Radio-Frequency Spectroscopy on
the Mass-Imbalanced 6Li-40K
Fermi-Fermi Mixture
DISSERTATION

by

Dipl.-Phys. Michael Jag
submitted to the Faculty of Mathematics, Computer Science,
and Physics of the University of Innsbruckin partial fulfillment of the requirements
for the degree of Doctor of Philosophy (PhD)

Advisors:
Univ.-Prof. Dr. Rudolf Grimm,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences

Innsbruck, July 2016

SummaryUltracold quantum gases are ideal systems to experimentally tackle fundamental problems
in both many- and few-body physics. The parameters characterizing such systems can becontrolled and readout to an unprecedentedly high degree of precision. In this thesis, we discuss our experimental investigations of mixtures of two ultracold gases of different fermionic
atom species, 6 Li and 40 K. The studies can be assigned to two main research themes. These
are the fundamental many-body system consisting of an impurity in a Fermi sea on the one
hand, and Li-K few-body physics on the other hand.
To address the many-body physics of an impurity in a Fermi sea experimentally, we realize
a mixture of a few K atoms and a large, dense cloud of fermionic Li atoms. The interaction
between the impurity atoms and the atoms of the Fermi sea can be tuned by means of amagnetic Feshbach resonance. We then employ radio-frequency spectroscopy of the K atoms
in various interaction regimes, ranging from strongly repulsive to strongly attractive, to reveal
the energy landscape of the impurities in the Fermi sea. In the regime of strong interactions,where the scattering length exceeds the inter-particle separation, the impurity together withthe excitation of the Fermi sea are described as a quasiparticle termed polaron, following
Fermi liquid theory. We determine the energy, the residue, and the lifetime of such polarons
in our system. Then, by applying a spin-echo technique, we investigate the decoherence of
impurities in a Fermi sea as a function of the interaction strength. For moderate interaction
strengths, our measurements provide a determination of the quasiparticle scattering rate. For
near-resonant interactions, the decoherence rates are almost an order of magnitude larger.
In another series of experiments, we probe the response of a Fermi sea to a rapidly introduced impurity on timescales much shorter than the Fermi time. Employing a Ramsey-type
interferometry technique, we track the formation dynamics of repulsive and attractive polaron in real time. For resonant interactions, we observe the quantum interference arisingfrom the simultaneous excitation of the corresponding repulsive and attractive many-bodybranches.
To investigate the physics of Li-K few-body systems, we create dense samples of Li-K dimersfrom almost balanced mixtures of Li and K atoms. We then perform radio-frequency spectroscopy to investigate the elastic interactions of heavy K atoms with light-heavy Li-K dimers.
Contrary to the atom-dimer interactions in a mass-balanced Fermi-Fermi mixture, our measurements reveal a strong K-LiK attraction in a regime of Li-K repulsion. This atom-dimer
attraction is speculated to facilitate the access to the realization of exotic many-body statesand phases.
Whether it is possible to experimentally realize such phases, strongly depends on the lifetimeof the atom-dimer mixture, which is typically limited by the lifetime of the dimer cloud.
Therefore, we also investigate the lifetimes of dimer clouds as a function of the magnetic
detuning from a Feshbach resonance. Our measurements allow us to discriminate between
different contributions to the decay of our dimer sample and to determine accurate rates forthe spontaneous decay as well as for the decay arising from collisions between two dimers or
between one dimer and one atom.

iii

Zusammenfassung
Ultrakalte Quantengase sind ideale Systeme, um fundamentale Probleme sowohl der Vielteilchen- als auch der Mehrteilchen-Physik experimentell in Angriff zu nehmen. Die Parameter,
die solche Systeme charakterisieren, können zu einem beispiellos hohen Grad an Präzision
kontrolliert und ausgelesen werden. In dieser Arbeit werden Experimente mit Gemischen
zweier Gase verschiedener fermionischer Atom-Spezies, 6 Li und 40 K, diskutiert. Unsere Untersuchungen können zwei größeren Forschungsthemen zugeordnet werden. Diese sind zum
einen das grundlegende Vielteilchen-System, bestehend aus einem Fremdatom und einem
Fermi-See, und zum anderen die Li-K Mehrteilchen-Physik.
Um uns der Vielteilchen-Physik eines Fremdatoms im Fermi-See zu widmen, erzeugen wir
ein Gemisch von wenigen K-Atomen und einer großen, dichten Wolke von Li-Atomen. Wirkönnen die Wechselwirkung zwischen den Fremdatomen und den Atomen des Fermi-Sees
mittels einer magnetischen Feshbach-Resonanz variieren. Wir führen dann RadiofrequenzSpektroskopie an den Kalium-Atomen in mehreren Wechselwirkungsregimes, von stark repulsiv bis stark attraktiv, durch und legen so das Energiespektrum des Kalium-Atoms im
Lithium-Fermi-See offen. Im stark wechselwirkenden Regime, wo die Streulänge größer ist
als der Teilchenabstand, wird das Fremdatom, zusammen mit der Anregung des Fermi-Sees,
Landaus Theorie der Fermi-Flüssigkeiten folgend, als Quasiteilchen beschrieben, das Polaron
genannt wird. Wir bestimmen die Energie, das Residuum und die Lebensdauer der Polaronen in unserem System. Des Weiteren untersuchen wir die Dekohärenz der Fremdatome
im Fermi-See als Funktion der Wechselwirkungsstärke, unter Verwendung einer Spin-EchoTechnik. Für mäßige Wechselwirkungsstärke stellen unsere Messungen eine Bestimmung der
Quasiteilchen-Streurate dar. Bei nahresonanter Wechselwirkung bestimmen wir nahezu eineGrößenordnung höhere Dekohärenzraten.
In einer weiteren Messreihe untersuchen wir, auf Zeitskalen viel kürzer als die Fermi-Zeit,
wie unser System auf ein rasch eingeführtes Fremdatom reagiert. Unter Verwendung einer
Ramsey-Interferometrie-Technik verfolgen wir die Entstehung von repulsivem und attraktivem Polaron in Echtzeit. Bei resonanter Wechselwirkung beobachten wir die Quanteninterferenz, die durch gleichzeitiges Anregen des repulsiven und attraktiven Vielteilchen-Zweiges
zustande kommt.
Für die Untersuchungen der Physik in Li-K Mehrteilchen-Systemen erzeugen wir dichte Li-KDimerproben aus einem beinahe ausgewogenen Gemisch von Li- und K-Atomen. Wir führen
dann Radiofrequenz-Spektroskopie durch, um die elastische Wechselwirkung zwischen KAtomen und Li-K-Dimeren zu untersuchen. Im Gegensatz zur Atom-Dimer-Wechselwirkung
in einem Gemisch zweier Gase gleichschwerer fermionischer Atome, zeigen unsere Messungen eine starke K-LiK-Anziehung in einem Regime, wo sich Li- und K-Atome gegenseitig
abstoßen. Die beobachtete Atom-Dimer-Anziehung in unserem Gemisch soll es vereinfachen,
exotische Vielteilchenzustände und -phasen experimentell zu realisieren.
Ob es möglich ist, solche Phasen experimentell zu realisieren, hängt stark von der Lebensdauer des Atom-Dimer-Gemischs ab, die typischer Weise durch die Dimer-Lebensdauer begrenzt ist. Daher untersuchen wir die Lebensdauer von Dimer-Wolken als Funktion deriv

magnetischen Verstimmung von einer Feshbach-Resonanz. In unseren Messungen können
wir zwischen verschiedenen Beiträgen zum Zerfall einer Dimerprobe unterscheiden und wir
erhalten die Raten für den spontanen Zerfall sowie für den Zerfall durch Stöße zwischen zwei
Dimeren oder zwischen einem Dimer und einem Atom.

v

Contents
Summaryiii

Zusammenfassungiv

1 Introduction
1.1 The interest in ultracold Fermi gases . . . . . . . . . . . . . . . . . . . . . . .
1.2 Short history of ultracold Fermi gases . . . . . . . . . . . . . . . . . . . . . .
1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
4

2 Introduction to the Experimentalist’s Toolbox
2.1 Scattering-formalism basics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Scattering amplitude and cross section . . . . . . . . . . . . . . . . . .
2.1.2 Scattering in a spherical square-well potential . . . . . . . . . . . . . .
2.1.3 Scattering in a spherical square-well potential with a delta-function
barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Tuning interactions near a Feshbach resonance . . . . . . . . . . . . . . . . .
2.3 Description of the mean-field energy . . . . . . . . . . . . . . . . . . . . . . .
2.4 Elastic atom-dimer interactions near a Feshbach resonance . . . . . . . . . . .
2.5 Basics of radio-frequency spectroscopy in our Li-K system . . . . . . . . . . .
2.5.1 Rabi-flopping: two-level system without interactions . . . . . . . . . .
2.5.2 Two-level system with one (weakly) interacting state - measuring themean-field energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
7
9
11
13
16
18
22
23
24

3 Publication: Metastability and Coherence of Repulsive Polarons in a Strongly
Interacting Fermi Mixture
29
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Supplementary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.2 Polaron peak in the spectral response . . . . . . . . . . . . . . . . . . 41
3.5.3 Molecule-hole continuum . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.4 Decay rate of the repulsive polarons . . . . . . . . . . . . . . . . . . . 46
3.5.5 Decay of repulsive polarons to molecules . . . . . . . . . . . . . . . . . 47
3.5.6 Rabi oscillations and polaron quasiparticle residue . . . . . . . . . . . 50
3.6 Correction: Molecule-hole continuum . . . . . . . . . . . . . . . . . . . . . . . 51
vii

4 Publication: Decoherence of Impurities in a Fermi Sea of Ultracold Atoms
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Optical trap setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Parameters of the 154.7 G Feshbach resonance . . . . . . . . . . . . .
4.5.3 Light shift of the Feshbach resonance . . . . . . . . . . . . . . . . . . .
4.5.4 Determination of the Li atom number . . . . . . . . . . . . . . . . . .
4.5.5 Determination of Li atom number: summary . . . . . . . . . . . . . .
4.5.6 Absorption imaging near 154.7 G . . . . . . . . . . . . . . . . . . . . .
4.5.7 Heating due to molecule formation . . . . . . . . . . . . . . . . . . . .

53
54
54
55
60
61
61
62
64
67
74
74
75

5 Preprint: Ultrafast Many-Body Interferometry of Impurities Coupled to a Fermi
Sea
79
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 Supplementary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.2 Role of physical processes on different time scales . . . . . . . . . . . . 90
5.5.3 Role of interaction during finite-length rf pulses . . . . . . . . . . . . . 94
5.5.4 Universal features of impurity dynamics and relation to orthogonality
catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5.5 Experimental and data analysis procedures . . . . . . . . . . . . . . . 101
5.5.6 Linearity of rf response . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Publication: Observation of a Strong Atom-Dimer Attraction in a Mass-Imbalanced
Fermi-Fermi Mixture
109
6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Supplemental material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5.1 Light shift of the Feshbach resonance . . . . . . . . . . . . . . . . . . . 117
6.5.2 Preparation of the atom-dimer mixture . . . . . . . . . . . . . . . . . 119
6.5.3 Determination of the temperatures and the densities . . . . . . . . . . 120
6.5.4 Importance of higher partial wave scattering and comparison to theequal-mass case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Lifetime of 6 Li-40 K Dimers near a Feshbach Resonance125
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Experimental procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.1 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2.3 Dimer detection and dimer temperature determination . . . . . . . . . 129
viii

7.3

7.4

Measurements of dimer decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3.1 Spontaneous dissociation of Li|1⟩K|3⟩ dimers . . . . . . . . . . . . . . 130
7.3.2 Dimer-dimer collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3.3 Atom-dimer collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8 Outlook

139

Bibliography141

Acknowledgements

159

ix

Chapter

1

Introduction

1.1 The interest in ultracold Fermi gases
Ultracold Fermi gases are pristine many-body systems. Realized under certain conditions,these systems can show aspects of behavior that find their direct counterpart in condensedmatter systems of tremendous interest [Ben14]. These condensed-matter systems include,e.g., high-temperature superconductors (high-Tc SC), the so-called quark-gluon plasma, astate our universe is assumed to be gone through shortly after the big bang, or neutronstars.
The reason for the enormous interest in these condensed matter systems is rooted in theirimportance. Gaining knowledge about the early stage and the evolution of the universe or,in the case of high-Tc SCs, being able to build wires that can conduct current at room temperature without losses, are strong driving forces to do research. The discovery of the latter
would, for example, be an amazing scientific achievement with huge potential applicationon the energy market. However, all of the mentioned condensed-matter systems have incommon that essential features of them are not understood. Huge efforts are made to obtain theoretical descriptions reproducing such features, and rather frequently new proposals
emerge. For the theorists it is crucial to have their proposals experimentally tested.
Due to the astonishing analogies between the condensed-matter systems mentioned above
and ultracold Fermi gases (though they have very different densities and temperatures)the latter offer an ideal testbed to benchmark potential theoretical descriptions or trigger
their development. This process is referred to as quantum simulation: One can realize
ultracold Fermi gases, offering excellent control of almost all relevant system parameters,
under well defined conditions and use them to simulate a condensed-matter system [Blo12].
The ultracold Fermi gas is then typically investigated under variation of a system parameterand its behavior can be checked against the theoretical prediction. As a result, one cangive crucial input for the theorist, i.e. in an ideal case one can falsify or verify a theoretical
approach.
Usually such testing procedures require continuous control of the system’s parameters: theinteractions between fermions, the temperature, the density of fermions, the dimensionality12

1.2. SHORT HISTORY OF ULTRACOLD FERMI GASESof the confinement (1D, 2D, 3D), the mass ratio of different fermions, and the population ofvarious states of the fermions. We will discuss in the course of this thesis, how we can experimentally access and tune these parameters of ultracold, trapped Fermi-Fermi mixtures.

1.2 Short history of ultracold Fermi gases
At the time when the work presented in this thesis was started, there was already a large
variety of groups working on ultracold Fermi gases. Here, we give a short summary of theachievements with ultracold atomic Fermi gases prior to the thesis.
It was a few years after the first realization of a Bose-Einstein condensate (BEC) of ultracold 87 Rb and 23 Na in 1995 [Dav95, And95] that a degenerate gas of fermionic atoms
was experimentally realized [DeM99]. Initially, the experiments used stable fermionic alkali
isotopes, which are 40 K [DeM99, Roa02] on the one hand, and 6 Li [Tru01, Sch01, Gra02,
Had02, Joc03b] on the other hand. In the mean time, the list of elements of which ultracold
degenerate Fermi gases were realized was extended by 3 He∗ [McN06], 171 Yb [Fuk07a], 173 Yb[Fuk07b], 87 Sr [DeS10, Tey10], 161 Dy [Lu12], 167 Er [Aik14], and 53 Cr [Nay15].
New fields of research were opened up by establishing mixtures of fermions in two different spin states of the same atomic species where interactions can be tuned utilizing
Feshbach resonances [Chi10]. With such systems, Fermi gases under strongly interacting
conditions [O’H02, Bou03] and, in particular, under the strongest possible, resonant interactions, the so-called unitary Fermi gas, could be investigated as prominently demonstrated
by the measurement of the heat capacity of the unitary Fermi gas [Kin05]. Moreover, Feshbach resonances were utilized to create dense samples of diatomic Fermi-Fermi molecules[Reg03, Str03, Cub03, Joc03a]. Such molecules, made up of two fermionic constituents, arein fact bosons and follow bosonic quantum statistics. Soon after the first production ofdense molecule samples, this was spectacularly demonstrated by experimentally realizing
a BEC of Fermi-Fermi molecules [Joc03b, Gre03, Zwi03, Bou04]. These studies sparked
experimental efforts on investigating the pairing properties in the crossover from the BEC(bosonic) to a weakly attractive two-component Fermi gas, described by Bardeen-CooperSchrieffer (BCS) theory (fermionic). Experimentally, the first BEC-BCS crossover studies
were reported in Refs. [Bar04b, Reg04b]. Already shortly after, measurements of the collective modes [Kin04, Bar04a] showed evidence for superfluidity. Vivid proof for superfluidity
in the crossover regime was then delivered by the observation of vortices in a strongly interacting Fermi gas [Zwi05]. Later, numerous experiments addressed the investigation ofsuperfluid properties in Fermi gases and, to list some of the prominent examples, yielded theobservation of critical velocities [Mil07], a quenched moment of inertia [Rie11], and second
sound [Sid13].
Fermi gases consisting of spin-state mixtures with an imbalanced population were started
to be explored only a few years later by the Hulet (Rice) and the Ketterle (MIT) groups.
They addressed the question of survival of superfluidity in these mixture versus interaction
strength [Zwi06] and demonstrated the phase separation of such mixtures into a superfluid
core surrounded by remnant majority atoms [Par06, Shi06]. These observations triggered

1.2. SHORT HISTORY OF ULTRACOLD FERMI GASES

3

further investigations of the phase diagram of the unitary Fermi gas [Shi08] as well as themeasurements of the equation of state, describing its thermodynamic properties both in theunitarity regime (Ref. [Nas10] and references therein) and in the BEC-BCS crossover [Nav10].
Experiments addressing the physics in the almost fully population-imbalanced Fermi mixture- few impurities in a Fermi sea - were carried out only a few years prior to this thesis work
[Sch09]. For strong interactions, the impurities and the excitations it causes to the fermionicenvironment form a quasiparticle termed polaron [Lan56]. This system is one of the simplest
and most basic ones in condensed-matter theory, where it finds its analog in polarons formedby electrons exciting the crystal lattice the electrons are traveling through.
The list of spectacular results obtained using ultracold Fermi gases is by far not exhausted.
Already the ones mentioned above, however, expose a wealth of physics that can be addressedwith them. These experiments were using homo-nuclear mixtures of fermionic gases, i.e.
mixtures of either 6 Li or 40 K atoms in two different spin-states, where the components
necessarily have equal mass.
Mass-imbalanced Fermi-Fermi mixtures
A very important and experimentally vastly untackled question is the effect of a mass imbalance between the components on the physics of a Fermi-Fermi mixture. There are many
theoretical proposals to investigate its effect both in the many- and in the few-body regime,
see e.g. Ref. [Gio08] and references therein for an overview. Exotic phenomena and superfluid phases are predicted to exist, which gives rise to more complex and interesting phase
diagrams compared to mass-balanced systems [Isk06, Par07, Bar08]. Such phases include theFulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase [Ful64, Lar65, Mat11], the Sarma [Sar63],
and the breached-pair phase [Gub03, For05] or a stable crystalline phase [Pet07]. In the
few-body regime, novel long-lived trimer and few-body cluster states are predicted to exist[Kar07, Lev09, SN13, Blu12].
Experimentally, this intriguing new physics can be addressed with a mixture of two gases ofdifferent fermionic atom species. Furthermore, such mixtures add another tuning knob to
the toolbox to investigate Fermi gases. By adding species-selective optical lattice potentials,
such systems allow for the investigation of Fermi-Fermi mixtures in mixed dimensions [Nis08].
Similarly, employing an optical lattice potential of unequal depth for the two species, one
can tune the imbalance of the species’ effective masses simply by varying the intensity of thelattice laser beam.
To our knowledge, currently there are only two combinations of different fermionic species
that experimental groups are working on. These are the combinations of 6 Li with 171/173 Yb 1
in the Gupta group at the University of Washington [Han13] and in the Takahashi groupat Kyoto University [Har11] and of 6 Li with 40 K in four groups worldwide, including theDieckmann group in Singapore (formerly Munich) [Cos10], the Zwierlein group at MIT[Wu11], the Salomon group in Paris [Sie15], and us.
1

The applicability of interaction tuning by means of a magnetic Feshbach resonance in this mixture, however,is questionable: The resonances are predicted to be narrow, with a width ∆ < 3 mG, and to appear athigh magnetic fields, > 1000 G, due to the missing hyper-fine structure of ground-state Yb [Bru12]. The
demanded magnetic field stability to exploit such resonances for interaction tuning is far beyond what has
been realized in experiments and, well possible, out of reach.

4

1.3. THESIS OVERVIEW

Status of the Li-K mixture and the FeLiKx lab in 2011
When the author of the present thesis joined the team of the FeLiKx lab, situated at theInstitute for Quantum Optics and Quantum Information (IQOQI), to start my PhD work in
2011, the toolbox to investigate strongly interacting 6 Li-40 K Fermi-Fermi mixtures was justfilled. Cooling and preparation procedures were developed [Tag08, Wil09, Spi10b, Spi10a,
Wu11], as well as the stability of the mixture and possibilities to tune the inter-species
interactions were characterized [Wil08, Spi09, Tie10, Cos10, Spi10a, Nai11]. Furthermore,
there were first results on the creation of bosonic molecules from the Fermi-Fermi mixture
by means of Feshbach association [Voi09, Spi10a]. And in 2010, the first measurements on astrongly interacting mass-imbalanced mixture were performed by our group [Tre11a, Tre11b],
revealing a hydrodynamic behavior of the mixture during time-of-flight expansion. In early
2011, the team had just started to perform first radio-frequency spectroscopic measurements
on a few fermionic 40 K atoms immersed in a Fermi sea of 6 Li atoms across an inter-species
Li-K Feshbach resonance [Koh12a]. This developed into one of the two main subjects ofthe present thesis, the investigation of the many-body system composed of an impurity
interacting with a Fermi sea.

1.3 Thesis overview
In this thesis, we discuss the experiments that we carried out since the author of the present
thesis joined the team in the laboratory. These experiments followed two lines of research:the many-body system of a K impurity in a Li Fermi sea (Chapter 3, 4, and 5), and Li-K
few-body physics (Chapter 6 and 7), which we investigate in experiments with mixtures offermionic atoms and dimers.
Chapter 2 gives a basic introduction to important theoretical ingredients of the experimentalist’s toolbox, which one needs to perform measurements such as the ones presented in thisthesis.
In Chapter 3 we report on the observation of repulsive and attractive polarons in our LiK experiment. Such quasiparticles are the main building blocks of Fermi-liquid theory.
We identify these quasiparticles by recording the energy spectrum of K impurities in a
Li Fermi sea using radio-frequency spectroscopy near the 155-G Li-K Feshbach resonance.
Additionally, we determine the lifetime of the repulsive polarons in our system. We find thequasiparticles to be remarkably long-lived, with a lifetime exceeding 200 µs even deep in thestrongly interacting regime. Driving Rabi oscillations between the strongly interacting anda non-interacting state of the K atom, we determine the residue of the quasiparticles acrossthe resonance.
The results are published in Nature 485, 615 (2012).
In Chapter 4 we outline our results on the investigations of the coherence of K impurities in
a Li Fermi sea across an interspecies Feshbach resonance. We use a spin-echo sequence todetermine the decoherence rate of the impurities from the decay of the contrast of interference
fringes. Combining the spin-echo sequence with an ultrafast interaction-switching technique,using laser light, empowers us to measure decoherence deep in the strongly interacting regime:

1.3. THESIS OVERVIEW

5

while executing the time consuming spin rotations away from resonance (where decoherence
mechanisms are slow) we switch the interactions to the desired strength of the interactions
only for the time between the radio-frequency pulses. On resonance, these measurements
reveal decoherence rates comparable to the inverse of the fastest timescale (the Fermi timeτF = ~/ϵF ) in our experiment.
The results are published in Phys. Rev. Lett. 115, 135302 (2015).
In Chapter 5 we report on our experiments where we probe the coherence of K impurities
in the Li Fermi sea on ultrafast timescales, faster than τF . Our method is a combinationof Ramsey interferometry with the ultrafast interaction switching technique similar to theone of the previous chapter. In the strongly interacting regime, these measurements allow
us to track the dynamics of the coherence of the impurity in the Fermi sea in real time.
On the repulsive/attractive side of the resonance, our measurements reveal the formation ofquantum many-body states, i.e. the build-up of repulsive/attractive polarons. For resonantinteractions we observe revivals of the Ramsey contrast, after a fast initial contrast decrease.
The results will be published soon; A preprint is available at arXiv:1604.07423.
In Chapter 6 we discuss our results on the investigation of the interactions of K atoms
with LiK dimers in the vicinity of our Li-K Feshbach resonance. We apply radio-frequency
spectroscopy to measure transition-peak shifts of K atoms when immersed in the dimer
cloud. We interpret the peak-shifts as a mean-field energy shift of the K atoms in the LiKdimer environment. We find that the interactions change their character as the resonanceis being approached from weakly repulsive, far from resonance, to strongly attractive, close
to resonance. This is in strong contrast to the mass-balanced case, where the atom-dimer
interactions are on the same order as the atom-atom interactions and are therefore expected
to remain repulsive across the entire range investigated. Furthermore, we determine thescattering rate from the widths of the transition peaks, and find a remarkable agreementwith the prediction from our mean-field approach.
The results are published in Phys. Rev. Lett. 112, 075302 (2014).
In Chapter 7 we report on the investigation of the lifetime of bosonic dimers formed in our
6 Li-40 K Fermi-Fermi mixture near a Feshbach resonance. We perform lifetime measurements
using both trapped, high-density as well as expanded, low-density samples after release fromthe trap. The combination of these measurements allows us to discriminate between the
spontaneous and the collisional dimer decay. Furthermore, we determine the dimer decay
due to Li-LiK collisions in a trapped atom-dimer mixture. Our measurements reveal a morethan three-fold (five-fold) decrease in decay due to dimer-dimer (atom-dimer) collisions as
the Feshbach resonance is approached. This observation can largely be explained by theincreased fermionic character of the halo-dimers together with Pauli blocking.
This chapter contains the experimental part of a manuscript in preparation.
Finally, in Chapter 8 the thesis concludes with an short- to medium-term outlook on research
topics that will be addressed with the FeLiKx machine.

Chapter2

Introduction to theExperimentalist’s Toolbox

2.1 Scattering-formalism basics
In this section, we first summarize the basic concepts of elastic scattering of neutral atoms
as it can be found in many textbooks, e.g. in Ref. [Lan81]. Staying close to Ref. [Wal10],
we then consider a very simple case of scattering in a square-well potential, which we thenextend in order to prepare for the discussion of scattering near a so-called narrow Feshbachresonance, as laid-out in more detail in Ref. [Pet13].

2.1.1 Scattering amplitude and cross section
The Hamiltonian of the relative motion of two particles (1 and 2) with momentum p1,2 , mass
m1,2 , and at location r1,2 interacting via a potential V (r1 − r2 ) is given byH=

p2+ V (r).
2µ2.1Here, we used the relative momentum p = ( mµ1 p1 − mµ2 p2 ), the reduced mass µ = m1 m2 /(m1 +m2 ), the relative coordinate r = r1 − r2 , and the interaction potential V (r) with lim V (r) =r→∞
0.
The Schrödinger equation for the relative motion of the particles is 2p+ V (r) ψk (r) = Ek ψk (r),2µ2.2

and we are now looking for solutions (eigenfunctions) with (eigen)energies Ek = ~2 k 2 /(2µ)
that have the following formψk (r) ∝ eikz + f (k, θ, φ)
7

eikr
r

2.3

8

2.1. SCATTERING-FORMALISM BASICS

far away from the scattering event (i.e. for large |r|, where V (r) → 0). Such a collisional
state is a superposition of two waves: The first part is the incoming plane wave, which, forsimplicity, we assume to be propagating along the z axis. The second part, including theso-called scattering amplitude f (k, θ, φ), is the scattered wave. The scattering amplitude
f (k, θ, φ) generally depends on the energy of the collision ∝ k 2 and the polar (θ) and theazimuthal (φ) angles between incoming and outgoing wave.
Considering only spherically symmetric potentials, V (r) = V (r), the problem is simplified
significantly. Expanding ψk (r) in terms of its radial and angular parts Rl (k, r) and Ylm (θ, φ)(the spherical harmonic functions), respectively, we obtainψk (r) =∞ XlXRl (k, r)Ylm (θ, φ).

2.4

l=0 m=−l2Then, we separate p2 in Eq. (2.2) into a radial and angular part, p2 → p2r + Lr2 with
∂
the radial and angular momentum operator pr = −i~ ∂r+ 1r and L, respectively. Exploiting L2 Ylm (θ, φ) = ~2 l(l + 1)Ylm (θ, φ) we obtain the equation for the radial part of the
wavefunction
 2


~
d2
2 d
l(l + 1)
− 2−+
+ V (r) Rl (k, r) = ERl (k, r).
2.5
2µ
dr
r dr
r2
Furthermore, for V (r) = V (r), the scattering problem becomes invariant under rotation
around the z axis, which means that the scattering amplitude in Eq. (2.3) becomes independent of φ, i.e. f (k, θ, φ) → f (k, θ) and ψk (r) = ψk (r, θ, φ) → ψk (r, θ). Applying thesesubstitutions together with the partial-wave expansion of the plane wave eikz to Eq. (2.3)
yields the so-called partial-wave expansion of the scattering amplitude. The details of thederivation are outlined (e.g.) in reference [Dal98] and here we only give the result,∞f (k, θ) =1 X
(2l + 1)(e2iδl − 1)Pl (cos θ),
2ik

2.6

l=0

where δl denotes the scattering phase shift and Pl are the Lengendre polynomials. This
expression holds for scattering of distinguishable atoms. For identical fermions or bosons
the summation is only over all odd, l = 1, 3, 5, . . . (p, f, h, . . . waves), or even, l = 0, 2, 4, . . .
(s, d, g, . . . waves), respectively.
In ultracold atomic gases, at ultra-low temperatures T , typically atoms can not overcome
the potential barrier Ecb = ~2 l(l + 1)/[2µr2 ] ≫ kB T (kB is the Boltzmann constant) in acollision and will just be reflected by it. It is often said that higher partial-wave interactions
are “frozen out”. Therefore, the scattering in the potential V (r) will go to zero for l > 0 andEq. (2.6), in this case, reduces to the s-wave (l = 0) scattering amplitude [Lan81]
f (k, θ) = f0 (k) =e2iδ(k) − 1
1
=
,
2ik
k cot δ(k) − ik

2.7

where δ(k) is the momentum-dependent s-wave scattering phase shift, which fully describes
the scattering process.

2.1. SCATTERING-FORMALISM BASICS

9dσ
From the scattering amplitude one can derive the (differential) cross section σ ( dΩ)dσ
= |f |2dΩ

Iand σ =

dσ
dΩ,
dΩ
4π

2.8

where dΩ is the infinitesimal solid angle. For the case of pure s-wave scattering of two
distinguishable atoms, the cross section is given by
σl=0 = 4π|f0 (k)|2 .

Another very useful expression for the cross section is provided by the optical theorem[Lan81], which relates σ to the imaginary part of the forward scattering amplitude Imf (k, 0):σ=

4πImf (k, 0).
k

2.9

2.1.2 Scattering in a spherical square-well potential
In this section we investigate the scattering of two particles in a spherical square-well potential at low energy (k → 0). Staying close to the description in Ref. [Wal10], we give an
intuition for the tuning of the interaction strength (rather s-wave scattering length a) by
varying the depth of a square-well potential and, thereby, the energetic detuning of a boundstate from the collisional threshold of the two colliding particles.
We can simplify Eq. (2.5) to a 1D Schrödinger equation if we substitute Rl (r) by χ(r)/r− χ′′ (r) + V (r)χ(r) = Eχ(r).

2.10

Let us now assume V (r) to be the following square-well potential of range r0
V (r) =(
~2 κ2Umin = − 2µ00

for r < r0
for r > r0

,

2.11

where Umin = −~κ20 /[2µ] is the depth of the well. Outside p
and inside the potential well,
the solution of Eq. (2.10) is χout ∝ sin(kr + δ) and χin ∝ sin( κ20 + k 2 r), respectively. The
phase shift δ is obtained from the boundary condition to the solutions, continuity of the
wavefunction and its derivative at r = r0 ,qq
χ′out (r0 )/χout (r0 ) = χ′in (r0 )/χin (r0 ) → k cot(kr0 + δ) = κ20 + k 2 cot( κ20 + k 2 r0 ). 2.12

Because it is somewhat instructive, we solve Eq. (2.12) for the scattering phase shift δ and
get


kr0
p
 .
δ = −kr0 + arctan  p2
k 2 + κ0 r0 cot
k 2 + κ20 r0102.1. SCATTERING-FORMALISM BASICS/ @2energy/ @1

0

7h2 520
27
0

1

2

3

4

r/r0

Figure 2.1: Radial wavefunction χ1 and χ2 within and outside the range r0 of the potential of depth~2 κ20~2 k 22µ . The collision energy 2µ is near zero and shown as the horizontal dashed line.

The first term −kr0 is just a background contribution to the phase shift, linear in k. The
second term, however, shows the tunability of the phase shift by varying the potential depth∝ κ20 : While the collision energy at ultralow temperatures is typically small compared to theof thedepth of the potential wells hosting many bound states, k 2 ≪ κ20 , the contributionpsecond term to δ is small. However, increasing the potential depth such that κ20 + k 2 r0 ≈(n + 21 )π, i.e. the potential turns deep enough to host another bound state, the second term
can become infinitely large.
Doing some calculus we obtain the cotangent of the scattering phase shift δp pk tan (kr0 ) tan
κ20 + k 2 r0 + κ20 + k 2
 pp
cot δ =.
k tan
κ20 + k 2 r0 − κ20 + k 2 tan (kr)

To obtain information about the momentum dependence of scattering in our model potential,
we Taylor expand k cot δ(k) for low momenta k → 0 up to second order in k. This procedure
is called the effective-range expansion. Its details, for this simple potential, are presented inRef. [Wal10] and here we just give the result:tan(κ0 r0 )2
3
3 r0 −+ κ0 r0 
κ0
1
1 2 
+
2.13
k cot δ = −
k
r
1
−

2  ,
0
0 r0 )
2tan(κ0 r0 )2
r0 − tan(κ3κ0 r0 r0 −κ0κ0or, equivalently,
1 1
k cot δ(k) = − + k 2 reff ,a 2
where we used the definition of the s-wave scattering length aa ≡ − lim

1

k→0 k cot δ(k)= r0 −tan(κ0 r0 )
,
κ0

2.14

2.15

2.1. SCATTERING-FORMALISM BASICS

11

and introduced the effective range as
3a + κ20 r03
reff = r0 × 1 −3κ20 r0 a2

2.16

for the square-well potential.
Let us discuss Eq. (2.14) by rewriting it using a(k) = −[k cot δ(k)]−1 ,a(k) =a.
1 − 21 reff k 2 a

2.17

From this equation we see, that the correction of the second term in the denominator tothe momentum-dependent scattering length a(k) becomes important only for k 2 & 1/|reff a|.
Hence, the effective range reff can be viewed as a measure for the momentum dependenceof scattering in a given scattering potential. Also we see from the effective range expansion,that the effect of momentum dependence on the scattering length gets stronger for largera.
However, for large a ≫ r0 , the effective range of the square-well potential (Eq. (2.16)) always
yieldsa≫r
reff → 0 r0 .
That means, the model does not allow an effective range different than r0 for diverging a.
This important fact disqualifies the square-well potential as a model potential to catch thescattering physics in the vicinity of a so-called narrow Feshbach resonance giving rise to astrong momentum dependence of the scattering amplitude, where typically reff ≫ r0 . To
properly simulate the latter, one can extend the simple square-well potential by a deltafunction barrier at r = r0 , as we will see in the next section.

2.1.3 Scattering in a spherical square-well potential with a delta-function
barrier
Staying close to the description of Dmitry Petrov in Ref. [Pet13], we extend our previous
square-well model potential by a delta-function barrier to mimic a weak coupling between
the short-range and long-range parts of the wavefunction(~2 κ2Umin = − 2µ0 for r < r0
V (r) = gδ(r − r0 ) +.
2.18
0for r > r0
This potential is shown in Fig. 2.2. We now proceed analogously to the preceding section to
extract the s-wave scattering length a and the effective range reff from the effective range
expansion.
The boundary condition for the solution of the 1D Schrödinger equation, Eq. (2.10), for asquare-well potential with a delta-function barrier is given byqq22
k cot(kr0 + δ) − κ0 + k cot( κ20 + k 2 r0 ) = g,
2.19

12

2.1. SCATTERING-FORMALISM BASICS

energy

g

/ @2

/ @1
0

7h2 520
27
0

1

2

3

4

r/r0

Figure 2.2: Radial wavefunctions χ1 and χ2 and the potential V (r) with the delta-function barrier
2 2of height g. The collision energy ~2µk is shown as the horizontal dashed line.

and we obtain in analogy to the previous section
p
p
k 2 + κ20 + tan
k 2 + κ20 r0 [g + k tan (kr0 )]
p

h
p
 p
i.
cot δ(k) =k tan
k 2 + κ20 r0 − tan (kr0 ) g tan
k 2 + κ20 r0 + k 2 + κ20

2.20

The Taylor expansion of k cot δ for small momenta k up to second order only has terms oforder 0 and 2. As we compare to Eq. (2.15), the 0th-order term can be identified as −1/a,yielding
1.
2.21
a = r0 −g + κ0 cot (κ0 r0 )In Figure 2.3 we show the dependence of the scattering length a on the depth of the potential
well. By varying the depth of the well we can tune the scattering length to a resonance,
whenever a bound state crosses zero energy. Weakening the coupling between the short-range
and the long-range part of the wavefunction by introducing the delta-function barrier g > 0,
we narrow down the resonance. To illustrate this effect, we compare the case of scatteringin a potential without barrier (g = 0) to the case of scattering in a potential with a barrierof height gr0 = 10.
From the 2nd-order term of the expansion of Eq. (2.20) we obtain the effective range
reff = r0!

3r0 a 1 + ag + ag 2 r0 − gr02 (1 + 2ag − gr0 ) + r04 κ20
1−.
3a2 κ20 r02

2.22

Eq. (2.21) and (2.22) agree with Eq. (2.15) and (2.16), respectively, for g → 0, as to be
expected.
As stated in the previous section, the effect of reff on the scattering amplitude is importantwhen |k 2 | ≥ 1/|reff a|. For these large values of a the effective range (Eq. (2.22)) can be well

2.2. TUNING INTERACTIONS NEAR A FESHBACH RESONANCE

g=0

a(r0 )

4

2

0

0

-2

-2

-4
0

1

gr0 = 10422

well depth (50 r0 =:)

13

-4
0

1

2

well depth (50 r0 =:)Figure 2.3: Scattering length versus depth of the scattering potential. Left: Scattering in a potential without a delta-function barrier. Right: Scattering in a potential with a delta-function barrier
decoupling the short- and the long-range solutions for r < r0 and r ≫ r0 , respectively. The figure is
reproduced from Ref. [Pet13].

approximated by its value for a → ∞ [Pet04a]g (1 + gr0 )a→∞reff → −2R∗ = r0 × 1 −,κ20 r0where we introduced the positive length parameter R∗ [Pet04a]. With this, the effectiverange expansion, Eq. (2.14), can be written in another common form,1
k cot δ(k) = − − R∗ k 2 + . . . .
a

2.2 Tuning interactions near a Feshbach resonanceIn the field of ultracold quantum gases, Feshbach resonances [Fes58, Fes62, Fan61] are acommonly used tool to tune interactions between atoms. The concept of Feshbach resonances
is discussed in detail in the review [Chi10]. Here, we only provide an intuitive picture, whichis then used to introduce the quantities relevant for interaction tuning near a magnetic
Feshbach resonance.
A magnetic s-wave Feshbach resonance is a resonance of the s-wave scattering length a ina collision of two atoms. This resonance appears when a magnetic field B is tuned across aresonance value B0 , where the the scattering state is strongly affected due to the presenceof a bound state the colliding atoms couple to. Figure 2.4 illustrates the scenario using two
molecular potential curves. Consider two atoms with a small relative momentum k → 0
2 2
colliding in the energetically open channel Vbg (r), Vbg (r → ∞) < E = ~2µk → 0. As

2.2. TUNING INTERACTIONS NEAR A FESHBACH RESONANCE

E
A

0
Vbg (r)

7bg B
7cl B
magnetic field

Vcl (r)a(B)energyEclenergy14rmagnetic field

Figure 2.4: Simplified illustration of interaction tuning close to a Feshbach resonance. The left panel
shows the open and the close channel as well as a bound state detuned by an energy Ecl far awayfrom the collisional energy threshold at zero energy E = 0. The top right panel shows the shift ofthe collisional threshold and of the bound state in the presence of a magnetic field B. At B = B0
the bound state reaches the threshold and a Feshbach resonance occurs. The lower right panel shows
the behavior of the s-wave scattering a length around the Feshbach resonance at B0 (dashed verticalline) given by Eq. (2.23).

discussed in the preceding section, Vbg will impose a phase shift to the solution for the
scattering wavefunction, giving rise to the background scattering length abg . Varying themagnetic field B, shifts Vbg along the energy axis by µbg B, where µbg is the magnetic
moment of the atoms. The background scattering length abg remains essentially constant,as the shape of Vbg is hardly affected by the magnetic field.
The potential curve Vcl (Vcl (r → ∞) > E) in Fig. 2.4 may correspond to the molecularpotential of the two atoms in a different (spin) state representing the closed channel. Thispotential hosts a bound state at energy Ecl (B) < 0 and shifts in presence of a magnetic field
B by µcl B, where µcl is the magnetic moment of the closed channel. Typically µcl and µbg
are not identical, instead one can define the differential magnetic moment δµ = µcl − µbg .
Therefore, assuming δµ > 0, the bound state at Ecl (B) < 0 shifts towards the threshold forthe two atoms with zero kinetic energy at E = 0 in the presence of a magnetic field B. Then,a finite coupling between the open and the closed channel will strongly affect the solution
for the scattering wavefunction and, especially, it will cause a resonance for the scattering
length a at B = B0 , where the bound state reaches the threshold (cf. Sec. 2.1). Aroundsuch a Feshbach resonance the dependence of the scattering length a on magnetic field B,
as derived by Moerdijk et al. [Moe95], is given by the simple expression∆Ba(B) = abg 1 −,2.23
B − B0where ∆B and B0 are the width and the center of the Feshbach resonance.
A coupling between the open and the closed channel will, indeed, as well affect the solutionof the bound-state’s wavefunction and therefore its energy Eb (B) ̸= Ecl around the Feshbacha (1000a0 )2.2. TUNING INTERACTIONS NEAR A FESHBACH RESONANCE

15

2
0
-2

Eb (h#kHz)

0

-200

-400
-200

-100

0

100

200

B ! B0 (mG)

Figure 2.5: Scattering length and dimer binding energy around a Feshbach resonance with R∗ =2650 a0 , ∆B = 0.88 G and abg = 63.0 a0 corresponding to the parameters of the 155-G Feshbach resonance between Li|1⟩ and K|3⟩. The upper graph shows the scattering length according to Eq. (2.23).
The lower graph shows the molecular binding energy Eb from Eq. (2.25) as the solid black line. The
solid red line corresponds to the linear extrapolation of the binding energy ∝ δµ at large detuningsR∗ ≫ a. The solid blue line shows the universal binding energy ∝ a−2 .

resonance. The binding energy Eb (B) can be derived from the poles of the scattering amplitude, as pointed out, for example, in Ref. [Pet04a]. Let us first express the binding energy in
2 2
terms of κ, such that Eb (B) = ~2µκ , where µ is the reduced mass of the two colliding atoms.
We then apply the expansion in Eq. (2.14) to Eq. (2.7), set k → iκ and require1− R∗ κ2 − κ = 0,a2.24

with the range parameter R∗ = ~2 (2µδµabg ∆B)−1 [Pet04a]. Solving the quadratic equation,
Eq. (2.24), for κ, we obtain the binding energy~2
Eb (B) = −
8µR∗2

!2
4R∗
1+
−1 .
a(B)

s

2.25

Note that in the model of Ref. [Pet04a] the two free atoms are assumed to be non-interacting
other than through the short-range coupling to the closed channel and the magnetic-field∆dependence of the scattering length a in the above equation follows a(B) → abg B−B.
0
In Fig. 2.5 we plot the scattering length and the binding energy of the Feshbach molecule inthe vicinity of the Li|1⟩-K|3⟩ Feshbach resonance at B0 ≈ 155 G, respectively. The latter isfully characterized by abg = 63.0 a0 , ∆B = 0.88 G, and R∗ = 2650 a0 (a0 = 5.29 × 10−11 mis the Bohr radius). For small detunings B − B0 from the resonance, where a ≫ R∗ , the~2binding energy is quadratic in 1/a, Eb (B) = − 2µa(B)2 , shown as the blue line in the Fig. 2.5.
Here, the radial wavefunction of the molecule is ∝ exp(−κr) = exp(−r/a) and the meaninter-atomic separation is hence also order a, largely exceeding the range of the molecularpotential [K0̈6]. Due to this fact, these dimers are also called halo dimers or universal
dimers and their wavefunction is essentially independent of the molecular potential details.

