Spin dynamics in a tunable Heisenberg model
realized with ultracold atoms
byPaul Niklas Jepsen
B.Sc., Christian-Albrechts Universität zu Kiel (2010)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2022
c Massachusetts Institute of Technology 2022. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
December 31, 2021

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Phyics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deepto Chakrabarty
Associate Department Head of Physics

2

Spin dynamics in a tunable Heisenberg model
realized with ultracold atoms
byPaul Niklas Jepsen
Submitted to the Department of Physics
on December 31, 2021, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Spin is the elementary unit of magnetism. The interactions between many spins give amaterial its magnetic properties. This thesis focusses on one of the simplest magnetic
materials: A one-dimensional chain of spins with tunable nearest-neighbor interactions. This system is described by the Heisenberg model, a paradigmatic model,
which has been studied for almost a century now. But so far, experiments on spin
dynamics have been mostly limited to isotropic spin-spin interactions.
In this work we use ultracold atoms to implement the first quantum simulator forthe anisotropic Heisenberg model, with fully adjustable anisotropy of nearest-neighborspin-spin interactions (also called the XXZ model). We study spin dynamics in previously unexplored regimes far away from equilibrium, as well as stable spin patterns
far away from the ground state, which are even exact many-body eigenstates. For this
we utilize quantum quenches from initial far-from-equilibrium spin-helix patterns:
Spin transport: By using a longitudinal spin-helix pattern, which involves a modulation in the population of spin up and spin down atoms, we see a drastic impact on
the transport properties, when the anisotropy is varied. When spins are coupled only
along two of three possible orientations (the XX model), we find ballistic behavior ofspin dynamics, whereas for isotropic interactions (the XXX model), we find diffusive
behavior. More generally, for positive anisotropies, the dynamics ranges from anomalous superdiffusion to subdiffusion, whereas for negative anisotropies, we observe acrossover in the time domain from ballistic to diffusive transport.
Spin dephasing: A transverse spin-helix pattern is sensitive to additional decay
mechanisms: Anisotropic spin couplings break spin-rotational symmetry. Transverse
spin components are no longer conserved and can decay not only by transport, butalso by fast, local dephasing. However, even for isotropic interactions, we observedephasing due to a new effect: an effective magnetic field created by superexchange,
which has its origin in the mapping from the Hubbard model and which has not been
observed before.
Bethe phantom states — excited many-body eigenstates of the Heisenberg model:
For a given anisotropy, there exists one special winding angle, such that the transverse3

spin helix is an exact many-body eigenstate of the Hamiltonian. We find this eigenstate experimentally, by varying the winding angle and measuring the decay rate,
which reveals a pronounced minimum. In a next step, we then use the sensitivityof the Bethe phantom states as a tool, to actually measure the anisotropy directly.
We then find that the anisotropy can be strongly affected by nearest-neighbor off-site
interactions, which have never been observed before for particles, which only interact
with contact interactions.
Our new quantum simulator platform with tunable interactions opens up possibilities for many new studies which are likely to provide new insight into the rich
dynamics of Heisenberg spin models and beyond.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Phyics

4

To my family

5

6

Acknowledgments
All the research presented in this thesis has been a huge team effort. I feel very
lucky for having been part of such an amazing team. And there are countless people
supporting me on the way.
Wolfgang is truely the best advisor I could have ever wished for. I feel reallygrateful. Wolfgang is incredibly good at physics and solving problems, but also at
explaining it and communicating the science in well understandable language. It isamazing to talk about physics with Wolfgang, and I will really miss it. Thank you
for teaching me your way of thinking about physics, for letting me learn in my own
pace and giving us the freedom to do research the way we want. Thank you foryour patience. Thank you for letting me stay in the lab to do one more experimentwith Eunice and Hanzhen. Thank you for your support during times when I was
struggling. No matter if you gave me amazing physics advice or amazing life advice.
Grad school was the best time of my life so far. Thank you, for making me succeed
in grad school. And the reason why the science worked out so well in the end, is you.
Jesse, Ivana and I have been the core team of the BEC5 lab for many years.
Thank you for this amazing time together. We have built this machine from scratch
and got amazing science results together in the end! But there was an intermediatetime, when we did struggle, when nothing was working yet. We went through a lot
together. Through exciting times and through tough times. We grew into a really
good team! And I really miss working with you. You drove me to perform at mybest. Thank you, that I could learn so much from both of you. And thank you formaking me a better team player! I think there used to be a time, when there was
nobody in the world who I spend more time with than you!
Jesse always found the perfect balance between making something perfect andalso getting the job done. You always came up with new out of the box ideas. Thankyou for implementing the in-situ polarization-rotation imaging. That was a gamechanger. Thank you for staying in lab with us as a postdoc after defending yourthesis!
7

With Ivana’s get-it-done attitude, she was always the driving force in keeping the
momentum going in our team. And I would like to emphasize: You are really anamazing experimentalist: When something stopped working, Ivana was the fastest tofind the correct knob to turn and fix it. So many times something was not working,and you would just throw in a random lens somewhere and the problem was fixed.
Sometimes you would fix something quick and dirty in 5 minutes, for what I would
spend a whole day. And I think I actually learned a lot from your way of working.
Otherwise I would have never finished the last experiments. Also: Thank you for
putting us on the spin 𝑆 = 1/2 route! That was a game changer! For the longest time
we were working on 𝑆 = 1 experiments. With your experiment about the tilted Mott
insulator, we really moved away from 𝑆 = 1. And this opened up the stage for the
anisotropic spin-1/2 Heisenberg model!
Thank you to Wen Wei and Eugene for this fantastic theory-experiment collaboration. Wen Wei, you really explained the Heisenberg model to us experimentalists.
Thank you for also contributing so much to actively writing and editing the papers.
I was unexperienced and you and Wolfgang showed me how it is done.
Thank you to Hanzhen and Eunice for being amazing new teammates. I had ablast with you this summer (2021), taking data non-stop 24 hours a day and 7 daysa week. It was a huge team effort again, and it turned out so well, because both ofyou gave your best. You are so highly motivated, it is amazing to see how much you
have learnt and driven the experiment further, since you joined the lab.
Yair, thank you for helping us in the lab and putting our experiment back together,in a time when the machine was falling apart during the pandemic.
Thank you to all past member of the BEC5 team, especially to Will Lunden,
Yichao Yu, Michael Messer, David Weld, Graciana Puentes and Alex Keesling. Thankyou for being an important part of our team in the past and helping us build themachine. A lot of your contributions are still crucial parts of the experiment today!
Special thanks to my thesis committee! Martin Zwierlein, when we started building our BEC5 lab, you were starting to build you Fermi2 lab, so that we had friendly
competition going on. I have always admired your results, while BEC5 was stuck8

with problems for a long time. But once we had results on spin physics, you havealways been such a big fan of our research. And that meant a lot to me and it has
been very motivating. Thank you for even inviting me to present our work at thestudent/postdoc triple feature. That was a big honor! And thank you for being
so generous, lending us your green laser, which enabled our last experiment on the
Bethe phantoms. Kyoshi Masui, thank you for serving on my thesis committee andfor being so interested in our work! And special thanks for your patience. Thanks toall of you, Martin, Kyoshi, and Wolfgang, for being so understanding that instead ofwriting my thesis, I preferred to stay in the lab for one more experiment.
I would especially like to thank Catherine Modica. You have always been there
for me for advice of any kind, especially in difficult times. You are really there for all
the physics students. And you are a big part for me feeling home at MIT.
Thank you to the whole Ketterle group, present and past members. And thankyou to the whole Center for Ultracold Atoms community as a whole. It is really likea cold-atoms-family. And a super special environment to be part of.
There is a special mentor I would like to thank: Dave Pritchard. He was Wolfgang’s postdoc advisor, so he is my grandadvisor, so-to-speak. Dave it not just areally good physicist but also a really good sailor! Thanks to you and your wife
Andy, for the countless times on the water. It is really interesting to talk with you
about physics and also anything else, and you definitely explained to me the physics
of sailing! You have always been there for guidance of any kind. As a junior student,
when I was new to MIT, it made me settle into the MIT physics community a lotsmoother, knowing that you are around.
Misha Lukin has been also a very special mentor to me. Misha is not just a really
good physicist. He is also a really good hockey player! And I get to play with him
most Saturdays. Thank you, Misha, for all the special mentorship on the ice. You
were always there to talk about physics and hockey. Sometimes I would bring ourlatest results to hockey to discuss them with you afterwards. Thank you for being
such a special part of my journey through grad school. It definitely kept me going
and motivated. Once Misha said to Wolfgang: “I‘m looking forward to seeing Niki9

improve at physics as much as he has improved at hockey.”
And I would like to thank all of my hockey friends! Hockey has been a really
important part of my grad school life: John, Yury, Alexei, Alex, Erik, Theo, Konstantin, JJ, Marco, Christoph, Blake, Eva, Melody, Mallory, Nick, Russian Nick,
Timur, George, Andrew, Matt, Lee, Yiqi, Simone, Brian, Sherry, Keith, Kylie, Alan,
Nolan, Henry, Aubry, Jon, Nick, Liz, Brett, ... and so many more!
And especially I would like to thank the MIT Hockey team! You made me a better
team player, thank you for showing me another world at MIT, thank you for pushingme to my physical limits. I had no idea, how exhausted one could be. Special thanks
to our coach and my good friend Nick! And also to coach Dave and coach Tommyfor making it an amazing time!
Timur and Lawrence, you have been my best friends in grad in school and myroom mates. Living with you made me really feel home in Boston. When things were
not working in the lab, you were always there for me and kept me going! Sam andRussian Nick also were a big part of my time at MIT. Melody, I want to thank you
for giving me self-confidence in the lab, at a time when I needed it most. Thank youthat you were there for me every day when data taking was intense. You were part of
the journey, through my PhD. And it was a special time! Tobi and Lilin and Matt,
thank you for so many kite surf adventures. Thank you for building me back up in atime when I was struggling. This year (2021) and the last experiment would not havegone this well for me, without you. It made a huge difference. Thanks for always
being there for me.
Michael and Nancy, thank you for being my American parents. You have always
been welcoming into your family and it always made a huge difference for me to knowthat you are there. Paul and Judy, thank you for being my west coast parents. Alp
and Hatice always had an eye out for me, too, and making sure I’m doing fine. Thankyou all for your love and support. I would also like to thank my friends in Germany.
Especially Dominik and Arne. You have been such an important part of my life andI can always rely on you.
Most importantly, I would like to thank my family, my parents Karin and Søren,10and my brother Maui. You have worked so hard to give me every opportunity tosucceed. I can count on you every moment and you would do anything for me. You
have never missed out on any opportunity to visit me in Boston. I could have notdone a PhD like this, without your support.

11

12

Contents
1 Introduction

15

1.1

Building blocks of condensed matter systems . . . . . . . . . . . . . .

15

1.2

Ultracold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.3

Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2 The Heisenberg model

25

2.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.2

Previous experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3 Li7 in optical lattices

31

3.1

Realization of the Heisenberg model . . . . . . . . . . . . . . . . . . .

31

3.2

Far-from-equilibrium spin-helix states . . . . . . . . . . . . . . . . . .

38

3.3

Polarization-rotation imaging . . . . . . . . . . . . . . . . . . . . . .

40

3.4

Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.5

Optical resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4 Spin transport

49

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2

Tunable superexchange dynamics . . . . . . . . . . . . . . . . . . . .

52

4.3

XX model (Δ = 0) and XXX model (Δ = 1)

. . . . . . . . . . . . . .

53

4.4

Positive anisotropies (Δ > 0) . . . . . . . . . . . . . . . . . . . . . . .

59

4.5

Negative anisotropies (Δ < 0) . . . . . . . . . . . . . . . . . . . . . .

63

13

4.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Transverse spin dynamics

67
71

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.2

XX model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.3

XXX model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.4

Imaging the effective magnetic field . . . . . . . . . . . . . . . . . . .

79

5.4.1

Observation as a beatnote . . . . . . . . . . . . . . . . . . . .

79

5.4.2

Observation through spin precession. . . . . . . . . . . . . .

81

5.4.3

Spectroscopic observation . . . . . . . . . . . . . . . . . . . .

84

Dephasing mechanisms for the transverse spin helix . . . . . . . . . .

84

5.5

6 Bethe phantom states

87

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.2

Stability of transverse spin-helix states . . . . . . . . . . . . . . . . .

90

6.3

Observations of phantom helix states . . . . . . . . . . . . . . . . . .

92

6.4

Extension to higher dimensions . . . . . . . . . . . . . . . . . . . . .

96

6.5

Direct measurement of the anisotropy . . . . . . . . . . . . . . . . . .

99

7 Summary and Outlook

105

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A Spin transport in a tunable Heisenberg model
realized with ultracold atoms

115

B Transverse spin dynamics in the anisotropic Heisenberg model
realized with ultracold atoms

137

C Catching Bethe phantoms and quantum many-body scars:
Long-lived spin-helix states in Heisenberg magnets

14

157

Chapter 1
Introduction
1.1

Building blocks of condensed matter systems

Atoms are the building blocks of ordinary matter. And the interactions between many
atoms define its physical properties. For example, in the absence of any interactions,
all the atoms would just form a non-interacting gas, the so-called ideal gas, which
actually turns out to be a good approximation for many dilute gases we know. Butinteractions are definitely necessary to make atoms form a liquid or a solid, especially for forming a lattice structure in crystals (see Figure 1-1a). Such systems hostlattice vibrations (phonons) which are governed by a dispersion relation and whichenable propagation of sound waves (non-equilibrium dynamics), or contribute to heat
conduction (near-equilibrium) and to a material’s heat capacity (equilibrium).
Charges are the building blocks of electricity. In a solid, the interactions of theelectrons with the ion cores and their lattice structure determine a material’s electricalproperties, such as the electrical resistivity defining whether a material is an insulator,
a conductor, or a semi-conductor (see Figure 1-1b). The periodic potential for theelectrons create a band structure, describing the range of energy that an electron may
have (lying within a band) or may not have (band gaps). Electrons are filling up all
available energy levels starting from the lowest one, forming the so-called Fermi sea
and the electrical properties are typically defined by the electrons on the top of thatFermi sea: by the electrons at the Fermi surface.
15

a

Atoms

MattergasbliquidCharges+

Electricity

insulator

c

solid

conductor

Spins

Magnetismaligneddisorderedanti-aligned

non-magnetic

Figure 1-1: Building blocks of condensed matter systems. Atoms (a), charges(b), and spins (c) are the building blocks for ordinary matter, electricity, and magnetism.
Spins are the building blocks of magnetism. If all spins in a material are aligned(ferromagnet), the material is magnetized and it creates a magnetic field on its own.
If the spins are disordered and point in random directions, their magnetic fields cancel out (see Figure 1-1c). But we could force the spins to align by applying an
external magnetic field, which is sufficiently strong. In some materials, spin-spin
interactions keep the spins aligned, even if the external magnetic field is removed(ferromagnetism). There are materials, where spins are non-interacting (paramagnetism) and independent from each other. An external field is required for spins toalign, so without a field their orientation is random. And in many materials, spinseven favor to anti-align with their neighbors (antiferromagnetism). Some materials
also do not even have any spins at all; they are non-magnetic. Otherwise, all in all,the magnetic properties of a material are defined by the interplay of many spins andthe interactions between them.
The interactions between spins can have different microscopic origins. We notethat these interactions typically do not involve a magnetic field, as one might naively
think. Of course there is a direct interaction of a pair of spins via their magnetic
16

a

Typical condensed matter systemsbCold atoms experiment

Figure 1-2: Condensed matter systems built from ultracold atoms. a, In
typical condensed matter systems, lattice vibrations (phonons), electron motion, andspins exist all simultaneously and interact with each other. b, In order to study
magnetism in its pure form, we build a one-dimensional spin-chain using ultracold
atoms and engineer tunable spin-spin interactions between nearest neighbors. There
are no lattice vibrations (photons) and no electron motion.

dipole field; however, this interaction is negligible compared to much stronger interactions of different origin [1]. Ferromagnetic spin-spin interactions typically emerge
due to Coulomb exchange (also called potential or direct exchange), whereas antiferromagnetic spin-spin interactions can be mediated through kinetic exchange fromtunneling, also called superexchange.
In the end, we are always dealing with lot of particles and lot of interactions! Thisis the subject of condensed matter physics and solid state physics or, more generally,
many-body physics. The typical problem is that in most materials, all of the abovebuilding blocks and interactions appear simultaneously.
Figure 1-2a shows many atoms arranged in a lattice structure by their interactions.
But this structure is not rigid, atoms can vibrate around their equilibrium position:
The lattice can host sound waves. That is illustrated by a snap shot of a large
amplitude lattice vibration with a single wavelength 𝜆 along a diagonal direction.
17

Its amplitude is plotted in an exaggerated way. The wavelength also corresponds
to a certain value of the vibration frequency. The relation between wavelength andvibration frequency is described by a dispersion relation 𝜔 = 𝜔(k) for the phonons,
where k is the wavevector with magnetude |k| = 2𝜋/𝜆. At any finite temperature,
every lattice is filled with a lot of these lattice vibrations. Thermally excited lattice
vibrations are actually one big contributor to storing heat in a solid, i.e. to a material’s
heat capacity.
On top, there is a Fermi sea of electrons which interact with the lattice and hence
also with the lattice vibrations. The electrons feel the periodic potential from theion cores, leading to a band structure (dispersion relation for the electrons), which isdifferent from the quadratic dispersion for free electrons. And since the lattice hosts
lattice vibrations, the periodic potential is not static, and that leads to electronphonon interactions. Electrons naturally interact with each other due to Coulombinteractions; these are electron-electron interactions. Furthermore, thermally excitedelectrons are a second big contributor to a material’s heat capacity
And finally, if the material is magnetic, there are spins, which also interact withthe lattice vibrations, and with the motion of the electrons. Typically, it is eventhe electrons spin, which make a material magnetic. However, the atomic nucleus
also carries a spin (but with typically 103 times smaller magnetic moment). And
the orbitally motion of bound electrons around the nucleus can also contribute a net
magnetic moment which is nonzero.
In order to study magnetism in its pure form, ideally we would like to investigate
a system with only localized spins and the interactions between the spins — nothingelse, no Fermi sea of electrons, no lattice vibrations (phonons), and ideally the simplest
spin arrangement possible, which is still non-trivial: a single chain (as in Figure 1-2b).

1.2

Ultracold atoms

Using ultracold atoms we can build artificial materials like this, which are simplified
and idealized systems. We also call this a quantum simulation. We simulate a real18

condensed matter system, with an artificial system, which obeys the same laws ofphysics. By now, a number of such platforms exist [2–7], with varying capabilities. In
our lab, we use the tools of atomic physics and build condensed matter systems frombottom up. Using laser light, magnetic fields and radio-frequency fields, utilizing
techniques such as laser cooling and evaporative cooling, we bring atoms down totemperatures around < 1 𝜇K, freezing out the motional degree of freedom. Theseultracold atoms can then be arranged in crystal structures using optical lattices oreven arbitrary controlled patterns using spatial light modulators or optical tweezer
arrays [8].
Tunable interactions can be engineered between the atoms to implement a wholevariety of different Hamiltonians. A strength of quantum simulators is that one canoften start by studying a simplified system (e.g. a non-interacting system) and thenmore complexity can be added (e.g. weak or strong interactions) in a continuous
and well-controlled fashion, to see how the systems properties change. In typicalcondensed matter experiments, interactions are fixed.
Additionally, we can study quantum dynamics far away from equilibrium. In atypical condensed matter system this would be really hard or impossible. For example,
an energetically highly excited spin pattern would immediately thermalize with thebath of phonons, which absorb its energy. However, with an isolated spin system wecan study the pure spin dynamics in regimes of the Hilbert space, which have neverbeen studied before. So far, most research in condensed matter physics has been
limited to ground states and elementary excitations close to the ground state, and, at
higher energies, thermal states at finite temperature 𝑇 . With our quantum simulator,
we can study highly excited pure states and the following coherent time evolution(quantum dynamics), and even highly excited eigenstates of the Hamiltonian, whichcannot decay, because the spin system is not coupled to a bath. This is especially true
for the most excited state! We can study the spin dynamics under the pure Heisenberg
model, a paradigmatic model for magnetism, which has been studied extensively fornear-equilibrium physics.
Lastly, one-dimensional systems are not only appealing, because they have the19Figure 1-3: Heisenberg spin chain with nearest neighbor interactions realized withultracold atoms. After imprinting spin patterns far away from equilibrium, we study
the dynamical behavior of this interacting spin system. The background shows a real
image of the spins, revealing an initially high contrast periodic modulation of the(blue) spin |↑⟩ atoms.
simplest geometry. Quantum fluctuations are also much more pronounced than inhigher dimensional systems. 1D is the most quantum out of all possible geometries.
Similarly, spin-1/2 is the “shortest” and hence the most quantum out of all spins,
quantized into two states |↑⟩ and |↓⟩. This makes the spin-1/2 Heisenberg modeleven integrable, i.e. there are as many conserved quantities as degrees of freedom.
That allows for especially rich dynamics compared to higher dimensions, where thevast amount of degrees of freedom leads to self-thermalization, where the system acts
as its own bath, and any dynamics eventually asymptote to a “boring” thermal state.
In summary, we can say that a quantum simulator is already a simple kind ofquantum computer. We can control and program the arrangement of the atoms.
And we can control and program the interactions between them. However, it is
not a universally programmable quantum computer. Nevertheless, it is a quantum
20

computer set up to solve one particular problem. And for us, this is to study the
Heisenberg model, i.e. to solve Schrödinger’s equation for the time evolution of certain
initial states under the Heisenberg Hamiltonian. This is an analog computer.

1.3

Magnetism

Spin is the elementary unit of magnetism. The interactions between many spins give amaterial its magnetic properties. This thesis focusses on one of the simplest magnetic
materials: A one-dimensional chain of spins with nearest-neighbor interactions (Figure 1-3). And these interactions are tunable: By varying these spin-spin interactions,
we can control the magnetic material and its properties. This system is describedby the Heisenberg model, a paradigmatic model, which has been studied for almosta century now. But nevertheless it is still under active experimental and theoreticalinvestigation, especially its spin transport properties [9–13].
For spin-1/2, the one-dimensional Heisenberg model has been solved analyticallyalmost 100 years ago with the Bethe ansatz [14]. This exact solution explains lowlying excitations very well (single magnons and few-magnon bound states). However,for dynamics far away from the ground state, the Bethe ansatz becomes very technical and it hardly provides any physical intuition, because such dynamics then has tobe decomposed into magnons and magnon bound states in a complicated way. Numerical simulations for one-dimensional systems are possible, but the computation
time already starts to become unfeasible for large chain lengths with more than 50
spins. Experimentally, it is very difficult to find condensed matter systems, which aredescribed by pure one-dimensional Heisenberg chains. Typical lattice geometries are
more complicated. Additionally, spin degrees of freedom couple to lattice vibrations(phonons) and electron motion, which are present in most solids. That makes it hard
to study magnetism in its pure form. This is especially true for spin dynamics far
away from equilibrium, where coupling to a bath (e.g. bath of phonons) leads tostrong dissipation and fast thermalization. In order to study pure coherent quantum
dynamics of a many-body system, it needs to be isolated from all other degrees of21

Figure 1-4: Spin dynamics far away from equilibrium. The red and blue magnetic domains of opposite magnetization (indicated by compass needles) fade awaywith time, as seen from left (an earlier image) to right. The bottom row shows real images of the blue domains, here each about 5 atoms wide. Depending on the spin-spin
interactions, the domains are observed to fade away due to spin transport (domainsmix) or spin dephasing (magnetization gets destroyed locally). Under certain conditions, the spin pattern is stable and does not decay.
freedom: A system which only consists of spins, and interactions between them. On
top of that, in most condensed matter systems, the explicit form of the interactions
between the spins are fixed and predefined by the material.
In this work we use ultracold atoms to implement the first quantum simulator forthe anisotropic Heisenberg model, with fully tunable anisotropy for nearest-neighborspin-spin interactions. Previous experiments were limited to isotropic spin-spin interactions [15–27]. And we study spin dynamics in previously unexplored regimes
far away from equilibrium (Figure 1-4). Dynamical behavior gives new insights into
the properties of such a Heisenberg quantum magnet. We do this, by starting witha far-from-equilibrium spin-helix pattern (Figure 1-5) and see how this spin-patternrelaxes over time. This thesis is about a series of three experiments.
22

a

Sz
λ

Sx

b

Sz
z

Sx

ϕ

z

Figure 1-5: Spin-helix patterns with adjustable wavelength 𝜆 (and wavevector
𝑄 = 2𝜋/𝜆 and winding angle 𝜙 = 𝑄𝑎) are excitations far away from equilibrium. a, A
longitudinal spin helix involves a population modulation of spin |↑⟩ and |↓⟩. It can
only decay by spin transport, because spin |↑⟩ and |↓⟩ atoms have to travel throughthe spin chain in order to mix. b, A transverse spin helix is a pure phase modulation,and it can also decay by fast, local dephasing. However, for one special winding
angle 𝜙, the transverse spin helix is an exact many-body eigenstate of the Heisenberg
model, far away from the ground state, and does not decay.
1. Spin transport [28]: By using a longitudinal spin-helix pattern (Figure 1-5a),which involves a modulation in the population of spin |↑⟩ and spin |↓⟩ atoms,
we see a drastic impact on the transport properties, when the anisotropy isvaried. When spins are coupled only along two of three possible orientations (theXX model), we find ballistic behavior of spin dynamics, whereas for isotropicinteractions (the XXX model), we find diffusive behavior. More generally, forpositive anisotropies, the dynamics ranges from anomalous superdiffusion to
subdiffusion, whereas for negative anisotropies, we observe a crossover in thetime domain from ballistic to diffusive transport. Those are two strikingly
different crossover phenomena: Anomalous diffusion for positive anisotropies
and behavior reminiscent of a classical gas for negative anisotropies.
2. Spin dephasing [29]: A transverse spin-helix pattern (Figure 1-5b) is sensitive23

to additional decay mechanisms: Anisotropic spin couplings in the Hamiltonianbreak spin-rotational symmetry. Transverse spin components are no longer
conserved and can decay not only by transport, but also by fast, local dephasing.
However, even for isotropic interactions, we observe dephasing due to a neweffect: an effective magnetic field created by superexchange, which has its origin
in the mapping from the Hubbard model and which has not been observed
before. This field leads to additional dephasing mechanisms: there is dephasing
due to (i) inhomogeneity of the effective field from variations of lattice depth
between chains; (ii) a twofold reduction of the field at the edges of finite chains;and (iii) fluctuations of the effective field due to the presence of mobile holes
in the system. The latter is a new coupling mechanism between holes andmagnons.
3. Bethe phantom states — highly excited many-body eigenstates of the Heisenberg model [30]: For a given anisotropy, there exists one special winding angle(Figure 1-5b), such that the transverse spin-helix is an exact many-body eigenstate of the Hamiltonian. We find this eigenstate experimentally, by varying
the winding angle and measuring the decay rate, which reveals a pronounced
minimum. In a next step, we then use the sensitivity of the Bethe phantom
states as a tool, to actually measure the anisotropy directly. We then find thatthe anisotropy can be strongly affected by nearest-neighbor off-site interactions,which have never been observed before for particles, which only interact withcontact interactions.

