PHD Thesis Tobias Schuster

Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Diplom-Physiker: Tobias Schuster
Born in: Kirchheimbolanden, Germany
Oral examination: 20. June, 2012
2
Feshbach Resonances and Periodic
Potentials
in Ultracold Bose-Fermi Mixtures
Referees:
Prof. Dr. Markus K. Oberthaler
Prof. Dr. Selim Jochim
4
Meinem Papa gewidmet.
6
Zusammenfassung
In der vorliegenden Arbeit werden die Möglichkeiten, stark korrelierte Quantensysteme anhand der Bose-Fermi Mischung von Natrium und Lithium experimentell umzusetzen, untersucht. Zu diesem Zweck wurden die intraspezies Wechselwirkung von Natrium sowie die
interspezies Wechselwirkung zwischen Natrium und Lithium erforscht. Durch eine quantitative Analyse des Natrium Feshbach Resonanz Spektrums konnten wir dessen Wechselwirkungseigenschaften präzise bestimmen und insbesondere die Genauigkeit der Energie des
letzten gebundenen Zustands um einen Faktor 50 erhöhen. Zur Untersuchung der interspezies
Wechselwirkungseigenschaften haben wir einen allgemeingültigen Ansatz entwickelt, dessen
Anwendbarkeit wir anhand der Natrium-Lithium Mischung zeigen konnten: Vorzeichen und
Größe der Streulänge wurden bestimmt und dieser Wert verwendet, um die Energie des
letzten gebundenen Zustands zu bestimmen, was als Ausgangspunkt für die Erklärung des
gemessenen Feshbach Spektrums diente. Im Spezialfall Natrium-Lithium konnten wir 23 der
26 beobachteten Resonanzen als d-Wellen einordnen. Dieses unerwartete Ergebnis wurde
durch eine Coupled-Channels Rechnung bestätigt, deren Resultate für die Wechselwirkungseigenschaften auch mit den unabhängig experimentell ermittelten Größen übereinstimmen.
Als ergänzende Herangehensweise, um stark korrelierte Systeme zu untersuchen, haben wir
unseren experimentellen Aufbau durch ein optisches Gitter ergänzt. Wir konnten Methoden,
um die Bandbesetzung zu bestimmen und kohärent zu manipulieren, demonstrieren und
den interspezies Energieübertrag auf verschiedene Weisen untersuchen. Die Gitterfrequenz
wurde mit besonderem Augenmerk auf die erreichbare Genauigkeit bestimmt. Der ermittelte
Wert in Kombination mit den Ergebnissen der Wechselwirkungsanalyse zeigt, dass Fröhlich
Polaronen in der ultrakalten Bose-Fermi Mischung von Natrium und Lithium erfolgreich
über den Anstieg in effektiver Masse detektiert werden könnten.
7
Abstract
This thesis investigates the possibilities to experimentally realize strongly correlated quantum systems by means of the sodium-lithium Bose-Fermi mixture. For that purpose, we
have studied the intraspecies interactions of sodium as well as the interspecies interactions
within the mixture. With a quantitative analysis of the sodium Feshbach spectrum, we were
able to refine its scattering properties and in particular improve the accuracy of the last
bound state’s energy by a factor of 50. For determining the interspecies scattering properties, we developed a widely applicable approach, which we demonstrated by means of the
sodium-lithium mixture: We obtained sign and magnitude of the scattering length experimentally and used this value to determine the last bound state energy, which served as the
starting point for the explanation of the measured Feshbach spectrum. In the particular
case of sodium-lithium we could assign 23 of 26 observed resonances as d-waves. This unexpected result was confirmed by a coupled-channels calculation, which also yielded scattering
properties in good agreement with our experimental findings.
As a complementary approach to study strongly correlated systems by directly tuning
the interactions, we implemented an otical lattice into our system. Methods to map out
the band population as well as coherently manipulate it were demonstrated and interspecies
energy transfer was studied in different ways. We investigated the determination of the
lattice frequency with regard to the achievable precision. The value obtained combined with
the results from the interaction analysis show that in the ultracold Bose-Fermi mixture of
sodium-lithium Fröhlich polarons could be observed succesfully via detecting the increase in
effective mass.
8
Contents
1. Introduction
13
I. Feshbach Resonances
17
2. Scattering of Ultracold Atoms
2.1. Hamiltonian . . . . . . . . . . . . . . .
2.2. Scattering Length . . . . . . . . . . . .
2.3. Mean-Field Energy . . . . . . . . . . .
2.4. Simple Model of a Feshbach Resonance
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19
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3. Experimental Setup to Measure Feshbach Resonances
3.1. Alkali atoms in magnetic fields . . . . . . . . . . . . . . . . . .
3.2. Preparation of Different Spin Channels . . . . . . . . . . . . .
3.2.1. Ultracold Atoms in the Optical Dipole Trap . . . . . .
3.2.2. Rapid Adiabatic Passage . . . . . . . . . . . . . . . . .
3.2.3. Stern-Gerlach . . . . . . . . . . . . . . . . . . . . . . .
3.3. Loss Mechanisms in Trapped Ultracold Atom Samples . . . . .
3.3.1. One-Body Losses . . . . . . . . . . . . . . . . . . . . .
3.3.2. Two-Body Losses . . . . . . . . . . . . . . . . . . . . .
3.3.3. Three-Body Losses . . . . . . . . . . . . . . . . . . . .
3.3.4. Broad and Narrow Resonances . . . . . . . . . . . . . .
3.4. Interpretation of Loss Curves . . . . . . . . . . . . . . . . . .
3.4.1. Atom Density Distributions in Traps . . . . . . . . . .
3.4.2. Loss Analysis of Trapped Homonuclear Atom Samples
3.4.3. Loss Analysis of Trapped Heteronuclear Atom Samples
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29
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4. Sodium Intraspecies Feshbach Resonances
4.1. s-wave Resonance Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Moerdijk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2. Asymptotic Bound-State Model . . . . . . . . . . . . . . . . . . . . .
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9
4.2. Higher Partial Waves . . . . . . . . . . . . . . . . . .
4.3. Results of the Coupled-Channels Calculation . . . . .
4.3.1. Resonance Widths . . . . . . . . . . . . . . .
4.3.2. Scattering Length for Different Spin Channels
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5. Sodium-Lithium Interspecies Feshbach Resonances
5.1. Previous Knowledge about NaLi Scattering Properties
5.2. Scattering length determination . . . . . . . . . . . . .
5.2.1. Absolute value of the scattering length . . . . .
5.2.2. Sign of the Scattering Length . . . . . . . . . .
5.2.3. Last Bound State Energy from a . . . . . . . .
5.3. Feshbach Spectrum . . . . . . . . . . . . . . . . . . . .
5.4. Assignment and Fit of Resonances . . . . . . . . . . . .
5.4.1. Quantum Numbers . . . . . . . . . . . . . . . .
5.4.2. Triple Features . . . . . . . . . . . . . . . . . .
5.4.3. Resonance Widths . . . . . . . . . . . . . . . .
5.5. Concluding Remarks . . . . . . . . . . . . . . . . . . .
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66
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II. Bose-Fermi Mixtures in Periodic Potentials
91
6. Design and Implementation of the Optical Lattice
6.1. Design of the SSODT . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Dimensionality criteria . . . . . . . . . . . . . . . . . . .
6.1.2. Fundamentals of Optical Lattices . . . . . . . . . . . . .
6.1.3. Full QM Description of a Particle in a Periodic Potential
6.2. Characterization of the Optical Lattice . . . . . . . . . . . . . .
6.2.1. Phase Lattice for Sodium . . . . . . . . . . . . . . . . .
6.2.2. SSODT for Lithium . . . . . . . . . . . . . . . . . . . . .
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93
. 94
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. 99
. 104
. 104
. 106
7. Brillouin Zone Mapping
109
7.1. Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2. Lattice Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3. Brillouin Zone Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8. Controlled Excitations in the Optical Lattice
8.1. Quantum Mechanical Picture of Oscillations . . . .
8.2. Interspecies Energy Transfer . . . . . . . . . . . . .
8.3. Coherent Transfer Between Different Bloch Bands .
8.4. Determination of the Interspecies Scattering Length
8.5. Rabi Oscillations . . . . . . . . . . . . . . . . . . .
10
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115
115
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119
121
125
9. Conclusion and Outlook
127
A. List of Constants
131
B. All NaLi Resonance Data
133
Bibliography
148
11
12
1. Introduction
Der Weg entsteht im Gehen.
Reinhold Messner
Since the first experimental realization of Bose-Einstein-Condensation in 1995 [1, 2], the
research field of ultracold atomic gases has been growing rapidly. Whereas the first experiments investigated rather weakly interacting systems, after some years there were several
developements making it possible to realize strongly interacting systems, which are in the
focus of interest of today’s ultracold atom research [3].
A system is called strongly interacting if its interaction energy Eint is much larger than its
kinetic energy Ekin , i. e. Eint /Ekin ≫ 1. In contrast to this case, for a simplified description
of gases one often considers the ideal gas, which is by definition non-interacting. To take also
interactions into account, most gases can – outside of critical points – be correctly described
by the van der Waals equation. This regime of weak interactions is both experimentally
and theoretically well understood. To be able to investigate more complicated systems,
tools have been developed for tuning ultracold quantum gases into the strongly interacting
regime, where the strong correlations make a correct theoretical description more challenging.
On the one hand, Feshbach resonances [4, 5] make it possible to get a direct handle on the
interaction energy Eint . On the other hand, the use of optical lattice systems [6] gives
access to experimentally tune the ratio Eint /Ekin , e. g. by lowering the dimensionality of the
system [7, 8].
These developments paved the way to realize the long-standing proposal of a quantum simulator [9] in the field of ultracold atom research: In 1982, R. P. Feynman showed that the
computation time ’classical’ computers require to simulate a quantum system scales exponentially with the size of its Hilbert space. To circumvent this problem, he proposed to use
a so-called quantum simulator, on which the Hamiltonian of the system of interest could be
emulated. A famous example, where in an ultracold atom system a problem of condensed
matter has been tackled and solved this way, are imbalanced Fermi gases: It was well-known
that for sufficiently strong interactions, ultracold Fermi gases of two spin states will show
superfluidity induced by a BCS-like pairing mechanism [10, 11, 12]. But it had not been
clear, which imbalance between the two spin states is necessary to destroy this superfluidity
13
completely. In ultracold atom systems, where one – in contrast to condensed matter systems
– is able to set an arbitrary imbalance at will, this Clogston-Chandrasekhar limit [13, 14] has
been mapped out [15], being an impressive demonstration of practicability of the quantum
simulator.
In the extreme limit of imbalance, where one impurity of spin down is immersed in a sea of
spin up Fermions, a quasiparticle, the so-called Fermi polaron is formed [16]. One can think
of it as a particle, which drags a cloud of particle-hole excitations with it due to its strong
interaction with the atoms in the other spin states. This gives rise to an increase in effective
mass m∗ > m, which could be measured experimentally by mapping out the resulting shift in
oscillation frequency [17]. Until recently, experiments have only investigated Fermi polarons
composed of equal [16] and different [18] fermionic atom species. The polaron concept has
first been introduced in condensed matter theory [19], where it describes an electron travelling
through a (ionic) crystal. The coulomb interaction results in a displacement of the atomic
cores, i. e. the lattice sites, which can be described by lattice phonons. The quasiparticle
formed by the coupled electron-phonon system is called polaron and can be attributed an
effective mass m∗e > me being bigger than the bare electron mass me .
For ultracold atomic gases, there are proposals to simulate this polaronic Hamiltonian using
an impurity atom immersed in a bosonic background [20, 21]: In comparison to the condensed
matter system, the phonons correspond to the Bogoliubov modes of the condensate and the
role of the electron is taken by the impurity atom. As long as one only considers the single
polaron case, there is no difference between bosonic and fermionic impurities. To express
the strength of the coupling between impurity and phonons, the commonly used parameter
is denoted α, which in the case of an ultracold atom system reads
α=
a2IB
.
aBB ξ
(1.1)
Here, aIB denotes the interspecies scattering length between impurity√ and background
atoms1 , aBB the intraspecies scattering length of the bosons and ξ = 1/ 8πnaBB the condensate healing length with n being the bosonic atom’s density. We see that α, whose
value is fixed for a condensed matter system, can be tuned by changing e. g. the scattering
lengths aIB or aBB . Of special interest is the strongly interacting regime α > 1, where
there are different theoretical approaches to solve the problem, e. g. variational Feynman
path-integral [21] and quantum monte carlo treatment [22]. Those calculations, which have
been performed for different experimental situations, could be validated by experiment, in
particular their results for the effective mass m∗ . The impact of a deep understanding of the
polaron problem outreaches the field of ultracold atom systems: As they are considered to
1
Depending on the context, in this thesis the abbreviations aIB (Impurity-Boson) and aBF (Bose-Fermi)
are used equivalently, as our impurity 6 Li is a fermion.
14
play an important role in the understanding of high-Tc superconductors [23], results of such
measurements would also be highly interesting for the condensed matter community.
So far, the investigation of polarons in Bose-Fermi mixtures has only been done in one
dimension [24]. To be able to study them in our experiment, which uses bosonic sodium
and fermionic lithium, we first have to understand the tuning properties of aBB and aIB .
Moreover, the experimental tools to detect a small increase in effective mass m∗ > m have
to be developed. The implementation of a species selective optical lattice does not√only give
the possibility to measure precisely the oscillation frequency of lithium ω ∗ ∝ 1/ m∗ , but
also provides the versatility to investigate physics in lower dimensions.
Contents of this Thesis
The first part of this thesis deals with Feshbach resonances, i. e. the tuning properties of
aBB and aIB are investigated in detail. For that purpose, we start with basic scattering
theory in chapter 2, where important terms as the scattering length are introduced and
its connection to repulsive and attractive interactions within a simple mean-field picture is
derived. In chapter 3, we give a short introduction to our experimental setup, with a focus on
the preparation of different spin state combinations. Loss mechanisms in ultracold samples
are treated theoretically in detail and the results are applied to interpret experimentally
obtained loss curves. As a main result of this chapter, the difference between the sodiumlithium singlet and triplet interspecies scattering lengths is determined.
In chapter 4, Moerdijk and asymptotic bound-state model are introduced and applied to
develop an understanding of the sodium intraspecies Feshbach resonance spectrum. Starting
with a model only involving s-wave resonances, we extend our theory to higher partial waves
and compare our results to a coupled-channels calculation.
Chapter 5 first presents measurements performed to understand the sodium-lithium interspecies Feshbach resonance spectrum. We obtain sign and absolute value of aIB experimentally and relate it to the least bound state energy, which finally serves as a guidance
to develop an explanation of the measured Feshbach resonances, most of which are assigned
d-wave resonances. The results of a coupled-channels calculation confirm the experimental
data and our assignment of resonances with the asymptotic bound-state model.
In the second part of this thesis, Bose-Fermi mixtures in periodic potentials are investigated.
In order to design a species selective optical dipole trap in form of an optical lattice in
chapter 6, we derive different criteria which our system has to fulfil. Starting with analytic
approximations, we extend our picture to a full quantum-mechanical treatment of the periodic potential. First results of the effects of the periodic potential on sodium and lithium
are shown and analyzed quantitatively.
As a tool to investigate the states in the optical lattice in detail, we introduce Brillouin
15
zone mapping in chapter 7, a technique which allows to map out the population of the
different Bloch bands. By adding a sodium background, we are able to optimize the lithium
lattice loading process and thus prepare atoms only in the lowest Bloch band. Starting from
that state, we show in chapter 8 how we can excite the lithium atoms in a controlled way
and analyze the resulting state in terms of Bloch states. The interspecies energy transfer in
the Bose-Fermi mixture is analyzed quantitatively, with the dynamics of the process giving
the absolute value of aIB in agreement with the results of the first part of the thesis. Finally,
we show our control of the optical lattice by demonstrating Rabi oscillations.
The Outlook gives a perspective on the physics beyond this thesis, which can be investigated
on basis of the results presented in the following.
16
Part I.
Feshbach Resonances
The first part of this thesis deals with Feshbach resonances in the NaLi mixture. In an
introduction to basic scattering theory of ultracold atoms the important concept of the
scattering length is introduced. We show how we prepare different spin mixtures and analyze
their losses both theoretically and experimentally.
By means of the Feshbach spectrum of sodium, we introduce the Moerdijk and ABM models
to explain our findings. For the sodium-lithium mixture, we present different measurements
to obtain information about sign and magnitude of the scattering length. Guided by these
results, we develop a model explaining all 26 NaLi resonances observed experimentally.
18
2. Scattering of Ultracold Atoms
In this chapter, we will investigate scattering of ultracold atoms from the theoretical point
of view, starting with the van der Waals potential as an example of a typical interatomic
potential. The scattering length a is introduced and exemplarily calculated using the box
potential as a simple model. With the results, we connect the sign of a to repulsive and
attractive interactions and show how they come into play in mean-field theory. The chapter
concludes by explaining the underlying principles of a Feshbach resonance used to tune the
scattering length.
2.1. Hamiltonian
As scattering processes of two atoms can be highly complex processes [25], we first want to
investigate in which regime our experiments take place, i. e. which approximations can be
made in order to simplify the description of interactions.
The interaction potential U(r) between two alkali atoms, which have one electron in their
outer shell, can be divided into two regions: For small interatomic distances, the potential
is repulsive due to Pauli blocking of the electrons and the repulsive interaction between the
positively charged nuclei. As this part strongly depends on the two electrons’ spin state,
which can be a singlet S = 0 or triplet S = 1, one can assign each S a different potential
US (r) as depicted in figure 2.1 a). The underlying reason for the different potentials is the
required asymmetry of the two electrons’ wavefunction: A spin triplet (singlet) is symmetric
(antisymmetric), thus the spatial wavefunction must be antisymmetric (symmetric). This
means, that the electrons have a higher probability to be found between the atoms in a spin
singlet, thus they bind the two positively charged atom nuclei, which form the molecule,
more strongly and so the singlet potential is deeper than the triplet. As an example, in
the case of Na2 there are νX = 64 singlet and νa = 14 triplet states [26], when there is no
magnetic field applied.
The long-range part of the potential, which is the same for both singlet and triplet, is
given by the induced dipole-dipole interaction between the atoms, scaling as
U(r) = −
C6
r6
(2.1)
with the van der Waals coefficient C6 given by the details of the atoms’ electronic configu-
19
ration. The associated length scale scale of this potential is given by
RvdW
1
=
2
2µC6
~2
1/4
,
(2.2)
where µ denotes the effective mass of the two interacting atoms. With typical values for the
van der Waals coefficient C6 , we get an RvdW ranging from 10a0 to 100a0 for the alkalis [5],
where a0 denotes the Bohr radius1 .
This value can be compared to other typical length scales of our system. Dealing with
atom densities ranging from n = 1012 cm−3 in a thermal cloud to n = 1015 cm−3 in a BEC,
the ratio of RvdW to the mean interatomic distance RvdW · n1/3 is well below 0.1, which we
call dilute.
0.2
0
b)
a3Σ+u
−0.2
−0.4
X1Σ+g
−0.6
−0.8
−1
2
4
6
8
interatomic distance r [a0]
combined potential V(r) [mK]
potential U(r) [104K]
a)
10
5
0
V(r) for l=0
V(r) for l=1
V(r) for l=2
coll. energy
l=1 rot. barr.
l=2 rot. barr.
−5
−10
−15
50
100
150
200
interatomic distance r [a0]
Figure 2.1.: a) Van der Waals potential US (r) for electrons in a singlet S = 0 and triplet S = 1
configuration (values taken from [26]). We see that the singlet state, spectroscopically la3 +
beled X 3 Σ+
g , is much deeper than the triplet state a Σu . b) Effective interatomic potential
V (r) for different angular momenta l. Note the different energy and distance scales in a)
and b).
q
2
2π~
, which
Another length scale is given by the thermal de Broglie wavelength λT = mk
BT
gets at the onset of quantum degeneracy at temperatures of about 1 µK on the order of the
interparticle spacing λT = O(0.1...1) µm [27]. Thus the details of the interaction potential
can not be resolved in cold collisions and so a quantum mechanical treatment of the scattering
processes is unavoidable.
1
In the following, we will use the abbreviations for constants as listed in appendix A and will for reasons
of brevity not explicitly introduce them unless required.
20
As to be found in every basic quantum mechanics textbook [28], a partial wave expansion2 of the atoms’ scattering wave function yields for its radial part Rl (r) with angular
momentum l
2µ
2
(2.3)
Rl′′ (r) + Rl′ (r) + 2 [E − V (r)] Rl (r) = 0 ,
r
~
with
~2 l(l + 1)
(2.4)
V (r) = U(r) +
2µr 2
being the sum of bare interaction potential and rotational term, as sketched in figure 2.1 b).
The resulting rotational barrier has a height of
Urot
1
=√
2C6
~2 l(l + 1)
3µ
3/2
(2.5)
which is for the p-states in sodium and lithium 1 mK and 8 mK, respectively. With our
atomic samples having temperatures of around 1 µK, we see that for a basic understanding
of the collisions it is sufficient to consider s-wave scattering only.3
2.2. Scattering Length
To gain first insight into the mechanisms of scattering, it is sufficient to replace the real
potential by some model potential with wavefunctions that can be calculated easily. As we
will see later, despite of this simplification the basic physics is still preserved.
A potential with well known energy eigenfunctions is the square well potential
−V0 for 0 ≤ r ≤ r0
(2.6)
U(r) =
0 for r > r0
Plugging this potential into eq. (2.3) and substituting χl (r) = r · Rl (r), we get the onedimensional Schrödinger equation
χ′′l + [ε − Ve (r)]χl (r) = 0
(2.7)
2 2
with ε = 2µE/~2 and Ve (r) = 2µV (r)/~2. With E = ~2µk , the solution of eq. (2.7) for V0 = 0,
i. e. free particles, reads
χl (r) = c · sin (kr + η0 ).
(2.8)
2
The kind of partial wave expansion applied here is only valid in the case of a spherically symmetric
potential, which is justified in our case due to our atoms not having a permanent dipole moment. Note
that there are also numerous cold atoms experiments where the dipolar interactions are in the focus of
interest [29, 30] and thus scattering has to be analyzed differently.
3
As we will see later in chapter 4 and 5, this is not completely true, as e. g. the dipole-dipole interaction
couples l = 0 to l = 2 states.
21
bound state energy [a.u.]
scattering length a [abg]
4
a>0
2
0
a=0
−2
a<0
bound state
−4
0
1
2
3
wavevector k0 [π / r0]
4
5
6
Figure 2.2.: Scattering length a according to eq. (2.11) in dependence
of the wavevector k0 , latter
√
being obtained from the potential depth V0 as k0 = 2µV0 /~. The wavefunctions for the
three highlighted points are shown in figure 2.3. Schematically shown is also the energy of
the universal bound state (black line) emerging for a > 0.
In the following we want to show how we can understand the underlying scattering physics
from the parameter η0 .
As a first easy example, we choose V0 = −∞, which is the so-called hard-sphere potential,
meaning that the two atoms scatter like classical billiard balls. As wavefunction, we get
0
for 0 ≤ r ≤ r0
(2.9)
χ0 (r) =
c · sin (k(r − r0 )) for r > r0
We note that η0 = −k · r0 to make the wavefunction continuous at r = r0 . By defining
a = lim −η0 /k, we introduce the scattering length a, which is one of the most important
k→0
parameters to characterize the strength of interactions in ultracold quantum gases, as we
will see later. In this simple case of the hard-sphere potential, we see that a exactly coincides
with r0 , the range of our potential, i. e. the scattering length reduces the complex quantum
scattering processes to one parameter used to describe classical scattering.
~2 k 2
In the following, we will consider the case of a potential well with finite depth V0 = 2µ0 > 0.
As the wavefunction is now allowed to penetrate into the range of the potential, it reads
c1 sin (k+ r)
for 0 ≤ r ≤ r0
(2.10)
χ0 (r) =
c2 sin (kr + η0 ) for r > r0
2
with k+
= k02 + k 2 . Here we already set the phase of the wavefunction inside the potential
to zero, as χl (0) = 0 is required due to the regularity of Rl (0). Moreover, the wavefunction
has to be continuously differentiable at r = r0 and thus we get for the scattering length
a = r0 −
22
tan k0 r0
k0
(2.11)
atomic wavefunction χ0(r)
a>0
1
0.5
a<0
0
a=0
a<0
a>0
bound state
−0.5
a=0
−1
−500
0
500
interatomic distance r [a0]
0
5000
10000
interatomic distance r [a0]
Figure 2.3.: Wavefunctions χ0 (r) for the points highlighted in figure 2.2, the left graph being a
zoom-in for small interatomic distances r. We see that for attractive/repulsive interactions
a ≶ 0, the wavefunction is being dragged into/out of the origin (green/red line), whereas
for a = 0 (blue line) the wavefunction looks almost as there is no interatomic potential
present. Furthermore, the weakly bound universal halo dimer (black dashed line) is shown.
The graphs have been obtained using typical experimental parameters for collision energy
corresponding to T ∼ 1 µK and potential range r0 = 60a0 .
which is visualized in figure 2.2 with the corresponding wavefunctions shown in figure 2.3.
We see that the points where k0 r0 = π/2 + nπ are of special interest, as a is diverging
there. To get more detailed insight into that special configuration, we consider a bound state
2 2
of the square well under consideration, i. e. a state with εb = − ~2µk < 0. The solution of the
one-dimensional Schrödinger equation (2.7) now reads
c1 sin (k− r) for 0 ≤ r ≤ r0
(2.12)
χ0 (r) =
c2 e−kr
for r > r0
2
where k−
= k02 − k 2 . The requirement of continuous differentiability now yields
k− cot (k− r0 ) = −k, which in the limit of weakly bound states k/k0 → 0 is readily solved
by k0 r0 = π/2 + nπ, which we identify as being coincident with the values of diverging a.
Calculating the binding energy for a ≫ r0 , we get using eq. (2.11)
εb = −
~2
.
2µa2
(2.13)
The range of validity of eq. (2.13) is called the universal regime, as the state and thus the
physics can be described by one parameter, the scattering length a, only. This can also
be understood intuitively by considering the wavefunction of this weakly bound state for
large r,
r
2 −r/a
χ0 (r) =
e
,
(2.14)
a
23
which describes a weakly bound halo dimer [5]. In figure 2.3 we see that in this state the
probability to find the atoms is mostly located beyond the classical turning point r0 of the
potential, and in this range a is the only parameter needed to describe the wavefunction.
Furthermore, the meaning of the scattering length is highlighted: At temperatures around
1 µK as achieved in our experiment, the atomic wavefunction is not able to resolve the
structure of the potential, but its effect can be captured by the phase shift η0 which the
atoms aquire during a collision. Note that although a diverges, η0 stays finite due to the
finite temperatures and resulting finite k we deal with.
2.3. Mean-Field Energy
With a potential being attractive for all r0 and V0 , it is not obvious how to get repulsive interactions between the atoms. As shown in introductory books to ultracold atom theory [31],
the full potential V (~r) can be replaced by a contact interaction
V (~r) =
4π~2 a
δ(~r) ,
m
(2.15)
which gives rise to an interaction energy Eint ∝ a. One sees how the sign of the scattering
length a connects to an energy change due to collisions and thus gives rise to repulsive or
attractive interactions.
In the following, we want to develop a picture how this result, which is rigorously derived
by means of quantum field theory, can be understood in an intuitive picture. For that
purpose, we consider the two particles being trapped in an external, infinitely high box
potential of size R, where their wavefunction according to eq. (2.10) reads for r ≫ r0
R0 (r) =
c
sin (k(r − a)).
kr
(2.16)
With the boundary condition of R0 (R) = 0, we get for the energies in the interacting and
noninteracting case
2
ñπ
~2
(2.17)
E=
2µ R − a
and
~2
E=
2µ
ñπ
R
2
,
(2.18)
respectively, with ñ ∈ N. From a Taylor expansion of the difference of eqs. (2.17) and (2.18),
we get an energy shift by interactions of
δE =
24
~2 (ñπ)2
a.
µ R3
(2.19)
Now, we want to relate this energy shift in the two-body problem with the corresponding
many-body-system containing N atoms. From the normalization condition of the wavefunction,
Z
ρ(r) 3
d r = 1,
(2.20)
ρ0
ρ0
| {z }
Veff
we get by using eq. (2.16) and ρ(r) = |R02 (r)| for the effective volume occupied by the two
particles Veff = πñ2 2 R3 . Inserting this into the result for the energy shift eq. (2.19) and noting
that for identical particles with mass m = 2µ there are N 2 /2 pairs of atoms in our system,
we end up with a total energy increase due to interactions of
∆E =
2π~2 N 2
a,
m V
(2.21)
which translates into a chemical potential of
µ=
4π~2
∂E
=
na.
∂N
m
(2.22)
We see that at positive scattering lengths a, it costs energy to add a particle to the system
of density n, which corresponds to a repulsive interaction. From figure 2.3 we see that this
is the case when there is a bound state just below threshold, i. e. the incoming atom can be
considered as being pushed out of the origin.
We note that in the case of mixtures of distinguishable particles, e. g. fermions nF in a
bath of bosons nB , eq. (2.22) reads
µF =
∂E
2π~2
=
nB aBF .
∂NF
µBF
(2.23)
Here, µBF = mB mF /(mB + mF ) denotes the effective mass and the subscripts B and F
denote bosons and fermions, respectively. The approximation made in deriving the chemical
potential due to interactions is the so-called mean-field approximation: Instead of considering
the whole many-body problem, one considers only the one-body problem of a single atom in
the external mean field of the others. This approximation breaks down if e. g. interactions
become strong, i. e. not only two-, but also three-body processes gain importance.
2.4. Simple Model of a Feshbach Resonance
To lead over to an experimentally more realistic situation, we have to go from a single- to a
two-channel description of the problem. As sketched in figure 2.4 a), the incoming atoms in a
certain spin state called entrance or open channel interact via the corresponding background
potential Vbg (r). If there is a closed channel which the entrance channel couples to and
25
energy
Vc
Vbg
closed channel
open channel
scattering length a [abg]
b) 3
a)
2
1
0
∆
abg
B0
−1
−2
−3
−2
interatomic distance
0
2
magnetic field B [∆]
Figure 2.4.: a) During a collision, two atoms in the open channel (green) can be coupled to a
closed channel (red). The coupling can be resonantly enhanced by tuning the bound state
energy of the closed channel appropriately. b) Scattering length a as described by eq. (2.24)
in dependence of the magnetic field. Additionally shown are the energies of open (green)
and closed (red) channel having different magnetic moments.
the associated potential Vc (r) has a bound state, this can give rise to a Fano-Feshbach
resonance [32, 33] if the bound state’s energy is tuned near the collision energy of the
incoming atoms. The tuning can e. g. be done by an external magnetic field, if open and
closed channel have a difference in magnetic moments δµ 6= 0. To develop an intuition for the
magnetic tuneability with the simplest possible model, we reduce the two-channel problem
to a one channel model with a potential of width r0 depth V0 = δµ · B. This way, we get a
potential of magnetically tuneable depth an can thus refer to the previously derived results.
Denoting B0 as the field at which the bound state is tuned in resonance, we can expand
eq. (2.11) around that value and get
∆
a = abg 1 −
.
(2.24)
B − B0
Here, we introduced the background scattering length abg = r0 and the width of the resonance
∆ = −~2 /(2µr02 δµ). Although here it has formally only been derived for the case of a box
potential, eq. (2.24) provides the general form of a so-called magnetically tuneable Feshbach
Resonance [5], which is visualized in figure 2.4b). Note that eq. (2.24) has been derived
using k02 ∝ B resulting in the resonance width ∆ being independent of the potential depth
V0 ∝ k02 , although in figure 2.2 the widths of the different resonances are obviously not all
the same, which is due to the x-axis being given there in units of k0 and not in dependence
of the magnetic field B as in figure 2.4.
In chapter 4 and 5 we will learn more about Feshbach resonances by means of the exam-
26
ples sodium and sodium-lithium. The knowledge about the characteristics of the scattering
lenght a will prove important to analyze density profiles as well as scattering dynamics of
trapped ultracold samples.
27
28
3. Experimental Setup to Measure
Feshbach Resonances
Unlike the title of this chapter might suggest, we will not give an introduction to the technical
details of our machine, which can be found in previous diploma and PhD theses [34, 35, 36, 37,
38]. We will rather explain the different experimental techniques, which will be used later to
adress fundamental physical questions. Especially for a deep understanding of the Feshbach
resonance spectra, profound knowledge of the atoms’ nature is crucial: The characteristics of
their different spin states in an external magnetic field will be studied by means of a derivation
of the Breit Rabi formula. Furthermore, we will show how we can transfer between those
states and check the results afterwards using a Stern-Gerlach technique, both being of crucial
importance for recording the Feshbach spectrum in chapter 5.
Important for our experiments is also a deep understanding of loss processes, which we will
first analyze in detail theoretically in different regimes. To understand the experimentally
measured loss curves, we then give an overview of atomic density distributions in traps,
which we apply to the homonuclear sample. Using the theory derived before, we are able to
extract information about the sodium-lithium scattering lenght from loss measurements of
heteronuclear samples.
3.1. Alkali atoms in magnetic fields
As known from basic atomic physics [39], the state of an atom can be characterized by its
quantum numbers, which read for the alkalis having one electron in the outer shell:
ˆ n: The main quantum number characterizes the electron’s energy state.
ˆ l: The orbital angular momentum of the electron
ˆ s: The electron spin, s = 1/2 for the alkalis
ˆ j: The total electron angular momentum j = |l − s|, ..., |l + s|
ˆ i: The nuclear spin
ˆ f : The atoms’ total spin f = |i − j|, ..., |i + j|
29
Moreover, each of the angular momentum quantum numbers k ∈ {l, s, j, i, f } can be assigned another quantum number mk being the projection of the angular momentum on the
quantization axis, where latter can e. g. be given by an external magnetic field B. Most
generally, the state of the alkali atom can be written as |n, l, s, j, i, f, mF i, but as we are
dealing with alkali atoms (s = 1/2) in the ground state1 (n = 2(3) for Li(Na), l = 0), it is
sufficient to characterize the state of the atom with fixed i by |f, mf i in the limit of small B
or |ms , mi i in the limit of large B, respectively. One should keep in mind that mf = ms + mi
is a conserved quantity for almost all Hamiltonians we deal with in the following.
With the assumptions made, the Hamiltonian relevant for the description of an alkali atom
in a magnetic field consisting of the hyperfine Hamiltonian Hhfs and the Zeeman Splitting
HZeeman reads [39]
Hhfs + HZeeman =
µB B
ahfs ~
~s · i + (gs ms + gi mi )
.
2
~
~
(3.1)
Here, ahfs denotes the hyperfine constant, gs and gi the gyromagnetic ratio of electron and
nucleus, respectively. In the following, we want to diagonalize this Hamiltonian and discuss
the resulting energy eigenstates from an experimental point of view.
For the socalled stretched states |f = i + 1/2, mf = ±f i, the Hamiltonian is already diagonal with the eigenvalues E = ahfs 2i ± (gs /2 + igi )µB B. For all other states, the analysis
is more involved. First, we rewrite the term ~s · ~i as
~s · ~i = sz · iz +
1
(s+ i− + s− i+ )
2
(3.2)
with the spin lowering and raising operators s± = sx ± isy and i± = ix ± iiy . Considering
the properties of these operators [40],
q
j± |j, mj i = ~ (j ∓ mj )(j ± mj + 1) |j, mj ± 1i
(3.3)
with j = i, s, we see that Hamiltonian eq.(3.1) only interconnects the states |ms , mi i =
|1/2, mf − 1/2i and |ms , mi i = |−1/2, mf + 1/2i. Thus, determining the eigenenergies reduces to the diagonalization of