16

2.3. DESCRIPTION OF THE MEAN-FIELD ENERGY𝑛𝜓𝑖𝑚1 ℏ𝑘1

𝜓𝑡
ℏ𝑘1
𝑚2

𝑑
𝑧

Figure 2.6: A particle of mass m1 crosses a thin slice filled with scatterers of mass m2 . The initialplane wave ψi propagates along the z-axis. It crosses a slice filled with homogeneously distributed
scatterers with a number density n. The transmitted wave ψt . The figure is adopted and modified
from Ref. [Dal98].
2~
The magnetic field region, where Eb is well approximated by − 2µa(B)2 is called the universal
region.

At large detunings, where |R∗ | ≫ a(B), the dependence of the binding energy of the scat~2
tering length changes to Eb (B) ≈ 2µa(B)R∗ and becomes linear, with a slope given by thedifferential magnetic moment δµ = µcl − µbg between the uncoupled closed channel and thebackground channel. This linear extrapolation of the binding energy is shown as the redsolid line in Fig. 2.5. Such dimers are called closed-channel dimers. As we see from thediscussion, Eq. (2.25) nicely interpolates between these two regimes.

2.3 Description of the mean-field energy
In general, the interactions of a particle with a cloud of distinguishable particles, which wewill call scatterers in the following, is a many-body problem. The idea of the mean-field
theory is, to reduce this many-body to a one-body problem, by treating all interactions of
the particle with the scatterers as an effective interaction, or a field the particle is travelingthrough. In a regime of weak particle-scatterer interactions, i.e. for n|a|3 ≪ 1, where n is
the density of the scatterers and a the particle-scatterer scattering length, this can be done.
We want to follow the refractive-index approach of Ref. [Dal98] to derive the (mean-field)energy of a particle scattering in a dilute cloud of scatterers. This approach is based on
the analogy of this physics to scattering of long-wavelength light in a dilute medium and isstrongly leaned on the corresponding description in Ref. [Jac99].
Let us consider a particle of mass m1 traveling along the z-axis with momentum ~k1 througha thin slice of thickness d filled with scatterers. This scenario is illustrated in Fig. 2.6. The
scatterers of mass m2 shall be homogeneously distributed, described by the constant number
density n. Furthermore the scatterers are distinguishable from the incident particle. For
simplicity, we will assume that the scatterers remain at rest (~k2 = 0) and that they are
sufficiently dilute, such that one can treat each collision between incident particle and any one

2.3. DESCRIPTION OF THE MEAN-FIELD ENERGY

17

scatterer independently. For low momenta ~k1 → 0, this is usually written as the conditionn|a|3 ≪ 1.
The incident state is a plane wave ψi (z) ∝ eik1 z . The transmitted wave is a superpositionof the incident wave and of all the scattered waves (similar to Eq. (2.3)) resulting fromcollisions.
Zf (k) ik|r−rs | ik1 (z+zs ) 3
ik1 z
2
ψt (z) = e
+n
e
e
d rs
2.26
slice |r − rs |The integration is done over the entire slice rs = [xs , ys , zs ] and we used the relative momentum ~k = ~ mµ1 k1 , where µ = m1 m2 /(m1 + m2 ) is the reduced mass. This exactly solves toψt (z) = A exp(ik1 z) [Dal98], with
A=1+i

2πnd
2πnd
f (k, 0) ≈ ei k f (k,0) ,k

2.27

where f (k, 0) is the forward-scattering amplitude (Eq. (2.6)) and where the latter is correct
to first order in d.
It is very insightful to separate the forward-scattering amplitude into its real part, Ref (k, 0),
and imaginary part, Imf (k, 0). Furthermore using the optical theorem (Eq. (2.9)), σ =[4π/k]Imf (k, 0), we obtain for the transmitted wave
12πndψt (z) = e− 2 ndσ ei k Ref (k,0) ψi (z).

2.28

Hence, the transmitted wave ψt (z) has a decreased amplitude and a shifted phase withrespect to the incident wave ψi (z), described by the imaginary and the real part of theforward-scattering amplitude, respectively.
We can interpret these observations in a rather straight-forward way. The forward-propagating
wave amplitude is attenuated by elastic collisions with cross section σ causing the incident
wave to be scattered out of the forward direction. The phase shift, on the other hand, arises
from the momentum of the particle being changed by δk1 = 2πnk Ref (k, 0) inside the slice.
2π~2 n
And the change of kinetic energy µ Ref (k, 0) of the particle inside the slice can then be
related to a mean-field potential
U =−2π~2 nRef (k, 0),µ2.29that the scatterers pose for the particle. To summarize, while the particle is in the slice filledwith scatterers, its energy is changed ∝ nRef (k, 0) and its amplitude of forward propagation
is diminished ∝ exp(nσ) after it passed the slice. In Chapter 6, we use these findings
to interpret our experimental results on the atom-dimer interactions obtained by radiofrequency spectroscopy. Since these atom-dimer interactions have strong contributions fromhigher partial waves, l > 0, we stress explicitly the dependence of the mean-field energy
and elastic scattering cross section on the forward-scattering amplitude including all partial
waves.
However, let us reduce these findings to the ~k → 0-regime of pure s-wave scattering tosimplify the following discussion. Here, the s-wave scattering amplitude is f (k, 0) → (−a−1 −182.4. ELASTIC ATOM-DIMER INTERACTIONS NEAR A FESHBACH RESONANCE
2ik)−1 , and Eq. (2.29) reduces to the well-known expression Us = 2π~µ n a. In the following
we discuss two interesting limits within the regime, where n|a|3 ≪ 1 is fulfilled: strong(ka ≈ 1) scattering in a dilute gas (n1/3 ≪ k) and weak (ka ≪ 1) scattering in a dense gas(n1/3 ≫ k).
Particles traversing a dilute gas (n1/3 ≪ k) impinge one scatterer at a time. If one was to calculate a macroscopic cross section of the entire medium it was just the single-scatterer crosssection times the number of scatterers. In such media, under strong-scattering conditions, i.e.
ka ≈ 1, the total amount of wave amplitude scattered sideways can be large. The describedsituation is similar to visible light entering the atmosphere, which is described by Rayleighscattering [Jac99]. Here, electromagnetic waves scatter from the electric dipoles of moleculesin the atmosphere and the scattering cross section is ∝ k 4 . Therefore, short-wavelength lightis scattered most and the sky appears blue.
Considering weak scattering (ka ≪ 1) in a dense medium, where n1/3 ≫ k, the scattering iscollective. To describe this our approach from above would need to be adapted by allowing
the incident particle to scatter from several scatterers and essentially simultaneously. But, we
can learn already from the analog of light scattering in a dense medium, such as visible lightscattering in a dense gas or in water. Here, the light dominantly propagates into the forward
direction, which can be understood from the interference of scattered waves originating fromnearby scatterers, which is dominantly destructive (constructive) into the sideways (forward)
direction [Hec02]. Interestingly, a higher density of the medium increases the suppression ofside-ways scattering in the regime of a long-wavelength particle probing a dense medium.
We have now discussed a few examples of a particle scattering in a medium in the weaklyinteracting regime, where the mean-field approach is valid. In several Chapters of thisthesis we go beyond this mean-field regime. In Chapter 3 we present measurements of theinteraction energy of particles in a dense Fermi gas in the strongly interacting regime. In
the strongly interacting regime the particle essentially interacts with the entire Fermi sea
at once and the Fermi sea can be excited in terms of particle-hole excitations. The particle
including the excitations propagates as a quasiparticle termed polaron through the Fermisea, dramatically different from the physics described in this chapter. The coherence of thesequasiparticles is discussed in Chapter 4 where we also identify its collisional properties.

2.4 Elastic atom-dimer interactions near a Feshbach resonanceIn the previous sections we have introduced the formalism of ultracold scattering of atoms aswell as the tunability of the interactions between atoms in trapped atomic clouds by meansof a magnetic Feshbach resonance. We have shown that, on the repulsive side of a Feshbach resonance, there always exists an energetically lower-lying dimer state, into which aninteracting atom-atom mixture can decay. This decay, which will be discussed in Chapter 3
for the case of a Li-K mixture, limits the lifetime of the mixture. Experimenting with an
repulsively interacting Fermi-Fermi mixture on timescales on the order of or larger than thislifetime, knowledge of the interactions of the dimers, i.e. the decay products, with the atoms
as well as between the dimers is crucial. While the inelastic processes will be discussed in2.4. ELASTIC ATOM-DIMER INTERACTIONS NEAR A FESHBACH RESONANCE19

1.5

add =aaad =a

3

2

1

0

5

10

1

0.5

1

5

10

M=m

M=m

Figure 2.7: Atom-dimer s-wave scattering
length aad versus mass imbalance of the dimer
components for a ≫ r0 . Replotted data from
Ref. [Pet03].

Figure 2.8: Dimer-dimer s-wave scattering
length add versus mass imbalance of the dimer
components for a ≫ r0 . Replotted data from
Ref. [Pet05].

detail in Chapter 7, the purpose of this section is to summarize the elastic interactions forthe case of a Fermi-Fermi mixture, which are partially also discussed in Chapter 6.
For a theoretical description of atom-dimer (dimer-dimer) scattering in fermionic mixtures,
one has to solve a three-body (four-body) problem. This can be rather easily done for thecase of a scattering length a largely exceeding the range of the interatomic potential, a ≫ r0 .
Here, the radial wavefunction of the weakly bound, open-channel (Feshbach) dimer state isgiven by [K0̈6]
 r
1
exp −
for r ≫ r0 .
2.30
φ(r) = √a
2πaThis is the so-called “halo-dimer” state, where the wavefunction of the dimer is characterized
purely by the atom-atom s-wave scattering length a. The spatial extend of the halo-dimeris extremely large, on the order of a. In this regime, the three- and four-body problems ofelastic atom-dimer and dimer-dimer collisions, respectively, are exactly solvable [Pet04b] andthe amplitudes of elastic s-wave interactions between atoms and dimers and among dimers
depend solely on a.
The elastic atom-dimer interactions in the s-wave channel are characterized by an atomdimer scattering length aad . The details of its calculation are presented in Ref. [Pet03]. In
Figure 2.7, we show the behavior of aad as a function of the mass imbalance M/m for M > m,re-plotted from Ref. [Pet03]. The s-wave atom-dimer interactions are always repulsive andbecome stronger for an increasing mass imbalance. Whereas in homo-nuclear Fermi mixturesaad ≈ 1.2 a, it reaches approximately 2 a for mK /mLi .
Analogously, the elastic s-wave dimer-dimer interaction are characterized by the dimer-dimer
scattering length add . The details of its derivation are given in Refs. [Pet04b] and [Pet05]. In
Figure 2.8 we re-plot the results from these calculations published in Ref. [Pet05]. Similarly
to atom-dimer scattering, the s-wave interaction is always repulsive and strengthens withmass imbalance, being add = 0.6 a for dimers composed of equal mass fermions and 0.9 a
for the case of the Li-K mixture. Without going into detail, we mention, that the presented

202.4. ELASTIC ATOM-DIMER INTERACTIONS NEAR A FESHBACH RESONANCE
theoretical results for aad , and add were all obtained in the a/R∗ = 0 limit, i.e. for thewide-resonance case. The effect of a finite R∗ on the atom-dimer and dimer scattering
process is discussed in Ref. [Lev11]. A considerable value for R∗ decreases the open-channel
fraction of the dimer wave function and effectively weakens the atom-dimer and dimer-dimer
interactions, yielding lower values for aad , and add , respectively.
The discussed l = 0 contribution to the atom-dimer interaction is, however, not at all
sufficient to describe the scattering process. Under typical experimental conditions, also
when atom-atom scattering for l > 0 is suppressed, higher partial-wave contributions to
atom-dimer scattering can be strong and even dominate the interactions. This important fact
is the subject of the publication presented in Chapter 6. A very complete summary for atomdimer interactions, especially for our Li-K Fermi-Fermi mixture, is given in reference [Lev11].
Here, we recall its most important findings, and to prepare the ground for Chapter 6, wegive an intuition for the angular-momentum dependence in atom-dimer collisions.
Let us consider a heavy-heavy-light three-fermion system. While for large separations r,
atom and dimer can be described by separate wavefunctions, in an atom-dimer collision (i.e.
for r → 0) we have to solve the three-body Schrödinger equation to retrieve the (three-body)
wavefunction describing the system. A good qualitative understanding, however, can alreadybe gained from considering the problem in the Born-Oppenheimer approximation. Applying
the latter, we can write the wavefunction as the product of a wavefunction φ(r) describing
the two heavy fermions of mass M separated by r and a light-atom (mass m) wavefunction
ψ, and assume that the latter adiabatically adjusts itself to the distance r between the heavy
fermions. The light-fermion wavefunction ψ is a superposition of the wavefunction of thelight fermion bound to either of the heavy ones, ψ1 and ψ2 , respectively. This superposition
can be symmetric or antisymmetric, ψ = ψ1 ± ψ2 , as depicted in Fig. 2.9. For r → 0,
the curvature of the anti-symmetric light atom wavefunction ψ1 − ψ2 (ψ1 + ψ2 ) is increased(decreased) with respect to the far separated system, giving rise to an increased (decreased)energy of the respective state. In this picture, the symmetry of the collision of the atomwith the dimer (say l = 0 or 1, i.e. s or p wave) is projected onto the two-heavy-atom
wavefunction.
Finally, the total wavefunction ∝ ψφ(r) must be restricted to be antisymmetric with respect
to the permutation of the two heavy fermions. This gives rise to an angular-momentum
dependent interaction: If the atom-dimer collision occurs in an even (odd) partial wave,
i.e. s, d, . . . (p, f, . . . ), the heavy-fermion wavefunction is symmetric (anti-symmetric) andthe light-atom is forced into the anti-symmetric (symmetric) state. In the symmetric (antisymmetric) state the potential U+ (U− ) arising from the light-fermion exchange is positive(negative), giving rise to a repulsive (an attractive) interaction.
The exact quantitative analysis [Lev11] shows that U+ ∝ −1/[mr2 ]. For l > 0, this attractive exchange potential competes with the centrifugal barrier Ucb ∝ l(l + 1)~2 /M r2 . With
increasing mass imbalance M/m the net attraction U− /Ucb increases and is strongest in thepartial wave l = 1. In Fig. 2.10 we show the p-wave atom-dimer potentials Vl=1 = U− + Ucb
for the mass imbalance of the Li-K system (M/m = 6.7) as well as for larger mass imbalances. For a mass ratio M/m → 8.2 the potential Vl=1 develops a well deep enough to hosta three-body bound state [Kar07], and when M/m ≥ 13.6, the potential becomes overall

2.4. ELASTIC ATOM-DIMER INTERACTIONS NEAR A FESHBACH RESONANCE21

𝜓1 − 𝜓2

𝜓1

𝜓2

𝜓1 + 𝜓2heavy fermion
light fermion
probability distrib.

Figure 2.9: Scenario of an atom-dimer collision. When atom and dimer are separated, r → ∞,the light-atom wavefunction is ψ1 or ψ2 , respectively, corresponding to the light atom being boundto either of the heavy ones. As atom and dimer approach each other, the wavefunction is thesuperposition ψ = ψ1 ± ψ2 , that can be symmetric or anti-symmetric. Depending on the symmetry of
the collision, the light atom is forced in either of the states at lowered (ψ1 + ψ2 ) or increased (ψ1 − ψ2 )energy, respectively.

0.5

2ma # Vl=1

c.b.

Li-K

0

M
m = 8:2-0.5
0Mm = 13:6

1

2
r=a

3

4

Figure 2.10: Atom-dimer p-wave potentials for various mass imbalances and for R∗ ≪ a. The dottedline is the p-wave centrifugal barrier ∝ m−2 . The solid, dashed, and dashed-dotted lines correspond to
the p-wave potentials for the Li-K system (M/m = 6.7), M/m = 8.2, and M/m = 13.6, respectively.
The data is collected and replotted from Ref. [Lev11].

attractive ∝ −r−2 , giving rise to an infinite number of bound states leading to the Efimov
effect [Efi70]. All the curves shown correspond to R∗ = 0. A large R∗ effectively weakens
the atom-dimer exchange interactions and, therefore, has a similar effect as a decreased mass
ratio [Lev11].
In Fig. 2.11 we show the phase shifts as a function of collision energy in units of dimer binding
energy Ecoll /Eb in the three lowest partial waves for the Li-K system, where M/m ≈ 6.7,
as the solid lines. For collision energies Ecoll ≈ 0.2Eb the p-wave attraction dominates
the scattering process, such that summing over all partial waves the net interactions are
attractive. Also for mass-balanced systems, M = m, the contribution of p-wave interaction
to the total scattering cross section is not negligible, as the phase shifts for the mass-balanced
case show (dashed lines in Fig. 2.11). The collisional phase shifts acquired in the p-wave
channel are still on the order of ∼ 10% of the s-wave phase shift in the vicinity of a Feshbach

222.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

/l =:

0.5

0

-0.5
0

0.5
Ecoll =Eb

1

Figure 2.11: The phase shifts acquired in an atom-dimer collision in the three lowest partial waves
for R∗ = 0. The black, red, and green lines correspond to the phase shift in an s-, p-, and d-wave
collision, respectively. Solid lines are for the Li-K mass imbalance, whereas the dashed lines are forthe mass-balanced case. The data is replotted from Ref. [Lev11].

resonance, where the collision energy Ecoll can become a significant fraction of the dimer
binding energy Eb .

2.5 Basics of radio-frequency spectroscopy in our Li-K system

In every publication presented in this thesis we use radio-frequency (rf) pulses to manipulate
the hyper-fine states of either the lithium-6 or potassium-40 atoms, respectively. We will
therefore use this section to review the most essential findings from treating the problem
of a two-level system coupling to an rf photon. This will allow us to introduce the methodof rf spectroscopy, a tool used every day in our lab to, e.g., determine magnetic fields orinteraction shifts of transition lines. We will conclude this section giving an example for ameasurement of a mean-field energy shift of K atoms immersed in a Li cloud.
In Figures 2.12 and 2.13 we show the hyper-fine manifolds of the ground states of 6 Li and
40 K, respectively, for magnetic fields B up to 200 G. We label these states counting themwith rising energy. The Feshbach resonance we exploit for interaction tuning is located at
B0 ≈ 155 G and occurs between potassium in the third-to-lowest state K|3⟩ and lithium in
its lowest hyper-fine state Li|1⟩ (red in Fig. 2.13 and 2.12, respectively). The neighboring
hyper-fine states, i.e. Li|2⟩ and K|2⟩ and K|4⟩ (blue in Fig. 2.13 and 2.12, respectively) are
separated by E/h ≈ 68 MHz from Li|1⟩ and E/h ≈ 39 MHz and E/h ≈ 41 MHz from K|3⟩,
respectively. Note that these radio frequencies correspond to wavelength of a few meters andthe momentum transferred to an atom in a absorption process is negligible. Thus, these rf
transitions can be interpreted as pure spin-flip operations.

100

-400

0

-500

j8i

-600

j6i

E/h (MHz)E/h (MHz)2.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

-100
j3i

-200
0

-300
-400

j2i

-1000
0

0

500

100

j7i
j5i

-700

j4i

0

j3i
j2i
j1i

-800 -1000

j1i

1000

-2000

200

-900

23

0

500

0

1000

100

200

B/G

B/G

Figure 2.12: Hyper-fine structure of theground-state manifolds of Li versus magnetic
field B. The inset shows the hyperfine structure of the 6 Li over a larger magnetic field
range.

Figure 2.13: Hyper-fine structure of the F =9/2 manifold of the 40 K ground-state versus
magnetic field B. The inset extends over alarger magnetic field range.

2.5.1 Rabi-flopping: two-level system without interactions
The setup to discuss the so-called Rabi problem is illustrated in Figure 2.14(a). In our Li-K
system, such a two-level system is realized, e.g., by identifying the “ground state” |0⟩ as the
K atom residing in state K|2⟩, and the “excited state” |1⟩ as the K atom residing in K|3⟩. We
can couple these states by rf photons of a frequency ν ≈ ν0 , near-resonant to the hyper-finesplitting hν0 between the states K|2⟩ and K|3⟩ at a magnetic field B, and off-resonant to allother neighboring hyper-fine states.
Measuring all energies relative to the energy of the |0⟩ state, the Hamiltonian, describing thesystem, is given by
H = H0 + Vrf =




~Ω
0
e−i2πνt
hν0 0
,
+
0
0 0
2 ei2πνt2.31

where H0 describes the unperturbed atom, Vrf is the atom-light interaction term. Furthermore, Ω = d01~·E is the Rabi frequency, where d01 is the magnetic dipole matrix elementand E is the electric field vector. Then we look for the solution of the Schrödinger equationdψ = Hψ that has the form ψ(t) = c0 (t)|0⟩ + c1 (t)ei2πνt |1⟩. Using the rotating-wave
i~ dt
approximation (assuming δ = ν − ν0 ≪ ν and Ω ≪ ν) we end up with
d
i~
dt

 

c1
2πδ
= ~ Ω i2πδtc02eΩ −i2πδt
2e

0

 
c1
.
c0

2.32

It is very interesting to discuss the solution of these coupled equations for the case, where
one state is initially unoccupied. The occupation probabilities of the states |1⟩ and |0⟩ aregiven by |c1 (t)|2 and |c0 (t)|2 , respectively. Assuming that an atomic sample was initially, at
t = 0, prepared in the |0⟩ state, we can derive the occupation of the |1⟩ state versus time t,

24

2.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

(b)

(a)

energy/ℎ

𝜈

𝑃 |1

𝛿

|1

𝜈0

𝑡 (arb. u. )|0Figure 2.14: Two-level system coupling to radio-frequency photons of frequency ν. (a) Illustrationof the ground and excited states |0⟩ and |1⟩, respectively, separated by an energy hν0 that are coupledby radio-frequency photons of frequency ν with a detuning δ = ν − ν0 . (b) Oscillations of the excitedstate population in the presence of coupling of the |0⟩ and |1⟩ by a radio-frequency ν = ν0 +δ for δ = 0,Ω/[2π], and 2Ω/[2π] represented by the solid black, dashed blue, and dotted red line, respectively.

during which the sample is in a field of radio-frequency photons,Ω2
P|1⟩ (t) = |c1 (t)| = 2 × sin2
Ωeff2

where Ωeff =




Ωeff
t ,
2

2.33

p
Ω2 + (2πδ)2 .

In Figure 2.14(b) we show examples of these oscillations of the excited-state population
P|1⟩ (t) for three different detunings δ, as described by Eq. (2.33). These oscillations occur
at a frequency, the effective Rabi frequency Ωeff , which has the lowest value for δ = 0, whereΩeff = Ω. A full population transfer from the |0⟩ to the |1⟩ state can only be accomplishedfor δ = 0. The shortest possible resonant, δ = 0, pulse flipping the entire (half of the)population of the atoms from one state to the other has a duration of π/Ω (π/[2Ω]) and iscalled a π (π/2) pulse.
We typically determine our magnetic field by measuring the Zeeman energy ~ω between two
hyperfine states at a high magnetic field by rf spectroscopy and using the Breit-Rabi formula[Bre31]. We prepare a sample of several thousand K|2⟩ atoms at a magnetic field B andapply an rf pulse with a given intensity and a duration corresponding to less or equal thanthe π-pulse duration. We perform this experiment for several radio frequencies ν around thehyper-fine transition frequency ν0 to the state K|3⟩ to determine the frequency of maximum
transfer. Applying the Breit-Rabi formula [Bre31] we then determine the magnetic fieldcorresponding to the measured hyper-fine splitting hν0 between the states K|2⟩ and K|3⟩.

2.5.2 Two-level system with one (weakly) interacting state - measuring themean-field energy
In this subsection, we extend the two-level system of the previous subsection by allowing thestate |1⟩ to be an interacting state. We consider only interactions between K|3⟩ atoms and2.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

(a)

(b)

𝜈MF

K|3

𝜈

𝜈0

25

𝜈

𝜈𝑖

K|2

without Li|1

with Li|1

Figure 2.15: Sketch of the interaction-shift of the excited state |1⟩. (a) In the absence of interactions,with zero population of the Li|1⟩ state, the K|2⟩ and K|3⟩ levels are separated by an energy hν0corresponding to the Zeeman splitting between these two states. (b) When the K atoms reside in
a Li|1⟩ environment, which causes a mean field for the interacting K|3⟩ atoms, the excited stateenergy shifts. Comparing the transition frequencies νi to ν0 , we can determine the mean-field energy
EMF = hνMF = h(νi − ν0 )co-trapped Li|1⟩ atoms. The Li|1⟩-K|3⟩ interactions can be tuned by means of the Feshbachresonance at a magnetic field of about 155 G, as introduced earlier.
In the magnetic field region of at least ±2 G around the center of this Feshbach resonance,interactions in all neighboring spins-state combinations are very weak, described by respective small background scattering lengths. For the combinations of Li|1⟩-K|2⟩, Li|1⟩-K|4⟩, and
Li|2⟩-K|3⟩ these background scattering lengths abg are all around 65 a0 [Nai11]. This property of the Li-K system is turns out to be extremely valuable, as it is one of these states that
we use as the initial (final) state from (to) which we transfer, when performing spectroscopy,
and it allows us to interpret any changes in the spectral response, as we tune the magnetic
field near the Li|1⟩-K|3⟩ Feshbach resonance, to arise from Li|1⟩-K|3⟩ interactions. This is instrong contrast to, e.g., the widely used Li spin-state mixture, where the high-field Feshbachresonances between any combination of the lowest three spin states strongly overlap andinteractions are generally strong (> 1000 a0 ) [Zür13, Chi10].
In our lab, we frequently use rf spectroscopy to measure interaction shifts of hyperfine transitions. Therefore, we give a few more details about how we perform such measurements andwhat needs to be considered, when analyzing and interpreting spectroscopy data. We do thisby considering an example under typical experimental conditions. To simplify the discussion,
we assume a very low temperature and a rather large detuning from resonance. Here, theforward scattering amplitude Ref (k) is well approximated by −a and the interactions aretreated within mean-field theory.
For the case of a weakly interacting Li|1⟩-K|3⟩ mixture (|a3 nLi | ≪ 1), the energy of a K|3⟩2
atom propagating through a Li|1⟩ atom cloud shifts by EMF = 2π~µnLi a (cf. Eq. (2.29)).
Here, nLi is the Li number density and µ = mLi mK /[mLi + mK ] is the reduced mass. The
radio frequency for driving transitions from K|2⟩ to K|3⟩ (or vice versa) will then be shiftedby EMF /h (h is Planck’s constant). This is illustrated in Figure 2.15. Generally, comparing
the transition frequencies in the absence (e.g. when Li is in the state Li|2⟩), Fig. 2.15(a),and presence, Fig. 2.15(b) of interactions, ν0 and νi , respectively, one directly retrieves the26

2.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

nLi (r) and nK (r)1nK (r)nLi (r)

0.5

0
-30

-20

-10

0

10

20

30

distance from trap center r (7m)Figure 2.16: Normalized lithium and potassium in-trap number densities as a function of the radial
distance r from the trap center. We show the number densities of a central cut, perpendicular to
the axial direction of our cigar-shaped atom clouds. To obtain the data, we assumed 104 K atoms
in thermal equilibrium with a degenerate gas of 3 · 105 Li atoms at a temperature T = 400 nK in
an optical dipole trap. The optical confinement is characterized by the radial (r) and axial (a) trap
frequencies of Li and K, νLi,r = 600 Hz, νLi,a = 75 Hz, νK,r = 330 Hz, and νK,a = 42 Hz, respectively.

mean-field energy shift as EMF = hνMF = h(νi − ν0 ).
In the experiment, we perform rf spectroscopy of an ensemble of K atoms, whose numberdensity distribution in the trap is described by nK (r). These K atoms are trapped together
with a Li cloud (number-density distribution nLi (r)) in an optical dipole trap. This meansthat the K atoms experience an inhomogeneous Li number distribution, nLi → nLi (r) depending on their position r in the trap. In Figure 2.16 we show the K (red solid line) andLi (blue solid line) number densities of a central cut, perpendicular to the axial direction ofa cigar-shaped cloud, as a function of the distance r from the trap center and for typicalexperimental conditions1 .
Applying a hypothetical rf π pulse of infinite duration at a frequency ν, the rf response, i.e.
the fraction of K atoms transferred into the K|3⟩ state, is given byZ
A(ν) ∝trapnK (r)δ (ν − ν0 − νMF ) d3 r =~nLi (r)a d3 r.
nK (r)δ ν − ν0 −
µ
trap

Z

2.34

This response is shown as the red crosses in Figure 2.17 for the same conditions as in Fig. 2.16
and for a ≈ −1000 a0 . The low-frequency onset at approximately −2.3 kHz originates fromatoms in the trap center at high Li density. The spectral tail reaches up to ν − ν0 = 0
and stems from atoms in the wings of the thermal distribution ofR the K atoms. From thespectral response we can derive the average mean-field shift ν̄MF = nK (r) ~nLiµ (r) a d3 r, whichcorresponds to the first moment of A(ν). For our example the average mean-field shift isν̄MF = 1.79 kHz, shown as the vertical black dashed line in Fig. 2.17.
1

We assumed a thermal distribution of 104 K atoms at a temperature T = 400 nK mixed and thermalizedwith a degenerate Li gas of 3 · 105 atoms confined in an optical trap, characterized by radial (r) and axial(a) trap frequencies of νLi,r = 600 Hz, νLi,a = 75 Hz, νK,r = 330 Hz, and νK,a = 42 Hz, respectively.

fraction transferred (arb. u.)

2.5. BASICS OF RADIO-FREQUENCY SPECTROSCOPY IN OUR LI-K SYSTEM

1

27

MF
MF (convolved)
MF (fourier)

0.5

0
-5

-4

-3

-2

-1

0

1

2

8 ! 80 (kHz)

Figure 2.17: Calculated spectral response in our system in a regime, where the mean-field approximation is valid. The red crosses correspond to the spectral response of our system to the applicationof an infinitely long rf pulse, whereas the blue solid line is obtained using an rf pulse of 1 kHz Gaussian
width. Both spectra have the same first moment of about ν̄MF = ν − ν0 = 1.79 kHz (black dashedvertical line). The black dotted line corresponds to a Gaussian (1 kHz width) centered at ν̄MF .

In a real experiment, the rf-pulse duration is finite. This causes a Fourier broadening of
the spectral response, which is inversely proportional to the duration. Indeed, an rf pulse√
with a Gaussian intensity envelope of 1/ e duration τ (full-width-half-maximum (FWHM)
1
duration τ1/2 ) translates into a Gaussian shaped response with a width of ∆ν = 2πτ
(FWHM
2 ln 2of ∆ν1/2 = πτ ) in the frequency domain. Applying an rf pulse of a finite duration, we
1/2
therefore retrieve a spectrum AG (ν) that is the convolution of the spectral response A(ν)(Eq. (2.34)) with such a Gaussian,Z′ 2
1 (ν−ν )AG (ν) ∝ A(ν ′ )e− 2 ∆ν 2 dν ′ .
2.35

The spectrum AG (ν) obtained with an rf pulse of duration τ = 160 µs is plotted in Figure 2.17
as the blue solid line. It is centered at the frequency ν̄MF and has a width only slightly larger
than the Fourier width, in this case 1 kHz. This can be seen by comparing AG (ν) to thespectral shape of the 160-µs rf pulse centered at ν̄MF , shown as the black dotted line in
Fig. 2.17. Only for the case that the Fourier width of the rf pulse, ∆ν, is much larger
than the width of the spectral response A(ν), one can determine the average mean-fieldenergy shift in our inhomogeneous Li-K system by observing the shift of the peak-transfer
frequency.
Without getting into detail, we mention, that there are further effects leading to broadeningof spectral lines, besides broadening due to inhomogeneity and Fourier broadening arisingfrom the finite-length of the rf-pulse. The interacting state can have a finite lifetime causing
Lorentzian broadening of the transition line due to several processes: Most prominently,
these are decay of the interactring state into a third (other hyper-fine or molecular) state at
a rate 1/τd or momentum-changing scattering of K atoms with the Li environment at a rate
1/τs . These processes essentially gives rise to an “effective lifetime” τ = (1/τs + 1/τd )−1 ofthe interacting state, causing Lorentzian broadening of the line with a FWHM of (2πτ )−1 .

Chapter3

Publication: Metastability and
Coherence of Repulsive Polarons
in a Strongly Interacting Fermi
Mixture1

Nature 485, 615 (2012)
submitted 26 November 2011, accepted 9 March 2012
DOI: 10.1038/nature11065

C. Kohstall1,2 , M. Zaccanti1 , M. Jag1,2 , A. Trenkwalder1 , P. Massignan3 , G. M. Bruun4 , F.
Schreck1 , and R. Grimm1,2
1Institut für Quantenoptik und Quanteninformation (IQOQI), Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
ICFO - Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona),
Spain
4
Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark1

The author of the present thesis performed the measurements and analyzed the data together with C.K.,
M.Z., and A.T. under the supervision of F.S. and R.G. and contributed to writing the manuscript.

29

30

3.1. ABSTRACT

3.1 Abstract

Ultracold Fermi gases with tunable interactions provide a test bed for exploring the manybody physics of strongly interacting quantum systems [Blo08, Gio08, Rad10, Che10]. Overthe past decade, experiments have investigated many intriguing phenomena, and precise
measurements of groundstate properties have provided benchmarks for the development oftheoretical descriptions. Metastable states in Fermi gases with strong repulsive interactions[Dui05, LeB09, Con09, Jo09, Pil10, Cha11, San12] represent an exciting area of development.
The realization of such systems is challenging, because a strong repulsive interaction in an
atomic quantum gas implies the existence of a weakly bound molecular state, which makes
the system intrinsically unstable against decay. Here we use radio-frequency spectroscopy tomeasure the complete excitation spectrum of fermionic 40 K impurities resonantly interactingwith a Fermi sea of 6 Li atoms. In particular, we show that a well-defined quasiparticle
exists for strongly repulsive interactions. We measure the energy and the lifetime of this‘repulsive polaron’ [Pil10, Mas11, Sch11], and probe its coherence properties by measuring
the quasiparticle residue. The results are well described by a theoretical approach that takes
into account the finite effective range of the interaction in our system. We find that whenthe effective range is of the order of the interparticle spacing, there is a substantial increasein the lifetime of the quasiparticles. The existence of such a long-lived, metastable manybody state offers intriguing prospects for the creation of exotic quantum phases in ultracold,
repulsively interacting Fermi gases.

3.2 Introduction

Landau’s theory of Fermi liquids [Lan56], and the underlying concept of quasiparticles, iscentral to our understanding of interacting Fermi systems over a wide range of energy scales,
including liquid helium-3, electrons in metals, atomic nuclei and quark–gluon plasma. In
the field of ultracold Fermi gases, the normal (non-superfluid) phase of a strongly interacting system can be interpreted in terms of a Fermi liquid [Lob06, Sch09, Nav10, Nas11]. In
the population-imbalanced case, quasiparticles known as Fermi polarons are the essentialbuilding blocks and have been studied in detail experimentally [Sch09] for attractive interactions. Recent theoretical work [Pil10, Mas11, Sch11] has suggested a novel quasiparticle
associated with repulsive interactions. The properties of this repulsive polaron are of fundamental importance to the prospects of repulsive many-body states. A crucial question
for the feasibility of future experiments is the stability against decay into molecular excitations [Pek11, Mas11, San12]. Indeed, whenever a strongly repulsive interaction is realized bymeans of a magnetically tuned Feshbach resonance [Chi10] (FR), a weakly bound molecularstate is present into which the system may rapidly decay.

3.3. MAIN RESULTS

0.5

31

Li

E+
K |1〉

E/HF

0

Em

RF

E–

Li

–0.5

Em – HF

K |0〉

–1
–2

–1

0
–1/NFa1

2

Figure 3.1: Energy spectrum of an impurity in the Fermi sea. The energies, E+ (red line) andE− (green line), of the two polaronic branches are plotted as functions of the interaction parameter,−1/κF a. The shaded area between the dashed lines representing Em and Em − ϵF (see text) shows
the continuum of molecular excitations. The vertical lines at 1/κF a = ±1 indicate the width of
the strongly interacting regime. The inset illustrates our radio-frequency (RF) spectroscopic scheme
whereby the impurity is transferred from a noninteracting spin state, |0⟩, to the interacting state, |1⟩.