24

Chapter 2
The Heisenberg model
2.1

TheoryThe Heisenberg model describes localized spins with nearest neighbor interactions.

𝐻=𝐽

∑︁
⟨𝑖𝑗⟩

S𝑖 · S𝑗 = 𝐽∑︁(𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 + 𝑆𝑖𝑧 𝑆𝑗𝑧 )

(2.1)

⟨𝑖𝑗⟩

Here S𝑖 = (𝑆𝑖𝑥 , 𝑆𝑖𝑦 , 𝑆𝑖𝑧 ) are the spin vectors, and the sum is over all pairs of nearest
neighbors ⟨𝑖𝑗⟩. In the classical limit, the interaction energy between the two spins
S𝑖 and S𝑗 is proportional to the overlap S𝑖 · S𝑗 = 𝑆 2 cos 𝜃𝑖𝑗 , where 𝜃𝑖𝑗 is the angle
between S𝑖 and S𝑗 , and 𝑆 is their length (or in other words: the projection of S𝑖
onto S𝑗 ). For 𝐽 < 0, neighboring spins favor alignment (ferromagnetic model). For
𝐽 > 0, they favor antialignment (antiferromagnetic model). For 𝐽 = 0, spins are noninteracting (paramagnetic model). For finite 𝐽 ̸= 0, this is the isotropic Heisenberg
model, where spin-spin interactions are isotropic and characterized by a single parameter: the coupling constant 𝐽. The scalar product S𝑖 · S𝑗 is rotationally symmetricand the interaction energy between two spins only depends on the angle 𝜃𝑖𝑗 between
them. But there is no preferred overall direction in spin-space.
The Heisenberg model has been used extensively to study its ground state, equilibrium physics at a temperature 𝑇 (in contact with a bath), and phase transitions.
But dynamics far from equilibrium have been left relatively unexplored. If a given
25

Si

Sj

Figure 2-1: Heisenberg model with classical spins. A spin S𝑖 experiences atorque from its neighbor S𝑗 , given by 𝐽 S𝑗 × S𝑖 , forcing it to precess around the axis
parallel to S𝑗 (dashed line). At the same time, it experiences a torque from its otherneighbor. This leads to a set of coupled equations of motion for all spins.

spin S𝑖 is not aligned (or anti-aligned) with its neighbor S𝑗 , it experiences a torque𝐽 S𝑗 × S𝑖 towards the aligned (𝐽 > 0) or anti-aligned (𝐽 < 0) configuration exerted
on it by its neighbor S𝑗 . Since spins carry intrinsic angular momentum, S𝑖 is thenforced to precess around the axis parallel to S𝑗 , such that 𝑑S𝑖 /𝑑𝑡 = 𝐽 S𝑗 × S𝑖 is its
equation of motion (see Figure 2-1). Simultaneously, also S𝑗 experiences a torqueexerted by S𝑖 , so the spins are actually forced to precess around each other. For thefull spins system, every given spin S𝑖 experiences a torque from each of its neighbors,
leading to a set of coupled equations of motion 𝑑S𝑖 /𝑑𝑡 = (𝜕𝐻/𝜕S𝑖 ) × S𝑖 , which yield𝑑S𝑖 /𝑑𝑡 = 𝐽 (S𝑖−1 + S𝑖+1 ) × S𝑖 in the case of a one-dimensional spin chain, where every
spin has two neighbors. These are the so-called Landau-Lifshitz equations of motion
for classical spins. For a large number of spins, the collective dynamical behavior far
from equilibrium can be very complicated.
In the anisotropic Heisenberg model, the longitudinal and transverse spin couplings 𝐽𝑧 and 𝐽𝑥𝑦 can be different.
𝐻=∑︁[𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 )]

(2.2)

⟨𝑖𝑗⟩

And the anisotropy Δ := 𝐽𝑧 /𝐽𝑥𝑦 is defined as the ratio between the two. By changing
this value, we can vary the relative importance of the two terms in the Hamiltonian.
For |𝐽𝑧 | > |𝐽𝑥𝑦 | alignment (or antialignment) is favored in the 𝑆𝑧 -direction, whereas for|𝐽𝑥𝑦 | > |𝐽𝑧 | this favored direction is within the 𝑆𝑥 -𝑆𝑦 plane. Δ ̸= 1 implies that there
is a preferred direction in spin-space, so spin-rotational symmetry of the Heisenberg
26

+Jxy/2

-Jz/4

+Jz/4

Figure 2-2: Heisenberg model with quantum spins. The transverse spin coupling
𝐽𝑥𝑦 leads to spin exchange (left), where two neighboring sites exchange their spins.
The longitudinal spin coupling 𝐽𝑧 is a spin-spin interaction which favors alignment
for 𝐽𝑧 < 0 (middle) and anti-alignment for 𝐽𝑧 > 0 (right).

Hamiltonian is broken. In all previous ultracold-atoms experiments the anisotropy
was fixed at Δ ≈ 1 [15–19, 25–27]. We, however, can adjust the anisotropy freely,
going beyond previous experiments. This leads to new interesting physics which isthe subject of this thesis.
For classical spins, the Landau-Lifshits equations 𝑑S𝑖 /𝑑𝑡 = (𝜕𝐻/𝜕S𝑖 ) × S𝑖 still apply, but the dynamics is more complicated due to the anisotropic couplings. However,quantum mechanical spins follow the rules of quantum mechanics (Figure 2-2). They
are now described by spin operators S𝑖 = (𝑆𝑖𝑥 , 𝑆𝑖𝑦 , 𝑆𝑖𝑧 ). For spin quantum number
𝑆 = 1/2, spin |↑⟩ and |↓⟩ are the spin states pointing in the ±𝑆𝑧 -direction. And moregenerally, a spin state pointing along the direcrion of n = (cos 𝜃, sin 𝜃 cos 𝜑, sin 𝜃 cos 𝜑)
is given by |𝜓(𝜃, 𝜑)⟩ = cos(𝜃/2) |↑⟩+sin(𝜃/2)𝑒−𝑖𝜑 |↓⟩, with polar angle 𝜃 and azimuthal
angle 𝜑. With spin |↑⟩ and |↓⟩ as the basis states, we can interpret the two terms inthe Hamiltonian in the following way:
The first term 𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 is still a spin-spin interaction. For negative values 𝐽𝑧 < 0
it favors alignment between neighboring spins. For positive values 𝐽𝑧 > 0 it favors
anti-alignment.
𝐽𝑧|↑𝑖 ↑𝑗 ⟩4
𝐽𝑧
𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 |↓𝑖 ↓𝑗 ⟩ = + |↓𝑖 ↓𝑗 ⟩4𝐽𝑧𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 |↑𝑖 ↓𝑗 ⟩ = − |↑𝑖 ↓𝑗 ⟩4𝐽𝑧𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 |↓𝑖 ↑𝑗 ⟩ = − |↓𝑖 ↑𝑗 ⟩4
𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 |↑𝑖 ↑𝑗 ⟩ = +

27

(2.3)
(2.4)
(2.5)
(2.6)

The second term 𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 ) = 𝐽𝑥𝑦 (𝑆𝑖+ 𝑆𝑗− + 𝑆𝑖− 𝑆𝑗+ )/2 is a spin-exchange term,a coupling term, which allows two neighboring sites to exchange their spins. This
term is the origin of spin transport.
𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 ) |↑𝑖 ↑𝑗 ⟩ = 0(2.7)

𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 ) |↓𝑖 ↓𝑗 ⟩ = 0
𝐽𝑥𝑦|↓𝑖 ↑𝑗 ⟩𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 ) |↑𝑖 ↓𝑗 ⟩ =2
𝐽𝑥𝑦
𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 ) |↓𝑖 ↑𝑗 ⟩ =|↑𝑖 ↓𝑗 ⟩
2

(2.8)
(2.9)
(2.10)

That is a pure quantum effect. We note that, in contrast, in the classical limit, a state|↑𝑖 ↓𝑗 ⟩ would not evolve. Both spins feel zero torque, because they are anti-parallel.
However, in quantum mechanics, both spins have quantum fluctuations in the 𝑆 𝑥 and𝑆 𝑦 direction, which are zero only on average. These quantum fluctuations exert atorque on each spin’s neighbor and cause them to spin-flip.
+Jz+Jxy/2

Figure 2-3: Mapping to spin-less fermions. By mapping spin |↑⟩ to a fermion |1⟩
and spin |↓⟩ to an empty site |0⟩, we can interpret 𝐽𝑥𝑦 as a hopping rate for fermions,and 𝐽𝑧 as a nearest-neighbor interaction.
In this spin |↑⟩ and |↓⟩ basis, the Hamiltonian features a competition between two
terms. The first term 𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 is an interaction term between |↑⟩ and |↓⟩ spins. The
second term 𝐽𝑥𝑦 (𝑆𝑖+ 𝑆𝑗− + 𝑆𝑖− 𝑆𝑗+ )/2 is a kinetic term of |↑⟩ and |↓⟩ spins. In fact (atleast in 1D) we can understand this system even more intuitively in terms of spinless fermions, by mapping spin |↑⟩ onto one fermion and spin |↓⟩ onto an empty site
(Figure 2-3), by the so-called Jordan-Wigner transformation [31]. Then 𝐽𝑥𝑦 can beinterpreted as the hopping matrix element for the fermions, and 𝐽𝑧 as the nearestneighbor interaction. In our experiment we can switch-off this interaction (𝐽𝑧 = 0),or we can make it repulsive (𝐽𝑧 > 0), or we can also make it attractive (𝐽𝑧 < 0).
28

bProbabilityaof a single spin excitation in the Heisenberg model. a, A single spin is flipped at the centre site of a 1D spin chain. Each5neighbours through the superexchange couplingJ . b,c, Coherent evolution showing the polar angles of the spins after
measurement, the spin-flipped atom is observed on sitej with the probability[1− cos( θj ) ]/ 2, where θj is the polar angle of

c
0 ms

40 ms

60 ms

80 ms
10

Wavefront position (lattice sites)

http://science.sciencemag.org/ on October 9, 2019by applying a mapping technique26,( 27) ing that even though strong spin currents arethe spin dynamics with an applied bias on theErcombined with a Stern-Gerlach filter (Fig. present in the system, no net mass flow can bedouble wells for a short lattice depth of 15
2C). A maximum spin imbalance of 60 to observed for our initial state. We fit the traces and the same depths for the long and transverse
70% was observed for our initial state for Nz(t) with a sum of two damped sine waves lattice as before. Starting with an initial anticorresponding to a probability of 80 to 85% with variable frequenciesw1,2/2p and amplitudes ferromagnetically ordered state, as above, we
for having prepared the desired state, | . We A1,2. For the damping we assume Gaussian first let the system evolve in symmetric double
believe that this measured value is mainly characteristics with 1/
e-damping constantsg1,2. wells (D = 0) for t0 = 4.5 ms until the first node
reduced as a result of our detection method.The results of the fit are displayed in Fig. 4. For Nz(t) = 0 of the pspinimbalance is reached for the
ffiffiffi
Direct spin-exchange processes emerge during Vshort ≥ 15 E r (Fig. 4A, inset), we can identify state (|t + i|s )/1.0
2 (Fig. 5). After freezing out) and can lead to a only a single frequency component correspond-the relative phase betweens |and |t by ramping
2
mixing of the spin configuration and thus a ing to the superexchange oscillation withJ 4/hU up the potential barrier, a defined potential bias
reduction of the measured Néel order and full relative amplitude (Fig. 4B). We are able D is applied and a second evolution sequence
to observe this frequency down to 4.8(4) Hz at with hold time t = t − t0 is initialized by
0.5
ramping down the short lattice again to 15
Er.
The subsequent detection follows the scheme
described above.
Figure 5A shows the evolution of the spin
imbalanceNz(t) in symmetric double wells to0.2
gether with the time traces for two different
bias energiesD0.1> U, yielding an effective coupling of J ex(D) ≈ −J ex(D = 0) and−J ex(D = 0)/2,
respectively. The sign reversal ofJ ex due tothe introduction
0.2of the bias is directly visibleby the change in slope of the spin imbalancet=t0. It should be0.1
noted that the now negative signof J ex for bosons does not imply a violation of the
Lieb-Mattis theorem, because the new groundstate in this regime
0.2 is the spin triplet state ,0|and the superexchange couples the first andsecond excitedstatess | and |t , which have
0.1
reversed order forD > U (Fig. 5B, left inset).
The introduction of a nonzero tilt leads to an-100exincreased sensitivity of the exchange-5frequency(lattice sites)
to fluctuations due to the inhomogeneities Positionin the

8

free

6
4
2
0
0

bound
50
Time, t (ms)100

Figure 2-4: Previous experiments near the ground state. a, Superexchange with
two spins [7]. Spin-imbalance (blue) and mass imbalance (gray) was measured as afunction of time. b, One single magnon in a spin-chain [17]. The initially localized
wavefunction (at time 𝑡 = 0) of the magnon spreads out as a function of time (shownfor 𝑡 = 40 ms, 60 ms, 80 ms). c, Two interacting magnons form a bound state [18]. The
wavefront positions are plotted as a function of time for both the travelling boundstate (green) and for free magnons (blue), revealing a velocity ratio of 2.

2.2

Previous experiments

Spin exchange 𝐽𝑥𝑦 has first been directly observed in ultracold atoms in a double-wellpotential by the Bloch group [7]. Two opposite spins swap places at the spin-exchange
rate 𝐽𝑥𝑦 /ℎ (Figure 2-4a). Using a quantum gas microscope with single site resolution, the same group even directly observed spin exchange in a whole spin chain [17].
A single spin-flip travels with ballistic wavefronts (Figure 2-4b). Such a single freemagnon, initially localized to one lattice site, then moves in both directions simultaneously through the spin chain, undergoing a quantum random walk. Interactions 𝐽𝑧
do not play a role here, because this is a single free magnon |↓⟩ moving in a background of |↑⟩ spins. Flipping one single spin in the |↑⟩-background to |↓⟩, has anenergy cost of 2 · 𝐽𝑧 /4, no matter where the spin |↑⟩ is located.
For two spin-flips this is different. Interactions 𝐽𝑧 can now lead to a two-magnon
bound state [18]. For 𝐽𝑧 < 0, magnons are attractively bound. Flipping two spins in
the |↑⟩-background to |↓⟩ costs an energy of 4 · 𝐽𝑧 /4, when they are more than one
lattice site apart from each other. But if they are neigbors, the energy cost is only
2 · 𝐽𝑧 /4. The two-magnon bound state still travels with ballistic wavefronts. The two29

a Sinusoidal spin modulationb Linear spin modulationc Stepwise spin modulation
1

(i)

(ii)

0

(iii)

Figure 2-5: Previous experiments far away from equilibrium. a, A sinusoidal spin
modulation (spin-helix) with initially 100% contrast (left) fades away (right) due tospin transport [19]. b, A linear spin modulation (created by a magnetic gradient)displays a spatial spin imbalance in opposite sides of the sample, which mixes due
to spin transport [15]. c, A stepwise spin modulation (sharp domain wall) which is
initially perfectly sharp (top) smoothens out (bottom) due to spin transport acrossthe interface [33].
magnons just move together (Figure 2-4c). But the velocity is smaller, since it now
requires two spin-exchange processes for the bound state to move by one lattice site.
(The first magnon has to move first. Then the second magnon has to follow.) This isa higher-order process involving an intermediate with higher energy. Theory predicts
that the bound magnons move a factor of 2 slower, or a factor of 2Δ for generalanisotropies [32], and that is what is being observed experimentally, too. Actually,
there is also a small probability that the two-magnon bound state breaks up, which
has been studied experimentally by Ref. [18], as well.
For many spin-excitations however, interactions change the transport behavior
into diffusion. This was observed both for a sinusoidal spatial spin modulation in
Ref. [19] in 2014 (Figure 2-5a), as well as a linear spatial spin modulation in theZwierlein group [15] in 2019 (Figure 2-5b). A recent experiment by the Bloch group[33] in 2021 (Figure 2-5c) studied the dynamics for a sharp domain wall, where thespin modulation is a step function, which shows superdiffusive behaviour.
All of these experiments realize the isotropic Heisenberg model, where the anisotropy is fixed at Δ = 1. The following chapter explains how we implement a Heisenberg
model with tunable anisotropies, using the two lowest hyperfine states of 7 Li as spin|↑⟩ and |↓⟩ states in an optical lattice, and controlling their interactions with Feshbachresonances.
30

Chapter 3
Li7 in optical lattices
3.1

Realization of the Heisenberg modelTwo component bosons in an optical lattice are described by the Hubbard model𝐻=−∑︁(𝑡 𝑎†𝑖𝜎 𝑎𝑗𝜎 + H.c.) +⟨𝑖𝑗⟩𝜎∑︁ 𝑈𝜎𝜎𝑖𝜎

2𝑛𝑖𝜎 (𝑛𝑖𝜎 − 1) +∑︁𝑈↑↓ 𝑛𝑖↑ 𝑛𝑖↓

(3.1)

𝑖

characterized by the tunneling matrix element 𝑡 and the on-site interactions 𝑈↑↑ ,𝑈↑↓ , and 𝑈↓↓ . In the Mott insulating regime with one atom per lattice site, the motional degree of freedom is frozen out and atoms are localized to individual latticesites [34]. The remaining degree of freedom forms an effective spin model [35–38],
where spin-spin interaction are mediated by second-order tunneling [7] (see Figure 31). In the limit of 𝑡 ≪ 𝑈↑↑ , 𝑈↑↓ 𝑈↓↓ , the tunneling term 𝑉 = −†⟨𝑖𝑗⟩𝜎 (𝑡 𝑎𝑖𝜎 𝑎𝑗𝜎 , + H.c.)∑︀

can be treated as a perturbation. The Hubbard model 𝐻 can then be mapped to aneffective spin Hamiltonian 𝐻eff , by the Schrieffer–Wolff transformation [39, 40], withmatrix elements
1 ∑︁
1
1⟨𝑚| 𝐻eff |𝑛⟩ = 𝐸𝑛 𝛿𝑛𝑚 +⟨𝑚| 𝑉 |𝑛⟩+⟨𝑚| 𝑉 |𝑘⟩+⟨𝑘| 𝑉 |𝑛⟩+...
2 𝑘
𝐸𝑚 − 𝐸𝑘 𝐸𝑛 − 𝐸𝑘
(3.2)
(︂

)︂

where |𝑛⟩, |𝑚⟩ and |𝑘⟩ are the eigenstates of the unperturbed Hamiltonian 𝐻0 =
∑︀

𝑖𝜎

∑︀
𝑈𝜎𝜎
𝑛𝑖𝜎 (𝑛𝑖𝜎 − 1) + 𝑖 𝑈↑↓ 𝑛𝑖↑ 𝑛𝑖↓ .
2The zeroth- and first-order terms are zero. The
31

second-order terms are called superexchange, or second-order tunneling, because it
involves two tunneling matrix elements connecting an initial state |𝑛⟩ to a final state|𝑚⟩ via an intermediate state |𝑘⟩, which is detuned in energy by 𝐸𝑛 − 𝐸𝑘 and 𝐸𝑚 − 𝐸𝑘 ,respectively, appearing in the denominator. We show in the following that this effective Hamiltonian is the anisotropic Heisenberg model𝐻=∑︁[𝐽𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦 (𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 )] −∑︁ℎ𝑧 𝑆𝑖𝑧

(3.3)

𝑖

⟨𝑖𝑗⟩

in the presence of an effective magnetic field ℎ𝑧 .
Figure 3-1 displays a list of all nonzero effective matrix elements for a given pairof nearest neighbors. (a), (b) and (c) are diagonal matrix elements, where the initialand final state are identical |𝑛⟩ = |𝑚⟩. (d) is an off-diagonal matrix element, where|𝑛⟩ ̸= |𝑚⟩. In (a), both spins are aligned pointing up |↑; ↑⟩. One spin tunnels to its√√
neighbor with matrix element 2𝑡, this costs energy 𝑈↑↑ and it tunnels back with 2𝑡,
resulting in an effective matrix element of −2𝑡2 /𝑈↑↑ . The factor of 2 is the result ofBose enhancement. (Intuitively speaking, in this process there seem to be two possible
final states: The two spins |↑1 ; ↑2 ⟩ can either swap their places |↑1 ; ↑2 ⟩ → |↑2 ; ↑1 ⟩,or return to their original places |↑1 ; ↑2 ⟩ → |↑1 ; ↑2 ⟩. In quantum mechanics these
seemingly two final states are indistinguishable, hence the factor of 2 stems frombosonic counting statistics.) There is an additional factor of 2 accounting for the two
possible intermediate states: |↑↑; 0⟩ and |0; ↑↑⟩ (both spins meeting either on the left
site or on the right site). So the total effective matrix element is −4𝑡2 /𝑈↑↑ . In the caseof fermions, this matrix element is zero, because two fermions cannot be in the same
quantum state. Both particles are already in the same spin |↑⟩ state, so they cannot
share the same lattice site. This effectively results in 𝑈↑↑ ≈ +∞ (“infinite repulsion”).
A more rigorous argument is that the tunneling matrix element — instead of being√
Bose enhanced, yielding 2 · 𝑡 — is now Fermi suppressed, yielding 0 · 𝑡. (We notethat the two fermions can nevertheless share the same lattice site, if they occupy
different spatial wave functions, but this contribution is neglected in a lowest banddescription, where higher excited bands are energetically suppressed.) In (b), both32

Energy

Effective Heisenberg couplings

(a)

(b)

(c)

(d)

Figure 3-1: Mapping to the Heisenberg model. Spin interactions are mediated by superexchange: second-order tunneling between initial and final states |𝑛⟩and |𝑚⟩, involving an intermediate state |𝑘⟩, which is detuned in energy. The interactions energy scales as matrix element squared over energy. The diagonal matrix elements (a)-(c) mediate the spin-spin interaction 𝐽𝑧 . The off-diagonal matrix element (d) induces spin-exchanges 𝐽𝑥𝑦 . The energy difference between |↑↑⟩
and |↓↓⟩ (both aligned) can be interpreted as induced by an effective magnetic
field ℎ𝑧 = 4𝑡2 /𝑈↑↑ −4𝑡2 /𝑈↓↓ , which shifts their energies away from the averaged value−2𝑡2 /𝑈↑↑ −2𝑡2 /𝑈↓↓ (aligned configuration). Comparing this to the energy of |↑↓⟩ and|↓↑⟩ which is −2𝑡2 /𝑈↑↓ (anti-aligned configuration) defines the effective spin-spin coupling 𝐽𝑧 /2 = 2𝑡2 /𝑈↑↓ − (2𝑡2 /𝑈↑↑ −2𝑡2 /𝑈↓↓ ). There is also a contribution from direct
nearest-neighbor off-site interactions 𝑉↑↑ , 𝑉↑↓ , 𝑉↓↓ , which adds as a correction to theeffective superexchange matrix elements.

spins are aligned pointing down |↓; ↓⟩ and following the same reasoning as in (a), theeffective matrix element is now −4𝑡2 /𝑈↓↓ . In (c), the spins are anti-aligned |↑; ↓⟩. We
count two intermediate states again: |↑↓; 0⟩ and |0; ↑↓⟩ resulting in a factor of 2 suchthat the effective matrix element is −2𝑡2 /𝑈↑↓ . But there is no Bose enhancement (orno Fermi suppression), because the particles are in different spin states. Intuitively
speaking, there is only one possible final state: The two spins return to their originalplaces |↑; ↓⟩ → |↑; ↓⟩. If they swap their places |↑; ↓⟩ → |↓; ↑⟩, this is captured by33

the off-diagonal matrix element (d), also yielding −2𝑡2 /𝑈↑↓ for bosons. In the case offermions we get an additional factor of −1 because we interchanged the two particles,
yielding +2𝑡2 /𝑈↑↓ .
These superexchange mediated interactions between spins lead to an effective spinmodel, which is exactly described by the anisotropic Heisenberg model, discussed in
the previous chapter. So far, spin dynamics in optical lattice has been studied for repulsive on-site interactions 𝑈↑↑ , 𝑈↑↓ , 𝑈↓↓ > 0, because attractive interactions generally
cause instabilities. Repulsive interactions ensure that a Bose-Einstein condensate isstable, and that the Mott insulator is the ground state for strong interactions in a
lattice. Previous experiments were limited to isotropic interactions for the following
reasons:
For fermions, the Pauli exclusion principle enforces isotropic antiferromagnetism[15, 25–27]. The two superexchange matrix elements (a) and (b) for aligned spins|↑; ↑⟩ and |↓; ↓⟩ are zero due to Fermi suppression. Superexchange interactions thenaffect only anti-aligned spins |↑; ↓⟩ in (c) lowering their energy to yield −2𝑡2 /𝑈↑↓and hence favoring this anti-aligned configuration (at least for typical systems, whereinteractions are repulsive 𝑈↑↓ ). Since in a Heisenberg model, the aligned (|↑; ↑⟩ and|↓; ↓⟩) and anti-aligned (|↑; ↓⟩ and |↓; ↑⟩) configuration are split in energy by 𝐽𝑧 /2,
the resulting longitudinal coupling is 𝐽𝑧 /2 = +2𝑡2 /𝑈↑↓ . Fermionic statistics also resultin a positive sign for spin-exchange (d) yielding 𝐽𝑥𝑦 /2 = +2𝑡2 /𝑈↑↓ . The result isisotropic and antiferromagnetic spin couplings (for the typically repulsive interactions
𝑈 := 𝑈↑↓ > 0):
𝐽𝑧 = 𝐽𝑥𝑦 = +

4𝑡2
> 0.
𝑈

(3.4)

For bosons, the most commonly used atom, 87 Rb, has almost equal singlet andtriplet scattering lengths (resulting in 𝑈↑↑ ≈ 𝑈↑↓ ≈ 𝑈↓↓ ), implying effectively isotropic
spin physics [16–19]. The two superexchange matrix elements (a) and (b) for aligned
spins |↑; ↑⟩ and |↓; ↓⟩ are now Bose enhanced by a factor of 2, compared to the antialigned spins |↑; ↓⟩ in (c), and hence favoring the aligned configuration (at least for
typical systems, where interactions are repulsive 𝑈↑↑ ≈ 𝑈↑↓ ≈ 𝑈↓↓ > 0), with longitu34

dinal coupling 𝐽𝑧 /2 = −2𝑡2 /𝑈↑↓ . Bosonic statistics also results in a negative sign for
spin-exchange (d) yielding 𝐽𝑥𝑦 /2 = −2𝑡2 /𝑈↑↓ . The result is isotropic and ferromagnetic spin couplings (for the typically repulsive interactions 𝑈 := 𝑈↑↑ = 𝑈↑↓ = 𝑈↓↓ > 0):
𝐽𝑧 = 𝐽𝑥𝑦 = −

