.1.3 BEC in chip traps . . . . . . . . .
3.2 Experimental methods . . . . . . . . . .
3.2.1 Trap bottom stability . . . . . . .
3.2.2 Trap frequency measurement . . .
3.2.3 Atom number determination . . .
3.2.4 Temperature calibration . . . . .
3.3 Lifetime close to surface . . . . . . . . .
3.3.1 Introduction . . . . . . . . . . . .
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ii
Contents
3.4
3.5
3.3.2 Lifetime measurement . . . . .
Local disorder potentials . . . . . . . .
3.4.1 Previous experiments . . . . . .
3.4.2 Thermal atoms close to surface
3.4.3 BECs close to surface . . . . . .
Optimized atom chip geometries . . . .
3.5.1 Single gold layer chips . . . . .
3.5.2 Chips with two isolated layers .
3.5.3 Direct electron-beam writing . .
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4 BEC as magnetic field microscope
4.1 Mapping two-dimensional magnetic field landscapes
4.1.1 Position control . . . . . . . . . . . . . . . .
4.1.2 Magnetic potential reconstruction . . . . . .
4.2 Comparison to state-of-the-art sensors . . . . . . .
4.2.1 Common magnetic field sensors . . . . . . .
4.2.2 Sensitivity of BEC to magnetic fields . . . .
4.3 Reconstruction of the current density . . . . . . . .
4.4 Probing other local potentials . . . . . . . . . . . .
5 Exploring low-dimensional BECs
5.1 Theory of 1d BECs . . . . . . . . . . . . . . . . . .
5.2 Cross-over between 1d and 3d BEC . . . . . . . . .
5.2.1 Ballistic expansion of a BEC . . . . . . . . .
5.2.2 Expansion measurements of BECs . . . . . .
5.3 Trapping geometries for 2d BECs . . . . . . . . . .
5.3.1 Introduction to dipole traps . . . . . . . . .
5.3.2 Diffraction of a BEC from an optical lattice
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6 Outlook: matter-wave interferometry
109
6.1 RF-induced double-well potential . . . . . . . . . . . . . . . . . . 109
6.2 Coherent splitting of a BEC . . . . . . . . . . . . . . . . . . . . . 110
6.3 Future experiments: probing the phase-distribution of a 1d BEC . 112
7 Summary
115
A D2-line of Rubidium
117
B List of publications
119
C Acknowledgment
121
Bibliography
123
1 Introduction
Over the last decades laser cooling of neutral atoms has become a powerful technique. It boosted experiments in many laboratories all over the world, ranging
from highly technical to very fundamental studies: Atomic fountain clocks based
on laser cooled Cesium atoms have been built [1] and define today’s time standard. At the same time laser cooling enables to investigate fundamental questions
of quantum mechanics [2, 3] and paved the way to Bose-Einstein condensation
[4, 5, 6]. Today, not only the laser cooled atoms are under investigation by themselves, but these precisely controllable samples are used to model systems known
from other branches of physics: One example is the superfluid to Mott-insulator
transition [7].
The traps used in these experiments have been formed by structures located
outside of the vacuum chamber. This way the atomic samples inside the chamber
have been manipulated by means of dissipative laser fields, optical dipole traps,
or magnetic traps. In the past years several experiments miniaturized the field
generating structures and put them into the vacuum chamber close to the atoms.
One of the most prominent examples is the so called atom chip [8], composed of
a substrate sustaining micro-fabricated wires. The magnetic fields produced by
these current-carrying wires can be used to trap and manipulate atomic samples.
A variety of experiments has been performed: Atoms have been trapped and
guided [9, 10, 11] in complex multi-wire guides [12, 13], beam-splitters using
magnetic [14, 15, 16, 17] and electric [15] fields have been implemented, BoseEinstein condensates (BECs) have been produced in atom chip traps [18, 19],
and conveyor-belts for BECs have been realized [20].
This miniaturization is advantageous for several reasons: The highly integrated
atom chips allow for the construction of small setups which enable a robust and
stable operation [21]. This is even more essential if sensors based on these devices
are to be used outside of the laboratory. Moreover, the localization of atoms in
extremely steep traps is possible on an atom chip. These high trap frequencies in
combination with inter-trap distances on the order of a few microns could lead to
fast operation times of quantum-gates. This is desirable for quantum information
processing (QIP) and makes atom chips well suited for the implementation of QIP
[22]. Besides this technological purpose, localization of the cold atoms due to the
strong confinement in one or two dimensions has become comparable to their deBroglie wave-length, which is on the order of a micron at a temperature of 100nK.
As an example, transport and propagation of bosonic as well as fermionic atoms
in one-dimensional guides could be studied and compared to quantum transport
2
Introduction
in electron systems [23].
This thesis work was focussed on the generation and manipulation of onedimensional Bose-Einstein condensates (BECs) in magnetic micro-traps generated by an atom chip. To study these BECs a new apparatus has been set up
which is described in chapter 2. The aim was to create and manipulate BoseEinstein condensates of Rubidium-87 atoms close to the surface of an atom chip.
To keep the setup small and easy to handle, a single chamber vacuum system
bas been chosen which has been optimized for good optical access to the atomic
samples. By placing macroscopic wires directly underneath the atom chip, the
need for external coils has been reduced to a minimum: An optimized U-shaped
current-carrying wire in combination with an external homogeneous magnetic
field is used to create the quadrupole field needed for a magneto-optical trap
(MOT). This way, up to 3 · 108 atoms can be captured a few millimeters above
the surface of the atom chip [24]. From this integrated mirror-MOT, the atoms
are being transferred via a magnetic trap generated by a macroscopic Z-shaped
wire into the final micro-traps. Here, Bose-Einstein condensation of up to 105
atoms has been achieved in various chip-traps down to a distance of a few microns
from the atom chip surface (chapter 3).
Based on these BECs a highly sensitive magnetic field sensor has been invented
[25] which is discussed in chapter 4. It enables to measure magnetic field variations with a field sensitivity of 4nT at a spatial resolution of 3µm. Thus, field
mapping in a regime has become possible which cannot be covered by commonly
used field sensors. Moreover this field microscope allows to probe magnetic field
variations ∆B in the presence of large offset-fields B up to ∆B/B = 10−5 . Therefore, it has been used to map the magnetic field produced by a micro-fabricated
current-carrying wire. From this two-dimensional magnetic field landscape the
current-density distribution in the wire can be reconstructed which is in particular interesting in the context of disorder potentials: Disorder potentials originate from a meandering current-flow inside a wire, leading to fluctuations of the
magnetic potential along the micro-trap. In many experimental groups, these
unwanted and uncontrollable potential modulations excluded experiments closer
than ∼ 100µm to the surface of the trapping wire. Due to a superior fabrication
technique [26] of the atom chip used in the experiment discussed in this thesis,
these disorder potential fluctuations are reduced by a factor of 100 [27]. Thus,
experiments in an up to now not accessible regime have become possible: The
surface of the trapping wire can be approached down to a distance of 30µm still
maintaining a homogeneous and unperturbed BEC.
These extremely smooth potentials have been used to generate and investigate
one-dimensional BECs (chapter 5). In this regime the atoms are confined strongly
in two directions so that excitations in these directions are effectively frozen
out. The cross-over from three-dimensional BECs to the weakly-interacting onedimensional regime has been monitored in detail by measuring the transverse size
of the expanded BECs. Good agreement to theory has been found. Moreover the
3
detailed shape of the profile has been compared to calculated profiles obtained
by a simulation of the two-dimensional Gross-Pitaevskii equation. These onedimensional BECs microns above the surface of the atom chip are an ideal starting
point for further experiments since the elongated trapping potential can easily be
modulated by close-by chip wires. For example a steep and highly localized dip
can be added to the relaxed longitudinal potential to study growth dynamics of a
one-dimensional BEC [28, 29]. These experiments are directly linked to growth of
three-dimensional BECs which is currently under investigation [30]. Potentially
effects beyond mean-field theory could be studied this way.
To study the phase properties of a one-dimensional BEC a coherent beamsplitter has been demonstrated which relies on adiabatic potentials generated by
structures on the atom chip (chapter 6). Based on this beam-splitter a interferometer has been built and phase preserving splitting of a one-dimensional BEC
has been shown [31].
4
Introduction
2 Optimized U-MOT setup for
BEC production
In this chapter an integrated setup for robust and precise loading of various atom
chip traps will be presented. A wire-based magneto-optical trap (MOT) allows
simplified trapping and cooling of a large number of atoms near a material surface.
With a modified U-shaped current-carrying copper structure more than 3 · 108
Rubidium-87 atoms can be collected in a mirror MOT without using quadrupole
coils (section 2.3). These atoms are subsequently loaded to a Z-wire trap where
they are evaporatively cooled to a Bose-Einstein condensate (BEC) close to the
surface (section 2.4). A brief introduction into the basic concepts of magnetic
trapping of neutral atoms will be given focussing in particular on features of
micro-traps (section 2.2). In the first section of this chapter the experimental
setup will be described (section 2.1).
The content of sections 2.3 and 2.4 of this chapter has been published as:
Optimized magneto-optical trap for experiments with ultracold atoms near surfaces, S. Wildermuth, P. Krüger, C. Becker, M. Brajdic, S. Haupt, A. Kasper,
R. Folman, and J. Schmiedmayer, Phys. Rev. A, 69, 030901(R).
2.1 Experimental setup
Today, production of BECs can be routinely achieved in many laboratories all
over the world. Standard techniques of laser cooling of neutral atoms are used
to collect large numbers of atoms in a MOT. These pre-cooled atoms are transferred to magnetic or optical traps where Bose-Einstein condensation is achieved
by evaporative cooling. These experiments are performed inside an ultra-high
vacuum (UHV) chamber.
Since general introductions to laser cooling and trapping are available in several
textbooks [32] only specific details of the experimental apparatus will be discussed
in this section. Several aspects of the setup have already been described in
numerous diploma theses [33, 34, 35, 36]. In the beginning of this section the
compact laser system will be discussed (section 2.1.1). The single-chamber UHVsetup will be introduced in section 2.1.2. To guarantee a sufficient vacuum for
BEC production an optimized Rubidium-dispenser operation in a pulsed-mode
will be highlighted. Section 2.1.3 deals with the absorption imaging system which
will be described in detail. It allows to image atomic clouds located only microns
6
Optimized U-MOT setup for BEC production
Figure 2.1: Schematic drawing of the
laser-stabilization section of the laser setup.
The shown setup is put onto a 60 × 90cm2
bread-board for thermal, mechanical, and
electric isolation and is covered by a closed
wooden box. The upper half contains a selfmade master-slave laser combination providing the light for the repumping transition. In the lower half the commercially
available TA-100 laser-system consisting of
a master laser and a tapered amplifier is
shown providing the light for the cooling
transition. As can be seen in the drawing
the light required for spectroscopy is taken
directly from the master laser so that the
full laser power of the slave (amplifier) can
be used in the experiment.
above the atom chip surface from all three directions. In section 2.1.4 the central
part of the apparatus will be introduced: the atom chip and its mounting. Section
2.1.5 deals with the stand-alone instrumentation and control system.
2.1.1 Laser system
The laser setup used in the experimental apparatus can be divided into three
sections: The first section consists of two diode-lasers setups including a spectroscopy setup for each laser which is required for frequency stabilization. In the
second stage these two light-beams are being split up and frequency shifted by
acusto-optic modulators. This light is sent into the third section directly or by
means of optical fibers and is guided into the vacuum chamber where the experiments are performed. In the following, each section will be briefly discussed.
Laser stabilization
To achieve laser-cooling of neutral atoms intense light close to a strong atomic
transition is needed. In Rubidium-87 the D2-line (λ ∼ 780nm) provides a strong
cooling transition. This near infrared light can easily be generated by semiconductor laser-diodes which allow for small and robust laser setups [33, 35]. These
laser-diodes have to be frequency-stabilized to a linewidth below the natural
linewidth of this Rubidium transition which is ∆Γ ∼ 6MHz. By monitoring the
absorption of the laser light in a Rubidium vapor cell an error signal can be generated being proportional to the deviation of the laser-frequency from the atomic
resonance. This electronic signal can be used in a feedback loop to stabilize the
2.1 Experimental setup
7
Figure 2.2: Hyperfine structure of the
D2-line of Rubidium-87. The frequency of
the splitting between the hyperfine levels
is given [37]. On the right the F -number
of the hyperfine level is indicated and in
parenthesis the Landé gF -factor for each
level is given.
laser frequency to an accuracy of ∆Γ/Γ < 10−8 .
For the operation of a magneto-optical trap (MOT) the cooling transition from
the hyperfine state 52 S1/2 , F = 2 to 52 P3/2 , F = 3 is used (Fig. 2.2). This light
is generated by a commercially available laser system (TA-100)1 schematically
depicted in Figure 2.1: Light from a master laser is sent into an amplifier which
provides an output power of 450mW [33]. About 10% of the light from the master
laser can be used in a Doppler-free absorption spectroscopy setup [38]. Here the
laser light is split up into two paths: One is only used to monitor the absorption signal on an oscilloscope whereas the second beam is used for a frequencymodulation spectroscopy [39]. Following this technique the light frequency is
modulated by an electro-optic modulator (EOM) with a frequency of 20MHz
to generate the error-signal. Consequently a higher regulation bandwidth (compared to the modulation of the repumper-laser in the kHz-range discussed below)
is achieved and additionally technical noise (‘1/f -noise’) is reduced. This laser
setup has been well characterized including a high resolution measurement of the
Rubidium-85 and Rubidium-87 spectra and a detailed analysis of the linewidth
of the laser-frequency [33].
As can be seen in Figure 2.2 atoms can leave the cooling cycle and fall to the
2
5 S1/2 , F = 1 ground state. Due to the large hyperfine splitting (∼ 6.8GHz) of the
Rubidium-87 ground state these atoms cannot be excited by the cooling light. To
pump the atoms back into the cooling cycle a second laser setup is used providing
laser light on the 52 S1/2 , F = 1 to 52 P3/2 , F = 2 transition. Therefore a self-made
master-laser is used to provide light for a Doppler-free absorption spectroscopy
1
Tapered amplifier laser system TA-100, Toptica Photonics AG, Martinsried, Germany
8
Optimized U-MOT setup for BEC production
Figure 2.3: Schematic drawing of the
frequency preparation section of the laser
setup. Laser light from the TA-100 and the
repumper enters this section in the lower
left corner and its frequency is shifted by
several AOMs. The MOT light leaves this
section in the upper right corner whereas
light used for imaging, optical pumping,
and monitoring the repumper is coupled
into single-mode fibers. The depicted section has a size of ∼ 0.6 × 1.3m2 .
and to feed a slave-laser. The error-signal is generated by modulating the laserdiode current in the kHz-range. The laser-lock electronics and the laser-driver
have been build in-house. Details of the design [40, 41] and the setup [35] have
been discussed extensively. The master-laser is operated at a fairly low output
power of 15mW where best long-term stability has been found. The slave-laser
provides an output power of 50mW.
Both lasers are not stabilized to the desired absorption line in the Rubidium
spectrum but are locked to a cross-over peak [38] close by2 . A detailed spectrum
of Rubidium-87 can be found in appendix A. The cooling laser is stabilized to the
F = 1 and F = 3 cross-over peak and the repumping laser is stabilized to the F =
1 and F = 2 cross-over. To shift the frequency of the lasers (close) to the desired
Rubidium transition acusto-optic modulators are used. The corresponding setup
will be discussed in the next paragraph.
2
These additional peaks arise from the pump and the probe beam being resonant to different
transitions of a velocity class v 6= 0 whereas the standard Lamp-peaks appear when pump
and probe beam are resonant to the same transition of a velocity class v = 0. Thus the
cross-over peaks appear at frequencies (ω1 + ω2 )/2 where ωi is the frequency of the atomic
transition. This provides additional reference frequencies which can be used to stabilize the
lasers.
2.1 Experimental setup
9
Frequency preparation
The laser light which is stabilized to the cross-over peaks can be precisely shifted
by acusto-optic modulators (AOM) to the desired detuning with respect to the
atomic transition frequency. In the AOM a radio-frequency causes density modulation in a crystal at which the laser light is diffracted and thus frequency shifted
depending on the diffraction order. Typically the first diffraction order is used
and the light is shifted by 50 − 150MHz with an efficiency of 70 − 80%. In the
setup shown in Figure 2.3 the AOMs are used in single-pass and in double-pass
configuration. Double-pass configuration is advantageous if the frequency shift
of the laser light has to be changed during the experiment, because the changing
deflection angle is compensated since the beam is retro-reflected. The single-pass
AOMs are usually operated at a fixed frequency. In the shown setup the light
provided by the TA-100 is divided into three paths: One is shifted by a doublepass AOM operated at 96MHz to be 20MHz red-detuned with respect to the
cooling transition. The second beam is shifted in a double-pass by 106MHz to be
directly on the atomic resonance for imaging purposes. The last beam is shifted
in a single-pass by −55MHz to be resonant to the optical pumping transition.
Finally the light provided by the repumping laser is shifted in a single-pass by
79MHz to be resonant to the repumping transition. The non-diffracted light (zero
order) is used to monitor the frequency of the slave laser in a Fabri-Perot cavity.
In addition these AOMs provide a very fast switching of the laser light in less
than 1µs with an extinction ratio of 10−3 for the single-pass configuration (10−6
for the double-pass configuration). These fast switches are accompanied by slow
mechanical shutters (a few ms switching time) to block the light completely. The
light for the MOT-beams leaves this section by regular mirrors whereas light for
imaging the atoms, optical pumping and for monitoring the repumper is coupled
into single-mode fibers.
MOT section
The cooling and the repumping light is passed through a telescope to enlarge the
beam diameter to 2cm (Fig. 2.4). Here, nearly half of the cooling light is cut
off by a pinhole because an almost homogeneous beam shape has been found to
lead to the highest atom numbers in the MOT. The cooling light is split into four
beams of 30mW each and is directed into the vacuum chamber. Two beams enter
the chamber horizontally. The two other beams are directed into the chamber
under an angle of 45 degrees from below and are reflected into each other at the
atom chip surface. The repumping light (5mW per beam) is overlapped at a
polarizing beam-splitter cube with these latter beams. To generate the correct
circular polarization needed for MOT-operation, four quarter-wave plates are put
into the beams before they enter the chamber.
The whole laser setup and the vacuum chamber is put onto a table (2 × 1.5m2 )
10
Optimized U-MOT setup for BEC production
Figure 2.4: Schematic drawing of the
MOT section of the laser setup. The cooling and the repumping light beam are enlarged to a diameter of 2cm by the telescopes. The light is split up in four separate laser beams: Two are passing horizontally through the vacuum chamber whereas
the remaining two are reflected at the chip
surface under an angle of 45 degrees. The
repumping light is overlapped with the latter two beams. The region enclosed by the
dashed box is located ∼ 35cm above the
plane of the laser table.
which is hovering on inflatable table-legs for vibrational isolation. Additionally
the laser-setup is put onto a 90 × 60cm2 bread-board which is vibrationally and
electrically isolated from the table by a rubber-mat. A wooden box covers the
laser-setup to avoid thermal drifts and air turbulence. The frequency-preparation
stage is divided from the vacuum chamber by a black wall, so that no stray-light
can enter the vacuum chamber. Most of the beam-path is covered by cardboard3
since the atom number in the MOT is sensitive to air-turbulence in the path of
the laser beams.
2.1.2 Vacuum chamber
The vacuum chamber consists of a stainless steal4 octagon with a height of 7cm
and a diameter of 20cm. This main-chamber is attached to a 5-way cross where
the pumps and the vacuum gauge are attached to (Fig. 2.5a). The main-chamber
can be optically accessed through high quality windows5 : A big window from
the bottom (free diameter 11cm) and five windows (free diameter 30mm) at
the side have been sealed with Helicoflex gaskets6 . These windows have high
optical quality in terms of flatness and are anti-reflection coated for the Rubidium
wavelength. On the other three ports at the side of the octagon CF-flanges with
3
These highly sophisticated cardboard constructions have been carried out by Berth Origami
Hoffer.
4
Steal with low magnetization, 316LN (1.4429 ESU)
5
Two sizes have been used: A diameter of 132mm (47.6mm) at a thickness of 8.5mm (6mm)
made of UV-FuS (BK7). The flatness of the surface is better than lambda/10 and the
windows have been coated with antireflection coating for λ = 780nm on both sides for
angles of 0 − 45deg (0deg). The reflectance has been found to be smaller than 1% (0.25%).
Lens Optics GmbH, Allershausen, Germany
6
Type HLV290B. Garlock GmbH, Neuss, Germany
2.1 Experimental setup
11
Figure 2.5: (a) A schematic drawing of the vacuum chamber. The main chamber
consists of a stainless steal octagon which allows for good optical access through seven
side windows (c) and through a big window at the bottom (b). One port at the side
is used to insert the Rubidium dispensers. The chip mounting is put into the chamber
from the top. The chamber is attached to a five way cross where the pumps and the
vacuum gauge have been attached.
a regular copper sealing have been used. Here, two standard windows close the
chamber and at one port the dispenser source is attached.
The Rubidium source is build up as follows: Four copper feedthroughs of a
diameter of 6mm enter the chamber. Two of them hold three Rubidium dispensers
in parallel. These dispensers are heated up by a current-flow and Rubidium which
is chemically bound in the dispenser-core is evaporated. The outer end of the
copper-rods is water-cooled to allow for fast cooling of the dispensers. This is
appropriate because the background pressure should be as low as possible in the
magnetic trapping phase. Therefore the dispensers are operated in a pulsed mode:
While the MOT is loaded the dispensers are heated and the current is switched
off during the magnetic trapping phase. With this mode of operation a lifetime
of 45s of the magnetic trap can be achieved (section 2.4.2). The other two copper
rods have been connected by a copper bar inside the vacuum chamber. The
current passing through the dispensers is led through this bar in the opposite
direction to compensate the generated magnetic field. Since this bar and the
dispensers are separated by ∼ 1cm and the distance to the MOT is ∼ 4cm this
field compensation is sufficient.
To reduce the time needed for MOT loading the dispenser cycle has been optimized: In the early phase of the experiment a current of 25A was pushed through
the dispensers and the resulting MOT-loading time was 50s. The fluorescence
signal of the MOT can be seen in Figure 2.6a as a red curve and indicates the
atom number trapped. The blue curve shows the glowing of the three dispensers
monitored by a photo-diode which indicates the temperature of the dispensers.
12
Optimized U-MOT setup for BEC production
Figure 2.6: Optimization of the dispenser cycle. (a) The dispensers are operated
at a constant current of 25A for 50s. They are switched off 5s before the magnetic
trapping phase starts (indicated by the dashed line). It can be seen that the glowing of
the dispensers (blue curve) drops to zero within this cooling time and good vacuum is
guaranteed. The MOT-fluorescence light is shown as red curve. (b) Here the dispensers
are operated at a high current of 32.5A for the first 11.5s to reach the same temperature
compared to (a), which is indicated by the amount of glowing of the dispensers. After
that the dispenser current is reduced to load the MOT and switched off 5s before
magnetic trapping starts. The measured curves have been normalized to the maximum
value obtained in the optimized cycle. Note, that the loading time of the MOT has
been almost reduced by a factor of two.
To reach the same dispenser temperature faster a higher current of 32.5A can
be pushed through the dispensers (Fig. 2.6b). Great care has to be taken since
the dispensers do not stand this high current in continuous operation. After the
same temperature has been reached the current is reduced to 25A to saturate the
atom number in the MOT. The dispensers are switched off 5s before the magnetic
trapping phase starts (black dashed line) and fast cooling can be seen in the corresponding photo-diode signal. The reduction of the MOT-fluorescence is caused
by a reduction of the laser-light intensity which is needed to avoid exited-state
collisions. Almost all atoms can be held in the MOT for these 5s. Using this
optimization strategy the MOT-loading time has been reduced by almost a factor
of two down to 27s. Further minor optimization by using even higher currents
has lead to the experimental cycle discussed in section 2.4.
The fast recover of a good vacuum is not only provided by the water-cooling of
the dispensers but also by a large (400l/s) ion-pump attached to the 5-way cross.
Since ion-pumps need strong permanent magnets for operation the pump has
been put far away from the main chamber. Additionally it could be covered by a
µ-metal shield. To allow for this option two pieces of µ-metal have been put into
the vacuum tube which connects the ion-pump to the 5-way cross [35]. On one
other port of the 5-way cross a Titanium sublimation pump has been attached.
2.1 Experimental setup
13
To avoid direct evaporation of the Titanium onto the vacuum gauge located at
the opposite port, a semicircular iris has been put in front of this gauge. The
chip mounting is inserted through the top port of the 5-way cross so that the
atom chip is hanging up-side down at the height of the small side windows.
2.1.3 Imaging system
Building high quality objectives has a long tradition and elaborated schemes containing up to 10 different lenses have been invented. These objectives are mainly
used in photography, offer a good imaging quality even far off the optical axis,
and are for example corrected for chromatic abberations. For the specialized application needed in the context of cold atom experiments it is often advantageous
to use self-made imaging systems. Since only monochromatic light is used, lenses
designed for this wavelength with small f-numbers can be used allowing for a
spatial resolution of a few microns. Additionally phase-plates can be easily put
into the beam path to enable phase-contrast imaging. Simple diffraction limited
imaging systems can be build using two lenses. Basic principles will be given in
this section as well as a description of the actual setup. A detailed discussion can
be found in a separate publication [42].
In the following sections the technical details of the imaging systems are highlighted whereas the quantitative analysis of the pictures taken by these imaging
systems can be found in section 3.2.3. In that section the determination of the
atom number of an atomic cloud will be described for the case of absorption
imaging.
Introduction
If a lens with a focal length f and a diameter a is illuminated by parallel light
this light is focussed in the focal plane of the lens down to a spot – the Airy-disk.
The radius of this disk is given by r = 1.22 λ f /a where λ is the wavelength of
the light. Two points in the object plane can be distinguished if the center of the
first spot’s Airy-disk is falling onto the border of the seconds spot’s Airy-disc.
Thus the diffraction limited resolution can be defined to be equal to r. Typically
the quantity f /a is referred to as f-number.
To reach a diffraction limited resolutions smaller than 10µm several lenses have
been tested using a ray-tracing software7 . Optical parameters of different lens
types of various manufactures are implemented in this program. This allows to
simulate the imaging properties of single lenses and lens-systems containing several lenses. Best performance has been found for monochromatic glass-doublets
which are available with focal lengths down to 100mm8 resulting in a f-number of
7
8
ZEMAX, Focus Software Inc., Tucson AZ, USA
Melles Griot, Bensheim, Germany: Diode Laser Glass Doublets 06LAI011 with f = 100mm
(a = 30mm) and 06LAI013 with f = 145mm (a = 40mm).
14
Optimized U-MOT setup for BEC production
Figure 2.7: Schematic drawing of the
imaging section of the laser setup. Imaging is performed longitudinally and transverse with respect to the elongated atomic
samples (see indicated Z-shaped wire in the
center of the vacuum chamber). The longitudinal imaging light is overlapped with
the MOT-beams using two beam-splitters.
On the axis of the transverse imaging optical pumping light is added. Here, two cameras can be chosen depending on the position of the flip-mirror: A high-resolution
imaging using a frame-transfer camera or
an overview imaging using a standard CCDcamera combined with an objective.
3.3. This leads to a diffraction limited resolution of ∆x = 3.1µm for λ = 780nm.
To exploit this high spatial resolution a section of the length ∆x has to be imaged onto one pixel of the length ∆y of the CCD-camera. These pixel-sizes are
typically larger than 10µm for CCD-cameras. Therefore the focal length f2 of
the second lens has to be chosen large enough such that the magnification of the
imaging system matches the desired spatial resolution as well as the pixel-size
of the camera. Even larger magnifications can be useful to avoid aliasing-effects
caused by the periodicity of the pixels on the CCD-chip.
In this configuration the object to be imaged is placed in the focal plane of
the first lens and the CCD-camera is put into the focal plane of the second lens.
The distance between the two lenses can be adjusted at will and only effects the
field of view. To guarantee optimal image quality the optical axis of the imaging
system has to be carefully adjusted which is done by high precision lens holders
mounted onto a precisely machined aluminium bar [42].
Imaging setup
In the experimental setup four imaging systems have been integrated to image
atomic clouds from all directions. A schematic drawing can be seen in Figure 2.7. A high resolution imaging was set up to image elongated clouds along
the transverse direction (indicated by the Z-shaped wire in the center of the
vacuum chamber). Here two different magnifications can be chosen using a flip
mirror. This allows to use high spatial resolution imaging onto a frame-transfer
2.1 Experimental setup
f1 (mm)
f2 (mm)
Airy-disk size (µm)
magnification
camera pixel-size (µm)
object-size/pixel (µm)
spatial resolution (µm)
15
longitudinal
150
1200
5.7
9.3
20
2.15 ± 0.05
5.7
transversal
100
400
3.1
3.88 ± 0.02
13
3.35 ± 0.02
3.35 ± 0.02
vertical
145
700
3.5
4.8 ± 0.2
20
4.2 ± 0.2
4.2 ± 0.2
overview
100
55
3.1
0.55
8.6
16
16
Table 2.1: Parameters of the four different imaging system. The focal length f1 and
f2 of the first and second lens has been taken from the specifications of the manufacturer. The diffraction limited resolution given by the Airy-disk of the first lens has been
calculated. The magnification for the combination of the two lenses has been measured
(calculated from the focal lengths when no error is present). The CCD-camera’s pixel
size has been taken from the data-sheet of the camera. Using this size and the magnification, the size of the object which is imaged onto one pixel has been calculated.
The spatial resolution is either limited by this object-size or, if magnification is large
enough, is assumed to be given by the size of the Airy-disk.
CCD-camera9 with a small field of view (around 3.5 × 3.5mm2 ) and low spatial
resolution imaging with large field of view (around 12 × 12mm2 ) using a standard
CCD-camera10 . The latter imaging is used as an overview imaging to look at the
atoms trapped in the MOT or in the initial stage of magnetic trapping in the
copper-Z trap. Here, only a standard objective is used as second lens.