3.3 Main results
Our system consists of impurities of fermionic 40 K atoms immersed in a large Fermi sea of 6 Li
atoms, which is characterized by an effective Fermi energy of ϵF = h × 37 kHz and a temperature of T = 0.16 ϵF /kB (Methods), where h and kB respectively denote the Planck and the
Boltzmann constants. For a particular combination of spin states [Nai11] (quantum numbers
F = 9/2, mF = −5/2 for 40 K and F = 1/2, mF = +1/2 for 6 Li), the mixture has a FR centred at B0 = 154.719 G (Methods). The FR enables us to tune widely the s-wave interaction,
parameterized by the scattering length, a, using a magnetic field, B. The√ interaction strength−1
is described by the dimensionless parameter −1/κF a, where κF = ~2mLi ϵF = 1/2, 850a0
is the Fermi wavenumber. Here ~ = h/2π, a0 is the Bohr radius and mLi is the mass of a
6 Li atom. Near the centre of the FR, the linear approximation −1/κ a ≈ (B − B )/20 mG
0F
holds. The momentum dependence of the interaction is characterized by the effective range,
which we express in terms of the parameter R∗ as defined in Ref. [Pet04a] (Methods andSupplementary Information). For our relatively narrow FR, R∗ = 2, 700a0 and the corresponding value of κF R∗ = 0.95 indicates that the finite effective range will have an important
influence on the interaction with the Fermi sea.
In Fig. 3.1, we illustrate the basic physics of our impurity problem in the T = 0 limit, showing
the energies of different states as functions of the interaction parameter. The situation is
generic for any impurity in a Fermi sea, but quantitative details depend on both the mass
ratio of the two atomic species and the width of the FR. The theoretical curves are based

32

3.3. MAIN RESULTS

on a generalization of an approach presented in Refs. [Pun09, Mas11] to our case of a finite
effective range.
The spectrum has two quasiparticle branches, which do not adiabatically connect when the
FR is crossed, and a molecule–hole continuum (MHC), indicated by the shaded area in
Fig. 3.1. The interaction-induced energy shifts of the two branches (E+ > 0 and E− < 0)are generally described in a many-body picture by dressing the impurities with particle–holeexcitations. Far away from the FR centre, this simplifies to a mean-field shift proportional
to a. The lower branch of the system (E− ; green line) corresponds to the attractive polaron,
which has recently received a great deal of attention theoretically [Che10, Lob06, Com07,
Pun09, Sad11] as well as experimentally [Nas09, Sch09, Nav10]. This polaronic branch remains the ground state of the system until a critical interaction strength is reached, where thesystem energetically prefers to form a bosonic 6 Li40 K molecule by binding the 40 K impurity
to a 6 Li atom from the Fermi sea [Pro08, Pun09, Sad11]. The MHC arises from the fact thatan atom with an energy between 0 and ϵF can be removed from the Fermi sea to form themolecule. This continuum thus exists in an energy range between Em and Em − ϵF (Fig. 3.1,
dashed lines), where Em is the energy of a dressed molecule including the binding energy ofa bare molecule in vacuum and a positive interaction shift. The attractive polaron can decay
into a molecular excitation if this channel opens up energetically (E− ≥ Em − ϵF ).
The upper branch (Fig. 3.1, red line) corresponds to the repulsive polaron [Pil10, Mas11,
Sch11] with an energy E+ > 0. Approaching the FR from the a > 0 side, E+ gradually
increases and reaches a sizeable fraction of ϵF . However, the polaronic state becomes increasingly unstable as it decays to the lower-lying states (attractive polaron and MHC).
Close to the FR centre, the repulsive polaronic state becomes ill-defined as the decay rate
approaches E+ /~.
To investigate the excitation spectrum of the impurities, we use radio-frequency spectroscopy[Chi04, Shi07, Ste08]. We initially prepare the 40 K atoms in a non-interacting spin state,|0⟩ ≡ |F = 9/2, mF = −7/2⟩ and then, with a variable frequency, νrf , drive radio-frequency
transitions into the resonantly interacting state |1⟩ ≡ |F = 9/2, mF = −5/2⟩. Our signal is
the fraction of atoms transferred, measured as a function of the radio-frequency detuning,νrf − ν0 , with respect to the unperturbed transition frequency, ν0 , between the two spinstates. This excitation scheme provides access to the full energy spectrum of the system.
In particular, it allows us to probe the metastable repulsive polaron as well as all statesin the MHC. We furthermore take advantage of the coherence of the excitation process bydriving Rabi oscillations. This is an important practical advantage, because it allows very
fast and efficient transfer of population into a short-lived quasiparticle state by applicationof π-pulses. Moreover, we find that measurements of the Rabi frequency directly reveal
quasiparticle properties (see below).
In Fig. 3.2, we show false-colour plots of our signal as detected for different values of thedetuning parameter, ∆ = h(νrf − ν0 ), and for variable interaction strength, −1/κF a. Figure 3.2a displays a set of measurements that we optimized for the signal strength and spectral
resolution of the polaronic excitations by using moderate radio-frequency power. The two
insets show the polaron peaks on top of a background due to additional excitations in the
Fermi sea (Supplementary Information). The spectrum in Fig. 3.2b was optimized for detec-

0.5

Δ /εF

0

0.6

–1/κFa = 2

Signal

a

0
–0.3
0.6

0
0

0.1

–2

b

Δ /εF

0

0.1

–1/κFa = −0.8Signal

–0.5

–0.2

Δ /εF

0.3

0.4

–1

0
−1/κF a

1

2

1

0.5

Δ /εF

0

1

–0.5

–1

Signal

–1.5

0
–2

–1

0
−1/κF a1

2

Figure 3.2: Spectral response of 40 K impurities in a 6 Li Fermi sea. The false-colour plotsshow the fraction of 40 K atoms transferred from the noninteracting spin state, |0⟩, to the interacting
state, |1⟩, for different values of the radio-frequency detuning parameter, ∆ = h(νrf − ν0 ), and forvariable interaction strength, −1/κF a: low radio-frequency power (a); high radio-frequency power(b). For comparison, the lines correspond to the theoretical predictions for E+ , E− , Em , and Em − ϵFas shown in Fig. 3.1. In a, the two insets show the signals for −1/κF a = −0.8 and 2, respectively,
corresponding to vertical cuts through the signal data.

33

34

3.3. MAIN RESULTS

10–2

1

–1/κF a = −1.45

Signal10–1

ħ Γ /εF

–0.6
0
10–3

10–4

0

–1.05

1
2
Hold time (ms)

–2

–1

0

−1/κF aFigure 3.3: Decay rate of the repulsive polaron. The data points display the measured decay
rates, Γ, as extracted by exponential fits to decay curves; the error bars indicate the fit uncertainties.
Sample decay curves are shown in the inset. The solid lines represent theoretical calculations of thetwo-body decay into the attractive polaron (blue line) and the three-body decay into the MHC (redline).

tion of the molecular excitations. Here we had to use a much higher radio-frequency power(greater than that in Fig. 3.2a by a factor of 100) because of the reduced Franck–Condon
wavefunction overlap. For the polaronic branches, the high radio-frequency power leads tohighly nonlinear saturation behaviour.
Our data show both polaronic branches, and the measured energies of the branches are inexcellent agreement with theory. The attractive polaron is found to disappear in the stronglyinteracting regime. This behaviour, which is different from that observed in 6 Li spin mixtures[Sch09], is consistent with the crossing of E− and Em − ϵF at −1/κF a ≈ +0.6 as we expectfor our system. By contrast, the repulsive polaron extends far into the strongly interacting
regime. The spectrum has a sharp peak that fades out near −1/κF a ≃ −0.3 (Supplementary
Information). The low radio-frequency power produces only a weak MHC signal (Fig. 3.2a),whereas for high radio-frequency power the MHC signal is strong (Fig. 3.2b). For weaker
interactions on the a > 0 side of the FR (−1/κF a < −1), the molecular signal decreases
because of the reduced Franck–Condon overlap. Outside the strongly interacting regime, thesituation corresponds to the radio-frequency association of bare molecules (Supplementary
Information).
To investigate the decay of the repulsive branch, we apply a radio frequency pulse sequence
to convert repulsive polarons in state |1⟩ back into non-interacting impurities in state |0⟩
after a variable hold time (Methods). This back-conversion depends sensitively on the radiofrequency resonance condition and thus allows us to discriminate 40 K atoms in the polaronic
state from those forming molecules. In Fig. 3.3, we present the experimental results. The3.3. MAIN RESULTS

35

inset shows three sample curves taken for different values of the interaction parameter.
The main panel displays the values extracted for the decay rate, Γ, from the decay curves
using exponential fits. The data reveal a pronounced increase in decay as the FR centre isapproached, which is in good agreement with theoretical model calculations [Mas11] (Fig. 3.3,
solid lines; Supplementary Information). The decay populates the MHC and may occur
in a two-step process whereby the repulsive polaron decays via a two-body process into
an attractive polaron (Fig. 3.3, blue line) that in turn decays into a molecular excitation.
Alternatively, the repulsive polaron may decay directly into the MHC in a three-body process(Fig. 3.3, red line). Very close to the FR centre, for −1/κF a = −0.25, we find that ~Γ/ϵF ≈0.01, which corresponds to a 1/e lifetime of about 400 µs. By comparing this decay ratewith the corresponding energy shift, E+ = 0.30 ϵF , we obtain ~Γ/E+ ≈ 0.03 ≪ 1, which
demonstrates that the repulsive polaron exists as a well-resolved, metastable quasiparticle
even deep in the strongly interacting regime.
The lifetime observed for the repulsive branch is remarkably long, when compared withfindings in recent experiments on 6 Li spin mixtures[San12]. The latter mass-balanced system
features a broad FR with a negligible effective range (R∗ → 0). Our theoretical approachallows us to give a general answer to the question of how mass imbalance and the widthof the FR influence the lifetime. We find that the mass imbalance has only a minor role
[Mas11] and that the dominant effect results from the finite effective range. In the strongly
repulsive regime, our system allows us to obtain the same amount of repulsive interactionenergy as would a hypothetical system with a broad FR (R∗ → 0), but with an almost
ten-fold increased lifetime (Supplementary Information).
Apart from energy and lifetime, the polaron is characterized by its effective mass, m∗ , and its
quasiparticle residue, Z (0 ≤ Z ≤ 1). The difference between m∗ and the bare mass [Mas11]
does not produce any significant features in our radio-frequency spectra. The residue quantifies how much of the non-interacting
particle is contained in the polaron’s wavefunction,√
which can √be written as Z |1⟩ plus terms describing excitations in the Fermi sea. The
pre-factor Z directly manifests itself in the Rabi frequency, Ω, that describes the coherent
radio-frequency coupling between the non-interacting state and the polaronic state (Supplementary Information).
In Fig. 3.4, we show the experimental data on Rabi oscillations for variable interaction
strength. The sample curves in Fig. 3.4a demonstrate both the interaction-induced change inthe frequency and a damping effect. We apply a simple harmonic oscillator model (includinga small increasing background) to analyse the curves, which yields the damping rate, γ,
and the frequency, Ω. The damping strongly increases close to the FR centre, but does notshow any significant dependence on the unperturbed Rabi frequency,Ω0 (Fig. 3.4b). We notethat the population decay rates, Γ, measured for the repulsive branch (Fig. 3.3) stay well
below the values of γ, which suggests that collision-induced decoherence is the main damping
mechanism.
Figure 3.4c shows the measured values for the Rabi frequency normalized to Ω0 . Theinteraction-induced reduction of Ω/Ω0 is found to be independent of the particular valueof Ω0 (comparison√ of blue squares and red dots; see also Supplementary Information). The
solid lines show Z as calculated within our theoretical approach for both the repulsive po-

a

1.0

Signal

–0.5

0.5
–1.25
REF

b

10–1

ħ γ/εF

0.0
0.0

10–2

0.1
Pulse length (ms)

0.2

10–3

Ω / Ω0

c

1.0

0.8

0.6

0.4
–3

–2

–1

0

1

2

3

−1/κF aFigure 3.4: Rabi oscillations and the quasiparticle residue. a, Sample Rabi oscillations (for−1/κF a = −1.25 (magenta points) and −1/κF a = −0.5 (green points)) with harmonic oscillator
fits (solid lines) demonstrate the two effects of the interaction with the Fermi sea: damping and areduction of the Rabi frequency. The black curve (REF) is a reference curve recorded without 6 Li.
b, c, Damping rates, γ (b), and the normalized Rabi frequencies, Ω/Ω0 (c), as measured for two
different values of the radio-frequency power: Ω0 = 2π × 6.5 kHz (blue) and 2π × 12.6 kHz (red).
√ The
error bars indicate the fit uncertainties. The solid lines represent the theoretical behaviour of Z forthe repulsive polaron (left) and the attractive polaron (right).

36

3.3. MAIN RESULTS

37

laron and the attractive polaron. The comparison
with the experimental data demonstrates√
a remarkable agreement with the relation Z = Ω/Ω0 . Our results therefore suggest thatmeasuring the Rabi frequency is a precise and robust way to determine the quasiparticle
residue, Z, and thus provides a powerful alternative to methods based on the detection of
the narrow quasiparticle peak in the spectral response [Sch09, Pun09].
In general, our set of spectroscopic methods applies to any resonantly interacting spin or
species mixture that can be efficiently radio-frequency-coupled to a weakly interacting one.
Such a situation is the rule rather than the exception in ultracold atomic systems [Chi10]. In
mixtures of different species, we anticipate a large variety of suitable systems. For the massbalanced case of a spin mixture, the well-established 40 K system [Ste08] would be an obvious
choice. We also point out that narrow FRs are more common in the field than broad ones
[Chi10]. Therefore, effects of the finite effective range similar to those described in our work
will govern the behaviour of most systems that can be realized in the laboratory. The benefitof an increased lifetime could help overcome the problem of decay into molecular excitations[Pek11, San12] in the experimental investigation of metastable many-body states that rely on
repulsive interactions, such as phase-separated states of two fermionic components [Dui05,
LeB09, Con09, Jo09, Cha11, San12].

Methods summary
A cloud of 2 × 104 40 K atoms is confined in an optical dipole trap together with a 6 Li Fermi
sea of 3.5 × 105 atoms, at a temperature of T ≈ 290 nK. The K atoms reside in the centreof the much larger Li cloud and thus sample a nearly homogeneous Li environment. The
relevant energy scale is given by the Li Fermi energy averaged over the K density distribution,
ϵF = h × 37 kHz = 1.8 µK.
The FR is fully characterized by the parameters [Nai11] B0 = 154.719 G (centre), ∆B =0.88 G (width), abg = 63.0 a0 (background scattering length), and δµ/h = 2.3 MHz/G (differential magnetic moment). The scattering length can be calculated from the standard
expression [Chi10] a(B) = abg (1 − ∆B/(B − B0 )), and the range parameter [Pet04a] is obtained as R∗ = ~2 /(2mr abg δµ ∆B) = 2700 a0 , where mr is the reduced mass of the K–Li
pair. We note that the common textbook definition of the effective range corresponds tore = −2R∗ , for a → ±∞.
The radio-frequency pulses used in measuring the data in Figs 3.2 and 3.3 were Blackmanshaped to avoid side lobes in the spectrum. The pulses were 1 ms (Fig. 3.2a) or 0.5 ms(Fig. 3.2b) long, and the radio-frequency power was adjusted such that π-pulses (Fig. 3.2a)or 5π-pulses (Fig. 3.2b) were realized in the absence of interactions with the Fermi sea. For
the measurements in Fig. 3.3, a first π-pulse (duration between 150 and 500 µs) was used
to drive the impurity from state |0⟩ into state |1⟩, selectively creating repulsive polarons. A
second, 60-µs, π-pulse transferred the population remaining in state |0⟩ into a third state. A
third pulse, equal to the first one, transferred the state-|1⟩ polarons back into state |0⟩, where
they were finally measured using spin-state-selective absorption imaging. The measurements
shown in Fig. 3.4 were performed with simple square pulses.

38

3.4. METHODS

Acknowledgements

We thank A. Sidorov for contributions in the early stage of the experiments, and T. Enss, S.
Giorgini, W. Ketterle, J. Levinsen, C. Lobo, D. Petrov, A. Recati, R. Schmidt, J. Song, C.
Trefzger, P. Zoller, W. Zwerger, M. Zwierlein, and in particular M. Baranov for many stimulating discussions. We acknowledge support by the Austrian Science Fund FWF throughthe SFB FoQuS. M.Z. is supported within the Lise Meitner program of the FWF. P.M. isindebted to M. Lewenstein for support through the ERC Advanced Grant QUAGATUA.

3.4 Methods
Experimental conditions

Our system consists of 2 × 104 40 K atoms and 3.5 × 105 6 Li atoms confined in an optical
dipole trap. The trap is realized with two crossed beams derived from a 1, 064-nm singlemode laser source. The measured trap frequencies for Li and K are respectively νr = 690 and
425 Hz radially and νz = 86 and 52 Hz axially; this corresponds to a cigar-shaped sample
with an aspect ratio of about eight. The preparation procedure is described in detail in
Ref. [Spi10a].
The Fermi energies, according to the common definition for harmonic traps,p
EF = h 3 6N νr2 νz , are EFLi = h×44 kHz = kB ×2.1 µK and EFK = h×10.4 kHz = kB ×500 nK.
At a temperature of T ≈ 290 nK, the 6 Li component forms a deeply degenerate Fermi sea
Li ≈ 0.14) whereas the 40 K component is moderately degenerate (k T /E K ≈ 0.6).
(kB T /EF
B
F

Effective Fermi energy

The 40 K atoms sample a nearly homogeneous 6 Li environment. This is because the opticaltrapping potential for 40 K is about twice as deep as for 6 Li and the 40 K cloud is confined tothe centre of the much larger 6 Li Fermi sea [Tre11a]. This allows us to describe the systemin terms of the effective Fermi energy, ϵF , defined as the mean Fermi energy experienced by
the 40 K atoms. We find that ϵF = h × 37 kHz, with two effects contributing to the fact that
this value is about 15% less than EFLi . The finite temperature reduces the Li density in thetrap centre, leading to a peak local Fermi energy of h × 40 kHz. Moreover, the 40 K atoms
sample a small region around the trap centre, where the density and local Fermi energy are
somewhat lower than in the centre. The distribution of Fermi energies experienced by the
40 K cloud, that is, the residual inhomogeneity of our system, can be quantified in terms ofa standard deviation of h × 1.9 kHz.

3.4. METHODS

39

Concentration

The mean impurity concentration (mean number density ratio, nK /nLi ) is about 0.4, if bothspin states of the population of K atoms are considered. This may be too large a-priori
to justify the interpretation of our data in terms of the low-concentration limit. We findthat this interpretation is nevertheless valid, as we take advantage of several facts. Under
strongly interacting conditions, only a fraction of the K atoms are transferred into spinstate |1⟩ (Fig. 3.2), which reduces the concentration of interacting impurities. A recent
quantum Monte Carlo calculation of the equation of state of a zero-temperature 6 Li–40 K
Fermi–Fermi mixture [Gez09] further supports our interpretation in the low-concentration
limit: the strongest interaction in the mass-imbalanced system is expected when there areabout 4 times more 40 K atoms than 6 Li atoms, and for concentrations up to a value of 1
the interaction energy per 40 K atom is expected to remain essentially constant. To support
our basic assumption with experimental data, we also measured radio-frequency spectra
for variable numbers of 40 K atoms, confirming that finite concentration effects remained
negligibly small in the relevant parameter range.

Interaction control through Feshbach resonance

The FR used for interaction tuning is discussed in detail in Refs. [Nai11, Tre11a]. It is present
for 6 Li in the lowest spin state (F = 1/2, mF = +1/2) and for 40 K in the third-to-lowest spinstate (F = 9/2, mF = −5/2). The latter represents our interacting state, |1⟩. The neighbouring state with mF = −7/2 serves as state |0⟩; here the interspecies scattering length(+65 a0 ) is so small that any interaction can be neglected to a good approximation. The
tunable scattering length for state |1⟩ in the Fermi sea is well described by the standard formula, a = abg (1 − ∆B/(B − B0 )), with abg = 63.0 a0 , ∆B = 0.88 G, and B0 = 154.719(2) G.
The value given for B0 refers to the particular optical trap used in the present experiments,
as it includes a small shift induced by the trapping light. The value therefore deviates somewhat from the one given in Refs. [Nai11, Tre11a]. In free space, without the light shift, theresonance centre is located at 154.698(5) G. The uncertainties given for the resonance centre,
B0 (2 mG in the trapped case and 5 mG for free space), correspond to standard deviations,obtained from analysing molecule association spectra for various trap settings.
The character of the resonance is closed-channel dominated [Chi10]. Following the definition[Pet04a] of a range parameter, R∗ = ~2 /(2mr abg δµ ∆B), where mr = mLi mK /(mLi + mK )is the reduced mass and δµ/h = 2.3 MHz/G is the differential magnetic moment, the resonance is characterized by R∗ = 2, 700 a0 . For a → ±∞, this parameter corresponds to thecommon textbook definition [Chi10] of the effective range, re = −2R∗ . Our value for R∗
coincidentally lies very close to 1/κF = 2, 850 a0 , which means that the strongly interacting
regime corresponds roughly to the universal range of the resonance. Our system therefore
represents an intermediate case (κF R∗ = 0.95), where the behaviour is near universal, butwith significant effects arising from the finite effective range.

40

3.5. SUPPLEMENTARY INFORMATION

Details on radio-frequency pulses
To measure the data in Fig. 3.2, we used Blackman pulses [Kas92] to avoid side lobes in the
spectrum. For Fig. 3.2a, the pulses were 1-ms long (spectral width, 0.7 kHz ≃ 0.02 ϵF /h) andthe radio-frequency power was adjusted such that π-pulses would be realized in the absenceof interactions with the Fermi sea. For the data in Fig. 3.2b, the radio-frequency power wasincreased by a factor of 100 and the pulse duration was set to 0.5 ms. This resulted in pulses
with an area of 5π without the Fermi sea. For the lifetime measurements in Fig. 3.3, we useda sequence of three Blackman pulses. The first pulse (duration between 150 and 500 ms)
was set to drive the non-interacting impurity from spin state |0⟩ (mF = −7/2) into state |1⟩(mF = −5/2); here the frequency was carefully set to resonantly create repulsive polarons
and the pulse area was set to fulfill the π-pulse condition. The second pulse was a short(60-ms) cleaning pulse, which removed the population remaining in |0⟩ by transferring it toanother, empty, spin state (mF = −9/2). The third pulse had the same parameters as thefirst one and resonantly transferred the population of the polaronic state in |1⟩ back to the
non-interacting state, |0⟩, where it was finally measured by spin-state-selective absorption
imaging. The measurements of Rabi oscillations in Fig. 3.4 were performed with simple
square pulses.

3.5 Supplementary information
3.5.1 Theoretical framework
The theoretical results presented in the main text and in this Supplementary Information
are obtained from a model that describes the behaviour of a single impurity embedded in a
Fermi sea with tuneable s-wave interaction near a Feshbach resonance with arbitrary effective
range. Two different wavefunctions are needed, depending on whether one is interested in thepolaron [Che06, Com07] or molecule [Mor09, Pun09, Com09] properties. The quasiparticle
parameters for the polaron (energy E+ and E− , residue Z, effective mass) and the molecule
properties can be found either variationally, or diagrammatically using the ladder approximation. Both approaches yield identical results, which closely match independent Monte-Carlocalculations [Pro08]. The properties of the repulsive polaron, which is intrinsically unstable
due to the presence of the molecule-hole continuum (MHC) and of the attractive polaron,
are obtained from the self energy. In particular, the interaction induced energy shift andthe decay rate are given by the real part and twice the imaginary part of the self energy,
respectively [Mas11].
Previous treatments [Che06, Com07, Mor09, Pun09, Com09, Pro08, Mas11] were based ona universal scattering amplitude, describing broad Feshbach resonances. To include effectsof the finite effective range we employ a many-body T-matrix given by [Bru05a, Mas08b]−1mr
T (K, ω) =− Π(K, ω).
3.1
2π~2 ã(K, ω)
Here ~K = pK + pLi is the total momentum with pLi and pK the momenta of Li and K,

3.5. SUPPLEMENTARY INFORMATION41

mr = mLi mK /(mLi + mK ) the reduced mass, Π(K, ω) the Li-K
 pair propagator in the pres∆Bence of the Fermi sea, and ã(K, ω) ≡ abg 1 − B−B0 −ECM /δµ an energy-dependent lengthparameter, with abg , ∆B, B0 , and δµ being the background scattering length, the width,
the center, and the relative magnetic moment of the Feshbach resonance. ECM (K, ω) =~ω − ~2 K2 /(2M ) + ϵF , with M = mLi + mK , is the energy in the center of mass reference
frame of the colliding pair. In vacuum and close to resonance, the scattering amplitude ofour model has the usual low energy expansion− fk−1 = a−1 + ik − re k 2 /2 + . . . ,3.2

with the relative momentum ~k = (mLi pK −mK pLi )/M . The effective range is approximatedby re ≈ −2R∗ (1−abg /a)2 , where we introduce the range parameter R∗ = ~2 /(2mr abg ∆Bδµ),
see Ref. [Pet04a].
Within the one particle-hole approximation, the energies E± of the repulsive and attractive
polarons at zero temperature are given by the two solutions of the implicit equation
E± =

p2K
+ Re[Σ(pK , E± )],
2mK

3.3

where the self-energy describing the interactions of the impurity with the bath reads [Com07]
Σ(pK , ω) =X

T

pK + pLi , ω +pLi <kF

p2Li
− ϵF
2mLi


.

3.4

The residue of polarons with vanishing momentum, which we plot in Fig. 3.4c of the main
paper, are given by
"
 #−1

∂Σ(pK = 0, ω).
3.5
Z± = 1 − Re∂ωE±
A more detailed theoretical analysis of this model is given in Ref. [Mas12].

3.5.2 Polaron peak in the spectral response
The spectra in Fig. 3.2a of the main text show a narrow, coherent peak on top of a spectrally
broad, incoherent background. Here, we investigate these two spectral parts in more detail.
Note that the background is actually better visible in Fig. 3.2b, but these spectra do notallow for a quantitative comparison of the two parts because of the strong saturation of thenarrow polaron peaks.
The narrow peak stems from the attractive or repulsive polarons, which correspond to welldefined energy levels, provided that the lifetime of the quasiparticle exceeds the pulse duration. As a consequence, the lineshape is expected to be Fourier limited except for the rapidly
decaying repulsive polarons very close to resonance. In contrast, the background is spectrally
wide, on the order of εF . The main contribution to the background stems from the MHC.
Another contribution may arise from the excitation of additional particle-hole pairs in the42

3.5. SUPPLEMENTARY INFORMATION
a

−2
−1
)
0 κ aF
/(

−1

2
3

b

0
∆/εF

−0.5

0.5

0

Signal1
1heightof narrow peak

0.5

norm. areaof background

1

0.5

c

d

0
1

0

0

∆

peak

/ε

F

0.2

−0.2
−7 −6 −5 −4 −3 −2 −1

0

1

2

3

4

5

6

−1/(κF a)

Figure 3.5: Double Gauss analysis of the low-power spectra. The data are the same as presented
in Fig. 3.2a plus additional data in an extended range of 1/(κF a). (a) The Gauss function fitting
the wide background is shaded grey. The Fourier-limited Gauss function, fitting the narrow peak, iscoloured red (green) along the repulsive (attractive) polaron branch. We identify the narrow peakwith one of the polaron branches only if its maximum signal exceeds a threshold value of 0.085,
corresponding to two times the standard deviation of the noise in our data. Any smaller peak may
be caused by fitting to a noise component. The lower panels show (b) the maximum signal of thenarrow peak with the dashed line indicating the threshold, (c) the area under the wide Gauss function
normalized to its maximum value, and (d) the detuning at the center of the narrow peak, provided
that the peak signal exceeds the threshold, compared to the theoretical calculation of E+ and E−(red and green line). The error bars indicate the fit uncertainties.

Fermi sea when transferring to a quasiparticle with a momentum that is different from themomentum of the impurity in the initial state.
We distinguish between the narrow peak and the wide background by means of a double
Gauss fit. Vertical cuts through Fig. 3.2a are presented in Fig. 3.5a together with the fit
curves. The width σp of the Gauss function fitting the narrow peak is fixed to the one
associated with the Gaussian fit of the Blackman pulse line shape used in the experiment,
σp = 0.7 kHz= 0.019 εF /h. We constrain the width σb of the Gauss function reproducing the3.5. SUPPLEMENTARY INFORMATION

43

background to 3×0.019 εF /h < σb < 0.5 εF /h. The lower bound avoids the misinterpretationof the narrow peak as background and the upper bound, corresponding to the maximal widthof the continuum as obtained from the spectra in Fig. 3.2b, avoids unphysically large valuesof σb when the background signal is weak. We find that the narrow peak dominates forweak positive and negative interaction strength while the wide background dominates inthe strongly interacting regime. This trend is shown in Fig. 3.5b and Fig. 3.5c, where we
present the maximum signal of the narrow peak and the area of the background, respectively.
Note, that the signal in Fig. 3.5b is proportional to the area of the narrow peak since σpis kept constant. Figure 3.5d shows the detuning at the center of the narrow peak, whichcorresponds to the energy of the quasiparticles. The measured energies agree remarkably wellwith the calculation. The slight mismatch between theory and experiment may be attributed
to systematic errors in the determination of εF and B0 .
The area of the wide background exhibits a maximum close to −1/(κF a) = 0, but it shows anasymmetry as it falls off significantly slower on the attractive (a < 0) side, see Fig. 3.5c. We
attribute this asymmetry to the narrow character of the Feshbach resonance. The interaction
becomes resonant when the real part of the inverse scattering amplitude, given in Eq. (3.2),
2
is zero. This leads to the resonance condition a−1
res = re k /2, where ares is the value of thescattering length at which the interaction becomes resonant. In the limit of a broad resonancewith re = 0, this condition is fulfilled for any k at the center of the resonance, where thescattering length diverges. However, at a narrow resonance with re < 0 the condition requiresa negative ares for k > 0. The mean square momentum in the Fermi sea is 3/5 × κ2F , leading
to a mean square relative momentum of 3/5 × (40/46 × κF )2 . Using this value for k 2 , andinserting re ≈ −2R∗ in the above resonance condition, we obtain −1/(κF a) = 0.43. This
represents an effective shift of the Feshbach resonance center, as we average over all momentaof the Fermi sea [Ho12]. The magnitude of this shift agrees well with the observed asymmetry.
Moreover, we find that many features at our narrow resonance appear to be shifted, e.g. thepolaron-to-molecule crossing. However, the narrowness has many more implications andcannot simply be reduced to this shift. We will come back to this point in the context of the
lifetime of the repulsive polaron, see Sec. 3.5.4.
The repulsive polaron peak is clearly visible up to −1/(κF a) ≈ −0.3 while the attractive
polaron peak vanishes already at −1/(κF a) ≈ 0.9, see Fig. 3.5b. The fading out of thequasiparticle peak towards the strongly interacting regime approximately coincides with theposition where the quasiparticle branches merge into the MHC. This shows that the polaronstate is hardly observable as soon as it becomes degenerate with molecule-hole excitations.
The MHC is not strictly limited to the range from Em to Em − εF , as discussed in moredetail in Sec. 3.5.3. It extends below Em − εF because of finite temperature effects. It
also extends slightly above Em because of additional excitations in the spectral function of
the molecules [Sch11]. As a consequence, for finite temperature, the attractive polaron canbecome degenerate with molecule-hole excitations for values of the interaction parameter
above the calculated polaron-to-molecule crossing. This explains that the observed sharp
peak is observed to disappear already at −1/(κF a) ≈ 0.9, which lies somewhat above thezero-temperature polaron-to-molecule crossing predicted at 0.6.
It is interesting to consider the data analysis presented in Fig. 3.5b and c in relation to thecommon method of extracting the quasiparticle residue Z from the spectral weight of the44

3.5. SUPPLEMENTARY INFORMATION

1−1/(κ a) = −1.9
F

0.5

0

−2

Signal−1/(κF a) = −1.6

−1

0

∆ /εF

0.5

0

−2

−1/(κ a) = −1.3
F

−1

0

∆ /ε

F

0.5

0

−2

−1

0

∆ /εF

Figure 3.6: Molecule association spectra for different values of the interaction parameter. The signal
is the fraction of transferred atoms as a function of the rf detuning. The data correspond to verticalcuts through Fig. 3.2b. The dashed line is the line shape model for zero temperature and the solidline for finite temperature. The upper threshold of the theoretical spectra corresponds to Em .

narrow peak [Din01]. Close to resonance, we are in the linear response regime and our data
can be interpreted in terms of this method. Our data suggests that this method leads to asignificant underestimation of Z. For example at −1/(κF a) ≈ 0.9, where the narrow peakof the attractive polaron vanishes, our theory still predicts Z ≈ 0.7. This underestimation isconsistent with the one reported in Ref. [Sch09], see also related discussion in Ref. [Pun09].
A plausible explanation may be that such a method does not probe the polaron states alone,
but also the molecule-hole excitations, which are degenerate with the polaron state. Our
alternative method of measuring the residue via the Rabi frequency, as presented in the main
paper, offers the advantage of being much less affected by the molecule-hole contribution.
In fact, only the coherent part of the quasiparticle is expected to produce Rabi oscillations,
see Sec. 3.5.6.

3.5.3 Molecule-hole continuum
The spectra presented in Fig. 3.2b of the main text reveal the MHC. This continuum arises
from processes where the rf field associates a K impurity and a Li atom out of the Fermi sea
to a molecule. Here we present a simple model for the spectral line shape, which allows us
to interpret the data up to −1/(κF a) ≈ −1, see Fig 3.6.

3.5. SUPPLEMENTARY INFORMATION45

For modeling the line shape, we consider two-body processes in which the rf field associates
one K and one Li atom to a molecule. Higher-order processes, involving more than two
particles, are neglected in this model but are briefly discussed at the end of this section. Let
us first consider the association of Li and K with momenta pLi = pK = 0. This results ina molecule at rest plus a Fermi sea with a hole in the center. The energy of this state isdetermined by the binding energy of the molecule and by the interaction of the moleculewith the Fermi sea. It is given by Em and sets the onset of the MHC from the right (thetop) in Fig. 3.6 (Fig. 3.2b). In general, Li and K have finite initial relative momentum ~k,leading to an initial relative kinetic energy in the center of mass frame Er = ~2 k 2 /2mr .
The energy conservation of the association process is expressed in the Dirac δ function in
Eq. (3.6). As a consequence, the molecule spectrum extends downwards to energies below
Em . We now consider an ensemble of K and Li atoms. Our experimental conditions are wellapproximated by a thermal cloud of K in a homogeneous Fermi sea of Li (see Methods).
FD (E ), with
The momentum distribution of Li is given by the Fermi-Dirac distribution fLi
Li2ELi = pLi /2mLi . The one of K is approximated by the Maxwell-Boltzmann distribution
fKMB (EK ), with EK = p2K /2mK . The latter distribution does not change its momentum
dependence with position, thus, no integration over space is needed to obtain the spectral
response
Z Z
FDS(∆) ∝
d3 pLi d3 pK fLi(ELi ) fKMB (EK ) F(k) δ(−Em + Er + ∆),3.6
where F(k) is the Franck Condon overlap of the initial wavefunction with the molecule
wavefunction. In our case the interaction in the initial state is negligible and F(k), as
given in Ref. [Chi05], reduces to F(k) ∝ (Er /Eb3 )1/2 (1 + Er /Eb )−2 . The parameter Eb isthe binding energy of a molecule in vacuum at a resonance with finite
effective range andpreads Eb = ~2 /(2mr a∗2 ) with the parameter [Pet04a] a∗ = −re /( 1 − 2re /a − 1). In thecalculation of F(k), we do not account for interactions with the Fermi sea. Because of thisapproximation, we apply the model only for −1/(κF a) < −1. For fitting the model line
shapes to the experimental data, adjustable parameters are the individual heights of thespectra and the center of the Feshbach resonance. The latter parameter is required to be thesame for all data sets in Fig 3.6. Independently determined parameters are kB T /εF = 0.16
and εF = h × 37 kHz. The model (solid lines) reproduces our data remarkably well. It
allows us to pinpoint the resonance position to B0 = 154.719(2) G. This determination of
B0 relies on our theoretical model to calculate Em . To test this model dependence, we
replace Em simply by the binding energy of the molecule in vacuum plus the mean fieldenergy, considering the corresponding atom-dimer scattering length [Lev11]. Using thissimple model, the fit yields a resonance position that is 1 mG higher, which shows that themodel dependence causes only a small systematic uncertainty. Moreover, the statistical fit
uncertainty and the field calibration uncertainty are about 1 mG each.
For T = 0 and all other parameters unchanged, the model provides the dashed lines in
Fig 3.6. The spectra show a sharp drop at ∆ = Em − (40/46) εF , which corresponds to theassociation of an impurity at rest and a majority atom at the Fermi edge. In an equal-mass
mixture this process would occur at ∆ = Em − (1/2) εF . Thus, the width of the MHC in
the two-body approximation is much larger for a heavy impurity than it is for an equal-mass
impurity and it is even narrower for a light impurity.
The true zero temperature ground state is actually at the energy Em − εF , a molecule at46

3.5. SUPPLEMENTARY INFORMATION

rest formed from a K atom at rest and a Li atom at the Fermi edge. However, to reach thisstate, momentum conservation requires a higher-order process, i.e. the scattering of at least
one additional Li atom from and to the Fermi surface. Such processes are not included in
the model presented here, which only considers the direct association of two atoms by an rf
photon.
In the strongly interacting regime the spectral function of the molecule shows additional
excitations above the molecular ground state [Sch11]. This leads to an extension of the
MHC spectral response above Em , of which we find clear indications in our data. The lower
panel in Fig 3.6 shows finite signal above Em and the extension above Em is very evident in
the strongly interacting regime, see Fig. 3.2b.

3.5.4 Decay rate of the repulsive polarons
We analyze the decay of the repulsive polarons by assuming that they decay into well defined
attractive polarons or well-defined molecules. In this quasiparticle picture, the decay is
associated with the formation of a particle-hole pair in the Fermi sea to take up the released
energy. In this sense, the decay into the attractive polaron is a 2-body process and the decay
into the molecule is a 3-body process. We calculate the decay rate for these two channels
by including them into the polaron self energy using a pole expansion of the K propagator
writing G(k, ω) ≃ Z+ /(~ω − E+ − ~2 k 2 /(2mK )) + Z− /(~ω − E− − ~2 k 2 /(2mK )) and a pole
expansion of the T-matrix writing T (k, ω) ≃ Zm g 2 /(~ω − (Em − εF ) − ~2 k 2 /(2M )). Here,Z± is the quasiparticle residue of the repulsive and attractive polaron respectively
and Zmp
the quasiparticle residue of the molecule. The factor g 2 = 2π~4 /(m2r a∗ 1 − 2re /a) is theresidue of the vacuum T-matrix for a general resonance. The details of this approach are
given in Refs. [Mas11, Bru10], the only difference being that here we include the effectsof the finite effective range. The imaginary part of the self energy gives the decay rate of
the wavefunction and we thus take twice the imaginary part to calculate the population
decay. The 2-body decay into the attractive polaron and an additional particle-hole pair iscalculated numerically to all orders in the T-matrix by inserting the pole expansion for the
K propagator in the self energy in the ladder approximation. For the 3-body decay into a
molecule and an additional particle-hole pair, we include terms containing two Li holes in
the K self energy [Bru10], and an expansion to second order in the T-matrix relevant for−1/(κF a) ≪ −1 yields
!5

3
Z+
64κF amLi 3/2
a
ϵF
~κF
p
p
ΓP M ≃
1+
. 3.7
√
2
3
∗
∗
45π mK mLi
M
2(E+ − Em + ϵF )a 1 − 2re /a ~For simplicity, we have taken Zm = 1, which is an appropriate assumption for −1/(κF a) ≪−1. The effect of the narrow resonance on the decay rate enters through the quasiparticle
residue Z+ , the energies E+ , E− , Em and directly through the effective range re . This decayrate has the same a6 dependence as the three-body decay in vacuum in the limit of a broadresonance derived in Ref. [Pet03]. The numerical prefactor however differs since we haveincluded the effects of the Fermi sea in a perturbative calculation.
The results for the decay rates of repulsive polarons are shown in Fig. 3.7. The experimental

3.5. SUPPLEMENTARY INFORMATION

47

Figure 3.7: Decay rates of repulsive K polarons in a Fermi sea of Li atoms, shown as a functionof interaction strength (left) and of the energy of the repulsive polaron (right). Blue and red lines
represent the two- and three-body contributions, respectively, while data points are the experimental
findings as also shown in Fig. 3.3 of the main text. The results for the moderately narrow resonanceunder study here (solid lines) is compared with the theoretical results obtained for the universal limitof a very broad resonance (dashed lines). The experimental values of E+ are obtained by interpolationof the narrow peak position data ∆peak , see Fig. 3.5d.

data agree well with the theoretical results obtained for our narrow resonance (continuouslines) as already shown in Fig. 3.3 in the main text. For comparison, we also show the
decay rates one would obtain in the limit of a broad resonance (dashed lines). We findthat as magnitude of the effective range increases with respect to the interparticle spacing,
the dominant two-body decay is strongly suppressed. This suppression is mainly due to alarge reduction of the attractive polaron residue Z− . Instead, the weaker three-body decay
increases, which we attribute to the reduction of the polaron-molecule energy difference
E+ − Em + ϵF . Taking both decay rates together, the decay rate is at least an order ofmagnitude smaller at our narrow resonance as compared to the case of a broad resonance.
It is important to note that this strong suppression of the decay at a given −1/(κF a) cannot
be simply attributed to the effective resonance shift at our narrow Feshbach resonance as
discussed in Sec. 3.5.2. When taking this shift into account, a suppression factor of fiveto ten remains. To highlight this point, we choose a representation that is independent of
the interaction parameter and that gives the dependence on the polaron energy, a direct
manifestation of strong interactions. The right panel shows the same data and calculations
as a function of E+ . Also for a given E+ , the repulsive polaron at our narrow resonanceturns out to be much more stable than the repulsive polaron at a broad resonance.