4𝑡2
<0
𝑈

(3.5)

Anisotropic spin couplings 𝐽𝑧 ̸= 𝐽𝑥𝑦 require bosons with on-site interactions 𝑈↑↑ ,𝑈↑↓ , 𝑈↓↓ , which are not all the same. If the intra-species interactions differ (𝑈↑↑ ̸= 𝑈↓↓ ),this implies that both aligned configurations |↑; ↑⟩ and |↓; ↓⟩ in (a) and (b) also
have different energies −4𝑡2 /𝑈↑↑ and −4𝑡2 /𝑈↓↓ . This energy splitting can be interpreted as being caused by an effective magnetic field ℎ𝑧 = 4𝑡2 /𝑈↑↑ − 4𝑡2 /𝑈↓↓ whose
finite value shifts the two states symmetrically in opposite directions away fromtheir average value −2𝑡2 /𝑈↑↑ − 2𝑡2 /𝑈↓↓ . Comparing this energy, in turn, to the value−2𝑡2 /𝑈↑↓ of the anti-aligned configuration |↑; ↓⟩ in (c), determines which configuration is favored. And the energy difference yields the longitudinal spin coupling𝐽𝑧 /2 = 2𝑡2 /𝑈↑↓ − 2𝑡2 /𝑈↑↑ − 2𝑡2 /𝑈↓↓ . The spin-exchange coupling (d) 𝐽𝑥𝑦 /2 = −2𝑡2 /𝑈↑↓is entirely determined by 𝑈↑↓ . This, in general, results in anisotropic spin couplings
4𝑡2
4𝑡2
4𝑡2
−
−
𝑈↑↓ 𝑈↑↑ 𝑈↓↓
4𝑡2
= −
𝑈↑↓
4𝑡2
4𝑡2
=
−
𝑈↑↑ 𝑈↓↓

𝐽𝑧 =

(3.6)

𝐽𝑥𝑦

(3.7)

ℎ𝑧

(3.8)

and a finite effective magnetic field ℎ𝑧 . In particular, the transverse coupling can
be ferromagnetic (𝐽𝑥𝑦 < 0) for repulsive 𝑈↑↓ > 0, and antiferromagnetic (𝐽𝑥𝑦 > 0) for
attractive 𝑈↑↓ < 0. We realize wide tunability of 𝐽𝑧 and 𝐽𝑥𝑦 by using 7 Li atoms, whose
Feshbach resonances we have characterized in previous work [41], which tune the onsite interactions 𝑈↑↑ , 𝑈↑↓ , and 𝑈↓↓ as a function of an externally applied magnetic field(see Figure 3-2). Additionally, lithium with its light mass has the advantage of fast
spin dynamics (set by second-order tunneling [7]), decreasing the relative importanceof heating and loss processes compared to heavier atoms.
35

a

300

Scattering length [a0]

200
100
0
-100
-200
-300

Anisotropy Jz/Jxy

b

2
1
0
-1
-2
820

840

860

880

Magnetic field [G]Figure 3-2: Tunability of the anisotropy. a, Scattering lengths 𝑎↑↑ (blue), 𝑎↑↓(purple) and 𝑎↓↓ (orange) measured in previous work using interaction spectroscopy[41]. Here 𝑎0 is the Bohr radius. b, We interpolate the data points for the scattering
lengths in a using hyperbolic fits (solid lines) and calculate values for the anisotropy
Δ = 𝐽𝑧 /𝐽𝑥𝑦 . The excluded region (|𝑎↑↑ | > 700 𝑎0 ) is around a Feshbach resonance inthe |↑⟩ state near 845.4 G.
Ultracold neutral atoms (at low densities) interact with contact interactions.
Therefore, neighboring spins interact mainly through superexchange, where they experience the contact interaction as an on-site interaction through second-order tun36

neling. On-site interactions are due to the wavefunction overlap of two atoms sharing
the same lattice site𝑈𝜎𝜎′ =

4𝜋~2 𝑎𝜎𝜎′ ∫︁ 3
𝑑 𝑟 |𝑤(r)|4
𝑚

(3.9)

where 𝑤(r) is the (lowerst-band) Wannier function of an atom localized to a latticesite. However, this wavefunction is not perfectly localized, but also extend slightly
into the neighboring site, which is one lattice spacing 𝛿r = (0, 0, 𝑎) away. This leads toa direct nearest-neighbor off-site interaction, simply due to wavefunction overlap [42]:𝑉𝜎𝜎′ =

4𝜋~2 𝑎𝜎𝜎′ ∫︁ 3
𝑑 𝑟 |𝑤(r)|2 |𝑤(r + 𝛿r)|2𝑚(3.10)

These direct interactions are typically really small, as on-site interactions 𝑈𝜎𝜎′ areabout 4 orders of magnitude smaller than off-site interactions 𝑉𝜎𝜎′ . They can nevertheless become the dominating form of spin-spin interactions close to a Feshbachresonance, when superexchange is suppressed by a large value of 𝑈𝜎𝜎′ and offsite interactions 𝑉𝜎𝜎′ are strong. Therefore we include also mention them in Figure 3-1 andinclude them as a correction to the anisotropy in Figure 3-2.
The magnitude of superexchange can be varied over two orders of magnitude
by changing the lattice depth, which mainly affects tunneling 𝑡 and scales the entire Hamiltonian. In our experiment we control the anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦 by using the two lowest hyperfine states of 7 Li as spin states |↑⟩ := |𝐹 = 1, 𝑚𝐹 = +1⟩ and|↓⟩ := |𝐹 = 1, 𝑚𝐹 = 0⟩ and Feshbach resonances to tune the relative interactions 𝑈↑↑ ,𝑈↑↓ , 𝑈↓↓ as a functions of the applied magnetic field 𝐵 (Figure 3-2), while keeping
𝐽𝑥𝑦 > 0 (antiferromagnetic). We measured these values [41, 43] with high precision
and accuracy using interaction spectroscopy of two atoms per site in a deep Mott
insulator. In principle we can also realize purely ferromagnetic couplings by changing the sign of 𝐽𝑥𝑦 using a constant force to tilt the lattice, as we demonstrated
in previous work [44]. The ability to tune the anisotropy Δ over a wide range of
positive and negative values allows us to explore dynamics beyond previous experiments [15–19,25–27] in which Δ ≈ 1. Special cases include the XX model with Δ = 0,
which can be mapped onto an effective model of free spin-less fermions. Further37

more we can interpolate between Δ = 0 and 1 continuously, effectively switching on
interactions between these effective fermions. The isotropic point (Δ = 1) is then in
the strong-coupling regime. And in addition, we can also go to the regime whereinteractions dominate (Δ > 1). More generally we can compare positive anisotropies(Δ > 0) to negative anisotropies (Δ < 0). Or, in the effective fermionic language, we
can compare repulsive interactions to attractive interactions.

3.2

Far-from-equilibrium spin-helix states

The experiments start by adiabatically loading a Bose-Einstein condensate of about
4.5 × 104 7 Li atoms initially in the |↑⟩ state into an optical lattice (spacing 𝑎 = 532 nm),driving the system into the Mott insulating regime with one atom per site, with astrong repulsive scattering length 𝑎↑↑ = +307 𝑎0 to avoid doubly occupied sites. Two
deep lattice beams (𝑉𝑥 , 𝑉𝑦 = 35 𝐸𝑅 ) in the 𝑥 and 𝑦 direction separate the lattice sites
into independent one-dimensional spin chains. The third lattice beam (𝑧 direction)
controls the superexchange couplings (𝐽𝑧 and 𝐽𝑥𝑦 ) within chains. The typical chain
length is 44 lattice sites. We always perform experiments simultaneously in an ensemble of ∼ 1000 chains and measure an averaged result. For the preparation of thespin helix [19–21], superexchange coupling is switched off (𝐽𝑧 ≈ 0, 𝐽𝑥𝑦 ≈ 0) by a deep𝑧 lattice beam with a depth of 𝑉𝑧 = 35 𝐸𝑅 .
First, we ramp the external magnetic field 𝐵 to the right value required to adjust
the anisotropy Δ. Then we prepare the initial state in the following way: The atoms
are initially spin-polarized in the |↑⟩ state (Figure 3-3a):|𝜓↑ ⟩ =

∏︁|↑⟩𝑖

(3.11)𝑖

A global 𝜋/2-pulse of ∼ 75 𝜇s rotates all spins into the 𝑆𝑥 -𝑆𝑦 plane of the Bloch-sphere(Figure 3-3b)|𝜓→ ⟩ =

∏︁√(|↑⟩𝑖 + |↓⟩𝑖 ) / 2(3.12)𝑖

By applying a magnetic field gradient in 𝑧 direction, the spins precess around this
38

a

Sz

Sx

b

Sz
z

Sx

c

Sz
z

Sx

d

Sz
z

Sx

a

z

z

x
ImagingyFigure 3-3: Spin-helix preparation. a, All spins are initially pointing |↑⟩. b,
A global 𝜋/2-pulse rotates them into the 𝑆𝑥 -𝑆𝑦 plane of the Bloch sphere. c, An
applied magnetic field gradient winding the spins up into a transverse spin-helix. d,
An additional 𝜋/2-pulse turns this into a longitudinal spin-helix.

39field at slightly different rates, so that they wind up into a spin-helix (Figure 3-3c)|𝜓𝑥𝑦 (𝑄)⟩ =

∏︁ (︁

)︁ √
|↑⟩𝑖 + 𝑒−𝑖𝑄𝑧𝑖 |↓⟩𝑖 / 2(3.13)𝑖

The resulting wavevector 𝑄 = 2𝜋/𝜆 is controlled by the strength of the gradient andthe winding duration. The resulting state is a transverse spin helix. Its windingorientation is in the 𝑆𝑧 -𝑆𝑥 plane of the Bloch sphere. With an additional 𝜋/2-pulse
we can rotate this state into a longitudinal spin helix (Figure 3-3d)|𝜓𝑧𝑥 (𝑄)⟩ =∏︁

(sin(𝑄𝑧𝑖 /2) |↑⟩𝑖 + cos(𝑄𝑧𝑖 /2) |↓⟩𝑖 )(3.14)𝑖whose winding orientation is in the 𝑆𝑥 -𝑆𝑧 plane of the Bloch sphere.
We then initiate the time evolution for this far-from-equilibrium state, by ramping
down the 𝑧-lattice beam to a value 𝑉𝑧 between 9 and 13 𝐸𝑅 in order to switch onsuperexchange couplings 𝐽𝑧 and 𝐽𝑥𝑦 to a controlled value. We let the system evolvefor a variable evolution time 𝑡 up to ∼ 500 spin-exchange times ~/𝐽𝑥𝑦 . For detection
we freeze the dynamics by rapidly increasing the lattice depth back to 35 ER.

3.3

Polarization-rotation imaging

We measure the resulting spatial spin distribution of the spin |↑⟩ state in-situ withstate-selective polarization-rotation imaging. The imaging system is transverse to thechains and the optical resolution is about 6 lattice sites. The resulting images showa characteristic stripe pattern (Figure 3-3) which arises from the residual sinusoidal
modulation of population in |↑⟩ in each chain.
The optical density of the atomic ensemble is too high (> 14) to allow for in-situ
observation of the modulation of ⟨𝑆 𝑧 ⟩ via absorption imaging. Instead, we employ
dispersive imaging, which uses the phase accumulated by the transmitted light inorder to form an image of the atomic density distribution. When light at frequency
𝜔𝐿 is detuned from the atomic resonance 𝜔0 by many natural linewidths Γ, it picks
up an approximate phase 𝜃 ≈ −2𝛿/Γ × OD(𝑦, 𝑧, 𝛿), where OD is the optical density at40

linearλ/4
waveplate analyzer

B

z

+

σ

y
x

Linear polarization
at 45°

Atoms

Phase shift θ
for x-component

Elliptical polarization

Circular analyzer

Image

Figure 3-4: Polarization-rotation imaging. For imaging we address a 𝜎 + transition only (and the 𝜋-transition is really far detuned). That means, when theimaging beam is coming from the side (𝑦-direction), the atoms can only interact witha linearly 𝑥-polarized beam (because then it has overlap with 𝜎 + -polarization), but
the atoms are transparent for linearly 𝑧-polarized beam (𝜋-polarization). If we detuned from the imaging transition by a few natural linewidths, the atoms act as a
waveplate, with a fast and a slow axis. The column density is proportional to a phaseshift of the slow axis. So, if the incoming beam is at an angle of 45∘ , its polarizationis rotated. The polarization rotation can then be turned into an amplitude modulation by using a polarizer. (This setup is analogous to an electro-optical modulator
configured as an amplitude modulator.)

detuning 𝛿 = 𝜔𝐿 − 𝜔0 , while absorption is suppressed by a sufficiently large detuning 𝛿.
In order to form an image, the phase-shifted light must be interfered with a reference
beam. In this work, we make use of the fact that the optical transition we use for
imaging is driven only by a single polarization component; after passing through theatoms, the shifted and unshifted components are combined on a polarizer. A judicious
choice of input and output polarizers yields an interference signal 𝐼 on the camera
which is 𝐼 = 𝐼0 (1 − sin 𝜃)/2 [45, 46] (see Figure 3-4).
The imaging light is initially linearly polarized at 45∘ , i.e. there are two equal
orthogonal polarization components in 𝑥 and 𝑧 direction with a (normalized) polarization vector as e0 = √12 (1, 1). The first one interacts with the atomic imaging tran41

60

Position y [a]-2Column density [a ]

-20

0

30

20
-20

0

20
-20
Position z [a]

0

20

0

Figure 3-5: Measured column density of atoms in the |↑⟩ state after imprinting alongitudinal spin-helix along the (horizontal) 𝑧-direction with wavelength 𝜆 = 10.4 𝑎.
Right: Every pixel is a local measurement of the column density (number of atoms perunit area). The 𝑦 and 𝑧 axis are displayed in units of lattice spacings 𝑎 = 0.532 𝜇m.
Left: Visualization of the chain length distribution within a cut through the centerof the spherical Mott insulator. Its diameter determines the maximum chain length
𝐿max = 44 𝑎. The optical imaging resolution is about 6 lattice sites. The pixel size ofthe camera is 16 𝜇m × 16 𝜇m. This image was taken with a magnification of 𝑀 = 50.
So every pixel represents a region in space of size 0.32 𝜇m = 0.6 𝑎.
sition (and experiences a phase shift 𝜃) and the second one does not. This rotates
the polarization vector to e = √12 (𝑒𝑖𝜃 , 1). With a circular analyzer the polarization
gets projected onto 𝜎 + = √12 (𝑖, 1) to yield the fraction of the transmitted intensity as|e · 𝜎 + |2 = 14 |𝑖𝑒𝑖𝜃 + 1| = 12 (1 − sin 𝜃). The phase shift 𝜃 is related to the column densityas𝜃(𝑦, 𝑧) = −

∫︁
2𝛿
2𝛿
𝜎0
OD(𝑦, 𝑧) = −𝑑𝑥 𝑛(𝑥, 𝑦, 𝑧)ΓΓ 1 + (2𝛿/Γ)2
2𝛿
𝜎0
= −
𝑛2D (𝑦, 𝑧)Γ 1 + (2𝛿/Γ)2(3.15)(3.16)and 𝜎0 = 6𝜋(𝜆/2𝜋)2 /2 is the resonant scattering cross-section (reduced by 2, because
𝑥-polarized light only has 50% overlap with 𝜎 + ). This way we can measure the column
density 𝑛2D (𝑦, 𝑧) in-situ, detuned, with dispersive imaging, without scattering light(as shown in Figure 3-5).
42

b

c

60

d

Position, y [a]-2aColumn density [a ]

-20

30

0

0

-1

Linear density [a ]

20
1,500

0
-20

0

20

-20

0

-20
20
Position, z [a]

0

20

-20

0

20

Figure 3-6: Contrast measurement using the linear density. The images a-dshow the distribution of atoms in the |↑⟩ state. The images are projected (integrated) along the 𝑦-direction from 𝑦 = −30 𝑎 to +30 𝑎 to obtain the linear density (number of atoms per unit length). The resulting 1D distributions are fittedwith 𝑓 (𝑧) = 𝑔(𝑧) · [1 + 𝒞 cos(𝑄𝑧 + 𝜃)]/2 (solid line), where 𝑔(𝑧) is a Gaussian envelope(dashed line), between 𝑧 = ±54 𝑎. Examples a-d were measured at different evolution times (a) 𝑡 = 0 ~/𝐽𝑥𝑦 , (b) 𝑡 = 2.3 ~/𝐽𝑥𝑦 , (c) 𝑡 = 6.3 ~/𝐽𝑥𝑦 , (d) 𝑡 = 12.0 ~/𝐽𝑥𝑦 , foranisotropy Δ ≈ 0 and wavelength 𝜆 = 10.4 𝑎 (see chapter 4). The obtained contrast
𝒞(𝑡) is shown in Figure 4-3. In general, we also normalize by the initial measured
contrast 𝒞(0) to correct for finite optical imaging resolution. This is important for
shorter wavelengths 𝜆 close to the optical resolution of 3 𝜇m, where the measured
contrast 𝒞(𝑡) is reduced compared to the real contrast 𝑐(𝑡) = 𝒞(𝑡)/𝒞(0).

3.4Image analysis

The imaging system already averages over chains distributed along the 𝑥 direction (theimaging direction). We project the images along the 𝑦 direction to average over thissecond transverse direction. Then we obtain the one-dimensional spin distribution
averaged over all spin chains. We analyze the spin-helix pattern by obtaining thehelix contrast 𝒞 through a fit
𝑓 (𝑧) = 𝑔(𝑧) · [1 + 𝒞 cos(𝑄𝑧 + 𝜃)]/2.

(3.17)as shown in Figure 3-6. Here g(z) is a Gaussian envelope function which accounts
for the spatial distribution of all atoms 𝑛 = 𝑛↑ + 𝑛↓ . 𝜃 is a random phase that varies
from shot to shot, due to small magnetic bias field drifts.
Later on, we extended our fitting technique to the whole two-dimensional image43

-1

Linear density [a ]

1500

0

60

Position y [a]-2Column density [a ]

-20

0

30

20
-20

0

20
-20
Position z [a]

0

20

0
750
-1
Linear density [a ]0Figure 3-7: Contrast measurement using the column density. The central image
represents raw data and shows the distribution of atoms in the |↑⟩ state. The left panel
is the two-dimensional fit 𝑓 (𝑦, 𝑧) = 𝑔(𝑦, 𝑧)·[1+𝒞 cos(𝑄𝑧+𝜃)]/2. The image is projected(integrated) both along the horizontal 𝑧 direction (right panel) and vertical 𝑦 direction(top panel) from 𝑦, 𝑧 = −42 𝑎 to +42 𝑎 to obtain the linear densities (number of |↑⟩atoms per unit length) for the measured data (points) and the fit (solid line). The
dashed line in the 𝑦-projection shows the (parabolic) envelope function for the spatial
density distribution in the cloud.
using the fit function
𝑓 (𝑦, 𝑧) = 𝑔(𝑦, 𝑧) · [1 + 𝒞 cos(𝑄𝑧 + 𝜃)]/2(3.18)

as demonstrated in Figure 3-7. Here the envelope function 𝑔(𝑦, 𝑧) also takes into
account that the atoms are actually distributed in a homogeneous sphere of radius 𝑅(the Mott insulator) such that𝑔(𝑦, 𝑧) = 𝐴√︁

1 − 𝑦 2 /𝑅2 − 𝑧 2 /𝑅2 · 𝐻(1 − 𝑦 2 /𝑅2 − 𝑧 2 /𝑅2 )(3.19)

with 𝐻(𝑟) is the Heaviside function.

3.5

Optical resolution

The longitudinal spin helix exhibits a 𝑆𝑧 -magnetization profile (i.e. in the spin |↑⟩population), which is a sinusoidal modulation. That makes it the perfect tool toprobe the modulation transfer function (MTF) directly in reciprocal space. The
44

+2
+1
0
Imaging beam

-1
Atoms

-2
1st lens

Figure 3-8: Optical resolution limit. The optical resolution is determined by the
first lens in the imaging system. The atoms form a sinusoidal phase grating, because
the longitudinal spin helix exhibits a 𝑆𝑧 magnetization profile (i.e. in the spin |↑⟩population), which is a sinusoidal modulation. That results in higher diffraction orders. (That is analogous to driving an electro-optical modulator to create a sinusoidal
phase modulation in the time-domain, which adds higher-order sidebands to a laser
beam.) And the angle between the orders depends on the wavevector 𝑄 of the phasemodulation. We can only resolve the grating with our imaging system, if at least thefirst orders are captured by the first lens. We use an air-spaced doublet (Thorlabs
ACA254-050-A) with a clear aperture of 20 mm and a focal length of 50 mm resulting
in a numerical aperature of NA ≈ 0.2. This yields a resolution of 𝜆/NA = 3.3 𝜇m.
Note that here the prefactor is exactly equal to 1, because we are not imaging a point
source, but a sinusoidal modulation.

optical resolution of our imaging system (NA ≈ 0.2) was determined to have a cutoff at modulation wavelength 𝜆 ≈ 3.0 𝜇m = 5.6 𝑎 (Figure 3-8). The reduction of themodulation transfer function MTF(𝑄) near the cut-off reduces the observed contrast
𝒞(𝑡) = MTF(𝑄) · 𝑐(𝑡) compared to the real contrast 𝑐(𝑡). Assuming that the experimental preparation sequence for the initial spin helix state achieves full contrast
𝑐(0) = 1 for any wavevector 𝑄 (based on careful pulse calibration and characterization), we can use 𝒞(0) as a direct measurement of MTF(𝑄) and determine the realcontrast as 𝑐(𝑡) = 𝒞(𝑡)/𝒞(0).

45

3.6

Summary

We have shown how to control the spin-exchange coupling 𝐽𝑥𝑦 using the lattice depth,
and how to tune the anisotropy Δ := 𝐽𝑧 /𝐽𝑥𝑦 with an external magnetic field using
Feshbach resonances. We can wind longitudinal and transverse spin-helix states withvariable wavevectors 𝑄. The imaging system can resolve wavevectors ranging from𝑄 = 0 to 𝑄 = 2𝜋/6𝑎 and measure their contrast. By lowering the lattice depth afterstate preparation, and thereby switching on spin couplings, we can now study theirevolution as a function of time. Figure 3-9 summarizes the experimental sequence.
Chapter 4 focusses on longitudinal spin helix states and spin transport. Chapter 5 isabout transverse spin helix states and spin dephasing. In Chapter 6 we find special
spin-helix states which are very stable and (in an ideal system) are even exact many-

Lattice loading

Spin helix preparation

Spin dynamics

B’
Gradient
and RF

Magnetic
field

RF

B

Bload

Bmeasure

Lattice
depth

Vx
Vy
Vz

Figure 3-9: Experimental sequence. a, Ultracold 7 Li atoms in the |↑⟩ state are
loaded into a deep optical lattice (𝑉𝑥 , 𝑉𝑦 , 𝑉𝑧 = 35 𝐸𝑅 ) with a strong repulsive scattering length 𝑎↑↑ = +307 𝑎0 (selected by an appropriate magnetic field 𝐵load ) to avoid
doubly occupied sites. The anisotropy Δ is selected by ramping to the magnetic field
to and appropriate value 𝐵measure , tuning the on-site interactions 𝑈↑↑ ,𝑈↑↓ ,𝑈↓↓ throughFeshbach resonances. b, A spin-helix state is prepared with RF pulses and a magnetic
field gradient. c, Time evolution is initiated by decreasing the lattice depth 𝑉𝑧 to avalue between 9 and 13 𝐸𝑅 , switching on interactions between spins along the 𝑧 direction. After a variable evolution time 𝑡, spin dynamics is frozen by rapidly increasing
the lattice depth back to 𝑉𝑧 = 35 𝐸𝑅 und the spatial spin distribution is imaged.
46

body eigenstates of the Hamiltonian. These special transverse spin-helix states canhave very short wavelengths. In order to image their contrast, we utilize a special
unwinding step just before imaging, allowing us to access the whole range of possible
wavevectors.

47

48

Chapter 4Spin transport

Using a longitudinal spin-helix pattern, which is a modulation in the population ofspin |↑⟩ and |↓⟩ atoms, we see a drastic impact on the transport properties, when theanisotropy is varied. When spins are coupled only along two of three possible orientations (the XX model), we find ballistic behavior of spin dynamics, whereas for
isotropic interactions (the XXX model), we find diffusive behavior. More generally,for positive anisotropies, the dynamics ranges from anomalous superdiffusion to subdiffusion, whereas for negative anisotropies, we observe a crossover in the time domainfrom ballistic to diffusive transport. These are two strikingly different crossover phenomena: Anomalous diffusion for positive anisotropies and behavior reminiscent of aclassical gas for negative anisotropies.

This chapter supplements work reported in the following publication [28](included in Appendix A)

P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler, W. Ketterle
Spin transport in a tunable Heisenberg model realized with ultracold atoms
Nature 588, 403-407 (2020)
49

b

Jxy/2

+Jz/4

-Jz/4

Anisotropy Jz/Jxy

a

2
1
0
-1
-2
820

840
860
Magnetic field [G]880

Figure 4-1: Tunability of the XXZ model with 7 Li [28]. a, The Hamiltonian ischaracterized by two energy scales: Transverse spin coupling 𝐽𝑥𝑦 (spin exchange) andlongitudinal spin coupling 𝐽𝑧 (spin-spin interactions). b, Anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦 as afunction of applied magnetic field. The solid line is a fit to experimental data points
which are calculated from measured values 𝑈↑↑ , 𝑈↑↓ , 𝑈↓↓ (see chapter 3).