1
ahfs
(mf
2
− 12 ) + g2s + gi (mf − 21 ) µB B
q
1
(i + 12 )2 − m2f
2
1
2
−ahfs
(mf
2
q
(i + 12 )2 − m2f


+ 12 ) + − g2s + gi (mf + 21 ) µB B
(3.4)
During the MOT cooling stage and the final absorption imaging, of course the atoms will also be in their
excited state, as described in [37]. But for the physics of interest in this thesis treating the atoms in their
ground state is sufficient.
30
With x =
(gs −gi )
µ B,
ahfs (i+1/2) B
the resulting eigenenergies read
ahfs (i + 1/2)
ahfs
+ gi mf µB B ±
E(B) = −
2
2
r
mf
+ x2 .
1 + 2x
i + 1/2
(3.5)
This is the celebrated Breit-Rabi-Formula [41] giving the magnetic field dependence of the
energy of an atom in state |s, i, f, mf i or |s, i, ms , mi i, where the appropriate choice of
quantum numbers depends on the strength of the magnetic field B applied. For the atoms of
interest for this thesis, sodium and lithium with hyperfine constants aNa
hfs = 885.8130644 MHz
and aLi
=
152.1368407
MHz,
the
Breit-Rabi-Diagrams
are
plotted
in
figure 3.1. Especially
hfs
for the understanding of magnetic trapping, microwave (MW), radiofrequency (RF) transfers
and Feshbach resonances a profound knowledge of those energy diagrams is essential. As it is
obvious that each state |ms , mi i at high fields (Paschen-Back regime) adiabatically connects
to a certain |f, mf i at low fields (Zeeman regime), we will for the sake of simplicity in the
following denote the states according to their low-field nomenclature.
6
Lithium
energy [MHz]
Sodium
mi
400
1
0
−1
200 f=3/2
f=2
2000
ms=+1/2
ms=+1/2
0
3/2
1/2
−1/2
−3/2
0
ms=−1/2
−200 f=1/2
−400
mi
4000
0
ms=−1/2
−2000
−1
0
1
50
100 150 200
magnetic field [G]
250
−4000
f=1
0
500
1000
1500
−3/2
−1/2
1/2
3/2
2000 2500
magnetic field [G]
Figure 3.1.: Breit-Rabi-diagrams of lithium and sodium according to eq.(3.5). At low fields (Zeeman regime), where |f, mf i are good quantum numbers, the different hyperfine manifolds
are split by ahfs (i + 1/2), which is 228 MHz and 1.7 GHz for lithium and sodium, respectively. At high fields (Paschen-Back regime), |ms , mi i are the quantum numbers which
characterize the atoms best: We see that the two main branches with slope ±1.4 MHz/G
are given by the electron spin orientation ms = ±1/2 and the substructure reveals the
different mi states, which are offset by ahfs /2 from each other.
31
3.2. Preparation of Different Spin Channels
3.2.1. Ultracold Atoms in the Optical Dipole Trap
As the preparation of an ultracold sodium-lithium mixture in an optical dipole trap (ODT)
has already been described in detail elsewhere [38], we will in the following only shortly
summarize the most important steps here and focus on the new items being of direct relevance
for this thesis.
In the magnetic trap (MT), sodium and lithium are both trapped in their respective
stretched states Na|2, 2i and Li|3/2, 3/2i as in this combination spin-changing collisions (see
section 3.3.2) are not possible and thus sympathetic cooling works most efficiently [42, 43].
When the atoms are cold enough, both species are loaded into a crossed beam optical dipole
trap (ODT), which has been realized in two different ways during the course of this thesis:
ˆ For all measurements presented in our paper on sodium intraspecies Feshbach Resonances [26], we used optical fibers to guide the light to our experimental setup.
Due to the limited power capability [44], we had to use high trapping frequencies
of ω Na /2π = (310, 130, 310) Hz to provide sufficient trap depth of ∼ 10 µK to hold the
atoms.
ˆ As our old ODT was both limited in trap depth due to power issues as well as in sodium
atom number due to density-induced three-body losses (see section 3.4.2), we decided
to change our setup and send the infrared beams forming the ODT potential free space
to the experiment. The resulting trapping frequencies are ω Na/2π = (74, 78, 145) Hz at
similar trap depths as before, i. e. three-body losses are not limiting any more, resulting
in 4 · 106 thermal Na atoms after loading from the MT into the ODT or an almost pure
condensate of 1·106 atoms after an additional stage of evaporative cooling, respectively.
ˆ Due to imperfections in the alignment of the ODT and a residual curvature in our
magnetic field, we had to change the power balancing in the ODT beams for some of the
resonances at high fields (see chapter 5 and appendix B), resulting in ω̄ Na/2π = 102 Hz.
Throughout this thesis, the coordinate system will be oriented as follows: The x-axis points
in the direction of the optical lattice (chapter 6), the external magnetic field is applied in
y-direction and the camera takes pictures in the x−y plane (for visualization, see figure 6.1).
Gravity acts along the z-axis, thus our trap frequencies in this direction are chosen highest
in order to counteract the gravitational sag and thus maximize the overlap between the two
atomic clouds.
In both ODT configurations, the trapping frequencies for lithium are given by ω Li =
2.11ω Na, as can be easily seen by the formulas given in section 6.1.2, where optical dipole
potentials are treated in detail. The oscillation measurements presented in figure 5.1 of
section 5.2 confirm the theory: From the respective fits, we get ωyLi /2π = (154.2±1.4) Hz and
ωyNa/2π = (74.9 ±0.4) Hz, which yields a trapping frequency ratio of 2.06 ±0.02. Considering
32
the large amplitude oscillation used for that measurement, which makes the atoms also probe
the anharmonic regime of the trap, the agreement between theory and experiment is good.
3.2.2. Rapid Adiabatic Passage
Sodium atoms in the |2, 2i state are not a good starting point for making a BEC as this
state heavily suffers from three-body losses [45]. Thus we want to drive the transition
|2, 2i → |1, 1i, as the three-body loss coefficient L3 of the final state is an order of magnitude
lower than the one of the initial state (see measurements section 3.4.2). A robust way to
achieve this transfer is using a rapid adiabatic passage (RAP) [46], which we will explain in
the following.
transition frequency [MHz]
20
15
Li |3/2, 3/2> → |3/2, 1/2>
Li |3/2, 1/2> → |3/2,−1/2>
Li |3/2,−1/2> → |3/2,−3/2>
Li |1/2, 1/2> → |1/2,−1/2>
Na |1,1> → |1, 0>
Na |1,0> → |1,−1>
RF ω/2π = 16.9MHz
10
5
0
0
5
10
15
20
magnetic field [G]
25
30
35
Figure 3.2.: RF-transitions between different spin states in lithium and sodium as used in our
experiment. The antenna resonance frequency ω is shown as dotted black line. Note that
the transitions Li |3/2, 1/2i → Li |3/2, −1/2i and Li |1/2, 1/2i → Li |1/2, −1/2i almost lie
exactly on top of each other and are not resolved in this graphical representation.
In short, the RAP is based on the dressed states of a two-level system with resonance
frequency ω0 in an electric field oscillating with ω. The idea is to adiabatically keep the
atom in the dressed state when sweeping the detuning ∆ = ω − ω0 over resonance, thereby
transfering the atom between the two bare states, which resemble the dressed states in the
limit of large detuning. On resonance, where the detuning ∆ = 0, the Rabi frequency is
denoted as Ω. If one sweeps the oscillation frequency2 over the two-level system’s resonance
at a constant sweep rate such that ∆(t) = αt, according to the celebrated Landau-Zener
formula [47] the probability P for a diabatic transition is
Ω2
P = e−2π |α| .
2
(3.6)
Equivalently, one can also sweep the resonance frequency ω0 instead of ω, which can be easily done e. g. by
applying a small external magnetic field which shifts the levels and thus ω0 due to the Zeeman effect.
33
MW transfer |2,2> → |1,1>
RF transfer |1,1> → |1,−1>
1.2
P1=1.6 W
P2=5 W
1
P3=10 W
0.8
0.6
0.4
0.2
1
Na|1,−1> fraction
Na|2,2> fraction
1.2
0.8
0.6
F=2
0.4
0.2
RF
F=1
0
0
0
50
100
150
sweep time [ms]
200
MW
0
5
10
sweep time [ms]
15
Figure 3.3.: Landau-Zener sweeps of the sodium |2, 2i → |1, 1i and |1, 1i → |1, −1i transitions,
for three different MW power settings Pi . The experimental data are fitted using eq. (3.6),
yielding the Rabi frequencies of Ω1 /2π = (228 ± 9) Hz, Ω2 /2π = (473 ± 21) Hz, Ω3 /2π =
(668 ± 44) Hz for the MW- and ΩRF /2π = (12.8 ± 1.5) kHz for the RF-transition. The inset
shows a schematic of the transfers between and within the different hyperfine manifolds.
We see that |α| ≪ Ω2 must be fulfilled in order to make the passage adiabatic, i. e. a high
Rabi frequency is advantageous. The word rapid refers to the passage not only being used
in the case of RF- and MW-, but also optical transitions. There, the passage has to be done
rapidly in comparison to the timescale given by spontaneous decay processes. For a more
detailed description the reader may be referred to [48], where a comprehensive overview of
RAPs using a Bloch Sphere picture is given.
Experimentally, the RAP on the Na |2, 2i → Na |1, 1i transition is realized by sweeping an
offset magnetic field around 1 G at fixed MW frequency ω/2π = 1775.3 MHz. Simultaneously,
we do the transfer Li |3/2, 3/2i → Li |1/2, 1/2i by irradiating ω/2π ≈ 231.6 MHz, such that
the resulting final state of the mixture does not suffer from spin-changing collisions. Note
that this transfer has to take place after loading the atoms into the ODT, as the final states
are high-field seekers and thus not magnetically trappable.
After some optional additional evaporative cooling in the ODT, we prepare the atoms in
the spin channel we want to do the experiments, e. g. Feshbach spectroscopy, in. For this
purpose, we built an antenna being impedance matched at 16.9 MHz.3 Depending on the
magnetic field sweep range, we can drive the transitions shown in figure 3.2.
Figure 3.3 shows the results of the MW- and RF-transfer of Na |2, 2i → Na |1, 1i and
Na |1, 1i → Na |1, −1i, respectively. With the frequency sweep ranges of 511 kHz and
3
This frequency is mostly determined by the geometric constraints for the antenna: It has been designed
such that it does not diminuish the optical access when being placed near the atoms, i. e. on the glass
cell.
34
11.21 MHz corresonding to the respective ramps in magnetic field, we get a Rabi frequency
of ΩMW /2π = (668 ± 44) Hz and ΩRF /2π = (12.8 ± 1.4) kHz, respectively.4 It has to be
stressed that in the current configuration the antenna, which is also being used for evaporation in the magnetic trap, is not impedance matched, which severely limits the Rabi
frequency ΩMW . Thus we can currently not exceed a Rabi frequency of ΩMW = 0.67 kHz on
the Na |2, 2i → Na |1, 1i transition without major changes of the MW setup.
Recently, MW induced Feshbach resonances have been proposed for sodium [49]. Their
width depends on the strength of the applied radiation and can be estimated as follows: With
~Ω ∝ ∆µBMW , where ∆µ is the magnetic moment of the transition and BMW the magnetic
field strength of the microwave radiation, we see that this strongly limits BMW . 0.4 mG.
The corresponding width of the microwave induced Feshbach resonance is ∼ 1 mHz [49],
which would require a magnetic field stability on the order of ∼ 0.5 nG for tuning, a value
being experimentally not feasible. Thus, at least an impedance matching circuit and the
resulting increase in Ω would be essential to observe MW induced Feshbach resonances.
3.2.3. Stern-Gerlach
Having applied the RAP on the atoms, we have to check whether the transfer has been
succesful. For that purpose, we use a Stern-Gerlach type experiment: The common way to
image our atoms is absorption imaging [27], usually after a time-of-flight (TOF) expansion
in order to reduce the high optical densities. If we consider the expansion of a thermal cloud
at temperature T , the 1/e2 radius of the cloud σ(tTOF ) reads after a time tTOF of TOF
σ 2 (tT OF ) = σ 2 (0) +
kB T 2
t
m T OF
(3.7)
where m denotes the mass of the atoms and σ(0) is the in-situ width of the trapped atomic
cloud (for a derivation, see section 3.4.1). We see that the normal TOF expansion does
not differ between different spin states. This changes if we apply a magnetic field gradient
B ′ = ∂B/∂y during a time tTOF −tD of the expansion. If we denote µ as the atoms’ magnetic
moment in their respective spin state, the shift of the cloud center reads
µB ′ 2
tTOF − t2D .
∆x =
2m
(3.8)
We see that ∆x ∝ t2TOF , while σ ∝ tTOF , i. e. by choosing long TOFs we are able to seperate
the different spin components. The external magnetic field is not applied during the whole
expansion time tTOF , but only during tTOF −tD , as the fintune coil used to create the gradient
produces a magnetic field B(y) = B0 + B ′ y with B0 = 10.4 G and B ′ = 3.9 G/cm. Thus the
offset field B0 results in a transition shift for sodium of 27 MHz= 2.7Γ, where Γ = 10 MHz
4
Note that the transition Na |1, 1i → Na |1, −1i is a two-photon transition via the Na |1, 0i state, so the
single transition Rabi frequency for Na |1, 1i → Na |1, 0i or Na |1, 0i → Na |1, −1i is expected to be higher.
35
Sodium
|3/2,3/2>
|1/2,1/2>
|1/2,−1/2>
|3/2,−3/2>
|2,2>
|1,1>
|1,0>
|1,−1>
integrated atom density
−500
Lithium
0
position [µm]
500
−1000
−500
0
500
1000
position [µm]
Figure 3.4.: Stern-Gerlach expansion of sodium (left) and lithium (right), showing pictures of
different experimental preparations superimposed. The respective spin states of the condensed sodium atoms are nicely seperated due to their low expansion energy, whereas the
different lithium spin states overlap and are thus offset for reasons of clarity. The calcuLi
lated profiles have been obtained using eqs. (3.7) and (3.8) with tNa
TOF = 11 ms, tTOF = 6 ms,
TLi = 500 nK and TNa = 100 nK, latter chosen to yield a good guide to the eye. Due to
the short time-of-flight, the sodium clouds of the F = 1 hyperfine manifold are optically
dense.
is the linewidth of the transition used for imaging. Thus, if we would not turn off the
finetune coil before imaging, the sodium atoms would scatter about a factor of thirty less
imaging light than at vanishing magnetic field B = 0. Experimentally we choose a delay
time of tD = 2 ms to ensure all fields to have reached a sufficiently low value, which yields
the displacements given in table 3.1 by means of eq. (3.8).
36
Na spin state
theor. displ. ∆x[µm]
exp. displ. ∆x[µm]
Li spin state
theor. displ. ∆x[µm]
exp. displ. ∆x[µm]
|2, 2i
551
603(4)
|3/2, 3/2i
579
530(10)
|1, 1i
-289
-341(6)
|1/2, 1/2i
-300
-280(12)
|1, 0i
-17
-33(3)
|1, −1i
264
270(2)
|1/2, −1/2i
58
64(10)
|3/2, −3/2i
-579
-553(3)
Table 3.1.: Theoretically calculated and measured displacements of different spin states after
Li
Stern-Gerlach separation. The TOF-durations chosen are tNa
TOF = 11 ms and tTOF = 6 ms.
The values were obtained without using any free parameters. Considering the high sensitivity on the order of about 100 µm/ms on the TOF-duration tTOF and especially on the
delay time tD , the agreement of theory and experiment is good.
In order to visualize the effect of the Stern Gerlach-expansion, in figure 3.4 we show
absorption images of sodium and lithium in different spin states. The preparation of the
condensed sodium atoms can be checked by eye, whereas determining quality of the lithium
preparation requires a fit to the data.
3.3. Loss Mechanisms in Trapped Ultracold Atom Samples
Having prepared a atoms in a certain spin state, one can map out a loss curve of the
sample to determine its lifetime, e. g. in dependence of an external magnetic field applied.
In the following, we will explain which useful information about the scattering length can
be concluded from such measurements.
3.3.1. One-Body Losses
Loss processes, which only involve a single trapped atom, are called one-body losses. There
are mainly two mechanisms leading to such a process: As the vacuum of an experimental
apparatus is never perfect, there is always the probability that an atom at room temperature
scatters with a trapped cold atom, which makes latter immediately leave the trap.5
Another process which can lead to the loss of atoms are uncontrolled changes in the trap
frequency: If the trapping potential features noise at multiples of the trap frequency, this
can cause excitations of the atom to a higher harmonic oscillator state, which finally results
in loss. As we will show in detail in chapter 8, using this kind of excitation process can also
be used in a controlled way to study relaxation of the resulting excited state.
5
This is actually how vacuum problems in our experimental setup show up: We observe strongly reduced
atom lifetimes if e. g. the atomic beam shutter is broken and thus the hot atom beam scatters with the
trapped atoms.
37
Typically, the atoms in the MT feature a lifetime of ∼ 30 s and in the ODT ∼ 30 s as well.
Together with a calculated photon scattering rate of Γsc ∼ 0.01 1/s from the IR-beams, this
sets an upper bound of Γbg < 0.03 1/s on the one-body loss rate induced by collisions with
the background gas.
3.3.2. Two-Body Losses
Spin-Exchange Processes
The interaction potential of two alkali atoms U(r) can to a good approximation be assumed
to be spherically symmetric. During the collision, the two valence electrons of the colliding
atoms can either be in a singlet S = 0 or triplet S = 1 state. As explained in chapter 2, the
associated potentials Ut (r) and Us (r) are similar for large, while differing substantially for
small interatomic distances. Thus, we can write the interaction potential as
Us + 3Ut
+ (Us − Ut )s~1 · s~2
(3.9)
4
with Ps and Pt being the singlet and triplet projection operators, respectively. Writing
U = Us Ps + Ut Pt =
1
s~1 · s~2 = s1,z · s2,z + (s1,+ · s2,− + s1,− · s2,+ )
2
(3.10)
we see that the interaction described by eq. (3.9) conserves the total spin S and its zcomponent MS , but can exchange the spin of the atoms by flipping one from up to down
and the other vice versa. Therefore, this interaction is also called spin-exchange interaction.
In the following, we want to investigate the characteristics of the spin-exchange interaction
on the example of a sodium-lithium mixture. The most trivial case is the one of a mixture
of atoms in their respective stretched states, i. e. Na|2, ±2i +Li|3/2, ±3/2i. Here, there are
simply no states for an electron spin exchange available, or more mathematically speaking,
the atoms are in a pure triplet state, so these mixtures are stable against spin exchange.
For other spin mixtures, as already shown in the derivation of the Breit-Rabi-Formula, we
can decompose the atoms’ state as
|f, mf i = α |ms = 1/2, mi = mf − 1/2i + β |ms = −1/2, mi = mf + 1/2i ,
(3.11)
where i and s, which are fixed for a given atom, have been omitted in this notation for
simplicity. If no external field B is applied, α and β are simply given by the well-known
Clebsch-Gordan coefficients, and in the case of high magnetic fields we end up in a state
where electron and nuclear spin are completely decoupled, i. e. either α = 1 or β = 1,
depending on the state (see figure 3.1). In the intermediate regime, one can get α and β by
diagonalizing the Breit-Rabi Hamiltonian matrix and thus determining its eigenvectors.
In figure 3.5, we plot the energy of some selected spin mixtures with different MF =
Na
mf + mLi
f . As already pointed out, the spin exchange interaction can not change MF , which
strongly limits the transitions allowed between different spin states.
38
energy [GHz]
−1.1
−1.2
|1,−1> + |1/2,−1/2>
|1, 0> + |3/2,−3/2>
|1,−1> + |1/2, 1/2>
|1, 0> + |1/2,−1/2>
|1, 1> + |3/2,−3/2>
|1, 0> + |1/2, 1/2>
|1, 1> + |1/2,−1/2>
|1, 1> + |1/2, 1/2>
−1.3
−1.4
MF = 3/2
−1.5
MF = 1/2
MF = −1/2
−1.6
MF = −3/2
0
50
100
150
magnetic field B [G]
200
250
Figure 3.5.: Energies of spin state combinations with different MF . Spin exchange transitions
are only possible within states of the same MF manifold.
In terms of losses, the Na|1, 1i +Li|1/2, 1/2i state is obviously the best choice, as it does
not cross with any other state having MF = 3/2 and is additionally the energetically lowest
of all.
More interesting is the MF = 1/2 state decomposed analog to eq. (3.11)
|ii ≡ |1, 1iNa + |1/2, −1/2iLi = αiNa |1/2, 1/2iNa + βiNa |−1/2, 3/2iNa
+αiLi |1/2, −1iLi + βiLi |−1/2, 0iLi
(3.12)
From figure 3.5 we can see that there exists a state with equal MF = 1/2 but being composed
as
|f i ≡ |1, 0iNa + |1/2, 1/2iLi = αfNa |1/2, −1/2iNa + βfNa |−1/2, 1/2iNa
+αfLi |1/2, 0iLi + βfLi |−1/2, 1iLi .
(3.13)
We see that the two states are connected via ~sNa · ~sLi yielding
1
hf |~sNa · ~sLi |ii = βf∗Na αf∗Li αiNa βiLi ≡ ηspin
2
(3.14)
Of course transitions from |ii to |f i can only take place if they are energetically allowed,
i. e. in the magnetic field range up to B = 80 G. In the following, we will derive a quantitative
formula of the associated transition rate.
Motivated6 by the derivation of eq. (2.22) in chapter 2, we replace Us − Ut in eq. (3.9)
by 2π~2 (as − at )/µ, where as and at refer to the scattering lengths for singlet and triplet
6
This is of course not a strict derivation. The interested reader may find the rigorous proof in [50], yielding
the same result.
39
potential, respectively. Recalling Fermis golden rule
Pi→f =
2π
|hf |U|ii|2 ρ
~
(3.15)
giving the transition rate from an initial state i to the final state f when interacting via
a potential
U, we get by noting that the density of states for free particles7 reads ρ =
µ √
2µE with E being the energy difference between initial and final state
2π 2 ~3
p
Pi→f = 4π 2E/µ(as − at )2 |ηspin |2 .
(3.16)
For typical parameters of cold atom experiments, we can estimate Pi→f ∼ 10−12 cm3 /s.
Due to the high energy gain in a spin-exchange collision E ∼ O(MHz), which can be seen
in figure 3.5 as the vertical distance between two lines of equal MF , both participating
atoms will be lost from the trap. Therefore, we can relate the spin exchange rate Pi→f to
experimentally observable trap loss
ṅLi = ṅNa = Pi→f nLi nNa
(3.17)
with nLi and nNa being the densities of lithium and sodium, respectively. We see that the
measurement of spin-exchange losses should reveal information about the scattering lengths’
difference, as we will further investigate quantitatively by means of experimental data below.
Dipolar Processes
The assumption of the electrons interacting only via a spherically symmetric potential U(r)
is not completely correct. As we will see later in chapter 4, the magnetic dipole-dipole
interaction Udd between the electron’s spins can despite of its small energy be of crucial
importance for understanding the experimental observations. The associated potential can
be written as [50]
µ0 (2µB )2
[~s1 · ~s2 − 3(~s1 · ~r)(~s2 · ~r)] ,
(3.18)
Udd =
4πr 3
where ~r denotes the spatial vector between the two electrons. We can see from an expansion
in spherical harmonics [51] that this interaction drives transitions with ∆l = ±2, 0 (except
l = 0 → l = 0), taking the required angular momentum from the spins such that MS + ml =
8
Li
mNa
s + ms + ml , is conserved .
In order to estimate the importance of the dipole interactions for two-body losses, we want
to compare it to the previously considered spin-exchange collisions. Normalizing eq. (3.18)
7
Although experimentally the atoms are placed in a trap, after a spin exchange collision they will have
gained so much energy E ≫ Utrap that they can be treated as free particles.
8
Including also the nuclear spin in the treatment, one can more generally say that MI + MS + ml is
conserved. Compared to the dipole-dipole interaction of the spins, the one of the nuclei is by a factor of
µi /µs smaller and thus negligible.
40
for a finite volume and comparing it with eq. (3.9), where we again replace the potentials by
the appropriate scattering lengths, we get
µ0 µ2B µ
Udd
≈ 2
.
Uex
~ (as − at )
(3.19)
For this rough estimate, we neglected constants and the spin dependencies, which yield
factors on the order of 1. With typical scattering lengths on the order of several 10a0 , we
dd
ex
get as an estimate for Udd /Uex ∼ 10−2 , or for the associated transition rate Pi→f
≈ 10−4 Pi→f
∼ 10−16 cm3 /s. We thus see that being in a spin channel where spin-exchange takes place,
we do not need to take losses due to the dipole interaction into account; these might only
be of interest e. g. in the stretched states, where spin exchange is not possible.
3.3.3. Three-Body Losses
Another very important loss mechanism in ultracold atom samples are the three-body-losses,
e. g. they make the transfer Na|2, 2i →Na|1, 1i necessary as already pointed out in section 3.2.
In the energetically lowest spin state Na|1, 1iLi|1/2, 1/2i, which does not suffer from twobody losses, loss mechanisms involving more than one atom are described by three-body
processes. To be able to understand them better, we will explain the underlying mechanisms
in the following.
When two alkali atoms collide, they interact via a combination of singlet and triplet potential, each of which gives rise to bound molecular states. Therefore, with a finite probability
the two atoms can be scattered into one of those bound states, thereby gaining the associated binding energy |εb |. For forming a molecule, two atoms do not have enough degrees
of freedom to fulfil momentum and energy conservation (which is trivially fulfilled in the
two-body processes described above in section 3.3.2). Thus they need a third atom which
carries away a certain fraction of the bound state energy η|εb| in form of kinetic energy. To
further proceed, we have to distinguish two cases:
1. Away from any resonance, the binding energy εb is way larger than the typical trap
depth of U0 = O(10µK). Thus, all three involved atoms are immediately lost from the
trap.
2. Near a resonance, for the sake of simplicity we assume the universal formula for the
~2
9
bound state energy eq. (2.13) εb = − 2µa
The typical trap can hold the
2 to be valid.
atom and the molecule for scattering lengths a on the order of 500a0 and more, as only
then the resulting dimer is so weakly bound that the energy release is small enough.
But as this molecule is in a vibrationally highly excited state, the next collision with
an atom will result in a relaxation to lower lying states and thus again, three atoms
9
In section 3.3.4 we will see to what extent this assumption is justified for different types of resonances.
41
will be lost from the trap, but in contrast to the first case one atom having gained the
recombination energy −ηεb will stay trapped and thus effectively heat the sample.
So we see that in both scenarios three atoms will be lost from the trap. Moreover, besides the
recombination heat, inherent to the three- and two-body loss processes there is the socalled
anti-evaporation: As the losses in an atom sample of density n scale as ṅ = −Lm nm with m
being 2 for the two- and 3 for the three-body loss, atoms are preferably lost in the middle
of the trap, where the density is biggest. The atoms there have an energy lower than the
sample average and thus the process is just the contrary of evaporative cooling: There we
spill the most energetic atoms, while two- and three-body losses remove the coldest atoms
and thus effectively heat the sample.
A dimensional analysis shows that the three-body loss coefficient L3 scales near a Feshbach
resonance as a4 [52], which is fundamental for doing Feshbach spectroscopy: Experimentally,
we are sensitive to the enhanced losses and identify the position of maximum loss as the
resonance position.10
3.3.4. Broad and Narrow Resonances
In our simplified treatment, we have so far considered the effects of two-body interactions,
which are elastic scattering and losses, as independent processes. As to be read in detail
in [5], one can incorporate the two-body losses into the imaginary part of the scattering
length a, which then reads
∆δµ
a = abg 1 +
(3.20)
−E0 + i(γ/2)
where E0 is the threshold resonance position, most commonly tuned by a magnetic field, δµ
is the difference in magnetic moments of open and closed channel and γ the decay rate for
the bound state into all available loss channels. In figure 3.6, eq. (3.20) is visualized: The
difference of loss maxima between two-and three-body losses are an interesting feature, which
we will revisit in chapter 5. The plot of imaginary vs. real part of a shows the advantage of
using ’broad’ resonances, i. e. experimentally speaking such with a large width ∆.11 Noting
that the functional form shown here is a circle with center (x, y) = (abg , abg ∆δµ/γ) and
radius abg ∆δµ/γ, we see that not only for large widths ∆, but also for large background
scattering lengths abg , the tuneability of a at the same loss rate b is higher.
Motivated by the discussion about the tuneability, one can define the resonance strength
parameter
abg δµ∆
sres =
,
(3.21)
RvdW EvdW
10
Of course, physics is not that easy, but a detailed treatment of three-body losses being a field of ultracold
atom research on its own [53] is far beyond the scope of this thesis.
11
A more rigorous definition of the term ’broad’ will be given at the end of this chapter.
42
4
b)
3
12
10
Im(a) = losses
Re(a), Im(a) [abg]
a)
2
1
8
6
4
2
0
0
−1
−2
0
energy E0 [∆]
2
−5
0
5
10
Re(a) = scattering length
Figure 3.6.: a) Real (blue) and imaginary (green) part of the scattering length a according to