3.5.5 Decay of repulsive polarons to molecules
The decay of the repulsive polarons, shown in Fig. 3.3 of the main text, is measured byapplying a special three-pulse scheme (see Methods). In this section we exploit the flexibility

48

3.5. SUPPLEMENTARY INFORMATIONof this scheme to study the decay to lower-lying energy states in more detail. At a given
interaction strength −1/(κF a) = −0.9, we demonstrate that the repulsive polarons decay tomolecules by showing that an rf spectrum taken after decay perfectly matches a reference
spectrum of molecules.
To populate the repulsive polaron branch, as done for the measurements of the decay rate, wetune the energy of the first pulse to E+ , corresponding to ∆ = 0.16 εF at −1/(κF a) = −0.9.
The pulse duration (tp = 0.06 ms) and the intensity are set to correspond to a π-pulse in thenoninteracting system. The second pulse removes the remaining non-transferred atoms bytransferring them to a third spin state. In contrast to the decay measurement presented in
the main text, we here use much more rf power for the third pulse to be able to efficiently
dissociate molecules. For this purpose, we set tp = 0.3 ms and the pulse area corresponds toa 3π-pulse in the noninteracting system. By varying the rf detuning, we record spectra forzero hold time (black squares) and for a hold time of 2 ms (red dots), see Fig. 3.8a. The peakat small positive detuning shows the back-transfer of repulsive polarons. The corresponding
signal decreases with hold time, signalling the decay of the repulsive polaron. In addition, awide continuum in a range of negative detunings rises with increasing hold time. Such a wide
continuum involves coupling to high momentum states, signaling a short distance between K
and 6 Li. To confirm that this continuum stems from molecules, we compare it to a reference
spectrum of the dissociation of molecules (blue diamonds). We find a perfect match. To
take such a reference spectrum, only the detuning of the first rf pulse is changed to directly
associate molecules in the MHC instead of populating the repulsive polaron branch. We
achieve a good association efficiency with ∆ = −0.54 εF and tp = 0.5 ms.
To study the evolution of the molecule population, which is fed by the decay of the repulsive polarons, we set the detuning of the third pulse to the peak of the molecule signal at∆ = −1.3 εF and record the signal as a function of the hold time, see Fig. 3.8b. A simple
exponential fit yields a rate of about 1 ms−1 = 0.0043 εF /~, which is in good agreement withthe measured decay rate of the repulsive polaron at −1/(κF a) = −0.9. The finite signal atzero hold time may have two origins. One contribution is some decay during the finite pulse
durations of the three pulses, which are not included in the hold time. Another contribution
may be the high momentum tail of the repulsive polarons as discussed in Ref. [Sch09].
Note that we do not find any second sharp peak at negative detuning, which would indicate
the population of the attractive polaron branch. In case the repulsive polaron decays to theattractive polaron, the absence of the attractive polaron peak implies a very rapid subsequent
decay of the attractive polaron to the MHC. Such a fast decay of the attractive polaron to
the MHC is consistent with the very small signal of the attractive polaron peak throughout
the regime of strong interaction as discussed in Sec. 3.5.2.
Let us briefly discuss the possible role of inelastic two-body relaxation in the Li-K mixture,
which is energetically possible as K is not in the lowest spin state. This process was identified
in Ref. [Nai11] as a source of losses. However, this relaxation is about an order of magnitude
slower than the measured decay rate of the repulsive polaron and thus does not affect ourmeasurements.

Figure 3.8: Decay of repulsive polarons to molecules at −1/(κF a) = −0.9. (a) The black squares(red dots) show the spectrum right after (2 ms after) the repulsive polaron has been populated. The
blue diamonds show the dissociation spectrum of molecules for reference. The signal is the fractionof atoms transferred from the interacting spin state |1⟩ to the noninteracting spin state |0⟩. Notethat the polaron peak at positive detuning is highly saturated and thus its signal is not proportional
to the number of polarons. (b) The rf energy detuning is fixed to ∆ = −1.3 εF and the signal isrecorded versus hold time. The error bars indicate the statistical uncertainties derived from at least
three individual measurements.

49

50

3.5. SUPPLEMENTARY INFORMATION

Figure 3.9: Linear increase of the Rabi frequency Ω with the unperturbed Rabi frequency Ω0 . The
left (right) panel shows the driving to the repulsive (attractive) polaron. The solid lines are linear
fits without offset and demonstrate the proportionality Ω ∝ Ω0 .

3.5.6 Rabi oscillations and polaron quasiparticle residue
For high rf power, the signal is well beyond linear response and the K atoms exhibit coherent
Rabi oscillations between the spin states |0⟩ and |1⟩. In this regime the oscillations are so
fast, that the polaron decay plays a minor role and can be ignored to a first approximation.
The Rabi frequency depends on the matrix element of the rf probe between the initial state|0⟩ and the final state |1⟩. Since the probe is homogenous in space, it does not change thespatial part of the atomic wavefunction and it can be described by the operator [Mas08a]
PR̂ ∝ Ω0 q (â†1q â0q + h.c.) where â†iq (âiq ) creates (annihilates) a K atom with momentum
q in spin state i and Ω0 is the unperturbed Rabi frequency of the |0⟩ to |1⟩ transition in
the non-interacting case. Considering for simplicity an impurity at rest, the initial noninteracting state is given by |I⟩ = â†0q=0 |FS⟩ where |FS⟩ is the Li Fermi sea. The final
polaronic state at zero momentum can be written as [Che06]|F ⟩ =√

Zâ†1q=0 |FS⟩ +X

φp,q â†1q−p b̂†p b̂q |FS⟩ + . . .

3.8q<~κF <pwhere b̂†q (b̂q ) creates (annihilates) a Li atom with momentum q. The second term contains a
Fermi sea with at least one particle-hole excitation and thus is orthogonal
to an unperturbed√
Fermi sea. Therefore the matrix element reduces to ⟨F |R̂|I⟩ = Z Ω0 and we obtain the
Rabi frequency√
3.9Ω = Z Ω0 .
We neglect the momentum dependence of the quasiparticle residue and do not perform athermal average over the initial states, which we expect to be a good approximation sinceT ≪ ϵF /kB .
In Fig. 3.9 we plot the observed Rabi frequency Ω as a function of the unperturbed Rabi
frequency Ω0 . We find that the proportionality Ω ∝ Ω0 holds over a wide range of rf power.
The measurements presented in the main text, taken at Ω0 = 2π × 6.5 kHz and 12.6 kHz, aresafely within this range.

51−1/𝜅F 𝑎 = -2.0

−1/𝜅F 𝑎 = -1.9

−1/𝜅F 𝑎 = -1.7

−1/𝜅F 𝑎 = -1.6

−1/𝜅F 𝑎 = -1.4SignalSignal3.6. CORRECTION: MOLECULE-HOLE CONTINUUM−1/𝜅F 𝑎 = -1.3

Δ/𝜀F

Δ/𝜀F

Figure 3.10: Molecule association spectra for different values of the interaction parameter. The
signal is the fraction of transferred atoms as a function of the rf detuning. The black diamonds
correspond to the experimental data. Left: The solid (dashed) lines correspond to the correctedline-shape model for finite (zero) temperature, where the Feshbach resonance center is shifted to a
1.5 mG higher field. Right: Replotted from Fig. 3.6 with an error in the line-shape model, see text.

3.6 Correction: Molecule-hole continuum
Our attention was called to an error in the line-shape model (Eq. (3.6)), after publication of
the work presented in this section. As the factor F(k),we inserted the energy-normalized
R
Franck-Condon factor from Ref. [Chi05]
fulfillingdEF(E)= 1. This factor includes an√additional density-of-state ρ(Er ) ∝ Er , that must be divided out, in the integration overmomentum in Eq. (3.6). Hence, the Franck-Condon factor used in this equation should readF(k) ∝ (1/Eb3 )1/2 (1 + Er /Eb )−2 .
Revisiting our data analysis we find only a minor effect on the results presented in thissection. The result of our reanalysis of the data shown in Fig. 3.6, i.e. fitting the correctedline-shape model to the molecule-association data, is shown in the left column of Fig. 3.10.
From this reanalysis, we determine the center of the Feshbach resonance to be at B0 =154.7205(20) mG, a slightly larger magnetic-field value compared to our previous erroneous
analysis, which yielded 154.719(2) mG. To take this correction of the Feshbach resonancecenter into account, all experimental data should be shifted to about 0.08 lower values of theinteraction parameter −1/(κF a), i.e. −1/(κF a) → −1/(κF a) − 0.08.

Chapter4

Publication: Decoherence of
Impurities in a Fermi Sea of
Ultracold Atoms1

Physical Review Letters 115, 135302 (2015)
submitted 10 June 2015, published 22 September 2015
DOI: 10.1103/PhysRevLett.115.135302
Editor’s suggestion

M. Cetina1,2 , M. Jag1,2 , R. S. Lous1,2 , J. T. M. Walraven1,3 , R. Grimm1,2 , R. S. Christensen4 ,and G. M. Bruun4
1

Institut für Quantenoptik und Quanteninformation (IQOQI), Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands
4
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark1

The author of the present thesis contributed to developing the experimental procedures together with
M.C., performed the measurements with M.C. and R.S.L., and analyzed the data together with M.C., and
R.S.L. under the supervision of R.G. All authors contributed to writing the manuscript.

53

54

4.1. ABSTRACT

4.1 Abstract
We investigate the decoherence of 40 K impurities interacting with a three-dimensional Fermi
sea of 6 Li across an interspecies Feshbach resonance. The decoherence is measured as afunction of the interaction strength and temperature using a spin-echo atom interferometry
method. For weak to moderate interaction strengths, we interpret our measurements interms of scattering of K quasiparticles by the Fermi sea and find very good agreement with a
Fermi liquid calculation. For strong interactions, we observe significant enhancement of thedecoherence rate, which is largely independent of temperature, pointing to behavior that isbeyond the scattering of quasiparticles in the Fermi liquid picture.

4.2 Introduction
Many-body fermionic systems with strong interactions play a central role in condensedmatter, nuclear, and high-energy physics. The intricate quantum correlations between
fermions challenge our understanding of these systems. Mixtures of ultracold fermionic gases
offer outstanding opportunities to study strongly interacting fermions experimentally. Since
the turn of the century, the excellent control over the strength of the interaction and thecomposition of these mixtures has allowed investigations addressing the broad spectrum fromfew-body to many-body phenomena [Blo08, Gio08]. Tuning of the interaction is achieved
using Feshbach resonances [Chi10]. The composition is varied by selecting internal statesor by mixing different atomic species. This development has led to many exciting results
concerning the quantum phases of fermionic mixtures, their excitations, superfluid behavior,and the equation of state [Zwi15].
In two-component fermionic systems with a large population imbalance, the minority atoms
have been shown to form quasiparticles termed Fermi polarons, even for surprisingly large
coupling strengths [Sch09, Koh12b, Kos12, Mas14]. These are long-lived states describedby Fermi liquid theory [Bay04]. Their lifetime is limited by scattering against the majority
atoms, which is suppressed by Pauli blocking as the temperature approaches zero [Lan57,
Lan56]. Although the quasiparticle scattering rate has been determined in two-dimensional
electron gases [Ber95, Mur95, Slu96], measurements in well-defined three-dimensional (3D)
fermionic systems have remained an experimental challenge.
Intriguing questions are related to the behavior of impurities and, more generally, Fermi
mixtures in the strongly interacting regime [Mas14, Nas11, Sag15]. For investigating an
impurity in a Fermi sea, Refs. [Goo11, Kna12] suggested a time-domain method that is
applicable for a wide range of interaction strengths. This approach can be regarded as ameasurement of the coherence of a superposition of internal states of the impurity atoms
using interferometery [Cro09]. Atom coherence has previously been used to probe many-bodydemagnetization in fermionic systems [Bar14] and impurity scattering in bosonic systems[Sce13].
In this Letter, we report on measurements of decoherence of K atoms immersed in a Fermi

4.3. MAIN RESULTS

55

sea of Li using the method proposed in Ref. [Kna12], in the regime of strong population
imbalance. We tune the interaction between the Li and K atoms using an interspecies
Feshbach resonance (FR). For weak to moderately strong interactions, we interpret themeasured decoherence in terms of scattering of K quasiparticles by the Li Fermi sea. We findvery good agreement with a Fermi liquid calculation. This provides a determination of thequasiparticle scattering rate in a clean 3D fermionic system. We extend our measurements
to strong Li-K interactions and find decoherence rates that are comparable to the fastest
dynamics available in our system. These rates do not increase with temperature, which isan indication of zero-temperature quantum dynamics in a fermionic many-body system.

4.3 Main results
The starting point of our experiments is an evaporatively cooled, thermally equilibrated
mixture of typically 3 × 105 6 Li atoms and 1.5 × 104 40 K atoms, trapped in a crossed-beam1064-nm optical dipole trap under conditions similar to those in Ref. [Koh12b]. The Li cloudis degenerate, with kB T /ϵF as low as 0.15, where T is the temperature and ϵF ≈ h × 35 kHz
is the average Li Fermi energy sampled by the K atoms. Because of the Li Fermi pressure
and the more than two times stronger optical potential for K, the K cloud is much smaller
than the Li cloud [Tre11a], and therefore samples a nearly homogenous Li environment, witha standard deviation in the local Li Fermi energy of less than 0.1 ϵF . In spite of the smaller
size of the K cloud, the concentration of K in the Li sea remains low, with n̄K /n̄Li ≈0.3,
where n̄K (n̄Li ) is the average K (Li) number density sampled by the K atoms. The K
ensemble is correspondingly non-degenerate, with kB T /EFK > 0.9, where EFK is the peak K
Fermi energy.
We tune the interaction between the K and Li atoms using an interspecies FR between the
Li atoms in the lowest Zeeman sub-level Li|1⟩ and K atoms in the third-lowest sub-level K|3⟩
[Nai11]. We quantify the interactions√between Li and K by the dimensionless interaction
parameter −1/κF a, where κF = ~−1 2mLi ϵF is the Li Fermi wavenumber with mLi the
Li mass, and a is the s-wave interspecies scattering length. The latter can be tuned asa = abg [1−∆/(B −B0 )] by applying a magnetic field B, where B0 ≈ 154.7 G is the resonancecenter, abg = 63.0 a0 (a0 is Bohr’s radius) and ∆=880 mG [Nai11]. The relatively narrow
nature of our FR causes significant momentum dependence of the interspecies interaction.
We characterize this effect by the length parameter R∗ [Pet04a, Koh12b]. In our experiments
κF R∗ is approximately 0.9, corresponding to an intermediate regime where the interaction
is near-universal with substantial effective-range effects.
We probe the decoherence of the K atoms using a radio-frequency (rf) interferometric technique, as illustrated in Fig. 4.1. The K atoms are initially prepared in the second-lowest
Zeeman sub-level K|2⟩ while the Li atoms remain in the Li|1⟩ state throughout the experiment. On the time scale of our measurements, the interactions between these atoms,
characterized by the s-wave scattering length a12 ≈ abg , can be neglected. We apply a π/2
rf-pulse (typically 10 µs-long) to prepare the K atoms in an equal superposition of the K|3⟩
and K|2⟩ states. After a variable interaction time τ , we apply a second π/2 rf-pulse before
determining the numbers N2 and N3 of atoms in the K|2⟩ and K|3⟩ states using absorption

56

4.3. MAIN RESULTS
τ /2

K|2〉

τ /2

π

π

π

2

2

N3

K|3〉

φ

Li|1〉
1.0

N3 / (N3 + N2)

N2

τ=
10
110
210
410
1010
µs

0.5

0.0
0.00

0.25

0.50

0.75

1.00

φ / 2πFigure 4.1: Interferometeric method for measuring the decoherence of K in a Li Fermi sea. The
upper illustration shows a schematic of the rf pulse sequence. The atoms in the K|3⟩ state interact
with a Fermi sea of Li|1⟩ atoms, as indicated by the shaded region. The graph shows the fraction of
the K atoms transferred to the K|3⟩ state as a function of the relative phase of the final π/2 rf pulse
for various interaction times τ and for −1/κF a=2.1, T = 0.16ϵF /kB .

imaging (see Supplemental Material). To decrease the sensitivity to the magnetic field noise
and to the inhomogeneities in the atom densities, we perform a spin echo by splitting theinteraction time into two equal halves separated by a π rf-pulse.
Shifting the phase of the rf oscillator by φ between the π and the second π/2 pulses causesa sinusoidal variation in the fraction f = N3 /(N2 + N3 ) of the K atoms transferred to K|3⟩,
as shown in Fig. 4.1. We quantify the coherence of the state of the K atoms by the contrast
C = (fmax −fmin )/(fmax +fmin ) of these oscillations. The interaction of the K atoms with the
Li cloud causes an exponential decrease in the observed contrast with increasing interaction
time τ , as shown in Fig. 4.2(a). The interaction also shifts the rf transition frequency anddecreases the rf coupling between the K|2⟩ and K|3⟩ states [Koh12b], which we account forby adjusting the rf frequency and the duration of our rf pulses. In this way, we measure
the decoherence of K atoms for −1/κF a < −0.8 and −1/κF a > 1.4. Near the center ofthe resonance, the fast loss of contrast during the rf pulses limits the applicability of thismethod.
To measure the decoherence of K in the strongly interacting regime, we use laser light torapidly displace our magnetic FR [Bau09b, Bau09a, Cla15]. Optical control of our FR
allows us to apply the rf pulses away from the FR and then rapidly bring the atoms into
resonance for the duration of the interaction time2 τ . This method circumvents the loss ofcontrast during the rf pulses and allows us to probe the K decoherence across the full range ofinteraction parameters. The displacement of our FR arises from the laser-induced differential
AC Stark shift between the free-atom level and the molecular state involved in the FR. The
2

For measurements on the attractive (repulsive) side of the FR, we shift B0 upwards (downwards). For
measurements near the resonance, we verify that the direction of the shift of B0 does not affect the result.

4.3. MAIN RESULTS

57
1-1 / κF a = 2.1(1)0.1

contrast C

0.01

(a)
0

200

400

600

1

800

1000

-1 / κF a = 0.1(1)

(b)

0.1
0

5

10

15

20

interaction time τ (µs)

2530Figure 4.2: Contrast C as a function of interaction time τ . In (a), we show results for moderately
attractive interspecies interactions (−1/κF a=2.1), corresponding to Fig. 4.1. In (b), we probe thesystem in the strongly interacting regime (−1/κF a=0.15) for T = 0.20 ϵF /kB by rapidly shifting theinteraction parameter from 2.2 to 0.15 during the interaction time. The solid lines are exponential
fits to the points with τ > 7 µs. The dotted line is an extrapolation to τ =0.

AC Stark shift is induced by the 1064-nm trapping light, as we investigated in Ref. [Jag14].
Although the differential shift here amounts to only 10% of the total trapping potential,
using a high-intensity beam with up to 65 kW/cm2 , we can displace B0 by up to 40 mG inless than 200 ns – all while preserving the harmonic trapping potential (see Supplemental
Material). This displacement corresponds to a change in the interaction parameter of up to±2.1 on a timescale of 0.05 τF , where τF = ~/ϵF ≈ 4.5 µs is the Fermi time.
In Fig. 4.2(b), we show the dependence of the contrast C on the interaction time τ near thecenter of our FR. The contrast starts to decay after an initial delay of approximately τF .
This delay can be explained in terms of quantum evolution of the system with an interactionenergy bounded from above by ϵF [Kna12]. For τ > 1.6τF ≈ 7 µs, the decrease in contrast
is well-described by an exponential decay. The fitted rate γcoh =0.28(2)τF−1 is comparable tothe inverse Fermi time, indicating that our experiment cannot be fully described by the scattering of quasiparticles in the Fermi liquid picture, which assumes long-lived quasiparticles[Bay04].
In Fig. 4.3 we show the dependence of the fitted rate γcoh on the interaction parameter.
We present data with two decades of dynamic range and demonstrate a dramatic resonant
enhancement of the decoherence rate, reaching values up to 0.4 τF−1 . The data do not exhibit
any clear dependence on n̄K /n̄Li across the full range 0.17≤ n̄K /n̄Li ≤0.43. In addition
to the statistical errors indicated by the error bars, the data are subject to variations of
kB T /ϵF , κF R∗ and n̄K /n̄Li with standard deviations of 0.01, 0.02 and 0.07 about their meanvalues of 0.16, 0.93 and 0.27, respectively. The calibration of the Li atom number introduces
a 6% systematic uncertainty in ϵF and τF , as well as a corresponding 3% uncertainty inκF . Our total error budget includes further 3% systematic errors in a and R∗ arising from58

4.3. MAIN RESULTS

γcoh τ F

0.1

0.01

-2

-1

0

1

-1 / κF a

2

3

Figure 4.3: Decoherence rate of K in a Li Fermi sea as a function of the interaction parameter foran average temperature T =0.16 ϵF /kB (see text). The measurements with (without) rapid shiftingof the FR are shown as the red circles (black squares). The measurements from Fig. 4.2 are indicatedby open symbols. The solid upper (blue) and lower (black) lines correspond to the prediction of the
Fermi liquid theory with and without medium corrections, respectively. The dashed lines incorporate corrections due to decay to Feshbach molecules. The shaded areas show the 1σ effect of theexperimental uncertainties on the theoretical predictions.

the uncertainty in ∆B, and a ±0.05 error in 1/κF a resulting from an uncertainty in thedetermination of B0 of ±1 mG (see Supplemental Material).
For weak to moderate interactions, there are well-defined K quasiparticles, and we now showthat the evolution of the contrast C on timescales much longer than τF can be related to themean quasiparticle scattering rate γs . Each scattering event provides which-way informationthat distinguishes between the two paths in the interferometer in Fig. 4.1 and thus erases
the interference effect. At any given time, the interaction affects only one of the two paths,
decreasing the probability for the system to stay in this path at the rate γs . Since our signal
arises from the interference of the amplitudes in the two interferometer paths, we expect theinteraction to lead to a decrease of the observed contrast at the rate γs /2.
From Fermi liquid theory, the scattering rate γp1 of a K quasiparticle with momentum p1 isgiven by [Bay04]
ZZ
mr pr
γp 1 =
dp̌2 dΩ|T |2 [fpLi2 (1 − fpK3 − fpLi4 ) + fpK3 fpLi4 ].
4.1
4π 2Here T is the scattering matrix for the scattering of K atoms with Li atoms with momenta
p1 and p2 respectively to momenta p3 and p4 . We have defined dp̌2 = d3 p2 /(2π)3 , andΩ is the solid angle for the direction of the outgoing relative momentum. The distribution
Li/K
Li/K
= [eβ(Ep −µLi/K ) + 1]−1 with the chemical potentials µLi/K for the Li /K
functions are fp
atoms respectively. The dominant medium effects can be shown to enter in the scatteringmatrix T via ladder diagrams, whereas the quasiparticles can be assumed to have the ideal
K/Li
gas energy dispersion Ep= p2 /2mK/Li [Bru05b, Ens12]. The details of the calculation of
γp1 are described in [Chr15]. In addition, we account for the reduced quasiparticles residue

4.3. MAIN RESULTS

59

Z by multiplying the collision rate by Z calculated from the ladder approximationR [Mas14].
To obtain the mean scattering rate γs , we calculate the thermal average γs = dp̌fpK γp .
To include the effects of the trap, we use effective Fermi energies, which are obtained byaveraging the local Fermi energy over the density of the K atoms in the trap. This approachis justified since the K atoms only probe a small region of the Li gas, and because themomentum distribution of the K atoms is nearly classical.
On the repulsive side of the FR, we need to consider additional effects arising from the decayof the atoms into the molecular state that underlies our FR. The rate Γ of this process wascalculated and confirmed by measurements in Ref. [Koh12b], reaching values as high as 0.02
τF−1 close to resonance. Since the decay to molecules provides which-way information, it willcontribute at least Γ/2 to the measured decoherence rate. The decay also releases energy andcreates holes in the Li Fermi sea, increasing the value of kB T /ϵF during our measurement
to 0.20 (1) (see Supplemental Material).
In Fig. 4.3, we plot the calculated decoherence rate γs /2 as a function of the interaction parameter. The lower solid line is obtained by using the vacuum scattering matrix Tvac [Chr15]
in (4.1), whereas the upper solid line is obtained by using a T matrix which includes medium
effects using the ladder approximation. The dashed lines include the effects of decay into
the molecular state. The calculated decoherence rate agrees with the experimental values
very well for −1/kF a &1.5 and for −1/kF a . −1. This gives strong evidence that the observed decoherence indeed is due to quasiparticle collisions. The significant asymmetry of thedecoherence rate around 1/kF a = 0 arises from the narrow nature of the FR [Chr15]. The calculated decoherence rate is larger when medium effects are included in the T matrix. This isdue to pair correlations, which can increase the collisional cross section significantly [Chr15].
We see that the inclusion of these medium effects on the scattering matrix improves theagreement with the experimental data. For stronger interactions, the calculation does not
fit the experiment, which is expected since there are no well-defined quasiparticles in theunitarity regime [Koh12b]. Our model agrees with the observed absence of a dependenceof γcoh on n̄K /n̄Li since the K cloud is close to the classical regime where fpK3 ≪ 1 and themomentum distribution of the K atoms is solely determined by the temperature.
Further insight into the nature of the observed decoherence can be gained by varying thetemperature of our atom mixture, which we accomplish by changing the endpoint of ourevaporative cooling. We show the dependence of the measured decoherence rate on temperature in Fig. 4.4. In addition to the statistical errors shown by the error bars, the data
are subject to small variations of −1/κF a, κF R∗ and n̄K /n̄Li with standard deviations of
0.05, 0.03 and 0.1, respectively. Our total error budget also includes the above-mentioned
systematic uncertainties in ϵF , κF , a and R∗ .
Away from the FR, the measured decoherence rates are in very good agreement with thepredictions of the Fermi liquid theory. The linear dependence of γcoh on temperature inthis regime arises from the high relative mass of the K atoms, causing the Li-K scattering
to resemble scattering by fixed impurities. This is similar to the situation in metals where
the scattering of electrons by fixed nuclei gives rise to the well-known linear dependence ofthe nuclear decoherence rates on temperature [Kor50]. The red circles in Fig. 4.4 represent
the measurements for resonant interactions. The rates obtained in this regime are more

60

4.4. CONCLUSION

0.4γcoh τF

0.3

0.2

0.1

0.0
0.0

x2.5
0.2

0.4

kBT / εF

0.6

0.8

Figure 4.4: Decoherence rate of K in a fermionic Li cloud as a function of temperature. The datafor −1/κF a=0.2, κF R∗ =0.94, n̄K /n̄Li =0.2 (−1/κF a=2.4, κF R∗ =0.89, n̄K /n̄Li =0.3) measured with
(without) rapid shifting of the FR is shown as red circles (black squares). The solid blue and black lines
correspond to the predictions of the Fermi liquid theory for −1/κF a=2.4 with and without medium
corrections, respectively. The shaded areas show the 1σ effect of the experimental uncertainties on
the theoretical predictions.

than an order of magnitude higher than the off-resonant rates, and do not increase withtemperature.

4.4 Conclusion
In conclusion, we established that for weak to moderate interaction strengths, the decoherence of K in a Li Fermi sea is dominated by quasiparticle scattering. Our observations forstrong interactions cannot be explained solely by quasiparticle scattering and indicate decoherence processes which persist at zero temperature. This offers an exciting opportunity to
explore the many-body quantum dynamics of an impurity submerged in a Fermi sea.

Acknowledgements

We thank F. Schreck, C. Kohstall, I. Fritsche, P. Jeglič, Y. Ohashi, R. Schmidt, E. Demler,and, especially, M. Parish, J. Levinsen and M. Baranov for helpful discussions. We thankP. Massignan for sending us data for the quasiparticle residue and decay. We acknowledge
funding by the Austrian Science Fund FWF within the SFB FoQuS (F40-P04) and supportof the Villum Foundation via Grant No. VKR023163.

4.5. SUPPLEMENTAL MATERIAL

61

4.5 Supplemental material
4.5.1 Optical trap setup
We perform our measurements in dipole traps formed by two single-frequency, 1064-nm
laser beams produced by a solid-state laser system (Innolight Mephisto 42NE MOPA). The
beams intersect at an angle of 16◦ . To avoid standing wave effects, the beams are offset infrequency by 5 to 10 MHz. To shift the Feshbach resonance (FR), we keep one of the two
beams (“D”) static and use acousto-optic modulators (AOMs) to rapidly change the otherbeam. Changing only the intensity of this latter beam would excite strong oscillations of theatomic cloud. We counter this by switching from a beam with a low peak intensity and small
size (“S”) to a beam with a large intensity and large size (“V ”) propagating in the same
direction. The waists, positions and intensities of the “S” and “V ” beams are adjusted so as
to match the centers and curvatures of the resulting optical potentials, preventing collective
excitations of the atomic cloud.
This method for shifting the FR poses two technical challenges. First, the overlap between
the optical axes of the “S” and “V ” beams needs to be maintained with an accuracy thatis much better than the smallest extent of the atomic clouds (about 5 µm). Second, tomaximize the shift of the resonance, the curvature of the optical potential due to the larger
beam needs to be minimized relative to its intensity. We address both of these challenges
using the optical system shown in Fig. 4.5.
To ensure beam pointing stability, we couple the laser light into Panda polarization-maintaining fiber patchcords (Thorlabs P3-980PM-FC) with 6 µm core diameter. For the “S” and“D” beams, we use 5-m long fibers. When using a 5-m long fiber for the “V ”-beam, weobserved a saturation of the fiber output power at 1.1 W, together with a sharp increase of
the reflected power from the fiber. These effects were not observed with up to 10 W of output
power from the fiber when a 2-m long fiber was used. We interpret these observations in termsof stimulated Brillouin scattering [Ruf04], whose threshold power is inversely proportional
to the fiber length.
To prevent the degradation of the fiber ends, the “V ” beam is operated with brief (< 0.3 ms)
and infrequent (< 1/min) pulses. Although, in the P3-980PM-FC patchcords, the fiber is
attached to the ferrule using an epoxy adhesive, we did not observe any degradation of thefiber transmission after one year of operation with peak powers up to 10 W.
To maintain relative pointing stability of the “S” and “V ” beams, the outputs of the “S”
and “V ” fibers (e−2 divergence half-angle = 74(3) mrad) are combined on a polarizing
beamsplitter. At this location, the “S” and “V ” beams have waists of 0.31(1) mm and0.75(1) mm, respectively. In the same plane, “S” and “V ” beams are converging with radiiof curvature of 370(20) mm and 390(20) mm, respectively. The output of the beamsplitteris projected with 7.6× demagnification onto the atoms using a telescope composed of thelenses labeled as L1, L2, L3 and L4 in Fig. 4.5. The distance between the lenses L2 and L3
is adjusted so as to obtain a nearly collimated “S” beam after the lens L3 with a 1.2(1)-mm
waist.

62

4.5. SUPPLEMENTAL MATERIAL

CCD beam profiler

AOMV

all measures in mm

f=8, t: PCX, asph., TL#: C240TME
5AOM

f = focal lengtht = type
TL# = Thorlabs Item#87±1
f=150, t: PCX

S

47°

L1
19±1 30±1L287.5±0.7

L3

117±1

82±2

62.5±1.0

iris

PBS
f=3.3
t: DCX, asph.
TL#: A414TM

L4

59±1

155.3±0.5imaging plane

f=250, t: PCX
27±1

f=100
t: PCX

f=11
t: PCX, asph.
TL#: C220TME

f=150
t: PCX

f=150
t: PCX

Figure 4.5: Illustration of the optical setup for producing and switching optical dipole traps ofdifferent sizes with highly stable relative position.

Higher spatial frequency components in the beams (arising e.g. from stray reflections) increase the curvature of the optical potential relative to its depth. To mitigate this problem, we spatially filter the “S” and “V ” beams by passing them through a 5.3-mm diameter graphite aperture in a plane that is Fourier-conjugate to the location of the atoms(Fig. 4.5).
We measured the sizes of the “S” and “V ” beams at the location of the atoms by deflecting
the beams using an auxiliary mirror and then focusing them onto a CCD beam profiler,
as shown in Fig. 4.5. The 1/e2 radii of the “S” and “V ” beams were determined to be38(2) µm, and 91(3) µm, respectively. The size of the “D” beam at the location of theatoms was determined to be 48(2) µm.

4.5.2 Parameters of the 154.7 G Feshbach resonance
The FR that we employ for tuning the interactions in our system occurs between 6 Li atoms
in their lowest internal state, denoted Li|1⟩ (F = 1/2, mF = +1/2), and 40 K atoms in their
third-to-lowest state K|3⟩ (F = 9/2, mF = −5/2). We parametrize the Li|1⟩-K|3⟩ scattering
length near the 154.7 G FR by the usual expression [Chi10]∆a (B) = abg 1 −.
4.2
B − B0For the background scattering length abg and the resonance width ∆, we use the values
abg = 63.0 a0 and ∆ = 0.880 G from the coupled-channel calculation in [Nai11].
The narrow nature of the Li|1⟩-K|3⟩ FR causes a significant variation of Li-K scattering
across the range of the collision energies encountered in our experiment [Chi10]. We obtain quantitative information on this effect from measurements of the molecular binding
energy. We measure the binding energy using two methods: radio-frequency (rf) association

4.5. SUPPLEMENTAL MATERIAL

63

Figure 4.6: Molecular binding energy near the 154.7 G Li|1⟩-K|3⟩ Feshbach resonance measured
using rf association (red points) and magnetic field modulation spectroscopy (blue points). The lower
panel shows the residuals of a fit to the data with the model based on Eq. (4.3) with δµ and B0 as
free parameters.

[Jag14] and magnetic field modulation spectroscopy [Lan09]. The combined results of thesemeasurements are shown in Fig. 4.6.
In order to parametrize the dependence of the molecular binding energy Eb on the magnetic
detuning B − B0 near the dissociation threshold, we first introduce the wavenumber κ, interms of which Eb = ~2 κ2 /2µ, where µ = mLi mK / (mLi + mK ) is the reduced mass. We
then express the magnetic detuning B − B0 as a function of κ by a Taylor expansion up tosecond order of the form
B − B0 = −abg ∆κ −

~2 κ2
.
2µδµ

4.3

The coefficient abg ∆ in front of the linear term is determined by the well-known universal
relation between κ and a near the resonance. We fit Eq. (4.3) to the data from Fig. 4.6 with
δµ and B0 as free parameters, while fixing abg and ∆ to the values from Ref. [Nai11]. From
this fit, we obtain δµ/h = 2.35 (2) MHz/G. Our fitting model is equivalent to the predictionof the two-channel model from [Pet04a] with R∗ = ~2 / (2µabg δµ∆). We note that this model
neglects the background scattering term in Eq. (4.2).
Our measurements of the binding energy allow us to determine the momentum dependence64

4.5. SUPPLEMENTAL MATERIAL

of Li-K collisions near the resonance. For small values of the collision momentum ~k, we canuse the well-known effective range expansion to write the inverse scattering amplitude as
1 1
f (k)−1 = − + reff k 2 − ik .
a 2
Since the existence of the bound state implies that the scattering amplitude has a pole at
k = iκ, we obtain
1 1
0 = − reff (iκ)2 + i (iκ) .
a 2
Substituting the expression (4.2) for a (B), we can then write

−1
∆
1
−1
abg 1 −
+ reff κ − κ = 0 .
B − B02
Then, substituting the expression (4.3) for the detuning B − B0 , we obtain
!−1
∆
1
a−1
1−
+ reff κ − κ = 0 .
bg
~2 κ2
2
−abg ∆κ −2µδµTaylor-expanding this near κ = 0, we get



~2
reff
+
κ2 + O κ3 = 0 ,−abg +
2
2abg µδµ∆whence we obtainreff = 2abg −~2.
abg µδµ∆4.4

We can summarize the parameters of the Li|1⟩-K|3⟩ FR with three independent parameters:
abg = 63.0 a0 ,∆ = 0.880 G ,δµ/h = 2.35 (2) MHz/G .

From this, we can derive the valuesR∗ = 2650 (25) a0 ,reff

= −5175 (50) a0 .

4.5.3 Light shift of the Feshbach resonance
As we pointed out in the main text, as well as in Refs. [Koh12b, Jag14], the optical trap
induces a differential light shift between the atom pair state and the molecular state giving
rise to the FR near 154.7 G. This leads to a light-induced shift of the FR. To produce thesesignal4.5. SUPPLEMENTAL MATERIAL

0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
-200

65

154.679 G

154.698 G

154.717 G

-150

-100

(υ-υ0) (kHz)-50

0Figure 4.7: Data from the molecular rf-association spectroscopy for trap 3. The red points weretaken with a 0.5-ms rf pulse with an intensity set to the value matching the π-pulse condition in
the absence of interactions (no Li|1⟩ present). The green points were recorded with more than 30×increased rf intensity. The dashed lines indicate the binding energy Eb (B) as determined with the
20% criterion (see text).

shifts in the experiments presented in the main text, we use a near-infrared laser as discussed
in Section 4.5.1 in four different trap settings (see Table 4.1).
To determine the resonance center B0 for a given trap setting, we follow the experimental
procedure outlined in the Supplemental Material of Ref. [Jag14]. For the data analysis,
we use the updated binding energy model presented in the preceeding Section. For eachtrap setting, we perform rf association spectroscopy of the Feshbach molecules. We start by
preparing a nonresonant mixture of Li atoms in the state Li|1⟩ and K atoms in their secondto-lowest state K|2⟩ several tens of mG below B0 . At this field, we apply an rf pulse (durationof a few 100 µs) at a variable frequency ν, several kHz below the unperturbed K|2⟩→K|3⟩
transition frequency ν0 . This pulse drives Li|1⟩-K|2⟩ atom pairs into the Li|1⟩K|3⟩ dimer
state. To determine the number of dimers associated, we subsequently dissociate the dimers
into pairs of Li|1⟩ and K|3⟩ atoms by a magnetic field ramp (duration of a few 100 µs)
to a magnetic field above 154.8 G. By recording absorption images we then determine thepopulations N2 and N3 of the K spin states K|2⟩ and K|3⟩, respectively.
Plotting the signal, given by N3 /(N3 + N2 ), against the rf detuning ν − ν0 , we resolve themolecule association spectrum. In Fig. 4.7 we show sample spectra recorded for one of thetrap settings used in the experiments. We determine the energy of the molecules relative tothe energy of noninteracting K|3⟩ atoms from the onset frequency of the molecular association
spectra. As the onset frequency, we use the upper rf frequency at which the fraction of atoms
transferred is roughly 20% of its peak height. We have checked that, within the errors of ourmeasurements, this criterion agrees with the result obtained by fitting the line-shape modelof Ref. [Chi05] to the spectra, as was done in Ref. [Koh12b]. This procedure is applied atvarious magnetic fields for each trap setting used in the experiments.

66

4.5. SUPPLEMENTAL MATERIAL

0
20
40

E b (kHz)60
80
Data in Trap 1

100

B 0 = 154.7195 G

Data in Trap 2a120

B 0 = 154.7465 G

Data in Trap 3140

B 0 = 154.7405 G

Data in Trap 4160180154.600

B 0 = 154.7785 G

154.650154.700

154.750

B (G)Figure 4.8: Determination of the FR center B0 by rf association of dimers. The points show theexperimentally determined molecular binding energies Eb (B) for four trap settings. The solid curves
are fits of the binding energy according to Eq. (4.3). The gray shaded areas indicate the typical error
range of our fit analysis.

The interaction between the formed LiK molecules and the Li atoms leads to an energy shiftof the molecular state. We use the mean-field model from [Jag14] to predict the corresponding
shift in the onset frequency in the rf association measurements as +2.0(4) kHz. To determine
the molecular binding energy in the absence of the Li cloud, we subtract this small offset
from the onset frequencies determined above.
We fit the binding energy according to Eq. (4.3) to the data, with B0 as the only freeparameter (see Fig. 4.8). The other parameters are fixed to the values from Section 4.5.2.
This procedure allows us to determine the resonance center in each trap setting with anuncertainty of ±1.5 mG. The accuracy of our determination of the resonance position islimited by the uncertainty in the FR parameters in the model for the binding energy. The
FR centers determined for our four trap settings of Fig. 4.8 are given in Table 4.1.
To record the data shown in Figs. 3 and 4 of the main text we switch between trap settings
1 and 2b as well as between settings 3 and 4 within less than 200 ns. In our experiments,
switching between trap 1 and 2b (3 and 4) changes the interaction parameter 1/κF a by
1.2 (2.1), without changing the harmonic potential in which the atoms are trapped. When
we recorded the data for the FR center determination, the data in trap 2a of Fig. 4.8 wasrecorded with 11% higher trapping-light powers compared to trap 2b, in which measurementsof the main text were taken. Therefore, the FR center B0,2 is shifted 11% more relative tothe center of the FR in the absence of the 1064-nm light. We determine the latter to be
at 154.699(1) G by extrapolating the FR centers for various beam intensities to the zero
trapping-light intensity. The correct value of the FR center of trap 2b, as used in theexperiments of the main text, is therefore given by 154.699 G + (B0,2 − 154.699 G)/1.11 =154.7420(15) G.

4.5. SUPPLEMENTAL MATERIAL
Trap
1
2b
2a
3
4

B0
(G)
154.7195
154.7420
154.7465
154.7405
154.7785

νr,K
(Hz)
415
415
580
580

67
νa,K
(Hz)
56
56
80
80

νr,Li
(Hz)
650
650
945
945

νa,Li
(Hz)
88
88
130
130

PS
(W)
0.175
0
0
0.380
0

PD
(W)
0.380
0.380
0.425
0.815
0.815

PV
(W)
0
4.4
4.85
0
7.6

Table 4.1: Typical trap parameters for the various trap settings. In our experiments, we switch fromtrap 1 to 2b i.e. from trap 3 to 4 by switching the powers PS and PV of the S and V beams within
less than 200 ns.

The determination of the relative shifts of the FR centers of two trap settings can be donewith an even higher accuracy. As an example: We record association spectra at a magnetic
field B3 (B4 ) in trap 3 (4) with the FR center at B0,3 (B0,4 ). The magnetic fields are chosensuch that the spectra are taken at roughly the same detuning B3 − B0,3 ≈ B4 − B0,4 . We
then compare these spectra from trap 3 and trap 4, and overlap their association onsets byshifting one of them, say the one in trap 4, with respect to the other, trap 3, along thefrequency axis by δν. This frequency shift δν can be translated into a magnetic detuning
shift δB0 by comparing it to the slope of the binding-energy at that detuning dEb /dB. Then
δB0 can be extracted from δB0 = hδν/[dEb /dB]. Finally we can derive the relative shift of
the FR centers in trap 3 and trap 4 to be B0,4 − B0,3 = B4 − B3 − δB0 . We estimate theaccuracy of this relative FR center determination to be on the order of ±0.5 mG.
Table 4.1 shows the typical resonance positions, trap frequencies and laser powers for thetraps used in the measurements in the main text. The trap frequencies are determined byobserving oscillations of the atomic clouds. We observe variations in the FR centers of lessthan 1.5 mG and trap frequencies of less than 4% over weeks of measurement time, which
we ascribe to variations of the 1064 nm trapping-laser power and drifts of the relative beam
positions. We account for these variations in the data analysis.

