Dissertation
submitted to the
Combined Faculty of Natural Sciences and Mathematics
of Heidelberg University, Germany
for the degree ofDoctor of Natural Sciences

put forward by
Jan Hendrik Willibald Becher
Born in: Limburg a.d. Lahn
Oral examination: 13.05.2020

Characterizing Few-Fermion Systems
with Momentum Correlations

Supervisors:
Prof. Dr. Selim Jochim
Prof. Dr. Francesca Ferlaino
Referees:
Prof. Dr. Selim Jochim
JProf. Dr. Fred Jendrzejewski

Abstract
In this thesis I report on the experimental study of fermionic systems in two complementary experimental settings, i.e. large ensembles of dipolar erbium atoms andfew-particle systems of fermionic lithium trapped in optical tweezers.
Compared to alkali atoms, dipolar quantum gases offer new avenues to explore dueto their long-range anisotropic interactions and a rich internal structure. Such richness comes at the cost of complexity and therefore requires precise investigations of
the atomic properties as well as the development of new experimental methods. In
the first part of this thesis we present measurements of the anisotropic light shiftof erbium atoms and compare the results to semiempirical electronic-structure calculations. Measurements of scalar and tensor polarizabilies of the ground and one
excited state show good agreement with calculated values. We furthermore present
the first experimental realization of a two-component strongly-interacting Fermi gas
with dipolar interactions. We identify several intra and interspin Feshbach resonances
at low magnetic field and precisely map out the scattering length across one broadresonance.
The second part of this thesis is dedicated to the characterization of small fermionicsystems with momentum correlation measurements. Starting with systems of two or
three indistinguishable fermions, we detect and discuss second and third-order momentum correlations that arise from quantum statistics alone. We then extend thestudy of correlation functions to interacting systems and develop a scheme to constrain
large parts of the density matrix. Based on these constraints we reconstruct physicaldensity matrices via Bayesian inference. We finally use the reconstructed states toaddress the influence of exchange symmetry on particle-particle entanglement in systems of identical fermions. Using the simple notion of an Antisymmetric Negativity
we are able to separate entanglement from antisymmetrization from entanglement
induced by interaction.

iZusammenfassung
In dieser Arbeit berichte ich von der experimentellen Untersuchung fermionischer Systeme in zwei komplementären Umgebungen: große Ensembles dipolarer Erbiumatome
und Wenigteilchensysteme aus fermionischem Lithium.
Im Gegensatz zu Alkaliatomen ist die Wechselwirkung zwischen Erbiumatomen langreichweitig und anisotrop, was zusammen mit ihrer reichhaltigen inneren Struktur
neue Möglichkeiten bietet, interessante Vielteilchenphänomene experimentell zu untersuchen. Dazu ist jedoch eine genaue Kenntnis über die atomare Struktur von
Erbium und die Entwicklung neuer experimenteller Methoden nötig. Im ersten Teildieser Arbeit stellen wir Messungen der anisotropen Stark-Verschiebung von Erbiumatomen vor und vergleichen die Ergebnisse mit semiempirischen AtomstrukturBerechnungen. Messungen der Skalar- und Tensorpolarisierbarkeit des Grundzustands und eines ausgewählten angeregten Zustands stimmen gut mit berechneten
Werten überein. Des Weiteren präsentieren wir die erste experimentelle Realisierung
eines stark wechselwirkenden zweikomponentigen Fermigases mit dipolarer Wechselwirkung. Wir identifizieren mehrere intra- und interspin Feshbach-Resonanzen bei
niedrigem Magnetfeld und bestimmen präsize die Streulänge in der Nähe einer ausgewählten Resonanz.
Der zweite Teil dieser Arbeit befasst sich mit der Charakterisierung kleiner fermionischer Systeme durch Impulskorrelationsmessungen. Ausgehend von Systemen aus
zwei oder drei ununterscheidbaren Fermionen messen und diskutieren wir Impulskorrelationen zweiter und dritter Ordnung, die sich allein aus der Quantenstatistik
ergeben. Anschließend erweitern wir die Studie der Korrelationsfunktionen auf wechselwirkende Systeme und entwickeln ein Schema, um den Großteil der Dichtematrix
einzuschränken. Basierend auf diesen Bedingungen rekonstruieren wir physikalische
Dichtematrizen durch Bayes’sche Inferenz. Schließlich verwenden wir die rekonstruierten Zustände, um den Einfluss der Austauschsymmetrie auf Teilchen-TeilchenVerschränkung in Systemen identischer Fermionen zu untersuchen. Unter Benutzung
einer antisymmetrischen Negativität können wir Verschränkung von Antisymmetrisierung von Verschränkung trennen, die durch Wechselwirkung induziert wird.

iii

Danksagung
Es gibt viele Menschen, die zum Gelingen dieser Arbeit beigetragen haben. An dieser
Stelle möchte ich mich bei jedem dafür bedanken.
Besonderer Dank gilt Francesca Ferlaino und Selim Jochim für die gute Betreuungwährend meiner gesamten Promotion und für die Möglichkeit, das erste Jahr meiner
Doktorarbeit in Innsbruck zu verbringen.
Des Weiteren bedanke ich mich recht herzlich bei JProf. Fred Jendrzejewski für die
Übernahme der Zweitkorrektur sowie bei Prof. Björn Malte Schäfer und Prof. Jörg
Evers für die Bereitschaft, Teil meines Prüfungskommitees zu sein.
Diese Arbeit wäre natürlich nicht ohne Arbeitskollegen möglich, die mir im Labor
jederzeit tatkräftig unter die Arme gegriffen haben. Danke an das ERBIUM Team:
Simon, Dani, Gabriele, Giulia, Lauriane, Manfred und mittlerweile auch Alex undan das RARE Team: Max, Philipp, Claudia, Arno und Gianmaria. Vielen Dank an
die gesamte Gruppe in Heidelberg: Luca, Marvin, Keerthan, Lukas, Ram-Janik, Laurin, sowie Philipp L und an alle früheren Doktoranden, mit denen ich noch im Labor
zusammenarbeiten durfte. Ganz besonderer Dank gilt Ralf und Philipp P. für die
viele Unterstützung, die Geduld beim Diskutieren und die letzten zwei gemeinsamen
Jahre im Labor. Ein besonderes Dankeschön nochmal an Ralf, Philipp L., Laurin und
Luca für das Korrekturlesen meiner Arbeit und die vielen hilfreichen Kommentare.
Ich bedanke mich ebenfalls bei allen Koautoren, mit denen ich direkt zusammenarbeiten konnte und ohne die sämtliche Publikationen so nicht möglich gewesen wären.
Besonderer Dank gilt den Koautoren Maxence Lepers, Enrico Sindici und Andrew
Daley.
Zu guter Letzt bedanke ich mich bei all meinen Freunden für die schöne und unvergessliche Zeit in Heidelberg und meiner gesamten Familie für den Rückhalt unddie bedingungslose Unterstützung während meines gesamten Studiums.

vContents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Danksagung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i
iii
v

Preamble
Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3

I Ultracold Quantum Gases
A Playground for Quantum Physicists

5

1 Introduction
1.1 Magnetic Quantum Gases . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Characterization of Fermi-Hubbard-like States . . . . . . . . . . . . . .

79102 Realizing Ultracold Quantum Gases
2.1 Cooling and Trapping of Neutral Atoms . . . . . . . . . . . . . . . . .
2.1.1 Vaccum Chamber . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Dissipative Cooling . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Conservative Potentials and Evaporative Cooling . . . . . . . .
2.2 Interaction between Neutral Atoms . . . . . . . . . . . . . . . . . . . .

13
13
13
15
16
18

II

23

Interacting Dipolar Fermi Gases

3 Properties of Erbium
3.1 Atomic Properties of Erbium . . . . . . . . . . . . . . . . . . . . . . .
3.2 Energy Spectrum and Optical Setup . . . . . . . . . . . . . . . . . . .
3.3 Magnetic Properties of Erbium . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Erbium in an External Magnetic Field . . . . . . . . . . . . . .

25
25
26
28
28

4 Anisotropic Polarizability of Erbium
4.1 Anisotropic Polarizability . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 AC-Polarizability of Erbium . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Measurement of Ground-State Polarizability . . . . . . . . . . .
4.2.2 Anisotropic Ground-State Polarizability . . . . . . . . . . . . .

31
32
35
38
40

vii

CONTENTS

4.2.3

Excited-State Polarizability . . . . . . . . . . . . . . . . . . . .

41

5 A Strongly Interacting Fermi Gas of Dipolar Atoms
5.1 Preparing a Spin Mixture of 167 Er . . . . . . . . . . . . . . . . . . . .
5.2 Feshbach Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
46
49

6 Conclusion and Outlook

55

III Characterizing Fermi Systems via Momentum-Correlation
Measurements
57
7 Experimental Tools in the Laboratory
7.1 6 Li - Atomic Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Deterministic Preparation of Few-Fermion Systems . . . . . . . . . . .
7.3 Fermi-Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Spin-resolved single-atom imaging . . . . . . . . . . . . . . . . . . . .
7.4.1 Single-Particle Coherences . . . . . . . . . . . . . . . . . . . . .
7.5 Momentum Correlation Functions . . . . . . . . . . . . . . . . . . . . .

59
59
61
64
69
73
75

8 High-Contrast Interference of Ultracold Fermions
8.1 HBT Effect between Identical Fermions . . . . . . . . . . . . . . . . .
8.2 Third-Order Momentum Correlations between Identical Fermions . . .
8.2.1 Wick’s Theorem and Field Correlation Functions . . . . . . . .

79
81
85
94

9 Density-Matrix Reconstruction
9.1 Momentum Correlations of the Interacting Double Well . . . . . . . .
9.1.1 Three Interacting Fermions in a Triple Well . . . . . . . . . . .
9.2 Bayesian Quantum State Estimation . . . . . . . . . . . . . . . . . . .
9.3 State Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Spin-Rotated Triplet State in a Double Well . . . . . . . . . . .
9.3.2 Interacting Singlet State in a Double Well . . . . . . . . . . . .
9.3.3 Interacting Triple Well . . . . . . . . . . . . . . . . . . . . . . .

97
97
101
105
107
107
109
111

10 Identical-Particle Entanglement117
10.1 A Brief Introduction to Entanglement . . . . . . . . . . . . . . . . . . 118
10.2 Particle-Particle Entanglement . . . . . . . . . . . . . . . . . . . . . . 120
10.3 Generalization of Entanglement . . . . . . . . . . . . . . . . . . . . . . 121
10.4 Entanglement Quantification . . . . . . . . . . . . . . . . . . . . . . . 123
10.5 Particle-Particle Negativity . . . . . . . . . . . . . . . . . . . . . . . . 124
10.6 Antisymmetric Negativity . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.7 Application of AN to Experimental States . . . . . . . . . . . . . . . . 128
11 Summary and Outlook

137

12 Bibliography

141

Appendices

163

CONTENTS

A Reconstructed Density Matrices165

B Negativity Bounds

173

CONTENTS

Preamble
The work presented in this thesis was conducted in two different ultracold-atom laboratories. I spent the first year of my graduation at Innsbruck University, where I
worked in the erbium laboratory under the supervision of Prof. Dr. Francesca Ferlaino. During this time, we pursued two major projects. The first project was theinvestigation of effects of the anisotropic polarizability of erbium atoms and its potential application for state-dependent manipulation or trapping of an atomic sample.
The second project focused on preparing a spin mixture of fermionic erbium in an
optical lattice and to map out the scattering length across an s-wave Feshbach resonance, which might enable future studies of BEC-BCS crossover physics with ultracold
dipolar atoms. After the first year in Innsbruck, I continued to work at Heidelberg
University under the supervision of Prof. Dr. Selim Jochim. During that time we developed new methods to investigate and characterize few-fermion systems of fermionic
lithium trapped in an array of optical tweezers. By analyzing single-particle resolved
momentum-density correlation functions, we are able to reconstruct physical density
matrices of systems containing two or three mobile, interacting fermions via Bayesianinference. We apply this scheme to a range of different experimental states and use
the reconstructed density matrices to investigate entanglement properties of fermionic
identical-particle states.
Due to the very different kind of experiments in Innsbruck and Heidelberg, they will
be presented in separate parts in this thesis. The thesis is therefore divided into threemajor parts. The first part gives a general introduction to the field of ultracold atoms.
The second part presents the results obtained during my first year in Innsbruck andfinally the last part presents results from my time in Heidelberg.

Outline of this Thesis
Introductory Part: This part provides an introduction to the field of ultracold
atoms and embeds the presented experiments within the respective research field. We
briefly introduce the main aspects and tools that make ultracold atoms a versatile
platform to investigate a range of different physics. We discuss how to trap and coolan ensemble of atoms and how laser light can be used to create optical potentials forneutral atoms. We also comment on how interactions between neutral atoms can betuned in the laboratory. While most concepts presented in this part apply to both
species, erbium and lithium, this part is also used to briefly compare properties, where
they are distinctly different.
1

OUTLINE

Erbium Part: This part presents the work done in Innsbruck. We start by giving
an overview on the main atomic properties of erbium, including the discussion of thestructure of energy levels, available optical transitions and magnetic properties. This
sets the stage to present the main results from my first year. We start by discussing
the concept of anisotropic light shifts and present measurements of the anisotropic
polarizability of erbium both in the ground and in one excited state. We compare
the measurements to electronic structure calculations and find very good agreement
despite the complicated structure of electronic energy levels. The following chapter
presents our work on the realization of a strongly interacting two-component Fermi
gas of dipolar atoms. We make use of a lattice projection technique to suppress inelastic collisions during the initialization of the spin-mixture and map out an s-wave
Feshbach resonance at around 680 mG via modulation spectroscopy in an optical lattice. This measurement presents the first realization of a dipolar Fermi gas withtunable s-wave interactions.
Lithium Part: This part presents the work done in Heidelberg. We start with abrief chapter on the main properties of lithium and the tools that we have available in the laboratory to initialize and characterize few-particle Fermi systems. In
the experiment we can measure single-particle and spin resolved momentum distributions of systems containing few mobile, interacting particles in arrays of opticaltweezers. This technique enables the measurement of momentum-density correlation functions, which we use to characterize the system. We start by investigating
momentum-density correlations in systems involving two or three non-interacting indistinguishable atoms, where we measure and discuss strong second and third-ordercorrelations induced by exchange statistics. In the following chapter we develop amethod to analyze measured momentum-density correlation functions of states consisting of mobile, interacting particles and to construct physical density matrices via
Bayesian inference. We apply this scheme to a range of different states and finally
use the reconstructed density matrices to investigate particle-particle entanglement
in systems with identical particles. To this end we employ the notion of an Antisymmetric Negativity to remedy conceptual problems that arise when individual particles
are treated as subsystems for entanglement characterization.

2

LIST OF PUBLICATIONS

List of Publications
This thesis is based on the work published in the four publications:• Measurement of Identical Particle Entanglement and the Influence ofAntisymmetrisation, arXiv:2002.11207
J.H. Becher, E. Sindici, R. Klemt, S. Jochim, A.J. Daley, and P.M. Preiss• High-Contrast Interference of Ultracold Fermions, Phys. Rev. Lett.
122, 143602 (2019)
P.M. Preiss, J.H. Becher, R. Klemt, V. Klinkhamer, A. Bergschneider, N. Defenu, and S. Jochim• Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms,
Phys. Rev. Lett. 121, 093602 (2018)
S. Baier, D. Petter, J.H. Becher, A. Patscheider, G. Natale, L. Chomaz, M.J.
Mark, and F. Ferlaino• Anisotropic polarizability of erbium atoms, Phys. Rev. A 97, 012509(2018)
J.H. Becher, S. Baier, K. Aikawa, M. Lepers, J.-F. Wyart, O. Dulieu, and F.
Ferlaino
Further publications, not covered within this thesis:• Density Oscillations Induced by Individual Ultracold Two-Body Collisions, Phys. Rev. Lett. 122, 083401 (2019)
Q. Guan, V. Klinkhamer, R. Klemt, J.H. Becher, A. Bergschneider, P.M. Preiss,
S. Jochim, and D. Blume• Experimental characterization of two-particle entanglement throughposition and momentum correlations, Nature Phys. 15, 640–644 (2019) A.
Bergschneider, V.M. Klinkhamer, J.H. Becher, R. Klemt, L. Palm, G. Zürn, S.
Jochim, and P.M. Preiss• Spin-resolved single-atom imaging of 6 Li in free space, Phys. Rev. A
97, 063613, (2018)
A. Bergschneider, V.M. Klinkhamer, J.H. Becher, R. Klemt, G. Zürn, P.M.
Preiss, and S. Jochim• Observation of roton mode population in a dipolar quantum gas, Nature Phys. 14, 442–446 (2018)
L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier, J.H. Becher,
M. J. Mark, F. Wächtler, L. Santos and F. Ferlaino

3

4

Part I

Ultracold Quantum Gases
A Playground for Quantum
Physicists

5Chapter 1

Introduction
About one and a half centuries ago, a series of peculiar experimental observations
such as the spectrum of hydrogen and the discovery of the Balmer series [1], thediscovery of the photoelectric effect by Heinrich Hertz [2] or the Zeeman-splitting
effect by Pieter Zeeman [3] required physicists to question the validity of classicalphysics on microscopic scales. The seminal work of many physicists, including Einstein, Planck, Born or Pauli, ultimately led to the development of quantum mechanics
in the early 20th century, which revolutionized our understanding of the microscopic
world. The key ideas were to quantize the energy of states, to ascribe both a wave
and a particle character to quantum mechanical objects and to allow superposition
states. Although being heavily debated at that time, already from the very beginning,quantum mechanics was able to explain most experimental observations such as blackbody radiation or the famous Stern-Gerlach experiment, it enabled the developmentof the Bohr model for the atom, and facilitated the understanding of the periodic
table of elements. Due to the rapid understanding of quantum mechanics, it only
took a few decades until it caused the first technological breakthroughs, for example
the development of the first photomultipliers in the 1930s [4], the understanding anddevelopment of semiconductor devices like the transistor in 1948 [5] or the realizationof the Maser and the Laser in the 1950s and 1960s [6]. Quantum mechanics therefore was crucial for the technological development of today’s world. Until the endof the 20th century quantum mechanical concepts have become essential for many
core technologies in everyday life and can be found in many devices, ranging frommedical applications like MRI or cancer treatment to industrial applications such as
telecommunication, cutting metal, sensing and robotics or designing and fabricating
modern computers and smartphones.
While those technological breakthroughs rely on single-particle effects, there is arange of intriguing quantum phenomena that involve many strongly correlated electrons or atoms. In fact, in the regime where the (deBroglie) wavelengths of individual
constituents of a system becomes comparable to the spacing between them, quantum
effects become dominant. One of the most prominent examples of the peculiar behavior of quantum-mechanical many-body systems was the discovery of superconductivity
in 1911, a phenomenon which was baffling theorist for a long time [7]. While being

7

CHAPTER 1. INTRODUCTION

extensively studied in condensed matter systems, phenomenological theories could
capture the basic physics of superconductors and superfluids, but still today, thequantum-mechanical description of systems containing only a few strongly correlated
atoms is a challenging task for theory. For instance the exact diagonalization of asystem of N atoms with two internal degrees of freedom requires the diagonalizationof a 2N × 2N matrix, which already exceeds available computational power for about
20-30 atoms.
Initially inspired by the idea of Richard Feynman [8], experimental physicists
pursued the road of quantum simulation: Instead of trying to simulate a quantum
mechanical system on a classical computer, it might be much more efficient to realizea system in the laboratory that evolves under the same Hamiltonian as the system
of interest. One hence simulates a quantum system with another quantum system,that is more accessible in the laboratory. There are many different possible platforms
for quantum simulation, such as ensembles of neutral atoms [9], photonic [10] or ionicsystems [11]. The idea of engineering a quantum system has led to the developmentof a huge research field, including experiments on ultra-precise optical clocks [12],the development of nuclear clocks [13], the realization of a quantum computer withe.g. superconducting qubits [14], or the study of hybrid systems such as excitonpolaritons condensates [15], levitated microspheres as ultra-precise force sensors [16]or optomechanical systems [17]. Research interests do no longer only include quantum simulation. Instead, the unprecedented control and precision that can today
be achieved in the laboratory enables fundamental tests of physics including loophole
free Bell tests [18], the search for the electron electric-dipole moment [19] or for exotic
forces beyond the standard model [20], the variation of fundamental constants, thedetection of gravitational waves [20] or the realization of measurement devices beyond
the standard quantum limit [21].
The cornerstone for quantum simulation with neutral atoms was laid by the first realization of a Bose-Einstein condensate (BEC) in 1995 [22, 23] and the first degenerate
Fermi gas in 1999 [24]. Followed by many succeeding ground-breaking experiments,
neutral atoms, trapped in optical traps, have proven to be a promising candidate forquantum simulation [9, 25] and for the search of exotic phases of matter. Over theyears, many powerful theoretical and experimental tools have been developed bothto describe and to manipulate ensembles of interacting atoms or molecules. Manydifferent species, including both fermionic and bosonic isotopes are routinely cooled
to quantum degeneracy and trapped in arbitrary optical potentials such as quasicrystalline optical lattices [26] or arbitrary three-dimensional arrays of optical tweezers [27]. Using (orbital) Feshbach resonances [28, 29], interactions can be tuned overa wide range, enabling the realization of strongly interacting and strongly correlated
systems. Recent experimental advances made it possible to engineer systems withsingle-particle control and readout [30], which opened up a new era for cold quantum
gases: Large genuinely entangled systems can be readily created in the laboratory[31], advancing concepts for quantum computing and quantum information processing. Low-entropy states in optical lattices serve as toy models to simulate the doped
Hubbard model [32]. The ability to bring electronically more complex atomic species(such as Dy, Er, Yb, Sr) or molecules to quantum degeneracy enables the study of8

CHAPTER 1. INTRODUCTION

exotic phases with no equivalent in natural matter and recently led to the observationof supersolidity in a gas of magnetic atoms [33–35]. Arrays of optical tweezers mightlead to a new generation of atomic clocks and new applications in metrology [36].
Despite enormous progress in the field of ultracold atoms, there are still many
challenges, of which we address two within this thesis. Experimental progress of laser
cooling and trapping made it possible to cool molecules [10] or atoms with a complicated electronic structure to quantum degeneracy, which offers new avenues to explore,
due to properties including long-range anisotropic interactions, state-dependent lightshifts, ultra-narrow optical transitions or a rich internal structure. However, in such
systems much more care has to be taken to initialize interesting states. A precise
understanding of the atomic structure and magnetic properties is therefore inevitablein such systems. In the first part of the thesis we measure the ac stark shift of erbium
atoms at different wavelengths to verify the validity of electronic-structure calculations and we develop methods to map out the interaction in a two-component Fermi
gas of magnetic erbium atoms without being affected from inelastic dipolar collisions.
In contrast, the preparation of interesting states is well under control in quantum
gas experiments with alkaline atoms. However, a major challenge in such experiments is the experimental characterization of the final state due to the lack of proper
many-body observables. Currently many new ways to characterize strongly correlated systems are proposed and tested both theoretically and experimentally. In thelast part of this thesis, we develop and explore a new method to fully characterizesystems consisting of few mobile and interacting fermions, trapped in an array of
optical tweezers. This method is complemetary to traditional tomography methods
and presents an alternative approach to characterizing strongly correlated, fermionicsystems in the laboratory.
The following two sections elaborate on the state of the art in the two respective
research areas and embeds the studies presented in this thesis within the two fields ofmagnetic quantum gases and state characterization in Hubbard systems.

1.1

Magnetic Quantum Gases

From the first observation of Bose-Einstein condensation until 2005, all species that
had been Bose-Einstein condensed interacted predominantly via short-range isotropic
potentials. The first BEC of Chromium atoms in 2005 [37] allowed to study effectsof the long-range, anisotropic dipole-dipole interaction on the behaviour of ultracold
quantum gases. This led to the observation of many dipolar effects in BECs, including
the suppression of the inversion of ellipticity during expansion [38], the observationof a d-wave collapse [39] or the spontaneous demagnetization of a dipolar gas at low
magnetic fields [40]. The successful realization of a Bose-Einstein condensate of themost magnetic element Dysprosium in 2011 [41] and shortly after the first BEC of
Erbium atoms in 2012 [42] opened a new frontier to study the interplay between
short-range isotropic and long-range anisotropic interactions in systems of ultracoldbosonic atoms.
One of the most prominent examples for the impact of dipolar interactions on the9CHAPTER 1. INTRODUCTION

behaviour of ultracold atoms was the observation of the formation of droplets [43],stabilized by quantum fluctuations [44, 45]. This manifestation of a beyond meanfield effect was followed by the observation of a roton mode in a dipolar BEC [46]and ultimately led to the observation of supersolididy, a state of matter that features
both superfluid properties and a crystalline structure [33–35].
While bosonic dipolar quantum gases experienced a lot of attention, it took until
2012 until the first realization of a dipolar degenerate Fermi gas with Dysprosium[47], followed by Erbium in 2014 [48] and Chromium in 2015 [49]. Dipolar Fermi
gases raise prospects for studying many different physical phenomena such as liquid
crystal phases [50], p-wave, topological or anisotropic superfluid phases [51–53] or unconventional magnetism [54]. Due to their large quantum numbers in the electronic
ground state, magnetic atoms possess a very rich internal structure and are therefore
interesting candidates to study spinor physics with long-range dipolar interaction, to
use the internal states as synthetic dimensions [55, 56] or to investigate the influenceof dipole-dipole interaction on BEC-BCS crossover physics [57] or the BKT transitionin two dimensional systems [58].
The results that we present in this thesis report on the measurement of the anisotropic
light shift of erbium atoms and the first realization of a two-component Fermi gas withtunable interaction. The measurements on the anisotropic light shift agree very wellwith electronic-structure calculations and show that despite the complex electronic
structure of erbium, it is well understood in terms of its atomic properties. Such understanding facilitates the application of state-dependent manipulation and trapping.
We furthermore realize the first two-component strongly interacting dipolar Fermi-gas
with tunable s-wave interaction, which paves the way to exploring many interestingareas in dipolar Fermi systems.

1.2

Characterization of Fermi-Hubbard-like States

With the advent of experiments operating on the single-particle level [30, 59–63],
the field of ultracold quantum gases entered a new era of control. Experimentalists
can readily initialize strongly correlated systems in the laboratory and probe themwith spin and single-particle resolution [30, 64–67]. To fully characterize a preparedstate in the laboratory however, the reconstruction of the full many-body density
matrix is necessary. Unfortunately the number of entries of the density matrix scales
quadratically with the number of available states and therefore scales exponentiallywith the system size. Traditional tomography of a many-body state thus also requires
exponentially many measurements. A straightforward application to larger systems
hence quickly becomes impractical. Despite considerable effort and progress in developing new techniques for the characterization of many-body states [68–71], theefficient characterization of Fermi-Hubbard like states, involving mobile, interacting
particles, is still very demanding. The required protocols are technically challenging
to realize for large systems since they require a large number of unitary operations onneighboring lattice sites. That’s why throughout the community many new, complementary approaches are proposed to characterize a many-body state, including the10

CHAPTER 1. INTRODUCTION

measurement of correlation functions [72, 73], the study of entanglement properties[74, 75], the application of random unitary operations [76, 77] or the use of machinelearning [32, 78] or neural networks [79] to classify experimental data.
In this thesis we explore a new scheme for characterizing small Fermi-Hubbard likesystems via the measurement of single-particle and spin resolved momentum-density
correlation functions. We develop a scheme that allows us to reconstruct physical density matrices of small systems based on experimental results by employing a Bayesianquantum state estimation. In particular, our measurement protocol does not dependon the system size and only involves two complementary measurements, one in real
space and one in momentum space. This is fundamentally different to other tomography protocols, where the number of operations scales with the system size [68]. We
successfully apply our scheme to systems consisting of two or three mobile, interacting
particles in an array of optical tweezers. As a first application, we reconstruct density
matrices of a range of different states and discuss entanglement properties of fermionic
identical-particle states. Furthermore we believe that the experimental methods developed within this thesis are not restricted to Fermi-Hubbard like states but will alsofind applications for the characterization of strongly correlated mesoscopic continuous
systems.

11

CHAPTER 1. INTRODUCTION

12Chapter 2

Realizing Ultracold Quantum
Gases
Quantum mechanical effects become dominant at low temperatures, where the thermal deBroglie wavelength exceeds the interparticle spacing. Studying quantum mechanical effects in dilute samples of neutral atoms hence requires to cool down thesystem into the quantum degenerate regime, where the temperature becomes smaller
than characteristic energy scales in the system. For Fermi systems, this characteristicenergy scale is given by the Fermi temperature TF . While the Fermi temperature
for metallic superconductors is on the order of 105 K, a typical ensemble of fermionic
atoms in an optical dipole trap has a Fermi temperature of only about 1 µK, due tothe very low densities on the order of 1013 cm−3 [80]. In order to realize such systems
in the laboratory, atoms have to be trapped and cooled down from room temperature
to the nK regime. As will be explained in this chapter, this is commonly achieved
with magneto-optical traps and evaporative cooling in optical dipole traps. Once theroute to quantum degeneracy is established, an important tool to tune interactions
are Feshbach resonances, which we will also briefly introduce in this chapter.

2.1

Cooling and Trapping of Neutral Atoms

In this section we briefly present the main tools that are used in our experiments tocool a hot beam of atoms down to quantum degeneracy. Experimental details havebeen discussed in detail in previous works, see e.g. [81, 82] for the erbium machineand [83–85] for the lithium machine.

2.1.1

Vaccum Chamber

Experiments with ultracold gases absolutely require ultra-high vacuum on the order of
10−11 mbar since otherwise collisions with the background gas might limit the lifetimeof the cold sample. Figure 2.1 shows a schematic of both vacuum chambers, the erbiummachine in Innsbruck and the lithium machine in Heidelberg. Both experiments
have a very similar design. The experiment starts with a high-temperature oven,13

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

Figure 2.1: Vacuum chamber. (a) Vacuum chamber of the erbium machine. (b) Vacuum
chamber of the lithium machine. Both experiments have a similar design, including oven, Zeeman slower and main chamber. Vacuum is maintained on the
10−11 mbar level by a differential pumping stage with two titanium sublimation
pumps and two ion pumps. The main conceptual difference between the twomachines is the additional transversal cooling stage in the erbium experiment.
(a) taken and adapted from [81], (b) taken and adapted from [83].

where the respective species is heated up to approximately its melting temperature.
At that temperature atoms start to evaporate from the surface, which leads to anincrease of the vapor pressure and creates a hot gas of atoms. This gas is first of
all precollimated with an initial transversal cooling stage in the erbium machine andonly a simple tube in the lithium machine. The oven connects to the Zeeman slower
and finally to the main chamber, where the experiments take place. Optical accessto the science chamber is possible via eight (erbium) or six (lithium) viewports fromthe sides and two larger viewports from the top and the bottom, which are reentrant
in the lithium machine, allowing a high numerical aperture of up to N A = 0.65. In
both experiments two titanium sublimators and two ion pumps are used to reach andmaintain the ultra-high vacuum at a pressure on the order of 10−11 mbar.

14

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

2.1.2

Dissipative Cooling

The atoms that exit the high-temperature oven are collimated to form an atomic beam
towards the main chamber. At that stage of the experiment, atoms typically havevelocities of around 500 − 1000 m/s. The first two cooling steps employ near resonant
laser light to create a dissipative force on the atoms, which first slows them down in
the Zeeman slower and enables trapping them in a magneto-optical trap (MOT).
Zeeman-Slower:
The atomic beam is guided through the Zeeman-Slower, a tube surrounded by magnetic field coils, which create a spatially varying magnetic offset field along the pathof the atoms (spin-flip configuration in the erbium experiment, decreasing field configuration in the lithium experiment). A near-resonant, circularly polarized laser beam
is colinearly aligned with the counter propagating atomic beam. Fast atoms, which
are initially Doppler shifted from the bare atomic resonance, become resonant to thelaser beam via the Zeeman-shift in the spatially varying magnetic field such that they
absorb photons with a directed momentum ”-kph ” along the atomic beam direction.
The excited atom spontaneously emits a photon in a random direction so that after
one cycle the atom lost on average one photon momentum. The lower Doppler shift ofthe slowed atom is compensated by the varying magnetic field along the beam path.
Atoms are kept resonant throughout the whole Zeeman slower and arrive in the science chamber with velocities of on average only a few meters per second. The lithium
Zeeman-slower uses the D2 line with a natural linewidth of Γ = 2π × 5.8724 MHz [86].
The erbium Zeeman slower is operated on the broadest line at λZ = 401 nm with anatural linewidth of Γ = 2π × 29.7 MHz [81]. After the atoms are slowed down, they
can be captured in a magneto-optical trap (MOT).
Magneto-Optical Trap:
A MOT consists of six laser beams, typically realized by three retro-reflected, reddetuned laser beams with circular polarizaion, and a magnetic gradient field, producedby two coils close to anti-Helmholtz configuration. In fact in the erbium experiment,
the MOT only requires five beams; due to its high mass, the top beam is not necessaryfor efficient operation of the MOT [87]. The laser beams cause a velocity-dependent
radiation pressure on the atoms and the magnetic field gradients prevents the atoms
to diffuse out of the trap by inducing a position dependent restoring force on theatoms. In the lithium experiment we have to add an additional repumper-frequency
due to the decay of atoms into the F = 3/2 manifold in the ground state. A repumper
is not necessary in the erbium experiment, where the five-beam, narrow-line MOT
automatically spin-polarizes the atoms in the lowest state.
Due to the repeated process of scattering photons from the MOT beams, the temperature of the MOT is intrinsically limited by the natural linewidth of the transition, it
is operated on. This so-called Doppler temperature, i.e. the lowest temperature that15

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

can be achieved in the MOT, is given by [88]
TD =

ℏΓ
2kB

(2.1)

with ℏ being the reduced planck constant, Γ the scattering rate and kB the Boltzmann
constant. The lithium MOT is operated on the D2 line with a natural linewidth ofΓ = 2π × 5.8724 MHz, which yields a Doppler temperature of TD = 141µK. The
erbium MOT profits from the electronically more complex structure of erbium andemploys a relatively narrow line at λM = 583 nm with a natural linewidth of Γ ≈2π × 190 kHz, achieving a Doppler temperature of TD ≈ 5 µK [81, 89].

2.1.3

Conservative Potentials and Evaporative Cooling

In order to enter the quantum degenerate regime, the temperature has to be further
reduced. To this end, the atoms are transferred from the MOT into a crossed-beam
optical dipole trap (ODT) and then evaporatively cooled to quantum degeneracy.
Optical dipole traps rely on the polarizability of an atom and are one of the workhorsesof cold-atom experiments. Far detuned laser beams are not only used for evaporative
cooling of atoms but also for the generation of external potentials and to confine
atoms e.g. in optical lattices, optical tweezers or to reduced dimensions. A briefsummary on the optical setup for the generation of optical potentials for both erbium
and lithium will be given in Part II and Part III, respectively.
Far-Off-Resonant Traps
In order to generate a conservative potential for trapping the atomic sample, far-offresonant laser beams are used. Depending on the sign of the detuning with respect
to the optical resonance, such laser beams either create a repulsive or an attractive
potential for neutral atoms.
⃗ (rWhen a neutral atom is placed into an oscillating electric field E⃗ , t), the atom getspolarized with its induced dipole moment ⃗p given by⃗.
⃗p = αE(2.2)

α is called the ac-polarizability and generally is a 3 × 3 tensor, taking into accountthe anisotropy of the induced dipole moment. However, anisotropic effects vanish forfar-detuned laser beams and we therefore neglect them in this part of the thesis. We
elaborate on anisotropic light shifts in Chapter 4 for the case of erbium atoms. In
the isotropic case, the induced dipole moment ⃗p is parallel and proportional to theamplitude of the electric field and α is a complex number. The energy of the induced
dipole moment ⃗p in the electric field of a laser beam is then given by [90]11 ⃗
⃗ E (r⃗ )⟩ = −ℜ(α)I(r⃗ ),Udip (r⃗ ) = − ⟨p
2
2ϵ0 c

(2.3)

with the real part of the polarizability ℜ(α) and the intensity of the laser beamI = ϵ01c |E0 |2 . Note that the imaginary part of the polarizability is related to the16CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

off-resonant scattering of photons
Γsc = −1ℑ(α)I.
ℏϵ0 c

(2.4)

Considering the atom as a classical oscillator within Lorentz’s model [90], the polarizability can be calculated for a two-level atom by
α = 6πϵo c3Γ,ω02 (ω02 − ω 2 ) − iω 3 Γ(2.5)

with the laser frequency ω and the resonance frequency ω0 . Γ is the on-resonancescattering rate and can be calculated using the dipole transition element between the
ground state |g⟩ and the excited state |e⟩
Γ=

ω03
2
|⟨e| µ |g⟩| .
3πϵ0 ℏc3(2.6)

Using Eq. (2.5), we can directly calculate the real and imaginary part of α and canwrite down explicit equations for the dipole potential Udip and the scattering rate Γsc ,)︃(︃
3π 2 Γ
1
1+
I(r⃗)
2ω03 ω0 − ω ω0 + ω)︃2(︃ )︃3 (︃3π 2 Γ2 ω
1
1
+
I(r
⃗ ).
2ℏω03 ω0
ω0 − ω ω0 + ω

Udip (r⃗)= −Γsc (r
⃗)

=

(2.7)

The potential experienced by the atom is hence proportional to the intensity of thelaser beam, while the sign is given by the detuning δ = ω0 − ω. For red detuned laser
beams (δ > 0) the atoms experience an attractive potential and can be trapped inregions of maximum intensity. For blue detuned traps (δ < 0), atoms experience a
repulsive potential and get expelled from high-intensity regions. Since the scattering
rate Γsc scales with 1/δ 2 , the detuning is typically chosen very large. This in turnrequires the application of high-power lasers to realize deep optical traps with very
low number of photon-scattering events.
The concept of optical trapping can be exploited to create complex spatial potentials
by shaping the form of the laser beam. Prominent examples are interfering laser beams
to create optical lattices (as we use in the erbium machine), or optical tweezers (as we
use in the lithium machine). However, in principle arbitrary optical potentials can becreated by using either liquid crystal spatial light modulators or digital micro mirror
devices to control either the phase or the amplitude of a light field.
Note that the given formulas are valid only for two-level systems. For real atoms,
the dipole potential is given by a sum over all optical transitions, weighted by the
coupling strength of each line (cf. Chapter 4). Typically in alkali atoms at infrared
wavelengths, the ground-state polarizability is almost entirely given by the D1 and theD2 line [91]. As we will discuss in Chapter 4, this is different in erbium atoms, where
the accurate calculation of the polarizability is hindered by a lack of spectroscopic
data.