On the longitudinal axis a more complicated scheme has to be used since the
MOT-beams enter the chamber on this axis, too. Therefore MOT-beams and
imaging beams are overlapped with polarizing beam-splitter cubes. This limits
the smallest separation of the first lens from the center of the vacuum chamber
to ∼ 140mm. The 4th imaging system is not shown in the Figure since it is
almost perpendicular to the chip surface. The fiber coupler and the camera11
are mounted underneath the vacuum chamber and the imaging laser beam is
reflected at the chip surface. As an example, Figure 2.8a shows the structures on
the chip imaged by this imaging-system. Dark regions correspond to areas where
no gold is located and thus no light is reflected. A shadow of an elongated atomic
cloud can be seen centered above the 100µm-wide wire and the corresponding
absorption picture is shown in (c). All parameters characterizing these imaging
systems have been combined in Table 2.1.
9
MicroMAX:1024BFT, back-illuminated camera with quantum efficiency of 72% for λ =
780nm, Roper Scientific, Duluth GA, USA
10
TM6AS, Jai Pulnix Inc., Sunnyvale CA, USA
11
For the vertical and longitudinal imaging the same camera has been used: Spectroscopy
CCD-camera with 1300 × 400 pixel. Princeton Instruments, Trenton NJ, USA
16
Optimized U-MOT setup for BEC production
Characterization of the imaging systems
The precise determination of the spatial resolution and magnification of an imaging system is a complicated task since the dimension of the imaged atomic cloud
is not known and no other object in the vacuum chamber can be used for calibration. Therefore the following procedure has been carried out to calibrate the
transversal imaging system: The first lens has been put (approximately) one focal
length away from the center of the vacuum chamber and the CCD-camera has
been put one focal length of the second lens away from the second lens. Since both
lenses are mounted in a solid, high-precision holder to keep the optical axis fixed,
they can only be moved simultaneously. By moving this lens-system an optimal
position has been found where a small atomic cloud is optimally focussed. After
this, the whole setup has been moved to a different table keeping the distance
between camera and second lens fixed. Here, different test-targets have been put
in front of the first lens.
To determine the magnification of the transversal imaging system a micro chip
fabricated on a transparent sapphire substrate has been used. It contained two
10µm-wide gold wires which are separated by a center-to-center distance of 2mm.
Since the distance between these two wires is very precisely known due to fabrication of the chip a good magnification measurement can be performed. The
magnification was found to be 3.88 ± 0.02. Calculating the spot size which is
imaged onto one pixel of the CCD-camera shows that the resolution is not limited by the Airy-disk size of the first lens but by the magnification. This yields
a spatial resolution of 3.35 ± 0.02µm. Additionally a 4µm-wide wire has been
imaged in absorption. The width of the image of this wire never exceeded one
camera pixel which indicates a spatial resolution smaller than the wire width.
These measurements are in good agreement with calculated values obtained with
the ray-tracing software. Also the stability against small misalignments has been
tested which can occur when the imaging system is put back into the experiment.
In the case of the vertical imaging the determination of the spatial resolution
and magnification is not problematic since the chip structures can be imaged
directly. For the longitudinal imaging the magnification can be determined by
imaging atomic clouds at different heights above the chip surface. These distances
are already known from the transversal imaging and calibration of magnification
is possible. All parameters characterizing these imaging systems have been combined in Table 2.1.
Imaging atomic clouds close to a surface
The imaging light beam for the transversal and longitudinal imaging has to be (almost) parallel to the atom chip surface. Usually it is inclined slightly (∼ 20mrad)
with respect to the chip surface so that the beam is reflected at this mirror surface. This leads to two images of one atomic cloud since a real and a mirror
2.1 Experimental setup
17
Figure 2.8: (a) A picture of the chip structures taken with the vertical camera system.
An elongated atomic cloud can be seen centered above the 100µm-wide wire. The
corresponding absorption picture is depicted in (c). An example for a transverse image
is shown in (b). Since the imaging beam is slightly inclined with respect to the reflecting
chip surface (indicated by the dotted line) two images of the atomic cloud can be seen.
This allows to precisely measure the distance of the cloud to the chip surface. An
absorption picture taken along the longitudinal direction can be seen in (d).
image is produced [43]. These two images can be seen in Figure 2.8b and allow
a direct measurement of the distance of the atomic cloud to the chip surface.
Due to diffraction at the chip surface the intensity profile of the imaging light
beam becomes rather complicated and the image of the atomic cloud is obstructed
at certain positions by these features. This can be seen in Figure 2.9top where
every column corresponds to the sum (in longitudinal direction) over a picture
similar to Figure 2.8b. For each column the current in the wire and thus the
distance to the chip surface have been varied. Figure 2.9bottom shows a wavefront propagation calculation where the surface distance and the atomic density
have been used as the only fitting parameters. The smoothing in the measured
data is due to limited spatial resolution of the imaging system.
Stability and noise
For absorption imaging two pictures have to be taken: One contains the shadow
of the atomic cloud and the second is a reference image containing the light beam
intensity. Since these pictures have to be divided in order to calculate the atom
number per pixel (see section 3.2.3 for a detailed discussion of the determination of
atom number) even small shifts in the position of structures in the imaging beam
lead to high disturbances in the resulting picture. To estimate this background
18
Optimized U-MOT setup for BEC production
Figure 2.9: Every column of the shown
plot (top) corresponds to the sum (in the
longitudinal direction) over a picture similar to Figure 2.8b. The atoms move away
from the surface as the current in the chip
wire is increased. A wave-front propagation
simulation (bottom) reproduces the structure of the experimental data. The smoothing of the data is due to the limited resolution of the imaging system.
noise the noise-level can be measured in a region where no atoms are present (see
next paragraph). To minimize the noise several optimizations had to be done:
Low noise (in terms of loudness) light beam shutters have been used to switch
the imaging light which are vibration isolated to the laser table. Furthermore
all optical elements have been attached in a very stable way onto the laser table
to avoid vibrations and all stray light at optical elements in the beam path has
been eliminated. Covering the complete beam path with tubes of card-board not
only prevents optical elements from being soiled with dust but additionally avoids
disturbing air turbulence. Additionally the fans operating in the CCD-cameras
for cooling are operated in a pulsed mode and are switched off several seconds
before the pictures are taken. Good overlap of the two pictures can be achieved
if the delay time between both pictures is reduced as much as possible. Therefore
a frame-transfer camera can be used: These cameras shift the first picture onto
the CCD-chip into a masked region and are able to take the second picture after
a few ms. After both pictures have been taken the low-noise readout of the chip
is started which can take several seconds. On the transverse imaging such a
camera has been used with a delay of only 80ms between the two pictures. Even
shorter delays did not lead to a better image quality. But they can be in principle
achieved if the atoms are moved away from the chip by magnetic-field gradients
or are far detuned from the imaging resonance by homogeneous magnetic-fields.
An atomic cloud can be imaged by the reflected part of the imaging light beam
if it is close to the chip. This is usually the case for in-situ imaging where atoms
are still captured in the magnetic trap. If the trap is switched off atoms expand
ballistically and fall away from the chip due to gravity. Here the image is taken by
a part of the imaging light beam which passes the chip without being reflected
at the chip surface. For these two regions the noise level has been monitored
for 70 images: In the direct part a Gaussian shaped noise distribution with a
noise-level of σ = 0.64atoms/pixel has been found and no pronounced structures
in the images could be observed. For the reflected part much more noise has
been found ranging from less than 1atom/pixel to up to 4.5atoms/pixel. Here a
2.1 Experimental setup
19
regular pattern of parallel stripes has been visible which is caused by the vacuum
window of the CCD-camera (see below). This noise-level fluctuates from shot
to shot whereas the noise-level of the direct part was found to be constant over
time. Thus, the fluctuating noise-level in the reflected part can be attributed to
vibrations of the chip mounting. Note, that these noise-levels do not necessarily
display the noise-level found in an atomic cloud since the imaging beam structure
imprinted onto the shadow of the atomic cloud does not drop out when the first
picture is divided by the reference picture.
The parallel strips seen in the pictures originate from the interference of light
being reflected between CCD-chip and vacuum window of the CCD-camera.
These fringes can be avoided if the camera is tilted with respect to the incoming
light. This results in a region of a certain width (depending on the tilting angle)
on the CCD-chip where no fringes are present. Of course the field of view is
severely limited by this method.
If dense atomic clouds or BECs have to be imaged in-situ, great care has to be
taken when the atom number is to be extracted. In atom chip traps the transverse
size of the cloud easily becomes smaller than the length which is imaged onto one
CCD-camera pixel. In this case a non saturated picture can be seen even if the
cloud is optical dens and absorbs nearly all light. Thus a refined model has to be
developed to extract the atom number in this case.
20
Optimized U-MOT setup for BEC production
Figure 2.10: (a) Photograph of the assembled chip mounting right before inserting
it into the vacuum chamber. The chip is glued to a ceramics block where the copper
structures are embedded in (b). The broad U-shaped structure is isolated from the Hshaped structure on top of it by a thin Kapton-foil (d). The chip has been bonded to the
small pins at the border of the ceramics block (e). Each chip-wire has been connected
by several bonding wires. In (c) a numerical simulation of the current density in the
broad U-shaped wire can be seen. Darker regions indicate high current density. An
almost homogeneous distribution in the broad plate is achieved.
2.1.4 Chip mounting
The chip mounting has to provide electric connections from the outside of the
vacuum chamber to the copper structures underneath the chip and to the chip
itself (Fig. 2.10a). This is done by several high current feedthroughs12 and by
a 35-pin feedthrough13 . These feedthroughs are welded to a regular DN100CFvacuum14 flange onto which the complete chip holder is being build. The height
of this holder is ∼ 31cm and the atom chip is glued to15 the ceramics block on top
of this mounting. This ceramics block is shown in Figure 2.10d and the copper
structures embedded into this block can be seen. The broad U-shaped wire used
to generate the quadrupole field for the MOT is electrically isolated from the
H-shaped structure used to generate a Ioffe-trap by a thin (thickness of 50µm)
Kapton foil (section 2.3). The chip is put on top of these structures and the
connection-pads at the side of the chip are wire-bonded to small pins at the side
of the ceramics block (Fig. 2.10e). Here, up to 20 bonding wires are required
for a single connection since these thin free-standing wires form ideal fuses. At
the lower end of the pins a Kapton-isolated cable is attached to connect the pin
to the feedthrough. Each material used in this mounting has been tested to be
ultra-high vacuum prove. This mounting has been used together with the atom
12
Caburn-MDC
Caburn-MDC
14
Caburn-MDC
15
A UHV-prove
13
GmbH, Berlin, Germany, high current feedthrough (150A, 5kV), MC5-150C
GmbH, 35 conductor pins, IFA35
GmbH, CFBL 150
glue has been used: Epo-Tek 920, Polytec GmbH, Waldbronn, Germany
2.1 Experimental setup
21
Figure 2.11: Photograph of the improved
chip-holder. The copper structure has been
modified so that current can be pushed
through the broad plate in the commonly
used U-shaped path and, in addition, in a
Z-shaped path to form the relaxed trap for
initial loading of atoms (see section 3.5).
The small copper structure on top of the
broad plate has been cut into three separate
pieces: The central Z-shaped wire is similar to the wire used for BEC production.
The two parallel outer copper bars can now
easily be used to provide longitudinal confinement and to shift atomic clouds to the
experimental site. At the border of the ceramics block pins for connecting the atoms
chip can be seen.
chip depicted in Figure 3.1 for all experiments discussed in this thesis. Detailed
engineering drawings and assembling guidelines can be found in [34].
For putting the electron-beam written atom chip (section 3.5) into the vacuum chamber an improved chip holder has been build (Fig. 2.11). This new
holder is based on the previous design and adopts several techniques. Here, the
improvements will be briefly discussed: The copper structure has been changed
to allow to drive (in addition to the U-shaped current path) a Z-shaped current
path to generate a shallow magnetic Ioffe-trap (section 2.3). Furthermore the
H-shaped structure has been cut into three pieces: The central Z-shaped wire is
maintained but the outer copper bars can be controlled independently. They will
be used to generate longitudinal confinement in the chip traps and if the current
in these wires is unbalanced an atomic cloud or a BEC can easily be moved to
the experimental site. Note, that this design allows for all configurations which
have already been used for BEC creation with the old mounting. In addition the
Macor material has been replaced by Shapal because Shapal has more favorable
thermal properties16 and can be machined more easily. To exploit the full capability of the 35-pin feedthrough the chip design has been changed: There are nine
pads in every corner of the chip which allow for more elaborate structures on the
chip (section 3.5). Furthermore the pads and thus the bonding pins and wires on
the central axes have been removed so that unhindered imaging is possible over
a range of 6mm in the transverse and 4mm in the longitudinal direction.
16
−1
Macor is a machinable glass ceramic and has a thermal conductivity of 1.5W(Km) . Shapal
−1
consists of machinable aluminium nitride and has a thermal conductivity of 100W(Km) .
Data-sheets can be found on URL: http://www.goodfellow.com
22
Optimized U-MOT setup for BEC production
2.1.5 Computer control and data acquisition
Controlling the experimental sequence is a complex task since high timing resolution (20µs) is necessary and, at the same time, the duration of the experimental
cycle is comparable long (40s). This task is tackled with a stand-alone instrumentation and control system17 providing 24 analog output channels, 32 digital
TTL-channels, and 8 analog input channels. This system contains the necessary
IO-cards and a processor unit and can be modularly extended if more channels
are needed. The stand-alone unit is connected to a regular computer via an
opto-coupled local-area network (LAN) interface. This decouples the noisy computer environment from the experimental control. The user-interface is based on
Matlab18 and transforms the input-data into an appropriate file-format before
it is sent to the control system. Extensive tests and optimization of the timing
resolution have been carried out as well as detailed characterization of the noise
spectrum of the computer control [36].
The computer for experimental control is connected to four more computers
via a small local network: Two computers are used to control the CCD-cameras
(section 2.1.3), one is used as a data-server, and the fourth is used for realtime analysis and automatic storage of the data. The latter runs two Matlabprograms simultaneously to acquire and to process the data. Here, it is necessary
that the recorded data can be linked to the parameters of the corresponding
experimental run. Therefore the computer control saves all parameters defining
an experimental cycle in a file labelled with a global counter. The acquisition
program receives network communication from the experimental control and from
the CCD-camera program in use to match the taken picture to the actual global
counter. Both, the parameter file and the picture data is put onto the dataserver for further analysis. The program used for realtime analysis calculates the
atom column-density and from this the atom number (section 3.2.3) and sample
temperature (section 3.2.4). If an in-situ image is taken, the distance to the chip
is derived by a double-Gaussian fit (section 4.1.1).
The Matlab-program controlling the experiment can be used to scan (several)
parameters automatically. For this two different options are available: One can
scan parameters from a start to a stop value in steps of constant size. The second
option allows to read in a parameter file which is then passed through so that
arbitrary scans can be performed. Thus, an automatic data taking over-night is
possible where experimental conditions are best since less sources of noise and
vibrations are operated in the building.
17
18
ADwin-Pro-System, Jäger Computergesteuerte Messtechnik GmbH, Lorsch, Germany
Matlab 7.1 Release 13, The MathWorks Inc., Natick MA, USA
2.2 Magnetic wire traps
23
2.2 Magnetic wire traps
Starting with the first demonstration of laser cooling of neutral atoms in the 1970s
([44] and references therein) a breathtaking race started to reach colder and colder
sample temperatures. The rewarding goal was to reach the quantum degenerated
regime which had been already proposed for noninteracting particles in 1924
[45, 46]. Dissipative light forces can collect atoms from a vapor at ∼ 1000K and
cool them down to ∼ 20µK allowing at the same time to trap the atomic cloud
at one spot. But for Bose-Einstein condensation (BEC) to occur, even colder
temperatures (∼ 100nK) are needed. Therefore a mechanism had to be found
where the final temperature is not limited by the recoil energy of a single scattered
photon: Atoms had to be trapped in the dark only held by magnetic fields.
Combining this new trapping technique with further cooling by evaporation of
atoms from this trap eventually succeeded in creating a BEC in dilute alkali gases
in 1995 [6, 4, 5].
In this section the basic principles of magnetic trapping of neutral atoms will
shortly be reviewed (section 2.2.1), focusing on magnetic traps generated by
planar wire structures mounted on a surface (section 2.2.2). All general features
of such traps can be understood assuming idealized infinitely thin wires. This
assumption holds as long as the distance of the magnetic trap to the wire center is
larger than the dimension of a realistic wire. Since this condition is not fulfilled in
most of the experiments discussed in this thesis, finite size effects will be discussed
at the end of this chapter (section 2.2.3).
2.2.1 Magnetic trapping of atoms
In an inhomogeneous magnetic field B a neutral atom with the magnetic dipole
moment µatom experiences a force F = ∇(µatom · B). This force has been first
demonstrated in the famous experiment by Stern and Gerlach in 1924 by splitting
a beam of silver atoms [47]. This quantized deflection is due to the quantization
of the magnetic moment of the atom µatom = mF gF µB where mF is the quantum
number identifying the magnetic sublevel, gF is the g-factor associated to the
atomic hyperfine level and µB is the Bohr magneton. Constant development of
tools using this interaction to manipulate atomic beams has led to magnetic field
geometries allowing to trap atoms. Today, a wide range of different magnetic
traps is available [48, 49]. Two basic field geometries will be reviewed in the
following: the quadrupole trap and the Ioffe trap.
A quadrupole field can be created by simply using two coils carrying equal but
counter-propagating currents in anti-Helmholtz configuration (Fig. 2.12a). The
magnetic field is zero at the center between the coils due to symmetry. Since
the extension of a trapped atomic cloud is much smaller than the size of the
coils only the configuration of the magnetic field close to the center of the trap
has to be taken into account. The magnetic field close to the center is given
24
Optimized U-MOT setup for BEC production
Figure 2.12: Schematic coil configurations used for magnetic trapping. (a) Quadrupole trap; Two coils carrying counter-propagating currents (radius R separated by a
distance D) generate a magnetic field which has a zero at the center between the coils.
The field geometry is quadrupole-like close to the minimum of this magnetic trap.
If D = R the system is a so called anti Helmholtz-configuration. (b) Ioffe trap; Four
straight wires generate a two-dimensional quadrupole field confining atoms in the radial
direction. To close the axial direction two so called pinch coils carrying co-propagating
currents are used. This results in a finite magnetic field at the minimum of the trap.
by B(x, y, z) = B · (x; y; −2z). The axial gradient (z-direction) has twice the
size than the radial gradient (x- and y-direction) and at any line through the
minimum of the trap the gradient is fixed. Such a trap has been used to trap
atoms for the first time in 1985 [50]. Typically the depth of a magnetic trap is
limited to a few mK.19 Therefore magnetic trapping is usually combined with precooling of atoms by laser cooling: A magneto-optical trap collects atoms from a
thermal background gas and cools them down to a few mK followed by an optical
molasses decreasing the temperature of the atomic sample to a few µK [32].
In order for the quadrupole trap to work, the atomic moment must be oriented
to the magnetic field vector in a way that the atom is drawn towards regions of
low field. This is usually ensured by preparing the atoms in the correct spin state
during an optical pumping phase before capturing them in the magnetic trap.
But the well defined spin state has to be preserved even when the atoms are
moving in the magnetic field of the trap which can change its direction in a very
complicated way. This adiabatic motion is ensured if the change of the magnetic
field in the restframe of the atom is small compared to the Larmor precession
ωLarmor = µatom B/h̄ of the atomic moment: ωLarmor À (dB/dt)/B. Obviously
this condition is violated for B → 0 close to the minimum of the quadrupole
trap. Here atoms change their spin state in an uncontrolled way – so called
Majorana spin flips [51] – causing losses of atoms from the trap. These losses
increase with decreasing temperature T of the atomic sample scaling like T −2
[52, 53], prohibiting to reach the quantum degenerate regime. Various schemes to
19
To compare the thermal energy of an atomic sample given by Ethermal = kB T to the potential
energy Emagnetic = µB it is convenient to express magnetic field strengths in temperature
units: 1G magnetic field equals 67µK for Rubidium-87 in the |F = 2, mF = 2i state.
2.2 Magnetic wire traps
25
overcome this problem have been realized experimentally: time-orbiting potential
(TOP) traps [53], optically plugged quadrupole traps [6], and magneto-static
traps in Ioffe configuration [54].
Today, the most common way to avoid losses due to Majorana spin flips is to
trap the atoms in a Ioffe trap. This trap has been originally used for plasma
trapping [55]. Based on this geometry magnetic trapping of neutral atoms has
been proposed [56] and experimentally demonstrated [57, 58]. The magnetic field
of a Ioffe trap can be created by a coil geometry depicted in Figure 2.12b. Four
straight current-carrying bars generate a two-dimensional quadrupole field. Since
the magnetic field is translation invariant along the z-axis a closed trap cannot
be formed this way. An additional gradient along this axis can be generated by
two end-cap coils closing the trap. If these coils carry co-propagating currents
a so called Ioffe trap is formed, guaranteing B 6= 0 all over the trap. For cold
atomic samples and all BECs the magnetic potential is well approximated by an
anisotropic three-dimensional harmonic oscillator potential characterized by its
axial and radial frequencies:
s
ωi =
µatom d2 B
m dx2i
(2.1)
where m is the mass of the trapped atom. For typical Ioffe traps the radial frequency is larger than the axial frequency resulting in a cigar-shaped atomic cloud.
The magnitude of the trap frequencies indicates the size of the harmonic area of
the field which becomes linear at the outer trapping regions. The remaining loss
rate in this Ioffe trap due to spin flips scales like Γloss ∼ ωrad exp(−ωLarmor /ωrad )
and is negligible for typical experimental parameters: A typical value of the field
strength at the trap minimum is 1G corresponding to a Larmor-frequency of
2π · 700kHz in the case of Rubidium-87 in the F = 2 state. Thus, trap frequencies can be several tens of kHz without losing atoms due to Majorana spin flips
[59].
2.2.2 Basic wire traps
The side-guide
In the first BEC experiments and in many apparatuses used today magnetic traps
are created by current-carrying structures which are located outside of the ultrahigh vacuum (UHV) chamber. This natural ansatz limits the obtainable magnetic
field gradients and the spatial resolution with which the magnetic trapping field
can be structured. The idea to use microscopic magnetic traps [60] generated by
microfabricated wires on a surface which is put directly into the vacuum chamber
has been realized in several groups [8]. This allows to minimize the separation
between the trap generating structures and the cold atomic samples or BECs
down to a few microns.
26
Optimized U-MOT setup for BEC production
Figure 2.13: (a) Schematic picture of the side-guide configuration. The magnetic field
of a current-carrying wire is superimposed with a homogenous magnetic offset field (bias
field). These two fields cancel at a distance r0 from the wire and form a quadrupole
like field configuration close to the trap minimum (inset). Atoms can be guided along
the wire in a two-dimensional quadrupole guide. (b) The magnetic potential in a plane
perpendicular to the wire is shown. The calculation assumes an infinitely thin and long
wire carrying a current of Iwire = 2A and a bias field of Bbias = 20G. The contours
are equally spaced by Bbias /10. The two one-dimensional plots (black curves) show
cuts through the minimum of the magnetic trap. If a homogeneous field component
Bpar = 4G in the direction of the wire is added the zero at the minimum is lifted to a
finite value leading to a harmonic field configuration (red curves).
First experiments used freestanding wires to trap atoms orbiting around the
wire [61, 62]. Atoms with sufficient velocity and therefore angular momentum
circle around the wire in stable orbits. Here, atoms in magnetic sub-states which
are drawn towards high magnetic fields – so called high field seekers – can be
guided along the wire.
To achieve high robustness and design flexibility one wants the wires to be
mounted on a surface. Obviously the atom guide needs to be at the side of the
wire requiring that the current-carrying field source is outside of the trapping
region. In this case it is only possible to create local minima of the magnetic
field (maxima are forbidden by the Earnshaw theorem [63, 64]) allowing to trap
atoms in a weak field seeking state.
Figure 2.13 shows how such a local minimum can be generated by superimposing the magnetic field created by a straight wire carrying a current Iwire with an
external homogenous magnetic field Bbias oriented perpendicular to the direction
of the wire (bias-field). These two fields cancel at a certain distance r0 from
the wire leading to a two-dimensional confinement of the atoms. This side-guide
geometry forms a tube for atoms which can be guided along the wire. Using
this idea neutral atom traps [65] and guides [61] have first been realized using
freestanding wires. Integrated planar wire traps mounted on a microfabricated
2.2 Magnetic wire traps
27
surface – the so called atom chip – have been successfully demonstrated afterwards [9, 66, 67, 10]. If a homogenous field component Bioffe parallel to the wire
is added the zero of the field at the minimum of the side-guide is removed and
the two dimensional quadrupole trap then has got a harmonic potential close to
its minimum.
The basic scaling laws of the side-guide can be calculated from looking at
an infinitely thin and infinitely long current-carrying wire. For evaluating BiotSavart’s law it is convenient to set µ0 = 4π, mF gF µB = 1, h̄ = 1 and kB = 1. In
these units the field of a wire just becomes B = 2Iwire /r where B is measured in
G, Iwire in A (mA) and r in mm (µm). The following scaling laws can be derived:
distance from the wire is given by r0 = 2Iwire /Bbias
gradient close to minimum of the quadrupole field is given by ∂B/∂r =
2
Bbias
/2Iwire = Bbias /r0 = 2Iwire /r02
frequency of the harmonic
potential close
√
√ the minimum of the guide is given
2
by ω ∼ Bbias
/I mBioffe = Bbias /r0 mBioffe where m is the mass of the
trapped atom
The proportionality constant for ω is 2π · 12.7kHz in the case of Rubidium-87.
These simple scaling laws allow to estimate the achievable trap gradients of
wire traps: Assuming a finite wire with a diameter R the closest distance of the
atoms to the wire center is limited to ∼ R, thus
∂B/∂r ∼ Iwire /r02 ≤ Iwire /R2 ∼ j
(2.2)
This means that the maximally achievable gradient is given by the current density
j which the wire can tolerate. In [26] it has been shown that wires with smaller
cross-section can tolerate higher current densities. This reveals that miniaturizing the current carrying structures not only offers higher spatial resolution for
structuring the magnetic potential but also provides higher trap gradients.
U- and Z-trap
The magnetic guide provided by a wire in side-guide configuration can easily
be extended to a three-dimensional trap. To achieve this, an additional field
gradient in the direction parallel to the side-guide wire is needed. This gradient
can be generated by bending the side-guide wire into an U-shape or into a Z-shape
(Fig. 2.14) since the field components stemming from these end-caps cannot be
compensated by the bias-field. If the wire is bent in a U-shape the contributions
of the two end-caps cancel at the center of the trap and a quadrupole like field
configuration is generated. In the Z-shape case these two contributions add up
and a non-zero field minimum together with a harmonic trap shape is formed.
These trapping potentials have been studied extensively in [68, 69, 8].
28
Optimized U-MOT setup for BEC production
Figure 2.14: A side-guide can be closed in the direction parallel to the wire by adding
a magnetic field gradient along this direction. This can be achieved by bending the
wire into a U- or into a Z-shape.
Those two structures allow to generate quadrupole and Ioffe traps on an atom
chip with only one conducting layer. Here, a drawback is that the longitudinal
and transverse confinement cannot be controlled independently since the current
in the central side-guide wire and in the end-caps is the same. If the fabrication
technology of an atom chip allows two electrically isolated layers this problem
can be avoided: An H-like structure can be used allowing to generate the twodimensional trapping potential with the central wire and to add an independent
longitudinal field gradient by running a parallel or a anti-parallel current in the
outer end-cap wires. This situation is analogous to the Ioffe trap depicted in
Figure 2.12b and allows the versatile control of all trapping parameters.
These two elementary building blocks can be used for the generation of quadrupole fields needed for a magneto-optical trap (section 2.3) and as a Ioffe trap for
atom trapping and BEC generation, respectively (see section 2.4.2). Furthermore, fabrication of isolated two-layer chips has been achieved and the actual
wire design of these chips is discussed in chapter 3.5.2.
Multi-wire guides and traps
The experiments described in this thesis use this basic side-guide potential to
manipulate BECs and cold thermal clouds. The end-caps are either provided by
bending the wire or by additional wires next to the side-guide wire (see section
3.1.1 for details of the chip design).
Numerous experiments have demonstrated more complex wire traps and guides
[8]. For example, the bias-field can be generated by wires mounted on the atom
chip [70], multi-wire guides [12] and beam splitters [14] have been realized, and
conveyor belts [11, 20] have been used to transport thermal atomic clouds and
BECs. As an example, one important guiding geometry should be described
2.2 Magnetic wire traps
29
Figure 2.15: Fluorescence picture of a
thermal atomic cloud guided in a spiral shaped guide. The picture is taken
perpendicular to the chip surface. The
guide is formed by two wires with counterpropagating currents and a homogenous
offset-field perpendicular to the chip surface
(inset).
Figure 2.16: Fluorescence picture of
a thermal atomic cloud which is guided
through an interferometer potential. The
picture is taken perpendicular to the chip
surface. Due to a barrier in the guide
only the atoms with high longitudinal velocity pass through the split interferometer
arms. The guide is formed by four currentcarrying wires and a homogeneous offsetfield (inset).
enabling omni-directional guiding on the whole two-dimensional surface of the
atom chip [12, 13, 71]. Such an atom fiber is depicted in Figure 2.15: Thermal Lithium-7 atoms are guided along a spiral-shaped curved path at varying
curve radii. This direction independent potential is formed by two wires carrying
counter-propagating currents and a homogeneous offset-field pointing perpendicular to the chip surface. The magnetic field of the wires and the vertical offset
field cancel at a line between the wires above the chip surface.