4.5.4 Determination of the Li atom number
An accurate determination of the number NLi of Li atoms in our experiment and the corresponding Fermi energy is an important, non-trivial task. Here we present four different
methods to determine the number of Li atoms with an error of less than 10%.

Absorption imaging on a nearly closed transition

One method for determining the number of atoms is absorption imaging. In this method, thespatial dependence of the fraction A of the light absorbed by the atomic cloud is recorded

68

4.5. SUPPLEMENTAL MATERIAL

using a camera and used to obtain a measure of the atom number
2π  u 2 X
NI = − 2ln [1 − A (X, Y )] ,
3λ M
X,Y

where λ is the light wavelength, M is the magnification of the imaging system, u is thecamera pixel size, and A (X, Y ) is the absorbed light fraction as measured by the camera
with X and Y the camera pixel indices. If atoms at rest are imaged using a weak lightpulse that resonantly excites a closed atomic transition, NI will be equal to the true atom
number. In this section, we will present reference experimental conditions that approximate
this situation and use these conditions to obtain a measure NIhigh−B that is close to the Li
atom number NLi . We will then discuss the remaining systematic effects and thereby relate
NIhigh−B to NLi .
We approximate a closed transition by imaging Li atoms in the second-lowest Zeeman state(Li|2⟩, mJ = −1/2, mI = 0) using σ − light near λ = 671 nm that resonantly excites themto the second-lowest Zeeman state of the 2 P3/2 manifold (Li|2′ ⟩, mJ = −3/2, mI = 0) at
the magnetic field of 1150 G. The dominant branching from this transition is due to thespontaneous decay of Li|2′ ⟩ to the Li|4⟩ (mJ = 1/2, mI = −1) state. We calculate thecorresponding branching ratio as 0.12% using the dipole selection rules and the expression of
the relevant states in the (mJ , mI ) basis. Under our imaging conditions, the other branchingratios are more than 100 times smaller.
We record the images of the atoms using a back-illuminated CCD camera (Andor DV-434)with a pixel size of u = 11 µm. We determine the magnification M = 2.93 (5) of our imaging
system by imaging the interference pattern formed by two 671-nm laser beams intersecting
at an angle of 29.0 (4) mrad at the location of the atoms onto our camera.
We check the purity of the polarization of the imaging light by imaging an optically dense
Li cloud. We obtain optical depths greater than 3, corresponding to a sum of the intensitiesof the unabsorbed σ + and π light components that is less than 5% of the intensity of theσ − imaging light. We minimize the effect of the polarization errors by ensuring that the
maximal optical depth of the imaged cloud is smaller than 0.4, implying a relative error in
the determined atom number of less than 6%.
We minimize saturation effects by using a low light intensity I ≈ 0.05 Is , where Is =2.5 mW/cm2 is the saturation intensity of the Li D2 transition.
While mechanical effects exerted by light on atoms form the basis of laser cooling, the effectof these forces on absorption imaging of atoms is usually neglected. However, for lightatoms, this effect can be significant. For Li, the scattering of a single photon of the 671-nm
imaging light imparts a momentum ~kL = 2π~/λ to the atom, leading to a Doppler shiftof the imaging transition by δrec = ~kL2 /mLi = 0.025 ΓD2 , where ΓD2 = 36.897 µs−1 is the
spontaneous emission rate from the Li 2 P3/2 state [McA96]. For imaging Li atoms, we choose
a reference set of conditions. The imaging pulse duration is set to 18 µs, the light intensity toI ≈ 0.05 Is and the detuning δ0 is adjusted to obtain the maximal value of NI . Under these
conditions, we expect the radiation pressure to lead to a mean laser detuning during the4.5. SUPPLEMENTAL MATERIAL

69

Figure 4.9: Li atom number determined by absorption imaging in the presence of a magnetic fieldof 1150 G as a function of the duration of the imaging light pulse. The lines show a multivariate fit
to the model of Eq. (4.6). The dashed line indicates the Li atom number that would be obtainedusing a weak resonant light pulse.

imaging pulse that is smaller than 0.1 ΓD2 , corresponding to a small effect on the measured
atomic absorption.
We experimentally investigate the mechanical effects of the light on the atoms by varying thelaser detuning δ and the duration of the imaging pulse. Fig. 4.9 shows the number NI as afunction of the duration t of the imaging pulse for different laser detunings. The red, orange,
green and cyan points correspond to (δ0 − δ) / (2π) = 9 MHz, 7 MHz, 5 MHz and 3 MHz,
respectively. The single blue point corresponds to the reference imaging conditions.
For pulse durations that are significantly longer than our reference pulse duration, we observea large effect of the radiation pressure. We model this effect by the following set of differential
equations:

dNγ
dt
dη
dt
dδ
dt

ΓD2
s
η,
2 1 + s + δ 2 / (Γ/2)2dNγ= − (1 − r)
η,
dt
dNγ
= δrec,dt=

4.5

where s = I/Is , the parameter r is the branching ratio to the Li|2⟩ state, and Nγ is themean number of photons scattered per atom, and η is the fraction of atoms remaining in

70

4.5. SUPPLEMENTAL MATERIAL

the nearly closed two-level system. The time t = 0 corresponds to the start of the imaging
pulse, with η (0) = 1.
In the limit of r = 1 and small s, these equations can be solved by separation of variables toyield:
δ (t) =

ΓD2
g (δrec st) ,2Nγ (t) = [g (δrec st) − g (0)] /δrecΓD2 /2,

where g (u) satisfies
g (u)2 − g (0)2 g (u)4 − g (0)4+=u2
4with g (0) = 2δ (0) /Γ. The number NI is be proportional to the mean number of photons
scattered per unit time Nγ /t. Therefore, we write
NI= NI0 [g (δrec st) − g (0)] / (δrec st) ,4.6

where NI0 is the value of NI determined in the limit of a weak, resonant light pulse.
We fit the model of Eq. (4.6) to the data from Fig. 4.9 with δ0 , NI0 and s as free parameters
and obtain δ0 = −2π × 1.2 (1) MHz, s = 0.043 (2), and NI0 = 2.02 (4) × 105 . The fitted valueof NI0 is indicated by the dashed line in the Fig. 4.9.
The above model implies that, under the reference conditions, δ0 /2π = −1 MHz and
NIhigh−B /NI0 = 0.96 (1), with each atom scattering a mean number of 12 photons during theimaging pulse. Using Eq. (4.5) to include the effects of saturation and branching ratio, we obtain a relation between NIhigh−B and the true Li atom number NLi as NIhigh−B = 0.91 (2) NLi .
Including the effects of the imperfect light polarization and the uncertainty in the magnification, we obtain NIhigh−B = 0.89 (5) NLi .

Fitting of Li Fermi profiles
Another method to determine the Li atom number is to image a degenerate Li atom cloudafter releasing the atoms from a trap. In the zero-temperature limit, the spatial extent of
the imaged cloud is determined by the trap frequencies, the time elapsed after release fromthe trap and the atom number.
To account for the finite temperature of the Li atoms in our experiment, we use a nondegenerate sample of K atoms to measure the temperature of the Li atoms. We prepare
a mixed sample of approximately 2.5×105 Li|2⟩ atoms and 2×104 K|1⟩ atoms at 1150 G
in a deep crossed-dipole trap with frequencies frK = 408 (1) Hz, frLi = 649 (3) Hz, fzK =56.1 (4) Hz, fzLi = 89.1 (2) Hz. After waiting 0.5 s for the Li and K to thermalize, we release
the atoms from the trap. By imaging the non-degenerate K cloud after 5.5 ms time of flight,
we determine the temperature of the atoms to be T = 370 (15) nK.

Figure 4.10: Fraction of the imaging light absorbed by a Li atomic cloud after release from an
optical trap (a), together with residuals of a fit to the data (b).

71

72

4.5. SUPPLEMENTAL MATERIAL

We image the Li atoms after tTOF =2.5 ms time of flight using the reference imaging pulse
described in Section 4.5.4. By averaging 45 absorption images, we obtain the image shown in
Fig. 4.10. We fit the obtained absorption data to a function of the form A + e−σn2D , where
!!
2m T2
kB
mLi u2 (X − X0 )2 mLi u2 (Z − Z0 )2Li
n2D = − 2 3 Li Li2 −q exp −−4π ~ fr2χr M 2 kB T
2χz M 2 kB T
is the 2D
profile with Li the polylogarithm function andq Fermi atom density
2
Li tχr,z = 1 + 2πfr,z
TOF . The fit parameter A accounts for technical offsets in the absorption data while X0 and Y0 fit the location of the cloud center on the camera image. The
fit parameters q and σ correspond to the atoms’ fugacity and the light absorption crosssection. The fit residual is shown in Fig. 4.10b. Taking into accounts the uncertainties in
the magnification, the temperature and the trap frequencies, we obtain σ = 0.82(9) × 6π/kL2 ,corresponding to NIhigh−B = 0.82(9) NLi .

LiK molecule dissociation and K number determination
Another method to determine the Li atom number is to associate LiK Feshbach molecules andto compare the number of K and Li atoms after dissociating the molecules. The advantageof this method is that the K atoms can be imaged on a closed transition and that, being
6.6 times heavier than the Li atoms, the K atoms are much less affected by the radiationpressure of the imaging light.
We associate the Li|1⟩K|3⟩ molecules by a magnetic ramp across the FR and then thoroughly
clean our trapped sample from any remaining free Li and K atoms by a combination of radiofrequency and laser light pulses; see [Jag14] for details. The obtained molecule samples are
essentially pure, consisting of approximately 1.5 × 104 molecules and less than 300 remaining
free Li and K atoms.
We subsequently dissociate the molecules by an inverse ramp across the FR and determine
number of the free Li|1⟩ and K|3⟩ atoms via absorption imaging at a magnetic field near
154.7 G. For imaging the Li|1⟩ atoms, we use the parameters from Section 4.5.6. We convert the atom number NIlow−B determined using these parameters to the number NIhigh−Bof Li atoms obtained by absorption imaging at 1150 G using the imaging ratios from Section 4.5.6.
We relate the number of atoms determined by absorption imaging of K|3⟩ atoms near 154.7 G
to the true K atom number in two steps. First, we transfer an independent sample of K atoms
from the state K|1⟩ to the state K|3⟩ using two consecutive resonant rf pulses. Imaging theseatoms before and after the transfer allows us to relate the number of K|3⟩ atoms determinedlow−B
by absorption imaging after molecule dissociation to the number NI,K
of K|1⟩ atoms
that would be measured by absorption imaging near 154.7 G. In the second step, using alow−B
similar procedure to the one described in Section 4.5.6, we compare NI,K
to the numberhigh−BNI,K
of K|1⟩ atoms determined by absorption imaging at 1150 G using a weak laserlight pulse. We then find for the relative atom number of K and Li determined at 1150 G:
high−B
NIhigh−B = 0.92(5) NI,K.

4.5. SUPPLEMENTAL MATERIAL

73

Accounting for the saturation of the K imaging light (s ≈ 0.05) and the errors in the lighthigh−Bpolarization and the magnification of the K imaging system, we can relate NI,K
to thehigh−B
true K atom number as NI,K
= 0.92 (5) NK .

Finally, assuming the real numbers of K and Li atoms after dissociation to be equal, we
obtain NIhigh−B = 0.85(7) NLi .

Measurement of rf shifts

A different method to determine the Li number is to use rf spectroscopy to measure theinteraction energy of the K atoms with the Li cloud. By comparing the measured data to thepredictions of a dressed quasiparticle model [Mas12] with accurately determined parametersof the FR, one can determine the mean Fermi energy of Li sampled by the K atoms. From
the knowledge of the Li temperature and trap frequencies, one can then determine the Li
atom number.
We prepare a sample of about 2.7×105 Li atoms and 2×104 K atoms at the temperature of
T = 290(15) nK in a crossed-beam optical dipole trap with trap frequencies frK = 395(2) Hz,
frLi = 632(3) Hz, fzK = 50.0(5) Hz, fzLi = 80(1) Hz. We use a Blackman-shaped rf π-pulse totransfer the K atoms from the K|2⟩ to the K|3⟩ state at various magnetic fields B near the154.7 G Li|1⟩-K|3⟩ Feshbach resonance. We compare the rf frequency at which we obtainmaximal transfer of the K atoms when the Li atoms are in the Li|1⟩ state (f ) to the frequency
for maximum transfer with the Li atoms in the Li|2⟩ state (f0 ).
Fig. 4.11 shows the frequency difference f − f0 as a function of the magnetic field near
154.7 G. We verify that f0 remains unchanged in the absence of the Li atoms. Therefore,h(f − f0 ) corresponds to the difference E3 − E2 of the mean interaction energies of the Katoms in the K|3⟩ and K|2⟩ states with Li atoms in the Li|1⟩ state.
We assume a uniform distribution of the Li atoms across the K cloud and use the mean Li
Fermi energy ϵF = ~2 κ2F /2mLi sampled by the K atoms as a free parameter. We calculate theinteraction energy E3 between the K|3⟩ atoms and Li|1⟩ atoms from a two-channel polaronmodel [Mas12] with the resonance parameters determined in Section 4.5.2: abg = 63.0 a0 ,reff = −5175 a0 , ∆B = 0.880 G. Since the interaction between the K|2⟩ atoms and Li|1⟩
atoms is weak, we may approximate E2 by the mean-field expression E2 = 2πabg ~2 κ3F / 6π 2 ,where a21 = 63 a0 is K|2⟩-Li|1⟩ scattering length [Nai11]. We use the position B0 of the
Feshbach resonance as the second free parameter. By fitting the above model to our data,
we find ϵF = h × 31.8 (4), B0 = 154.715 (1) G.
From the knowledge of the trap frequencies and the Li atom temperature, we can use ϵF todetermine the Li atom number as NLi = 275 (15) × 103 . Simultaneous with the above measurements, we record the number of Li atoms NIlow−B determined using absorption imaging
near 154.7 G as described in the Section 4.5.6. We use the conversion between NIlow−B and
NIhigh−B described in the same section to find NIhigh−B = 0.91 (7) NLi .

74

4.5. SUPPLEMENTAL MATERIAL

Figure 4.11: The measured shifts of the K|2⟩ →K|3⟩ rf transition frequency due to the presence of
Li|1⟩ atoms as a function of the magnetic field (dots) together with a fit using a quasiparticle model(see text).

4.5.5 Determination of Li atom number: summary
We summarize the results of our measurements of the Li atom number by the obtained
ratios NIlow−B /NLi . These ratios are subject to errors that are largely uncorrelated between
the different methods, with the notable exception of the error in the determination of themagnification of the imaging system. To obtain the best estimate for the ratio NIhigh−B /NLi ,we fix the magnification to a certain value M0 and calculate the mean µ (M0 ) and variance
V (M0 ) of the four results weighted by their inverse uncorrelated variances. We repeat thesame procedure with M0 sampled from a normal distribution whose mean and variancecorrespond to our experimental determination of M . We add the variance of µ due to thevariation in M0 to V to obtain:
NIlow−B /NLi = 0.86 (5) .
In the main text, we use a more conservative error estimate of 16% for the relative uncertainty
in the Li atom number determination. This corresponds to a relative uncertainty in the Fermi
energy of 6%.

4.5.6 Absorption imaging near 154.7 G
We commonly determine the Li atom number near the Li|1⟩-K|3⟩ FR at 154.7 G. We do thisby absorption imaging of Li|1⟩ atoms using σ − light on the Li|1⟩ (mI = 1, mJ = −1/2)→

4.5. SUPPLEMENTAL MATERIAL75

Li|3′ ⟩ (mI = 1, mJ = −3/2) transition. The dominant loss channel for this imaging transitionis the spontaneous decay from the Li|3′ ⟩ state to the Li|5⟩ (mI = 0, mJ = +1/2) state. We
calculate the corresponding branching ratio as 4% using the method from Section 4.5.4. The
other loss processes are at least three orders of magnitude less probable. We use an 8-µs
long imaging pulse whose intensity corresponds to s = 0.26 and whose frequency is adjusted
to obtain the maximal value of NI .
We calibrate this imaging method relative to the imaging of Li atoms in the |2⟩ state at themagnetic field of 1150 G as follows. We first prepare a Li|2⟩ sample at the field of 1150 G
and image it as described above. Then, in a separate experiment, we ramp the magnetic
field to 154.7 G, followed by a ramp back to 1150 G. We verify that these ramps lead tothe loss of less than 5% of the atoms. In the third experiment, we execute only the first
field ramp and then use rapid adiabatic passage to transfer the Li atoms into the Li|1⟩ statewith efficiency larger than 98%. We image these atoms as described above to obtain theatom number NIlow−B . We use the relationship between NIlow−B and the average of the atom
numbers recorded at the magnetic field of 1150 G with and without the double field ramp
to determine NIhigh−B = 2.4 (1) NIlow−B .

4.5.7 Heating due to molecule formation
As explained in the main text, on the repulsive side of the FR, the K atoms can pair with
Li atoms to form molecules. By removing Li atoms from the Fermi sea and releasing energy,
this pairing process leads to heating with a corresponding increase in kB T /ϵF .
To estimate the effects of this decay to molecules, we approximate our system by a uniform
system with Li and K densities n̄Li and n̄K , respectively. Before the impurity atoms aretransferred into the interacting state, the system is in thermal equilibrium and we may write
the number and energy densities of the Li atoms asn̄Li = −f3/2 (−q0 ) /λ3dB

4.7

and
uLi = −f5/2 (−q0 ) ×

3 kB T0,
2 λ3dB

4.8

where q0 and T0 are the fugacity and the temperature of the Li atoms before the start of the1/2
is the thermal de Broglie wavelength of the Li atoms
experiment, λdB = 2π~2 /mkB T0
and f3/2 i.e. f5/2 are the polylogarithm functions [Hua87]. The Fermi energy ϵF of the Liatoms is related to n̄Li as
2/3~2
6π 2 n̄Li
4.9
ϵF =
2mLi
and to the fugacity as
kB T
ϵF


=

2/3
4
−1
√
.
3 π f3/2 (−q0 )

4.10

76

4.5. SUPPLEMENTAL MATERIAL

The heating due to molecule formation progresses during the interaction time. To estimate
the effect of this heating on our measurements of the decoherence rate, we choose a typical−1
interaction time τD = γcoh
. We also assume that each decay event removes a single Li atomfrom the Fermi sea. We also assume that only 50% of the K atoms participate in the decay
since the other 50% are in the non-interacting state during the echo sequence. This implies
that the fraction l = ΓτD n̄K /2n̄Li of the Li atoms will be converted to molecules. As the
decay rate Γ, we take the sum of the two-body and three-body decay rates from [Koh12b].
We further assume that each decay to molecules releases energy αEF equal to the difference
between the repulsive polaron energy and the middle of the molecule-hole continuum from
[Koh12b]. Since the Li is much lighter than LiK, we assume that the full energy released tothe decay is delivered to the Li sea. Finally, we assume that remaining Li atoms thermalize
with each other.
Under these conditions, we may express the number density n̄′Li , the energy density u′Li and
the Fermi energy ϵ′F of the Li atoms during the experiment as:n̄′Li = (1 − l) n̄Li ,u′Li = (1 − l) uLi + lαnLi ϵF,0 ,ϵ′F= ϵF (1 − l)2/3 .

Using Eqns. (4.7-4.10), we obtain−f3/2 (−q1 ) (T1 /T0 )3/2 = − (1 − l) f3/2 (−q0 ) ,
5/3−f5/2 (−q1 ) (T1 /T0 )5/2 = − (1 − l) f5/2 (−q0 ) + l −f5/2 (−q0 )(π/6)1/3 α ,where q1 and T1 are the average fugacity and temperature of the Li atoms during the experiment. Using Eq. (4.9), we can then obtain the average degeneracy parameter kB T1 /ϵ′F .
Fig. 4.12 shows the predicted typical degeneracy parameter kB T1 /ϵ′F of the Li atoms in the
spin-echo measurements as a function of the interaction parameter, for the parameters from
Fig. 4.3 of the main text.

Figure 4.12: The predicted degeneracy parameter of the Li atoms during the echo sequence computed
for kB T0 /ϵF = 0.16, n̄K /n̄Li = 0.27 and the typical interaction time τD , as a function of the interaction
parameter. The line shows a second-order interpolation between the data points.

77

Chapter5

Preprint: Ultrafast many-bodyinterferometry of impurities
coupled to a Fermi sea1

Preprint, arXiv:1604.07423
submitted 19 February 2016

M. Cetina1,2 , M. Jag1,2 , R. S. Lous1,2 , I. Fritsche1,2 , J. T. M. Walraven1,3 , R. Grimm1,2 , J.
Levinsen4 , M. M. Parish4 , R. Schmidt5,6 , M. Knap7 , and E. Demler6
1

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands
4
School of Physics and Astronomy, Monash University, Victoria 3800, Australia
5
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
6
Department of Physics, Harvard University, Cambridge, MA 02138, USA
7
Department of Physics, Walter Schottky Institute and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany

1

The author of the present thesis contributed to developing the experimental procedures together with M.C.,
performed the measurements, and analyzed the data with M.C., R.S.L. and I.F. under the supervision of
R.G. All authors contributed to writing the manuscript.

79

80

5.1. ABSTRACT

5.1 Abstract
The fastest possible collective response of a quantum many-body system is related to its
excitations at the highest possible energy. In condensed-matter systems, the correspondingtimescale is typically set by the Fermi energy. Taking advantage of fast and precise control ofinteractions between ultracold atoms, we report on the observation of ultrafast dynamics ofimpurities coupled to an atomic Fermi sea. Our interferometric measurements track the nonperturbative quantum evolution of a fermionic many-body system, revealing in real time theformation dynamics of quasiparticles and the quantum interference between attractive andrepulsive states throughout the full depth of the Fermi sea. Ultrafast time-domain methods
to manipulate and investigate strongly interacting quantum gases open up new windows on
the dynamics of quantum matter under extreme non-equilibrium conditions.

5.2 Introduction
Non-equilibrium dynamics of fermionic systems is at the heart of many problems in scienceand technology, from the physics of neutron stars and heavy ion collisions to the operation ofelectronic devices. The wide range of energy scales, spanning the low energies of excitations
near the Fermi surface up to high energies of excitations from deep within the Fermi sea,challenges our understanding of the quantum dynamics in such fundamental systems. The
Fermi energy EF sets the shortest response time for the collective response of a fermionicmany-body system through the Fermi time τF = ~/EF , where ~ is the reduced Planck
constant. In a metal, i.e. a Fermi sea of electrons, EF is in the range of a few electronvolts,
which corresponds to τF on the order of 100 attoseconds. Dynamics in condensed mattersystems on this timescale can be recorded by attosecond streaking techniques [Kra09] andthe initial applications were demonstrated by probing photoelectron emission from a surface[Paz15]. However, despite these spectacular advances, the direct observation of the coherent
evolution of a fermionic many-body system on the Fermi timescale has remained beyond
reach.
In atomic quantum gases, the fermions are much heavier and the densities far lower, whichbrings τF into the experimentally accessible range of typically a few microseconds. Furthermore, the powerful techniques of atom interferometry [Cro09] now offer the exciting opportunity to probe and manipulate the real-time coherent evolution of a fermionic quantum
many-body system. Such techniques have been successfully used, e.g. to measure bosonic
Hanbury-Brown-Twiss correlations [Sim11], demonstrate topological bands [Ata13], probe
quantum and thermal fluctuations in low-dimensional condensates [Gri12, Had06], and tomeasure demagnetization dynamics of a fermionic gas [Kos13, Bar14]. Impurities coupled toa quantum gas provide a novel and unique probe of the many-body state. Strikingly, they
allow direct access to the system’s wave function when the internal states of the impurities
are manipulated using a Ramsey atom-interferometric technique [Goo11, Kna12].
We employ dilute 40 K atoms in a 6 Li Fermi sea to measure the response of the sea to asuddenly introduced impurity. For near-resonant interactions, we observe coherent quantum

5.3. MAIN RESULTS

81

Figure 5.1: Illustration of the experimental setup and procedure. (A) Li (blue) and K (red) atoms
are held in a crossed-beam optical dipole trap. The magnetic field coils (gold) and the rf coil (blue) areused to manipulate the atoms. (B) An rf π/2 pulse is used to prepare the K atoms in a superpositionof internal Zeeman states as shown on a Bloch sphere. A second rf pulse is used to probe the final
state.

many-body dynamics involving the entire 6 Li Fermi sea. We also observe in real time theformation dynamics of the repulsive and attractive impurity quasiparticles. In the limit of
low impurity concentration, our experiments confirm that an elementary Ramsey sequenceis equivalent to linear-response frequency-domain spectroscopy. We demonstrate that our
time-domain approaches allow us to prepare, control, and measure many-body interacting
states.

5.3 Main results
Our system consists of a small sample of typically 1.5 × 104 40 K impurity atoms immersed
in a Fermi sea of 3×105 6 Li atoms (Sec. 5.5 and [Cet15]). The mixture is held in an optical
dipole trap (Fig. 5.1A) at a temperature of T = 430 nK after forced evaporative cooling.
Because of the Li Fermi pressure and a more than two times stronger optical potential
for K, the K impurities are concentrated in the central region of the large Li cloud. Herethey experience a nearly homogeneous environment with an effective Fermi energy of ϵF =kB × 2.6µK (Sec. 5.5), where kB is Boltzmann’s constant. The corresponding Fermi timeτF = 2.9 µs sets the natural time scale for our experiments. The degeneracy of the Fermi
sea is characterized by kB T /ϵF ≈ 0.17. The concentration of K in the Li sea remains low,with n̄K /n̄Li ≈ 0.2, where n̄Li (n̄K ) is the average of the Li (K) number density sampled by
the K atoms (Sec. 5.5).
The interaction between the impurity atoms in the internal state K|3⟩ (third-to-lowest Zeeman sublevel) and the Li atoms (always kept in the lowest Zeeman sublevel) is controlled

Contrast |S(t)|82

5.3. MAIN RESULTS
1

1

0.5

0.5

' (t) / 2:

0
1

A

B

0
1
0.5

0.5
0

E

0.0

C
0

D
10

t / =F

20 0

-0.5
10

t / =F20F
05t / =F

10

15

Figure 5.2: Impurity dynamics in the Fermi sea. (A and C) Contrast |S(t)| and phase ϕ(t) of
the interference signal depending on the interaction time t in the repulsive polaron regime for X =−0.23(6), with the rf pulse applied at X1 = −3.9. (B and D) Same quantities in the attractive
polaron regime for X = 0.86(6) and X1 = 5.8. (E and F) Same quantities for resonant interactions(X = 0.08(5), X1 = 4.8). The solid blue lines show the results of the TBM calculations. The solid(dashed) red lines show the results of the FDA calculations at the measured (at zero) temperature.
The shaded regions reflect the combined experimental uncertainties in X, kB T and ϵF . The errors
in the experimental data are typically smaller than the symbol size.

using a rather narrow (Sec. 5.5) interspecies Feshbach resonance near a magnetic field of
154.7 G [Nai11, Cet15]. We quantify the interaction
with the Fermi sea by the dimensionless√parameter X ≡ −1/κF a, where κF = ~−1 2mLi ϵF is the Li Fermi wavenumber with mLi
the Li mass, and a is the s-wave interspecies scattering length. While slow control of X isrealized in a standard way by variations of the magnetic field, fast control is achieved using
an optical resonance shifting technique [Cet15]. The latter permits sudden changes of X by
up to about ±5 within a time shorter than τF /15 ≈ 200 ns.
Our interferometric probing method is based on a two-pulse Ramsey scheme (Fig. 5.1B),
following the suggestions of Refs. [Goo11, Kna12]. The sequence starts with the impurity
atoms prepared in the spin state K|2⟩ (second-to-lowest Zeeman sublevel), for which thebackground interaction with the Fermi sea can be neglected. A first, 10-µs-long, radiofrequency (rf) π/2-pulse drives the K atoms into a coherent superposition between this noninteracting initial state and the state K|3⟩ under weakly interacting conditions (interaction
parameter X1 with |X1 | ≈ 5). Using the optical resonance shifting technique [Cet15], thesystem is then rapidly quenched into the strongly interacting regime (|X| < 1). After an
evolution time t, the system is quenched back into the regime of weak interactions and asecond π/2-pulse is applied. The population difference N3 − N2 in the two impurity states ismeasured as a function of the phase of the rf pulse [Cet15]. The contrast |S(t)| and the phase
ϕ of the resulting sinusoidal signal is finally determined as a function of t. In the limit of low
impurity concentration, the complex function S(t) = |S(t)|e−iϕ(t) can be interpreted as theoverlap of the interacting and the non-interacting components of the system’s wavefunction[Goo11]. The squared amplitude |S(t)|2 is then equivalent to the common definition of a
Loschmidt echo [Los76, Jal01].

5.3. MAIN RESULTS

83

We first consider the interaction conditions where polaronic quasiparticles are known toexist [Mas14]. Figures 5.2A-D show the evolution of the contrast and the phase measured in
the repulsive and the attractive polaron regimes, where X = −0.23(6) and X = +0.86(6),respectively. For short evolution times of up to about 4τF , we observe both contrast signals to
exhibit a similar initial parabolic transient, which is typical of a Loschmidt echo [Jal01]. For
longer times, this connects to an exponential decay of the contrast and a linear evolution of
the phase. In Ref. [Cet15], we showed that the long-time decay of the contrast in this regime
can be interpreted in terms of quasiparticle scattering. Here, the linear phase evolution
corresponds to the energy shift of the quasiparticle state, for which we obtain +0.29(1)ϵFfor the repulsive case in Fig. 5.2C and −0.27(1)ϵF for the attractive case in Fig. 5.2D.
Remarkably, while the long-time behavior reflects the quasiparticle properties, the observed
initial transient reveals the ultrafast real-time dynamics of the quasiparticle formation.
On resonance, for the strongest possible interactions, the quasiparticle picture breaks down.
Here our measurements, displayed in Fig. 5.2E and 5.2F for X = 0.08(5), reveal the striking
quantum dynamics of a strongly interacting fermionic system forced into an extreme nonequilibrium state. The contrast |S(t)| shows pronounced oscillations reaching zero, while thephase ϕ(t) exhibits plateaus. The revivals of the contrast |S(t)| indicate partially reversible
entanglement between the internal state of the impurity and the Fermi sea [Goo11]. This
process involves the whole Fermi sea and occurs on the fastest timescale available to thecollective dynamics of a fermionic system.
To further interpret our measurements we employ two different theoretical approaches: thetruncated basis method (TBM) (Sec. 5.5) and the functional determinant approach (FDA)[Kna12]. The TBM models our full experimental procedure assuming zero temperature andconsidering only single particle-hole excitations. This approximation, known as the Chevy
ansatz [Che06], has been successfully used to predict the properties of quasiparticles in coldgases [Mas14]. The predictions of the TBM are represented by the blue lines in Fig. 5.2. Thismethod accurately describes the initial transient, as well as the period of the oscillations ofS(t) on resonance. While the zero-temperature TBM calculation naturally overestimates thecontrast in the thermally dominated regime (t > 6τF ), it accurately reproduces the observed
linear phase evolution and thus the quasiparticle energy. The FDA is an exact solution for a
fixed impurity at arbitrary temperatures taking into account the non-perturbative creation of
infinitely many particle-hole pairs. The FDA calculation is represented by the solid red lines
in Fig. 5.2. We see remarkable agreement with our experimental results, which indicatesthat the effects of impurity motion remain small in our system. This observation can be
explained by the fact that our impurity is sufficiently heavy so that the effects of its recoilwith energies of about 0.25 ϵF (Sec. 5.5) are masked by thermal fluctuations. To identify
the effect of temperature, we performed a corresponding FDA calculation for T = 0 andshow the results as the dashed lines in Fig. 5.2. Here, we see a slower decay of |S(t)|, which
follows a power law at long times (Sec. 5.5) under the idealizing assumption of infinitely
heavy impurities.
Time-domain and frequency-domain methods are closely related, as is well known in spectroscopy. In the limit of low impurity density, where the interactions between the impurities
can be neglected, S(t) is predicted to be proportional to the inverse Fourier transform of thelinear excitation spectrum A (ω) of the impurity [Noz69]. To benchmark our interferometric84

5.3. MAIN RESULTS

A(!) / =F

3

A

1.5

B

2

1

1

0.5

0

C

0
-1

-0.5

0

(!-! ) =
0

0.5
F

-0.5

0

(!-! ) =
0

0.5
F

1

-1

0

(!-! ) =0

1
FFigure 5.3: Rf spectroscopy of an impurity in the Fermi sea. (A and B) show the rf spectra forthe repulsive (X = −0.23(6)) and the attractive (X = 0.86(6)) interactions, respectively. (C) shows
the rf spectrum for resonant interactions (X = 0.08(5)). The spectral data are normalized to unit
integral. The gray lines correspond to the numerical Fourier transform of the S(t) data from Fig. 5.2.
The width of the gray curve reflects the combined experimental errors in the S(t) data.

method, we measure A (ω) using rf spectroscopy similar to our earlier work [Koh12b], butwith great care to ensure linear response (Sec. 5.5). The measured excitation spectra areshown in Fig. 5.3. In the repulsive and attractive polaron regimes, we observe the characteristic structure of a peak on top of a broad pedestal [Mas14]. While the peak determines
the long-time evolution of the quasiparticle, the pedestal is associated with the rapid dynamics related to the emergence of many-body correlations. For resonant interactions, therf response is broad and nearly symmetric about ω0 , implying that the zero crossings ofS(t) are accompanied by jumps in its phase by π, as is seen in Fig. 5.2E and 5.2F. Basedon the observed spectral response, we interpret the oscillations of S(t) in Figs. 5.2E and5.2F as arising from simultaneous excitations of the two branches of our many-body systemcorresponding to the two humps in the rf spectrum.
A detailed comparison of our time- and frequency-domain measurements reveals the powerful capability of our approach to prepare and control many-body states. This is revealedin Fig. 5.2, where we show the Fourier transform of the S (t) data from Fig. 5.2 as thegray curves. We observe that time-domain measurements where the rf pulses are appliedin the presence of weakly repulsive interactions (Fig. 5.3A) emphasize the upper branchof the many-body system while in the attractive case (Fig. 5.3B,C), the lower branch isemphasized. We explain this observation by the action of the rf pulses to prepare weaklyinteracting polaron states (Sec. 5.5). Compared to the non-interacting initial state used in
the frequency-domain spectroscopy, these polarons have an increased wavefunction overlap
with the corresponding strongly interacting repulsive and attractive branches, leading to theobserved shift in the spectral weight. Our measurements demonstrate that the control overthe initial state of many particles can be used to precisely manipulate quantum dynamics
in the strongly interacting regime. This unique capability of time-domain techniques opens
up a wide range of applications, including the study of the dynamical behavior near thephase transition from a polaronic to a molecular system [Mas14] and the creation of specificexcitations of a Fermi sea down to individual atoms [Dub13].
Our interpretation of the results in Figs. 5.2 and 5.3 relies on the assumption that our
fermionic impurities are sufficiently dilute so that any interactions between them can beneglected. We can extend our experiments into a complex many-body regime where the5.4. CONCLUSION

85

1

Contrast |S(t)|

0.3

0.2

0

0

5

10

15

0.1

0
5

t / =F

10

15

Figure 5.4: Observation of induced impurity-impurity interactions. Resonant dynamics of the contrast is shown for X = −0.01(5), X1 = 5.2, ϵF = kB × 2.1(1) µK, kB T /ϵF = 0.24(2) and different
impurity concentrations n̄K /n̄Li . The black, green, and blue squares correspond to n̄K /n̄Li = 0.53,
0.33, and 0.20, respectively. The red circles correspond to the linear extrapolation of the complex S(t)data to the limit of a single impurity, taking into account the errors in the data. The inset reproduces
this extrapolation together with the highest-concentration data points. The red line shows the resultof the FDA calculation. The shaded region reflects the combined experimental uncertainties in X,
kB T and ϵF .

impurities interact both with the Fermi sea and with each other, by increasing the impurity
concentration (Sec. 5.5). Figure 5.4 shows the time-dependent contrast measured for kB T =0.24(2)ϵF and n̄K /n̄Li =0.20, 0.33, and 0.53. An extrapolation of the S(t) data to zeroconcentration (open red circles) lies close to the data points for n̄K /n̄Li =0.20, which is
the typical concentration in our measurements, and agrees with the FDA calculation. Thisconfirms that the physics that we access in the measurements with a small sample of fermionic
impurities is close to that of a single impurity, which we posit to be a consequence of thefermionic nature of the impurities. When the impurity concentration is increased, we findthat the contrast for t > 5τF is decreased and the period of the revivals of |S(t)| is prolonged.
We interpret this as arising from effective interactions between the impurities induced by
the Fermi sea [Mor10, Yu10]. Such interactions between fermionic impurities are predicted
to lead to novel quantum phases [Zwe12].

5.4 Conclusion
Our results demonstrate the power of many-body interferometry to study ultrafast processes in strongly interacting Fermi gases in real time, including the formation dynamics ofquasiparticles and the extreme non-equilibrium dynamics arising from quantum interference
between different many-body branches. Of particular interest is the prospect of observing
Anderson’s orthogonality catastrophe (Sec. 5.5 and [Kna12]) by further cooling the Li Fermi
sea [Har15] while pinning the K atoms in a deep species-selective optical lattice [LeB07].

86

5.5. SUPPLEMENTARY MATERIALS

Acknowledgements
We thank M. Baranov, F. Schreck, G. Bruun, N. Davidson and R. Folman for stimulating
discussions. We acknowledge support by the Austrian Science Fund FWF within the SFB
FoQuS (F40-P04). R. S. was supported by the NSF through a grant for the Institute for
Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory. M. K. acknowledges support from Technical Universityof Munich - Institute for Advanced Study, funded by the German Excellence Initiative andthe European Union FP7 under grant agreement 291763. E. D. acknowledges support from
Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR Quantum Simulation MURI,
the ARO-MURI on Atomtronics, and support from Dr. Max Rössler, the Walter HaefnerFoundation, the ETH Foundation, and the Simons Foundation.

5.5 Supplementary materials
5.5.1 Theoretical description
In this section, we summarize the approaches that we developed to theoretically model
the results of our interferometric Ramsey experiments. We first discuss the microscopic
model that we use to describe the narrow Feshbach resonance of the Li-K mixture, andthen we outline how we calculate the time evolution of the system within two approaches:
the Truncated Basis Method (TBM) and the Functional Determinant Approach (FDA). In
this section, we assume that a ‘perfect quench’ is performed, where the impurity is initiallynon-interacting with the Fermi sea and there are no interactions during the radio-frequency(rf) pulses. A discussion of the role played by interactions during the rf pulses is deferred to
Section 5.5.3.

Narrow Feshbach resonance model for Li-K mixtures
In our experiment, the K impurities are concentrated in the central region of the Li Fermi
gas where they experience a nearly uniform Li environment (see Section S5.A). Hence weconsider in our model K impurities that are immersed in a Li Fermi gas of uniform density.
The Li-K mixture is prepared at magnetic fields near a closed-channel dominated Feshbachresonance between the Li|1⟩ and K|3⟩ states that occurs near 155 G. The narrow character ofthis resonance is a consequence of the limited strength of the coupling of atoms in the open
channel to a closed-channel molecular state. To describe this system we use the two-channel
Hamiltonian
X
X
X
Ĥ =
ϵk,Li ĉ†k ĉk +ϵk,K dˆ†k dˆk +[ϵk,M + ϵM (B)] b̂†k b̂k
k

k

k



g X
+√
χ(k) b̂†q ĉq/2+k dˆq/2−k + dˆ†q/2−k ĉ†q/2+k b̂q ,
V k,q

5.1

5.5. SUPPLEMENTARY MATERIALS

87

where the first line defines the non-interacting Hamiltonian Ĥ0 . Here, V is the total system
volume, ĉ†k (ĉk ) creates (annihilates) a Li fermion with momentum ~k and single-particle~2 k2
energy ϵk,Li = 2m
, and dˆ†k (dˆk ) creates (annihilates) a K impurity atom in the K|3⟩ state
Li
2 2

~ k
with dispersion ϵk,K = 2m
, where we define k ≡ |k|. The closed-channel molecule is createdKk(annihilated) by b̂†k (b̂k ). It has the dispersion ϵk,M = 2(m~K +m, and a bare energy relativeLi )to the scattering threshold, ϵM (B) = δµ(B − Bc ). Here δµ is the differential magnetic
moment between the open and closed channels, and Bc denotes the threshold crossing of thebare molecular state [Chi10].
2 2

Close to the Feshbach resonance, the scattering length a diverges and the interaction between
the K impurities and the Li atoms is predominantly mediated by exchange of the closedchannel molecule. We therefore neglect the background scattering potential in the open
channel [Nai11]. The strength of the coupling between the open and closed channels is given
by g, and we take a form factor χ(k) = 1/[1 + (r0 k)2 ], which accounts for the finite extent
r0 of the closed-channel wave function ∼ e−r/r0 /r.
The parameters of the model δµ, Bc , g, and r0 are fully determined by known experimental parameters. First, the differential magnetic moment has recently been measured to beδµ = h×2.35(2) MHz/G [Cet15]. Second, close to resonance, the scattering length may be
parametrized as


∆B
∆B
a = abg 1 +
≈ abg
,
5.2
B0 − B
B0 − B
where B0 is the center of the Feshbach resonance with width ∆B = 0.880 G and backgroundscattering length abg = 63.0 a0 [Nai11]. To connect with our model, we consider the on-shell
two-body scattering amplitude f (k), which for the Hamiltonian 5.1 is given by [Sch12]−1µred 2~2 k 2g 2 µred2f (k) =g χ(k) −+ ϵM (B) −
,
2π~2
2µred
4π~2 r0 [1 − ikr0 ]2

5.3

where µred = mLi mK /(mLi + mK ) is the reduced mass and k is the relative scattering wave
vector. Using the low energy expansion f −1 (k) ≈ −a−1 + 21 reff k 2 − ik, with reff the effective
range, we thus identify
1

a=

,
1
∗
2
2r0 + 2R µred δµ(B − Bc )/~reff = −2R∗ + 3r0 − 4r02 /a,

5.4
5.5

where R∗ ≡ ~4 π/(µ2red g 2 ) is the range parameter of the Feshbach resonance [Bru04, Pet04a].
Comparing Eqs. (5.2) and (5.4) yields~2,
2µred abg δµ∆B
1
B0 − Bc = ∆Babg /r0 .
2
R∗ =

5.6
5.7

Equation (5.6) relates R∗ , and thus the coupling constant g, to the known experimental
parameters. The extent of the closed-channel wave function r0 in turn follows by comparing

88

5.5. SUPPLEMENTARY MATERIALS

Eq. (5.7) to the theoretical prediction from quantum defect theory [Gór04, Szy05], B0 −Bc =abg ∆B/ā, where ā = 0.955lvdw and lvdw = 40.8 a0 is the van der Waals length [Nai11]. Thus
we obtain r0 = ā/2. Finally, B0 was obtained in Ref. [Cet15], allowing the determination ofBc .