17

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

Evaporative cooling
Once atoms are trapped in the optical dipole trap, they can be evaporatively cooled
to quantum degeneracy. Evaporative cooling [88, 92] relies on continuously removing
the highly energetic atoms from the trap by lowering the optical potential. At thesame time, interactions between the atoms in the trap lead to fast rethermalization.
Hence energy is constantly removed from the atomic sample at the cost of decreasing
the atom number. Evaporative cooling became a standard technique to cool atomic
samples to quantum degeneracy. Since interactions between ultracold atoms usually
are only of s-wave character, identical fermions do not interact and hence evaporatively cooling an alkaline Fermi gas requires a second internal state [80]. In thelithium machine this is achieved by cooling a balanced mixture of two of the threelowest hyperfine states. For fermionic erbium, this is not necessary due to universal
dipolar scattering [48] as we will briefly discuss in the next section.
Typical experimental atom numbers at the end of the evaporation are about 40.000
atoms per spin state at a temperature of 0.1 TF in the lithium machine and 40.000
spin-polarized atoms in the lowest state at temperatures of about 0.2 TF in the erbiummachine.

2.2

Interaction between Neutral Atoms

Collisions between two neutral atoms are governed by two different kinds of interactions. Atoms with a spherically symmetric electronic configuration, such as alkali
atoms, only interact via the short-range van-der-Waals (vdW) interaction; an attractive potential, that drops with the sixth power of the distance r between the
two atoms, r−6 . In addition, atoms with a permanent magnetic dipole moment alsointeract via the anisotropic magnetic dipole-dipole interaction [81],
U (r
⃗)

=

UvdW (r⃗) →Udd (r
⃗)=UvdW (r⃗ ) + Udd (r⃗)
C6, for r → ∞r−6
Cdd (e⃗ 1 · ⃗e2 )r2 − 3(e⃗ 1 · ⃗r)(e⃗ 2 · ⃗r)
.
5
4π
r

(2.8)
(2.9)
(2.10)

Here, C6 is the vdW C6 coefficient, Cdd = µ0 µ1 µ2 is the dipolar coupling constantwith the vacuum permeability µ0 and the magnetic moments of the two colliding
atoms µi = mJ gJ µB with magnetic quantum number mJ , Landé factor gJ and the
Bohr magneton µB . ⃗ei is the orientation of the magnetic moment of each atom, as
illustrated in Fig. 2.2 (a).
Short-Range Interaction
The vdW interaction arises due to the interaction between
the electrons of the two
(︂
)︂1/4
2µC6
atoms and is characterized by the vdW length β6 =[93] with the reducedℏ2
mass µ of the two interacting atoms. This length is typically on the order of afew hundreds of picometers and therefore much shorter than the typical interparticle
distance in the atomic sample. For describing the scattering physics of ultracold gases,
18

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

Figure 2.2: Dipolar interaction. (a) Geometry of dipolar interaction between two randomly aligned dipoles. (b) Interaction between two dipoles that are polarized
by an external magnetic field. (c) Angle dependence of dipolar interaction Udd
between two polarized dipoles. The interaction vanishes at the magic angle ofθm ≈ 54.7◦ . It is attractive for a head-to-tail alignment (θ < θm ) and repulsive
for a side-by-side configuration (θ > θm ).

one typically expands the relative wave function of the two colliding atoms where at
ultralow temperatures only the lowest partial wave contributes. UvdW can then be
replaced by a contact-like delta-function interaction (for derivations see e.g. [93, 94])
UvdW (r) = g0 δ(r)

d
r,
dr

(2.11)

2

with the interaction strength g0 = 4πℏ
m as . Here as is the s-wave scattering length,
which is defined via the phase shift an atom experiences in the interaction potential
upon scattering. as is strictly zero for identical fermions due to their relative wave
function being zero at r = 0 because of the Pauli-exclusion principle.
It is important to note that the vdW interaction is isotropic for atoms with a spherically symmetric electronic configuration, as it is the case for lithium. However, in thecase of atoms with large permanent dipole moments, interactions show anisotropic
effects [95, 96]. In particular, two colliding bosonic erbium atoms (with total angular momenta J1 = J2 = 6) have 49 non-degenerate molecular Born-Oppenheimer
potentials and hence 49 different C6 coefficients for 12 different collisional channels
in
out
out
Ω = minJ1 + mJ2 = mJ1 + mJ2 (for more information, see [81]), giving rise to chaotic
scattering physics, which was observed both in dysprosium and erbium. [97, 98].
Long-Range Dipolar Interaction
The magnetic dipolar interaction between two atoms decays with r−3 and is fundamentally different from the vdW interaction, both because of its range and its
anisotropic character. In the experiment, the atoms are typically polarized by an
external magnetic field, so that Eq. (2.12) simplifies toUdd =µ0 µ2 1 − 3 cos2 θ,4π
r3(2.12)

with θ being the angle between the vector ⃗r between the two atoms and their polarization axis, see Fig. 2.2 (b). As illustrated in (c) in the same figure, the interaction between two dipoles can be either attractive (θ ∈ [0◦ , 54.7◦ ]) or repulsive
19

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES(θ ∈ [54.7◦ , 90◦ ]), and vanishes at the magic angle of θm = 54.7◦ . This anisotropicinteraction is the main difference between alkali or alkali-earth atoms and strongly
magnetic lanthanide atoms such as Er or Dy. It is responsible for many peculiar effects that were not observed with alkali atoms before, such as a d-wave collapse [39],chaotic scattering [97], and beyond mean-field effects like the formation of droplets[43, 99], the observation of a roton mode [46] and supersolidity [33–35].
Scattering Cross Sections - Alkali vs. Dipolar
As already mentioned in the discussion around Eq. (2.11), our experiments are typically conducted in regimes where only the s-wave scattering length as is relevant
for the short-range physics. This implies that identical fermions do not interact via
the short-range vdW interaction potential at ultracold temperatures. In contrast,
identical fermions can interact via the long-range dipolar interaction. In fact it turns
out that at ultralow temperatures the scattering cross section of dipolar collisions
approaches a universal value [81, 100, 101]. The scattering cross sections both foralkali and and magnetic atoms can be summarized as⎧⎨0
id. fermions
σalkali =(2.13)⎩4πas otherwise⎧⎨ 32π a2
id. fermions
15 D
σdipole =.
(2.14)
16π
2⎩
a + 4πas otherwise
15

D

In the case of erbium, this allows even identical fermions to scatter and to thermalize
and hence it is possible to evaporatively cool a single-component Fermi-gas of erbium atoms to quantum degeneracy [48] whereas for alkali atoms two spin states arenecessary for efficient evaporative cooling.
Tuning the Interaction: Feshbach Resonances
One very helpful tool, which makes ultracold quantum gas experiments a versatile
platform to study a range of different physical phenomena, is the ability to tune swave interactions over a wide range via Feshbach resonances (FRs). A particularlyinteresting regime is the regime of a diverging scattering length. In this so called unitary regime, the scattering length dominates over all other length scales in the systemand therefore solely dictates the behavior of the atomic ensemble [103]. Extensive literature on FR in ultracold atomic gases is available [28] and will not be discussed here.
The effect relies on the coupling of colliding atoms to molecular states and therefore
strongly depends on the molecular structure of the colliding atoms. While 6 Li featuresa very broad s-wave FR between each two of the three different spin states (see upper
panel in Fig. 2.3), erbium shows hundreds of FRs with higher partial wave character[98, 104] and only very sparse knowledge about the scattering lengths in erbium (andalso in dysprosium) is available. Recently, we were able to map out one specific swave FR between the two lowest hyperfine states of fermionic 167 Er via modulation
spectroscopy in an optical lattice [102], see lower panel in Fig. 2.3. Both resonances,20

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

Figure 2.3: Feshbach Resonances. The upper plot shows the scattering lengths between
each two of the three lowest spin states of 6 Li. Each combination shows one
Feshbach resonance with a diverging scattering length. The lower plot shows
one example of an s-wave Feshbach resonance for 167 Er [102] between its twolowest spin states (note the different scaling of the x-axis).

in lithium and erbium, show qualitatively different behaviour. While lithium featuresa single FR, that spans over hundreds of Gauss, the shown FR of 167 Er has a widthof only a few mG and additionally many further close overlapping FRs (not shown in
the plot, refer to [102] or Chapter 5). However, the required relative magnetic fieldstability for a precise tuning of the scattering length in both systems is similar, paving
the route to the study of BEC-BCS crossover physics also in dipolar quantum gases.
Tuning Dipolar Interaction
In the erbium machine we furthermore have the ability to also tune the magnetic
dipole-dipole interaction. To this end we employ a set of three coils around theexperimental table to freely rotate the magnetic field axis [81]. In an elongated dipole
trap, this enables to tune the mean interaction energy in the atomic cloud frompositive (dipoles mainly aligned head-to-tail ) to negative (dipoles mainly aligned sideby-side). In an optical lattice this can be used to switch off the dipolar interaction
along one particular axis by rotating the magnetic field such that the angle to thelattice axis is θm = 54.7 ◦ .

21

CHAPTER 2. REALIZING ULTRACOLD QUANTUM GASES

22

Part IIInteracting Dipolar Fermi
Gases

23

Chapter 3

Properties of Erbium
The rare-earth atom erbium is mostly known from its application in optical amplifiersin the telecommunication regime [105]. In an erbium-doped fiber amplifier, the trivalent cation Er3+ is optically pumped at around 980 nm or 1440 nm and radiates ataround 1530 nm via stimulated emission. Moreover, recent experiments have shownthat Erbium might also play an important role in the development of integrated photonic devices [106, 107].
For ultracold-atom experiments however, erbium is particularly interesting becauseof its large magnetic moment of 7µBohr , which comes along with strong dipolar interaction (see Sec. 2.2), but also because of its very rich and complex electronic levelstructure, offering hundreds of optical transitions with linewidths ranging from tens ofMegahertz down to linewidths of only a few Hertz. Such richness of course comes withthe disadvantage of complexity, which sometimes also aggravates the development of
new methods in the laboratory. First experiments on laser cooling erbium were already done in 2006 [108] but it took until 2012 to produce the first Bose-Einsteincondensate of erbium [42] and until 2014 to cool a fermionic isotope to quantum degeneracy [48].
In this chapter we present the main atomic properties of erbium and give a brief
overview of the optical setup that we have available in the laboratory.

3.1

Atomic Properties of Erbium

Erbium has an atomic number of Z = 68 and features six stable isotopes with naturalabundances as shown in Fig. 3.1 (a). In the experiment we can readily switch between
the four most abundant isotopes, three bosonic isotopes 166 Er, 168 Er, 170 Er and one
fermionic isotope 167 Er. Erbium is located in the lanthanide block of the periodic
table and therefore has a rather complicated electronic structure: the 6s shell getsfilled with 2 valence electrons before the 4f shell starts to get populated with theremaining 12 electrons. There are hence two vacancies in the 4f shell, resulting in aso called submerged shell structure. The electronic configuration of the ground stateis typically written as[Xe]4f12 6s2 ,25

CHAPTER 3. PROPERTIES OF ERBIUM

Figure 3.1: Properties of erbium. (a) Natural abundances of stable isotopes of erbium.
In the experiment, we can readily switch between four most abundant isotopes:
166
Er, 168 Er, 170 Er (bosonic) and 167 Er (fermionic). (b) Isosurface plots of theelectron wave functions (red = positive, blue = negative) of the 4f orbitals
for different ml quantum numbers (general set) and the anisotropic electron1∑︁density of erbium|Ψml |2 .
ml =−3where [Xe] represents the electronic configuration of the ground state of Xenon. The
contour plots in Fig. 3.1 (b) illustrate the electronic 4f wave functions, of which the
ml = 3 and ml = 2 states are only occupied with a single electron. These two electron
vacancies are responsible for an anisotropic electronic wave function around the core(see bottom right panel in Fig. 3.1 (b)). In contrast to alkali and earth-alkali atoms
that are commonly used in cold-atom experiments, erbium offers very large quantum
numbers in its ground state, L = 5, S = 1 and J = 6. All bosonic isotopes have anuclear spin of I = 0, and therefore no hyperfine structure, while the fermionic isotope
167
Er has a nuclear spin of I = 7/2, and hence a total angular momentum ranging
from F = J + I = 19/2 to F = J − I = 5/2 in the ground-state manifold, which we
illustrate in Fig. 3.2 (b). The anisotropic nature of the electronic configuration andthe large quantum numbers in the ground state are responsible for the main atomic
properties that make erbium interesting for cold-atom experiments.

3.2

Energy Spectrum and Optical Setup

Currently, the NIST database reports 674 energy levels for neutral Erbium with total
angular momenta quantum numbers ranging from J = 1 to J = 12 [109] and 232
observed spectral lines [110] between 877 nm to 336 nm. Recent theoretical work
predicts many further energy levels and dipole allowed transitions [111], that have
not yet been observed experimentally. A significant attribute of the spectrum isthe very broad transition at 401 nm with a linewidth of around 2π × 30 MHz [81]but also a plurality of narrow lines in the kHz or even Hz-regime. Neutral erbium
has an ionization threshold of 49262 cm−1 , corresponding to a wavelength of 203 nm.
Fig. 3.2 (a) shows the spectrum of energy levels of erbium as a function of the total
angular momentum and indicates all laser frequencies that are used in our experiment.
For ground-state erbium, only the transitions starting from the J = 6 ground state26

CHAPTER 3. PROPERTIES OF ERBIUM

Figure 3.2: Energy spectrum of erbium. (a) Level scheme of erbium in the experimentally relevant region for all angular-momentum states J ∈ [1, 12]. Blueand red levels indicate different level parity. The arrows represent all laser
frequencies that we use in the experiment. There are two resonant lasers, one
for Zeeman-slower, transversal cooling and imaging (401 nm) and the other for
the MOT (583 nm). The three off-resonant lasers are used for optical dipole
traps (1064 nm and 1570 nm) and the optical lattice (532 nm). (b) Zoom-in on
the fermionic ground state with hyperfine splitting and 8 different sublevels
F ∈ {19/2...5/2}. Figure taken and adapted from [82].

are relevant. There is a variety of different optical transitions to choose from but
besides the linewidth also branching ratios to metastable states and the availabilityof laser sources is important for the usefulness in the experiment. Suitable transitions
for an efficient cooling of erbium have been discussed in the literature [112, 113] anda few lines have been already experimentally characterized [114, 115].
As already mentioned before, we use the broadest transition at λ = 401 nm with alinewidth of Γ ≈ 2π × 30 MHz for the Zeeman slower, the transversal cooling andfor imaging of the atomic sample. The laser light is derived from a Toptica TASHG laser module and locked on the spectroscopic signal of a hollow cathode lamp.
The MOT profits from the relatively narrow line at λ = 583 nm with a linewidth ofΓ ≈ 2π ×190 kHz. The light is generated by a Radiant Dyes Dye laser, which is locked
onto a ULE cavity. We are furthermore using three off-resonant lasers for creating
optical dipole traps and for our optical lattice. We employ two crossed-beam optical
dipole traps, one at λ = 1064 nm (Mephisto MOPA, Coherent), which we primarily
use for the bosonic isotopes, and the other at λ = 1570 nm (NKT Photonics), which
is used for the fermionic isotope (cf. [81, 82, 116]). As described in [82, 117] werealize a three-dimensional optical lattice with a cuboid unit cell with lattice constants(ax , ay , az ) = (266 nm, 266 nm, 532 nm). To this end we interfere two laser beams at awavelength of 532 nm with their retroreflected beams in the horizontal plane, whichyields the lattice spacing ax , ay = λ/2 = 266 nm and similar with a laser beam at a27

CHAPTER 3. PROPERTIES OF ERBIUM

wavelength of 1064 nm along the z-axis.

3.3

Magnetic Properties of Erbium

As already mentioned before, due to the submerged 4f shell, most lanthanide atoms
possess very large quantum numbers in their electronic ground state. Compared toalkali atoms this causes substantially larger dipolar effects. The magnetic moment µof an atomic state with magnetic quantum number mF can be calculated asµ = mF gF µB(3.1)

with µB being the Bohr magneton and gF the Landé-factor for the given state. gF is
given byF (F + 1) − I(I + 1) + J(J + 1)
gF = gJ,(3.2)
2J(J + 1)
with gJ given by
gJ = 1 + (gS − 1)

J(J + 1) − L(L + 1) + S(S + 1).
2J(J + 1)(3.3)

Here gS is the electronic Landé-factor gS ≈ 2.0023. Note that in the case of bosonicerbium without hyperfine structure (I = 0) the above equations evaluate to gF = gJ .
For the ground state of erbium we find the theoretic values gJ = 1.16705 and gF =0.63157 × gJ = 0.73503 for the fermionic isotope, respectively. The experimentally
reported value is slightly smaller, gJ = 1.163801(1) [118]. For the state with maximal
mF , this hence results in a magnetic moment ofµEr = 6.982606(6)µB .

(3.4)

As a comparison, lithium in its electronic ground state (at low magnetic field) featuresa magnetic moment of µLi = 2/3µB and hence the dipole-dipole interaction between
two erbium atoms is a factor of (µEr /µLi )2 ≈ 110 stronger than the dipolar interaction
between two lithium atoms, see Eq. (2.12). Note that the magnetic moment depends
on the magnetic sublevel mF , of which there are 20 in the ground state, ranging
from −19/2 up to +19/2. Furthermore gJ depends on the quantum number of therespective state and is different in excited states.

3.3.1

Erbium in an External Magnetic Field

When an atom in subjected to an external magnetic field B, energy levels with different magnetic moments split up due to the Zeeman-effect. In the case of bosonicerbium with zero nuclear spin I, the total angular momentum reads as F = J = L+S
and hence no hyperfine splitting occurs. As a consequence, in fields relevant for theexperiments (i.e. 0-100 G, far below the Paschen-Back regime), the energy splitting
is linear since the magnetic energy shift is much weaker than the spin-orbit coupling.

28

CHAPTER 3. PROPERTIES OF ERBIUM

Figure 3.3: Zeeman splitting. (a) The bosonic isotopes has F = 0 and splits linearly into
13 levels mJ ∈ {−6...6}. (b) The fermionic isotope with F = 7/2 features 20
Zeeman sublevels mF ∈ {−19/2...19/2}. Although the splitting looks perfectly
linear in the graph, there is a small quadratic correction, as explained in themain text.

The energy levels split up according to
EZBo = µB = mJ gJ µB B.

(3.5)

Fig. 3.3 (a) shows a plot of the splitting of the ground state of bosonic erbium. The
13 magnetic sublevels mJ ∈ {−6, ..., 6} are degenerate at zero magnetic field andsplit linearly in an external magnetic field without any quadratic contribution to theenergy shift. The energy spacing between two adjacent levels is independent of themagnetic quantum number mJ ,∆EZBo = gJ µJ B = 1.628879 MHz/G × hB

(3.6)

with the Planck constant h. This comes with some complications when it comes to theinvestigation of spin-mixtures of bosonic erbium, since the coupling of two adjacentlevels e.g. with a radio-frequency pulse always couples all 13 magnetic levels.
The scenario slightly changes for fermionic erbium where the hyperfine splitting comes
into play. Also here, all 20 magnetic sublevels are degenerate at zero magnetic fieldand split linearly in small magnetic fields. However, when the magnetic energy ison the same order as the hyperfine coupling, the total angular momentum quantum
number F ceases to be a good quantum number since the nuclear spin I and theelectronic angular momentum J start to decouple. This leads to a deviation of theenergy shift from its linear behavior. We plot the Zeeman-shift for all 20 sublevels
in Fig. 3.3 (b). The quadratic correction to the linear shift is so small, that it cannot
be seen in the figure (it will be discussed again and plotted in Chapter 5). In theexperimentally relevant regime, this correction can be sufficiently captured by adding
a quadratic term to Eq. (3.5),
EZFe = mF gF µB B + z(F 2 − m2F )B 2 .

(3.7)

29

CHAPTER 3. PROPERTIES OF ERBIUM

The quadratic correction vanishes for the two stretched states mF = ±19/2 andis maximal for the mF = 0 state. The linear term can directly be calculated asgF µB = h × 1.0287 MHz/G, whereas the coefficient z is determined with a fit to theenergy spectrum at low magnetic field [82] to bez = −h × 12.76(1) Hz/G2 .

(3.8)

Although this quadratic correction is very weak, it causes the energy difference of two
adjacent magnetic sublevels to be no longer degenerate. Instead it is given by∆EZFe = EZFe (mF ) − EZFe (mF + 1) = gF µB B + z(mF + 1)B 2 .

(3.9)

Hence at sufficiently large magnetic fields, two adjacent levels can be coupled without
coupling to other mF state. This will become relevant in Chapter 5 for the initialization of spin mixtures.

30

Chapter 4

Anisotropic Polarizability of
Erbium
Optical trapping of ultracold atoms relies on the ability of an atom to be polarized
by the oscillating field of a laser beam. The basics of optical trapping were alreadypresented in Section 2.1 and can be understood from a simple oscillator model [119]:
The closer the wavelength of the laser is to the wavelength of an optical transition,
the larger is its polarizability. The polarizability diverges on resonance and its signdepends on the detuning of the laser with respect to the optical transition.
The common approach to calculate the polarizability of an atom at a given laser
frequency ω is hence to sum over all optical transitions, weighted by the detuningof the laser to the optical transition and by the strength of the optical transition,i.e. the dipole matrix element. In the case of alkali atoms, where the ground-statepolarizability is almost completely determined by the D1 and D2 transitions, thegood spectroscopic knowledge of those transitions enables an accurate calculation of
the ground-state polarizability [91]. In contrast, for electronically more complex elements, such as erbium or dysprosium, there are hundreds of optical transitions fromthe ground state with linewidths ranging from many MHz down to only a few Hz.
Moreover, reliable spectroscopic data is only available for very few of those transitions, aggravating a direct calculation of the polarizability.
To tackle this problem, electronic-structure calculations [120] are used to model electronic properties of the atom and to extract all parameters of optical transitions
so that the polarizability can again be calculated with a sum-over-state formula[96, 111, 121]. Hence an experimental investigation of the atomic polarizability
presents an excellent testbed to verify the validity of such electronic-structure calculations and in the best case even to feed back important information that improvetheir predictive power.
Furthermore there is also an increasing experimental interest of gaining a better understanding of the atomic polarizability, in particular for atoms, where the polarizability can exhibit a significant anisotropic character. This is typically the case foratoms with an anisotropic electronic configuration around the core as it is the case forlanthanide atoms with a submerged 4f shell. Applications where a precise knowledge

31

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

of the polarizability is required are e.g the determination of magic wavelengths forclock transitions [122, 123], the use of tuneout wavelengths in cold-atom experiments[124–127] or the realization of state-dependent potentials [128–132].
This chapter presents an experimental investigation of the anisotropic polarizability of erbium atoms and compares the measured values to calculations based onsemiempirical electronic-structure calculations [111]. We briefly introduce the concept of anisotropic polarizability and the main formulas that are used for our calculations. We then present measurements of the scalar and the tensor contribution ofboth the ground-state and the 583 nm excited-state polarizability for three different
wavelengths, that are particularly relevant for optical trapping. We also theoretically
explore the usability of near-resonant light shifts for a precise tuning of optical potentials via vector and tensor polarizabilities. The content of this chapter was publishedin [133].

4.1

Anisotropic Polarizability⃗ gets polarized parallel (or anAn isotropic medium subjected to an electric field E
tiparallel) to the applied electric field. The response of the medium to the field is
described by the polarizability α and the induced dipole moment ⃗p reads as⃗.
⃗p = αE(4.1)

For an isotropic medium such as an alkali atom in its electronic ground state, α isa complex scalar. However, in the case of multi-electron lanthanide atoms with asubmerged-shell structure, the atomic polarizability can exhibit a tensorial structure,
due to the anisotropic electronic configuration. In that case an electric field mightalso induce a dipole moment ⃗p⊥ with a component perpendicular to the electric field.
To better understand those two contributions, it is instructive to consider the totalenergy shift experienced by the atom in an electric field,
U=

1⃗† ⃗
E PE .
2(4.2)

Here P is the 3 × 3 polarizability tensor, whose diagonal entries describe the scalar
response of the atom (p⃗ ) and whose off-diagonal entries can induce anisotropic effects(p⃗ ⊥ ). We can decompose the polarizability tensor P into the scalar polarizability
tensor As , containing only the diagonal entries, the vector polarizability tensor Av ,which is the antisymmetric part of P and the tensor polarizability tensor At , thesymmetric part of the off-diagonal entries. With this decomposition, we identify thethree following contributions to the total light shift
Utot

32

= Us + Uv + Ut
1⃗†⃗,=
E [As + Av + At ]E
2

(4.3)

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

that we call the scalar, vector, and tensor light shift, Us , Uv , and Ut , respectively.
Let us now consider the case of an atom with angular momentum J and magnetic⃗ (rquantum number mJ , placed in a laser field of intensity I(r⃗ ) = ϵ02c |E⃗ )|2 , polarizationcvector u and frequency ω = 2π λ . ϵ0 is the vacuum permitivity, λ the wavelength ofthe laser and c the speed of light. Following the derivations in Refs. [111, 134] theabove expression can be developed and an explicit equation for the three terms canbe written down as
U (r⃗ , ω)=Us + Uv + Ut
[︂
1
mJ
= −
I(r
⃗ ) αs (ω) + |u∗ × u| cos θkαv (ω)
2ϵ0 c2J]︂3m2 − J(J + 1) 3 cos2 θp − 1+ J×αt (ω)
J(2J − 1)
2
1
I(r⃗ )αtot (ω).
=: −
2ϵ0 c

(4.4)

We call αtot the total polarizability, i.e. the sum over the scalar, the vector (linear inmJ ) and the tensor (quadratic in mJ ) contribution to the total light shift. While thescalar term only depends on ω and the spectral properties of the atomic species, thelatter two also depend on the internal state of the atom |J, mj ⟩ and also the polarization and propagation axis of the laser field. As illustrated in the inset of Fig. 4.1,
θk (θp ) is the angle between the propagation (polarization) axis and the quantization
axis, set by the external magnetic field. More precisely, θk is the angle between thevector ⃗v = u∗ × u and the quantization axis. The three polarizability coefficients(K){αs , αv , αt } can be expressed in terms of the so called coupled polarizability αJ as

αs (ω)αv (ω)αt (ω)= − √︁

1(0)3(2J + 1)

αJ (ω)

√︄

2J
(1)
α (ω)
(J + 1)(2J + 1) J√︄

2J(2J − 1)(2)
α (ω),
3(J + 1)(2J + 1)(2J + 3) J

=

=

(4.5)

(K)

where αJ can be calculated with a sum-over-state formula [135] and in constant-sign
convention is given by(K)

αJ (ω)=√

2K + 1 ×

∑︂

1
J

}︄

(−1)J+J′(J ′ )

{︄

K
J′

1
J

|⟨J ′ ||d||J⟩|2 ×

[︃
]︃
1
1
(−1)K
ℜ
+
.
ℏ
∆−′ − iγJ ′ /2 ∆+− iγJ ′ /2′
⏞ JJ⏟⏟ J J
⏞

(4.6)

=:A

33

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

with K ∈ {0, 1, 2}. The sum runs over all excited states, whereas the dipole selection
rules are incorporated into the Wigner 6j symbol (in curly brackets). |⟨J ′ ||d||J⟩|2
is the reduced dipole transition element, γJ ′ the natural lifetime of the excited stateand ∆±J ′ J = ωJ ′ J ± ω is either the difference or the sum of the transition frequency
ωJ ′ J to the excited state and the laser frequency ω. Under the assumption that
γJ ′ /2 << ∆±J ′ J , i.e. that the laser in far detuned to the optical transition, we can
express the real part of the last line in Eq. (4.6) asℜ[A] =

2ωJ ′ J.
ωJ2 ′ J − ω 2(4.7)

Once all intrinsic atomic properties are know, the calculation of the polarizability
hence boils down to the threefold evaluation of the sum in Eq. (4.6) and can be doneat any wavelength ω. The imaginary part of the above expression is directly related
to the off-resonant photon-scattering rate, it reads asℑ[A] = γJ ′

ωJ2 ′ J + ω 2

2.

(ωJ2 ′ J − ω 2 )(4.8)

From an experimental point of view, a good understanding of the photon scattering rate is crucial to realize dipole traps with low heating rates and low atom loss.
Scattering rates can be calculated in a similar way by employing a slightly modified
sum-over-state formula (cf. [111]).
From Eq.
{︄ (4.6), a few}︄interesting cases can be discussed. First of all, the Wigner
j1 j2 j3
6j symbol
is invariant under permutation of any two columns andJ1 J2 J3
under exchanging the entries within any column [136]. Furthermore, the conservationof angular momentum is guaranteed by the constraint that all triples {(k1 , k2 , k3 ), k ∈{j, J}} have to fulfill triangle rules, |k2 − k3 | ≤ k1 ≤ k2 + k3 and have to sum up to
an integer. Together with the explicit formulas for the total light shift we find thefollowing special cases:
1. Alkali Atoms: Alkali atoms have J = 1/2 in their electronic ground state.
Hence αt = 0 and the tensor light shift Ut is zero. As a consequence, alkali
atoms do not have a quadratic light shift in their electronic ground state anda state-dependent light shift can only be induced via the vector light shift Uv
or by a detuning of the laser light that is comparable to the hyperfine splitting[137].
Ut = 02. Linear Polarization: For linear polarization of the laser beam, the polarization vector u is real and therefore u∗ = u. The cross product u∗ × u evaluates
to zero and the vector light shift Uv vanishes.
Uv = 03. Perpendicular Propagation: For the case where the angle between the prop34

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.1: Calculated atomic polarizability. The plot shows the total polarizability
for an erbium atom in its lowest Zeeman sublevel (mJ = −6) with θp = θk =90◦ as a function of wavenumber and wavelength of the light field. With
the shown configuration (inset), Uv vanishes and Ut is maximally ’negative’.
Each divergence of the total polarizability indicates an optical transition (finite
amplitude of peaks are due to finite number of calculated points). There is amostly blue-detuned region towards shorter wavelengths and a red-detuned
region towards longer wavelengths as indicated by the back ground color.

agation axis of the laser beam and the quantization axis (magnetic field) isθk = 90◦ , we have cos(θk ) = 0 and the vector light shift vanishes.
Uv = 04. Magic Angle: If the angle between the polarization vector and the quanti√
zation axis is such that cos(θp ) = 1/ 3, i.e. θp ≈ 54.7◦ , the tensor light shift
vanishes even though αt ̸= 0.
Ut = 05. Dependence on mJ : For finite vector and tensor polarizability, the total lightshift is state dependent and depends quadratically on mJ . This enables almost
arbitrary tuning of the relative light shift between adjacent hyperfine states.
If not explicitly stated otherwise, all shown calculations are performed for themJ = −6 state.

4.2

AC-Polarizability of Erbium

As mentioned above, the calculation of the polarizability requires a good knowledge ofthe electronic spectrum and lifetimes of optical transitions. While the NIST database[138] only lists 232 observed optical transitions for neutral erbium, of which only
a few contain the relevant information for the calculation of the polarizability, webase our calculations on a more detailed list from semiempirical electronic-structure
calculations [111, 133]. The full list of states that was used for the calculations of theground-state polarizability can be found as supplemental information of Ref. [133].

35

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.2: Different contributions to the total light shift. The four plots show thecalculated polarizability for the same setting as in the previous figure but foreach plot only lines with the indicated strength are taken into account. The
lower plot does not show calculations of the polarizability but only indicatesthe narrow transitions with vertical lines.

For the calculations shown within this thesis, we replace the 401 nm and 583 nm lines
with experimentally measured values from Ref. [81]. In the whole chapter, the results
are given in atomic units (a.u.). The conversion to SI units is given by
2

α[Hz/Wmm ] =1.6488 × 10−35
α[a.u.].
2hϵo c(4.9)

Figure 4.1 shows the calculation of the total polarizability αtot for an atom in thelowest state mJ = −6 as a function of the light’s wavenumber and wavelength forthe scenario depicted in the inset of the same figure, i.e. for linearly polarized lightwith θk = θp = 90◦ . As argued above, in this configuration the vector light shift Uv
vanishes completely and the prefactor to the tensor light shift is maximally negative,
1
αtot (θp = 90◦ , θk = 90◦ ) = αs − αt .
(4.10)
2
Each divergence of the polarizability indicates an optical transition. The finite heightof some peaks is caused by a limited number of calculated data points. The main
contribution to the total polarizability comes from the broadest optical transition(2π × 29.7 MHz [81]) at around 401 nm. Together with a few further relatively stronglines in the same region, it determines the main shape of the polarizability. In the region towards longer wavelengths the polarizability is predominantly positive, enabling
the generation of attractive optical potentials, whereas in the region towards shorter
wavelengths the polarizability is mainly negative, giving rise to repulsive optical potentials. On top of the broad transitions, there are hundreds of further transitions
with linewidths ranging from the MHz down to the Hz-regime.
To better visualize this complex spectrum and in order to separate the contribution of36

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.3: State dependent light shift. (a) Total polarizability in the proximity of the
MOT transition for the four lowest hyperfine states mJ ∈ {−6, −5, −4, −3}
for two different polarization angles θp = 0◦ (top) and θp = 90◦ (bottom).
(b) Total polarizability at 586.56 nm for all hyperfine states and three different
polarization angles. (c) Total polarizability as a function of θp for the 7 lowest
hyperfine states at 586.56 nm.

broad and narrow lines, Fig. 4.2 shows 4 plots with the same calculation as for Fig. 4.1
but for each plot only transitions within a certain linewidth region were taken into
account. The upper plot shows that there are 4 optical transitions with linewidths
above 10 MHz. All those transitions are located around 400 nm. The second panel
shows that there are a few lines in the range of 1 MHz to 10 MHz, primarily located in
the ultra-violet region. As shown in the third panel, by far the most transitions havea linewidth in the range of 1 kHz to 1 MHz. Some of them are in the visible region
but also here the biggest part is located in the ultra-violet regime. Narrow transitions
below 1 kHz are sparsely distributed over the whole plotted range as shown in thelowest panel. Note that this panel only indicates the positions of the narrow transitions and does not show the calculated polarizability.
As shown e.g. with Dy [125, 139] or Sr [140], narrow lines are particularly promising candidates for state-dependent trapping and manipulation of cold atoms due tolarge tensorial and vectorial light shifts. To show how the anisotropic light shift can be
exploited for generating state-dependent potentials, we discuss possible applicationsof near-resonant laser light in a region with overlapping resonances. We choose thetransition on which we operate our magneto-optical trap with λ = 582.84 nm, J ′ = 7,
and a measured linewidth of Γ = 186 kHz [81]. Figure 4.3 (a) shows the calculation of
αtot for θk = 90◦ for two polarization angles θp = 0◦ (top) and θp = 90◦ (bottom) for
4 different spin states, mJ ∈ {−6, −5, −4, −3}. The two resonances at λ1 = 578.5 nm
and λ2 = 587 nm have total angular momentum of J1′ = 5 and J2′ = 6, which explains
the asymmetric behavior around the MOT transition (e.g. the presence of the magic
wavelength below the MOT transition but not above). However, a change of thepolarization angle by 90 degrees has an enormous effect on the total light shift. This37

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

behavior is summarized in (b) and (c) at the example wavelength λ∗ = 586.56 nm,
which is indicated by the grey line in (a). For each angle θp the total polarizability depends quadratically on the magnetic quantum number mJ . This quadratic behavior
disappears at the magic angle of θp0 ≈ 54.7◦ , where the tensor light shift Ut vanishes(see Eq. (4.9)). For the particular wavelength λ∗ , the light shift for each spin stateeven flips its sign at the magic angle. This becomes even clearer in the plot in (c),where we plot the total polarizabilities for 7 different hyperfine states as a functionof the polarization angle θp . The state dependency of the polarizability vanishes at
θp0 and the different hyperfine levels even flip their energetic order. In fact the wavelength λ∗ was chosen such that αs = 0 and hence the total light shift vanishes at themagic angle. By using also the vector light shift or the magnetic field dependence ofthe coupling to excited states with different g-factors gJ , one can almost arbitrarily
tune the relative light shift [141].

