Fachbereich
Physik
Momentum-Resolved Optical Lattice Modulation
Spectroscopy on Bose-Fermi Mixtures
Impulsaufgelöste Modulations-Spektroskopie an
Bose-Fermi-Mischungen in optischen Gittern
Diploma Thesis
Bastian Hundt
Universität Hamburg
MIN-Fakultät, Department Physik
Institut für Laser-Physik
July 2011
Abstract
Research on ultracold quantum gases has developed into one of the most dynamic fields
in physics. Breakthroughs on a multitude of topics occur at daily basis. A cornerstone of
this process was the development of optical lattices. The high degree of control over the
experimental parameters makes ultracold quantum gases in optical lattices an ideal tool
to simulate fundamental theories of solid state physics. But also fundamental processes of
many-body physics are studied extensively. A central role is played by quantum phases and
quantum phase transitions. Interaction and tunneling governs the behavior of the systems of
interest. Momentum resolved spectroscopy on fermionic, bosonis and Bose-Fermi mixtures
gives access to the underlying physics with high precision.
A novel spectroscopy method has been developed at the “Bose-Fermi mixtures” experiment
in the group of K. Sengstock in Hamburg. In the context of this thesis, the method based
on modulating the lattice depth has been developed and characterized. The band structure
of ultracold fermionic Potassium is probed with high accuracy. A significant influence of
the harmonic confinement altering the band structure is observed and the lattice depth
experienced by the atoms can be deduced very precisely. This allows the calculation of
fermionic tunneling energies. The method has then been employed on mixtures of fermions
and bosons. A central result of this thesis is the observation of a shift in the lattice depth
experienced by fermions of up to 20%. This corresponds to a decrease in fermionic tunneling
of up to 30%. The decrease depends on the bosonic occupation of the lattice sites and is
explained in terms of an effective potential created by attractive interactions.
The developed method and the obtained data can be helpful to further understand the
bosonic Mott-insulator to superfluid transition. Employing the method on interacting
fermionic mixtures could lead to the observation of new phenomena. Publication of the
results is in preparation.
Observing such new phenomena requires efficient and reliable detection of the atomic ensembles. To assure high quality absorption detection, this thesis presents a new optical
design for the detection setup used at the “Bose-Fermi mixtures” experiment. The new
design is developed using state of the art optical design software. Using optimization algorithms, a diffraction limited lens system, subject to very small optical errors is designed,
tested and integrated into the experimental setup.
Zusammenfassung
Die Forschung an ultrakalten Quantengasen hat sich in den letzten Jahren zu einem der dynamischten Forschungsfelder der Physik entwickelt. Dazu beigetragen hat im großen Maße
die Realisierung von ultrakalten Quantengasen in optischen Gittern. Die außergewöhnlich
gute Kontrolle über eine Vielzahl physikalischer Parameter macht sie zu einem perfekten
Werkzeug zur Simulation sowohl festkörpertheoretischer Fragestellungen als auch fundamentaler Fragen der Physik von stark wechselwirkenden Vielteilchensystemen. Von zentraler
Bedeutung sind die auftretenden Phasen und Phasenübergänge. Diese werden sowohl in rein
bosonischen oder rein fermionischen Systemen als auch in Bose-Fermi Mischungen durch
die Wechselwirkungs- und Tunnelenergie bestimmt. Impulsaufgelöste Spektroskopie an solchen Systemen erlaubt dabei einen hochpräzisen Zugang. Am „Bose-Fermi-Mischungs“Experiment in der Gruppe von K. Sengstock wurde ein neuartiges, spektroskopisches Verfahren entwickelt, das bei voller Impulsauflösung das Anregungsspektrum fermionischer
Atome in optischen Gittern untersucht.
Im Rahmen dieser Arbeit wurde ein Verfahren basierend auf der Modulation der Gittertiefe entwickelt und charakterisiert. Daraufhin konnte die Bandstruktur von fermionischem
Kalium in optischen Gittern mit hoher Präzision vermessen werden. Es wurden Daten bis
einschließlich des vierten Bandes gewonnen. Dabei konnten signifikante Einflüsse des harmonischen Einschlusses auf die Bandstruktur beobachtet werden. Aus den vermessenen
Bandstrukturen ist die Gittertiefe präzise abgeleitet worden. Dies erlaubt auf der einen
Seite eine genaue Kalibrierung des optischen Gitters, auf der anderen Seite eine exakte Bestimmung der Tunnelenergie. Die entwickelte Methode wurde anschließend auf Mischungen
von Fermionen und Bosonen angewendet. Als zentrales Ergebnis dieser Arbeit wurde eine
Mischungsverhältnis abhängige Zunahme der fermionischen Gittertiefe beobachtet, die im
Rahmen eines effektive Potentials erklärt werden kann. Für ein hohes Mischungsverhältnis
zwischen Rubidium und Kalium wurde dabei eine Abnahme der fermionischen Tunnelenergie von 30% ermittelt.
Die entwickelte Methode und die durchgeführten Messungen könnten sich beim Verständnis
des bosonischen Mott-isolator zu superfluid Phasenübergangs in Bose-Fermi Mischungen als
wertvoll erweisen. Die Anwendung auf wechselwirkende fermionische Systeme verspricht die
Beobachtung neuartiger Phänomene. Eine Veröffentlichung der gewonnenen Ergebnisse ist
in Vorbereitung.
Im Rahmen dieser Arbeit wurde darüber hinausgehend ein neuer optischer Aufbau für die
am Experiment verwendete Detektion mittels Absorptionsmessung entwickelt und implementiert. Dabei wurde das vorhandene Linsensystem durch einen neuen optischen Aufbau
ersetzt, der sich durch eine starke Reduktion der Abbildungsfehler und ein hohes Maß an
Flexibilität auszeichnet.
Referenten
Referent:
Prof. Dr. Klaus Sengstock
Universität Hamburg
Fakultät für Mathematik, Informatik und Naturwissenschaften
Department Physik – Institut für Laser-Physik
„Quantengase und Spektroskopie“
Koreferent:
Prof. Dr. Henning Moritz
Universität Hamburg
Fakultät für Mathematik, Informatik und Naturwissenschaften
Department Physik – Institut für Laser-Physik
„Quantenmaterie“
Erklärung zur Eigenständigkeit
Ich versichere hiermit, dass ich die Diplomarbeit ohne fremde Hilfe selbstständig verfasst
und nur die angegebenen Quellen und Hilfsmittel benutzt habe.
Mit einer späteren Ausleihe meiner Arbeit bin ich einverstanden.
Hamburg, den 05. Juli 2011
Bastian Hundt
Contents
1 Introduction
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
2.1 Experimental Setup and Capabilities . . . . . . . . .
2.2 Optical Lattice Potentials . . . . . . . . . . . . . . .
2.3 Non-Interacting Atoms in Optical Lattice Potentials
2.4 Ultracold Bosons in Optical Lattices . . . . . . . . .
2.5 Ultracold Fermions in Optical Lattices . . . . . . . .
2.6 Band Mapping . . . . . . . . . . . . . . . . . . . . .
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5
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15
3 Lattice Modulation Spectroscopy
3.1 Momentum Resolved Lattice Modulation Spectroscopy
3.2 Chracterization of Modulating the Lattice Amplitude .
3.3 Band Structure of Spin-Polarized Fermions . . . . . .
3.4 Spectroscopy on Pure Bosonic Samples . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
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19
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34
35
4 Lattice Modulation Spectroscopy with Bose-Fermi Mixtures
4.1 Ultracold Mixtures of Bosons and Fermions . . . . . . . . . . . . . . . . . .
4.2 Lattice Modulation Spectroscopy on Bose-Fermi Mixtures . . . . . . . . . .
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
37
38
41
5 Absorption Imaging with a Diffraction Limited Objective
5.1 Absorption Imaging . . . . . . . . . . . . . . . . . . .
5.2 Image Formation, Resolution and Abberrations . . . .
5.3 Raytracing and Computer Aided Optical Engineering
5.4 Old Detection Setup . . . . . . . . . . . . . . . . . . .
5.5 Design of a New Diffraction Limited Detection System
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
43
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6 Conclusion and Outlook
69
List of Figures
71
List of Tables
72
Bibliography
77
1
1 Introduction
A major breakthrough of our understanding of nature was the development of quantum
mechanics at the beginning of the 20th century and following this the formulation of different
types of quantum-statistics: Fermi-Dirac statistics for particles subject to Pauli’s exclusion
principle and Bose-Einstein statistics describing bosonic particles. The latter is the basis of
A. Einstein’s proposal of Bose-Einstein condensation from 1924 [1], which has been realized
in the Groups of E.A. Cornell [2] and W. Ketterle [3] in 1995. The first realization of a
macroscopic, many-body quantum state allowed unprecedented insights into fundamental
processes of nature and was awarded with the Nobel Prize in Physics 2001.
With the realization of a quantum degenerate Bose gas also the interest in ultracold gases
of fermions arose. But due to technical and physical complications it took until 1999 to
realize a quantum degenerate Fermi sea. The breakthrough for reaching bosonic quantum
degeneracy has been evaporative cooling. Due to Pauli blocking this technique could not be
used to prepare a spin-polarized sample of fermions. The first quantum degenerate sample
of Potassium was realized with a spin mixture in 1999 by B. DeMarco and D. S. Jin [4].
Using Feshbach resonances [5] the BEC-BCS crossover could be realized [6] triggering the
hope for a deeper understanding of high-temperature superconductivity.
Advances in preparation and the high degree of control over ultracold quantum gases led to
proposals for using these gases as a tool to simulate other physical systems. Most notably the
development of optical lattices allowed for the realization of the Hubbard Hamiltonian which
plays a crucial role in solid-state theory [7]. A major breakthrough was the observation of
the bosonic Mott insulator to superfluid phase transition in 2002 [8], where the transition
from a strongly interacting and localized state to a weakly interacting delocalized state was
realized. The recent simulation of classical magnetism [9] and quantum magnetism [10]
in such systems showcases again the remarkable possibilities offered by quantum gases.
Fermions in optical lattices are exceptionally well suited for simulating solid-state physics
due to the resemblance to electrons in a crystal. Such a system was realized in 2005 and
led to the observation of a crossover from a metallic phase to a band insulator [11]. In 2008
a fermionic Mott-insulator was observed using a spin-mixture of 40 K [12, 13]. The search
for antiferromagnetic ordering of fermionic quantum gases in optical lattices is ongoing.
With the realization of single species quantum gases it soon was realized that mixtures
between different species offer new research possibilities. Especially the mixture of atoms
obeying different quantum statistics led to new phenomena governed by intra- and interspecies interactionss. Using Bose-Fermi mixtures, it was possible to observe a collapse of a
fermionic gas induced by bosons [14]. The creation of ultracold heteronuclear molecules [15]
with a permanent dipole moment which are subject to long range, anisotropic interactions [16] and give access to ultracold chemistry [17]. Recently overlapping fermionic and
2
1 Introduction
bosonic Mott-insulators were prepared, showing - among other phenomena - e.g. a complete
phase separation [18].
During the first experiments using mixtures of bosons and fermions in optical lattices, a
unexpected effect was observed. While examining the bosonic Mott-insulator to superfluid
phase transition an admixture of fermions led to a decrease of the bosonic visibility [19–21].
This effect has been studied with great interest from theoretical and experimental side. It
has been argued that adiabatic heating [22] results in a loss of coherence. Another model
explains the visibility shift in terms of interaction-induced self-trapping of the bosons [23].
Until now it could not be decided which effect leads to the visibility shift.
50
pure fermions
NRb / NK = 2
Emod / h [kHz]
45
NRb / NK = 10
40
7.6 EKr
35
8.1 EKr
9.2 EKr
0
0.2
0.4
0.6
0.8
1
q [kBZ]
Figure 1.1: Fermionic dispersion relation of the third band with an admixture of bosons. A
mixture of 40 K and 87 Rb has been prepared in an optical lattice. Modulation spectroscopy yields the
dispersion relation of fermions. In red the pure fermionic band structure is shown. Green shows a
moderate amount of bosons added to the system. Blue depicts the case of high admixtures of bosons.
To gain further insight into the interplay between fermions and bosons, and to possibly understand the phase transition better, it is necessary to probe the tunneling rate of ultracold
atoms exposed to a background field of another species. A tool to probe and access the
underlying physical system are spectroscopical measurements.
In the context of this thesis a novel spectroscopy method was developed at the “Bose-Fermi
mixtures” experiment in Hamburg in the group of K. Sengstock. Employed on fermionic
samples as well as Bose-Fermi mixtures in an optical lattice, it allows for momentum resolved
spectroscopy. Based on modulation of the lattice depth, the method allows for a precise
energy transfer to the atomic ensemble which excites atoms into higher bands. The occupation of the different bands is revealed applying the band mapping technique, allowing for
the extraction of the full band structure. During the course of this thesis lattice modulation
spectroscopy was characterized and employed on a mixture of fermionic Potassium-40 and
bosonic Rubidium-87. As a central result the response of the system to lattice modulation
with different atom number ratios is obtained. For large bosonic fillings a substantial shift
3
of the band structure is observed (see Fig. 1.1) which can be explained within an effective
potential approach. This shift results in a decrease of the fermionic tunneling energy of up
to 30% and depends on the bosonic lattice occupation.
Structure of this thesis
The second chapter of this thesis starts with an introduction to the experiment. After the
description of creation and preparation of ultracold atoms in optical lattice potentials, the
effect of periodic potentials on non-interacting particles is reviewed. The inclusion of interactions leads to the description of bosons in optical lattices in terms of the Bose-Hubbard
model. Fermions can be described by a similar model. The knowledge of the behavior of
fermions in an optical lattice allows us to characterize the band mapping technique which
is of fundamental importance for the spectroscopy method introduced in the next chapter.
The third chapter describes the method of lattice modulation. Starting with a general
introduction we will obtain expressions for the transition probabilities between the lowest
Bloch-band and higher bands. The method is further characterized to ensure accuracy
and the avoidance of beyond linear-response effects. Thereafter spin-polarized fermions
are studied. Obtaining the band structure for various lattice depths the influence of the
harmonic confinement is studied. As a last step, bosons are examined in this context.
Chapter four concentrates on a mixture of bosons and fermions. The mixture is described
theoretically and an interaction-induced effective potential mechanism is introduced. Lattice modulation spectroscopy is employed and the band structure for different atom number
ratios is measured.
The fifth chapter concentrates on a more technical aspect of the experimental sequence
which is absorption detection. First, absorption imaging after time of flight (TOF) is
introduced. Then the process of image formation is explained and the errors, introduced
by optical elements in the optical path of the detection laser, are explained. A new optical
setup specifically engineered to minimize these errors is designed and evaluated.
At the end of this thesis, proposals for further applications of the introduced spectroscopy
method are given.
4
1 Introduction
5
2 Ultracold Bose-Fermi Mixtures in Optical
Lattices
2.1 Experimental Setup and Capabilities
All the experiments performed in the context of this thesis have been carried out at the
“Bose-Fermi mixture” experiment in the group of Klaus Sengstock at the University of
Hamburg. A sketch of the setup is shown in Fig. 2.1.
2D-MOT
Pumping Stage &
Vacuum System
LA
3D-MOT
D2
Figure 2.1: Experimental setup. The 3D-magneto-optical-trap (MOT) is loaded from a 2D-MOT
which are connected via an differential pumping stage separating the upper vacuum system from the
lower. The MOT/lattice laser beams denoted by D1, D2 and LA are tilted with respect to the detection
axis.
The experimental setup allows the trapping and preparation of ultracold mixtures of fermionic
6
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
40 K
and bosonic 87 Rb [24, 25]. The atoms are loaded from a 2D magneto optical trap (MOT)
into a 3D-MOT. After cooling in an optical molasses and preparing the ensemble in one
hyperfine state the atoms are transferred to a magnetic trap where evaporative cooling
is performed. Potassium is sympathetically cooled due to collisions with Rubidium. The
mixture is then loaded into an optical dipole trap which operates at the magic wavelength
(808 nm) to compensate for different gravitational sags. The atoms reach quantum degeneracy where 87 Rb forms a Bose-Einstein-Condensate (BEC) and 40 K forms a Fermi-sea while
reaching temperatures of T ≈ 0.1Tf where Tf is the Fermi-temperature. Depending on the
experiments to conduct, one species can be removed by shining in resonant laser light for
a short amount of time. Using radiofrequency (rf) or microwave pulses a high degree of
control over the internal atomic states is possible. The interactions between the atoms can
be tuned using intra- or interspecies Feshbach resonances. A cubic optical lattice can be
superimposed. The lattice is generated by using three orthogonal pairs of laser beams denoted in Fig. 2.1 with D1, D2 and LA and operated at a wavelength of λL = 1030 nm. The
radius of the atoms trapped is to small to be detected directly thus the atoms are released
from all trapping potentials and allowed to expand freely. By shinning in resonant light a
shadow of the density of the cloud is imaged onto a charge coupled device (CCD) camera.
From the imaged shadow the density distribution of the expanded atoms is derived. This
allows extraction of momentum distributions, correlation analysis etc.
2.2 Optical Lattice Potentials
Atoms exposed to a laser field which is not on resonance with an atomic transition will
experience a dipole force due to the coupling between the atomic energy levels and the
photon field of the laser beam. A red detuning of the laser ωL with respect to the atomic
resonance ω0 results in a force driving the atoms to the points of maximum intensity of the
laser light. Blue detuning will push the atoms to the intensity minima.
An optical lattice is generated by standing light waves formed by counter-propagating laser
beams with the same frequency. For red detuning the resulting standing wave confines
the atoms at the intensity maxima which occur at alat = λL /2. Three pairs of counterpropagating beams orthogonally aligned will result in a 3D-cubic-lattice. The resulting
potential around the intersection of the beams can be expressed as (see [26])
V (x, y, z) = Vlat (x, y, z) + Vharm (x, y, z)
≈ V0 cos2 (kx x) + cos2 (ky y) + cos2 (kz z)
(2.1)
m 2 2
2 2
2 2
+
(ω x + ωy y + ωz z ).