eq. (3.20). The parameters chosen are (abg , ∆δµ, γ) = (1, 1, 0.5). Shown in red is the
scaling of the three-body losses ∝ a4 . b) Tuning of the losses ∝ Im(a) with the real part
of the scattering length Re(a). Shown are the curves with (abg , ∆δµ, γ) = (1,1,0.5) (blue),
(1,3,0.5) (green) and (3,1,0.5) (red). One sees that for a given loss Im(a), resonances with
larger ∆ and abg are advantageous for tuning.
2
where RvdW and EvdW = ~2 /(2µRvdW
) are the van der Waals radius and energy, respectively.
If sres ≫ 1, we call the resonance broad. Intuitively, one can see that in this case the
microscopic details of the potential do not matter, as the defining length and energy scales are
given by abg and δµ∆, respectively. Molecules being formed in the range of ±∆ around such a
resonance have a very small closed channel admixture in their wavefunction [5] and thus their
binding energy can be described by the universal expression eq. (2.13) having the scattering
length a as only parameter. In contrast, narrow resonances show universal physics only
around a very small fraction of their (already narrow) width ∆, but nevertheless they have
recently found an application in the experimental investigation of the Fermi Polaron [18].
3.4. Interpretation of Loss Curves
In the following, we will extract information from experimentally recorded loss curves, starting from a short overview about atom density distributions in traps.
3.4.1. Atom Density Distributions in Traps
In order to quantitatively understand loss processes occuring in ultracold atom systems, we
have to analyze their density distributions. For N distinguishable particles with mass m at
temperature T in a trap with frequency ω, the Maxwell-Boltzmann distribution reads
43
p~ 2
mω 2~x 2
Np exp −
.
n(~x, ~p) = NNx exp −
2kB T
2mkB T
(3.22)
2
mω
1
The normalization factors read Nx = ( 2πk
)3/2 and Np = ( 2mπk
)3/2 . Integrating out
BT
BT
the momentum part, we are left with the in-situ density distribution, which we can obtain
via absorption imaging [27]. For thermal atoms at a temperature of T = 1 µK in our optical
dipole trap with frequencies12 (ωxNa, ωyNa , ωzNa)/2π = (74, 78, 145) Hz and (ωxLi , ωyLi, ωzLi )/2π =
Na
Na
(157, 154, 305) Hz, the in-situ widths of the profiles are (σ0,x
, σ0,y
) = (41, 39) µm and
Li
Li
(σ0,x , σ0,y ) = (38, 38) µm. As this result is very sensitive on the precise knowledge of the
trap frequencies and an easy determination of σ is complicated by the high optical density
of the trapped clouds, it is more senseful to image the atom distribution a time tTOF after
the turn-off of the confining potential. The new distribution then reads
Z
n(~x) = d3 r0 d3 p0 n(~r0 , ~p0 )δ 3 (~r0 + ~p0 tTOF − ~x)
(3.23)
~x 2
= NNx (tTOF ) exp − 2
2σ (tTOF )
with
σ 2 (tTOF ) =
t2TOF kB T
kB T
+
= σ 2 (0)(1 + ω 2t2TOF ).
mω 2
m
(3.24)
and the new normalization Nx (tTOF ) = 1/(2πσ(tTOF ))3/2 . From eq. (3.24) we see the advantage of time-of flight compared to in-situ images: To deduce temperature, the in-situ
measurement requires precise knowledge of the trapping frequency ω. Moreover, high optical densities and trap anharmonicities can be problematic for the in-situ image analysis.
In contrast, as long as tTOF ≫ 1/ω, the TOF-picture only requires precise knowledge of the
experimentally well determined tTOF to deduce the temperature T .
So far, our treatment has only considered classical particles, but when dealing with ultracold quantum gases, naturally also the statistics come into play. Weakly interacting bosons
at low temperatures can be described by the famous Gross-Pitaevskii equation [54, 55], which
is basically a Schrödinger equation with the nonlinear term eq. (2.22) motivated in section 2
and reads
~2 2
−
∇ ψ(r) + V (r)ψ(r) + U0 |ψ(r)|2 ψ(r) = µψ(r).
(3.25)
2m
Here, V (r) denotes the external confining potential, U0 = 4π~2 a/m the interaction energy
the chemical potential. The wavefunction ψ(r) is normalized such that
Rand3 µ = ∂E/∂N
2
d r|ψ(r)| = N with N being the total number of condensed atoms. The exact solution
of the Gross-Pitaevskii equation (GPE) requires numerics, but in the experimentally often
12
In the course of this thesis, these values have not been constant, but adapted to the respective experimental
requirements. The value given here has been used to obtain the optical lattice data presented in chapter 8.
44
relevant case of a kinetic energy being small in comparison to potential and interaction
energy, we obtain the density of trapped atoms as
µ − V (r)
2
,0 .
(3.26)
n(r) = |ψ(r)| = max
U0
In this socalled Thomas-Fermi approximation, the radii Ri of the cloud in a harmonic confining potential V (r) = mωi2 x2i /2 read
s
2µ
.
(3.27)
Ri =
mωi2
To express the density distribution in terms Rof experimentally accessible parameters, we
take into account the normalization condition dV n(r) = N and thus get for the chemical
potential
2/5
152/5 Na
µ=
~ω̄ ,
(3.28)
2
ā
√
where we introduced the mean trap frequency ω̄ = 3 ωx ωy ωz and the associated harmonic
p
oscillator length ā = ~/(mω̄). Equation (3.28) shows some important characteristics of
the trapped interacting Bose gas: In comparison to the density distribution of a classical gas
(3.22) we see that its peak density does not simply scale with N, but with N 2/5 , i. e. it is only
weakly dependent on the atom number, a dependency which will be important later when
investigating loss processes. The parameter Na/ā is a measure of the ratio of interaction
and kinetic energy and can thus be used to check the overall validity of the Thomas-Fermi
approximation. At the surface of the condensed cloud, the Thomas-Fermi approximation
is not giving a correct description (see figure 3.7), as due to the low density the kinetic
energy term gains importance and thus one can not neglect it any more. The whole GrossPitaevskii equation (3.25) loses its validity in the regime of strong interactions, where a mean
field description of the atoms’ wavefunction is not justified any more.
In a time-of-flight picture, the Bose condensate simply rescales its charateristic shape of
the inverted parabola (3.26), which has in the first experiments been used as a signature for
a macroscopic occupation of the ground state [1]. While the exact mathematical description
of the BEC expansion process is rather complex [56], for analyzing a TOF-picture of a partly
condensed Bose gas we simply fit it with a bimodal distribution, i. e. a Maxwell-Boltzmann
distribution (3.23) with a superimposed inverted parabola. The width of the gaussian gives
us the temperature T and by integrating over the two distributions, we can also determine
the condensate fraction η. Knowing the critical temperature for the BEC transition in a
harmonic trap [50],
kB Tc = 0.94~ω̄N 1/3
(3.29)
we get via the relation for a harmonically trapped bose gas
3
Nc
T
η=
=1−
N
Tc
(3.30)
45
2
1.5
y position [mm]
2.5
1
experimental data
Thermal
BEC
total
1
1.5
2
x position [mm]
Ttherm
= 328(33)nK
x
Ttherm
= 273(27)nK
y
ηx = 0.414
ηy = 0.399
Tcond
= 327(6)nK
x
Tcond
= 330(6)nK
y
2.5
Figure 3.7.: Time-of flight picture of a partly condensed Bose gas. The temperature can be either
therm ) or from the condensate fraction
determined from the size of the thermal cloud (Tx,y
cond
(Tx,y ). The results for the two different axes agree within their error bars, as expected
for a thermalized gas. An interesting feature of the fit is the edge of the BEC, where
the fit (black line) and the experimental data (red) disagree due to the Thomas-Fermi
approximation losing its validity there.
a complementary measure of the temperature.
In figure 3.7, we see a typical picture of a partly bose-condensed cloud after a time of
flight of tTOF = 31 ms. With the fitting procedure described above, we get a condensate
fraction of 40.6(1.5)% (as error, we take the difference between the results of x- and y-axis).
Knowing the trap frequencies and the total atom number, we can derive the temperature
using eqs. (3.29, 3.30) and get Txcond = 327(6)nK and Tycond = 330(6)nK, where the error is
mainly given by the uncertainties in the trap frequencies. A different method to get the
46
temperature is to extract it from the size of the thermal cloud using eq. (3.24), which in our
case results in Txtherm = 328(33)nK and Tytherm = 273(27)nK. The big errors can be explained
by considering the quadratic dependence of T on σ, which itself has a fit uncertainty of
typically 3%. Moreover, the magnification of our imaging system is not known to an accuracy
of better than 4%, which in combination makes this temperature determination less precise
than the method making use of the intrinsic BEC properties. But e. g. for small condensate
fraction, temperature determination via the size of the thermal cloud is advantageous.
3.4.2. Loss Analysis of Trapped Homonuclear Atom Samples
The kind of losses most easy to understand and analyze are one-body losses obeying the
equation ṅ = −γn, which can be easily solved by N(t) = N0 exp(−γt), with N(t) and N0
being the total particle number after time t and at t = 0, respectively. This solution is
independent of the distribution n(r) of the trapped atoms.
For loss processes involving more than one trapped atom, deriving the loss equation
is not as straightforward. In general, m-body losses obey ṅ = −Lm nm for thermal and
ṅ = −Lm nm /m! for condensed bosonic atoms, where the factor m! stems from the indistinguishability
of particles in the BEC. As we usually measure the total particle number
R
N = dV n(r), we first have to integrate the whole differential equation in space, yielding
in the case of Ṫ = 0
Ṅ = −αN λ+1
(3.31)
which can be solved by
N(t) =
1/λ .
1 + αλN0λ t
process
thermal cloud
2-body
N(t) = 1+BN20N0 t
3/2
mω 2
B2 = L2 4πk
BT
3-body
N0
(1+(ζ+2)B3 N02 t)1/(ζ+2)
3
mω 2
√
B3 = L3 2 3πk T
B 0
N(t) =
N0
(3.32)
BEC
N(t) =
N0
2/5
(1+ 52 A2 N0 t)5/2
A2 = L2 420π21a2 aho
N(t) =
A3 =
15a
aho
7/5
N0
4/5
(1+ 54 A3 N0 t)5/4
L3 15120π21(aaho )3
15a
aho
9/5
Table 3.2.: Scaling of different loss processes in homonuclear systems of thermal and condensed
atoms. Note that for the case of three-body losses in a thermal cloud, the model takes into
account a temperature increase of ζkB T with ζ ≈ 1 per lost atom due to recombination
heating [57].
47
The peak density of a thermal cloud scales linearly with N, whereas according to eq. (3.28)
for a BEC it only grows with N 2/5 yielding different λ. The results of the integration for the
different cases can be found in table 3.2.
In figure 3.8, we show typical loss curves of trapped sodium clouds. Experimentally, we
prepare thermal atoms in the |2, 2i or |1, 1i state in the ODT, which we can bose-condense
by forced evaporative cooling. The trap loss is recorded in dependence of the hold time t as
shown for a |2, 2i BEC and a |1, 1i thermal cloud.
5
5
3
exp.
R2=0.936
2.5
BEC
R2=0.989
2
thermal R2=0.978
Na|1,1>
3
2
1
0
0
x 10
b)
data
4
Na|2,2>
6
x 10
a)
5
10
hold time [s]
15
data
R2=0.902
R2=0.977
R2=0.996
1.5
1
0.5
0
0
10
20
hold time [s]
30
Figure 3.8.: Loss curves for a |2, 2i BEC (a) and a |1, 1i thermal cloud (b). Note the different
time scales on the t-axes. Each dataset is fitted with an exponential, a BEC and a thermal
cloud loss curve (see table 3.2). One can see by eye that the exponential fit is not the
correct description, but to get a quantitative handle on the fit quality one can consider the
R2 value, which for good fits is supposed to take a value → 1. For the loss curves above,
R2 indicates that the BEC fit is superior compared to the thermal fit when the atoms are
condensed and vice versa.
From the loss curves, we can extract the 3-body loss coefficient L3 . From the respective fits
6
6
|1,1i
|2,2i
to sodium BECs in |2, 2i and |1, 1i, we get L3 = 3.3 · 10−28 cms and L3 = 2.1 · 10−29 cms .
The two loss coefficients differ by one order of magnitude, in agreement with Ref. [45], but
the absolute values are off. A possible reason is that for long hold times the cloud is partly
condensed and partly thermal, as described in detail below. This complexity could be
circumvented by extracting L3 from loss measurements of a thermal cloud. But that is also
not straightforward, as the L3 determined this way strongly scales with the temperature T 3 ,
whereas the BEC loss curve has only ω and a as parameters, which are both known with
high precision.
A useful observation is that the BEC data shows a constant temperature T determined
from the thermal cloud size. The heating effect, which we would expect due to losses of
the coldest atoms as discussed in section 3.3.3 is counteracted by the finite trap depth and
48
a)
1
condensate fraction η
thermalization of the high density atom sample, i. e. by plain evaporation. Having this
in mind, an interesting feature which can be observed when recording the losses from a
condensed cloud is the temporally decreasing condensate fraction η as shown in figure 3.9.
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1
b)
0
0
5
10
hold time [s]
15
exp. data
fit η=1−αN
α=(99±4)⋅103
0
5
10
inverse atom number 106/N
Figure 3.9.: Decrease of the condensate fraction η of a sodium |2, 2i BEC vs. time t (a) and
inverse atom number 106 /N (b). The insets in a) shows the sodium density profiles for
high and low condensate fraction η. The errorbars in a) are obtained averaging over the
datapoints of two runs, whereas in b) they originate from taking the mean of the fits in
the two directions x and y.
But rather than plotting the condensate fraction versus the time t, it is physically more
meaningful to choose for the x-axis the inverse atom number 1/N, as according to eqs. (3.29)
and (3.30) we expect the scaling
η =1−
kB T
0.94~ω̄
3
1
.
N
(3.33)
3
kB T
Plotting and fitting the data accordingly, we get 0.94~ω̄
= (99±4)·103, which translates to a
cond
temperature of T
= (197 ± 11) nK, with the error mainly determined by the uncertainties
in the trapping frequencies. This result is in excellent agreement with the value we get from
the fit to the thermal cloud T therm = (180 ± 93) nK, but has a way higher precision.
To conclude the subsection investigating the loss features of homonuclear samples, we want
to stress that the value obtained for L3 can not be attributed a high accuracy: As we have
seen in the last paragraph, the condensate fraction is strongly decreasing over time, which
makes our model, which for simplicity assumes a pure condensate, inaccurate. For a more
precise determination of L3 , one could either measure with a pure thermal cloud or start
with bigger condensates and just take the initial atom loss for fitting into account, where
η changes only slightly. From the theoretical side, one could also implement a numerical
49
model taking both the BEC and the thermal cloud into account, but this is far beyond the
scope of this thesis.
Important for all further measurements are the long timescales O(30 s) of losses in a |1, 1i
BEC, which makes it possible to observe interspecies effects in the lifetimes of heteronuclear
samples, as we will see in the following.
3.4.3. Loss Analysis of Trapped Heteronuclear Atom Samples
Having understood the loss spectra of sodium in different hyperfine states, we will in the
following investigate what happens if we add lithium. The fermion is non-interacting due to
Pauli blocking and the freezing out of p-wave scattering at ultracold temperatures. Nevertheless, lithium can show homonuclear losses due to p-wave resonances at certain magnetic
fields [58, 59], which are however not interfering with our measurements presented in the
following.
In general, the loss equation for lithium in the mixture reads
ṅLi = −ΓnLi − LNaLi
nLi nNa − LNaLi
nLi n2Na .
2
3
(3.34)
Due to the fermionic statistics of lithium, a three-body loss event always involves two sodium
and one lithium atom, as mathematically described in eq. (3.34). Experimentally, we usually
have nNa ≫ nLi and thus we will in the following make the assumptions that during the loss
ˆ no sodium is lost, i. e. nNa = const.
ˆ the temperature of the sample is constant, which means that we can write nLi =
NLi f (x, T ) with f (x, T ) being the distribution of thermal atoms in the trap (we neglect
small effects due to a possible slight lithium degeneracy)
A constant temperature is ensured by the trap depth of our ODT in combination with a
good interspecies thermalization rate, but the assumption of constant sodium number gives
rise to systematic errors in our evaluation, which will be taken into account where necessary.
Integrating eq. (3.34) over space and solving the differential equation, we get for the lithium
atom number
NLi (t) = NLi (0) · e−(Γ+Γ2 +Γ3 )t
(3.35)
R
(n−1)
with Γn = Ln dV f (x, T )nNa . We see that the lithium loss is – under the simplifying
assumptions made – purely exponential.
As most simple case we prepare both species in their respective ground state, Na|1, 1i
and Li|1/2, 1/2i. As shown before in section 3.3.2, there can be no inelastic two-body loss
from that state and so Γ2 = 0. Thus the lithium loss shown in figure 3.10 is only given
by background Γ1 > 0 and three-body losses Γ3 > 0 with the sodium. To account for the
three-body losses with lithium, we plot the sodium numbers as NNa − 2NLi and fit the data
with the three-body loss curve of a BEC. In comparison with a pure Na|1, 1i sample shown
50
in figure 3.8 b), we see qualitative agreement of the loss curves, which decrease with similar
time constants.
5
4
x 10
b)
data
exp. fit
Li|1/2,1/2>
4
3
2
1
0
0
5
10
15
hold time [s]
20
Na|1,1> − 2⋅Li|1/2,1/2>
a)
x 10
8
data
BEC fit
6
4
2
0
0
5
10
15
hold time [s]
20
Figure 3.10.: Losses in a Li |1/2, 1/2i Na |1, 1i mixture. The lithium (a) is fitted with a purely
exponential loss yielding a time constant of τ = (24.6 ± 3.5) s. The sodium atom number
(b) is first rescaled to account for the three-body loss and then fitted with the condensate
loss formula from table 3.2.
More interesting is the case of Na|1, 1iLi|1/2, −1/2i. As we already saw in figure 3.5, this is
the energetically lowest spin state with MF = 1/2 for B > 80 G, i. e. L2 (B > 80 G) = 0, but
in the low field range Na|1, 0iLi|1/2, 1/2i is a possible spin exchange loss channel. Recalling
the formula for the spin exchange loss rate eq. (3.16)
p
L2 = Pi→f = 4π 2E/µ(as − at )2 |ηspin |2
(3.36)
we see that we roughly expect maximal loss where the energy between the two states is
maximized (B ≈ 40 G). A more precise calculation also taking the energy dependence of
ηspin into account yields, that Pi→f is maximized for B ≈ 34 G.
Figure 3.11 a) shows lithium loss curves recorded while applying an external magnetic
field B = 0 G and B = 40 G, respectively. The effect of spin-exchange losses is lower than
naively expected, as also reflected in the fit results, which agree within their error bars.
Therefore, instead of measuring single loss curves, we map out the remaining number of
atoms after a fixed time t = 20 s in dependence of the applied magnetic field B. We clearly
see in figure 3.11 that for fields below 80 G there are less atoms
for higher fields,
p left than
2
which we can quantify using as fitfunction N(B) = N0 exp(−c E(B)ηspin
(B)) with N0 and
c being free parameters. Assuming Γ and Γ3 of eq. (3.35) as independent of the magnetic
field, we get
NLi (B = 34 G) = NLi (B > 80 G) · e−Γ2 (B=34 G)t
(3.37)
51
4
number of Li|1/2,−1/2> atoms
a)
4
b)
x 10
3.5
8
B=0G
τ = (19 ± 6) s
B = 40 G
τ = (17 ± 6) s
6
x 10
3
2.5
2
4
1.5
2
1
exp. data
fit
0.5
0
0
10
20
30
0
0
hold time [s]
50
100
150
magnetic field B [G]
Figure 3.11.: a) Loss curves of Li |1/2, −1/2i for B = 0 G (blue) and B = 40 G (red) with a
Na |1, 1i background. Each point is an average of three experimental runs. The solid
lines are exponential fits to the data, yielding the same time constant within the error
bars. b) Remaining number of lithium atoms after a fixed hold time of t = 20 s. Each
point is an average of four to five experimental runs. The red solid line is a fit to the
data taking the magnetic field dependence of E and ηspin for the spin exchange process
Li |1/2, −1/2i Na |1, 1i → Li |1/2, 1/2i Na |1, 0i into account as detailed in the text. Note
that a) and b) have been obtained with different sodium background densities.
with NLi (B > 80 G) = (2.1 ± 0.1) · 104 and NRLi (B = 34 G) = (12 ± 2) · 103 from our fit. After
integrating the density distribution Γ2 = L2 d3 xf (x, T )nNa , we get
L2 = (7.8 ± 4.6) · 10
3
−15 cm
.
(3.38)
s
The large uncertainty stems from the systematics in sodium number and condensate fraction,
which decrease during t = 20 s by a factor of ∼ 2.3 (see figure 3.10) or from 50% to 20%,
respectively. Moreover, the uncertainty in temperature determination contributes to the
statistical error. Note that in the calculation of L2 , we had to take the gravitational sag
∆z = g · ((1/ωzLi )2 − (1/ωzNa)2 ) ≈ 9 µm into account, which reduces the value of the overlap
integral of lithium with the condensate to ∼ 70% and with the thermal cloud to ∼ 80%, thus
leading to an overall overlap decrease of ∼ 75%. With the energy difference E = 6.03 MHz
from the Breit-Rabi formula and the spin factor
1
1
ηspin (B = 34 G) = βf∗Na αf∗Li αiNa βiLi = · 0.7259 · 0.4339 · 0.4802 · 0.7382
2
2
(3.39)
which we obtain from diagonalization of the Breit-Rabi-Matrix eq. (3.4), we obtain the important result
|as − at | = (5.9 ± 1.7) a0
(3.40)
52
for the difference between triplet and singlet scattering length. This small value explains
why it was not easy to recognize the effect of two-body loss in the loss curves directly and
why our result for L2 is so untypically small.
In this chapter, we have developed an understanding of the behaviour of different atomic spin
states in magnetic fields and shown how to manipulate and analyze them. The timescales
obtained by measuring loss curves for different atomic samples were shown to be so long
that e. g. thermalization measurements could be easily done without the detrimental effects of
substantial losses. In chapter 5 we will see how our experimental findings about the scattering
length can contribute to the understanding of the NaLi Feshbach resonance spectrum.
53
54
4. Sodium Intraspecies Feshbach
Resonances
To develop an understanding of Feshbach resonances, we will first treat the homonuclear
case, which could be used to tune α due to its dependence on the condensate intraspecies
scattering length. To interpret the spectrum of s-waves obtained in our experiment, we first
introduce the Moerdijk model, which will turn out not to be sufficient to explain all resonances. Therefore, the asymptotic bound-state model (ABM) as an extension is introduced
and a complete understanding of the s-wave resonance spectrum is developed and an analysis of the higher partial wave resonances together with the results of a coupled-channels
calculation are given. Finally, we show how to obtain the scattering length for an arbitrary
spin channel from the triplet and singlet scattering lengths as and at .
4.1. s-wave Resonance Spectrum
Summing up what we introduced in the last chapters, we see that the Hamiltonian for two
atoms interacting with an external magnetic field applied can be written as
H = T + Hhf + HZ (B) + U(r)
(4.1)
with T = −~2 ∇2 /(2µ) being the particles’ relative kinetic energy,
Hhf =
aα
aβ
~sα · ~iα + 2 ~sβ · ~iβ
2
~
~
(4.2)
the hyperfine interaction and
HZ = [(gsα szα + giα izα ) + (gsβ szβ + giβ izβ )]
µB B
~
(4.3)
the Zeeman term. The interatomic potential can be projected onto its singlet and triplet
part by the operators1 PX and Pa yielding
U(r) = UX (r)PX + Ua (r)Pa .
1
(4.4)
Note that in comparison to the previous chapter we changed the indices indicating the total spin from s
to X for the singlet and t to a for the triplet to be consistent with the notation of molecular physics and
our paper [26].
55
Using that Hamiltonian, we want to investigate the characteristics of Feshbach resonances
in a certain atomic system, starting with determining their position. From chapter 2, we
know that a Feshbach Resonance occurs2 when a molecular state crosses the free atom
threshold. Thus, we have to calculate the eigenstates of eq. (4.1) and determine the crossings
of their respective energy with the free atom threshold, latter simply being given by the
Breit-Rabi formula eq. (3.5).
For the diagonalization of eq. (4.1), we first have to choose a suitable basis set. Denoting
the total molecular spin as f~ = f~α + f~β and the total molecular angular momentum as
F~ = f~ + ~l with ~l being the orbital angular momentum of the atoms’ relative motion, we
see that these quantities are only conserved for B = 0. Their respective projection mf =
MS + MI = ms,α + ms,β + mi,α + mi,β and MF = mf + ml are conserved for all magnetic
fields B, and furthermore MS is a good quantum number for high magnetic fields B which
yield HZ (B) ≫ Hhf . Thus the choice of |l, ml i |S, MS , I, MI i as basis set is appropriate.3
As an ansatz, we choose to diagonalize Hamiltonian eq. (4.1) starting from the molecular
levels, a method related to deperturbation theory commonly used in the context of molecular
spectra [60]. We can write the molecular wave function |νl, σi = |ΨS,l
ν i |σi as a product of
spin part |σi and spatial part |ΨS,l
i
being
dependent
on
the
spin
S = 0, 1, vibrational
ν
quantum number ν and the atoms’ relative angular momentum l.
The matrix elements of the Hamiltonian of our homonuclear system, i. e. α = β in the
previous equations, read
Hν ′ l′ σ′ ,νlσ = εS,l
ν δν,ν ′ δl,l′ δσ,σ′ + µB B(gs MS + gi MI )δν,ν ′ δl,l′ δσ,σ′
SS ′
′
+ahf δl,l′ ην,ν
s1~i1 + ~s2~i2 |σi /~2 .
′ (l) hσ |~
(4.5)
Here, δij denotes the Kronecker δ, εS,l
ν the binding energy of the rovibrational level (ν, l) of
the singlet S = 0 or triplet S = 1 potential. The last term is responsible for coupling of
singlet and triplet levels, as we will further elucidate by means of a complete explanation
of the sodium intraspecies Feshbach resonance spectrum. From the orthogonality of the
molecular wavefunctions within the singlet and triplet manifold, respectively, it is clear that
S,S
S,l
S,l
the Franck-Condon factor ην,ν
′ (l) = hΨν |Ψν ′ i = δν,ν ′ , i. e. different vibrational molecular
states having the same electron spin S can not be coupled directly.
To gather some first insight into the working mechanisms of our model, we restrict ourselves
to the simple case of just one rovibrational molecular level of each spin state involved,
described by the quantum numbers |νa , l = 0, S = 1i and |νX , l = 0, S = 0i. Motivated by
our experiments, we choose as atomic states sodium atoms in the |1, 1i state, which thus
defines the basis states the Hamiltonian has to be diagonalized in. Recalling that MF =
ml + MS + MI = 2 has to be conserved, we get with iNa = 3/2 the possible states listed in
2
Of course this is just an approximation, but in the frame of our work, which almost exclusively deals with
narrow resonances, this describes the situation very well.
3
At this point, we could as well have chosen |S, MS , mi,1 , mi,2 i, which we will use later in the treatment of
Feshbach resonances of distinguishable particles.
56
l
ml
S
MS
I
MI
(SI)f
MF
0
0
0
0
2
2
(02)2
2
0
0
1
-1
0
1
1
3
3
3
1
3
2
1
1
(13)2
(13)3
(13)4
(11)2
2
2
2
2
Table 4.1.: List of l = 0 states for each vibrational singlet (S=0) and triplet (S=1) state with
MF = ml + MS + MI = 2 due to the choice of the atomic spin state. The last two colums
show the quantum numbers |f, MF i of the coupled basis, which adiabatically connects to
the corresponding states |MS , MI i at high fields.
table 4.1 by means of combinatorics. A further restriction for all states is that l + S + I
must be even to yield a symmetric bosonic wavefunction. In the specific case of s-wave
scattering, where l = 0 and thus the spatial wavefunction is symmetric, this means that the
spin wavefunction needs to be symmetric as well, i. e. S + I must be even.
The first part of eq. (4.5) can be easily evaluated, but the hyperfine structure term needs
some more effort. To get a handle on it, we can rewrite its spin part as
i
1h
(~s1 + ~s2 )(~i1 + ~i2 ) + (~s1 − ~s2 )(~i1 − ~i2 )
~s1~i1 + ~s2~i2 =
2
(4.6)
1 ~ ~ ~ ′ ~′ =
S·I +S ·I
2
where we have defined S~ ′ ≡ ~s1 − ~s2 and I~′ ≡ ~i1 − ~i2 .
4.1.1. Moerdijk Model
The term S~ ′ · I~′ induces a coupling of singlet and triplet states. In order to simplify the
problem for a first understanding we will neglect it, i. e. only consider the problem of decoupled singlet an triplet states. This approximation is known as Moerdijk model [61] and is
valid for small singlet-triplet coupling, which can be assumed in our case as we will see later.
Noting that we can write
~ · I~ = Iz Sz + 1 (I+ S− + I− S+ )
(4.7)
S
2
with the spin raising and lowering operators eqs. (3.3) introduced in the previous chapter,
we can write the Hamilton matrix using the states of table 4.1 as basis and get
 I=2

HS=0
0
0
I=3
0 .
HS=1
HMoerdijk =  0
(4.8)
I=1
0
0
HS=1
57
ahf
I=2
I=1
a
Here, HS=0
= εX
ν + 2µB Bgi and HS=1 = εν + µB B(gs + gi ) + 2~2 are single matrix elements,
I=3
whereas HS=1
contains off-diagonal elements and can be written as