Using this idea, a symmetric interferometer potential in the spatial domain
can be formed (Fig. 2.16). The guiding potential is generated by four wires
carrying counter-propagating currents (see inset) [33]. Splitting is realized by
just increasing the distance between the two innermost wires resulting in two
separated atom-fibers. The magnetic potential has been tested with thermal
Rubidium-87 atoms which have been pushed through the splitting area by an
additional longitudinal gradient. One clearly sees (Fig. 2.16) that only the atoms
with high longitudinal velocity pass through this area. A detailed description of
the experimental realization can be found in [72, 73].
30
Optimized U-MOT setup for BEC production
Figure 2.17: Magnetic field of a 100µm
wide and 10µm tall wire calculated according to equation 2.3. The absolute value
of the magnetic field is shown by equally
spaced contour lines whereas its direction
is indicated by arrows.
2.2.3 Finite size effects
To derive the scaling laws given in section 2.2.2 an infinitely thin and long wire
has been assumed. In real life this assumption is not valid and the finite size
of the wires lead to a deviation from the 1/r dependence of the magnetic field,
limiting the achievable maximal magnetic field and thus the trap gradients. In
the case of a wire with a circular cross-section the field outside of the wire is
equivalent to the field of an infinitely thin wire located at the center of the round
wire. The highest possible magnetic field is given by the field strength on the
surface of the wire.
On our atom chips the wires are of square cross-section. Due to the fabrication
process the width W is usually larger than the height H of the wire. H can be
in the order of 0.2 − 5µm whereas W can be several hundreds of microns. For a
chosen H the smallest possible width of a wire is approximately half of its height
[74]. The magnetic field B of such a square wire which is assumed to be infinitely
long can be calculated analytically:
"
Ã
!
x−
x+
Bx (x, y, W, H) = j 2y− arctan
− arctan
y−
y−
!
Ã
x−
x+
− arctan
+ 2y+ arctan
y+
y+
!
Ã
!#
Ã
2
2
2
x2+ + y+
x− + y −
+ x+ ln 2
+ x− ln 2
2
2
x− + y +
x+ + y −
By (x, y, W, H) = −Bx (−y, x, H, W )
(2.3)
(2.4)
where x± = x ± W/2 and y± = y ± H/2 have been introduced. The wire extends
from −H/2 to +H/2 in y-direction and from −W/2 to +W/2 in x-direction.
The current density j = I/(W H) is assumed to be constant over the entire
cross-section of the wire. As an example the field configuration of a rectangular
wire (cross-section 100 × 10µm2 ) is depicted in Figure 2.17. In section 4.1.1 an
experiment will be discussed in which a BEC is positioned at a constant height
of y = 10µm above such a rectangular wire at different x-positions. For the exact
calibration of measured transverse positions and for the precise calculation of the
trap frequencies this refined model has to be taken into account.
2.2 Magnetic wire traps
31
Figure 2.18: Left: The magnetic field strength is plotted in dependence of the distance
to the wire surface. The magnetic field has been calculated for different wire heights
according to equation 2.3. The wires are normalized to an equal cross-section. Thus,
the magnetic field strength is normalized to current and current density. The dotted
curves represent the field strength of an infinitely thin wire (H = 0). For comparison
the field dependence for the infinitely thin wire has been plotted (black dashed line).
Right: The field gradient is plotted versus the distance to the wire surface. Here the
cross-sections of the wires are indicated (dashed lines).
To illustrate the effect of the finite wire size the magnetic field of different
rectangular wires has been plotted in Figure 2.18 for x = 0. Highest magnetic
field values and thus highest gradients can be achieved when using tall and small
wires. For comparison the magnetic field of an infinitely thin and long wire has
been plotted (black dashed line). It can be seen that these finite size effects come
into play if the atomic cloud is positioned at a distance d from the wire which is
in the same order of magnitude as the dimensions of the wire.
In the discussion above the current density has been assumed to be constant
over the wire cross-section. This assumption holds if the length L of the wire
is much larger than W and H. This condition is usually fulfilled for wires on
the atom chip where L ∼ 2mm and W, H ≤ 200µm. But if these wires have
junctions or crossings the current leaks into the additional material (Fig. 2.19a
and b). The resulting non-homogeneous current density can severely alter the
trapping potential leading to barriers, dumps, or even leaks in the trap.
In the case of macroscopic wire traps (typically H, W = 1mm and L ≥ 2mm
see Fig. 2.22) the current density distribution becomes very complicated and a
numerical simulation is needed. This calculated current density can be used to
refine a stick model where a wire of a finite size is interpolated by many infinitely
thin wires. This method enables the calculation of the full three-dimensional
trapping potential of a macroscopic wire trap. These calculations using a few
32
Optimized U-MOT setup for BEC production
Figure 2.19: Numerical calculations of the current density in a bulk conductor are
shown. Current leaks into non current-carrying arms of a junction (a) or a wire crossing
(b). In the case of a Z-shaped wire (c) the effective length of the central side-guide wire
is reduced compared to the center-to-center distance of the two current-carrying leads.
thousand wires can be performed in a reasonably short time on standard computers [33].
2.3 Designing magnetic potentials: the U-MOT
Starting point of BEC-experiments is a magneto-optical trap (MOT) which collects atoms from a background vapor and cools them to a temperature of a few
µK. This tool has been demonstrated in 1987 for the first time [75] and has become a standard tool in atom optics laboratories described in many textbooks
(see [32] for an introduction). As apparatuses become more complex an integration of the field generating coils is advantageous. Following this idea an integrated
wire-based MOT [24] will be presented in this section.
In section 2.3.1 the concept of a mirror MOT will be briefly introduced. The
further integration of the magnetic field generating coils into the atom chip holder
will be discussed in detail (section 2.3.2) and measurements on this novel integrated U-MOT will be shown (section 2.3.3).
2.3.1 Mirror MOT
A MOT requires laser light forces from all directions. If a MOT is to be placed
close to a surface either the diameter of the laser beams has to be small enough
compared to the distance of the atoms to the surface or the surface has to be
transparent or reflecting. The problem of a material object (partially) obstructing the access of the six laser beams used in a conventional MOT has also been
circumvented by producing the MOT [18] or even the BEC [76] elsewhere and
transfer it to the chip by means of dynamic magnetic fields [18] or optical tweezers
[76]. The alternative is to directly load a mirror MOT [9, 77, 10] only millimeters
2.3 Designing magnetic potentials: the U-MOT
33
Figure 2.20: Schematics of a mirror MOT.
Two laser beams are reflected at a gold surface. Due to the change of helicity after reflection the quadrupole axis needs to be oriented at 45 degrees with respect to the mirror surface. The quadrupole field is generated by a current-carrying plate underneath
the mirror and a homogeneous offset-field.
The second beam pair (perpendicular to the
shown plane) needed for MOT operation is
not depicted.
away from a reflecting surface that acts as a mirror (Fig. 2.20). In this configuration at least one of the MOT beams is reflected of the mirror. In a simple
version used in many microchip experiments two of the regular six MOT beams
are replaced by reflections of two beams impinging upon the mirror at an angle
of 45 degrees. To ensure the correct quadrupole field orientation with respect to
the helicity of the laser beam pairs, the quadrupole axis has to coincide with one
of the 45 degrees laser beams. Up to now, this had to be considered a drawback
since the coils usually employed to provide the quadrupole field are bulky, dissipate a large amount of power, and deteriorate the optical access to the MOT
itself and to the region where the experiments are carried out. As experimental
setups are likely to grow more complex in the future, including quadrupole coils
in the setup will present a major obstacle. Apparatus involving cryostats aiming
at a significant reduction of thermal current noise in conducting surfaces are just
one example.
2.3.2 Optimization of magnetic field
As has been pointed out in section 2.2.2 traps with a quadrupole (Ioffe) like shape
can be generated using wires which are bend in a U (Z) shape together with a
homogeneous offset-field. The drawback of this method is that the exact field
geometry is only reproduced close to the minimum of the trap. To generate a
large volume quadrupole (Ioffe) field one can for example alter the wire shape and
benefit from effects of the finite size of the wire [34]. This optimization procedure
will be discussed in the following.
Large volume quadrupole field
Figure 2.21a shows the field configuration generated by a pair of coils in common
anti-Helmholtz configuration in comparison to the magnetic field of a regular Utrap (Fig. 2.21b). A MOT based on a simple U-shaped wire cannot be used for an
efficient collection of a large number of atoms (for example, from the background
34
Optimized U-MOT setup for BEC production
Figure 2.21: Vector plots of different field configurations. The solid (dashed) lines
indicate the axes of the approximated (ideal) quadrupole fields. (a) Ideal quadrupole
field, (b) regular U-wire quadrupole field with non-tilted bias-field, (c) optimized U-wire
quadrupole field. The bottom parts of (a)-(c) show a cross-section of the wire (black),
the substrate, and the surface above it (gray). (d) Angular deviations from the ideal
quadrupole axes are plotted as a function of the distance from the reflecting surface
along the two 45 degrees light beam paths [dashed lines in (b) and (c)]. The solid
(dashed) line corresponds to the regular (optimized) U-wire configuration. The zero
point of the position axes is chosen to be the center of the quadrupole field (field zero).
The broad wire U clearly approximates the ideal quadrupole field better throughout a
larger spatial region than the thin wire U. The parameters chosen in these examples
were IU-wire = 55A, Bk = 14.5G (12.8G) in the plane parallel to the wire and B⊥ = 0G
(3.0G) perpendicular to the wire for the regular (optimized) U.
2.3 Designing magnetic potentials: the U-MOT
35
Figure 2.22: Photograph of the copper
wire structure fitted in a macor ceramics
block. The broad U-wire is isolated to the
H-shaped structure by a thin Kapton foil.
The innermost Z-shaped wire can be used
to create a magnetic ioffe-trap. The broad
U-wire forms the quadrupole field for the
integrated U-MOT.
vapor). This is caused by the fact that the regular U-wire field is only a true
quadrupole field near the field center (point of vanishing field). Further out
there is a non-vanishing angle between the quadrupole axes and the field lines
(Fig. 2.21d). This angle increases at larger distance from the field zero, i.e. the
MOT center, and eventually the direction of the field vector is even reversed. As
the operation principle of a MOT relies on the correct orientation of the fields
with respect to the polarization of the laser light in each beam, the effective
capture region of the trap and thus the loading rate and the maximum number
of atoms in the MOT are limited. Consequently, the regular U-MOT has to be
loaded from a quadrupole coil MOT in order to collect a large number of atoms.
In addition to the shape of the quadrupole field two more constraints have to
be taken into account: Firstly sufficient field gradients needed for MOT operation
have to be guaranteed. Typically a standard MOT is operated at gradients of
10 − 20G/cm. Secondly the center of the quadrupole field has to be located at
the center of the region where the MOT laser beams overlap. In our experimental
apparatus the laser beam diameter is 20mm and is of Gaussian shape. Therefore
the distance of the MOT center to the chip surface has to be 4 − 6mm leading to
a distance between the U-wire center and the MOT of 6 − 8mm.
For optimizing these three conditions several parameters can be changed: The
current in the U-wire can be varied as well as the strength of the external offsetfield. The bent field lines in the case of the simple U wire can be attributed to the
fact that the thin wire produces a field whose field lines are circles. The simplest
way to overcome this is to fan out the current flow through the central part of
the U by replacing the thin wire by a broadened plate. Inclining the bias-field
with respect to the plane formed by the outer leads of the U improves the field
configuration further. If the plane itself is inclined and if the shape of the current
flow through the plate is adjusted properly, the resulting field will approximate
an ideal quadrupole field more closely.
We chose to set the last two possibilities aside in our experiment, mainly because they lead to only marginal improvements compared to the wide U and are
more difficult to implement. Figure 2.21c shows the field vectors of the quadru-
36
Optimized U-MOT setup for BEC production
Figure 2.23:
Photograph of atoms
trapped in a MOT created by using the
optimized U-wire. The minimum of the
quadrupole field is located ∼ 5mm below
the reflecting surface of the atom chip. This
magnetic field in combination with four
laser beams in a configuration depicted in
Figure 2.20 forms the MOT.
pole field obtained with a modified planar U-shaped wire. The various parameters
(geometry, wire current, and bias-field) were optimized numerically to achieve the
required field gradients and height of the MOT below the mirror surface. A comparison of the field configuration of the U-wire quadrupole field with the ideal
field shows no significant differences in the planes not shown in Figure 2.21. Only
the field gradients deviate from those obtained in a conventional quadrupole configuration: In the direction parallel to the central bar of the U-shaped wires,
the gradients are weak while those in the transverse directions are of approximately equal magnitude. The gradient ratios for the regular (optimized) U-wire
are ∼ 1 : 4 : 5 (∼ 1 : 3 : 4), for the ideal quadrupole field, 1 : 1 : 2. Gradient
ratios, however, are not critical for a MOT operation. In fact, this can even be
an advantage because the aspect ratio of the MOT cloud is better matched to
the magnetic microtraps.
Large volume Ioffe trap
Using the integrated U-MOT a large number of atoms can be cooled to ∼ 20µK
in the close vicinity of the chip surface. To transfer these atoms into the pure chip
traps, it is advantageous to use a large volume Ioffe trap as an intermediate step.
This trap can be provided by a macroscopic copper wire (cross-section ∼ 1mm)
which allows to trap more than 108 atoms and even cool them to a BEC [78, 43].
The first chip holder used in the experiments discussed in this thesis consisted
of a broad U-shaped wire and a H-like structure allowing to generate a Ioffe
trap (Fig. 2.22). Connecting the innermost leads (center-to-center distance of
4mm) leads to a Z-shaped current path. This trap has been used for BEC creation
(section 2.4.2) and transfer of atoms to the chip potentials (section 3.1.2). Typical
operation parameters are Iwire = 60A, a bias-field of Bbias = 27.6 and a bf-field of
Bbf = 16.4G decreasing the trap bottom to 8.7G. The resulting trap frequencies
are ωtr = 2π · 69Hz and ωlo = 2π · 24Hz and the minimum is formed at a distance
of 1.6mm from the chip surface.
During transfer of the atoms from the molasses to this magnetic trap the atoms
2.3 Designing magnetic potentials: the U-MOT
37
Figure 2.24: Calculated magnetic potential of the thin (blue curve) and the broad
(black curve) copper-Z. Left: Magnetic potential versus distance from the chip (mirror
surface is at y = 0). Middle: Magnetic potential in the transverse direction parallel to
the chip surface (x-direction). Right: Magnetic potential in the axial direction parallel
to the broad plate. The dashed line indicates the depth of the magnetic trap due to
gravity derived from the left plot. The inset shows a photograph of the copper structure
embedded in a marcor block.
are heated up by an order of magnitude. This is caused by the small capturing
volume of the trap (Fig. 2.24 blue curves) compared to the typical size of the
molasses of x × y × z = 1.5 × 1.5 × 3mm3 . To enable a better transfer the new
chip holder (inset of Fig. 2.24) provides a large volume Ioffe trap: the broadened
plate can be used as a Z-shaped wire. This allows to push higher wire currents
and thus to form the minimum of the trap further away from the chip surface.
Additionally the slope of the magnetic trapping potential for distances close to
the chip surface gets more shallow. The calculated potential shown in Figure
2.24 (black curves) corresponds to a wire current of Iwire = 120A, a bias-field of
Bbias = 34 and a bf-field of Bbf = −2G lifting the trap bottom to 4.8G. The
resulting trap frequencies are ωtr = 2π · 35Hz and ωlo = 2π · 6.5Hz and the
minimum is formed at a distance of 3.7mm from the chip surface.
This trap can be used to load the atoms more efficiently from the molasses and
successively compress them to the final Z-trap formed by the small copper-Z in
the center of the chip holder.
38
Optimized U-MOT setup for BEC production
Figure 2.25: Top: The number of atoms
is plotted versus the tilting angle between
the bias-field and the plane of the broad Ushaped wire. For these measurements the
current in the U-wire was 55A, the biasfield strength 13G. Bottom: Corresponding
vector field plots for three different angles
(0, 13, and 26 degrees) as indicated by the
arrows. The maximum number of atoms is
trapped at a bias-field angle of 13 degrees
where the shape of the field is closest to
an ideal quadrupole field. Note that this
optimal trap position is not centered above
the wire center.
2.3.3 Measurements on U-MOT
We operate the MOT at a U-wire current of 60A and a bias-field of 13G, i.e. at
magnetic field gradients of 5G/cm (20G/cm) along the axis of weakest (strongest)
confinement. In order to confirm the effect of the improved quadrupole field on the
MOT, we have compared a MOT formed by a thin U-shaped wire (see Fig. 2.22)
with the broad wire U-MOT. The change in geometry yielded an improvement
factor of 10 in atom number in the MOT in our experiment. Inclining the biasfield further enhances this number by a factor of 2 − 3. Figure 2.25 shows the
measured atom number in the MOT as a function of the angle between the biasfield and the plane of the modified U-shaped wire. The corresponding quadrupole
fields for specific angles are shown. The dependance of the number of atoms on
the quality of the approximation of a true large volume quadrupole field is clearly
visible: The MOT contained the highest number of atoms (3 · 108 ) for the optimal quadrupole field that is obtained at a 13 degrees inclination of the bias-field.
To compare the results of the U-MOT, we carried out test experiments with a
conventional six beam MOT before introducing the atom chip assembly in the
apparatus. Neither the loading rates (3 · 107 atoms/s) nor the maximum number
of trapped atoms exceeded those measured with the U-MOT under similar UHV
conditions. Thus we conclude that a modified U-MOT can replace a conventional MOT completely. This integrated MOT serves as a starting point for BEC
creation discussed in detail in section 2.4.2.
2.4 BEC production close to surfaces
39
2.4 BEC production close to surfaces
Typically experiments with trapped ultra-cold atomic clouds and BECs are performed in the following way: Thermal atoms from an oven or dispenser-source
are collected and cooled in a magneto-optical trap. After transfer to a magnetic
or a dipole trap the atomic cloud is cooled to BEC. The density distribution
is imaged destructively after manipulating the BEC. This experimental cycle is
repeated constantly. Typical timescales for a single run of the experiment range
from 30s to a few minutes. The experimental cycle used in our setup consists of
four stages:
• Laser-cooling: Collection of Rubidium-87 atoms from a background vapor
using the integrated U-MOT and further cooling by an optical molasses to
reach sub-doppler temperatures.
• Pre-cooling: Transfer to macroscopic magnetic traps provided by copper
structures and cooling by forced evaporation in these traps.
• Chip experiments: Manipulation of ultra-cold clouds or BECs in microscopic magnetic traps provided by micron-sized chip wires.
• Imaging: Destructive imaging of the atomic density distribution onto a
CCD-camera using high-resolution optics.
Usually the third stage of the cycle changes significantly depending on the actual
chip experiment performed. Therefore this stage will be discussed separately in
chapter 3.1. The other three stages have been standardized remaining unchanged
and will be discussed in this section in detail.
2.4.1 Experimental cycle
In the laser-cooling stage the integrated U-MOT is loaded from a background
gas generated by a dispenser source. This stage of the experimental cycle lasts
22s which are divided in three parts of 10s, 7s, and 5s duration: In the first
10s of the MOT-loading a current of 35A is pushed through three dispensers in
parallel to quickly heat them up. After this fast increase of the temperature of
the dispensers the current is reduced to 25A for 7s to constantly load atoms into
the MOT. After this the current in the dispensers is switched off to allow the
background pressure to drop to a value where magnetic trap lifetimes are long
enough to enable efficient forced evaporative cooling. This intermediate stage
lasts for 5s and atoms can be held in the MOT almost without any loss if the
intensity of the cooling light is reduced from 25mW per beam to 12mW to avoid
exited-state assisted collisions. The detuning of the laser light below the cooling
transition is kept at 20MHz during the whole loading phase. Details on the
optimization of this pulsed dispenser operation can be found in section 2.1.2.
40
Optimized U-MOT setup for BEC production
Figure 2.26: Timing scheme of the experimental cycle: The first stage is used for lasercooling the atoms in an integrated U-MOT followed by an optical molasses. After that
the atomic cloud is magnetically captured in a macroscopic copper-Z trap and further
cooled by forced evaporation. This pre-cooled thermal cloud can be transferred to chip
traps and various experiments can be performed. As a result the density distribution
of the atomic cloud is imaged by a CCD-camera system. To avoid thermal drifts the
total time of the cycle is fixed to 40s always.
The MOT loading is carried out at the center of the region where all laser-beams
overlap (see Fig. 2.20). To transfer the atomic cloud from this position 4 − 6mm
below the chip surface to the position of the minimum of the magnetic trap
(1−2mm) generated by the copper-Z structure the MOT has to be moved towards
the surface. Therefore the bias-field20 controlling the distance of the MOT center
to the chip surface has to be increased from 13G to 17G resulting in a distance
of ∼ 2mm while the current in the U-wire is kept constant. This additionally
increases the gradients of the MOT leading to higher atomic densities. Note, that
the weak gradients on the long axis of the MOT quadrupole field lead to a better
matching to the aspect ratio of the elongated magnetic copper-Z trap. Since the
exact position of the MOT is sensitive to the local shape and intensity-balance
of the laser beams it is usually necessary to additionally match the position of
the MOT in the plane of the chip adjusting the up-field and the bf-field. This
20
Homogeneous offset-fields superimposed with magnetic fields of current-carrying wires are
used to form magnetic traps. The bias-field is parallel to the chip surface and perpendicular
to the central wire (side-guide) of a U- or Z-trap. The bf-field is parallel to the chip surface
and parallel to the central wire. The ud-field is perpendicular to the chip surface. For
details of wire traps see section 2.2.2.
2.4 BEC production close to surfaces
41
Figure 2.27: Timing scheme for optical
pumping the atoms to the F = mF = 2
state at the beginning of the magnetic trapping phase. After 10ms of molasses-cooling
the laser-cooling phase ends at time-step
t = 0 and the pre-cooling phase begins
(compare to Fig. 2.26).
process is referred to as mode-matching and is carried out within the last 100ms
of the laser-cooling stage.
Directly after compressing the MOT the three offset-fields created by external
coils are switched off within 100µs (< 1µs for the current in the copper U). At
the same time the frequency of the cooling light is shifted to be 50MHz below
the cooling transition and the light intensity is kept almost constant. This light
configuration is applied to the atoms for 10ms and displays an optical molasses
[79] resulting in a final temperature of the atomic cloud of 20µK being well below
the Doppler-temperature TD ∼ 140µK. The density of the atomic cloud after
molasses cooling has been found to be ∼ 5 · 1010 atoms/cm3 . Further optimization
of the molasses-cooling has been set aside since the atomic cloud is heated up to
∼ 350µK due to gain of potential energy when it is transferred to the magnetic
trap. This increase in temperature is caused by the small capture volume of the
magnetic trap compared to the size of the atomic cloud after molasses cooling
(section 2.3.2).
After the molasses-cooling the atomic cloud has to be captured in a magnetic
trap which is formed by a Z-shaped copper wire underneath the chip. To trap the
highest possible atom number the atoms have to be transferred to the F = mF =
2 state using optical pumping. This is performed by switching on the bias-field
500µs before the other fields which are needed for the magnetic trap have been
switched on. While the bias-field strength is increasing linearly to 0.5G (after
0.5ms), the atoms are exposed to a 200µs short flash of circular polarized light
propagating along the bias-field direction (detailed timing see Fig. 2.27). This
optical pumping leads to a gain of a factor ∼ 4 in atom number compared to
the captured atom number without optical pumping. To avoid large momentum
transfer to the atoms, light intensity of only 50µW/cm2 have to be used. After
the optical pumping phase the light used for repumping the atoms from the F = 1
hyperfine-state of the ground-state is switched off and the atoms remain trapped
in a magnetic trap.
The time needed for switching on the magnetic fields has to be short enough
so that no atoms drop out of the capturing area due to gravity. In our case the
42
Optimized U-MOT setup for BEC production
limit on switching the fields is set by the power-supply driving the bias-field. To
generate a magnetic trap of reasonable depth it needs to push 15A, corresponding
to 30G, through a pair of coils in Helmholtz-configuration. This final current can
be reached within 4.5ms if the power-supply is pre-charged: The power-supply is
forced to put out its maximum voltage while a switch opens the electric circuit.
After 10ms of molasses the switch closes the electric circuit again and highest
possible rise-time is achieved by applying the maximal voltage to the coils. When
the desired value of the current is reached, the power-supply is switched from its
maximal voltage to the desired one. The currents in the copper-Z and in the bfcoils needed to form the magnetic trap have to be adjusted to match the shape
of the rising bias-field current (see Fig. 2.27).
This procedure allows to capture 2 · 108 atoms in the magnetic trap. These
atoms can be further cooled and manipulated. Usually the atoms are pre-cooled
within 10s by radio-frequency (rf) induced evaporative cooling to a temperature
of 10µK resulting in an atom number of 3 · 106 . These atoms can be transferred
to a pure chip trap without any loss in atom number. The discussion of the
optimization of this transfer process can be found in section 3.1 Typically experiments in the final chip trap last for 1 − 3s and as a result a picture of the atomic
density distribution is obtained.
To image the atomic density distribution the atoms are exposed to a resonant
laser beam and a picture of the shadow of the atomic cloud is recorded. The
duration of this flash of light varies from 20µs to 1ms depending on the used
imaging system (see section 2.1.3 for a description of the different cameras and
optics). To extract quantitative results a second picture of the laser beam without
absorbing atoms has to be taken allowing to calculate the optical density of the
atomic cloud (see section 3.2.3). Since background light disturbs the data-analysis
an additional third picture without any imaging light can be taken to subtract
the background intensity from the two pictures. To achieve high timing precision
the exposure time tex is controlled by an acusto-optic modulator switching the
imaging laser beam rather than controlling tex by the CCD-camera itself. The
delay time between these three picture is only limited by the time needed for the
camera to readout the CCD-chip. Therefore the time between two pictures can
vary from a few ms for a frame-transfer camera to a few seconds for a low-noise
readout of the full CCD-chip.
In our apparatus it is important to always keep the time for a full experimental
cycle constant. This is guaranteed by waiting the full 18s even if the final picture
is taken directly after the transfer of the atoms to the magnetic trap. This
restriction is necessary to avoid temperature drifts in the coils because of changing
duty-cycle. Also the current-ramp of the Rubidium dispensers has been optimized
to a certain amount of heating and cooling giving at the same time optimal MOTloading and long magnetic trap lifetimes. Therefore the cycle time is fixed to 40s.
2.4 BEC production close to surfaces
43
Figure 2.28: Lifetime measurement in the
copper-Z trap without applying rf-cooling.
A drop of the temperature of the atomic
cloud can be observed in the first 25s
(from 350µK to 50µK) leading to an overexponential decay. After equilibrium has
been reached exponential loss due to background gas collisions can be observed. The
inset shows a measurement with a precooled cloud at 25µK yielding a lifetime of
τ = 43 ± 4s.
2.4.2 BEC in a copper-Z trap
To efficiently capture the molasses-cooled atoms in the copper-Z trap the magnetic trap is formed at moderate gradients (Iwire = 60A, Bbias = 27.6G,
Bbf = 16.4G). This results in a shallow trap (ωtr = 2π · 69Hz and ωlo = 2π · 24Hz)
with a high trap-bottom of 8.7G at a distance of 1.6mm from the chip surface.
This trapping geometry is optimized to fit the size and position of the atomic
cloud. After loading the atoms to this trap the bias-field is increased linearly
within 1s to 41.4G to increase the gradient. At the same time the trap-bottom
of the magnetic trap is reduced to ∼ 1G by increasing the bf-field to 25G. This
results in a strong increase of the region of linear slope of the potential resulting
in higher rethermalisation rates compared to a harmonic trapping potential [80].
The trap frequencies of this trap are ωtr = 2π · 380Hz and ωlo = 2π · 32Hz.
In this trap rf-induced evaporative cooling [52, 53] is applied to the atoms.
This technique couples different mF -states by a rf-transition which is in the MHzrange. The inhomogeneous field of the magnetic trap is truncated at a magnetic
field value of B[G] = ν[MHz]/0.7 leading to an energy-selective loss of atoms.
Hot atoms evaporate from the truncated trap and if rethermalization-rates are
high enough the phase-space density of the remaining sample increases. Typically evaporation stops if the ratio of trap-depth to temperature of the atoms
(truncation parameter) is around 10 [80]. If the rf-frequency is reduced further,
while the sample cools, an increase in phase space of several orders of magnitude
can be achieved.
For this cooling technique to work the lifetime of the atomic cloud in the
magnetic trap must be much longer than the time needed for rethermalization.
Since high gradients can be achieved in wire traps and thus rethermalization goes
fast, constraints to the vacuum are not as strict as in standard BEC experiments.
Figure 2.28 shows the atom number in the copper-Z trap monitored over a time
of 40s. Following a first over-exponential decay due to hot atoms leaving the trap
thermal equilibrium is reached after 25s and a lifetime of τ = 43 ± 4s can be
deduced.
44
Optimized U-MOT setup for BEC production
Figure 2.29: Phase space density has been
monitored over the entire 10s of the evaporative cooling phase in the copper-Z trap
(Iwire = 60A and Bbias = 41.4G). The insets
show the density distribution of a atomic
cloud after 14ms of time-of-flight expansion. Here, onset of BEC can be observed as
the isotropic shape of a thermal cloud (a)
changes to an elongated shape for a BEC
(b).
Usually the radio-frequency is radiated by an antenna outside of the vacuum
chamber. Since we use current-carrying structures close to the atoms the radiofrequency can be coupled to the copper-Z wire using a bias-T. This efficient
coupling allows to connect the rf-source21 without further amplification to the
copper-Z wire. In the copper-Z trap we start at an rf-value of 20MHz and decrease
the radio-frequency within a certain time t to a final value of 0.6 − 1.5MHz
depending on the set trap-bottom. We tested linear rf-ramps of length t = 5−25s
and found most efficient cooling for t = 10s. This allows to either produce a
thermal cloud of 3 · 106 atoms at T = 10µK which can be transferred to the chip
traps (see section 3.1) or a BEC. The evolution of the phase space density nλdB
during rf-cooling can be seen in Figure 2.29. Finally nλdB exceeds the critical
value of ∼ 2.6 for BEC. The inset shows this transition as a change of aspect
ratio in the density distribution of a ballistically expanded cloud: The thermal
cloud expands isotropic while the BEC gets an elongated shape.