4.2.1

Measurement of Ground-State Polarizability

We experimentally access the polarizability of the ground state by measuring thedepth of the optical potential, which is induced by an off-resonant laser beam. To
this end we measure trap frequencies and relate them to the total polarizability αtot .
For a laser beam propagating along the x-axis with total power P and Gaussian2
2z 2
radial intensity profile I(y, z) = I0 exp(− 2yσ 2 − σ 2 ) with beam waists σy/z along the
y

z

, the
two orthogonal directions perpendicular to the propagation axis and I0 = πσ2Py σzdepth of the induced optical potential is given by
U0 = −

P
αtot .
ϵ0 cπσy σz(4.11)In harmonic approximation the depth U0 can be related to the trap frequencies ωi ,i ∈ {z, y} via
√︄
ωi =

−4U0
,
σi2 m

(4.12)

with the atomic mass m. Combining the two above equations yields the expression
σy σz σi2.
(4.13)
4P
Hence to deduce the total polarizability from trap frequency measurements, a precise
characterization of the trapping beam is necessary. In particular the beam waists σ
have to be measured as precise as possible, because of the σ 4 scaling of αtot . We measure the power in the beam with a power meter and stabilize the light power with a
PID loop onto a photo diode to avoid power drifts in the course of the measurement.
The most reliable measurement of the beam waists was done using the knife edge
method [142] or using a beam profiler camera mounted on a piezo-controlled translation stage. This allows us to measure the beam waist at multiple points along thebeam propagation axis. Figure 4.4 shows a typical such measurement from which we√︁extract the beam waist σi , i ∈ {y, z}, by fitting the data with σ(z) = σi 1 + (z/zR )2 ,αtot = ωi2 ϵ0 cπm

38CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.4: Trapping-beam characterization. (a) Typical beam waist measurement oftrapping beam. We plot the radial size of the beam along the propagation axis
and fit the data to extract the beam waists σi along both axes. (b) Typical
trap frequency measurement with damped sine fit.

with the Rayleigh range zR = πσi2 /λ. To reduce deviations between the trapping
beam and the beam for the measurement of the waist to a minimum, we implement a
mirror directly in front of our vacuum chamber and reflect the trapping beam in orderto perform the beam-waist measurement as close to the atomic position as possible.
Despite a careful characterization of the beam waist, we assume a systematic uncertainty of 2 µm due to aberrations and a finite ellipticity of the beam. Because ofthe σ 4 scaling this is the major contribution to the uncertainty of the polarizability.
Trap frequencies are measured by exciting a center-of-mass oscillation of the atomic
cloud in the optical trap and monitoring the position of the atoms over time. For
the measurement of the total ground-state polarizability we choose θk = θp = 90◦ ,such that the vector light shift vanishes (Uv = 0) and the tensor contribution is maximally negative. In that case, the total polarizability is given by Eq. (4.10). Figure 4.4
shows a typical trap frequency measurement in a single-beam optical dipole trap ofwavelength λ = 1064.5 nm. We extract the trap frequency by fitting a damped sinefunction.
From the measurements of the beam waists and the trap frequencies we compute thetotal light shift and deduce the total polarizability for three different wavelengths,532.26 nm, 1064.5 nm, and 1570.0 nm. The following table summarizes the experimental results and gives the calculations for the same wavelengths for θk = θp = 90◦ .
λ (nm)
532.261064.51570.0exp
αtot
430 ± 8st ± 80sys
166 ± 3st ± 61sys
163 ± 9st ± 36sys

theo
αtot
317
176
162

As mentioned above, the large systematic uncertainty is caused by the uncertaintyof the beam waist. We find a good overall agreement between the measured andcalculated polarizabilty, in particular the measurement of the two wavelengths in theinfrared region match very nicely with the theoretical predictions. As can be seenin Fig. 4.2 the light shift at those wavelengths is not significantly affected by many
optical transitions and mainly determined by the broad optical transitions around
400 nm so that a reliable prediction can be made. However the polarizability at

39

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

532.26 nm significantly deviates from the theoretical value even beyond the systematic
uncertainty. We attribute this to the presence of many strong and narrow transitionsin the proximity of that wavelengths and small uncertainties of their parameters canhave a large effect on the calculated polarizability. Nevertheless, the overall agreement
is very satisfying.

4.2.2

Anisotropic Ground-State Polarizability

Although the above measurements showed a good overall agreement between measurement and theory for the total light shift, they do not tell anything about different
contributions to the light shift. With the vector light shift being zero for θk = 90◦ ,we perform an additional measurement to isolate the scalar and the tensorial contributions to the total light shift. To this end we measure trap frequencies for different
angles θp by either rotating the magnetic field or the linear polarization of the trapping beam with a waveplate, while keeping θk = 90◦ . In order to eliminate systematic
uncertainties that may arise from the beam waist measurement, we investigate therelative change of the total light shift and quantify this change via the parameterκ(θp ) withκ(θp )==
J=6,mJ =−6=

U − UsUt
ω(θp )2 − ω(θp0 )2
=
=
Us
Us
ω(θp0 )23m2J − J(J + 1) 3 cos2 θp − 1 αt×.
J(2J − 1)2αs
3 cos2 (θp ) − 1 αt.
2αs(4.14)√

Here θp0 is such that cos(θp0 ) = 1/ 3, i.e. the angle where the tensor light shift
vanishes, Ut (θp0 ) = 0. Note that for the maximally polarized state, mJ = ±J, theprefactor in the second line evaluates to 1 so that the peak-to-peak variation of κ isgiven by κ0 = 1.5 × ααst , from which also an absolute value for the tensor polarizability
can be deduced. Figure 4.5 summarizes the measurements for the two wavelengths532.26 nm and 1064.5 nm. For both wavelengths there is a sinusoidal dependence ofthe total light shift on the polarization angle θp . We fit the data with Eq. (4.14) toextract κ0 and use κ0 and the measured scalar polarizabilities to calculate αt . The
dotted lines in the plot are the expected curves for κ(θp ) using the calculated scalar
and tensor polarizability, αs and αt respectively. From the fit we find the following
values for κ0 and for αt .
λ (nm)
532.261064.5κexp
0 (%)−5.3 ± 1−1.8 ± 0.8

κth
0 (%)
-9.2
-4.7

αtexp
−15 ± 3st ± 6sys
−1.9 ± 0.8st ± 1.2sys

αtth
-19
-5.4

For both wavelengths we measure slightly smaller tensorial effects than predicted by
the calculations, but the agreement is still satisfying. The deviations might be dueto the very small changes in trap frequencies, which might be affected by additional
systematic effects in the measurements.
As a comparison, the erbium tensor polarizability for 1064.5 nm is similar but slightly
40CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.5: Anisotropic ground-state polarizability. The plot shows the anisotropy
parameter of the light shift κ for 532.26 nm (blue) and 1064.5 nm (red). The
measured values show a sinusoidal dependence on the polarization angle θp .
Solid lines are fits to the data, dotted lines are theory curves. For both wavelengths the measured anisotropy is slightly smaller than expected from theory.

larger than the dysprosium tensor polarizability, which was recently measured tobe {αs , αt } = {184.4(2.4), 1.7(6)} [143]. Given the complexity of the erbium atomic
spectrum, the overall agreement between theoretical predictions and the experimental
values for all investigated wavelengths is remarkable.
However, we further test the electronic-structure calculations and finally investigate
the 583 nm excited-state polarizability, where anisotropic effects are expected to belarger than in the ground state.

4.2.3

Excited-State Polarizability

Anisotropic effects are expected to be stronger in excited states due to the even moreanisotropic electronic wave function. We therefore investigate the light shift of the
583 nm excited state (MOT transition, 17157 cm−1 , J ′ = 7) to further test the calculated spectrum of erbium. For the chosen state we cannot perform trap frequency
measurements due to its short lifetime. We therefore extract the light shift by measuring the shift of the bare atomic resonance frequency. Experimentally this is done
by applying a short laser pulse of circularly polarized 583 nm light to the atomic sample in the optical dipole trap. This polarization couples the |J = 6, mF = −6⟩ groundstate to the |J = 7, mF = −7⟩ excited state. As shown in Fig. 4.6 (a) the trap induces
a different light shift in the ground and the excited state and therefore shifts the resonance frequency. By scanning the laser frequency around the atomic resonance wefind a resonant atom loss whenever the laser frequency matches the sum of the atomic
transition, the ground-state light shift and the excited-state light shift. To calculate
the ground-state light shift we use the measured ground-state polarizability and neglect its angle dependence since the anisotropic effects are two orders of magnitude
smaller than in the excited state. Figure 4.6 (b) shows the measured light shift of theexcited state for θk = 90◦ as a function of θp for the two wavelengths λ = 1064.5 nm
and λ = 1570 nm. The scalar polarizabilities of both wavelengths have opposite signand therefore cause a repulsive potential for λ = 1064.5 nm and an attractive po-41

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.6: Measurement of excited-state polarizability. (a) Instead of performing
trap frequency measurements, we measure the shift of the bare transition frequency from the ground to the excited state, induced by the optical dipole
trap. The light shift of the excited state can either be repulsive (upper panel)or attractive (lower panel) and depends on the angle θp as indicated by thedifferent shades of the excited state. (b) Measurement of total polarizability as
a function of θp for the two wavelengths 1064.5 nm (top) and 1570.0 nm (bottom). Solid lines are a fit to the data. Color gradient of solid lines correspondto color shades in (a).

tential for λ = 1570 nm. However, the tensor polarizability have the same sign, sothat they tune equally with θp . The total light shift shows a strong dependence onθp , which is much stronger than the measured anisotropies in the ground state. The
tensor polarizability αt is even on the same order as αs , so that the total light shift
can be substantially changed by changing the angle θp . We fit the data with Eq. (4.4)
and extract the scalar and tensor polarizability. For the two wavelengths we find thefollowing results.
λ (nm)
1064.51570.0αsexp (a.u.)
66.6 ± 0.5st ± 28sys−203 ± 9st ± 50sys

αsth (a.u.)91-254

αtexp (a.u.)−11.3 ± 0.5st ± 2sys
−141 ± 9st ± 19sys

αtth (a.u.)
-18
-68.5

Again, the large systematic uncertainties are caused by the systematic errors of theground state polarizability, that in turn are caused by the uncertainty of the beamwaist measurements. Nevertheless, considering the complex electronic structure of
erbium, the agreement between the measured and calculated values is surprisingly
good. In particular, as we show in Fig. 4.7, the two investigated wavelengths are
surrounded by many broad and narrow optical transitions. This is different in the
ground state, where there are only very few transitions in the infrared.
All in all we find good agreement between calculated and measured values of thepolarizability for all investigated wavelengths both for the ground and the excitedstate. In particular the measurement of the ground-state scalar polarizability of theinfrared wavelengths agrees excellently with the predicted values. We measure a small
but finite anisotropic effect in the ground state and showed that the 583 nm excited42

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

Figure 4.7: Calculated excited-state polarizability. The plot shows the scalar (blue),
vector (grey) and tensor (red) polarizability of the 583 nm excited state. There
are many optical transitions in the proximity of the two experimentally investigated wavelengths of 1064.5 nm and 1570 nm. Given this complex structure,
the agreement between measured and calculated polarizabilities is surprisingly
good. The shown calculations for the excited state polarizability were done by
M. Lepers.

state exhibits much stronger anisotropic light shifts, which are caused by the many
optical transition in the proximity of the investigated wavelengths. Although there arestill large uncertainties on the measured values, the results prove a good understandingof the level structure of erbium. Our results do not only test semiempirical electronicstructure calculations [111] but might also enable the use of near resonant light inorder to exploit large vector and tensor light shifts for generating mJ -dependent
optical potentials, to control the energy landscape on the different hyperfine levels[141] and to use the quadratic light shift to initialize and control spin dynamics [144].

43

CHAPTER 4. ANISOTROPIC POLARIZABILITY OF ERBIUM

44

Chapter 5

A Strongly Interacting Fermi
Gas of Dipolar Atoms
Despite the significant number of machines with utracold dipolar gases [37, 41, 42,
47, 48, 145–147], only very few experiments with fermionic species were conducted[139, 148–150]. In particular, those experiments only investigated spin-polarizedFermi-systems and did not exploit the spin degree of freedom. In fact, dipolar atoms,
such as Er or Dy, posses large quantum numbers in their ground state and therefore
offer a huge number of spin states. For instance, fermionic 167 Er has a total angularmomentum of F = 19/2 and hence 20 hyperfine states. Such a system offers thepossibility to explore spin physics with many different internal states and to study
the effect of dipolar interaction in spinor systems.
At least in the case of erbium atoms, the reason for the lack of experiments on
fermionic spinor gases is threefold. On the one hand the investigation of fermionicsystems with strong dipolar interaction is hindered by complex scattering dynamics[98] and the presence of many higher-order Feshbach resonances even between identical spin states. For spin mixtures, the number of Feshbach resonances is expected
to be even higher [139], eventually leading to Ericson fluctuations [151] at high magnetic field, which makes it even harder to find a magnetic field region with a long
lifetime and high collisional stability. A further complication is the spontaneous relaxation of the atomic spin since the dipole-dipole interaction only conserves the total
angular momentum, but not the spin [148, 152]. Finally, in the case of fermionicerbium, the weak hyperfine coupling further aggravates the deterministic preparationof multi-component spin systems. In the magnetic-field region at around 1 G, where
our experiments are conducted, the quadratic Zeeman splitting is almost negligible
and hence a standard spin initialization via radio-frequency (rf) coupling cannot be
straightforwardly applied. For negligible quadratic Zeeman splitting, rf coupling ofneighboring spin states would immediately couple to all further spin states due to thefact that all adjacent levels are equally spaced in energy (see Fig 5.1 (a)).
This chapter presents the first steps towards solving the described problems andstudying spinor physics with fermionic erbium atoms. To this end we make use of45

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.1: Quadratic Zeeman effect. (a) Schematic of erbium level scheme at low andlarge magnetic field. (b) Zeeman shift for all 20 magnetic sublevels of fermionic167
Er. The plot in the inset shows the relative splitting between two lowest
adjacent pairs of states, δ(∆E) = (E17/2 − E15/2 ) − (E19/2 − E17/2 ). Note thedifferent y-axis for main plot and the inset.

a new method to initialize a long-lived two-component spin mixture and perform
Feshbach spectroscopy by employing a lattice-projection technique. We identify andcarefully map out a comparatively broad Feshbach resonance between the two lowest
hyperfine states of 167 Er. The results may enable the study of strongly interacting fermions with dipole-dipole interaction and the investigation of the BEC-BCS
crossover in a dipolar quantum gas. The results of this chapter are published in [102].

5.1

Preparing a Spin Mixture of 167 Er

As already mentioned and presented in Sec. 3.1, fermionic 167 Er has a nuclear spin of
I = 7/2 and hence a total angular momentum quantum number of F = 19/2 with
20 magnetic sublevels mF ∈ {−19/2...19/2}. In an external magnetic field, thesesublevels split up due to the Zeeman effect. In contrast to the bosonic isotopes of
erbium with I = 0, the hyperfine coupling induces a quadratic Zeeman splitting in
the fermionic isotope 167 Er according to Eq. (3.7). However, in the magnetic fieldregion, where most experiments with erbium are conducted (i.e. around one to a few
Gauss), the quadratic splitting is typically smaller than magnetic-field noise such thatit cannot be resolved by radio-frequency coupling of adjacent states. It can therefore
not easily be exploited for a deterministic initialization of a spin mixture. Similiar
ideas such as the coupling of two next-to-nearest spin states via a Raman scheme
also suffer from the small quadratic Zeeman shift. It is hence unavoidable to performthe spin preparation at high magnetic field, where the quadratic correction to the
Zeeman shift is large enough to be able to exclusively couple two neighboring spinstates. This requires that the relative energy splitting between the states, δ(∆E) =(E17/2 − E15/2 ) − (E19/2 − E17/2 ) is much larger than the magnetic field noise and theFourier limit of the applied rf pulse. For instance, as shown in Fig. 5.1 (b), in orderto get a relative splitting on the order of 50 kHz, a magnetic field of around 40 G isneeded. Unfortunately an evaporation in the low-field region (around 600 mG) implies

46

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.2: Spin preparation and band mapping. (a) In order to prevent heating andatom loss during the ramp to high magnetic field before the spin preparation,
we initialize a spin-polarized band insulator. We then ramp to high magnetic
fields and apply a resonant rf-pulse to induce spin mixing. (b) Band mappingin the horizontal plane. Red arrows indicate the calculated size of the Brillouinzone. (c) Three examples of different spin mixtures of the lowest two hyperfine
states, without populating higher spin states. Shown examples have δs ∈{1, 0.02, −0.94} (from left to right).

that we have to ramp the magnetic field across hundreds of Feshbach resonances1 inorder to start the spin-preparation. Furthermore, once the spin-mixture is initialized,
we need to ramp the magnetic field back to low values across even more Feshbachresonances. This procedure will cause heating of the atomic sample and significant
atom loss and hinders the preparation of a cold spin mixture in our setup.
In order prevent heating of the atomic sample during the preparation, we employ
a lattice-projection technique that suppresses atomic collisions. As illustrated inFig. 5.2, we start by evaporatively cooling a spin-polarized gas in the lowest hyperfine
state in our optical dipole trap to a temperature of about 0.15TF . We then adiabatically ramp up a deep three-dimensional optical lattice [117] with lattice spacings(ax , ay , az ) = (266, 266, 532) nm and prepare a band insulator by populating only thelowest band. This effectively freezes out any collisions between atoms. We verify
the preparation of a band insulator by band mapping, see Fig. 5.2 (b), a standard
technique that is commonly used in cold-atom experiments [153]. By comparing
absorption images to expected profiles computed from the first Brillouin zone, we estimate the populations in higher bands to be < 5% along the z-axis and even smallerin the xy plane, as it is below our detection limit. With this system we ramp the magnetic field to 40.51 G, where the quadratic splitting between the three lowest states isδ(∆E) = 42.6 kHz. During this ramp we do not see significant atom loss or heating.
In principle the high quadratic splitting enables the initialization of any arbitrary spin
mixture [144], but we only consider scattering properties of mixtures of the two lowest hyperfine states |mF = −19/2⟩ and |mF = −17/2⟩ in this work. We initialize thespin-mixture by applying a resonant rf pulse and realize different spin compositions
by varying the duration of the pulse. As shown in Fig. 5.2 (c), this procedure solely
couples the two lowest spin states and does not populate any higher spin states. We
N−Nquantify the spin composition with the parameter δs = −19/2Ntot −17/2 , ranging fromδs = 1 (fully polarized in the mF = −19/2 state) to δs = −1 (fully polarized in the
mF = −17/2 state). Fig. 5.2 (c) shows three examples for δs ∈ {1, 0.02, −0.94}.
While the atomic gas is stable in the lowest hyperfine state mF = −19/2, there
1 See Extended Data Figure 1 in Ref [98]47CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.3: Collisional stability of erbium spin mixtures. (a) There are five possible
scattering processes driven by dipolar interaction that can occur between two
identical particles. They can be both spin conserving (δM = 0) or spin nonconserving (δM = ±1, ±2). Spin non-conserving processes are usually strongly
suppressed by energy conservation. (b) Lifetime measurements of three different spin mixtures in a deep optical lattice. δs = 1 and δs = −0.92 in theleft plot, δs = 0.6 in the right plot. Blue (red) data points correspond to
mF = 19/2 (17/2). Solid lines are exponential fits to the data.

are different types of collisions for higher hyperfine states, that might influence thelifetime of the atomic sample. In principle there are 5 different scattering processes that can happen between two identical dipolar atoms [40, 144]. As illustrated in Fig. 5.3 (a), both magnetization-conserving (δM = 0) and magnetizationnon-conserving (δM ̸= 0) collisions are allowed. Since these processes are driven by
the long-range DDI, they can even occur between 2 atoms on neighboring sites. However, collisions that do not conserve magnetization are typically strongly suppressedin a magnetic field because of the energy mismatch of the initial and final state due tothe Zeeman shift. However, energy conversation can in principle be guaranteed e.g.
by scattering with a third particle. Eventhough this process should be strongly suppressed in the lattice we experimentally test the collisional stability of a spin mixtureof the two spin states, mF = −19/2 and mF = −17/2. We perform this measurement
in a region without a Feshbach resonance, at a magnetic offset field of B = 3.99 G.
Figure 5.3 (b) shows the decay of the atom number over time for three different spincompositions with δs ∈ {1, 0.6, −0.92}. We fit an exponential decay to the data andextract surprisingly long lifetimes of τ19/2 = −31(3) s and τ−17/2 = 12.2(7) s. In
particular we find no strong dependence of the lifetime on the initial spin mixture.
However, we always observe a shorter lifetime for the mF = −17/2 state. Since we
do not measure an increase of atom number in the mF = −15/2 state, we conclude
that no magnetization-conserving scattering processes (δM = 0) occur. We instead
attribute the reduced lifetime of the mF = −17/2 state to a small but continuoustransfer of atoms into higher bands via intensity noise on the optical lattice. Two
atoms in the mF = −17/2 state on the same lattice site can then decay via spinrelaxation (δM = −2), where the released Zeeman energy is large enough to escapefrom the lattice. This process is not possible for the mF = −19/2 state.

48

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.4: Large Feshbach scan. The three plots show a Feshbach scan for three different spin mixtures, δs = 1 (top), δs = −0.54 (middle) and δs = 0.4 (bottom).
The dark grey shaded region illustrates the broadest loss feature that occurs
in both spin states. We therefore perform a finer Feshbach scan in the lightgrey shaded region, which is presented in Fig. 5.5.

The results of this section show that we can initialize the system in arbitrary spincompositions with lifetimes of several seconds. This systems provides a good starting
point for the investigation of scattering physics between spin states. The next section
presents our measurements on Feshbach loss spectroscopy in an ODT and modulation
spectroscopy in a deep optical lattice.

5.2

Feshbach SpectroscopyWith the ability to prepare arbitrary spin mixtures, we can investigate scattering
properties both between identical fermions in the mF = −17/2 state but also between
the two lowest hyperfine states. To this end we make use of our lattice-projection
technique to ramp the magnetic field through the dense Feshbach spectrum. As
described above, we initialize the spin mixture in the optical lattice at high magnetic
field. We then ramp the magnetic field to the value that we want to probe andwait for the magnetic field to stabilize. We then release the atomic sample backinto the dipole trap and hold the spin mixture for th = 500 ms before measuring thespin-resolved atom number. At a Feshbach resonance, the scattering length diverges,
three-body recombination rates increase substantially and hence the lifetime of thesample decreases, causing a dip in the atom number during the Feshbach scan [28].
Figure 5.4 shows a Feshbach scan in the region from zero to two Gauss for threedifferent spin mixtures, δs ∈ {1, −0.54, 0.4}. The different spin combinations enable
the assignment of occurring loss features to the spin states. The upper pot (δs = 1)
only indicates Feshbach resonances between identical atoms with mF = −19/2. Thoseare resonances of higher partial wave character, depending on temperature and opticalconfinement [154]. With the two further plots (δs = −0.54 and δs = 0.4), we canidentify Feshbach resonances between two identical atoms in the |mF = −17/2⟩ state,such as the narrow feature at around 0.8 G or the very broad feature around 0.35 G
or identify interspin resonances e.g. at 1.1 G. The light-grey shaded sector showsa broad region without loss feature, close to a relatively broad loss feature in both49

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.5: High-resolution Feshbach scan. The three plots show a high resolution
Feshbach scan for different spin mixtures of the two lowest hyperfine states,|mF = −19/2⟩ (blue) and |mF = −17/2⟩ (red) with δs = 1 (top), δs = −0.6
(middle), and δs = 0 (bottom). The shaded regions indicate loss featuresthat only appear in the respective spin (red and blue) or in both spin states(orange).

spin components, highlighted by the dark-grey shaded region. This points towards
a promising candidate of a broad interspin Feshbach resonance, which we further
investigate by performing a second Feshbach scan with higher resolution within thelight-grey shaded region. This scan is presented in Figure 5.5, again for three different
spin combinations, δs = 1 (top), δs = −0.6 (middle), and δs = 0 (bottom). Thanks
to the employed lattice-projection technique, the loss features do not suffer frommagnetic field ramps at all and do not show any artificial broadening. We are therefore
also sensitive to very narrow resonances on the order of a milligauss. The measurement
reveals 7 further narrow intra-spin resonances (blue and red-shaded regions in top
and middle plot) and 5 inter-spin resonances (orange-shaded regions in bottom plot),from which the one at 680 mG stands out because of its much larger width of about
50 mG. We choose that resonance for further investigation and perform modulation
spectroscopy in our optical lattice to extract the inter-spin scattering length.
We start by preparing a balanced spin mixture (δs = 0) in the optical lattice. As
illustrated in Fig. 5.6 (a) we sinusoidally modulate the depth of the optical potential
at frequency νmod . If the modulation frequency is resonant to the onsite interactionof two atoms, i.e. hνmod = Uo , the system is resonantly excited and doubly occupied
lattice sites are created. Those doublons have a shorter lifetime and get lost fasterfrom the optical lattice than singly occupied sites. After modulation of the lattice
depth we therefore hold the sample in the lattice and wait for the doublons to decay.
After this hold time we detect the atom number via spin-resolved absorption imaging
after Stern-Gerlach separation of the two spin states and find a resonant loss whenever
the above resonance condition is met.
For dipolar gases there are two contributions to the onsite interaction [117], the50

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.6: Modulation spectroscopy in the optical lattice. (a) We load a band
insulator into a 3D optical lattice and prepare a balanced spin mixture withδs = 0. Modulation of the lattice depth creates double occupancies if theresonance condition is met. Double occupancies have a shorter lifetime andcan be detected as a loss of atoms. (b) Exemplary scan of the modulation
frequency νmod . The dip in the total atom number indicates atom losses dueto the resonant creation of doubly occupied sites. Solid line is a Gaussian fit
to the data.

contact interaction Uc , and the dipole-dipole interaction, Udd , with
Uo
Uc
Udd

= Uc + Udd
∫︂
4πℏa
4
=
dr
⃗ |ϕ(r⃗ )| ,mEr∫︂∫︂
2⃓ ′ ⃓2µ0 µ1 µ22 1 − 3 cos (θ⃗r −r⃗′ ) ⃓
dr⃗ dr⃗ ′ |ϕ(r⃗ )|ϕ(r⃗ )⃓ .
=′ 34π|r⃗ − ⃗r |(5.1)

Here a is the inter-spin s-wave scattering length, that we want to determine from themeasurement. mEr is the mass of erbium, ϕ(r⃗ ) is the onsite Wannier wave function,µ0 is the vacuum permeability, µ1 and µ2 the magnetic moment of the first and second
atom, respectively, and θ⃗r−r⃗′ is the angle between the alignment axis of the dipole withrespect to their interparticle axis (see Sec. 2.2 on dipolar interaction). Both sign andstrength of Udd depend on the shape of the Wannier wave function. Following [117] we
1/4
define the aspect ratio as AR = lz /lxy , with li = di /(πsi ), where di is the harmonic
oscillator length and si the lattice depth in units of the photon recoil energy Ei /Erec .
Due to our lattice geometry and available laser powers in the laboratory, the Wannierfunction has a prolate shape (AR > 1). As illustrated in Fig. 5.7 (a), for dipolesaligned along the z-axis, the dipoles mainly experience a head-to-tail configuration,in which they interact attractively. This causes the dipole-dipole onsite interactionzenergy to be negative, Udd< 0. By rotating the magnetic field into the xy plane, thedipoles mainly align side by side and interact repulsively, which leads to a positivez
dipole-dipole onsite energy Udd> 0. By performing modulation spectroscopy for thosexyz
two configurations, we get the equalities |Uoxy | = |Uc + Udd| and |Uoz | = |Uc + Udd|,
xyz
where Udd and Udd contribute to the total energy with different signs. By comparing
those two energies we can directly extract the amplitude and the sign of Udd and use
the result to determine the shape of the Wannier wave function, with which we finally
extract the scattering length as . Figure 5.7 shows the experimentally obtained values
for the scattering length in the proximity of the observed loss feature. We fit a simple

51

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.7: Feshbach resonance in 167 Er. (a) The dipole interaction energy Udd depends on the aspect ratio (AR) and the alignment of the dipoles. For a fixedAR ̸= 1, the magnitude and sign of Udd can be changed, which allows us to
extract the sign of the scattering length a by comparing interaction energies
for two angles. (b) Scattering length as a function of magnetic field. The solidline is a single-channel fit to the experimental data (see main text). Dashedgrey vertical lines indicate further narrow resonances (cf. Fig. 5.5).

single-channel model to the data [28, 155], taking into account a second resonancearound 480 mG (cf. Fig. 5.4),(︃a(B) = abg 1 −∆∆′−B − B0B − B0′

)︃
,

(5.2)

with the background scattering length abg , the widths ∆ and ∆′ , and the resonancepositions B0 and B0′ . We find a background scattering length of abg = 93(9)a0 , wherea0 is the Bohr radius. The lower resonance is located at B0 = 480(4) mG with a widthof ∆ = 33(6) mG. For the other resonance we extract a position of B0′ = 687(1) mG,
and a slightly broader width of ∆′ = 57(6) mG, consistent with the loss feature in the
Feshbach scan in Fig. 5.4. The grey dashed vertical lines in Fig. 5.7 (b) indicate thefour further narrow resonances, which we observed in Fig. 5.5 and which we do nottake into account for the fit. Assuming a differential magnetic moment of δµ = 3µ0 (aswas measured in bosonic Er2 [156]), we can estimate the order of the effective range
to be r∗ = ℏ/mEr abg δµ∆ ≈ 600 a0 for the resonance at 480 mG and r∗ ≈ 1000 a0
for the resonance at 687 mG. This value of the effective range yields an intermediate
dimensionless parameter of γ ∗ = π8 kF1r∗ ≳ 1 [157].
In a final measurement we investigate the lifetime of the atomic sample acrossthe resonance at 687 mG. Figure 5.8 (a) shows the decay of the atom number overtime for positive scattering lengths of a = 880(140) a0 at a magnetic offset field of
680 mG and for negative scattering length a = −1500(500) a0 at a magnetic offset
field of 690 mG. We fit the data with an exponential decay and extract 1/e-lifetimesof τ+ = 150 ms and τ− = 1200 ms, respectively. The longer lifetime on the repulsive
site of the resonance (a < 0) is a direct consequence of the Pauli principle, which
suppresses three-body losses in a two-component Fermi gas [158]. On the attractive(BEC) site, the existence of a weakly bound molecular state supports three-body
52

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

Figure 5.8: Collisional stability across the resonance. (a) Two exemplary lifetime
measurements below (top) and above (bottom) the resonance for a 50-50 mixture of the mF = −19/2 (blue) and mF = −17/2 (red) spin state. Solid lines
are exponential fits to the data. (b) Initial decay rate across the 687 mG Feshbach resonance as a function of the dimensionless coupling constant 1/(kF a).
Solid line only serves as guide to the eye.

recombination and leads to a faster decay of the atom number. This behavior canalso be seen in Figure 5.8 (b), where we plot the initial decay rate Ṅ /N0 as a functionof the dimensionless coupling constant 1/(kF a). We extract the initial decay rate witha linear fit to the initial decay of the atom number. We find a large collisional stabilityon the repulsive site, where the lifetime can reach a few seconds. On the attractive
site, the initial decay is more than one order of magnitude larger. This asymmetry isan essential feature for the BEC-BCS crossover and hence the investigated resonancemight be a promising candidate to study such physics in systems with anisotropic
long-range interaction.
In this chapter we presented the first realization of a two-component Fermi gas
with dipolar and tunable s-wave interaction. Our preparation scheme does not suffer from inelastic dipolar scattering and can easily be extended to prepare arbitrary
spin compositions [144]. Even though the measured resonances are only of intermediate strength and much more narrow than typical Feshbach resonances in systems
with alkali atoms (in terms of the magnetic field), the presented resonance can beused to freely tune the interaction between the two lowest hyperfine states in 167 Er,|mF = −17/2⟩ and |mF = −19/2⟩. With a magnetic field stability of below 1 mG inthe experiment, we can precisely tune the scattering length to the desired value. Our
results therefore enable the investigation of BEC-BEC crossover physics in dipolar
quantum gases but also motivates the search for further broad resonances at higher
magnetic fields, or between further combinations of spin states. Furthermore suchmeasurements could facilitate the understanding of complex spin dynamics [144] andenable the study of rich magnetic behaviors in systems with long-range interaction
and the search for exotic phases of matter by exploiting the many spin states as

53

CHAPTER 5. A STRONGLY INTERACTING FERMI GAS OF DIPOLAR
ATOMS

synthetic dimensions [159].

54

Chapter 6

Conclusion and Outlook
This part of the thesis presented the results that were obtained within my first year
in Innsbruck at the erbium experiment.
The first project was the investigation of the anisotropic light shift of erbium atoms.
We presented measurements on the anisotropic ac-polarizability of the ground state
and the 583 nm excited state of erbium and compare the experimental results tosemiempirical electronic-structure calculations. Despite the complicated electronic
configuration in erbium and a huge number of optical transitions in the proximity of
the investigated wavelengths we find good agreement between theory and experiment.
There are few investigations in dysprosium experiments [125, 143], however, a similar
direct comparison of experiment and theory is still pending. Our results prove agood understanding of the erbium level structure and a good knowledge of existing
optical transitions beyond the NIST database. Such an understanding facilitates thedesign of new experimental schemes and raises prospects for realizing state-dependent
optical potentials via the application of near-resonant laser light. It also enables stateselective manipulation by using narrow lines for spin-orbit coupling [139] or to realize
synthetic dimensions with the large number of internal spin states[55].
The second project investigated the preparation of a strongly interacting dipolar Fermi
gas. We presented a scheme for the deterministic preparation of two-component spinmixtures in a Fermi gas of erbium atoms. Despite spin non-conserving processes
we measure a high collisional stability and find that the lifetime is independent of
the initial spin-composition. We performed Feshbach loss spectroscopy in the region
from 0 − 2 G, where we detect a range of inter and intra-spin Feshbach resonances.
We map out and characterize one particularly broad inter-spin Feshbach resonancearound 680 mG via modulation spectroscopy in an optical lattice. Our measurement
constitutes the first realization of a two-component dipolar Fermi gas with tunableinteraction and enables the future study of strongly interacting dipolar Fermi gases.

Outlook
The results obtained within this thesis directly enable a range of interesting experiments with dipolar Fermi gases. As shown in a recent experiment in our group [144],

55

CHAPTER 6. CONCLUSION AND OUTLOOK

state-dependent light shifts can be used to quickly control dipolar exchange dynamics
by compensating the quadratic Zeeman shift with a quadratic light shift. This paves
the way to the study of rich magnetic behaviors in dipolar Fermi gases involving ahuge number of internal degrees of freedom.
A further interesting topic is the study of transport properties in optical lattices andthe influence of dipolar interaction [160]. Based on energy conservation, a systeminteracting via nearest-neighbor dipolar interaction is expected to form dynamicallybound nearest-neighbor dimers, which decrease or even completely suppress expansion
and transport in a dipolar system. Fermi gases are particularly promising candidates
to study such effects [161] since optical lattices with unity filling can be readily realized.
Furthermore the ability to tune the short-range interaction and to realize a strongly interacting Fermi gas allows to investigate the effect of dipolar interactions on fermionicpairing and the BEC-BCS crossover. In particular, similar as was done in bosonic
dipolar gases, the study of systems where the interaction can be tuned from purely
dipolar to a regime where it is dominated by s-wave interaction might reveal interesting new phases or dynamics caused by the interplay of short-range isotropic andlong-range anisotropic interactions.

56Part III

Characterizing Fermi Systems
via Momentum-Correlation
Measurements

57

Chapter 7

Experimental Tools in the
Laboratory
With our setup in the Heidelberg experiment we have the ability to deterministically prepare mesoscopic Fermi-Hubbard-like systems by trapping few atoms in anarray of few optical tweezers [94, 162, 163]. By using a single-atom and spin sensitive fluorescence imaging scheme, which we developed in our group over the past few
years [67, 85], we can investigate those few-fermion systems both in real space and in
momentum space. By investigating the system in two conjugate basis (i.e. position
and momentum) we have direct access to coherence properties of the prepared stateand can therefore extract information that is not accessible when only looking in real
space. As we will discuss in the following chapters, we can use this information tocharacterize few-fermion systems and to constrain large parts of their density matrix.
Based on these constraints, we can reconstruct physical density matrices of systems
containing two or three mobile, interacting atoms by performing a Bayesian quantum
state estimation. In this first chapter we present the basic tools that we have available in the laboratory. We start by introducing the main properties of lithium andbriefly discuss how we deterministically prepare few-particle systems in the laboratory. We then briefly discuss which states we can prepare in the laboratory and how
we investigate them using our single-atom sensitive imaging scheme.