4.1

Introduction

Studying spin transport in one-dimensional spin chains was the first way we demonstrated fully tunable anisotropy of the spin-spin couplings (Figure 4-1). The transport
properties change even drastically when the anisotropy is varied. Our initial goal has
simply been to demonstrate that we implement the anisotropic Heisenberg model.
This has been a long standing goal in the field of ultracold atoms and quantum simulation. Previous studies were all limited to the isotropic case. For large signal to noise
ratio, we then chose to utilize full contrast spin-helix patterns (Figure 4-2), which are
additionally also straight forward to prepare experimentally. This, in turn, enables us
to study spin transport far from equilibrium in previously unexplored regimes. These
spin-helix states exhibit maximum possible contrast, perturbing the system even as
far from equilibrium as possible, and far away from the linear-response regime. A
longitudinal spin helix (Figure 4-2) is an ideal probe to study spin transport, because
it is a spatial modulation of the population in spin |↑⟩ and |↓⟩. So spin transport
50

Sz
λ
Sx
z

a

z

x
ImagingyFigure 4-2: Spin helix realized from two hyperfine states (spin |↑⟩ and |↓⟩) [28].
The spin S winds in the 𝑆𝑧 -𝑆𝑥 plane as a function of position 𝑧 in the spin chain.
Deep optical lattices along 𝑥 and 𝑦 create an array of independent spin chains. The
𝑧-lattice is shallower and controls spin transport along each chain.
is the only way it can decay by. There is nothing else in the Hamiltonian, which
couples spin |↑⟩ to |↓⟩, except the spin-exchange term in the Heisenberg model (Figure 4-1a), which drives motion of spins. In order to study transport behavior, wemeasure how the decay timescale 𝜏 depends on the modulation lengthscale 𝜆. Notefor ballistic motion time grows linearly with distance (𝜏 ∝ 𝜆), whereas for diffusion
it grows quadratically (𝜏 ∝ 𝜆2 ). A lot of our results were unexpected and asking for
more and more data to explain them.
First of all, we confirm a similar experiment from the Bloch group [19] with spinhelix states in the isotropic system (Δ = 1), which yields diffusive transport (𝜏 ∝ 𝜆2 ).
In the XX model (Δ = 0), on the other hand, we measure ballistic transport (𝜏 ∝ 𝜆),
51

a

1

b

Contrast c(t)

0.8
0.6
0.4
0.2
0

0

10

20
30
Time t [ms]

40

50

0

10
Time t [ћ/Jxy]20

Figure 4-3: Superexchange dynamics. a, Spin helix contrast 𝑐(𝑡) for 𝜆 = 10.4 𝑎measured at three different lattice depths 9 𝐸𝑅 (orange), 11 𝐸𝑅 (blue), 13 𝐸𝑅 (yellow).
b, Decay curves collapse when times are expressed in units of the corresponding spinexchange times ~/𝐽𝑥𝑦 = 0.64 ms, 1.71 ms, 4.30 ms, respectively.
and we find oscillations yielding a linear dispersion relation, as might be expected,
since this system can be mapped to non-interacting fermions. This inspired us to revisit the isotropic case and we actually find small oscillations with a linear dispersion
relation there, too, which we consider somewhat unexpected. For arbitrary positive
anisotropies we find anomalous diffusion (𝜏 ∝ 𝜆𝛼 ), with exponents ranging from superdiffision (1 < 𝛼 < 2) to subdiffusion (𝛼 > 2); however, we can always find (mostly
small) oscillations with linear dispersion relations. For arbitrary negative anisotropies,
we see a crossover from ballistic to diffusive transport in the time-domain.

4.2

Tunable superexchange dynamics

The initially full contrast spin-helix state decays due to spin transport and we measure the modulation contrast 𝑐(𝑡) as a function of time (shown in Figure 4-3 for
𝜆 = 10.4 𝑎). We show that we can control the spin-exchange rate 𝐽𝑥𝑦 , by repeating
the measurement at three different lattice depths (Figure 4-3a). If the time axis isrescaled in units of the corresponding spin-exchange times ~/𝐽𝑥𝑦 , which we can calculate based on Hubbard parameters [28], we see that they collapse nicely on top of each
52

a

1

b

Δ=0

Δ=1

Contrast c(t)

0.8
0.6
0.4
0.2
0

0

10
Time t [ћ/Jxy]

20

0

20

40
Time t [ћ/Jxy]60

Figure 4-4: XX model and XXX model. Spin helix contrast 𝑐(𝑡) for 𝜆 = 10.4 𝑎measured at three different lattice depths 9 𝐸𝑅 (orange), 11 𝐸𝑅 (blue), 13 𝐸𝑅 (yellow)
in the XX model where Δ = 0 (a) and in the XXX model where Δ = 1 (b).
other (Figure 4-3b). This demonstrates that we observe transport by superexchange
and not some other process. Furthermore, using the external magnetic field, we canalso control the anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦 , and we see in Figure 4-4 how the decay curvechanges, comparing Δ = 0 and Δ = 1.

4.3

XX model (Δ = 0) and XXX model (Δ = 1)

We first compare two special cases: the isotropic model where Δ = 1 (the so-called
XXX model) which has been studied before [19] and the XX model where Δ = 0,
which can be mapped onto non-interacting fermions. Even for one given wavelength
𝜆 there is already a big effect. The shape of the spin-helix decay 𝑐(𝑡) is qualitatively
very different for both cases, as shown in Figure 4-4 for 𝜆 = 10.4 𝑎. Even without
any fit function, we can already learn a lot from the data itself. In the following, we
do a side-by-side comparison of (in the fermionic language) non-interacting fermions(Δ = 0) to strongly-interacting fermions (Δ = 1). For Δ = 0, in addition to an overall
decay, we observe a damped oscillation. The revival occurs after about ∼ 10 spinexchange times ~/𝐽𝑥𝑦 for a wavelength of 𝜆 of about ∼ 10 lattice sites 𝑎, intuitively
suggesting a velocity of one lattice site 𝑎 per spin-exchange time ~/𝐽𝑥𝑦 . This can be
53

a

b

Δ=0

Δ=1

λ = 15.7 a

λ = 15.7 a

λ = 13.4 a

λ = 13.4 aContrast c(t)λ = 11.8 a

λ = 11.8 a

λ = 10.4 a

λ = 10.4 a

λ = 9.4 a

λ = 9.4 a

λ = 8.5 a

λ = 8.5 a

λ = 7.8 a

λ = 7.8 a

λ = 7.2 a

λ = 7.2 a

λ = 6.7 a

0

10

20
Time t [ћ/Jxy]

λ = 6.7 a

30

0

c

75
Time t [ћ/Jxy]

150

d

1

Contrast c(t)

0.8
0.6
0.4
0.2
0
0

0.5
1
1.5
2
Rescaled time t [(ћ/Jxy)(λ/a)]

2.5

0

0.2
0.4
0.6
2
Time t [(ћ/Jxy)(λ/a) ]0.8

Figure 4-5: Linear and quadratic scaling. Spin helix contrast decay 𝑐(𝑡) for several
wavelengths 𝜆 in the XX model where Δ = 0 (a) and in the XXX model where Δ = 1(b). All decay curves collapse into a single curve if the time axis is rescaled by 𝜆 (c)and by 𝜆2 (d), respectively.
understood with a simple model, where two counter-propagating waves constructively
interfere again after traveling a relative distance of one wavelength at a propagation
velocity of one lattice site per super-exchange time. However, for the isotropic case of
Δ = 1 (strongly-interacting fermions), we observe qualitatively very different behavior:Interactions make the decay a lot slower and the oscillations are strongly suppressed.
To learn more about transport behavior, we now vary the wavelength 𝜆, as shown
54

100

Decay time τ [ћ/Jxy]

a

b

Δ=0

Δ=1

30
ballistic
α = 1.00(5)diffusive
α = 1.87(4)

10

3

0.05

0.1

0.15
Wavevector Q [2π/a]Quantum random walk

0.05

0.1

0.15

Classical random walk

Figure 4-6: Power-law scalings of the decay time constants 𝜏 as a function of
wavevector 𝑄. a, In the XX model (non-interacting fermions) the power law scaling
is linear, indicating ballistic transport, reminiscent of the quantum random walk ofa single magnon [17], moving to the left and the right coherently with equal matrix
elements. In the XXX model (strongly-interacting fermions) the power law scaling isclose to quadratic, indicating diffusive transport, consistent with a classical random
walk, moving to the left and the right incoherently with equal rates.
in the waterfall plots in Figure 4-5. For smaller wavelengths 𝜆, the decay gets faster.
But for the isotropic case (Δ = 1), the decay time-scale depends much more strongly
on the wavelength 𝜆 than in the free-fermion case (Δ = 0). In each case, all decay
curves collapse into a single curve, if the time axis is rescaled by 𝜆 (for Δ = 0) or by
𝜆2 (for Δ = 1). Linear scaling indicates ballistic transport (as one might expect for
non-interacting fermions), whereas quadratic scaling indicates diffusion (as one might
expect for strongly-interacting fermions).
Diffusion has already been observed in a previous experiment [19], which also
revealed quadratic scaling. Diffusion also suggests a purely exponential decay, being a solution to the diffusion equation. Ballistic transport in the XX model has
not been observed before, as previous work was limited to the isotropic case. Furthermore, we find that the ballistic case shows not only an overall decay, but also a
55

b

non-interacting fermions

E(k) = Jxy cos(ka)E(k)

kFrequency, ω [Jxy/h]avF = a/(ћ/Jxy)

0.15

0.1

v = 0.76(1) vF

0.05
v = 0.35(1) vF

vF :=

1 ∂E
a
=
ℏ ∂k
ℏ/Jxy

0

Fermi velocity

00.1
Wavevector Q [2π/a]

0.2Figure 4-7: Dispersion relations and velocities. a, Fermi velocity 𝑣𝐹 = 𝑎/(~/𝐽𝑥𝑦 )of a half-filled band in the non-interacting XX model. b, Measured oscillation frequencies follow linear dispersion relations 𝜔(𝑄) = 𝑣𝑄. For non-interacting fermions in
the XX model (Δ = 0) the velocity 𝑣 = 0.76(1) 𝑣𝐹 (red) is close to the Fermi velocity𝑣𝐹 (dashed line). For strongly-interacting fermions in the XXX model (Δ = 1) thevelocity 𝑣 = 0.35(1) 𝑣𝐹 (blue) is reduced by roughly a factor of 2.

a1

λ = 10.4abc

λ = 7.2a

λ = 6.3aContrast, c(t)

0.8
0.6
0.4
0.2
0

0

20

40

60 0

10
20
30
Time, t [ћ/Jxy]

0

10

20

Figure 4-8: Oscillations in the XXX model. Decay and weak oscillation at
the isotropic point Δ ≈ 1 measured for different wavelengths 𝜆. Solid lines are fits𝑐(𝑡) = (𝑎0 + 𝑏0 cos 𝜔𝑡) 𝑒−𝑡/𝜏 + 𝑐0 and dashed lines show the overall decay 𝑎0 𝑒−𝑡/𝜏 + 𝑐0 ,around which the oscillations take place. The oscillations become more pronounced
for short wavelengths 𝜆, because the decay time (𝜏 ∝ 𝜆2 ) decreases with smaller wavelength more strongly than the oscillation period (𝑇 ∝ 𝜆).

56

Bound magnon hopping

Free magnon propagationb

Wavefront position (lattice sites)a Initial state

8

free

6
4
2
0
0

bound
50
Time, t (ms)100

Figure 4-9: Free and bound magnons. a, Bound magnons can travel together
or break up into two free magnons, as studied experimentally [18] for Δ = 1. a,
The velocity 𝑣bound of the bound magnon state (green) is reduced by a factor of 2,
compared to the velocity 𝑣free of the free magnons (blue). This is consistent with atheoretical prediction 𝑣free /𝑣bound = 2Δ for general anisotropies [32].

(damped) oscillating part. This raises the question, how finite interactions suppress
these oscillations. In fact we can then notice, that small oscillations are even visible
in the isotropic case! If we look very closely, the decay actually deviates from purely
exponential decay ever so slightly (see Figure 4-4).
This is a great example, how studying a tunable anisotropic Heisenberg model(Δ ̸= 0) helps us learn something even about the isotropic system (Δ = 1). We would
probably have never noticed the small oscillation in the isotropic case, if we had
not seen the well pronounced oscillations for Δ = 0. It is easy to miss, if you are
not looking for it. But it also gets more pronounced at shorter wavelengths (seeFigure 4-8). The way we initially noticed this oscillation, was actually by using a fit
function.
Initially, we simply used an exponential fit function 𝑐(𝑡) = 𝑎0 𝑒−𝑡/𝜏 + 𝑐0 for the
decay curves at the isotropic point (Δ = 1) like in Ref. [19], since an exponentiallydecaying amplitude is even a solution to the diffusion equation. But this definitely
does not work for the decay curves in the XX model, which show oscillations as well.
So we looked for a fit function which would capture the dynamics of both the XX
model and the XXX model. We fit the data with the sum of a decaying part with57

time constant 𝜏 and a damped oscillating part with frequency 𝜔:𝑐(𝑡) = (𝑎0 + 𝑏0 cos 𝜔𝑡)𝑒−𝑡/𝜏 + 𝑐0

(4.1)

with 𝑎0 , 𝑏0 , 𝑐0 , 𝜔 and 𝜏 as adjustable parameters. This is plotted in Figure 4-4 asa solid line. We can see the oscillations more clearly, by plotting only the decaying
part 𝑎0 cos(𝜔𝑡)𝑒−𝑡/𝜏 + 𝑐0 as a dashed line, which is so-to-speak the average trajectory,
and the oscillations take place around it. In the following we study the decay time𝜏 and the oscillation frequency 𝜔 as function of the wavevector Q. The decay time𝜏 = 𝜏 (𝑄) yields the transport behavior, whereas the oscillation frequency 𝜔 = 𝜔(𝑄)yields a dispersion relation.
For the XX model (Δ = 0), the decay time 𝜏 shows a linear scaling with inverse
wavevector: a power-law fit 𝜏 ∝ 𝑄−𝛼 yields an exponent of 𝛼 = 1.00(5), indicating
ballistic transport (Figure 4-6a). This is reminiscent of the single (free) magnon
which travels with ballistic wavefronts in a spin chain according to a quantum random
walk, where a particle moves coherently with equal matrix elements in both directions
simultaneously.
For the XXX model (Δ = 1), a power-law fit of the decay constant 𝜏 versus 𝑄
yields an exponent of 𝛼 = 1.87(4), which is close to 2, indicating diffusive transport(Figure 4-6b). This is reminiscent of a single magnon which travels within a spin
chain according to a classical random walk, where it moves incoherently with equal
rates in both directions.
The oscillation frequencies 𝜔 as a function of the wavevector 𝑄 yield dispersion
relations. Interestingly, we measure linear dispersion relations 𝜔(𝑄) = 𝑣𝑄 for both
Δ = 0 and 1, yielding 𝑣 = 0.76(1) 𝑣𝐹 and 𝑣 = 0.35(1) 𝑣𝐹 , where 𝑣𝐹 = 𝑎/(~/𝐽𝑥𝑦 ) is the
Fermi velocity (see Figure 4-7). Interaction slow this velocity down by a factor of ≈ 2.
Perhaps this is related to the observations in the bound-magnon paper [18], whereinteractions reduce the velocity of a bound magnons compared to a free magnon, also
by a factor of 2, as shown in Figure 4-9. The theoretical prediction of the velocity
ratio is 2Δ for general values of the anisotropies [32].
58

4.4

Positive anisotropies (Δ > 0)

We gradually tune spin-spin interactions from Δ = 0 (non-interacting fermions) to
Δ = 1 (strongly-interacting fermions) to study how the behavior of transport changes
from ballistic to diffusive. Additionally, we study Δ > 1, where spin-spin interactions
dominate. In a classical interacting gas, for example, one would expect ballistic behavior on lengthscales smaller than the mean-free path and diffusion on larger lengthscales; and the mean-free path is determined by the interaction strength. However,
we find transport which is neither ballistic nor diffusive at all measured modulation
length-scales 𝜆. For various different anisotropies Δ we measure the 𝑄-dependenceof the spin helix decay time 𝜏 (Figure 4-10a), and we find that the data is very welldescribed by power laws 𝜏 = 𝜇𝑄−𝛼 , at least in the measured regime 𝜆 = 6 to 30 𝑎. The
obtained exponents 𝛼 are shown in Figure 4-10b.
We observe the following general behavior: As we increase the anisotropy startingfrom Δ = 0, initially the exponent stays close to 𝛼 = 1, but the coefficient 𝜇 increases.
Above a critical anisotropy of about 𝛼 = 0.5 however, it is the exponent which increases. We identify the following regimes. Ballistic regime: As we go from Δ = 0
to 0.5, transport stays ballistic but the characteristic velocity. Superdiffusive regime:
Between Δ = 0.5 to 1 the exponent increases smoothly from 𝛼 = 1 to 𝛼 = 2. For
example we measured 𝛼 = 1.48(4) at Δ = 0.78. Subdiffusive regime: For Δ > 1 theexponent continues to increase smoothly to values 𝛼 > 2. For example at Δ = 1.58,
we measured an exponent of 𝛼 = 2.83(14). This observation is in contrast to a recent theoretical prediction [11], which rather suggests a sharp transition: ballistic(𝛼 = 1) for all Δ < 1, diffusive (𝛼 = 2) for all Δ > 1 and superdiffusive (𝛼 = 1.5) onlyfor Δ = 0. But this is calculated for a partially polarized domain wall, which is amixed (i.e. thermal) state. We, however, study a fully polarized spin-helix, which
is pure state far from equilibrium. Nevertheless, we have some evidence, that forvery long wavelengths 𝜆 ≫ 𝑎 we would recover such a sharp transition with spin-helix
states as well [28].
Remarkably, we find linear dispersion relations for all positive anisotropies Δ ≥ 0
59

100

b

3
sub-diffusive

30

Exponent α

Decay time τ [ћ/Jxy]a10diffusive2

super-diffusive
ballistic

1
3

0.05
0.1
Wavevector Q [2π/a]

0.15

0

+1
Anisotropy Δ+2

Figure 4-10: Power law scalings of decay time constants 𝜏 for positive anisotropies
Δ > 0. a, Experimental results for Δ = −0.12 (orange), 0.35 (orange), 0.78 (yellow),
1.01 (blue), 1.58 (green). Lines are power law fits. b, Fitted power law exponents fromexperiments (filled symbols) and theory (open symbols), obtained from simulationsof the defect-free XXZ model (blue). For Δ ≥ 0 we observe anomalous diffusion: The
exponent increases smoothly from ballistic (red) to superdiffusive (yellow) to diffusive(blue) to subdiffusive (green).
(Figure 4-11). With increasing anisotropy Δ, the oscillations become smaller and thevelocity decreases. We already saw that at Δ = 1 the velocity is reduced by about afactor of about 2Δ. This is analogous to the velocity of bound-magnons comparedto free magnons. It seems this prediction also applies qualitatively for spin-helix
states for general anisotropies close to Δ = 1. But we find it remarkable, that wemeasure linear dispersion relations at all, in the regime where the decay time 𝜏 hassuperdiffusive, diffusive and subdiffusive scaling.
We note that while oscillations are difficult to discern by eye (e.g. in Figure 4-8a),
especially for large anisotropies Δ and small wavevectors 𝑄, the fitted oscillations
frequencies 𝜔 all fall very well on linear dispersion relations, which we regard as
evidence for their significance. The linear scaling 𝜔(𝑄) = 𝑣𝑄 even persists in the
superdiffusive, diffusive and subdiffusive regimes, where the power law scaling of the
decay time constant 𝜏 ∝ 𝑄−𝛼 is strongly nonlinear.
But since the oscillations are small away from the ballistic regime, for all positive
anisotropies, the decay curves can still be approximately collapsed into a single curve,
60

0.1

0.8

b

Velocity, v [vF]

Frequency, ω [Jxy/h]

a

0.6

0.05

0

0

0.4

0.1
Wavevector, Q [2π/a]

0.2

0

0.5
1
Anisotropy, Δ

1.5

Figure 4-11: Dispersion relations. a, The oscillation frequencies follow linear dispersion relations 𝜔(𝑄) = 𝑣𝑄 shown for Δ = −0.12 (red), 0.35 (orange), 0.78 (yellow),
1.01 (blue) and 1.27 (light blue). b, The obtained velocities 𝑣 decrease with increasing anisotropy Δ. It seems that close to Δ = 1 the velocity is reduced to 𝑣0 /2Δ (blueline) compared to the velocity 𝑣0 measured for Δ = 0.
if we rescale time by 𝜆𝛼 (Figure 4-12) This shows that this power-law scaling indeed
applies for all evolution times 𝑡 (which we could measure experimentally, i.e. early
and intermediate times). This collapse suggests universal behavior, at least for the
decaying part of the decay curve, described by the decay time 𝜏 . However, thiscollapse is not quite perfect, because we always find small oscillations around theoverall decay, and these oscillations have a different scaling with wavelength.

61

a

1

b

Δ = 0.5

Δ = 0.35

Δ = 0.7

Δ = 0.55

0.8
0.6

α=1

α=1

0.4
0.2
0
0

c

1

2

3

1

0

1

2

1

2

3

d

Δ = 0.85

Δ = 0.78

0

4

0

1

2

Δ=1

Δ = 1.01

Contrast, c(t)

0.8
0.6

α = 1.5

α=2

0.4
0.2
0
0

e

0.5

1

1

1.5

0

0.5

0

0.4

f

Δ = 1.25

Δ = 1.27

0.2

0.6

0

0.2

0.4

Δ = 1.5

Δ = 1.58

0.8
0.6

α = 2.5

α = 3.5α=3

α = 4.5

0.4
0.2
0
0

0.1

0.2

0.3

0

0.02

0

0.05

0.1

0.15

0

0.005

0.01

Rescaled time, t [(ћ/Jxy)(λ/a) ]αFigure 4-12: Collapse of decay curves for positive anisotropies Δ > 0. All decay
curves 𝑐(𝑡) for different wavelengths 𝜆 = 15.7 𝑎, 13.4 𝑎, 11.7 𝑎, 10.4 𝑎, 9.4 𝑎, 8.5 𝑎, 7.8 𝑎,
7.2 𝑎, 6.7 𝑎 collapse very well into a single curve for all evolution times 𝑡, when timeunits are rescaled by 𝜆𝛼 , where the exponent 𝛼 is a function of anisotropy Δ, bothfor experiment (points) and theory (solid lines). Experimental points were measured
for lattice depths 9 𝐸𝑅 (red), 11 𝐸𝑅 (blue), 13 𝐸𝑅 (yellow). a, b, Ballistic regime(𝛼 = 1), c, superdiffusion (𝛼 = 1.5), d, diffusion (𝛼 = 2), e, f, subdiffusion (𝛼 = 2.5, 3for experiment and 𝛼 = 3.5, 4.5 for numerical simulations. In f, experiments covereda reduced range 𝜆 ≤ 10.4 𝑎). The experimentally measured oscillation frequencies 𝜔follow a linear dispersion relation for all anisotropies Δ ≥ 0 (Figure 4-11) and have ascaling behavior different from the decay rates. However, such oscillations are small
outside the ballistic regime 𝛼 ≈ 1, and therefore only lead to a small deviation fromthe collapse behavior.

62

4.5

Negative anisotropies (Δ < 0)

So far we looked at positive anisotropies Δ ≥ 0. Now we show our results for negative
anisotropies Δ > 0. What happens, when we switch the sign of the interactions? For
Δ < 0, the effective fermions are also interacting. But now spin-spin interactions 𝐽𝑧
and spin exchange 𝐽𝑥𝑦 have opposite signs. This sign change introduces qualitatively
very different transport behavior. We studied the spin helix decay at three different
negative anisotropies Δ = −1.02, −1.43, and −1.79. But for a long time we could notunderstand the data. We almost decided not to publish these datasets. The previous

a

b

Δ≤0

1

Δ≥0

Contrast, c(t)

0.8
0.6
0.4
0.2
0

0

10

20

30

0

Time, t [ћ/Jxy]

10

20

30

Figure 4-13: Decay behavior as a function of anisotropy ranging from (a) negative to (b) positive, for one fixed wavelength 𝜆 = 10.4 𝑎. Using Δ ≈ 0 as a reference
point, we show how the temporal profile of the decay curve 𝑐(𝑡) changes when we
introduce positive or negative interactions. Every data point is an average of two
measurements at lattice depths 9 𝐸𝑅 and 11 𝐸𝑅 . a, from bottom to top: Δ = −0.12(red), −0.59 (pink), −0.81 (yellow), −1.02 (blue), −1.43 (green), −1.79 (purple). b,from bottom to top: Δ = −0.13 (red), 0.08 (purple), 0.35 (pink), 0.55 (orange), 0.78
(yellow), 1.01 (blue), 1.27 (light blue), 1.58 (green). Regardless of the sign, for increasing |Δ| the decay always slows down and the revivals damp out more quickly.
However, there is a big difference in how this slowdown happens: b, For increasing
positive interactions Δ > 0, the initial rate of decay increases continuously. a, In contrast, for all negative interactions Δ < 0, the initial rate of decay stays constant (andis ballistic), coinciding with the Δ ≈ 0 case. It is only after a critical time 𝑡0 thatthe decay suddenly starts slowing down (and becomes diffusive) for times 𝑡 > 𝑡0 . This
critical time 𝑡0 decreases with increasing negative interaction strength |Δ|.
63

a

Δ = -1.02

1b

Δ = -1.43c

Δ = -1.79

Contrast c(t)

0.8
0.6
0.4
0.2
0

0

10

20

0

10

20

0

10

20

Time t [ћ/Jxy]

Figure 4-14: Temporal crossover from ballistic to diffusive transport for negative anisotropies Δ < 0. Spin helix contrast 𝑐(𝑡) for 𝜆 = 10.4 𝑎 and Δ = −1.02 (a),−1.43 (b), and −1.79 (c) (filled circles) measured at lattice depths 11 𝐸𝑅 (blue) and
13 𝐸𝑅 (yellow). A piecewise fit (solid line) is linear at short times (I) and exponential
at long times (II) with a crossover at 𝑡 = 𝑡0 (vertical dotted line). For 𝑡 < 𝑡0 the decay
coincides well with the non-interacting case Δ ≈ 0 (open circles and dashed line).
fit function (which we used extensively for Δ ≥ 0) does not work anymore for negative
anisotropies Δ < 0. Only one thing was clear to us right from the start, right fromtaking the first such dataset: Positive and negative anisotropies are very different.
But we only really understood what is going on, when we plotted the data as shownin Figure 4-13; at one fixed wavelength 𝜆 for varying anisotropies Δ.
Starting with zero interactions as a reference point, we show how the temporalprofile of the decay curve changes when we introduce positive or negative interactions.
On the positive side (Δ ≥ 0), we see how stronger interactions remove the oscillationand turn the decay more and more into an exponential. On the negative side (Δ ≤ 0),stronger interactions also remove the oscillation. But for short enough times, all thecurves lie right on top of each other and, in particular, on top of the ballistic curve.
It seems like, for short enough times, the system does not feel the interactions yet.
But after enough time each curve suddenly starts to deviate from the ballistic caseand decays much more slowly for later times.
Motivated by this, we tried a piecewise fit for the two regimes: (I) a linear function
1 − 𝑡/𝜏I at short times and (II) an exponential 𝑒−𝑡/𝜏II at long time, with the respective
time constants 𝜏I and 𝜏II and the crossover time 𝑡0 as free parameters (see Figure 464

aDecay time τ [ћ/Jxy]b

Δ = -1.02

30

Δ = -1.43c

Δ = -1.79αΙΙ = 2.37(14)αΙΙ = 2.15(16)10αΙΙ = 2.11(21)

3

10

6
3

αΙ = 1.08(6)αΙ = 0.88(3)

0.05

0.1

0.15

0.05

αΙ = 0.96(10)

0.1

0.15

0.05

0.1

0.15

Wavevector Q [2π/a]Figure 4-15: Power law scalings of decay time constants 𝜏I and 𝜏II for negative
anisotropies Δ < 0. Experimental results are shown for Δ = −1.02 (a, blue), −1.43(b, green), −1.79 (b, purple). Lines are power law fits. For Δ < 0 we observe behavior
reminiscent of a classical gas: Transport is ballistic at (I) short times (triangles) anddiffusive at (II) longer times (squares).