√
3ahf
3ahf
εaν + µB B(−g
+
3g
)
−
0
i
2~2
2~2
√
√s


I=3
3ahf
5ahf
a
HS=1
=
 . (4.9)
ε
+
2µ
Bg
i
ν √ B
2~2
2~2
5ahf
ahf
a
εν + µB B(gs + gi ) + 2~2
0
2~2
Matrix (4.8) can be diagonalized for every magnetic field B yielding the result graphically
presented in figure 4.2. It is indicated that by a change of the singlet and triplet binding
a
energies εX
ν and εν , one can shift the molecular spectrum and thus its crossings with the free
atom threshold, which determine the position of Feshbach resonances.
Connection to Open and Closed Channel
Before we proceed, we want to discuss how the states retrieved by the diagonalization of
eq. (4.8) are connected to the so-called open and closed channel introduced in chapter 2
and often found in literature. Therefore, we want to recall that we work in the basis
|S, MS , I, MI i, i. e. the matrix (4.8) is
HMoerdijk = hS, MS , I, MI |H|S, MS , I, MI i .
(4.10)
Each of the scattering atoms is prepared in an atomic hyperfine state |ji = |f1 , mf,1 i |f2 , mf,2 i.
In the case of |f1 , mf,1 i = |f2 , mf,2 i = |1, 1i, we have MF = 2 and thus the appropriate basis
set to determine open and closed channel reads
|f1 , mf,1 i |f2 , mf,2 i = |1, 1i |1, 1i
|2, 1i |1, 1i
|2, 1i |2, 1i
|2, 2i |1, 0i
|2, 2i |2, 0i
,
,
,
,
.
(4.11)
Via Clebsch-Gordan coefficients, those five states can be connected to the |f, Mf i states
in table 4.1, which can then be written as functions
P of our basis states |S, MS , I, MI i as
4
shown at the end of this chapter . Expanding 1 =
|ji hj|, we can rewrite the Hamiltonian
j
Matrix:
HMoerdijk = hS, MS , I, MI |1|H|1|S, MS , I, MI i
!
!
X
X
|j ′ i hj ′ | |S, MS , I, MI i
= hS, MS , I, MI |
|ji hj| |H|
=
X
j,j ′
4
j
j′
hS, MS , I, MI |ji hj|H|j ′i hj ′ |S, MS , I, MI i = UHU „
Note that this last transformation is dependent on the actual value of the magnetic field.
58
(4.12)
We see that for |ji = |j ′ i = |1, 1i |1, 1i, HP P ≡ hj|H|ji defines the open channel. Noting the
closed channel as Q, H can thus be written as
H=
HP P HP Q
,
HQP HQQ
(4.13)
where the part HQQ can be diagonalized leaving HP P unaffected. This representation is
particularly helpful to determine the widths of resonances [62]. Moreover, we get insight
into the character of the open channel: Unlike for simplicity often stated, the open and
closed channel are never pure singlet or triplet states5 . As to be seen by eq. (4.12), open
and closed channel are always a magnetic field dependent linear combination of singlet and
triplet and can be retrieved by the appropriate unitary transformation of HMoerdijk.
Application of the Moerdijk Model
As a first application of the Moerdijk model, we choose the |1, 1i channel with resonances
at 851.0 G and 905.1 G, which have already been known for a long time [4]. Additionally,
due to our capability to apply external fields of up to 2.2 kG [38], we were able to measure
a resonance at 2054.2 G. In the |1, −1i channel, we obtained a resonance at 1500.1 G. When
extending the Breit-Rabi formula eq. (3.5) to negative magnetic fields, we see that e. g. B < 0
in the |1, 1i channel corresponds to B > 0 in the |1, −1i channel. Thus the resonance at
1500.1 G in |1, −1i can be conveniently expressed as −1500.1 G resonance in |1, 1i. The loss
curves presented in figure 4.1 show the number of atoms left after holding them for a certain
time th at a magnetic field B. As we will see later, the hold time th strongly correlates to
the resonance width.
By means of our data we will test the accuracy of the Moerdijk model. As an additional
input we take data from magnetic field dependent molecular spectroscopy [63] telling us
that the binding energy of the last singlet bound state is εX
ν = −10738GHz. Plugging this
number in and doing a least-squares fit with the triplet energy εaν as free parameter, we
get the fit results listed in table 4.2 and graphically represented in figure 4.2. The overall
resonance structure is well reproduced, but the numerical predictions deviate by ∼ 3 G from
the experimentally obtained values.
So far, the existence of the singlet state has no influence on the calculated resonance
positions, as we are describing them using a pure triplet state which does not couple to a
singlet state, a restriction which we will release in the following.
5
Obviously if both atoms are in their respective stretched state |f = i + s, mf = f i, the open channel does
have pure triplet character. But this case can be left out of the discussion as atoms in this state do not
show magnetically tuneable Feshbach resonances.
59
6
2.0
3.0
7
a)
t
h
= 100ms
b)
8
t
h
= 10ms
c)
t
h
= 100ms
0.6
t
h
e)
= 25ms
2.5
t
h
= 25ms
0.5
1.5
5
6
5
NAtoms [10 ]
d)
2.0
0.4
4
1.0
1.5
0.3
4
3
1.0
0.2
2
0.5
2
0.5
1
0
0
1498 1500 1502
0.1
0.0
0.0
1200 1202 1204 1206
848
850
852
854
902
0.0
904
906
908
2052 2054 2056
magnetic field [G]
Figure 4.1.: Feshbach spectroscopy of Na |1, −1i (a-b) and Na |1, 1i (c-e) showing the number
of atoms left after holding them for a certain time th at a magnetic field B. Each data
point is an average over one to four experimental runs with the error bar representing the
1σ statistical uncertainty. The solid lines are Gaussian fits to determine the resonance
positions. Due to the loss feature asymmetry caused by the relatively large width of
resonance d), only the datapoints below the position of maximal loss have been taken into
account for the fit.
4.1.2. Asymptotic Bound-State Model
To include singlet-triplet coupling, we have to take the term ∝ S~ ′ · I~′ of eq. (4.6), which we
~ · I,
~ we can write
have neglected so far, into account. As already done for the term S
1 ′ ′
S~ ′ · I~′ = Iz′ Sz′ +
I+ S− + I−′ S+′ .
2
(4.14)
With the help of the common expansion of triplet and singlet states |S, MS i in |ms,1 , ms,2i,
|1, 1i = |1/2, 1/2i
√
|1, 0i = 1/ 2(|1/2, −1/2i + |−1/2, 1/2i)
|1, −1i = |−1/2, −1/2i
√
|0, 0i = 1/ 2(|1/2, −1/2i − |−1/2, 1/2i)
60
(4.15)
we can easily verify the relations
Sz′ = sz,1 − sz,2 = |1, 0i h0, 0| + |0, 0i h1, 0|
√
S+′ = s+,1 − s+,2 = 1/ 2(|0, 0i h1, −1| − |1, 1i h0, 0|)
√
S−′ = s−,1 − s−,2 = 1/ 2(− |0, 0i h1, 1| + |1, −1i h0, 0|)
(4.16)
~ operators with
As already anticipated, we see that the S~ ′ operators act like the normal S
the difference that they couple between singlet and triplet states. The decomposition of the
I~′ operators is more involved, as it is a priori not obvious how they act on our basis states
|I, MI i. By writing
X
|i1 , mi,1 , i2 , mi,2 i hi1 , mi,1 , i2 , mi,2 |I, MI , i1 , i2 i
(4.17)
|I, MI , i1 , i2 i =
mi,1 +mi,2 =MI
with hi1 , mi,1 , i2 , mi,2 |I, MI , i1 , i2 i being the well known Clebsch-Gordan coefficients, we expand |I, MI i in the new basis |i1 , mi,1 , i2 , mi,2 i that our operators
Iz′ = iz,1 − iz,2
I±′ = i±,1 − i±,2
(4.18)
can act on6 . With the hyperfine part of the Hamiltonian written as
′
SS
′
Hhf = ahf δl,l′ ην,ν
s1~i1 + ~s2~i2 |σi /~2
′ (l) hσ |~
SS ′
′ 1 ~ ~
S · I + S~ ′ · I~′ |σi /~2 ≡ Vhf+ + Vhf−
= ahf δl,l′ ην,ν
′ (l) hσ |
2
the matrix of Hamiltonian eq. (4.5) denotes
HS=0 Vhf−
HABM =
Vhf− HS=1
with
I=3
HS=1
0
HS=1
I=1 ,
0
HS=1
(4.19)
(4.20)
(4.21)
where latter incorporates already Vhf+ as offdiagonal terms (see eq. (4.9)).
The term Vhf− is coupling singlet and triplet with a coupling strength proportional to
S,S ′
S ′ ,l
S,l
the overlap parameter ην,ν
′ (l) = hΨν |Ψν ′ i. It can be evaluated analytically if the state’s
binding energy is so small that the classical turning point rc lies beyond the van-der-Waals
6
The same expansion could already have been done for S~′ , but there the number of states involved is still
manageable such that one can as well do it by hand. Moreover, the expansion of S~′ is applicable to all
alkali atom mixtures, as they all have one outer electron each.
61
radius RvdW . Then, we have a halo state, which has only a very small fraction of its total
probability density inside the classically allowed region, and thus η can be calculated using
eq. (2.10) yielding a value close to 1. More generally, the van-der-Waals shape −C6 /r 6 can
be used to describe the potential as long as r ≫ rX , where the exchange radius rX is defined
by the distance where the magnitude of the exchange interaction eq. (3.9) equals the one
of the van der Waals interaction. States having an rc ≫ rX are called asymptotic bound
states and their Franck-Condon factors are calculable by integrating the radial Schrödinger
equation using the aymptotic shape −C6 /r 6 of the potential only [62, 64].
In our case of sodium, the triplet state with a binding energy of 5 GHz has its classical
turning point for rc = 22a0 , which has to be compared to typical7 values of the exchange
radius rX ≈ 15a0 [65]. So we are certainly not in the regime where rc ≫ rX and therefore
it is difficult to obtain the Franck-Condon factors η via calculations, thus they will serve as
free parameters in the following, unless there is other sources to determine their values from.
ABM with one Singlet and one Triplet State
The width of the avoided crossing between the triplet νa = 14 and singlet νX = 64 states [63]
0,1
≈ 0.85. With this additional information at hand, one can diagonalize
gives an overlap η64,14
matrix eq. (4.20) and do a least-squares fit with the triplet energy εaν as free parameter,
yielding the fit results listed in table 4.2 and graphically represented in figure 4.2.
2
energy [GHz]
0
−2
ε1,0
14
−4
−6
−8
−10
−12
−2500 −2000 −1500 −1000 −500
0
500
1000 1500 2000 2500
magnetic field B [G]
Figure 4.2.: Different theoretical descriptions of measured resonance spectrum (green dots). The
resonance positions can be adjusted by varying the triplet binding energy ε1,0
14 to determine
the crossing of the molecular states (blue: Moerdijk, red: ABM) with the atomic threshold
(black dashed). The difference between the two models, i. e. the singlet-triplet coupling, is
evident at the avoided crossings.
7
In our paper [26], we define an outer radius Rout = 21a0 , which can as well serve as a measure of the
exchange radius rX ≈ Rout .
62
We see that the more realistic ABM model has not improved our fit with respect to the
Moerdijk model, even made it worse. Besides the rms deviation from the fit result, one can
exemplarily as in indicator also take distance in magnetic field between the two resonances
around 900 G, ∆exp = 54.1 G. Comparing that to the results of the Moerdijk and the ABM
model, ∆Moerdijk = 59.9 G and ∆ABM = 45.2 G, one can see that so far the theoretical
description does not fit our experimental findings which have an uncertainty of not more
than δB = 0.5 G8 .
ABM with Virtual States
Our model so far lacks also the explanation of the resonance at −1202.6 G, which has been
first observed in Ref. [66]. To explain this resonace and improve on the overall fit, again
motivated by [63] we take a so-called virtual singlet and triplet state, i. e. states with positive
a
binding energies εX
ν and εν , into account. As also to be seen in figure 4.3 a), despite they
are continuum states for B = 0, for B 6= 0 they can become bound and are thus essential to
describe both number and positions of the Feshbach resonances observed.
parameter
Moerdijk
ABM1
ABM2
ε0,0
64 [MHz]
-10738
-10738
-10738
exp. values
Moerdijk
ABM1
ABM2
ε1,0
14 [MHz]
-4974.1
-5000.7
-4990.5
0,1
η64,14
—
0.85
0.85
-1500.1 G -1202.6 G
-1502.2 G
—
-1500.8 G
—
-1500.1 G -1202.6 G
ε0,0
65 [MHz]
—
—
1347
851.0 G
847.7 G
852.8 G
850.9 G
ε1,0
15 [MHz]
—
—
1999.9
905.1 G
907.6 G
898.0 G
905.1 G
0,1
η64,15
—
—
0.758
2054.2 G
2053.6 G
2056.5 G
2054.2 G
0,1
η65,14
—
—
0.467
0,1
η65,15
—
—
0.990
rms deviation
4.7 G
7.7 G
0.13 G
Table 4.2.: Results of the least-square fitting procedures using the Moerdijk model and the ABM
with one singlet and triplet state (ABM1) and two (ABM2), respectively. The parameters
varied for fitting are printed bold. The results are visualized in figures 4.2 and 4.3.
With only five resonances at hand, a fit varying all eight free parameters is not meaningful.
Moreover, e. g. the energy of the νX = 64, 65 singlet states and their overlap parameters with
the triplet states cannot be determined independently, as there is no measured resonance
directly connected to the singlets. Thus, we take as additional information the data given in
0,0
0,1
[63, 67] to get the values of ε0,0
64 , ε65 and η64,14 . The optimization procedure only runs on the
1,0
S,S ′
triplet bound state energies ε1,0
14 and ε15 as well as the Franck-Condon overlaps ην,ν ′ . With
all five measured resonances included, the least squares fit with two singlet and triplet states
8
Depending on the exact magnetic field values and the coils used to apply it, our rf spectroscopy shows
widths of 80 mG to 300 mG. The central value B0 around which the field fluctuates can be determined
with a much higher precision though, i. e. the value of δB = 0.5 G should only be considered as an upper
bound.
63
a)
5
energy [GHz]
νa=15, νX=65
0
νa=14
−5
νX=64
−10
−2500−2000−1500−1000 −500
0
500 1000 1500 2000 2500
magnetic field B [G]
b)
c)
−1.8
energy [GHz]
energy [GHz]
−1.2
−1.4
−1.6
−1.8
−2
−1.9
−2
−2.1
−2.2
820 840 860 880 900 920 940
−1500 −1400 −1300 −1200
magnetic field B [G]
magnetic field B [G]
Figure 4.3.: a) Description of the entire s-wave resonance spectrum using two molecular singlet and triplet states. b) The additional singlet is essential to explain the resonance at
−1202.6 G. c) By the avoided crossing, the unperturbed resonance, which would show up
at 910.1 G (red dashed-dotted line), shifts by -5 G to the experimentally observed value of
905.1 G.
′
S,S
and their respective overlap parameters ην,ν
′ yields the results listed in table 4.2. We see a
drastic improvement in the rms error of the fit and an astonishing agreement of experimental
data and theoretical explanation with deviation ≤ 0.1 G. Zoom b) of figure 4.3 shows how the
the MS = 1 level of the virtual triplet state gives rise to the observed −1202.6 G resonance,
which can thus be used to pin down the energy of the virtual state ε0,0
15 . Moreover, especially
zoom c) nicely illustrates how the indirect coupling between the triplet states νa = 14 and
νa = 15 mediated via the νX = 65 singlet shifts the position of the 905.1 G resonance from
64
the uncoupled position 910.1 G by −5 G to the observed experimental value. Thus, the
theoretically calculated difference in magnetic field between the two MS = +1 resonances
now matches the experimentally obtained one.
To finish the chapter dealing with sodium s-waves, we finally want to discuss the resonance
in state |1, −1i at 1202.6 G, which is special in several ways. It is the only s-wave resonance
requiring a virtual triplet state to be explained. As to be seen in figure 4.3 b), this also leads
to a very small difference in magnetic moments of the atomic threshold and the molecular
state, i. e. there is a very nice tuning with the magnetic field according to eq. (2.24) where we
have seen that ∆ ∝ 1/δµ. A more involved theoretical calculation [5] confirms this result.
200 decrease
interactions
100
0
−100
−200
bound state energy [a.u.]
scattering length a [a0]
300
−300
1199 1200 1201 1202 1203 1204 1205
magnetic field B [G]
Figure 4.4.: Sodium |1, −1i intraspecies resonance at 1202.6 G. Due to the negative width ∆ < 0,
interactions are first decreased when increasing the magnetic field B. Thus a = 0 (vertical
red line) can be achieved without the detrimental effects of three-body losses and molecule
formation, latter being enabled by the appearance of a bound state (schematically shown
as green line).
Moreover, we can also see that the sign of δµ and thus ∆ must be different for the 1202.6 G
resonance compared to all others, which is also depicted in figure 4.4 depicting the results
of the coupled-channels calculation [68] yielding ∆ = −1.473 G. We can see that in this case
of a negative width an increase in magnetic field first leads to a decrease in interactions and
one can even tune to a = 0, i. e. realize the textbook example of a non-interacting Bose
gas without having to cross a resonance.9 For even higher fields, the strongly interacting
regime is entered with scattering lengths a < 0. A resonance with ∆ < 0 thus has the
advantage that we can tune to weak interactions without having to cross the resonance. In
the reverse case of ∆ > 0, tuning to a = 0 requires a ramp over the resonance, which leads
9
Experimentally, being confronted with magnetic field instabilities δB = O(30 mG), one can still tune to
a = abg δB/|∆| ≈ 1a0 .
65
to three-body losses and – during the ramp back to zero field, where the atoms are imaged
– molecule formation.
4.2. Higher Partial Waves
To explain the wealth of sodium intraspecies resonances measured [26], just including swaves is not sufficient. Therefore, we also have to take the higher partial waves l 6= 0 in
eq. (4.5) into account. In our case of two identical bosons colliding, the spin wavefunction
must be symmetric, which then requires the spatial wavefunction to be symmetric as well,
i. e. l = 0, 2, 4, .... One could think that the observation of higher partial wave resonances is
highly supressed due to our temperature of T ≈ 1 µK and a rotational barrier of 6 mK as
calculated in chapter 2. But as we have seen in chapter 3, the dipole-dipole interaction (3.18)
couples l = 0 with l = 2 states. Evaluating eq. (3.18) for r = RvdW , we get an associated
energy of 3.8 MHz, which translates into a shift of the Feshbach resonance positions on
the order of 1 G. Thus we can justify not to include the dipole-dipole interaction in the
Hamiltonian, as its effect does not alter the Feshbach spectrum by much. Nevertheless,
in the following we will take advantage of it by including d- and g-wave resonances in our
considerations, which are coupled to the atoms having collision energies that only give rise
to s-wave scattering. The dipole-dipole interaction circumvents the very weak tunneling of
the scattering atoms through the high centrifugal barrier and thus makes higher partial wave
resonances observable.
f
M
= 1
2
M
= 1
S
M
4
S
0
2
0
M
1,3
S
= 0
0
2
−2
-2
4
2
1,3
−4
2
= 1
M
4
0
2
2
3
S
M
S
= 0
-4
= 0
E/h (GHz)
molecular potential U(r) [GHz]
S
l=4 l=2 l=0
-6
−6
M
2
S
M
M
−8
S
0
interatomic distance r
50
S
= -1
= -1
= -1
100
-8
150
200
B (mT)
Figure 4.5.: Moerdijk model of the νa = 14 triplet state with MF = 2 and l = 0 (thick blue
lines), l = 2 (thin red lines) and l = 4 (dotted green lines). The energy of the atoms
prepared in |1, 1i + |1, 1i is shown as dashed black line. The experimentally measured
Feshbach resonances are depicted by solid squares (s-waves), circles (d-waves) and triangles
(g-waves). On the left, a sketch of the molecular triplet potential containing the l = 0, 2, 4
states indicates the molecular bound states which the Feshbach resonaces stem from.
66
Binding energies and overlap parameters for s-waves:
ε0,0
64
−110001
0,1
ε1,0
η64,14
(0)
14
-4991(1)
0.851
ε0,0
65
14003
0,1
0,1
0,1
ε1,0
η64,15
(0) η65,14
(0) η65,15
(0)
15
2014(10)
0.79
0.47
1.0
Rotational splittings and overlap parameters for d-waves:
0
D64
1556(38)2
1
D14
1309(2)
0,1
η64,14
(2)
0.851
0
D65
504(38)2
1
D15
465(34)2
0,1
0,1
0,1
η64,15
(2) η65,14
(2) η65,15
(2)
0.44
0.47
—
G115
—
0,1
0,1
0,1
η64,15
(4) η65,14
(4) η65,15
(4)
—
—
—
Rotational splittings for g-waves:
G064
5064(45)2
G114
4214(2)
0,1
η64,14
(4)
—
G065
—
Table 4.3.: Results of the ABM least-square fit including s-, d- and g-waves. All energies are given
in MHz. Additionally to our measured resonances, we made use of molecular spectroscopy
data, which are marked i , where i = 1, 2, 3 refers to publication [63],[69],[67].
When l = 2 is allowed, the number of possible states MF = mf + ml increases from five
shown in table 4.1 to 22+5, which are listed in [26]. Restricting ourselves to the Moerdijk
model with the triplet νa = 14 only, we see from eq. (4.5) that the only two new parameters
are the binding energies εS,l
ν for l = 2, 4. Doing a least-squares fit with our experimental
1,4
data10 , we get ε1,0
=
−4976
MHz, ε1,2
14
14 = −3679 MHz and ε14 = −765 MHz. The resulting
molecular spectrum shown in figure 4.5 together with the fit values in table 4.4 demonstrate
that the Moerdijk model is sufficient to explain the resonance spectrum with deviations from
the experimental values not exceeding 3 G.
As already done in the case of s-waves, we can include singlet-triplet coupling and use two
singlets and triplets each to get a more precise description of our data. To reduce the large
number of free parameters, we take the input of molecular spectroscopy experiments [69, 63,
67] and determine the rest of the parameters via a least-squares fit, whose results are shown
S,0
in table 4.3. Here we defined the rotational splitting for d-waves as DνS = εS,2
and
ν − εν
S,0
analog for g-waves GSν = εS,4
−
ε
.
Interestingly,
the
shape
of
the
potential
is
reflected
in
ν
ν
the rotational splittings: As the singlet potential is deeper and thus steeper as the triplet, its
rotational splitting is bigger due to the lower centrifugal distortion and the resulting smaller
moment of inertia.
Note that our fit is not unique, i. e. some of the parameters are strongly correlated such
0,1
that they can not be varied independently. For instance, η65,15
(l) and ε0,l
65 are correlated,
0,1
0,l
such that choosing a different η65,15 (l) results in a different ε65 . Nevertheless, with the fit
done here we can describe the experimental data with deviations of less than 0.5 mG as to
be seen in table 4.4.
10
At the time the paper was published, we had only measured resonances at fields B > 0, so this fit only
takes those into account.
67
4.3. Results of the Coupled-Channels Calculation
In order to further improve on the fit results and to calculate resonance widths, a CC
calculation based on Hamiltonian 4.5 with the full interaction potentials has been performed.
In addition to our experimental data, also conventional spectroscopy data [70, 69, 63, 71,
67, 72, 73, 74] has been included in the analysis. As this work was done by our colleagues
E. Tiesinga (NIST) and E. Tiemann (Universität Hannover), we will just summarize their
methods and main results.
Modifications of the Hamiltonian
In eq. (4.2), we assumed a constant hyperfine constant aα,hf . But for small interatomic
distances, this will not hold any more due to the strong electronic distortions of one atom
by the other. Thus, one makes the ansatz
cf
,
(4.22)
aα (R) = aα,hf 1 + (R−R0 )/∆R
e
+1
where the CC calculation determines the constants as cf = −0.029, R0 = 11.0a0 and ∆R =
1.0a0 . Although just being a small correction, it shifts the resonances by up to 0.5G, more
than our experimental uncertainty.
Moreover, we discussed the effects of spin-spin interaction, but did not take it into account
in our Hamiltonian and just made use of the fact that it induces the coupling to observe
higher partial waves in an ultracold atom sample. Doing some spin-algebra on eq. (3.18), we
get the spin-spin interaction
~ = 2 λ(R)(3S 2 − S 2 ) ,
VSS (R)
Z
3
with
3
λ(R) = − α2
4
1
+ aSO e−bSO R
3
R
(4.23)
.
(4.24)
The first term in λ(R) stems from the magnetic dipole-dipole interaction eq. (3.18), whereas
the second is a second-order spin orbit contribution [67]11 . With λ(R) and R given in atomic
units, α is the fine structure constant.
Model potential, calculation and results
The modeling of the potential can be done by splitting it up in three parts: A short range
repulsive interaction USR (R), an intermediate region containing the potential minimum
UIR (R), and a long range part ULR (R), which has in leading order the typical van-derWaals shape −C6 /R6 . This potential inserted into the one-dimensional radial Schrödinger
11
As the calculation shows, the value of aSO has an estimated uncertainty of 100%.
68
Exp.
l f
0 4
0 2
0 3
2 4
2 4
2 4
2 4
2 2
2 4
2 2
2 3
4 3
4 3
4 3
mf
2
2
2
4
3
2
1
2
0
1
3
B0exp (mT)
85.10(2)
90.51(4)
205.42(4)
49.36(2)
53.66(2)
58.63(2)
64.48(3)
66.28(3)
71.56(1)
72.71(1)
159.00(3)
50.80(2)
50.88(2)
51.09(2)
Moerdijk
ABM
Bc (mT) Bc (mT)
84.82
85.10
90.81
90.52
205.45
205.44
49.34
49.40
53.57
53.63
58.52
58.58
64.41
64.47
66.38
66.28
71.54
71.60
73.03
72.72
159.02
159.13
50.76
50.80
50.89
50.89
51.12
51.09
CC
B0 (mT) ∆(mT)
85.114
9.7[-4]
90.517
0.104
205.501
0.012
49.344
1.7[-4]
53.650
3[-5]
58.615
5[-6]
64.477
3[-7]
66.283
4[-7]
71.556
2[-8]
72.718
4[-7]
159.011
2[-4]
50.775 < 5[−7]
50.859 < 5[−7]
51.065 < 5[−7]
Table 4.4.: Overview of the experimentally and theoretically obtained results on Feshbach resonances in Na prepared in the lowest hyperfine substate |f, mf i = |1, 1i. Experimentally
the position of maximum loss B0exp is given, which is determined by Gaussian fits to the
loss features. The errors reflect the one standard deviation statistical uncertainty in the
magnetic field calibration, determined by the FWHM of the RF calibration spectra and
from the profile fit. The middle columns show the results for the Moerdijk and ABM models). The last two columns show the Feshbach resonance position B0 and width ∆ from
the CC calculation. The brackets in the last column give the exponent to the power ten.
To characterize the bound state, the quantum numbers l, f, mf are used; note that for the
l=4 states the mf quantum numbers can not be assigned due to the strong mixing by the
dipole-dipole interaction.
equation 2.7 yields the wavefunction, which is used to determine the scattering length a
in dependence of the applied magnetic field B. Analog to chapter 2, the Feshbach resonance positions are determined as the points where a diverges. With the help of additional
photoassociation [71] and spectroscopic data [63, 72, 73, 74], one can adjust the potential
parameters in an iterative way. As our resonances are caused by the least bound state, this
procedure primarily improves the long-range shape of the potential, in particular we increase
the precision of the least bound state energy by a factor of 50 from 15 MHz to 0.3 MHz.
4.3.1. Resonance Widths
Having calculated the wavefunctions, one can extract the scattering length a(B) and thus
get also the width ∆ of each resonance, conveniently defined as the distance in magnetic
69
field between divergence and zero crossing of a. In order to get information about the width
ratio of s- and d- wave resonances, one can for a very course estimate consider the associated
energy scales: The dipole-dipole interaction evaluated at r = RvdW = 45a0 yields 3.8 MHz
representing the coupling of s- to d-states. The coupling giving rise to the s-wave resonances
is connected to the hyperfine constant ahfs = 886 MHz, i. e. we would expect the width of sand d-wave resonances to scale as the squared coupling ratio (886/3.8)2 ≈ 5 · 104 . Table 4.4
shows that this estimate gives some rough order of magnitude of the ratio of resonance
widths.
Figure 4.6 depicts how the hold times used to map out a resonance scale with the respective
calculated resonance widths. Although there was no systematics done concerning the hold
time (we just chose it such that the resonance loss feature was not saturated), we see a clear
connection of hold time and width in the double-logarithmic plot. This serves as an elegant
experimental confirmation of the calculated resonance widths, most of which are too narrow
to be resolved due to the technical noise of the magnetic field B [38].
2
inverse hold time [1/s]
10
1
10
0
10
−5
10
0
10
calculated resonance width ∆ [G]
Figure 4.6.: Relation of inverse hold time and calculated resonance width ∆. Although the values
were not corrected for e. g. different densities, there is still a clear correlation.
With the resonances measured at magnetic fields B > 0, it was also possible to predict
resonances at negative fields, in particular the position of the −1202.6 G resonance measured
after publication was predicted to be at −1207.5(5.0) G. The agreement between prediction
and measurement is astonishing, as the resonances used to fit the spectrum in our paper
[26] are only shifted weakly by the influence of the virtual triplet state, which gives rise to
the described resonance at negative fields. The high predictive power of the model potential
shows the great applicability of Feshbach resonances to map out the long-range shape of the
potential in combination with molecular spectroscopy data.
70
4.3.2. Scattering Length for Different Spin Channels
Singlet and triplet potential can be each assigned a scattering length as and at , respectively.
From the coupled channels calculations, one gets as = 18.81(80)a0 and as = 64.30(40)a0.
When prepared in a certain spin state, the atoms have (unless they are in the stretched
state) both singlet and triplet character. In order to get the background scattering length
a for a certain spin channel, one thus has to figure out its singlet and triplet fraction. We
thus write
X
hf1 , f2 , f, mf |f1 , f2 , mf,1 , mf,2 i |f1 , f2 , f, mf i
(4.25)
|f1 , f2 , mf,1 , mf,2 i =
f,mf =mf,1 +mf,2
with hf1 , f2 , f, mf |f1 , f2 , mf,1 , mf,2 i being the well-known Clebsch-Gordan coefficients. In
order to describe the overlap between the state |f1 , f2 , f, mf i and the molecular state being
characterized by |S, I, MS , MI i (see table 4.1), one has to decompose
X
hS, I, MS , MI |f1 , f2 , f, mf i |S, I, MS , MI i
(4.26)
|f1 , f2 , f, mf i =
S,I,mf =MS +MI
The factor hS, I, MS , MI |f1 , f2 , f, mf i describes the coupling of four spins and can be
written as


s1 i1 f1 
p
hS, I, MS , MI |f1 , f2 , f, mf i = (2S + 1)(2I + 1)(2f1 + 1)(2f2 + 1) s2 i2 f2
(4.27)