7.1

6

Li - Atomic Properties

Lithium is the lightest alkali atom with atomic number Z = 3. It has a single valence
electron and compared to the previously discussed erbium a simple electronic structurein the ground state. There are two stable isotopes which are both commonly used incold atom experiments: bosonic 7 Li with a natural abundance of 92.4% and a nuclear
spin of I = 3/2 and fermionic 6 Li with a natural abundance of 7.6% [164] and a nuclear
spin of I = 1. In our experiment we use the fermionic isotope 6 Li.
Figure 7.1 (a) shows the level structure and the hyperfine splitting of 6 Li in the groundand the first excited state. The electronic ground state (22 S1/2 ) splits into two submanifolds (F = 1/2 and F = 3/2) with a splitting of ∆ = h×228.2 MHz. The 2 P excited59

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.1: Lithium level structure. (a) Level structure and hyperfine splitting of 6 Li
at zero magnetic field (not to scale). (b) Zeeman splitting of the different mF
states for the two experimentally relevant states, i.e. the ground state 22 S1/2
and the upper excited state 22 P3/2 . Note the different x-axes in the top andbottom plot. Figure taken and adapted from [86].

state splits into the two manifolds 2 P1/2 and 2 P3/2 with a fine structure splittingof around ∆F S = h × 10.056 GHz, which in turn split up into five sub-manifolds
with different total angular momenta F . The D1 and D2 lines have wavelengths of
λ1 = 670.992421 nm and λ2 = 670.977338 nm, respectively, with a natural linewidthof Γ = 5.8724 MHz [86]. In our experiments however, we only drive transitions within
the D2 line, i.e. to the 22 P3/2 excited state. At zero magnetic field, the hyperfine
splitting of that state is smaller than the D2 linewidths and can hence not be resolved.
It is interesting to note that compared to other alkali atoms the fine structure splittingof ∆F S = 10.056 GHz of the excited state is very small. Even though this can havetechnical advantages, e.g. for the implementation of grey-molasses cooling [165], it
might also complicate the application of strong Raman couplings within the groundstate [166].
When lithium is subjected to an external magnetic field, the different mF sublevels ofeach F -manifold split up due to the Zeeman effect. Generally this splitting is linear at
small magnetic fields, but once the magnetic interaction energy becomes comparable
to the hyperfine interaction energy, F ceases to be a good quantum number and thestates rearrange in groups according to their mJ value (see Fig.7.1 (b) or [85, 86]
for a more detailed description of the different levels). For the 22 P3/2 excited statethis already happens at very low magnetic field (around 1 Gauss) due to the small
hyperfine constant of the excited state of A22 P3/2 = −1.155 MHz [167]. In contrast,
60

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

for the ground state much larger magnetic fields are necessary to decouple the nuclear
moment from the electronic angular momentum. As described in [67, 85, 168], thishas important consequences for our free-space momentum imaging scheme, due tosmall but finite branching rations for the decay from the excited state into the threehighest states (labeled |4⟩, |5⟩, |6⟩ in Fig. 7.1 (b)) of the ground state.
For our experiments, we use two of the three lowest hyperfine states of the groundstate, labeled |1⟩, |2⟩ and |3⟩, according to their energy at high magnetic field. At highmagnetic field, the splitting between two adjacent states is on the order of 80 MHz
so that they can be individually addressed via the D2 -line. Experiments presented
in this thesis are all done with a |1⟩-|3⟩ mixture, because (i) state |3⟩ is the only
state that features a completely closed optical transition [168], which is beneficial for
imaging, and (ii) to maximize the energy difference between the two states in orderto suppress off-resonant scattering during the imaging process.

7.2

Deterministic Preparation of Few-Fermion Systems

In order to deterministically prepare few-fermion systems it is essential to prepare
atomic samples with very low temperature and entropy. This section outlines the main
steps of the experimental procedure. A more detailed description of the experimental
setup and procedure can be found e.g. in [84, 85, 94, 169]. For the setup of thevacuum chamber, see Chapter 2 or given references.
Optical Setup
As described in the first part of the thesis, cooling an atomic sample requires bothresonant and off-resonant laser light. For dissipative cooling in the Zeeman slower
and the MOT we derive near-resonant laser light at λres = 671 nm from a Toptica
tapered amplifier. We employ two further Toptica DL100 external cavity diode lasers
at the same wavelength for absorption and fluorescence imaging. All visible lasers arebeat-offset locked to a reference laser which in turn is locked to the D2 line (to the
F = 3/2 → 22 P3/2 transition) via modulation transfer spectroscopy. Fast frequency
jumps during the imaging process (e.g. for state-resolved imaging) are achieved bycontrolling the DC modulation input of the DL100 with a Red Pitaya [94].
For experiments in this thesis, we use two kinds of dipole traps. Evaporation is performed in a crossed-beam optical dipole trap. Laser light for this trap is derived from a1064 nm, 100 W Yb-doped fiber amplifier, YLR-200-LP-WC, IPG Photonics. The deterministic preparation (see below for details) is done with optical tweezers, for which
we use light from a 1064 nm, 1 W Innolight Mephisto laser. The tweezer array is realized with an acousto optical deflector (AOD) [170], operated at low radio-frequency(rf) powers to keep nonlinearities in the crystal as small as possible. Generation ofmultiple tweezers is achieved by sending multiple rf tones to the AOD. We focus
the tweezer light through our high-resolution objective [171], sitting in the reentrant
viewport on top of our main chamber. The objective has a design numerical aperture61

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.2: Experimental sequence for deterministic preparation of few-fermion
systems. The top row shows the different blocks of the full experimental
sequence, including the initial MOT phase, followed by the evaporation andthe experimental part. Below the top row there is a zoom-in on the preparation, manipulation and imaging sections. We also schematically illustrate
the sequence for the tweezers, the magnetic offset field and the magnetic field
gradient during the experimental part.

of N A = 0.6 with an an effective focal length of 20.3 mm. We use the same objective
for fluorescence imaging as will be described in Sec. 7.4.
Experimental Sequence
The experiment is controlled by an ADwin Pro II from Jäger Messtechnik, using 2×32
DIO channels, 16 analog out and 8 analog input channels. Digital channels can be
set with 0.5 µs time steps, analog channels have an update rate of 400 kHz, which
is fast enough for the implementation of real-time PIDs for active stabilization ofexperimental parameters. Figure 7.2 shows a schematic of the experimental sequence.
The experiment starts with the loading of a magneto-optical trap (MOT), in which
typically 107 atoms are captured. At the end of the MOT phase, all atoms get pumped
into the F = 1/2 state. We then compress the MOT and transfer it into a crossed-beam
optical dipole trap (ODT), where the ramping on of the offset magnetic field producesa spin-imbalanced mixture of atoms in states |1⟩ and |2⟩. Spin balance is establishedby a high-power radio-frequency pulse. Atoms are eventually transferred from state |2⟩
to |3⟩ by an adiabatic Landau-Zener passage. After a short phase of plain evaporation,
the power of the ODT is lowered to force evaporation either at low (300 G) or high(685 G for |1⟩ − |3⟩) magnetic field (refer to Sec. 2.2 for scattering lengths). At theend of the evaporation we typically have 40.000 atoms per spin state at temperaturesof below 0.5 TF . We then ramp on one or, depending on the experiment, multiple
tweezers at the same time, that can be individually controlled by controlling theradio-frequency tones that are sent into the AOD. As illustrated in Fig. 7.3, once thetweezer is filled with atoms, we first switch off the crossed-beam ODT and only retain
a completely filled tweezer. At that stage there are only a few hundreds of atoms left.
We deterministically reduce the atom number to a few atoms by applying a magnetic

62

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.3: Deterministic preparation of few fermions. (a) After the evaporationin our large crossed-beam optical dipole trap, we transfer the atoms into a
dimple trap, created by a tightly focussed laser beam. (b) We switch off thelarge reservoir trap and retain a few hundred atoms in the dimple trap. (c) We
lower the optical potential and apply a magnetic field gradient to deterministically spill atoms above a certain trap level out of the trap. (d) We obtain adeterministic atom number and achieve fidelities on the order of 96% per atom.

field gradient along the longitudinal axis of the tweezer and by subsequently lowering
the optical potential. As indicated in the figure this procedure spills out all atoms
above a certain trap level [162]. The preparation of single atoms works with a fidelityof about 96 %. Depending on the experiment, we either separately prepare multiple
tweezers with single atoms or prepare a fixed number of atoms in the first tweezer andinitialize the final system with an adiabatic passage to the total ground state (see e.g.
[163]). After the final state has been prepared, we eventually induce interactions bytuning the magnetic field across the Feshbach resonance. We finally investigate thestate by two different means: fluorescence imaging in a microMOT (only atom-number
resolved) or free-space single-atom and spin resolved imaging.
MicroMOT
A first and very robust method to investigate a few-atom state is the microMOT. In
order to count the total atom number after the experimental sequence, we recapture
the atoms from the tweezers into our MOT and collect the fluorescence signal on
a CCD camera for one second [162]. To get a good signal-to-noise ratio on thecamera, we compress the MOT as much as possible by using a much higher magnetic
field gradient and less detuning than we use in the initial MOT, hence the namemicroMOT. With this method we can count the total atom number up to about
15 atoms with very high fidelity. In combination with the above described spilling
technique, the microMOT is well suited to perform spectroscopic measurements [172–174]. While the microMOT cannot provide any information about the momenta of
the atoms it can in principle be applied to extract information beyond the total atom
number. We can e.g. measure the spin and site-resolved populations by removing
all atoms of one spin state and switching off all wells except one prior to imaging.
However, such sequences can get very tedious so that we developed a complementary
imaging scheme that gives access to both real and momentum-space spin-resolved
single-particle positions.

63

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Free-Space Imaging
In order to extract more information from the state than just the total atom number, we employ a single-atom and spin-resolved free-space imaging scheme [67, 85].
Briefly explained, we illuminate atoms with resonant laser light for 20 µs and collecton average around 20 fluorescence photons per atom with our high-NA objective onto
an EMCCD camera. Successively taking two images of different spin states also provides spin-resolution. In particular, this imaging scheme works without any confining
potentials so that also the momenta of single atoms can be accessed. The results presented later in the thesis were obtained with this imaging scheme. We will therefor
give a more detailed description in Sec. 7.4. However, for all technical details, pleaserefer to [67, 85, 168].

7.3

Fermi-Hubbard Model

A system consisting of few fermions trapped in an array of optical tweezers can be
described by the Fermi-Hubbard model [163]. This model was initially used to describeinteracting fermions in flat energy bands [175] but is readily applied to ultracold atoms
in optical lattices [176, 177]. Despite its simplicity, the Fermi-Hubbard (FH) model
still raises open questions and is seen as a promising candidate to explore physics
related to high-TC superconductivity [178]. This section discusses the basic aspectsof the FH model, that are relevant for our experiments and discusses the states that
we can realize in the laboratory.
The Hubbard model is a simple model to describe a system in a periodic potential
with nearest-neighbor hopping and onsite interaction. It only considers the lowest
energy band and furthermore assumes that different spatial modes are orthogonal toeach other. For a two-component Fermi gas in a periodic potential, we can write theFermi-Hubbard Hamiltonian as
Ĥ F H = −J)︂∑︂ (︂ †
∑︂∑︂ĉiσ ĉjσ + ĉ†jσ ĉiσ + Uµi (ni↑ + ni↓ ),n̂i↑ n̂i↓ +i⟨i,j⟩,σ

⏞

⏟⏟

Hkin

⏞

⏞

(7.1)

i

⏟⏟

Hint
(†)

⏞

⏞

⏟⏟

Hext

⏞

with fermionic annihilation (creation) operators ciσ on site i with spin σ and number
operators n̂iσ = c†iσ ciσ . Figure 7.4 (a) schematically illustrates the different termsof this Hamiltonian. The kinetic term Ĥ kin describes hopping of fermions between
neighboring sites ⟨i, j⟩ with hopping element J. The interaction term Ĥ int describes
the interaction of two fermions with opposite spin on the same lattice site, contributing
the onsite interaction energy U . Finally the last term takes into account and external
confinement or an energy offset µi on individual sites of the lattice, as illustrated
in Fig. 7.4 (b) for the case of a double-well potential. In the illustrated case of adetuned double well, we have µ1 = −µ2 = µ and a doubly-occupied left well, where
the repulsive interaction between the two opposite spins contributes the energy U to
the total energy of the system.
For small systems, it is instructive to explicitly write out the FH Hamiltonian. For

64

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.4: Fermi-Hubbard model. (a) Illustration of the FH model for a twocomponent Fermi gas on a two-dimensional optical lattice. The evolution of thesystem is governed by only two parameters, the tunnelling element J and theonsite interaction U . (b) Two interacting atoms in an imbalanced double-wellpotential with µ1 = −µ2 .

later purposes we discuss the cases of two atoms in a double well and three atoms ina triple well.
2 Atoms in a Double Well
For the case of two identical atoms, i.e. if the two atoms have the same spin, there is
only one possible state within the Fermi-Hubbard model. In the notation |·, ·⟩, where
the first entry is the spin population in the left well and the second entry the spin
population in the right well, the state of two identical atoms reads as |↑, ↑⟩. In thatstate there is exactly one fermion per site since the two atoms cannot occupy the same
well due to the Pauli-exclusion principle. Hence, the Hamiltonian is simply Ĥ = 1,
since tunneling is forbidden (J = 0) and there is no onsite interaction U .
The case of two interacting atoms with different spin (labeled |↑⟩ and |↓⟩) is illustrated
in Fig. 7.4 (b). With the same notation as above, we can identify four basis states as|Ψ⟩∈{|↑↓, ·⟩ ; |↑, ↓⟩ ; |↓, ↑⟩ ; |·, ↑↓⟩} .

(7.2)

An alternative and useful representation of the same states (that we will use later) is
what we will call the particle representation. Instead of giving the population of thefirst or second well, the first or the second entry in |·, ·⟩ indicate the well in which thespin-up, or the spin-down particle are located, respectively. In this representation we
can rewrite the above states as|Ψ⟩∈{|LL⟩ ; |LR⟩ ; |RL⟩ ; |RR⟩} .

(7.3)

The first and the last state are doubly occupied sites and therefore come along withan energy penalty of U . The occupation of those states is hence favored (suppressed)
for attractive (repulsive) interaction. There is no energy penalty associated with thetwo other states, but they are connected to the doubly occupied states via a single
tunnelling event J. We can write the Hamiltonian in its matrix representation, using65CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.5: Spectrum of balanced double well. The top panel shows the coefficientsof the four basis states in the ground state, see Eq. (7.5). The lower plot shows
the four eigenenergies of the balanced double well as function of the interaction
U/J. The attractively interacting ground state (blue solid line) approaches thestate with the superposition of two doubly occupied sites. On the repulsive
site, each atom becomes localized on an individual site.

Hij = ⟨Ψi | Ĥ |Ψj ⟩, as⎛
U + 2µ1⎜ −J⎜
Ĥ DW = ⎜
⎝ −J
0

−J
0
0
−J

⎞
0
−J ⎟
⎟
⎟.
−J ⎠
U + 2µ2

−J
0
0
−J

(7.4)

We can analytically diagonalize this Hamiltonian to find the four eigenstates and theassociated eigenenergies, which we plot in Fig. 7.5 for the balanced double well, i.e.
for µi = 0. In our experiments we are primarily interested in the ground state |Ψgs ⟩,which is highlighted in the figure.
The ground state |Ψgs ⟩ is explicitly given by1|Ψgs ⟩ =

√︁

with

U
NU J =+
4J

2 + 2NU2 J(|LL⟩ + |RR⟩ + NU J (|LR⟩ + |RL⟩))
√︄

(︃
1+

U
4J

(7.5)

)︃2
.

(7.6)

The prefactors of both the doubly and singly occupied states are plotted in the upper
panel of Fig. 7.5. The population of doubly occupied sites is favored (suppressed) for
attractive (repulsive) interaction and the coefficients already saturate at an interactionof U/J ≈ ±8. To gain some understanding of the states, let’s take a look at the threelimiting cases of U = 0, U/J → ∞, and U/J → −∞. In those cases, the state from66

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Eq. (7.5) takes the form
|Ψ0 ⟩1(|LL⟩ + |RR⟩ + |LR⟩ + |RL⟩)
2
1
1√ (|L⟩ + |R⟩) ⊗ √ (|L⟩ + |R⟩)
2
2
1
√ (|LR⟩ + |RL⟩)
2
1√ (|LL⟩ + |RR⟩) .
2

=
=

|Ψ∞ ⟩

→

|Ψ−∞ ⟩

→

(7.7)
(7.8)
(7.9)

At U = 0 the state factorizes into a product of the two single-particle states, whereas
the strongly interacting states are maximally entangled Bell states. For later purposes we explicitely write out the density matrices of the three states, which can becalculated by ρ = |Ψ⟩ ⟨Ψ|. In the basis from Eq. (7.3) the density matrices are given
by

⎛

1
⎜ 0
1
⎜
ρ−∞ = ⎜
2⎝ 0
1

0
0
0
0

0
0
0
0

⎞
⎛
1
1
⎜ 1
0 ⎟
1
⎟ 0
⎜
⎟,ρ = ⎜
4⎝ 1
0 ⎠
1

1

1
1
1
1

1
1
1
1

⎞
⎛
1
0
⎜ 0
1 ⎟
1
⎟ ∞
⎜
⎟,ρ = ⎜
2⎝ 0
1 ⎠
1

0

0
1
1
0

0
1
1
0

⎞
0
0 ⎟
⎟
⎟.
0 ⎠
0
(7.10)

3 Atoms in a Triple Well
In the trivial case of three identical atoms, there is again only a single state available,|↑, ↑, ↑⟩. We therefore consider the configuration with two spin-up and one spin-down
atom (or vice versa). In that case, we identify the 9 following states as basis states,|Ψ⟩∈{|↑, ↑, ↓⟩ ; |↑, ↓, ↑⟩ ; |↓, ↑, ↑⟩ ;|↑↓, ↑, ·⟩ ; |↑↓, ·, ↑⟩ ;|↑, ↑↓, ·⟩ ; |·, ↑↓, ↑⟩ ;|↑, ·, ↑↓⟩ ; |·, ↑, ↑↓⟩},(7.11)

where similar to the double well, |·, ·, ·⟩ describes the occupation of the first, the second
and the third well, respectively. Alternatively, we can again write these states in theparticle representation as|Ψ⟩∈{|L, C, R⟩ ; |L, R, C⟩ ; |C, R, L⟩ ;|L, C, L⟩ ; |L, R, C⟩ ;|L, C, C⟩ ; |C, R, C⟩ ;|L, R, R⟩ ; |C, R, R⟩},

(7.12)

where the first two entries describe the spatial mode (left (L), center (C) and right (R)well) of the two spin-up particles and the third entry the spatial mode of the spin-down

67CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

particle. Note that this representation identifies individual particles as subsystemsof the state and is therefore not unique, since e.g. the state |L, C, R⟩ describes thesame spin-configuration as the state |C, L, R⟩. Due to the indistinguishability of thetwo spin-up atoms special care must be taken to ensure a proper antisymmetrizationof the state. The subtleties of treating individual particles as systems rather thanspatial modes will be discussed in Chapter 10.
In the above basis, the Hamiltonian of the balanced triple well (i.e. µi = 0) in its
matrix representation takes the form
⎛
0
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Ĥ TW = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0
0

h.c.

0
0
0

0
0
0
U

0
−J12
−J12
−J23
U

−J23
−J23
0
−J12
0
U

0
−J12
−J12
0
0
0
U

−J23
−J23
0
0
0
0
0
U

⎞
0
0 ⎟
⎟
⎟
0 ⎟
⎟
0 ⎟
⎟
⎟
0 ⎟,
⎟
0 ⎟
⎟
−J23 ⎟⎟⎟−J12 ⎠(7.13)

U
where J12 is the tunnel coupling between the left and the center well and J23 thetunnel coupling between the center and the right well.
Figure 7.6 shows the nine eigenenergies of this Hamiltonian for the homogeneous triple
well (J12 = J23 = J) as a function of the interaction strength U/J. On the repulsive
site, U > 0, the three lowest states are only separated by an energy on the orderof the super-exchange energy 4J 2 /U , which vanishes for strong interactions. This
energy scale will become important in Chap. 9 when we discuss the experimentallyreconstructed density matrix of the ground state of the homogeneous triple well.
Also for the triple well we are primarily interested in the ground state, since we will
discuss its entanglement properties in Chap. 10. Similar to the double well, doubly
occupied sites are favored for attractive interaction, whereas they are suppressed in thestrongly repulsively interacting ground state. However, on the attractive (repulsive)
site, the spin-down particle (the doublon) delocalizes across the triple well, so thatfor U/J → ±∞ the ground state takes the following form,|Ψ∞ ⟩ =
|Ψ−∞ ⟩ =1
2
1√ |↑, ↑, ↓⟩ + √ |↑, ↓, ↑⟩ + √ |↓, ↑, ↑⟩
6
6
6
1
1
1
|↑↓, ↑, ·⟩ + √ |↑↓, ·, ↑⟩ + √ |↑, ↑↓, ·⟩ +
2
8
8
1
1
1
√ |·, ↑↓, ↑⟩ + √ |↑, ·, ↑↓⟩ + |·, ↑, ↑↓⟩ .
2
88(7.14)(7.15)

Using the experimental techniques that we discussed above, we can deterministically initialize such FH-like states. To verify the successful preparation of thosestates, it is necessary to fully characterize their quantum state. Since experimentallyrealized states usually suffer from technical noise, it is particularly relevant to access
68CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.6: Spectrum of homogeneous triple well. The plot shows the nine eigenenergies of the homogeneous triple well (J12 = J23 = J) as a function of theinteraction strength U/J. The blue solid line is the energy of the ground state,
which favors doubly occupied sites on the attractive site and suppresses themon the repulsive site.

coherence properties of the state. As mentioned before, coherences can be accessedby investigating the system in two conjugate bases, such as in real and in momentum
space. In our experiment this is achieved with our spin-resolved free-space, single
atom imaging, which will be presented in the next section.

7.4

Spin-resolved single-atom imaging

Apart from the deterministic preparation of small systems in an array of opticaltweezers, the key feature of our experiment is the possibility to image single atoms
in a spin-resolved way. In particular, there is no need for confining potentials during
the imaging process and we can readily apply the scheme after time-of-flight to detect the momenta of single particles. Our imaging technique hence provides access toboth the single-atom resolved in-situ and momentum distribution of the system. This
gives access to coherence properties by sampling the momentum wave-function of individual particles or by calculating momentum correlation functions of few-fermion
systems. In this section the working principle of our imaging scheme is described andthe measurement of single-particle coherences is presented to exemplify the usefulnessof accessing single particle momenta. A detailed characterization of the performanceof the imaging scheme can be found in [67, 85, 168].
To image a single atom, we apply two counter-propagating, horizontally polarized, resonant laser beams in an alternating scheme, see Fig. 7.7 (a). The alternating
scheme is chosen to avoid interference between the two imaging beams. We pulse eachbeam with a periodicity of 200 ns and a duty cycle of 40 % for a total imaging time of
20 µs and focus the scattered light onto an Electron Multiplying CCD-camera (EMCCD, ANDOR iXon DV887, back illuminated). At an intensity of s0 = I/Isat ≈ 8.5

69

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.7: Single-atom imaging. (a) Resonant pulses of laser light are successively
applied to the atom from both sides in an alternating scheme and the scatteredphotons are collected with our high-NA objective. Driving σ+ transitions andmaking use of the dipolar radiation pattern, we focus on average 20-30 photon
per atom on an EMCCD camera. (b) Histogram of single-atom fluorescencesignal (see text). The grey vertical line indicates the cutoff above which we
identify an event as an atom.

and a natural linewidth of γ = 2π × 5.8724 MHz, the atomic scattering rate on res−1
onance is Γsc = (1 + s−1
γ/2 = 0.45γ. Hence an atom scatters in total around0 )330 photons during an illumination time of 20 µs. The horizontally polarized imaging
light is polarized perpendicular to the magnetic field and can therefore only drive σ+or σ− transitions. Tuning the laser to resonance we selectively drive the σ+ transition
into the lowest branch of the excited state (mJ = −3/2) to suppress radiative decay
from the excited state into other hyperfine states of the ground state [168]. With thisscheme, state |3⟩ features a completely closed optical trantision. In order to close theoptical transitions of states |1⟩ and |2⟩ we add a second laser frequency to the imaging
light (see [67, 85] for exact transitions). Taking advantage of the anisotropic dipolar radiation pattern for σ-transitions, we estimate that about 10% of the scatteredphotons get detected by the camera. On average we hence only detect around 20-30
photons per atom. In this low photon regime it is essential to use an EMCCD-camera,
operated in photon-counting mode. In this mode (EM mode), a single electron on thecamera sensor gets stochastically amplified in an extra gain register before being readout. This causes the signal from a single electron to be well above the electronic readnoise, enabling the detection of single photons. However, this stochastic amplification process makes it impossible to clearly distinguish a single photon from multiple
photons on the same pixel [94]. This is why the magnification of the imaging system
has to be chosen such that the signal from a single atom gets distributed over several
pixels.
We process the images with three simple steps. The individual steps are exemplarily illustrated in Fig. 7.8. First of all, to get rid of electronic noise, the raw image
that we get from the camera is binarized, i.e. we introduce a cutoff, above which
we identify the count on a single pixel as an electron (see [85, 94]) and convert the
raw image into an image containing only zeros (no electron) or ones (at least one
electron). These electrons can be caused both by light or electronic noise. We apply
a Gaussian low-pass filter (LP) to the binary image to filter out background noise,
which appears with lower spatial frequencies than the actual atomic signal. We thensearch for local maxima in the lowpass-filtered images and plot the distribution of the70

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.8: Image processing. In EM mode the camera introduces counting noise dueto the stochastic amplification of electrons on the sensor. We therefore need toprocess the images with three simple steps. The raw image from the camera(RAW) is first binarized using a threshold value above which a pixel is identified
with at least one electron (Binary). A Gaussian lowpass filter (LP) smoothes
the background noise which appears with a different spatial frequency than theatomic signal. Local maxima above a threshold are identified as positions of
single atoms (10 atoms in the shown example).

local maxima in a histogram. An example of such a histogram is shown in Fig. 7.7 (b).
This histogram shows a clear bimodal distribution with a peak at very low amplitude,
corresponding to the maxima in the background noise, and a distinct peak at higher
amplitude, caused by the atomic signal. Based on that histrogram we define a cutoff
above which we identify a single atom, see grey line in Fig. 7.7 (b). We choose thiscutoff such that the amount of false positive and false negative events are identical.
Obviously the identification of a single atom depends on the chosen size of the regionin which one searches for an atom, since the number of false positive events scales linearly with the size of the region of interest (ROI). For very small ROIs, e.g. imaging
a single atom in a well localized optical tweezer, the identification of a single atom
works with a fidelity of over 99 % [67]. For larger ROIs, e.g. after expansion of theatoms in a wave guide potential, we hence have to increase the cutoff for identifying
an atom at the cost of detection efficiency.
The performance of our imaging scheme is mainly limited by the two following
things:• Spurious charges (such as residual background light or clock-induced charges(CICs)) on the sensor set a lower bound on the signal above which an atom canbe identified. Reducing the number of CICs is therefore absolutely crucial to
improve the imaging performance. We currently plan to do this by replacing
the current Andor EMCCD camera with a Nuvu camera, which has proven tohave significantly lower CICs at similar read-out settings [179].
• Diffusive motion of the atoms during the imaging process due to scattering ofphotons limits the spatial resolution. This effect is particularly bad for lithium
because of its small mass and limits our resolution to about 30 µm. Note that
this is substantially larger than our in-situ well spacing, which is typically on
the order of a few µm so that we have to employ a small trick to measure thesingle-particle resolved insitu population, as we explain below. However, thelarge resolution typically does not matter for imaging after time-of-flight, since71CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

we typically expand the atoms in our dipole trap by several hundreds of µm.
The diffusive motion is expected to be almost fully suppressed for heavier atoms
such as rubidium or erbium.
Spin Resolution
Spin resolution is implemented by successively taking two images of the two different
spin states. To this end we make use of the fast-kinetics mode of our camera, whichallows us to take two images within a few µs. For the experiments with the double
and triple well, where we expand the atoms in our ODT, the fast kinetics mode shifts
the first image by 40 pixels within 20 µs so that the atomic signal gets shifted outof the imaging region. At the same time, we quickly ramp the laser frequency of
the imaging laser to be resonant to the second spin component and briefly wait for
the laser frequency to settle. We subsequently image the second spin state and thenread out the whole camera chip. The time between the beginning of the first and the
beginning of the second image is hence only 50 µs and therefore negligible on the timescale of a few milliseconds, during which we expand the atoms in the ODT.
In-Situ Imaging
As already mentioned above, the diffusive motion of the atoms during the imaging
process effectively enlarges the point spread function of the imaging system to around
30 µm. Since this is much larger than typical distances between adjacent tweezers, wehave to employ a small trick to resolve the site-resolved insitu density. As we illustrate
in Fig. 7.9 (a) we first project the prepared state by instantly increasing the depth of
the individual tweezers and thus abruptly switching off the tunnel coupling between
the wells. We then map each well to a distinct momentum by a sudden displacementof each tweezer and a evolution in the respective tweezer by a quarter of the trap
period, Ttweezer /4. During the evolution in the displaced tweezer, an atom in well
i is mapped onto the momentum ki . We switch off the optical tweezers and let theatoms evolve freely in our crossed-beam ODT, acting as a one-dimensional waveguide.
We then detect the atoms after time-of-flight. Fig. 7.9 (b) shows the sampled in-situ
distribution in the triple well with exactly this scheme: we map the two outer wells
to ±k and keep the center well switched on during imaging. The figure shows theaveraged signal after about 1000 realizations. The scheme can in principle also be
applied to larger systems by mapping different wells to different momenta
Momentum-Space Imaging
Following the ideas of [180], we access the single-particle momentum distribution by
expanding the system in our crossed-beam optical dipole trap (ODT) for a quarterof the trap frequency TODT /4. Our tweezers are aligned along the long axis of the
ODT with a small residual angle that we estimate to be on the order of 1 ◦ . Afterthe evolution in the ODT, the initial momentum of the atoms is mapped onto theposition and vice versa. Longitudinal trap frequencies of the ODT are typically on
the order of 40 Hz so that a typical evolution before detection is on the order of 6 ms.
Note that the expansion in the ODT only provides the 1D momentum along the long72

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.9: In-situ imaging. (a) To measure the in-situ distribution in an array of tweezers we have to project each tweezer onto a distinct momentum and subsequently
apply our momentum imaging technique. We do this mapping by instantly
displacing each tweezer, followed by the in-trap evolution of the atomic wavefunction for a quarter of the trap period T /4. We then let the atoms expand
in a wave guide potential and detect them after time-of-flight. (b) The plotshows the sampled in-situ distribution of a triple-well potential, averaged overaround 1000 repetitions.

axis of the ODT. For the measurement of a 2D momentum, an expansion in a 2D
confining potential is necessary.
To illustrate the usefulness of our imaging scheme, let’s discuss the example of asingle atom in the ground state of a double-well potential. By sampling the singleatom momentum distribution we can measure the coherence of the single-particle
wave function.

7.4.1

Single-Particle Coherences

Examining a quantum state only in one basis does not give access to its coherence
properties. Accessing off-diagonal entries of the density matrix requires a complementary measurement in second basis. By investigating the system in real and momentum
space, we have access to off-diagonal entries of the density matrix. We illustrate thisby detecting the coherence of a single-particle superposition state.
Suppose we have created a single atom in a coherent superposition state, such as the
single-particle ground state of a balanced double-well potential, where the atom is in
the superposition of being either in the left or the right well,1|Ψ⟩ = √ (|L⟩ + |R⟩) .
2(7.16)

Repeatedly measuring the in-situ density results in finding the atom in |L⟩ for 50%of the measurements and in |R⟩ for the remaining 50% of the measurements. An
incoherent mixture of an atom in |L⟩ or |R⟩ would yield exactly the same result.
Based on the in-situ measurement it is hence impossible to decide, whether the atom
is in a coherent superposition-state of the two spatial modes, ρc , or in the fully mixed
state, ρm . We can therefore not distinguish between the two states described by the73CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.10: Single-particle coherences. Measurements in a conjugate basis provide
additional information about the coherence of the quantum state. The plotsshow sampled real and momentum space densities of a single atom, initiallyprepared in the ground state of a balanced double-well potential. The contrastof the interference fringes in the momentum density is directly related to thepurity of the initial state.

density matrices
(︄
c

ρ =

1/2

1/2

1/2

1/2

)︄

(︄
or ρ

m

=

1/2

0

0
1/2

)︄
,

(7.17)or between any state between those, ρ = rρc + (1 − r)ρm with r ∈ [0, 1]. Having also
access to the momentum of the atom allows to investigate the system in the conjugate
basis and to directly measure the off-diagonal entries of the density matrix. As explained above, we switch off the double-well potential at t = 0 and let the atom freely
evolve in our crossed-beam ODT. After a quarter of the trap period t = TODT /4 theinitial momentum of the atom is mapped onto its position [180]. During this evolution the two components of the coherent superposition state start to overlap andinterfere, showing density modulations in the far field (in analogy to Young’s double
slit experiment). The contrast of these interference fringes is directly related to thecoherence and hence the purity of the initial state. If the initial state was the incoherent mixture ρm , no interference fringes in the momentum density would form,
whereas the coherent superposition state ρc shows full-contrast modulations.
Figure 7.10 shows the measurement of the real space and momentum space density ofa single atom prepared in the ground state of a double-well potential, a superposition
state as given by Eq. (7.16). The state was repeatedly initialized and either probed inreal or momentum space. The in-situ measurement shows that the atom is detectedin the left and right well with equal probability. The sampled momentum distribution shows strong modulations, a signature of the coherent nature of the state. The
periodicity of the momentum-space modulation is given by the real space distance a
between the two wells, klat = π/a74

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

7.5

Momentum Correlation Functions

In the previous section we discussed how we can use our imaging scheme to extract
coherence properties of a single-particle state, which allows us to fully determine
the single-particle density matrix. However, the full characterization of a many-bodyquantum state necessarily also involves the analysis of many-particle observables. Obviously the atomic real or momentum space densities ⟨nx ⟩ or ⟨nk ⟩, respectively, aresingle-particle observables and therefore do not contain any information beyond thesingle-particle level. In order to gain information about the many-body nature of astate, one possible approach is to look at correlation functions. A lot of pioneering
work has already been done in this direction such as the characterization of the superfluid to Mott-insulator transition [181] or the observation of pair-correlated fermions[182] via noise correlations in the atomic density or the description of the system via
higher-order correlation functions [72, 73, 183, 184].
The tools that we have in our laboratory are ideal to study possible prospects for thecharacterization of such systems via the measurement of correlation functions both inreal and in momentum space. Having access to the single-particle resolved real-spaceand momentum-space density gives us the possibility to investigate a system in termsof higher-order correlation functions, which is equivalent to solving the many-bodyproblem step by step [72]. In a strongly correlated system, each order of correlation
function might contain new information that is not accessible via the lower order correlation functions and might therefore indicate interesting physics. We therefore now
briefly introduce the basic concepts of measuring and analyzing correlation functions,in particular by distinguishing connected and disconnected correlations.
The nth order correlation function of an arbitrary operator Ô(x) is defined via
ξ (n) (x⃗ ) = ⟨Ô(x1 )Ô(x2 )...Ô(xN )⟩,(7.18)

where ⟨A⟩ denotes the mean over the operator A. Experimentally the mean is typically realized by either averaging over many experimental realizations or by averaging
over a large ensemble of atoms. Detecting a non-zero nth order correlation function
ξ (n) does not necessarily imply that the system features true nth order correlations,since lower-order correlations can also cause structure in the nth order correlation
function. To investigate true higher-order correlations one therefore has to subtract
all lower-order contributions. This is done by decomposing the correlation function
into two parts, the connected and the disconnected correlation function,(n)(n)ξ (n) = ξcon
+ ξdis .

(7.19)(n)

The disconnected correlation function ξdis contains all lower-order correlations up(n)to order n − 1. It is hence the connected correlation function ξcon , that contains(n)information about true nth order correlations. ξdis can be calculated as a sum over
all lower-order correlation functions, weighted with combinatorial factors that can be75

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

constructed by [72]
⟨︄
(n)
ξcon
=

∑︂
π

(−1)|π|−1 (|π| − 1)!
⏞
⏟⏟
⏞
A(π)

⟩︄

∏︂

∏︂

B∈π

i∈B

Ô(xi ) .