2 x
V0 is the potential depth which is conveniently expressed in terms of the recoil energy
Er = ~2 kL2 /2m. V0 is proportional to the power of the laser beams which allows a direct
manipulation of the lattice depth in the experiment. The second term in (2.1) represents the
additional harmonic confinement due to the spatial intensity variation of the laser beams.
Instead of using all three pairs of beams, just overlapping two pairs or one pair will result
in a 2D-lattice or 1D-lattice. These different lattice geometries are depicted in Fig. 2.2. At
2.3 Non-Interacting Atoms in Optical Lattice Potentials
7
Figure 2.2: Different Lattice Geometries. By using different geometries and multiple numbers of
counter propagating laser beams it is possible to create a multitude of lattice geometries in various
dimensions. Shown here is the case of pairs of counter-propagating beams creating confining lattice in
1D (pancake structure), 2D (tube structure) and 3D (cubic lattice structure).
the Bose-Fermi mixtures project we use a laser which is far red detuned with respect to the
atomic transitions which are at 767 nm for 40 K and 780 nm for 87 Rb.
2.3 Non-Interacting Atoms in Optical Lattice Potentials
Considering optical lattices the question arises which eigenstates and energies will be possessed by non-interacting particles. This problem can be solved in a similar way in which
the problem of electrons in a periodic potential created by ions in solid-state-physics is
solved (see [27–29]). Considering the cubic lattice potential (2.1) without the additional
harmonic confinement, inserting this potential into the time independent Schrödinger equation and making use of the fact that the potential is separable with respect to its spatial
coordinates, yields a one dimensional equation of the following form
!
p̂2
+ Vlat (z) ψ(z) = Eψ(z).
2m
(2.2)
Bloch’s theorem [27] states that the eigenstates of a particle in a periodic potential can be
written as
ψq(n) (z) = eiqz u(n)
q (z)
(2.3)
8
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
(n)
where uq (z) has the periodicity of the underlying lattice. The index n is called band index
and q is the quasimomentum. The quasimomentum q is restricted to the first Brillouin zone
−kBZ < q ≤ kBZ where R = 2π/alat = 2kL = 2kBZ is the reciprocal lattice vector. Because
Figure 2.3: First Brioullin Zone of a Cubic Lattice. Here the first Brillouin zone of a square lattice
geometry in reciprocal space is shown. The blue dots represent the atoms in reciprocal space.
the potential as well as the eigenstates are periodic with respect to the lattice spacing alat
a Fourier expansion is intuitive:
ψq(n) (z) = eiqz
X (n)
cK,q eiKz
K
Vlat (z) =
X
vK eiKz .
K
(2.4)
(2.5)
Equation (2.4) is called a Bloch state and is denoted by |n, qi in Dirac notation. The index
K in the sums has to be considered for all available reciprocal lattice vectors. Inserting
equations (2.4) and (2.5) into (2.2) will result in a eigenvalue problem of the form (see [27])
!
X
~2
(n)
(n)
(q + K)2 − Eq(n) cK,q +
vK 0 −K cK 0 ,q = 0.
2m
K0
(2.6)
The potential is proportional to cos2 (kz z) which can be rewritten to
cos2 (kz z) =
1 2ikz z
e
+ e−2ikz z + 2 .
4
(2.7)
The only nonzero contributions of the Fourier expansion (2.5) are thus v0 = −V0 /2 and
v1 = v−1 = −V0 /4 (see [28, 29]). Using this result and neglecting the v0 contribution
equation (2.6) becomes
4Er
V0 (n)
V0 (n)
(n)
(n)
(q + K)2 cK,q −
cK−1,q −
cK+1,q = Eq(n) cK,q .
2
R
4
4
(2.8)
This equation can be solved numerically by restricting the calculation to a finite amount of
different K. All calculations in this thesis are done with K = [−10R, 10R] and assume the
same intensity of the laser beams forming the periodic potential. This calculation yields
2.3 Non-Interacting Atoms in Optical Lattice Potentials
9
V0 = 5ErK ~12.5ErRb
60
60
K
87
n=4
50
50
40
40
n=5
n=4
n=3
30
Rb
30
(n)
Eq / h [kHz]
40
20
n=3
20
n=2
n=2
10
10
n=1
0
−1
0
1
n=1
0
−1
0
q [kBZ]
1
q [kBZ]
Figure 2.4: Band structure for 40 K and 87 Rb. These band structures are calculated for the same
intensity of the laser beams which form the periodic potential. Because of the larger mass of 87 Rb and
the different wavelengths of the atomic transitions 87 Rb has a smaller recoil energy. The gray shaded
areas depict the energies where the bands of both species overlap.
(n)
values for the energies Eq
(n)
as well as the Bloch coefficients cK,q which will be used later to
(n)
calculate the transitions probabilities between different bands. Plotting the energies Eq
for different quasimomenta q and different band indices n will reveal a single particle band
structure shown in Fig. 2.4. 87 Rb experiences a mRb /mK ≈ 87/40 ≈ 2.18 deeper potential
because of different recoil energies. The atomic transitions for the two species are separated
by 13 nm making an additional correction necessary which results in a factor of about 2.5
[30]. The relation between the potential depths of 87 Rb and 40 K is given by
Rb
EK
r ≈ 2.5 Er .
(2.9)
This difference in the lattice depth experienced by the two species results in much smaller
bandwidths (maximum to minimum energy of one band) and larger bandgaps (energy difference between the energy minimum of one band to the maximum energy of the next lower
band) for 87 Rb in comparison to 40 K. Furthermore bands with different band indices of the
two species overlap each other as shown by the gray shaded area in Fig. 2.4. For exmaple
the region around 35 kHz where the third band of 40 K coincides with the fourth band of
87 Rb.
10
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
2.4 Ultracold Bosons in Optical Lattices
In section 2.3 the Bloch states were introduced to describe non-interacting single particles in
an optical lattice potential. While it allows the calculation of eigenenergies and eigenstates
and thus gaining insights into systems governed by the kinetics in periodic potentials, it
completely neglects the fact that atoms interact with each other. For bosonic 87 Rb-atoms
this interaction is repulsive thus the interplay between the interaction and the tunneling
energy governs the behavior of the system. This interplay results in interesting effects like
a quantum phase transition between the superfluid phase and the Mott-insulator phase [8]
which will be described after a short presentation of the model used widely to describe this
phases and the transition between them.
2.4.1 The Bose-Hubbard Model
The behavior of interacting ultracold bosonic atoms in an optical lattice can be described
in terms of a Bose-Hubbard model [7]. The Bose-Hubbard model considers only tunneling
between adjacent lattice sites (tight-binding approximation) and restricts itself to the lowest
Bloch band (see [7]). Further only on-site interactions are considered. The Bose-Hubbard
Hamiltonian is derived from a full Hamiltonian given in second quantization [7, 31] and by
changing from a Bloch basis to a Wannier basis. In terms of three dimensional Bloch states
a Wannier state is expressed as
wn (~r − ~ri ) = √
1 X (n)
ψ (~r) e−iq~ri .
NL q q
(2.10)
Here NL is the total number of lattice sites and ~ri are the lattice site positions. The Wannier
states are maximally localized at each lattice site. The Bose-Hubbard Hamiltonian now can
be written as
ĤBH = −J
X †
b̂i b̂j +
hi,ji
X
UX
i n̂i
n̂i (n̂i − 1) −
2 i
i
(2.11)
The first term describes the energy associated with tunneling (i.e. kinetic energy) from a
lattice site i to a neighboring lattice site j. Lattice sites i and j are restricted to be nearest
neighbors. The tunneling matrix element J is given by
J=
Z
!
w1∗ (~r
~2 2
− ~ri ) −
∇ + Vlat (~r) w1 (~r − ~rj ) d~r
2m
(2.12)
The second term in (2.11) accounts for the interaction energy between the atoms where n̂i
counts the numbers of bosons at lattice site i. U is expressed as
U=
4πas ~2
m
Z
|w1 (~r)|4 d~r.
(2.13)
Here as is the s-wave scattering length between the bosons. The last term in (2.11) contributes an energy offset i for each lattice site. This offset accounts for the fact that the
2.4 Ultracold Bosons in Optical Lattices
11
0.3
0.2
U
U
U, J [Er
Rb
]
J
0.1
J
0
0
5
10
15
Rb
Lattice Depth [Er ]
Figure 2.5: Bose-Hubbard model. In the Bose-Hubbard model bosons are allowed to tunnel between
nearest neighbor lattice sites and thereby gain kinetic energy. The model also includes the effect of
interaction which in the case depicted here increases the potential energy of the system. The effect of
harmonic confinement is not shown. The graph on the right shows the behavior of U and J depending
on the lattice depth.
laser beams forming the optical lattice have a Gaussian intensity distribution which yields
an additional harmonic confinement (see Sec. 2.2). As outlined above the interplay between
U and J leads to the emergence of a quantum phase transition as described in the following
sections.
2.4.2 Superfluid Phase
When the tunneling parameter J is much larger than the interaction U (which is the case
in a shallow lattice), the bosonic gas tries to minimize its energy by tunneling. Because U
is small compared to J the system is able to occupy lattice sites with more than one atom.
This state is called a superfluid state where every atom is maximally delocalized over the
entire lattice. A result of this delocalization is the emergence of interference peaks at ±2~k
Figure 2.6: Superfluid Phase. The emergence of coherence peaks is characteristic for an ultracold
bosonic gas in an optical lattice potential.
during TOF expansion. This can be seen in Fig. 2.6. Here, the bosonic 87 Rb atoms were
prepared in a superfluid state. Then the lattice has been switched off instantly and the
atoms are allowed to expand freely.
12
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
2.4.3 Mott-Insulator Phase
Figure 2.7: Mott insulator phase. The destroyed coherence between the atoms at the different lattice
sites result in a broad distribution.
In contrast to the aforementioned situation, the case where U J results in a phase called
Mott insulator which occurs at higher lattice depths. In this case it is very costly for the
system to allow atoms to occupy a lattice site where another atom is located. So the system
tries to minimize its energy by distributing the atoms evenly over the lattice. This results
in an integer filling of the lattice sites. The fixed atom number per lattice site results in a
uncertainty in the phase destroying the phase coherence.
2.4.4 Superfluid to Mott Insulator Transition
The two phases explained above are connected via a quantum phase transition. By changing the lattice depth it is possible to change the ratio between U and J and observe the
Superfluid to Mott insulator phase transition [32]. In Fig. 2.8 typical TOF pictures for the
phase transition are shown.
Increasing Lattice Depth
Figure 2.8: Phase transition between superfluid and Mott-insulator..
The lattice depth where a transition between the two phases occurs can be estimated from
a mean-field approach [32]. For the 3D cubic lattice used at the Bose-Fermi mixtures
experiment, the transition is evaluated to a lattice depth of around 13.6 ERb
r . One interesting
feature of the Mott insulator to superfluid phase transition is the build up of insulating
regions which are separated by superfluid domains. This so called wedding cake structure
is shown in Fig. 2.9. A wedding cake structure has been observed experimentally with
rf-spectroscopy [33] and with single site resolution microscopy [34, 35].
2.5 Ultracold Fermions in Optical Lattices
13
(a) phase diagram
(b) wedding cake structure
3
n= 3
superfluid
2
µ/U
n= 2
n= 2
n= 1
1
n= 1
Mott insulator
0
0
0 .0 1
0 .0 2
0 .0 3
J/U
Figure 2.9: Phase diagram for bosons in an optical lattice. In an experimental situation the Mottinsulating phases (blue) with fixed numbers of atoms per site n, are separated by Superfluid regions
(grey). In a the phase diagram of a Bose-Hubbard model is shown. b shows the wedding cake structure.
2.5 Ultracold Fermions in Optical Lattices
In the previous section interacting bosons in optical lattices where studied. While the
behavior of bosons is governed by the repulsive interaction and tunneling, fermions are
subject to Pauli’s exclusion principle and behave according to Fermi-Dirac statistics.
2.5.1 The Fermi-Hubbard Model for Spin-Polarized Fermions
It is possible to describe a single component fermi gas in an optical lattice with a Hubbard
type model as done before for bosons. Ultracold fermions prepared in the same spin state do
not interact via s-wave scattering because of Pauli’s exclusion principle. P-wave scattering
is highly suppressed [36] so interactions are neglected. The resulting Hamiltonian is
ĤFH = −J
X †
ĉi ĉj −
hi,ji
X
i n̂i .
i
(2.14)
The first term describes tunneling between adjacent lattice sites while the former again
describes the harmonic confinement of the atoms. For more details see Sec. 2.4.1.
2.5.2 Phase Diagram of Spin-Polarized Fermions
The exclusion principle does not allow more than one fermion per lattice site in the same
quantum state. According to Fermi-Dirac statistics, fermions fill up all available energy
14
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
states up to the fermi-energy beginning with the state with the lowest energy. So intuitively
there are two situations one has to consider when looking at a fermi gas in an optical lattice.
The first situation arises when the fermi-energy (i.e. chemical potential) is smaller than
the bandwidth of the occupied band. This allows the atoms to minimize their energy
by tunneling between lattice sites. This state belongs to a metallic phase because of the
compressibility of the ensemble. If the fermi-energy is larger than the bandwidth of the
lowest band, the fermi gas will occupy all available energy states up to the first bandgap.
Thus the whole first Brillouin zone is occupied. This state is called band insulator.
The dominant energy scales in a noninteracting fermi gas are the characteristic trap energy [13]
Et = Vt
γN
4π/3
2/3
(2.15)
and the tunneling matrix element J which is connected to the bandwidth of a one dimensional lattice according to
(2.16)
4J = max (Eq(n) ) − min (Eq(n) ).
Vt denotes the minimal energy offset between two adjacent lattice sites, γ the aspect ratio of
the trap, and N the number of atoms in the lattice. The characteristic trap energy denotes
the Fermi energy of a noninteracting cloud in the limit of no tunneling [13].
0.8
N=100000
0.7
energy [EKr ]
N=70000
12J
0.6
band insulator
N=40000
0.5
0.4
metal
Et
0.3
0.2
0.1
0
6
8
10
12
14
16
18
20
lattice depth [EKr ]
Figure 2.10: Phases of the non-interacting fermi-gas. In red the tunneling matrix element J is
plotted for the 3D case (i.e. three times 4J). The dots represent the characteristic trap energy Et with
different numbers of particles.
In Fig. 2.10 a phase diagram for non-interacting fermions is shown. All measurements
are performed in a three dimensional lattice, the bandwidth has to be considered in three
dimensions: 3 · 4 J. The red curve shows 12 J depending on the lattice depth. The dots
show the characteristic trap energy for different amounts of fermions. The point at which
2.6 Band Mapping
15
the characteristic trap energy gets larger than the tunneling bandwidth is the point at which
all fermions are in a Band-insulating phase. It is important to note that a fraction of the
atomic sample can be in a band insulating phase while others are in a metallic phase. This
is because the trap energy depends on the position in the trap so the 12J boundary can be
reached at much lower lattice depths. Using the band mapping technique (see Sec. 2.6) it
is possible to distinguish the two phases. In a shallow lattice (for example V0 ≈ 3 EK
r ) we
expect to be in a metallic phase. We expect to see a washed out first Brillouin zone with
a pronounced center towards small momentum components. In the case of deep lattices
the Brillouin zone should be evenly filled and we expect to see sharp edges at the borders.
Fig. 2.11 shows ultracold fermionic Potassium loaded into a cubic optical lattice of varying
lattice depths. Because of the cubic lattice geometry, also the first Brillouin zone is cubic.
The lattice in our experimental setup is tilted by 45 degrees with respect to the detection
axis so we expect to see a rectangular Brillouin zone elongated in the direction of the tilting
(see Fig. 2.1 and Fig. 2.11). The above explained behavior is observed. A shallow lattice
leads to a pronounced center and a washed out Brillouin zone. Increasing the lattice depth
results in a evenly filled Brillouin zone corresponding to fermions in a Band-insulating
phase. It is not possible to find a distinct point where the phase transition occurs. We
attribute this to imperfect Band-mapping and the fact that parts of the fermions can be in
a band-insulating phase while other are not (see above).
orientation of the 1.BZ
V0 = 3ErK
V0 = 6ErK
V0 = 9ErK
V0 = 12ErK
Figure 2.11: Phase transition from metallic phase to band insulating phase. Shown is the orientation of the first Brillouin zone on the current experimental setup. On the right the resulting TOF
pictures are presented for different lattice depths.
2.6 Band Mapping
Band-Mapping Principles
One experimental challenge is to observe and detect the momentum resolved population
of occupied bands. Because q is restricted to the first Brillouin zone, and can always be
transformed into this zone, the quasimomentum is not an eigenstate of the real momentum
operator. This means that turning off the lattice will project the quasimomenta onto real
momenta. Information about the population of bands is lost. This problem can be circumvented by ramping down the lattice with a ramp duration chosen such that no interband
transitions are possible [37, 38] which means that the atoms will be mapped onto real momenta. The band mapping time (i.e. ramp down time) tBM has to be slower than the time
16
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
45
40
35
E / h [kHz]
30
25
Eq(n)
20
15
ħ2k2
2m
10
5
0
−3
−2
−1
0
1
2
3
k [kBZ]
Figure 2.12: Band mapping. Ramping down the lattice maps quasimomenta onto real momenta.
associated with the energy of the bandgap, but has to be faster than the time it would take
the atoms to redistribute their momentum within a band due to trap dynamics. With this
the criterion for effective band mapping is given by [39]
EBG h/tBM ~ωt .
(2.17)
Here EBG is the energy of the smallest bandgap (between bands to be observed) and ωt is
the trap frequency. Useful band mapping times for our experimental conditions are in the
order of a couple of hundred µs to some ms as discussed in the next section. These times
are consistent with the observations made by other groups [39].