14). This fit function reproduces the data well. And we observe that for stronger
interaction |Δ| the crossover time (between the two regimes I and II) moves to earlier

2

Diffusion coefficient, D [a /(ћ/Jxy)]

1.5

1

0.5

0

-2

-1.5

-1

Anisotropy, Δ+1Figure 4-16: Diffusion coefficients for the diffusive long-time regime (II) obtained from experiment (filled symbols) and theory (open symbols). For negative
anisotropies Δ < 0, values were determined from quadratic power law fits 1/𝜏 = 𝐷𝑄2
to the data points in Figure 4-15 (experiment) and to the time constants obtained
from Figure 4-17a-c (theory) for the diffusive regime (II). Note that for Δ ≥ 0 thesystem is only diffusive for Δ = +1, as shown in Figure 4-6b (experiment).
65

a

1

b

Δ = -1c

Δ = -1.5

Δ = -2

0.8
0.6
0.4

Contrast, c(t)

0.2
0
d

1

e

Δ = -1.02

f

Δ = -1.43

Δ = -1.79

0.8
0.6
0.4
0.2
0

0

1

2

0

1
2
Rescaled time, t [(ћ/Jxy)(λ/a)]

0

1

2

Figure 4-17: Collapse at short times for negative anisotropies Δ < 0. All decay
curves 𝑐(𝑡) for different wavelengths 𝜆 collapse into a single curve at early times,
when time units are rescaled by 𝜆 (indicating ballistic behavior). For later times the
decay is diffusive with different scaling. a-c, Theory (from top to bottom: 𝜆 = 31.3 𝑎,
23.5 𝑎, 18.8 𝑎, 15.7 𝑎, 13.4 𝑎, 11.7 𝑎, 10.4 𝑎, 9.4 𝑎, 8.5 𝑎, 7.8 𝑎, 7.2 𝑎, 6.7 𝑎, 6.3 𝑎). The
dotted lines are exponential fits 𝑒−𝑡/𝜏II to the diffusive regime and the time constants
𝜏II are used to ob obtain numerical power-law exponents in Figure 4-18c and diffusion
coefficients in Figure 4-16 . d-f, Experiment (from top to bottom: 𝜆 = 18.8 𝑎, 13.4 𝑎,
10.4 𝑎, 8.5𝑎 , 7.2 𝑎, 6.3 𝑎) shown for a lattice depth of 11 𝐸𝑅 . The dashed line indicatesthe ballistic case Δ ≈ 0 (see Figure 4-5c).
times. We obtain the time constants 𝜏I and 𝜏II for short time and long time behavior
separately and plot them as function of wavevector Q in Figure 4-15. For short times
we obtain ballistic scaling. For long times, we obtain diffusive scaling. For stronger
interactions, diffusion slows down. For weaker interactions, diffusion speeds up. The
diffusion coefficient is changes when the anisotropy Δ is varied (Figure 4-16). Thisbehavior is reminiscent of a classical gas: ballistic motion at short times (shorter thanthe mean-free time) and diffusive at longer times.
Furthermore, we can now see that all of our data for different wavelengths 𝜆66

collapses for the initial ballistic regime, if we rescale the time axis by 𝜆, as shownin Figure 4-17d-f. The deviation from the collapsed curve indicates a crossover todiffusion where the scaling is different. This observation is reproduced by numericalsimulations in Figure 4-17a-c.

4.6

Summary

For positive anisotropies Δ 𝑔𝑒𝑞 0, we see anomalous diffusion. The exponent 𝛼 varies
smoothly. With increasing interaction strength |Δ| we go through a ballistic, superdiffusive, diffusive, and subdiffusive regime. This is confirmed by numerical simulations.
Decay curves for different wavelengths 𝜆 (almost) collapse into a single curve, if timeis rescaled by 𝜆𝛼 , indicating close to universal behavior. But there are small oscillations with linear dispersion relations 𝜔(𝑄) = 𝑣𝑄. For negative anisotropies Δ , 0, we
see behavior reminiscent of a classical gas. At short times the dynamics is ballistic.
At longer times the dynamics is diffusive. This is confirmed by numerical simulations,as well. The diffusion coefficient varies smoothly as function of anisotropy Δ. Thesefindings are summarized in Figure 4-18.
We still do not have an intuitive physical explanation, why positive and negative
anisotropies cause such different behavior. Why does the sign of interactions matter somuch? Is there a difference in resonant and off-resonant scattering? Is the dynamics
on the positive side more coherent and on the negative side more incoherent? Here
we summarize the open questions:
1. Why do we observe linear dispersion relations 𝜔(𝑄) = 𝑣𝑄 for all positive anisotropies? We already mentioned that this might be related to the propagationof bound-magnons with a reduced velocity by a factor of 2Δ, compared to thevelocity of free magnons.
2. Why do we observe anomalous diffusion 𝜏 ∝ 𝑄−𝛼 for all positive anisotropies?
We think, we have evidence, that in the continuum limit 𝜆 ≫ 𝑎 there is a sharp
crossover from 𝛼 = 1 to 𝛼 = 2 as a function of anisotropy. We think, that the
67

a
30
Decay time τ [ћ/Jxy]

b

Δ≤0
II

10

Δ≥0

30

3

10

10

6

I

3

3

0.05

0.1

0.15
0.05
Wavevector Q [2π/a]

0.1

0.15

c

3

II

Exponent α

sub-diffusive
diffusive2

super-diffusiveIballistic

1
-2

-1

0
Anisotropy Δ+1+2

Figure 4-18: Power law scalings of decay time constants 𝜏 for anisotropies Δ
ranging from (a) negative to (b) positive. Experimental results are shown in a for
Δ = −1.02 (blue), −1.43 (green), −1.79 (purple) and in b for Δ = −0.12 (red), 0.35
(orange), 0.78 (yellow), 1.01 (blue), 1.58 (green). Lines are power law fits. c, Fitted
power law exponents from experiments (filled symbols) and theory (open symbols),
obtained from simulations of the defect-free XXZ model (blue) and 𝑡-𝐽 model with
5 % hole fraction (red). a-c, For Δ ≥ 0 (circles) we observe anomalous diffusion: The
exponent increases smoothly from ballistic (red) to superdiffusive (yellow) to diffusive(blue) to subdiffusive (green). For Δ < 0 we observe behavior reminiscent of a classicalgas: Transport is ballistic at (I) short times (triangles) and diffusive at (II) longer
times (squares).

68smooth crossover, which we observe in the experiment, is related to the relatively
short wavelengths between 𝜆 = 6 𝑎 and 30 𝑎 compared to the lattice spacing 𝑎,which we use in the experiment. In that case, the origin of anomalous diffusion
is the finite lattice spacing 𝑎 (see Methods section in our publication [28]).
3. Why is there a temporal crossover from ballistic to diffusive motion for negative
anisotropies?
4. Why is the behavior at the isotropic point Δ = 1 not superdiffusive with 𝛼 = 1.5?
This was predicted by Ref. [11] for a sharp domain wall state, however which
has smaller than full contrast (i.e. a mixed, thermal state). Surprisingly, theBloch group [19] recently measured superdiffusion (𝛼 = 1.5) even for a fullypolarized sharp domain wall. I therefore expect that we can definitely observe
superdiffusion with a spin-helix at the isotropic point (Δ = 1), too, if we startwith a partially polarized spin helix, even if it still has 50% contrast.

69

70

Chapter 5
Transverse spin dynamics
A transverse spin helix is sensitive to more decay mechanisms than a longitudinal spin
helix. Anisotropic spin couplings in the Hamiltonian break spin-rotational symmetry.
Transverse spin components are no longer conserved and can decay not only by transport, but also by fast, local dephasing. However, even for isotropic interactions, weobserve dephasing due to a new effect: an effective magnetic field created by superexchange, which has its origin in the mapping from the Hubbard model and which has
not been observed before. This field leads to additional dephasing mechanisms, because our system is not a perfect, homogeneous, and infinitely long spin-chain: there
is dephasing due to (i) inhomogeneity of the effective field from variations of lattice
depth between chains; (ii) a twofold reduction of the field at the edges of finite chains;and (iii) fluctuations of the effective field due to the presence of mobile holes in thesystem. The latter is a new coupling mechanism between holes and magnons.

This chapter supplements work reported in the following publication [29](included in Appendix B)

P. N. Jepsen, W. W. Ho, J. Amato-Grill, I. Dimitrova, E. Demler, W. Ketterle
Transverse Spin Dynamics in the Anisotropic Heisenberg Model Realized with Ultracold Atoms
Phys. Rev. X 11, 041054 (2021)
71

5.1

Introduction

In the previous chapter, we were able to show that the anisotropy profoundly changes
the nature of transport of longitudinal spin components after a quantum quench froma so-called longitudinal spin helix (Figure 5-1b). We observed ballistic, subdiffusive,
diffusive and superdiffusive behavior controled by the anisotropy. The decay rate isgiven by power laws𝛾 ∝ (𝑄𝑎)𝛼 ,(5.1)

where the exponent 𝛼 depends on the anisotropy. Another way to probe the anisotropyof the Hamiltonian is to rotate the initial state and look for a change in behavior of
the time evolution. Here we begin with a transverse spin helix (Figure 5-1a), whichis related to the longitudinal spin helix (Figure 5-1b) by a 𝜋/2-rotation. And we
observe even more dramatic effects of the anisotropy. A longitudinal helix is an excellent probe of transport poperties because, involving a modulation in the populationof |↑⟩ and |↓⟩, it can only decay by rearrangement of physical spins over distances
comparable to the modulation wavelength 𝜆. A transverse helix, however, is a modulation entirely in the phase sector, so a loss in contrast can also be caused by spinsprecessing at different rates around the 𝑆𝑧 axis. A transverse spin helix can therefore
decay not only by transport, but also by local dephasing. Actually, away from theisotropic point (Δ = 1), these two decay mechanism become indistinguishable.
In the classical limit, any transverse spin helix for any anisotropy Δ is stationary, since the torques exerted by neighboring spins on a given spin cancel exactly(Figures 5-2 and 5-3). Therefore, what we study here are the effects of quantum fluctuations on their stability (Figure 5-3). And we find that the decay rate is describedby𝛾 = 𝛾1 |Δ − cos(𝑄𝑎)| + 𝛾0 .

(5.2)

The first term 𝛾1 |Δ − cos(𝑄𝑎)| captures pure spin dynamics, which we study for the
XX model (Δ = 0) and XXX model (Δ = 1) in the next two Sections 5.2 and 5.3.
But we identify an additional isotropy breaking term in Hamiltonian, which even
72

a

Sz
λ

Sx

b

Sz
z

Sx

a

z

z

x
ImagingyFigure 5-1: Geometry of the experiment. The initial state is a transverse (a) or
longitudinal (b) spin helix where the spin vector winds within the 𝑆 𝑥 -𝑆 𝑦 plane (a) or𝑆 𝑧 -𝑆 𝑥 plane (b). The transverse helix (a) is a pure phase modulation of spin |↑⟩ and|↓⟩ states, whereas the longitudinal helix (b) also involves population modulation.
Deep optical lattices along the 𝑥 and 𝑦 directions create an array of independent spin
chains. The 𝑧 lattice is shallower and controls spin dynamics along each chain.
exists for isotropic interactions: an effective magnetic field created by superexchange,
which arises from the mapping from the underlying Hubbard model. This field induces
additional dephasing mechanisms which are captured by a nonzero background decayrate 𝛾0 > 0. We find three different ways to observe this field directly (Section 5.4).
73

τR⃗

a

τR⃗

b

φ
τL⃗

φ
τL⃗

Figure 5-2: Classical spins. a, The spin in the middle experiences a torque 𝜏 𝑅(blue) from the spin to its right, towards the aligned (𝐽𝑥𝑦 < 0) or anti-aligned (𝐽𝑥𝑦 > 0)
configuration. At the same time, it experiences an opposite torque 𝜏 𝐿 (red) from thespin to its left. b, In a transverse spin helix, the torques from both neighbors cancel
exactly 𝜏 𝐿 + 𝜏 𝑅 = 0, because the angles 𝜙 = 𝑄𝑎 between neighboring spins are equal.
A classical transverse spin helix is stationary.
Lastly, we discuss how this effective magnetic field induces dephasing (Section
5.5). If this effective magnetic field was uniform, it should not influence the dynamics
but only lead to uniform spin precession on top of the overall spin dynamics. Sucha uniform magnetic field term could formally be removed from the Hamiltonian,by transforming into a corotating frame. However, finite chain length, holes in thesystem, and inhomogenous lattice beams lead to overall dephasing, because theseeffects make this effective field non-uniform. In an ideal homogeneous system (oneinfinitely long spin chain without holes) these dephasing mechanisms would be absent
and 𝛾0 = 0. On one hand, a finite 𝛾0 > 0 is due to imperfections of the system. On
the other hand, such an imperfect system allows to detect the effective magnetic
field directly. Furthermore, mobile holes could even be used to study hole-magnon
coupling.

5.2

XX modelWe realize a very anisotropic system by tuning to the non-interacting point (Δ = 0).
As before, we confirm that the dynamics is driven by superexchange in the Heisenberg
model, by repeating the experiment at two different lattice depths 9 𝐸𝑅 and 11 𝐸𝑅 .
Figure 5-4a shows that the decay of a transverse spin helix is generally fast for thewhole range of wavelengths 𝜆 = 6 𝑎 to 30 𝑎 which we studied. Varying the wavelength
74

a

τR⃗

b

τL⃗

τR⃗

τL⃗

Figure 5-3: Quantum spins. a, Quantum fluctuations of the spins lead to a fluctuating torque which is only zero on average ⟨𝜏 𝐿 + 𝜏 𝑅 ⟩ = 0. A quantum transverse
spin helix decays as a result of quantum fluctuations. b, For Δ = 0, i.e. for vanishinglongitudinal coupling 𝐽𝑧 = 0, the quantum fluctuations of the neighboring spins do
not couple to the spin in the middle, if spins are at an angle of 𝜙 = 90∘ .
shows that for very short wavelengths the decay gets even slower! This is in stark
contrast to the linear Q-dependence of a longitudinal spin helix at Δ = 0, where spinhelix states with shorter wavelengths decay faster. This shows, in particular, that the
decay is not caused by transport, but by local dephasing.
We were very surprised, that the decay slowed down for shorter wavelengths. In
particular, since only the three largest values of 𝑄 in our dataset showed this effect,
where we were already pushing our imaging system to its resolution limit. At thattime, we did not even know if this effect is real, especially since this was only a ∼ 20%
reduction in the decay rate. Only much later we found out that we were alreadyseeing the signature of an exact many-body eigenstate [47, 48], which is realized at
𝑄𝑎 = 𝜋/4, when neighboring spins are at an angle of 90∘ . This is the topic of chapter
6, where we extended our observable range of wavevectors 𝑄 all the way to 𝑄𝑎 = 𝜋.
Nevertheless, shortly after this first measurement, our theory collaborator Wen
Wei Ho already found out that the decay rates should scale as 𝛾 ∝ | cos(𝑄𝑎)|. Ourdata did not show such strong dependence on 𝑄. Nevetheless, the data can still be
explained, if we assume that there exists one more large background dephasing rate
𝛾0 , which is 𝑄-independent. It took us a long time to justify to ourselves, that thisis appropriate. But we also observe such a 𝑄-independent background decay rate 𝛾0much more directly in the isotropic model (Δ = 1), because there the longitudinal
and transverse spin helix should be behave exactly the same.
75

a 1

c

Δ≈0

Δ=0

Contrast c(t)0.80.6
Q = 0.160

0.4
0.2

Jxy/2

Q = 0.032

02

4Time t [ћ/Jxy]

6

8

b
0.3

Energy [Jxy]

0

1
+–
2

0

Jxy/2

Decay rate γ [Jxy/ћ]transverse
1
-–
2

0.2

0.1
longitudinal

0

0

0.05
0.1
0.15
0.2
Wavevector Q [2π/a]

0.25Figure 5-4: Spin dephasing and spin transport for the XX model (Δ ≈ 0).
a, Spin relaxation. Transverse spin-helix contrast 𝑐(𝑡) for 𝑄𝑎 = 2𝜋 × 0.032, 0.138
and 0.160 (bright to dark orange). The curves for different lattice depths 11 𝐸𝑅 (∘)
and 13 𝐸𝑅 () collapse when times are rescaled in units of the corresponding spinexchange times ~/𝐽𝑥𝑦 = 1.71 ms (∘) and 4.30 ms (). The transverse spin decays ona time scale of a few spin-exchange times which increases for smaller wavelengths
𝜆 = 2𝜋/𝑄. b, Wavevector dependence. The initial decay rates (orange points;
with values from a highlighted) follow a cosine dependence 𝛾(𝑄) = 𝛾1 cos(𝑄𝑎) + 𝛾0(solid orange line) with a constant background rate 𝛾0 = 0.20(2) 𝐽𝑥𝑦 /~ (dashed orange line). This is in strong contrast to the longitudinal spin helix (purple) whichshows linear scaling with 𝑄 (indicating ballistic transport). A spin echo (𝜋-pulse
at time 𝑡/2) reduces the background rate to 𝛾0 = 0.13(3) 𝐽𝑥𝑦 /~ (dashed blue line).
c, Energy levels of the Heisenberg Hamiltonian for two spins in a double-well potential. For Δ ̸= 1, the triplet states are split, causing dephasing for an 𝑥 polarizedstate |→→⟩ = (|↑⟩ + |↓⟩)(|↑⟩ + |↓⟩) = |↑↑⟩ + (|↑↓⟩ + |↓↑⟩) + |↓↓⟩.

76

c

Δ=1

a

Δ≈1

1

Contrast c(t)0.80.6
spin echo

0.4
0.2

no echo

0

10

b

20
Time t [ћ/Jxy]

30

0.3

transverse helix

Decay rate γ [Jxy/ћ]

Jxy

0

0.2

0.1
longitudinal helix

0

0.05
0.1
Wavevector Q [2π/a]

0.15

0

Figure 5-5: Spin dephasing and spin transport for the XXX model (Δ ≈ 1).
a, Spin relaxation. Transverse spin-helix contrast 𝑐(𝑡) for 𝑄𝑎 = 2𝜋 × 0.056 without
spin echo (orange) and with spin echo (blue). The difference shows the presence of
inhomogeneous dephasing. The curves taken for different lattice depths 11 𝐸𝑅 (∘) and
13 𝐸𝑅 () collapse when times are rescaled in units of the corresponding spin-exchange
times ~/𝐽𝑥𝑦 = 2.55 ms (∘) and 6.42 ms (). b, Wavector dependence. Decay rates
for the transverse helix (orange and blue points; with values from d highlighted) andlongitudinal helix (purple points), for Δ ≈ 1. The orange (no echo) and blue (withecho) solid lines are fits 𝛾(𝑄) = 𝐷𝑄2 + 𝛾0 assuming two contributions to the decay
rate: one quadratic term (indicating diffusive transport) with diffusion constant 𝐷(taken from the longitudinal spin dynamics shown in purple), the other 𝑄-independent
𝛾0 (shown by the dashed lines). c, Energy levels of the Heisenberg Hamiltonian
for two spins in a double-well potential. For Δ = 1, the triplet states are degenerate.
Therefore, the 𝑥 polarized state |→→⟩ = (|↑⟩+|↓⟩)(|↑⟩+|↓⟩) = |↑↑⟩+(|↑↓⟩+|↓↑⟩)+|↓↓⟩
is an eigenstate.

77

5.3

XXX model

When Δ = 1, we realize the isotropic Heisenberg model. Here, the time evolutionof the transverse and longitudinal spin-helix should be exactly the same, just as thetransverse and longitudinal spin coupling are the same 𝐽𝑥𝑦 = 𝐽𝑧 .
However, they are not the same, but at least they are similar. In particular, thelongitudinal helix exhibits a purely diffusive scaling with wavevector 𝑄. Its decayrate obeys 𝛾(𝑄) = 𝐷𝑄2 , where 𝐷 is the diffusion constant (as shown in the previous
chapter and our publication [32]), whereas the transverse helix has an additional 𝑄independent decay rate of 𝛾0 = 0.096(10) 𝐽𝑥 𝑦/~, with the net decay rate fitted well by𝛾(𝑄) = 𝐷𝑄2 + 𝛾0 (Figure 5-5). So the decay separates into two parts: One transportchannel, which scales as 𝑄2 , and one dephasing channel, which is 𝑄-independent.
So for the XXX model, the transverse and longitudinal spin helix behave in a very
similar way and only differ by a background decay 𝛾0 . In contrast, for the XX model,
their behavior is very different, as expected for a very anisotropic system (Δ = 0).
The dynamics of the longitudinal spin helix can be described by power laws forboth the XX model (𝛾 ∝ 𝑄) and the XXX model (𝛾 ∝ 𝑄2 ). Both cases have a common
description. On the other hand, the transverse spin helix shows a cosine dependence
for the XX model (𝛾 = 𝛾1 | cos(𝑄𝑎)| + 𝛾0 ) and diffusive transport for the XXX model(𝛾(𝑄) = 𝐷𝑄2 + 𝛾0 ). That is a much more striking difference than in the longitudinal case. Nevertheless, we can still find a common description. Wen Wei showedthat the expression 𝛾(𝑄) = 𝛾1 | cos(𝑄𝑎)| + 𝛾0 for Δ = 0 can be generalized to arbitrary
anisotropies Δ in the following way
𝛾(𝑄) = 𝛾1 |Δ − cos(𝑄𝑎)| + 𝛾0

(5.3)

which, for Δ = 1, even reduces to the (diffusive) quadratic case𝛾(𝑄) = 𝛾1 [1 − cos(𝑄𝑎)] + 𝛾0 ≈𝛾1(𝑄𝑎)2 + 𝛾0
2

(5.4)

if the wavelength is long 𝜆 ≫ 𝑎 compared to the lattice spacing. We will study the78

first part of the expression, 𝛾1 |Δ − cos(𝑄𝑎)|, extensively in Chapter 6.
The remaining part of this chapter focusses on the background decay rate 𝛾0 ,which is present for any value of Δ. It is induced by the effective magnetic field ℎ𝑧
which is created by superexchange.

5.4

Imaging the effective magnetic field

When we measured our first decay curve for a transverse spin-helix for the isotropic
model Δ = 1 (similar to Figure 5-5a), we really expected to get the same result as forthe longitudinal spin helix. And we were very confused, when we saw fast decay, withthese pronounced oscillations instead. What was going on? At that point, we had
no idea that this oscillation at frequency Ω was exactly the signature of the effective
magnetic field ℎ𝑧 = ~Ω. And it was right in front of us! But one thing we figuredout pretty quickly: Wolfgang immediately realized that the only symmetry breaking
term in the Hamiltonian can be the effective magnetic field ℎ𝑧 . And we showed thata spin echo removes the oscillations and decreases the decay rate. By repeating theexperiment at two different lattice depths (9 and 11 𝐸𝑅 ) and thereby varying thespin-exchange time ~/𝐽𝑥𝑦 , Figure 5-5a clearly shows that the time dependence of thiseffect scales with superexchange. This indicates that we are indeed seeing a magnetic
field which is created by superexchange, and not a real magnetic field. This is alreadyan indirect observation of the effective magnetic field. Over time we found different
ways to measure it directly and confirm that it is indeed the effective magnetic fieldgiven by ℎ𝑧 = 4𝑡2 /𝑈↑↑ −4𝑡2 /𝑈↓↓ .