(7.20)The sum runs over all possible partitions π of {1, 2, ...n}, the first product over all
blocks B in π and the second product over all elements i in B. To better understandthe formula, we explicitly give two examples for n = 2 and n = 3.
Example for n = 2
1. n = 2 → There are two partition: π1 = {1, 2} and π2 = {1}{2}2. |π1 | = 1, |π2 | = 2
3. A(π1 ) = 1, A(π2 ) = −1
4. There is one block in π1 and two blocks in π2 , each with one element(2)

5. ξcon = ⟨Ô(x1 )Ô(x2 )⟩ − ⟨Ô(x1 )⟩⟨Ô(x2 )⟩
⏞
⏟⏟
⏞
ξdis

Example for n = 3
1. n = 3 → There are five partitions:{π1 ; π2 ; π3 ; π4 ; π5 } = {{1, 2, 3}; {1, 2}{3}; {2, 3}{1}; {3, 1}{2}; {1}{2}{3}}2. |π1 | = 1, |π2 | = 2, |π3 | = 2, |π4 | = 2, |π5 | = 3
3. A(π1 ) = 1, A(π2 ) = A(π3 ) = A(π4 ) = −1, A(π5 ) = 2
4. There is one block in π1 , two blocks in π2,3,4 and three blocks in π5
5. There are three elements in the block in π1 , two elements in the first block andone element in the second block in π2,3,4 , and one element in each block in π5 .
(3)

6. ξcon = ⟨Ô(x1 )Ô(x2 )Ô(x3 )⟩⎫−⟨Ô(x1 )Ô(x2 )⟩⟨Ô(x3 )⟩ ⎪⎪⎪−⟨Ô(x2 )Ô(x3 )⟩⟨Ô(x1 )⟩ ⎬(3)
− ξdis−⟨Ô(x3 )Ô(x1 )⟩⟨Ô(x2 )⟩ ⎪⎪⎪⎭+2⟨Ô(x1 )⟩⟨Ô(x2 )⟩⟨Ô(x3 )⟩As will become clear in Chap. 8, the choice of the operator influences whether there
are higher-order correlations or not and that even systems with pairwise exchange
statistics can exhibit higher-order momentum-density correlations.
In order to illustrate the concept of connected and disconnected correlation functions, we look at the state of two non-interacting fermions in a balanced double-wellpotential, that we theoretically already discussed in Sec. 7.3,
1
1
|Ψ0 ⟩ = √ (|L⟩ + |R⟩) ⊗ √ (|L⟩ + |R⟩).
2
2
76(7.21)

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

Figure 7.11: Connected and disconnected correlation functions. (a) Measured
first and second-order correlation functions ξ (1) (k) and ξ (2) (k) of two noninteracting atoms in the ground state of a double-well potential. (b) Dis(2)connected correlations function ξdis (k), computed from measured first-order(2)
correlation function ξ (1) (k). (c) Connected correlation function ξcon (k), com(2)(2)
puted as the difference between the measured ξ (k) (a) and ξdis (k). The
connected correlation function vanishes since all correlations are purely causedby single-particle coherences.

As described in detail in [62, 85], we initialize this state in the laboratory as the noninteracting ground state of a tunnel-coupled double-well potential with well spacinga. We measure the spin-resolved single particle momenta k↑ and k↓ by expanding thesystem in our ODT for a quarter of the trap period, ttof = TODT /4 before imaging.
We postselect the data set to the correct atom number, N↑ = N↓ = 1 and calcu(1)(1)
late both the average momentum distributions ξ(↑) (k) = ⟨nk↑ ⟩ and ξ(↓) (k) = ⟨nk↓ ⟩(2)and the second-order momentum correlation function ξ(↑↓) (k) = ⟨nk↑ nk↓ ⟩.

The

(1)
plots in Fig. 7.11 (a) show the measured momentum densities ξ(↑/↓) (k) together with(2)
the computed momentum correlation function ξ(↑↓) (k). As explained in Sec. 7.4.1,(1)
ξ(↑/↓) (k) shows interference fringes that indicate single-particle coherences. Accord-

ing to Eq. (7.20) we define the disconnected and connected correlation functions asξdis (k)(2)=⟨nk↑ ⟩⟨nk↓ ⟩

(2)
ξcon
(k)

=

⟨nk↑ nk↓ ⟩ − ⟨nk↑ ⟩⟨nk↓ ⟩,(7.22)

which we can both compute from the measured first and second-order momentum
correlation functions. We plot the disconnected correlation function in Fig. 7.11 (b)and the connected correlation function in (c). This example nicely shows that eventhough there is a clear structure in the computed second order correlation function,these correlations are purely caused by single-particle coherences and no second-ordercorrelations are present in the system. The connected correlation function hence
vanishes and does not show any structure. Only if the nth order connected correlation
function does not vanish, the system exhibits true nth order correlations. As we will
discuss in the next chapter, such correlations depend on the operator that we choose
to investigate and can be purely caused by quantum statistics.

77

CHAPTER 7. EXPERIMENTAL TOOLS IN THE LABORATORY

78

Chapter 8

High-Contrast Interference of
Ultracold Fermions
The analysis and measurement of correlation functions finds applications in almost
all fields of physics. One of the most prominent examples thereof was the famous ideaof Hanbury-Brown and Twiss (HBT) to measure the angular diameter of a brightstar by correlating the intensity measured from two distant detectors [185, 186], seeFig. 8.1 (a) for illustration. They were able to show that intensity correlations of abright star measured with two spatially separated detectors depend on their mutual
distance and that the periodicity of the correlated signal was given by the angularspread of the star. By analyzing the correlation function as a function of the detector
spacing, the angular diameter can be resolved even beyond the optical resolution of
the two detectors. While this classical effect is caused by the wave nature of theelectromagnetic field, there exists an analogous quantum mechanical effect, in which
the same type of correlations can be observed.
Let us consider two absolutely identical fermions in a double-well potential, with one
fermion per site. Furthermore, suppose that we have two detectors at variable positions x1 and x2 , as illustrated in Fig. 8.1 (b). At t = 0 we release the two fermions
from the double-well potential and monitor the coincidence events between the two
detectors, i.e. the events in which we detect one particle in each detector at the sametime. For those events, there are two possible paths the atoms can take from theirinitial position to the detectors. However, due to the indistinguishable nature of theatoms, the observer cannot tell, which of the two possible paths were taken. Hencequantum mechanics dictates that each atom takes both paths at the same time, whichleads to correlations in the relative wave function of the two atoms. While the individual signals of the two detectors do not contain any signature of those correlations,they manifest themselves in the correlated signal of the two detectors. Indeed, as we
will discuss in detail later in this chapter, the two paths interfere either constructively
or destructively so that for some detector distances d = x1 − x2 no coincidences occur, while for others they are favored. One finds a sinusoidal dependence of thosecoincidences on d, where the frequency is solely given by the initial separation of thetwo atoms.

79CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.1: Hanbury-Brown Twiss effect. (a) The HBT effect was originally proposedin radio astronomy to resolve the angular diameter of a star beyond the opticalresolution of a telescope by analyzing intensity correlations from two distantdetectors. (b) In a quantum mechanical system, similar correlations occur
when there is no which-way information and different paths interfere.

These correlations are caused by the exchange symmetry of the two fermions andthey are a pure quantum-mechanical effect. In particular it is a prominent example, that shows how correlation functions can be used to extract information beyond
single-particle observables, and, as in the case of radio astronomy, it does not sufferfrom a limited optical resolution of the detectors. The HBT effect is therefore also
an important tool in other fields of physics, as for example in high-energy nuclear
and particle physics [187]. Together with similar effects, such as Hong-Ou-Mandelinterference [188], the HBT effect has been extensively studied in photonic systems[189–191] and finally led to the development of quantum optics [192]. More recently
such correlations also became relevant for ultracold quantum gas experiments, operating with individual particles [60, 61, 63, 193–195] and have proven to be ideal tools
for the charaterization of states but also for studying many-body interference effects
and their influence on the dynamics and entanglement properties of many-body states.
In this chapter we present momentum-density correlation measurements on systems consisting of two or three identical, non-interacting fermions. In analogy to
the HBT effect, we observe strong sinussoidal correlations in the second-order correlation function and show that three identical fermions can exhibit true third-ordercorrelations in the momentum density. These experiments combine the on-demand
preparation of highly indistinguishable particles with high-quality measurements ofcorrelation functions and might enable the future study of quantum-optics experiments with massive particles such as fermionic ghost imaging or Bell tests [196–200].
Parts of this chapter have been published in [61].

80CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

8.1

HBT Effect between Identical Fermions

Quantum-mechanical correlations of the HBT type are caused by exchange statistics and therefore may arise between any two indistinguishable atoms. In order toobserve the HBT effect in a quantum-mechanical setting, we experimentally prepare
two identical fermions in a double-well potential. To this end we independently prepare two atoms in two spatially separated optical tweezers with large initial separation
ai . Even though the two identical fermions are strictly non-interacting, this ensuresthat the two particles are independently initialized and that observed correlations
are purely caused by quantum statistics. After this preparation we approach the twotweezers to the final distance of af < ai from which we release the atoms into our
optical dipole trap (ODT), serving as a one-dimensional wave guide potential as explained in Sec. 7.4. After a fixed time of flight of a quarter of the ODT trap period,
ttof = TODT /4, we detect the positions of the two atoms using our single-atom sensitive imaging scheme and post select the data set to the correct atom number, N ↑ = 2,
with post-selection rates of about 80%. The positions x1 and x2 after time of flightcorrespond to the initial momenta k1 and k2 . From this data set we compute the momentum correlation function ξ (2) (k) = ⟨nk1 nk2 ⟩, which we show in Fig. 8.2 (a). The
momentum correlation function shows almost full-contrast oscillations in the relative
coordinate k1 − k2 .
To understand why we observe oscillations in the second-order correlation functions we choose for now to describe the state within the framework of first quantization. We therefore introduce labels {1, 2} to identify the particles via |X⟩1 and |X⟩2 ,X ∈ {L, R}. We obtain the first-quantized wave function by antisymmetrizing overthe degree of freedom in which the particles are distinguishable,1|Ψ⟩ = √ (|L⟩1 |R⟩2 − |R⟩1 |L⟩2 ) .
2

(8.1)

For fermions trapped in optical tweezers, we can explicitly write down the singleparticle spatial wave function as |X⟩i = ψ(xi − xX ), with X ∈ {L, R}, i ∈ {1, 2}
and the onsite wave function ψ(x). Using the harmonic oscillator ground state forψ(x) and plugging everything into Eq. (8.1), we illustrate the time evolution of the2
relative wave function |Ψ(x1 , x2 )| = ⟨Ψ|Ψ⟩ between the two atoms in Fig. 8.2 (b).
Note that x1 and x2 are not two orthogonal real-space dimensions but are the onedimensional coordinates of particle 1 and 2, respectively. The atoms are initiallylocalized within the two tweezers and start expanding at t = 0. During the expansion
the relative wave function develops fringes in the relative coordinate x1 − x2 . We will
now explicitly perform this transformation and calculate the second-order momentumdensity correlation function ξ (2) (k).
∫︁Using ϕ(k) = ψ(x)eikx dx and basic rules of the Fourier transform we write thesingle-particle states in momentum space as|L⟩i→ g(ki )eiki xL|R⟩i→ g(ki )eiki xR ,

(8.2)

81

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.2: Fermionic Hanbury-Brown Twiss experiment.
(a) Measured
momentum-density correlation function ξ (2) (k) for a system of two identicalfermions in two optical tweezers (Number of datapoints: 16600). Size bar
indicates real-space dimension of appearing correlation features. (b) Simulated time evolution of fermionic wave function |Ψ(x1 , x2 )|2 released from twotightly confined optical tweezers. (c) Normalized correlator C (2) (d) for correlation function shown in (a). Red solid line is a fit to the data points. Grey
points around d = 0 are excluded from analysis because they are below theresolution of our imaging scheme.

with an envelope function g(k) = F[ψ(x)], i.e. the Fourier transform of the singleparticle real-space wave function. We will from now on drop the particle subscripts
in the state representation and write the states as |X⟩1 |Y ⟩2 → |XY ⟩, identifying thefirst entry as the first particle and the second entry as the second particle, respectively.
∑︁
Setting xL = −xR = a/2 and using n̂k′ = k δ(k ′ − k), we can explicitly calculate
the second-order momentum-density correlation function,
ξ (2) (k)==⟨nk1 nk2 ⟩∑︂⟨Ψ| δ(k1 − p)δ(k2 − q) |Ψ⟩
p,q

=
(8.2)

=

82

)︁
1 (︁
2g(k1 )2 g(k2 )2 − ⟨LR|RL⟩→k − ⟨RL|LR⟩→k4
g(k1 )2 g(k2 )2(1 − cos ((k1 − k2 )/klat ))
2

(8.3)

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

with the lattice momentum klat = π/a. One typically normalizes the correlation
function by dividing out the envelope functions,ξn(2) (k) =1⟨nk1 nk2 ⟩= (1 − cos ((k1 − k2 )/klat )) .
g(k1 )2 g(k2 )2
2

(8.4)

Using Eq. (7.20), we can finally also write down the connected, normalized correlation
function as
(2)
ξcon
(k)

1
= − cos ((k1 − k2 )/klat ).
2

(8.5)

The momentum-density correlation function hence exhibits true second-order correlations in the relative momentum of the particles, k1 − k2 , multiplied by an envelope
function, which is given by the Fourier transform of the single-particle real-space wave
functions, exactly as we observe in Fig. 8.2 (a). This is completely analogous to theclassical HBT effect. In the spirit of their initial ideas, one could even deduce the initial spacing between the two wells a, which cannot be resolved optically, by extracting
the periodicity of the correlation function. One should note that the same resolution of
the initial spacing can be achieved with first-order interference experiments. We presented such an experiment in Sec. 7.4.1, where we discussed single-particle coherences.
However, interference experiments are typically technically much more demanding oreven impossible in the case of stellar astronomy.
To quantify the oscillation in the relative momentum and to extract its contrast
and periodicity we define the normalized correlator
C (2) (d) =∫︂⟨nk nk+d ⟩dk.
⟨nk ⟩⟨nk+d ⟩(8.6)

This integration is performed along the center-of-mass momentum, i.e. along the
diagonal k1 + k2 of the correlation function. Using Eq. (8.3), we find C (2) (d) =1 − cos(d/klat ). We compute C (2) (d) from the experimentally measured correlation
functions and show the result in Fig. 8.2 (c). The correlator exhibits almost fullcontrast oscillations (limitations of contrast are discussed below). We exclude thepoints around d = 0 (grey points in the plot) from the further analysis because of theinability to detect two identical particles below the resolution of our imaging scheme[67]. We fit the correlator with a damped cosine of the formC f (d) =

1
2

(︃

(︃
1 − erf|d| − sw)︃)︃ (︂)︂d2y0 − ce− 2σ2 cos (πd/klat ) ,(8.7)

with s, w, y0 , σ, c and klat as free fit parameters. The error function takes the detection
hole around d = 0 into account. The Gaussian envelope with width σ accounts foran eventual loss of contrast c for large particle separation d due to technical noise.
From the fit to the correlator (solid red line in Fig. 8.2 (c)) we obtain a contrast ofc = 79(2)%.
As argued above, the observed correlations are purely caused by fermionic ex83

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.3: Reduction of measured contrast. We measure the contrast of the correlation function ξ (2) (k) for different separations between the optical tweezers.
While the frequency of the modulation increases due to an increasing klat , thecontrast of the modulation decreases for increasing separation. As illustrated
in the insets, we attribute the reduced contrast to a reduced mode overlap
along the expansion axis.

change statistics and are present between any two indistinguishable fermions. This isfundamentally different to the case where correlations arise due to the fact that thesystem is in a superposition state, as we discussed in Sec. 7.4.1, and as we will further
discuss in Chap. 10. The observed correlations between identical fermions thus do
not suffer from decoherence and perfectly indistinguishable fermions should yield acontrast of 100%. However, there are a few experimental reasons that can reduce theobserved contrast:
1. Detection: We post select our images to the correct atom number N = 2.
However, this also includes images where e.g. only one atom was prepared anda second was detected as a false-positive event due to imaging noise. For such
events, the detected positions are uncorrelated and therefore reduce the contrast.
This second-order effect is very small and we estimate it to be negligible since
it involves e.g. a bad preparation, (typically on the order of 5-10%) and afalse-positive detection (typically on the order of 1%) at the same time.
2. Imaging Resolution: The finite imaging resolution blurs the correlation function. As discussed in [67, 85], the position-uncertainty is on the order of 2 pixels
and thus much smaller than the oscillatory feature of the correlation function.
We hence exclude that this effect significantly reduces the measured contrast.
3. Excitation to Higher Tweezer Levels: For the case where one of the two
atoms does not occupy the lowest level within its tweezer, no correlations arise
after time of flight, since the particles are no longer indistinguishable. Witha ground-state preparation fidelity of about 97 % we expect a reduction of thecontrast of at most 5 %. This is a significant contribution, but cannot explain
the reduction of the contrast by 20 %.
4. Bad Mode Overlap: Despite careful alignment of all optical components,
there is a finite angle between the axis connecting the two tweezers and the axis84

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONSof the ODT, in which we perform the expansion. As illustrated in Fig. 8.3, thiseffectively induces some degree of distinguishability between the two particles,
because of a reduced mode overlap along the expansion axis. The larger the
initial separation between the atoms, the larger the loss of contrast, as we verify
with a range of measurements. The contrast linearly decreases with increasing spacing. We fit the measurement with a linear function and extrapolate acontrast of c0 = 99 (2) at a = 0. This extrapolation might overestimate theexpected contrast but is in good agreement with the reduction of the contrast
that we expect from the above mentioned effects.
Since both the tweezer axis and the angle of the dipole trap cannot be changed in
the experiment, an improvement of the contrast is not straightforward. However, an
expansion in a two-dimensional trap would remove this issue so that in the future wehope to be able to achieve even higher contrasts for arbitrary well spacings. Nevertheless, the presented system realizes a high-quality, on-demand single-fermion source
and the high contrast of c = 79(2)% is sufficient to explore a range of quantum optics
experiments with massive particles [199] or to address many-particle interference infermionic systems [195].
In the next step we increase the systems size and study a system with threeidentical fermions in a triple-well potential. This allows us to investigate higher-ordermomentum correlations and the influence of exchange statistics on the third-ordermomentum correlation function.

8.2

Third-Order Momentum Correlations between
Identical Fermions

We have seen that two indistinguishable fermions exhibit strong second-order correlations in the momentum density. These correlations are purely caused by the quantum
statistics and by the antisymmetrization of the wave function and they even exist for
two fermions that have been prepared in completely separate optical tweezers. Using
our single-atom sensitive imaging scheme we are able to measure the second-order correlation function with very high quality. In this section we investigate correlations ina system of three identical fermions. We detect and discuss strong connected thirdorder correlations in their momentum density. This minimal example of fermionicmany-particle interference shows that a many-body system of identical particles ismore than just the sum over its constituents [195, 201].
Before starting to discuss the measurements we first evaluate the relevant correlation functions, for which we switch to the second quantization formalism. For thecase of N fermions trapped in a one-dimensional array of N optical tweezers with85

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.4: Interference between many fermions. (a) Two identical fermions released
from individual traps feature a sinusoidal momentum correlation function due
to interference of possible paths they can take to the detector. (b) and (c)For multiple sources, the measured second-order correlation function is a sum
over all mutual correlation functions, resulting in sharp dips at the lattice
momenta for large systems. (d) If a systems exhibits true third-order correlations, the correlation function contains more information than second-ordercorrelation functions and cannot be constructed from lower-order correlation
measurements.

positions xi we define the real and momentum-space field operators as†Ψ̂σ (x)=N∑︂Φx (x − xi )c†σ,ii=1†

Ψ̂σ (k)=N∑︂Φk (k)e−ikxi c†σ,i ,

(8.8)

i=1

with the onsite wave function Φx (x), its Fourier transform Φk = F[Φx ](k) andfermionic creation operators c†σ,i on site i with spin σ. The sum is running over†

all wells i. Using the density operator n̂ = Ψ̂ Ψ̂ we can calculate all relevant correlation functions. Since we are considering identical fermions, we drop the spin label
σ for the following calculations. The first moment of the momentum density is givenbyξ (1) (k)

=

⟨nk ⟩

=

⟨Ψ† (k)Ψ(k)⟩∑︂2|Φk (k)|eik(xi −xj ) ⟨c†i cj ⟩⏞ ⏟⏟ ⏞
i,j

=

δij

=

2

N |Φk (k)| .

The normalization is such that
∫︂ ∞
∫︂ ∞
2
⟨nk ⟩dk = N|Φk (k)| = N.
−∞

86

−∞

(8.9)

(8.10)

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

The second-order correlation function is calculated in the same way,
ξ (2) (k)= ⟨nk1 nk2 ⟩= ⟨Ψ† (k1 )Ψ† (k2 )Ψ(k2 )Ψ(k1 )⟩
2

2

2

2

2

2

= |Φk (k1 )| |Φk (k2 )|∑︂
eik1 (xi −xn )+ik2 (xj −xm ) ⟨c†i c†j cm cn ⟩⏞ ⏟⏟ ⏞
i,j,m,nδin δjm −δim δjn= |Φk (k1 )| |Φk (k2 )|)︂∑︂ (︂1 − ei(k1 −k2 )(xi −xj )i,j= |Φk (k1 )| |Φk (k2 )|
(︂
)︂
∑︂
ij
(2 − 2 cos (k1 − k2 )/klat,(8.11)⟨i,j⟩ij
with the lattice momenta klat
= π/(xi − xj ). The remaining sum runs over all combinations of wells ⟨i, j⟩ and is valid for any particle and well number N . Hence, as illustrated in Fig. 8.4 (a-c), in the case of non-interacting fermions, the full second-ordercorrelation function is the sum over all mutual pairs of wells. E.g. for the commensurately spaced triple well there are three combinations, ⟨i, j⟩ ∈ {⟨1, 2⟩, ⟨2, 3⟩, ⟨1, 3⟩}
and each combination contributes with its own spatial frequency, given by the inverse of the respective well spacing π/(xi − xj ). The correlation function is hence theFourier synthesis of all appearing well spacings. For a larger array of wells, the sum
gets more terms and for a homogeneously spaced array ultimately results in distinct
dips at the lattice momenta, as was observed via noise-correlation measurements in
optical lattices [194].

The measurement of the second-order correlation function is analogous to the
double-well case, that we discussed in the previous section. We start by preparing
three identical fermions in three independent tweezers with large initial spacings andthen approach the tweezers to the final configuration. We expand the atoms in our
ODT for ttof = TODT /4 and measure the single-particle momenta. We post selectthe data set to the correct atom number N = 3 with post-selection rates of about
60 %. We realize two different well configurations, namely the commensurate triplewell with equal spacing (a12 = a23 = 2 µm) and an incommensurately spaced triple
well with spacing a12 = 1.6 µm, a23 = 1.5a12 = 2.4 µm. These two configurations
are illustrated in the upper insets in Fig. 8.5 (a) and (b). The same figure shows themeasured second-order momentum correlation function and also the theoretically expected full-contrast calculations in the lower insets. We also compute the integrated
correlator C (2) (d), which we plot below the correlation functions. Compared to the
double-well case, the momentum correlation function of both configurations show
additional structure, which is caused by the additional spatial frequencies from theadditional pairs of wells. To verify the validity of Eq. (8.11), we also measure mutual
second-order momentum correlation functions of all well combinations by preparing
atoms only in two of the three wells (see Fig. 8.5 (c)), from which we extract contrast

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CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.5: Second-order correlations in the triple well. (a-b) Measured and expected (inset) second-order correlation functions for three identical atoms in a
triple well for two different spatial configurations. The plots below show the
integrated correlator C(d) together with a curve one would expect from measurements of mutual second-order correlations (the solid line in not a fit to thetriple-well data). (c) Measurement of all mutual correlation functions for two
atoms in a triple well.

and periodicity using Eq. (8.7). We feed the results into Eq. (8.11) and plot the resulting correlator as solid red line on top of the data points in the lower panels in
Fig. 8.5 (a-b). We find excellent agreement between the data computed from the threeatom system and the sum over the data from the two-atom systems. This agreement
indicates that indeed the main mechanism for the loss of contrast is the bad mode
overlap for larger well spacings as we already discussed in the previous section and nota poor fidelity of the state preparation. The results also show that we can measurehigh-quality momentum-correlation functions also for larger systems and that the observed contrast is currently only limited by the alignment of the tweezer-axis withrespect to our optical dipole trap. The presented methods are hence perfectly suited
to detect microscopic ordering in small systems where the optical resolution is not
sufficient to detect single atoms [181] or where ordering appears on sub-wavelengthscales [202, 203].

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CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

The fact that the full second-order correlation function is only a sum over pairwise
correlations, naively suggests that the third-order momentum correlation function
does not contain additional information. This is however not the case, as we see will
see in the following. We start by explicitly calculating the third-order momentum
correlation function, analogously to the calculation of the second-order correlation
function above,ξ (3) (k)= ⟨nk1 nk2 nk3 ⟩= ⟨Ψ† (k1 )Ψ† (k2 )Ψ† (k3 )Ψ(k3 )Ψ(k2 )Ψ(k1 )⟩
2

2

2

2

2

2

= |Φk (k1 )| |Φk (k2 )| |Φk (k3 )|∑︂
eik1 (xi −xs )+ik2 (xj −xl )+ik3 (xm −xn ) ⟨c†i c†j c†m cn cl cs ⟩⏟⏟⏞⏞i,j,m,n,l,s(∗)= |Φk (k1 )| |Φk (k2 )| |Φk (k3 )|∑︂ (︂
1 + eik1 (xi −xj )+ik2 (xj −xm )+ik3 (xm −xi )i,j,m+eik1 (xi −xm )+ik2 (xj −xi )+ik3 (xm −xj )−eik1 (xi −xm )+ik3 (xm −xi )−eik1 (xi −xj )+ik2 (xj −xi )−eik2 (xj −xm )+ik3 (xm −xj )
2

2

)︂
2

= |Φk (k1 )| |Φk (k2 )| |Φk (k3 )|∑︂ (︂
1 + ei(k1 −k3 )xi +i(k2 −k1 )xj +i(k3 −k2 )xmi,j,m+ei(k1 −k2 )xi +i(k2 −k3 )xj +i(k3 −k1 )xm)︂−ei(k1 −k3 )(xi −xm ) − ei(k1 −k2 )(xi −xj ) − ei(k2 −k3 )(xj −xm ) . (8.12)
The expression (∗) in the third line is only non-zero, if the three creation operators
cancel the effect of the three annihilation operators. One finds the expression(∗) = δis δjl δmn + δin δjs δml + δil δjn δms − δin δjl δms − δil δjs δmn − δis δjn δml . (8.13)
The result of Eq. (8.12) depends on the precise shape of the triple-well potential. We
explicitly write down the third-order correlation function for the two studied cases.
For the commensurately spaced triple well we find⟨n̂k1 n̂k2 n̂k3 ⟩ =2nk1 nk2 nk3(327− 2 cos(2π(k1 − k2 )) − cos(4π(k1 − k2 ))− 2 cos(2π(k1 − k3 )) − cos(4π(k1 − k3 ))− 2 cos(2π(k2 − k3 )) − cos(4π(k2 − k3 ))+ 2 cos(2π(2k1 − k2 − k3 ))+ 2 cos(2π(2k2 − k1 − k3 ))+ 2 cos(2π(2k3 − k1 − k2 ))).

(8.14)

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CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.6: Calculated third-order momentum correlation function of commensurately spaced triple well. The contour plots show isosurfaces of thethird-order momentum correlation function (red: positive, blue: negative). Integration over any of the three momenta of the full correlation function (a)or the disconnected correlation function (b) gives the second-order correlation
function, as indicated with the projection on the three planes. The integral
over any of the three momenta of the connected correlation function (c) vanishes, since nth order connected correlations cannot be accessed from n − 1order correlations.

The correlation function of the incommensurately spaced triple well evaluates to⟨n̂k1 n̂k2 n̂k3 ⟩ =2nk1 nk2 nk3(3
27− cos(2π(k1 − k2 )) − cos(3π(k1 − k2 )) − cos(5π(k1 − k2 ))− cos(2π(k1 − k3 )) − cos(3π(k1 − k3 )) − cos(5π(k1 − k3 ))− cos(2π(k2 − k3 )) − cos(3π(k2 − k3 )) − cos(5π(k2 − k3 ))+ cos(2π(3k1 + 2k2 − 5k3 ))+ cos(2π(2k1 + 3k2 − 5k3 ))+ cos(2π(−5k1 + 2k2 + 3k3 ))+ cos(2π(−5k1 + 3k2 + 2k3 ))+ cos(2π(3k1 − 5k2 + 2k3 ))+ cos(2π(2k1 − 5k2 + 3k3 )).

(8.15)

To make the expression more readable and to keep the normalization from Eq. (8.10)
n2
we substituted |Φk (ki )| by Nki and rescaled the momenta by the lattice momentum
klat = π/a12 , that we define in terms of the smallest well spacing.
The third-order correlation function depends on all three momenta and can hence not
be plotted in a straightforward way. We illustrate the correlation function ⟨nk1 nk2 nk3 ⟩for the commensurately spaced triple well in Fig. 8.6 (a) as a three dimensional contour
plot. Integrating the third-order correlation function along any of the three momenta
yields the second-order momentum correlation functions ξ (2) (k) as is indicated in the
figure by the projection onto the three planes. Similar to the second-order cases, thecontour plot indicates that there is no structure along the diagonal of the correlation90

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.7: Measured and calculated correlator of the triple well. The figure shows
the measured (left) and calculated (right) momentum correlator C (3) (d1 , d2 )for two triple-well configurations as illustrated in the insets. Number of data
points for commensurately/incommensurately spaced triple well: 36100/55300

function. We hence define the normalized correlator as∫︂⟨nk nk+d1 nk+d2 ⟩C (3) (d1 , d2 ) =dk,⟨nk ⟩⟨nk+d1 ⟩⟨nk+d2 ⟩(8.16)

which again corresponds to an integral along the axis of the center-of-mass momentum
k1 + k2 + k3 .
To compute the third-order correlator from experimental data we use the samedata set as we used for the calculation of the plots shown in Fig. 8.5. Figure 8.7
shows both the measured and calculated C (3) (d1 , d2 ) for both the commensuratelyspaced (a) and the incommensurately-spaced (b) triple well. The correlator showsa very rich structure, in particular, the correlator for the incommensurately-spaced
triple well clearly shows the presence of three distinct spatial frequencies. Besides
the reduced contrast in the experimentally measured correlator, which we alreadydiscussed in the previous section, the agreement with the calculated correlator is very
good.
91

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

As was already discussed in Sec. 7.5, for the investigation of true nth order correlations
we have to consider the connected correlation function and subtract all lower-ordercorrelations from the full correlation function. Looking at Eq. (8.12) - Eq. (8.15) andusing Eq. (7.20) for n = 3 as well as the calculation of the second-order correlation
function we can rewrite the third-order correlation function as⟨nk1 nk2 nk3 ⟩ = ⟨nk1 nk2 nk3 ⟩con + ⟨nk1 nk2 nk3 ⟩dis ,(8.17)

with the disconnected correlation function defined as⟨nk1 nk2 nk3 ⟩dis= s(N )(2⟨nk1 ⟩⟨nk2 ⟩⟨nk3 ⟩)p(N ) (⟨nk1 ⟩⟨nk2 nk3 ⟩ + ⟨nk2 ⟩⟨nk3 nk1 ⟩ + ⟨nk3 ⟩⟨nk1 nk2 ⟩) .

+(8.18)−1)(N −2)
−2)
6
= 27
and p(N ) = N (N
= 13 are scaling coHere s(N ) = N (N −1)(NN3
N 2 (N −1)
efficients, that account for a proper normalization of the correlation function andensure particle-number conservation (see below for additional information). The disconnected correlation function contains at most second-order correlations and canhence be calculated from experimental second-order measurements. The connected
correlation function can then be calculated as the difference between the full and thedisconnected correlation function, which explicitly evaluates to⟨nk1 nk2 nk3 ⟩con=1(cos(2π(2k1 − k2 − k3 )) + cos(2π(2k2 − k3 − k1 ))2+ cos(2π(2k3 − k1 − k2 )))(8.19)

for the commensurate case and⟨nk1 nk2 nk3 ⟩con=1(cos(π(3k1 + 2k2 − 5k3 )) + cos(π(2k1 + 3k2 − 5k3 ))4+ cos(π(5k1 − 2k2 − 3k3 )) + cos(π(5k1 − 3k2 − 2k3 ))+ cos(π(3k1 − 5k2 + 2k3 )) + cos(π(2k1 − 5k2 + 3k3 )))(8.20)

for the incommensurate triple well.
The connected correlation function contains all cosine terms with three momenta in
the argument. These are exactly the terms that do not show up in the lower-ordercorrelation functions, as they involve the momenta of all three atoms. Similar to thefull correlation function, we illustrate the disconnected and the connected correlation
functions as a three dimensional contour plot in Fig. 8.6 (b) and (c), respectively. Integration over any of the three momenta of the disconnected correlator again yields
the second-order correlation functions, whereas the integral vanishes for the connected
correlator. It is hence exactly this three-dimensional structure that shows the correlations that are not accessible from second-order measurements. They should howevershow up in the connected correlator, which we define similar to Eq. (8.16) for the connected and the disconnected correlation functions. Figure 8.8 shows the measured and(3)(3)
calculated correlators Cdis and Ccon for the two well configurations. Apart from the92CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

Figure 8.8: Disconnected and connected correlations in the triple well. The figure
shows the measured (top) and calculated (bottom) disconnected and connected(3)(3)
third-order correlators Cdis (d1 , d2 ) and Ccon (d1 , d2 ) for the commensurately(a) and incommensurately (b) spaced triple well. For better visibility we plot
(3)
2Ccon (d1 , d2 ) and adapt the colorscales of the measured correlation function as
indicated by the different colorbars on the right.

reduced contrast, the measured correlators agree very well with the calculated correlators. The results show that three indistinguishable fermions exhibit true third-ordercorrelations in the sense that the measured combination of two momenta has a direct
impact on the momentum of the third atom. It is interesting to note, that the correlators C (3) (d1 , d2 ) behave similar to the correlation functions ξ (3) (k) upon integration,i.e. integration over any of the two coordinates d1 or d2 results in the second-ordercorrelator C (2) (d) for the full and the disconnected correlators but vanishes for theconnected correlator.
Particle-Number Conservation
From the explicit calculations of the third-order momentum correlation functions
in Eq. (8.14) and Eq. (8.15), it seems intuitive to define the connected correlation
function as the sum over all cosine terms with all three momenta in the argument.
However, this definition does not agree with the definition of the connected correlation function in Eq. (7.20). Indeed from that definition of the connected third-ordercorrelation function, we would find significant connected third-order correlations onlyby the conservation of total particle number, cf. [204].
Intuitively speaking, detecting one out of N uncorrelated atoms at position x reduces
the possibility of detecting another particle at position x′ to 1 − N1 . This effect vanishes for large particle number. In contrast, in systems of identical fermions, thePauli-exclusion principle prohibits the detection of a second particle at position x.
We hence include the two scaling coefficients s(N ) and p(N ) in Eq. (8.18), that ac-

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CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

count for particle-number conservation and a proper normalization of the correlation
functions. Normalization of the correlation functions is such that∫︂⟨n̂k1 ⟩dk1 = N∫︂ ∫︂⟨n̂k1 n̂k2 ⟩dk1 dk2 = N (N − 1)∫︂ ∫︂ ∫︂⟨n̂k1 n̂k2 n̂k3 ⟩dk1 dk2 dk3 = N (N − 1)(N − 2).
(8.21)
N=3

N=3

−2)
−1)(N −2)
6
By choosing s(N ) = N (N −1)(N= 27
and p(N ) = N (N
= 13 we ensure
N3
N 2 (N −1)
that correlations from number conservation are removed, such that uncorrelated par(3)ticles give Ccon (d1 , d2 ) = 0 for any N and our definition of the disconnected correlation
functions agrees with the standard form for N → ∞.

8.2.1

Wick’s Theorem and Field Correlation Functions

It is important to mention that what was called nth order momentum-correlation†
functions are nth order correlations in the momentum-density operator n̂ = Ψ̂ Ψ̂
and therefore correlation functions of (2n)th order in the quantum-field operators
Ψ̂. Vacuum expectation values of field operators can always be decomposed into
second-order correlation functions and hence no correlations beyond second order are
expected to occur [205]. Using Wick’s theorem [206], we can compute the momentumdensity correlation functions in terms of the field operators as follows,⟨n̂k1 n̂k2 ⟩ = ⟨nk1 ⟩⟨nk2 ⟩ − Gk1 ,k2 Gk2 ,k1 ,(8.22)††

where we used ⟨n̂k1 ⟩ = ⟨Ψ̂k1 Ψ̂k1 ⟩ and define the propagator Gk1 ,k2 = ⟨Ψ̂k1 Ψ̂k2 ⟩, i.e.
the diagonal and off-diagonal entries of the one-body density matrix, respectively.
Since Gk1 ,k2 = G∗k2 ,k1 , the connected second-order momentum-density correlation
function reads as2⟨n̂k1 n̂k2 ⟩con = −Gk1 ,k2 Gk2 ,k1 = − |Gk1 ,k2 | .

(8.23)

Hence we only measure the absolute value of the propagator and do not access its
phase, which might be accessible by an additional first-order interference measurement.
With the above definitions we can expand the third-order momentum-density corre-

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CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONS

lation function as⟨n̂k1 n̂k2 n̂k3 ⟩ = ⟨n̂k1 ⟩⟨n̂k2 ⟩⟨n̂k3 ⟩−⟨n̂k1 ⟩Gk2 ,k3 Gk3 ,k2 − ⟨n̂k2 ⟩Gk1 ,k3 Gk3 ,k1 − ⟨n̂k3 ⟩Gk1 ,k2 Gk2 ,k1+Gk1 ,k2 Gk2 ,k3 Gk3 ,k1 + Gk1 ,k3 Gk3 ,k2 Gk2 ,k1= ⟨n̂k1 ⟩⟨n̂k2 ⟩⟨n̂k3 ⟩−⟨n̂k1 ⟩|Gk2 ,k3 |2 − ⟨n̂k2 ⟩|Gk1 ,k3 |2 − ⟨n̂k3 ⟩|Gk1 ,k2 |2+Gk1 ,k2 Gk2 ,k3 Gk3 ,k1 + Gk1 ,k3 Gk3 ,k2 Gk2 ,k1= −2⟨n̂k1 ⟩⟨n̂k2 ⟩⟨n̂k3 ⟩+⟨n̂k1 ⟩⟨n̂k2 n̂k3 ⟩ + ⟨n̂k2 ⟩⟨n̂k1 n̂k3 ⟩ + ⟨n̂k3 ⟩⟨n̂k1 n̂k2 ⟩+2ℜ{Gk1 ,k2 Gk2 ,k3 Gk3 ,k1 }.