Mapping of the First Brillouin Zone with Fermion
We can use the knowledge about the appearance of the first Brillouin zone discussed in
Sec. 2.5.2 to determine useful band mapping times. We therefor prepare 40 K in a deep
optical lattice (V0 ≈ 12 EK
r ) and vary the time in which the lattice is ramped down between
200 µs and 10 ms. The resulting pictures are presented in Fig. 2.13. On the left side are the
resulting TOF images shown. On the right the row- and column sums of these pictures are
presented. The row sum shows the characteristic triangular structure because of the tilting
of the camera axis with respect to the lattice axis (see Sec. 2.1 and Sec. 2.5.2). A mapping
time of 200 µs results in a slightly rounded Brillouin zone as seen in the corresponding
column sum. Mapping times on the order of 1 ms to 3 ms result in very homogeneous and
sharp edged column sums and very pronounced triangular shapes in the row sum. After 5 ms
a small peak starts to appear in the column sum, which further develops at a mapping time
of 10 ms. For even longer mapping times the atomic distribution would transform back to
an undisturbed fermi cloud governed by the optical dipole trap. From this series of pictures
2.6 Band Mapping
17
it is clear that mapping times longer than 5 ms are not practicable because the resulting
momentum distribution is starting to be dominated by the momentum distribution of the
trap. Good mapping times for our experimental setup thus are on the order of 1 ms to 3 ms
whereby a reasonable mapping is also possible for 200 µs.
TOF image
row sum
column sum
200µs
1ms
3ms
5ms
10ms
Figure 2.13: Result of Different Band Mapping Times. Different band mapping times (i.e. ramp
down time of the lattice) result in different mapping of the first Brillouin zone of a fermi gas loaded into
an cubic optical lattice.
18
2 Ultracold Bose-Fermi Mixtures in Optical Lattices
19
3 Lattice Modulation Spectroscopy
The high control of optical lattice systems together with advances in generation and preparation of ultracold fermions and bosons is a promising combination of tools to understand
and explore fundamental processes and interaction phenomena of many body physics. Spectroscopical research on ultracold quantum gases has been proven as a key method to access
the underlying physics.
Different spectrosopical methods have been developed to access the available momentum
states. It has been used to examine fundamental properties like the excitation spectrum a of
single component BEC [40]. The method has been used to probe the Bogoliubov excitation
spectrum of a bosonic superfluid in an optical lattice [41, 42]. Very recently, it could be
used to detect the Amplitude Mode in the crossover region between a Superfluid and an
Mott insulator [43]. For fermions Bragg spectroscopy has been used to probe the BEC-BCS
crossover regime [44]. Photo emission spectroscopy based on radiofrequency (rf) pulses,
has been used to observe the pairing gap of strongly interacting fermions and to probe the
single particle excitation spectrum of fermions near the BEC-BCS crossover [45, 46].
In optical lattices, modulating the lattice depth will transfer energy to the system. This
has developed into an easy to implement, multipurpose spectroscopy method. It has been
used to study the excitations of bosonic atoms to higher bands [47], the two dimensional
Mott-insulator to Superfluid transition [48] and is proposed to study the spectral function
of ultracold bosons in optical lattices [49]. The detection of a gapped excitation mode
in the fermionic Mott-insulating phase [12, 50] has been carried out by lattice modulation
spectroscopy. Great effort is put into the realization of antiferromagnetic ordering and modulation spectroscopy is again proposed as a mean to detect and observe antiferromagnetic
ordering [51, 52].
In the framework of this thesis a novel spectroscopy method has been developed. It is
based on lattice amplitude modulation and band mapping, and allows momentum resolved
spectroscopy of ultracold atoms in optical lattices. The method allows us to precisely determine the potential depth experienced by the atoms and from that accurate determination
of tunneling energies and rates.
In this chapter, first lattice modulation spectroscopy is described in a general way. Expressions for allowed transitions and their strength are theoretically derived. The influence
of modulation amplitude and time is examined. Based on these considerations the band
structure of non-interacting fermions is probed with momentum resolution, showing the
accuracy of the presented method. Lastly the band structure for bosons is examined.
20
3 Lattice Modulation Spectroscopy
3.1 Momentum Resolved Lattice Modulation Spectroscopy
Emod / h [kHz]
The described method of lattice modulation spectroscopy consists of two experimental techniques: The first is the modulation of the lattice amplitude, which transfers atoms to higher
bands creating particle-hole pairs. This creation of particle-hole pairs conserves the quasimomentum (see Fig. 3.1). Forming an band insulator, fermions occupy the whole first
Brillouin zone. Bosons fill the first Brillouin zone when trapped in a deep lattice being
in a Mott-insulator phase. Because of the different curvature of the bands, an specific
amount of imprinted energy will have a resonance at one specific momentum. The second
technique reveals the occupation of higher bands and is the band mapping procedure (see
Sec. 2.6). The spectroscopical information about the system under study is extracted by
varying the lattice modulation frequency and recording the position of the transferred atoms
after time-of-flight (TOF).
Particle
Hole
−1
0
1
q [kBZ]
Figure 3.1: Effect of modulating the lattice amplitude. Modulating the amplitude of the optical
lattice will excite atoms from the ground state to higher bands creating particle-hole pairs. The transfer
will preserve the quasimomentum q reflected by the straight lines.
Experimental Procedure
The experiments performed always start with an ultracold sample of fermionic 40 K in the
hyperfine state F = |9/2, 9/2i or a mixture of 40 K and bosonic 87 Rb (in hyperfine state
F = |1, 1i) in an optical dipole trap. Thereon an optical lattice is superimposed. Afterwards
the amplitude of two lattice laser beams is modulated. This is done by varying the power
of the laser beams which form the corresponding axis of the optical lattice. The modulation is applied for 1 ms with a relative amplitude of 15% avoiding beyond linear response
effects. The frequency of the amplitude modulation determines the imprinted energy onto
the system. Thus the variation of the frequency allows addressing of all bands. Most of
3.1 Momentum Resolved Lattice Modulation Spectroscopy
21
the measurements performed in the context of this thesis have been done while transferring
atoms to the third band. This band has the highest transition probability (see Sec. 3.2) and
thus give best signal to noise ratios. After modulating, the lattice is ramped down during
lattice ramp up
100ms
latt. mod.
1ms
latt. ramp down
200µs
time of flight
20ms
Figure 3.2: Experimental sequence of lattice modulation spectroscopy. Modulation spectroscopy
consists of two key procedures namely the modulation of the lattice and mapping of the population of
bands to reveal the momentum distribution by a fast lattice ramp down.
200 µs in order to map the quasimomentum onto real momenta. The band mapping time
of 200 µs is chosen to prevent dynamics induced by the harmonic trapping potential (see
Se. 6). Afterwards the atoms are released from all trapping potentials and expand freely for
typically 20 ms thus revealing the momentum distribution. The whole procedure is sketched
in Fig. 3.2.
Effective Band Structure
It is important to note that the first band is within our typically used lattice depths almost
but not entirely flat. So the energy required to transfer atoms at a certain momentum to
an higher band will be the difference between the energies of the higher band and the first
band at that momentum. Hence an effective band structure is observed as shown in 3.3.
This effective band structure is directly connected to the real band structure thus in the
following the effective band structure will be denoted as the band structure.
Eq(n) / h [kHz]
40
Band Str.
Eff. Band Str.
30
20
10
0
−1
0
q [kBZ]
1
Figure 3.3: Effective band
structure. Lattice modulation
spectroscopy transfers atoms
from the first band to higher
bands. The first band is slightly
curved even for higher lattice
depths. This results in an effective band structure shown here.
In light orange the original band
structure for 40 K at a lattice
depth of V0 = 7.5 EK
r is shown.
In red the effective band structure incorporating the energy
offset from the first band is depicted.
22
3 Lattice Modulation Spectroscopy
Deducing the Momentum from TOF Images
without modulation
with modulation
Figure 3.4: TOF images obtained after lattice modulation and band mapping. Modulating the
amplitude of the lattice results in creation of particle-hole pairs. Mapping the quasimomentum to real
momenta and letting the atoms expand freely allows extraction of the population. The left side shows
the images obtained without any modulation. The right side shows a typical picture where the lattice
has been modulated in one lattice axis. The sharp peaks left and right of the first Brillouin zone are
atoms transferred to the third band. The distance between the center of the first Brillouin zone and the
excited atoms is directly related to the quasimomentum (see text).
In a typical experimental situation the above outlined procedure results in TOF images
shown in Fig. 3.4 (fermionic atoms). The left side shows the sequence without any modulation resulting in pictures of the first Brillouin zone. On the right side the effect of
modulating the lattice is shown. The modulation is performed in one axis of the lattice.
Atoms are transferred from the first Brillouin zone to a higher band (here the third) at
a specific momentum (sharp peaks to the left and right). The momentum of the excited
atoms is related to the distance observed on the image from a CCD camera by
p=
(n − 1 + q)~kL m
ttof .
mK
rpixel
(3.1)
Here p is the distance between the center of the first Brillouin zone and the transferred atoms
on the camera in pixel. rpixel is the size of a pixel on the camera and m the magnification
of the detection optics. n is the band, q the quasimomentum and ttof the time the atoms
expand freely. All parameters are known and p can be determined very accurately. Using
this relation it is possible to reconstruct the momentum q from the TOF images. This data
combined with the frequency of the lattice modulation gives access to the band structure
of the lattice. By modulating another lattice axis or modulate more than one axis it is
possible to address bands in other directions.
3.2 Chracterization of Modulating the Lattice Amplitude
This section concentrates on induced transitions between different bands. In the first part
we will derive expressions for the strength of the coupling between different bands. The
second part characterizes the transfer to higher bands for different modulation amplitudes
and different modulation times experimentally.
3.2 Chracterization of Modulating the Lattice Amplitude
23
3.2.1 Theory of Interband Transitions
To understand the effect of modulating the amplitude of the lattice one has to consider the
effect of an oscillating amplitude of the lattice potential. We can incorporate the modulation
by adding a time-dependent oscillating perturbation to the Hamiltonian (2.2)
p2
(3.2)
+ V0 cos (kz z)2 [1 + cos (ωt)] .
2m
Here is a small number representing a small perturbation. In a periodic potential it
is intuitive to use the Bloch states introduced in Sec. 2.4 as the eigenstates of the nonperturbed system:
H=
hz|n, qi = ψq(n) (z) = eiqz
X (n)
cK,q eiKz
(3.3)
K
with
(3.4)
H0 |n, qi = Eq(n) |n, qi.
where Ĥ0 is the non-perturbed Hamiltonian. To answer the question which transitions
between the eigenstates of the system are possible we calculate the transition probabilities
between two Bloch states. Therefore we express the perturbation as
(3.5)
V̂ (z, t) = V0 cos (kz z)2 cos (ωt).
Using Fermi’s golden rule [53] the transition probability between two bloch states with
momenta q, q 0 and band indices n, m is
(3.6)
nm
0 2
Wqq
0 ∝ |hn, q|V̂ (z)|m, q i| f (T, ω0 − ω).
Here f (T, ω0 − ω) is a function describing the response of the coupling on the modulation
time T and the modulation frequency ω. ω0 is the resonance frequency of the transition.
For long modulation times, f (T, ω0 −ω) can be reduced to a Kronecker delta. The remaining
part of the perturbation is now conveniently expressed as
2
p
H0 − 2m
V (z)
= cos (kz z)2 =
.
V0
V0
(3.7)
Using this and inserting into (3.6) yields
nm
0
2
0 2
Wqq
0 ∝ | hn, q|Ĥ0 |m, q i −hn, q|p̂ |m, q i| .
|
{z
(3.8)
}
0
Here ω = ω0 and constant factors have been neglected. The first term is zero for n 6= m
and/or q 6= q 0 . With p̂2 ∝ ∂z2 and inserting the Bloch states (3.3)
nm
Wqq
0 ∝ |
X (n) (m) Z
0
0
e−iqz e−iKz ∂z2 eiq z eiK z dz |2
cK,q cK 0 ,q0
K,K 0
∝|
X (n) (m)
0
0 2
cK,q cK 0 ,q0 (q + K )
K,K 0
Z
0
0
−i(q−q )z −i(K−K )z
2
|e {z } e|
{z
} dz | .
δq,q0
δK,K 0
(3.9)
24
3 Lattice Modulation Spectroscopy
Using the Kronecker delta this becomes
Wqnm ∝ |
X (n) (m)
cK,q cK,q (q + K)2 |2
(3.10)
K
This equation incorporates one very important fact: modulating the amplitude of the lattice only couples states with the same q. This means that no momentum transfer other
than integral multiple of the reciprocal lattice vector are allowed. In this thesis the transi-
Wnm
q [a.u.]
1.0
1↔5
1↔4
1↔3
1↔2
1.0
0.5
0.5
0
-1
1↔5
1↔4
1↔3
1↔2
0
q [kBZ]
1
0
-1
0
q [kBZ]
1
Figure 3.5: Interband transition probabilities due to lattice modulation. Shown are the transitions
probabilities Wqnm between n = 1 and m = 2, 3, 4, 5. Calculated for a lattice depth of 2.5 EK
r on the left
on
the
right.
and 7.5 EK
r
tion probabilities using experimental parameter have been calculated using Eqn. (2.3) and
Eqn. (3.10). The transitions possibilities for transitions between n = 1 and n = 2 to n = 4
are shown in 3.5. The transition to the third band is for every momentum much higher
than for transitions to all the other bands. A very distinct feature is observed for even
band indices at momentum zero. Here a transition is always forbidden. The curves for even
bands will approach zero for every q for deep lattices. Thus in a deep lattice, modulation of
the amplitude does not induce transitions between the first and any even numbered band.
In a simple picture this can be understood by approximating a deep lattice site with an
harmonic oscillator. The eigenstates of the harmonic oscillator have alternating parities.
Lattice modulation is a even parity operation [47] thus only couples states with the same
parity. Hence transitions between the first (i.e. lowest) eigenstate of the harmonic oscillator
and the second, fourth etc. are not allowed. This explains the suppression of transitions to
even numbered bands in deep lattices.
3.2.2 Time and Amplitude of the Modulation
For effective spectroscopy several criteria have to be fulfilled. First a proper signal to noise
ratio has to be obtained. Thus the number of excited atoms has to be sufficient. On the
other hand it is necessary to keep the overall perturbation of the system small. A large
perturbation can lead to beyond linear-response effects, for example final-state interaction
and higher order processes.
3.2 Chracterization of Modulating the Lattice Amplitude
25
The behavior of fermions subject to lattice modulation is studied by changing the amplitude
of the modulation and varying the modulation time. Both affect the number of atoms
transferred to higher bands (here the third). In Fig. 3.6 the relative amount of excited
atoms is shown. The amount is normalized to the total number of atoms. The data has been
experimentally obtained for 40 K loaded into an optical lattice with a lattice depth of 7.5 EK
r .
Fitting a gaussian to the momentum distribution of the excited atoms yield an estimate for
the width ∆k of the momentum distribution of the atoms in the third band. The momentum
width is thereby deduced taking the FWHM of the Gaussian. Longer modulation times as
8
8
excitation fraction [%]
(b)
excitation fraction [%]
(a)
6
4
2
0
5
0
10
15 20 25 30 35
modulation intensity [%]
40
6
4
2
0
45
(c)
0
0.2
0.2
Δk [kBZ]
0.25
Δk [kBZ]
1 .5
1
modulation time [ms]
2
1
1.5
modulation time [ms]
2
(d)
0.25
0.15
0.1
0 .5
0.15
0
5
10
15 20 25 30 35
modulation intensity [%]
40
45
0.1
0
0.5
Figure 3.6: Number of atoms and momentum width for different modulation amplitudes and
modulation times. The amount of excited atoms depend on the modulation time and the amplitude
of the modulation. Experimentally obtained data is shown here for different durations and relative
amplitudes of modulation. Data is obtained for 7.5 EK
r . Shaded are the parameters chosen for the
following measurements. The black dotted curves are a guide to the eye. For a and c a modulation time
of 1 ms is used. In b and d a realtive modulation amplitude of 15 % is applied.
well as higher modulation amplitudes yield more excited atoms. Decreasing either the
modulation time or the modulation amplitude will reduce the width of the momentum
distribution. Considering Fig. 3.6a and Fig. 3.6b one can see a decrease in the slope of the
curves. We attribute this behavior to a decrease in the number of atoms available in the
first band which leads to a reduced number of atoms transferred to the third band. Higher
order processes and transfers from the third band to higher bands could be also possible.
To examine the coherence properties of lattice modulation, a high modulation intensity
26
3 Lattice Modulation Spectroscopy
(40%) has been used while varying the modulation time. This measurement is presented
in Fig. 3.7. Damped Rabi oscillations between the numbers of atoms in the first Brillouin
zone and the third are observed. Rabi oscillations are a effect which is not described by
linear-response theory. This means that a large modulation amplitude will substantially
perturb the system which could result in a disturbed spectroscopy signal.
atom fraction [%]
100
80
first band
60
40
third band
20
0
0
0.5
1
1.5
2
modulation time [ms]
Figure 3.7: Coherence of the lattice modulation process. Lattice modulation drives coherent Rabi
oscillations. The blue curve shows the amount of atoms in the first band. The green curve shows the
relative amount of atoms the the third band. This data is obtained using a large modulation amplitude
of 40%.
The decrease of the slope in the curves for the number of transferred atoms and the emergence of Rabi oscillations for high modulation amplitudes are both signatures of beyond
linear-response effects. To keep these effects small but still get a usable signal, a modulation time of 1 ms and a modulation amplitude of 15 % has been chosen for all measurements
when not mentioned otherwise.