S I f
with the last factor being a Wigner 9j symbol, which can either be expressed in terms of 6j
symbols or directly be calculated.12 Table 4.5 shows the results of applying the procedure
presented above exemplarily on different spin channels of sodium, with the experimentally
most relevant result
3
13
a|1,1i = as + at .
(4.28)
16
16
spin state
decomposition
CC calculation
|1, 1i + |1, 1i
55.77(36)a0
54.54(20)a0
|1, 0i + |1, 0i
52.03(36)a0
52.66(40)a0
|1, 1i + |1, −1i
50.08(37)a0
50.78(40)a0
Table 4.5.: Background scattering values for different spin channels obtained with a simple decomposition as described in the text and with the CC calculation.
We see good, but not perfect agreement between our method and the full coupled channels calculation. The differences stem from our simplified description using either the atomic
12
Care has to be taken with http://www.svengato.com/ninej.html, which does not give the analytical result
correctly. Only numerical, but correct results are given by http://plasma-gate.weizmann.ac.il/369j.html.
71
|f1 , f2 , f, mf i or molecular |S, I, MS , MI i basis, which is not applicable in general, as scattering occurs at distances where neither of them provides the appropriate physical description.
Thus, in order to get the correct background scattering length, the full CC calculation is
needed.
In this chapter, we have introduced the ABM model and shown how it can be used to analyze
Feshbach spectra. With this knowledge at hand, we will explain the NaLi Feshbach spectrum
in the next chapter.
72
5. Sodium-Lithium Interspecies
Feshbach Resonances
As explained in the introduction, one of the main goals of our experiment is to investigate
a2IB
polarons, whose nature is determined by the parameter α = aBB
. In the last chapter
ξ
we learned about the tuning properties of aBB , but due to the quadratic dependence one
would rather like to have a handle on aIB . Therefore, in the following we will investigate
the Feshbach spectrum of the NaLi mixture, starting from the experimental and theoretical
knowledge at the time when our experiment started [75, 76]. Our measurements of sign and
magnitude of the scattering length provide important input for the following analysis of the
Feshbach spectrum, which turns out to differ completely from the expectations. Using the
ABM, we identify 23 of the 26 measured Feshbach resonances as d-waves, which is confirmed
by a coupled-channels calculation yielding scattering lengths in excellent agreement with the
experimentally obtained values. Important features in the resonance spectrum are discussed
by means of the atomic and molecular structure.
5.1. Previous Knowledge about NaLi Scattering Properties
From the sucess of sympathetic cooling in the early NaLi experiments [42, 43], one can
already infer that the triplet scattering length between sodium and lithium must be sufficiently high.1 Moreover, there were also three experimentally known interspecies Feshbach
resonances for this system [75]. Based on those, Gacesa et al. made predictions of further
resonances including the widths [76], which are summarized in table 5.1. Naturally, the
resonance at B0th = 1186 G having a width ∆ = 8.7 G, seemed from the experimental point
of view very appealing to us, such that we designed our antibias coils in a way they could
create such high fields.
As minor results of that paper, also the singlet and triplet scattering lengths together with
the respective vibrational level binding energies were published, as shown in table 5.1.
There are three issues about [76] which are not consistent:
ˆ From the theoretical point of view, having similar triplet and singlet scattering lengths,
but very different binding energies, is not possible, as C6 , which mainly determines
1
By the use of the word ’sufficiently’ it is implied that we can not make a quantitative statement, but in
this case |a| & 10a0 might be a good estimate.
73
B0exp [G]
dB/dt [G/s]
B0th [G]
∆ [G]
746.0 ± 0.4
15
746.13
0.044
singlet S = 0
triplet S = 1
759.6 ± 0.2
0.3
759.69
0.310
795.6 ± 0.2 —
10
—
795.61
1096.68
2.177
0.153
scattering length a [a0 ]
15.9±0.3
12.9±0.6
—
—
1185.70
8.726
—
—
1766.13
0.156
last bound state energy εSν [MHz]
-1.6±0.2
-5720±16
Table 5.1.: Measured and predicted Feshbach resonance positions B0exp , B0th and widths ∆ of
Na |1, 1i Li |1/2, 1/2i. Singlet S = 0, triplet S = 1 scattering lengths and last bound state
energies (from [76]). Ramp speed dB/dt and B0exp have been taken from [75].
a together with the binding energies, is the same for both singlet and triplet (the
potentials show the same long-distance shape). Therefore, either one of the bound
state energies εSν , the scattering lengths a or several of these parameters must be
wrong.
ˆ From our understanding of sodium intraspecies Feshbach resonances, the statement
”We have also checked that there are no higher partial wave resonances” [76] is highly
questionable. With s-wave resonances at similar fields as in the NaNa system and a
similar rotational splitting, with the resonance assignment of Ref. [75] there must be
higher partial wave resonances at magnetic fields below the already measured resonances.
ˆ From the experimental point of view, we saw in chapter 4 that resonance widths ∆
and inverse hold times to observe them stronly correlate (figure 4.6). Our inverse hold
times are essentially the same as the ramp speeds dB/dt in Ref. [75], so it is highly
suspicious that these experimental values are in sharp contradiction to the width values
of theory, as to be seen in table 5.1.
After all, more experimental measurements were needed to clear up those issues. In the
following, we will therefore describe how we investigated the NaLi problem2 , starting from
scattering length measurements, continuing with mapping out the Feshbach spectrum, which
is finally explained using the ABM, while for the resonance widths a CC calculation completes
the picture.
2
We will not follow a historic order, but rather explain things such that they are presented in a logical,
inductive way.
74
5.2. Scattering length determination
One result of the theoretical predictions which has to be experimentally confirmed is the
value of the scattering length a. From the sucess of sympathetic cooling we know that |a|
should be sufficiently large, but using this kind of thermalization process to determine a
quantitatively is not straightforward, both in experiment as in evaluation. Therefore, in the
following we will analyze the damping of oscillations in the ODT to determine |a| and get
the sign of the scattering length from the comparison of suitable absorption images.
5.2.1. Absolute value of the scattering length
Let us first assume both the lithium and sodium sample as pointlike. When they are excited
to oscillate in y-direction with their respective trap frequencies ωyLi and ωyNa, where ωyLi > ωyNa ,
a simple trigonometric argument shows that their ways cross with the rate 2ωyLi . Next, want
to determine the probability P of a scattering event in such a crossing of sodium and lithium.
As sodium is in the majority, we consider the case of one lithium atom moving through a
sodium background with density
Na
nNa (x, y, z) = η · nNa
c (x, y, z) + (1 − η) · nth (x, y, z)
(5.1)
composed of condensate nc and thermal background nth . The single lithium atom traveling
along the y-direction can be described by a δ-distribution nLi (x, y, z) = δ(x−x0 )δ(y)δ(z−z0 ).
Thus the probability P that the lithium atom is scattered by the sodium cloud during one
complete penetration along the y-direction reads
Z
Z
Na
P = σ dy n (x0 , y, z0) = σ dV nNa (x, y, z)δ(x − x0 )δ(z − z0 ) ,
(5.2)
where we introduced the interspecies scattering cross section σ = 4πa2 . Important for the
scattering events with the condensed atoms to take place and thus our derivation to be valid
is that the lithium atoms’ velocity exceeds the superfluid critical velocity of the condensate
vc ≈ 1 cm/s, which is well fulfilled for the following measurements.
To describe the experimental situation more accurately, we take into account the lithium
atom cloud’s spatial distribution, which we assume to be thermal3 . Therefore it can be
factorized as
nLi (x, y, z) = f (x)f (y)f (z).
(5.3)
Analog to eq. (5.2), the scattering probabilty can be written as
Z
P = σ dV nNa (x, y, z) · f (x)f (z) .
3
(5.4)
In reality, we have T /TF . 1, but by comparing the density distributions we checked that at the temperatures of our experiment the difference between thermal and Fermi distribution is still negligible.
75
a)
b)
Li after t Li
= 5ms
TOF
1000
Li density nLi(y)
Na density nNa(y)
potential V(y)
fit yielding τ = (39 ± 13)ms
Na after t Na
= 31ms
TOF
nNa, nLi, V(y)
cloud center after TOF [ µm]
We see that eq. (5.4) is independent of the density distribution of lithium in y-direction f (y),
an approximation which is valid as long as the oscillation amplitudes are large enough such
that after each crossing the clouds have completely penetrated each other. When a lithium
atom scatters with a sodium atom, it does not oscillate in phase any more with the rest,
i. e. it will on average end up at momentum p = 0 and does not contribute any more to the
coherent oscillation signal. Thus, the lithium center of mass oscillation amplitude decreases
with the time constant
Z
Li
Li
1/τ = 2ωy · P = 2ωy σ dV nNa (x, y, z) · f (x)f (z) .
(5.5)
500
0
−500
0
10
20
30
40
−100
time after displacement [ms]
−50
0
50
100
y position [µm]
Figure 5.1.: a) Simultaneous oscillation of Na |2, 2i (blue) and Li |3/2, 3/2i (red) along the ydirection of the ODT. The fits (solid lines) are exponentially decaying sine-functions yielding ωyLi /2π = (154.2 ± 1.4) Hz and ωyNa /2π = (74.9 ± 0.4) Hz. b) Cut through the theoretically calculated cloud shapes for t ≈ 7 ms. We see that the assumption of completely
penetrating clouds is well justified for the high oscillation amplitudes reached by displacing
the atoms with a magnetic gradient.
Experimentally, we prepared sodium in |2, 2i and lithium in |3/2, 3/2i, i. e. a pure triplet
state, displaced them with the gradient of our finetune coils and let them oscillate. From
the time-of flight pictures we can determine the damping of the lithium oscillation, which
we get as τ = (39 ± 13) ms from the fit in figure 5.1. With the oscillation amplitudes in
p-space, we can check if our picture of mutually penetrating clouds is correct. As to be seen
in figure 5.1, where a cut through the theoretically calculated cloud shapes is plotted for
t ≈ 7 ms, our approximation is justified.
With T = (225 ± 15) nK, N Na = (1.80 ± 0.24) · 105 and a condensate fraction of η =
0.4 ± 0.1, we get for the scattering probability per penetration P = (7.9 ± 0.7)1013 mσ2 . With
ωyLi = 2π · 154 Hz from the fit, we finally get
|at | = (69 ± 13)a0 .
76
(5.6)
This result differs substantially from the one presented in [76], as will be further elucidated
later. In chapter 8, we present a complementary measurement, yielding |aBF | = (70 ± 12)a0
and thus confirming the value obtained here, but also giving us an absolute value only.
Therefore, next we want to try and get a handle on the sign of the scattering length, which
our oscillation damping measurement is not sensitive to.
5.2.2. Sign of the Scattering Length
As already discussed in section 3.4.1, the in situ distributions of trapped atoms can not
be considered a reliable tool for quantitative data analysis, but nevertheless they can be
used to extract information about the sign (though not the absolute value) of the scattering
length. In chapter 2 we derived that a fermion immersed in a bath of bosons of density nB
experiences an additional mean-field-potential
µF =
2π~2
∂E
=
nB aBF
∂NF
µBF
(5.7)
due to interactions. We see that for the case of repulsive (attractive) interspecies interactions
aBF > 0 (aBF < 0), this potential is also repulsive (attractive). Thus when a lithium
distribution with a sodium background is compared to a distribution of lithium without
background, we expect the peak densities to be decreased (enhanced) by the mean-field
potential of the condensate.
Experimentally, we prepare a bose-fermi mixture of sodium and lithium in our ODT and
can – if required – remove the sodium selectively by a resonant light pulse provided by the
same source which we use for absorption imaging. Before we can proceed with the image
analysis, we have to check whether this removal has an unwanted effect on lithium, e. g. a
heating of the cloud due to scattering with the expelled sodium atoms.
If exposed to resonant light, sodium atoms are excited at a rate of typically Γ = 2π ·
10 MHz. This means that their absorption process happens way faster than their photoninduced movement starts, a fact which is essential for absorption imaging to work properly:
We usually image the atom density distribution such that the atoms do not have time to
rearrange for the picture. The sodium atom has such a big recoil momentum from the
absorbed photons that any collision with a lithium atom will lead to both atoms being lost
from the trap due to momentum and energy conservation. With the numbers from the
previous section, for a complete penetration of the two atomic clouds this happens with
a probability P = (7.9 ± 0.7)1013 mσ2 . Thus inserting the interaction cross section from
eq. (5.6) we get P ≈ 1.4%. As we start from lithium and sodium being trapped in the same
space, only about half a penetration takes place, i. e. the scattering probability per lithium
atom is reduced to less than 1%. So we see that the sodium removal does not affect the
lithium temperature, but only reduces its number by an amount which is way lower than
our statistical uncertainty of ∼ 10%.
77
TOF distribution
in situ distribution
lithium density [a.u.]
0.05
w/ Na
0.04
w/o Na
difference 0.03
0.1
0.02
0.05
0.01
0
0
0
100
200
y−position [pix]
300
−0.01
0
100
200
y−position [pix]
300
Figure 5.2.: Lithium density distributions with (blue) and without (red) sodium background.
Each distribution is an average over five shots, with every picture normalized to one. The
w/o Na
w/ Na
= (6.8 ± 1.1) · 104 for the
= (5.6 ± 0.8) · 104 and NIS
lithium atom numbers NIS
w/o Na
w/ Na
in-situ profiles and NTOF = (7.0 ± 1.6) · 104 and NTOF = (7.7 ± 1.4) · 104 for the timeof-flight measurements agree within their mutual error bars. The attractive interspecies
interaction is obvious when the difference (green) is plotted. As the expansion is taking
place with the sodium condensate still present, also in the respective TOF picture the
lithium density distribution is more narrow than without background.
In figure 5.2 we show lithium density distributions with and without sodium background,
both for in situ and time-of-flight imaging. The mutual deviations are visible by eye and are
getting even more pronounced when looking at the plotted difference of the distributions:
The sodiums clearly drags atoms to the center, which are then missing in the wings. This
clearly shows that aBF < 0. In principle, one could also extract quantitative information
from the TOF-pictures, but due to lithium having a temperature of T /TF < 1 a correct
description of the expansion is challenging as it is neither purely classical nor purely quantum
and moreover the sodium background is also expanding during TOF in a non-trivial way.
Therefore, we take the absolute value of at from the previous section to get
at = −(69 ± 13)a0 .
(5.8)
We thus see that a neither agrees with Ref. [76] in sign nor magnitude, which therefore
raises high doubts about the – anyway partly incorrect – last bound state energy, which also
determines the predicted Feshbach spectrum. To guide the further analysis, in particular of
the Feshbach spectrum, an easy connection between a and the last bound state energy εSν is
established in the following.
78
5.2.3. Last Bound State Energy from a
In the case of NaNa Feshbach resonances, where we only dealt with one hyperfine state, we
had defined the bound state energy with respect to the free atom threshold, i. e. εatoms (B =
0) ≡ 0. In the following, we will use the ABM model to analyze Feshbach resonances measured in different hyperfine states which thus have different εatoms (B = 0). Therefore, in the
following the bound state energy εSν will be defined with respect to the atomic hyperfine multiplet barycenter, e. g. in Na|1, 1iLi|1/2, 1/2i states with εSν > εatoms (B = 0) = −1.26 GHz
are not bound. This convention makes the different hyperfine state combinations easier
comparable with each other.
With the values of a from the previous section and from chapter 6, the possible range of
a covers −82a0 to −56a0 . For the moment we do not make a difference between singlet and
triplet scattering lengths as and at , as according to section 3.4.3 their values differ by less than
the experimental uncertainty of a. With the additional knowledge of C6 = 1467(2)a. u. [77],
one can calculate the last bound state energy εSν = −~2 κ2 /(2µ) + εatoms (B = 0), where µ
denotes the reduced mass, as follows [78]. Introducing
r
4 2C6 µ
β6 =
= 2RvdW
(5.9)
~2
as the characteristic length scale of the van-der-Waals potential, we can determine the scattering length a as
1
,
(5.10)
a = 0.480β6 1 +
tan(πF (β6 , κ))
where
β6 κ 0.443 − 0.0973β6κ + 0.146(β6 κ)7/3 + (β6 κ)3 (−0.280 + 0.461(β6 κ)2/3 )
F (β6 , κ) =
.
2.91 + 2.24(β6κ)4
(5.11)
Our scattering length measurements thus set lower and upper limits of the bound state energy
to -10.4 GHz< εSν <-9.7 GHz, respectively, differing completely from εS=1
= −5720 MHz and
ν
Nevertheless,
we
will
in
the
following
first
try to explain the
εS=0
=
−1.4
MHz
in
Ref.
[76].
ν
measured Feshbach spectrum starting from latter values.
5.3. Feshbach Spectrum
In order to apply the ABM to describe the Feshbach spectrum of heteronuclear NaLi resonances, we proceed as already described in chapter 4. Instead of taking |S, MS , I, MI i as a
Li
base, we choose |S, MS , mNa
i , mi i, which is more appropriate for distinguishable particles.
Apart from that change, which makes calculations even slightly easier, all steps can be taken
analog to the NaNa case.
79
S
MS
mNa
i
mLi
i
1
-1
3/2
1
1,0 1,0
0
0
3/2 1/2
0
1
1
1
1
1
1
1
3/2 1/2 -1/2
-1
0
1
Table 5.2.: List of l = 0 states for the vibrational singlet (S = 0) and triplet (S = 1) state with
Li
MF = mNa
f + mf = 3/2.
To start with, we consider the channel Na|1, 1iLi|1/2, 1/2i, the only one for which NaLi
Feshbach resonances had been known before our experiment started. To describe them within
the framework of ABM, we first determine the spectrum of possible quantum numbers of
Li
Na
Li
molecular states |S, MS , mNa
i , mi i with MF = MS + mi + mi = 3/2, which can give rise
to Feshbach resonances in that particular spin channel (table 5.2).
First, we try to reproduce the results of Ref. [76], a CC calculation, with the help of
the ABM. Therefore, we do a least square fit to the theoretically predicted s-wave resonance positions, three of which had already been measured in experiment. This yields
εS=0
= −4131.3 MHz and εS=1
= −5714.5 MHz for the singlet and triplet binding energies,
0
0
respectively, latter being consistent with the result of Ref. [76]. The two new predicted res1
Na|2,2> + Li|1/2,−1/2>
Na|1,0> + Li|3/2,3/2>
Na|1,1> + Li|1/2,1/2>
0
energy [GHz]
−1
−2
−3
−4
−5
−6
−7
−2000
−1500
−1000
−500
0
500
magnetic field B [G]
1000
1500
2000
Figure 5.3.: ABM representation of the Feshbach spectrum in the MF = 3/2 channel, with the
three measured resonances [75] (green circles) interpreted as s-waves [76]. The resulting
binding energies are εS=0
= −4131.3 MHz and εS=1
= −5714.5 MHz for singlet and triplet,
0
0
respectively. The free atom states are depicted as thick lines (see legend), the molecular
l = 0 states as thin blue lines. In the region from 1 kG to 1.3 kG, no resonances could
be found (red circle). The inset shows the resonances in two different channels for B =
[−1150; −850] G, the structure of which prevents an easy explanation of the Feshbach
spectrum using s- and p-waves only.
80
S
MS
ml
mNa
i
mLi
i
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
2
2
2
1
1
0
3/2 1/2 -1/2 3/2 1/2 3/2
-1
0
1
0
1
1
S
MS
ml
mNa
i
mLi
i
1,0 1,0
0
0
0
-1
1/2 3/2
1
1
1,0 1,0
1,0 1,0 1,0 1,0 1,0
0
0
0
0
0
0
0
2
2
2
1
1
1
0
1/2 -1/2 -3/2 3/2 1/2 -1/2 3/2
-1
0
1
-1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
1
0
0
0
-1
-1
-2
-1/2 -3/2 1/2 -1/2 -3/2 3/2 1/2 -1/2 3/2 1/2 3/2
-1
0
-1
0
1
-1
0
1
0
1
1
Table 5.3.: List of l = 2 states for the vibrational singlet (S=0) and triplet (S=1) state with
Li
MF = MS + mNa
i + mi + ml = 3/2.
onances of singlet character, also depicted in figure 5.3, are predicted by ABM to be at
1097.3 G and 1185.3 G, respectively, also in excellent agreement with the CC predictions.
To locate the predicted resonances, we did Feshbach spectroscopy in a wide range around
the predicted positions, but could not confirm any of them. On the contrary, we found a lot of
unpredicted resonances depicted in figure 5.3. As to be seen in channel Na|1, 1iLi|1/2, 1/2i,
their mere quantity already exceeds the number of possible s-waves. Taking additional pwaves into account is also not an option, as to be seen in the inset: This way, we could
fit the two resonances in Na|1, 1iLi|1/2, 1/2i with an s- and a p-wave, but this would leave
the resonance in Na|1, 0iLi|3/2, 3/2i unexplained. Taking additionally into account that the
binding energies of ∼ −5 GHz assumed for such a scenario differ a lot from the ∼ −10 GHz
obtained above from experimental data, we tried to do an assignment of the resonances using
mostly d-waves.
Li
Therefore, the number of possible states |S, MS , mNa
i , mi i is increased substantially, as
Li
the constraint now is MF = MS + ml + mNa
i + mi = 3/2 with |ml | ≤ l = 2. For MS = −1,
the number of possible states rises from one to six, and in total now we have 35 instead of
8 molecular states (see table 5.3).
5.4. Assignment and Fit of Resonances
The most important steps of the procedure to fit the measured Feshbach resonances, whose
positions and experimental raw data are given in table 5.5 and Appendix B, can be summarized as follows:
ˆ From the l = 0 state bound state energy estimate −10.4 GHz< εS0 < −9.7 GHz we can
exclude s-wave resonances at magnetic fields B < 1.3 kG. The only s-wave resonances
81
ABM
CC
εS=1
[GHz]
εS=0
[GHz] εS=1
[GHz]
εS=0
[GHz]
η
0
0
2
2
-9.3521
-9.4505
-5.8509
-5.9493
0.9817
-9.35335(50) -9.3838(50) -5.85180(30) -5.95634(40) —
Table 5.4.: Triplet S = 1 and singlet S = 0 bound state energies obtained from ABM and CC.
. Due
as well as the good agreement in εS=0
and εS=1
Note the excellent agreement in εS=1
2
0
2
S=0
to the positions of our Feshbach resonances not being sensitive on ε0 , this value differs
most from the results obtained by the CC calculation.
we can measure with our experimentally achievable magnetic fields have MS = ±1,
resonances from MS = 0 molecular states could only appear around B ≈ 3 kG.
ˆ From the measurements of resonances in the MF = 5/2 channel, which are not sensitive
on the d-state singlet energy εS=0
, we can deduce the d-state triplet energy εS=1
. Here,
2
2
the three resonances in the MF = −3/2 channel already discussed above are helpful
for an unambiguous assignment of the resonances, as figure 5.5 a) shows.
ˆ Knowing that the singlet and triplet l = 0 bound state energies are roughly the same,
we can also assume the same rotational splitting for both, as the related part of the
potential is mainly given by C6 , which is the same for both singlet and triplet potential.
Thus we can infer that with the s-wave bound state energies εS=1
≈ εS=0
, we also get
0
0
S=1
S=0
for the d-wave energies ε2 ≈ ε2 .
ˆ With those assumptions, we first try and assign as many resonances as possible as
d-waves and assume the rest, i. e. resonances which are more than ∼5 G off a crossing
of molecular states and atomic threshold, as s-waves.
With our results almost not being sensitive on the singlet l = 0 bound state energy as
pointed out above, instead of taking 3 fit parameters each for l = 0 and l = 2 states, we
rather fit the d-state energies εS=0
and εS=1
, a Franck-Condon factor η and the rotational
2
2
S=1
S=1
splitting ε2 − ε0 . The results of our weighted least squares fit procedure are summarized
and compared to the results of a CC calculation (see below) in tables 5.4 and 5.5. From our
ABM analysis so far, the 3 resonances assigned s-waves could be p-waves as well, but the
CC calculation using the full interaction potentials can determine the rotational splitting
and thus confirms our assumption.
Figure 5.4 shows an overview of the resonance assignment for the different spin channels.
Although they do not deliver information about the quality of the fit, these picture are
important to get an impression of the overall molecular state and resonance structure. For
B = 0, the molecular states show a splitting of 1.77 GHz, the sodium hyperfine splitting. If we
zoom further in (see figure 5.5 b), we see that theses two branches show a splitting of 228 MHz,
the lithium hyperfine splitting. By calculating the total molecular spin f~ = f~Na + f~Li , we
can also explain the substructure of the molecular levels visible in the zoom. Each state f
82
MF = 1/2 channel
Na|1,−1> + Li|3/2,3/2>
Na|1,0> + Li|1/2,1/2>
Na|1,1> + Li|1/2,−1/2>
energy [GHz]
0
−2
−4
−6
−2000 −1500 −1000 −500
0
500
1000
1500
2000
MF = 3/2 channel
Na|2,2> + Li|1/2,−1/2>
Na|1,0> + Li|3/2,3/2>
Na|1,1> + Li|1/2,1/2>
energy [GHz]
0
−2
−4
−6
−2000 −1500 −1000 −500
0
500
1000
1500
2000
MF = 5/2 channel
Na|1,1> + Li|3/2,3/2>
energy [GHz]
0
−2
−4
−6
−2000 −1500 −1000 −500
0
500
magnetic field B [G]
1000
1500
2000
Figure 5.4.: Near-threshold molecular spectra from ABM for different MF , showing the molecular
states l = 0 (blue) and l = 2 (red) states, and the observed s- and d-wave Feshbach
resonances as blue and red dots, respectively. Atomic thresholds are depicted as thick lines
and labeled in the legend. Zooms of the two boxes in the MF = 3/2 figure are shown in
figure 5.5. The horizontal green line depicts the sodium hyperfine splitting. A negative
magnetic field corresponds to a sign change in MF .
83
has several Zeeman sublevels |mf | ≤ f , which are for a certain spin channel MF only shown
if MF = mf + ml can be fulfilled (see table 5.3).
energy [GHz]
a)
−2.7
(−1/2,0,−2) (1/2,−1,−2)
(−1/2,−1,−1)
b)
−5.1
−2.8 (−3/2,1,−2)
(−3/2,0,−1)
−2.9
(−3/2,−1,0)
−3
−5.2
−3.1
−5.3
−3.2
fLi=3/2
fNa=2
fLi=1/2
3/2
5/2
−5.4
−3.3
−1200 −1100 −1000
magnetic field B [G]
−900
f
7/2
5/2
3/2
1/2
−10
0
10
magnetic field B [G]
Figure 5.5.: Zooms of the MF = 3/2 state of figure 5.4. a) The molecular states (red lines),
Li
which all have MS = 1, are labeled with their quantum numbers (mNa
i , mi , ml ). The
green horizontal lines show the splitting due to different mi,Na as obtained from the BreitRabi formula, the magenta horizontal lines have a length of aLi
hfs /2. b) For small B, the
molecular states can be characterized by their quantum numbers f and mf . The horizontal
magenta line depicts the lithium hyperfine splitting.
5.4.1. Quantum Numbers
The zoom in figure 5.5 shows how the resonances in the MF = 3/2 channel, which could
not be explained in an s-wave scenario, naturally arise from the l = 2 molecular states. We
can also extract important information about the structure of the resonances: As also to be
seen in figure 5.4, the molecular states for a certain total electron spin projection MS have a
coarse structure given by states with a sodium nuclear spin mNa
∈ {−3/2, −1/2, 1/2, 3/2}.
i
The offset between those states can be easily calculated by the Breit-Rabi formula. Each line
consists of up to three states with different lithium nuclear spin
bundle with a certain mNa
i
Li
mi ∈ {−1, 0, 1}, whose mutual offset is aLi
hfs /2, as we are far in the Paschen-Back regime of
the atomic hyperfine structure.
With this explanation in mind, it is straighforward to extract the quantum numbers of the
molecular states with MS = ±1 causing Feshbach resonances. For the MS = 0 states, which
can not be assigned S = 0 or S = 1 due to the strong mixing of singlet and triplet, more
care has to be taken: By tracing back the molecular state from the resonance position B0
to B = 0, we can determine the quantum numbers f and mf and thus ml = MF − mf from
Li
Na
Li
a picture like figure 5.5. Being in an MS = 0 state, we have mf = mNa
f + mf = mi + mi ,
84
Li
where mNa
i and mi can be determined by tracing the molecular state in ABM representation