(8.24)We recover the same structure that we obtained in Eq. (8.18) with the following
definition of the connected correlation function⟨n̂k1 n̂k2 n̂k3 ⟩con = 2ℜ{Gk1 ,k2 Gk2 ,k3 Gk3 ,k1 }.

(8.25)

It is exactly this product of propagators that we cannot access in the experiment,
since we are not sensitive to the phase of Gki ,kj . We therefore detect true thirdorder momentum-density correlations but emphasize that the third-order momentumdensity correlation function would completely factorize into second-order field correlation functions, if we had access to the phase of the respective propagators.
Third-order connected correlations have recently also been studied in photonic
systems [207, 208] in the context of multi-particle quantum interference. Following
there ideas we can rewrite the propagator as Gki ,kj = rij eiϕij , where rij ∈ [0, 1] is zerofor orthogonal states and one for perfectly indistinguishable states. In the measured
second-order correlation functions, rij plays the role of the reduced contrast due topartial distinguishability, see Sec. 8.1. The phase ϕij drops out in the above secondorder formulas and does not play a role in second-order interference experiments.
However, rewriting the connected third-order correlation function in the same way,⟨n̂k1 n̂k2 n̂k3 ⟩con = 2ℜ{ReiΦ } = R cos Φ,(8.26)

with R = r12 r23 r31 and the so called triad phase Φ = ϕ12 + ϕ23 + ϕ31 , immediately
makes clear that the phase ϕij does play a role for the observed third-order correlations. A non-zero triad phase reduces the contrast of the connected third-ordercorrelation function beyond the reduction that is caused by partial indistinguishability due to bad mode overlap.
In this chapter we presented momentum-correlation measurements on systems
containing two or three identical particles trapped in an array of optical tweezers. The
observed correlations are purely caused by fermionic exchange statistics. We haveshown that the pairwise exchange symmetry induced true third-order correlations95

CHAPTER 8. HIGH-CONTRAST INTERFERENCE OF ULTRACOLDFERMIONSin the momentum-density of three identical particles. The understanding of suchcorrelations is crucial for the investigations of mesoscopic systems, where additionalcorrelations may arise due to interactions. Our measurements prove that our methods
can produce high-quality momentum-density correlation functions of small systems
and that they are therefore perfectly suited to study complex correlations in strongly
correlated systems, to detect ordering beyond the optical resolution of the imaging
scheme or to study many-particle interference and quantum optics with systems of
massive fermions.

96

Chapter 9

Density-Matrix
Reconstruction
So far we investigated correlations between identical fermions in a double-well andtriple-well potential and showed that fermionic statistics and the antisymmetrizationof the wave function induce complex higher-order correlations in the momentum density. The obtained results also proved that we can apply our experimental tools tomeasure high-quality momentum-density correlation functions. In this chapter, we develop new methods to analyze momentum correlation functions in systems containing
two or three mobile, interacting particles. We present a scheme to relate momentum
correlation functions to off-diagonal entries of the density matrix and reconstruct
physical density matrices based on Bayesian inference. We start by discussing the
double-well case and then generalize the scheme to the triple well.

9.1

Momentum Correlations of the Interacting Double Well

In contrast to the measurements that we discussed in the previous chapter, we now
consider the interacting double well. The basic physics and relevant states were already discussed in Sec. 7.3. As explained in [62, 85, 94, 163] and illustrated in Fig. 9.1,
we experimentally prepare the ground state of the double well by deterministicallyloading a single tweezer with two atoms, followed by an adiabatic passage to the balanced ground state by ramping on the second tweezer. During this passage the two
atoms are non-interacting and stay in the total ground state of the system, resulting
in the state described by Eq. (7.7). We then exploit the interspin Feshbach resonanceand ramp the magnetic offset field to tune the interaction to the desired value of U/J.
In the chosen basis, the final state can be described by a 4 × 4 density matrix. As
described in Sec. 7.4 we either probe the in-situ density or the single-particle momentum distribution by releasing the two atoms into our optical dipole trap, acting as a
one-dimensional wave guide, and detect the position of the atoms after time of flight.
After post selection to the correct atom number N↑ = N↓ = 1 with a post-selection

97

CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.1: Ground-state preparation of the interacting double well. We start by
preparing two non-interacting atoms in the ground state of a single opticaltweezer. We adiabatically ramp on a second well to the balanced double well.
We finally tune the interaction by ramping the magnetic offset field to thedesired value of U/J. The two interacting atoms in the ground state of the
double well are described by a 4 × 4 density matrix.

rate of about 80%, we compute the second-order in-situ and momentum space cor(2)(2)
relation function ξ(↑↓) (x) = ⟨nx↑ nx↓ ⟩ and ξ(↑↓) (k) = ⟨nk↑ nk↓ ⟩. Figure 9.2 presents
both correlation functions for three selected interaction strengths: attractive interaction (U/J = −5.87), zero interaction (U/J = 0) and strong repulsive interaction(U/J = 18.5).
The in-situ measurements are presented in a spin-resolved way, i.e. in each run wemeasure one of the states Ψi ∈ {|LL⟩ , |RL⟩ , |LR⟩ , |RR⟩}, where the first (second)
entry is the location of the spin-up (spin-down) particle. Each quadrant in the upper
panel in Fig. 9.2 represents one of those states. The shown data is the sampled in-situ
distribution from around 1000 realizations. Summation of each quadrant yields theprobability of projecting the state on the respective basis state ⟨Ψi |Ψgs ⟩ ⟨Ψgs |Ψi ⟩ =p2i , i ∈ {LL, LR, RL, RR}, as indicated in the figure. Those numbers directly givethe diagonal entries of the density matrix ρii . As discussed in Sec. 7.3, in particularin Fig. 7.5, the Hamiltonian favors (suppresses) doubly occupied sites for attractive(repulsive) interactions, which we observe in the in-situ data set by the induced asymmetry for the two interacting cases as compared to the non-interacting case (see [62]for more quantitative discussion of experimental results).
As argued in Sec. 7.4.1, diagonal entries of the density matrix do not contain information about coherence properties so that a pure state cannot be distinguishedfrom a maximally mixed state without measuring off-diagonal entries of the density
matrix. Following the proposals in [209, 210] we access information about the offdiagonal entries of the density matrix via the analysis of the momentum correlation
functions. The key idea is to relate distinct pattern in the momentum correlation
function to entries of the density matrix, which is achieved by expanding a generalstate to momentum space, as we will now work out in detail for the double-well case.
Similar to the discussion around Eq. (8.11) we again start by defining the general real
and momentum space wave functions of the two spin states as†Ψ̂σ (x)=N∑︂Φx (x − xi )c†σ,ii†

Ψ̂σ (k)=N∑︂Φk (k)e−ikxi c†σ,i .

(9.1)

i

Here Φ(x) is the onsite Wannier function, which is well approximated by a Gaussian
within the experimental precision. We assume that the onsite wave functions on98CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.2: In-situ and momentum correlations of the interacting double well.
The upper panel shows the measured spin-resolved insitu correlation functions
for three different interaction strength U/J = −5.87 (left), 0 (center) and 18.5
(right). The lower plots show the spin-resolved momentum correlation function
for the same U/J. For vanishing interactions the correlation functions factorize
and no second-order correlations are present. For attractive (repulsive) interactions, double occupancies are favored (suppressed) and correlations in theabsolute (relative) momenta arise.

different sites are orthogonal. Φk = F[Φx ](k) is the Fourier transform of Φ(x) andc†σ,i are fermionic creation operators on site i with spin σ. The sum runs over all wells
i. Neglecting the envelope Φx from now on, we can write down the opposite-spin
second-order momentum-density correlation function as⟨n̂k1 ↑ n̂k2 ↓ ⟩ ==††⟨Ψ̂↑ (k1 )Ψ̂↑ (k1 )Ψ̂↓ (k2 )Ψ̂↓ (k2 )⟩∑︂
eik1 (xl −xk )+ik2 (xn −xm ) ⟨c†↑k c↑l c†↓m c↓n ⟩(9.2)

k,l,m,nwith k, l, m, n ∈ {L, R} for the double-well case. To relate the above expression to
the density matrix, we will rewrite the expression in its matrix representation. To
this end we evaluate the term in the brackets ⟨·⟩ with respect to all combinations ofbasis states, using⟨nk1 ↑ nk2 ↓ ⟩ij= ⟨ii′ | nk1 ↑ nk2 ↓ |jj ′ ⟩∑︂=
eik1 (xl −xk )+ik2 (xn −xm ) ⟨0| c↑i c↓i′ c†↑k c↑l c†↓m c↓n c†↑j c†↓j ′ |0⟩k,l,m,n(9.3)

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

with i, i′ , j, j ′ the respective entries of the four basis states i, j ∈ {LL, LR, RL, RR}.
We exemplarily evaluate this expression for {ii′ , jj ′ } = {LR, LL},
⟨LR| n̂k1 ↑ n̂k2 ↓ |LL⟩ =∑︂eik1 (xl −xk )+ik2 (xn −xm ) ⟨0| c↑L c↓R c†↑k c↑l c†↓m c↓n c†↑L c†↓L |0⟩k,l,m,n

=

∑︂

=

∑︂

=

2eik2 a ,eik1 (xl −xk )+ik2 (xn −xm ) δLn δRm δkl

k,l,m,neik1 (xk −xk )+ik2 (xL −xR )

k

(9.4)

where the factor 2 in the last line is due to the normalization of the wave function.
In the last line we furthermore used a = xL − xR with the well spacing of the doublewell potential a. We obtain the full momentum-density correlation operator in matrix
form as⎞⎛1e−iak2e−iak1e−ia(k1 +k2 )⎟⎜1e−ia(k1 −k2 )e−iak1
⎟
⎜
(9.5)
(nˆ︂
⎟.
k1 nk2 ) = ⎜
−iak
2
⎠
⎝
1
e
h.c.

1

From the operator expression above, we can calculate the momentum-density correlation function by taking the trace of the operator, multiplied by the density matrix,⟨nk1 nk2 ⟩ = Tr(ρ(nˆ︂
k1 nk2 ))=∑︂ρji (nˆ︂
k1 nk2 )ij

i,j= pLL + pLR + pRL + pRR{︁}︁
+2ℜ (ρ13 + ρ24 )eiak1{︁}︁+2ℜ (ρ12 + ρ34 )eiak2{︂}︂+2ℜ ρ14 eia(k1 +k2 ){︂}︂+2ℜ ρ23 eia(k1 −k2 )(9.6)

For our analysis we rewrite this expression into the quadrature representation, which
will also become useful for the generalization to larger systems,⟨nk1 nk2 ⟩ = 1+ 2ℜ(ρ13 + ρ24 ) cos (ak1 )+2ℑ(ρ13 + ρ24 ) sin (ak1 )+ 2ℜ(ρ12 + ρ34 ) cos (ak2 )+2ℑ(ρ12 + ρ34 ) sin (ak2 )+ 2ℜ(ρ14 ) cos (a(k1 + k2 ))+2ℑ(ρ14 ) sin (a(k1 + k2 ))+ 2ℜ(ρ23 ) cos (a(k1 − k2 ))+2ℑ(ρ23 ) sin (a(k1 − k2 )).

(9.7)

The four diagonal entries of in Eq. (9.6), pLL , pLR , pRL , and pRR , are the projection
probabilities of the state onto the four basis states, that we get from the in-situ measurements and sum up to 1. ρij are the off-diagonal entries according the basis states
as defined above (see also Fig. 9.1). The result shows that each oscillatory pattern inthe momentum-correlation function can be related to specific elements of the density100CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.3: Illustration of fitting functions for momentum correlation functions.
Each pattern can be related to an entry or a sum over two entries of the density
matrix. Horizontal and vertical stripes correspond to single-particle coherenceswhereas diagonal stripes are caused by two-particle coherences.

matrix, e.g. the contrast of a cosinusoidal oscillation in the relative momentum k1 −k2corresponds to the real part of ρ23 whereas a phase shifted oscillation indicates a nonzero imaginary part of ρ23 . Note that not all entries of the density matrix can be
uniquely identified, because some of them only appear as a sum, such as (ρ13 + ρ24 ).
In total there are oscillatory patterns with four different frequencies, as we illustrate
in Fig. 9.3, together with a Gaussian envelope that we omitted in the calculations.
As mentioned before, this envelope is given as the Fourier transform of the real-space
wave function (cf. Eq. (9.1)). By decomposing the measured momentum-density correlation function into the set of trigonometric functions according to Eq. (9.7), we
extract a set of 8 equations for the 6 complex off-diagonal entries of the density matrix (i.e. 6 real and 6 imaginary coefficients). In particular, we can directly determine
the anti-diagonal elements ρ14 and ρ23 . However, the two patterns with oscillationsin k1 or k2 only appear with the sum of two entries so that only the sum of thoseentries can be constrained. Note that those patterns correspond to the single-particle
coherences, that we already discussed in Sec. 7.4.1. A determination of those entries
is possible e.g. via a rotation in the {L, R} basis [209], or by performing correlation
measurements in the near field [211]. However, we do not perform such additional
measurements, but rather employ a Bayesian quantum state estimation [212]. Thisapproach reconstructs a set of physical density matrices, based on the constraints that
we obtain from the momentum-density correlation function. Before discussing thisapproach in Sec. 9.2, we generalize the analysis of the momentum-correlation function
to the case of three interacting, mobile particles in a triple-well potential.

9.1.1

Three Interacting Fermions in a Triple Well

Before going into the technical details of how to analyze the triple-well correlation
functions, we present a more intuitive way of how to relate the entries of the density
matrix to trigonometric functions that appear in the momentum correlation function.
As illustrated in Fig. 9.4 (a) for the double well and in (b) for the triple well, one cananalyze how many tunnel events are necessary to transform one basis state Ψi intoanother basis state Ψj . For example for ρ24 of the 4 × 4 matrix of the double well, thestate Ψ2 = |LR⟩ connects to Ψ4 = |RR⟩ via a single tunnel event of the first atomfrom left to right. This process contributes with cos(k1 a) to the total momentum
correlations function, since the first atom moved the distance a in positive direction.

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.4: Density-matrix reconstruction. Instead of doing the maths to relate patterns in the momentum correlation function to entries of the density matrix,
we can also analyze, which states contribute to the entry and by how manytunneling events they are connected. (a) Illustration of assignment scheme
for the double well with spacing a. (b) Illustration of the assignment scheme
for the triple well with spacings a12 , a23 , and a13 = a12 + a23 . Due to theindistinguishability of two atoms, there are always two possible tunneling configurations. Labeling the particles is crucial to associate the correlations tothe correct momenta. However, also the antisymmetric correlations k1 ↔ k2
appear due to the indistinguishability of the two identical particles.

Similarly for ρ21 , the second atom hops to the left well and therefore contributeswith cos(−k2 a). The anti-diagonal entries of the density matrix correspond to processes, that involve the movement of both atoms, e.g. ⟨LR| ↔ |RL⟩ contributes withcos(k1 a − k2 a), since the first particle hops to the right (”+k1 ”) and the second to theleft (”−k2 ”), respectively. We can apply the same scheme to the triple-well case, buthave to take care of the spacing between the wells, which might not be commensurate.
Furthermore in the case of indistinguishable particles, one also has to take the antisymmetrized form into account. Based on that intuitive scheme we will define a set of
trigonometric basis functions into which we decompose the highest-order momentum
correlation function.
Suppose we have prepared three atoms in three optical tweezers in the laboratory
and we want to characterize the exact state of the system. The only interacting
configuration consists of two spin-up and one spin-down atom (or vice versa). As
discussed in Sec. 7.3, there are hence nine different basis states, which we define as|Ψ⟩∈{|↑, ↑, ↓⟩ ; |↑, ↓, ↑⟩ ; |↓, ↑, ↑⟩ ;|↑↓, ↑, ·⟩ ; |↑↓, ·, ↑⟩ ;|↑, ↑↓, ·⟩ ; |·, ↑↓, ↑⟩ ;|↑, ·, ↑↓⟩ ; |·, ↑, ↑↓⟩},(9.8)

where the first, second and third entry describe the spin occupation of the left, centerand right well, respectively. Within the lowest-band approximation of the FermiHubbard model (see Sec. 7.3) fermions with opposite spin can occupy the same well102

CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

but two identical spins in the same well are forbidden by the Pauli-exclusion principle.
Similar to the double-well case, we investigate the system in real space by repeatedly
imaging the spin-resolved in-situ distribution. We directly extract the nine diagonal
entries of the density matrix pi , which are given by the probability of projecting thestate onto each of the nine basis states.
We then measure the spin-resolved single-particle momenta from which we construct(3)
the third-order momentum-correlation function ξ(↑↑↓) (k1 , k2 , k3 ) = ⟨nk1 ↑ nk2 ↑ nk3 ↓ ⟩.
As argued above, we know that each off-diagonal entry of the density matrix contributes to the correlation function with specific trigonometric patterns. Motivated(3)by Eq. (9.7) and Fig. 9.4 we decompose ξ(↑↑↓) into a general set of trigonometric basis
functions
∑︂
(3)
⃗) =
ξ(↑↑↓) (kci cos(2πq⃗ i⃗k ) + si sin(2πq⃗ i⃗k ),
(9.9)
iwhere ⃗k = (k1 , k2 , k3 )T and the different ⃗q i s are vectors that contain the appearing real
space distances between spatial modes. In the general case of an incommensurately
spaced triple well, with a12 ̸= a23 , each of the three components in ⃗q i can takeany of the seven values {−a13 , −a12 , −a23 , 0, a23 , a12 , a13 }. In total this gives rise to
73 = 343 different ⃗q i s, including the zero-vector (0,0,0), which does not generate any
pattern. Due to the symmetry of the cosine and sine functions (cos(x) = cos(−x) andsin(x) = − sin(−x)), half of the ⃗q i s are redundant, reducing the total set of possible⃗q i to 342/2 = 171 (hence 171 ci and 171 si ).
Following the discussion around Eqs.(9.5 - 9.7) we expand the third-order momentum(3)⃗density correlation operator(︂ ξ̂ (k)︂) and calculate the correlation function of a general⃗ ) using Mathematica [213]. The full momentumdensity matrix ρg as Tr ρg ξ̂(kdensity correlation function can be written in a similar form as Eq. (9.9),
(3)

⃗) =
ξ(↑↑↓) (k∑︂

Kij ℜ(ρj ) cos(2πq⃗ i⃗k ) + Nij ℑ(ρj ) sin(2πq⃗ i⃗k ).

(9.10)

i,jHere K and N are coefficient matrices and ⃗ρ is a vectorized form of the density
matrix. Equating the two equations (9.9) and (9.10), we obtain a set of coupledlinear equations for the off-diagonal entries of the density matrix
Kℜ(ρ
⃗)

= ⃗c

Nℑ(ρ
⃗)

= ⃗s.

(9.11)The size and the rank of the two matrices M and N depend on the shape of thetriple well. Let us consider the incommensurately spaced triple well with two spin-up
and one spin-down particles. In the case where two of the three atoms are indistinguishable, the number of possible trigonometric pattern can be further reduced. We
choose k1 and k2 for the atoms in |↑⟩ and k3 for the atom in |↓⟩.
Pauli Exclusion: Due to the Pauli-exclusion principle, no states which involve
two identical fermions on the same lattice site are involved. Combinations such as⃗q i = (−a13 , −a12 , X) or ⃗q i = (−a12 , a23 , X) for any X are not allowed, because they
are caused by processes where the two indistinguishable particles either hop to the103CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

same well or start from the same well (cf. Fig. 9.4). This reduces the number of= 108. The factor 1/2 is due to the ⃗q i s that have alreadypossible ⃗q i to 171 − 3·6·7
2
been removed as argued above.
Diagonal Entries: There are 3 ⃗q i s on the diagonal (such as ⟨↑↑↓ | ↑↑↓⟩ → cos((k1 −k2 )a12 ), which do not contribute to the sine terms, because all diagonal entries are
real. Since we constrain the diagonal entries with the in-situ measurements, we exclude them from the further analysis. There are hence 105 ⃗q i for the off-diagonals.
Indistinguishable Duplicates: Within the 2 · 105 = 210 trigonometric pattern
there are six of the form (0, 0, X). The remaining 204 entries contain 102 duplicates
due to the indistinguishability of the first two atoms in the sense that they can beobtained from each other by exchanging k1 and k2 .
For the incommensurately spaced triple well we are finally left with 108 trigonometric pattern that constrain the off-diagonal entries of the density matrix and 9 constraints for the diagonal entries from the in-situ measurement. Combining the matrices K and N into a single block diagonal matrix B and the real and imaginary entriesof the vectorized density matrix into a single vector of the form ⃗b = (ℜ(ρ⃗ ), ℑ(ρ⃗ ))T ,
we can write Eq. (9.11) as(︄ )︄
⃗ = ⃗c ,Bb(9.12)⃗s⃗ ) = 81 × 1 and dim(Bb⃗ ) = 117 × 1. Obviously these 117
with dim(B)=117 × 81, dim(bequations are not linearly independent. In fact we find that the matrix B has a rankof rank(B) = 69 and hence constrains 69 of 81 entries of the density matrix.
With similar arguments we can deduce the matrix Bc also for the commensurate triple
well and find dim(Bc ) = 61 × 81 and rank(Bc ) = 37. The reason for the lower rankof Bc compared to the incommensurate case is that there is one spatial frequency lessin the commensurate case (since a12 = a23 ) and therefore less unique constraints forthe density matrix. However, the set of equations in Eq. 9.12 is based on experimental measurements and therefore also contains experimental noise so that the set of
equations of Eq. 9.11 effectively has full rank. As we will explain later, we use the fullset of equations for the further analysis.
The set of equations that we can extract from the momentum-density correlation
function and the in-situ measurements are not sufficient to fully constrain the density
matrix. This applies both to the double-well and the triple-well case. Even if we could
constrain the full density matrix, the result would most likely not describe a physicalstate due to the density matrix being not positive semidefinite because of statisticaluncertainties of all entries. Based on the reconstructed set of equations B, we choose
to reconstruct the full density matrix via a Bayesian quantum state estimation as we
describe in the following section.

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

9.2

Bayesian Quantum State Estimation

Finding a density matrix that describes measurement outcomes is called a quantum
state estimation. As nicely outlined in Ref. [212] there are two typical approaches
to estimate a quantum state: Maximum Likelihood Estimation (MLE) and BayesianMean Estimation (BME). In both cases one wants to find a state ρ that would haveproduced the observed data M. One therefore defines the likelihood L(ρ) = p(M|ρ)
as the probability of measuring M if the prepared state was ρ.
MLE finds the most likely state ρMLE , i.e. the state that simply maximizes the likelihood function L(ρ). However, this approach has serious flaws, in particular since it
might predict rank-deficient states with zero eigenvalues (with no error bars), which
can never be justified with a finite number of experiments. The idea of state estimation however is not only to explain the obtained measurements but to find a statefrom which one can also predict future measurements.
The approach of BME is different. Here, one does not only obtain the most likely
state, but considers all states that are consistent with the observed measurements.
Since there are many such states, it also naturally comes with error bars and therefore
does not suffer from rank-deficiency. In this section we explain how we can performa Bayesian mean estimation based on the previously discussed correlation measurements.
With the presented analysis of correlation functions there are two remaining issues:
1. We cannot constrain the full density matrix, since some patterns in the momentum correlation function cannot be unambiguously related to a single entry ofthe density matrix. A full state tomography would require additional measurements [209, 211], however, for larger systems these measurements might become
technically impossible.
2. The decomposition of the correlation function into trigonometric basis functions,as we discussed in the previous section, determines each entry of the density
matrix with its own statistical uncertainties. These statistical uncertainties
might lead to unphysical properties of the density matrix, such as negative
eigenvalues.
In order to obtain a physical density matrix that is most compatible with our measurements, we employ a Bayesian quantum state estimation. We follow the scheme
outlined in Ref. [212] and divide the estimation into three major steps.
1. Generate a Likelihood Function from Experimental Data, L(ρ)
The likelihood function represents all the information about the system that we
can extract in the experiment. From a Bayesian point of view, the likelihood
function L(ρ) = p(M |ρ) is the probability of observing the outcome M , conditioned on the real density matrix ρ. Assuming Gaussian distributed errors, thelikelihood function is given byL(ρ) =∏︂i)︄(Mi − Tr(M̂ i ρ))2
√︁
exp −
.
2σi2
2πσi2
1

(︄

(9.13)

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

This function compares measured values Mi to the expectation values of theobservables M̂ i for a given ρ, weighted by the experimental uncertainties σi .
2. Prior and Posterior Distribution
We assume no prior knowledge about the states and therefore choose an uninformative prior π(ρ)dρ. For d dimensional pure states a suitable measure is theHaar measure dd ρ. Mixed states are taken care of by starting with the Haar
measure in d + n dimensions and tracing out the ancillary dimensions. dn ρis hence the Haar measure while dd ρ is the Hilbert–Schmidt measure. Multiplying the prior with the likelihood function yields the posterior distribution
L(ρ)π(ρ)dρ
3. Mean Posterior Distribution
The best description of the knowledge of the state is the integral over the poste∫︁rior distribution ρBME = ρL(ρ)π(ρ)dρ. In order to compute this function over
our integral measure, we apply the Metropolis-Hastings (MH) algorithm, whichcalculates the average of a function ρ over the integration measure. Comparedto other Monte-Carlo techniques, that randomly sample over the integration
measure, the MH algorithm makes local, biased jumps: Before jumping from ρL(ρ)π(ρ)
to ρ′ , the likelihood ratio r = L(ρ′ )π(ρ′ ) is computed. If r > 1, the algorithm′
jumps from ρ to ρ , but if r < 1 the algorithm jumps with probability r to the
new state. This jumping rule ensures that the algorithm spends more time atregions with more likely states.
To apply the above algorithm to the reconstruction of density matrices of experimentally realized states, we use the constraints that we obtain from the analysis of
the momentum-density correlation function (see Eq. (9.12)) to define the likelihood
function. As described, we write the basis states in a vectorized form ⃗ρ with 10 realand 6 imaginary entries for the double well and 45 real and 36 imaginary entries
for the triple well. We combine the two matrices K and N(︄into)︄ a single matrix B,⃗c
and combine the constraints ⃗c and ⃗s into the vector ⃗χ =together with their
⃗s
(︄ )︄
⃗c
uncertainties δχ⃗ =. We obtain the likelihood function as⃗sL(ρ) =

√︁
i

∑︁
(︃
)︃
(χi − i Bij ρj )2
exp −
.
2δχ2i
2πδχ2i1∏︂(9.14)The MH algorithm step by step samples the posterior distribution L(ρ)π(ρ)dρ overthe integration measure and in each step generates a physical density matrix. While
the mean over all those states ρBME is the best description of the state based on theexperimental observations, the full posterior distribution of density matrices can beused for further analysis and especially for the calculation of uncertainties. In thenext section we apply the Bayesian quantum state estimation to all states that werealized in the laboratory.

106

CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

9.3

State Reconstruction

We are now ready to apply the reconstruction scheme to experimental data. In total we investigate three different classes of states to which we apply the described
schemes; a polarized, spin-rotated triplet state in the double well (i), a set of interacting singlet states in the double well with different interaction strengths (ii), andtwo three-particle states in the triple well (iii). In this section we separately describe
and discuss the density-matrix reconstruction of each investigated state. The explicit
form of the reconstructed density matrices of each state and the uncertainties are
given in Appendix A. In the next chapter we will use the obtained density matrices toinvestigate entanglement properties in systems containing indistinguishable particles.

9.3.1

Spin-Rotated Triplet State in a Double Well

The first state that we investigate is a non-interacting state of identical, polarized
fermions. As we will see, this state features strong second order momentum correlations in each spin configuration and will be relevant for the discussion of particleparticle entanglement in the next chapter. Since the basis states are different from theinteracting double well, it has to be treated in a slightly different way. However, wechoose to start with the discussion of that state because it is nicely suited to illustrate
the working principle of the Bayesian reconstruction. To initialize the state, we startwith the spin-polarized state |↑, ↑⟩, which we discussed in Chapter 8, and which is illustrated again in Fig. 9.5 (a). By applying a resonant radio-frequency (rf) pulse, we rotate the spin of each atom into the equatorial plane of the Bloch sphere, i.e. we rotate√πeach of the two atoms into the coherent superposition |↑⟩ −→ |→⟩ = 1/ 2 (|↑⟩ + |↓⟩).
This procedure realizes the stateΨ=

1(|↑⟩L + |↓⟩L ) ⊗ (|↑⟩R + |↓⟩R ) ,2(9.15)

where we omitted to explicitly write out the antisymmetrized form for better readability. If we keep measuring in the σz basis, the state shows spin fluctuations and the density matrix becomes a 4×4 matrix with the four basis states {|↑, ↑⟩ , |↑, ↓⟩ , |↓, ↑⟩ , |↓, ↓⟩},
see Fig. 9.5 (a).
We investigate this state only in momentum space and do not explicitly measure
the in-situ density. Instead we constrain the diagonal entries of the density matrix
from the occurrences of spin combinations in the momentum measurements. As described before, to compute momentum-density correlation functions, we release theatoms into the ODT and measure the single-particle momentum distributions aftertime-of-flight. Upon imaging the two atoms get equally projected onto one of the four
spin combinations. The occurence of each configuration directly gives the diagonal
entries of the density matrix. While the measurement of |↑, ↑⟩ and |↓, ↓⟩ is uniquely
related to one diagonal entry, we cannot uniquely assign the appearance of differentspin combinations to any of the states |↑, ↓⟩ and |↓, ↑⟩, and therefore only extract thesum of the two entries p↑↓ + p↓↑ .
Figure 9.5 (b) shows all measured momentum correlation functions between all possi107

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Figure 9.5: Reconstruction of spin-rotated state. (a) Initialization of state via radiofrequency rotation. We start with the state |↑, ↑⟩ and apply a radio frequency
π pulse. In the σz basis this enlarges the density matrix to a 4 × 4 matrix. (b)Measured momentum correlation functions with occurrences of spin combinations (inset) from which we infer ρ↑↑ , ρ↑↓ + ρ↓↑ , and ρ↓↓ . ρ23 is given by by thecontrast of ⟨nk↑ nk↓ ⟩, which we obtain by fitting the correlator with a cosine
function. Total number of datapoints for all three correlation functions: 14500(c) Sampled posterior distribution of off-diagonal entries. The unconstrained
entries explore the full space, while the constrained entry ρ23 (red points) only
navigates within the experimental uncertainties.

ble spin combinations, ⟨nk↑ nk↑ ⟩, ⟨nk↑ nk↓ ⟩, and ⟨nk↓ nk↓ ⟩. The state features identical
momentum correlations within each spin combination. As we discussed in Chapter 8
for the state |↑, ↑⟩, these correlations are only due to the exchange symmetry of two
identical fermions. To determine the contrast of the modulation of the correlation
functions we fit the correlator C (2) (d) with a damped cosine function (see Eq. (8.6)
and Eq. (8.7)). Within the statistical uncertainty all correlation functions show an
identical contrast of 62(2) %. As illustrated in Fig. 9.5 (a), the contrast of the fit to⟨nk↑ nk↓ ⟩ constrains the entry ρ23 of the density matrix. All remaining off-diagonal
entries cannot be constrained. Although we have very limited constraints on the density matrix in this case, we apply the Bayesian quantum state estimation as outlinedin the previous section. The likelihood function, that we define as given in Eq. (9.14),
hence only constrains the diagonal entries and ρ23 , all remaining off-diagonal entries
do not appear in the likelihood function and are therefore not constrained at all.
Figure 9.5 (c) shows the sampled posterior distribution of all off-diagonal entries withℜ(ρij ) on the x-axis and ℑ(ρij ) on the y-axis, respectively (5000 steps of the MetropolisHastings algorithm). While all unconstrained entries (grey points) freely navigate
around and randomly explore the full physical Hilbert space, ρ23 only explores theHilbert space within the experimental uncertainties. The diagonal entries are notplotted in the same graph since they are real numbers and therefore only explore
the real axis. Instead we plot the the posterior distribution of the diagonal entries
as histograms in Fig. 9.6 (a). The plot in (b) of the same figure shows exemplary
histograms of the posterior distribution of the real part of four off-diagonal entries,
including ℜ(ρ23 ). The histograms nicely show that all entries that are directly constrained by the likelihood function are Gaussian distributed, i.e. ρ11 , ρ44 , ρ23 , and108

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Figure 9.6: Posterior distribution of Bayesian state estimation. (a) Posterior distribution of all diagonal entries and of the sum ρ22 + ρ33 , which is directly
constrained by the likelihood function. (b) Posterior distribution of the realpart of four exemplary off-diagonal entries, including ρ23 . All quantities thatare directly constrained by the likeklihood function are Gaussian distributedand sharply peaked. (c) ρBME for the rotated triplet state. The height of thebars is given by |ρBME | and the phase of each entry is encoded in the color.
(d) 68% credible interval of the posterior distribution of all entries.

ρ22 + ρ33 . Entries that are not directly constrained, such as ρ22 and ρ33 , have a large
spread, but explore the Hilbert space in a correlated way, as their sum is constrained.
All unconstrained off-diagonal entries spread across the whole allowed Hilbert space.
The mean over the posterior distribution ρBME represents the best knowledge of theexperimentally realized state. However, note that the full information about uncertainties is encoded in the posterior distribution of states, that are sampled by the
Bayesian estimation. We show ρBME for the discussed state in Fig. 9.6 (c) and the
68% credible interval of each entry in (d). The height of the bars gives the absolutevalue of each entry |ρBME | and the color encodes the phase ϕ(ρBME ) as indicated by
the phase wheel in the figure. The density matrix shows that all unconstrained entries
have large uncertainties, as is expected from the posterior distribution.

9.3.2

Interacting Singlet State in a Double Well

We now turn to the discussion of states involving two interacting atoms in a the groundstate of a double-well potential. The measurements of the correlation functions werein parts already shown in Sec. 9.1 and discussed in detail in Ref. [62]. As also discussed
in Sec. 9.1, we extract 12 values from the analysis of the experimental data: 4 real
numbers from insitu measurements that constrain the diagonal entries and 8 numbersthat constrain real and imaginary party of all off-diagonal entries, of which two are
only given as the sum of two off-diagonal entries. Note that even though this does109

CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.7: Posterior distribution of interacting double well. The upper plots show
the sampled posterior distribution of all off-diagonal entries for the interacting
double well for three different interaction strengths, U/J = −5.9 (a), U/J = 0(b), and U/J = 18.5 (c). The histograms below show the distribution of thereal part of all off-diagonal entries. Colored entries are directly constrained bylikelihood function.

not fully constrain the density matrix, all entries appear in the likelihood function
and are therefore constrained in some way. This is qualitatively different from thestate that we discussed before, since we now expect the posterior distribution of eachentry to be a peaked function. In total we apply the Bayesian state estimation to 8
different data sets with interaction strength U/J ∈ {−5.9, −2.9, 0, 2.5, 5, 8, 11.6, 18.5}.
However, we will only discuss the three cases of U/J ∈ {−5.9, −0, 18.5} in this section. We will nevertheless use the reconstructed states of all interaction strengths in
the next chapter.
Fig. 9.7 shows the sampled posterior distribution of all off-diagonal entries together
with histograms over the real part of all off-diagonal entries for all three interaction
strengths. The colored entries are the anti-diagonal entries of the diagonal entries,
which are directly constrained by the measurements. The posterior distribution of
the diagonal entries is not plotted. We sampled the posterior distribution with 5000
steps of the Metropolis-Hastings algorithm. Both the posterior distribution and thehistograms show that all entries are peaked functions and indeed, all constrained
quantities (i.e. ρ23 , ρ14 , ρ12 + ρ34 , and ρ13 + ρ24 ) are Gaussian distributed. The
spread of each entry is given by experimental uncertainties. The mean estimate ρBME
for all three interaction strengths is shown in Fig. 9.8 (a) together with the standard
deviation of all entries in (b). The shown density matrices show exactly the structure,
which we discussed in Sec. 7.3, i.e. on the attractive side, U/J < 0, doubly occupied
wells are favored, whereas on the repulsive side, U/J > 0, doubly occupied sites aresuppressed. In the non-interacting state, U/J = 0, all entries of the density matrix are
equally high. For detailed discussion of the obtained density matrices, see Ref. [62].
As mentioned above, we also reconstruct density matrices for the five remaining interaction strengths, that are not shown here. However, we will use the results in the next
chapter for the investigation of entanglement properties in identical-particle states.
As a consistency check of the reconstructed density matrices, we calculate the cor110

CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.8: Reconstructed density matrices of interacting double well. (a)(b) ρBME and standard deviation of each entry for three different interaction strengths, U/J = −5.9 (left column), U/J = 0 (middle column), andU/J = 18.5 (right column). Height indicates the absolute value |ρBME | andcolor indicates the phase, according to the color wheel. Note the different zscales for the mean value and the standard deviations. (c) Comparison of themeasured correlation functions to the correlation function that we reconstructfrom ρBME .

relation functions that one would measure of a state described by the density matrix
ρBME . To this end we use Eq. (9.7) and plug in the entries of ρBME and the lattice
momentum klat that we obtained by fitting the correlation functions. Figure 9.8 (c)shows the measured and the reconstructed momentum-density correlation functions
for the same states as above. We find very good agreement.