21
Agilent 33220A 20MHz Function/Arbitrary waveform generator, Agilent Technologies Inc.,
Palo Alto CA, USA
3 Micromanipulation of BECs on
atom chips
Bose-Einstein condensates close to the surface of a micro-fabricated device (atom
chip) allow for a variety of different experiments [8]. This is due to the high design flexibility of the trapping potentials, allowing for a manipulation of atomic
samples by means of magnetic, electric, and optical fields. In this section the
manipulation of Bose-Einstein condensates with magnetic fields generated by
current-carrying wires mounted on a silicon substrate will be discussed. In section 3.1 the wire geometry of the used atom chip will be described in detail
and the generation of a Bose-Einstein condensate in various chip traps will be
discussed. Section 3.2 deals with the characterization of these micro-trap and
with the measurement of properties of the trapped atomic samples. In section
3.3 lifetime close to the surface will be an issue and the influence of material
below the top-most gold layer of the chip will be discussed. Characterization of
reduced disorder potentials close to an atom chips fabricated by evaporation of
the gold layer is outlined in section 3.4. More complex atom chips consisting of
two isolated gold layers and a high spatial resolution section fabricated by direct
electron-beam writing will be introduced in section 3.5.
3.1 BEC in atom chip traps
3.1.1 Chip wire design
The atom chip used in the experiments presented in this thesis consists of an evaporated gold layer on top of a silicon surface. The gold layer has been structured
by photolithography (see section 3.5.1 for details) and wires of several widths
(2 − 100µm) at a height of 3.1µm are defined by ∼ 10µm-wide gaps.
An overview of the central region of this atom chip can be seen in Figure
3.1. Ioffe traps based on a Z-shaped wire as discussed in section 2.2.2 can be
generated by a 100µm-wide (C-D) and a 50µm-wide (C-E) Z-shaped wire. These
traps will be referred to as ‘100µm-trap’ and ‘50µm-trap’. Two 10µm-wide Lshaped wires (A-F and B-F) allow to form a side-guide potential which can be
closed at the open end by additional U-shaped wires next to it. Typically the
outermost wire (L-M) is used to provide this end-cap. This trapping geometry
will be referred to as ‘10µm-trap’. The other U-shaped wires can be used to
46
Micromanipulation of BECs on atom chips
Figure 3.1: Design of the inner region of the atom chip. Different wire geometries
are defined by gaps where no gold has been evaporated (black lines). Wire-widths are
ranging from 2µm to 100µm at a layer thickness of 3.1µm. The two Z-shaped wires
(red; C-D and C-E) and the two L-shaped wires (blue; A-F and B-F) in combination
with the U-shaped wire (blue; L-M) providing an end-cap can be used to form Ioffe
traps (see section 2.2.2). The additional three U-shaped wires (green; G-H, H-J and
K-L) can be used to structure the longitudinal trapping potential of the elongated Ioffe
traps. The shown section of the chip corresponds to an area of 3.85 × 2.35mm2 .
Figure 3.2: Magnetic potential in the transverse (a) and longitudinal (b) direction of
the 10µm-trap calculated at a wire current of 0.45A, a bias-field of 14G, and a bf-field
of 1.1G. An elongated trapping potential (aspect ration ∼ 325) at a distance of 62µm
from the chip surface is formed. The dotted parabola in (a) represents the harmonic
approximation of the trapping potential close to the minimum of the trap.
3.1 BEC in atom chip traps
47
Figure 3.3: The longitudinal potential of
the 10µm-trap can be modified by applying a small current Imod to the U-shaped
wire (H-J) at the side of the trapping wire.
The black curve represents the unmodified potential yielding a trap frequency of
ωlong = 2π · 8Hz. By applying currents
Imod = 5(10, 25, 50)mA the confinement
can be increased significantly to ωlong =
2π · 44(58, 91, 130)Hz.
structure the longitudinal potential of the elongated magnetic traps.
As an example the magnetic potential of the 10µm-trap (A-F) is shown in Figure 3.2 which has been calculated using the stick-model described in section 2.2.3
taking finite-size effects into account. The L-shaped wire and the U-shaped wire
both carry a current of 0.45A to achieve a symmetric confinement in the longitudinal direction (z-direction). Remaining small asymmetries can be attributed
to the partially cancelling currents flowing inwards and outward in the neighboring leads of the U-shaped wire. With a bias-field1 of 14G trap frequencies of
ωtrans = 2π · 2.6kHz and ωlong = 2π · 8Hz are achieved. Due to this large aspect
ratio of the wire traps BECs formed in these traps show phase fluctuations along
the longitudinal direction – so called quasi-condensates (see sections 3.1.3 and
5.1).
To be able to generate BECs with aspect ratios smaller than 20 at high transverse trap frequencies, a small dip in the longitudinal potential can be created by
pushing an additional current in one U-shaped wire (H-J). The direction of the
current has to be configured such that the magnetic field of the U-shaped wire
subtracts from the Ioffe-field of the trap at the position close to the minimum of
the unperturbed 10µm-trap (Fig. 3.3). Using this dip atomic clouds can easily be
compressed longitudinally and trap frequencies ωlong > 150Hz can be achieved.
This allows to control the transverse and longitudinal confinement individually.
An example for BEC creation in a similar trap can be found in section 3.1.3.
1
Magnetic wire traps are created by the superposition of the magnetic field of a currentcarrying wire and a homogeneous offset-field perpendicular to the wire (section 2.2.2). The
offset-field being perpendicular to both the wire and the chip surface is referred to as udfield. The bias-field is the field component being perpendicular to the wire but parallel to
the chip surface. The remaining field direction parallel to the wire is denoted as bf-field.
48
Micromanipulation of BECs on atom chips
Figure 3.4: Left: The temperature of a thermal atomic cloud has been monitored
while the atoms are transferred from the copper Z-trap into a pure chip Z-trap (blue
data points). Due to decreasing trap-bottom an adiabatic cooling can be observed.
For comparison the transverse trap frequency has been calculated for each time-step
(solid curve) and show a similar scaling. Right: Schematics of the ramp timing. The
chip current in the 100µm-wide Z-shaped wire is ramped from 1A to 2A, the copper-Z
current is ramped from 60A to zero, the bias-field is decreased from 41.4G to 18.4G,
and the bf-field is ramped from 24.6G to −3.9G reversing its direction.
3.1.2 Loading of pure chip traps
After pre-cooling the atoms in the copper-Z trap, they have to be transferred to
a trap generated by currents in the chip wires. To transfer atoms from an initial
trap to a second trap generated by different wires it is necessary to guarantee
a smooth passage of the atoms from one trap minimum to the other. If the
distance of the minimum of the trap to the wires is larger than the separation of
the wires, efficient transfer can usually be achieved by using the following scheme:
The current in the wires forming the second trap has to be ramped to its final
value while the initial trap is still switched on. After that the currents creating
the initial trap are ramped to zero. During this transfer, the trap gradients and
the position of the minimum have to be adjusted by changing the homogeneous
offset-field needed for trapping.
In our experiment the following transfer process has been tested: The 100µmwide chip wire is carrying a current of 1A over the whole pre-cooling phase. This
leads to a slightly more efficient pre-cooling process due to higher gradients and a
higher barrier to the chip-surface compared to the pure copper-Z trap. Within 1s
the chip current is increased from 1A to 2A, the copper-Z current is ramped from
60A to zero, the bias-field is decreased from 41.1G to 18.4G, and the bf-field is
ramped from 24.6G to −3.9G reversing its direction (Fig. 3.4right). The chosen
parameters lead to an almost equal compression in the final chip trap compared
to the initial copper-Z trap. To understand this transfer in detail one has to
3.1 BEC in atom chip traps
49
Figure 3.5: Using a Stern-Gerlach type
experiment the relative atom number in the
two magnetic sub-states mF = +2 and
mF = +1 has been measured for different transfer velocities. A thermal cloud has
been transferred from the copper Z-trap to
a pure chip Z-trap. For faster transfer velocities non-adiabatic spin flips can be observed. The inset shows a time-of-flight picture of the atomic density distribution for
a ramp time tramp = 0.4s and the corresponding normalized one-dimensional density profile. The blue line is a guide to the
eye.
take into account that the chip-Z can easily be misaligned with respect to the
broad copper-Z. Additionally the homogeneous offset-fields might be misaligned
with respect to the chip-Z. In our setup both is the case: The chip is shifted
by ∼ 200µm and rotated by an angle of ∼ 20mrad with respect to the copper
structure underneath the chip. Furthermore a fraction of ∼ 8% of the bias-field
are pointing into the direction of the Ioffe-field of the magnetic trap and decrease
the trap-bottom. Putting these correction into the calculations of the magnetic
trapping potentials results in a correct description of our experimental findings.
For example, the temperature of a thermal cloud has been monitored while it is
transferred (blue data points in Fig. 3.4left). The observed adiabatic cooling can
be well understood when comparing its scaling behavior to the varying transverse
confinement of the trap (solid line).
To test whether a transfer is adiabatic the following Stern-Gerlach type experiment has been carried out: When releasing the atoms from the trap the
current in the chip wire is left on during expansion of the cloud. This results in
a magnetic gradient which spatially separates different mF -states. For our magnetically trapped Rubidium-87 atoms the two trapped states (mF = +1, +2) can
be observed. In the inset of Figure 3.5 an absorption pictures of such separated
clouds can be seen: The lower cloud has been more strongly accelerated by the
magnetic field gradient than the upper cloud allowing to determine the relative
atom numbers in the two states by a double-Gaussian fit. From the data obtained
with this method a clear signature of non-adiabatic spin-flips can be seen when
the trapping potentials are changed too fast (Fig. 3.5). This allows to test and
optimize all ramps to avoid non-adiabatic transitions.
Using the simulations of the trapping potentials and the Stern-Gerlach method
to avoid non-adiabaticity, the transfer to the chip traps has been optimized. As
an example, the timing of an improved transfer to the compressed 100µm-wire
50
Micromanipulation of BECs on atom chips
Figure 3.6: Optimized ramp timing.
Atoms are first being transferred to an
intermediate, relaxed chip-trap before the
trap gradients are increased. This ramping scheme has been used for different traps
– for example the 100µm-Z-wire and the
10µm-trap. Only the final strength of the
bias-field and bf-field have to be adopted to
match the desired trap-gradients and trapbottom. Note that the bf-field reverses its
direction (see main text).
trap is sketched in Figure 3.6 (final current of 2A in the chip wire at an bias-field
of 18.4G). The total duration of the transfer has been decreased to 300ms and
this scheme has successfully been used for different final chip traps. The main
idea is to divide the transfer into two steps: First of all the atoms are loaded to a
very relaxed intermediate chip trap within 100ms. This trap is kept at constant
parameters for a time of 100ms to ensure that all magnetic fields arrive at their
final value (Bbias = 7G and Bbf = −4.9G resulting in ωtrans = 2π · 85Hz). In
a second step, the trap is compressed within 100ms to the desired transverse
trap frequency and the trap-bottom is adjusted for an evaporation cooling stage
(Bbias = 18.4G and Bbf = −0.6G resulting in ωtrans = 2π · 1kHz). Note that the
homogenous external bf-field has to be configured in a way that it increases the
field strength at the minimum of the chip trap whereas in the copper-Z trap it
decreases field strength. This is necessary because the slightly misaligned biasfield decreases the trap bottom. To allow for this degree of freedom two pairs of
coils are used to generate the bf-field: One pair carrying a controllable current of
0−30A decreases the trap bottom while a second pair of coils constantly increases
the trap-bottom by 6G. This results in an effective bf-field which can be varied
from −6G to 54G where the sign indicates the direction.
This ramping scheme has been used for most of the experiments discussed in
this thesis. The chip traps formed by the 100µm-wide and 50µm-wide Z-shaped
wires and the 10µm-trap can be efficiently loaded this way. In the compression
stage of the chip trap (last 100ms) a homogeneous magnetic offset-field pointing
perpendicular to the chip surface (ud-field) can be ramped up in addition to the
bias-field to rotate the trap to the side of the wire – see for example experiments
discussed in section 4.4. This transfer procedure has been tested for thermal
atomic clouds at temperatures ranging from 1 − 20µK. For hotter samples a
significant loss of atoms has been observed. Usually 2 · 106 thermal atoms at a
temperature 10µK can be transferred to the final chip trap without any loss.
3.1 BEC in atom chip traps
51
Moreover this ramping procedure has been successfully used to transfer a BEC
from the copper-Z trap to the pure chip trap. But this possibility to transfer a
BEC has been set aside, mainly for two reasons: Firstly a BEC is very sensitive to
even smallest non-adiabaticities in the transfer which results in a large fraction of
the atoms being in the thermal component after transfer. Secondly the duration
of transfer has to be subtracted from the lifetime of the BEC (typically ∼ 500ms)
reducing the time available for experiments significantly. The transfer of thermal
atoms to the desired chip trap and cooling to BEC therein has proven to be more
robust and handy than to transfer a BEC from the copper-Z. This final cooling
stage to BEC will be discussed in the next section.
3.1.3 BEC in chip traps
In the final stage of the experimental cycle atomic samples are cooled in a pure
chip trap to a temperature just above the critical temperature for condensation
or to a BEC if desired. This procedure is similar for all chip traps and BECs with
up to 105 atoms have been created in the 100µm-trap as well as in the 10µmtrap. Starting point for these experiments are 2 · 106 atoms at a temperature of
5 − 20µK depending on the compression of the specific chip trap.
As an example the cooling procedure to a BEC in the 10µm-trap will be discussed here. While the atoms have been transferred to the chip trap the radiofrequency (rf) used for evaporative cooling has been held on a constant value of
20MHz. This avoids atoms from being removed from the trap accidently if the
trap-bottom is changing while transfer. After the compression phase the starting
value of the rf is adjusted to match the temperature of the transferred atoms.
Typically the rf is ramped linearly from 3MHz to a final value of 1 − 0.6MHz
depending on the chosen value of the trap-bottom. Fast evaporative cooling to
a BEC within 500ms is possible because of quick rethermalization of the atomic
cloud caused by high transverse trap frequencies (ωtrans > 2π · 1kHz). Optimal
cooling in terms of stability and final atom number in the BEC is achieved at a
duration of the evaporation of 1.5s.
Usually the aspect ratio λ = ωtrans /ωlong of the chip traps in our experiment
is in the order of 50 but can easily be several thousand. In these elongated
traps the regime of so called quasi-condensates is usually reached. Here, the
phase of the condensate is only coherent over a certain length which is smaller
than the longitudinal size of the system. This means that the phase coherence
length is smaller than the longitudinal length of the condensate (see section 5.2 for
experiments characterizing this one-dimensional regime). Although the coherence
length depends on temperature and atom number, changing the aspect ratio of
the trap strongly changes the longitudinal phase coherence properties of the BEC.
These quasi-condensates exist already in three-dimensional (3d) systems, as
has been demonstrated in an experiment in Hannover [81]. In this experiment
the cross-over from pure 3d-behavior to a quasi-condensate has been measured to
52
Micromanipulation of BECs on atom chips
Figure 3.7: Density profiles of BECs after
ballistic time-of-flight expansion. (a) A homogeneous density profile indicates phase
coherence over the entire length of the condensate. The density is well below the limit
for a one-dimensional Thomas-Fermi BEC.
(b) Here stripes in the density distribution are a signature for a phase-fluctuating
quasi-BEC. The density is high enough for
the system to be three-dimensional.
happen at λ ≈ 20. The indication for phase fluctuations are stripes in the density
distribution of the BEC after ballistic expansion. As an example the density
distribution of a 3d quasi-condensate can be seen in Figure 3.7b. The contrast,
size, and position of this fringe pattern varies randomly from shot to shot. This
can be seen in Figure 5.5 where 20 individual pictures of quasi-condensates have
been averaged and a homogeneous profile is recovered.
To guarantee a constant phase over the entire length of the BEC small aspect
ratios can be used. These traps can be formed by an additional current-carrying
wire next to the trapping wire generating a dip in the longitudinal potential
(Fig. 3.3). Using this dip, BECs in a trap with λ = 2π · 1.9kHz/2π · 0.25kHz = 7.6
have been generated and a homogeneous density profile in time-of-flight expansion
has been found (Fig. 3.7a). Since the peak density is ∼ 120atoms/µm, this BEC
has been formed in the 1d Thomas-Fermi regime.
These examples clearly show that chip trap are ideally suited to investigate lowdimensional systems: True BECs and phase-fluctuation BECs can be produced in
both, one-dimensional and 3-dimensional trapping geometries. Section 5.2 deals
with experiments characterizing the cross-over between 3d BECs and 1d BEC by
measuring the size of these BECs after ballistic expansion.
3.2 Experimental methods
53
3.2 Experimental methods
In the previous section the production of cold thermal samples and BECs in
various chip traps has been described. Robust and stable operation can only be
achieved if all parameters of the trapping potential are well known and precisely
controllable. Therefore the characterization of the trap-bottom stability will be
discussed in section 3.2.1 followed by a discussion of methods used to measure the
transverse and longitudinal frequency of a magnetic trap (section 3.2.2). Section
3.2.3 briefly illustrates the determination of the atom number from an absorption
image. Finally, the calibration of the temperature of atomic samples will be
discussed in section 3.2.4, both for time-of-flight and in-situ measurements.
3.2.1 Trap bottom stability
An important quantity characterizing a Ioffe trap for atomic clouds or BECs is
the magnetic field strength at the minimum of the trap (trap-bottom). In the
case of a Z-shaped wire trap the trap-bottom is generated by the leads of the
bend wire and thus depends on the current in the wire. Using a homogeneous
external magnetic field the trap-bottom can be raised or lowered to match the
desired value. Typically the trap bottom is in the range of 0.7G − 2G in the final
micro-trap.
The stability of the radio-frequency (rf) cooling process depends on the stability of the trap-bottom. Consequently the trap-bottom’s stability is of extreme
importance. Starting at a rf-frequency high above the trap-bottom, the final rffrequency removes atoms slightly above the trap-bottom from the trap. Thus,
the stability of the trap bottom2 sets a limit on the precision and reliability of
BEC preparation at a specific chemical potential µ.
To assess the stability of the trap-bottom in the chip traps the following experiment has been carried out: An one-dimensional BEC has been prepared at a final
rf-value where only a small condensed cloud remained in the trap. Over several
tens of experimental cycles at constant settings a small drift in the trap-bottom
occurred and finally the rf removed all atoms from the trap. After about 60min
the BEC reappeared. From the corresponding images the chemical potential µ
of the BEC has been computed (see section 4.1.2). In Figure 3.8 each data point
corresponds to the chemical potential measured as a result of one experimental
run. The left gray shaded region corresponds to a time where no atomic cloud
was visible and thus enables to determine the fluctuations caused by the imaging
system. The mean value of this region has been set to be the zero of the vertical
axis. After a certain time the BEC was visible again (right gray shaded region).
Note that every data-point is above the zero-level (dashed line) and a mean chemical potential of µ/µatom = 113 ± 13µG has been computed (µatom = µB mF gF is
2
The stability of the rf-source is much better than the stability of the trap bottom and will
be neglected in this discussion.
54
Micromanipulation of BECs on atom chips
Figure 3.8: The chemical potential µ of a BEC prepared in a mirco-trap has been
monitored over 160 experimental runs at constant parameters. The final value of the
rf-ramp is chosen such that a slightly drifting trap-bottom leads to a complete removal
of the BEC from the trap (left gray shaded region). After a certain time interval a
small BEC was again visible in the trap (right gray shaded region) and a non-zero value
of the chemical potential of µ/µatom = 113 ± 17µG has been measured (lower inset).
The data taken at the beginning of the sequence where no atoms have been observed
on the pictures is used to define the zero-level of the vertical axis. The right plot shows
the Gaussian curves obtained by fitting each histogram of the data-points in the two
shaded regions. The width of the resulting Gaussian curve is σzero = 34µG for the
zero-level and σc = 69µG for the condensed cloud. To plot the histograms the data has
been binned into 8 (6) containers of a width of 20µG (40µG), respectively. Note that
in the right region no data point falls below the zero-level.
3.2 Experimental methods
55
the magnetic moment of an atom). From the histograms fitted by a Gaussian
distribution one clearly sees that this region is well separated from the zero-level.
The width of the corresponding Gaussian curve is σ = 69µG. Thus, a trap-bottom
stability of ∆B ∼ 0.069mG has been found for a time interval of ∼ 30min.
This result has been obtained at an absolute trap-bottom of Btb = 1G leading
to a relative stability of ∆B/Btb = 6.9 · 10−5 . If this number is compared to the
current of Iwire = 340mA flowing in the wire a relative stability of the current can
be estimated to be ∼ 25µA. It should be annotated that the currents in the coils
contributing to the trap-bottom have not been stabilized directly but have been
operated in a voltage-controlled mode. Therefore thermal drifts – for example
of the room temperature – result in slight changes of the resistivity of the coils
and thus cause long-time drifts of the magnetic fields and in particular of the
trap-bottom.
Fluctuations of the trap-bottom can occur because all electronic devices used
in the experiment to generate magnetic fields pick up 50Hz-noise stemming from
the power-line. As a solution the experimental cycle could be phase-locked to the
frequency of the power-line. Since the stability of the frequency of the power-line
is not better than 10−3 a locking at the beginning of the cycle with a typical
length of 40s would be not sufficient. Therefore a synchronization stage just
before the final rf-cooling stage would be necessary which needs drastic changes
in the instrumentation and control system.
Other sources of magnetic field fluctuations could be avoided by shielding the
experimental site by µ-metal shields. Two sources of these fluctuations should
be discussed in the following: The magnetic field of a current in a wire can
be calculated by B = 2 · I/r (section 2.2.2). Usually currents in power-cables
cancel, but an unbalanced current flow of 30mA already causes a magnetic field
fluctuation of 60µG at a distance of 1m. Another source of fluctuating magnetic
fields is given by computer screens. The MPR-II-standard3 specifies the magnetic
field variations in the frequency band from 5Hz - 2kHz to be below 2.5mG at a
distance of the operator (∼ 30cm). If this field drops off as r−2 (dipolar source)
a magnetic field fluctuation of 25µG at a distance of 3m can be estimated.
3.2.2 Trap frequency measurement
Precise knowledge of all parameters of the trapping potential is needed for most
experiments. To check the agreement of calculated magnetic potentials and traps
used in the experiment a method to measure the longitudinal and transverse trap
frequency is needed. Longitudinal frequencies are usually smaller than 100Hz and
can easily be measured by observing center-of-mass oscillations of thermal clouds
and also BECs. Transverse frequencies which can be several tens of kHz can be
3
Abbreviation for Statens Mät och Provråd (Swedish), national testing laboratory for instruments
56
Micromanipulation of BECs on atom chips
Figure 3.9: Damped center-of-mass oscillations of a thermal cloud in a magnetic
trap can be used to determine the longitudinal trap frequency. The atoms have
been displaced from the trap center and
harmonic oscillations are excited. The three
sets of data are taken for different initial displacements and have been fitted by damped
oscillations (Equ. 3.1) yielding a trap frequency of ωlong = 2π · (17.2 ± 0.7)Hz.
measured by parametric heating of atoms induced by a radio-frequency. Both
technics will be briefly discussed and one example will be given for each method.
Longitudinal trap frequency
In Figure 3.9 a damped center-of-mass oscillation of a thermal cloud for three
different initial amplitudes can be seen. The trap has been generated by a Zshaped copper wire used in the initial stage of the experiment (see section 2.4.2).
A thermal atomic cloud has been displaced relative to the minimum of the trap in
the longitudinal direction right before it has been transferred from the molasses
into the copper-Z potential. Fitting a damped oscillation
µ ¶
l(t) = l0 exp
t
sin (tωlong )
τ
(3.1)
to the data yields a longitudinal trap frequency of ωlong = 2π·(17.2±0.7)Hz which
is in good agreement with the simulated value. From the damping rate of the
longitudinal oscillations one can roughly estimate the timescale needed for atoms
to rethermalize along this direction. The fitted exponential decay results in a
time-constant of τ = 39 ± 5ms resulting in a rethermalization rate of Γ ∼ 26s−1 .
An example for measuring the trap frequencies in a similar trap by center-of-mass
oscillations of a BEC can be found in [82].
Transverse trap frequency
Transverse trap frequencies are usually larger than 2π · 200Hz and can be up to
several tens of kHz in micro-traps. Therefore a direct observation of center-ofmass oscillations is very hard since the amplitudes are small and a high time
resolution would be needed. In addition harmonic and linear oscillations would
3.2 Experimental methods
57
Figure 3.10: A measurement of the transverse trap frequency can be performed by
monitoring the loss of atoms from the trap
which is induced by parametric heating.
This heating is caused by a rf-field which is
applied while a constant rf-shield removes
all heated atoms from the trap. A clear
resonance behavior can be observed and
a Gaussian fit yields a trap frequency of
ωtrans = 2π · (388 ± 3)Hz.
be mixed because of the transverse shape of the trap (harmonic trap bottom
combined with linear potential at the outer regions). An easy to use technique is
provided by parametric heating of trapped atoms. For a modulation frequency ν
a resonance is expected at ν = νt and higher harmonics where νt is the frequency
of the trap.
We tested two ways of detecting these resonances: One can monitor the heating
of the atomic cloud for different radio frequencies (rf) or, alternatively, the loss
of atoms from the trap can be monitored when a rf-knife constantly removes the
heated atoms. We implemented the second way for routinely measuring trap
frequencies. In Figure 3.10 the resonance for ν = νt can be seen which has been
measured for atoms trapped in the copper-Z trap (see section 2.4.2). To avoid
broadening of the resonance, amplitude and duration of the rf-pulse used to heat
the atoms have to be adjusted carefully so that atoms are not completely removed
from the trap at resonance. This technique has been applied to various chip traps
and the measured resonance frequencies have been found to be in good agreement
with trap frequencies which have been calculated.
58
Micromanipulation of BECs on atom chips
3.2.3 Atom number determination
The atom number and the spatial distribution of an atomic cloud or BEC can
easily be probed by absorption imaging. The atoms are exposed to resonant laser
light propagating into the z-direction which is imaged onto a charge-coupled
device (CCD) camera (see section 2.1.3 for details of the imaging system). The
transmitted intensity It (x, y) is related to the incoming laser intensity I0 (x, y)
and the column density n(x, y) by Beer’s lay and can be used to compute n(x, y)
[83]:
³ ´2
Ã
!
1 + 2∆
I0 (x, y)
Γ
n(x, y) =
ln
(3.2)
σ0
It (x, y)
where σ0 = 3λ2 /(2π) is the resonant cross-section for light absorption (σ0 =
291 · 10−15 m2 in the case of Rubidium-87 [32]), Γ is the natural linewidth of the
transition and ∆ is the detuning from resonance.
To deduce the total atom number from the column density the imaging area
per pixel A has to be known. The total atom number is given by
N =A
X
pixel
n(x, y) =
∆xCCD ∆yCCD X
n(x, y),
f2
pixel
(3.3)
where ∆x and ∆y represent the size of one pixel of the CCD-camera and f is
the magnification of the imaging system. These formulas are valid for intensities
of the imaging laser beam much below the saturation intensity of the atomic
transition (Isat = 1.6mW/cm2 in the case of Rubidium-87 [32]). Furthermore the
spontaneous scattering of photons into the solid angle of the detection system as
well as reabsorption of scattered photons has been neglected. An example of the
two-dimensional column density of a BEC obtained by this method is depicted
in Figure 3.11a.
The longitudinal one-dimensional density profile has been computed
(Fig. 3.11b). A bimodal distribution consisting of a BEC component in the center
and thermal wings at the outside is visible. To determine the atom number in
both components the theoretically expected density profile has to be known. It
can be obtained by integrating the density profile of an expanded cloud for two
cases, being above or below the critical temperature TC of phase transition to
BEC [84]:

Ã
x
nth (x, t) = nth (0) exp −
lx (t)

Ã
!2 
 , T > TC
x
nc (x, t) = nc (0)max 1 −
λx (t)
!2
(3.4)
2
, 0 , T < TC .
(3.5)
Here nth (0) and nc (0) are the peak column densities and lx (t) and λx (t) are the
half lengths of the distribution. One-dimensional density profiles obtained by
3.2 Experimental methods
59
Figure 3.11: Image of a BEC after timeof-flight expansion. (a) From absorption
images the collum density has been computed. Atom number is given in atoms per
pixel. One pixel corresponds to an area of
11.2µm2 . (b) The one-dimensional density
profile (black curve) can be obtained by integrating along the x-direction. The typical
bimodal distribution of a BEC can be seen.
Fitting nth (x, t) + nc (x, t) to the data results in a Gaussian part (blue) describing
the thermal component and a parabola fitting the BEC component. Atom numbers
deduced from these fits are Nth = 2.6 · 104
and Nc = 1.9 · 104 .
integrating experimental images along one direction can be fitted using these
equations. As a first approximation the bimodal distribution can be described by
the sum of nth (x, t) + nc (x, t) and the relative atom numbers in both components
can be obtained. To deduce more precise results on the atom numbers from a
bimodal distribution a more advanced analysis has to be done: The separation
of the thermal and condensed phase have to be taken into account which leads
to a deviation from the simple model.
3.2.4 Temperature calibration
Time-of-flight technique
The temperature of an atomic cloud is defined by its velocity distribution. When
the atoms are released from a trap via a sudden non-adiabatic switch off of the
magnetic trapping potential their momentum distribution is conserved and they
expand ballistically according to their velocity in the trap. By imaging the density
profile after a certain time t the momentum distribution has been transferred to
a spatial distribution. This time-of-flight technic can be used to determine the
temperature T of the atomic cloud [83].
The atomic density profile is imaged for different expansion times tex and using
equation 3.4 a half-length lth (tex ) for each time-step is obtained (Fig. 3.12). The
60
Micromanipulation of BECs on atom chips
Figure 3.12: Expansion measurements of
a thermal atomic cloud for the longitudinal
(a) and the transverse (b) direction have
been fitted according to equation 3.6. The
resulting temperatures of Ty = 1.2 ± 0.1µK
(Tx = 2.0 ± 0.1µK) in the longitudinal
(transverse) direction can be understood by
hydrodynamic expansion (see main text).