5.4.1

Observation as a beatnote

If we assume that the ensemble experiences two pronounced values of the effective
magnetic field, then the time evolution of the cloud-averaged contrast c(t) will show
a beat note at a frequency which corresponds to the difference of the two valuesof the effective magnetic field. This is the case for our atom clouds, which feature aMott insulator plateau surrounded by a dilute shell of individual atoms [28] which are
79

pinned to their lattice sites by the gradient of the harmonic trapping potential [44].
Many of these individual atoms do not have neighbors for spin exchange and, therefore, do not feel an effective magnetic field, while those in the Mott insulator plateau
do. The observed beat frequencies Ω = 0.90(1) 𝐽𝑥𝑦 /~ (Figure 5-6) agree well with thepredicted value of the superexchange-generated effective magnetic field ℎ𝑧 = 0.89 𝐽𝑥𝑦 .
As expected, the beat note is more pronounced by spatially selecting the outer partsof the cloud (Figure 5-6a, blue) and disappears with a spin echo (Figure 5-6b).

abspin-echo cdContrast c(t)

1
0.8

1

λ = 23.5 a

0.6

Effective magnetic field hz [Jxy]

1.5

0.5

0.4
0.2

0

0

λ = 10.4 a

0

10

20

0
Time t [ћ/Jxy]

10

20

30

846
848
Magnetic field B [G]Figure 5-6: Absolute measurement of the effective magnetic field value hz
as a beat note between the spin precession frequencies in the inner and outer parts ofthe cloud. a, Transverse spin-helix contrast 𝑐(𝑡) for 𝜆 = 23.5 𝑎 (all filled symbols) andΔ ≈ 1. For measurements averaged over the whole atom cloud (purple), they were
performed at 11 𝐸𝑅 (∘) and 13 𝐸𝑅 (). The contrast at the center of the atom cloudfor radii 𝑟 ≤ 8.5 𝑎 (orange filled symbols) decays slower with less pronounced oscillations, whereas the contrast in the spatial wings for radii 𝑟 ≥ 20 𝑎 (blue filled symbols,
including ≈ 7 % of the atoms in the cloud) decays faster with more pronounced oscillations. Data points for diamonds are an average of measurements at 11 𝐸𝑅 and
13 𝐸𝑅 . Open diamond symbols represent data for 𝜆 = 10.4 𝑎 and show the same oscillation frequency but decay faster due to spin transport. Lines are fits described in the
Appendix of the paper [29]. Curves are offset from each other for clarity. The dottedlines indicate their respective zeros. b, The oscillations disappear with a spin-echo.
c, Different regions used to obtain a contrast measurement. d, The beat frequency Ωvaries as a function of the externally applied magnetic field 𝐵. The measured valuesfor ~Ω (points) follow the theoretical prediction (without any adjustable parameter)
for the effective magnetic field ℎ𝑧 (solid line) which is tuned by varying the scattering
lengths via Feshbach resonances.
80

5.4.2

Observation through spin precession

The fact that our lattice depth is slightly inhomogeneous between the chains (dueto its Gaussian cross section) causes inhomogeneity in the effective magetic field,
too. The ensemble of spin helices then precess at slightly different rates. We cantherfore observe the effective field directly by purposefully displacing the 𝑧 latticebeam vertically. In this way we create a controlled gradient in the effective field acrossthe ensemble of chains. This leads to a gradient in precession speed which causes the
stripes in the recorded images to rotate. In Figure 5-7 we plot the rotation angle as afunction of time. The data points for two lattice depths collapse, if time is plotted in

25

Rotation angle φ [degrees]

20
15
10
5
0
-5

0

2
Time t [ћ/Jxy]

4

6

8

10

Figure 5-7: Direct observation of the effective magnetic field through spin
precession. Rotation angle of the stripe pattern as function of evolution time 𝑡 (filled
symbols) for two lattice depths 11 𝐸𝑅 (blue) and 13 𝐸𝑅 (yellow) without (solid line)
and with (dashed line) spin echo pulse at 𝑡 = 5 ~/𝐽𝑥𝑦 .
81

3

1.5

10

2

1

5

1

0.5

0

0

0

-5

-1

-0.5

B = 848.53 G

04Time t [ћ/Jxy]

845

8

847
Magnetic field B [G]Effective magnetic field hz [Jxy]

15

2

b 4

B = 846.37 G

Rotation rate φ̇ [degrees/(ћ/Jxy)]

Rotation angle φ [degrees]a849

Figure 5-8: Tunability of the effective magnetic field. a, Rotation angle as afunction of evolution time 𝑡 for different magnetic fields fields 𝐵 = 846.37 G (red),
847.17 G (blue), 847.59 G (light blue), 848.00 G (orange), 848.17 G (yellow), 848.34 G(purple), 848.53 G (green). b, Angular velocities obtained from linear fits in b compared to predicted effective magnetic fields ℎ𝑧 (solid line) with the scale factor between
the two 𝑦 axes as a fitting parameter yielding 𝜑˙ ~/𝐽𝑥𝑦 = 0.037 ℎ𝑧 /𝐽𝑥𝑦 = 2.1∘ ℎ𝑧 /𝐽𝑥𝑦 ,representing the (uncalibrated) gradient of the lattice depth. Times are normalized
by the spin-exchange time ~/𝐽𝑥𝑦 for the central part of the atom cloud. The scaling
factor is consistent with a displacement of the 𝑧-lattice beam (1/𝑒2 radius of 125 𝜇m)by an amount of 29 𝜇m = 55 𝑎.

units of spin-exchange time, which confirms that the rotation is superexchange driven
dynamics. And a 𝜋-pulse reverses the direction of the rotatio, so that the spin helices
can rephase, if applied after half of the evolution time. That confirms nicely, that the
spin echo works.
The slope in Figure 5-7 (rotation speed) is a direct measurement of the effective field strength. We now use our Feshbach resonances to tune the value of theeffective magnetic field ℎ𝑧 . And then we measure it using the rotation speed andsee if it matches our prediction. Figure 5-8a shows how the rotation speed varies
with increasing Feshbach field. In particular, we see how the rotation speed getssmaller and even changes sign. The position of the zero-crossing in the effective field
is in good agreement with our prediction. The predicted curve in Figure 5-8b has
82

0.8
0.6

Population

0.4
0.2
0
0.8
0.6
0.4
0.2
0

-0.6

-0.4

-0.2
0
0.2
Detuning δ/2π [kHz]

0.4

0.6

Figure 5-9: Spectroscopic observation of the effective magnetic field ℎ𝑧 asa shift in the spin-flip frequency for ℎ𝑧 < 0 (bottom panel) compared to ℎ𝑧 ≈ 0 (toppanel). Shown is the fraction of atoms in each state as a function of the final detuning𝛿 of a 22 ms sweep of the RF frequency, starting at 𝛿 = +30 kHz with all atoms in
the |↑⟩ state (closed circles) and no atoms in the |↓⟩ state (open squares). The
detuning is relative to the single-particle transition frequency. The power of the
RF drive is also ramped to zero after the frequency sweep to make the transition
sharper. A nonzero detuning 𝛿 for equal spin populations compensates for the effective
magnetic field ℎ𝑧 which shifts the curves for 11 𝐸𝑅 (bottom panel) lattice depth
compared to 35 𝐸𝑅 (top panel) where ℎ𝑧 ≈ 0. For the sweep experiment, we chosethe second-lowest (closed circles) and third-lowest (open squares) hyperfines statesof 7 Li due to the smaller sensitivity to external magnetic fields (due to a smallerdifferential magnetic moment). At 1025 G, the scattering lengths are approximately𝑎↑↑ ≈ 𝑎↑↓ ≈ −50 𝑎0 and 𝑎↓↓ ≈ +350 𝑎0 , leading to an estimate for effective magnetic
field of ℎ𝑧 ≈ −1.14 𝐽𝑥𝑦 ∼ ℎ×(−100 Hz) which is consistent with our observation.

only been multiplied by one free parameter, which accounts for the amount of beamdisplacement.
83

5.4.3Spectroscopic observation

A spatially uniform effective magnetic field can always be transformed away in asuitable rotating frame. However, even then, it can still be observed as a shift ofthe spin-flip resonance. We compare decoupled atoms in a deep lattice (𝑉𝑧 = 35 𝐸𝑅 ),where the effective magnetic field is ℎ𝑧 = 0, and interacting atoms in a shallow lattice(𝑉𝑧 = 11 𝐸𝑅 ), where the effective field ℎ𝑧 has a finite value. We determine the resonance position with adiabatic frequency (Landau-Zener) sweeps, to find the ending
frequency where half of the atoms are transferred from spin |↑⟩ to |↓⟩.
The sweep detuning corresponds to an external 𝐵𝑧 field (in the rotating frame)
and the Rabi frequency to a transverse 𝐵𝑥 field, realizing a Heisenberg model withmagnetic fields. Starting from a fully polarized state with all atoms in the |↑⟩ stateand large detuning of the radio frequency, we reduce the 𝐵𝑧 field adiabatically andobserve the spin imbalance. The spins are balanced when the detuning compensates
for the effective magnetic field created by superexchange. With this method, we canobserve the effective field for different lattice depths (Figure 5-9).

5.5Dephasing mechanisms for the transverse spin
helix

We identified three different mechanisms how the effective magnetic field leads todephasing and summarize them here:
1. Effect of inhomogeneity in the effective magnetic field between chains: This
arises from slight variations in the lattice depth from the Gaussian nature of
the laser beams, and can be eliminated by applying a spin-echo pulse at half ofthe evolution time t, as we can see in Figures 5-5 and 5-6.
2. Effect of finite chain lengths: For a 1D chain, the effective magnetic field is
reduced to ℎ𝑧 /2 at the ends, half the value in the bulk (Figure 5-10). Forℎ𝑧 ̸= 0, this nonuniformity hence cannot simply be transformed away by going84

into an appropriate corotating frame. Although this reduced field is localized
at the ends of the chain, it cannot be eliminated with an echo protocol, since its
effect propagates along the chain via spin dynamics as illustrated in Figure 5-11
3. Effect of mobile holes: Every hole acts as if it carries a magnetic field −ℎ𝑧 /2
experienced by spins next to it. A mobile hole then leads to a fluctuating
magnetic field in space and in time.

hz/2

hz

hz

hz

hz

hz/2

hz/2

t

hz

hz/2

t

Figure 5-10: Dephasing from the edges of the chains and mobile holes. The
effective magnetic field ℎ𝑧 is created by superexchange. If a spin has only one neighbor(instead of two), the value of the effective field is reduced by a factor of two. This is
the case for the atoms at the ends of a finite spin chain. Also a mobile hole reduces
the effective magnetic field of its two neighboring by a factor of two. Then the holeacts as if it carries a magnetic field −ℎ𝑧 /2. This leads to a coupling between spin andmotional degrees of freedom (hole-magnon coupling).
1b + 2–

Decay rate γ [Jxy/ћ]816

c 0.03z

Lattice site index i

Magnetization 〈S 〉

a1

0

10

20

30

40

0

1
-–
20

Time t [ћ/Jxy]

10

20

30

40

0.02

0.01
5

10
20
Chain length L

30

40

Figure 5-11: Dephasing from the edges of the chains. a,b, Dynamics of an 𝑆𝑥 polarized (𝑄 = 0) state for a 𝐿 = 16 finite chain, with anisotropy Δ = 1 and ℎ𝑧 = 0.8 𝐽𝑥𝑦reflecting the parameters of the experiment. Fig. (a) shows how the initially homogeneous phase of the locally-measured transverse spin ⟨𝑆𝑖𝑥 (𝑡)⟩ gets distorted at theedges, and how this perturbation propagates through the chain, leading to a loss of∑︀overall contrast of the total transverse magnetization 𝑖 ⟨𝑆𝑖𝑥 (𝑡)⟩ (b). A fit (dashedlines) to cos(𝜔𝑡 + 𝜑)𝑒−𝛾𝑡 /2 determines the decay rate 𝛾 which is shown as a functionof system size 𝐿 in panel c. A log-log plot shows the scaling 𝛾 ∝ 𝐿−1 for large 𝐿.

85

86

Chapter 6
Bethe phantom states
Exact solutions for quantum many-body systems are rare. Yet, for or a given interaction anisotropy, there exists one special winding angle, such that the transverse
spin-helix is an exact many-body eigenstate of the Heisenberg Hamiltonian. We findthis eigenstate experimentally, by varying the winding angle and measuring the decayrate, which reveals a pronounced minimum. This finding confirms the recent prediction of phantom Bethe states, exact many-body eigenstates carrying finite momenta
yet no energy. We then use the sensitivity of these phantom helix states as a tool,
to measure the interaction anisotropy directly. This reveals that the anisotropy canbe strongly affected by nearest-neighbor off-site interactions, which have never been
observed before for particles, which only interact with contact interactions.

This chapter supplements work reported in the following publication [30](included in Appendix C)

P. N. Jepsen, Y. K. Lee, H. Lin, I. Dimitrova, Y. Margalit, W. W. Ho, W. Ketterle
Catching Bethe phantoms and quantum many-body scars: Long-lived spin-helix
states in Heisenberg magnets
arXiv:2110.12043 (2021)
87

6.1

Introduction

The previous chapter showed that a transverse spin helix is really sensitive to theanisotropy Δ and that its decay rate can be described by𝛾 = 𝛾1 |Δ − cos(𝑄𝑎)| + 𝛾0

(6.1)

The background decay rate 𝛾0 accounts for effects such as finite chain length and holes
in the spin chain, which we studied extensively in the previous chapter. These effects
would be absent (𝛾0 = 0) in an ideal spin chain, infinitely long and without holes, suchthat every spin has two neighbors. This chapter focusses on studying the decay due
to pure spin dynamics 𝛾1 |Δ − cos(𝑄𝑎)| in detail. Especially, we experimentally findthe spin helix states fulfilling the so-called phantom condition
Δ = cos(𝑄𝑎)(6.2)

where the decay rate vanishes. This phantom helix state is even a perfect many-bodyeigenstate of the anisotropic Heisenberg model with anisotropy Δ [47, 48]. In thiseigenstate, neighboring spins are just at the right angle 𝜙 = 𝑄𝑎 = arccos Δ, such thatfor each spin, the interactions with its two neighbors cancel exactly (Figure 6-1). This
angle depends on the value of the anisotropy Δ.
In the isotropic system (Δ = 1), this special angle is 𝜙 = 0∘ and the spin helix
trivially reduced to the ferromagnetic state, where all spins are aligned (within the𝑆𝑥 -𝑆𝑦 plane). Indeed, we have already seen in the previous chapter, that the decayrate is minimum at 𝑄 = 0 (Figure 5-5b). In the XX model (Δ = 0), this special angle
increases to a value of 𝜙 = 90∘ , forming a spin helix with a wavelength of 4 latticesites (𝜆 = 4 𝑎). This is too short for observing this state directly, because our imaging
resolution is limited to wavelengths 𝜆 > 6 𝑎. Nevertheless, we have already seen thesignature if this many-body eigenstate, as the decay rates started to decrease withincreasing angle 𝜙 = 𝑄𝑎 in Figure 5-4b.
In this chapter we extend our range of observations to arbitrary wavevectors 𝑄,
88

Sz

Sx

ϕ

z

Figure 6-1: Spin-helix state in the 𝑆𝑥 -𝑆𝑦 plane with winding angle 𝜙. It is anexact eigenstate of the anisotropic Heisenberg model, if its winding angle fulfills thephantom condition Δ = cos 𝜙.
all the way up to 𝑄 = 𝜋/𝑎, where neighboring spins are anti-aligned. Such wavelength(𝜆 = 2 𝑎) is way too small for our imaging system to resolve and would require singlesite resolution. In order to measure the modulation contrast nevertheless, we utilize
a novel unwinding step (applying an inverse magnetic gradient), just before imaging(see Figure 6-2).
aWinding Gradientb
XXZ InteractionSy

λ = 2π/Q

c

Sz

Sx

Unwinding GradientdFigure 6-2: Preparation and observation of spin-helix states. An initiallyspin-polarized state in the 𝑆 𝑦 direction (a) is wound into a spin helix with variable
wavevector 𝑄 using a magnetic field gradient, here illustrated for 𝑄𝑎 = 𝜋/2 (b). This
state evolves under the XXZ Heisenberg Hamiltonian (c). After unwinding the remaining spin modulation to a resolvable wavevector (d), the local 𝑆 𝑦 -magnetization
is imaged in-situ. Only the 𝑆 𝑥 and 𝑆 𝑦 components of the spin are shown.
89

6.2

Stability of transverse spin-helix states

In classical physics, in chains with purely classical spins, every transverse spin helix
is stable, for any wavevector 𝑄, and even for any anisotropy Δ. The reason is simple.
We have already mentioned this before in Chapter 5, but it is so important for this
Chapter 6, that we will explain the argument again here. Every spin experiences atorque from each of its two neighbors (Figure 6-3). But since the two angles between
this spin and its two neighbors are exactly equal (𝜙 = 𝑄𝑎) in a spin helix, the two

a

τR⃗

τR⃗

b

φ
τL⃗

φ
τL⃗

Figure 6-3: Classical spins. a, The spin in the middle experiences a torque 𝜏 𝑅(blue) from the spin to its right, towards the aligned (𝐽𝑥𝑦 < 0) or anti-aligned (𝐽𝑥𝑦 > 0)
configuration. At the same time, it experiences an opposite torque 𝜏 𝐿 (red) from thespin to its left. b, In a transverse spin helix, the torques from both neighbors cancel
exactly 𝜏 𝐿 + 𝜏 𝑅 = 0, because the angles 𝜙 = 𝑄𝑎 between neighboring spins are equal.
A classical transverse spin helix is stationary.

a

τR⃗

b

τL⃗

τR⃗

τL⃗

Figure 6-4: Quantum spins. a, Quantum fluctuations of the spins lead to a fluctuating torque which is only zero on average ⟨𝜏 𝐿 + 𝜏 𝑅 ⟩ = 0. A quantum transverse
spin helix decays as a result of quantum fluctuations. b, For Δ = 0, i.e. for vanishinglongitudinal coupling 𝐽𝑧 = 0, the quantum fluctuations of the neighboring spins do
not couple to the spin in the middle, if spins are at an angle of 𝜙 = 90∘ , so that sucha quantum transverse spin helix is stationary.
90torques cancel exactly 𝜏𝐿 + 𝜏𝑅 = 0. The classical spin helix is stationary! Also, thelongitudinal spin coupling 𝐽𝑧 in the 𝑆𝑧 direction does not matter for classical spins in

ϕ = 0°

ϕ = 90°a1

ϕ = 180°bcContrast c(t)

0.8
0.6
0.4
0.2
0

0

2

0

4

2
4
Time t [ћ/Jxy]

0

2

4

d
d

Decay rate γ [Jxy/ћ]

0.6

0.4

0.2

0

0

π/2
Winding angle ϕπFigure 6-5: Decay of transverse spin-helix states. a-c, Contrast decay 𝑐(𝑡) measured for Δ ≈ 0 at two different lattice depths 9 𝐸𝑅 (red) and 11 𝐸𝑅 (blue), for threewinding angles 𝜙 = 𝑄𝑎. All spins aligned 𝜙 = 0∘ (a), neighboring spins perpendicular
𝜙 = 90∘ (b), all spins anti-aligned 𝜙 = 180∘ (c). Decay curves collapse in each casewhen times are normalized in units of ~/𝐽𝑥𝑦 . d, The spin-helix lifetime is significantly larger for 𝜙 = 90∘ compared to 𝜙 = 0∘ and 𝜙 = 180∘ . The decay rate 𝛾 shows apronounced minimum for 𝜙 = 90∘ .
91

a transverse spin helix, because the spins have a vanishing z-component 𝑆𝑧 = 0. Only
the transverse coupling 𝐽𝑥𝑦 matters. The anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦 , therefore, does notplay a role then.
But in quantum mechanics, the spins have quantum fluctuations, random quantum
noise, which is uncorrelated (Figure 6-4a). The two torques 𝜏𝐿 and 𝜏𝑅 only cancel on
average ⟨𝜏 𝐿 + 𝜏𝑅 ⟩ = 0. Only the expectation value for the total torque is zero, butwith a finite variance ⟨(𝜏𝐿 + 𝜏𝑅 )2 ⟩ = 0. Therefore, the spin in the middle experiences
a fluctuating torque, and thus the decay of this spin-helix state is due to quantum
fluctuations. A classical spin helix would be stable. That is the difference between
classical magnetism and quantum magnetism. In particular, there are also quantum
fluctuations in the 𝑆𝑧 direction, which is the reason why the anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦
starts to play a role.
For example, if we choose the value Δ = 0 (XX model) we can avoid the effectof the quantum fluctuations, by making the angle 𝜙 = 𝑄𝑎 between neighboring spins
𝜙 = 90∘ (Figure 6-4b). The quantum fluctuations are then either in 𝑆𝑧 direction, butwith Δ = 0 the 𝐽𝑧 coupling is switched off. Or the quantum fluctuations are alignedwith the neighboring spin and, therefore, do not cause a torque. This state should bestable, even in the presence of quantum fluctuations. Therefore, we looked for thisstable state experimentally.

6.3

Observations of phantom helix states

We tune the anisotropy to Δ = 0, and we systematically vary the winding angle 𝜙between neighboring spins, while measuring the decay rate 𝛾. And indeed, Figure 6-5
shows a pronounced minimum in the decay rate when neighboring spins are at anangle of 90∘ . This is the signature of the helix phantom state! The fastest decay
occurs for the ferromagnetic state, when all spins are aligned within the 𝑆𝑥 -𝑆𝑦 plane,
and for the Néel state, when all spins are anti-aligned.
When we increase the anisotropy Δ, we observe how the minimum shifts to asmaller angle 𝜙. For an anisotropy of Δ = +1/2 the special is at 𝜙 = 60∘ (Figure 692

Δ ≥ +1

Δ ≤ -1

Decay rate γ [Jxy/ћ]

1.2 a
0.8

e
Δ > +10.4
0

Δ < -1Δ ≈ +10

Δ ≈ -1

π 0

π/2

0°

Decay rate γ [Jxy/ћ]

π

180°

Δ = +1/2

Δ=0

b

0.6

π/2

Δ = -1/2

c

d

0.4
0.2
0

0

π/3

60°

2π/3

π 0

π/2

90°

π 0

π/3

2π/3

π

120°

Winding angle ϕ

Figure 6-6: Observation of phantom helix states. Decay rate 𝛾 as a function of
wavevector 𝑄, shown for fitted anisotropies Δ ranging from positive (a-b) throughzero (c) to negative values (d-e) measured at lattice depths of 9 𝐸𝑅 (red) and 11 𝐸𝑅(blue). The decay rate minimum occurs at a winding angle 𝜙 which increases smoothly
from 𝜙 = 0 for Δ ≈ +1 (a) to 𝜙 = 180∘ for Δ ≈ −1 (e). The long-lived spin pattern in
the 𝑆𝑥 -𝑆𝑦 plane is illustrated below each panel. Fits 𝛾(𝜙) = 𝛾1 |Δ − cos 𝜙| + 𝛾0 (lines)
are used to find 𝜙 which minimizes the decay rate 𝛾 and to find the anisotropies Δ
shown in Figures 6-10 and 6-11. The curves above were measured at applied magnetic
fields of 𝐵 = 847.887 G (Δ > 1) and 847.286 G (Δ ≈ 1) (a), 845.760 G (b), 842.905 G(c), 839.376 G (d), 833.004 G (Δ ≈ −1) and 827.287 G (Δ < −1) (e).

936b). When we decrease the anisotropy Δ, the minimum shifts to a larger angle 𝜙.
For Δ = −1/2 we find the special angle at 𝜙 = 120∘ (Figure 6-6d). The solid lines area fit function which our theory collaborator Wen Wei Ho derived [29], originally in ashort-time expansion:
𝛾(𝜙) = 𝛾1 |Δ − cos 𝜙| + 𝛾0

(6.3)

We treat the scaling constant 𝛾1 , the background decay rate 𝛾0 , and the anisotropy Δ
as free fit parameters. This fit function is rigorously valid only in the short-time limit𝑐(𝑡) = 1 − 𝛾 2 𝑡2 + · · · . So we have always taken it with a grain of salt. Nevertheless,
it is amazing how well it describes the data, for arbitrary anisotropies ranging all theway from large negative values Δ < −1 to positive values Δ > + 1. For example, thefit function even works for Δ = 1, the isotropic point, where
𝛾(𝜙) = 𝛾1 (1 − cos 𝜙) + 𝛾0

(6.4)

and the ferromagnetic state (𝜙 = 0∘ ) is the most long-lived state (Figure 6-6a). For
𝜆 ≫ 𝑎, we recover the quadratic scaling for diffusive transport, if the wavelengths are
long, as we already showed in the previous chapter: 1 − cos 𝑄𝑎 ≈ (𝑄𝑎)2 /2. However,when the wavelengths 𝜆 become so short that they are comparable with the latticespacing 𝑎, diffusive transport breaks down, and we see that the curve is actually acosine 𝛾 = 𝛾1 (1 − cos 𝑄𝑎) + 𝛾0 and not a parabola 𝛾 = 𝛾21 (𝑄𝑎)2 + 𝛾0 . The most unstable spin helix is the Néel state (𝜙 = 180∘ ), where spins are anti-aligned. This is adirect experimental demonstration that this classical antiferromagnetic state is not
the ground state of the quantum antiferromagnetic Heisenberg Hamiltonian, which
we actually implement here. For Δ = −1 it is exactly the other way round:𝛾(𝜙) = 𝛾1 (1 + cos 𝜙) + 𝛾0

(6.5)

Here, the Néel state is an eigenstate state (Figure 6-6e). And the ferromagnet is themost unstable spin helix. Interestingly, for Δ > +1, the fit function still gives thecorrect prediction which fits the data (Figure 6-6a). The curve just shifts up by a94

Decay rate γ [Jxy/ћ]bc

Δ ≈ 0.5

Δ≈0

Δ ≈ -0.5d

0.6

Polar angle

0.4

θ = 90°θ = 75°
θ = 60°

0.2
0

0

π/3

2π/3

π 0

π/2

π 0

π/3

2π/3

π

Azimuthal angle ϕθϕFigure 6-7: Phantom helix states for several polar angles. In addition to thepurely transverse helices (polar angle 𝜃 = 𝜋/2, circles), we also show the decay of spin
helices with polar angles 𝜃 = 5𝜋/12 (green triangles) and 𝜃 = 2𝜋/3 (orange triangles).
constant amount (compared to the Δ = 1 case):𝛾(𝜙) = 𝛾1 (1 − cos 𝜙) + 𝛾1 (Δ − 1) + 𝛾0

(6.6)

But now the ferromagnetic state (𝜙 = 0∘ ) is not a many-body eigenstate anymore.
Even in an ideal system (infinite spin chain without holes), where the background decay is 𝛾0 = 0, the ferromagnetic state still decays with a finite decay rate of 𝛾1 (Δ − 1).
And the same is true for Δ < −1 (see Figure 6-6e), where the curve also shifts up by
a constant amount (compared to the Δ = −1 case):𝛾(𝑄) = 𝛾1 (1 + cos 𝑄𝑎) − 𝛾1 (Δ + 1) + 𝛾0

(6.7)

Here the Néel state (𝜙 = 180∘ ) still decays with a finite decay rate of −𝛾1 (Δ − 1), in
addition to the background decay 𝛾0 , so it is not an eigenstate anymore, either.
For anisotropies |Δ| > +1, there exist no solutions to the phantom condition95

Δ = cos 𝜙, at least not for any real winding angle 𝜙. There are solutions only for
imaginary winding angles 𝜙. Hence, there are corresponding many-body eigenstates
which are no spin-helix states but they rather look like smooth domain walls [47],where the domain-wall width is related to Δ. These smooth domain-wall statesmight be an interesting direction for future experiments.
So far, we only looked at transverse spin-helix states which are fully lying within
the 𝑆𝑥 -𝑆𝑦 plane. We have been varying only the azimuthal angle 𝜙 = 𝑄𝑎, but so far
the polar angle has always been 𝜃 = 90∘ , such that the 𝑧-magnetization was ⟨𝑆𝑧 ⟩ = 0.
We can easily tilt the spin helix out of plane, decreasing the polar angle, and measure
its effect on the decay rate 𝛾. In Figure 6-7 we repeat the same experiment forseveral polar angles 𝜃, tilting the spin helix out of plane by just 15∘ (green) and 30∘(orange). We observe that the decay slows down quite a lot. But the minimum inthe decay rate is always at the same azimuthal angle 𝜙. This shows that there isa whole family of phantom helix states for a given value of the anisotropy Δ. It
is straightforward to show, that these states are exact eigenstates of the anisotropic
Heisenberg Hamiltonian [30]. These phantom helix states were originally discovered
theoretically in 1D as a coherent superposition of phantom Bethe states, special
degenerate solutions to the Bethe ansatz equations [47, 48].