Truncated Basis Method
To model a mobile impurity as in the experiment, we consider an approximate wave function
for the zero-momentum impurity that incorporates the scattering of a single particle out of
the Fermi sea:XX|ψα ⟩ = α0 dˆ†0 |FS⟩ +αq b̂†q ĉq |FS⟩ +αk,q dˆ†q−k ĉ†k ĉq |FS⟩.
5.8
q

k,q

Here, the first term on the right hand side describes the product state of the impurity K
atom at zero momentum and the ground state of the non-interacting Li Fermi sea |FS⟩ =Q†|k|<kF ĉk |0⟩, where kF is the Fermi momentum, which is related to the Fermi energy by
ϵF = ~2 kF2 /(2mLi ). The last two terms correspond, respectively, to the impurity binding a Li
atom to form a closed-channel molecule, and the impurity exciting a particle out of the Fermisea, in both cases leaving a hole behind. When using the TBM, we focus on zero temperaturein order to capture the purely quantum evolution of the impurity. For convenience, within
this model we also take r0 → 0, which formally requires taking the bare crossing Bc → ∞to keep a finite. This approximation is justified, as R∗ exceeds r0 by about two orders ofmagnitude.
Truncated wave functions of the form 5.8 have been used extensively in the study of Fermi
polarons in ultracold atomic gases, starting with the work of Chevy [Che06]. While most of
the previous work has focused on equilibrium properties, recently it has been proposed thatthese wave functions may be extended to dynamical problems using a variational approachto obtain the equations of motion [Par13], for instance to calculate the decay rate of excitedstates.
Here, we adapt the use of truncated wave functions for the Fermi polaron to the calculation of
the dynamical response of the impurity to an interaction quench. For a perfect quench and atzero temperature, the quantity measured in experiment corresponds to the overlap between
the interacting and non-interacting states of the system, i.e., we have [Goo11, Kna12]
S(t) = ⟨ψ0 (t)|ψint (t)⟩ = ⟨ψ0 | eiĤ0 t/~ e−iĤt/~ |ψ0 ⟩ .

5.9Here |ψ0 ⟩ ≡ dˆ†0 |FS⟩ is the initial non-interacting state of energy E0 , and ψint (t) is the state
after a quench at time t = 0 from zero to finite impurity interactions with the Fermi sea.
Formally expanding in a complete set of states for the single impurity problem, the Ramsey
signal 5.9 then becomesX
S(t) =|⟨ψ0 |φj ⟩|2 e−i(Ej −E0 )t/~ ,5.10
j5.5. SUPPLEMENTARY MATERIALS

89

where |φj ⟩ is an eigenstate of the interacting Hamiltonian with energy Ej . However, thisrequires one to solve the entire problem which is generally not possible for a mobile impurity.
Thus, within the Truncated Basis Method (TBM), we restrict the Hilbert space to wave
functions of the form 5.8 and diagonalize the Hamiltonian within this truncated basis. As
we shall see, this truncation permits an extremely accurate description of the initial quantum
dynamics of the impurity.
For small t, we expand e−iĤt/~ to findS(t) ≈ 1 − (t/τF )2(1 + mLi /mK )2,3πkF R∗

5.11

with τF the Fermi time. This reveals that the short-time dephasing dynamics of S(t) iscompletely determined by the two-body properties, which are captured exactly by the TBM.
As we will see below, the TBM describes the impurity behavior also beyond the two-bodytimescale since higher order correlations and multiple particle-hole excitations take longer
to build up. Indeed, for a mobile impurity and for sufficiently weak attraction where theattractive polaron is the ground state, the TBM correctly describes the long-time behavior
S(t) → |α0 |2 e−iεp t/~ . Here, |α0 |2 is the polaron residue (squared overlap with the noninteracting state) and εp is the polaron energy, which are both accurately determined usinga wave function of the form 5.8 [Vli13].
With the TBM we consider zero temperature in order to isolate the quantum dynamics
of the impurity. To better model the experiment, in principle one can extend the TBM to
finite temperature by taking the initial state to be a statistical thermal distribution involving
multiple impurity momenta. However, a more convenient approach at finite temperature is
described in the next section.

Functional Determinant ApproachAt times t substantially exceeding τF , the full description of the impurity dynamics requires the inclusion of multiple particle-hole pair excitations as well as the effect of finite
temperature, both of which present a theoretical challenge. In order to study and describe
both effects, we employ the Functional Determinant Approach (FDA) [Lev96, Lev93, Kli03,
Kna12].
In the FDA the impurity is treated as an infinitely heavy object. In this limit, the FDA
provides an exact solution of the dynamical many-body problem at arbitrary temperatures
and times. The justification of the infinite mass approximation, which will be discussed in
more detail in Section 5.5.2, is rooted in two observations. First, in our experiment, the massof the K impurities is much larger than that of the Li atoms (mass ratio mK /mLi ≈ 6.7) whichconstitute the surrounding Fermi gas. Therefore, the recoil energy gained by the K impurities
due to the scattering with a Li atom is small. We estimate the typical recoil momentum kR byaveraging over all possible scattering processes on the Fermi surface, yielding kR = 4kF /3.
mLi
From that we obtain an estimate for the typical recoil energy ER = 16
9 mK ϵF ≈ 0.25ϵF ,which determines a typical time scale τR = ~/ER ≈ 4τF , up to which one expects recoil to905.5. SUPPLEMENTARY MATERIALS

have a minimal effect on the many-body quantum dynamics, cf. Section 5.5.2. Second, attimes exceeding the thermal time scale τT = ~/(kB T ), which in our experiment is given byτT ≈ 6τF , thermal effects due to the averaging over various statistical realizations become
relevant. The resulting thermal fluctuations disrupt the coherent quantum propagation of
the impurity, and hence, for times t > τT , mask the effect of recoil [Ros99].
To a good approximation, we may thus take the limit of infinite impurity mass, which admits
the mapping of Eq. (5.1) onto the bilinear HamiltonianX
X
Ĥ = ϵM (B)m̂† m̂ +ϵk ĉ†k ĉk + g
χ(k)[m̂† ĉk + m̂ĉ†k ].
5.12
k

k

Here, m̂† is the creation operator of the localized closed channel molecule and the interaction
is described by the annihilation of a Li atom converting the empty impurity molecular state
into an occupied one. By taking the limit mK → ∞ we obtain a modified reduced mass µ′red =mLi , which differs by a factor of 40/46 from the experimental one. This needs to be taken
into account when identifying the microscopic parameters. To ensure, in particular, thatthe off-diagonal coupling g in Eq. (5.12) remains of the same strength as in the experiment,a reduced resonance parameter R′∗ = (40/46)2 R∗ has been used, which we do for all data
shown in the main text. Using these identifications, the model Eq. (5.12) also accurately
describes the short-time dynamics as given by Eq. (5.11), cf. Fig. 5.2 in the main text.
The calculation of time-resolved, many-body expectation values such as Eq. (5.9) at arbitrary
temperature presents a theoretical challenge. However, for the model (5.12), we are able tocalculate the time-resolved Ramsey response in an exact way using the FDA [Kli03, Kna12].
This is based on the observation that for bilinear Hamiltonians thermal expectation values
in the many-body Fock space can be reduced to determinants in the single-particle space byvirtue of the identityTr[ρ̂ eŶ1 eŶ2 . . .] = det[1 − n̂ + n̂ eŷ1 eŷ2 . . .].

5.13

Here Ŷ1 , Ŷ2 , . . . are many-body operators, ŷ1 , ŷ2 , . . . are their single-particle counterparts, ρ̂is the many-body density matrix describing the state of the system, and n̂ = 1/[eβ(ĥ0 −µ) + 1]
is the occupation operator defined in the single-particle space, with µ the fermion chemical
potential. A specific example for Eq. (5.13) is the perfect quench Ramsey response, which
at finite temperature is given by [Kna12]
S(t) = Tr[ρ̂ eiĤ0 t e−iĤt ] = det[1 − n̂ + n̂ eiĥ0 t e−iĥt ].

5.14

P
Here, Ĥ0 = k ϵk ĉ†k ĉk is the free Hamiltonian of the Li Fermi gas and Ĥ is the Hamiltonianin the presence of impurity scattering given in Eq. (5.12), while ĥ0 and ĥ are their singleparticle counterparts. A numerical evaluation of Eq. (5.14) then only requires a calculation of
the single particle orbitals and energies in order to obtain the single-particle determinant.

5.5.2 Role of physical processes on different time scales
The combination of both our theoretical approaches allows us to accurately model the physics
at various time scales in our experiment. Making use of the fact that the FDA and the TBM

5.5. SUPPLEMENTARY MATERIALS

91

differ distinctly in their treatment of multiple particle-hole excitations, the impurity mass,and finite temperature, we can use a comparison of their predictions to determine the role ofthese processes and effects in the many-body non-equilibrium dynamics of our experiment.
To keep the analysis transparent, in this section we still assume that a perfect quench is
performed.

Multiple particle-hole excitations
In order to analyze the role of multiple particle-hole excitations, we first consider the limitof a fixed (infinitely heavy) impurity at zero temperature. In this scenario, the FDA yields
the exact solution of the impurity problem. Since, in this case, the TBM only differs fromthe FDA by its neglect of multiple particle-hole excitations, a comparison of the predictionsof the two methods allows us to isolate the effect of these excitations.
1

1

0.5

0.5

A

0

B

E

0

1

1
0.5

0.5

0

0

C
0

-0.5

D
5

10

15

20 0

5

10

15

20

-1

F
0

5

10

15

20

Figure 5.5: Effect of multiple particle-hole fluctuations. Taking the idealizing limit of zero temperature and infinite impurity mass, we compare the Ramsey response for a perfect quench (top:amplitude, bottom: phase) obtained exactly with the FDA (red, long dashed) to the one obtainedwith TBM (blue, short dashed) for (A, C) X = −0.23, (B, D) X = 0.86, and (E, F) X = 0.08. For
this comparison, we take r0 = 0 and kF R′∗ = 1.1(40/46)2 .

In Fig. 5.5 we display the predictions for the Ramsey response using the two theoretical
approaches. We find that both theoretical predictions agree extremely well at short times.
In particular, for both the amplitude and phase of S(t), our results imply that multiple
particle-hole excitations start to influence our observables at a time scale of around 6τF , andonly become prominent beyond 10τF . Thus, at shorter time scales, multiple particle-holeexcitations can be neglected when predicting the results of the Ramsey measurements.
We note that the fixed impurity scenario is a worst-case scenario for the TBM: At T = 0,
the infinitely heavy impurity is subject to the orthogonality catastrophe with an associatedpower-law decay of the Ramsey contrast at long times [And67]. This decay, which arises due
to an infinite number of particle-hole fluctuations and which leads to a vanishing quasiparticle

92

5.5. SUPPLEMENTARY MATERIALS

weight, is exactly incorporated in the FDA. By contrast, in the long-time limit, the TBM
predicts the saturation of |S(t)| to a constant value (see Fig. 5.5), corresponding to a spurious
finite residue. However, for a mobile impurity at zero temperature, recoil becomes relevant.
These recoil effects lead to the absence of the orthogonality catastrophe [Ros99], and thus
to an increased accuracy of the TBM in the case of finite impurity mass.
Generally, one expects that the relevant time scale for multiple particle-hole excitations is
closely related to the Fermi time τF . As discussed above, we find that such excitations
become relevant for a description of S(t) only at around 6τF or beyond. This observation
can be understood in a twofold way. First, in the equilibrium case it was found that contact interactions in the Fermi polaron problem lead to an approximate cancellation of terms
involving identical fermions, thus suppressing the emergence of multiple particle-hole fluctuations [Com08]. Our observation may hence be interpreted as a generalization of thesefindings to the non-equilibrium case. Second, the spectrum of the Fermi polaron problem
features a dominant contribution involving the excitation of fermions from the bottom of
the Fermi sea to the Fermi surface [Kna12]. As discussed in Ref. [Kna12], these excitations
manifest themselves as oscillations with period 2πτF in the Ramsey contrast |S(t)|. Such abottom of the band excitation is also present in the truncated wavefunction 5.8, and indeed
the remarkable agreement of the TBM with the exact solution from the FDA up to the time
2πτF suggests that this effect can be captured by single-particle hole excitations.

Impurity mass
1

1

0.5

0.5

A

0

B

E

0
1

1

0.5
0

0.5

0

C
0

-0.5

D
5

10

15

20 0

5

10

15

20

-1

F
0

5

10

15

20

Figure 5.6: Effect of the impurity motion on the short-time dynamics. Amplitude (top) and phase(bottom) of the perfect quench zero temperature Ramsey response S(t) as a function of time for (A,C) X = −0.23, (B, D) X = 0.86, and (E, F) X = 0.08. We compare the results of the TBM obtained
for mK = (40/6)mLi and kF R∗ = 1.1 (solid) with the TBM results for fixed impurities mK → ∞ andkF R′∗ = 1.1(40/46)2 (dashed).

As discussed in the main text, our experimental findings are well described by the static
impurity approximation, although the impurity has finite mass. To quantify the effect of the5.5. SUPPLEMENTARY MATERIALS

93

finite impurity mass, we study here the case of zero temperature. This allows us to isolate theeffect of finite impurity recoil from the influence of thermal fluctuations, which will become
dominant beyond times τT ≈ 6τF , as discussed in the section below. In order to estimate
at which time scale recoil becomes important, we make use of the capability of the TBM
to describe impurities of arbitrary mass. Furthermore, our analysis in Sec. 5.5.2 shows that
the TBM yields highly accurate results for the short-time dynamics of S(t). Accordingly,
in Fig. 5.6 we display the Ramsey response for a static impurity and for the experimentally
relevant impurity mass, both calculated within the TBM. We see that for both amplitude andphase, the impurity motion only results in a small difference in the Ramsey signal at times
t . 4τF . Physically, this time scale corresponds to the effective recoil time τR associated with
Li collisions on K atoms, which we estimated in Sec. 5.5.1 to be τR ≈ 4τF , in agreement with
our findings here. At times exceeding τR , we find that the dynamics is indeed affected by thefinite impurity mass. However, at such times, thermal fluctuations dominate the behavior
in experiment, as we now discuss.

Temperature
1

1

0.5

0.5

A

0

B

E

0

1

1
0.5

0.5

0

0

C
0

-0.5

D
5

10

15

20 0

5

10

15

20

-1

F
0

5

10

15

20

Figure 5.7: Effect of finite temperature on the impurity dynamics. We compare the Ramsey signal(upper panels: amplitude, lower panels: phase) for an infinitely heavy impurity obtained from anexact FDA calculation at zero (long dashed) and finite temperature (solid curves). The ordering of
the graphs is as in the main text: (A, C) X = −0.23, T /TF = 0.17, (B, D) X = 0.86, T /TF = 0.16,
and (E, F) X = 0.08, T /TF = 0.18. We assume a perfect quench and choose r0 = 0 as well as
kF R∗ = 1.1, i.e., kF R′∗ = 1.1(40/46)2 .

At long times, the time evolution reduces to a simple exponential decoherence of S(t). The
time scale at which this crossover to exponential decay takes place is given by the thermal
time scale τT . In our experiment, where T /TF ≈ 0.15, this corresponds to τT ≈ 6τF and,hence, we observe both regimes within the dynamical range probed in our experiment.
In this section, we use finite-temperature FDA calculations to gauge the role of temperaturein the impurity dynamics. To this end we compare the results for the Ramsey signal at zero

94

5.5. SUPPLEMENTARY MATERIALS

and finite temperature for the experimentally realized parameters. The results are shownin Fig. 5.7. We indeed find that at times ∼ 6τF the time evolution at finite temperaturestarts to deviate from the purely quantum behavior. Finite temperature leads to an exponential decoherence of the Ramsey signal and has the consequence that thermal fluctuations
dominate over the impurity motion at times t & 6τF [Ros99]. Hence they mask the effect of
impurity recoil as discussed in Sec. 5.5.1.
Overall, the conditions in our experiment give rise to three competing time scales. Multiple
particle hole excitations become relevant for our measurement of S(t) at around 6τF , therecoil time is τR ≈ 4τF , and the thermal scale is set by τT ≈ 6τF . A comparison of thesescales reveals the reason for the remarkable agreement between the FDA and experiment:
Recoil is only weakly probed at short times t < τR , while its effect is washed out by thethermal fluctuations at long times t > τT ≈ τR .

5.5.3 Role of interaction during finite-length rf pulses
In this section, we analyze the role of the ‘imperfect’ interaction quench in our experiments,
where residual interactions are present during the rf pulses. Furthermore, we discuss how
our findings pave the way towards the use of our experimental techniques to exert control
over many-body states in real time.

Idealized versus realized Ramsey scenario
Thus far, we have assumed the idealized scenario of a perfect two-pulse Ramsey scheme. In
this case, the initial spin state of the impurity (K|2⟩ in the experiment) is non-interacting with
the Li Fermi sea and there are no interactions during the applied rf π/2 pulses. Each pulse
then yields a perfect rotation on the Bloch sphere,√ e.g., the initial state K|2⟩ is transformed
into the spin-state superposition (K|2⟩ + K|3⟩)/ 2. For such a perfect Ramsey sequence,
the measured Ramsey signal S(t) gives the overlap between the time-evolved interacting andnon-interacting states of the system [Goo11, Kna12], yielding Eqs. (5.9) and (5.14) for zeroand finite temperature, respectively. In this idealized scenario, the Fourier transform of S(t)corresponds to the excitation spectrum of the system in linear response [Mah00],
Z ∞dt iωtA(ω) = Ree S(t),
5.15
π
0where ω is the frequency of the applied field.
In our experiments, however, residual interactions are present during the π/2 pulses, whichtake a finite time to be completed. As shown in the illustration of our experimental sequencein Fig. 5.8, the state K|3⟩ can already interact with the Li cloud during the π/2 rotation,
which potentially affects the observed dynamics of the system. Specifically, this stage of theexperiment is performed at a detuning from the Feshbach resonance which corresponds to a
weak interaction strength X1 between the impurities and the Fermi sea (cf. Section 5.5.3 andFig. 5.8). After preparing the superposition state of the impurity spin, we quench the system

5.5. SUPPLEMENTARY MATERIALS
interaction
parameter

95

X1

X1

X

φrf

interacting state

non-interacting state

π/2
rotation
10.0 μs

trf

π/2
rotation
1.5 μs

1.5 μsvariable time

twait

twait

t

measurement
N3

N2

10.0 μs

trf

Figure 5.8: Schematic of the experimental Ramsey procedure. The K atoms start out in the hyperfine
state K|2⟩, which is effectively non-interacting with the Fermi sea. A 10 µs (3.4τF ) long square π/2pulse is applied in the presence of weak interactions between the K|3⟩ atoms and the Li atoms,
quantified by the interaction parameter X1 . We then use optical control of our Feshbach resonanceto rapidly (in less than 200 ns (0.08 τF )) quench the system into the strongly interacting regime(interaction parameter X). After a variable interaction time t we optically shift the interaction
strength back to X1 , and then close the Ramsey sequence by a second π/2 pulse. We vary the phaseof this pulse by shifting the phase of the rf source by φrf before the second pulse is applied.

to strong interactions (interaction parameter X) by optically shifting the Feshbach resonance[Cet15]. We previously focussed on the complex non-equilibrium dynamics resulting from thestrong interactions X during the time t. In the following, we analyze the effect of the residualinteraction X1 during the finite-duration π/2 spin rotations. In particular, we investigate
the impact of these weak interactions during the rf pulses on the Ramsey response S(t) andthe spectrum A(ω) as obtained from the Fourier transform Eq. (5.15).

Modelling of rf pulses within TBM
In this section, we extend our modelling of the zero-temperature impurity dynamics within
the TBM to directly simulate the entire experimental procedure, as illustrated in Fig. 5.8.
In order to model the rf pulses, we explicitly include both K|2⟩ and K|3⟩ spin states, as wellas the rf field. This modifies the Hamiltonian, Eq. (5.1), to Ĥ = Ĥ + Ĥrf with the additionalterm
 XΩ X  iφrf ˆ† ˆe dk,2 dk,3 − e−iφrf dˆ†k,3 dˆk,2 +(ϵk,K + ~(ωrf − ω0 ))dˆ†k,2 dˆk,2 .
5.16
Ĥrf =2ikkHere, we have used the rotating wave approximation. Ω corresponds to the strength of therf field, φrf is the variable phase of the second rf pulse, and dˆ†k,σ creates a particle in thestate K|σ⟩ with momentum ~k. Note that dˆ† ≡ dˆ† in the original two-channel Hamiltoniankk,3

5.1. The interactions during the rf pulses cause a shift in the transition frequency between
the K|2⟩ and K|3⟩ states from the bare transition frequency ω0 to ω0 + ε1 /~, where ε1 is thepolaron energy at interaction parameter X1 . As described in Sec. S5.B, we account for thisshift by adjusting the frequency of our rf pulses to ωrf = ω0 + ε1 /~.
According to the last term in Eq. (5.16), the shift in the frequency of the rf source from ω0965.5. SUPPLEMENTARY MATERIALS

to ωrf causes the observed signal to accumulate an additional phase (ωrf − ω0 )t during theinteraction time t. To account for this, we introduce the phase φ = φrf + (ωrf − ω0 )t.
We then determine |S(t)| and the phase ϕ(t) by noting that the Ramsey signal (N3 −N2 )/(N3 + N2 ) corresponds to a sine-wave function of φ plus an offset, i.e., it takes theform F (t) + |S(t)| cos(φ − ϕ(t)) with F (t) a real, φ-independent function. This mirrors theexperimental procedure, where F (t), |S(t)|, and ϕ(t) appear as fit-parameters for the Ramsey
signal, see Sec. S5.B.

Figure 5.9: Role of the residual interactions within TBM. We present the zero-temperature response
S(t) and the corresponding spectrum A(ω) for the perfect quench (dashed blue) and the actual
experimental sequence shown in Fig. 5.8 (solid blue). As in the main text, we take kF R∗ = 1.1 andthe interaction parameters: (A, C) X = −0.23, X1 = −3.9, (B, D) X = 0.86, X1 = 5.8, and (E,F) X = 0.08, X1 = 4.8. For comparison, in (B, D, E, F), we represent by black dotted lines the
scenario where the initial state before the quench is approximated as a weakly attractive polaron —see Sec. 5.5.3 for details. The spectra have been convolved with the experimental Fourier-limited rf
spectral lineshapes, which are Gaussian-shaped with width σ, where στF = 0.03 for X = 0.86, −0.23,
and στF = 0.1 for X = 0.08.

Within the TBM, we determine the approximate eigenstates and eigenvalues of Ĥ within
the more general class of truncated wavefunctions:X|ψrf ⟩ = α0,3 dˆ†0,3 + α0,2 dˆ†0,2 |FS⟩ +αq b̂†q ĉq |FS⟩
q

+

X

αkq,3 dˆ†q−k↓ ĉ†k ĉq + αkq,2 dˆ†q−k,2 ĉ†k ĉq



|FS⟩.

5.17

kq

To model the experimental quench sequence illustrated in Fig. 5.8, we apply a series of time
evolution operators to the initial state consisting of a K|2⟩ atom and the Li Fermi sea. At
the end of the sequence we then extract the number of K atoms in states K|2⟩ and K|3⟩,
respectively. We include explicitly the rf pulses, the wait times, and the interaction time tduring which the system is strongly interacting. The results of this procedure are displayedin Fig. 5.2 of the main text. Here, we account for slight additional experimental decoherenceby scaling the prediction for |S(t)| as described in Section S.5C.

5.5. SUPPLEMENTARY MATERIALS

97

In the upper panels of Fig. 5.9 we compare the Ramsey response obtained by simulating theactual experimental sequence (solid line) with that of the perfect quench scenario (dashedline). We see that the residual interactions X1 in experiment can indeed influence thequantum evolution of the impurity. The difference in the responses can be straightforwardly
explained by assuming that the main effect of X1 is to produce a weakly interacting initialstate. Specifically, for weak attractive interactions X1 > 0, the Ramsey response can beapproximated as
S(t) ≃ Z ⟨ψX1 | e−iĤt/~ |ψX1 ⟩ ,5.18

where |ψX1 ⟩ is the ground state of the Hamiltonian 5.1 at interaction parameter X1 , andZ is the corresponding polaron residue. Note that we cannot formally construct a similar
expression for the repulsive case X1 < 0, since the repulsive polaron is a metastable state,involving multiple eigenstates of the Hamiltonian.
Referring to Fig. 5.9, the excellent agreement between the approximation 5.18 and the full
Ramsey signal provides strong evidence that the residual interactions X1 produce a weakly
attractive initial state. This is further supported by the spectrum A(ω) shown in the bottom
panels, where we see that the residual interactions enhance the attractive polaron peaks for
X = 0.08 and 0.86. A similar enhancement of the repulsive polaron peak is observed for
X = −0.23. Hence we conclude that the explicit modelling of the impurity dynamics using
the full Hamiltonian Ĥ = Ĥ + Ĥrf is not essential for the description of the dynamics during
the initial π/2 spin rotation and instead one can fully describe the time evolution using the
Hamiltonian 5.1.
1.0

1.0

0.5

0.5

0

A
0

4

B
5

10

15

20 0

5

10

15

20

0

D

C

0

E

5

10

15

20

F

3

1

2
1
0
-1

- 0.5

0

0.5

- 0.5

0

0.5

1

0

-1

0

1

Figure 5.10: Role of the residual interactions in the Ramsey sequence at finite temperature. Upper panels: we compare the perfect quench Ramsey response (dashed) with a simulation of theexperimental sequence (solid). Lower panels: we compare the linear-response excitation spectrum(dashed) with the Fourier transform of the signal obtained using the experimental sequence (solid).
As in the main text, we take kF R∗ = 1.1 and the interaction parameters: (A, C) X = −0.23,
X1 = −3.9, (B, D) X = 0.86, X1 = 5.8, and (E, F) X = 0.08, X1 = 4.8. The temperatures are
T /TF = 0.166, 0.158, 0.177, respectively.

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5.5. SUPPLEMENTARY MATERIALS

Modelling of experimental procedure at finite temperature within FDA
The interplay between the residual interactions and finite temperature presents a further
theoretical challenge. In the following, we use the FDA to simulate the experimental protocol(Fig. 5.8) at finite temperature. To achieve this, we exploit the finding from Sec. 5.5.3 thatthe detailed dynamics of the rf-driven oscillations between the K|2⟩ and K|3⟩ states can beignored when calculating S(t). Thus, we assume
that the initial π/2 rotation effectively√
produces a spin superposition (K|2⟩ + K|3⟩)/ 2, independently of the residual interaction
X1 of the impurity in the state K|3⟩ with the Fermi sea. To account for the dynamics dueto the weak interaction X1 , we then let the system evolve under this interaction for a holdtime th = trf /2 + twait , which models the dynamics at weak interaction X1 as the result ofa sudden switch-on of this interaction at the midpoint of the π/2 pulses. After the holdtime th , the final quench to the strong interactions X is performed. For the measurementof the Ramsey contrast, this sequence is reversed. Theoretically, this yields the modified
time-dependent overlaphiS(t) = Tr ρ̂ eiĤ0 (2th +t) e−iĤ1 th e−iĤX t e−iĤ1 th ,5.19
where Ĥ1 and ĤX denote the Hamiltonian 5.1 at interaction strength X1 and X, respectively.
Using the FDA, the expression Eq. (5.19) is evaluated exactly according to Eq. (5.13) at theexperimental temperature. As can be inferred from Eq. (5.19), this simplified model of theexperimental protocol corresponds to a sequence of interaction quenches.
In the upper panel of Fig. 5.10 we compare the result for |S(t)| at the experimental temperatures obtained for the experimental sequence (solid lines) to the result for an idealized, i.e.,
perfect quench, Ramsey sequence (dashed lines). Similarly to the case of zero temperature,
we see that the time evolution at X1 has an experimentally observable effect on the dynamics. In particular, it generates an additional decoherence of the Ramsey signal already at
t = 0, as well as an enhancement of the oscillations in |S(t)| for resonant interactions – see
Fig. 5.10E.
For the calculation of the FDA results shown in Fig. 5.2 of the main text we use the sameprocedure as described above. We account for slight additional experimental decoherence byscaling the prediction for |S(t)| as described in Section S.5C. We also note that the phase
ϕFDA (t) of the Ramsey signal S(t) = |S(t)|e−iϕFDA (t) , as determined from Eq. (5.19), differsfrom the experimentally measured phase ϕ(t) due to the detuning of the rf frequency from
ω0 . They are related by ϕ(t) = ϕFDA (t) − (ωrf − ω0 )(2twait + trf ). Similar to the previous
section and to the experiment, we take ωrf − ω0 = ε1 /~.
As outlined in Section 5.5.3, in the idealized Ramsey scenario the Fourier transform A(ω)
of S(t) is equivalent to the rf absorption in linear response, cf. 5.15 [Kna12]. Similarly to
our T = 0 analysis in Sec. 5.5.3, we now study the effect of the residual interactions X1
on the spectral decomposition of S(t). To this end we compare the two signals A(ω) forthe perfect quench with the result obtained for the experimental sequence as modelled by
Eq. (5.19). We show the comparison of the spectra obtained in the idealized (dashed) andexperimentally realized scenario (solid) in the lower panel of Fig. 5.10. As for our T = 0
results discussed above, we find only a small difference between the two finite-temperature5.5. SUPPLEMENTARY MATERIALS

99

spectra. Therefore, in agreement with the experimental observation, cf. Fig. 5.3 in the main
paper, under the condition of |X1 | ≈ 5 we see that the weak interactions during the rf pulses
have an observable but small effect on the predicted spectra.
In accordance with the results from the TBM shown in Fig. 5.9, we find from the evaluationof Eq. (5.19) that weak interactions X1 lead to a small shift of spectral weight into thecorresponding dominant polaron branches. This shift of spectral weight is also observed
experimentally, see Fig. 5.3 of the main text.

Stronger interactions during rf pulses: illustration of quantum state preparation

A(!) / =F

1.5

A

B

C

D

1
0.5
0

A(!) / =F

1.5(!-!0) =F

1
0.5
0
-1

-0.5

0

(!-!0) =F

0.5

1

-1

-0.5

0

0.5

1

(!-!0) =F

Figure 5.11: Control of the spectral decomposition of many-body quantum states. Upper panel: We
compare the experimentally measured rf spectrum at the interaction parameter X (green squares) to
the Fourier transform of S(t) obtained using the measurement procedure illustrated in Fig. 5.8 withinitial interaction parameter X1 (gray shading). Lower panel: we compare the theoretical prediction
from the FDA for the linear-response excitation spectrum (green) to the Fourier transform of thesignal obtained by simulating the experimental sequence according to Eq. (5.19). (A, C) X = 0.14,
X1 = −2.2, kF R∗ = 1.09, T /TF = 0.174. (B, D) X = −0.25, X1 = 1.7, kF R∗ = 1.1, T /TF = 0.17.

The shift of spectral weight towards the attractive or repulsive branches of the spectrum,
cf. Figs. 5.9 and 5.10, may be interpreted as follows: The residual interactions present
during the initial π/2 impurity spin rotation serve to produce an interacting many-bodyquantum state. As such, this procedure can be viewed as an adiabatic preparation of an
attractive or repulsive polaron. Compared to the noninteracting state, this polaron has anincreased wavefunction overlap with the corresponding branch of the strongly interacting
system. When the system is then quenched into the regime of strong interactions, theincreased overlap results in the corresponding shift of the spectral weight. An intriguingquestion is then whether such an approach can provide a novel way to experimentally control
the spectral decomposition of quantum states.

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5.5. SUPPLEMENTARY MATERIALS

To investigate this possibility, we increase the interaction during the π/2 rotations, corresponding to decreasing |X1 |, and determine the effect on A(ω). In the upper panel ofFig. 5.11 we show the spectra obtained by linear-response rf spectroscopy (green squares).
Similar to Fig. 5.3 of the main paper, we compare this result to the Fourier transform of theRamsey signal S(t) (gray shading), as obtained from the experimental sequence described inFig. 5.8. We also compare our experimental result to the prediction from the FDA, where
the dynamics has been modeled as described by Eq. (5.19). As in the main text, we findexcellent agreement between experiment and theory. Indeed, both feature a strong shift ofspectral weight to regions of the spectrum that are adiabatically connected to the dominantpolaron branches at interaction X1 . Furthermore, when comparing A(ω) in Fig. 5.11, withthe spectrum for |X1 | ≈ 5 in Figs. 5.9 and 5.10, it is clear that the amount by which thespectrum is shifted can be controlled by the strength of the interaction during the rf pulses.
This strongly supports the assertion that the initial interactions can be used to precisely
control the many-body dynamics. Our experimental techniques thus allow for a precise,dynamic control of the spectral decomposition of quantum states in future experiments.
The excellent agreement between theory and experiment also demonstrates that our theoretical approaches can be used to explore experimental ramps in combination with interferometric protocols in order to find, for instance, optimized spin and interaction trajectories.

5.5.4 Universal features of impurity dynamics and relation to orthogonality
catastrophe
For impurities localized in space, which, for instance, can be achieved by species-selective
three dimensional optical lattices, our experimental setup allows one to study universal
features exhibited by the Anderson orthogonality catastrophe [And67]. The orthogonality
catastrophe was originally studied in the context of x-ray absorption spectra in metals,
where high-energy x-ray photons create atomic core holes by photo emission of inner-shellelectrons [Mah00]. These core holes produce a scattering potential for the electrons in the
conduction band, leading to characteristic power-law edges in the absorption spectra withan exponent that is universally determined by the scattering phase shift at the Fermi surface[And67]. However, impurities, phonons, residual interactions between the electrons, and alack of knowledge of microscopic parameters makes it difficult to unambiguously determine
the universal features of the orthogonality catastrophe in typical solid state materials [Oht90].
In contrast, the Hamiltonian in our experiment is well characterized on all relevant energy
scales, and therefore the full dynamic response of the system can be reliably calculatedby theory and probed by the ultrafast experimental techniques demonstrated in this work.
This enables one to obtain fundamental insights into universal features of the orthogonality
catastrophe, which are difficult to access in other systems.
To illustrate how the orthogonality catastrophe would manifest itself in an ultracold atomicgas experiment, the response of infinite mass impurities calculated using the FDA for theperfect quench scenario is shown in Fig. 5.12. First, at short times and for a range parameterof the Feshbach resonance R∗ > 0, we see that the Ramsey contrast decays quadratically for
all scattering parameters and temperatures considered, in accordance with Eq. (5.11). The
main universal feature associated with the orthogonality catastrophe is expected in the long-5.5. SUPPLEMENTARY MATERIALS

101

time dynamics at T = 0: Here, the Ramsey response is predicted to exhibit power law tails,
which depend only on the scattering phase shift at the Fermi surface [And67, Kna12]. Thisis explicitly verified in Fig. 5.12A where we fix the scattering phase shift at the Fermi surface
but change the scattering parameters. While the response at intermediate times depends on
the scattering parameters, we see that the long-time evolution approaches a universal powerlaw that only depends on the phase shift at the Fermi surface. We note that the long-time
dynamics is universal: It is the same for a system with a broad resonance where R∗ = 0(solid line in Fig. 5.12A), as it is for our system with a finite range parameter (dashed anddotted lines).
When the temperature is non-zero, as in the experiment, thermal fluctuations alter the powerlaw dephasing dynamics at sufficiently long times. Instead, exponential tails due to thermal
decoherence appear as another universal feature of the dynamics [Kor50, And67, Yuv70,
Kna12]. The exponential tails are illustrated in Fig. 5.12B. The effects of thermal decoherence
could be countered by employing the recently developed cooling methods [Har15], opening
the door to observing the orthogonality catastrophe in a cold-atom system.
Finally, we note that in our experiment temperature becomes relevant at a time scale similar
to those associated with recoil and multiple particle-hole excitations. It is a challenge fortheoretical approaches to exactly account for both recoil and higher order particle-hole excitations [Ros99]. However, experiments at lower temperatures which take advantage of thetunability of the impurity mass using optical lattices would be ideally suited to probe thecompetition between these effects. Such ultracold-atom experiments would hence provide
important insight into this long standing theoretical question.

5.5.5 Experimental and data analysis procedures
In this section we discuss the procedures used to record and analyze the data presented inthis work. We detail the cooling and preparation of our atomic samples, the details of the rf
pulses used in our Ramsey sequences, the methods used to analyze the data and the methodthat we use to vary the concentration of the K atoms.

Sample preparation
The atomic samples are prepared by forced evaporation of Li atoms from a Li-K mixture
held in an optical trap, where the K atoms are sympathetically cooled by the Li environment.
This preparation procedure is described in detail in Refs. [Tre11a, Spi10a]. At the end of theforced evaporation, the Li and K atoms are transferred into an optical trap composed of twocrossed 1064-nm laser beams, as described in Ref. [Cet15]. The measured radial and axial
trap frequencies of the Li atoms are fr,Li = 941(5) Hz and fz,Li = 134(1) Hz, respectively.
The measured radial and axial trap frequencies of the K atoms are fr,K = 585(3) Hz and
fz,K = 81(1) Hz, respectively.
At the end of the preparation procedure, the Li and the K atoms are in their lowest Zeeman

102

5.5. SUPPLEMENTARY MATERIALS

states Li|1⟩ and K|1⟩. Before the Ramsey sequence, the K atoms are transferred to the K|2⟩
state using an rf pulse. Following this rf transfer, the Li and K atoms are thermalized byholding them for 750 ms in the crossed-beam optical trap. While the interaction between
the Li|1⟩ and K|2⟩ atoms, characterized by the scattering length a12 = 63a0 [Nai11], issufficient to ensure thermalization during this hold time, it can be neglected during theRamsey experiments. The temperature of the atoms is determined by releasing the atoms
from the trap and observing the free expansion of the K cloud.
Due to the Li Fermi pressure and the more than two times stronger optical potential forK, the K cloud is much smaller than the Li cloud [Tre11a], and therefore samples a nearlyhomogeneous Li environment. Because of the small variation of the Li environment sampled
by the K atoms, we introduce the effective Li Fermi energy ϵF as
Z
1
ϵF =EF (r)nK (r)d3 r .
5.20
NKHere, nK (r) is the local K number density at position r in the trap, and~2 6π 2 nLi (r)EF (r) =
2mLi

2/3
5.21

is the local Li Fermi energy as determined by the local Li number density nLi (r). We quantify
the small inhomogeneity of the Li environment experienced by the K atoms by the standard
deviation of the local Li Fermi energyσ(EF ) =

1
NK

Z

2

3

(EF (r) − ϵF ) nK (r)d r

1/2
.