Due to the strong transversal confinement
resulting in a small initial size, the transverse expansion is already asymptotic at the
time the first data point has been taken.
In the case of the longitudinal expansion
this asymptotic behavior is indicated by
the dashed line. For each time-step three
expansion picture have been analyzed and
the statistical errors of the average value is
smaller than the size of the depicted symbols.
Figure
3.13:
Longitudinal onedimensional density profile of a thermal
cloud imaged in-situ (black). Based on the
exact knowledge of the current-carrying
structures the magnetic trapping potential
has been computed. The corresponding
density profile (blue) has been fitted to
the data resulting in a temperature of
1.8 ± 0.1µK.
time dependance of this isotropic expansion is given by
v
!
Ã
u
u
2kB Ti 2
t
2
t
li (t) = li (0) +
m
i ∈ {x, y}
(3.6)
where m is the mass of one atom, kB is Boltzmann’s constant and Ti are the
temperatures in x and y direction. For a collisionless expansion of an atomic
cloud in thermal equilibrium Tx should equal Ty . Hydrodynamic expansion will
be discussed at the end of this section. For large expansion times and/or small
initial size li (0) the expansion can be described by
s
li (t) ≈
2kB Ti
t
m
i ∈ {x, y}.
(3.7)
3.2 Experimental methods
61
This asymptotic behavior can clearly be seen in Figure 3.12. The longitudinal
extension ly (0) of the trapped cloud is 180µm whereas the transverse extension
lx (0) is below the spatial resolution of ∼ 3.5µm of the imaging system. From
this initial size of the atomic cloud the frequency ωi of the harmonic trapping
potential can be calculated:
1
mωi2 li (0)2 = kB Ti
2
i ∈ {x, y}.
(3.8)
In-situ temperature measurement
Even in the magnetic trap itself the atomic clouds density profile shows a characteristic shape depending on its temperature. By imaging this distribution in-situ
a temperature measurement can be performed. Figure 3.13 shows such an onedimensional longitudinal profile imaged in-situ which is connected to the trapping
potential Utr (y) by
Ã
!
Utr (y)
ntr (y) = ntr (0) exp
.
(3.9)
k B Ty
To relate the in-situ profile to the temperature exact knowledge of the longitudinal trapping potential is needed. By using the stick model described in section
2.2.3, the magnetic field of the chip traps used in the experiment can be calculated. Temperatures derived by this method have been compared to temperature
measurements using the time-of-flight technique and show good agreement.
Hydrodynamic expansion
As an example of these two techniques, measurements of the temperature of an
elongated thermal cloud (trapping frequencies ωtrans = 2π · 1.9kHz and ωlong =
2π · 6Hz) have been performed. The atomic cloud has been held for ∼ 500ms
in the trap only applying little cooling by a constant rf-shield to reach thermal
equilibrium. The in-situ measurement (Fig. 3.13) yields a temperature of the
atomic cloud of Ttr = 1.8±0.1µK. If the atomic density n0 at the center of the trap
is high enough that atoms collide during the initial phase of expansion, kinetic
energy can be transferred from the longitudinal direction of expansion to the
transverse direction. This shows up as cooling in the longitudinal direction and
heating in the transverse direction which can indeed be seen in the data obtained
by time-of-flight measurement (Fig. 3.12): Ty /Ttr = 0.67 and Tx /Ttr = 1.11.
To distinguish the two regimes of collisionless and
expansion the
√ hydrodynamic
2 −1
mean free path for a uniform gas given by λ0 = ( 2n0 8πa ) (where a = 5.2nm
is the s-wave scattering length of Rubidium-87 [85]) has to be compared to the size
of the cloud. From the one-dimensional density profile (Fig. 3.13) the maximal
density at the trap center can be estimated to be ∼ 23atoms/µm3 if a transverse
extension of the cloud of 1.7µm is assumed. This results in a mean free path of
λ0 = 39µm. Comparing λ0 to the longitudinal and transverse size ly and lx allows
62
Micromanipulation of BECs on atom chips
to estimate whether the expansion is of hydrodynamic or collisionless character.
For the longitudinal direction one calculates λ0 /ly = 39µm/180µm = 0.22 which
indicates hydrodynamic expansion whereas for the transverse direction λ0 /ly =
39µm/0.9µm = 43, thus collisionless expansion is expected.
A more detailed discussion of hydrodynamic expansion of thermal atomic
clouds and their experimental observation can be found in the literature [86, 87].
3.3 Lifetime close to surface
3.3.1 Introduction
Most experiments discussed in this thesis take place in the vicinity (< 50µm)
of the atom chip. Close approaches to the chip structures are desirable because
high trap frequencies can be achieve this way. This enables for example fast
gate operation times which are needed for quantum information processing [22].
Additionally, long lifetimes are necessary if dynamic properties of systems are to
be studied – for example guiding and formation of one-dimensional condensates
(chapter 5). Cold thermal atoms or even BECs at T ∼ 100nK in the vicinity of
the trap generating structures at room-temperature result in an extreme temperature gradient of up to 3 · 108 K/m. In this regime unintentional coupling cannot
be avoided and causes loss of atoms from the trap, heating of atomic samples,
and decoherence of matter waves. These effects have been extensively studied
theoretically as well as experimentally and a nice overview can be found in [73].
In the frame work of this thesis the loss of Rubidium-87 atoms from a microtrap due to thermally induced spin flips will be discussed. Other processes – for
example spilling over, Majorana spin-flips, 2-body and 3-body collisional losses,
stray light – and technical noise induced losses can be neglected or circumvented
by appropriate design of the chip traps [73].
The spin of magnetically trapped atoms at a distance h above the metallic
surface of an atom chip can be flipped by thermally induced currents in the
conducting material. This leads to loss of atoms since only low-field seeking
states can be trapped. The corresponding loss rate scales as Γ ∼ 1/h for a
bulk and as Γ ∼ l/h2 for a layer ofqa thickness l [88, 8]. This approximation
is valid for l ¿ h ¿ δ where δ = 2ρ/(µ0 ωLarmor ) is the skin-depth (ρ is the
resistivity of the material). The skin-depth is the penetration depth of highfrequency radiation into a metal. For our experiment a typically used magnetic
field value at the trap minimum is B = 1G. This corresponds to a Larmorfrequency ωLarmor ∼ 2π · 700kHz resulting in a skin-depth of δ ∼ 90µm. Since
the thickness of the chip layers is smaller than 5µm, the assumption given above
is fulfilled. If h becomes smaller than the thickness l the scaling behavior for a
bulk is recovered.
In addition to the interpretation of the skin-depth as penetration-depth it can
3.3 Lifetime close to surface
63
Figure 3.14: (a) An atomic cloud held by
a trapping wire can be positioned far away
from this wire but close to the surface of
the atom chip. This can be achieved by
using a homogeneous offset-field which is
oriented almost perpendicular with respect
to the chip surface. (b) The lifetime of an
atomic cloud has been measured in dependance on the distance to the chip surface.
For distances smaller than 10µm the lifetime is limited by surface evaporation. The
shown theory curves are discussed in the
main text.
be interpreted as the thickness of the part of the material which couples to the
atoms. If the thickness of this material is much smaller than the skin-depth
the material lying in a layer underneath this thin top layer might also couple
to the atoms. Therefore a detailed investigation of effects of sub-surface chip
layers has been carried out [89]. In conclusion these sub-surface layers do not
contribute to the loss rate as long as their resistivity is much larger than the
resistivity of the conducting top-layer. The atom chip used in the experiment
meets this requirement since the resistivity of the gold layer (ρAu = 2.2 · µΩcm)
is about four orders of magnitude smaller than the resistivity of the doped silicon
substrate (ρSi = 17 · mΩcm) which thickness is 700µm. The isolation layer (SiO2 )
and the adhesion layer (Ti) between gold and substrate can be neglected in this
estimation because of their small thickness (500nm and 35nm, respectively). For
details of the fabrication process see section 3.5.1.
3.3.2 Lifetime measurement
To measure the lifetime in dependance on the height h above the chip surface
(gold thickness 1.8µm) the trap geometry depicted in Figure 3.14a has been
used. The minimum of a magnetic Z-trap (section 2.2) formed by the 100µmwide wire (C-D depicted in Fig. 3.1) can be moved close to the surface of the
atom chip. But at the same time it is kept far away from the current-carrying
wire (distance d) since a magnetic offset-field close to perpendicular to the chip
surface is used. This allows to keep the trap frequency constant over a wide
range of h and additionally reduces effects of technical noise of the wire current.
The further away the trap is moved from the trapping wire the shallower the
trapping potential gets. Due to attractive surface potentials the barrier towards
64
Micromanipulation of BECs on atom chips
the surface is reduced and atoms are lost. This effect limits the closest approach
to the surface and thus the smallest achievable distance to the surface depends
on the steepness of the trapping potential.
To sort out the lifetime-data which is limited by surface evaporation, the atomnumber in the trap was not only monitored by in-situ measurements but additionally time-of-flight expansion has been performed. This allows to follow the
evolution of the temperature of an atomic cloud. A decrease of temperature over
time is a clear indication for surface evaporation. Additionally the atom-numbers
have been found to be equal for both methods. This method guarantees that the
loss rate is only dependent on surface-induced losses and background collisions.
The data depicted in Figure 3.14b has been taken at a distance of d = 280µm
from the center of the 100µm-wide wire which displays a good compromise between being dominated by technical noise for small d and losing atoms due to
evaporation towards the surface for large d. Here, the lifetime data taken for
d < 10µm is limited by surface evaporation. Numerical calculations taken from
[89] are depicted as lines. The model fits the experimental data within a factor of
two if the top-most structure is assumed to be a gold layer on the doped silicon
substrate and an additional height independent loss rate is assumed (red curves).
This background loss-rate is the only free parameter. It can be seen that the
measured lifetime is enhanced compared to the lifetime predicted by a model assuming a bulk (green curves). The results of the calculations do not change if the
complete chip mounting including Macor block and copper structures (see Fig.
2.22) is included. The remaining discrepancy between experiment and theory
could be caused by technical noise in the wire current and offset-fields.
The new multi-layer chip described in section 3.5.3 consists of a gold layer of
a thickness of 110nm. The innermost part of the chip has been structured by
direct electron-beam writing allowing for a structure size smaller than 1µm. The
proposed experiments described in that section use a similar technique as depicted
in Figure 3.14a: High transverse confinement is generated by a large current in
the trapping wire whereas the manipulation of the longitudinal potential is done
by the high resolution wires at the side of the trapping wire. Here, the atoms
stay more than 20µm away from the thick trapping wire and lifetime is only
limited by the close-by thin gold layer. Taking the above mentioned theory
which reasonably agrees with the experimental findings, a lifetime of ∼ 1s can
be estimated for these traps at a surface distance of d = 1µm above the thin
gold layer. Similar results have been found in an experiment using micro-traps
generated by permanent magnets for the creation of BEC above a 400nm thick
metallic layer [90].
3.4 Local disorder potentials
65
3.4 Local disorder potentials
Magnetic wire traps are formed by subtracting the magnetic field produced by
a current-carrying wire and a large homogeneous offset-field (side-guide configuration, see section 2.2.2). The remaining field at the minimum of the guide is
given by the angle between the field of the wire and the bias-field. If the current
direction differs from a straight flow, this meandering current can cause a varying
magnetic field value along the minimum of the guide.
These disorder potentials have been observed in many experimental groups
[91, 76, 92, 93]. In their setups the investigated chips have been produced by an
electroplating technique resulting in a relatively rough wire surface compared to
the surface quality obtained by evaporation and lithography in our group. For
comparison a scanning-electron pictures of an evaporated (a) and an electroplated
(b) wire is depicted in Figure 3.18. Therefore it is not surprising that disorder
potentials are reduced by a factor of 100 using our fabrication technique [27]. A
brief description of this technique can be found in section 3.5.1. In the mentioned
experiments with electroplated wires, disorder potentials lead to a breaking up
of cold thermal clouds and BECs already at a surface distance of d ∼ 100µm.
This severely limits the distance to the surface and thus the achievable trap
frequencies.
In section 3.4.1 a brief overview of recent experiments with electroplated chips
will be given. In section 3.4.2 an upper bound for potential roughness for evaporated chip will be deduced from experiments using thermal atomic clouds. Finally
section 3.4.3 summarizes measurements performed with BECs where the scaling
behavior of the disorder potential versus distance to the wire surface has been
measured. These results suggest strongly that local properties of the wire play
a more important role than effects of the wire edges. The detailed experimental
procedure for calibrating the distance of the atomic clouds to the chip surface and
the method to deduce the magnetic field variations from the density fluctuations
of the BEC can be found in chapter 4.
3.4.1 Previous experiments
The perturbing disorder potentials have been attributed to inhomogeneous magnetic field components ∆B in the direction parallel to the current-carrying wire
creating the trapping field B [91, 76, 92, 93]. It has been suggested that such field
components could be derived from fabrication inhomogeneities, surface roughness
[94, 95] and residual roughness of the wire borders [96]. The model of Wang et al.
[96] provides a full quantitative explanation of the disorder potentials found near
electroplated gold wires. In the following a brief overview of previous experiments
will be given:
In experiments performed in Tübingen by Kraft et al. [97] the disorder potential has been measured by observing the breaking up of a BEC. The strength
66
Micromanipulation of BECs on atom chips
of the disorder potential ∆B relative to the bias-field B at a surface distance of
d = 109µm was ∆B/B = 2 · 10−4 . These perturbations originate from irregular
current flow in the wire rather than from permanent magnetic inhomogeneities
which could be nicely shown by reversing the direction of the current and the
bias-field which lead to an inversion of the disorder potential.
In experiments performed at the MIT by Leanhardt et al. [98] a fragmented
BEC in a micro-trap (surface distance d = 85µm) was compared to a BEC held
in a dipole trap at the same position in space. In the latter case no breaking up
of the condensate was observed implying that the disorder potential arises from
the presence of current flow in the wire.
In experiments performed in Sussex by Jones et al. [99] the disorder potential close to a macroscopic copper wire was measured (core diameter 370µm
surrounded by an 55µm aluminium layer with 10µm ceramic outer coating).
With thermal atoms the strength of the disorder potential has been found to
be ∆B/B = 3.3 · 10−3 at a distance of d ∼ 25µm to the surface of the conductor.
A quantitative analysis of the disorder potential produced by a current-carrying
micro-wire has been presented by Estève et al. [93] from the Orsay group. They
could show that the measured potential fluctuations agree with the calculated
potential variation derived from scanning-electron microscope pictures of the wire
edges. They found a ∆B/B = 1.5 · 10−3 at an atom to surface distance of
d = 33µm.
3.4.2 Thermal atoms close to surface
The residual potential roughness has been probed for various trapping geometries
based on a 100µm-wide (C-D in Fig. 3.1) and two 10µm-wide (A-F and B-F)
wires at atom-surface distances down to 3µm. The global parameters of the
atomic cloud like atom number and temperature have been determined by a
ballistic expansion of the cloud in time-of-flight measurements.
With thermal atoms we always observe smooth longitudinal absorption profiles inside the trap (Fig. 3.15a), independent of the wire used to form the trap
and the position of the atomic cloud. For the closest approach of d = 3µm, a
cloud of T = 1µK remains un-fragmented within our detection resolution, even
when summing up many realizations of the experiment to reduce measurement
noise. Assuming that the atomic density profile follows the Boltzmann distribution n ∼ exp (−V /kB T ), an upper limit on the residual magnetic field roughness
∆B/B < 2 · 10−4 can be put where B is the field produced by the wire at that
distance.
3.4.3 BECs close to surface
BECs are a much more sensitive probe of potential roughness than thermal atoms.
The relevant energy scale, given by the chemical potential µ, can be orders of
3.4 Local disorder potentials
67
Figure 3.15: In-situ absorption images of atomic clouds positioned at a chip surface
distance of ∼ 5µm above a current-carrying wire (cross section 3.1 × 10µm2 ). Parts of
the images do not fully represent the atom density distribution since the imaging light
beam was obstructed by bonding wires (hatched regions). (a) Thermal atoms show
no fragmentation. (b) BECs display a much higher sensitivity and residual disorder
potentials cause a fragmentation of the cloud. (c) A longitudinal displacement of the
BEC by tuning the trapping potential shows that the disorder potential is stable in
position.
Figure 3.16: Longitudinal potential profiles measured with BECs at a constant distance of d = 10µm from the surface of the
100µm-wide wire. The different traces were
measured at different currents and are normalized to the respective trapping fields.
The bias-field (10G, 20G, 30G; black, blue,
red curve) was adapted in order to keep d
constant. The inset shows a histogram of
the deviations of the curves. The width of
the distribution (σ ∼ 6 · 10−6 ) is similar to
the shot to shot variations of different realizations of the same experiment with equal
wire currents.
magnitudes smaller than the temperature of thermal atoms. Figure 3.15b and
3.15c show typical absorption images of fragmented BECs at a distance d = 5µm
from one of the 10µm wide wires (B-F in Fig. 3.1). Altering the longitudinal confinement by varying the current in the U-shaped wire (L-M) creating the end-cap
leads only to an overall displacement of the cloud while the local disorder potential variations remain stable in their position. Over many months of experiment
no change was observed in the position of the fragments.
To assess whether the observed disorder potentials are magnetic in origin, the
wire current has been varied while the bias-field has been adapted so that the
BECs were trapped at a fixed distance to the wire. Magnetic disorder potentials
stemming from an irregular current flow should scale linearly with the current I
in the wire. Figure 3.16 shows an example of relative potential variations ∆B/B
reconstructed from BECs positioned at d = 10µm for three different currents
68
Micromanipulation of BECs on atom chips
Figure 3.17: Spectral power density of the
disorder potentials near the 100µm-wide
wire for three different spatial frequencies.
The open symbols correspond to data where
the detected singnal is limited by the chemical potential of the BEC. The solid lines
are best fits according to local fluctuating
current path model, the dashed lines show
best fits to the model outlined in [96]. For
both models the only fitting parameter is
the strength of current path fluctuation at
the respective spatial frequency k.
(see chapter 4 for details of the reconstruction of the potential and the position
calibration). To qualify the consistency between the measurements, the shot to
shot variations of ∆B/B with equal currents have been compared to those with
different currents. Equal widths of the residual differences between the graphs
have been found. Thus, one can conclude that within the statistical similarity of
the ∆B/B distributions (∼ 8 · 10−6 ), any current independent sources of disorder
potentials such as electrostatic patch effects [100] can be excluded at the scale of
10−13 eV for d > 5µm.
In order to study the source of the irregular current flow the variations of the
disorder potential with d have been measured. Wires of two different width,
10µm and 100µm, were used. The main observation is that the scaling of the
amplitude and the frequency spectrum of the disorder potential with d for the
two wires are very similar. For d < 50µm this would not be the case if the edge
fluctuations were dominating as can be derived from an edge fluctuation model
outlined in [96].
For the 100µm-wide wire, Figure 3.17 shows potential spectral densities (PSD)
of the disorder potential at three different spatial frequencies k. In the examined
d-range, the potentials scale more strongly with d than they would for dominating edge fluctuations [96] for all frequency components (dashed curves). This
clear difference in the slope of the experimental data compared to the wire edge
model can be interpreted as an indication that local current path deviations are
important. Such deviations can occur due to inhomogeneous conductivity in the
wire or top surface roughness [94, 95]. The simplest model taking local sources of
current path deviations into account is a current flowing along a narrow irregular
path below the atoms. This is equivalent to using the wire edge model mentioned
above but with a very small wire width and equal current. Such a model gives
reasonable agreement in the slope of the PSD as d is increased (solid curve in
Fig. 3.17) as can be expected as long as d is small compared to the relevant
period 1/k. Applying this method over the full spectrum (k > 1/200µm−1 ), we
3.5 Optimized atom chip geometries
69
obtain a local current flow fluctuation spectrum that scales as ∼ 1/k 2 . Microscopically well characterized wires will have to be fabricated and tested to develop
a more refined model explaining the disorder potentials caused by local current
deviations.
From this data and the simple local model the root-mean-square (rms) strength
of the relative disorder potential can be estimated and scaled to different heights.
At a surface distance of d = 10µm the root-mean-square ∆B/B = 3 · 10−5
(< 10−5 ) for spatial frequencies k > 1/200µm−1 (k > 1/50µm−1 ) can be found.
At d > 30µm, where disorder potentials near electroplated wires have been measured, ∆B/B is significantly smaller than the measurement sensitivity in our case
(6 · 10−6 ). This corresponds to a reduction by about two orders of magnitude
compared to the results outlined in section 3.4.1.
3.5 Optimized atom chip geometries
Using standard semiconductor technology for texturing a gold layer on a substrate
offers the opportunity to structure the field generating wires on a (sub)micron
scale, thus allowing to tailor the magnetic potential with the same spatial resolution. These subtle wires can stand high current densities larger than 107 A/cm2
[26] which allows to reach extreme trap-gradients since the achievable trapgradients scale with the current-density in the wire (Equ. 2.2.2): Trap frequencies
of up to ω ∼ 2π · 1MHz and corresponding ground state sizes of ∼ 10nm are possible. This allows to reach high tunnelling rates between neighboring trapping
sites. Thus, fast gate operation times needed for scalable quantum information
processing are possible.
In this section the fabrication process will be described, starting with the features of the standard single gold layer chips (section 3.5.1) used in many experiments in our group [14, 15, 12]. Section (3.5.2) illustrates a fabrication scheme
which allows for two isolated gold layers on one chip. This gives full flexibility for
designing the trapping potentials since wire crossings become possible – for example longitudinal and transverse trap frequency can be controlled independently.
In the last section (3.5.3) the integration of direct electron-beam writing will be
sketched leading to spatial resolutions down to 20nm. The following sections
only briefly mention the fabrication technique itself but rather concentrates on
the features of the chosen wire designs. A detailed description of the fabrication
process and an elaborate characterization of wire quality can be found in [26, 74].
3.5.1 Single gold layer chips
Up to now our atom chips consisted of a single gold layer of a height of 1−5µm at
wire widths of 1 − 200µm. These chips have been fabricated by standard microfabrication technology: A very flat silicon substrate of size 2 × 2.5cm2 isolated
70
Micromanipulation of BECs on atom chips
Figure 3.18: (a) Scanning-electron microscope (SEM) picture of an edge of a microwire fabricated by evaporation. The top surface shows up as bright gray shaded region.
A typical grain size of 50− 100nm can be estimated. (b) For comparison a SEM picture
of a wire fabricated by electroplating is shown [95]. (c) Picture of a section of the whole
atom chip (compare to Fig. 3.1) used in the experiments taken by a light microscope.
The gold wires (yellow) are defined by 10µm-wide gaps (black) where no gold has been
evaporated.
on the top by a thin (smaller than 200nm) layer of SiO2 is covered with photoresist which is structured by photolithography. After evaporating an adhesion
layer (∼ 50nm of Ti) and the final gold layer (1 − 5µm) the remaining resist is
removed in a lift-off process. This way 10µm-wide gaps defining the wires have
been produced (Fig. 3.18c). Almost the full chip is covered by a high-quality
gold layer serving as a mirror for the integrated mirror-MOT.
The substrate has been optimized to serve as a good heat sink offering high
thermal conductivity and capacity to efficiently dissipate the ohmic heat produced
by the current in the wire. To guarantee good thermal coupling to this heat-sink
the thermal contact resistivity between gold wires and substrate has to be small
implying the use of a preferably thin electric isolation layer. This way high
current densities larger than 107 A/cm2 can be achieved. These chips have been
tested to stand voltages larger than 300V over a gap of 10µm which allows for
manipulation of atomic clouds by electric fields [15].
The scanning-electron microscope (SEM) picture in Figure 3.18a allows to estimate the typical grain-size of an evaporated gold wire to be 50−100nm. Comparing this to a wire fabricated by electroplating (Fig. 3.18b), a much rougher wire
surface and less well-defined edge are found. Exceptionally high-quality wires
are essential because smallest inhomogeneities in the bulk of the wire and on the
edges can lead to uncontrolled deviations of the current-flow and therefore to
disorder potentials in the trapping potential (section 3.4).
The chip used in the experiments presented in this thesis has been fabricated
by the procedure discussed above except the fact that the wire-thickness has been
enlarged by evaporating a second gold layer on top of the first gold layer. To be
robust against slight misalignments in the fabrication process the width of all
3.5 Optimized atom chip geometries
71
wires in the top layer has been reduced by 2µm. This top layer has been added
onto every wire with a width between 10µm and 100µm (of the bottom layer).
Most areas of the chip which serve as a mirror remained at the thickness of the
single layer of 1.6µm whereas the total height of the enlarged wires is 3.2µm.
This top layer can be seen in Figure 3.18c: The gold area in the upper left and
lower right corner of the photograph show a different reflection at their edges
compared to the other wires because of the missing second layer. On the other
wires depicted on this photograph the edge of the bottom layer as well as the
edge of the top layer can be seen.
3.5.2 Chips with two isolated layers
To achieve full flexibility in designing the magnetic trapping potentials it is necessary to overcome the limitations set by single-layer fabrication. Allowing wires to
cross gives more freedom in manipulating cold atomic clouds and BECs. This way
trapped-atom interferometers discussed in [101] would become fully controllable
without any limitations.
To build atom chips consisting of two layers a commonly used technique using
polyamide as an isolation layer between the two conducting gold layers has been
adopted. Due to the high viscosity of polyamide, this layer is typically 0.5 − 1µm
thick. Since good electric isolation is linked to good thermal isolation, layer
thickness is a crucial issue because high current density in the wire can only be
achieved if the ohmic heat produced in the wire is efficiently removed into the
substrate. Therefore the isolation layer does not cover the whole chip surface but
is only put onto areas where two wire cross.
In Figure 3.19 an overview of the chip design can be seen. The wires shown in
red belong to the top layer which thickness is 1.3µm to allow for large currents
of ∼ 1A. These four wires will be used to create the transverse confinement in
the micro-traps and could generate a side-guide of a length of up to 2cm. One
of these wires is 100µm-wide to allow for efficient transfer of thermal atoms from
the macroscopic copper-Z trap to the chip-traps (details of the transfer can be
found in section 3.1.2). The other three wires are 10µm-wide and can be used
to achieve high transverse confinement. This top layer is crossed by the bottom
layer at three positions: At the center of the chip a region of 0.7 × 0.7mm2 in the
bottom layer has been designed to be structured in an arbitrary way. Therefore
a small polyamide layer isolates the two layers in the center (hatched region in
Fig. 3.19right). Additionally the top wires are crossed by four 500µm-wide wires
(blue) which can be used to create the longitudinal confinement in the microtraps. The innermost (outermost) pair of these wires depicted in blue has a
center-to-center spacing of 2mm (4mm).
In this geometry the constrained heat removal through the polyamide layer
has not been a severe limitation: The 100µm-wide wire stands a current of 2A
in continuous operation and the 10µm-wide wires can be operated at up to 0.7A
72
Micromanipulation of BECs on atom chips
Figure 3.19: Left: Overview of the design of the two-layer chip. A region of 2 ×
2.5cm2 is shown. The top layer contains the thick wires used for confining the atomic
clouds transversely (red). The remaining part of the chip surface is covered by the
bottom layer containing thin wires to connect the central part of the chip (green),
wires for longitudinal confinement (blue), and regions producing the mirror surface
(yellow, magenta). The wires can be connected using the pads at the border of the
chip. Right: Detailed picture (0.8 × 0.7mm2 ) of the central part of the chip. The
isolation layer between the top layer (red) and the bottom layer (green, yellow) can be
seen as a hatched area.
continuously. These parameters slightly exceed the currents in the chip wires
used in the experiments discussed in this thesis.
To generate confinement along the side-guides created by the wires in the top
layer, the 500µm-wide wires in the bottom layer can be used. Since the crosssection of these wires is ∼ 500µm × 150nm = 50 × 1.5µm2 and the contact
area to the substrate is large, continuous operation at 1 − 2A can be expected
to be possible. An alternative way of creating the longitudinal confinement is
provided by two straight copper wires mounted underneath the chip surface (see
Fig. 2.11). These wires have a center-to-center spacing of 12mm, their center
position is located 1.2mm below the chip surface and they can be operated at
60A. By unbalancing the currents in these wires the trap minimum can be freely
positioned along the longitudinal direction to move an atomic cloud or BEC to
the experimental site.
To compare the achievable trapping geometries of this new chip to the traps
successfully used in the experiments discussed in this thesis the following calculations have been performed as an example: The regular Z-shaped 100µm-wide
wire operated at a current of 1A creates a trap frequencies of ωtrans = 2π · 2kHz
and ωlong = 2π · 25Hz at a distance of d = 100µm to the surface (the bias-
3.5 Optimized atom chip geometries
73
field is fixed at 18.7G and the trap-bottom is set to 1G). Using the new geometry, the longitudinal trap-frequency can be independently adjusted from zero to
ωlong = 2π · 45Hz, pushing a current of up to 1A in the innermost closing wires,
or even to ωlong = 2π · 82Hz using 60A in the copper wires.
To tap the full potential of this new chip design, stiff traps can be formed
at the side of the wires providing the longitudinal confinement, similar to the
experiments discussed in chapter 4. Here the atomic cloud or BECs stays far
away (d > 20µm) from the thick wires which cause losses due to spin-flips induced by thermally fluctuating currents in the conducting material (compare to
lifetime measurements in section 3.3). Additionally losses due to technical noise
are reduced. At the same time the atoms hover microns or even closer above the
thin bottom layer which can be used to manipulate the atomic samples. These
thin layers allow life-times in the order of seconds [90] microns above the surface.
These rotated traps are extremely stiff, for example a transverse confinement of
2π · 40kHz can be achieved at moderate 0.5A in the 10µm-wide wire at a distance
of 20µm to the surface of this wire. This stiffness allows to approach the surface
even in the presence of attractive surface potentials.