3.3 Band Structure of Spin-Polarized Fermions
Lattice modulation spectroscopy can now be employed to measure the spectrum of a spinpolarized sample of fermionic 40 K. In Fig. 3.8 the experimentally obtained data for a lattice
depth of 5 EK
r is shown.
Each horizontal line corresponds to one modulation frequency. The data is obtained by
taking the column sum of the pictures obtained after TOF (see for example Fig. 3.4). The
frequency is varied between 0 kHz and 55 kHz in steps of 500 Hz. The color represents
the amount of atoms. The amount of atoms is normalized hence red is close to one and
white zero. The x-axis shows the momentum calculated for a given time-of-flight (here
20 ms) in terms of the reciprocal lattice vector. Starting at 0 kHz the first Brillouin zone
is clearly seen. No transfer to higher bands happen before reaching around 13 kHz. This
corresponds to the first bandgap between the first and the second band (compare with
Sec. 3.3). Increasing the energy (i.e. frequency) further, the first atoms are excited to the
3.3 Band Structure of Spin-Polarized Fermions
27
n=4
50
Emod / h [kHz]
40
30
n=3
20
n=2
10
0
-3
-2
-1
0
1
2
3
k [kBZ]
Figure 3.8: Band structure for 40 K. The band structure for fermions at a lattice depth of 5EK
r
is depicted here. Shown is the energy imprinted onto the system in terms of the lattice modulation
frequency depending on the obtained momentum distribution. A red color expresses a high amount of
atoms where blue and lighter colors represent few or no atoms. Visible are the second, third and fourth
band together with the corresponding bandgaps.
second band. A very distinct feature arises here. Comparing with the bandstructure it can
be seen that the second band has a negative slope. Thus the energy difference between the
first and the second band is smaller for higher quasimomenta so the first atoms which are
transferred to the second band are at q at the edge of the Brillouin zone. Imprinting higher
energies result in particle-hole pairs at lower momenta. This behavior is clearly seen in
the first Brillouin zone (corresponding to the reduced zone schema). The second bandgap
between the second and third band occurs at around 22 kHz. The third band is again clearly
seen while it is not possible to observe the third bandgap which is in the order of 300 Hz
which is below the resolution of the measurement.
Band Structure of Fermions for Different Lattice Depths
This measurement has been repeated for different lattice depths shown in Fig. 3.9.
The bandwidths decrease with deeper lattices while simultaneously the widths of the bandgaps
increase. A number of distinct features are observable. For small lattice depths the number
of atoms excited to the second and third band is nearly the same. At small momenta more
atoms are transfered to the third band than to the second while for higher momenta this
28
3 Lattice Modulation Spectroscopy
(a) 1.5 EK
r
(b) 5 EK
r
50
50
40
40
30
30
20
20
10
10
0
−3
−2
−1
0
1
2
3
0
−3
−2
(c) 8 EK
r
(d) 11 EK
r
50
50
40
40
30
30
20
20
10
10
0
−3
−2
−1
0
1
2
3
0
−3
−2
−1
0
1
2
3
−1
0
1
2
3
Figure 3.9: Band structures for different lattice depths. Shown is the data obtained from lattice
modulation spectroscopy with fermions for different lattice depths. With increasing lattice depths the
bandgap widths are increasing while the bandwidths are decreasing. The amount of transferred atoms
to even bands is getting smaller while the width of the excited atom momentum distribution get larger.
behavior turns around. This is consistent with the considerations from Sec. 3.2.1 in particular the left side of Fig. 3.5. At higher lattice depths it is expected that the amount of atoms
excited to the second band is decreased which is again in agreement with the theoretical
prognosis. Smaller bandwidths (i.e. flat bands) together with Fourier broadening of the
imprinted energy yields a broader momentum distribution of the transferred atoms in the
third band as well as in the second. This is very nicely shown in the band structure plot 3.9
for 11 EK
r .
Reduced Zone Scheme
A more detailed view on the band structure of fermions is obtained by subtracting the
atomic distribution in the first Brillouin zone without any excitation (energies less than
8 kHz) from the complete dataset shown in Fig. 3.9. The data obtained in such a way is
shown in Fig. 3.10. The red areas show regions where more atoms are present in comparison
29
particles
3.3 Band Structure of Spin-Polarized Fermions
n=4
50
30
n=3
0
Emod / h [kHz]
40
20
n=2
holes
10
0
-3
-2
-1
0
1
2
3
k [kBZ]
Figure 3.10: Detailed band structure for 40 K. The data has been normalized to the amount of atoms
present in the first Brillouin zone. In red colored areas atoms are present. The blue areas represent holes.
Clearly visible is a reduced zone scheme in blue while in red the momentum distribution is shown. Lattice
depth 5EK
r . The black dotted circles mark regions where the band-mapping procedure failed.
to the case without any excited atoms. The blue area corresponds to regions where less
atoms are present. The dark blue region between +kBZ and −kBZ can be interpreted as
the reduced zone scheme which is now clearly visible. This area represents holes. The
dark red areas outside of the first Brillouin zone are the excited atoms (i.e. particles).
The picture shows imperfections in the band-mapping process marked with black circles.
These imperfections are not limiting because their signal is so low that no influence on the
momentum distribution is observable.
Dispersion relation for the third band
By using equation (3.1) a detailed analysis of the energy-momentum relation for the third
band is accessible. In Fig. 3.11 the lattice modulation frequency is shown depending on the
momentum for a lattice depth of 11.3 EK
r . The red points represent the mean momentum
for the excited atoms. The red line shows the band structure for the third band calculated
as shown in Sec. 3.3. The black dotted line shows a numerical simulation which incorporates the lattice potential as well as the harmonic confinement induced by the lattice laser
beams. The light and dark shaded area shows a ±5% and ±10% deviation of the band
structure shown by the red line. The bar graph on the right side shows the fraction of
atoms transferred to the third band.
30
3 Lattice Modulation Spectroscopy
experimental data
55
lattice + confinement
for 11.3 EK
r
Emod / h [kHz]
lattice for 11.3 EKr
50
45
± 5%
40
± 10%
0
0.2
0.4
0.6
0.8
q [kBZ]
1 0%
10%
excited fraction
Figure 3.11: Detailed analysis for fermions in the third band. Here the third band has been analyzed.
Red points denote the mean momentum for a lattice depth of 11.4EK
r . The red curve shows a fitted
band structure where fitting has been limited to 0.4kBZ and 0.8kBZ . The black dashed line shows a
simulation of the system incorporating the periodic potential and the lattice potential. The shaded areas
show 5% and 10% deviations from the band structure at 11.4EK
r .
The red line shows the third band for a lattice depth of 11.3ErK . This depth has been
deduced by a fit to the experimental data. The experimental data shows deviations in the
shape of the curves between the expected band structure (i.e. red line) and the experimental data at high momenta and at momenta below 0.4 kBZ which is consistent with the
simulation. The fit thus has been limited to the region between 0.4 kBZ and 0.8 kBZ . From
this fit a lattice depth of 11.3EK
r ± 0.1% is deduced. The error presented her is the standard
deviation of the fit. It is important to note that the variation of the lattice depth through
experimental imperfections is of the order of 2%. This precision thus governs the findings
presented here.
This analysis is performed for all data shown in Fig. 3.9. For each dataset a fit has been
performed. The result is shown in Tab. 3.1.
Image
3.9a
3.9b
3.9c
3.9d
V0,fit [EK
r ]
1.4 ±0.5%
5.0 ±0.6%
8.7 ±0.5
11.3 ±0.1%
J [EK
r ]
0.164
0.0658
0.026
0.0143
Table 3.1: Fitted lattice depth for spin polarized fermions with resulting tunneling energies.
Presented are the lattice depths obtained from the band structures in Fig. 3.9. From the obtained lattice
depths, the tunneling energy is calculated.
3.3 Band Structure of Spin-Polarized Fermions
31
The lattice depth data shown in Tab. 3.1 allows us to determine the tunneling energy of
the fermionic sample. Using (2.12) the tunneling energy is calculated.
Effect of harmonic confinement and finite modulation time
The black dotted line in figure 3.11 shows a numerical simulation which includes the effect of
harmonic confinement due to the Gaussian intensity distribution of the laser beams and the
finite modulation time. The simulation has been performed for the deduced lattice depth
of 11.3 EK
r and shows three distinct features which all can be seen in the experimental data.
First at small momenta and energies an overall shift to higher momenta is seen. Excitation of
atoms at momenta near q = 0 is suppressed and excitations are possible before reaching the
expected energy momentum relation. At low to medium momenta a shift to higher energies
is observed while at high momenta a shift to higher energies is visible while excitations
of atoms at high momenta are suppressed. The two effects leading to deviations are now
explained more thoroughly.
Finite modulation time. Exciting atoms with less modulation (i.e. energy) than required
can be understood by considering the finite modulation time of the perturbation by lattice
modulation. In first-order perturbation theory, the probability of transferring an atom to
an excited state is given by
(3.11)
P (νmod ) ∝ sinc2 [tpulse (ν0 − νmod )].
Here νmod is the modulation frequency and ν0 the resonance frequency. In figure 3.12a the
perturbation of the system depending on time is shown. The atomic response is plotted in
figure 3.12b. This broadening mechanism is called Fourier broadening. From figure 3.12b
(a)
(b)
1
atomic response [a.u.]
perturbation amplitude [a.u.]
1
0 .6
0 .2
− 0 .2
− 0 .6
−1
0.8
0.6
0.4
0.2
0
−1
0
time [ms]
1
38
40
42
frequency [kHz]
Figure 3.12: Broadening due to finite modulation time. Due to the finite modulation time (a) the
transitions probability (atomic response) behaves as shown in b.
it can be deduced that the response peak is broadened with a full width at half maximum
32
3 Lattice Modulation Spectroscopy
(FWHM) of approximately 1kHz. Further sidebands develop at ±1.5kHz around the center
frequency. Due to sidebands and broadening the atoms can be transferred to another band
is reached earlier for low momenta and is left later for high momenta.
Harmonic confinement A second effect which leads to deviations from the theoretical
expected band structure is the harmonic confinement induced by the lattice laser beams
and the optical dipole trap. The lattice depth is varied by changing the power of the laser
beams thus a different lattice depth leads to a different harmonic confinement. This spatial
harmonic confinement induces a harmonic confinement in reciprocal space. The confinement
alters the density of states (DOS) of the system [54, 55]. In reciprocal space this means that
more states are available at medium to higher momenta and few states are available for low
momenta. This affects the DOS and depends strongly on the lattice depth. This feature is
seen when comparing the beginning of the third band for a very shallow lattice (Fig. 3.9a)
and a deep lattice (Fig. 3.9d). In the shallow lattice it is possible to excite atoms at q = 0
while in the deep lattice we observe no transfer. So a shift of the density of states to higher
energies has occurred. Because in the experiment mean momenta are detected the result is
a deviation to higher momenta for low energies.
The effect of harmonic confinement and the effect of broadening due to finite modulation
times is depicted in Fig. 3.13.
(a) shallow lattice
Emod / h [kHz]
Emod / h [kHz]
(b) deep lattice
−1
0
q [kBZ]
1
−1
0
1
q [kBZ]
Figure 3.13: Effect of harmonic confinement and finite modulation time. The density of states of
the first band is altered significantly due to the harmonic confinement from the optical dipole trap and
the lattice laser beams. A deep lattice results in a smaller density of states for low and high momenta.
This results in a shift to higher momenta. The dotted arrows represent the Fourier broadening.
3.3 Band Structure of Spin-Polarized Fermions
33
Lattice Calibration
The precise determination of lattice depths makes the method of lattice modulation spectroscopy with fermions exceptionally well suited to determine the laser power necessary to
create a lattice with a specific depth. The standard method [56] to calibrate an optical lattice has been the use of a BEC loaded into the lattice. Successive modulation of the lattice
amplitude imprint energy onto the system. The modulation frequency is chosen such that
a transfer into the third band is possible. Because the BEC has a momentum distribution
sharply peaked at q = 0 a resonance at the corresponding energy is expected. On the
resonance, the sample is substantially heated which can be observed as shown in 3.14.
(a)
(b)
rel. atom number central peak
1
0.75
0.5
0.25
15
15.5
16
16.5
17
17.5
18
modulation frequency [kHz]
Figure 3.14: Principles of lattice calibration using bosons. After preparing bosons in an optical
lattice, the lattice amplitude is modulated. When the modulation frequency is such that a transfer to
the third band is possible energy is imprinted onto the system. This energy can be observed as shown in
a. Varying the modulation frequency yields a resonance as shown in b. The resonance position is then
compared to the theoretically possible transition energies and thus a lattice depth is deduced.
The interaction between the bosons lead to a shift of the resonance position. Further
the influence of the harmonic confinement (see 3.3) leads to errors. By employing lattice
modulation spectroscopy with fermions, these problems can be circumvented. This method
is now standard at the "Bose-Fermi mixtures" experiment in Hamburg.
34
3 Lattice Modulation Spectroscopy
3.4 Spectroscopy on Pure Bosonic Samples
In this section the method of lattice modulation spectroscopy is employed using a pure
bosonic sample. Bosons are not subject to Pauli’s exclusion principle thus the lowest band
is not filled in the same way it is for fermions. To allow momentum resolved spectroscopy
the bosons have to occupy higher momentum states. This is achieved by preparing 87 Rb in
a deep lattice (around 16 ERb
r ) thus being in a Mott insulating phase. Due to the spatial
localization the atoms essentially are found in a Fock state thus all momentum components
are occupied.
The result of this measurement is shown in Fig. 3.15. On the y-axis again the energy given
by the modulation frequency of the lattice is shown. Each row shows the column sum of
the momentum distribution obtained after time of flight imaging. It is nicely seen that the
atoms do not homogeneously fill up the first Brillouin zone. Especially the middle to higher
momentum parts are only populated with few atoms. We attribute this to interaction effects
between the bosons. The interaction leads to a redistribution of the atoms resulting in a
shift to lower momenta. For low energies no atoms are transferred to higher bands due to
50
Emod / h [kHz]
40
30
20
10
0
−5
−4
−3
−2
−1
0
1
2
3
4
5
k [𝛑/ a]
Figure 3.15: Band Structure for bosons. Shown is the band structure for bosons. Lattice depht
V0 = 16.5 ERb
r . Nicely seen is the inhomogenous filling and the broad excitation.
the first bandgap. A population of the second band is not observed which is consistent with
the findings of Sec. 3.2.1 where for deep lattices less coupling between the first and even
numbered bands is expected. At higher modulation frequencies the third band is populated.
The momentum distribution is very broad and only very few atoms are excited resulting in
3.5 Summary
35
a very weak signal. Some but very few atoms are transferred to the fourth band so no clear
signal can be obtained for higher bands then the third.
The third band is further analyzed by extracting the momentum of the excited atoms for
different energies. Plotting yields a band structure for the third band analogous to what
has been shown for fermions. The result is shown in Fig. 3.16. Fitting a single particle
32
experimental data
lattice for 16.4 ERb
r
Emod / h [kHz]
30
28
26
24
± 5%
22
± 10%
20
0
0.2
0.4
0.6
q [𝛑/ a]
0.8
1 0%
10%
excited fraction
Figure 3.16: Detailed band structure for bosons. Shown is the third band. Obtained using lattice
modulation spectroscopy, large deviations from the ideal band structure are observed.
band structure to the data (again limiting to the region of 0.4 kBZ to 0.8 kBZ ) yields a lattice
depth of 16.6 ERb
r ± 1%. The theoretical expected curve is shown in red. Deviations are
again observed at low to medium and high momenta. The shifts are again induced by the
harmonic confinement, the finite modulation time and the interaction. The small and broad
signal results in a one order of magnitude higher error in the fit. This high error and the
big deviations as wells as the low signal do not allow to extract useful informations from
the measurement.
3.5 Summary
In this chapter the principles of lattice modulation spectroscopy were presented. It has been
shown that amplitude modulation of an optical lattice allows the creation of particle-hole
pairs while conserving the quasimomentum. Band mapping reveals the quasimomentum
distribution. This techniques allow sensitive probing of the band structure for fermionic
samples. A significant influence of the harmonic confinement from the lattice laser beams
and the finite modulation time has been observed. The dispersion relation for bosons
could be extracted but because of the deep lattice and the inhomogenous filling of the first
36
3 Lattice Modulation Spectroscopy
band precise measurements are difficult. The obtained data for fermions on the other hand
allows a precise measurement of the lattice depth experienced by the fermionic sample. This
information gives direct access to the tunneling energy and is crucial for most experiments
performed with optical lattices. The next chapter of this thesis shows the application of
modulation spectroscopy to a mixture of bosons and fermions.
37
4 Lattice Modulation Spectroscopy with
Bose-Fermi Mixtures
Mixtures of ultracold atoms obeying different quantum statistics have been developed into
a powerful tool for simulating solid-state systems, as well as giving access to fundamental
quantum mechanics. The interactions between bosons and fermions strongly influence the
behavior of many-body systems.
One of the first observations while studying Bose-Fermi mixtures in optical lattices, was a
decrease in the bosonic visibility when mixed with fermions [19–21]. This observation has
not been fully understood until today. Various theoretical proposals have been made. One
reason for the observed shift could be an adiabatic heat-up of the mixture in the lattice [22].
Other models incorporate higher bands and yields an effective potential approach [23] reducing the tunneling rate of the bosons. This ansatz has been further developed, incorporating
non-linear corrections to tunneling, renormalized two-body interactions and effective three
body interactions [57].
In this thesis, lattice modulation spectroscopy is used to study the excitation spectrum of
fermions with an admixture of bosons. The high precision of this spectroscopy method
allows to determine the lattice depth with high accuracy. Because the tunneling rate is directly connected to the depth of the lattice the method is very promising to study variations
in the tunneling rate of fermions with an admixture of bosons.
In this chapter the band structure of fermions is probed with an admixture of different
amounts of bosons. First, a theoretical description of Bose-Fermi mixtures in optical lattices
is given. Afterwards the experiments conducted are presented and the results obtained are
shown.