6.4

Extension to higher dimensions

The Bethe ansatz works only in 1D. This raises the question if they are a phenomenon
tied exclusively to 1D and integrable systems. We find that they are not: We caneasily see that they are a more general phenomenon. Spin helix eigenstates also exist
in higher dimensions and in other non-integrable systems, as Wen Wei Ho realized [30],and we will now show a few examples.
Spin helix eigenstates can be found on a square lattice as shown in Figure 6-8.
This example is a perfect many-body eigenstate for anisotropy Δ ≈ 0.771, because
the spins are oriented in such a way, that any line through this lattice forms a onedimensional phantom helix state with winding angle of 𝜙 ≈ 39.6∘ . Or in other words,
96

any spin in this lattice experiences an interaction from each of its four neighbors. But
the two interactions from each pair of opposite neighbors cancel exactly.
Phantom helix states can be found even for other lattice geometries. For example,
a triangular lattice supports such a solution, too, but only for anisotropy Δ = −1/2
and hence a winding angle of 𝜙 = 120∘ , as shown in Figure 6-9 (left). Again, any line
through this lattice forms a one-dimensional phantom helix state with winding angle

Figure 6-8: Phantom spin helix for the square lattice.

Figure 6-9: Phantom spin helices for triangular and kagome lattices.
97Anisotropy Δ

+1
+0.5
0
-0.5
-1
Magnetic field B [G]Scattering length [a0]

+200

0

-200
835

840

845

Magnetic field B [G]Figure 6-10: Tuning the anisotropy with magnetic fields around the narrow
Feshbach resonance in the |↑⟩ state. The bottom panel shows the scattering lengths𝑎↑↑ (blue), 𝑎↑↓ (purple) and 𝑎↓↓ (orange) measured in previous work using interaction
spectroscopy. Near the resonance, the anisotropy (top panel) follows the 𝑎↑↑ scattering
length.
of 𝜙 ≈ 120∘ . The phantom helix state in the triangular lattice is also known as the 120
degrees Néel ordered state. It is the ground state of the classical antiferromagnetic
Heisenberg model with Δ = +1. We have just seen that it is also an eigenstate of thequantum Heisenberg model with Δ = −1/2. The kagome lattice for Δ = −1/2 and𝜙 ≈ 120∘ is another such example, as shown in Figure 6-9 (right).
So the spin-helix eigenstates, which we found experimentally in one dimension,
are a more general phenomenon not tied to 1D or integrability, but instead thisphenomenon relies on a delicate cancellation of interactions.
98

+1

0
-0.5

Anisotropy Δ

+0.5

-1

0

Scattering length [a0]

+200

-200
900

1000

1100

Magnetic field B [G]Figure 6-11: Tuning the anisotropy with magnetic fields around the broad
Feshbach resonance in the |↑⟩ state. The bottom panel shows the scattering lengths𝑎↑↑ (blue), 𝑎↑↓ (purple) and 𝑎↓↓ (orange) measured in previous work using interaction
spectroscopy. Near the resonance, the anisotropy (top panel) follows the 𝑎↑↑ scattering
length.

6.5

Direct measurement of the anisotropy

Now that we understand the phantom helix states, we can turn the whole thingaround! We can use these states as a tool to actually measure the anisotropy Δ, basedon the decay rate minimum we find. Until now we had no way to measure it directly.
and had to calculate it, based on our measurements of the on-site interactions 𝑈↑↑ ,𝑈↑↓ , and 𝑈↓↓ . However, we had no tool to confirm, that this prediction is correct —until now! So we have gone full circle. Measuring the anisotropy Δ with the phantom
helix states is a way to check, if our superexchange- and Hubbard-model description is99

Anisotropy Δ

+1
+0.5
0
-0.5
-1

On-site interaction [kHz]+500

Magnetic field B [G]

0

-500

-1000

-1500
841

843

845

Magnetic field B [G]847

Figure 6-12: Tuning the anisotropy with magnetic fields around the narrow
Feshbach resonance in the |↑⟩ state. The bottom panel shows interaction spectroscopy
and almost perfectly follows the theory in the Thomas Busch paper [49].
even correct or complete. This is especially true close to the two Feshbach resonancesin the |↑⟩ state, where the on-site interactions 𝑈↑↑ between two |↑⟩ atoms diverge.
Figure 6-10 and 6-11 compare the measured anisotropies Δ to the standard model
for superexchange (solid line) using previously measured scattering lengths [43]. The
black dashed line includes the bond-charge correction to tunneling [50] and a small adjustment of the background scattering length of 𝑎↑↓ , which was not tightly constrained
100

+0.5
0
-0.5

Anisotropy Δ

+1

-1

0

-500

On-site interaction [kHz]

+500

-1000

860

880

900

Magnetic field B [G]

920

-1500
940

Figure 6-13: Tuning the anisotropy with magnetic fields around the broad Feshbach resonance in the |↑⟩ state. The bottom panel shows interactions spectroscopy
and almost perfectly follows the theory in the Thomas Busch paper [49].
by previous measurements. Major deviations near the two Feshbach resonances at845.506 and 893.984 G (vertical dotted lines) are evidence for off-site interactions.
Away from the Feshbach resonances, the measurement roughly agrees with thegeneral trend of the prediction. We think that the discrepancy is due to some slight
uncertainties in the measured on-site interactions. But near the Feshbach resonances,
we measure something qualitatively new! This effect can only be explained by nearest
101neighbor off-site interactions 𝑉 . Since, superexchange scales as 𝑡2 /𝑈 , it is supposed tobe suppressed and should not explode, when the on-site interactions get really large,
because 𝑈 appears in the denominator. However, two neighboring atoms, even if they
only have contact interactions, also have a direct off-site interaction V, because thewave function leaks into the neighboring lattice site. But the overlap intergral
𝑉 =4𝜋~2 𝑎 ∫︁ 3
𝑑 𝑟 |𝑤(r)|2 |𝑤(r + 𝛿r)|2
𝑚

(6.8)

is really small for two Wannier functions 𝑤(r) localized on two neighboring latticesites, one lattice spacing 𝛿r =(0, 0, 𝑎) apart. Typically, off-site interactions are 4 ordersof magnitude smaller than on-site interactions
4𝜋~2 𝑎 ∫︁ 3𝑈=
𝑑 𝑟 |𝑤(r)|4
𝑚

(6.9)

where the wave function overlap is a lot larger. For this reason, the common believe
has been that off-site interactions are completely negligible and unobservable. However, in spin physics with one atom per site, both effect appear with opposite scaling!
Away from the Feshbach resonance superexchange 𝐽 = 4𝑡2 /𝑈 dominates. And close
to the Feshbach resonance off-site interactions 𝑉 dominate. This is the first observation of nearest-neighbor off-site interactions for particles, which only have contact
interactions.
However, it turns out that the current model of off-site interactions in equation 6.8
is insufficient near a Feshbach resonance. In order the explain the large effects on theanisotropy, we would require a scattering length 𝑎↑↑ ∼ 6000 𝑎0 . That is much larger
than the theoretical limit: the harmonic oscillator length ∼ 500 𝑎0 (for on-site trapping
on a lattice site). Therefore, near the Feshbach resonances we are observing muchstronger off-site interactions than theory allows. In order to characterize the Feshbachresonances better really close to unitarity, we perform interaction spectroscopy ina deep Mott insulator (lattice depth 35 𝐸𝑅 corresponding to ≈ 500 kHz band gap
between 0th and 2nd band) with two atoms per site, from the weakly interacting
state |↑↓⟩ to the strongly interacting state |↑↑⟩, in Figures 6-12 and 6-13. This is102

a direct measurement of 𝑈↑↑ − 𝑈↑↓ . With the combined knowledge of the on-site
interaction 𝑈↑↑ , and the anisotropy Δ (which is an indirect measurement of the offsite interaction 𝑉↑↑ ) we hope to develop a correct model for off-site interactions neara Feshbach resonance. This will be the topic of a forthcoming publication.

103

104

Chapter 7
Summary and Outlook
We have used ultracold atoms to implement the Heisenberg model with tunableanisotropy. This is the first quantum simulation of the anisotropic Heisenberg model.
We studied spin dynamics in previously unexplored regimes far-away from equilibrium. Some of our results are unexpected. There are many open questions which our
new quantum simulator platform allows to be addressed in the future.

7.1

Summary

We studied both quantum many-body dynamics far-away-from equilibrium as well
as stable spin patterns far away from the ground state, which are exact many-bodyeigenstates of the anisotropic Heisenberg model. Longitudinal spin-helix patterns
decay by spin transport, where the decay rate is well described by power laws𝛾 ∝ (𝑄𝑎)𝛼

(7.1)

where the transport exponent 𝛼 is a function of the anisotropy Δ. Transverse spinhelix patterns decay also by local spin dephasing, such that the decay rate has adifferent form:
𝛾 = 𝛾1 |Δ − cos(𝑄𝑎)| + 𝛾0
105

(7.2)

For the special case of isotropic interactions (Δ = 1), both descriptions agree for longwavelengths 𝜆 ≫ 𝑎, since 𝛾1 |1 − cos(𝑄𝑎)| ≈ 𝛾21 (𝑄𝑎)2 . The background rate 𝛾0 accounts
for dephasing induced by the effective magnetic field enabled by finite chain length,
mobile holes in the chain, and inhomogeneity between chains due to the finite sizeof the optical lattice beams, which would be zero for an ideal system. For a given
anisotropy Δ, there exists one special spin helix with a winding angle 𝜙 = 𝑄𝑎 fulfilling
the phantom condition
Δ = cos(𝑄𝑎)(7.3)

where the decay rate vanishes (at least in an ideal system where the backgrounddecay is 𝛾0 = 0). This phantom helix state is a perfect many-body eigenstate of theanisotropic Heisenberg model with anisotropy Δ.

We have confirmed the connection of the Heisenberg model to the underlying
Hubbard model. We showed that we can control 𝐽𝑧 and 𝐽𝑥𝑦 with the lattice depth
and Feshbach resonances. This has a dramatic impact on spin transport. We also
found a way to directly measure the anisotropy Δ = 𝐽𝑧 /𝐽𝑥𝑦 using the phantom helix
states. We identified an effective magnetic field which is created by superexchange
and showed that this is a real part of this mapping, and we demonstrated that one
can measure this field directly in three different ways: with a beatnote, throughspin-precession, and spectroscopically. The effective magnetic field varies with the
Feshbach resonances in the expected way. On top of superexchange, we found a direct interaction between spins (zeroth-order in tunneling), a nearest-neighbor off-site
interaction, and studied how it modifies the interaction anisotropy.

7.2

Discussion

Recently, there has been strong interest and a big debate about transport in the
Heisenberg model at the isotropic point (Δ = 1). The questions is, if transport behavior is described by diffusion (𝛼 = 2) or by superdiffusion (𝛼 = 1.5). There was a106

very recent result by the Bloch group [33], which might be considered unexpected.
For Δ = 1 they studied a quench from a sharp domain wall with variable contrast, andthey find superdiffusion (𝛼 = 1.5) for all values of the contrast, both for a partiallypolarized domain wall (as predicted by Refs. [11, 12]) and also for a fully polarizeddomain wall (in contradiction to Ref. [51] which predicts diffusion). Interestingly,
for a (fully polarized) spin-helix state we find diffusion (𝛼 = 2) both experimentallyand with numerical simulations. This also confirms a previous experiment [19] (alsoboth experimentally and with numerical simulations). However we note, that our
obtained scaling exponent 𝛼 depends slightly on how we treat the observed (ballistic) oscillations. If we ignore the oscillations and fit pure exponentials 𝑎0 𝑒−𝑡/𝜏 to the
decay curves 𝑐(𝑡), then the exponent is very close to (𝛼 = 2). If we account for theoscillations in the fit function (𝑎0 + 𝑏0 cos 𝜔𝑡)𝑒−𝑡/𝜏 , this reduced the observed exponent to 𝛼 = 1.87(4). Furthermore, our experiment might have been performed at an
anisotropy which was slightly higher than Δ = 1, since Δ is only calibrated up to±0.1. So the true value of the exponent 𝛼 at the isotropic point might even be lower
than 𝛼 = 1.87(4). To address these questions, the following two experiments would
be especially interesting:
1. Spin-helix states with 50% contrast. I expect this to be superdiffusive with
𝛼 = 1.5.
2. A sharp domain wall (both 100% and 50% contrast).
With our quantum simulator platform, both experiment can be done as a functionof Δ, tuning the anisotropy, as well as a function of interchain coupling, to study
the crossover from 1D to 2D, which is a continuous knob on an integrability breaking
term.

7.3

Outlook

Generally, there are of course many more possible experiments. They mainly fall into
the following categories:
107∙ Different initial states: sharp domain walls, smooth domain walls, partiallypolarized (i.e. mixed) states∙ Different Hamiltonians: higher spatial dimensions (2D or 3D), higher spin quantum numbers (e.g. 𝑆 = 1)
∙ Different observables: Single site resolution with a quantum gas microscope
Spin dynamics in higher spatial dimensions
Our experimental protocol is not restricted to studying one-dimensional chains; it caneasily be extended to two or three spatial dimensions, by using not only one shallow
lattice to control superexchange coupling in a single direction, but by using two or
three shallow lattices. Studying coupled one-dimensional chains is interesting as acontinuous crossover from 1D to 2D. Inter-chain coupling can be understood as anintegrability breaking term. An experimental complication in higher spatial dimensions is that deeper lattices are required to keep atoms localized on individual latticesites, than compared to one dimension, which slows down superexchange dynamics.
Quantum many-body dynamics in two or three dimensions is completely out of range
for numerical simulations, at least for reasonable system sizes. Preparing phantom
helix states in higher dimensions would realize an exception to the usual expected
thermalizing dynamics in non-integrable systems.

A sharp domain wall
We studied spin transport with quantum quenches from a far-from-equilibrium spin
helix
∏︁

(sin 𝑄𝑧𝑖 /2 |↑⟩𝑖 + cos 𝑄𝑧𝑖 /2 |↓⟩𝑖 )

(7.4)

𝑖

which involves a sinusoidal modulation of the 𝑆𝑧 -magnetization (population of spin|↑⟩ and |↓⟩ atoms) with wavevector 𝑄. That is a measurement in momentum space,at a single value of 𝑄. In contrast, a quench from a sharp domain wall|· · · ↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓ · · ·⟩
108

(7.5)

between two magnetized domains, is a measurement in real space. This scenario has
been extensively studied theoretically [51–54]. The modulation of the 𝑆𝑧 magnetization is a step function. This situation is reminiscent of mixing two different classicalgases, which are initially separated in two opposite sides of the system. This scenario
has been studied theoretically for the fully polarized domain wall [51] and the isotropic
case (Δ = 1), revealing diffusive behavior (however with a ballistic light-cone). However, for a sharp domain wall with partially polarized domains (infinite temperature)transport behavior changes into superdiffusion (𝛼 = 1.5) for Δ = 1 [11, 12], which canbe understood from generalized hydrodynamics involving local equilibriation of conserved quantities [55, 56]. This was experimentally confirmed by Ref. [33], although
they even find superdiffusion for the fully polarized domain wall. Our platform can
extend these experimental studies to arbitrary anisotropies. We cannot optically resolve domain walls, which are sharper than 6 lattice sites. Upgrading our setup to aquantum gas microscope with single site resolution, would address this issue. Nevertheless, our current resolution might be good enough to interesting dynamics.

Benthe phantom states for anisotropies |Δ| > 1
For anisotropies |Δ| ≤ 1, the transverse spin-helix state with wavevector 𝑄 satisfying
the phantom conditioncos(𝑄𝑎) = Δ(7.6)

is an exact many-body eigenstate of the Hamiltonian — the Bethe phantom state.
However, for anisotropies |Δ| > 1 there exists no such spin helix, since the phantom
condition does not have a solution for any real wavevector 𝑄. Nevertheless, there
are solutions for imaginary wavevectors Q, which means that, instead, the Bethe
phantom state takes a different form, which is a smooth domain wall with a finite
width [47], which depends on the anisotropy Δ. It would be interesting to study thestability of domain wall states as a function of their width, where we expect to finda pronounce minimum in the decay rate for one specific width.

109

Partially polarized spin-helix
So far, we studied the relaxation of spin-helix states starting with a 100% modulation,i.e. a pure state. Theoretical studies have shown in the case of sharp domain walls[11, 12, 51], that the dynamical behavior can change significantly, if the two domains
are not fully polarized. We can also create spin-helix states with <100% modulation
depth, by winding a spin-helix not starting from a fully polarized state, but from apartially polarized state. A partially polarized state can be created with the following
protocol.
1. We start with spin |↑⟩ on every site.
𝜌 = |↑⟩ ⟨↑|(7.7)

2. We tilt the spins to a polar angle 𝜃 with an RF pulse so that the state iscos(𝜃/2) |↑⟩ + sin(𝜃/2) |↓⟩ on every site. This is a pure state with density matrix
𝜌 = cos2 (𝜃/2) |↑⟩ ⟨↑| +11sin(𝜃) |↑⟩ ⟨↓| + sin(𝜃) |↓⟩ ⟨↑| + sin2 (𝜃/2) |↓⟩ ⟨↓| (7.8)
22

3. We dephase the 𝑆𝑥 and 𝑆𝑦 components and turn this into a mixed state, bylowering the lattice and switching on superexchange, while the anisotropy is
tuned to Δ = 0, for example. The transverse components of that state should
decay after a few spin-exchange times ~/𝐽𝑥𝑦 . Afterwards we ramp the lattice
back up to 𝑉𝑧 = 35 𝐸𝑅 and end up with a mixed state on every lattice site, withdensity matrix
𝜌 = cos2 (𝜃/2) |↑⟩ ⟨↑| + sin2 (𝜃/2) |↓⟩ ⟨↓|(7.9)

4. Starting from this partially polarized state, we can now wind a spin helix,
exactly with the same procedure as we did earlier, where we started with a fullypolarized state.
We can conveniently use anisotropic spin couplings (Δ ̸= 1) to dephase transverse spincomponents. The experiment in Ref. [33] is limited to Δ = 1, which makes this much110

harder. They actually used single site resolution to apply randomized light shifts toindividual spins, which varied from run to run, to dephase the transverse components
manually.

Single site resolution with a quantum gas microscrope
With a quantum gas microscope it would be possible to directly image holes in thesystem. And this would open the door to studying the boson t-J model with single
site resolution. The lattice depth allows us to control the dynamics of holes (tunneling) and spins (superexchange; second-order tunneling) separately. The study ofhole-magnon coupling may be interesting for other system where such couplings arepresent such as high temperature superconductors [57–59].

Mapping out hole dynamics in the frequency domain
Even without single site resolution it should be possible to study hole dynamics andhole-magnon coupling. We have shown that a spin-echo can rectify the fluctuationsof hole dynamics [29]. Since holes carry a localized magnetic field which couples tospin dynamics, this leads to enhanced dephasing. By using a series of echo pulses
at frequency 𝜔, one could map out the frequency spectrum of the effective magnetic
field and hence follow the hole dynamics in the frequency domain, using concepts
from dynamic decoupling [60].

Higher spin quantum numbers
A spin-1 Heisenberg model can be realized with two atoms per site and realizing
three spin states |⇑⟩ := |↑↑⟩, |∘⟩ := |↑↓⟩, and |⇓⟩ := |↓↓⟩. This system is restricted
to realizing isotropic spin couplings (Δ = 1). But it includes an additional term,
the so-called single-ion anisotropy 𝑢, as it is commonly called in condensed mattersystems:
𝐻=𝐽∑︁S𝑖 · S𝑗 + 𝑢∑︁(S𝑖 )2(7.10)𝑖⟨𝑖𝑗⟩

The single-ion anisotropy 𝑢 = (𝑈↑↑ + 𝑈↓↓ )/2 − 𝑈↑↓ can be tuned through Feshbach res111

onances in our system. It would be interesting to create a spin-Mott state | ∘ ∘ ∘ · · · ∘⟩which we can do easily using a 𝜋-pulse |↑↑⟩ → |↑↓⟩, while |↑↓⟩ → |↓↓⟩ can be far detuned from resonance (𝑈↑↑ − 𝑈↑↓ ) ̸= (𝑈↑↓ − 𝑈↓↓ ) causing an interaction blockade [41].
This spin-Mott is the ground state for 𝑢 ≪ 𝐽. Starting from this state, we can study
a quantum quench by suddenly switching on spin interactions 𝐽 to a finite values (byramping down the lattice depth), while the single-ion anisotropy is set to 𝑢 = 0 or
𝑢 > 0. An experimental complication might be, that there is 3-body loss, which limits
the coherence. This time constraint might inhibit adiabatic state preparation. But it
might be enough time to observe interesting spin dynamics far away from the groundstate.

A transverse spin-helix has a macroscopic spin current
For transverse spin-helix states, so far, we have only studied the transverse magnetization 𝑆𝑥 . However, a transverse spin helix intrinsically has a macroscopic spin current
along the spin chain, which over time should lead to an increasing spatial imbalanceof spin |↑⟩ and spin |↓⟩ population. This effect should be observable by measuring the𝑆𝑧 magnetization profile after a quench from an initial transverse spin-helix state, asa function of time.

Extended Hubbard model
The discovered nearest-neighbor off-site interactions 𝑉 close to a Feshbach resonancecan be used to implement extended Hubbard models:𝐻 = −𝑡

∑︁

∑︁ 𝑈

⟨𝑖𝑗⟩

𝑖

( 𝑎†𝑖𝜎 𝑎𝑗𝜎 + H.c.) +2𝑛𝑖 (𝑛𝑖 − 1) + 2𝑉

∑︁

𝑛𝑖 𝑛𝑗

(7.11)⟨𝑖𝑗⟩

For example, analogous to a Mott-insulator, where strong on-site repulsion 𝑈 can suppress double occupancies and can localize atoms on individual lattice sites if 𝑡 ≪ 𝑈 ,strong nearest-neighbor off-site interactions 𝑉 prevent atoms to be nearest neighbors,i.e. this can lead to a density modulation.

112

In summary, our new quantum simulator platform with tunable interactions opens
up the possibilities for many new studies which are likely to provide new insight into
the rich dynamics of Heisenberg spin models, t-J models with hole-magnon couplingsand Hubbard models with offsite interactions.

113

114

Appendix A
Spin transport in a tunable
Heisenberg model realized withultracold atoms
P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler, W. Ketterle
Spin transport in a tunable Heisenberg model realized with ultracold atoms
Nature 588, 403-407 (2020)115

Article

Spin transport in a tunable Heisenberg
model realized with ultracold atoms
https://doi.org/10.1038/s41586-020-3033-y
Received: 18 May 2020

Paul Niklas Jepsen1,2,3 ✉, Jesse Amato-Grill1,2,3, Ivana Dimitrova1,2,3, Wen Wei Ho3,4,
Eugene Demler3,4 & Wolfgang Ketterle1,2,3

Accepted: 25 September 2020
Published online: 16 December 2020
Check for updates

Simple models of interacting spins have an important role in physics. They capture theproperties of many magnetic materials, but also extend to other systems, such as
bosons and fermions in a lattice, gauge theories, high-temperature superconductors,
quantum spin liquids, and systems with exotic particles such as anyons and Majorana
fermions1,2. To study and compare these models, a versatile platform is needed.
Realizing such systems has been a long-standing goal in the field of ultracold atoms.
So far, spin transport has only been studied in systems with isotropic spin–spin
interactions3–12. Here we realize the Heisenberg model describing spins on a lattice, withfully adjustable anisotropy of the nearest-neighbour spin–spin couplings (called the
XXZ model). In this model we study spin transport far from equilibrium after quantum
quenches from imprinted spin-helix patterns. When spins are coupled only along two of
three possible orientations (the XX model), we find ballistic behaviour of spin dynamics,
whereas for isotropic interactions (the XXX model), we find diffusive behaviour.
More generally, for positive anisotropies, the dynamics ranges from anomalous
superdiffusion to subdiffusion, whereas for negative anisotropies, we observe acrossover in the time domain from ballistic to diffusive transport. This behaviour is incontrast with expectations from the linear-response regime and raises new questions inunderstanding quantum many-body dynamics far away from equilibrium.