In conclusion these two layer chips offer the opportunity to explore two different
regimes: The longitudinal confinement can be extremely small allowing to investigate very elongated one-dimensional systems (chapter 5). On the other hand,
manipulation of atomic samples on the (sub)micron scale close to the surface at
long trap life-times becomes possible.
3.5.3 Direct electron-beam writing
In the last section the structures generating the longitudinal and transverse trapping potentials have been discussed. Here, the structures for manipulating atomic
clouds and BECs on the micron-scale will be addressed.
The bottom layer is divided into two parts: The center region (0.7 × 0.7mm)
of the chip and the remaining chip surface. The center region (Fig. 3.19right)
is fabricated in a first lithography step by direct electron-beam (e-beam) writing
which allows high spatial resolution (up to 20nm depending on the desired wire
height). The outer region is added in a second regular (chapter 3.5.1) lithography
step connecting the central e-beam written part to the pads at the border of the
substrate. A detailed description of this fabrication technique will be available
in a separate publication [74].
The central region has been tailored as follows (Fig. 3.20left): One set of five
700nm-wide wires has been separated by gaps of a width of 300nm to allow for
a modulation of the longitudinal trapping potential with a periodicity of 1µm.
The other set consists of six 3µm-wide wires separated by gaps of a width of 1µm
allowing a periodicity of 4µm. These wires offer the possibility to modulate the
longitudinal confinement on a micron scale. At the same time three wires of each
set can be used to split elongated atomic clouds along the transverse direction,
74
Micromanipulation of BECs on atom chips
Figure 3.20: Left: Detailed picture of the design of the region structured by direct
electron-beam writing. Two 10µm-wide wires (red) in the top layer creating the transverse confinement can be seen. The bottom layer contains two sets of wires (green):
700nm-wide wire with a gap of 300nm in between and 3µm-wide wire with a gap of 1µm
in between. The hatched region displays the area where polyamide has been put to
isolate the two conducting layers. Note, that unhindered surface approaches will still be
possible since the surface is covered by polyamide only partially. Black color indicates
regions where no gold has been evaporated. Right: At the other side of the thick wires
a different wire design has been chosen: Three wires are widened to a total width of
20µm to study current-path deviations in thin (∼ 110nm) gold layers. The meandering
wire structure at the top allows to create lattice potentials using electro-static fields
(see main text).
since they bend in a Z-shape and run parallel to the thick wires in the top
layer. As can be seen in Figure 3.20, polyamide for isolating the layers (hatched
region) has only been put underneath – and 10µm around them to be robust
against misalignment – the thick wires so that unhindered surface approaches
are possible.
On the other side of the thick wires a different wire design has been chosen:
Three of the 3µm-wide wires are enlarged to a width of 20µm. Here currentpath deviations in micro-fabricated wires could be investigated using the BEC
magnetic field microscope (see chapter 4 for details of this microscopy method).
These wires are more than one order of magnitude thinner than the wire studied
in section 4.3. Therefore insight into the origin of current-path irregularities in
micro-fabricated wires can be expected from these measurements.
Previously, electro-static fields of up to ∼ 300V have been successfully used to
trap and split thermal atomic clouds held at a distance of ∼ 100µm to the surface
[15]. To manipulate BECs close to the surface much lower voltages are required
as has been demonstrated in the measurements discussed in section 4.4. To make
use of electro-static fields the following geometry can be used (Fig. 3.20right):
One thin wire meanders upwards accompanied by two neighboring wires. These
3.5 Optimized atom chip geometries
75
Figure 3.21: Scanning electron microscope picture of the region written directly
by an electron-beam. The two 10µm-wide
wires in the top layer can be seen on top of
the isolating polyamide layer (black). The
sub-micron wires in the bottom layer of
a thickness of 110nm are clearly resolved
(compare to Fig. 3.20left).
structures enables to generate a lattice potential along the longitudinal trapping
direction, if an elongated trap is centered over these wires and an electro-static
field is applied to the thin wires. This results in lattice periods of 1µm, 2µm,
or 4µm depending on which wires are charged. The number of potential wells
is 300, 150, or 75, respectively. Similar chip based lattice potentials created by
magnetic means have recently been used to couple atoms out of a BEC [102].
76
Micromanipulation of BECs on atom chips
4 BEC as magnetic field
microscope
Making use of the high sensitivity of Bose-Einstein condensates to variations of
the trapping potential – for example a changing magnetic field landscapes – a
highly sensitive microscope has been built. A sensitivity of 4nT at a spatial
resolution of 3µm has been achieved [25]. This enables field measurements in
a regime which is not accessible by today’s field sensors. In section 4.1 the
operation principle of this microscope will be described, followed by a comparison
to state-of-the-art magnetic field sensors (section 4.2). Section 4.3 deals with the
reconstruction of the local current density in a micro-fabricated wire. The novel
microscope is well suited for this application since it can measure field variations
∆B in the presence of a high offset-field B up to a ∆B/B = 5 · 10−5 . At the end
of this chapter measurements of other local potentials will briefly be discussed by
means of electric patch effects (section 4.4).
The content of this chapter has been published as: Bose-Einstein condensates: Microscopic magnetic-field imaging, S. Wildermuth, S. Hofferberth,
I. Lesanovsky, E. Haller, L. M. Andersson, S. Groth, I. Bar-Joseph, P. Krüger,
and J. Schmiedmayer, Nature 435, 440 (2005).
4.1 Mapping two-dimensional magnetic field
landscapes
A Bose-Einstein condensate (BEC) held by a current-carrying trapping wire (section 2.2.2) can be arbitrarily positioned above an independent sample to be
probed (Fig. 4.1). Locally varying potentials – for example magnetic disorder
potentials (see section 3.4) – generate a modulation of the atomic density of the
BEC along the longitudinal direction which can be imaged onto a CCD-camera.
From these images the longitudinal potential and thus the modulation strength
of the local potential can be derived.
In section 4.1.1 the precise positioning of the BEC relative to the probed sample
will be discussed and details on the calibration process will be given. Afterwards
the reconstruction of the longitudinal potential from the density variations will
be addressed in section 4.1.2 and the achievable field sensitivity will be deduced.
78
BEC as magnetic field microscope
Figure 4.1: Left: Schematic drawing of the operation principle of the BEC magnetic
field microscope. Right: A two-dimensional cut in a plane perpendicular to the trapping
wire is sketched. The magnetic field of the current-carrying wire in combination with
a homogeneous offset-field form a minimum of the magnetic field where atoms can be
trapped (section 2.2.2). By controlling the ratio between bias-field and ud-field and
thus the angle of the offset-field with respect to the chip surface and by tuning the
magnitude of the offset-field and the current in the wire, the BEC can be positioned
over a wide range: The BEC can be located directly above the trapping wire (i) as well
as above an independent sample (ii).
4.1.1 Position control
The atomic density can be imaged by absorption imaging (section 2.1.3 and 3.2.3)
on two perpendicular axes: A high resolution imaging almost parallel to the chip
surface determines the distance to the chip surface and an overview imaging close
to perpendicular to the chip surface allows to measure the transverse position.
These two imaging systems allow to precisely position the BEC over a wide parameter range above the chip structures to be investigated. Detailed description
of the optics and the CCD-cameras can be found in section 2.1.3.
The laser beam of the high resolution imaging is slightly inclined with respect
to the chip surface. This results in two pictures of the same atom cloud on
the CCD-camera (insets in Fig. 4.2). Half of the distance between these two
pictures of the atomic cloud is a direct measure of the distance d of the cloud
to the chip surface. This direct position measurement is obstructed at certain
heights by interference stripes in the imaging laser beam due to diffraction at
the chip surface (Fig. 4.2b). These effects are reproduced by a simulation of the
wave-front propagation close the a reflecting surface (section 2.1.3). Furthermore
the two clouds cannot be resolved for distances d < 7µm because of the limited
spatial resolution of the imaging optics. Note, that the spatial resolution in the
transverse direction of the BEC is smaller than in the longitudinal direction. This
is due to the atom chip and its mounting which reduce the optical aperture of the
4.1 Mapping two-dimensional magnetic field landscapes
79
Figure 4.2: Distance of an atomic cloud
to the chip surface can be derived from the
two mirror images (insets (a) and (c)). The
data has been fitted by a model taking the
exact wire shape into account. From this fit
small distances (< 7µm) can be estimated
which cannot be resolved by the imaging
system (d). Due to diffraction of the imaging laser beam at the chip surface pictures
are obstructed at certain heights (b).
lens along the transverse direction by approximately a factor of two compared to
the longitudinal direction.
But even in these two cases a precise position calibration is possible: If one
measures the distance for different currents in the wire at a fixed bias-field one
can fit the obtained data – the so called height-scan – by a wire-model taking the
exact shape of the wire (see section 2.2.3) into account. This leads to a precision
of 1µm for the measurement of d and allows to calibrate the strength of the
bias-field with an accuracy of 50mG [28].
To determine the transverse position an imaging system almost perpendicular
to the chip surface can be used (section 2.1.3). Typically the quality of the
resulting pictures is limited by stray-light stemming from the edges of the wires.
To estimate the precision of transverse positioning the following measurement
has been carried out: A thermal cloud has been positioned at different transverse
positions at a constant height d = 10µm above the surface of the 100µm-wide
Z-shaped wire. For each position the corresponding bias-field (known by the
calibration described above) has been plotted in Figure 4.3 (red data points).
At the same time the ud-field has been determined by measuring the current
in the field generating coils (blue data points). From the measured height and
transverse position in combination with the wire current both components of the
offset-field have been calculated taking the exact wire-shape (see section 2.2.3)
into account (black curve). Good agreement can be seen, leading to a precision
of the ud-field of 0.1G and thus a transverse resolution of 1µm. If the cloud is
positioned close to an edge of a wire (or even above the non-reflecting silicon
surface) a direct measurement of the transverse position is not possible anymore:
The dashed lines in Figure 4.3a indicate the position of the wire borders. Here,
measured and calculated offset-fields deviate strongly. If the cloud is positioned
further away (above a neighboring wire) a reliable position measurement is again
possible (see data point at ∼ 85µm).
In conclusion, a precise control of the position of the BEC is possible which even
exceeds the spatial resolution of the imaging systems. To achieve this the exact
wire-shape has to be taken into account for the calculation of its magnetic field.
80
BEC as magnetic field microscope
Figure 4.3: Thermal atomic clouds have been positioned at various transverse positions maintaining a constant height of d = 10µm above a 100µm-wide wire. The edges
of the wire are indicated by the two dashed lines. The bias-field (red) and the ud-field
(blue) have been determined for each transverse position by measuring the current
flowing in the field generating coils. From the current in the wire and the measured
positions (height and transverse displacement) the magnetic offset-fields expected for
this positions have been calculated according to equation 2.3 and are depicted as black
line. Good agreement can be seen in the residuals (b and c) as long as the atomic
cloud is above a reflecting part of the chip. If the cloud is positioned close to the edge
of the wire or even above the non-reflecting silicon surface reliable transverse position
measurement is not possible [see left dashed line in (a)]. Since the deviations are strong
not all residuals are shown in the plot (b) and (c). If the cloud is displaced further to
be imaged above a neighboring wire (data point at ∼ 85µm) good position calibration
is again possible.
4.1.2 Magnetic potential reconstruction
In the elongated chip traps, BECs with an aspect ratio of up to ∼ 2000 form
in the one-dimensional Thomas-Fermi regime (1dTF) [103]. Here, the chemical
potential µ of the BEC exceeds the longitudinal (weak confining direction) energy
level spacing (µ À h̄ωlo ) but is smaller than the transverse (strong confining
direction) energy level spacing (µ < h̄ωtr ). Details on low-dimensional BECs can
be found in chapter 5. In the 1dTF-regime the longitudinal potential energy is
connected to the longitudinal one-dimensional density n1d by
B0 − B(z) = 2h̄ωtr
ascat
n1d (z)
mF gF µB
(4.1)
where ascat is the s-wave scattering length (ascat = 5.2nm in the case of Rubidium87 [85]) and B0 is a homogeneous offset field.
Figure 4.4 shows a section of the longitudinal density derived from an insitu absorption image (inset). On the left axis the density is expressed in
atoms/µm being below the longitudinal density limit for the 1dTF-regime which
4.1 Mapping two-dimensional magnetic field landscapes
81
Figure 4.4: Longitudinal density profile
derived from an in-situ image (inset) of an
one-dimensional BEC. Since the imaging
laser beam is slightly inclined with respect
to the reflecting chip surface two images of
the atomic cloud can be seen.
Figure 4.5: Longitudinal potential reconstructed from in-situ absorption images of
a fragmented one-dimensional BEC. The
three curves have been shifted by 0.12mG
(0.24mG) since the BEC has been prepared
at different chemical potentials µ by adjusting the RF-knife. The typical structure
of the longitudinal potential is precisely resolved up to the chosen µ.
is ∼ 200atoms/µm. The calculation of the column density and thereby the atom
number from an absorption picture has been described in section 3.2.3. To calibrate the atom number, absorption images of the BECs have been compared for
both, in-situ imaging in the trap and time-of-flight imaging of the expanded cloud.
Comparing both methods, good agreement of the determined atom numbers has
been found.
From such a density trace the longitudinal magnetic field can be calculated
using equation 4.1. In Figure 4.5 the reconstructed longitudinal potential can be
seen. The BEC has been prepared at different chemical potentials by adjusting
the final value of the radio-frequency used for evaporative cooling. Therefore the
potential landscape is only resolved up to the chemical potential. The blue (black)
curve has been shifted by 0.12mG (0.24mG) with respect to the red cure, precisely
resolving the typical structure of the longitudinal potential. These measurements
have been carried out in a trap with a transverse confinement of ωtr = 2π · 3kHz,
corresponding to a ground state size of 200nm.
This reconstruction of the longitudinal potential is correct if a constant chemical potential is assumed over the considered stretch. This condition is only
fulfilled for an equilibrium state of the system. The longitudinal density variations have been monitored over the entire lifetime of the BEC and have been
found to be stable if the analyzed section is limited to a length of ∼ 200µm.
Similar observations have previously been made in optical double well potentials
[104]: If strong barriers separate the different fragments of a BEC, variations of
82
BEC as magnetic field microscope
Figure 4.6: A section of the reconstructed
longitudinal potential is shown for three different experimental runs. The pixel-wise
deviations between these curves have been
plotted in the histogram (inset). Thus, the
single shot sensitivity of the magnetic field
microscope is given by the width of the
Gaussian fit: σ = 4nT = 40µG. This corresponds to ∆B/B = 4 · 10−6 if the potential
variations are normalized to the offset-field
B creating the trap (right axis).
the chemical potential are maintained longer than the lifetime of the BEC.
Consequently, a stretch of a length of 200µm has been chosen to estimate the
sensitivity of the field measurement. In Figure 4.6 the longitudinal potential
reconstructed from three individual experimental runs is depicted. The pixelwise deviations between the three curves have been plotted in a histogram (inset
of Fig. 4.6). The width of the Gaussian fit is σ = 4nT corresponding to the
single shot sensitivity of the magnetic field measurement. These curves have
been measured at a distance d = 10µm above a wire carrying a current of 170mA
at a bias-field of 10G. This leads to a relative field resolution of ∆B/B = 4 · 10−6
if the longitudinal magnetic-field variations are scaled to the global offset-field.
4.2 Comparison to state-of-the-art sensors
4.2.1 Common magnetic field sensors
In this section a brief introduction to today’s magnetic field sensors is given.
A comparison of typical field sensitivity and spatial resolution can be seen in
Figure 4.7.
The operation principle of a Magnetic Force Microscope (MFM) is directly
related to atomic force microscopy which has been invented by Binning et al. in
1986 [105]. Micro-fabrication technology can be used to fabricate cantilevers
which have a force constant smaller than the spring constant of an atom bonded
to a surface. The defection of this cantilever can be monitored, for example by
light beam defection or by means of interferometers, resulting in an image of the
force between cantilever and surface. This allows to image the topography of the
surface without perturbing the atomic structure. Two modes of operation are
possible: In operation under constant-force a feedback loop stabilizes the force
between sensor and surface during a scan. Here the deflection of the cantilever
and thus the force remains constant. Furthermore the cantilever can be oscillated
close to its resonance frequency and the resonance amplitude can be used to probe
4.2 Comparison to state-of-the-art sensors
83
force gradients close to the surface.
To be sensitive to magnetic forces a tip made of ferromagnetic material is required: etched micro-wires can be used or thin magnetic films can be evaporated
on an atomic-force microscope tips placed on a cantilever. Consequently, the
precise structure of the magnetic tip is not known. Therefore quantitative field
measurements are hard and interpretation of MFM-images require detailed theoretical calculations. Today, precise calibration of MFM-tips [106] is still under
investigation. A typical field resolution of 0.5mT at a spatial resolution smaller
than 50nm can be obtained [107, 108].
Scanning Hall-Probe Microscopy (SHPM) has been firstly used by Broom
et al. in 1962 ([109] and references therein) and relies on the commonly known
Hall-effect: Electrons moving in a current-carrying wire are deflected towards the
side of the wire if a magnetic field perpendicular to the wire is applied. This
leads to a voltage drop between the two opposite sides of the wire. Similarly to
this bulk effect the defection of electrons happens in a two-dimensional electron
gas. Already in 1971 a field sensitivity of 10µT at a spatial resolution of 4µm
has been demonstrated [109]. Using today’s etching technics to structure semiconductor materials, the size of the active area of a micro Hall-probe can easily
be sub-micron. Sensitivities of 5µT at a spatial resolution of ∼ 100nm can be
achieved [110, 111]. Usually the Hall-sensor is mounted on a piezoelectric scanner of a scanning tunnelling microscope. Using this device surface topograph and
magnetic field can be measured simultaneously.
Operation of a Superconducting Quantum Interference Device (SQUID)
is possible due to two effects: flux quantization in superconducting current loops
and the Josephson effect.
The Josephson effect was theoretically predicted by B.D. Josephson in 1962
[112, 113] and describes the tunnelling of Cooper-pairs through a weak link between two superconducting electrodes. This weak link can be realized by a ∼ 10Å
thick oxide-layer leading to a barrier between the superconducting materials. Describing this system by two weakly coupled macroscopic wave functions leads to
a current I flowing through the barrier depending on the phase difference ∆φ
of the two coupled systems according to I = I0 sin ∆φ, where I0 is the maximal
current given by the actual design of the barrier.
A SQUID consists of a ring of superconducting material interrupted by two
Josephson junctions. These two weak links allow only for a small current circulating around the ring, much lower than the critical current of the non-intersected
ring. If a magnetic field perpendicular to the ring is applied this magnetic flux
ΦB causes a phase shift of ∆φ = 2πΦB /Φ0 of the electron-pair wave-function
along the ring. Since ΦB not necessarily equals an integer multiple of the fluxon
Φ0 = h/2e = 2.07 · 10−7 Gcm2 , a small current i in the ring is induced leading to
a phase shift across the two Josephson junctions so that ∆φ(B) + 2∆φ(i) = n2π.
This small current i has a periodic dependance on the applied magnetic field,
with a periodicity given by Φ0 . Consequently, this current can be used to measure
84
BEC as magnetic field microscope
Figure 4.7: Comparison of field sensitivity and spatial resolution of state-of-the-art
magnetic field sensors. MFMs and Hall-probes operate at a high spatial resolution but
poor field sensitivity, whereas SQUIDs and atomic magnetometers allow for high field
sensitivity at low spatial resolution. The magnetic field sensor using a BEC enables to
measure magnetic fields in the intermediate region which cannot be covered by other
sensors. The demonstrated resolution and sensitivity used to probe the field variations
above the trapping wire itself was 4nT at 3µm (Fig. 4.6). If the BEC is moved to an
independent sample (section 4.4) and thus the trap’s transverse frequency is relaxed
the sensitivity is expected to be smaller than 1nT. The black line gives an estimate on
the potential sensitivity of the BEC-microscope. A detailed explanation of the three
regimes (a, b, and c) is given in the main text.
small magnetic fields and today a field sensitivity of 2pT at a spatial resolution
of 10µm can be achieved [114, 115, 116].
In Atomic Magnetometers the Larmor-frequency of spin-polarized atoms is
used as a measure for an applied magnetic field. Most of these magnetometers
use a thermal vapor of alkali-metal atoms. Ultra-high field sensitivity smaller
than 1fT at a spatial resolution of 2mm can be reached [117]. The sensitivity is
mainly limited by interatomic collisions and by collisions of the atoms with the
walls of the vapor-cell. Efforts are still being made to integrate these devices to
reach better spatial resolution [118].
4.2.2 Sensitivity of BEC to magnetic fields
The sensitivity to magnetic fields demonstrated in the experiments discussed in
this chapter depends on the transverse confinement of the magnetic trap. A
reduction of the transverse confinement leads to a reduction of the chemical
potential of the BEC and thus to a higher sensitivity. Of course this sensitivity
has to be adjusted to the strength of the variations to be probed in order to
optimally map the desired potential: In the case of the magnetic field landscape
4.2 Comparison to state-of-the-art sensors
85
above the trapping wire itself, a sensitivity of 4nT at a spatial resolution of 3µm
has been demonstrated (Fig. 4.6). If an independent wire carrying a small current
of ∼ 5mA is to be probed the transverse confinement has been lowered to 690Hz
(section 4.4) and a field sensitivity smaller than 1nT can be expected. These two
cases have been plotted in Figure 4.7 in comparison to state-of-the-art magnetic
field sensors. It can be seen that the BEC-microscope allows for measurements in
the intermediate range which cannot be accessed by the commonly used sensors.
The spatial resolution in the transverse direction ρ0 given by the ground state
size does not need to be the same as the spatial resolution in the longitudinal
direction z0 given by the imaging system. The sensitivity of the field measurement
can be further optimized if the atomic density is detected atom shot noise limited.
Rewriting equation 4.1 leads to a potential sensitivity of the BEC-sensor of
∆B = γ
∆N
,
ρ20 z0
(4.2)
where ∆N is the minimal atom number resolved by the imaging system and
γ=
2h̄2 ascat
.
gF mF µB m
(4.3)
This coefficient contains all atomic properties and in the case of Rubidium-87
atoms in the F = mF = 2 state its numeric value becomes γ = 8.63 · 10−29 Tm3 .
In the measurements presented in Figure 4.6 a single-shot-sensitivity of 4nT has
been found. These measurements were performed at a transversal trap frequency
of 2π · 3kHz. Thus, the minimal atom number resolved by the imaging system
can be calculated. Equation 4.2 yields ∆N ∼ 6atoms for the parameters of that
measurement.
Evaluating equation 4.2 for typical experimental parameters allows to estimate
the potential field sensitivity versus spatial resolution range where the BECmicroscope could compete with today’s magnetic field sensors (see black line in
Fig. 4.7). For this purpose a shot noise limited detection of N = 100 atoms
has been assumed (∆N = 10). Three regimes can be distinguished which are
depicted in Figure 4.7 labelled by (a), (b), and (c). They will be discussed in the
following three paragraphs:
(a) The spatial resolution ∆x of the optical imaging system can be increased
up to ∼ 500nm if a transition in the blue is used (around λ = 421.67nm of
the Rubidium 52 S1/2 − 62 P1/2 resonance [119]). Consequently, the spatial
resolution of the magnetic field measurement in the intermediate range from
∆x = 0.5 − 10µm can be assumed to be given by the imaging system: A
longitudinal section of the BEC of the length z0 = ∆x has to be imaged onto
one pixel of the CCD-camera and at the same time the transverse ground
state size ρ0 has to be equal to ∆x. This corresponds to a transversal
86
BEC as magnetic field microscope
trap frequency of ω = 2π · (460 − 1.1)Hz. In this regime (Fig. 4.7a),
∆B scales most strongly with the spatial resolution of the measurement:
∆B = γ∆N/∆x3 .
(b) The generation of BECs in very shallow traps of up to ρ0 = 10µm ground
state size (ω = 2π · 1Hz) has been successfully achieved but further decrease
of trap frequency seams to be technically challenging [120]. Consequently
higher magnetic field sensitivity can only be gained by decreasing the resolution ∆x in the longitudinal direction at a fixed transverse confinement of
2π · 1Hz (Fig. 4.7b). This leads to an anisotropic spatial resolution of the
sensor and the sensitivity scales as ∆B = γ∆N/(ρ20 ∆x) keeping ρ0 fixed.
(c) In the high spatial resolution limit where only moderate magnetic field sensitivities can be achieved the spatial resolution in the longitudinal direction
is limited by the imaging system to ∆x = 500nm. But the ground state
size ρ0 in the transverse direction can be much smaller – for example 20nm
at a transverse frequency of 2π · 300kHz. The necessary accuracy of the
positioning of the trap can be achieved by precisely controlling the currents in the trapping wire and in the coils. This positioning resolution can
be below 10nm as has been demonstrated in a experiment which transports a BEC in a conveyor-belt [20]. Thus, the field sensitivity scales as
∆B = γ∆N/(ρ20 ∆x) keeping ∆x fixed.
The sensitivity of the BEC-microscope can be further increased if a different
atomic species is used. Atoms with higher mass increase the sensitivity and, for
example Cesium-133, has been successfully Bose-condensed [121]. Additionally
Cesium allows to exploit Feshbach resonances which enable to adjust the s-wave
scattering length ascat at will. In the case of Cesium, ascat becomes zero around an
easily manageable magnetic field value of 17.0G and the linearized slope around
this zero-crossing of ascat can be estimated to be ∼ 6nm/G [122]. One finds the
relation
1 aCs
γCs '
· scat
· γRb
(4.4)
4.6 aRb
scat
and by decreasing the scattering length by one order of magnitude compared to
Rubidium the BEC-microscope would already gain a factor of 50 in sensitivity
if ∆N and the spatial resolution are kept constant. Since the stabilization of
magnetic fields even on the level of 1mG is easy, a surpassing field sensitivity
could be expected.
4.3 Reconstruction of the current density
87
4.3 Reconstruction of the current density
In general, it is not possible to transform a magnetic field image into the corresponding current density in a unique way ([123] and references therein). However,
if one assumes the current density to be confined to a two-dimensional plane the
inversion is possible. Mostly the magnetic field component perpendicular to the
current sheet is the quantity to be measured since it can be easily mapped by
MFMs [124] and SQUID-sensors [125].
In the case presented here one magnetic field component parallel to the current
sheet can be measured. In Figure 4.8a such a two-dimensional field map can be
seen: A current of 340mA in a gold wire (cross-section 100 × 3.1µm2 ) flows into
the z-direction. An elongated BEC has been positioned at 28 equally spaced
transverse positions (x-direction) maintaining a fixed height of y = 10µm above
the wire surface. Since the BEC is sensitive to variations of the Ioffe-field of the
trap the magnetic field component in z-direction Bz has been mapped.
The magnetic field B(r) can be calculated from the current density j(r) by
Biot-Savart’s law
0
0
µ0 Z j(r ) × (r − r ) 3
B(r) =
dr
(4.5)
4π
|r − r 0 |3
where µ0 = 4π·10−7 Tm/A is the permeability of free space. Assuming the current
density to be confined in the x, z-plane at y = 0, the magnetic field component
Bz (x, z) at a distance y above the current sheet can be expressed as
0
0
0
0
0
µ0 Z jx (x , 0, z )y − jy (x , 0, z )(x − x ) 0 0
Bz (x, z) =
dx dz .
4π
[(x − x0 )2 + y 2 + (z − z 0 )2 ]3/2
(4.6)
If a current sheet is assumed, only jx contributes to the magnetic field component
Bz (x, z). Thus, jx can be reconstructed using the deconvolution theorem:
jx (x, z) = F −1 {B̄(kx , kz )ek|y| }(x, z)
(4.7)
where B̄(kx , kz ) = F{Bz (x, z)}(kx , kz ) and F indicates a two-dimensional Fourier
transform.
To estimate the spatial resolution ∆j of the reconstructed current-density two
effects have to be taken into account: (1) The magnetic field has been measured
at a finite distance of y = 10µm above the surface resulting in a smoothing of
the magnetic field variations. (2) The field has been mapped on a grid given
by the spatial resolution of the imaging system in the z-direction and by the
transverse positioning of the BEC in the x-direction. Here, two regimes have to
be considered: If the spatial resolution ∆B of the magnetic field measurement is
larger than the distance y to the wire, one finds ∆j ≈ ∆B. In the opposite limit
(y ≥ ∆B) the resolution of the current density can be found as follows: A pointlike current-density component of a finite width ∆j can be linked by Biot-Savart’s
88
BEC as magnetic field microscope
Figure 4.8: (a) Two-dimensional map of the longitudinal magnetic field component
Bz taken at a distance of y = 10µm above the 100µm-wide and 3.1µm-tall wire. The
wire carried a current of 340mA at an offset-field of 20G resulting in a current density
of jz = 1.06 · 105 A/cm2 in the wire. The magnetic field has been measured at 28
equally spaced positions by shifting an elongated BEC in the transverse direction (xdirection) maintaining a constant height above the wire. (b) The jx -component of
the current density has been reconstructed according to equation 4.7. Local variations
of the direction of the current-flow correspond to angular deviations of 0.01 − 1mrad.
(c) From the reconstructed current density jx the longitudinal magnetic field component
has been calculated by Biot-Savart’s law (Equ. 4.5) to test the applied deconvolution
methodology. The resulting field map is in good agreement with the measured magnetic
field shown in (a).
4.3 Reconstruction of the current density
89
Figure 4.9: Scaling behavior according to equation 4.8 of the spatial resolution ∆j of
the reconstructed current density in the wire. (a) Dependence on the distance y to the
wire. (b) Dependence on the spatial resolution ∆B of the magnetic field measurement.
Note that equation 4.8 is only valid for y ≥ ∆B, otherwise ∆j is approximately given
by ∆B.
law to the resolution ∆B of the corresponding magnetic field, yielding
∆j ≈ y +
3 ∆B 2
.
10 y
(4.8)
In Figure 4.9 ∆j has been plotted for the parameter range accessible in the
experiment. In the case of the reconstructed current density of the 100µm-wide
wire, the spatial resolution is limited by the large distance to the wire.