4.1 Ultracold Mixtures of Bosons and Fermions
Bose-Fermi-Hubbard Hamiltonian
Mixtures of bosons and fermions in an optical lattice are described by a Hubbard type
Hamiltonian consisting of the single species Hamiltonians (see Sec. 2.4 and Sec. 2.5) and an
38
4 Lattice Modulation Spectroscopy with Bose-Fermi Mixtures
interspecies interaction term [58]:
ĤBF = − JB
X †
b̂i b̂j +
hi,ji
− JF
X †
ĉi ĉj −
X
UX B B
B
n̂i (n̂i − 1) −
B
i n̂i
2 i
i
X
X
(4.1)
i
hi,ji
+ UBF
F
F
i n̂i
B
n̂F
i n̂i .
i
Here the indices B and F denote the species. The interaction between bosons and fermions
is given by
UBF = gBF
Z
|w1B (~r)|2 |w1F (~r)|2 d~r
(4.2)
with
gBF =
2π~2 aBF
.
µ
(4.3)
mF
µ = mmBB+m
is the reduced mass of two interacting atoms. aBF is the s-wave scattering
F
length between the bosonic and the fermionic atoms. For a mixture of 40 K and 87 Rb the
interaction is attractive with a scattering length of aBF = −205a0 [59].
Effective Potential due to Interactions
The attractive interaction gives rise to mutual effective potentials, determined by the density
distribution ρ of the other species [23]:
VBeff (~r) =V (~r) + gBF ρF (~r)
VFeff (~r) =V (~r) + gBF ρB (~r).
(4.4)
The interplay between the two potentials can lead to a phenomenon called self trapping [23].
The fermions mediate a trapping between bosons. For higher bosonic fillings the experienced
potentials become deeper. The trapping is mutual hence a deeper potential is experienced
by both species. In Fig. 4.1 the effect of an effective potential depending on the occupation
number is shown. The wedding-cake structure of a bosonic Mott-insulator results in site
dependent occupation numbers. This in turn results in site dependent, effective potentials
for the atoms.
4.2 Lattice Modulation Spectroscopy on Bose-Fermi Mixtures
The method of lattice modulation spectroscopy introduced in the chapter 3.1 is now employed to probe the fermionic band structure with an admixture of bosons.
4.2 Lattice Modulation Spectroscopy on Bose-Fermi Mixtures
39
Figure 4.1: Interactions lead to an effective potential. Depicted is an optical lattice with harmonic
confinement. In red fermions are shown. Small blue circles represent bosons. The effective potential
created by attractive interaction is shown in blue for different boson occupations. In grey the potential
without bosons is shown.
Experimental Procedure
The principle experimental procedure has been outlined in chapter 3.1. To tune the ratio
between the atom numbers, different loading times of the MOT at the beginning of the experimental sequence are employed. This results in atom numbers of up to 2 × 105 Rubidium
atoms forming a BEC and up to 1 × 105 Potassium atoms at 0.1 TF . The mixture of spinpolarized fermions and bosons is prepared in an optical lattice. A lattice depth of 7.5 EK
r
is chosen for all experiments. The bosons experience a deep lattice (approximately 19 ERb
r )
and thus form a Mott-insulating phase due to strong localization. The lattice is modulated
and the quasimomentum distribution is mapped onto real momenta. Both species are detected consecutively. The momentum distribution as well as the amount of 40 K atoms is
extracted from the TOF images. Absorption imaging of the 87 Rb atoms yield the number
of bosonic atoms.
Spectroscopy of Bose-Fermi Mixtures
The fermionic band structure has been probed for different ratios of fermions and bosons.
The resulting dispersion relations are shown in figure 4.2. The unperturbed band structure
is shown in red. Fitting a band structure to the data yields a lattice depth of 7.6 EK
r .
Admixture of about twice as much bosons as fermions results in the curve shown in green.
A slight deviation starting at low momenta is seen over the whole third band. A lattice
depth of 8.1 EK
r is deduced. Large amounts of bosons result in even higher deviations from
the original band structure shown in blue. The lattice depth is evaluated to 9.2 EK
r which
is a 20% deviation from the pure fermionic system. Some very interesting features are
observable. First while the green curve shows only a small deviation from the pure band
structure adding a large amount of bosons alters the shape of the resulting band structure
significantly (blue curve). Further comparing the distribution of the amount of excited
atoms, it can be seen that the amount of transferred atoms is smaller and the distribution
of excited atoms has been altered significantly.
We explain this behavior by the an effective potential explained in Sec. 4.1. The data for
the green curve with an ratio NRb /NK ≈ 2 leads to a mean occupation of smaller than
40
4 Lattice Modulation Spectroscopy with Bose-Fermi Mixtures
50
pure fermions
NRb / NK = 2
Emod / h [kHz]
45
NRb / NK = 10
40
7.6 EKr
35
8.1 EKr
9.2 EKr
0
0.2
0.4
0.6
0.8
1
q [kBZ]
0%
10%
excited fraction
Figure 4.2: Fermionic dispersion relation of the third band with an admixture os bosons. A
mixture of 40 K and 87 Rb has been prepared in an optical lattice. Modulation spectroscopy yields the
dispersion relation of fermions. In red the pure fermionic energy-momentum relation is shown. Green
shows an moderate amount of bosons mixed into the system. Blue depicts the case of an atom number
ratio of about 10.
two. The bosons will build a Mott-insulator thus starting to localize the fermions. This
localization results in a slightly increased lattice depth. Increasing the atom number ratio
to NRb /NK ≈ 10, bosons will build up an extended wedding-cake structure with a maximal
occupation number of three bosons per lattice site. This leads to localization of the fermions
resulting in a deeper lattice. Deviations at small momenta are attributed to the effect of
the harmonic confinement as discussed in Sec. 3.3. For large atom number ratios, this effect
is increased due to the wedding-cake structure of the bosons which create a inhomogenous
overlap between the fermions and the bosons. The localization yields a reduced tunneling
rate which is dependent on the occupation number.
NRb /NK
0
2
10
V0,fit [EK
r ]
7.6
8.1
9.2
J [EK
r ]
0.034
0.0301
0.0237
Table 4.1: Fermionic tunneling energy for different admixtures of bosons. Higher admixtures of
bosons lead to a deeper lattice for the fermions. From the lattice depth the tunneling energy is calculated.
With the obtained lattice depth and equation (2.12) the tunneling energy J can be calculated. The result of this calculation is shown in Tab. 4.1. The tunneling energy is reduced
by 30%.
4.3 Summary
41
4.3 Summary
In this chapter, lattice modulation spectroscopy has been employed to probe the band
structure of fermionic samples with an admixture of bosons. A significant increase in the
lattice depth experienced by fermions is observed. This behavior can be explained in terms
of an effective potential which localizes the atoms. This localization leads to a decreased,
site-dependent tunneling rate. The decrease has been evaluated to 30% for an atom number
ration of NRb /NK ≈ 10.
42
4 Lattice Modulation Spectroscopy with Bose-Fermi Mixtures
43
5 Absorption Imaging with a Diffraction
Limited Objective
Efficient and reliable detection of atomic densities is a central requirement in all quantum
gas experiments. A number of methods have been developed and are in use.
One of the most widely used technique is a combination of time-of-flight (TOF) expansion
with absorption imaging. After conducting the experiments, the atomic cloud is released
from all confining potentials and allowed to expand freely. A resonantly tuned laser beam
illuminates the atoms. The atoms absorb the light resulting in a shadow. This light
distribution is imaged onto a CCD camera. The imaging onto the CCD is carried out by
a high performance lens system. It is necessary that this lens system images the resulting
shadow with as much detail as possible (i.e. high resolution with as much light collected as
possible) with very few errors introduced (imaging only limited by diffraction).
In this diploma thesis, a new high performance lens system has been designed, build and
tested. The design process is based on theoretical considerations concerning the optical
requirements of absorption imaging. We will show the relevant optical errors introduced
by lenses. With the help of computer aided optical engineering tools, the present detection
setup at the “Bose-Fermi mixtures” experiment is evaluated. Using automatic optimization
algorithms, a complex new design for the detection setup is presented. After mechanical
realization, optical tests reveal the performance of the system.
5.1 Absorption Imaging
Detection according to the absorption imaging scheme is carried out by illumination of the
atomic cloud with a resonant laser beam and successive imaging of the resulting shadow
onto a CCD camera. The Beer-Lambert law is used to calculate the atomic density from the
obtained picture. It relates the intensity of a light beam propagating through a medium
with the distance passed. Consider a light beam with intensity I through a medium of
density ρ in the x-direction. The Beer-Lambert law is then [60]
dI
= −σeff (I) ρ I.
dx
(5.1)
Here σeff (I) is the intensity dependent absorption coefficient
σeff (I) =
σ0
1 + I/Isat + (2δ/Γ)2
(5.2)
44
5 Absorption Imaging with a Diffraction Limited Objective
3
~ω
with σ0 = 3λ2 /2π and Isat = 12πc
2 Γ the saturation intensity. Detection is carried out on
resonance, thus the detuning is δ = 0. This equation is integrated along the direction of
the detection laser beam passing the atomic cloud. It becomes
ln
I0 I0 − I
= σ0 ρ
+
I
Isat
(5.3)
where I0 is the initial intensity and I is the intensity after passing the ensemble. The right
side of the equation is the optical depth of the cloud and is proportional to the number of
atoms.
In the experiment, three pictures are taken. The first picture is the absorption image IA
where the shadow of the atoms is imaged. It represents the intensity after passing the
cloud. A short amount of time later (on the order of milliseconds), a second picture is
taken called reference image IR . This image is taken without any atoms present. It is used
as an intensity reference. The third image taken is a dark picture ID where the detection
beam is blocked. To account for stray light this image is subtracted from the reference and
the absorption image. Using this intensities, relation (5.3) reads
ln
IR − ID IR − IA
+
= σ0 ρ.
IA − ID
Isat
(5.4)
Taking these three images allows precise determination of the density distribution of the
atomic cloud. The imaging of the three pictures is carried out by a lens system and a CCD
camera. A schematic of the detection system in use is shown in Fig. 5.1. The detection
laser coming from the left illuminates the atoms. The resulting light is collimated by a lens
an afterwards focussed onto the camera. A central part of the following chapter is the setup
of the lenses used to create the image on the camera.
imaging system
detection laser
g
atomic sample
b
CCD camera
Figure 5.1: Schematic of absorption imaging. Shown is a schematic of the absorption detection
system used at the Bose-Fermi mixtures project. A resonant laser is passing through the atomic sample.
Light is absorbed by the atoms and the shadow is imaged onto a CCD camera.
5.2 Image Formation, Resolution and Abberrations
This part focusses on the process of image formation. We will see how the maximum
achievable resolution is determined and which errors occur due to deviations from paraxial
optics.
5.2 Image Formation, Resolution and Abberrations
45
5.2.1 Resolution
A lens forms an image of an object onto an image surface. The finite size of the lens leads
to diffraction at the edges. Hence the aperture (i.e. diameter for which light can pass
the optical assembly) is the fundamental limit for the achievable quality of an image. An
optical system only limited by diffraction is called diffraction limited. The resolution (i.e.
smallest observable structures) of such a system is determined by the diffraction pattern on
the image surface. For a spherical aperture a point source creates an intensity distribution
on the image surface called an Airy disk (see Fig. 5.2a). The intensity distribution of an
Airy disk in one dimension is [61]
I(θ) = I0
2J1 (kD/2 sin θ)
.
kD/2 sin θ
(5.5)
I0 is the total intensity, J1 (x) the first order Bessel function, k = 2π/λ and D is the diameter
of the aperture. θ is the angle between the optical axis and the point of interest on the
image surface.
Two point sources near each other (i.e. two features on an object to be imaged) create
two Airy disks. The overlap of the two Airy disks determines if the two point sources
are distinguishable. According to Rayleigh’s resolution criterion two points are considered
resolved when the central peak of one of the Airy disks coincides with the first diffracted
intensity maximum of the other disk (see Fig. 5.2d). According to this criterion, the maximum possible resolution of a system with focal length f and numerical aperture NA is given
by [62]
dRayleigh = 1.22
λ
.
2 NA
(5.6)
Here d is the minimal displacement of two objects that can be resolved. The numerical
aperture determines the amount of light accepted by the optical system.
The Sparrow criterion considers an object resolved as long as there is a minimum between
the central peaks of the two corresponding Airy disks (see Fig. 5.2c). With this the maximal
achievable resolution of a lens setup illuminated with incoherent light is [62]
dSparrow = 0.95
λ
.
2 NA
(5.7)
If coherent light is used for imaging, Rayleigh’s criterion breaks down because it only considers the distance between maxima and minima of the Airy disk and neglects interference
effects. The Sparrow criterion for coherent light is
dcoh = 1.46
λ
.
2 NA
(5.8)
46
5 Absorption Imaging with a Diffraction Limited Objective
(a)
(b)
Airy disk of one point source
two point sources - not resolved
(c)
(d)
two point sources
resolved according to Sparrow
two point sources
resolved according to Rayleigh
Figure 5.2: Resolving limits for an optical system. a A point source creates an Airy disk intensity
pattern. b two point sources near each other are not resolved. c Sparrow’s criterion assumes two point
sources as resolved, when there is a minimum between the two resulting Airy disks. d Rayleigh’s resolution
criterion requires that the first diffracted intensity maximum coincides with the central maximum of the
other Airy disk.
5.2.2 Aberrations
In paraxial optics only small angles between the light rays and the optical axis, and small
distances of these rays to the optical axis are considered. The height h of an object is then
connected to the height y 0 of the image with the following relation:
y 0 = mh.
(5.9)
Here m is the magnification of the optical system.
For large angles or large distances from the optical axis, errors are introduced into the image
. These deviations are called aberrations and degrade the resolution and overall image
quality significantly. The equation describing a light ray passing through an rotational
symmetric optical system incorporating aberrations is written as a power series of the
following form [63]:
y 0 = mh + A1 p
+ B0 p3 + B1 hp2 + B2 h2 p + B3 h3
+
5
X
Ci hi p5−i
(5.10)
i=0
+ ...
The distance of a light ray incident on the aperture of the optical system is p and is measured
from the optical axis. The linear term in p with the coefficient A1 describes defocus and is
5.2 Image Formation, Resolution and Abberrations
47
called a first-order aberration. Terms with coefficients Bi are called third-order aberrations
or Seidel aberrations.
The Seidel aberrations are considered the most important errors [64] and their coefficients
represent the following aberrations: B0 spherical aberration (SA3), B1 coma (CMA3), B2
astigmatism (AST3) and Petzval field curvature (PTZ3) and B3 distortion (DIS3).
Higher-order aberrations are highly suppressed [64] and will only play a minor role in the
following chapters.
The following section describes the monochromatic Seidel aberrations briefly. The section
follows [62] where more in-depth explanations can be found. We neglect chromatic aberrations because the detection is carried out at two very sharp defined wavelengths. Further
the detuning between the wavelengths is in terms of geometric optics very small (13 nm)
thus only minor chromatic effects are expected.
Spherical Aberration (SA3)
The spherical surface of a lens leads to different angles of incidence for light rays passing the
lens at distinct distances from the optical axis. The light rays experience different angles
of refraction after propagating through the optical element. See Fig. 5.3. A blurred image
is the result. SA3 depends only on p and not on the actual object size thus is the most
influential aberration.
Figure 5.3: Spherical Aberration. Spherical aberration occurs due to different refraction of light beams
that enter a lens further away from the optical axis in comparison with rays passing the lens near the
center. The result is a blurred image.
Coma (CMA3)
When light rays penetrate a lens under an angle with respect to the optical axis, light rays
hitting the lens further away from the optical axis are imaged onto a different point than
rays passing through the center of the lens. A point source imaged by an optical system
with coma seems to have a blurred tail similar to a comet. See Fig. 5.4.
48
5 Absorption Imaging with a Diffraction Limited Objective
image
Figure 5.4: Coma. Coma is the result of rays hitting a lens with an angle with respect to the optical
axis. The result is an image blurred and appearing to have a tail like a comet.
Astigmatism (AST3)
A point source which is not on the optical axis of the system will experience different
focal length for the rays passing the optical system parallel (sagittal rays) or perpendicular
(tangential rays) to the line between the point source and the optical axis. The result are
different focal points. An image appears defocused in one direction while focused in the
other.
tangential ray
sagittal ray
lens
point source - off axis
Figure 5.5: Astigmatism. A point source off-axis results in different focal points in the image plane.
Petzval Field Curvature (PTZ3)
Rays emitted from distinct parts of the object are focused onto a spherical image surface
due to different angles of incidence. See Fig. 5.6.
5.3 Raytracing and Computer Aided Optical Engineering
49
image surface
Figure 5.6: Petzval field curvature. Light rays from an extended object have different angles of
incidence on the imaging system. The result is a curved image surface.
Distortion (DST3)
When an off-axis point source is imaged closer to or further away from the optical axis,
then the image is said to be distorted. Straight lines will be projected as curved ones. See
Fig. 5.7. The reason for this aberration is variations of the magnification across the aperture
of the optical system.
object
pincushion distortion
barrel distortion
Figure 5.7: Distortion. Varying magnification across the aperture of the optical system results in
distortion. If the magnification around the center is larger than further away from the optical axis the
distortion is called barrel distortion. Smaller magnification around the center yields a pincushion like
distortion.
Summary
All the above presented aberrations degrade the performance of the detection setup. Aberrations are highly non-linear phenomena and can not be easily corrected. The help of
dedicated simulation tools is necessary. This will be the topic of the next paragraph.
5.3 Raytracing and Computer Aided Optical Engineering
The foregoing section showed that imaging systems will introduce aberrations (i.e. errors)
into the formed image. These aberrations degrade the performance of any optical system.