to B → ∞: There the different molecular states separate in bundles characterized by mNa
i ,
as to be seen in figure 5.4.
5.4.2. Triple Features
When examining the resonance positions in table 5.5, one will notice that around 1596 G and
1717 G, there are three resonances at almost equal magnetic fields characterized by similar
quantum numbers. More precisely, the resonances show up in the MF = 3/2, MF = 1/2 and
MF = −1/2 states when prepared in Na|1, 1i and Li|1/2, 1/2i, |1/2, −1/2i and |3/2, −3/2i,
respectively. Figure 5.6, where also the three resonances around 1717 G are depicted in one
graph, shows how such a triple feature arises: Three molecular lines characterized by their
Li
quantum numbers MS = 0, mNa
i , ml and mi = 0, ±1 each lead to a resonance when crossing
the atomic threshold characterized by mLi
i = 0, ±1.
b)
1
energy [GHz]
normalized Li atom number
a)
0.5
−5.1
MF=−1/2
−5.2
MF=1/2
−5.3
MF=3/2
−5.4
−5.5
−5.6
−5.7
0
1715
1716 1717 1718
magnetic field B [G]
1550
1600 1650 1700
magnetic field B [G]
1750
Figure 5.6.: Triple feature in the NaLi resonance spectrum. a) The three 1717 G resonances in
three different spin channels shown in one graph. b) The resonances (black dots) and the
respective molecular and atomic states (thin horizontal and thick diagonal lines, respectively). The atomic spin states are Na |1, 1i and Li |1/2, 1/2i (MF = 3/2), |1/2, −1/2i
(MF = 1/2) and |3/2, −3/2i (MF = −1/2). The green horizontal line shows the splitting
due to different mNa
= 3/2, 1/2 as obtained from the Breit-Rabi formula, the magenta
i
Li
horizontal lines have a length of aLi
hfs /2 and show the splitting between mi = 0, ±1.
The energetic difference between the three different hyperfine states in the Paschen-Back
regime is aLi
hfs /2, as depicted in the figure, which is the explanation for each single triple
feature. Even the difference in energy and thus magnetic field position of the resonances
can be understood: By recognizing that the 1596 G triple has mNa
= 3/2, whereas the
i
1717 G triple has mNa
=
1/2,
we
can
easily
calculate
the
energy
difference
via the Breiti
85
Rabi formula and with the slope of the atomic threshold of ∼ 2.8 MHz/G the difference in
resonance positions.
As one would suppose, due to their high similarity in quantum numbers, all three 1596 G
and 1717 G resonances show similar widths of 5 mG and 0.3 mG, respectively, which have
been obtained by a CC calculation (see table 5.5). Furthermore by looking at the ml quantum
number, one sees that the observation of one triple feature might be explained in an s-wave
scenario (ml = 0), whereas seing a second triplet nearby is a clear hint that higher partial
waves, i. e. d- or even g-waves (ml > 0 possible), have to be involved in the explanation of
the spectrum.
5.4.3. Resonance Widths
The resonance widths shown in table 5.5, which were obtained by a CC calculation, are all
very small. To be quantitative, one can determine their resonance strengths sres eq. (3.21)
and sees that all resonances are weaker than sres = 4 · 10−4 . This is due to a combination
of factors: First, the singlet and triplet interaction potentials are far from being resonant,
indicated by their scattering lengths which are of the same order as the van der Waals length
eq. (2.2) RvdW ≈ 35a0 . Second, the effective coupling between singlet and triplet states is
very small, indicated by |as − at | ≪ RvdW . A similar situation can be found in homonuclear
87
Rb [79], where the narrowness of the Feshbach resonances and the smallness of the loss rates
can be traced back to non-resonant and similar values of as and at [80]. The resulting long
lifetimes of the mixture in all spin channels are an advantage over e. g. the Na40 K mixture.
There, one has resonances nicely suited for tuning [81], but due to the large difference of
singlet and triplet scattering length |as − at | ≈ 900a0 [82] thermalization measurements with
long timescales involved are only possible in selected spin state combinations.
Due to the small resonance widths ∆ ≤ 10 mG in combination with magnetic field fluctuations δB ≫ 10 mG, the fitted loss feature width ∆B (see figure appendix B) is not a good
experimental measure of the resonance width ∆. To be able to compare different resonances
systematically, we took for each resonance a loss curve with a fixed magnetic field sweep
range of ∼ 1 G, an example of which is shown in figure 5.7 a). From an exponential fit, we
get the hold time τ , whose inverse shows correlations to the calculated resonances width.
The scatter though is far too high for a meaningful quantitative analysis.
The complete results of ABM fitting procedure, CC calculation and quantum number
assignment are summarized in table 5.5. The fact that almost all resonances are fitted with
less than 1 G deviation by only four ABM fit parameters is strongly supporting our scenario.
The exceptions can be explained by the CC calculation: It shows that the resonances in the
MF = 5/2 channel have a high inelastic loss rate to lower lying atomar channels, so as we
learned in section 3.3.4 their experimentally measured position of maximum loss might thus
not be coincident with the resonance position.
The resonances at 800.9 G and 1700.4 G were measured after our model was made, which
proves its accuracy and predictive power. The fact that the ABM yields way more resonances
86
4
x 10
inverse hold time [1/s]
12
Li atoms
10
τ = 266ms
8
6
4
2
0
0
0.5
time [s]
1
1
10
0
10
−1
10
0
10
1
10
resonance width [mG]
Figure 5.7.: Left: Example of a loss curve with fixed magnetic field sweep range (800.9 G resonance). Right: Fitted inverse hold time vs. resonance width from CC calculation. The red
dots, which are resonances from the lowest energetic atomic channel Na |1, 1i Li |1/2, 1/2i,
show slightly less scatter and a better correlation.
than we observed (see figure 5.4) is explained by the coupled channels calculation: Compared
to the observed resonances, the width of the ones not found is simply too narrow to be
detected by our apparatus.
The CC calculation yields for the scattering lengths as = −73(8)a0 and at = −76(5)a0 ,
which shows in sign, magnitude and mutual difference excellent agreement with the experimental values obtained above.
5.5. Concluding Remarks
With the chapter about NaNa resonances in mind, one might ask why it was so much more
difficult to explain the NaLi spectrum. First, much more data was known for NaNa from
molecular spectroscopy, whereas the NaLi data has just recently been obtained [83]. Moreover, the early sodium BEC experiments started in 1995 were relying on the fact aNaNa > 0,
otherwise condensates would have been unstable [84, 85]. Thus, the sign of the scattering
length
√ was known and the magnitude had to be consistent with the measured speed of sound
c ∝ an [86, 87], which pins down the molecular bound state energies quite well.
In contrast, in the case of NaLi there was only knowledge about good thermalization with
sodium [42, 43], but not even the sign was assumed correctly (Ref. [30] in [76]). That together
with only three measured resonances, which were interpreted as s-waves [75], was forming
an almost consistent picture [76]. This is the reason why it took us so long to see the need
of determining the scattering length a experimentally and completely changing our scenario
87
from s- to d-waves.
Although none of the resonances found is suitable for tuning due to their low widths, there
is still possibilities to get an α big enough to observe polaronic behaviour in our experiments:
According to our findings, aIB is a factor of 5 bigger than assumed before, thus we got a factor
of 25 increase in α ’for free’. With typical sodium numbers, we now have α ≈ 0.02, which
would result in an increase in the effective mass of lithium by 0.7%. The tunability could
then be achieved by changing aNaNa using the Feshbach resonance at 1202.6 G which has
been described in detail in chapter 4. At an experimentally feasible magnetic field stability
of 50 mG, tuning could reach α ≈ 0.028 and thus a 10% increase in effective mass. A possible
problem of this approach is the disappearance of Boguliubov modes for aNaNa → 0, which are
essential for the theoretical explanation of the polaron [21]. Thus another option is outlined
in our paper [88], where the CC calculation yields widths suitable for tuning the interaction
of 7 Li+Na, a mixture our apparatus could cool and trap after some changes in the laser
setup.
88
6
Li+Na
MF
Exp.
ABM
Coupled-Channels
B0exp (G) δB0ABM (G) δB0CC (G) ∆(mG)
Quantum Numbers
MS
mNa
i
mLi
i
l
ml
|2i+|1i
1/2
771.8(5)
822.9(5)
1596.8(4)
1716.7(3)
-0.7
0.5
0.2
0.0
-0.190
0.050
-0.314
0.231
10
0.5
5
0.2
1
1
0
0
3/2
1/2
3/2
1/2
0
0
0
0
2
2
2
2
-2
-1
-1
0
|1i+|3i
−1/2
1002.3(5)
1088.5(5)
-0.6
0.2
-0.209
-0.301
9
0.9
1
1
-1/2
-3/2
1
1
2
2
-2
-1
|3i+|1i
−1/2
800.9(2)
852.0(7)
1566.3(8)
1597.5(7)
1717.3(2)
-0.4
0.2
0.1
0.4
-0.2
0.096
-0.271
0.023
-0.144
0.038
10
0.5
0.03
6
0.3
1
1
1
0
0
3/2
1/2
-1/2
3/2
1/2
-1
-1
-1
-1
-1
2
2
0
2
2
-2
-1
0
-1
0
|1i+|1i
3/2
745.2(3)
759.0(3)
795.2(2)
1510.4(3)
1596.5(5)
1715.6(8)
1908.9(7)
2046.9(9)
-0.3
0.5
0.5
0.0
0.5
-0.3
0.4
0.5
0.175
0.022
-0.020
-0.024
0.009
0.034
-0.350
-0.608
10
0.02
0.5
0.04
5
0.3
0.04
4
1
1
1
1
0
0
0
0
3/2
3/2
1/2
-1/2
3/2
1/2
1/2
3/2
1
0
1
1
1
1
1
1
2
2
2
0
2
2
2
2
-2
-1
-1
0
-1
0
0
-1
|2i+|3i
−3/2
1031.7(3)
1117.3(6)
1902.4(6)
-0.3
0.0
-0.3
0.166
-0.511
-0.045
9
0.8
0.1
1
1
0
-1/2
-3/2
-3/2
0
0
0
2
2
2
-2
-1
0
|3i+|2i
−3/2
913.2(6)
1720.5(3)
-0.3
0.0
0.108
-0.103
9
0.06
1
1
1/2
-3/2
-1 2
-1 0
-2
0
|6i+|1i
5/2
1575.8(9)
1700.4(7)
0.9
1.5
-0.014
-0.040
—
—
1
1
3/2
1/2
1
1
2
2
-1
0
Table 5.5.: Overview of the experimentally and theoretically obtained results on Feshbach resonances, sorted by spin state and corresponding quantum number |MF |. In column “Exp.”
the position of maximum loss B0exp is reported, which is determined by Gaussian fits to
the loss features (s-waves in bold). As error we give the rms width of the RF calibration
signal. The other columns show the Feshbach resonance positions from the ABM fit, and
the positions and widths ∆ from coupled-channels calculation. The theoretical positions
are given by their deviation from the experimental value, i. e. δB0ABM ≡ B0exp − B0ABM and
δB0CC ≡ B0exp − B0CC . For the two resonances of asymptote |6i+|1i no widths are given
(see text). The assignment of each resonance in terms of the molecular quantum numbers
Li
Na
Li
|MS , mNa
i , mi , l, ml i with mi + mi + ml = MF is given in the last column. The spin
quantum number S takes the value S = 1 if MS = 1 and cannot be assigned for MS = 0
due to the strong mixing of singlet and triplet.
89
90
Part II.
Bose-Fermi Mixtures in Periodic
Potentials
In the second part of this thesis, we investigate atoms in a species selective optical lattice.
In order to be able to create one- and two-dimensional Fermi gases, we derive design criteria,
which we use to determine the parameters of the optical lattice. Starting with an analytic
harmonic oscillator approximation, we subsequently extend our calculations to describe the
full periodic potential using Bloch states. With that knowledge at hand, we are able to
explain the experimentally demonstrated phase lattice for sodium and the oscillations of
lithium.
As a tool to analyze the momentum distribution of atoms in the lattice, we implement the
technique of Brillouin zone mapping. As a first application, we use it to improve the loading
process of the lattice by having a sodium background present. In the following, we further
investigate the interspecies energy transfer from oscillating lithium to condensed sodium
atoms.
By applying a periodic shaking of the lattice, we are able to do a controlled transfer of
lithium atoms from first to second Bloch band. The subsequent relaxation in presence of
a sodium background is used to estimate the absolute value of the interspecies scattering
lenght aIB . Finally, we demonstrate and analyze a Rabi oscillation of lithium atoms in the
two-level system formed by the two lowest Bloch bands.
92
6. Design and Implementation of the
Optical Lattice
With our goal to engineer polarons in ultracold atom systems in mind [21], we have to think
about an appropriate experimental realization. As discussed in the first part of this thesis,
the tuneability parameters aIB and aBB are now well studied, such that we know how to
adjust α. A parameter characterizing the polaron is its effective mass, which scales in the
weakly interacting regime as m∗ = m · (1 +√0.3637α) [89]. As it is easy to measure masses
experimentally, e. g. via frequencies ω ∝ 1/ m [17], we want to design a potential in which
we can selectively excite oscillations of the lithium and map out their dependence on α.
The tool of choice for reaching that goal is the use of a species selective optical dipole
trap (SSODT) [90]. As we will show in the following, by tuning the laser, which creates the
SSODT, in close vicinity to the lithium transition, we obtain a trapping potential for lithium
only while the impact on sodium is negligible. This way, we can not only excite oscillations
of the lithium, but also drag it through the sodium background, which gives the possibility
to map out the superfuid critical velocity. Similar experiments have been conducted using a
blue detuned laser beam [91] and sodium atoms in different hyperfine states [92] for bosonic
condensates and a moving optical lattice for a superfluid Fermi gas [93]. Due to the precise
spatial control provided by dipole traps, lithium can this way even be used as a local probe
for condensate properties, an aim which is currently being pursued by experiments moving
ions through BECs [94].
There are numerous possibilities for the overall form of the SSODT potential: One could
cover the issues mentioned above using e. g. a single focused laser beam. We decided to
give our system more versatility by implementing the SSODT as a two-dimensional optical
lattice potential1 , such that we can form a stack of lithium pancakes each containing a twodimensional Fermi gas or even get an array of one-dimensional tubes. The interesting aspect
about lowering the dimensionality is that for interacting particles we get easily into the
strongly interacting regime, where Eint /Ekin ≫ 1. Thus, fermion pairing in two-dimensional
systems [95] and its evolution when dimensionality is increased to three [96] are objects of
current research.
1
At the time this thesis was written, only one axis had been implemented experimentally.
93
6.1. Design of the SSODT
6.1.1. Dimensionality criteria
Before starting with the design of the SSODT, we want to develop the criteria for a system to be considered one- or two-dimensional. Therefore, we recall the Fermi energy of N
harmonically trapped atoms
EF1d = Nω||
(6.1)
in one dimension and
EF2d =
√
2N · ω||
(6.2)
in two dimensions, respectively. The dimension in which the atoms are allowed to move is
characterized by the frequency ω|| , whereas the transversal direction in which motion shall be
frozen out, has frequency ω⊥ ≫ ω|| . For the gas to allow for a theoretical lower-dimensional
treatment, we have to fulfil the following criteria:
ˆ EF = µ(T = 0) ≪ ~ω⊥ : A new particle added must not occupy a transversally excited
state.
For N fermions in a lattice well, this reads N · ω|| ≪ ω⊥ for one-dimensional and
√
2N · ω|| ≪ ω⊥ for two-dimensional systems.
ˆ kB T ≪ ~ω⊥ : Temperature has to be so low that particles are not thermally excited to
higher transversal states.
ˆ J ≪ µ: In the case of an optical lattice, the tunneling energy J must be smaller than
the chemical potential µ to suppress hopping of atoms between the different lattice
sites [97].
Experimentally, we typically have NLi = 105 lithium atoms at a temperature of T =
0.2 µK. Thus, temperature imposes an ω⊥ /2π ≫ 4 kHz, while for the two-dimensional case
with at most N ≈ 700 atoms in the central sheet2 we get from the chemical potential
ω⊥ /2π ≫ 8.4 kHz, where we assumed ω|| /2π = 224 Hz as the geometric mean of the ODT
frequencies in y- and z-direction. Similarly, the tunneling condition yields J/2π ≪ 6 kHz.
In the following, we will derive some analytical formulas which will prove useful in the
succesive lattice design.
6.1.2. Fundamentals of Optical Lattices
Optical Dipole Forces
In the field of ultracold quantum gases, a common way to design potentials for the atoms is
to make use of the dipole force. In short, an atom with transition frequency ω0 and decay
2
This maximal value is reached for loading a T = 0 degenerate Fermi gas into a lattice formed by a
λ ≈ 670 nm retroreflected laser beam.
94
rate Γ feels due to its polarizability the dipole potential
Γ
Γ
3πc2 Γ
3πc2
+
I(~r) ≈
I(~r)
Vdip (~r) =
2ω03 ω − ω0 ω + ω0
2ω03 ∆
(6.3)
when exposed to a light field with intensity I(~r) oscillating at frequency ω [98]. The rotating
wave approximation on the right hand side is valid for detunings |∆| = |ω − ω0 | ≪ ω0 .
Spontaneous processes lead to a scattering rate
3 2
2
3πc2 ω
Γ
Γ
3πc2 Γ
Γsc (~r) =
+
I(~r) ≈
I(~r) .
(6.4)
2~ω03 ω0
ω − ω0 ω + ω0
2~ω03 ∆
Comparing eqs. (6.3) and (6.4), we see that Γsc /Vdip ∝ 1/∆, i. e. choosing a large detuning
helps us to reduce the scattering and thus heating of the atom sample. We further see that
the dipole potential is attractive for red detuning ∆ < 0 and repulsive for blue detuning
∆ > 0. In latter case, the atoms are trapped in an intensity minimum, which minimizes the
spontaneous scattering rate.3
As we want to have a species-selective lattice for lithium, we have to minimize its influence
on sodium, i. e. keep the lattice depth lower than the chemical potential VNa ≪ µ ≈ 100 nK.
As we can already anticipate, the lattice wavelength will have to be close to the lithium D2
transition λ = 670.977 nm, translating into I ≪ 2 · 105 W/m2 . If a detuning ∆ ≈ 1 nm is
chosen, the resulting lithium scattering rate in this light field is ΓSc ≈ 3/s, a value which
would limit possible experiments to a duration on the order of ∼ 100 ms.
Optical Lattices
A common way to create an optical lattice is to retroreflect a beam of wavelength λ and
thus get a standing wave potential with periodicity λ/2,
x V x
0
2
=
1 − cos 2π
,
(6.5)
V (x) = V0 sin 2π
λ
2
λ/2
where we get a lattice depth V0 by interfering a single beam of potential of depth V0 /4 with
its retroreflection. Approximating each lattice well as a harmonic oscillator V (x) = 21 mω 2 x2 ,
we get from a taylor expansion of eq. (6.5)
r
2π 2V0
ω=
.
(6.6)
λ
m
2 2
It is convenient to introduce the lattice recoil energy Er = ~2mk with the lattice wavevector
k = 2π/λ, which describes the kinetic energy gain of a lithium atom when absorbing a
3
Scattering can never be completely turned off due to the atom always having a thermal or quantum
mechanical distribution with finite spatial width ∆x, i. e. there will always be some light scattering.
95
photon from a lattice beam. Introducing V0 = sEr we can express ω conveniently as
ω=
2Er √
s.
~
(6.7)
With the intensity and detuning values from above, we get V0 ≈ 2.6Er with Er = 3.5 µK=
73 kHz for lithium and thus ω/2π ≈ 235 kHz, which fulfils the frequency conditions for
reaching lower dimensionality ω⊥ /2π ≫ 8.4 kHz easily.
So far, we were reducing our considerations to one lattice site only. Having several lattice
sites, their connection can be discribed by a tunneling matrix element J, which can for s ≫ 1
be approximated as [99]
√
√
4
4
3/4 −2 V0 /Er
√
Er = √ (s)3/4 e−2 s Er .
J=
(V0 /Er ) e
π
π
(6.8)
For s = 2.6, we get J/2π ≈ 14 kHz, i. e. it clearly violates the condition for no tunneling
J/2π ≪ 6 kHz. A way this can be understood intuitively is that lithium can due to its light
mass easily hop over the barrier between one lattice well and the next. A way to circumvent
this is thus to make the lattice spacing larger, which can formally be√
seen as increasing λ and
thus decreasing Er . Due to the exponential scaling of J with −1/ Er ∝ −λ, we decrease
the tunneling energy this way.
Experimentally, instead of creating the lattice by a retroreflected beam, we let two beams
interfere under an angle 2α as shown in figure 6.1 and thus effectively change the wavelength
λ → λ/ sin(α). Due to space limitations, we choose 2α = 35◦ , which results in a effective
λ = 2.2 µm or a lattice spacing of λ/2 = 1.1 µm, respectively. The formulas derived above
stay valid, with Er now being calculated with the new λ, i. e. it decreases by about one order
of magnitude to Er = 6.9 kHz.
The absolute lattice depth V0 does not change, but due to s = V0 /Er we get s = 28 and
thus for the tunneling coupling J/2π = 50 Hz at a vibrational frequency of ω/2π = 70 kHz,
i. e. our conditions for reaching lower dimensionality are well fulfilled. With the lithium
atoms being in the harmonic oscillator ground state, they scatter light at a rate Γsc ≈ 1/s
from the blue detuned light sheets.4 In figure 6.1 b) we can see how according to eq. (6.4)
the scattering rate increases when the lattice depth is varied at constant power by changing
the detuning.
4
For a red detuned SSODT, the overlap integral with the lattice light would be about a factor of 10
higher, which increases the scattering rate by the same factor. Additionally to the advantage of the lower
scattering rate, using a blue detuning drastically reduces the overall harmonic confinement induced by
the gaussian shape of the lattice beams compared to the red detuned configuration [101].
96
a)
14
120
12
100
10
80
8
60
6
40
4
20
2
35°
x
EOM
PBSC
photo
diode
PBSC
fiber
coupler
0
0
20
40
60
lattice depth s
80
scattering rate Γ [1/s]
y
frequency ω/2π [kHz]
z
b) 140
0
100
Figure 6.1.: a) Experimental setup to create the optical lattice potential. For technical details,
see Ref. [100]. b) Oscillation frequency ω (blue line) and spontaneous scattering rate Γsc
(red line) of lithium atoms in the blue detuned optical lattice in dependence of its depth s,
latter being tuned by changing the detuning ∆. The fixed light intensity chosen here is
such that the impact on sodium is kept sufficiently low, as described in the text.
Inhomogeneity
So far, we have not taken the beam shape into account: The SSODT is formed by Gaussian
beams, which are mathematically described by the intensity profile
2
2(x2 + y 2 )
w0
(6.9)
exp −
I(x, y, z) = I0
w(z)
w 2 (z)
p
for a beam propagating in z-direction. w(z) = w0 1 + (z/zR )2 is the waist, which takes
the value w0 at the beam focus and zR = πw02 /λ is called Rayleigh range. Normalization
determines I0 = 2P0 /(πw02) with P0 being the total beam power.
In our experimental setup, the lattice beam has a waist of w0 ≈ 400 µm at the fiber
outcoupler, which expands on its way to the glas cell to w ≈ 550 µm, i. e. we can neglect
the z-dependence of the intensity profile. Thus, two beams interfering under an angle 2α as
shown in figure 6.1 a) have the overall intensity envelope
2(y 2 + (x cos(α))2 + (z sin(α))2 )
I(x, y, z) ∝ exp −
.
(6.10)
w2
Plugging this result for intensity and thus potential depth V0 into the formula for ω eq. (6.6),
we see after a Taylor expansion that we get a frequency with spatial dependence
y2
(x cos(α))2 (z sin(α))2
ω(x, y, z) = ω0 1 − 2 −
−
.
(6.11)
w
w2
w2
97
For an estimate of the anharmonicity ∆ω experienced by the atoms, we can plug in the radii
(Rx , Ry , Rz ) = (48, 46, 25) µm of the T = 0 Fermi distribution and get
(∆ωx , ∆ωy , ∆ωz ) = (7 · 10−3 , 7 · 10−3, 2 · 10−4 )ω0
(6.12)
We see that we can expect to observe about 50 oscillations until the atoms in the center are
out of phase by π from the atoms at the cloud edge. This dephasing is independent of the
value of the lattice depth s, in contrast to the dephasing due to the finite bandwith J, as we
will see later.
Spatial Configuration and Implications for Polaron Experiments
At this point, we can develop a picture of our experimental situation: As to be seen in
figure 6.2, lithium occupies about 80 lattice sites, the number being simply determined by
the size of lithium density distribution of radius Rx ≈ 45 µm in the ODT. As we see in
the figure and know from eq. (3.27), the sodium condensate has a Thomas-Fermi Radius of
RT F ≈ 13 µm and thus only partly overlaps with the lithium in the lattice.
Li thermal nLi(x)
nNa, nLi, VLattice [a.u.]
1
Li T=0 nLi(x)
0.8
Na density nNa(x)
lattice potential
0.6
0.4
0.2
0
−50
−40
−30
−20
−10
0
10
20
30
40
50
x position [µm]
Figure 6.2.: SSODT potential (green) and cut through atomic density distributions in x-direction.
For plotting, we chose typical values (see section 8.4): At a temperature T = 240 nK, we
have 2.6 · 105 sodium atoms with a condensate fraction of η = 0.4 (blue line). The 7 · 104
lithium atoms at this particular temperature show a similar density distribution if assumed
to be distinguishable particles (red solid line) or T = 0 fermions (red dashed line).
For performing polaron experiments, it is of course desirable to have all lithium atoms
in contact with a BEC. This could be reached by opening the ODT after loading lithium
into the optical lattice, which means that in x-direction the lithium density distribution
is frozen whereas the extension of the sodium condensate can increase. As one can easily
−2/3
and thus we would have to decrease ωx by a
see from eqs. (3.27) and (3.28), RT F ∝ ωx
98
factor of ∼ 8 to have a condensed background gas in contact with each occupied lattice site.
A rather technical difficulty stems from the implementation of the ODT in our experiment,
i. e. simultaneously to ωx we also ramp down ωz , whose value is given by the same IRbeam. This would cause overlap problems due to the differential gravitational sag ∆z =
2
2
(g/ωz,Na
− g/ωz,Li
), which grows with decreasing trap frequencies.
An additional overlap problem could arise due the condensate Thomas-Fermi radius being
dependent on the intraspecies scattering lenght aBB . Combining eqs. (3.27) and (3.28), we
√
get RT F ∝ 5 aBB , i. e. decreasing aBB by one order of magnitude decreases RT F by 1.6, being
for aBB → 0 ultimately limited by the harmonic oscillator length aho = 2.4 µm. Reaching
this limit is - even apart from experimental challenges with regard to magnetic field stability
- problematic anyway, as in this non-interacting case the Bogoliubov modes vanish. Without
those phonon-like excitations, which are essential to explain the Fröhlich polaron, the validity
of the whole theoretical analysis of Ref. [21] has to be scrutinized.
6.1.3. Full QM Description of a Particle in a Periodic Potential
Bloch functions
So far, we have derived the design criteria for our SSODT in a simple harmonic oscillator approximation, i. e. we constrained our considerations to independent lattice wells with
vanishing tunneling coupling among each other. In the following, we will give a complete
theoretical treatment of the lattice Hamiltonian
H=
V0
p2
+
(1 − cos(2kx))
2m
2
(6.13)
describing an atom of mass m in the SSODT potential5 eq. (6.5) with periodicity d = π/k =
λ/2. The famous Bloch-Theorem [102] tells us that we can write the eigenfunctions of the
periodic Hamiltonian as
ψn,q (x) = eiqx/~un,q (x)
(6.14)
with the periodic function un,q (x) = un,q (x + d). We see that the eigenfunction ψn,q (x) does
not necessarily show the same periodicity as the Hamiltonian, as it can have an additional
phase factor eiqx/~. Here, n denotes the so-called bandindex and q ∈ [−~k; ~k] the quasimomentum, which will both be explained in detail later. Due to its periodicity, we can expand
the Bloch function un,q (x) in a Fourier series as
un,q (x) =
X
i2lkx
cn,q
l e
(6.15)
l∈Z
5
The inhomogeneity of the optical potential discussed above is neglected in the following to allow for an
analytical treatment.
99
with 2lk being the reciprocal lattice vectors. Analog, the potential reads
X
V0 V0 i2kx
e
+ e−i2kx .
−
Vl ei2lkx =
V (x) =
2
4
l∈Z
(6.16)
It is interesting to note that in the case of an optical lattice the only nonvanishing Fourier
components of the potential are V−1 , V0 and V+1 , whereas in the case of condensed matter
systems constraining the calculations to those three components is always an approximation.
Inserting the Fourier expansions and the Hamiltonian eqs. (6.15), (6.16) and (6.13) into the
Schrödinger equation Hψn,q (x) = En,q ψn,q (x), we get
"
#
X
X
(q + 2~kl)2
n,q
i(q/~+2kl)x
e
− En,q cl +
Vl′ cl−l′ = 0 .
(6.17)
2m
′
l∈Z
l ∈Z
We see that due to the orthogonality of plane waves the bracketed term must vanish for all
l and thus we can obtain the eigenenergies En,q and eigenvectors cn,q
by diagonalization of
l
the matrix