If the current density is reconstructed using equation 4.7 high frequency noise
in the magnetic field map has to be taken into account. This noise results from
the imaging process and leads to high frequency noise in the reconstructed current density. This causes artificial structures which do not represent the actual
current density distribution. Using the estimation of the expected current density resolution which has been discussed
above,
h
³ a filter
´i hfunction³can be
´idesigned.
kx −cx
kz −cz
−1
As this filter function, F (kx , kz ) = 1 + exp sx
1 + exp sz
has been
used. This filter function has been applied to B̄(kx , kz ) before computing jx using
equation 4.7. A reasonable choice of the filter parameters for the experimental
parameters is cx = 0.32µm−1 , cz = 0.22µm−1 , and sx = sz = 0.05µm−1 . These
parameters have been used to calculate jx (x, z) depicted in Figure 4.8b. As a
test of the applied deconvolution methodology, Bz has been calculated from the
reconstructed jx . To do this check, a calculation has been performed which does
not utilize Fourier-transformation. Biot-Savart’s law (Equ. 4.5) has been used to
directly calculate the magnetic field of the current density distribution (Fig. 4.8c).
Good agreement between this calculated magnetic field and the measured magnetic field map is found.
90
BEC as magnetic field microscope
This reconstruction method enables to estimate the deviations of the current
in the wire from a straight flow. Angular deviations of 0.01 − 1mrad are found. It
is obvious, that these deviations are caused by local effects. They cannot only be
understood by assuming a model [96] based on rough wire edges which has been
used to explain magnetic field fluctuations observed above electroplated wires
[93]. The origin of this irregular current-flow has to be investigated further to gain
a better understanding leading to a reduction of these current-path deviations.
Possible sources could be the roughness of the wire surface or local variations of
the resistivity of the wire. The following investigations should be done: Wires
fabricated out of different materials (gold, copper, and alloys – for example PdAu)
should be investigated. At the same time the thicknesses of the wire and the
evaporation parameters (temperature of the substrate, speed of evaporation) have
to be varied. A brief introduction to the fabrication process of the atom chip
can be found in section 3.5. As a first step the quality of a more than one
order of magnitude thinner gold wire could be studied if the new atom chip
described in section 3.5.3 is used. Furthermore current-flow in wires fabricated
of semiconductor materials such as GaAs [74] could be studied. In these wires
surface roughness would be reduced to a minimal amount due to epitaxial growth
of the current-carrying doped semiconductor.
In conclusion, the BEC-microscope has proven to be usable as a relevant technical application: The reconstruction of current-flow in wires has been demonstrated even in the presence of large offset-fields. Besides the investigation of
irregular current-flow in semi-conductor wires, an interesting continuative project
would be to map the current-density in a two-dimensional electron-gas or in superconducting material.
4.4 Probing other local potentials
In the above mentioned cases the magnetic field of the trapping wire itself has
been probed. But it is also possible to bring the BEC close to an independent
sample far away from the trapping wire (Fig. 4.1left). Here, no magnetic field
variations originating from current-flow deviations in the trapping wire are detected anymore. Only the potential variations caused by the sample are being
probed. This has been investigated by holding the BEC by the 100µm-wide Zshaped wire (C-D in Fig. 3.1) and rotating it above the 10µm-wide wire (B-F in
Fig. 3.1). This results in a distance of the BEC to the center of the trapping wire
of 125µm and in a transverse trap frequency of 2π · 690Hz. If the probed wire is
grounded no modulation of the density of the BEC down to distances of 5µm to
the wire surface could be seen. This corresponds to an upper bound in potential
roughness of less than 10−14 eV, corresponding to a temperature of 200pK (field
sensitivity of 300pT). As soon as a small current of ∼ 5mA is passed through the
wire, a characteristic profile is detected.
4.4 Probing other local potentials
91
Figure 4.10: Using an elongated BEC held
by a trapping wire above an independent
wire magnetic (blue) and electric (red) disorder potentials have been measured. The
dotted lines have been put at pronounced
potential minima of the potential caused by
electric fields to highlight the different pattern. The red curve has been artificially
shifted by an offset of 40nK for visibility.
Besides magnetic potentials this elongated BEC can be used to probe electric
potentials. It has been shown that adsorbed Rubidium atoms on the surface
severely alter the trapping potential at surface distances of ∼ 10µm [100]. These
partially ionized atoms on the surface cause an electric dipole field Ed which
polarizes a trapped Rubidium atom. In this case the interaction potential is given
by Uel = −αEd2 /2 where α is the polarizability (α = h · 0.0794Hz(cm/V)2 [37]
and for an convenient comparison to the strength of the magnetic interaction one
finds Uel [µK] ∼
= 190E 2 [V2 /µm2 ]). Although the electric field produced by a single
adsorbed atom is small, this field can be enhanced by applying a homogeneous
electric field Eperp perpendicular to the surface of the probed wire. The resulting
field is given by Uel = −(α/2) · (Ed + Eperp )2 ≈ −αEd · Eperp for large offset-fields.
As an example the reconstructed longitudinal potential stemming from electric
fields possibly created by adsorbed Rubidium atoms can be seen in Figure 4.10
(red curve). For this measurement the wire has been put on an electric potential
of 30V relative to the surrounding chip surface. For comparison, the magnetic
disorder caused by a small current of 30mA (blue curve) has been measured at the
same position above the 10µm-wide wire. The periodicity and pattern of these
curves do not show a correlation. This suggests a different origin of the underlying
disturbances. This needs not necessarily to be the case: The electric disorder
potential mainly depends on the surface topography of the wire. In contrast to
that, the magnetic disorder potential can be caused by the surface topography as
well as by variations inside the bulk – for example varying conductivity. It could
even be possible to use the information on surface topography gained from the
electric disorder potential, to refine the measurement of the magnetic disorder
potential. This way pure bulk effects might become visible.
92
BEC as magnetic field microscope
5 Exploring low-dimensional BECs
The investigation of low-dimensional systems has been performed in condensed
matter physics for several decades [126]. Rich physics has been discovered since
the nature of collective excitations and the properties of phase transitions depend
on the dimensionality of the investigated system. Bose-Einstein condensates of
neutral atoms held in magnetic or optical traps offer the possibility to perform
similar experiments in a much cleaner environment. Additionally, Feshbach resonances allow to tune the strength of interaction between atoms, even enabling
to change interaction from attractive to repulsive. Reducing the dimensionality
of these atomic systems increases the possible degenerated states. In three dimensions a true Bose-Einstein condensate is possible which has been intensively
studied by several groups in the last years [83]. If the condensate is confined
to two dimensions, two states become possible: the true condensate and the
so called quasi-condensate which is dominated by phase fluctuations along the
weakly confining axes. For a one-dimensional system of trapped bosons (discussed in section 5.1) additionally a Tonks-gas regime can be realized where the
bosons exhibit some fermionic properties.
In this chapter experimental geometries will be discussed which allow to create
two-dimensional and one-dimensional traps for neutral atoms with repulsive interaction. Since atom chips naturally generate elongated cigar-shaped traps the
focus will lie on the creation of a single one-dimensional (quasi) Bose-Einstein
condensate close to the surface of an atom chip. A detailed study of the crossover from one- to three-dimensional Bose-Einstein condensates will be presented
(section 5.2). After that a novel approach for the creation of two-dimensional
traps close to surfaces will be introduced, based on the combination of optical
lattice potentials and atom chips (section 5.3).
5.1 Theory of 1d BECs
One-dimensional (1d) systems are usually realized in cylindrically symmetric
traps offering strong confinement in the transverse direction and weak confinement along the longitudinal direction. If the relevant energy scale of a thermal cloud (kB T ) or a Bose-Einstein condensate (chemical potential µ) becomes
smaller than the energy h̄ωtrans associated with the transverse level-spacing, excitations in this direction are frozen out. In analogy to the 3d-case a Gross-
94
Exploring low-dimensional BECs
Pitaevskii equation (GPE) can be derived for 1d-systems [127]
"
#
h̄2 d2
−
+ Vtrap (z) + g1d N |ψ(z)|2 − µ ψ(z) = 0,
2m dz 2
(5.1)
where ψ(z) is the Bose-Einstein condensate (BEC) wave-function, µ the chemical
potential, Vtrap is the trapping potential and g1d is the effective one-dimensional
coupling strength. If the extension
of the ground state of the transverse harmonic
q
potential given by xtr = h̄/mωtr is much larger than the characteristic size of
the interatomic potential (xtr À a3d ), g1d can be expressed as [128]
g1d =
2h̄2 a3d
,
mx2tr
(5.2)
where a3d is the three-dimensional scattering length (a3d = 5.2nm in the case
of Rubidium-87 [85]). The one-dimensional scattering length a1d is connected to
the three-dimensional scattering length by a1d = x2tr /a3d .
In such a one-dimensional system a weakly and a strongly interacting regime
can be distinguished. To derive a quantity characterizing these regimes the interaction energy of the 1d-system Eint = n1d g1d can be compared to the kinetic
energy Ekin = h̄2 n21d /m of the ground state:
γ=
mg1d
Eint
= 2
.
Ekin
h̄ n1d
(5.3)
This parameter characterizes the behavior of trapped 1d-gases which is counterintuitive compared to the 3d-case. In the case of high densities the system is
weakly interacting (γ ¿ 1). This mean-field regime is well described by the GPE
and Bose-Einstein condensation is possible [129]. Here, the chemical potential
µ = n1d g1d of the BEC has to be smaller than the energy h̄ωtr . This leads to
the condition n1d a3d ¿ 1. Thus, the longitudinal 1d-density has to be (much)
smaller than ∼ 200atoms/µm for Rubidium-87 to stay in the one-dimensional
regime. When the longitudinal density is reduced (γ À 1) the system enters the
Tonks-Girardeau regime where the bosons exhibit some fermionic properties.
If the longitudinal confinement is
q assumed to be harmonic the longitudinal
ground-state size is given by xlo = h̄/mωlo . A dimensionless parameter α can
be introduced, comparing the longitudinal size of the trap to the one-dimensional
scattering length a1d :
xlo
xlo
= a3d 2 .
(5.4)
α=
a1d
xtr
−1/2
In the case of Rubidium-87 this quantity can be calculated by α ∼
= 10−3 · νtr νlo
where νi equals ωi /2π and its unit is Hz.
5.1 Theory of 1d BECs
95
Figure 5.1: Depending on the parameter
α and the atom number N different states
for trapped one-dimensional gases are possible. This diagram has been plotted following [130]. The dashed (solid) blue line
indicates the cross-over region between a
Thomas-Fermi BEC and a Gaussian BEC
(Tonks-Girardeau gas, respectively). The
gray lines have been plotted for constant γ.
Zero temperature and weakly interacting regime
The different possible regimes for a 1d-system at zero temperature are sketched
in Figure 5.1 [130]. The experiments discussed in section 5.2 are performed at
α ∼ 1 and probe the weakly interacting Thomas-Fermi regime. For α ¿ 1 the
strongly interacting regime is absent for any atom number N since the harmonic
confinement of the trap rather than interactions governs the motion of particles.
If N À α−1 the condensate forms in the one-dimensional Thomas-Fermi regime
where the chemical potential is µ À h̄ωlo . Reducing the atom number leads to
a reduction of the mean-field interaction which, at some point, becomes smaller
than the longitudinal level spacing h̄ωlo in the trap. This leads to a macroscopic
occupation of the ground state of the trap yielding an ideal gas condensate with
a Gaussian shape.
Zero temperature and strongly interacting regime
For α À 1 the atom number has to be sufficiently high for a Thomas-Fermi
condensate to form (N > α2 ). Reducing the atom number leads to the formation
of a strongly interacting Tonks-Girardeau gas. Recently experiments performed
in 2d-optical lattices, aimed at the formation of a Tonks-gas: For γ = 0.5 a
significant reduction of losses due to three-body collisions has been observed
[131] and with γ = 1 changes in the excitation spectrum of the system have been
reported [132]. Furthermore, the energy and the length of a Tonks gas have been
measured at up to γ = 5.5 [133]. All these experiments have been performed in
two-dimensional optical lattices forming several thousand individual traps at the
same time, thus circumventing the difficulties arising from detection of low atom
numbers.
In contrast to these systems, elongated magnetic traps generated by atom chip
potentials promise to allow for a single realization of a Tonks gas. In this well
defined system the momentum distribution, inelastic collisions and excitations
could be investigated. All of these properties should show strong evidence of
fermionization. Moreover fermionic nature can be exploited this way by gaining
96
Exploring low-dimensional BECs
Figure 5.2: Diagram of states for a finite
temperature 1d-system. For sufficiently low
T the quantum degenerated regime in entered (green line) where three states are
possible. The blue (black) curve corresponds to α = 30 (α = 10). Note, that the
right and the top axis have been plotted for
ωlo = 2π · 1Hz.
access to a 1d-system of trapped fermions represented by the Tonks-gas which
would be much harder to reach with real fermions.
Finite temperature and strongly interacting regime
High α is required for the formation of a Tonks-gas. In the following, the diagram
of states for a finite temperature 1d-system will be discussed and experimentally
achievable parameters will be given. This description is based on a paper by
Petrov et al. [130]. In Figure 5.2 the diagram of states is depicted for α = 10
(black lines) and α = 30 (blue lines). The Tonks-gas regime is entered if the atom
number is below 100 atoms for α = 10 and below 1000 atoms for α = 30. The
atom number axis and temperature axis giving T in units of h̄ωlo are valid for
any trap frequency combination whereas the density axis nTonks on the right and
the temperature axis on the top of the diagram have been calculated assuming
ωlo = 2π · 1Hz and ωtr = 2π · 10kHz (2π · 30kHz) for the black (blue) curve.
It can be seen that the condition for being one-dimensional given by kB T < h̄ωtr
is fulfilled in the shown temperature range since 2π · ω[kHz] ∼ 50[nK]. For
T < Td the quantum degenerate regime is entered, where Td ∼ N h̄ωlo (green
curve). In the case of sufficient small atom number N < α2 the system is in
the strongly interacting Tonks-gas regime. Here the longitudinal density profile
is significantly different from a Thomas-Fermi
profile as well as from a thermal
q
√
gas profile: nTonks (z) = ( 2N /πxlo ) 1 − (z/R)2 , where the half-length of the
√
distribution
is
given
by
R
=
2N xlo . From this the longitudinal density nTonks ∼
√
2N /(4xlo ) has been computed and is depicted on the right axis of Figure 5.2.
Due to this small densities the detection of the longitudinal profile of a single
Tonks-gas seems to be challenging. More sensitive imaging can be performed
along the longitudinal direction since the atomic column density is much higher.
In this context detection of single atoms with micro-optics devices mounted directly on the atom chip might be promising – for example fiber cavities could be
used [134]. Studying Tonks-gases on an atom chip with a single-atom detector
5.2 Cross-over between 1d and 3d BEC
97
has recently been proposed [135].
Finite temperature and weakly interacting regime
In the experimentally much more easily accessible regime of small α and large N
(or N > α2 ) one always has got a weakly interacting gas. Experiments which investigate the cross-over from this regime to the 3d-case have been performed and
will be discussed in the next section. This 1d-regime is characterized by the longitudinal phase coherence length Rph ∼ RTF (Tph /T ) ∼ N 2/3 /T of a condensate.
Here RTF is the Thomas-Fermi radius of the condensate and Tph = Td h̄ωlo /µ is
the characteristic temperature. The phase of the condensate is only coherent over
a certain length which is smaller than the longitudinal size of the system. This
means that the phase coherence length Rph is smaller than the longitudinal length
of the condensate. This so called quasi-condensate exists for Tph ¿ T ¿ Td .
Note, that the transition between the three possible states for a 1d-bose gas
– true-condensate, quasi-condensate, and Tonks gas – is smooth. The cross-over
region between true-condensate and quasi-condensate is depicted in Figure 5.2 as
a solid line.
5.2 Cross-over between 1d and 3d BEC
Atom chips offer the possibility to create very anisotropic traps. Trap frequency
aspect ratios can be up to several thousand allowing to investigate very elongated
BECs and thus to enter the 1d-regime. In several experiments using 2d optical
lattices the strongly interacting 1d-regime has been investigated (see previous
section). In contrast, this section will discuss the cross-over between weakly
interacting one-dimensional BECs and three-dimensional BECs.
In two experiments working with magnetically trapped atoms signatures of
this cross-over have already been observed, however never getting deeply into the
1d-regime: In an experiment with a Natrium-23 BEC by Görlitz et al. change
in the aspect-ratio of an expanding BEC has been observed for densities down
to n1d ∼ 0.2a−1
3d [136]. Schreck et al. studied the time-of-flight expansion of a
Lithium-7 BEC with n1d ∼ 0.5a−1
3d . They found good agreement with the time
evolution of a transverse single particle ground state wave function [137].
In the setup described in this thesis, it is possible to probe the entire crossover region down to longitudinal 1d-densities of 13atoms/µm. This is a factor 15
smaller than the limit given by 1/a3d . The transverse size for an expanded BEC
prepared at different longitudinal densities shows good agreement with theory and
a two-dimensional GPE simulation. Moreover the exact shape of the expanded
BEC has been probed in detail and the continuous evolution from a Gaussian
shape to a parabolic shape has clearly been observed.
98
Exploring low-dimensional BECs
5.2.1 Ballistic expansion of a BEC
The expansion of a single particle from a harmonic trap is governed by Heisenberg’s uncertainty principle. The initial width σ0 of the Gaussianqwave-function
is given by the ground state size of the harmonic oscillator xho = h̄/(mω). The
time-evolution of the size of the wave-function depends on the steepness of the
confining potential and follows [80]
q
σho (t) =
x2ho (1 + ω 2 t2 )
(5.5)
where ω is the trap frequency. This expression describes the expansion of a
BEC in the limit of low density where interactions do not play a role. In the
strongly interacting case, the Thomas-Fermi approximation can be used to find
an expression for the time-evolution of the ballistic expansion [138]. Here, the
transverse size follows a similar scaling law
q
ltrans (t) =
q
2
2
ltrans
(0)(1 + ωtrans
t2 ),
(5.6)
where li (0) = 2µ/(mωi2 ) is the Thomas-Fermi radius corresponding to the initial
size of the condensate in the trap, and l(t) is the half-length of the condensate
wave-function (equation 3.5 in section 3.2.3). This result is correct to zeroth
order of ² = ωlong /ωtrans ¿ 1 where no expansion in the longitudinal direction is
derived. Expansion in the longitudinal direction is only found if effects in second
order of ² are taken into account (equation (21) in [138]). For the experiments
discussed in this section a trap with trap frequencies ωtrans = 2π · 2.9kHz and
ωlong = 2π · 8Hz has been used, leading to ² ∼ 1/360. For a Thomas-Fermi
condensate of 106 atoms in this trap a longitudinal density of ∼ 3500atoms/µm
is found (ltrans (0) ∼ 0.8µm and llong (0) = 292.2µm). The size after 22ms of
ballistic expansion from this trap can be calculated using the above mentioned
theory to be 320µm in the transverse direction and 293.6µm in the longitudinal
direction. Thus, the expansion in the longitudinal direction can be neglected
(< 1%) in the further discussion.
For the intermediate density regime a 2d GPE simulation of the expansion
has been performed. As a result of these calculation, the atomic density distribution is obtained for a specific longitudinal density. These numerically found
distributions have been fitted by a Gaussian curve representing the single particle expansion and by a parabolic distribution resembling the Thomas-Fermi
expansion. In Figure 5.3 the root-mean-square deviations of the calculated distribution to both model functions have been plotted. The cross-over happens
around 190atoms/µm. To access this regime, experiments in two different traps
have been performed: To exploit the low-density regime (n1d < 70atoms/µm)
time-of-flight images of a disordered BEC have been analyzed. In a second experiment, higher densities have been generated in a longitudinally homogeneous
BEC.
5.2 Cross-over between 1d and 3d BEC
99
Figure 5.3: Transverse profiles of an expanded BEC have been numerically calculated and fitted by two test-functions. The
root-mean-square deviation of the residuals is shown. For low densities the profile is better approximated by a Gaussian
curve (red) whereas for higher densities
a parabolic curve (blue) is better suited.
A true Thomas-Fermi profile is recovered
for much larger densities (several thousand
atoms/µm) than shown in this plot.
5.2.2 Expansion measurements of BECs
Figure 5.4a shows an absorption image of a disordered BEC after 5ms of ballistic expansion (disorder potentials have been discussed in section 3.4). The BEC
has been formed at a distance of 10µm above the 100µm-wide Z-shaped wire
(C-D in Fig. 3.1) at a wire current of 340mA, resulting in ωtrans = 2π · 3.5kHz.
The longitudinal 1d-density n1d has been calculated (Fig. 5.4b) and clearly fulfills n1d < 190atoms/µm. For comparison the noise-level has been derived (red
curve) at a position where no atoms were present in the absorption picture shown
above (in section 3.2.3 the atom number calculation from absorption pictures has
been discussed). In the logarithmic plot (Fig. 5.4c) it can be seen that the
BEC stretches almost continuous over the entire length of 1mm. The detection
sensitivity can be estimated from this longitudinal profile to be ∼ 4atoms/µm.
This profile derived after time-of-flight expansion has been compared to a profile
taken by in-situ imaging. The longitudinal structure of the trapped BEC has
been found to be the same as the expanded one. Therefore the expansion in
the longitudinal direction is only a small correction and will be neglected in the
further analysis.
To determine the transverse width of the individual BECs the entire picture
has been divided into small transverse traces stretching over 2 longitudinal pixels.
These sections have been averaged and the resulting transverse density profile
has been fitted by a Gaussian test function. For each width the corresponding
longitudinal 1d-density n1d has been taken from the plot depicted in Figure 5.4b.
To distinguish data from background noise only fits with an amplitude larger
than three times the standard deviation of the fluctuations of the zero-level have
been included into the further analysis. This way 10 pictures have been analyzed,
their data has been averaged, and combined to Figure 5.6 after normalizing to
the width of a single particle expanding from the trap (red data points).
To probe higher densities a BEC has been formed at a distance of 50µm from
the chip surface where a homogeneous longitudinal profile is obtained. In Figure
5.5a the average over 20 absorption pictures of an expanded one-dimensional
100
Exploring low-dimensional BECs
Figure 5.4: (a) Absorption image of a fragmented BEC taken after 5ms of timeof-flight expansion. The BEC has been formed at a distance of 10µm from the wire
surface. (b) Longitudinal one-dimensional density profile (blue) derived from the absorption picture. The noise-level is shown in red. (c) To accentuate the noise-floor of
∼ 4atoms/µm the same data has been plotted logarithmic. A continuous BEC stretches
almost over a length of 1mm.
quasi-BEC can be seen. This BEC has been generated in a trap with ωtrans = 2π ·
2.9kHz and ωlong = 2π·8Hz and has been imaged after 22ms of ballistic expansion.
In the longitudinal 1d-density profile depicted in Figure 5.5b one clearly sees a
small thermal component which has been fitted by a Gaussian distribution (black
dashed curve). The resulting width and amplitude have been used to simulate an
isotropic 2d-thermal distribution which has been subtracted from the absorption
image. As a result the pure distribution of the BEC component is obtained (Fig.
5.5c). Small residual fluctuation at the outer parts of the distribution can be
estimated to be smaller than 10atoms/µm.
As has been discussed in the previous section, expansion in the longitudinal
direction of a BEC from such a trap is smaller than the resolution of the imaging
system. This has been checked by comparing the 1d-density distribution derived
from time-of-flight pictures to in-situ pictures. The same longitudinal profile
has been found. Therefore the transverse width of the BEC at every single
longitudinal position can be linked to the local 1d-density. Transverse traces have
been obtained by averaging over stretches of 2 longitudinal pixels and – as an
example – three such profiles are depicted in Figure 5.7. For high densities these
profiles deviate from the Gaussian shape but still have been fitted by a Gaussian
5.2 Cross-over between 1d and 3d BEC
101
Figure 5.5: (a) Average over 20 absorption pictures of an expanded onedimensional quasi-BEC created at a distance of 50µm above the chip surface. (b)
The longitudinal 1d-density profile shows a
small thermal component which has been
fitted by a Gaussian (black dashed line).
The resulting width and the amplitude have
been used to generate a 2d-thermal distribution which has been subtracted from the
image. (c) This way the pure BEC distribution is obtained. The width of the thermal component after 22ms of ballistic expansion yields a temperature of 373 ± 9nK.
The asymmetry is caused by a small gradient due to gravity.
distribution to avoid any bias when extracting the width. The Gaussian test
function underestimates the actual width of the distribution by a few percent. A
detailed discussion of the shape of the curves is given below. The resulting widths
of the Gaussian curve have been normalized to the single particle expansion and
have been plotted in Figure 5.6 (black data points).
To derive an analytic expression for the width of the BEC after ballistic expansion in dependance on the longitudinal 1d-density n1d a variational approach
has been performed: Assuming a 2d-harmonic oscillator potential the familiar
Gaussian shaped wave-function in obtained in the case of no interactions. Using
this test function with the harmonic oscillator lengths as variational parameters
the kinetic, potential, and interaction energy can been computed. Minimizing the
expression for the expansion energy (containing kinetic and interaction energy)
in dependence on the harmonic oscillator lengths yields
√
(5.7)
σ(t)/σho (t) = 4 1 + 4ascat n1d ,
where ascat is the three-dimensional s-wave scattering length, σ(t) is the transverse size of the BEC, and t is the time of expansion. This analytic expression has
previously been found by F. Gerbier [139] and shows excellent agreement with
a 2d GPE simulation. Fitting this equation to the data depicted in Figure 5.6
yields a value for the scattering length of ascat = 4.6 ± 0.2nm (blue curve). This
small error interval resulting from the fit is misleading because the main uncertainty results from the normalization of the data by σho (t). Here, the transverse
102
Exploring low-dimensional BECs
Figure 5.6: Transverse width of an expanded BEC normalized to the size of a
single particle expansion in dependence on
the longitudinal 1d-density. The red data
points have been taken from the experiments with a fragmented BEC whereas the
black data has been taken from the experiments with a homogeneous BEC. The
blue solid curve has been fitted to the data
according to equation 5.7 yielding ascat =
4.6 ± 0.2nm. The green dotted curve has
been plotted for ascat = 5.2nm.
trap frequency enters and changing its value results in a global shift of the data
depicted in Figure 5.6. If an uncertainty of 10% of the trap frequency is assumed,
an error of ∆ascat = 2nm can be estimated.
To further investigate the exact evolution of the shape of the expanded BEC
from a Gaussian curve to a parabolic shape a detailed analysis has been done: In
Figure 5.7(b-d) three transverse traces can be seen (gray data points). They have
been measured at three different longitudinal 1d-densities which are highlighted in
the longitudinal density profile shown in Figure 5.7a: 58, 152, and 248atoms/µm
have been marked by a red, green, and magenta open square. These longitudinal
densities correspond to the transverse traces depicted in Figure 5.7c, d, and b,
respectively. These profiles have been fitted by a Gaussian test-function (dashed
green curve) and by a parabolic test-function (dashed blue curve). Additionally,
the numerically obtained density profile (red curve) has been fitted to the data
and shows best agreement. The insets magnify the lower left edge and the top part
of the curves to allow for a better comparison. It can be seen that the almost
ideal Gaussian shape for the low density profile changes to a more parabolic
behavior when the density is increased. The true parabolic shape as expected for
a Thomas-Fermi condensate would be recovered for much higher densities than
investigated in these experiments.
The already mentioned experimental profiles have been fitted by numerically
calculated profiles covering the whole range from n1d = 1atom/µm up to n1d =
300atom/µm in steps of 25atom/µm. The root-mean-square deviations of the
residuals of these fits have been plotted in Figure 5.8left. The blue (black, and
red) data corresponds to a density of 248atoms/µm (152, and 58, respectively).
On the right the residuals of the fits are shown for three selected 1d-densities. It
can be clearly seen that best agreement to the experimental profiles is found for
matching 1d-density.
In conclusion, preparation of 1d-condensates in atom chip traps is possible
in a highly controllable and robust way. These atomic 1d-systems can be used
5.2 Cross-over between 1d and 3d BEC
103
Figure 5.7: (a) Longitudinal 1d-density of an expanded BEC. (b-d) For three densities
the corresponding transverse profiles are shown: 58 (c), 152 (d), and 248atoms/µm (b)
which have been highlighted by open squares (red, green, and magenta) in the longitudinal profile. The data shown in gray has been fitted by a Gaussian (dashed green
curve) and a parabolic (dashed blue curve) test function. Additionally the numerically
obtained profiles (solid red curve) have been fitted to the data. The insets magnify the
lower left edge and the top part of the distribution to allow for a better comparison.
Clearly the change from a Gaussian shape (c) to a parabolic shape (b) can be observed.
Note, that a true parabolic distribution expected in the Thomas-Fermi limit is obtained
only at much higher densities.
104
Exploring low-dimensional BECs
Figure 5.8: Numerically obtained transverse distributions have been fitted to three
measured transverse traces. The experimentally obtained profiles corresponded to a
longitudinal 1d-density of 58, 152, and 248atoms/µm (red, black, and blue curve, respectively). Left: The root-mean-square deviations of the residuals of these fits have
been plotted versus the 1d-density of the calculated profiles. It can be clearly seen that
best agreement to the experimental curves is found for matching 1d-density. Right: The
residuals of the fits of three calculated curves are shown for 50, 150, and 250atoms/µm.
In the top row (248atoms/µm) best agreement with the parabolic shape can be seen.
Being at the cross-over between 1d- and 3d-behavior the profile fitted in the middle
row (152atoms/µm) deviates from the Gaussian shape as well as from the parabolic
shape. In the lowest row (58atoms/µm) best agreement can hardly be seen for the low
density profile. But although the noise-level is fairly high, the rms-deviation in the left
plot reveals the expected behavior.
as an ideal starting point for further experiments – for example transport and
propagation in 1d-guides can be studied and compared to quantum transport in
electron systems [23]. Furthermore, issues of condensate growth and relaxation
can be investigated in detail since the precise manipulation of the longitudinal
magnetic potential of the trap is possible on a micron scale. This allows for example to add a steep and highly localized dip to the relaxed longitudinal potential
to study growth dynamics of a 1d-condensate [28, 29]. These experiments are
directly linked to growth of 3d-condensates which has been found to be not fully
understood [30]. Maybe effects which go beyond mean-field theory can be studied
this way.