50
5 Absorption Imaging with a Diffraction Limited Objective
Creating a new optical system has the goal to deliver the needed performance in the parameters needed for the later application. This could be exact color representation, wide
area of focus or high resolution. The ultimate goal is of course to create a design which is
diffraction limited in all the areas of application. This is not always possible. For example
lenses for digital-single-lens-reflex (DSLR) cameras are often not diffraction limited over the
whole range1 . Still they deliver high performance in their area of application.
In general such a high performance system (DSLR lens, high resolution optics, etc.) can not
be build out of one or two optical elements [65] and even for such small amounts of lenses
the number of adjustable parameters is huge (materials, thicknesses, curvatures, distances
etc.). Thus designing of an well corrected optical system is nowadays done using specialized
simulation and optimization programs. Popular applications are Code V by Synopsys,
ZEMAX by Radiant ZEMAX LLC. or OSLO by Lambda Research. All these programs
belong to the group of raytracers and incorporate functions for evaluation and design of
optical setups. In the framework of this thesis OSLO light has been used.
In this chapter the fundamentals of raytracing are briefly presented. Tools for evaluation of
optical designs are shown and lastly an introduction to automatic optimization of optical
designs is given.
5.3.1 Raytracing
Raytracing programs simulate the propagation of light rays through optical media and are
used in a multitude of applications. For example they are used to create high resolution
computer graphics. Generally optical raytracers are based on geometrical optics and use
Snell’s law for tracing [66]. A typical raytracing algorithm traces a given ray of light until
hitting the first surface. The angle of incidence is calculated and using Snell’s law the angle
of refraction is evaluated. The refracted ray is then traced until hitting the next surface.
This procedure is repeated until the ray hits the image surface. By tracing rays throughout
the whole aperture and under different angles, the properties of an optical design can be
analyzed.
5.3.2 Evaluation of Optical Designs
Raytracing allows the evaluation of optical systems with different techniques. First of all the
aberration coefficients introduced in equation (5.10) can be calculated. With this coefficients
it is possible to specifically optimize a system towards the needed performance. While this
is very powerful and necessary for optimization, other tools are better suited to evaluate the
overall performance (resolution, diffraction limitation, ...). Especially tools which represent
the overall optical performance graphically allow a fast and reliable evaluation. In the
following three tools are presented: the spot diagram gives a very intuitive picture of the
image formed by a lens setup. The wavefront analysis allows quantification of the amount of
errors introduced in comparison with an diffraction limited system. Finally the modulation
1
compare for example with http://www.pbase.com/samirkharusi/canon_mtf_curves
5.3 Raytracing and Computer Aided Optical Engineering
51
transfer function (MTF) incorporates the effect of diffraction and shows directly how well
small structures are imaged by the system.
Spot Diagram and Wavefront Analysis
The following two sections describe the spot diagram and the wavefront analysis.
Spot Diagram To generate a spot diagram a large number of rays is traced through an
optical system. After propagating through the system, the rays hit the image surface.
Each point where a ray hits the image surface is marked with a spot. This yields the spot
diagram. A typical spot diagram for a well corrected lens system is shown in Fig. 5.8. The
overall shape and density gives a very nice overview of the optical performance.
Figure 5.8: Spot Diagram. Tracing a large number of rays
through an optical system yields a spatial distribution of rays
hitting the image surface. This spot diagram was calculated
using OSLO for a well corrected lens system. The point
source has been placed off-axis resulting in a deformed spot
diagram. The black circle represents the Airy disk.
Calculating the size of the Airy-disk gives an impression how well the system is diffraction
limited. All spots lying within the calculated Airy-disk is a indication for an diffraction
limited lens setup. However, the spot diagram does not show the real intensity distribution
within the disk.
Wavefront Analysis To decide whether an lens system is diffraction limited and how
much the image is influenced by the aberrations is nicely seen in a wavefront analysis. A
perfect imaging system images a point source onto a point on the image surface. Thus light
waves leaving the imaging system form a wave with a spherical wavefront centered around
the point on the image surface. Aberrations will deform this wavefront. The amount of
deformation is measured as a peak-valley optical path difference (P-V OPD) as shown in
Fig. 5.9a. It can be shown that Rayleigh’s diffraction criterion is fulfilled as long as the P-V
OPD between the edge of the wavefront and the inner part is smaller than a quarter of a
wavelength [62]. For a heavily distorted wavefront, this criterion can be reformulated: to
fulfill Rayleigh’s criterion (i.e. being diffraction limited) the root mean square of the optical
path difference (RMS OPD) has to be smaller than 7% of λ.
In Fig. 5.9b a wavefront computed by OSLO is shown. A well corrected system with a point
source off-axis has been analyzed. The RMS OPD has been evaluated to 0.06λ and the P-V
OPD is 0.31λ. Hence the system under study is not diffraction limited but only by a small
margin.
52
5 Absorption Imaging with a Diffraction Limited Objective
(a)
(b)
OPD
0.23λ
y
distorted wavefront
-0.14λ
perfect wavefront
x
Figure 5.9: Wavefront analysis. The wavefront analysis shows the deviation of the wavefront from a
perfect one. a shows the optical path difference (OPD) between a perfect wavefront and a distorted.
In b shown is the wavefront for an well corrected optical system with a point source 1 mm off-axis
(y-direction).
Modulation Transfer Function
The wavefront analysis presented in Sec. 5.3.2 is based on raytracing data. Hence the
effect of diffraction is not incorporated. This shortcoming is circumvented by measuring
the modulation transfer function (MTF). The MTF measures the contrast of an image
depending on the reciprocal size of the object in front of the lens system. The reciprocal
size is measured in lines/mm. The object is considered to have a sinusoidal dark-bright
contrast. With that a line is one dark-bright cycle. The MTF is calculated by performing
a Fourier transform on the intensity distribution of the image. The modulation transfer
function (MTF) incorporates diffraction as well as refraction. The MTF is written as
MTF(f ) =
output(f )
input(f )
(5.11)
where input is the object in front of the optical system and output is the resulting image.
The input function is considered to be sinusoidal with ω = 2πf .
The diffraction limited (i.e. optimal) MTF for a lens setup with spherical aperture is [67]

λf
2
MTF(f ) = arccos
π
2 NA
−
λf
2 NA
s
1−
λf
2 NA
2

.
(5.12)
The diffraction limited MTF reaches zero at
2 NA
(5.13)
.
λ
This is the fundamental limit for the resolving power of an optical system and is called
cutoff frequency. The cutoff frequency resembles Sparrow’s resolution criterion (5.7).
f0 =
A MTF for a diffraction limited lens with λ = 780nm, NA = 0.1481 is shown in Fig. 5.10.
For this setup the cutoff frequency is f0 ≈ 380 lines/mm
Aberrations will reduce and deform the MTF. The universality of the MTF makes it one
of the standard tools to evaluate lens designs.
5.3 Raytracing and Computer Aided Optical Engineering
53
modulation transfer
1
cutoff
0.8
0.6
0.4
0.2
0
0
100
200
300
spatial frequency [lines/mm]
400
Figure 5.10: MTF of an diffraction limited lens. Shown is the MTF for an diffraction limited lens
with λ = 780 nm and NA = 0.1481. The green bar marks the cutoff frequency at f0 ≈ 380 lines/mm.
5.3.3 Optimization
The last sections showed the tools used to evaluate the following optical designs. As outlined
in the introduction to Sec. 5.3, the goal for a new lens design is to deliver the desired
performance in the areas of application. In most cases optical systems consist of more
than one or two elements, such that the number of variable parameters is large (materials,
curvatures, distances, thicknesses, ...). Hence optimization is in most cases carried out by
automatic optimization algorithms. The bases for an automatic optimization is an error
function (sometimes also: merit function). The method of optimization used throughout
this chapter is a damped least square (DLS) method with an error function φ of the following
form [66]:
φ(~c) =
m
X
wi fi2 (~c).
(5.14)
i=1
Here the vector ~c represents all parameters of the system. These are thicknesses of optical
elements, curvatures of surfaces, materials of lenses and index of refractions of space between
lenses. Also all parameters allowed to vary are stored in ~c. fi is called an operand and
returns a specific deviation from a target value. For example an operand could ensure
that the focal length of the system is a specific value. m is the number of operands. The
coefficients wi are weights to define the relative importance of the operands. The DLS
optimization tries to minimize φ. The primary task for a lens designer is to define a proper
error function, which incorporates the desired properties of a lens design. Unfortunately
because aberrations can only be minimized and not completely removed, the error function
has in general local minima. So during the optimization process it is necessary to manually
check for other solutions by varying the starting conditions and impose restrictions on the
variable parameters ~c. These restrictions are also necessary to make sure that a design is
practical. For example a 2 m thick lens is not often very reasonable. The operands and
weights used to develop the new detection objective is presented in Sec. 5.5.2.
54
5 Absorption Imaging with a Diffraction Limited Objective
5.4 Old Detection Setup
The setup used so far at the "Bose-Fermi mixture" project in Hamburg consists of two
achromatic lenses with focal length f1 = 120 mm and f2 = 250 mm as shown in Fig. 5.11.
The first lens collimates the incoming light while the second lens focusses the image onto the
camera. The atomic sample is prepared in a glass cell while detection is carried out outside
of the apparatus. Thus the wall of the glass cell has to be incorporated into the evaluation
process. This lens pair magnifies by approximately a factor of f2 /f1 = 250 mm/120 mm ≈
achromatic lens
f=120mm
atomic sample
11mm to inner cell wall
120mm
glass cell - 2mm thick
magnificaion ≅2.1
250mm
achromatic lens
f=250mm
CCD camera - PCO.pixelfly
Figure 5.11: Old detection setup. The so far used detection setup is based on two lenses which first
collimate the light and than focus it onto the camera.
2.1. The used camera is a PCO pixelfly with a pixel size of 6.45 µm × 6.45 µm. The object
side numerical aperture is NA ≈ 0.1638.
The system presented above has been simulated and analyzed using OSLO. To compare
with the later developed new lens design the aberration coefficients (see Sec. 5.2.2) are
presented in Tab. 5.1.
aberration
SA3
CMA3
AST3
PTZ3
DIS3
computed value
−0.3086
0.1627
−0.0879
−0.0015
0.0493
Table 5.1: Aberration coefficients for the two lens detection setup. Shown are the aberration
coefficients for the two lens detection setup computed by OSLO for a point source 1mm off axis.
A spot diagram for an on axis as well as an off axis point source is shown in Fig. 5.12.
The resulting diagram (Fig. 5.12a) shows a large amount of rays hitting outside of the Airy
disk. The radius of the Airy disk has been calculated to approximately 6 µm. The summed
root mean square radius of the spot diagram has been evaluated to 15 µm. The off axis spot
diagram (Fig. 5.12b) shows large amounts of coma and a much larger overall spot. The
Airy disk is barely visible as a tiny spot in the upper part of the image.
5.4 Old Detection Setup
55
(a) on optical axis
(b) 1mm off optical axis
Figure 5.12: Spot diagrams for the two lens detection setup. a shows the spot diagram for a point
source on the optical axis. The black circle represents the Airy disk. b shows the spot diagram for a
point source 1mm off axis. The Airy disk is barely observable in the upper part.
The resulting wavefront is shown in Fig. 5.13. The OPDs for the on axis source (Fig. 5.13a)
are: P-V OPD = 1.232λ and RMS OPD = 0.298λ. For the off axis source (Fig. 5.13b) the
OPDs are: P-V OPD = 2.355λ and RMS OPD = 0.475λ. Thus the system is even for an
on axis point source subject to large amounts of aberrations.
(a) on optical axis
(b) 1mm off optical axis
1λ
y
1λ
y
-1.9λ
x
-1.9λ
x
Figure 5.13: Wavefront analysis of the two lens detection setup. a shows the wavefront for a point
source on the optical axis. b shows the same point source moved 1mm off axis.
This is also consistent with the resulting MTF shown in Fig. 5.14. Because of geometrical
restrictions of the experimental setup (magnetic coils reduce the numerical aperture in one
direction) two curves are shown which represent two orthogonal axes (parallel and sagittal
axis). Both curves show huge deviations from an diffraction limited system.
5.4.1 Summary
In this chapter the old experimental setup used to perform absorption imaging has been
simulated. It was shown that the two lens setup introduces large errors into the image.
56
5 Absorption Imaging with a Diffraction Limited Objective
modulation transfer
1
diff. limited
parallel
sagittal
0.8
0.6
0.4
0.2
0
0
50
100
150
200
spatial frequency [lines/mm]
Figure 5.14: MTF of the two lens setup. Shown is the MTF for the two lens detection setup. Because
of geometrical restrictions (the magnetic coils obscure a small part reducing the numerical aperture in
one direction) the sagittal and parallel directions have slightly different MTFs. The diffraction limit is
shown in green.
Even an point source placed on axis (so only SA3 influences the image) is subject to large
aberrations such that diffraction limited imaging is not possible. Thus the maximal possible
resolution is not achieved and detail is lost.
5.5 Design of a New Diffraction Limited Detection System
In the foregoing chapter we have shown, that the old detection setup is subject to large
amounts of aberrations. The resolving capabilities are thus very limited. Based on these
considerations we decided to design and build a new optical setup which is subject to very
small aberrations.
5.5.1 Requirements for the New Design
The requirements for the new optical system are derived from typical experimental conditions. The atoms are, in most of the cases, not more than 3 mm displaced from the optical
axis. Hence the off axis performance is not the determining factor but of course should
still be good. Further we want to use the available NA. The available NA is limited by
geometrical restrictions to approximately NA ≈ 0.17. This numerical aperture allows a
absolute maximum resolution of 2.3 µm. The optical lattice used in the experiment has a
lattice site distance of 515 nm. Thus it will not be possible to achieve single site resolution.
Still the resolution should be as high as possible to perform high precision measurements
after TOF expansion. This in turn means that the design should be diffraction limited for
on axis sources as well as for sources placed slightly off-axis (up to 1 mm). For experimental
convenience the magnification of the system should be easily changeable.
5.5 Design of a New Diffraction Limited Detection System
57
5.5.2 Underlying Design and Optimization Process
The basis for the finally obtained lens design has been taken from [64] and [65]. The design
presented in these references is based on a setup consisting of four lenses which are used to
collimate the light from a source. These four lenses have curvatures of their surfaces chosen
such that aberrations introduced by one lens is compensated by another.
The focussing is then performed by a final optical element which in the following is called
focussing lens. The ratio of the focal lengths of the four lens objective and the focussing
lens determines the magnification. We decided to use an achromatic lens as the focussing
lens to minimize focal shifts for the two different detection wavelengths.
In a first step the principle design shown in [64, 65] has been adapted from one inch diameter
lenses to 50 mm lenses simply by approximately doubling the lens diameters. This doubling
essentially destroys the good optical performance of the original design. Nevertheless the
basic informations like curvatures (concave, convex) and distance ratios are useful for the
automatic optimization
The second step was to define an error function for optimization (see Sec. 5.2.2). The new
setup was expected to be a little bit bigger than the old two lens setup, so we decided to
increase the distance between the atomic sample and the first lens by 1.5 cm. This results in
a desired front focal length (FFL) of 135 mm. The FFL is the distance between the object
(atoms) and the first surface of the imaging system.
The effective focal length (EFL) is the optical focal length. It is not directly related to
the FFL. The EFL directly influences the achievable magnification, so an effective focal
length of 135 mm is the target focal length. To reach this target an operand to ensure
an corresponding EFL is added to the error function. The weighting factor for the EFL
operand has been chosen low. The low weighting factor allows the optimization algorithm
to slightly vary the EFL when through this variation an aberration can be lowered.
Because the atoms are not often displaced very much from the optical axis spherical aberration has been determined to be the most influential aberration. Therefore spherical aberration of third-order (SA3), fifth-order (SA5) and seventh-order (SA7) is incorporated into the
error function. The other Seidel aberrations should be minimized, but with lower priority
(weighting factors).
The resulting error function is shown in Tab. 5.2.
The so defined error function is minimized by allowing the variation of the curvatures of the
surfaces, the distances between the optical elements and the thicknesses of the elements.
For availability reasons BK7 has been chosen as the material of the lenses.
The third step it to replace the first virtual lenses (as seen from the atoms) with a commercially available lens by CVI Melles Griot2 . Again the system is optimized and the last lens
was replaced. This process is iterated for the remaining virtual lenses until all are replaced
by real counterparts.
2
see http://www.cvimellesgriot.com/
58
5 Absorption Imaging with a Diffraction Limited Objective
weight wi
0.1
1.0
0.3
0.3
0.3
0.3
0.2
0.2
operation
ensure 135 mm
minimize
minimize
minimize
minimize
minimize
minimize
minimize
operand fi
EFL
SA3
CMA3
AST3
PTZ3
DIS3
SA5
SA7
Table 5.2: Error function for new lens setup. Shown are the operands and weights used for all
automatic optimizations of the new lens design. The weighting factor is given for each operand. The
operand column shows which physical property of the system is manipulated and the operation column
shows in which way this is done. For example all aberrations should be minimized but the focal length
(EFL) should remain at 135 mm.
Finally, a last optimization iteration is performed, allowing the variation of the distances
between the real lenses. The in this way obtained design is shown in the next section.
5.5.3 New Design
This optimization resulted in a final design shown in Fig. 5.15. The CVI Melles Griot part
not to scale
glass cell - 2mm thick
atomic sample
11mm to inner cell wall
LPX277
123.9mm
LPX577
31.1mm
LDX373
40mm
lens diameters: 49mm
LPK161
21.1mm
interchangeable
achromatic lens
Figure 5.15: Diffraction limited detection setup. Shown here is the new design for a detection setup
at the Bose-Fermi mixtures experiment. The image is not to scale. Used lenses including their CVI Melles
Griot part numbers are shown as well as the distances between them.
numbers are depicted as well as the distances between the lenses. In Tab. 5.3 the precise
data for all lenses is shown. The materials, thicknesses and radii of curvatures (for the first
and second surface) are shown as well as the focal length of the corresponding element. All
lenses have a diameter of 49 mm ± 0/50 µm. The lens design has a numerical aperture of
0.1481 and an effective focal length of 135 mm. The FFL is 136.9 mm.