9.3.3

Interacting Triple Well

We finally realize two interacting three-particle states consisting of two spin-up andone spin-down particle. The first state that we investigate is a state with a biseparable
structure consisting of a spin singlet in a double well and a third independent particle
in a distant tweezer. Starting from that state we tried to realize the ground state of acommensurately spaced triple well by approaching the third well towards the double
well. We separately discuss the realization and analysis of these two states.

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

A Biseparable State in the Triple Well
We so far discussed two different origins of momentum correlations: fermionic exchange symmetry, as extensively discussed in Chapter 8, and interaction as discussed
in the previous section. Comparing the structure of observed momentum-density correlation functions, one sees that these two effects cause almost identical patterns. In
order to analyze those different types of contributions to the momentum correlation
functions and their influence on entanglement properties (as we will discuss in detailin the next chapter), an interesting candidate involving three particles is a state consisting of two interacting atoms in the ground state of a double well and an additional
third atom in an independent tweezer. If there is no tunnel coupling between thisindependent third well and the double well, the third particle only contributes to thecorrelation function via its exchange symmetry and indistinguishability to the otherspin-up particle.
As illustrated in Fig. 9.9 (a), we experimentally realize such a state by preparing two
atoms in one well and a single atom in a separate third well. Exactly as for the
preparation of the double-well ground state (cf Fig. 9.1), we adiabatically ramp on
the central well to obtain a balanced double well with a third distant tweezer, i.e. astate of the form |ψ⟩ = |↑⟩ ⊗ |DW ⟩. We ensure a negligible tunnel coupling to thethird well by choosing an incommensurate well spacing a12 = 1.5a23 (see figure) andby deliberately detuning the depth of the third well away from the tunnel resonanceto the central well. We do not observe any tunneling on experimental time scales andestimate a residual tunnel coupling of at most 0.1 Hz. After ramping to the balanced
double well we induce interactions by tuning the magnetic offset field to 630 G. At
that magnetic field we measure the interaction to be U/J ≃ 16.
To investigate the state we proceed exactly in the same way as we did for the doublewell measurements, i.e. we measure both the single-particle and spin resolved in-situ
populations and momentum density. We post select the data to the correct atom
number of N ↑ = 2, N ↓ = 1 with post-selection rates on the order of 70%. For thepresented data set we retain about 19400 shots. We extract the diagonal entries fromthe in-situ measurements and compute the third-order momentum-density correlation(3)
function ξ(↑↑↓) = ⟨nk1 ↑ nk2 ↑ nk3 ↓ ⟩. We plot the measured momentum-density correlator C (3) (d1 , d2 ) in Fig. 9.9 (b) (see Eq. (8.16) for definition of C). C (3) shows complex
structure with many different spatial frequencies.
(3)
As explained in Sec. 9.1.1, we decompose the correlation function ξ(↑↑↓) into a set of
trigonometric basis functions and obtain a 117 × 81 matrix B that constrains the entries of the density matrix. These 117 equations are not linearly independent. In fact
B has a rank of rank(B) = 69. We nevertheless define the likelihood function using
the full matrix B and apply the Bayesian state estimate. We sample the posterior
distribution with 50000 steps. This is a factor of 10 more compared to the double-welldata due to the significantly larger density matrix.
Figure 9.9 (c) shows the posterior distribution of all off-diagonal entries. Only the
major six entries are colored, while the remaining 30 entries concentrate around zeroand are shown in grey. Note that the distributions for ρ28 and ρ16 (blue points),
as well as the distributions for ρ26 and ρ18 (red points) are almost identical and lie

112CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.9: Reconstruction of incommensurately spaced triple well. (a) Illustration of preparation of biseparable three-atom state in which we entirely decouple the third well from the double well. (b) Measured momentum-density
correlator C (3) (d1 , d2 ). Number of data points: 19400 (c) Sampled posterior
distribution of all 36 off-diagonal entries. Only major six entries are colored.
Distributions for remaining 30 entries scatter around zero and are shown ingrey. Note that the distributions for ρ28 and ρ16 (blue points), as well as forρ26 and ρ18 (red points) lie almost on top of each other. (d) Reconstructed
momentum-density correlator. (e) Bayesian mean estimate ρBME (upper row)and standard deviations of all entries (lower row).

on top of each other. Due to the rather complicated structure of B, all entries ofthe density matrix appear in the likelihood function and are therefore constrained
to some degree. All constrained quantities are Gaussian distributed and, despite thereduced rank of B, all entries of the density matrix are sharply peaked with relatively
small standard deviations. The resulting mean estimate ρBME is shown in Fig. 9.9 (e)together with the standard deviation of all entries. Note the different z-scales of themean and standard deviation. We again check the resulting density matrix by computing the momentum correlator we would expect from a state described by ρBME .
We show the result in Fig. 9.9 and again find very good agreement. The reconstructed
density matrix ρBME has one major off-diagonal entry ρ12 , which is given by the twoparticle coherence within the double well. The four small entries at ρ16 , ρ18 , ρ26 andρ28 are single particle coherences within the double well, which are suppressed dueto the large repulsive interaction (see previous sections and chapter for discussion of
double well coherences). We illustrate the theoretically expected density matrix inthe lower left panel of Fig. 9.10 (lowest energy state at J12 = 0). Apart from thereduced magnitude of ρ12 the agreement between experimental and theoretical density matrices is very satisfying. We find the purity of the reconstructed state to bepexp = Tr(ρ2BME ) = 0.61(1), which we mainly attribute to a bad mode overlap along
the axis of expansion (as discussed in Sec. 8.1). Indeed if we include the measured lossof contrast for large distances between the tweezers (see Fig. 8.3) into our analysis,
we find almost perfect agreement between the measured and the expected coefficientsin B. Hence also for the interacting states, we assume the purity to be much higher
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Figure 9.10: Lowest states of interacting triple well. The plot shows the energy of
the three lowest eigenstates as a function of the tunnel coupling J12 between
the first and the second well at U/J ≃ 16. The energy splitting between
the lowest state is on the order of the super exchange energy 4J 2 /U . The
five density matrices are the corresponding eigenstates as indicated by thedifferent markers. In the experiment we first realized the lowest state at
J12 = 0 and for the second state we ramped the tunnel coupling from zero to
J12 = J23 .

and currently only limited by the expansion in the ODT.
Ground State of the Commensurately Spaced Triple Well
The plot in Fig. 9.10 shows the energies of the three lowest eigenstates of the triple
well as a function of the ratio of tunnel couplings J12 /J23 at an interaction strengthof U/J ≃ 16. We exemplarily show theoretical density matrices of the three states atdifferent tunneling strengths. The bi-separable state that we discussed above is theground-state of the triple well with J12 = 0 as indicated with the blue circle. It is
hence adiabatically connected to the ground state of the homogeneous triple well with
J12 /J23 = 1 (blue square). In a final measurement we made an attempt to realize sucha state by approaching the third well towards the central well. As indicated in the
figure, this effectively ramps the tunnel coupling J12 from zero to the homogeneous
triple well, J12 = J23 . The excitation gap between the lowest states is only on theorder of the super exchange interaction energy 4J 2 /U . For U/J = 16 and typicaltunneling rates around 100 Hz in our experiment, the splitting is only a few Hz, suchthat the purity of the experimental state might suffer a lot from too fast changes ofthe optical potentials.
To initialize the system we start from the biseparable state that we discussed above
and approach the third well. For the analysis we proceed as before by measuring(3)
both the in situ populations and the momentum-density correlation function ξ(↑↑↓) .
We show the measured momentum-density correlator C (3) (d1 , d2 ) in Fig. 9.11(b). We(3)
decompose the momentum correlation function ξ(↑,↑,↓) and obtain a 61 × 81 matrix B

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

Figure 9.11: Reconstruction of commensurately spaced triple well. (a) Illustrationof preparation of homogeneous triple well. (b) Measured momentum-density
correlator C (3) (d1 , d2 ). Number of data points: 10300 (c) Sampled posterior
distribution of all 36 off-diagonal entries. Only the major three entries arecolored. Distributions for remaining 33 entries scatter around zero and areshown in grey. (d) Reconstructed momentum-density correlator. (e) Bayesianmean estimate ρBME (upper row) and standard deviations of all entries (lowerrow).

with rank(B) = 37. As discussed in the previous chapter, the reduction of rank(B) as
compared to the incommensurately spaced triple well is due to the fact that there areless possible lattice momenta (since a12 = a23 = a). We define the likelihood function
as above by using the full matrix B as constraints and perform the Bayesian state
estimation. We again sample the posterior distribution with 50000 steps. The sampled posterior distribution of all off-diagonal entries is presented in Fig. 9.11 (c). Only
the major three entries are colored, ρ13 (red points), ρ12 (dark blue points), and ρ23(light blue points). All remaining entries scatter around zero and are shown in grey.
The histograms below confirm that all constrained values are Gaussian distributed.
The posterior distribution shows that the three major off-diagonal entries have a significant imaginary part, which can also be seen in the mean density matrix ρBME in (e).
In combination with the relatively high amount of doubly-occupied sites (ρ44 − ρ99 ),and a computed purity of p = Tr(ρ2 ) = 0.29(2) this indicates that the ramp to thefinal state was not performed in an adiabatic way. Indeed the chosen timing for the
ramp of 125 ms from J = 0 to the final configuration is fast compared to timescales
given by the super exchange energy. Such a quench results in a superposition state of
the lowest states, causing off-diagonal entries of the density matrix to become imaginary. Dephasing and random excitations to higher states due to technical noise lead toa substantial reduction of the purity of ρBME . The comparatively large uncertainties
on ρBME are mainly caused by the reduced rank of B but also by too little statistics
of about 10300 shots after post selection (71% post-selection rate). Nevertheless weagain verify the validity of the reconstructed density matrix by computing the momentum correlator that one would expect from the state ρBME . We show the result

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CHAPTER 9. DENSITY-MATRIX RECONSTRUCTION

in Fig. 9.11 (d) and find very good agreement.
Despite the very low purity of the reconstructed density matrix, this example
shows that our reconstruction scheme can readily be applied both to states involving
technical noise or poor statistics and in particular to mixed states. This is particularly
relevant for the characterization of experimental states since they always suffer fromdecoherence and reduced purity.
The density matrix contains the full information of a state. For instance its purity
characterizes the quality of the state in the sense of how mixed it is. Being able toreconstruct the density matrix of experimentally realized states is not only a valuable
tool to characterize the performance of an experiment, it also enables direct calculation of any observable including entanglement. In the next chapter we will use theexperimentally obtained density matrices to investigate entanglement properties in
identical-particle states. We are particularly interested in the entanglement between
particles rather than spatial mode entanglement, as we will motivate and discuss
throughout the chapter.

116

Chapter 10

Identical-Particle
Entanglement
Entanglement is one of the most fundamental properties of quantum mechanics. In
fact E. Schrödinger already considered entanglement to be ’the characteristic trait ofquantum mechanics, the one that enforces its entire departure from classical lines ofthought’ [214]. Initially causing many conceptual problems [215] such as the violationof locality and causality, entanglement turned out to be so fundamental that it can beused to fundamentally test quantum mechanics by ruling out any local hidden variable
theories via the violation of Bell’s inequalities [216]. On a more practical level, entanglement is seen as a major resource for quantum enhanced sensing and metrology[21, 217, 218], for quantum computation, but also for identifying and studying phasetransitions [74, 75] or identifying topological states [219–222]. Despite its usefulness
and wide range of applications, entanglement still seems to raise many conceptual
questions. One reason for the mysterious nature of entanglement might be the factthat entanglement is not only a property of a quantum state but entirely depends onthe choice of partitioning for which in turn there are various types one can choose
from.
A particularly useful choice of partitioning is to divide the system into spatial modes[223, 224] and to ask for entanglement between different spatial regions of the system. Scaling analysis of such entanglement has brought together many seemingly
unrelated physical systems such as black holes, quantum information science, andquantum many-body physics [75]. Furthermore the study of spatial mode entanglement helps to understand ground-state properties of the Hubbard model [59, 224–226]and enables entanglement steering in spatially separated Bose-Einstein condensates[227–229]. However, this is clearly not the only choice to partition a system and has
pathological limits in terms of extracting useful entanglement as a single particle delocalized in free space provides entanglement beween infinitely many spatial modes.
Another intuitive choice of subsystems is to consider individual particles. This obviously can cause conceptual problems, in particular when it comes to states of indistinguishable particles. The fact that particles are indistinguishable raises the question
whether they are all entangled with each other, which has led to a long discussion in117

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

the literature on the role of fermionic exchange symmetry on measured entanglement
properties [230–240].
In this chapter we use the reconstructed density matrices that we presented in theprevious chapter to investigate entanglement properties in states with indistinguishable particles. We are particularly interested in entanglement between particles rather
than spatial modes. We will discuss, that special care has to be taken for fermionicantisymmetrized wave-functions and introduce the simple notion of an antisymmetric
negativity for quantifying particle-particle entanglement of fermionic density matrices. We start by reviewing the basic concept of entanglement and its quantification.
The content of this chapter has been published in [241].

10.1

A Brief Introduction to Entanglement

In this section we try to give an intuition of how an entangled state looks like and try
to highlight where conceptual problems may arise. We do not rigorously discuss all
facets of entanglement, for which we refer the reader to the literature [242, 243].
In simple words, when two atoms are entangled with each other, they can no longer
be described as individual constituents of the whole system. They rather have to beconsidered together since any action on one atom immediately affects the state of theremaining atom. More formally, for two subsystems A and B with Hilbert spaces HA
and HB , respectively, a quantum state |Ψ⟩AB is called entangled with respect to thepartitioning A|B if it is not separable with respect to the tensor factorization HA ⊗HB ,i.e. if it cannot be written as a product state of the form |Ψ⟩AB ̸= |ψ⟩A ⊗ |ψ⟩B .
It is instructive to consider the following simple example of two atoms, which we canboth initialize in an arbitrary superposition of two internal states. Labeling those twostates as |↑⟩ and |↓⟩, we can write down the normalized general single-particle wave
function as|Ψ1 ⟩ =
|Ψ2 ⟩ =

√︁1 − α2 |↓⟩⏞ ⏟⏟ ⏞α′√︁β |↑⟩ + 1 − β 2 |↓⟩ .
⏞ ⏟⏟ ⏞α |↑⟩ +

(10.1)β′With the two above states we obtain a general two-atom wave function by tensor
multiplying the two states,|Ψsep ⟩ = |Ψ1 ⟩ ⊗ |Ψ2 ⟩ = (αβ |↑↑⟩ + αβ ′ |↑↓⟩ + α′ β |↓↑⟩ + α′ β ′ |↓↓⟩) .

(10.2)

This state is fully described by the two parameters α and β. However, it immediately
becomes clear that this wave function cannot describe all states that are allowed byquantum mechanics, for which one needs four parameters:|ΨG ⟩ = (a |↑↑⟩ + b |↑↓⟩ + c |↓↑⟩ + d |↓↓⟩) .

(10.3)

As illustrated in Fig. 10.1 (a), all two-particle states of the form |ΨG ⟩, that cannot be
written in the form of a separable state |Ψsep ⟩ = |Ψ⟩A ⊗ |Ψ⟩B are called entangled
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CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.1: Entanglement detection. (a) A state that cannot be factorized into its
subsystems is called entangled. An entanglement witness is a sufficient but
not necessary criterion for entanglement. (b) Illustration of spatial mode entanglement, which is used to characterize many-body systems by dividing asystem into spatial modes and asking for entanglement between these subsystems. (c) A single particle delocalized across the boundary of the subsystems
induces mode entanglement between systems A and B.

states.
A simple example of an entangled wave function based on the above example is thespin wave function of two atoms in a spin-singlet configuration,1|ΨS ⟩ = √ (|↑↓⟩ − |↓↑⟩) .
2(10.4)In order to write |ΨS ⟩ as a separable state in the form of Eq. (10.2), we need αβ =α′ β ′ = 0 but at the same time also αβ ′ = α′ β = √12 , which is impossible. For thestate |ΨS ⟩, a projective measurement on one atom can have both outcomes |↑⟩ or |↓⟩,
but would immediately affect the spin of the other atom, i.e. detecting the first atomin the |↑⟩ state, immediately projects the second atom into the |↓⟩ state. The definition of entanglement via the non-separability of the wave function is very general anddoes not only apply to internal modes of two atoms. There is obviously a range ofchoices of how to characterize entanglement. This leads to many different notions ofentanglement such as mode entanglement or particle-particle entanglement [223], butalso entanglement between different systems such as atom-photon, photon-phonon or
magnon-photon-phonon entanglement is possible [17, 244, 245].
As already mentioned above, a very useful choice for the characterization of ultracold
atom experiments is entanglement between spatial modes [59, 224, 229, 246]. The
basic idea of mode entanglement is illustrated in Fig. 10.1 (b). An ensemble of atoms
trapped e.g. in a two-dimensional optical lattice is spatially divided into two subsystems A and B. By looking for example at atom number fluctuations within the two
subsystems, entanglement can be detected by the violation of entanglement witnesses,a sufficient but not a necessary condition for entanglement (see Fig. 10.1 (a)). However, the detected mode entanglement entirely depends on the chosen boundary andeven a single atom, delocalized across two wells might induce mode entanglement, if
it is delocalized across the boundary of the subsystems, see Fig. 10.1 (c). In that sensea single atom in the ground state of a balanced double-well potential, as we discussed

119CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENTin Sec. 7.4.1, where the wave function in mode representation is given by1|Ψ⟩ = √ (|10⟩ + |01⟩) ,2(10.5)

with the atom being either in the first (|10⟩) or the second (|01⟩) well, induces spatial mode entanglement between the two wells. Hence a single atom delocalized
across many wells can in principle induce entanglement between arbitrarily many
spatial modes. Mode entanglement does not only exist between spatial modes. The∑︁momentum-space wave function of a single atom localized in space ( i c†ki |000...0⟩)is a superposition state of infinitely many momentum modes and can hence also beinterpreted as a mode-entangled state, where all momentum modes are entangled.
An alternative possibility is to consider entanglement between particles rather thanmodes. Obviously in the case of identical particles special care must be taken andconceptual difficulties arise when particles are fully indistinguishable.

10.2

Particle-Particle Entanglement

Entanglement between particles and the role of exchange symmetry has been extensively discussed in the literature [230–239, 242]. Taking individual atoms as subsystems immediately causes conceptual problems when two or more particles are indistinguishable. In order to uniquely identify single atoms as subsystems, particles have to
be labeled {1...N } and a first-quantized formalism has to be adopted. The conceptual
problems already become apparent by considering two identical fermions localized in
two spatial modes, such as the state that we extensively discussed in Chap. 8. Due tofermionic exchange symmetry, the first-quantized wave function of such a state has
to be antisymmetrized over the spatial degree of freedom,1|Ψ⟩ = √ (|L⟩1 |R⟩2 − |R⟩1 |L⟩2 ) ,2(10.6)

where the subscripts are particle labels and L and R the left and the right mode,
respectively. This state is not a separable state since it cannot be written as the
product of two single-particle wave functions. It even resembles the maximally entangled Bell-state and is formally an entangled state. However, this type of entanglement
only arises from antisymmetrizing the wave function and hence exists between any
two identical fermions. Although it might be argued that such entanglement is a mereartefact of the formalism of first quantization, it is still under debate whether it canactually be used for quantum information processing [247–249]. With the advent ofexperiments operating with individually controllable, indistinguishable particles [59–63, 224–226] the debate on the role of exchange symmetry on measured entanglement
properties has gained new urgency. In fact as we have demonstrated in Chap. 8 andChap. 9, fermionic exchange statistics can induce strong correlations in experimental
observables that look very similar to correlations in strongly interacting systems. It
is hence desirable to analyze experimental measurements in a way that entanglement
induced by (anti)symmetrization can be separated from other forms of entanglement.

120

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Before discussing how to quantify entanglement and how to isolate the different contributions to the detected entanglement, we need to introduce a few more concepts suchas mixed-state entanglement, separability and genuine multipartite entanglement.

10.3

Generalization of Entanglement

In this section we briefly comment on how to extend the concepts that we introducedin the previous section to mixed states and to larger systems that can be partitioned
into more than just two subsystems.
Mixed-State Entanglement
So far we only considered pure quantum states, i.e. states with a purity p = Tr(ρ2 )of one. However, experimentally characterized quantum states usually suffer fromexperimental noise and hence a reduced purity p < 1. In order to characterize experimental density matrices, the definition of entanglement has to be generalized tomixed quantum states. A mixed quantum state is called separable with respect tothe partitioning A|B if its density matrix can be written as a probabilistic mixture of
separable pure states [250],
∑︂
A|B
ρmix
pi ρsep,i ,(10.7)
sep =
i
A|B

∑︁

with
pi = 1 and the density matrix of a pure separable state ρsep,i , which can be
written as
A|Bρsep,i = |ψi ⟩ ⟨ψi |A ⊗ |ϕi ⟩ ⟨ϕi |B .
(10.8)
If the density matrix cannot be written in the form of Eq. (10.7), the state is called
entangled. For a general mixed density matrix there are of course infinitely many
decompositions into a probabilistic mixture of pure states and ruling out that there
is any decomposition of the form of Eq. (10.7) is a non-deterministic polynomial-time(NP)-hard problem [251].
Separability
The notion of entanglement becomes richer for systems that can be divided into
multiple subsystems. In such cases, there are different choices for the boundary
between two subsystems and both the detection but also the amount of detectedentanglement depends on the chosen boundary. Furthermore, partitioning the system
into more than two subsystems leads to the notion of multipartite entanglement (seebelow). To better understand the structure of possible states, one introduces theconcept of partial separability [252]. Let us consider a state with N modes and split
the system {1, ..., N } into k ≤ N subsystems. As illustrated in Fig. 10.2, we call thestate k-separable with respect to the split αk , if it can be written as a factorized stateover the partitioning αk , i.e.
|Ψ⟩ = |ψ⟩ ⟨ψ|α1 ⊗ |ϕ⟩ ⟨ϕ|α2 ⊗ ... |χ⟩ ⟨χ|αk .

(10.9)

121

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.2: Separability of quantum states. A state that can be factorized over the
k-fold partitioning αk is called k-separable. For each k with 1 < k < N there
are multiple possible partitionings as indicated on the right side for the caseof N = 4 and k = 2.

For a given k with 1 < k < N there are multiple possible partitionings αk . Notethat a k-separable state is always k − 1 separable while the opposite is not generally
true. For k = N we call the state fully separable. Since entanglement is defined
via the contrapositive of separability, the hierarchy of entanglement is inverted withrespect to separability. A fully separable state is not entangled at all whereas in a
non-separable state all possible bipartitions are entangled. Such a state is called agenuinely multipartite entangled state.
Genuine Multipartite Entanglement
To better understand genuine multipartite entanglement let us consider the following
state of three spins
|Ψ⟩ =
=

1
√ (|↑↑↓⟩ + |↑↓↑⟩)
2
1√ (|↑⟩ ⊗ (|↑↓⟩ + |↓↑⟩)) .
2(10.10)This state is separable with respect to the partitioning |·| · ·⟩ but not with respect tothe partitioning |· · |·⟩. The state is hence biseparable and not genuinely multipartite
entangled. However it is also not fully separable and therefore features bipartite entanglement within the partitioning |· · |·⟩. A famous example of a genuinely multipartite
entangled state is the GHZ-state [253, 254],
1√ (|↑↑↑⟩ + |↓↓↓⟩) .
2(10.11)1√ (|↑↑↓⟩ + |↑↓↑⟩ + |↓↑↑⟩) .
3(10.12)|ΨGHZ ⟩ =or the W-state [255],|ΨW ⟩ =

These state are not separable with respect to any possible bipartition and can therefore
not be factorized. They are therefore called genuinely multipartite entangled. In asimilar way as above, we call a mixed state genuinely multipartite entangled, if it canneither be written as a biseparable state with respect to any bipartition A|Ā [256]122CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

nor as a probabilitics mixture thereof
ρbs =

∑︂

pi ρAi |Āi ,(10.13)

i

with

∑︁

10.4

pi = 1 and separable states ρAi |Āi .

Entanglement Quantification

So far we only defined an entangled state as being a state that is not separable. However, for the general two-particle state of the form of Eq. (10.3) it seems to be intuitive
to find states that are more entangled than others. In order to understand how entanglement can be quantified, we follow Ref. [257] and first introduce the concept ofLocal Operations and Classical Communication (LOCC).
Suppose we have prepared a quantum system that we partition into two subsystems
A and B, which we transport into two distant laboratories. The two laboratories canperform local operations on their part of the system and communicate via a classicallink with each other. The set of all operations on the systems that only involve local
operations on the two subsystems and classical communication between the two laboratories is referred to as LOCC. If the two subsystems have not been entangled before,
LOCC cannot create any entanglement between them. Consequently, entanglement
is not allowed to increase under any LOCC operation. These two fundamental lawsof quantum information processing [258, 259] can be used to implement a hierarchyof entanglement.
If a state ρ can be transformed into state σ only by LOCC, ρ is at least as entangled
as σ. If the transformation is not possible, ρ is less entangled as σ [257]. However,care must be taken when comparing different classes of entangled states. E.g. in asystem of three atoms it is not possible to compare the entanglement of a GHZ state
to the entanglement of a W-state. The states cannot be converted into one anotherby LOCC but are both maximally entangled states [255].
In order to generalize this idea and to make it applicable to mixed states, one might
ask the question: how many copies of a state ρ are necessary to find an equal amountof entanglement in m copies of a different state σ, i.e. for which n can ρ⊗n be transformed into σ ⊗m only with LOCC. The ratio n/m is used for ordering the amount ofentanglement in the two states ρ and σ. Based on these ideas important entanglement
measures can be defined such as the Entanglement Cost [260, 261], the Distillable Entanglement [262, 263] or the Entropy of Entanglement [264, 265].
In contrast to this operationally motivated approach to quantifying entanglement
there is also an axiomatic approach to entanglement measures. Based on the above
considerations certain requirements for a proper entanglement measure can be formulated [257, 264, 266]:
1. An entanglement measure E(ρ) is a mapping from density matrices to realpositive numbers, ρ ↦→ E(ρ)
2. E(ρ) = 0 for any separable state ρ.

123CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

3. E does not increase on average under LOCC.
4. For a pure state ρ = |ψ⟩ ⟨ψ|, E reduces to the entropy of entanglement.
Apart from many others, a very useful axiomatically motivated entanglement measureis the (logarithmic) Negativity [267]. Since we will be using the Negativity on theexperimentally obtained density matrices, we briefly summarize its main properties.
Negativity
For a state, described by the density matrix ρ, the Negativity is defined as
N (ρ) =||ρΓA || − 1
2(10.14)

where ρΓA is the partial transpose of ρ with respect to subsystem A, and || · || denotes
the trace norm. Equivalently the negativity can be calculated as the sum of the abso∑︁lute values of all negative eigenvalues of ρΓA , i,λi <0 |λi |. The negativity is a convex∑︁∑︁
function, N ( pi ρi ) ≤pi N (ρi ) and does not increase under LOCC operations.
For a separable state ρs = ρA ⊗ ρB , one finds ρΓs A = ρTA ⊗ ρB = ρA ⊗ ρB = ρs andconsequently a negativity of 0. For 2 × 2 and 2 × 3 systems (i.e. systems of two
qubits or two qutrits), N > 0 is a sufficient and necessary condition for entanglement[268, 269]. For any larger system a finite negativity is only a sufficient but not a necessary condition for entanglement. The negativity is an entanglement measure, that iseasy to compute because it reduces to calculating the eigenvalues of a density matrix.
As we will see in the next section, when applied to states where we choose particles
as subsystems, the negativity detects entanglement due to the antisymmetrization of
the fermionic wave function.
PPT-mixtures
The negativity relies on the fact that separable states have a positive partial transpose(PPT) [268, 269]. While this condition is only sufficient and necessary for 2 × 2 or
2 × 3 systems, it is still a sufficient condition for larger systems. Separable states
have a positive partial transpose, so whenever a state does not have a positive partial
transpose, it is an entangled state. For later purposes we define a general mixed PPTstate as a state that can be obtained as a probabilistic mixture of pure PPT states[256],∑︂ρmix
pi ρppt .
(10.15)
ppt =
i

10.5

Particle-Particle Negativity

Let us again address the state of two identical fermions in a double-well potential,Ψ = √12 (|L1 R2 ⟩ − |R1 L2 ⟩) and apply the above definition of the negativity. We
mentioned already, that the state cannot be factorized due to antisymmetrization of
the wave function. In the first-quantized formalism, its density matrix ρ = |Ψ⟩ ⟨Ψ| is124

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

given by⎛0
1⎜⎜0
ρ = |Ψ⟩ ⟨Ψ| = ⎜
2 ⎝0
0

0
1
1
0

0
1
1
0

⎞
0
0⎟
⎟
⎟,
0⎠

(10.16)0

where we used the basis (|L1 L2 ⟩ , |L1 R2 ⟩ , |R1 L2 ⟩ , |R1 R2 ⟩). We perform the partial
transpose on one of the two atoms and get
⎛
0
⎜
1 ⎜0
ρΓ = |Ψ⟩ ⟨Ψ| = ⎜
2 ⎝0
1

0
1
0
0

0
0
1
0

⎞
1
0⎟
⎟
⎟,
0⎠

(10.17)0with the four eigenvalues λ1 = −1/2, and λ2,3,4 = 1/2. We calculate the Negativity to
be N (ρ) = 0.5 > 0. The negativity hence detects the state as non-separable and therefore as an entangled state. This result suggests that any two identical fermions areentangled because of their exchange symmetry. How should this result be interpreted?
According to Ghirardi, Marinatto, and Weber (GMW) [242] an antisymmetrizedstate should not be regarded as entangled if it can be obtained from antisymmetrizing
a separable state. In their work they define entanglement between two subsystems ofa composite system via the possession of a complete set of properties. This means that
if the two subsystems possess definite physical properties that are independent of theexistence of the other part, the two subsystems are non-entangled. While also withthis definition some conceptual problems arise for the case of identical particles, they
rigorously show that two identical fermions that are described by a pure normalizedstate |Ψ⟩ are non-entangled, if and only if |Ψ⟩ can be obtained by antisymmetrizinga factorized state.
For the wave function of two identical fermions discussed above, this immediately
implies that we should not consider the state as an entangled state, because it isobtained by antisymmetrizing the separable state,
1PA√ (|L1 R2 ⟩ − |R1 L2 ⟩) ,|Ψ⟩ = |L1 R2 ⟩ −−→2(10.18)where PA denotes the antisymmetrization operator. Note that the state before antisymmetrization is not unique, as also the state |R1 L2 ⟩ yields the same antisymmetricstate. While this trivial example nicely illustrates the idea of GMW, finding such astate e.g. for a mixed state consisting of three interacting particles is not a straightforward task. Hence, for the calculation of particle-particle entanglement special
care has to be taken in order to separate contributions to entanglement from antisymmetrization from other origins of entanglement such as interaction. In the next
section we therefore introduce the notion of an Antisymmetric Negativity, which isable to separate those two different origins of entanglement.

125

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.3: Schematic illustration of Antisymmetric Negativity. From the experimental results we obtain a normalized density matrix, which might not byphysical and contain negative eigenvalues due to systematical uncertainties.
The Bayesian inference method provides us with a representative set of physical density matrices that is compatible with our experimental uncertainties.
The AN searches for the optimal state σopt by minimizing the negativity ofthe optimization variable σ under the constraint that the projection on theantisymmetric subspace retrieves the initial state ρA . We call the negativityof σopt the antisymmetric negativity of ρA .

10.6

Antisymmetric Negativity

As described in the previous section, according to the GMW criteria [242] an antisymmetric state that is obtained from antisymmetrizing a factorized state should
be considered as a separable, non-entangled state. This suggests that entanglement
properties of identical-particle states should be addressed before antisymmetrization.
Based on this intuition we introduce the simple notion of an Antisymmetric Negativity, NA [241, 270], which determines the smallest entanglement that must alreadyexist in a state before (anti)symmetrization. For the mathematical details of the antisymmetric negativity, please see Ref. [270].
For an antisymmetrized, fermionic wave function ρA we consider the functional
EA (ρA ) = min {E(σ) : PA σPA = cρA } ,σ≥0(10.19)for any entanglement measure E. PA is the projection onto the antisymmetric subspace and the factor c = max{Tr(PA σ) : σ = σPPTm } is the maximal projection
probability of σ such that ρA may not be obtained by antisymmetrizing a PPTmixed state [270]. As derived in [270], c = 1/2 for the bipartite case with same local
dimensions such as our double-well states. Instead, c = 1/3 for the triple well, as
shown in [271]. We choose c = 1/2 for the discussion here. As illustrated in Fig. 10.3
the optimization variable σ is a normalized, positive-semidefinite quantum state that
does not obey any specific exchange symmetry and exists in a larger Hilbert spacethan the antisymmetric, physical state ρA . The optimization searches for the state
σopt with the smallest entanglement E(σopt ) under the constraint that its antisymmetric projection is 21 ρA .
Thus, for a given fermionic state ρA , if there exists a factorized state σ such that
PA σPA = 12 ρA , then EA (ρA ) = 0. If there is no such state, then EA (ρA ) > 0 and the126CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

identical-particle state is considered to be entangled. If we apply this formalism tothe state of two identical fermions, Eq. (10.18) immediately implies that EA (ρ) = 0
and therefore identifies the state as non-entangled.
This idea can be applied to any entanglement measure E but in the case of E being
the standard negativity N it can be explicitly calculated in the form of a semidefinite
program (SDP). We define the Antisymmetric Negativity (AN ) as{︃}︃1
NA (ρA ) = min N (σ) : PA σPA = ρA .
σ≥02

(10.20)Semi Definite Programming
Semi Definite Programming is a field of convex optimization and finds numerous
applications in quantum information theory [272, 273]. For Φ being a hermitianpreserving map between two complex Euclidean spaces A and B, with A and B
hermitian operators A ∈ Herm(A) and B ∈ Herm(B), a semidefinite program (SDP)
consists of the triple (Φ, A, B) associated with the following optimization problem,
minimize⟨A, X⟩

such that

Φ(X) = B
X ≥ 0,(10.21)

where ⟨., .⟩ is a linear function that is minimized under the constraints Φ(X) = B and
X being positive semidefinite, X ≥ 0. Following [270, 272], we can rewrite the trace
norm of a matrix as a SDP, which we can immediately extend to the negativity. We
find that the negativity of a matrix ρ is given by the optimal value of the SDP
minimize(Tr(M ) − 1)/2

such that−M ≤ ρΓ ≤ M.
M ≥ 0,(10.22)where ρΓ is the partial transpose of the partitioning for which the negativity is calculated. By adding further constraints to the above SDP we can ultimately formulate
the antisymmetric negativity as a SDP. For an antisymmetric fermionic state ρA wecall the bipartite antisymmetric negativity AN 2 the optimal value of the following
optimization,
minimize(Tr(M ) − 1)/2

such that−M ≤ X Γ ≤ M
M ≥0
X≥0
Tr(X) = 1
PA XPA = Tr(PA X)ρA
Tr(PA X) = c.

(10.23)127

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

This SDP minimizes the negativity of a positive-semidefinite, normalized state X,
whose projection on the antisymmetric subspace is cρA . Following [256, 274] we
generalize the AN to detect genuine multipartite entanglement. To this end we firstextend the standard negativity to the tripartite case. For a three-atom state we callthe genuine tripartite negativity N3 of a state ρ the optimal value of the following
SDP,
minimize(Tr(MA|BC + MB|AC + MC|AB ) − 1)/2

such that

MA|BC , MB|AC , MC|AB ≥ 0
ρ̃A|BC , ρ̃B|AC , ρ̃C|AB ≥ 0−MA|BC ≤ ρ̃ΓA|BC ≤ MA|BC−MB|AC ≤ ρ̃ΓB|AC ≤ MB|AC−MC|AB ≤ ρ̃ΓC|AB ≤ MC|ABρ = ρ̃A|BC + ρ̃B|AC + ρ̃C|AB ,(10.24)

where the indices denote the partitioning of the system, i.e. A|BC means that thepartial transposition operation is carried out on subsystem A. We finally extend N3
to the genuine tripartite antisymmetric negativity AN 3 via the optimal value of the
SDP,minimize(Tr(MA|BC + MB|AC + MC|AB ) − 1)/2

such that

MA|BC , MB|AC , MC|AB ≥ 0
ρ̃A|BC , ρ̃B|AC , ρ̃C|AB ≥ 0−MA|BC ≤ ρ̃ΓA|BC ≤ MA|BC−MB|AC ≤ ρ̃ΓB|AC ≤ MB|AC−MC|AB ≤ ρ̃ΓC|AB ≤ MC|ABρ̃ = ρ̃A|BC + ρ̃B|AC + ρ̃C|AB
PA ρ̃PA = Tr(PA ρ̃)ρA
Tr(PA ρ̃) = c.