5.3 Trapping geometries for 2d BECs
105
5.3 Trapping geometries for 2d BECs
Atom chips allow to generate a variety of differently shaped traps and guides
which are usually of 1d- or 3d-character. The construction of 2d-traps is not
obvious. Thus, a novel approach has been chosen: A red-detuned laser beam can
be reflected under a small angle (close to vertically) at the chip surface so that a
1d-optical lattice potential is formed. This stack of pancake shaped traps in the
vicinity of the atom chip can be used to experimentally investigate 2d-systems.
In this section a brief introduction to the chosen setup will be given. Using
thermal atoms loaded from a standard magnetic chip trap, the optical trap has
been characterized. Furthermore the diffraction of a BEC at the lattice potential
has been investigated and Bloch-oscillations have been observed. A detailed
summary of these experiments can be found in a separate publication [140].
5.3.1 Introduction to dipole traps
Optical dipole traps are used in many experiments and have enabled a lot of
exciting experiments – for example the formation of a molecular BEC [141].
Today these traps are a standard tool and have been discussed in several articles
(for a review see Grimm et al. [142]). Therefore only a brief introduction will be
given. The dipole potential Udip and the scattering rate Γscat are connected to
the laser intensity I by the following scaling laws:
Γ
I(r),
∆
µ ¶2
Γ
Γscat (r) ∼
I(r).
∆
Udip (r) ∼
(5.8)
(5.9)
Here Γ is the linewidth of the atomic transition and ∆ is the detuning from
the atomic resonance. It can be seen that a dipole trap is characterized by
two quantities: (1) The sign of the detuning determines whether the atoms are
attracted to regions of high field (red detuning, ∆ < 0) or the atoms are repelled
from regions of high field (blue detuning, ∆ > 0). Thus, the minima of a trap can
be formed in the region of high light intensity or at regions of low light intensity.
(2) The dipole potential scales as I/∆ whereas the scattering rate scales as I/∆2 .
Consequently, high light intensities at large detuning can be used to generate a
deep trapping potential and, at the same time, maintain small scattering rates.
Using this dipole force neutral atoms can be trapped in the focus of a single
red-detuned laser beam. Furthermore interfering one pair of laser beams leads
to the formation of a 1d-lattice potential. Higher dimensional lattices can be
generated by intersecting two or even three laser beam pairs. In our experiment
a single laser beam at a wavelength of λ = 782nm (detuning ∆ = 2nm) is reflected
under an angle of β ∼ 50mrad at the gold surface of the atom chip. This results
in an interference pattern up to a distance of ∼ 0.7mm from the chip surface.
106
Exploring low-dimensional BECs
Figure 5.9: Numerical simulation of the
intensity pattern of the reflection dipole
trap. A laser beam is reflected under
an angle of 50mrad at the gold surface
of the atom chip. A standing wave pattern is formed, generating a stack of pancake shaped traps. The lattice period in
the shown simulation has been exaggerated.
For light close to the Rubidium D2-line
(λ = 782nm) no lattice structure would be
visible on the depicted scale. At a beam
waist of 50µm and a laser intensity of 5mW
trap frequencies of ωtrans = 2π · 120kHz and
ωlong = 2π · 420Hz are achieved.
A numerical simulation of the resulting intensity pattern can be seen in Figure
5.9. Note, that the lattice period has been exaggerated in this picture. The exact
period d of this lattice is given by the angle of reflection and can be calculated by
d = λ/(2 cos β) ' 391nm. For a beam waist of 50µm at a laser intensity of 5mW
a potential depth of ∼ 50µK is obtained. The corresponding trap frequencies of
one trapping site closer than 200µm to the surface are ωtrans = 2π · 120kHz and
ωlong = 2π · 420Hz. Thus, the trap has a shape similar to a pancake providing
strong confinement in one direction.
5.3.2 Diffraction of a BEC from an optical lattice
To experimentally realize a dipole trap close to the atom chip surface a simple
laser setup has been used: A standard grating stabilized diode laser has been set
up to provide light at λ ∼ 782nm. This light has been passed through a heated
Rubidium gas cell to filter spontaneously emitted photons. To allow for fast
switching of the laser light an acusto-optic modulator (AOM) has been put into
the beam path, accompanied by a slow mechanical shutter entirely blocking the
light. This AOM could also be used for active intensity stabilization if needed.
This light is coupled into a single mode fiber and shine onto the atom chip from
below. Here the spot-size of the focussed laser beam has been measured to be
∼ 50µm by observing the stray light from the chip surface. This focus has been
adjusted onto the center of the 100µm-wide Z-shaped wire (C-D in Fig. 3.1).
The dipole trap has been loaded with thermal atoms at a temperature of ∼ 2µK
which have been pre-cooled by the standard experimental procedure (section 3.1).
During the final evaporative cooling stage in the chip trap, the dipole trap has
additionally been switched on and atoms could be loaded into the pure dipole
trap. In this trap a lifetime of 90 ± 17ms and a heating rate of 3.3 ± 0.9µK/s
5.3 Trapping geometries for 2d BECs
107
Figure 5.10: Transverse profiles of a
BEC for different pulse length of the
lattice potential have been combined to
one plot. It can be seen that 1st order
Bragg diffraction happens several times
if the pulse length of the standing light
potential is extended. A detailed description is given in the main text.
have been measured. This short lifetime and large heating rate are caused by the
near-resonant laser light used to generate the trap.
To exploit the 1d-optical lattice, diffraction of matter waves at this periodic
potential has been investigated. This is similar to experiments performed with
atomic beams at standing light fields [143] and Bragg spectroscopy on BECs [144].
For this experiment a BEC formed in a magnetic chip trap has been released from
the trap. While falling down due to gravity the optical lattice potential (wavevector k = 2π/λ) with a very low potential depth smaller than 1µK has been
switched on for a certain time tdiff . Thus, the potential depth is comparable to
the recoil energy of a single scattered photon (∼ 370nK in the case of Rubidium87) and Bloch-oscillations can be expected. Indeed these oscillations can be
seen in Figure 5.10. Every column of this plot corresponds to the longitudinal
profile of a BEC after ballistic expansion. These profiles have been obtained by
integrating the 2d-density distribution along the transverse direction. At tdiff = 0
the unperturbed BEC has fallen down for a distance of nearly 600µm. If the pulse
length tdiff of the light field is extended so that the BEC reaches the resonance
velocity vr while the lattice is switched on, Bragg-diffraction occurs. This happens
after ∼ 0.6ms of free fall at a velocity of vr = h̄k/m ∼ 5.9µm/ms. A momentum
of 2h̄k into the upward direction is transferred to the atoms. This can be seen in
the lower right plot in Figure 5.10 where the measured atomic density distribution
after time-of-flight imaging is shown (blue profile). Actually this distribution is a
momentum distribution and only atoms at medium velocity (located in the center
of this distribution) fulfill the resonance criterium and have been diffracted. The
distance between the dip in the broad distribution and the upper peak agrees
well with a momentum transfer of 2h̄k. If tdiff is further extended the atoms
can undergo 1st order diffraction several times (Fig. 5.10 upper right plot). The
theoretically expected position of the non-diffracted BEC (red profile) and of the
diffracted parts (green and yellow) are indicated by Gaussian profiles. The green
(yellow) profile corresponds to the position of a BEC which has been diffracted
once (twice).
108
Exploring low-dimensional BECs
Figure 5.11: (a) The Schrödinger equation has been simulated in the presence of a
periodic potential. A part of a freely falling
BEC is diffracted at this periodic potential. (b) If the potential is switched on for
a longer time atoms can undergo 1st order
diffraction a second time.
This behavior has been verified by a numerical simulation of the Schrödinger
equation in the presence of a periodic potential. In Figure 5.11a the vertical
position of a BEC has been plotted versus time. Since gravity points downwards
the typical parabolic curve expected for a free falling particle is recovered. Additionally a part of the BEC has been diffracted at the lattice potential which
has been switched on at t = 0 for a certain time-interval. If this time-interval
is extended atoms can undergo 1st order diffraction a second time as is shown
in Figure 5.11b. The experimental data depicted in Figure 5.10 has been taken
after a fixed time of expansion. This corresponds to a column at the right border
of the simulated diagrams. Qualitatively good agreement between theory and
experiment can be seen.
After this promising first experiments with dipole traps close to the surface of
an atom chip the laser setup is going to be upgraded: Using a Titanium-Sapphire
laser would allow for the same trap depth but for a factor of 100 less scattering
rate. This could be achieved if the detuning is enlarged to ∆ = 200nm and the
intensity is simultaneously increased by a factor of 100. This would enable to
hold atoms for a much longer time in the pure optical potential, thus allowing
to perform a final evaporative cooling step in the dipole trap itself. The goal
is to transfer ultra-cold thermal atoms or a BEC deterministically to a single
potential well of the lattice potential. As the separation of the potential wells is
given by λ/2 ∼ 400 − 500nm this seems to be possible since BECs in chip traps
with transverse ground state sizes smaller than 200nm can be routinely achieved.
This would allow to study the physics of a single 2d-system at a well defined atom
number in contrast to standard 1d-lattice experiments studying the properties of
a large number of wells. Additionally this pancake shaped trap can be structured
locally in the two relaxed directions by elements mounted on the atom chip – for
example by adding electro-static fields generated by chip structures.
6 Outlook: matter-wave
interferometry
In this thesis work one-dimensional (1d) Bose-Einstein condensates have been
investigated. Micro-traps have been successfully loaded which allow for the generation of very elongated Bose-Einstein condensates. Using these Bose-Einstein
condensates the cross-over from 1d Bose-Einstein condensates to 3d Bose-Einstein
condensates has been studied by observing changes in the momentum distribution
of the Bose-Einstein condensates. As a next step the phase along the longitudinal direction of these 1d Bose-Einstein condensates will be investigated and the
change of phase-coherence in the cross-over region to 3d will be studied. The
phase properties of such a system can be studied with a matter-wave interferometer.
Demonstrating a coherent beam-splitter for Bose-Einstein condensates as the
central building block of an integrated matter-wave interferometer has been a long
standing goal. Several schemes have been proposed and demonstrated [14, 15,
16, 17], none of which achieved coherent splitting. In momentum space coherent
splitting has been realized [145, 102] but for most applications spatially separated
paths are necessary. In this section a phase preserving beam-splitter in position
space will be presented [31]. Using adiabatic radio-frequency potentials a BoseEinstein condensate can be split over a wide range up to a distance of 80µm.
The interferometer cycle is completed by releasing the separated clouds from the
trap. From the interference pattern of the two overlapping density distributions
the contrast and the relative phase are recorded.
Here only a brief introduction will be given and the investigation of the phaseevolution along a one-dimensional Bose-Einstein condensate will be outlined. A
detailed characterization of the coherent splitting process itself [146] and future
experiments exploiting the rich capability of this interferometer [147] can be found
in separate publications.
6.1 RF-induced double-well potential
The wire geometry used for coherent splitting of a Bose-Einstein condensate
(BEC) is sketched in Figure 6.1. The BEC is trapped in an elongated magnetic
trap generated by a 50µm-wide Z-shaped wire (C-E in Fig. 3.1). A wire current
of 1.2A at an offset-field of 12G results in a transversal trap frequency of ωtr =
110
Outlook: matter-wave interferometry
Figure 6.1: An elongated BEC trapped
by a current-carrying wire can be rotated
above an independent wire carrying an alternating current. This additional current creates a rf-near-field which polarization is indicated by the black arrow. Due
to the different angle between rf-field and
local magnetic field of the trap a true
double-well potential (blue curve in inset) is only formed along the x-direction
whereas the potential along the y-direction
is just relaxed (black curve). The 2dquadrupole field of a side-guide configuration is sketched. Basic wire traps are discussed in section 2.2.2.
2π · 2.1kHz, corresponding to a ground state size of 230nm. The confinement in
the longitudinal direction is enhanced by pushing a small current through two
U-shaped wires (L-K and H-G in Fig. 3.1). This leads to an aspect-ratio of the
BEC of ∼ 400. This trap is positioned under an angle of 45 degrees directly
above a 10µm-wide wire (A-F in Fig. 3.1) at a distance of 80µm from the surface
of the atom chip. In this independent wire an alternating current of ∼ 60mA at
a frequency of ∼ 500kHz can be applied. This radio-frequency (rf) field couples
different magnetic sub-states similar to rf-induced evaporative cooling. But since
the rf-frequency is adjusted to be below the trap-bottom and the amplitude of
the rf-field is high enough to cause strong level-repulsion new adiabatic potentials
are formed [148, 149]. Additionally the angle between the rf-field and the local
trapping field has to be taken into account. Along the y-direction the rf-field is
always perpendicular to the magnetic field leading to a small relaxation of the
trapping potential (black potential shown in the inset). On the x-direction the
angle between rf-field and magnetic field varies resulting in a position-dependent
coupling strength. This way a true double-well potential can be formed along the
x-direction (blue potential shown in the inset).
6.2 Coherent splitting of a BEC
In this trapping geometry a BEC can be generated by the standard experimental
procedure discussed in section 3.1. To achieve splitting on a small distance (up
to 6µm) the amplitude of the rf-field is increased and the BEC is split smoothly
(adiabatically). By additionally increasing the rf-frequency to up to 4MHz the
splitting distance can be increased up to 80µm. To measure the distance between the two potential-wells the BECs are imaged in-situ (a description of the
6.2 Coherent splitting of a BEC
111
Figure 6.2: (a) By imaging the split condensates in-situ (insets) the double-well
separation has been measured. For three different trap configurations good agreement
to the theory (black curves) has been found. (b) After switching off the double-well
the BECs expand ballistically and overlap. The resulting interference pattern (insets)
can be used to extract phase and fringe spacing. The fringe spacing is connected to
the double well separation which is controlled by the rf-amplitude. For large splitting
(small fringe spacing) the data is well described by a two-point source model assuming
non-interacting particles (solid line). A numerical simulation of the Gross-Pitaevskii
equation describes the full data (dashed line).
imaging system can be found in section 2.1.3) along the longitudinal direction
(z-direction). The experimentally obtained trap separations for three different
trap-configurations are depicted in Figure 6.2a. Good agreement between data
(blue dots) and theory (black lines) can be seen. The insets shows the split BECs
for two different double-well separations.
After splitting the condensate the double-well potential is switched off abruptly.
After ballistic expansion for 14ms, the overlapping density distribution of the
BECs are imaged onto a CCD-camera, and a typical interference pattern is detected (see insets of Fig. 6.2b). This pattern contains information on the splitting distance d and on the relative phase Φ of the two condensates. By fitting
a Gaussian shaped envelope modified by a sinusoidal modulation the phase and
the fringe spacing ∆z can be extracted. This fringe spacing has been measured
over a wide range of rf-amplitudes (Fig. 6.2b). For large splitting and thus small
∆z the fringe spacing is given by ∆z = ht/md, where t is the expansion time and
m is the atomic mass. This equation has been plotted in the diagram (solid line)
and is only valid for large splitting since it assumes non-interacting particles expanding from a two-point source. For small splitting the inter-atomic interaction
has to be taken into account by a numerical simulation of the Gross-Pitaevskii
equation. The result of this calculation (dashed line) shows good agreement with
112
Outlook: matter-wave interferometry
Figure 6.3: Top: Differential phase
evolution of the split BECs has been
measured for four different experimental conditions. Sign and strength of
this phase evolution have been controlled by deliberately unbalancing the
double-well.
Bottom: The typical
phase spread shows a non-random behavior over the entire splitting process.
The dashed (solid) line represents the
limit for a deviation of three standard
deviations (one standard deviation).
the measured data.
The differential phase evolution of the split condensates has been measured
(Fig. 6.3top). The condensates have been separated within 20ms, corresponding
to an increasing distance between the wells. The phase and its spread have been
measured by repeating the experiment at the same settings up to 40 times. The
zero of the time axis (dashed gray line) is defined as the time at which the chemical
potential of the BEC equals the potential barrier between the two wells. For
smaller distances the relative phase is locked to zero whereas for larger distances
an evolution of the phase is observed. This phase evolution is in agreement
with a numerical simulation (dashed curves) for the experimental parameters.
The strength and the sign of this evolution can be controlled by deliberately
unbalancing the wells. The errorbars represent the statistical errors of the mean
value. In Figure 6.3bottom a typical distribution of the phase spread has been
plotted. A non-random spread is found over the entire splitting process. The
dashed black line indicates the limits for a deviation of three standard deviations
and the solid black line corresponds to one standard deviation.
6.3 Future experiments: probing the
phase-distribution of a 1d BEC
An integrated matter wave interferometer enables several interesting investigations. For example, the tunnelling regime can be investigated, similar to experiments which observed Josephson oscillations in optical lattice potentials [150].
Furthermore effects of the close-by room-temperature surface of the atom chip
can be studied by operating the interferometer at smaller distances to the surface.
Here, the fundamental question of decoherence induced by the surface could be
investigated.
6.3 Future experiments: probing the phase-distribution of a 1d BEC
113
Since one-dimensional BECs have been studied in this thesis work the investigation of these systems with an interferometer will be outlined. In the experiments
discussed above the coherence time has been found to be limited to ∼ 2ms. The
BECs which have been split up are of one-dimensional character. Therefore a
phase diffusion along the individual condensates can be expected after they have
been separated. Measuring the coherence time over the entire cross-over from 1d
to 3d would reveal details on the phase evolution in such systems. Experimentally the dimensionality of the BEC can be controlled by varying the density of
the BEC and additionally by changing the aspect-ratio of the trap. This latter
option can be realized by pushing currents through additional wires located at
the side of the elongated trap. As an example, a dip created in the longitudinal
trapping potential is depicted in Figure 3.3 and allows to vary the longitudinal
trap frequency over a wide range.
The varying phase along the condensates should show up in the interference
pattern of the expanded BECs if the density distribution is imaged perpendicular
to the chip surface. Here improvements of the imaging system on this axis are
needed to avoid stray light stemming from the wires which are directly located
underneath the condensates. In these pictures the change of the local phase
along the longitudinal direction of the BECs should show up as bending of the
interference strips. This allows to measure the phase profile along the elongated
BEC.
Furthermore, it would be interesting to study one-dimensional condensates at
a separation which still allows for tunnelling coupling of the two condensates.
In this regime two effects act into different direction: The phase fluctuations
along the condensates would cause a smearing out of the interference pattern. At
the same time the local coupling between the two condensates tents to lock the
relative phase. This regime has been investigated theoretically [151] and these two
competing effects have been found to strongly depend on the temperature of the
system. For low temperatures the local phase variation between the condensates
is expected to be small even in the presence of large phase fluctuations along the
longitudinal direction of the condensate.
7 Summary
This thesis work was focussed on the generation and manipulation of onedimensional Bose-Einstein condensates (BECs) in magnetic micro-traps generated by an atom chip. A new setup has been built to enable the formation of
such elongated BECs. In this apparatus the cross-over from three-dimensional
BECs to one-dimensional BECs has been studied in detail. As an application of
one-dimensional BECs a highly sensitive microscopic magnetic field sensor has
been invented.
To form one-dimensional BECs in a robust and stable way, a new experimental
setup has been build. This apparatus incorporates a novel wire-based magnetooptical trap (MOT) which allows to trap up to 3 · 108 Rubidium-87 atoms a few
millimeters above the reflecting surface of the atom chip. Due to this integratedMOT the need of external coils has been reduced to a minimal amount. Thus,
it was possible to optimize the design of the vacuum chamber to enable good
optical access to the experimental site.
Using this setup BECs of up to 105 atoms have been produced in various microtraps. These chip traps are very anisotropic: They offer strong confinement in
the two transverse directions whereas the longitudinal confinement is weak, thus
elongated BECs can be generated in these traps with aspect-ratios up to several
thousand. Using these BECs the cross-over from the one-dimensional to the threedimensional Thomas-Fermi regime has been probed by monitoring the transverse
size of the BEC after ballistic expansion for changing density. Good agreement to
theory has been found. Furthermore, the shape of the expanded BEC has been
compared to the shape obtained by a simulation of the two-dimensional GrossPitaevskii equation. The transition from a Gaussian shape to a parabolic shape
has clearly been observed. First experiments aiming at the formation of twodimensional traps close to the atom chip by integrating an optical lattice potential
into the setup have been started and Bloch-oscillations have been observed.
As an application of one-dimensional BECs a highly sensitive magnetic field
sensor has been build. This sensor enables field measurements in a region which
is not accessible for today’s state-of-the-art magnetic field sensors. A field sensitivity of 4nT at a spatial resolution of 3µm has been demonstrated. This sensor allows to probe magnetic field variations ∆B even in the presence of large
offset-fields B up to ∆B/B = 5 · 10−5 . Therefore it is ideally suited to probe the
magnetic field of a current-carrying wire: A two-dimensional map of the magnetic
field at a distance of 10µm above a 100µm-wide wire has been recorded. From
this potential landscape the current-density in the wire has been reconstructed.
116
Summary
This is in particular interesting in the context of disorder potentials which have
led to severe limitations in previous atom chip experiments performed by several
other groups. These disorder potential are caused by current-flow deviations inside the wire from a straight flow. With the novel magnetic field sensor the origin
of these current-flow deviations can be probed in detail in future experiments.
In the experiments discussed in this thesis the momentum distribution of onedimensional BECs in the cross-over region to three-dimensional behavior has been
studied. As a next step the phase-properties of the one-dimensional BECs will
be investigated. In this context a coherent beam-splitter for one-dimensional
BECs has been demonstrated. Based on this beam-splitter an interferometer on
the atom chip has been achieved. This is ideally suited to further study the
cross-over from one-dimensional to three-dimensional BECs and in particular to
investigate the phase-evolution along the longitudinal direction of the elongated
condensates.
A D2-line of Rubidium
A basic technique needed for the stabilization of the frequency of a laser is the
Doppler-free absorption spectroscopy [38]. Using the laser setup of the TA-100
which has been discussed in section 2.1.1, the spectrum of the D2-line of Rubidium
has been measured (Fig. A.1). The Doppler-valleys of both Rubidium isotopes
can be seen. Two of those (Rubidium-87) have been magnified and are depicted
in Figure A.2. The hyperfine state F 0 of the upper level of the transition has
been added to the transmission peaks. Additionally the cross-over peaks have
been labelled (compare to Fig. 2.2).
A variety of information on the Rubidium D2-line has been up together by
D. A. Steck. This compilation is available online [37].
Figure A.1: Frequency scan over the entire absorption spectrum of the D2-line of both
Rubidium isotopes. The broad Doppler-valleys correspond to F = 2 → F 0 of Rubidium87, F = 3 → F 0 of Rubidium-85, F = 2 → F 0 of Rubidium-85, and F = 1 → F 0 of
Rubidium-87 (from left to right).
118
D2-line of Rubidium
Figure A.2: The cooling (left) and the repumping (right) transitions used to laser-cool
Rubidium-87 have been monitored in detail. The hyperfine state F 0 of the upper level
of the transition has been added to the transmission peaks (dotted lines). Additionally
the cross-over peaks have been labelled.
B List of publications
In the framework of this PhD-thesis and the preceding diploma-thesis the following articles have been published:
• T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth,
I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, Matter-wave interferometry
in a double well on an atom chip. Nature Physics 1, 57 (2005).
• B. Zhang, C. Henkel, E. Haller, S. Wildermuth, S. Hofferberth, P. Krüger,
and J. Schmiedmayer, Relevance of sub-surface chip layers for the lifetime
of magnetically trapped atoms. Euro. Phys. J. D 35, 97 (2005).
• K. Brugger, P. Krüger, X. Luo, S. Wildermuth, H. Gimpel, M. W. Klein,
S. Groth, R. Folman, I. Bar-Joseph, and J. Schmiedmayer, Two-wire guides
and traps with vertical bias fields on atom chips. Phys. Rev. A 72, 023607
(2005).
• S. Wildermuth, S. Hofferberth, I. Lesanovsky, E. Haller, L. M. Andersson, S. Groth, I. Bar-Joseph, P. Krüger, and J. Schmiedmayer, Microscopic
magnetic-field imaging. Nature 435, 440 (2005).
• P. Krüger, L. M. Andersson, S. Wildermuth, S. Hofferberth, E. Haller,
S. Aigner, S. Groth, I. Bar-Joseph, and J. Schmiedmayer, Disorder potentials near lithographically fabricated atom chips. arXiv:cond-mat/0504686
(2004).
• S. Wildermuth, P. Krüger, C. Becker, M. Brajdic, S. Haupt, A. Kasper,
R. Folman, and S. Schmiedmayer, Optimized magneto-optical trap for experiments with ultracold atoms near surfaces. Phys. Rev. A, 69 030901(R)
(2004).
• X. Luo, P. Krüger, K. Brugger, S. Wildermuth, H. Gimpel, M. W. Klein,
S. Groth, R. Folman, I. Bar-Joseph, and J. Schmiedmayer, Atom fiber for
omnidirectional guiding of cold neutral atoms. Opt. Lett. 29, 2145 (2004).
• S. Groth, P. Krüger, S. Wildermuth, R. Folman, T. Fernholz, J. Schmiedmayer, D. Mahalu, and I. Bar-Joseph, Atom chips: fabrication and thermal
properies. Appl. Phys. Lett. 85, 2980 (2004).
120
List of publications
• P. Krüger, X. Luo, M. W. Klein, K. Brugger, A. Haase, S. Wildermuth,
S. Groth, I. Bar-Joseph, R. Folman, and J. Schmiedmayer, Trapping and
manipulating neutral atoms with electrostatic field. Phys. Rev. Lett. 91,
233201 (2003).
Currently two manuscripts are being prepared: The first manuscript describes the
details of imaging of atoms close to a reflecting surface. The second manuscript
summarizes the reconstruction of the current density in a micro-fabricated wire
described in section 4.3.
C Acknowledgment
Biorhythms and Work-Group Dynamics: An experimental Approach
K. Adamski,1 L.M. Andersson,1 I. Bar-Joseph,2 C. Becker,1 M. Brajdic,1 K. Brugger,1 R. Folman,1
D. Gallego Garcia,1 H. Gimpel,1 M.&M. Grauli,1 S. Groth,1 A. Haase,1 E. Haller,1 S. Haupt,1
B. Hessmo,1 S. Hofferberth,1 P. Krüger,1 I. Lesanovsky,1 A. Merkelbach,1 J. Rottmann,1
S. Schneider,1 T. Schumm,1 J.L. Verdú,1 D.&G. Wildermuth,1 S. Wildermuth,1 and J. Schmiedmayer1
1
2
Physikalisches Institut, Universität Heidelberg, 69120 Heidelberg, Germany
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: November 4, 2005)
The study of biorhythms is over 80 years old, with physical, intellectual, and emotional biorhythm
having been discovered along the way. There is a multitude of services and software being offered
with regards to biorhythms, like biorhythm charting, reading, forecast, the list is nearly endless.
Almost all of this information is based on personal biorhythm cycles. Although we consider these
personal services useful, that is not what this article is about. It is about what type of effect
biorhythms have on a group of people, or a team of physicists, all of whom have differing biorhythmic
cycles. We show how work-group dynamics and team-dynamics change when one adds or removes
certain members with unique properties. Furthermore we demonstrate that some people simply
don’t get along with each other, all due to biorythmic incompatibility! We present a scheme how
these problems can be overcome by biorythmic matching. So remember, teamwork is not an accident,
it is careful planning. That’s why it is called team-building!
Of course, this thesis would not have been possible
without the help of several people. This covers all aspects, be it work on the apparatus in the lab, discussion
of physics problems, or recreational activity. In the following I want to properly thank all of them:
• First of all, I thank the diploma-students who
helped in building and running the experiment:
Hartmut, Christiane, Sebastian, Mihael, Sebastian,
Elmar, Daniel, and Jörg.
• I thank Jörg for sharing his great insight into
physics with me and for all his inspiring ideas.
• Thanks to my parents who gave a model railway to
me. Thus, enabling my early success in building a
truly experimental setup. I’m kidding. Thank you,
for just being good parents!
• Thanks to Anne for shaking the cocktails and being
my lovely girlfriend.
• Hey Igor, thank you for helping out in case of my
limited knowledge of theory and numerics.
• Thanks to the two Post-Docs Ron and Mauritz who
I had the honor to work with.
• Thanks to the Israel-connection at the Weizmann
institute for (possibly) building the best atom chips
on the entire planet. Sönke, the master of a subtle
trade.
• I thank Simone for always keeping me in her mind.
• I thank the former PhD-students in our group
which taught me all the basics of vacuum, optics,
electronics, laser-cooling, partying, ... especially
Dr. Stephan Schneider, Dr. Karolina Brugger, and
Dr. Zapatisto Haase.
• Thanks to home-cinema Grauli for the opportunity
to watch very different movies ranging from unbelievably stupid to highly exciting.
• I thank Peter for friendship and four years of successful teamwork in the lab and beyond. Or may
it already be five years? My god, we have become
old!
• I thank the Feinmechanik Werkstatt of the
Physikalisches Institut for building the marvellous
vacuum chamber and several other components of
the experiment.
• I thank the Elektronikwerkstatt of the Physikalisches Institut which built all the laser-drivers and
a lot of electronics. Special thanks to Klaus Layer
the master of the 60A-switches.
• I thank Kai Adamski for standing in as first author
although contributing nothing.
• Thanks to Sebastian, Hartmut, Peter, Björn, José,
and Jörg for carefully proof reading parts of the
manuscript of this thesis.
• Special thanks to Sebastian who worked a lot to
run, maintain, and repair the experiment in the
last years.
• Merci beaucoup, Monsieur Schumm, for running
the experiment day and night and additionally at
the weekend. Truly, he is the only one who coherently splits a pitcher into several beer glasses.
Last but not least, I thank the Landesgraduiertenförderung of the Universität Heidelberg for financial support
by means of a scholarship.
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