For focussing an achromatic lens is chosen as explained above. To change the magnification
of the imaging system, the last lens is interchangeable. In Tab. 5.4 the different achievable
magnifications and the corresponding lenses are shown.
5.5 Design of a New Diffraction Limited Detection System
part
glass cell
LPX277
LPX577
LDX373
LPK161
material
silica glass
BK7
BK7
BK7
BK7
d [mm]
2
6
4
4.9
3
r1 [mm]
336.37
-129.68
59
r2 [mm]
103.74
194.51
336.37
-
f [mm]
200
375
325
-250
Table 5.3: Lenses used in the new detection setup. The table shows the first four lenses used to
collimate the light with their relevant properties such as the thickness d, curvatures r1 and r2 , material
and focal length f.
5.5.4 Evaluation of the New Design
The newly developed design is now analyzed using the tools presented above. The analyses
shown here has been performed for the twofold magnification. The results for the other
magnifications are summarized at the end of this section. First all the aberration coefficients
(see Sec. 5.2.2) are presented in Tab. 5.5 together with the difference to the aberration
coefficients of the old system (shown in Tab. 5.1). A vary large decrease (90 percent and
more) for SA3, CMA3, AST3 and DIS3 is observed. PTZ3 is reduced by nearly 30%.
(a) on optical axis
(b) 1mm off optical axis
Figure 5.16: Spot diagrams for the new detection setup. a shows the spot diagram for a point
source on the optical axis. The black circle represents the Airy disk. b shows the spot diagram for a
point source 1mm off axis.
magnification
1.1
2.2
3.0
4.4
5.1
part
LAO551
LAO656
LAO690
LAO807
LAO809
f [mm]
150
300
400
600
700
Table 5.4: Lenses used for focussing the collimated light onto the camera. Shown are the achromatic lenses used to focus the light onto the CCD camera. Depending on the desired magnification a
different lens can be mounted.
60
5 Absorption Imaging with a Diffraction Limited Objective
aberration
SA3
CMA3
AST3
PTZ3
DIS3
new system
−0.0190
+0.0038
−0.0017
−0.0011
−0.0007
old system
−0.3086
+0.1627
−0.0879
−0.0015
+0.0493
difference
−94%
−97%
−98%
−27%
−99%
Table 5.5: Aberration coefficients for the new lens detection setup. Shown are the aberration
coefficients for the new lens design computed by OSLO for a point source 1mm off axis. Further the
coefficients for the old system are shown together with the difference in percent.
The resulting spot diagram for a point source on axis as well as off axis is shown in Fig. 5.16.
The diagram shows a much more localized spot distribution in comparison to Fig. 5.12. Also
the deformation due to off axis effects is much smaller. Note that the spots for an off axis
point source are now very localized and most of the rays hit inside the Airy disk. This is
a huge improvement over the old system where the Airy disk is barely recognizable (see
Fig. 5.12b).
The wavefront analysis confirms this finding. It is shown in Fig. 5.17. The OPDs are:
P-V OPD = 0.1327λ and RMS OPD = 0.03595λ. For the off axis source (Fig. 5.13b) the
OPDs are: P-V OPD = 0.3122λ and RMS OPD = 0.05762λ. This shows that the new
design is very well diffraction limited on axis and thus is expected to reach the theoretically
possible resolution. Moving the point source 1 mm off axis delivers a much better result
than the old setup.
(a) on optical axis
(b) 1mm off optical axis
0.25λ
y
0.25λ
y
-1.33λ
x
-1.33λ
x
Figure 5.17: Wavefront analysis of the new detection setup. a shows the wavefront for a point
source on the optical axis. b shows the same point source moved 1mm off axis.
A MTF for the new setup is shown in Fig. 5.18 and supports the findings from the wavefront
analysis and the spot diagram. The resulting modulation is over a wide range near perfect
and shows again the huge improvements over the old design.
These evaluation steps have been performed for all magnifications. No substantial differences have been observed and all magnifications deliver a similar performance near to a
perfect result.
5.5 Design of a New Diffraction Limited Detection System
61
1
diffraction limit
modulation transfer
0.8
MTF new design
0.6
0.4
0.2
0
0
50
100
150
200
spatial frequency [lines/mm]
Figure 5.18: MTF of the two lens setup. Shown is the MTF for the new optics design. The diffraction
limit is shown in green whereas red shows the designed optical system.
Summary
In this part a new design for the optical components in the absorption detection setup is
presented. Starting with a basic design and a carefully chosen error function, the system
has been iteratively adapted and improved. Consisting of five optical elements, the new
design shows much better optical performance and is due to interchangeable magnifications
experimentally much more convenient.
5.5.5 Mechanical Realization
The physical setup of the new detection setup was engineered using computer aided design
software. A technical drawing is shown in Fig. 5.19. The basis for the system is a lens
tube with a diameter of 49.05 mm. This tube holds the first four lenses (see 5.5.3). To
achieve the desired distance between the lenses is maintained by precisely engineered distance rings with a diameter of 49.00 mm. The lenses are individually cut down to a diameter
of 49.00 mm ± 0/50 µm. The diameter of the tube, lenses and distance rings needs to be
precisely manufactured to achieve accurate axial alignment. The achromatic lenses used for
collimating are housed separately in a small tube system. This focussing tube is screwed
onto the large tube system and is aligned using precision alignment pins. This mounting
allows simple changing of the magnification (i.e. changing the focussing lens). The whole
optical assembly can be moved on a xy-translation stage.
All tubes are made of aluminum to reduce overall weight. The distance rings are made
out of stainless steel to improve stability. Brass is used to manufacture the middle part of
the xy-translation stage. This adds additional weight but allows frictionless moving of the
stage.
62
5 Absorption Imaging with a Diffraction Limited Objective
interchangeable lens housing
xy-translation stage
Figure 5.19: Mechanical setup of the new lens design. The mechanical realization of the new lens
setup is shown. It consists of a tube, housing the first four lenses used for collimation. The focussing lens
is suported separately and is mounted with screws and alignment pins. The whole system is mounted on
a xy-translation stage.
5.5.6 Optical Tests
To characterize the optical system, a USAF 1951 resolution test chart (shown in Fig. 5.20)
and a grid distortion test chart is used (see Fig. 5.26).
The resolution test chart consists of dark-bright line pairs of different sizes. The sizes are
specified and are identified by the surrounding numbers. Illumination of the chart is carried
out by a laser tuned to 780 nm. The chart is imaged onto a CCD camera using the new lens
system. The distance between the chart and the first surface of the lens system is chosen
according to the expected distance between the lens and the atomic sample later in the
experiment. A PCO pixelfly3 is used. This camera is of the same type as the one used later
in the experiment. This allows us to directly transfer the experiences made outside of the
apparatus to the situation where the lens is incorporated into the experimental setup.
As a benchmark, the maximum achievable resolution for incoherent light is calculated using
a NA of 0.1481:
dSparrow ≈ 2.7 µm.
(5.15)
As a first step the resolution test chart is imaged with a perfectly aligned setup. The
resulting pictures for the twofold and fivefold magnification are shown in Fig. 5.21. Because
3
pco.pixelfly, see http://www.pco.de/
5.5 Design of a New Diffraction Limited Detection System
63
Figure 5.20: USAF resolution test chart. The test chart consists of dark-bright line pairs of different
sizes.
the theoretically achievable resolution is 2.7 µm the twofold magnification makes it necessary
to distinguish structures off about 6 µm. This is approximately the size of one pixel of the
camera. This means that we are pixel limited and are not able to observe the resolution
limit. Obtained pictures are shown in Fig. 5.21a. The situation changes drastically when
(a) 2x magnification
(b) 5x magnification
Figure 5.21: Maximum achievable resolution. Shown are the images obtained for the 2x and 5x
magnification. The images for the 2x magnification is pixel limited. The 5x magnification shows a
resolution limit of 2.8 µm marked in red.
using the fivefold magnification which is shown in Fig. 5.21b. The last observable structure
is the fourth line pair in group seven. This corresponds to a resolution of 2.8 µm which is
indistinguishable to the expected limit. Thus the goal of diffraction limited imaging has
been reached.
Further characterization is now performed by moving the test chart from the optimal position. One key parameter is the depth of field (DOF) determining the distance which
the object can be moved without loosing resolution. To determine the DOF the distance
between the test chart and the first lens surface is varied and the resulting resolution is
recorded. The results are shown in Fig. 5.22. The graphs show that the region of maximum
64
5 Absorption Imaging with a Diffraction Limited Objective
(a) 2x magnification
(b) 5x magnification
17.5
12.4
11
13.9
resolution [µm]
resolution [µm]
15.6
12.4
11
9.8
8.8
7
5.5
4.4
−0.5
−0.3
−0.1 0 0.1
0.3
9.8
8.8
7.8
7
6.2
5.5
4.4
3.5
2.5
−0.3 −0.2 −0.1
0.5
0
0.1
0.2
0.3
test chart position [mm]
test chart position [mm]
Figure 5.22: Depth of field measurement. By varying the position of the test chart the depth of field
(DOF) is measured.
resolution is very small (below 0.1 mm) which on the one hand needs good alignment in
the experiment but on the other hand reduces the impact of light coming from out of focus
objects. Also it can be seen that the data obtained for the 5x magnification seem much
more reliable because the curve shows a much smoother overall slope. We attribute this to
the pixel limitation for the 2x magnification which makes reliable resolution measurements
complicated.
To determine how well the system has been assembled the DOF measurement has been
adapted. The test chart position is again varied but this time the amount of displacement
of the camera to refocus is recorded. The result is shown in Fig. 5.23.
(a) 2x magnification
(b) 5x magnification
150
measured
fit
10
camera position [mm]
camera position [mm]
15
5
0
−5
−10
measured
fit
50
0
−50
−100
−150
−15
−20
−3
100
−2
−1
0
1
2
test chart position [mm]
3
−200
−5 −4 −3 −2 −1 0 1 2 3 4 5
test chart position [mm]
Figure 5.23: Focus shift depending on test chart postition. Varying the position of the test chart
and successive focussing allows the determination of the magnification by fitting a theory curve to the
data points.
Using 1/f = 1/b + 1/g and m = b/g a relation between the object (i.e. test chart) shift ∆g
5.5 Design of a New Diffraction Limited Detection System
65
and the focus shift ∆b is deduced:
∆b =
m2 ∆g
.
1+ m
f ∆g
(5.16)
Here m is the magnification, f the focal length. f plays only a minor role for the fit thus
it has been set to values calculated by OSLO. The fit resulted in:
m2x,OSLO = 2.20
m2x = 2.19
(5.17)
m5x,OSLO = 5.07
m5x = 4.98
Comparing with the values calculated by OSLO (without the glass cell because the tests
have been performed outside of the experiment) yields deviations which are smaller than
2%. The remaining deviation is attributed to errors in finding the correct focus position
and minor misalignments of the whole setup. Overall the very good agreement between the
simulated and the measured parameters strongly suggests that the system is properly build
and aligned.
The field of view (FOV) is measured by varying the position of the test chart orthogonal to
the optical axis and recording the resulting resolution. The result is shown in Fig. 5.24. Seen
(b) 5x magnification
12.4
11
11
9.8
resolution [µm]
resolution [µm]
(a) 2x magnification
9.8
8.8
7.8
7
6.2
4.9
3.9
2.8
−3
8.8
7.8
7
6.2
5.5
4.9
3.9
2.8
−2
−1
0
1
2
test chart off axis position [mm]
3
−3
−2
−1
0
1
2
3
test chart off axis position [mm]
Figure 5.24: Field of view measurement. Variation of the transversal position of the test chart allows
the determination of the field of view.
is a dependence of the maximum observable resolution on the off axis distance. Comparing
with Sec. 5.5.4 we expect to remain diffraction limited close to 1 mm off axis shift. This is
not observed but the measurement is influenced by systematic errors. The finite size of the
resolution test chart is a major problem. Moving the chart into one direction, away from
the optimal position, it can happen that the originally observable structure is not resolved
anymore, but the next bigger structure is even further away and therefore also not observable. During the measurement the focus has not been realigned so that another possible
problem could be some amounts of Petzval field curvature. Nevertheless, the resolution
66
5 Absorption Imaging with a Diffraction Limited Objective
stays below 11 µm for a range of 6 mm which is the region in which the atoms typically are
allowed to expand. In realistic experimental conditions, the region where the position of
the atoms is varied stays below 1.5 mm which in turn means a resolution of below 4 µm.
At a mixture experiment detection has to be carried out with two wavelengths: one for each
species. We use 767 nm for Potassium and 780 nm for Rubidium. So we are interested in the
behavior of the system when changing the wavelength of the detection laser. Figure 5.25
shows the shift in focus for four different wavelengths. Seen is a shift of 2 mm between
6
focus shift [mm]
4
2
0
−2
−4
750
767
780
wavelength [nm]
795
Figure 5.25: Shift of focus for different wavelengths with 5x magnification. Changing the wavelength of the detection light results in a focus shift.
Potassium and Rubidium wavelengths. This shift is experimentally not limiting because
often only one species is detected using the new lens setup while the other species is detected
on another detection axis.
To probe the behavior of the system regarding Petzval field curvature and distortion the
grid distortion test chart is used. To image as many of the grid lines as possible (widest
view angle) the 1x magnification is used here. The obtained image is shown in Fig. 5.26.
Comparing the straight red lines with the lines imaged by the detection system, no Petzval
field curvature or distortion is observed.
The tests shown here have been performed on all available magnifications. All showed the
predicted behavior and performed similarly.
Summary
The optical tests performed in this chapter are based on a standardized resolution test chart.
The test results strongly suggest that the design worked the way it has been developed.
By reaching the theoretically possible resolution it is shown that diffraction is the limiting
factor and aberrations play only a minor role.
5.6 Summary
67
Figure 5.26: Grid chart imaged with 1x magnification. A periodic grid is imaged with the detection
system. No Petzval field curvature or distortion is observed. In red, straight and orthogonal lines are
shown.
5.5.7 The New Lens Build into the Experiment
The new lens system has been successfully integrated into the experiment. As a alignment
test, the magnification of the 1x and the 2x magnification is measured. This is done by
preparing a BEC in an optical dipole trap and performing a TOF measurement. By varying
the time between trap switch off and detection a free fall parabola is observed. The resulting
data is fitted and thereby the magnification can be deduced. In Tab. 5.18 the theoretically
obtained values, the values measured with the test chart and the magnifications obtained
in the experimental setup are shown.
m1x,OSLO = 1.11
m1x,measured = 1.10
m1x,freefall = 1.12
m2x,OSLO = 2.22
m2x,measured = 2.19
(5.18)
m2x,freefall = 2.28
The results agree well with the expected values. This allows two conclusions: First, the
alignment of the lenses within the optical assembly is accurate. Second, the alignment in
the experimental setup is precise.
5.6 Summary
This chapter showed the design and realization of a diffraction limited lens system to efficiently perform absorption detection. The system has been optimized using a dedicated
error function. Simulation and experimental testing showed the predicted behavior. The
objective has been built into the experiment and showed again good agreement between theoretical predictions and real behavior. Overall it was possible to build an high performance,
diffraction limited lens system with a maximal resolution of 2.8 µm. In the near future a
new camera with a high quantum efficiency (95%) will be build into the experiment. The
68
5 Absorption Imaging with a Diffraction Limited Objective
new lens system together with the high performance camera will improve the quality of the
detection especially for dilute fermionic clouds.
69
6 Conclusion and Outlook
In the context of this thesis a novel momentum resolved spectroscopy method has been
developed and employed. Based on modulating the lattice amplitude and successive band
mapping, the band structure of fermions in optical lattices could be measured with high
precision. A substantial effect of the underlying harmonic confinement has been observed,
significantly altering the spectra. The high precision of the method allowed the direct
observation of a shift in the potential experienced by fermionic atoms with an admixture
of bosons. The shift is explained in terms of an effective potential induced by attractive
interactions, leading to an occupation dependent decrease of the fermionic tunneling rate.
This measurements could be a cornerstone to understand the Superfluid to Mott-insulator
transition shift observed in Bose-Fermi mixtures.
While developing the spectroscopy method some effects could be observed which will be
studied in the near future. For example after creating a particle-hole excitation the dynamics
shown below has been observed. Further the lifetime of the created holes is much smaller
than the lifetime of the excitations. This features are attributed to the harmonic trapping
potential but have not been fully understood.
waiting time [ms]
20
0
momentum
In the future the method of optical lattice modulation spectroscopy could be employed
to study a multitude of systems. The use of Feshbach resonances between Rubidium and
70
6 Conclusion and Outlook
Potassium could further develop the understanding of interaction-induced effective potentials. By modulation with different frequencies multiple excitations can be created. The
dynamics of this excitations as well as collisions between such excitations promise interesting observations. Using mixtures of fermions in different hyperfine states allows sensitive
probing of the physics of interacting fermions. Again interatomic Feshbach resonances can
be used to change the interaction allowing experiments over a wide range of parameters.
For example the interaction allows the creation of bound state of excited atoms and holes.
This would allow the study of excitons.
These opportunities for research on fundamental physics show the versatile nature of BoseFermi mixture experiments. The flexibility of Bose-Fermi mixtures together with the high
degree of control in optical lattices promise further highly interesting studies in the near
future.
List of Figures
71
List of Figures
1.1
Fermionic band structure for a mixture of bosons and fermions . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
Experimental Setup . . . . . . . . . . . . . . . . . . . .
Different Lattice Geometries . . . . . . . . . . . . . . . .
First Brillouin Zone of a Cubic Lattice . . . . . . . . . .
Band Structure for Rubidium and Potassium . . . . . .