5.22

We also introduce the average Li and K number densities n̄Li and n̄K sampled by the Katoms asZ
1n̄Li,K =nLi,K (r)nK (r)d3 r .
5.23
NKIn contrast to the Li atoms, the K atoms in our measurements remain non-degenerate, withkB T /EFK (0) > 1.2, where EFK (0) is the local potassium Fermi energy in the center of the trap
when all K atoms are in the same internal state.
For all measurement presented in this work, Table 5.1 lists the total numbers of the Li andK atoms, their temperatures and trap-averaged densities, as well as the effective Li Fermi
energies and their standard deviations. Throughout our measurements, these parameters
remain nearly constant, with the exception of the measurements shown in Fig. 5.4. Here,in order to investigate the effect of the K concentration, the total number of the K atoms
is increased from about 1 × 104 to 2.5 × 104 . The attendant increase in the thermal load
during the Li evaporation results in a decrease of the Li atom number and an increase in thetemperature of the final atomic sample.
Note that, in contrast to our previous work [Cet15], our present experiments have been
optimized for large optically induced interaction shifts (|X − X1 | ≈ 5). These shifts are5.5. SUPPLEMENTARY MATERIALS
Figure(s)2A, 2C, 3A
2B, 2D, 3B
2E, 2F, 3C
5.11A
5.11B
4

NLi
(105 )
3.5(4)
3.3(4)
3.5(4)
3.1(4)
2.9(3)
2.35(30)

NK(104 )
0.95(10)1.0(1)
1.0(1)
1.0(1)
1.05(10)2.5(1)T(nK)
435(25)410(25)460(30)430(30)425(35)520(25)

103
ϵF /h
(kHz)
54.6(2.7)
53.9(2.4)
54.1(2.4)
52.0(2.9)
50.8(2.1)
44.2(2.3)σ(EF )
ϵF

%
7.4
7.1
7.7
7.7
7.7
10.4

n̄Li
12
10 cm−3n̄K
12
10 cm−3

8.9(7)
8.7(6)
8.8(6)
8.2(7)
8.0(6)
6.5(5)

1.8(3)
2.0(3)
1.7(3)
1.8(3)
2.0(3)
3.4(3)

Table 5.1: The total number of the Li atoms NLi , the total number of the K atoms NK , the sample
temperature T , the effective Li Fermi energy ϵF , the standard deviation σ(EF ) of the local Li Fermi
energy across the trap, the trap-averaged Li and K number densities n̄Li and n̄K in our measurements.

produced by switching one of the crossed trapping beams from a beam with a low peakintensity and small size to a beam with a large intensity and large size propagating inthe same direction. In our previous work [Cet15], as well as in the measurements shownin Fig. 5.11, the waists, positions and intensities of the two beams are adjusted so as toyield mode-matched trapping potentials, preventing excitations of the center-of-mass andbreathing collective modes of the atomic clouds. In the measurements presented in Figs. 2,
3 and 4, a larger beam intensity was used in order to produce a larger optical shift, resulting
in some excitation of the breathing modes.
The maximal interaction time in our Ramsey measurements of 60 µs is much smaller than
the shortest period of a collective oscillation (about 500 µs). We calculate that, during ourshort interaction time, the oscillations of the breathing modes cause at most a 6% variationof ϵF around its initial value specified in Table 5.1, without any significant effect on themeasurements presented here.

Rf pulses
We apply rf pulses in the Ramsey procedures by discretely gating a continously running rf
source. To record the atomic populations N3 and N2 as a function of the phase of the second
rf pulse, we change the phase of the rf source by a variable amount φrf before applying thispulse.
The weak interactions between the K atoms in the K|3⟩ state and the Li atoms corresponding
to the interaction parameter X1 cause the transition frequency between the K|2⟩ and theK|3⟩ states to differ from the transition frequency ω0 in the absence of the Li atoms. We
compensate for this effect by adjusting the frequency ωrf of the rf source to be resonantwith the K|2⟩−K|3⟩ transition at the time when the rf pulses are applied. For the data in
Figs. 2A, 2B, 2C, (ωrf − ω0 )τF is equal to +0.06, −0.07, −0.05, respectively. For the data in
Fig. 5.11C and 5.11D where the interaction of the K atoms during the rf pulses is stronger,(ωrf − ω0 )τF is equal to +0.11 and −0.16.
The shift in the frequency of the rf source from ω0 to ωrf causes the signal S(t) to accumulate

104

5.5. SUPPLEMENTARY MATERIALSan additional phase (ωrf −ω0 )t during the interaction time t. To account for this added phase,
we introduce the phase φ = φrf + (ωrf − ω0 )t.

Analysis methods
We determine the contrast |S(t)| and the phase ϕ(t) by fitting the Ramsey signal (N3 −N2 )/(N3 + N2 ) as a function of the phase φ to a sine wave with an offset i.e. F (t) +|S(t)| cos (φ − ϕ(t)). Decoherence during the rf pulses, as well as imperfections of the rf pulses
and the atom detection, cause the contrast for t = 0 to be slightly smaller than unity. When
comparing theoretical results from Figs. 5.9 and 5.10 to the experimental data in Fig. 5.2,
we account for this effect by scaling the theoretical predictions for |S(t)| by an overall factor
η. For each calculation, this factor is determined by fitting the prediction for |S(t)| to thethree data points with the the shortest interaction times. We obtain 0.92 < η < 1, whichcorresponds to an additional loss of contrast that is of the same order as the decoherence
during the rf pulses predicted by the FDA (see Fig. 5.10).
To compute the Fourier transform of the experimental S(t) data, we use piecewise linear
interpolations of log S(t) and ϕ(t) between the individual data points. Outside of the rangeof the data, we set S(t) = 0. To determine the error of the Fourier transform, we sample
the values of S(t) and ϕ(t) at each data point from Gaussian distributions whose means andstandard deviations correspond to the measured values and errors, respectively. We use thestandard deviation of the computed values of the Fourier transform for each value of ω asan estimate of the error indicated by the shaded areas in Figs. 3 and 5.11.

Varying the K concentration
We study the effects of the impurity concentration by varying the number of the stronglyinteracting K atoms. If this were done by changing the total number of the K atoms in
the experiment, the change in the thermal load on the Li atoms during forced evaporation
would result in a correlated variation in the number of Li atoms and the sample temperature(compare the settings for Fig. 5.2 and Fig. 5.4 in Table 5.1). To avoid these systematiceffects, in the measurements presented in Fig. 5.4, we keep the total number of the K atoms
constant and vary the fraction of the K atoms that participate in the Ramsey sequence. We
accomplish this by changing the intensity of the rf pulse that transfers the K atoms fromthe |1⟩ state to the |2⟩ state before the Ramsey procedure. During the subsequent 750 ms
preceding the Ramsey sequence, the K atoms collisionally thermalize with the much larger Li
cloud, resulting in an incoherent mixture of K|1⟩ and K|2⟩ atoms at a constant temperature.
When referring to these measurements, we use n̄K not for the average density of all K atoms,
but for the density of those K atoms that participate in the Ramsey sequence.
We minimized the small effects of long-time drifts in the temperature, the atom numbers
and the trapping potential by varying the experimental parameters in a specific order. For
each K concentration and interaction time, we recorded data for 4 different phases of thesecond rf pulse in order to obtain S(t). For each interaction time, the data with different K5.5. SUPPLEMENTARY MATERIALS

105

concentrations were recorded in immediate succession. The data sets for different interaction
times were then recorded in a random order.

5.5.6 Linearity of rf response
The response of atoms to an applied rf field is linear if the fraction of the atoms transferredfrom one state to another is proportional to the intensity of the field. Linearity can beensured by using a sufficiently weak rf pulse that is also much longer than the inverse widthof the relevant spectral features. The narrowest spectral features in the present work are
the polaron peaks in Figs. 3A and 3B with rms widths 0.06 ~/τF and 0.09 ~/τF , respectively.
To record these polaron spectra, we used Blackman-shaped rf pulses [Kas92] whose duration
trf = 300 µs ≈ 100 τF is much longer than the inverse widths of the polaron peaks.
We checked the linearity of the response by varying the intensity Irf of the applied rf field.
Fig. 5.13A shows the fraction of the K atoms transferred from the K|2⟩ to the K|3⟩ statein the repulsive polaron regime, under conditions similar to those in the measurements
shown in Fig. 5.3A. The frequency of the rf pulse is adjusted so that (ωrf − ω0 )τF = 0.3,
corresponding to peak response and resonant excitation of the repulsive polaron. The rf
intensity is measured in units of the intensity Iπ that results in a π-pulse for noninteracting
K atoms. For intensities up to the intensity Irf = 0.79 Iπ , which is used in the measurements
shown in Figs. 3A and 3B, we observe that the transferred fraction of the K atoms stays
essentially proportional to the intensity of the pulse.
In the linear-response regime, the atomic response is predicted to be proportional to theduration of the rf pulse. Fig. 5.13B shows the fraction of the K atoms transferred in therepulsive polaron regime by rf pulses with Irf = 0.79 Iπ , as a function of the pulse duration.
The frequency of the rf pulse is adjusted so that (ωrf − ω0 )τF = 0.3, in order to obtain
the peak response, as in Fig 5.13A. For pulses with duration up to 300 µs (indicated by
the dashed line), we observe that the transferred fraction of the K atoms stays essentially
proportional to the duration of the pulse.
Note that the maximal transferred fraction exceeds 0.5. We explain this observation by the
coupling of the initial non-interacting K state to multiple interacting K states by the rf pulse,which manifest themselves as the polaron peak and the broad pedestal in our spectra.
The spectra for resonant Li-K interactions shown in Figs. 3B, 5.11A, 5.11B were recorded
using Blackman-shaped rf pulses with duration of trf = 100 µs (approximately 35 τF ). The
intensity of these pulses was adjusted to 50% of that needed to produce π pulses for noninteracting K atoms. We verified the linearity of the rf response by comparing the spectra
recorded using this rf intensity to those recorded using the intensity needed to produce full
π pulses for noninteracting K atoms (Fig. 5.14). Our observations are in good agreementwith linear response.

1.0

1

0.8 A

B
1

0.6
0.1 0.1

0.4

0.1

0.5

1

5

10

50

0.1

100

0

50

100

1

10

100

Figure 5.12: Universal features of the dynamical orthogonality catastrophe. We show the Ramsey
contrast for an infinitely heavy impurity obtained within the FDA. (A) The zero-temperature Ramsey
contrast exhibits a power law decay, shown on a double logarithmic scale. We change the Feshbachresonance range kF R∗ and interaction parameter X in such a way that the scattering phase shift
at the Fermi surface is constant leading to a constant exponent of the power law tail. The data
corresponds to a fixed phase shift δkF = 1.4 with the choices (X, kF R∗ ) = (1, 1.12) (dashed red),(X, kF R∗ ) = (0.58, 0.56) (dotted blue), and (X, kF R∗ ) = (0.15, 0) (solid green). (B) Ramsey contrast
at various temperatures on a double logarithmic scale. We choose temperatures T /TF = 0 (blue),
0.05 (green), 0.15 (orange), 0.4 (red) at fixed values X = 1 and kF R∗ = 1.12. The inset shows thesame data on a logarithmic-linear scale to emphasize the appearance of exponential tails at finite
temperature.

1.0

fraction of K transferred

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

A

0.0

B

0.0
0

2

4

Irf / Iπ

6

8

0

1

2

trf (ms)

3

4

Figure 5.13: Linearity of the rf response in the repulsive polaron regime. (A) Fraction of the Katoms transferred from state K|2⟩ to the state K|3⟩ for X = −0.13(6) as a function of the intensity Irf
of an rf pulse with duration trf = 300 µs. (B) Fraction of the K atoms transferred for X = −0.23(6)as a function of the duration trf of the rf pulse for the rf pulse intensity Irf = 0.79 Iπ . Vertical dashedlines correspond to Irf = 0.79 Iπ and trf = 300 µs, respectively. The pulse frequencies are adjusted
to resonantly excite the repulsive polaron. The blue solid lines indicate linear fits to the data in theranges indicated by the same lines. The blue dashed lines show extrapolations of these fits.

106fraction of K transferred

0.6

0.4

#2
0.2

0
-1

-0.5

0

0.5

1

(! - !0) =F
Figure 5.14: Linearity of the rf response for resonant interactions. Fraction of the K atoms transferred from state K|2⟩ to the state K|3⟩ by an rf pulse with duration trf = 100 µs for X = +0.02(6).
For the black data points, the intensity of the rf pulse is adjusted to obtain a π-pulse in the absenceof Li atoms. The red data points correspond to a 50% lower intensity of the rf field.

107

Chapter6

Publication: Observation of aStrong Atom-Dimer Attraction ina Mass-Imbalanced Fermi-Fermi
Mixture1

Physical Review Letters 112, 075302 (2014)
submitted 20 November 2013, published 21 February 2014
DOI: 10.1103/PhysRevLett.112.075302
Editor’s suggestion

M. Jag1,2 , M. Zaccanti1,3 , M. Cetina1 , R. S. Lous1,2 , F. Schreck1 , R. Grimm1,2 , D. S. Petrov4 ,and J. Levinsen5,6
1

Institut für Quantenoptik und Quanteninformation (IQOQI), Österreichische Akademie der Wissenschaften2
Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
4
Université Paris-Sud, CNRS, LPTMS, UMR8626, Orsay, F-91405, France
5
TCM group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
6
Aarhus Institute of Advanced Studies, Aarhus University, DK-8000 Aarhus C, Denmark1

The author of the present thesis developed the experimental procedures together with M.Z., performed allmeasurements together with M.Z. and M.C., and analyzed the data together with M.Z., M.C., and R.S.L.
under the supervision of F.S. and R.G. All authors contributed to writing the manuscript.

109

110

6.1. ABSTRACT

6.1 Abstract
We investigate a mixture of ultracold fermionic 40 K atoms and weakly bound 6 Li40 K dimers
on the repulsive side of a heteronuclear atomic Feshbach resonance. By radio-frequency
spectroscopy we demonstrate that the normally repulsive atom-dimer interaction is turned
into a strong attraction. The phenomenon can be understood as a three-body effect inwhich two heavy 40 K fermions exchange the light 6 Li atom, leading to attraction in odd
partial-wave channels (mainly p-wave). Our observations show that mass imbalance in afermionic system can profoundly change the character of interactions as compared to thewell-established mass-balanced case.

6.2 Introduction
Ultracold fermions with tunable interactions provide remarkable possibilities to model the
many-body physics of strongly interacting states of quantum matter under well-controllable
conditions [Gio08, Blo08]. Fermionic superfluids, realized by combining two different spinstates of a fermionic atomic species and controlling their s-wave interaction through a Feshbach resonance [Chi10], have led to spectacular achievements. Beyond these experimentallywell-established fermionic systems, mass imbalance offers an additional degree of freedom,with interesting prospects for new many-body phenomena having no counterpart in the
mass-balanced case, such as novel quantum phases or superfluid states in various trapping
environments [Isk06, Bau09c, Gez09, vK11, Sot12, Cui13, Gub09, Mat11, Qi12, Dai12, Pet07,
Bar08, SC91, Ors10, Dal12, Nis08, Nis09b].
Striking effects of mass imbalance in fermionic systems already emerge at the few-body level.
A resonantly interacting three-body system of one light (↓) and two heavy (↑) fermions is
known to exhibit bound states depending on the mass ratio m↑ /m↓ . While Efimov trimer
states require large mass ratios (m↑ /m↓ > 13.6), for repulsive interactions, non-Efimovian
trimer states can exist in an intermediate regime (13.6 > m↑ /m↓ > 8.17) [Kar07]. Below
the critical value of 8.17, the last state turns into an atom-dimer scattering resonance in thep-wave channel [Kar07].
The 40 K-6 Li mixture serves as the prime system for current experiments on tunable massimbalanced Fermi-Fermi mixtures [Wil08, Cos10, Tre11a]. The corresponding mass ratioof m↑ /m↓ ≈ 6.64 lies well in the regime of near-resonant atom-dimer interactions [Lev09,
Lev11]: as the most prominent effect, theory predicts a substantial attraction resulting fromhigher partial waves (mainly p-wave) in a regime where one would naively, based on s waves
alone, expect a strong repulsion. This also makes the corresponding many-body problem
in a 40 K-6 Li mixture significantly more complicated and much richer than in the widely
investigated mass-balanced case.
In this Letter, we investigate the interaction between 40 K atoms and weakly bound 6 Li40 K
dimers near an interspecies Feshbach resonance (FR). We employ radio-frequency (rf) spectroscopy by using two different internal states of 40 K, one strongly interacting with the6.3. MAIN RESULTS

111

dimers and the other one practically non-interacting [Koh12b]. We observe line shifts andcollisional broadening and interpret these in terms of the real and imaginary part of theforward-scattering amplitude f (0) for atom-dimer collisions, calculated on the basis of thetheoretical approach of Ref. [Lev11]. The comparison between theory and experiment shows
excellent agreement and, in particular, demonstrates the predicted atom-dimer attraction on
the repulsive side of the interspecies FR.

6.3 Main results
The interaction of a heavy atom with a heavy-light dimer can be understood in the BornOppenheimer approximation, where the atom-dimer potentials are taken to be the eigenenergies of the light atom for a given separation R between the heavy ones. As in the usual
double-well problem with tunneling, the state localized near one heavy atom is mixed withthe state localized near the other; the symmetric and antisymmetric superpositions lead tothe attractive U+ (R) < 0 and repulsive U− (R) > 0 potentials, respectively. Note the analogy
to the well-known H+
2 cation, where the exchange of the electron leads to a symmetric boundstate and an antisymmetric unbound state [Pau28]. In our experiment, the heavy particles
are identical fermions, making the atom-dimer interaction channel dependent. The symmetric (antisymmetric) state corresponds to odd (even) values of the total angular momentum
l [Lev11]. In Fig. 6.1(a) we plot the total effective potentials U± + Ucb (solid lines) andthe bare centrifugal barriers Ucb = l(l + 1)~2 /m↑ R2 (dashed lines) for l = 0, 1, and 2 (i.e.,s-, p-, and d-wave channels) for typical experimental conditions. At distances on the orderof typical de Broglie wavelength, U± can be comparable to Ucb and we expect significant
interaction effects in non-zero partial waves.
The relevant quantity that characterizes the net effect of all partial waves is the atom-dimer
forward scattering amplitude [Sob72, Bar58b, Bar58a],∞X
sin 2δl (kcoll )sin2 δl (kcoll )f (0) =
(2l + 1)+i,
2kcoll
kcoll

6.1

l=0

√
where kcoll = 2µ3 Ecoll /~ is the wavenumber associated with the relative atom-dimer motion
and µ3 is the reduced atom-dimer mass. The phase shifts δl for the three lowest partial
waves have been computed in Ref. [Lev11] and here we extend the result to higher ones
since they give significant contributions (Supplemental Material). In Fig. 6.1(b) we show theresulting −Re f (0) as a function of the collision energy Ecoll for the same conditions as inFig. 6.1(a). In the limit of Ecoll → 0, the quantity −Re f (0) corresponds to the atom-dimers-wave scattering length. At Ecoll ≪ 0.1Eb , with Eb being the dimer binding energy, s-wave
scattering (dashed line) dominates and the net interaction is repulsive, −Re f (0) > 0.
For Ecoll & 0.1Eb , higher partial-wave contributions lead to a sign reversal of Re f (0),changing the character of the interaction from repulsive into attractive. This sign reversal
also appears if, at a fixed collision energy, the magnetic detuning from the FR center isvaried, see Fig. 6.1(c). In the realistic example of Fig. 6.1(c) the sign reversal takes place
at a magnetic detuning of B − B0 = −53 mG, where the binding energy is Eb /kB ≈ 3.1 µK,

(U± + Ucb) / kB (nK)

-Re f(0) / 1000 a0-Re f(0) / 1000 a0

400 (a)
300

p

d'

p'

200

d

B-B0 = -16 mG

10

15

s

100

s'

0
0
10

5
(b)

R (1000 a0)

B-B0 = -16 mG

5
0
-5

1

0

100

200

Ecoll / kB (nK)

300

400

(c)

0
-1

Ecoll/kB = 350 nK

-100

-80

-60

B-B0 (mG)

-40

-20

0

Figure 6.1: Interaction between 40 K atoms and 6 Li40 K dimers near the 155 G interspecies FR. (a)Total interaction potentials as a function of the distance R between the two K atoms for the s, p, andd channels (dashed curves with labels s′ , p′ , d′ refer to the unmodified centrifugal barriers). Here wehave chosen a magnetic detuning of B − B0 = −16 mG, corresponding to a s-wave scatering length of
a = 3528 a0 and to a dimer binding energy of Eb /kB = 600 nK. (b) Real part of the forward-scattering
amplitude f (0) as a function of the collision energy Ecoll (solid line) in comparison with the s-wave
contribution (dashed line). (c) Same as in (b), but as a function of the magnetic detuning B − B0 fora fixed collision energy Ecoll /kB = 350 nK. The dotted line indicates the dimer breakup threshold,
Ecoll = Eb .

112

6.3. MAIN RESULTS

113

corresponding to roughly ten times the collision energy Ecoll /kB = 350 nK. The theory lines
in Fig. 6.1(c) stop close to the FR center at the magnetic field detuning where |Eb | = Ecoll(dotted line), beyond which the inelastic channel of collisional dimer dissociation opens up.
The starting point of our experiments is an optically trapped, near-degenerate Fermi-Fermi
mixture of typically 4 × 104 40 K atoms and 1 × 105 6 Li atoms. The preparation proceduresare described in our previous work [Spi10a, Tre11a]. We choose a particular FR that occurs
between Li atoms in the lowest Zeeman sub-level Li|1⟩ (f = 1/2, mf = +1/2) and K atoms
in the third-to-lowest sub-level K|3⟩ (f = 9/2, mf = −5/2) [Nai11]. The s-wave interspecies
scattering length a can be magnetically tuned as a = abg [1 − ∆/(B − B0 )] with abg =63.0 a0 (a0 is Bohr’s radius) and ∆ = 880 mG [Nai11]. The resonance is rather narrow, as
characterized by the length parameter R∗ = 2700 a0 [Pet04a]. The position of the FR centernear B ≈ 154.7 G depends on the trap setting, as it includes small shifts induced by thetrapping light. For each trap setting we have calibrated the FR center B0 with ≤ 2 mG
accuracy (Supplemental Material).
We create an atom-dimer mixture by a Feshbach ramp across the resonance and by subsequent purification and spin-manipulation techniques (Supplemental Material). While thedimers are formed in the Li|1⟩-K|3⟩ spin channel, we initially prepare the free atoms in thesecond-to-lowest spin state K|2⟩ (f = 9/2, mf = −7/2), for which the interaction with thedimers is negligible. The total number of dimers and atoms is 1.5 × 104 and 7 × 103 , respectively. The interspecies attraction during the Feshbach ramp results in a collective oscillationof the dimer cloud, which we can take into account by introducing an effective temperature
Tef f (Supplemental Material). We use three different trap settings, for which Tef f = 165 nK,
232 nK, and 370 nK. This corresponds to mean dimer densities as experienced by the atomsof n̄D = 5.2 × 1011 cm−3 , 8.2 × 1011 cm−3 , and 1.4 × 1012 cm−3 , respectively.
To investigate the interaction between the K|3⟩ atoms and the Li|1⟩K|3⟩ dimers, we carryout rf spectroscopy. This can be done in two different ways, either by driving the K atoms
from the noninteracting state |2⟩ into the interacting state |3⟩ (method A) or vice versa(method B). With our K atoms initially prepared in the state |2⟩, we carry out method A
by applying a 1-ms rf pulse. For method B, we rapidly transfer the full K|2⟩ population into
K|3⟩ using a short 90-µs preparation pulse without spectral resolution, and then drive thespectrally resolving transition with a 1-ms pulse. Our signal in both cases is the fraction oftransferred atoms as a function of the rf detuning ν − ν0 with respect to the unperturbed
transition frequency ν0 , the latter being determined by the rf spectroscopy in the absence of
dimers.
Sample spectra, at a magnetic detuning of B−B0 = −20 mG, are shown in Fig. 6.2. The spectra recorded by methods A and B (circles and diamonds in Fig. 6.2) show both a broadening
and a peak shift, as compared to the spectra recorded in the absence of dimers (triangles).
Although the spectra very close to the FR center reveal asymmetries in their wings, which
depend on the method applied, their peak shifts and broadenings are consistent for bothmethods. In the range of detunings B − B0 studied in the present work the molecular dissociation signal is always well separated from the atomic line (inset of Fig. 6.2), and thus does
not affect the lineshape of the atomic signal.

114

6.3. MAIN RESULTS
0.6
0.5

0.5
0.4

signal

0

−40

−20

2

4

0.3

0

0.2
0.1
0
−8

−6

−4

−2

0

ν − ν0 (kHz)6

8

Figure 6.2: Sample rf spectra taken at B − B0 = −20 mG at Tef f = 232 nK. The red diamonds (bluecircles) show data recorded using method A (B). For reference, the gray triangles show data recordedin the absence of dimers together with a Gaussian fit (gray line). Inset: Spectrum at −17 mG overan extended frequency range. The molecular dissociation signal (open symbols), recorded with 30×increased rf power, is clearly separated from the atomic peak (filled symbols).

Figure 6.3 shows the widths and peak shifts 2 of the rf spectroscopic signal, recorded bymethod A, as a function of B − B0 for our three values of Tef f . When the FR center
is approached, the spectrum broadens and its peak shifts from a positive to a negative rf
detuning. With increasing temperature, the corresponding zero crossing shows a trend tomove towards larger detunings.
We interpret the obtained results in the framework of the impact theory of pressure-induced
effects on spectral lines, which assumes the collisions to be effectively instantaneous. This
theory predicts Lorentzian profiles centered near the unperturbed frequency ν0 whose line
shifts and broadenings are proportional to the real and imaginary parts of the thermally averaged atom-dimer forward scattering amplitude f (0) [Sob72, Bar58b, Bar58a], respectively.
The real part of f (0) shifts the energy of the K atoms, causing an average shift in the frequency of their peak rf response of δν = −~n̄D Re⟨f (0)⟩/µ3 , where ⟨f (0)⟩ denotes the thermal
average of f (0) over all atom-dimer collision energies Ecoll . The red solid lines in Fig. 6.3
show the theoretical results for δν for the respective molecule densities and collision energies.
The optical theorem relates the imaginary part of f (0) to the average elastic scattering rate
τ −1 as τ −1 = 4π~n̄D Im⟨f (0)⟩/µ3 . The resulting finite lifetime τ of the atoms’ wavepackets
causes Lorentzian broadening with a full-width at half-maximum (FWHM) 1/2πτ . The blue
solid lines in Fig. 6.3 show the predicted FWHM, including additional broadening due to thefinite duration of our rf pulse 3 .
The collisional broadening yields information on the elastic scattering rate. At typical detunings of B − B0 ≈ −20 mG, our data show an elastic atom-dimer scattering rate on the orderof 1/(100 µs). A comparison with the observed dimer decay rate of about 1/(5 ms) gives alower limit for the ratio of elastic to inelastic atom-dimer collisions of 50. We note that in2

3

To determine the peak shift and the width, we apply a double-Gaussian fit to the spectra. From the fit,
we identify the rf detuning of maximum signal and the width.
The finite duration of our rf pulse causes an additional Gaussian broadening of typically 1.2 kHz (FWHM).

FWHM (kHz)6.3. MAIN RESULTS

115
3.0
2.0
1.0

peak shift (kHz)

0.2
0.1
0
−0.1
−0.2
−0.3

Teff = 165 nK
−80

−40

232 nK0−80−40

B−B (mG)

370 nK
0

−80

−40

0

0

Figure 6.3: Widths (blue triangles) and peak shifts (red circles) extracted from the rf spectra as afunction of the magnetic field detuning B − B0 for the three different values of Tef f . The lines are thecorresponding theoretical predictions. To account for fluctuations in the dimer number of different
spectra, the widths and peak shifts are scaled to a dimer number of 15, 000, which is typical for all
spectra.

our system the dimers spontaneously dissociate on a timescale of about 10 ms [Nai11].

The comparison between the experimentally observed and the theoretically calculated line
shifts and broadenings shows remarkable agreement over the whole parameter range investigated. The somewhat asymmetric spectral wings are beyond the impact theory [Szu96] andthus cannot be reproduced. Indeed, a substantial contribution to the wings comes from thephoton emission/absorption events for which K atoms find themselves inside the atom-dimer
interaction range, i.e. during atom-dimer collisions, which are assumed instantaneous in theimpact theory. It is then understood that, for example, the left “attractive” wing of the
B-spectrum is larger than that of the A-spectrum. Since in the former case potassium atoms
are initially attracted by dimers, the probability to find them near dimers is enhanced. Effects that are beyond the impact theory become more pronounced as we approach the FR
because of the increased atom-dimer collision time.

We finally discuss the interaction strength in our mixture in terms of −Re⟨f (0)⟩, whichcharacterizes the interactions in a way that is analogous to a in the s-wave mean-field picture.
We use the experimental peak-shift data from Fig. 6.3 to extract −Re⟨f (0)⟩ and plot it
together with the corresponding theoretical results in Fig. 6.4. The sign reversal shows up
for values of a being somewhat below 2000 a0 , with the expected temperature dependenceof the zero crossing. For a ≈ 4000 a0 , the attractive interaction already corresponds toabout −2000 a0 . For even larger values of a, we would enter the more complicated regime ofcollisional dimer dissociation, which is beyond the scope of the present investigations. We
note, however, that rf spectra acquired more deeply in the strongly interacting regime show
strongly asymmetric lineshapes and have peaks shifted to even larger negative detunings.

116

6.4. CONCLUSION
3−Re 〈f(0)〉 (1000 a0)

2
1
0
−1
−2
−3
0

Teff =
165 nK
232 nK
370 nK12

3

a (1000 a0)

4

5

6

Figure 6.4: Real part of the atom-dimer forward-scattering amplitude as a function of the atomatom scattering length a for the three different values of Tef f . The symbols and the lines show thedata and the theoretical predictions from Fig. 6.3. For comparison, the dashed lines indicate therespective s-wave contributions. The theoretical lines stop at kB Tef f = Eb /2.

6.4 Conclusion
In conclusion, we have demonstrated a three-body phenomenon in a mixture of heavy andlight fermions, which leads to a sign reversal of the atom-dimer interaction near a FR,
turning repulsion into a strong attraction. The effect is due to higher partial-wave (mainly
p-wave) contributions, which are present even at very low collision energies in the nanokelvinregime. Remarkably, this few-body effect changes the character of the interaction without
introducing detrimental losses. In contrast to few-body phenomena of the Efimov type[Fer11], the centrifugal barrier still protects the atoms from approaching each other too
closely. The resulting collisional stability is a promising feature for many-body physics in
Fermi-Fermi mixtures.
Our work lays the ground for a wealth of future studies on mass-imbalanced fermionic mixtures in the strongly interacting regime. Asymmetric phases with coexisting dimers andheavy atoms are energetically favored in a way not present in mass-balanced systems [Qi12].
Related mechanisms in quantum-degenerate situations may lead to exotic new many-bodyeffects, including the emergence of imbalanced superfluids [Qi12], the condensation into nonzero momentum states [Mat11], and the appearance of p-wave superfluidity of heavy atoms
mediated by light atoms [Nis09a]. On the few-body side, a direct prospect for our K-Li
system is to confine the K atoms in an optical lattice, which is predicted to lead to theformation of stable trimer states [Pet07, Nis09b, Lev09].

Acknowledgements
We acknowledge funding by the Austrian Science Fund FWF with SFB FoQuS (F40-P04).
M.Z. was supported by the FWF within the Lise Meitner program (M1318), D.S.P. by theInstitut Francilien de Recherche sur les Atomes Froids (IFRAF), and J.L. acknowledges

6.5. SUPPLEMENTAL MATERIAL

117

support from the Carlsberg Foundation.

6.5 Supplemental material
6.5.1 Light shift of the Feshbach resonance
The Feshbach resonance (FR) that we employ for tuning the interactions in our system occurs
between 6 Li atoms in their lowest internal state, denoted Li|1⟩ (f = 1/2, mf = +1/2), and
40 K atoms in their third-to-lowest state K|3⟩ (f = 9/2, m = −5/2). This resonance has beenfinvestigated in detail in Ref. [Nai11]. The magnetic field dependent Li-K s-wave scattering
length is given by
∆a(B) = abg 1 −6.2
B − B0where abg = 63.0 a0 is the background scattering length, ∆ = 0.88 G is the width, and B0 is
the center of the resonance near 154.7 G.
As we already pointed out in Ref. [Koh12b], the optical trap induces a differential light shift
between the atom pair state and the molecular state. This leads to a light-induced shiftof the FR center. For the experiments presented in the main text, we use a near-infraredlaser with a wavelength of 1064 nm (single-mode operation) in three different trap settings.
Therefore, the center of the FR needs to be determined for each trap setting.
To determine B0 we perform radio-frequency (rf) spectroscopy of the Feshbach molecules.
For each trap setting, this is done in the following way: We prepare a nonresonant mixtureof Li atoms in state Li|1⟩ and K atoms in their second-to-lowest state K|2⟩ several tens ofmG below the approximate position of the resonance center. Here, we apply a strong 500-µs
rf pulse at a variable frequency ν, several kHz below the unperturbed K|2⟩→K|3⟩ transition
frequency ν0 . This pulse drives Li|1⟩-K|2⟩ atom pairs into the Li|1⟩K|3⟩ dimer state. To
determine the number of dimers associated, we subsequently dissociate the dimers into a
Li|1⟩ and a K|3⟩ atom by a 300-µs magnetic field ramp to 154.8 G. By recording absorption
images we then determine the populations N2 and N3 of the K spin states K|2⟩ and K|3⟩,
respectively.

By plotting the signal, given by N3 /(N3 + N2 ), against the rf detuning ν − ν0 , we resolve
the molecule association spectrum; see Fig. 6.5. The unperturbed transition frequency ν0 ,corresponding to the Zeeman splitting of the two states, is determined by rf spectroscopy in
the absence of Li|1⟩ (red points). We determine the binding energy of the molecules fromthe onset frequency of the molecular association spectra. As the onset frequency, we use

0.4

B = 154.597 G

signal0.2
00.4

154.629 G

0.2
0
0.4

154.658 G

0.2
0

−200

−150

−100

−50

0

(ν − ν0) (kHz)

Figure 6.5: Data from the molecular rf association spectroscopy in trap 2. Red circles were takenwith a rf power set to the value to match the π-pulse condition in the absence of interactions (noLi|1⟩ present) and is scaled by 0.5. Blue points were taken with a 30× larger rf power. The dashedlines indicate the binding energy Eb (B).

0
20
40

Eb (kHz)

60
80
100
120

Trap 1
Trap 2
Trap 3140160
180
154.58154.60154.62

154.64154.66

B
B (G)
(G)

154.68154.70154.72

Figure 6.6: Determination of the FR center B0 by rf association of dimers. The points show theexperimentally determined molecular binding energies Eb (B) for the three trap settings. The solid
curves are fits of a theoretical model (see text) to the experimental data.

118

6.5. SUPPLEMENTAL MATERIAL

119

the upper rf frequency at which the fraction of atoms transferred is roughly 10% of its peakheight. We have checked that, within the errors of our measurements, this criterion agrees
with the result obtained by fitting the line-shape model [Chi05] to the spectra, as was done in
Ref. [Koh12b]. This procedure is repeated for each trap power at various magnetic fields.
We then fit a model [Koh12b] for the molecular binding energy near our FR to the datawith B0 as the only free parameter; see Fig. 6.6. This procedure allows us to determine theresonance center in each trap setting with an uncertainty of ±2 mG. The accuracy of ourdetermination of the resonance position is limited by the uncertainty in the FR parameters[Nai11] used in the model for the binding energy. We determine the center of the FR of trap
1, 2, and 3 to be at the magnetic field of 154.704 G, 154.708 G, and 154.719 G, respectively.

6.5.2 Preparation of the atom-dimer mixture
To cool our atomic sample, we evaporate a Li|1⟩-Li|2⟩ spin mixture at a magnetic field near
1150 G on the attractive side of the 834-G Li|1⟩-Li|2⟩ Feshbach resonance in a single-beam
optical dipole trap [Spi10a]. During evaporation, a few 104 K atoms are sympatheticallycooled by the Li environment. The endpoint of evaporation is always set to the same final
value. After evaporation, we follow the scheme described in Ref. [Spi10a] to transfer theatoms into a crossed-beam optical dipole trap and reach a magnetic field of 154.8 G withtypically 106 Li atoms in state Li|1⟩ and 4 × 104 K atoms in state K|1⟩. We finally vary thetemperature of our sample by increasing the power of our crossed beams to adiabatically
recompress the trapped sample. This scheme allows us to maintain a similar population
imbalance and degeneracy for the three trap settings used.
To prepare for dimer association, we first create a weakly interacting Li|1⟩-K|3⟩ mixture at
B0 + 180 mG. A first rf pulse transfers ∼80% of the K|1⟩ population into state K|2⟩ and asecond rf pulse then transfers the total K|2⟩ population into the interacting state K|3⟩. The∼7 000 K atoms, which remain in the K|1⟩ state, later serve for the spectroscopy.
We associate dimers using a two-step magnetic field ramp. In a first 20-ms step we ramp
the magnetic field from B0 + 180 mG to B0 + 5 mG. This ramp is sufficiently slow for the
Li atoms to be attracted into the regions of high K density, increasing the density overlap
between the two clouds. We then associate the Li|1⟩K|3⟩ dimers via a 0.5-ms Feshbach ramp
to B0 − 17 mG. We note that, during these magnetic field ramps, two-body inelastic losses[Nai11] are negligible.
To obtain a pure sample of about 15 000 Li|1⟩K|3⟩ dimers, we remove all unbound atoms
from the states Li|1⟩ and K|3⟩. The Li|1⟩ atoms are removed by a sequence of rf and laser
pulses. This procedure consists of a first 250-µs rf pulse resonant with the free Li|1⟩→Li|2⟩
transition, followed by a 10-µs resonant light pulse, which selectively removes the Li|2⟩ atoms
from the trap. This scheme removes about 95% of the excess Li atoms without causing any
observable loss of KLi dimers. A second 250-µs rf pulse transfers the leftover 5% of Li|1⟩
atoms into the noninteracting Li|2⟩ state, where they remain without further affecting theexperiment.

120

6.5. SUPPLEMENTAL MATERIAL

TD (nK)

300

trf

250

TD

200
0

1

2

3

t wait (ms)45Figure 6.7: Radial oscillation of the dimer cloud after the magnetic field ramp and the removal of
the Li atoms. We plot the dimer temperature TD versus the wait time twait after the first rf cleaning
pulse to release from the trap. The filled circles are the experimental data, the solid line is a fitof a damped harmonic oscillation to the data. The shaded area indicates the time at which thespectroscopy rf pulses are applied and the dashed line marks the experimentally relevant averaged
dimer temperature T̄D .

Simultaneously with this “double-cleaning” of the unbound Li atoms, we remove the unboundK|3⟩ atoms in a similar way. Using a 90-µs rf pulse resonant with the K|3⟩→K|2⟩ transition,
followed by a second 145-µs rf pulse resonant with the K|3⟩→K|4⟩ transition, we empty the
K|3⟩ state with >99 % efficiency. The pulse lengths are chosen such that they are short,i.e. spectroscopically wide, compared to the frequency shifts due to atom-dimer and atomatom interactions but long, i.e spectroscopically narrow, compared to the binding energy
Eb = h×17 kHz (h is Planck’s constant) of the dimers, avoiding the dissociation of dimers.
In a final step, the ∼7 000 K atoms which resided in state K|1⟩ during the entire dimer association process, are transferred in the K|2⟩ state and thus prepared for the rf spectroscopy.
This is accomplished by a rf pulse which flips the K|1⟩ and K|2⟩ populations. We note thatthese K atoms remain unaffected by the dimer association since their interactions with theother components are negligible over the timescales of the experiment.
From here, we reach the specific magnetic field detunings B − B0 , at which the spectroscopy
is performed, by a 200-µs magnetic field ramp.

6.5.3 Determination of the temperatures and the densities
Here, we describe how we determine the temperatures and the densities of the atom cloud
and the dimer cloud. The resulting experimental parameters are summarized in Table 6.1.
Atom and dimer temperatures – The temperatures of our atom and dimer clouds are obtained
by Gaussian fits to absorption images of the clouds after a long time-of-flight of ttof = 6 ms.

6.5. SUPPLEMENTAL MATERIAL

121

Table 6.1: Parameters characterizing the three exploited trap settings. The table shows the effecticeatom-dimer temperature Tef f , the temperature of the K atoms, TK , and the average dimer temperature, T̄D . From the radial (axial) trap frequencies of K and Li, νr(a),K and νr(a),Li , we determine thetrap frequencies νr(a),D of the dimers. We also show the axial and radial in-situ Gaussian widths of
dimers (K atoms), σa,D(K) and σr,D(K) , respectively.

Trap
1
2
3
Trap
1
2
3

Tef f
(nK)
165(15)232(15)370(15)TK(nK)
138(5)
225(5)
345(5)T̄D(nK)
195(15)240(15)398(15)

νr,K(Hz)
197(5)
284(5)
415(5)

νr,D(Hz)
216(5)
310(5)
457(5)

νa,D(Hz)
27.0(10)39.3(10)59.0(10)σr,K(µm)
4.3(1)
3.8(1)
3.2(1)

σa,K
(µm)
33(2)
30(2)
25(2)

νa,K
(Hz)
25.5(10)36.4(10)54.0(10)σ̄r,D(µm)
4.4(1)
3.4(1)
2.9(1)

νr,Li(Hz)
314(5)
446(5)
671(5)

νa,Li(Hz)
34.0(10)54.6(10)85.0(10)

σa,D
(µm)
36(2)
33(2)
26(2)

With the measured radial Gaussian width σtof,K(D) the atom (dimer) temperature TK(D) is
given by2
kB TK(D) = mK(D) σtof,K(D) /ttof ,6.3
where mK(D) is the mass of the atom (dimer).
The magnetic field ramps and the removal of the surrounding Li shell, described in theprevious section, excite collective oscillations of the dimer cloud. We trace these oscillations
in momentum space as a function of a wait time twait after the cleaning procedure to releasefrom the trap. An example of such an oscillation is shown in Fig. 6.7. In order to characterize
the temperature at the time of the experiment, i.e. during the application of the 1-ms rf pulse(shaded area), we introduce the average temperature
Z
1
T̄D =
TD dt.
6.4
τrf
rf

Axial and radial sizes – To determine the densities of the atom (K) cloud and the dimer(D) cloud, we measure their Gaussian radial (r) and axial (a) widths σr,K(D) and σa,K(D) ,respectively. The axial widths are measured from a Gaussian fit to the axial profiles of in-situ
absorption images. Since the radial widths are on the order of our imaging resolution, they
can not be determined from in-situ images. We instead determine the radial widths of the
K atom cloud ass
kB TK
σr,K =,
6.5
mK (2πνr,K )2
where TK and νr,K denote the temperature and the radial trap frequency of the K atoms,
respectively. Accordingly we determine the average radial in-situ width of the dimers,skB T̄D
σ̄r,D =,
6.6
mD (2πνr,D )2

122

6.5. SUPPLEMENTAL MATERIAL

using the averaged dimer temperature T̄D , and the radial dimer trap frequency νr,D .
Trap frequencies of the dimers – We use the measured trap frequencies of the K and Li atoms
to determine the trap frequencies νr(a),D of the LiK-dimers. Since the dimers are weakly
bound over the magnetic field range investigated, their polarizabilities are approximately
given by the sum of the polarizabilities of the Li and the K atoms. We want to point out
that the differential light shift, shifting the FR center (see section 6.5.1), gives only a < 10%correction to the trap potential and is neglected. Therefore, to a good aproximation, thedimer trap frequencies are given byq22
νa(r),D = (mK νa(r),K+ mLi νa(r),Li)/mD ,
6.7
with mLi being the mass of a Li atom.
Mean Dimer Density – For a given dimer number, ND , the mean dimer density experienced
by the K atoms n̄D is given by
n̄D =

ND

q
,
2 + σ̄ 2 ) σ 2 + σ 2(2π)3/2 (σr,K
r,D
a,K
a,D

6.8

where we have assumed Gaussian-shaped atom and dimer clouds.
Effective temperature – Due to heating and oscillations caused by our preparation procedure, the dimer temperature TD in our system is different from the temperature of the
non-interacting K|2⟩ atoms that we use for rf spectroscopy. However, since our dimer andatom clouds are both non-degenerate, the energies of the atom-dimer collisions still assume a
Boltzmann distribution. Averaging this distribution over the oscillations of the dimer cloud
results in an effective atom-dimer collision temperature
Tef f = µ3 (TK /mK + T̄D /mD ),6.9

where µ3 = mK mD /(mK + mD ) is the atom-dimer reduced mass.