− V40
0
···
0
Tq,z − En,q + V20


− V40
Tq,z−1 − En,q + V20
− V40


..


V0
V0
0
−4
Tq,z−2 − En,q + 2
.

.


..
..


.
.
0
···
Tq,−z − En,q + V20
(6.18)
(q+2~kl)2
where Tq,l =
is the kinetic energy and the truncation parameter z ≥ l ≥ −z is
2m
chosen appropriately. Thus, for each quasimomentum q, we get 2z + 1 eigenvalues En,q ,
which are shown for different lattice depths V0 in figure 6.3. For V0 → 0, we see how the
dispersion relation for a free particle reduced to the first Brillouin zone is reproduced, where
we can retrieve the momentum p = q + 2m~k from quasimomentum q and reciprocal lattice
vector 2m~k with m ∈ Z chosen appropriately. n = 1, 2, ... is the so-called band index, a
denotation which becomes clear when looking at the energy levels for V0 > 0, which are
forming certain energy bands, also called Bloch bands. In the deep lattice limit V0 ≫ Er ,
these bands coincide with the equally spaced harmonic oscillator levels for small n, i. e. in
states which only probe the harmonic part of the sinusoidal potential.
The wavefunctions can be obtained by inserting the eigenvectors cn,q
resulting from the
l
diagonalization of eq. 6.18 into eqs. (6.14) and (6.15). In figure 6.4, we can see that in the
lowest Bloch band the wavefunction for q = 0 has no nodes, whereas for q = ±~k, i. e. at the
Brillouin zone edge, the wavefunction has nodes at all maxima of the optical lattice potential.
For the second band , all wavefunctions have a node at the potential minimum, as expected
for the first excited state, but the q = 0 and not the q = ±~k state has additionally nodes at
the potential maxima. These characteristics of the wavefunction explain the quasimomentum
dependence of the band energies En,q shown in figure 6.3.
100
V0 = 1 Er
V0 = 5 Er
V0 = 15 Er
V0 = 25 Er
30
energy En,q [Er]
25
20
15
10
5
0
−1
0
1
−1
0
1 −1
0
quasimomentum q [h/2π k]
1
−1
0
1
Figure 6.3.: Bloch band energies En,q obtained from diagonalization of matrix eq. (6.18) for different lattice depths V0 . The different Bloch bands are colour coded, whereas the respective
lattice depth V0 is shown as black dotted line. We see that for small lattice depth we approximately get the free particle dispersion relation in the reduced Brillouin zone scheme.
For deeper lattices, the bandgap starts opening in the lower bands first, until for deep
lattices the harmonic oscillator limit of (almost) flat bands is reached.
Wannier Functions
In deep lattices, instead of working with the Bloch wavefunctions delocalized over the entire
lattice it is more intuitive to use wavefunctions localized on one lattice site. For that purpose,
we introduce the Wannier functions
wn (x − xj ) = N −1/2
X
e−iqxj /~ψq,n (x)
(6.19)
q∈[−~k;~k]
with N being an appropriately chosen normalization factor. In the sum above, one has to be
careful to sum up all Bloch functions ψq,n (x) with equal phases enforced at x = 0 for getting
the correct w1 (x) [103]. Whereas the Bloch states ψq,n (x) are characterized by their quasimomentum q and are delocalized over the entire lattice, the Wannier functions wn (x − xj ) are
localized at x = xj . In figure 6.4, we can see a comparison of Bloch and Wannier functions,
latter being also approximated with the harmonic oscillator wavefunctions, which are a good
description for deep lattices. We see how our considerations to design the optical lattice,
which were done in the harmonic oscillator approximation, are justified: Starting from the
Bloch functions, which are the exact eigenfunctions of the lattice, we can do a basis change
to Wannier functions. In the deep lattice limit, those become identical with the harmonic
oscillator states.
101
Figure 6.4.: First Row: Wavefunctions ψq,n (x) for q = 0 and q = ~k, i. e. in the middle and at
the edge of the first Brilloin zone, respectively. Second row: Corresponding probability
densities |ψq,n (x)|2 . Third row: Wannier functions wn (x) in comparison with the harmonic
oscillator functions for comparable trap frequency. The larger spatial extent of the Wannier
functions due to tunneling is obvious. In all graphs the depth of the lattice, which is
depicted in green, is s = 5. For s ≈ 25, a typical value used in our experiments, the
difference between Wannier functions and harmonic oscillator states is not recognizable
any more.
102
Tunneling
When implementing lower-dimensional systems experimentally using an optical lattice, it is
crucial that there is no tunneling between the distinct systems localized on neighbouring
lattice sites i and j. The associated tunneling matrix element can be calculated as
J=
Z
~2 d 2
+ V (x) wn (x − xi ) .
wn (x − xj ) −
2m dx2
(6.20)
Here, the approximation of the Wannier functions by a simple harmonic oscillator is not
correct, as for the tunneling process the sidelobes of wn (x − xj ) on lattice site i become
important. From the results above, we can determine J in tight-binding approximation
s ≫ 1 by [104]
1
max(En,q ) − min(En,q ) ,
(6.21)
J=
q
q
4
i. e. it is proportional to the width of the Bloch band. Plotting eq. 6.21 together with the
approximate expression eq. (6.8) we see that for deep lattices s ≫ 1 numeric and approximate
analytic solution can be used likewise (figure 6.5 a).
b)
4
10
0
bandwidth ∆ω/ω
tunneling energy J [1/s]
a)
3
10
1st BB: J1
2
10
2nd BB: J2
0
10
20
lattice depth s [1]
10
−1
10
−2
10
30
0
20
40
lattice gap [kHz]
60
Figure 6.5.: a) Tunneling energies J1 and J2 of first (blue) and second (red) Bloch band, respectively, in dependence of the lattice depth s. The approximate formula for the first Bloch
band eq. (6.8) is shown as dashed blue line and is seen to be valid for s ≫ 1. We clearly
recognize that tunneling in the second band is always higher than in the first, which is explained by the higher spatial extent and thus overlap of the associated Wannier functions.
b) Bandwidth ∆ω in dependence of bandgap. For deep lattices, the bands get flat and the
bandgap can be approximated by the lattice frquency ω. The bandwidth ∆ω is defined
here as the sum ∆ω = 4(J1 + J2 ) ≈ 4J2 of first and second Bloch band bandwidth and is
responsible for dephasing of oscillations [105].
103
To finish the theoretical part of this chapter, in order to prevent misunderstandings in
nomenclature we want to discuss shortly the term of the ’effective mass’: In condensed
matter systems, the band structure gives rise to an effective mass
∗
m =
d2 En,q
dq 2
−1
(6.22)
of the particle moving through the periodic potential. It is a useful term when explaining
e. g. Bloch oscillations [106] induced by applying a force F = m∗ a. However, this is a pure
single particle effect appearing due to the band structure of the lattice. Throughout this
thesis however, the term ’effective mass’ is used to describe the mass increase of an impurity
immersed in a bosonic background, which obviously is a many-body effect and not connected
to the physics of periodic potentials.
6.2. Characterization of the Optical Lattice
Having designed the optical lattice, next we want to show how to characterize it by means
of experiments with ultracold atoms. For that purpose, we first demonstrate the effect of a
phase lattice on sodium and subsequently excite oscillations of lithium atoms.
6.2.1. Phase Lattice for Sodium
Before implementing the optical lattice for lithium, we used the dye laser as repumper for
sodium [38]. Therefore it was convenient to first try and see effects of the lattice on sodium,
before switching the dye from Rhodamine 6G to DCM to form a SSODT for lithium. For
the measurements reported here, we chose a detuning of ∆/2π ≈ 2 GHz and a lattice beam
power of ∼ 21 µW.
First, in order to see an impact of the lattice on the atoms, it would be desirable to observe
some structure being connected to the periodic potential, even better if this structure could
in some way be utilized to calibrate the lattice depth s. For that purpose, one can make
use of a phase lattice [101]: For a short time t, the lattice is pulsed on and in dependence of
this pulse duration, the diffracted momentum components 2l~k of the atoms after a TOFexpansion are recorded.
To analyze this process analytically, let us start with a sodium condensate described by the
wavefunction |Ψi = |q0 i, i. e. being in a certain momentum state. For simplicity, we assume
that q0 is in the lowest Bloch band. By making use of the fact that the Bloch wavefunctions
ψn,q form a basis set for a certain q = q0 , we can write
X
1=
|ψn,q0 i hψn,q0 |
(6.23)
n
104
fraction of diffracted atoms
a)
0.6
0.5
0.4
0.3
0.2
0.1
0
20
40
60
80
Pulse duration t [µs]
100
120
b)
1kRecoil
2kRecoil
Figure 6.6.: a) Sodium phase lattice for different pulse times t. The blue dots with errorbars
are an average over two experimental runs. The frequency of the oscillation obtained from
a sinusoidal fit (red line) is ω/2π = (9.5 ± 1.1) kHz, which translates to s ≈ 1.8. The
insets show TOF-pictures the data have been obtained from. b) Picture showing s-wave
scattering between |pi = |0i and |pi = |±2~ki atoms. To increase the visibility of the
scattered with respect to the diffracted atoms, the color scale has been chosen logarithmic.
and thus expand the condensate wavefunction as
X
|Ψ(t = 0)i =
|ψn,q0 i hψn,q0 |q0 i .
(6.24)
n
0 ∗
With the Bloch wavefunction eq. (6.14), we get hψn,q0 |q0 i = (cn,q
0 ) and thus the time
evolution of the condensate wavefunction reads
X
0 ∗
|Ψ(t)i =
e−iEn,q0 t/~(cn,q
(6.25)
0 ) |ψn,q0 i
n
105
if the lattice has been pulsed on for a time t. When it is turned off again, the wavefunction
is projected onto plane waves, which are the appropriate basis states to describe the free
atoms. The respective probability amplitude of the atom to end up in the momentum
state |q0 + 2l~ki reads
X
n,q0
0 ∗
).
(6.26)
hq0 + 2l~k|Ψ(t)i =
e−iEn,q0 t/~(cn,q
0 ) (cl
n
Making use of the fact that the condensate starts in a symmetric state |q0 i = |0i, it
can due to parity reasons only be excited to bands with odd n (see wavefunction plots in
figure 6.4). In the weak lattice limit, the atoms will thus only populate the q = 0 state of
the third band, i. e. it they will be diffracted into the momentum state |pi = |±2~ki, as can
be visualized from the reduction to the first Brillouin zone depicted in figure 6.3. Thus we
can simplify eq. (6.26) to
| h±2~k|Ψ(t)i |2 = α1 + α2 cos ((E3,0 − E1,0 )t/~) ,
(6.27)
where the αi are a function of the cn,q
l . The periodic diffraction of atoms into the momentum
states |±2ki can be mapped out in a TOF-series for different lattice pulse times t, as to be
seen in figure 6.6. From the frequency of the oscillation ω/2π = (9.5 ± 1.1) kHz, we get the
energy difference between first and third Bloch band (E3 − E1 ) for quasimomentum q = 0
and can thus determine the associated lattice depth s ≈ 1.8.
An interesting feature can be seen in figure 6.6 b). Besides the |pi = |±2~ki diffraction
peaks, there are also atoms in the region between |pi = |0i and |pi = |±2~ki. This can be
understood as follows: When being diffracted, some of the atoms in |pi = |±2~ki have to
travel through the remaining condensate at rest |pi = |0i. The resulting scattering process
of atoms from the two different velocity classes can be described in the center-of-mass system
with momentum |pi = |±~ki. The atom distributions after scattering, which appear in the
absorption image as circular discs centered around |pi = |±~ki, can be clearly identified as
the results of s-wave scattering [107].
In the following, we will present further possibilities how to characterize the optical lattice
by means of oscillations of lithium.
6.2.2. SSODT for Lithium
To follow the goal of our experiment to observe polaronic behaviour of lithium, we changed
the wavelength of the dye laser to form an SSODT with wavelength of ∼ 670 nm. After cooling the atoms in MT and ODT, we load them into the lattice, a process which is investigated
in detail in the next chapter. In short, we ramp up the SSODT with a sodium background,
such that the atoms have sufficient time for thermalization and a possible spatial rearrangement by tunneling. Depending on the kind of experiment, the sodium background is removed
subsequently by a resonant light pulse.
106
a)
ω/2π = (60.94±0.26)kHz
150
∆xTOF [µm]
100
50
0
−50
−100
not used for fitting
−150
50
100
150
200
250
oscillation time [µs]
300
350
400
−3
x 10
c)
9
50
8
40
7
6
∆xTOF [µm]
relative error ∆ω/ω
b)
linear fit
data
30
20
10
ω/2π = (22.2±4.3)kHz
5
0
4
50 60 70 80 90
number of datapoints used for fitting
0
0.5
EOM voltage ∆U [V]
1
Figure 6.7.: a) Oscillation of lithium in the lattice. The blue dots represent the center of mass
after TOF, the red line is a sinusoidal fit to the data for t < 222 µs. b) Relative frequency
fit error ∆ω/ω in dependence of the number of datapoints used for fitting. The dataset
minimizing the fit error ∆ω/ω has been used for the fit shown in a). c) Displacement
∆xTOF in dependence of EOM voltage ∆U (note that a different lattice than in a) and b)
was used!). With tTOF = 3 ms, we get ω/2π = (22.2 ± 4.3) kHz.
To characterize the lattice for lithium, we excite oscillations. For that purpose, the voltage
applied on the EOM6 (see figure 6.1) is suddenly switched by an amount ∆U. This induces
a phase shift between the two beams of ∆φ = φ′ ∆U, where φ′ = (0.414 ± 0.003)1/V in our
6
For simplicity, we ∆U given here is the voltage from our experiment control system. For inducing phase
shifts, this voltage is amplified by a factor of 20 by a high voltage amplifier and then directed to the
EOM.
107
setup [100]. The resulting spatial displacement of the lattice structure is
∆x =
nm
λ ′
φ ∆U = (72.5 ± 0.5)
∆U .
4π
V
(6.28)
By switching lattice and ODT off a certain time t after the displacement, we can map out
the oscillation in a TOF-series. The induced oscillation shown in figure 6.7 has a frequency
of ω/2π = (60.94 ± 0.26) kHz, which translates into s = 26.2. A feature which is not yet
understood is depicted in figure 6.7 b): ∆ω/ω increases, i. e. the fit quality decreases when
we take the datapoints for t > 222 µs into account. From the lattice inhomogeneity and the
confining ODT potential eq. (6.12), one would expect a timescale τ ≈ 100/ω ≈ 2 ms for this
process, which for longer times experimentally manifests as amplitude decrease. A possible
explanation could be an improper alignment of SSODT with respect to the ODT, which
would result in a faster dephasing than theoretically expected [105].
By fitting only the first ∼11 oscillations, we get for the frequency ω/2π = (60.94 ±
0.26) kHz, i. e. without further effort we can determine it with a relative accuracy of ∼0.4%.
This means that we are able detect an effective mass increase of m∗ = 1.008m, which can
be achieved by α = 0.02, a value that can easily be reached by reducing aBB only slightly
as the background value αbg = 0.018 is not much smaller.
In order to develop a deeper understanding for the oscillations, one can map out the
amplitude of the first maximum in dependence of the applied voltage jump ∆U. The maximal
atom velocity is v = ω∆x, which translates in TOF into ∆xTOF = vtTOF . From the fit of
the data in figure 6.7, we get ω/2π = (22.2 ± 4.3) kHz, which is in excellent agreement with
ω/2π = (22.6 ± 1.3) kHz determined from a direct measurement of the oscillation frequency.
In chapter 8, we will further investigate controlled excitations in the SSODT. For that
purpose, we apply the technique of Brillouin zone mapping, which will be described in detail
in the next chapter.
108
7. Brillouin Zone Mapping
In order to map out the momentum distribution of atoms in an optical lattice, a commonly
used tool is Brillouin zone mapping. In the following, we will explain its basic principles and
apply it to understand and improve the lattice loading procedure.
7.1. Basic Principles
As we have seen in the previous chapter, to atoms in different Bloch bands different momenta
can be assigned. Although the quasimomentum q can only take values within the first
Brillouin zone q ∈ [−~k; ~k], it can be adiabatically mapped onto a real momentum p =
q + 2m~k with m ∈ Z. The way this works is visualized in figure 7.1: Starting from a deep
lattice, its depth s is ramped down and the Bloch bands start more and more to resemble the
free particle dispersion relation. In this process the quasimomentum q is conserved and the
real momentum p is retrieved when the lattice has been turned off completely. A subsequent
TOF translates the obtained momentum distribution into a spatial distribution, which we
record via an absorption image.
Experimentally, the Brillouin zone mapping has to fulfill several requirements: On the one
hand, it has to be slow with respect to the vibrational frequencies, such that the ramp-down
is adiabatic and does not induce transitions between the different bands. On the other hand,
it has to be fast with respect to collision times changing the momentum distribution (which
is not an issue for fermions though). Moreover, as the confining ODT is turned off during
the Brillouin zone mapping, thermally driven expansion of the lithium has to be considered
as well.
We found that an exponential ramp ∝ exp(−t/τ ) with time constant τ = 125 µs and
duration 500 µs fulfils those requirements. With this tool at hand, we will in the following
analyze the lattice loading process.
7.2. Lattice Loading
Looking at the evolution of Bloch bands when changing the lattice depth (figures 6.3 and 7.1),
it is obvious that free atom states with energy E > ER can end up in an excited band
n > 1 when the lattice is ramped up, a process which can not be prevented by adiabatic
loading [108]. In our system, ER ≈ 320 nK and EF ≈ 760 nK, thus with the help of the
109
10
−2h/λ
energy [ER]
8
6
4
+2h/λ
2
0
−3
−2
−1
0
1
(quasi)momentum q,p [h/λ]
2
3
Figure 7.1.: Basic scheme of Brillouin zone mapping. The atoms (black dots) in the lattice of
depth s = 5 (green line) have a certain quasimomentum q, which is conserved during a
ramp down to e. g. s = 1.5 (red line). When the lattice is turned off completely, the atom’s
quasimomentum is mapped onto the free particle dispersion relation (blue line) by adding
a reciprocal lattice vector ±2~k = ±2h/λ to the quasimomentum q.
intuitive picture figure 7.3 a) we can estimate that even at T = 0 about 40% the atoms
will naturally end up in the second band. This theoretical prediction is confirmed by the
Brillouin zone mapping shown in figure 7.2 a), where 44% of the atoms are in excited states.
This energetically excited configuration is stable, as it can not relax due to the fermionic
nature of the lithium atoms.
This constraint is lifted if we have a sodium background present when the lattice is being
loaded: The lithium atoms in the excited states can scatter with the bosonic bath, thereby
relaxing to lower bloch bands, a process schematically shown in figure 7.3 a). In figure 7.2 b)
and c), we clearly see this effect, which we will investigate quantitatively in the following.
For the ramp-up of the lattice we choose an exponential ramp s(t) ∝ exp(−t/τ ) with time
constant τ and duration tr . Fixing τ = tr /4 is commonly done in ultracold atom experiments
[109, 110], but the ramp duration has to be adapted to our specific requirement that the
lithium in the lattice has to thermalize with the sodium background. Figure 7.3 b) shows
the result of loading the lattice with different ramp times: We see that for tr ≈ 100 ms,
the fraction of atoms in the second Bloch band has reached its steady state value.1 As a
comparison, the case of lithium only is shown, which clearly does not show any relaxation
from its metastable state.
1
Due to the finite overlap of lithium atoms and sodium background (see figure 6.2), it is not surprising
that some of the lithium atoms cannot relax to the first Bloch band due to the lack of sodium scattering
partners, especially at the trap edges in the outer part of the lattice.
110
w/o Na background
normalized atom number
8
w/ Na background
difference
8
2
6
6
1
4
4
2
2
−1
0
0
−2
−0.5
0
0.5
distance x [mm]
0
−0.5
0
0.5
distance x [mm]
−0.5
0
0.5
distance x [mm]
Figure 7.2.: Left: Brillouin zone mapping of lithium loaded into the lattice without sodium background. About 44% of the atoms end up in excited states, which appear in TOF as higher
Brillouin Zones depicted by different shadowing. Middle: If the lattice is loaded with a
sodium background present, 86% of the atoms are in the lowest Bloch band. Right: The
direct plot of the difference clarifies the effect of the bosonic background.
ωlattice
ωODT
fraction of atoms in 2nd BB
b)
energy
a)
0.35
0.3
0.25
0.2
0.15
0.1
position
w/Na, τ = 30ms
w/o Na
0
0.05
0.1
0.15
ramp time tr [s]
0.2
Figure 7.3.: a) Lattice site in harmonic approximation. Shown are the potential (green) with the
states of first (blue) and second (red) Bloch band, spaced by ωlattice . The energetic distance
between the transversally excited states is given by the ODT frequency ωODT . Note that
ωlattice ≫ ωODT and thus the picture is not to scale. b) Fraction of atoms in the second
band in dependence of the lattice ramp time tr . Without sodium (red dots), lithium stays
in the metastable state, whereas with sodium (blue dots), the atoms are allowed to relax
to the first Bloch band. The exponential fit (blue line) with a time constant τ = 30 ms
leads to the choice of tr = 100 ms.
111
In the next chapter, we will investigate the lithium relaxation process quantitatively in
detail. To understand it qualitatively, figure 7.3 a) is useful: With the help of sodium, the
lithium atom can relax from second (red) to first (blue) Bloch band. The fact that this
process is possible, i. e. the final state is not already occupied by another fermionic lithium
atom and thus Pauli blocked, is a peculiarity of the one-dimensional lattice system. A
manifold of transversal states from the ODT belongs to each Bloch band, as also depicted
in figure 7.3 a). This way, to a certain energy E > E2,q there exists one state of the first and
one of the second Bloch band. With typical values of ωlattice ≈ 50 kHz and ωODT ≈ 210 Hz,
we find that there are about 3 · 104 states in the first Bloch band with energy E lower than
the second band’s energy. This means that with typically not more than 3 · 103 atoms per
sheet, there is sufficient states for the fermionic atoms to relax to.
In the following, we will investigate how this configuration of states, which is specific to
our setup, leads to the steep Brillouin zone edges as we can see in figure 7.2, which are in
the case of a 3D lattice only obtained for certain fillings [111].
7.3. Brillouin Zone Edges
With some thousand atoms per lattice site, it is well justified to assume that the atoms’
quasimomentum distribution more or less uniformly covers the entire first Brillouin zone
−~k ≤ q ≤ ~k. This means that when we do a BZ mapping, from each lattice site atoms
start travelling with all those momenta q. Thus, the result of the bandmapping can be
explained by summing over in-situ distributions, each displaced by q with −~k ≤ q ≤ ~k, as
schematically depicted in figure 7.4.
To quantify our so far only theoretical considerations, we map out the steepness of the
Brillouin zone edges in dependence of different ODT frequencies, which change the slope of
the in-situ distributions. Experimentally, we choose a TOF of 5 ms for the Brillouin zone
mapping and obtain the prediction of the momentum distribution by summing over in-situ
pictures taken at the same ODT frequency ωODT . The resulting spatial distributions, which
have been calculated without free parameters, are linearly fitted at their respective Brillouin
zone edges. The results plotted in dependence of ωODT can be seen in figure 7.4 b)2 .
The linear fits of the Brillouin zone edge slopes, depicted as solid lines in figure 7.4 b),
are consistent within their uncertainties. Of course, agreement is not perfect due to several
reasons: As already pointed out in chapter 3, in-situ pictures are quantitatively only limited
trustworthy due to the high optical density. As the atom density increases with increasing
ωODT , due to saturation effects in imaging we expect a lower slope for those in-situ pictures
and the resulting sum. This explains why theory and experiment deviate more strongly for
high ωODT , as can be seen in figure 7.4 b). Another source of errors is the anharmonic part
of the in-situ distribution: Due to the ODT being formed by Gaussian beams of finite waist,
2
Due to the maximum slope of a Gaussian distribution being proportional to 1/σ ∝ ω, it is reasonable to
plot the slope of the BZ edge in dependence of ω.
112
the wings of the atom distribution are broader than expected in a pure harmonic potential.
This leads to the theoretically predicted shape of the Brillouin zone mapping having rather
smooth edges, whereas the experimentally obtained shape can be almost approximated by a
box as it mostly stems from atoms being trapped in the harmonic part of the ODT potential.
BZ mapping
in situ sum
in situ
b)
BZ edge steepness
lithium atom density
a)
−200
0
xTOF [µm]
200
0.6
0.8
1
ωODT/ωmax
ODT
Figure 7.4.: a) Schematic of the emergence of a typically observed spatial distribution in BZ
mapping: By summing over in-situ distributions with all different quasimomenta q (blue),
we get a theoretical predicition for the result of the BZ mapping (green). The result
shows astonishing similarity to the results obtained experimentally (red). b) Slope of the
Brillouin zone edges in dependence of the ODT frequency ωODT , color coding as in a).
Errorbars are due to taking the mean of slope of left and right BZ edge. The lines are
linear fits to the data with slopes aBZ = 24 ± 10 and aIS = 14 ± 4, agreeing within their
uncertainties.
In this chapter, we have introduced the tool of Brilloun zone mapping. By analyzing the
relaxation of excitations, we have applied it to optimize the loading procedure of the SSODT.
In the next chapter, we will go one step further and deliberately transfer atoms to higher
Bloch bands, a process which can be analyzed with the Brillouin zone mapping technique.
113
114
8. Controlled Excitations in the Optical
Lattice
In this chapter, we will investigate excitations of the atoms in the optical lattice in detail.
We start with a derivation of a quantum mechanical picture of oscillations in the harmonic
oscillator limit, which we extend to the full treatment using Bloch wavefunctions and compare
to experimental data. To get a quantitative understanding of the interspecies energy transfer
in a Bose-Fermi mixture, we analyze heating of the condensed Bose gas in dependence of
the lithium oscillation energy.
In order to be able to excite lithium to one specific Bloch band instead of preparing it
in a superposition as done in the case of oscillations, we apply an oscillatory displacement
of our lattice and are thus able to spectroscopically determine the bandgap. When the
lithium atoms are transferred to the excited state of the two-level system consisting of
the mechanically coupled first and second Bloch band, a sodium background can induce
their relaxation to the first Bloch band, as we have already seen in the previous chapter.
By employing a controlled excitation and quantitatively analyzing its relaxation, we can
determine the absolute value of the interspecies scattering length aIB . To give an outlook
on the rich spectrum of physics that can be investigated using this two-level system coupled
to a bath, we finally demonstrate Rabi oscillations between first and second Bloch band.
8.1. Quantum Mechanical Picture of Oscillations
In the limit of a deep lattice, where the Bloch wavefunctions can be approximated by simple
harmonic oscillator wavefunctions, an oscillation can be described analytically. To excite the
oscillation, we displace the wavefunction |ψ(x)i by ∆x, which can be described by applying
the displacement operator
i
e− ~ ∆xp̂ |ψ(x)i = |ψ(x − ∆x)i ,
(8.1)
which is well-known
q from basic quantum mechanics [28]. Introducing the harmonic oscillator
length scale x0 =
2~
mω
and the raising and lowering operators â„ and â, we can rewrite
√ ~mω „
∆x „
i
i
(â −â)
e− ~ ∆x·p̂ = e− ~ ∆x·i 2 (â −â) = e x0
= D(α) .
(8.2)
115
and introduced the displacement operator D(α) = exp(αâ„ − α∗ â),
Here, we defined α ≡ ∆x
x0
which is also used in Quantum Optics [112].
Starting with atoms in the harmonic oscillator ground state |ψ(x)i = |0i, we obtain its
decomposition in energy eigenstates |ni after the displacement
|ψ(x − ∆x)i =
=
X
n
X
n
=
X
n
=
X
n
|ni hn|ψ(x − ∆x)i
|ni hn|D(α)|0i
|ni hn|
e−
|α|2
2
X
n′
n
e−
|α|2
2
α
√ |ni .
n!
′
αn
√ |n′ i
n′ !
(8.3)
In the case of a lattice with finite depth s, we obtain the result analog to eq. (8.3) with |ni
in this case being the Bloch wavefunctions of the different bands. The overlap integrals can
be evaluated numerically using the eigenstates obtained in chapter 6. Note that in this case,
the expansion of the displaced state is weakly dependent on the quasimomentum q, an effect
which decreases for increasing lattice depth and vanishes in the harmonic oscillator limit.
In figure 8.1, oscillations with different amplitudes are shown for a lattice of depth s = 14.
In order to be able to compare theory with experiment, the theoretical data has been appropriately averaged over all quasimomenta q ∈ [−~k; ~k] of the first Brillouin zone. The
experimental data has been obtained by excitation of oscillations with different amplitude
and subsequent BZ mapping. We see qualitative agreement, but a quantitative discrepancy
which could e. g. be due to the imperfect preparation of the initial state, which still contains a considerable fraction of atoms in the second Bloch band (see datapoints for zero
displacement).
Very insightful is the comparison of the calculation involving the full set of Bloch states
with the harmonic oscillator approximation, which is for reasons of clarity in figure 8.1 only
shown for ∆x < 0. We see that for small displacements approximation and exact calculation
agree well for the three lowest bands. The fourth band, which is completely in the continuum
for s = 14, is for that reason qualitatively not correctly described by the harmonic oscillator
approximation.
8.2. Interspecies Energy Transfer
As we have the possibility to study not only lithium alone in the SSODT, but can also add a
sodium background, a question that comes up naturally is how thermalization between the
two works. In chapter 5, we have already studied the damping of oscillations in the case of
116
4th BB
overlap |<n | ψ (x−∆x)>|
3rd BB
2nd BB
1st BB
lattice
h.o.
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
displacement ∆x [lattice periodicity]
0.3
0.4
0.5
Figure 8.1.: Overlap of the displaced wavefunction |ψ(x − ∆x)i with the respective eigenstates
|ni. The first to fourth Bloch band are depicted in blue, red, green and black, respectively,
and have been offset for clarity, the respective zeros shown as dotted horizontal lines. The
experimental data (dots) have been obtained by Brillouin zone mapping and are averaged
over two experimental runs. The qualitative agreement with the theoretical calculation
using Bloch wavefunctions (solid lines) is obvious. For small oscillation amplitudes, the
harmonic oscillator approximation (dashed-dotted lines, for clarity only shown for ∆x < 0)
gives a good description for the three lowest Bloch bands. Note the symmetry in the case
of Bloch wavefunctions around a displacement of half a lattice period.
both species being excited. Working with an SSODT for lithium, we will thus study how
energy is transferred from the excited lithium atoms to the sodium condensate.
For that purpose, we prepare lithium in the SSODT with a sodium condensate background.
An oscillation of the fermions is excited and after a certain hold time th a TOF picture is
taken. Experimentally, we choose th = 1 s, a value which ensures that lithium and condensate
are well thermalized (see chapter 5). Figure 8.2 shows the resulting sodium condensate
fraction in dependence of the lithium oscillation amplitude. In the following, we will develop
a quantitative understanding of this energy transfer process.
The specific heat of an ideal Bose gas of NNa sodium atoms in a three-dimensional harmonic
trap is given by [31]
3
3
T
T
ζ(4)
≈ 10.8 · NNa · kB
(8.4)
NNa · kB
C =3·4·
ζ(3)
Tc
Tc
with ζ(x) denoting the Riemann zeta function. A temperature increase from Ti to Tf is
117
condensate fraction η
0.5
0.4
0.3
0.2
0.1
0
expt. data
fit
0
0.05
0.1
0.15
0.2
displacement ∆x [lattice periodicity]
Figure 8.2.: Sodium condensate fraction η in dependence of the lithium displacement. The experimental data (blue dots) are an average over two runs and over the condensate fraction
fitted in x- and y-direction. The red line is a fit based on the specific heat of the ideal
Bose gas as described in the text. The data were obtained in a lattice with frequency
ω/2π = 45 kHz.
connected with a heat input
∆Q =
Z
Tf
C(T ) dT .
(8.5)
Ti
Evaluating this integral and inserting it into the formula for the condensate fraction eq. (3.30)
3
η = 1 − TTc , we get
4 !3/4
Ti
∆Q
η =1−
+
.
(8.6)
2.7NNa kB Tc
Tc
Knowing the lattice frequency ω and the lithium atom number NLi , we can determine the
heat transfer
1
(8.7)
∆Q = NLi V0 sin2 (k∆x) ≈ NLi mLi ω 2 ∆x2
2
in dependence of oscillation amplitude, i. e. the displacement ∆x. Fitting the data in figure 8.2 with eqs. (8.6) and (8.7), we get when leaving the critical temperature as free parameter Tc = (355 ± 74) nK, with the error given by the uncertainties in NNa , NLi and the
fit. The corresponding ODT trapping frequency ω̄/2π = (118 ± 25) Hz is in good agreement
√
with the value 3 ωx ωy ωz /2π = 93 Hz which we obtain from the direct measurement of the
trapping frequencies.
From theoretical work [113] we know that for a harmonically trapped weakly interacting
Bose gas the specific heat scales as (T /Tc )α with α < 3 opposed to α = 3 in the noninteracting case. The scaling has experimentally been determined to be α = (2.7 ± 0.7) in
118
the case of rubidium [114], but leaving α as a free parameter for the fit of our data sample
results in a value being meaningless due to its large uncertainty.
This first measurement of lithium oscillating in a sodium background can be extended in
many ways: If the bosons form a pure condensate, the impurity should perform a frictionless
oscillation as long as the associated velocity is slower than the superfluid critical velocity vc .
Thus, a measurement of the energy transfer in dependence of the oscillation amplitude
can be used to determine vc . Moreover, if one is able to load just a few lattice sites, the
lithium can even be used as a local probe to map out its spatial, i. e. density dependence.
So far, the superfluid critical velocity has been determined experimentally by moving a blue
detuned laser beam [91] and sodium atoms in different hyperfine states [92] through a bosonic
condensate. The idea of dragging an impurity through a condensed background to determine
vc has just recently been implemented with an ion moved through a BEC [94].
In the following, we want to study the dynamics of the energy transfer from excited lithium
atoms to the sodium bath. Instead of exciting an oscillation of the lithium, which involves
several Bloch bands in the theoretical description, we rather want to drive the transition
between first an second band, which allows for a theoretical modelling as a two-level system.
8.3. Coherent Transfer Between Different Bloch Bands
From perturbation theory, we know that we can drive transitions between two states of a
system by applying an appropriate excitation at the associated frequency. Mathematically
speaking, the unperturbed time-independent Hamiltonian eq. (6.13)
H0 (x) =
p2
+ V0 sin2 (kx)
2m
(8.8)
experiences a time-dependent perturbation H ′(x, t) such that
H(x, t) = H0 (x) + H ′ (x, t) .
(8.9)
One sees that there is two obvious ways to achieve a time-dependent perturbation by
manipulating V (x): One can give the lattice depth V0 a time dependence, i. e. V0 →