Bose Hubbard Model . . . . . . . . . . . . . . . . . . . .
Superfluid Phase . . . . . . . . . . . . . . . . . . . . . .
Mott-Insulator Phase . . . . . . . . . . . . . . . . . . . .
Phase Transition between Superfluid and Mott-Insulator
Phase diagram and wedding cake structure for bosons in
Phases of the Non-Interacting Fermi-Gas . . . . . . . . .
Phase Transition from Metallic to Band Insulating . . .
Band Mapping Technique . . . . . . . . . . . . . . . . .
Band Mapping Times . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
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lattice
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7
8
9
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12
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13
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16
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3.1
3.2
3.3
3.4
3.5
3.6
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
Effect of modulating the lattice amplitude . . . . . . . . . . . . . . . . . .
Experimental sequence of lattice modulation spectroscopy . . . . . . . . .
Effective band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TOF images obtained after lattice modulation and band mapping . . . . .
Interband transition probabiities . . . . . . . . . . . . . . . . . . . . . . .
Number of excited atoms depending on amplitude and time of modulation
Fermionic band structure for 5EK
r . . . . . . . . . . . . . . . . . . . . . . .
Fermionic band structure for different lattice depths . . . . . . . . . . . .
Detailed band structure for 5ErK . . . . . . . . . . . . . . . . . . . . . . .
Detailed analysis for fermions in the third band . . . . . . . . . . . . . . .
Broadening due to finite modulation time . . . . . . . . . . . . . . . . . .
Effect of harmonic confinement and finite modulation time . . . . . . . . .
Lattice calibration using bosons . . . . . . . . . . . . . . . . . . . . . . . .
Band structure for bosons . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detailed band structure for bosons . . . . . . . . . . . . . . . . . . . . . .
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20
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27
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31
32
33
34
35
4.1
4.2
Effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Band structure for a mixture of bosons and fermions . . . . . . . . . . . . .
39
40
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Schematic of the absorption detection system
Explanations of the different resolution limits
Spherical aberration . . . . . . . . . . . . . .
Coma . . . . . . . . . . . . . . . . . . . . . .
Astigmatism . . . . . . . . . . . . . . . . . .
Petzval field cuvature . . . . . . . . . . . . .
Distortion . . . . . . . . . . . . . . . . . . . .
Example of a spot diagram . . . . . . . . . .
Example for a wavefront analysis . . . . . . .
44
46
47
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72
6 Conclusion and Outlook
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
MTF for an diffraction limited objective . . . . . . . . . . . .
Setup of the old detection syste, . . . . . . . . . . . . . . . .
Spot diagram for the old, two lens setup . . . . . . . . . . . .
Wavefront analysis for the old, two lens setup . . . . . . . . .
MTF for the old detection setup . . . . . . . . . . . . . . . .
New diffraction limited detection lens setup . . . . . . . . . .
Spot diagram for the new detection setup . . . . . . . . . . .
Wavefront analysis for new optical design . . . . . . . . . . .
MTF for the new detection setup . . . . . . . . . . . . . . . .
Mechanical realization of the new lens . . . . . . . . . . . . .
USAF resolution test chart . . . . . . . . . . . . . . . . . . .
Maximum achievable resolution with the new detection setup
Depth of field measurement . . . . . . . . . . . . . . . . . . .
Focus shift depending on the test chart position . . . . . . . .
Field of view measurement . . . . . . . . . . . . . . . . . . . .
Focus shift for different wavelengths. . . . . . . . . . . . . . .
Grid chart with 1x magnification . . . . . . . . . . . . . . . .
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53
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67
3.1
Fitted lattice depth for spin polarized fermions . . . . . . . . . . . . . . . .
30
4.1
Fermionic tunneling rate for different admixtures of bosons . . . . . . . . .
40
5.1
5.2
5.3
5.4
5.5
Aberration coefficients for the two lens detection setup
Error function for new detection optics . . . . . . . . .
Lenses used in the new detection setup . . . . . . . . .
Focussing lenses used in the new lens . . . . . . . . . .
Aberration coefficients for the new lens setup . . . . .
54
58
59
59
60
List of Tables
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Bibliography
73
Bibliography
[1] A Einstein. Quantentheorie des einatomigen idealen gases. Sitzungsbericht Preussische
Akademie der Wissenschaften, 1924.
[2] M H Anderson, J R Ensher, M R Matthews, C E Wieman, and E A Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science, 1995.
[3] K Davis, M Mewes, M Andrews, N van Druten, D Durfee, D Kurn, and W Ketterle.
Bose-Einstein Condensation in a Gas of Sodium Atoms. Physical Review Letters, 1995.
[4] B DeMarco and D. S. Jin. Onset of Fermi Degeneracy in a Trapped Atomic Gas.
Science, 1999.
[5] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach Resonances
in Ultracold Gases. arXiv, 2008.
[6] CA Regal and M Greiner. Observation of Resonance Condensation of Fermionic Atom
Pairs. Physical Review Letters, 2004.
[7] D Jaksch, C Bruder, J Cirac, C Gardiner, and P Zoller. Cold Bosonic Atoms in Optical
Lattices. Physical Review Letters, 81(15):3108–3111, oct 1998.
[8] M Greiner, O Mandel, T Esslinger, TW Hänsch, and I Bloch. Quantum phase transition
from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 2002.
[9] Julian Struck, Rodolphe Le Targat, Parvis Soltan-Panahi, Maciej Lewenstein, Patrick
Windpassinger, and Klaus Sengstock. Quantum simulation of frustrated magnetism in
triangular optical lattices. arXiv.org, 2011.
[10] J Simon, WS Bakr, R Ma, ME Tai, and PM Preiss. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature, 2011.
[11] Michael Köhl, Henning Moritz, Thilo Stöferle, Kenneth Günter, and Tilman Esslinger.
Fermionic Atoms in a Three Dimensional Optical Lattice: Observing Fermi Surfaces,
Dynamics, and Interactions. Physical Review Letters, 2005.
[12] Robert Jördens, Niels Strohmaier, Kenneth Günter, Henning Moritz, and Tilman
Esslinger. A Mott insulator of fermionic atoms in an optical lattice. Nature, 2008.
[13] U Schneider, L Hackermuller, S Will, Th Best, I Bloch, T A Costi, R W Helmes,
D Rasch, and A Rosch. Metallic and Insulating Phases of Repulsively Interacting
Fermions in a 3D Optical Lattice. Science, 2008.
74
Bibliography
[14] G Modugno, G Roati, F Riboli, and F Ferlaino. Collapse of a Degenerate Fermi Gas.
Science, 2002.
[15] C Ospelkaus, S Ospelkaus, L Humbert, P Ernst, K Sengstock, and K Bongs. Ultracold
Heteronuclear Molecules in a 3D Optical Lattice. Physical Review Letters, 2006.
[16] KK Ni, S Ospelkaus, MHG De Miranda, and A Pe’er. A High Phase-Space-Density
Gas of Polar Molecules. Science, 2008.
[17] S Ospelkaus, KK Ni, D Wang, and MHG De Miranda. Quantum-State Controlled
Chemical Reactions of Ultracold Potassium-Rubidium Molecules. Science, 2010.
[18] Seiji Sugawa, Kensuke Inaba, Shintaro Taie, Rekishu Yamazaki, Makoto Yamashita,
and Yoshiro Takahashi. Interaction and filling-induced quantum phases of dual Mott
insulators of bosons and fermions. Nature Physics, 2011.
[19] K Günter, T Stöferle, H Moritz, and M Köhl. Bose-Fermi Mixtures in a ThreeDimensional Optical Lattice. Physical Review Letters, 2006.
[20] S Ospelkaus, C Ospelkaus, O Wille, M Succo, P Ernst, K Sengstock, and K Bongs.
Localization of Bosonic Atoms by Fermionic Impurities in a Three-Dimensional Optical
Lattice. Physical Review Letters, 2006.
[21] Th Best, S Will, U Schneider, L Hackermuller, D van Oosten, I Bloch, and D S Lühmann. Role of Interactions in Rb87-K40 Bose-Fermi Mixtures in a 3D Optical Lattice.
Physical Review Letters, 2009.
[22] M Cramer, S Ospelkaus, C Ospelkaus, K Bongs, K Sengstock, and J Eisert. Do
Mixtures of Bosonic and Fermionic Atoms Adiabatically Heat Up in Optical Lattices?
Physical Review Letters, 2008.
[23] Dirk-Sören Lühmann, Kai Bongs, Klaus Sengstock, and Daniela Pfannkuche. SelfTrapping of Bosons and Fermions in Optical Lattices. Physical Review Letters, 2008.
[24] Silke Ospelkaus. Quantum Degenerate Fermi-Bose Mixtures of 40K and 87Rb in 3D
Optical Lattices. PhD Thesis, 2006.
[25] Christian Ospelkaus. Fermi-Bose mixtures — From mean-field interactions to ultracold
chemistry. PhD Thesis, 2006.
[26] DC McKay and B DeMarco. Cooling in strongly correlated optical lattices: prospects
and challenges. Reports on Progress in Physics, 2011.
[27] Neil W. Ashcroft and N. David Mermin.
senschaftsverlag, 2005.
Festkörperphysik.
Oldenbourg Wis-
[28] Markus Greiner. Ultracold quantum gases in three-dimensional optical lattice potentials. PhD Thesis, 2003.
[29] Dirk-Sören Lühmann. Multiorbital Physics in Optical Lattices. PhD Thesis, 2010.
Bibliography
75
[30] Jannes Heinze. personal communication, 2011.
[31] Christoph Becker. Multi component Bose-Einstein condensates. PhD Thesis, 2009.
[32] D van Oosten, P Van der Straten, and H Stoof. Quantum phases in an optical lattice.
Physical Review A, 2001.
[33] G K Campbell. Imaging the Mott Insulator Shells by Using Atomic Clock Shifts.
Science, 2006.
[34] Jacob F Sherson, Christof Weitenberg, Manuel Endres, Marc Cheneau, Immanuel
Bloch, and Stefan Kuhr. Single-atom-resolved fluorescence imaging of an atomic Mott
insulator. Nature, 2010.
[35] W S Bakr, A Peng, M E Tai, R Ma, J Simon, J I Gillen, S Folling, L Pollet, and
M Greiner. Probing the Superfluid-to-Mott Insulator Transition at the Single-Atom
Level. Science, 2010.
[36] T Esslinger. Fermi-Hubbard physics with atoms in an optical lattice. Condensed Matter
Physics, 2010.
[37] A Kastberg, WD Phillips, and SL Rolston. Adiabatic Cooling of Cesium to 700 nK in
an Optical Lattice. Physical Review Letters, 1995.
[38] Markus Greiner, Immanuel Bloch, Olaf Mandel, Theodor Hänsch, and Tilman
Esslinger. Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates.
Physical Review Letters, 2001.
[39] D McKay, M White, and B DeMarco. Lattice thermodynamics for ultracold atoms.
Physical Review A, 2009.
[40] J Stenger, S Inouye, A Chikkatur, D Stamper-Kurn, D Pritchard, and W Ketterle.
Bragg Spectroscopy of a Bose-Einstein Condensate. Physical Review Letters, 1999.
[41] PT Ernst, S Götze, JS Krauser, K Pyka, DS Lühmann, D Pfannkuche, and K Sengstock. Probing superfluids in optical lattices by momentum-resolved Bragg spectroscopy. Nature Physics, 6(1):56–61, 2009.
[42] N Fabbri, D Clément, L Fallani, C Fort, M Modugno, K van der Stam, and M Inguscio. Excitations of Bose-Einstein condensates in a one-dimensional periodic potential.
Physical Review A, 2009.
[43] U Bissbort, Y Li, S Götze, and J Heinze. Detecting the Amplitude Mode of Strongly
Interacting Lattice Bosons by Bragg Scattering. arXiv.org, 2010.
[44] G Veeravalli, E Kuhnle, P Dyke, and C Vale. Bragg Spectroscopy of a Strongly Interacting Fermi Gas. Physical Review Letters, 2008.
[45] Gretchen K. Campbell, Jongchul Mun, Micah Boyd, Patrick Medley, Aaron E. Leanhardt, Luis G. Marcassa, David E. Pritchard, and Wolfgang Ketterle. Observation of
the Pairing Gap in a Strongly Interacting Fermi Gas. Science, 2004.
76
Bibliography
[46] J T Stewart, J P Gaebler, and D S Jin. Using photoemission spectroscopy to probe a
strongly interacting Fermi gas. Nature, 2008.
[47] H Häffner, C McKenzie, A Browaeys, W D Phillips, and Rolst. A Bose-Einstein
condensate in an optical lattice. Journal of Physics B: Atomic, Molecular and Optical
Physics, 2002.
[48] T Stöferle, H Moritz, C Schori, and M Köhl. Transition from a Strongly Interacting
1D Superfluid to a Mott Insulator. Physical Review Letters, 2004.
[49] Rajdeep Sensarma, K Sengupta, and S Das Sarma. Momentum Resolved Optical
Lattice Modulation Spectroscopy for Bose Hubbard Model. arXiv.org, 2011.
[50] C Kollath, A Iucci, I P McCulloch, and T Giamarchi. Modulation spectroscopy with
ultracold fermions in an optical lattice. Physical Review A, 2006.
[51] Daniel Greif, Leticia Tarruell, Thomas Uehlinger, Robert Jördens, and Tilman
Esslinger. Probing Nearest-Neighbor Correlations of Ultracold Fermions in an Optical
Lattice. Physical Review Letters, 2011.
[52] Zhaoxin Xu, Simone Chiesa, Shuxiang Yang, Shi-Quan Su, Daniel E Sheehy, Juana
Moreno, Richard T Scalettar, and Mark Jarrell. The response to dynamical modulation
of the optical lattice for fermions in the Hubbard model. arXiv.org, 2011.
[53] Stephen Gasiorowicz. Quantenphysik. Oldenbourg Wissenschaftsverlag, 2005.
[54] Ana Rey, Guido Pupillo, Charles Clark, and Carl Williams. Ultracold atoms confined
in an optical lattice plus parabolic potential: A closed-form approach. Physical Review
A, 2005.
[55] Michael Köhl. Thermometry of fermionic atoms in an optical lattice. Physical Review
A, 2006.
[56] Jasper S. Krauser. Impulsaufgelöste Bragg-Spektroskopie an bosonischen Quantengasen in optischen Gittern. Diploma Thesis, 2009.
[57] Alexander Mering and Michael Fleischhauer. Multi-band and nonlinear hopping corrections to the 3D Bose-Fermi Hubbard model. arXiv.org, 2010.
[58] Alexander Albus, Fabrizio Illuminati, and Jens Eisert. Mixtures of bosonic and
fermionic atoms in optical lattices. Physical Review A, 2003.
[59] Francesca Ferlaino, Chiara D’Errico, Giacomo Roati, Matteo Zaccanti, Massimo Inguscio, Giovanni Modugno, and Andrea Simoni. Feshbach spectroscopy of a K-Rb atomic
mixture. Physical Review A, 2006.
[60] Andreas Bick. Detektion und Untersuchung von mehrkomponentigen Bose-EinsteinKondensaten in hexagonalen optischen Gittern. Diploma Thesis, 2010.
[61] Wolfgang Demtröder. Experimentalphysik 2. Springer Wissenschaftsverlag, 2006.
[62] WJ Smith. Modern optical engineering: the design of optical systems. McGraw-Hill
Professional, 2008.
[63] George Seward. Optical Design of Microscopes. SPIE Society of Photo-Optical Instrumentation, 2009.
[64] Franziska Herrmann. Entwicklung optischer Elemente für das Atom Trap Trace Analysis Experiment zur Spurengasanalyse von Kryptonisotopen. Diploma Thesis, 2009.
[65] W Alt. An objective lens for efficient fluorescence detection of single atoms. OptikInternational Journal for Light and Electron Optics, 2002.
[66] OSLO Optics Reference, 2005.
[67] G. D. Boreman. Modulation Transfer Function in Optical and Electro-Optical Systems.
SPIE Press, 2001.
78
Bibliography
Danksagung
Diese Arbeit wäre ohne die Hilfe einiger Personen nicht möglich gewesen. An dieser Stelle
möchte ich mich bei all jenen die zum gelingen dieser Arbeit beigetragen haben bedanken.
Großer Dank geht an Prof. Dr. Klaus Sengstock für die Möglichkeit ein lehrreiches,
spannendes und erlebnisreiches Jahr in seiner Gruppe verbringen zu dürfen. Insbesondere möchte ich mich für das entgegengebrachte Vertrauen, das Engagement und die alles
andere als selbstverständlichen Möglichkeiten wie z.B. Konferenz-Besuche bedanken.
Bedanken möchte ich mich weiter bei Prof. Dr. Henning Moritz für die freundliche übernahme der Zweitkorrektur und das anfängliche zur Verfügung stellen von OSLO.
Ganz besonders möchte ich dem gesamten „BFM“-Team danken: „meinen“ Doktoranden
Sören Götze, Jannes Heinze und Jasper Krauser für die Geduld und das Vertrauen was
ihr mir entgegengebracht habt. Bei „meinem“ PostDoc Christoph Becker möchte ich mich
für die großartige unterstützung das gesamte Jahr über bedanken. Dank geht auch an
„meinem“ Diplomanden-Kollegen Nick Fläschner für das allseits offene Ohr für sämtliche
Fragen. Dem gesamten „BFM“-Team danke ich für eine unglaublich tolle Atmosphäre und
die vielen tollen Erlebnisse im Labor und außerhalb.
Bei der gesamten Arbeitsgruppe Sengstock, sowie allen anderen Mitarbeitern des ILPs
möchte ich mich für die offenheit und hilfsbereitschaft in jeder Situation bedanken.
Last but not least: Einen riesen Dank möchte ich meiner Familie aussprechen: Ohne eure
Untersützung wäre ich jetzt nicht da wo ich bin und nicht das was ich bin. Danke!