6.5.4 Importance of higher partial wave scattering and comparison to theequal-mass caseIn this Section, we justify several important statements made in the main text. First, wehave argued that the range of the atom-dimer interaction is comparable with the typical de
Broglie wavelength and, therefore, quite a few partial waves are necessary to quantitatively
characterize the line shift. In Fig. 6.8, we display −Re f (0), the quantity which is thermallyaveraged in the main text to obtain the line shifts. The method of calculating the scattering
amplitude is described in Ref. [Lev09]. Remarkably, the real part of the forward-scattering
amplitude is seen to change sign at a collision energy much smaller than the binding energy,
even for a relatively large detuning of 21 mG. The second change of sign of −Re f (0) seenin Fig. 6.8(a) is attributed to the fact that δp exceeds π/2 above Ecoll ≈ 0.1Eb , the pointof the p-wave resonance. The p-wave contribution at larger collision energies then becomes
positive (repulsive) [see Eq. (6.1) of the main text]. However, this peculiar phenomenon

6.5. SUPPLEMENTAL MATERIAL

123

takes place only in a very close vicinity of the wide resonance limit as the p-wave phaseshift drops rather abruptly with R∗ /a [Lev09]. We also note how, as the collision energy
is increased, more and more partial wave channels are needed to accurately describe theforward-scattering amplitude. The calculation presented here includes the first 16 partial
waves, which is sufficient to obtain an essentially converged scattering amplitude at the dimer
breakup threshold.
As far as the equal mass case is concerned, the competition between the attraction in odd
partial waves and repulsion in even partial waves is also quite significant, yet much lesspronounced compared to the K-Li case. In Fig. 6.9 we display −Re f (0) as a function of
Ecoll for equal masses. Here the broad resonance case in Fig. 6.9(a) is relevant since it isreadily available in current experiments and since there the effect of higher partial waves
is most noticeable. We see that the forward-scattering amplitude does change sign in thiscase. However, in contrast to the K-Li case, this happens at a high collision energy close
to the dimer breakup threshold and, in fact, already for R∗ /a & 0.03 the crossing is nolonger on the scale. Thus, in the narrow resonance case illustrated in Fig. 6.9(b) and (c)the interaction is found to be repulsive below the dimer breakup threshold. In all cases thethermally averaged quantity −Re ⟨f (0)⟩ is positive.
Finally, let us also make a remark concerning the thermal averaging of the scattering amplitude which we use in the main text. In principle, the averaging procedure requires theknowledge of the phase shifts above the atom-dimer breakup threshold. However, we always
restrict ourselves to temperatures kB T . Eb /2 and we check that in this case the integration
result is insensitive to the exact extrapolation scheme. In practice we extrapolate the phaseshift δl (k) using the log function, which works very well when we calculate the phase shifts
above the breakup threshold in the Born-Oppenheimer approximation [Lev11].

2

HaL

HaL

1.0
-Re fH0La-Re fH0La

1
0
-1

0.5

0.0
-2
0.0

0.2

0.4
0.6
EcollEb

0.8

2

1.0

0.0

HbL

0.2

0.4
0.6
EcollEb

0.8

1.0

HbL

1.0
-Re fH0La-Re fH0La

1
0
-1

0.5

0.0
-2
0.0

0.2

0.4
0.6
EcollEb

0.8

2

HcL

0.2

0.4
0.6
EcollEb

0.8-Re fH0La

0
-1

1.0

HcL

1.0

1
-Re fH0La

0.0

1.0

0.5

0.0
-2
0.0

0.2

0.4
0.6
EcollEb

0.8

0.0

1.0

Figure 6.8: Scattering of a 40 K atom witha 6 Li40 K dimer. The quantitiy −Re f (0) is
plotted as a function of atom-dimer collisionenergy for (a) R∗ /a = 0 [B − B0 = 0], (b)R∗ /a = 1/2 [B − B0 = −10 mG], and (c)R∗ /a = 1 [B − B0 = −21 mG]. The lines
are including s-wave scattering only (black,dashed), including up to p-wave (blue, dotted), up to d-wave (purple, dot-dashed), andup to f -wave (gray, double dot-dashed). The
solid black line is −Re f (0) including the first16 partial waves.

0.2

0.4
0.6
EcollEb

0.8

1.0

Figure 6.9: Equal-mass case of atom-dimer
scattering. We plot −Re f (0) as a functionof collision energy for the homonuclear case,m↑ = m↓ . The conventions used for the lines
as well as the detunings in (a) to (c) are thesame as in Fig. 6.8. The solid black line includes the first 9 partial waves.

124

Chapter

7

Lifetime of 6Li-40K Dimers near a
Feshbach Resonance1

The present chapter contains the experimental part of a manuscript in preparation, authoredby M. Jag1,2 , M. Cetina1,2 , R. S. Lous1,2 , R. Grimm1,2 , J. Levinsen3 , and D. Petrov4 .
1Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
School of Physics and Astronomy, Monash University, Victoria 3800, Australia
4
LPTMS, CNRS, Université Paris Sud, Université Paris-Saclay, 91405 Orsay, France

7.1 Introduction
At ultra-low collision energies, radiative losses in collisions between ground-state neutral
atoms are suppressed and losses in collisions of pairs of such atoms chiefly occur by spontaneous relaxation into lower-lying atomic Zeeman states. This suppression of losses has
played a decisive role in achieving Bose-Einstein condensation in cold atomic gasses.
In the presence of strong interactions between atoms, such as those achievable using atomic
Feshbach resonances [Chi10], three-body losses play a significant role. Due to Pauli’s exclusion principle, three-body losses in fermionic gases can be strongly suppressed, resulting inlifetimes of strongly interacting fermionic gases that are measured in seconds [Cub03, Joc03a].
This remarkable property of tunable fermionic gases has led to spectacular achievements, including the realization of a BEC of molecules [Joc03b, Gre03] and the study of the BEC-BCS
crossover [Bar04b, Reg04b].
Feshbach resonances (FR) arise from the coupling of molecular bound states of pairs of atoms
to unbound scattering states. For weak atom-molecule couplings corresponding to narrow
1

The author of the present thesis developed the experimental procedures, performed the measurements
together with M.C. and R.S.L, and analyzed the data together with M.C. under the supervision R.G.

125

126

7.2. EXPERIMENTAL PROCEDURES

Feshbach resonances, close to the FR, the molecular state can be significantly populated.
Since the molecular bound states are distinguishable from free fermionic atoms, fermionicsuppression is predicted to be less effective in this case. Indeed, measurements of losses incollisions involving Feshbach molecules near the narrow 6 Li Feshbach resonance at 543 G are
consistent with the absence of fermionic suppression [Wan13]. In the intermediate regime ofa moderately narrow FR, the lifetimes of inter-species Feshbach molecules were investigatedin the 6 Li-40 K Fermi-Fermi mixture by [Voi09]. The lifetime of these dimers was observed
to increase near the FR, consistent with fermionic suppression.
In the following, we present detailed measurements of the lifetime of 6 Li40 K dimers near two
different inter-species FR. Improving on the earlier work by [Voi09], we distinguish between
spontaneous and collisional decay processes in our system and determine the loss-rate coefficient of a trapped dimer sample as well as an atom-dimer mixture. Near the Feshbachresonances, we find the atom-dimer and dimer-dimer collisional decay to be suppressed bymore than a factor of five and three, respectively. We compare our observations to theoreticalpredictions by Levinsen and Petrov [Lev11] and find very good agreement.

7.2 Experimental procedures
7.2.1 Feshbach resonances
We study the lifetime of Li-K dimers near two different interspecies Feshbach resonances.
The first resonance has been widely used in our previous work on Fermi-Fermi mixtures,
including the observation of the hydrodynamic expansion of a strongly interacting mixture[Tre11a], the investigation of polarons (Chapters 3 and 4), and the study of LiK-K atomdimer interactions [Jag14]. The resonance is found near 155 G in the combination of lithiumin its lowest Zeeman sub-level Li|1⟩ (f = 1/2, mf = +1/2) and potassium in its thirdlowest sub-level K|3⟩ (f = 9/2, mf = −5/2). The other resonance is found near 158 G
in the lowest spin-state combination of Li|1⟩ and K|1⟩ (f = 9/2, mf = −9/2). We use
this narrower resonance for comparison as it has the advantage of an absence of any Li-K
two-body losses.
The dependence of the Li-K s-wave scattering length a on the magnetic field B near a FR canbe described by the standard expression a(B) = abg [1 − ∆/(B − B0 )] [Chi10] with the relevant background scattering length abg , width ∆, and the resonance center B0 . In Table 7.1 wesummarize the values of these parameters for both resonances. To fully characterize the FRs,
we also present the differential magnetic moments δµ between the relevant open and closed
channels. From these parameters, we derive the length parameter R∗ = ~2 /(2mr ∆abg δµ)[Pet04a], characterizing the coupling strength between the open and the closed channels.
Here mr represents the Li-K reduced mass. The values for abg and ∆ have been obtained
from a coupled-channels calculation [Nai11]. The values for δµ as well as B0 for the Li|1⟩-K|3⟩
FR were experimentally determined, with a very high accuracy, as described in Chapter 4.
For δµ near the Li|1⟩-K|1⟩ FR we use the data obtained from a coupled-channels calculation

7.2. EXPERIMENTAL PROCEDURES
ChannelLi|1⟩K|3⟩
Li|1⟩K|1⟩

B0(G)
154.708(2)157.530(3)

127
abg
(a0 )
63.0
65.0

∆
(G)
0.88
0.14

δµ/h
(MHz/G)
2.35
2.3

R∗
(a0 )
2 650
16 500

Table 7.1: Parameters characterizing the two Feshbach resonances. We summarize the values from
Chapter 4, Ref. [Nai11] and the footnote2 for the position B0 , background scattering length abg , andwidth ∆ as well as the differential magnetic moment δµ. The values given for B0 include a small
shift (9 mG) induced by the trapping-laser light (see Chapter 4).

[Nai11] and for B0 we use the value of an independent experimental determination2 .

7.2.2 Sample preparation
Our procedure to prepare Li|1⟩K|3⟩-dimer samples is nearly identical to the one describedin Ref. [Jag14]. To produce Li|1⟩K|1⟩ dimer samples, we slightly adapt this procedure toaccount for the narrower character of the FR. In both cases, the starting point for our
experiments is an optically trapped and thermally equilibrated mixture of typically 105 Li
atoms and approximately 3 × 104 K atoms at a temperature of ∼ 370 nK and at a magnetic
field of 156.4 G. We reach these conditions by a preparation procedure described in detailin Ref. [Spi10a]. The cigar-shaped optical confinement of the atom mixture, realized by
two 1064-nm laser-light beams intersecting at an angle of about 16◦ , is characterized by
the radial and axial trap frequencies νr, K = 420(10) Hz and νa, K = 55(2) Hz for the K andνr, Li = 600(10) Hz and νa, Li = 90(3) Hz for the Li atoms. At this stage, all Li atoms are in
their lowest Zeeman sub-level Li|1⟩ and all K atoms are in their second-lowest sub-level K|2⟩
(f = 9/2, mf = −7/2).
From here on, the preparation steps differ depending on the Li-K spin-state combination fromwhich the dimers are created. Preparing for the creation of Li|1⟩K|3⟩ (Li|1⟩K|1⟩) dimersfrom these mixtures, we slowly ramp the magnetic field within 2 s to a value of 154.89 G(157.565 G), approximately 180 mG (35 mG) above the center of the FR. Here, we transfer
all K atoms into the K|3⟩ (K|1⟩) state by a radio-frequency rapid adiabatic passage.
We then associate approximately 104 LiK dimers by a Feshbach ramp [K0̈6, Chi10]. To
associate dimers from the Li|1⟩-K|3⟩ mixture, we do this in two steps, as illustrated in Fig. 7.1.
In a first step we ramp the magnetic field to B0 + 5 mG in 20 ms, which is sufficiently slow for
the Li atoms to be attracted into the regions of high K density, increasing the density overlap
between the two clouds. This is followed by the second step, in which we quickly ramp themagnetic field to B0 − 20 mG in 0.5 ms. In contrast, for the Li|1⟩-K|1⟩ mixture, we associate
the dimers by a single 2-ms Feshbach ramp to a magnetic field B = B0 − 16 mG. Due tothe more elaborate, two-step magnetic-field ramping, typical dimer numbers of Li|1⟩K|3⟩
samples are roughly 20% larger than typical dimer numbers of Li|1⟩K|1⟩ samples.
2

We determined B0 by measuring the energy shift of K atoms in a Li cloud and comparing it to thepredictions of a dressed quasiparticle model [Mas12]

B-B0 (mG)

20

(a)

0
-20
-40
-2.5

Li|1〉
K|3〉

-2.0

-1.5

-1.0

-0.5

0.0

0.5

K

|3〉
|2〉

Li

t (ms)

|1〉
|2〉

|2〉
|1〉
|2〉

|1〉

|3〉
|2〉
|1〉
|2〉

|2〉
|1〉
|2〉

|1〉

(b)

-550 -500 -450 -400 -350 -300 -250 -200 -150

t (µs)
Figure 7.1: (Color online) Scheme of the preparation of a pure Li|1⟩K|3⟩ dimer sample. (a) Startingfrom a magnetic field B = B0 +180 mG we approach the resonance by a first 20-ms ramp to B0 +5 mG.
Then, we associate dimers by a quick (0.5 ms) ramp across the FR to a magnetic field B0 − 20 mG.
Here, we remove unbound K and Li atoms from the sample. After this cleaning procedure we reachthe final magnetic field B, at which we perform the lifetime measurement, by a 200 -µs ramp (dottedline). (b) The cleaning procedure for both Li and K consists of radio-frequency pulses (solid black),
selectively transferring unbound atoms into another spin state, and successive removal of these atoms
from the trap by a resonant laser-light pulse (dotted red). This cleaning procedure is once repeated
to increase the purity of the dimer sample.

128

7.3. MEASUREMENTS OF DIMER DECAY

129

To obtain pure dimer samples we apply cleaning sequences to remove unbound atoms. In
both cases, for Li|1⟩K|3⟩ and for Li|1⟩K|1⟩ samples, this sequence consists of a combinationof radio-frequency (rf) and laser-light pulses. A roughly 100 µs-long rf π pulse selectivelytransfers the free Li atoms from state Li|1⟩ into state Li|2⟩. A subsequent 10-µs laser-lightpulse selectively removes the Li|2⟩ atoms from the trap. Simultaneously to this Li cleaning
procedure, we remove the unbound K atoms in a similar way. Applying two rf π pulsesof about 80 µs and 40 µs duration, we transfer the free K atoms from the initial to thesecond nearest neighboring state, and successively remove them from the trap by applying
a resonant laser-light pulse. As these cleaning procedures remove about 95% of the free Li
and K atoms, they are repeated one time to clean the respective states more thoroughly, seeFig. 7.1. We then quickly, within 200 µs, ramp the magnetic field to its variable final value,
at which we then perform the measurements.

7.2.3 Dimer detection and dimer temperature determination
We determine the dimer numbers from absorption images of Li and K atoms after dissociationof the LiK dimers into Li-K pairs by a reverse Feshbach ramp [K0̈6, Chi10]. For bothresonances we ramp the magnetic field B up to a value ≥ B0 + 50 mG within 10 µs. Afteran additional wait time of a few 10 µs, we simultaneously take absorption images of the
Li and the K clouds, from which we determine the numbers of Li and K atoms. In somemeasurements, which were carried out at an earlier stage of the experiments, we detected
only the number of Li atoms resulting after the reverse Feshbach ramp.
The temperature of the dimers is determined from Gaussian fits to absorption images of theclouds after a time-of-flight expansion period of tTOF = 4 ms. The procedure is discussed indetail in Ref. [Jag14]. From the measured radial width σr , we obtain the dimer temperature
TD from kB TD = mD (σr /tTOF )2 , where mD = mLi + mK is the mass of a Li-K dimer.
Typically, the temperatures of our dimer samples are about TD = 550 nK. This corresponds
to peak phase-space densities of about 0.1 for typical dimer number densities in our samples.
We explain the increased temperature of our dimer cloud compared to the temperature prior
to the dimer association (370 nK) by heating and oscillations caused by our preparation
procedure [Jag14].

7.3 Measurements of dimer decay
Our dimers created from atom pairs in an excited Zeeman state are subject to spontaneous
decay. In a sample of dimers, a dimer can furthermore decay in collisions with anotherdimer. When such a sample is mixed with an atomic cloud there will be additional dimer
decay arising from atom-dimer collisions. We perform several series of experiments on dilute,
expanded as well as dense, trapped dimer samples both pure and mixed with an atomic cloud.
The combination of the results of these measurements allows us to extract accurate rates atwhich the spontaneous and collisional processes occur.

130

7.3. MEASUREMENTS OF DIMER DECAY

N=1000

15

10

5

0

2

4

6

8

10

t (ms)
Figure 7.2: (Color online) Comparison of the dimer number evolution near the Li|1⟩-K|3⟩ andthe Li|1⟩-K|1⟩ FR. The blue squares show a typical decay curve of a Li|1⟩K|3⟩-dimer sample at
B = B0 − 296 mG. Fitting an exponential decay to the data yields the 1/e-lifetime τ = 5.8(4) ms.
The fit is represented by the blue solid line. The results from similar measurements with a Li|1⟩K|1⟩dimer sample at a magnetic detuning of −75 mG from the respective resonance center, are shownas the red triangles. Here, we observe the dimer number to remain essentially constant. A fit ofan exponential decay to the data (red solid line) is consistent with infinite lifetime. The error bars
represent 1σ uncertainties; in some cases they are smaller than the symbol size.

7.3.1 Spontaneous dissociation of Li|1⟩K|3⟩ dimers
A dimer created from atom pairs with at least one component in an excited Zeeman state canspontaneously decay via processes involving coupling between the two spins [Chi10]. Suchdecay has previously been studied theoretically and experimentally for the case of 85 Rb2
molecules [K0̈5, Tho05]. The spontaneous decay of Li|1⟩K|3⟩ dimers has been theoreticallyinvestigated in detail in Ref. [Nai11]. There, predictions for the lifetime of the dimers wereobtained from coupled-channels calculations, and we show these predictions in Fig. 7.3 as
the black solid line. While for magnetic detunings B − B0 of around a few hundred mG
the lifetime is around 6 ms, it is predicted to substantially increase near the resonance. This
increase can be attributed to the increasing halo character of the dimer wave function as the
FR is approached. This is accompanied by a decreased probability to find a pair of Li andK atoms within the short range where the spin coupling occurs [Nai11].
We investigate the lifetime of Li|1⟩K|3⟩ against spontaneous decay using dimer samples

7.3. MEASUREMENTS OF DIMER DECAY

131

with a very low number density, such that density-dependent collisional losses do not play asignificant role. We realize such dilute dimer samples by allowing the optically trapped dimer
cloud to expand after switching off the trap. After a variable expansion time t, we determine
the molecule number in the sample. Note that the 1064-nm light induces a shift of the FR
center B0 , as described in Chapter 4. When the optical trap is off, the FR center B0 of the
Li|1⟩-K|3⟩ resonance is found at 154.699 G, i.e. 9 mG lower than in the trap (Table 7.1). For
the Li|1⟩-K|1⟩ channel we assume the same small shift.
In Fig. 7.2 we show a typical decay curve of a Li|1⟩K|3⟩-dimer sample, recorded at a magnetic
detuning B −B0 = −296 mG (blue squares). For our analysis, we only consider data obtained
for t ≥ 1.5 ms, where the mean dimer number density in the sample is below 5 × 1010 /cm3 .
To these data we fit a simple exponential decay, N0 exp (−t/τ ), with the initial dimer number
N0 and the lifetime τ as free parameters. For the specific example of Fig. 7.2, this procedureyields τ = 5.8(4) ms and the fit result is shown as the blue solid line.
For comparison, we also show the evolution of the number of Li|1⟩K|1⟩ dimers recorded
75 mG below the center of the Li|1⟩-K|1⟩ resonance (red triangles). Here, the spontaneous
decay mechanism is absent. Indeed, we observe an essentially constant number of Li|1⟩K|1⟩
dimers, with the fit yielding the decay rate 1/τ = 0.008(7) s−1 . This result is essentiallyconsistent with an infinite lifetime and, at a 95% confidence level, provides a lower bound of
50 ms.
In Fig. 7.3, the blue circles show the experimentally determined lifetimes of the dimers against
spontaneous decay over a wide range of magnetic detunings. Our experimental results are inexcellent agreement with the theoretical prediction from Ref. [Nai11] over the whole magnetic
field range investigated. The lifetimes of the dimers we determine are generally longer than
5 ms. In particular, we observe the onset of the halo-dimer regime close to the FR, withdimer lifetimes surpassing 8 ms. Our results on the lifetime of the Li|1⟩K|3⟩ dimers in dilute
samples can be fully understood in terms of spontaneous dissociation.

7.3.2 Dimer-dimer collisions
In a second series of experiments, we investigate the collisional decay of a trapped dimer
cloud. In collisions with other dimers our shallowly bound dimers can relax into more deeply
bound states. The released binding energy is much larger than the depth of the trapping
potential, and thus the relaxation products are always lost from the trap. This two-bodydecay occurs at a rate βD n, which results as the product of the dimer-dimer two-body lossrate coefficient βD and the dimer number density n.
To experimentally determine the rate coefficient βD for these collisional decay processes,
we investigate the decay of a trapped sample of dimers. The initial number of typicallyN0 = 1.3×104 dimers corresponds to an initial number density N0 /Veff of about 1×1012 /cm3 ,2 )]3/2 is the effective volume of a thermalized sample, andwhere Veff = [(4πkB TD )/(mD ω̄D2
1/3ω̄D = 2π(νr, D νa, D )= 2π × 222 Hz is the mean dimer trapping frequency. After a holdtime t at a magnetic field B we measure the number of dimers, N (t), remaining in the

= (ms)

10

8

6
-800

-600

-400

-200

0

B ! B0 (mG)
Figure 7.3: (Color online) Lifetime of dimers against spontaneous decay near the Li|1⟩-K|3⟩ FR.
The data points show the experimental results and the black solid line represents the theoreticalprediction from Ref. [Nai11]. While the filled symbols result from decay curves, where both the Li
and the K component have been imaged after dissociation, the open symbols are based on detecting
K alone. The error bars represent the 1σ fit uncertainties.

132

N=N0

1

0.5

0

0

2

4

6

8

10

12

t (ms)
Figure 7.4: (Color online) Comparison of the decay of a trapped and an expanding dimer sample.
The blue squares show the measured dimer number in a trapped sample versus hold time t in the
trap. The red triangles show the dimer number determined in a dilute, expanding sample, 1.5 ms after
release from the trap. The blue and red lines correspond to the fit of our model to the data without
and with two-body decay (see text). For a direct comparison, the experimental data are normalized
to the initial dimer number N0 = 13000 (15300) obtained from the fit to the data acquired fromthe trapped (expanding) sample. The error bars represent 1σ uncertainties; in some cases they aresmaller than the symbol size.

133

134

7.3. MEASUREMENTS OF DIMER DECAY

sample. In Fig. 7.4 we show an example for a decay curve obtained at a magnetic detuningof −710 mG from the Li|1⟩-K|3⟩ FR (blue squares).
We model the decay with the common loss-rate equation
Ṅ /N = −1/τ − (βD /Veff )N.

7.1

Under the assumption that the sample remains in thermal equilibrium at the initial temperature TD , the differential equation has the solution
N (t) = Veff /τ.
Veff
βD + Nexp(t/τ
)
−
β
D
0τ

7.2

We fit Eq. (7.2) to the data to experimentally determine the loss rate coefficient βD . For
the fit, we fix τ to the corresponding theoretical value, as verified in the independent measurements presented before. Assuming a constant temperature of the sample, we neglectanti-evaporation heating [Web03]. We have checked that including the latter into our analysis, leads to slightly higher values for βD . We found the correction to stay well below 15%.
Furthermore, our determination of the loss coefficient is affected by a systematic error ofthe dimer number density. We estimate a combined systematic error of about 40%, arisingfrom largely uncorrelated estimated uncertainties of 25%, 7%, and 20% in the dimer number,the dimer trapping frequencies, and the dimer temperature, respectively. For the data ofFig. 7.4, the fit result is shown as the blue solid line. For comparison, we also show the decay
curve of a dilute dimer sample, where collisional loss is absent (red triangles), together withthe result of a fit of an simple exponential decay to this data (red line). This shows that, under typical experimental conditions, both collisional relaxation and spontaneous dissociation
give similar contributions to the total decay of the trapped dimer sample.
We take corresponding measurements at various magnetic detunings. Our experimental
results for βD , obtained with Li|1⟩K|3⟩ (Li|1⟩K|1⟩) dimer samples, are shown in Fig. 7.5 as
the blue circles (red squares). For the Li|1⟩K|1⟩ dimer sample we obtain values for the loss
rate coefficient βD of roughly 3×10−10 cm3 /s without significant dependence on the magnetic
detuning. For the Li|1⟩K|3⟩ dimer sample we obtain roughly the same value for detunings
B − B0 . −400 mG. The Feshbach molecules, at these large magnetic detunings, have a verysmall admixtures of the entrance channel and are thus strongly closed-channel dominated[Chin2010fri]. The decay of such molecules is largely independent of the exact state they arein [cite].
When approaching the Li|1⟩-K|3⟩ resonance, our experimental results (with the exceptionof one clear outlier 3 ) show an increasing reduction of collisional losses. For our data points
closest to resonance (about −30 mG detuning), this suppression effect amounts to more thana factor of three. Note that we cannot measure closer to resonance because of an onset ofcollisional dissociation [Joc03a].
We attribute this suppression of losses to the change of quantum statistics these dimers
follow as the FR is being approached. Closer to the resonance, the dimers have an in3

The data point at −142 mG clearly lies beyond the trend of the other data. Thoroughly re-checking thesettings of our experimental setup for this measurement yielded no hint for what could have caused thediscrepancy.

7.3. MEASUREMENTS OF DIMER DECAY

135-D (10!10 cm3 =s)

5
4
3
2
1
0
-800

-600

-400

-200

0

B ! B0 (mG)
Figure 7.5: (Color online) Measured loss rate coefficient for inelastic dimer-dimer collisions as afunction of magnetic detuning. The blue circles (red squares) correspond to the experimental results
obtained with samples of Li|1⟩K|3⟩ (Li|1⟩K|1⟩) dimers. The filled symbols correspond to results weobtained when determining the molecule number from both Li and K absorption images. Open circles(squares) represent fit results based on analyzing Li (K) images alone. The error bars represent the1σ fit uncertainties; in some cases they are smaller than the symbol size. We show the light blue anda light red line as guides to the eye.

creasing open-channel character and the fermionic nature of the dimer constituents becomesapparent in a collision. As it takes at least three particles, out of which two necessarily
are identical fermions, to come close for the dimer to decay into a deeper-bound state, collisional decay of open-channel dimers will be Pauli blocked. This fermionic suppression ofcollisional relaxation of Fermi-Fermi molecules has previously been observed in experimentswith homonuclear Li2 [Cub03, Joc03a] and K2 [Reg04a].

7.3.3 Atom-dimer collisions
In another set of experiments, we study the decay of dimers arising from their collisions with
Li atoms in a mixture of LiK dimers and Li atoms. Such decay occurs at a rate βLiD nLi ,which results as the product of the Li atom dimer loss coefficient βLiD and the Li density nLi .
The measurement of atom-dimer collisions is challenging because the corresponding decay
has to be discriminated against the background from spontaneous decay and dimer-dimer
collisional decay.

136

7.3. MEASUREMENTS OF DIMER DECAY

We realize Li atom dimer mixtures by adapting our preparation scheme presented in Sec. 7.2.2.
Here we start with the lithium component in a nearly balanced spin mixture of Li|1⟩ andLi|2⟩. The Feshbach ramp then produces a mixture of Li-K dimers, some remaining Li|1⟩
atoms, and the unaffected Li|2⟩ atoms. Then, at B = B0 − 20 mG, we apply only one radiofrequency π pulse flipping the populations of the Li|1⟩ and Li|2⟩ states. We subsequently
remove the Li|2⟩ atoms from the trap with a laser-light pulse. All other preparation steps,
in particular the K spin state cleaning, remain as described in Sec. 7.2.2. Applying thisprocedure, the number density distribution of the Li atoms in the trap, nLi , can be well approximated from the initial Li temperature and the Li trap frequencies using the well known
expressions for the density distribution of a trapped Fermi gas. Typically, we obtain samples
of & 9 × 103 dimers and & 6 × 104 Li atoms, corresponding to a mean dimer density of about
6 × 1011 /cm3 and a Li density averaged over the dimer distribution, ⟨nLi ⟩ (Chapter 5), of
about 1.5 × 1012 /cm3 . These conditions correspond to a thermal sample of dimers immersed
into a weakly degenerate Fermi sea, characterized by kB T /ϵF ≈ 0.55, where ϵF is the average
Fermi energy sampled by the dimers (Chapter 5).
To experimentally determine the rate coefficient βLiD , we again investigate the decay of
dimers from our sample. We ramp the magnetic field to a desired value B and, after avariable hold time t, we measure the number of dimers, N , remaining in the sample. For
each decay curve in the atom-dimer mixture we record a corresponding reference curve in apure dimer sample. The reference measurements, which independently provide the dimerdimer loss coefficient βD , have already been presented in the preceding section. To minimize
systematic errors by long-term drifts of the experiment the measurements in the atom-dimer
mixtures and the pure dimer samples are carried out in an alternating way.
We model the decay of dimers with a simple extension of the decay model from the previous
section. Our Li sample is much larger than the dimer sample, such that losses from the Li
sample can be neglected. In this case, the Li sample represents a constant bath and the lossof dimers arising from Li atom dimer collisions appears as a one-body loss, which we include
into our model by adding −βLiD ⟨nLi ⟩ to the right-hand side of Eq. (7.1). Hence, the solutionof our model is given by substituting τ −1 with βLiD ⟨nLi ⟩ + τ −1 in Eq. (7.2). We fit thissolution to our experimental data to determine the Li atom dimer loss coefficient βLiD . For
the fit, we fix τ to the corresponding theoretical value and the decay coefficient βD to thevalue we determined in the corresponding reference measurement on a pure dimer sample.
In Fig. 7.6, we show our results for the Li atom dimer loss coefficient βLiD at various magnetic
detunings. The blue circles (red squares) correspond to data acquired with a Li|1⟩K|3⟩(Li|1⟩K|1⟩) dimer sample. The error bars reflect the 1σ fit uncertainty of βLiD as well as thecontribution arising from the uncertainty in our determination of βD . We obtain atom-dimer
loss-rate coefficients of roughly 1.5 × 10−10 cm3 /s near the Li|1⟩K|1⟩ FR, where the molecules
have closed-channel character. The data obtained with Li|1⟩K|3⟩ dimers show a suppressionof atom-dimer collisional losses, which gets stronger as we approach the FR and the openchannel fraction of the dimers increases. The data point at a magnetic detuning of about−40 mG already shows a suppression by a factor of roughly five. From our measurements
closest to the FR, we determine a negative loss coefficient. We speculate, that this unphysicalresult is due to the repulsive mean-field interaction between the dimers and the Li atoms,
effectively increasing the cloud sizes and therefore decreasing the mean densities of the dimers

-LiD (10!10 cm3 =s)

2
1.5
1
0.5
0
-0.5
-500

-400

-300

-200

-100

0

B ! B0 (mG)
Figure 7.6: (Color online) Measured loss rate coefficient for inelastic Li atom dimer collisions as afunction of the magnetic detuning. The blue circles (red squares) correspond to the experimental
results obtained with samples of Li|1⟩K|3⟩ (Li|1⟩K|1⟩) dimers co-trapped with Li|1⟩ atoms. In theseexperiments, the dimer number was determined from the K absorption images only. The error bars
include the combined fit uncertainties (see text).

137

138

7.4. CONCLUSION AND OUTLOOK

and the Li atoms. Such an effect is beyond the assumptions of the model underlying our data
analysis and can therefore produce unphysical results. We estimate that all other values,
taken at larger detunings, do not suffer from such interaction effects.
We ascribe the observed suppression of atom-dimer collisional losses, similar to the suppression of dimer-dimer collisional losses presented before, to the increased fermionic characterof the dimers, as the FR is being approached.

7.4 Conclusion and outlook
We have experimentally determined the lifetime of dimers against spontaneous and collisional
decay. Our measurements of the spontaneous decay confirm the theoretical prediction from
Ref. [Nai11]. Furthermore, we showed that, approaching the resonance, the dimer decay
in atom-dimer and dimer-dimer collisions is being suppressed by a factor of five and three,respectively. The observed suppression qualitatively agrees well with the results from a fewbody theory [Lev11], which relates the collisional decay rate to the probability of finding thecolliding atoms within a short range.
We will, in close cooperation with Dima Petrov and Jesper Levinsen, the authors of Ref.
[Lev11], work on quantifying the agreement of their theory and our measurements. Preliminary results show good agreement and encourage to apply the theory to Fermi-Fermi
mixtures of other species. The obtained theoretical prediction would then allow to analyze
the prospect of such Fermi-Fermi mixtures for creating a BEC of dimers, which typically isa first step towards the investigation of the BEC-BCS crossover or the realization of exotic
quantum phases.

Chapter

8

Outlook

Here we shortly discuss a few research topics, which can be tackled using the FeLiKx machine.
These are immediate extensions of measurements done within this thesis work, which require
only small add-ons to the existing setup.
Many-body physics with FeLiKx - bosonic vs. fermionic impurities
For the experiments with the fermionic impurities, described in Chapters 3, 4, and 5, typically∼ 5 × 103 K impurities were immersed into the Fermi sea consisting of ∼ 3 × 105 Li atoms.
We could show, that with this system, we are very close to the regime of a single impurity
in a Fermi sea, c.f. Fig. 4 in Chapter 5. This is due to the fact that the impurities wereidentical fermions, 40 K, and therefore did not directly interact with each other.
Recently, we extended the K laser setup by a few optical components to also allow for thegeneration of laser light needed for laser cooling and imaging of the bosonic isotope 41 K.
Bose-Einstein condensates as pure as ∼ 80%, mixed with a highly degenerate Fermi sea of
6 Li, were realized so far. The 6 Li-41 K Fermi-Bose mixture, with both species prepared intheir lowest Zeeman state, exhibits a Feshbach resonance at a magnetic field around 335.3 G.
This resonance, with a width of 0.93 G and sitting on a background scattering length of+62 a0 , is very similar to the resonance of the 6 Li-40 K Fermi-Fermi mixture at a magnetic
field around 154.8 G 1 . As a first step, the stability of a mixture of a dilute boson cloud
immersed into a degenerate Fermi sea will be investigated across an inter-species Feshbachresonance, which is already a highly interesting research topic by itself [Yu11].
The major research block will be devoted to polarons, formed by bosons dressed by afermionic cloud, which are investigated using our well understood radio-frequency spectroscopy approaches in the time and frequency domain presented within this thesis. Using
bosonic 41 K instead of the fermionic 40 K as the impurities, where there is no Pauli blocking
preventing them to collide with each other, we expect strong effects arising from impurityimpurity interactions already for the low concentrations mentioned above. Further increasing
the impurity concentration, it will be interesting to see if the condensation of such polarons
can be observed [Yu11, Wu11]. The fortunate fact, that near the proposed Feshbach resonances interaction tuning can be achieved in such a similar manner, will facilitate the1

The 6 Li-40 K resonance appears between Li prepared in the lowest and K in the third-to-lowest state. Its
width and background scattering length are given by 0.88 G and +63 a0 , respectively.

139

140
investigation of the effect of quantum statistics on the impurity physics.
Few-body physics with FeLiKx - 40 K-40 K6 Li coupling to a trimer state
In Chapter 6, we discussed the attractive interaction between 6 Li40 K dimers and 40 K atoms
in a regime of repulsive Li-K interactions. This attraction is due to the presence of a p-wave
atom-dimer scattering resonance in our mass-imbalanced mixture (mK /mLi ≈ 6.6), which
originates from coupling to a virtual trimer state at positive energy [Lev09, Lev11]. For amass ratio above a critical value of 8.2 this state becomes a bound trimer state [Kar07].
We can increase our mass imbalance mK /mLi ≈ 6.6 above the critical value by adding a (e.g.
one-dimensional) optical lattice to the existing experimental setup (see e.g. Ref. [Zwe03]).
Due to the more than two times deeper optical potential, which 1064-nm light creates for the
K atoms compared to the Li atoms, the K-Li effective-mass ratio can be tuned by varying
the intensity of the laser beam forming the optical lattice. The needed laser light can rightaway be deduced from the source used for our current optical traps (Innolight Mephisto
42NE MOPA).
In such a (quasi-)two-dimensional geometry, a trimer state is predicted to exist in the immediate vicinity of the Feshbach resonance [Lev09]. Here, the trimer binding energy will only bea few percent larger than the dimer binding energy [Lev13] and, at typical temperatures inour experiments, already thermal excitations allow these trimers to dissociate into an atom
and a dimer. If a proof of the experimental realization of this three-body state in our system
will be provided using radio-frequency spectroscopy is questionable. However, signatures ofcoupling to this trimer state can be obtained by applying a method measuring few-bodycorrelations, such as Bragg spectroscopy [Vee08, Vee09].

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Acknowledgements
I want to thank the people who made my PhD time in Innsbruck possible and so enjoyable.
My supervisor Rudi Grimm, thank you for allowing me to be part of the great research
environment we have at the institutes. I am grateful for learning about your intuition andyour view on the physics in the many meetings we had. In particular, I will treasure ourMaria Waldrast conclaves - though the beer could have been better. I am especially gratefulfor your obliging support that made it possible to complete my PhD while still meeting myfamily obligations.
The former Co-PI and architect of the FeLiKx experiment Florian Schreck, I thank you forhaving built this brilliant machine with the team, supervising my initial activities in the lab
and having a good practical advice to absolutely everything that happened or was supposed
to happen in the lab.
Matteo Zaccanti, thanks for the first 20 months in and around the lab. Besides making me
fluent in Italian swearing, you really aroused my (and the others’) curiosity and were thedriving force for the results from the lab. Thank you for that, boss.
I thank my predecessors in the lab: Andreas Trenkwalder for his inexhaustible patienceduring the process of passing the lab over to me and Christoph Kohstall for many discussions
and for exchanging the views on new experimental insights. It really was a great pleasure towork with you!
I thank Marko Cetina for the four years we spent together in the lab. I absolutely admire
your eagerness to convey knowledge you gained in the lab and I am grateful for learningfrom you in numerous discussions. Jook Walraven I thank for the months he stayed with
us in Innsbruck. I value his input to the on-going experiments, the lectures he gave, andthe plenty of discussions we had. I also absolutely enjoyed working with the new team in
the lab: Isabella Fritsche, Bo Huang, and Rianne Lous, I very much appreciate the time we
spent together. I thank you for this time and I wish you a lot of success and nice results with
FeLiBoKx. Additionally, I thank Isabella Fritsche for making life in the office so bearable.
I also thank all the other members of the ultracold group at the IQOQI and at the university
for their advices, for their assistance, and for the discussions with them.
I thank the colleagues in the workshops, Gerhard Hendl, Andreas Strasser, Stefan Haslwanterand Bernhard Öttl, for the support and for extremely fast delivered solutions! I thank thecolleagues in the offices around Christine Götsch, Herr Knabl, Elisabeth Huck, Doris Corona,
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Valentin Staubmann, and David Jordan for their assistance with bureaucracy and IT mattersof any kind.
For proof reading of my thesis I thank Simon Stellmer, Leonid Sidorenkov and Bo Huang
and I thank Katharina for Jeden Tag mit Michael Jag!