V0 (1 + ε sin(ωex t)), which can be realized experimentally by modulating the lattice beam’s
intensity. In this case H ′(x, t) = V0 ε sin(ωex t) sin2 (kx) = H ′(−x, t), i. e.it is an even-parity
excitation which couples Bloch bands with ∆n = 2m with m ∈ Z.
Another way to induce an excitation is to shake the lattice, i. e. V (x) → V (x+x0 sin(ωex t)) =
V0 sin2 (k(x + x0 sin(ωex t))). In the harmonic oscillator limit, we can do a Taylor expansion
around x = 0 and get to leading order in x0 sin(ωex t)
H ′(x, t) = 2V0 · (kx0 )(kx) sin(ωex t) .
(8.10)
We see that in this case H ′(x, t) = −H ′ (−x, t), thus this odd-parity excitation couples
Bloch bands with ∆n = 2m + 1 with m ∈ Z, in particular first and second Bloch band.
119
Experimentally, we realize the driving using the EOM to periodically modulate the phase of
one beam with respect to the other and thus get a periodic displacement of the lattice wells.
A first example of the effect of the periodic shaking at the resonance frequency between
first and second Bloch band can be seen in figure 8.3 a): After a Brillouin zone mapping, we
clearly see that most atoms are transferred into the second Bloch band.
a)
b)
fraction in 2nd BB
trans. summed pic
0.35
0.3
0.25
0.2
0.15
0.1
−0.5
0
distance [mm]
0.5
FWHM =
(2.4 ±1.0)kHz
f =(51.10 ±0.45)kHz
40
50
60
excitation frequency [kHz]
Figure 8.3.: a) Excitation of lithium atoms to the second Bloch band by periodic lattice shaking.
b) Spectroscopy of lithium atoms in the SSODT. In contrast to a), the excitation amplitude
x0 is kept as low as 3.3 · 10−3 lattice periods in order to minimize the power broadening.
The shaking duration of ∼ 200 µs corresponds to about 10 periods needed to excite the
atoms.
Next, we want to improve our understanding of the excitation by doing spectroscopy of
the atoms in the lattice: We map out the fraction of atoms in the second Bloch band in
dependence of the excitation frequency at fixed amplitude x0 and duration of the shaking.
The resulting data are fitted with a Lorentzian lineshape, as to be seen in figure 8.3 b). The
fit determines ω/2π = (51.10 ± 0.45) kHz, i. e. we can determine the lattice frequency with
an accuracy of 9 · 10−3 . In chapter 6, we obtained a relative uncertainty of 4.3 · 10−3 when
determining ω by means of a spatial oscillation of Lihtium atoms, thus the spectroscopy
method as applied here can not (yet) compete with the ’simple’ oscillation measurement.
Another insightful parameter obtained by the fit is the width of the Lorentzian ∆ω = (2.4±
1.0) kHz. We see that ∆ω/ω = (47±20)·10−3, while we expect from the bandwidth ∆ω/ω =
39 · 10−3 , as graphically shown in figure 6.5. The agreement between theory and experiment
is good, advantaged by the large fit uncertainty. For a more accurate spectroscopy, one
expects to measure experimentally a slightly higher value of ∆ω/ω than expected from
the theoretically obtained band structure due to power broadening, an effect known from
quantum optics [46].
With the ability to transfer the lithium atoms from the first to the second Bloch band of
120
the SSODT, we will in the following analyze the subsequent relaxation process which can be
induced by a sodium background.
8.4. Determination of the Interspecies Scattering Length
One can evaluate the measurement of sodium induced lithium relaxation from the second
to the first BB in order to extract information about the interspecies scattering length.
Therefore, we start with the simple formula
Γ = nσv
(8.11)
describing the scattering rate Γ of particles with velocity v travelling through a medium of
density n. The scattering cross section σ = 4πa2BF gives us information about the absolute
value of the scattering length. In our experimental situation, in dependence of its lattice
site and transversal state each lithium atom probes a different bosonic background density
n = nNa . Therefore, if we attribute a relaxation rate Γi to each lattice site i, the number of
atoms in the second Bloch band N(t) will be described by
X
N(t) =
Ni e−Γi t ,
(8.12)
where Ni denotes the number of atoms in the second BB on lattice site i at time t = 0. As
already done in chapter 5, we approximate the envelope shape of the atoms in the lattice by
the factorizable Maxwell-Boltzmann distribution nLi (x, y, z) = Nf (x)f (y)f (z) and can thus
write
Z
X
−Γi t
N(t) =
Ni e
= dx Nf (x)e−Γ(x)t
Z
Z
(8.13)
≈ dx Nf (x) (1 − Γ(x)t) = N 1 − dx f (x)Γ(x)t
with the Taylor approximation being valid for small Γ(x)t. Experimentally, we measure and
fit a total relaxation
N(t) = Ne−Γt ≈ N (1 − Γt)
(8.14)
and thus by equating eqs. (8.13) and (8.14) we get
Z
Γ = dx f (x)Γ(x)
(8.15)
Noting that we have to weight the background boson density nNa in eq. (8.11) for a certain
lattice site with the two-dimensional fermionic density f (y)f (z), we finally arrive at
Z
Z
Z
Γ = dx f (x)Γ(x) = dx f (x) dy dz σvnNa (x, y, z)f (y)f (z)
Z
(8.16)
nLi (x, y, z)
.
= σv dV nNa (x, y, z)
N
121
As we experimentally deal with two-dimensional lithium clouds, the velocity v along the
x-direction will be given by the confinement, whereas in y and z it is determined by temperature. For those three independent directions, the velocities add quadratically, thus we
have to ask about the mean squared velocities. For a harmonic oscillator, the wavefunction
in the first excited state (i. e. in the second Bloch band of our lattice) reads in momentum
space
√
p2
2p
− 2m~ω
(8.17)
ψ(p) =
e
(π(m~ω)3 )1/4
and thus
2 3
pho = hψ(p)|p2 |ψ(p)i = m~ω .
2
Similarly, we get with the Maxwell-Boltzmann Distribution in a harmonic trap
f (p) = √
2
1
− p
e 2mkB T
2πmkB T
the thermal velocity for one dimension
Z
2
pth = dp p2 f (p) = mkB T .
(8.18)
(8.19)
(8.20)
For the experimental data presented in figure 8.4, the temperature is T = (240 ± 9) nK and
the lattice frequency ω/2π = 40 kHz, thus p2th /p2ho = 0.08, so most collisions happen in the
direction of the lattice. Moreover, the thermal velocity effects being almost negligibe also
justifies the assumption of the sodium background to be at rest, i. e. p
not to contribute to v.
The velocity associated to atoms in the second Bloch band vho = hp2ho i/m ≈ 6 cm/s is
much higher than the superfluid critical velocity of the condensate vc ≈ 1 cm/s, thus the
excited lithium atoms interact both with sodium atoms in the condensate and the thermal
cloud. Momentum and energy conservation yield that in a collision of lithium with a sodium
background atom, the fermion in the second band loses so much energy that it will most
probably end up in the lowest Bloch band. Therefore, as an approximation we can equalize
the collision rate Γ from eq. (8.14) with the relaxation rate we get from the fit to our
experimental data in figure 8.4, τ = (42 ± 12) ms1 . With an overlap integral of
Z
1
1
dV nNa (x, y, z)nLi (x, y, z) = (20.1 ± 2.3) · 1017 3
(8.21)
N
m
we finally get
|aBF | = (70 ± 12)a0 .
1
(8.22)
It is important to note that this timescale τ is much larger than the ∼ 200 µs needed to transfer the atoms
with a periodic lattice shaking from first to second Bloch band. This means that the transfer is not
influenced by the relaxation dynamics and thus both processes can be safely considered as independent.
122
number of atoms in BZ [104]
5
4
1st BZ: τ = (28±10) ms
2nd BZ: τ = (42±12) ms
3
2
1
0
0
0.1
0.2
0.3
hold time th [s]
0.4
0.5
Figure 8.4.: Relaxation of lithium |1/2, 1/2i atoms from second to first Bloch band induced by
a sodium |1, 1i background. The time constants τ are obtained from an exponential fit to
the data. τ is higher for the first than for the second Brillouin Zone due to an additional
contribution of atoms relaxing from the third Bloch band. For long hold times th ≈ 0.5 s,
an overall atom loss due to spontaneous processes is observed.
As the whole measurement has been done for sodium in |1, 1i and lithium in |1/2, 1/2i,
aBF is a linear combination of triplet and singlet scattering length, which are from the
measurements in chapter 3 known to fulfil |as − at | = (5.9 ± 1.7) a0 . This means that due to
the large uncertainty in aBF we can compare it directly to the result |at | = (69 ± 13)a0 from
the oscillation damping measurement in chapter 5 and see that they show great agreement.
Finally, we want to discuss the systematic errors in the determination of aBF presented
above:
ˆ Instead of considering the whole momentum distribution eq. (8.17) of atoms in the
second Bloch band, for simplicity we take the rms momentum eq. (8.18) as a measure
of the atoms’ motion.
ˆ Moreover, our result is obtained under the approximation that every interspecies scattering event leads to a relaxation of lithium from the second to the first Bloch band.
A refined model has to involve e. g. also the energy spread of the atoms in the second
band given by the two-dimensional Fermi energy due to the preparation process. This
width has to be compared to the energy loss in a collision with the sodium background,
since only collisions with an energy loss bigger than this width necessarily lead to a
relaxation to the lower band. In the measurement presented above, we had a Fermi
energy of ∼5 kHz and the lithium atoms’ kinetic energy due to their confinement in
the lattice was 30 kHz. If we assume s-wave scattering between sodium background
and lithium atoms, we see that only ∼64% of the lithium atoms scatter at an angle
which makes them lose more than the Fermi energy and thus relax to the lower band.
123
This additional prefactor leads to a 20% increase in the experimentally determined
scattering length, which is within the given error bars.
Those two issues can be overcome by making use of Fermi’s golden rule: One could integrate
the whole momentum distribution eq. (8.17), while accounting for energy and momentum
conservation by the appropriately chosen δ-distributions. This way, in principle the exact
rate of atoms relaxing due to scattering events can be determined.
Some complications, which should have a minor effect on the result, arise due to the
density distributions of the atoms:
ˆ For the lithium, we use a Boltzmann instead of a (finite temperature) Fermi distribution, which seems well justified by their small differences depicted in figure 6.2. Indeed,
using a T = 0 Fermi distribution for lithium changes our overlap integral eq. (8.21) by
less than 5% and thus our result for aBF by only 2%, which is way smaller than the
statistical error of 17%.
ˆ A small error stems from the fact that the effect of the lattice on sodium is not completely negligible. While the detuning is blue for lithium, the lattice is red detuned
and ∼ 40 nK deep for sodium. Thus the condensate background density is reduced
by about 4% on the lithium lattice sites, as one can show by integrating the GPE
eq. (3.25) numerically with the ODT and SSODT potential inserted.
ˆ Another effect on the density of sodium has the attractive interspecies interaction
aBF < 0, which enhances the sodium density at the lithium lattices sites. Even numerically calculating the resulting density profile of the Bose-Fermi mixture for a given
aBF 6= 0 is challenging on its own and would have to be done iteratively if – as in our
case – the interspecies scattering length aBF is unknown.
ˆ For simplicity, we assumed the sodium atoms to be at rest and not to contribute to the
velocity v in eq. (8.11). At least, sodium has a finite momentum due to occupying the
harmonic oscillator states of the ODT, and additionally the small periodic potential
from the SSODT increases their velocity v.
Also the dynamics of the decay process have been severely simplified:
ˆ When the lithium relaxes, it transfers energy to the sodium atoms, which has to result
in a decrease of the condensate fraction η, which we assumed to be constant throughout
the relaxation process.
ˆ In our derivation, the result for the relaxation rate eq. (8.15) is based on a Taylor
approximation valid for Γt ≪ 1. Nevertheless, due to the insufficient number of data
points for small times t, we use the entire experimentally measured relaxation curve
to determine Γ.
124
ˆ We cannot apply our model to data obtained at low temperatures, where the condensate fraction η → 1. In this case, due to the insufficient overlap of the atomic density
distributions (see figure 6.2) our model does not include the effect that the outer lattice
sites are not covered by a bosonic background and thus the atoms there cannot relax,
i. e. Γ = 0.
We see that there are still many more effects to be considered when trying to analyze
the relaxation process of atoms from first to second Bloch band quantitatively. The model
introduced here can at most serve as a starting point for future work. Despite all the approximations, the result for the scattering length fits the value obtained by the CC calculation
well.
8.5. Rabi Oscillations
Finally, we want to show the coherent character of the excitations induced by the periodic
shaking. For that purpose, we excite Rabi oscillations, i. e. monitor the oscillatory evolution
of atom number in first and second Bloch band. The Rabi freqeuncy Ω is as usual defined
via the relation
he|H ′ (x, t)|gi = Heg ≡ ~Ω sin(ωex t) .
(8.23)
With eq. (8.10) we thus get
~Ω = 2V0 (kx0 ) he|kx|gi =
√
2V0 (kx0 )(kaho ) ,
(8.24)
p
where aho = ~/mω denotes the harmonic oscillator length. For the evaluation of the dipole
matrix element he|x|gi, we have applied the approximation for deep lattices and used the
harmonic oscillator wavefunctions to be able to calculate analytic results.
Experimentally, we excite Rabi oscillations by applying a periodic shaking of the lattice at
resonance frequency ωex = ω with fixed amplitude x0 and variable duration. As to be seen
in figure 8.5, we obtain a damped sinusoidal oscillation between first and second Bloch band,
which we can fit to deduce the Rabi frequency Ω/2π = (3.58 ± 0.03) kHz. With ω/2π =
47.5 kHz and an excitation amplitude x0 = 14.5 nm, the theoretical prediction eq. (8.24)
yields Ω/2π = 3.77 kHz, i. e. it does not yet agree with the experimental observation.
As a possible source of the deviation, we identify the approximation done in the derivation
of eq. (8.10). There we just did a Taylor expansion up to first order in kx, but in the
measurement presented above, kaho = 0.54 and thus the higher order terms can not be
neglected. Including the next orders as well2 we get
2
H ′ (x, t) = 2V0 (kx0 )(kx − (kx)3 ) sin(ωex t) .
3
2
(8.25)
We only include terms oscillating at frequency ωex and disregard higher orders. This is the rotating wave
approximation, which is valid as long as the detuning is small, i. e. |δ| = |ωex − ω| ≪ ωex .
125
1
first Bloch band
second Bloch band
fraction in BB
0.8
0.6
0.4
0.2
0
Ω/2π = (3.58±0.03)kHz, τ =(1.2±0.2)ms
0
0.2
0.4
0.6
0.8
1
1.2
excitation time [ms]
1.4
1.6
1.8
2
Figure 8.5.: Rabi oscillations between first and second Bloch band. The periodic displacement of
the lattice was applied with a frequency ωex /2π = 47.5 kHz and an amplitude x0 = 14.5 nm.
The experimental data (dots) are fitted with a damped sinusoidal (solid lines), yielding a
Rabi frequency of Ω/2π = (3.58 ± 0.03) kHz.
The Rabi frequency including the energy correction reads
~Ω = 2V0 (kx0 )(k he|kx|gi −
√
2
he|(kx)3 |gi) = 2V0 (kx0 )(kaho − (kaho )3 ) .
3
(8.26)
The resulting new theoretically predicted Rabi frequency of Ω = 2.69 kHz even deviates more
from the experimentally obtained value. Thus a more elaborate theory, which e. g. involves
Wannier functions instead of harmonic oscillator wavefunctions, is needed to understand the
observed Rabi oscillations quantitatively.
In this chapter, we have studied excitations in the optical lattice. A quantum mechanical picture of spatial oscillations has been developed and the interspecies energy transfer
from oscillating lithium atoms to a condensed sodium background has been analyzed quantitatively. By periodically displacing the lattice, we are able to excite lithium atoms from
first to second Bloch band in a controlled way, which has enabled us to get a measure of
the interspecies scattering length aIB by analyzing the subsequent relaxation induced by a
bosonic background. Finally, the coherence of the excitations has been proven by exciting
Rabi oscillations, which could be explained in time-dependent perturbation theory. In the
Outlook, we will further elucidate which physics can be explored with the presented tools at
hand.
126
9. Conclusion and Outlook
Das, was hinter dir liegt, hast du erlebt, was vor dir liegt, ist unbekannt.
Das mag verunsichernd sein, macht es
aber auch interessant.
Walter Bonatti
With this thesis, we paved the way to measure polarons in ultracold atom systems, in
particular in the Bose-Fermi mixture of sodium and lithium. For that purpose, we analyzed
the tuning parameters in detail and developed possible detection schemes for an increased
effective mass m∗ . As outlined in the introduction, there are in principle two ways to change
the coupling parameter α, which characterizes the polaron: On the one hand, one can vary
the background intraspecies interaction aBB , on the other hand the dependence on aIB is
even stronger.
In our experiments, the background is formed by a condensate of sodium, whose scattering
properties were already known to a good accuracy before the Heidelberg experiment started.
With our measurements of the sodium Feshbach resonance spectrum, we gave valuable input
to a coupled-channels calculation, which improved the accuracy of the last bound triplet state
in the NaNa-potential by a factor of 50. Moreover, the refined model provides us with a
reliable prediction of the width of the 1202.6 G resonance, ∆ = −1.473 G, making it well
suited for tuning aBB .
To determine aIB and analyze its tuning properties, we had to spend a higher effort. A general procedure, which can be applied to any combination of atoms with unknown interspecies
scattering properties, has been developed and tested by means of the sodium-lithium mixture. The difference between singlet and triplet scattering length can be determined via
a loss measurement, whereas for obtaining the absolute value of the scattering length we
presented two different procedures, which yielded the same result. With this at hand, we
could set limits to the s-wave bound state energy and thus develop a model explaining all 26
Feshbach resonances, which were measured in the course of this thesis. Most of them turned
out to have d-wave character, and a coupled-channels approach yielded as largest width
∆ = 10 mG, which is not suited for tuning aIB . In agreement with our experimental results,
127
the background scattering length was theoretically calculated to be aIB = −(75±5)a0 , being
in its absolute value a factor of five higher than previously assumed.
For measuring the polaron effective mass m∗ , we have implemented a species selective optical
dipole trap in form of an optical lattice. Without further effort, we were able to determine
the lithium oscillation frequency with a relative accuracy of better than 0.5%, which brings
a measurement of the polaron effective mass in our system within reach. Moreover, with
the tool of Brillouin zone mapping, we are able to analyze excitations in the lattice. Rabi
oscillations induced by a periodic lattice shaking were analyzed this way, which paves the
way to a wealth of interesting physics, as will be outlined in the following.
Outlook
As already to be seen in the last chapter in figure 8.5, the Rabi oscillation damps out on a
time scale of about 2 ms. This behaviour can be caused by both dephasing and decoherence,
but by implementing an appropriate spin-echo experiment one could extract the T2 -time, a
well-known measure to characterize decoherence processes in NMR [115]. This timescale can
be related to the T1 -time, which we have measured as the time constant of lithium relaxation
from second to first Bloch band: If T2 = T1 /2, which is e. g. the case for a two-level atom with
spontaneous emission [112], our bath is Markovian, i. e. has no memory. In our experimental
situation of a trapped BEC as background, it is not a priori clear if the system is Markovian,
as the phonons, which carry the energy resulting from the relaxation of the two-level system,
do not simply vanish as the photons in the case of spontaneous emission.
But even the precise investigation of the relaxation dynamics of the two-level system can
give some deeper insight into the underlying physics: If the process would prove to be nonexponential, i. e. follow a power-law, this could be caused by an energy-dependent relaxation
rate as expected if correlations in the bath build up.
We see that investigating thermodynamic processes in our system could reveal a lot of
interesting physics. With a controlled local heat input on the lithium, one could try and
see collisionless superfluid internal convection in the sodium condensate, an effect which has
been theoretically predicted recently [116].
The geometry of our species selective optical dipole trap, being realized as an optical lattice,
also provides us with the possibility to study lower-dimensional systems. It has been shown
that in this case the amplitude of possible polaronic features is enhanced [117], which facilitates the experimental observation. Moreover, the use of confinement-induced resonances
in the low-dimensional system [118] provides a possibility to tune the polaronic coupling
strength α.
Recently, pairing of fermions in a two-dimensional configuration [119] and its evolution
from two to three dimensions [96] has been studied experimentally, but the properties of
128
the Fermi polaron in 2D has so far only been investigated theoretically [120]. The interlayer
coupling model of high-Tc superconductivity [121], which attributes an enhancement of superconductivity due to coupling between the two-dimensional layers, could be realized in our
system by allowing for a finite tunneling coupling between the different lattice sites.
Following a more complex proposal [122], one could also explore the rich phase diagram
of a Bose-Fermi mixture with both gases being confined to one dimension, after the lattice
setup has been extended appropriately.
Whereas the last experiments outlined are rather goals to pursue in the far future, measuring
an increase in polaron effective mass m∗ in our setup is feasible under the current conditions:
By tuning aBB with a Feshbach resonance, we can bring the coupling parameter α in a regime,
where the resulting increase in effective mass m∗ can be detected via a shift in oscillation
frequency.
129
130
A. List of Constants
The following table gives the values and abbreviations of constants used in this thesis. The
species specific constants have been taken from [123] and [124] for sodium and lithium,
respectively.
symbol
constant
value
a0
~
h
kB
µB
c
µ0
Bohr radius
Planck constant
Planck constant
Boltzmann constant
Bohr magneton
Speed of light
magnetic constant
5.29177249 · 10−11 m
1.05457266 · 10−34 Js
~ · 2π
1.380658 · 10−23 J/K
9.2740154 · 10−24 J/T
299,792,458 m/s
4π · 10−7 H/m
Sodium-specific constants
mNa
ωNa
ΓNa
aNa
hfs
gs
iNa
giNa
mass
D2 -line transition frequency
Natural linewidth
hyperfine constant
Electron spin g-factor
nuclear spin
nuclear g-factor
3.81754035 · 10−26 kg
2π · 508.8487162 THz
2π · 9.7946 MHz
885.81306440 MHz
2.0023193043622
3/2
-0.00080461080
Lithium-specific constants
mLi
ωLi
ΓLi
aLi
hfs
gs
iLi
giLi
mass
D2 -line transition frequency
Natural linewidth
hyperfine constant
Electron spin g-factor
nuclear spin
nuclear g-factor
9.9883414 · 10−27 kg
2π · 446.799677 THz
2π · 5.8724 MHz
152.1368407 MHz
2.0023193043622
1
-0.000447654
131
132
B. All NaLi Resonance Data
In the following, all NaLi resonances measured in the different spin channels are shown. All
pictures were taken in an ODT with ω usw. , a temperature of T ≈ 1µK and atom numbers
of NNa ≈ 106 and NLi ≈ 105 . Each loss curve is fitted with a Gaussian, which is shown as
solid line. The averaged lithium atom numbers with their statistical errors plotted on the
y-axis are normalized, the magnetic field [G] is the x-axis of each graph, inside which four
parameters characterizing the resonance are shown:
ˆ B0 : The resonance position
ˆ ∆B: The 1/e2 half width of the fit to the resonance data
ˆ τ : The hold time calculated from an exponential fit to the atom loss curve on resonance
ˆ ∆: The theoretical resonance width from a coupled channels calculation
d-wave resonances are plotted in red and s-wave resonances in blue.
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
τ = 105ms
∆ = 10mG
B0 = 1575.8G
0.2
0
∆B = 215mG
1574
1575
1576
1577
0.4
B0 = 1700.4G
0.2
0
∆B = 192mG
1699
1700
τ = 948ms
∆ = 3mG
1701
1702
Figure B.1.: MF = 5/2, Li |3/2, 3/2i + Na |1, 1i Feshbach resonances.
133
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
B0 = 745.2G
0.2
0
∆B = 292mG
744
745
τ = 57ms
∆ = 10mG
746
747
0.4
0
757
1.2
1
1
0.8
0.8
0.6
0.6
τ = 189ms
∆ = 500µG
B0 = 795.2G
0.2
0
∆B = 210mG
794
795
796
797
0.2
0
1
1
0.8
0.8
0.6
0.6
0.2
0
∆B = 254mG
1595
1596
τ = 382ms
∆ = 5mG
1597
1598
759
760
τ = 1312ms
∆ = 37µG
B0 = 1510.4G
1.2
B0 = 1596.5G
758
0.4
1.2
0.4
∆B = 174mG
0.2
1.2
0.4
τ = 2000ms
∆ = 21µG
B0 = 759G
∆B = 231mG
1509
1510
1511
0.4
τ = 1270ms
∆ = 260µG
B0 = 1715.6G
0.2
0
∆B = 359mG
1714
1715
1512
1716
1717
Figure B.2.: MF = 3/2, Li |1/2, 1/2i + Na |1, 1i Feshbach resonances (I).
134
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
τ = 3190ms
∆ = 42µG
B0 = 1908.9G
∆B = 174mG
0.2
0
1907
1908
1909
1910
0.4
τ = 84ms
∆ = 4mG
B0 = 2046.9G
∆B = 197mG
0.2
0
2045
2046
2047
2048
Figure B.3.: MF = 3/2, Li |1/2, 1/2i + Na |1, 1i Feshbach resonances (II).
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
τ = 668ms
∆ = 10mG
B0 = 771.8G
∆B = 213mG
0.2
0
770
771
772
773
0.4
1.2
1
1
0.8
0.8
0.6
0.6
τ = 129ms
∆ = 5mG
B0 = 1596.75G
0.2
0
∆B = 202mG
1595
1596
1597
1598
∆B = 64mG
0
821
1.2
0.4
τ = 1717ms
∆ = 493µG
B0 = 822.9G
0.2
822
823
0.4
τ = 734ms
∆ = 247µG
B0 = 1716.7G
0.2
0
∆B = 200mG
1715
1716
824
1717
1718
Figure B.4.: MF = 1/2, Li |1/2, −1/2i + Na |1, 1i Feshbach resonances.
135
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
∆B = 127mG
0.2
0
799
800
0.4
τ = 266ms
∆ = 10mG
B0 = 800.94G
801
1.2
1
1
0.8
0.8
0.6
0.6
τ = 1386ms
∆ = 31µG
B0 = 1566.3G
0.2
0
∆B = 214mG
1565
1566
1567
∆B = 56mG
0
850
802
1.2
0.4
τ = 2606ms
∆ = 467µG
B0 = 852G
0.2
1568
851
852
0.4
B0 = 1597.5G
0.2
0
∆B = 256mG
1596
1597
853
τ = 174ms
∆ = 6mG
1598
1599
1.2
1
0.8
0.6
0.4
B0 = 1717.3G
0.2
0
∆B = 199mG
1716
1717
τ = 820ms
∆ = 256µG
1718
1719
Figure B.5.: MF = −1/2, Li |3/2, −3/2i + Na |1, 1i Feshbach resonances.
136
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
B0 = 1002.3G
0.2
0
∆B = 53mG
1001
1002
τ = 190ms
∆ = 9mG
1003
1004
0.4
B0 = 1088.5G
0.2
0
∆B = 49mG
1087
1088
τ = 389ms
∆ = 938µG
1089
1090
Figure B.6.: MF = −1/2, Li |1/2, 1/2i + Na |1, −1i Feshbach resonances.
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
B0 = 913.2G
0.2
0
∆B = 93mG
912
913
τ = 491ms
∆ = 9mG
914
915
0.4
B0 = 1720.5G
0.2
0
∆B = 173mG
1719
1720
τ = 1096ms
∆ = 58µG
1721
1722
Figure B.7.: MF = −3/2, Li |3/2, −3/2i + Na |1, 0i Feshbach resonances.
137
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
τ = 200ms
∆ = 9mG
B0 = 1031.69G
0.2
0
∆B = 181mG
1030
1031
1032
0.4
B0 = 1117.3G
0.2
∆B = 82mG
0
1033
1116
1117
τ = 592ms
∆ = 846µG
1118
1119
1.2
1
0.8
0.6
0.4
B0 = 1902.4G
0.2
0
∆B = 154mG
1901
1902
τ = 1911ms
∆ = 102µG
1903
1904
Figure B.8.: MF = −3/2, Li |1/2, −1/2i + Na |1, −1i Feshbach resonances.
138
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Danksagung
Der Schlüssel zum Erfolg ist Kameradschaft und der Wille, alles für den
anderen zu geben.
Fritz Walter
An dieser Stelle möchte ich den Menschen danken, die – auf verschiedenste Art und Weise –
entscheidend zum Gelingen dieser Arbeit beigetragen haben.
ˆ Meinem Betreuer Herrn Prof. Dr. M. K. Oberthaler möchte ich recht herzlich dafür
danken, dass er nach dem Super-Gau“ ein NaLi 2.0 möglich gemacht hat und auch in
”
schweren Stunden versucht hat, uns durch seine Begeisterung für die Physik und seinen Optimismus (der uns in solchen Situationen dann meist recht unangebracht schien)
zu motivieren. Auch wenn seine fetzigen“ Ideen die Grundrichtung des Experiments
”
prägen, so hatte ich doch stets neben seinen Anregungen genug wissenschaftliche Freiheiten.
ˆ Herrn Prof. Dr. S. Jochim für die Zweitkorrektur dieser Arbeit. Die Tipps, insbesondere was das Lithium-Setup angeht, waren ähnlich wichtig wie die gemeinsamen Pizzaund Feuerzangenbowle-Abende mit seiner Gruppe.
ˆ Steven Knoop, der als Postdoc beim Wiederaufbau des NaLi Experiments großartiges
geleistet hat. Nur holländisch fest geschraubte (Opto-)Mechanik hat er dadurch mehr
als wett gemacht, dass er bei beiden Publikationen die treibende Kraft war, insbesondere bei der Lösung des NaLi Feshbach Resonanz Problems. Gemeinsame Grill- und
Fußballabende - letztere wie wohl auch bei der kommenden EM meist mit negativem
Ausgang für die Elftal - werden in guter Erinnerung bleiben.
ˆ Raphael Scelle, der das NaLi-Experiment während der vergangenen vier Jahre geprägt hat wie kein anderer. Ohne seine sorgfältige Arbeit beim Wiederaufbau wären
die meisten der Ergebnisse dieser Doktorarbeit nicht zustande gekommen. Aus (selten
auftretenden) Meinungsverschiedenheiten in gemeinsamen Diskussionen über experimentellen Aufbau und Physik war es stets möglich, gemeinsam mit ihm produktiv das
149
Beste für das Experiment zu erarbeiten. Einen angenehmeren Kollegen hätte ich mir
für die letzten dreieinhalb Jahre nicht wünschen können.
ˆ Jens Appmeier für die Pionierarbeit bei NaLi 1.0 und seinen Beitrag zu NaLi 2.0, wo
er im gemeinsamen Aufbau dank seines unerschöpflichen guten Humors über manche
Durststrecke hinwegtrösten konnte. Glücklicherweise sind seine Fähigkeiten im Umgang
mit der Grillzange besser als seine Kenntnis der ruhmreichen Fußballgeschichte des 1.
FC Kaiserslautern.
ˆ Arno Trautmann für den Interlock, den er zwar meist selbst in Anspruch nimmt, ohne
den das Experiment aber schon mehrfach wieder abgebrannt wäre. Sein Sinn für gute
Satire ist fast so ausgeprägt wie seine LATEX-Kenntnisse, die auch zum Entstehen dieses
Dokuments mehrfach dankbar in Anspruch genommen wurden.
ˆ Tobias Rentrop, der sich trotz der Widerspenstigkeit des Experiments in der Anfangsphase seiner Doktorarbeit nicht entmutigen lässt und durch seinen trockenen Humor
das Laborleben bereichert.
ˆ Den Bachelor-Studenten Andrea Bergschneider, Elisabeth Brühl und Mathias Neidig,
die weit über den Umfang ihrer Arbeit hinaus viel für das Experiment geleistet haben
und Pionierarbeit beim Lithium-Laser-System sowie dem Gitteraufbau erbracht haben.
ˆ Und zu guter Letzt, da das Ganze mehr ist als die Summe seiner Teile: Dem gesamten NaLi-Team, bei dem trotz sich ändernder Personalstruktur eine Sache stets gleich
geblieben ist: Der Zusammenhalt, ohne den es mir nicht möglich gewesen wäre, mich
jeden Tag aufs neue dem gemeinsamen Kampf gegen die Maschine auszusetzen und ihr
auch das ein oder andere Ergebnis abzuringen.
ˆ Gedankt sei hier auch den Fachkollegen aus Kollaborationen, die während dieser Arbeit
entstanden sind. Besonders hervorzuheben sind hier Prof. Dr. E. Tiemann, der mit
seiner Coupled-Channels Rechnung das NaLi Feshbach Resonanz Paper auf ein solides
Fundament gestellt hat und Prof. Dr. A. Komnik, mit dem wir uns gemeinsam in vielen
Diskussionen dem Polaron angenähert haben.
ˆ Dank auch an Herrn Prof. Dr. P. Bachert für die Übernahme des angewandten Teils
der Prüfung und aufschlussreiche Erklärungen zur NMR.
ˆ Den Mitgliedern des BEC(K)-Experiments, insbesondere Helmut für seine Expertise in
Elektronik und Tilman für hitzige, aber nichtsdestotrotz ergiebige Diskussionen über
Physik und vieles mehr. Und allen dafür, dass ständig Equipment aus unserem Labor
verschwindet und komischerweise im Nachbarlabor wieder auftaucht – die [...] sind
”
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raffiniert“ (V. A. Toni). Wenn nach B. Stromberg Büro ist Krieg“ gilt, so wäre Labor
”
mindestens Weltkrieg.
ˆ Den ATTA-Mitgliedern für ihre Standhaftigkeit im Kampf gegen die Überinterpretation
von Messdaten (t = (500 ± 500) Jahre).
ˆ Dem Aegis-Experiment, insbesondere Fabienne Haupert für ihre stete Hilfsbereitschaft
in Gruppen- und persönlichen Angelegenheiten.
ˆ Den Mitfahrern und -fallern bei Stützrad, insbesondere den Gründungsmitgliedern Eike Nicklas und Christian Groß, mit denen es bei Wind und Wetter gemeinsam auf
den Weißen Stein und den Königstuhl hoch ging. Des weiteren der Hochtourengruppe und Moritz Höfer, auf den man sich bei einem Spaltensturz hätte verlassen können.
ˆ Den Mitglidern der Werkstatt, insbesondere Herrn Lamade, Morris Weißer für seine
hervorragende Arbeit insbesondere beim Anfertigen der Spulenhalter, Herrn Herdt
für seine einfallsreichen Lösungen bei Mechnik-Problemen und Herrn Spiegel für seine
Nachsicht, wenn in der Diplomanden-Werkstatt mal wieder was unerklärlicherweise
fast von selbst kaputt gegangen war.
ˆ Den Mitgliedern der Elektronik-Abteilung, Jürgen Schölles als Ansprechpartner für
alles, Alexander Leonhardt für unzählige Netzteilreparaturen, Herrn Azeroth für sichere Hochstromeinrichtungen und Herrn Kiworra für die Beantwortung aller Fragen
zu Masseschleifen.
ˆ Den Team Assistants“ Dagmar Hufnagel und Christiane Jäger, die für einen rei”
bungslosen Ablauf organisatorischer Dinge innerhalb der Gruppe, insbesondere der
finanziellen Angelegenheiten sorgen.
ˆ Der HGSFP und insbesondere Frau Prof. Dr. S. Klevansky für die hervorragende Graduiertenausbildung, die die Fakultät für Physik der Universität Heidelberg bietet.
ˆ Der Klaus Tschira Stiftung für das großzügige Stipendium, das mir zur Verfüngung
gestellt wurde. Dank des erfreulich geringen Verwaltungsaufwands und der somit unkomplizierten Finanzierung meiner Promotion konnte ich meine Aufmerksamkeit fernab von Geldsorgen voll den Forschungsprojekten widmen.
ˆ Der Laserbrücke zwischen KIP und Neuenheimer Feld, die in ihrer Sinnhaftigkeit nur
durch ihre Tragfähigkeit überboten wurde. Und für deren Zustandekommen geht mein
besonderer Dank an alle, die die Courage haben, die Wahrheit und nicht immer nur
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ja“ zu sagen.
”
ˆ Martin Aeschlimann, der mir in schwierigen Situationen des Lebens stets beratend zur
Seite stand.
ˆ Michael Müller, der meinen Rücken immer wieder aufgerichtet hat, wenn er unter der
Last der Arbeit und Probleme eingeknickt ist.
ˆ Thomas Czarnecki für sein Mentoring in beruflichen Fragen.
ˆ Meinen Freunden, auf die ich mich in guten und schlechten Zeiten verlassen kann. Danke an Frederik für viele gemeinsame Mittagessen, Nicki für gemeinsames Kochen und
geteilten Humor, gracias a Antonia para bailar juntos y enseñarme español, Katharina
für mehr als fünf Jahre Freundschaft, Stephan für so viele gemeinsame Gespräche über
alle wichtigen Themen des Lebens und Martin für all das, was wir seit dem 1. Semester
erlebt haben.
ˆ Christoph und Bianca für gemeinsame Urlaube, geteilte Zeiten der Trauer und Momente des Glücks, und natürlich für eure Katzen. Und Christoph insbesondere dafür,
dass er mich 24 Jahre lang ertragen hat und trotzdem noch mit mir redet.
ˆ Meiner Mama dafür, dass sie meine Begeisterung für die Physik schon zu Schulzeiten
geweckt hat. Aber auch außerfachlich hat sie mich stets in jeder Hinsicht unterstützt
und ich habe mich in allen noch so verfahrenen Lebenslagen auf sie verlassen können.
Dank auch an ihren Freund Reiner, auf dessen Hilfe stets Verlass ist.
ˆ Meinem Papa, der das Ende dieser Doktorarbeit leider nicht mehr miterleben durfte.
Sein bewundernswerter Kampfgeist und seine Aufmunterung die Ohren steif zu hal”
ten“, mit der er mich auf Durststrecken jedweder Art zum Durchhalten motivierte,
werden unvergessen bleiben.
ˆ Meiner Freundin Anne, die mir weitaus mehr bedeutet, als dass ich es hier in Worte
fassen könnte.
Enden möchte ich mit den versöhnlichen Worten des großen Johannes Brahms: Und sollte
”
ich vergessen haben, jemanden zu beschimpfen, dann bitte ich um Verzeihung!“
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