(10.25)The latter equation can be readily extended to the N -partite case.
For the calculation of the AN we perform all optimizations using Qetlab [275] andCVX, a package for specifying and solving convex programs [276, 277].

10.7

Application of AN to Experimental States

In order to benchmark the usefulness of the AN we investigate the experimentallyrealized states that we discussed in Sec. 9.3. In particular we will demonstrate that(i) the AN can be computed from experimental data including noise; (ii) it identifies
states which exhibit correlations only due to quantum statistics as unentangled; (iii)
it identifies interaction-driven entanglement in two-particle systems, and (iv) it canbe extended to multipartite scenarios.
128

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENTAs already mentioned, we adopt a first-quantized formalism that allows us to treat
particles as subsystems. To this end we label the particles {1, ..., N } and convert alldensity matrices into their first-quantized representation. For the double-well data
this is achieved by mapping the 4 × 4 density matrices onto the 16-dimensional basis {|Xσ⟩ |Y τ ⟩}, for X, Y ∈ {L, R} and σ, τ ∈ {↑, ↓}. The 9 × 9 density matricesof the triple well are mapped onto the 216-dimensional basis {|Xσ⟩ |Y τ ⟩ |Zκ⟩}, for
X, Y, Z ∈ {L, C, R} and σ, τ, κ ∈ {↑, ↓}. This is done using the substitution ruleaccording to the slater determinant, i.e. |X⟩↑ |Y ⟩↓ → √12 (|X ↑⟩ |Y ↓⟩ − |Y ↓⟩ |X ↑⟩)for the double-well case and analogously for the triple-well case.
The states that we investigate can be grouped into three different classes. We startby discussing all spin-polarized states |↑, ↑⟩, |↑, ↑, ↑⟩ and |→, →⟩, which we discussed
in Chap. 8 and Sec. 9.3.1, respectively. We then continue with the discussion of two
interacting atoms in the ground state of a double well, see Sec. 9.3.2, and finallyinvestigate the two triple-well states, which we discussed in Sec. 9.3.3. For the doublewell states, we will try to give some explicit examples in order to better illustrate thecalculations. Expressions and density matrices for the triple well are typically toolarge to be explicitly shown.
Separable States
The first class of states that we investigate are spin-polarized states. Since they
consist of identical fermions, these are all strictly non-interacting states and hencecorrelations are purely caused by exchange symmetry. We discussed the three different
states |Ψ1 ⟩ = |↑, ↑⟩, |Ψ2 ⟩ = |↑, ↑, ↑⟩, and |Ψ3 ⟩ = |→, →⟩.
We explicitly performed the calculation of the negativity of the first state in Sec. 10.5
and showed that the negativity evaluates to N (ρ1 ) = 0.5. We also discussed that the
AN correctly identifies the state as non-entangled, AN (ρ1 ) = 0 (Eq. (10.18)). The
calculation of the second state is analogous. We first write down the first quantized,
antisymmetrized wavefunction,
|ΨA
2 ⟩ =

1
√ (|L ↑, C ↑, R ↑⟩ + |R ↑, L ↑, C ↑⟩ + |C ↑, R ↑, L ↑⟩6
− |R ↑, C ↑, L ↑⟩ − |L ↑, R ↑, C ↑⟩ − |C ↑, L ↑, R ↑⟩) ,(10.26)AA
and obtain the 36 × 36 first-quantized density matrix by calculating ρA
3 = |Ψ3 ⟩ ⟨Ψ3 |in the basis {|X ↑⟩ |Y ↑⟩ |Z ↑⟩}, with X, Y, Z ∈ {L, C, R}. For this state we explicitly
calculate the negativity to be N =1 for all possible partitionings due to the exchange
symmetry. The AN identifies this state as separable, AN (ρ2 ) = 0, since it is obviously obtained by antisymmetrizing a separable state.
As we discussed in Sec. 9.3.1, the third state |ψ3 ⟩, which was obtained by rf-rotationof the first state |ψ1 ⟩ into the equatorial plane of the Bloch sphere, shows spin fluctuations upon measuring and momentum correlations in all three possible spin combinations. These spin fluctuations are only induced by the rotation of the basis statesand do not change the structure of the state. However, the reconstructed density
matrix suffers from technical noise and therefore a reduced purity, so that the state129

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.4: Calculated negativities of rotated-triplet state. Histograms of calculated AN (a) and N (b) over the posterior distribution of density matrices (5000 states) of the rf-rotated state |Ψ3 ⟩. We find a median of+0.064
AN (ρ3 ) = 0.048−0.048
and N (ρ3 ) = 0.6+0.13−0.11 , where the errors indicate the
68% credible interval. The large spread of the negativites is caused by thelarge experimental uncertainties and the missing constraints for 5 of the 6
off-diagonal entries (see Fig. 9.5).

is well suited to test the AN on separable mixed states. Generally, a reduced purity
also reduces the detected entanglement due to the convexity of entanglement mea∑︁∑︁sures (E( (pi ρi )) ≤pi E(ρi )). As described in the previous chapter, the Bayesianreconstruction scheme provides us with a posterior distribution of physical density matrices, consisting of 5000 states, that represent the experimental measurements. We
convert each of these density matrices into their first-quantized representation andcalculate both N and AN for each state. Figure 10.4 shows histograms of the calculated negativities. The standard negativity is centered around 0.6 with a median ofN (ρ3 ) = 0.6+0.13−0.11 where the errors indicate the 68% credible interval. In contrast, theAN is peaked at zero with a median and credible interval of AN (ρ3 ) = 0.048+0.064−0.048 ,compatible with zero. The large spread of the calculated negativities is caused by
the poor reconstruction of that particular state, as we discussed in Sec. 9.3.1. Nevertheless, the AN removes the contribution of antisymmetrizing the wave function
and identifies the spin-rotated state as being compatible with a non-entangled state.
These three examples show that the AN identifies both pure and mixed separable
states as non-entangled.
Interaction-Induced Entanglement
To test the AN on states where entanglement is induced by interaction we investigate
the interacting singlet state in the double-well potential (see Sec. 9.3.2). We proceed
in a similar way as above: for each interaction strength we calculate both N and
AN for the whole posterior distribution that we obtain from the Bayesian quantum
state estimation, consisting of a representative set of 5000 density matrices each. In
addition we also calculate N and AN for theoretical density matrices for pure groundstates of the balanced two-well Fermi-Hubbard model, obtained by diagonalizing the
Hamiltonian in Eq. (7.4).
In Fig. 10.5 we illustrate the optimization of the SDP and the search of the AN witha single exemplary density matrix. We randomly chose a state of the posterior distribution from the Bayesian state estimation at an interaction strength of U/J = 18.5.
The state in (a) shows the 4 × 4 density matrix obtained by the state reconstruction.

130

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.5: Example of optimization. (a) Density matrix of a randomly chosen matrix from the posterior distribution of the Bayesian state estimation of theinteracting double well with U/J = 18.5. (b) First-quantized density matrix
for state in (a). For a clearer view, the basis states are only labeled from
1...16. (c) Density matrix of the optimal state σopt , of which the negativity
N (σopt ) is the antisymmetric negativity of ρA . (d) Antisymmetric projectionof the optimal state. (e) Difference between antisymmetrized state in (b) andantisymmetrized optimal state in (d).

Using the substitution rule as explained above, we convert the 4 × 4 density matrix
into its 16 × 16 first-quantized representation ρA , which we show in (b). ρA is thestate of which we want to compute the AN . The result of the optimization, σopt , isshown in (c). Although this state is normalized, i.e. Tr(σopt ) = 1, it is not a physicalstate since it does not necessarily satisfy fermionic exchange symmetry and therefore
allows the occupation of states that are forbidden by the Pauli-exclusion principle.
For instance the first entry ρ11 (which is non-zero in σopt ) is a state where two spin-up
particles occupy the same spatial mode. Antisymmetrizing σopt yields a physical statethat satisfies fermionic exchange symmetry. The state ρopt
A = PA σopt PA is shown in(d). We finally compute the difference between the states in (d) and (b) and show
the result in (e). This difference is basically zero, whereas the residual noise on theindividual entries is determined by the convergence parameters during the optimization. For the double-well case, the optimization only takes a few seconds per density
matrix.
Figure 10.6 presents the calculated negativities as a function of the interaction
strength U/J. All shown experimental values are the median with a 68% credibleinterval. In addition we also show one exemplary posterior distribution of N and
AN for an interaction strength of U/J = 18.5 in (b) of the same figure. For all
interaction strengths the experimental values are reduced compared to the theoretic
curves due to a reduced purity of the experimental density matrices. The solid lines
are the calculated negativities of the pure ground states. Both negativities show thesame qualitative behaviour, i.e. the states become stronger entangled for stronger interaction and are symmetric around the non-interacting state at U/J = 0. However,the standard negativity is always larger than the antisymmetric negativity and even131CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.6: Negativities of the interacting double well. (a) Calculated N (red)and AN (blue) for an interaction strength U/J ranging from −10 to 20.
Solid lines are the results from theoretical density matrices obtained fromdiagonalization of the Hamiltonian, data points are median and 68% credibleinterval of the negativities over the posterior distribution of the BME. (b)One example of the posterior distribution of N and AN for a repulsively
interacting state with U/J = 18.5.

N = 0.5 at U/J = 0, where the the theoretical and experimental AN both identify
the state as separable. Indeed the difference between both negativities is caused by
the antisymmetrizion of the density matrix over the spin degree of freedom. The AN
successfully removes this contribution to the negativity and only detects entanglement
due to interaction. It is interesting to note that the AN takes exactly the same values
as the spin-mode negativity [62], since the two atoms could be uniquely identified bytheir spin state.

Three-particle states
To generalize the concept of the antisymmetric negativity to genuine multipartite entanglement we consider three particles in a triple-well configuration. As presented inSec. 9.3.3, we consider the two experimentally realized states at U/J = 16 that we
illustrate again in the top panel of Fig. 10.7. The first state is the incommensurately
spaced triple well where we prepared an interacting singlet in a double well and added
a third independent atom in the third tweezer. The second state is the ground stateof the homogeneous triple well with J12 = J23 = J. We also theoretically investigate
the same pure states for variable interaction strengths U/J from −20 to 20.
As we argued in Sec. 9.3.3, the first state with J12 = 0 has a clear bi-separable
structure and features complex momentum correlations that are substantially modified both by interactions and antisymmetrization over the third independent particle.
The second state is in principle expected to show genuine multipartite entanglement
but due to a very low experimental purity of the state (see Sec. 9.3.3) we might not
be able to detect it.
The evaluation of the data is similar to the double-well case. However, in addition to N and AN we also calculate the tripartite negativity N3 and the tripartite
antisymmetric negativity AN 3 as we defined in the previous sections. Due to thesignificantly larger first-quantized density matrices (216 × 216), each calculation of132

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

AN 2 and AN 3 takes about 15-30min. Instead of calculating the antisymmetric negativity over the whole posterior distribution (containing 50.000 states), we choose arandomly sampled, representative subset of 1000 density matrices for the calculationof the experimental AN 2/3 .
Figure 10.7 summarizes all negativities for both states. We note that the calculationof the bipartite and tripartite negativity (solid line and light-colored dots in (a) of
the figure) yield exactly the same results due to the antisymmetrized structure of thestates. As a first characterization of the entanglement in the system, we can use thestandard negativity (plotted in (a)) as an entanglement witness, but we will see that it
requires the AN to provide a complete picture of multipartite entanglement. As indicated in the graph by the colored areas, there are three different regions, in which thenegativity reveals information about the entanglement structure of the state. These
three regions are based on the following two statements (see Appendix B for proofs):
1. The maximum negativity that can be obtained by antisymmetrizing a fully separable three-particle state is one.
2. The maximum negativity that can be obtained by antisymmetrizing a biseparable
state is 1.9428
Hence any state with a negativity above one cannot be obtained from antisymmetrizing a fully-separable state and therefore exhibits at most a bi-separable structure.
Any state with a negativity above 1.9428 is necessarily tripartite entangled, since itcannot be obtained by antisymmetrizing a bi-separable state. Indeed, as shown by
the solid red line in Fig. 10.7 (a), the first state saturates these bounds for U/J = 0
and for U/J → ±∞. The two histograms in Fig. 10.7 (a) show the results of the calculation of the negativity over the posterior distribution for both experimental states.
We detect a negativity of N = 1.39 ± 0.02 for the first and N = 1.14+0.05+0.04 for thesecond state, respectively. We conclude that none of the states is fully separable andboth exhibit at least bipartite entanglement.
As we plot in Fig. 10.7 (b) the AN of the first state evaluates to zero both for the theoretical and the experimental state despite the fact that the state is not fully separable.
This result is independent of the chosen partitioning. For each of the three possible
partitionings, i.e. 1|23, 2|31, or 3|12, the minimization finds a state where a single
atom can be factorized out. For instance for the partitioning 3|12 and U/J → ∞ the
optimization finds the state Ψopt = (|C ↑⟩1 |R ↓⟩2 + |R ↓⟩1 |C ↑⟩2 ) |L ↑⟩3 . This statefactorizes in the chosen partitioning and therefore features a negativity of zero. However, if we choose to investigate the partitioning 1|23, the optimization finds the stateΨopt = |L ↑⟩1 (|C ↑⟩2 |R ↓⟩3 + |R ↓⟩2 |C ↑⟩3 ) and returns an AN of zero. Not that
one cannot fine a state, that is separable in both partitioning at the same time, which
projects onto the antisymmetrized state. This factorization of the optimal state isindependent of the interaction strength so that we detect an AN of zero for all valuesof U/J. Indeed the AN is zero whenever the dimensions of the partitioning coincides
with dimensions of the partitioning in which the state is biseparable.
The ground state of the homogeneous triple well (solid blue line in (a)) even exceeds
the bound for biseparability for strong attractive and repulsive interactions. While133

CHAPTER 10. IDENTICAL-PARTICLE ENTANGLEMENT

Figure 10.7: Bi- and tripartite antisymmetric negativity in the triple well. (a)Calculated standard negativity for both illustrated states. The histograms
show the negativity of the posterior distribution of both experimentally realized states. Both experimental states exceed the bound for fully-separable
states. The calculation of the bipartite and tripartite negativity yield thesame values for each state due to the antisymmetrized structure of the density
matrix. (b) Calculated antisymmetric negativity. The ground state of the homogeneous triple well shows finite values for any finite interaction strength,
certifying the presence of genuine tripartite particle-particle entanglement.
AN of experimental states is consistent with zero due to low purity of theexperimental density matrices.

this is a signature of genuine multipartite entanglement, the standard negativity does
not reveal anything about the entanglement structure for weaker interactions. However, both the bipartite and tripartite antisymmetric negativity AN 2/3 of the samestate (solid blue line and light blue dots in Fig. 10.7 (b)) show that the state exhibits
genuine tripartite entanglement for any finite interaction strengths. Only at U/J = 0
the state can be obtained by antisymmetrizing a fully-separable state, so that we
detect an AN of zero. Unfortunately the AN of the experimentally realized stateis also 0, as shown in the histograms in Fig. 10.7 (b) due to the poor purity of thereconstructed density matrix.
To assess the relevance of the AN for experimental investigations, we analyze therobustness of the tripartite antisymmetric negativity AN 3 against white noise. To
this end we consider the ground state of the homogeneous triple well at an interactionof U/J = 16, |Ψ⟩, an gradually add white noise, ρnoise = (1 − r)1 + r |Ψ⟩ ⟨Ψ|, withr ∈ [0, 1]. We find a finite tripartite antisymmetric negativity AN 3 down to r = 0.75,
which corresponds to a purity of p = 0.61. This is significantly higher than the purity that we achieved with the second experimental three-particle state, but can bereasonably expected in experiments.

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In this chapter we introduced a new notion of entanglement to quantify particleparticle entanglement for fermionic density matrices, called Antisymmetric Negativity. We applied the AN both to theoretical and experimental states and showedthat it identifies separable states as non-entangled and that it can be used to separate entanglement originating from exchange symmetry from entanglement induced
by interaction. The AN can also detect genuine multipartite entanglement, in particular, it can be used to reveal the entanglement structure of a multipartite entangledstates, which is not accessible via the standard negativity alone. Our results enable
the quantitative study of entanglement and exchange antisymmetry in experimental
settings and help to provide further insight into the entanglement of indistinguishable
particles.

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136

Chapter 11

Summary and Outlook
In the second part of this thesis I explored the analysis of momentum correlation functions for the characterization of small, strongly correlated fermionic systems. Thisapproach to characterizing a many-body quantum state is complementary to traditional tomography methods and does not require complex measurement protocols.
We benchmark the applicability of our approach on a range of different states that werealize in the laboratory, including non-interacting systems, where correlations arise
purely by fermionic exchange symmetry, and systems, where additional correlations
may arise due to strong interactions.
In systems of non-interacting, indistinguishable particles we measure strong connected
second and third-order momentum-density correlations. The contrast of the measured
correlation functions indicates a high degree of indistinguishability of the individual
particles, which we think is currently only limited by the expansion in the ODT.
The measurements show both that our system realizes an on-demand source of highly
indistinguishable fermions and that momentum correlations are perfectly suited todetect ordering in mesoscopic fermionic systems. This enables the future study of
many-body interference or quantum optics experiments with massive particles.
In systems of distinguishable particles, correlations do not only arise due to exchange
statistics but also due to interaction. These two contributions to the momentum
correlation function are fundamentally different and have to be separated from eachother. By decomposing momentum correlation functions into a set of trigonometricbasis functions, we develop a scheme to constrain large parts of the density matrix
and to fully reconstruct physical states via a Bayesian quantum state estimation. We
reconstruct density matrices of various different states and discuss the contributionof antisymmetrization and interactions to measured correlation functions.
The reconstructed density matrices contain the full information about the states. In
the final chapter, we use these density matrices to address particle-particle entanglement in identical-particle states. When treating particles as subsystems, standard
entanglement measures lead to conceptual problems and detect entanglement due
to the antisymmetrized structure of the fermionic wave function. To remedy these
problems we employ the notion of an Antisymmetric Negativity that searches for thesmallest entanglement that must already exist before antisymmetrization. We show137

CHAPTER 11. SUMMARY AND OUTLOOK

Figure 11.1: Measurement of Pauli Crystal. The plots show the measurement of theconfigurational probability in a non-interacting Fermi gas of N = 3 (left)
and N = 6 (right) atoms [278]. Angular correlations are induced by Pauliexclusion principle.

that the antisymmetric negativity reveals multipartite entanglement structure that is
not accessible via the standard negativity alone.

OutlookThroughout the cause of this thesis, we developed new techniques and concepts tocharacterize strongly correlated fermionic systems via momentum correlation measurements. While we focused on the full reconstruction of density matrices, we anticipate that the presented ideas also find application for the characterization of larger
systems and the detection of many-body ordering or pairing in mesoscopic, fermionicsystems.
We recently upgraded our experimental apparatus to be able to generate two dimensional optical potentials [279]. This allows the expansion of states in the 2D plane andis expected to substantially enhance the measured contrast of momentum correlation
functions. Such an improvement enables to extend the study of Fermi-Hubbard-likestates to larger one-dimensional systems with higher purity but also to address problems related to many-body interference in arrays of fermionic atoms [280].
A two-dimensional optical potential furthermore extends the range of possible statesthat we can realize in the laboratory. Using similar preparation schemes as before we can deterministically initialize mesoscopic systems in closed-shell configurations. While such a system already provides interesting physics to explore on its
own [281, 282], we plan to investigate mesoscopic, interacting 2D systems via singleparticle resolved momentum-density correlations [182, 283]. Figure 11.1 shows themeasurement of the configurational probability in a non-interacting Fermi gas of 3
and 6 atoms (cf. Ref. [283]), a beautiful example, where fermionic exchange statistics
induces ordering, which can only be revealed by sing-particle resolved momentum
measurements. In particular by adding interaction and investigating such systems
across the BEC-BCS crossover we hope to see signatures of many-body pairing in thestrongly correlated regime [284]. Such signatures do not require a full tomographyof the quantum state but should be apparent in higher-order correlation functions.

138

CHAPTER 11. SUMMARY AND OUTLOOKWhile deep in the BEC or BCS regime, the system is expected to show only secondorder correlations in the in-situ or the momentum density, respectively, the signature
should change drastically in the strongly correlated regime and might also be visible
in higher-order correlation functions.
Due to the high degree of controllability in our setup, we can readily increase theparticle number of the two-dimensional system to bridge the gap from microscopic tomacroscopic Fermi systems. In future measurements we will investigate very dilute,
yet strongly correlated, low entropy states of a few 100 atoms. In this mesoscopic
regime the system is expected to show rich many-body behavior while both in-situ
and momentum correlation measurements on the single-particle level are still withinreach.
In addition we plan to further upgrade our experimental setup with a spatial light
modulator (SLM). An SLM can be used to shape complex optical potentials andwill enable the investigation of two-dimensional Fermi-Hubbard systems, including
the study of Nagaoka magnetism on a 2 × 2 plaquette [285] or to explore fractional
quantum hall physics using rotating traps [286].

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Appendices163

Appendix A

Reconstructed Density
Matrices
In this chapter we list the reconstructed density matrices of all states that were usedin this thesis. We also mention the number of steps of the Metropolis-Hastings (MH)algorithm, which is the number of states of the sampled posterior distribution. We
give the median of each entry together with the 68% credible interval.

Rotated Triplet State
Number of MH steps: 5000
Basis states: {|↑↑⟩ , |↑↓⟩ , |↓↑⟩ , |↓↓⟩}
⎛

0.307

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.002
0.290

0.123
0.152
0.208

h.c.

⎞
⎛
0.087
0
⎜
0.080 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.081 ⎠
0.195
h.c.

−0.1430−0.036−0.0010⎞−0.094−0.047 ⎟⎟⎟,0.044 ⎠
0
(A.1)

0.015
0.010
0

⎞
0.221
0.119 ⎟⎟⎟.
0.173 ⎠

with 68% credible interval
⎛

0.007

⎜
⎜
δℜ(ρ) = ⎜
⎝
h.c.

0.111
0.185

0.220
0.010
0.184

⎞
⎛
0.091
⎜
0.009 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.022 ⎠
0.004

0

h.c.

0.116
0

0
(A.2)

165

APPENDIX A. RECONSTRUCTED DENSITY MATRICESInteracting Double Well
Number of MH steps for each interaction strength: 5000
Basis states: {|LL⟩ , |LR⟩ , |RL⟩ , |RR⟩}
U/J = -5.9
⎛

0.456

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.102
0.038

0.077
0.028
0.031

h.c.

⎞
⎛
0.191
⎜
0.093 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.094 ⎠

−0.001
0

0

0.011
0.001
0

h.c.

0.0475⎞0.045
0.027 ⎟⎟⎟,
0.006 ⎠
0
(A.3)

with 68% credible interval
⎛

0.023

⎜
⎜
δℜ(ρ) = ⎜
⎝

0.029
0.008

0.035
0.008
0.007

h.c.

⎞
⎛
0.037
⎜
0.034 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.037 ⎠
0.023

0

0.039
0

0.005
0.004
0

h.c.

⎞
0.050
0.013 ⎟⎟⎟.
0.004 ⎠0
(A.4)

U/J = -2.9
⎛

0.389

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.129
0.095

0.129
0.073
0.086

h.c.

⎞
⎛
0.161
⎜
0.138 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.133 ⎠
0.4290

0

0.048
0

0.027
−0.0130⎞0.025−0.056 ⎟⎟⎟,0.006 ⎠

h.c.

0
(A.5)

with 68% credible interval
⎛

0.035

⎜
⎜
δℜ(ρ) = ⎜
⎝

0.043
0.019

0.045
0.019
0.018

h.c.

⎞
⎛
0.054
⎜
0.05 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.03 ⎠
0.036

0

0.034
0

0.037
0.001
0

h.c.

⎞
0.084
0.021 ⎟⎟⎟.
0.05 ⎠0(A.6)

U/J = 0
⎛

0.195

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.152
0.199

0.162
0.177
0.233

h.c.

⎞
⎛
0.177
⎜
0.211 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.241 ⎠
0.373

0

0.01
0

−0.013−0.0360h.c.

⎞−0.031
0.051 ⎟⎟⎟,
−0.019 ⎠0(A.7)

with 68% credible interval
⎛

0.016

⎜
⎜
δℜ(ρ) = ⎜
⎝
h.c.

0.045
0.016

0.054
0.022
0.016

⎞
⎛
0.049
⎜
0.058 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.026 ⎠
0.019

0

h.c.

0.035
0

0.002
0.01
0

⎞
0.043
0.05 ⎟⎟⎟.
0.0146 ⎠
0
(A.8)

166

APPENDIX A. RECONSTRUCTED DENSITY MATRICESU/J = 2.5
⎛

0.142

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.165
0.364

0.168
0.288
0.340

h.c.

⎞
⎛
0.110
⎜
0.190 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.187 ⎠
0.154

−0.09400−0.048−0.0070h.c.

⎞−0.0090.003 ⎟⎟⎟,
0.046 ⎠
0
(A.9)

with 68% credible interval
⎛

0.021

⎜
⎜
δℜ(ρ) = ⎜
⎝

0.038
0.031

0.054
0.030
0.029

h.c.

⎞
⎛
0.025
⎜
0.050 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.042 ⎠

0

0.022
0

0.040
0.004
0

h.c.

0.022

⎞
0.024
0.128 ⎟⎟⎟.
0.068 ⎠
0
(A.10)

U/J = 5.0
⎛

0.068

0.136
0.433

⎜
⎜
ℜ(ρ) = ⎜
⎝

⎞
⎛
0.053
⎜
0.149 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.140 ⎠

0.129
0.299
0.416

h.c.

0.084

0

0.004
0

−0.004−0.0070h.c.

⎞−0.004−0.009 ⎟⎟⎟,0.054 ⎠
0
(A.11)

with 68% credible interval
⎛

0.015

⎜
⎜
δℜ(ρ) = ⎜
⎝

0.045
0.034

0.047
0.034
0.033

h.c.

⎞
⎛
0.019
⎜
0.045 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.035 ⎠
0.017

0

0.043
0

0.052
0.042
0

h.c.

⎞
0.001
0.047 ⎟⎟⎟.
0.030 ⎠
0
(A.12)

U/J = 8.0
⎛

0.033

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.086
0.467

0.079
0.36
0.436

h.c.

⎞
⎛
0.025
⎜
0.136 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.126 ⎠
0.065

0−0.0330−0.040−0.037
0

h.c.

⎞
0.020
−0.004 ⎟⎟⎟,0.045 ⎠
0
(A.13)

with 68% credible interval
⎛

0.010

⎜
⎜
δℜ(ρ) = ⎜
⎝
h.c.

0.030
0.032

0.038
0.027
0.032

⎞
⎛
0.013
0
⎜
0.019 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.033 ⎠
0.014
h.c.

0.015
0

0.046
0.021
0

⎞
0.002
0.070 ⎟⎟⎟.
0.031 ⎠
0
(A.14)

167

APPENDIX A. RECONSTRUCTED DENSITY MATRICESU/J = 11.6
⎛

0.023

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.068
0.486

0.068
0.370
0.448

h.c.

⎞
⎛
0.019
⎜
0.105 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.092 ⎠
0.043

0−0.0300

0.015−0.0370h.c.

⎞−0.005−0.016 ⎟⎟⎟,0.049 ⎠
0
(A.15)

with 68% credible interval
⎛

0.007

⎜
⎜
δℜ(ρ) = ⎜
⎝

0.035
0.025

0.030
0.029
0.024

h.c.

⎞
⎛
0.014
⎜
0.027 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.032 ⎠
0.009

0

0.028
0

0.024
0.010
0

h.c.

⎞
0.008
0.006 ⎟⎟⎟.
0.053 ⎠
0
(A.16)

U/J = 18.5
⎛

0.016

⎜
⎜
ℜ(ρ) = ⎜
⎝

0.028
0.482

0.040
0.362
0.459

h.c.

⎞
⎛
−0.004⎜0.084 ⎟⎟⎜
⎟ , ℑ(ρ) = ⎜⎝0.066 ⎠
0.043

0

0.041
0

0.020
−0.014
0

h.c.

⎞
0.011
0.029 ⎟⎟⎟,
0.072 ⎠
0
(A.17)

with 68% credible interval
⎛

0.007

⎜
⎜
δℜ(ρ) = ⎜
⎝
h.c.

168

0.056
0.031

0.055
0.028
0.031

⎞
⎛
0.019
⎜
0.045 ⎟⎟⎜⎟ , δℑ(ρ) = ⎜⎝0.054 ⎠
0.011

0

h.c.

0.049
0

0.027
0.013
0

⎞
0.001
0.031 ⎟⎟⎟.
0.034 ⎠
0
(A.18)

APPENDIX A. RECONSTRUCTED DENSITY MATRICES

Incommensurately Spaced Triple Well
Number of MH steps: 50000
Basis states:{|↑, ↑, ↓⟩ , |↑, ↓, ↑⟩ , |↓, ↑, ↑⟩ , |↑↓, ↑, ·⟩ , |↑↓, ·, ↑⟩ , |↑, ↑↓, ·⟩ , |·, ↑↓, ↑⟩ , |↑, ·, ↑↓⟩ , |·, ↑, ↑↓⟩}
⎛

0.446

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
ℜ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.268 −0.005 0.005 0.005 0.076
0.465 −0.014 0.004 0.002 0.062
0.002 0.001
0 0.0020.002
0 0.001
0.002 0.001
0.034

h.c.

⎛

0

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
ℑ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.013 0.002
0.006
0.005 −0.013
00 −0.004 −0.006 −0.021
00 −0.001 −0.001
0 −0.001
0.002
0 −0.002
0−0.0090.001
0.001−0.001−0.001−0.001
0.006⎞
0.065 −0.005
0.077
0.007 ⎟⎟⎟
−0.0020 ⎟⎟0.001 −0.001 ⎟⎟⎟00.001 ⎟⎟
0.010
0.003 ⎟⎟
−0.0010.001 ⎟⎟⎟
0.034
0.003 ⎠
0.009

−0.005 −0.010
0.008
0.001
0 −0.002
0
0.002
0
0.002
−0.003 −0.007
0
0.004
0

⎞
0.007
0.001 ⎟⎟⎟
0.001 ⎟
⎟
0 ⎟
⎟
⎟
0 ⎟
⎟
0.004 ⎟⎟
0.002 ⎟⎟⎟
−0.002 ⎠h.c.

0with 68% credible interval
⎛

0.027

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
δℜ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
h.c.

⎞
0.014 0.008 0.004 0.007 0.009 0.012 0.019 0.004
0.027 0.009 0.001 0.009 0.029 0.005 0.010 0.007 ⎟⎟⎟0.001 0.001 0.001
0 0.002 0.003 0.002 ⎟⎟0.001
0 0.003 0.002 0.001 0.001 ⎟
⎟
⎟
0.001
0
0 0.003 0.002 ⎟⎟0.008 0.001 0.007 0.001 ⎟⎟0.002 0.004 0.002 ⎟⎟⎟0.008 0.005 ⎠
0.003

169

APPENDIX A. RECONSTRUCTED DENSITY MATRICES

⎛

0

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
δℑ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.003 0.005 0.007 0.002 0.015
0 0.002 0.007 0.005 0.009
0
0 0.001 0.004
0 0.001 0.002
0 0.003
0

0.002
0.008
0
0.001
0.002
0.005
0

0.027
0.021
0.002
0.003
0.002
0.002
0.001
0

⎞
0.014
0.010 ⎟⎟⎟
0.002 ⎟⎟
0.001 ⎟⎟⎟
0.002 ⎟ .
⎟0.005 ⎟⎟
0.004 ⎟⎟⎟
0.005 ⎠h.c.

0Commensurately Spaced (Homogeneous) Triple Well
Number of MH steps: 50000
Basis states:{|↑, ↑, ↓⟩ , |↑, ↓, ↑⟩ , |↓, ↑, ↑⟩ , |↑↓, ↑, ·⟩ , |↑↓, ·, ↑⟩ , |↑, ↑↓, ·⟩ , |·, ↑↓, ↑⟩ , |↑, ·, ↑↓⟩ , |·, ↑, ↑↓⟩}
⎛

0.197

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
ℜ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.010
0.162

−0.139
0.041
0.288

0.060
0

0.060
0.066
0

0.010
−0.0160.005
0.035−0.025−0.017
0.035
0.010
0.060

0.016
0.023
0.002
−0.0080.015
0.050−0.009
0.031
0.033
0.012
0.010
0.013
0.087

0.003
0.020
−0.0270.011−0.001
0.012
0.013
0.055

h.c.

⎛

0

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
ℑ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
h.c.

170

−0.0200.012−0.0230−0.0120.015−0.0080.009
0−0.0220.006−0.027−0.009
0.005
0

0.029
0.001
−0.034−0.003
0.015
0.011
0

−0.028−0.0080.010−0.0030.013
0.007−0.0150⎞0.023⎟
0.021 ⎟⎟
0.005 ⎟⎟⎟
−0.003 ⎟⎟0.012 ⎟⎟⎟
−0.004 ⎟⎟−0.015 ⎟⎟−0.002 ⎟
⎠
0.063

⎞
0.010
⎟
0.009 ⎟⎟
−0.031 ⎟⎟⎟0.012 ⎟⎟
0.007 ⎟⎟⎟
−0.012 ⎟⎟−0.015 ⎟⎟0.017 ⎟⎠
0

APPENDIX A. RECONSTRUCTED DENSITY MATRICES

with 68% credible interval
⎛

0.053

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
δℜ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.080 0.010 0.030 0.034
0.051 0.075 0.023 0.014
0.064 0.019 0.040
0.016 0.022
0.026

0.050
0.030
0.024
0.012
0.022
0.022

0.034
0.038
0.044
0.015
0.014
0.008
0.035

0.048
0.003
0
0.023
0.016
0.020
0.011
0.024

h.c.

⎛

0

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
δℑ(ρ) = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
h.c.

⎞
0.030
0.006 ⎟⎟⎟
0.017 ⎟⎟
0.004 ⎟⎟⎟
0.006 ⎟⎟
0.021 ⎟⎟
0.021 ⎟⎟⎟
0.010 ⎠
0.027

0.001 0.010 0.023 0.012 0.016 0.038
0 0.001 0.034 0.022 0.034 0.007
0 0.042 0.057 0.035 0.001
0 0.003 0.010 0.022
0 0.022 0.018
0 0.017
0

0.018
0.024
0.034
0.003
0.011
0.022
0.028
0

⎞
0.003
0.049 ⎟⎟⎟
0.052 ⎟⎟
0.009 ⎟⎟⎟
0.023 ⎟ .
⎟0.001 ⎟⎟
0.016 ⎟⎟⎟
0.028 ⎠
0171

APPENDIX A. RECONSTRUCTED DENSITY MATRICES

172

Appendix B

Negativity Bounds
We briefly proof the bounds of the negativity, which we used in the discussion aroundFig. 10.7.
1. The maximum negativity that can be obtained by antisymmetrizing a fully separable three-particle state is one.
A pure separable state can be written in the form
|Ψpsep ⟩ = |ϕ1 ⟩ ⊗ |ϕ2 ⟩ ⊗ |ϕ3 ⟩ ,(B.1)

with single-particle pure states |ϕi ⟩. We obtain its normalized, antisymmetrized
density matrix aspp| PA⟩ ⟨ψsep
PA |ψsep
,
(B.2)
ρA
p
p
sep =
Tr(PA |ψsep ⟩ ⟨ψsep | PA )and verify numerically that for any random single-particle state |ϕi ⟩ the negativity of the antisymmetrized wavefunction evaluates to N (ρAsep ) = 1 along
any possible bipartition due to the exchange symmetry of the first-quantizeddensity matrix. A general mixed fully-separable state can be obtained as theprobabilistic sum of pure fully-separable states,
ρ̃A
sep =

∑︂

pi ρA
sep,i .

(B.3)

i

Due to the convexity of the negativity, we find the inequality
N (ρ̃Asep ) ≤∑︂(︁)︁pi N ρA
sep,i = 1.

(B.4)

i

We therefore find the upper bound for the bipartite negativity of an antisymmetrized fully-separable state,
max
Nsep
= 1.

(B.5)

2. The maximum negativity that can be obtained by antisymmetrizing a biseparable
state is 1.9428

173

APPENDIX B. NEGATIVITY BOUNDS

We can write down a general biseparable state asp|ψbs⟩ = |ϕ⟩1 ⊗ |ϕ⟩23 ,(B.6)

with any pure two-particle state ϕ23 between atoms with label 2 and 3. For amaximally entangled two-particle state of the form)︁1 (︁|ϕ⟩23 = √ |Xσ⟩ |Y τ ⟩ + eiθ |Y τ ⟩ |Xσ⟩ ,
2

(B.7)

we explicitly calculate the negativity of its antisymmetrized density matrix ρbsA
to be N (ρbs)∼
1.9428.
Based
on
the
same
arguments
asabove,
wefindanA
upper bound of the negativity for any mixed, biseparable state ofmax
Nbs= 1.9428.

174(B.8)