Quantum dynamics is an active research area in many-body physics.
Even the linear-response (near-equilibrium) behaviour of many-bodysystems can be very complex. For example, spin transport inone-dimensional Heisenberg XXZ quantum spin chains, despite being
a topic that is decades old, is still under active investigation because of
the rich dynamics connected to integrability13–17. Dynamics in highly
out-of-equilibrium scenarios, such as from continual drives or quantum
quenches18–22, is even less well understood. It is hence highly desirable tohave a quantum simulator that can realize well isolated, programmableand controllable spin systems. By now, a number of such platforms
exist21–27, with varying capabilities.
Ultracold atoms in optical lattices offer an especially promising platform with which to realize tunable Heisenberg spin models27: in deep lattices where atoms become localized on individual sites, forming a Mott
insulator28, the dynamics of the remaining degrees of freedom is governed by effective spin–spin interactions realizing nearest-neighbour
Heisenberg XXZ spin models. For bosons, the most commonly used
atom, 87Rb, has almost equal singlet and triplet scattering lengths,
implying effectively isotropic spin physics4–7. For fermions, the Pauli
exclusion principle enforces isotropic antiferromagnetism3. Many
theoretical proposals have suggested ways to obtain richer spin models
in optical lattices29–32 over the past 20 years, and we now report here
the realization of the spin-1/2 Heisenberg model with fully adjustableanisotropy in the spin–spin interactions. This enables simulations ofanisotropic magnetic materials with easy-axis or easy-plane alignment.
1

The wide tunability is realized using 7Li atoms, the Feshbach resonancesof which we have characterized in previous work33. Additionally, lithium, with its light mass, has the advantage of fast spin dynamics (setby second-order tunnelling)26, decreasing the relative importance ofheating and loss processes compared to heavier atoms. We use this fast
and tunable platform to study far-from-equilibrium spin transport inpreviously unexplored regimes.
For many-body quantum simulation experiments, an ideal starting
point is a simple benchmark system to which more complexity can be
added. In this work, we first implement the XX model in one dimension,
which is exactly solvable by mapping to a system of non-interacting
fermions. We then tune the anisotropy to arbitrary values, which in thefermionic language corresponds to adding nearest-neighbour interactions. To implement the spin model, we use a system of two-component
bosons in an optical lattice, which is well described by the Bose–
Hubbard model. The two states, labelled |↑⟩ and |↓⟩, form a spin-1/2
system. In the Mott insulating regime at unity filling the effective
Hamiltonian is given by the spin-1/2 Heisenberg XXZ model29–32()

H = ∑ Jxy Six S xj + Siy S yj + Jz Siz S zj  ,


⟨ij ⟩

(1)

where nearest-neighbour ⟨ij⟩ couplings are mediated by super­exchange.
∼2
∼2
∼2
∼2
To leading order, Jxy = − 4t /U⇅ and Jz = 4t /U⇅ − (4t /U⇈ + 4t /U⇊) ,∼where t is the tunnelling matrix element between neighbouring sites,

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA. 2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA.
MIT-Harvard Center for Ultracold Atoms, Cambridge, MA, USA. 4Department of Physics, Harvard University, Cambridge, MA, USA. ✉e-mail: jepsen@mit.edu

3Nature | Vol 588 | 17 December 2020 | 403

a

Jxy /2b+Jz /4

Anisotropy, Jz /Jxy

Article2
1
0–1–Jz /4

–2
820

c

840
860
Magnetic field (G)

880

Sz
O
Sx
z

a

z

x
ImagingyFig. 1 | Tunability of the XXZ model with 7Li and implemenation of a spin
helix. a, The Hamiltonian (equation (1)) is characterized by two energy scales:
the transverse spin coupling Jxy (spin exchange) and the longitudinal spin
coupling J z (spin–spin interaction). b, Anisotropy Δ = J z /J xy as a function of
applied magnetic field. The solid line is a fit to experimental data points, which
are calculated from measured values U⇈, U⇅, U⇊ (see Methods and Extended
Data Fig. 1). c, Spin helix realized from two hyperfine states (spin |↑⟩ and |↓⟩).
The spin vector winds in the S z –S x plane as a function of position z in the spin
chain. Deep optical lattices along x and y create an array of independent spin
chains. The z lattice is shallower and controls spin transport along each chain.

and U⇈, U⇅, U⇊ are the on-site interaction energies. The transverse coupling Jxy induces spin exchange between neighbouring sites and is theorigin of spin transport. The longitudinal coupling Jz corresponds toa nearest-neighbour spin–spin interaction (Fig. 1a).
The magnitude of superexchange can be varied over two ordersof magnitude by changing the lattice depth, which scales the entire
Hamiltonian. We control the anisotropy Δ = Jz/Jxy via an applied magnetic
field that tunes the interactions through Feshbach resonances in two
hyperfine states (Fig. 1b) while keeping Jxy > 0 (antiferromagnetic).
The ability to tune the anisotropy over a wide range of positive andnegative values allows us to explore dynamics beyond previous experiments3–7 in which Δ ≈ 1.
An array of one-dimensional (1D) chains is implemented by a deep
optical lattice (of depth Vx, Vy = 35ER) in the x and y directions. The lattice depth Vz along the z direction controls the superexchange rate
within the chains (Fig. 1c). Here ER = π2ħ2/(2ma2) denotes the recoil
energy, where a is the lattice spacing, m the atomic mass and ħ the
reduced Planck constant. After preparing an identical spin helix7–9 withwavelength λ in each chain (see Methods), time evolution is initiated by
rapidly lowering Vz. The dynamics following this quench is governedby the 1D XXZ model (equation (1)) with an anisotropy Δ selected by anappropriate applied magnetic field. After an evolution time t of up to
404 | Nature | Vol 588 | 17 December 2020

500ħ/Jxy (well below the heating lifetime, approximately 1 s, of the Mott
insulator), the dynamics is frozen by rapidly increasing Vz and the atoms
are imaged in the |↑⟩ state via state-selective polarization-rotation
imaging with an optical resolution of about 6 lattice sites (see Methods).
Integrating the images along the direction perpendicular to thechains yields a 1D spatial profile of the population in the |↑⟩ state, averaged over all spin chains (see Extended Data Fig. 2). This is equivalent
to a measurement of the local magnetization ⟨Siz ⟩ = (ni ↑ − ni ↓)/2 =ni↑ − 1/2 which, as in Fig. 1c, shows a sinusoidal stripe pattern. We
determine the contrast C by a fit f (z) = g (z)[1 + Ccos(Qz + θ)]/2, whereQ = 2π/λ is the wavevector, g(z) is a Gaussian envelope function that
accounts for the spatial distribution of all atoms n = n↑ + n↓, and θ is arandom phase that varies from shot to shot, owing to small magnetic
bias field drifts. During the evolution time t the contrast C(t) decays,
and we study the dependence of c(t) = C(t)/C(0) on lattice depth Vz,
wavelength λ, and anisotropy Δ.
For all data, we measure spin dynamics at two or three different lattice depths Vz and verify that the decay curves c(t) collapse when timeis rescaled by the spin-exchange time ħ/Jxy (see for example, Fig. 2a).
This demonstrates that we observe transport by superexchange andnot some other process. To study transport behaviour, we measure how
the decay timescale τ depends on the modulation lengthscale λ. We
note that for ballistic motion, this decay constant grows linearly with
distance (τ ∝ λ), whereas for diffusion it grows quadratically (τ ∝ λ2).
Throughout this Article, we normalize time by the spin-exchange
time ħ/Jxy, length by the lattice spacing a, and velocities by the Fermi
velocity vF = a/(ħ/Jxy). These units are obtained from the experimentally
determined lattice depth using an extended Hubbard model and havean estimated systematic calibration error of about ±10%, in addition
to quoted statistical errors. The accuracy of Δ is estimated to be ±0.1
(see Methods). Unless noted otherwise, all error bars and uncertainties
herein are purely statistical and represent 1σ uncertainty of the fits.
Each data point for the contrast c(t) is obtained by simultaneously
fitting several images (usually six, but up to 15).

XX modelWe first study the case Δ = 0, which can be mapped by the Jordan–Wignertransformation34 to non-interacting spinless fermions undergoingnearest-neighbour hopping on a lattice. In this mapping, |↑⟩ corresponds to a site occupied by a fermion, and |↓⟩ to an empty site. Small
excitations around the Fermi sea at half-filling are spin waves withwavevector q and a linear dispersion relation ω(q) = vFq.
Figure 2a shows the decay of the contrast c(t) for Δ ≈ 0 (see Methods
for calibration of Δ). In addition to an overall decay, a local maximum
corresponding to a partial revival of the initial spin modulation appears
after about 12 spin-exchange times. We find the decay curves can bewell described by the sum of a decaying part with time constant τ and a(damped) oscillating part with frequency ω, resulting in a fitting function c(t) = [a0 + b0cos(ωt)]e−t/τ + c0, with a0, b0, c0, ω and τ as adjustableparameters (Methods and Extended Data Fig. 4 discuss the offset c0).
Numerical simulations, also shown in Fig. 2a, reproduce the major
features of the experimental dynamics very well (decay time τ andfirst oscillation), but differ in details, probably owing to a differencein hole fraction, uncertainties in Δ, or non-idealized initial-state preparation. By varying the wavelength λ = 2π/Q of the helix (Fig. 2b) we
obtain a dispersion relation ω(Q) for the oscillations (Fig. 2d). A linear
fit ω(Q) = vQ yields a characteristic velocity v = 0.76(1)vF, similar to thenear equilibrium dynamics, as expected for a non-interacting system.
The decay time constant τ also shows a linear scaling with inverse
wavevector: a power-law fit τ ∝ Q−α yields an exponent of α = 1.00(5), indicating ballistic transport (Fig. 3b, red). Indeed, if we plot c(t) in time units
rescaled by λ, all curves for different helix wavelengths collapse into a single
curve (Fig. 2c), showing that all aspects of the observed spin dynamics inthe XX model are ballistic and governed by one characteristic velocity.

b

c

e
Frequency, Z (×Jxy/h)

Δ≈0

1.0

Contrast, c(t)0.8
O = 15.7a0.6dFrequency, Z (×Jxy /h)

a

0.05

0.1

0
0
0.1
0.2
Wavevector, Q (×2π/a)

0.4

f

0
0
0.1
0.2
Wavevector, Q (×2π/a)

0.2
0O = 6.7a

0

10

20

0
Time, t (× /Jxy )

10

20

30

0

1.0
2.0
Rescaled time, t (×(/Jxy )/(O/a))

0

0.2
0.4
0.6
Rescaled time, t (×(/Jxy )/(O/a)2)

Fig. 2 | Ballistic and diffusive spin transport. a–d, XX model, ballistic
behaviour (Δ ≈ 0, non-interacting fermions). a, Spin-helix contrast c(t) for
λ = 10.4a measured at three different lattice depths 9ER (orange), 11ER (blue) and
13ER (yellow). Decay curves collapse when times are expressed in units of
ħ/J xy = 0.64 ms, 1.71 ms and 4.30 ms, respectively. The fit (black line) shows adecay with time constant τ = 5.5(2)ħ/J xy and a damped oscillation with period
T = 2π/ω = 13.7(2)ħ/J xy. The fit is typically applied to data points from all lattice
depths simultaneously, but analysing each lattice depth separately yields
identical results. Numerical simulations are also shown for the XX model∼(dashed line) and bosonic t –J model with 5% holes (dash-dotted line) and 10%
holes (dotted line). b, Decay curves for different wavelengths λ = 15.7a, 13.4a,
11.7a, 9.4a, 7.8a and 6.7a (offset for clarity). c, The decay curves in b collapse

into a single curve, if time is rescaled by λ (indicating ballistic transport) and
offsets c0 are subtracted. d, Oscillation frequencies (filled symbols) follow a
linear dispersion relation ω(Q) = vQ with velocity v = 0.76(1)vF and agree at the
10% level with numerical simulations (open symbols) yielding v = 0.85(1)vF.
Theoretical frequencies are obtained as the inverse of the first revival time.
Owing to damping, this may be an overestimate of 10%. e, f, XXX model,
diffusive behaviour (Δ ≈ 1, strongly interacting fermions). Oscillations arestrongly suppressed and time has to be rescaled by λ 2 (indicating diffusivetransport) for collapse. However this collapse is not perfect, because of thepresence of small oscillations (see Extended Data Fig. 3) that follow a linear
dispersion relation (f) with a velocity v = 0.35(1)vF.

XXX model

Positive anisotropies (Δ ≥ 0)

For finite Δ, the Jordan–Wigner transformation results in fermions withnearest-neighbour interactions. The isotropic case Δ = 1 corresponds tostrong interactions, which should generically turn fast ballistic transport into slow diffusive transport. Indeed, the decay slows down forincreasing wavelength λ much more dramatically than in the Δ ≈ 0 case(also illustrated in Extended Data Fig. 3a–c). A power-law fit of the decay
constant τ versus Q yields an exponent of α = 1.87(4), which is close to 2,
indicative of a diffusive process (Fig. 3b, blue). If time units are rescaledby λ2, then all contrast curves c(t) collapse very well into a single curve(Fig. 2e). However this collapse is not perfect, because of a (small) oscillating part that still obeys a linear dispersion relation ω(Q) = vQ (Fig. 2f).
Using 1/τ = DQ2, a diffusion constant can be determined as D =1
0.242(7)a2/(ħ/Jxy). Interpreting D = 2 δx 2 /δt as arising from a random
walk of step size δx (the mean free path) and time δt between steps,
and using v = δx/δt = 0.35(1)vF (obtained from the dispersion relationin Fig. 2f), we find δx = 1.39(5)a. A mean free path on the order of thelattice constant is analogous to the Mott–Ioffe–Regel limit for resistivity where simple quasi-particle pictures break down35,36, implying thatthe isotropic Heisenberg model is strongly interacting.
Our observation of diffusive behaviour and the value for the diffusion
coefficient are consistent with previous results7 (for zero hole fraction) on the 1D isotropic Heisenberg model with Jxy < 0. Our system isantiferromagnetic (Jxy > 0), and so this indicates that the overall sign of
the Hamiltonian is irrelevant, as expected from theoretical arguments
involving time-reversal symmetry (see Methods). However the small(ballistic) oscillatory component has not been previously reported.
We note that a two-dimensional Fermi–Hubbard system can also show
diffusive decay and ballistic oscillations37.
By tuning the interactions over a large range of Δ, we can study how
the transport behaviour changes. For an interacting gas of classicalparticles or quasiparticles, one would expect ballistic behaviour ontimescales shorter than the collision time and diffusion for longer
times. We indeed find this for Δ < 0, whereas for Δ ≥ 0 we observe
qualitatively very different behaviour (also illustrated in Extended
Data Fig. 5).

All measured decay curves c(t) are well described by the fitting function previously used. The observed oscillation frequencies ω follow
linear dispersion relations ω(Q) = vQ (Extended Data Fig. 3), whereas
the decay time constants τ show power-law scaling τ ∝ Q−α (Fig. 3) in thefollowing way: as the anisotropy is increased from Δ = −0.12 to Δ = 0.55,
the exponent stays close to α = 1 (‘ballistic regime’) and the characteristic velocity of oscillations decreases by a factor of about 1.6, tov = 0.47(1)vF. Between Δ ≈ 0.5 and 1 the exponent increases smoothly
from α = 1 to α ≈ 2 (‘superdiffusive regime’), for example, α = 1.48(4) at
Δ = 0.78 (Fig. 3b, yellow). For Δ > 1 transport slows down even more,
and the exponent also continues to increase smoothly to values α > 2(‘subdiffusive regime’), for example, α = 2.83(14) at Δ = 1.58 (Fig. 3b,
green). For each Δ ≥ 0, the measured decay curves collapse into a single
curve, if time units are rescaled by λα (Extended Data Fig. 6).
Power-law exponents between 1 and 2 (superdiffusion) are often
associated with Lévy flights or fractional Brownian motion where step
sizes are correlated38,39. Power-law exponents larger than 2 (subdiffusion) typically arise for transport through a disordered medium40,41 andhave also been recently observed in a tilted Fermi–Hubbard system42.
However, the XXZ Heisenberg Hamiltonian we study has no disorder.

Negative anisotropies (Δ < 0)Here the behaviour is qualitatively very different compared to positive
Δ of similar magnitude. We find a crossover in the time domain fromballistic to diffusive behaviour. For example, at Δ = −1.43 the initial decayof c(t) is fast and, in fact, coincides well with the non-interacting (ballistic) case Δ ≈ 0 (Fig. 4a), in stark contrast to the positive case Δ = +1.58(Extended Data Fig. 5). At t = t0 ≈ 2.8ħ/Jxy (dotted line) the decay suddenly
slows down. We therefore parameterize the decay curve c(t) by a piecewise fit with two timescales: (I) a linear function (1 − t/τI) at short timesand (II) an exponential e−t /τII at longer times, with respective time constants τI, τII. When the wavevector Q is varied, both τI and τII follow a
power law (Fig. 3a), but with different exponents: αI = 1.08(6) (ballistic)and αII = 2.15(16) (diffusive), respectively (Fig. 3c). In both experimental
Nature | Vol 588 | 17 December 2020 | 405

ArticleaΔ< 0Decay time, W (×/Jxy)

30

,,

10

b

Δ≥0

30

3
10
10
6

,

3

3

0.05

0.10

0.15
0.05
Wavevector, Q (×2π/a)

0.10

0.15

c
3

,,

Exponent, D

Subdiffusive
Diffusive

2

Superdiffusive,

Ballistic

1
–2

–1

0
Anisotropy, Δ+1+2

Fig. 3 | Power-law scalings. a, b, The power-law scalings of the decay time
constants τ for anisotropies Δ ranging from negative (a) to positive (b).
Experimental results are shown in a for Δ = −1.02 (blue), −1.43 (green) and−1.79 (purple) for the two timescales τI and τII; and in b for Δ = −0.12 (red),
0.35 (orange), 0.78 (yellow), 1.01 (blue) and 1.58 (green). Lines are power-law
fits. c, Fitted power-law exponents from experiments (filled symbols) andtheory (open symbols), obtained from simulations of the defect-free XXZ∼
model (blue) and t –J model with 5% hole fraction (orange). a–c can besummarized as follows. For Δ ≥ 0 (circles), we observe anomalous diffusion: theexponent increases smoothly from ballistic (red and orange) to superdiffusive(yellow) to diffusive (blue) to subdiffusive (green). For Δ < 0, we observe
behaviour reminiscent of a classical gas: transport is ballistic at short times(I; triangles) and diffusive at longer times (II; squares). (For diffusion
coefficients and theory see Extended Data Fig. 8.).

(Fig. 4b) and numerical results (Fig. 4c), all decay curves collapse forshort times before ‘peeling off’ at later times, if time is rescaled by λ.
Similar behaviour is observed for other negative anisotropies (Fig. 3a;also Extended Data Fig. 7), with the initial ballistic temporal decay (regimeI) almost independent of Δ. However, for larger |Δ|, we find that t0 (therange of regime I) is smaller, and the diffusion timescales τII in regime II
are longer. The diffusion coefficient decreases from D = 1.27(6)a2/(ħ/Jxy)
to 0.25(2)a2/(ħ/Jxy) between Δ = −1.02 and −1.79 (Extended Data Fig. 8c).
Figure 3c summarizes the different transport behaviours uncovered for the
anisotropic Heisenberg model and represents the main result of this paper.

Theoretical simulations
To validate our platform as a quantum simulator, we have carried out
numerical simulations of quench dynamics starting from a spin helix, using
a combination of exact diagonalization and tensor network methods(see Methods). We simulate the dynamics of the system without holes (XXZ∼
Hamiltonian), as well as with a small probability of holes (bosonic t –J
model), and compare the simulated contrast to experimental results. The
timescales of decay generally agree fairly well. A qualitative difference inthe decay curves is illustrated in Fig. 2a: the simulations always show an
406 | Nature | Vol 588 | 17 December 2020

initial quadratic decay (as expected from time-reversal symmetry, see Methods), and revivals are generally more pronounced. The initial quadratic
decay happens in the pure-spin simulations on the timescale of ħ/Jxy,∼whereas an addition of 5% to 10% holes reduces this to the timescale ħ /t∼∼7(where t is the tunnelling amplitude in the t –J model) and reduces
the amplitude of revivals. However, the presence of holes does not affectthe overall behaviour of the decay times of the spin contrast: the simulations∼of both the XXZ and the t –J model yield power-law scalings of time constants,
with exponents that agree reasonably well with experimental ones (seeFig. 3c and Methods section ‘Power-law scalings in the continuum limit’).

Discussion
Our work on spin transport illustrates the strength of a combined experimental and theoretical quantum simulation. Our quantum simulator
platform enables us to probe dynamical regimes that are difficult toachieve in numerical simulations, such as large system sizes or longtimes (Extended Data Figs. 6, 7), which require prohibitively large
computational resources. On the other hand, numerical simulations
provided valuable insight into effects that could not be studied experimentally, such as that of holes (Figs. 2a, 3c), different phases of thehelix, and the role of boundary conditions (Extended Data Figs. 9, 10).
Our observations are consistent with some theoretical predictions
for spin transport in the anisotropic Heisenberg model, but differ
sharply from others. For example, studies of quantum quenches frompure states involving a single domain wall43–46 have suggested ballistic
dynamics at Δ = 0 and diffusive dynamics at Δ = 1 (albeit with logarithmiccorrections), similar to our findings. In addition to diffusive transport,
a ballistic light-cone has been observed46 in numerics at Δ = 1, which
may be related to the small ballistic oscillatory component we haveobserved experimentally. In contrast to our findings, theoretical studies of long-time linear response of spin transport at high temperatures(that is, mixed states) have indicated a sharp transition from ballistic(Δ < 1) to diffusive (Δ > 1), with superdiffusive behaviour (α = 3/2) exactly
at the transition point Δ = 115,16, which can be understood from generalized hydrodynamics involving local equilibriation of conserved
quantities47,48. The situation we have studied is different, because theinitial spin-helix state is a pure state far from equilibrium. An accurate
description of coherent dynamics using the exact eigenstates from theBethe ansatz49 is a very challenging problem. Ultimately, the spin-helix
state will relax into a thermal state (or quasi-thermal generalized Gibbs
state), but probably outside the time window studied here. The richphenomenology observed in our experiments and dramatic differenceswith the cases studied in the literature calls for a deeper understanding of this dynamical regime, both theoretically and experimentally.
Our studies can be extended in many different directions: the role ofintegrability, which the XXZ Hamiltonian possesses, should be explored,
for example by adding next-nearest-neighbour integrability-breaking
terms realized through appropriate Rydberg dressing of atoms. We can
explore different initial conditions, including single domain walls43–46
and finite temperatures (by using partially polarized (mixed) states)and study the decay of transverse spin via transport and dephasing.
An interesting question is whether the power-law scalings change forvery large wavelengths that approach the continuum limit (see preliminary theoretical analysis in Methods and Extended Data Fig. 8).
We can furthermore realize Heisenberg models in two or three spatial
dimensions, or with purely ferromagnetic couplings by changing thesign of Jxy using a constant force to tilt the lattice50.

Online content
Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information,
acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code
availability are available at https://doi.org/10.1038/s41586-020-3033-y.

a

,

1.0

b

,,

c

Contrast, c(t)

0.8

0.6

O = 23.5a

0.4

0.2

0

O = 6.3a

Δ≈0
0

10

20
Time, t (× /Jxy)

Δ≈0
30

0

1

2
0
Rescaled time, t (×( /Jxy)/(O/a))1

Fig. 4 | Temporal crossover from ballistic to diffusive transport for negative
anisotropies Δ < 0. a, Spin-helix contrast c(t) for λ = 10.4a and Δ = −1.43 (filled
circles) measured at lattice depths 11ER (blue) and 13ER (yellow). A piecewise fit(solid line) is linear at short times (I) and exponential at longer times (II) with acrossover at t = t0 (vertical dotted line). For t < t0 the decay coincides well with the
non-interacting case Δ ≈ 0 (open circles and dashed line). b, Decay curves fordifferent wavelengths λ = 23.5a, 18.8a, 13.4a, 10.4a, 8.5a and 6.3a (average ofmeasurements at lattice depths 11ER and 13ER) collapse into a single curve atearly times if time is rescaled by λ (as for ballistic behaviour) and follow the
non-interacting case (thick dashed line). At later times, the decay is diffusivewith different scaling (see Fig. 3a). c, Numerical simulations for Δ = −1.5 and thesame wavelengths λ as in b show similar behaviour, although the range of timeprobed is more limited. Simulations could not be extended to longer times,
owing to the exponential increase in computation time.

1.

28. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in opticallattices. Phys. Rev. Lett. 81, 3108–3111 (1998).
29. Kuklov, A. B. & Svistunov, B. V. Counterflow superfluidity of two-species ultracold atoms
in a commensurate optical lattice. Phys. Rev. Lett. 90, 100401 (2003).
30. Duan, L.-M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold
atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).
31. García-Ripoll, J. J. & Cirac, J. I. Spin dynamics for bosons in an optical lattice. New J. Phys.
5, 76 (2003).
32. Altman, E., Hofstetter, W., Demler, E. & Lukin, M. D. Phase diagram of two-component
bosons on an optical lattice. New J. Phys. 5, 113 (2003).
33. Amato-Grill, J., Jepsen, N., Dimitrova, I., Lunden, W. & Ketterle, W. Interaction
spectroscopy of a two-component Mott insulator. Phys. Rev. A 99, 033612 (2019).
34. Jordan, P. & Wigner, E. Über das Paulische Äquivalenzverbot. Z. Phys. 47, 631–651(1928).
35. Ioffe, A. F. & Regel, A. R. Non-crystalline, amorphous and liquid electronic
semiconductors. Prog. Semiconduct. 4, 237–291 (1960).
36. Mott, N. F. Conduction in non-crystalline systems IX. the minimum metallic conductivity.
Philos. Mag. A 26, 1015–1026 (1972).
37. Brown, P. T. et al. Bad metallic transport in a cold atom Fermi–Hubbard system. Science
363, 379–382 (2019).
38. Dubkov, A. A., Spagnolo, B. & Uchaikin, V. V. Lévy flight superdiffusion: an introduction.
Int. J. Bifurc. Chaos 18, 2649–2672 (2008).
39. Andreev, A. È. et al. Correlation theory of processes with stationary random increments oforder n. Am. Math. Soc. Transl. 8, 87 (1958).
40. Agarwal, K., Gopalakrishnan, S., Knap, M., Müller, M. & Demler, E. Anomalous diffusion and
Griffiths effects near the many-body localization transition. Phys. Rev. Lett. 114, 160401 (2015).
41. Vosk, R., Huse, D. A. & Altman, E. Theory of the many-body localization transition in
one-dimensional systems. Phys. Rev. X 5, 031032 (2015).
42. Guardado-Sanchez, E. et al. Subdiffusion and heat transport in a tilted two-dimensional
Fermi–Hubbard system. Phys. Rev. X 10, 011042 (2020).
43. Antal, T., Rácz, Z., Rákos, A. & Schütz, G. M. Transport in the XX chain at zero temperature:
emergence of flat magnetization profiles. Phys. Rev. E 59, 4912–4918 (1999).
1
44. Gobert, D., Kollath, C., Schollwöck, U. & Schütz, G. Real-time dynamics in spin- chains2with adaptive time-dependent density matrix renormalization group. Phys. Rev. E 71,
036102 (2005).
45. Misguich, G., Pavloff, N. & Pasquier, V. Domain wall problem in the quantum XXZ chain
and semiclassical behavior close to the isotropic point. SciPost Phys. 7, 025 (2019).
46. Ljubotina, M., Žnidarič, M. & Prosen, T. A class of states supporting diffusive spin
dynamics in the isotropic Heisenberg model. J. Phys. A Math. Theor. 50, 475002(2017).
47. Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in integrable
quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016).
48. Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-of-equilibrium XXZ
chains: exact profiles of charges and currents. Phys. Rev. Lett. 117, 207201 (2016).
49. Caux, J.-S. & Mossel, J. Remarks on the notion of quantum integrability. J. Stat. Mech.
2011, P02023 (2011).
50. Dimitrova, I. et al. Enhanced superexchange in a tilted Mott insulator. Phys. Rev. Lett. 124,
043204 (2020).

2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.

Lee, P. A. From high temperature superconductivity to quantum spin liquid: progress instrong correlation physics. Rep. Prog. Phys. 71, 012501 (2007).
Broholm, C. et al. Quantum spin liquids. Science 367, eaay0668 (2020).
Nichols, M. A. et al. Spin transport in a Mott insulator of ultracold fermions. Science 363,
383–387 (2019).
Brown, R. C. et al. Two-dimensional superexchange-mediated magnetization dynamics inan optical lattice. Science 348, 540–544 (2015).
Fukuhara, T. et al. Quantum dynamics of a mobile spin impurity. Nat. Phys. 9, 235–241(2013).
Fukuhara, T. et al. Microscopic observation of magnon bound states and their dynamics.
Nature 502, 76–79 (2013).
Hild, S. et al. Far-from-equilibrium spin transport in Heisenberg quantum magnets. Phys.
Rev. Lett. 113, 147205 (2014).
Koschorreck, M., Pertot, D., Vogt, E. & Köhl, M. Universal spin dynamics in
two-dimensional Fermi gases. Nat. Phys. 9, 405–409 (2013); corrigendum 10, 170 (2014).
Trotzky, S. et al. Observation of the Leggett–Rice effect in a unitary Fermi gas. Phys. Rev.
Lett. 114, 015301 (2015).
Palzer, S., Zipkes, C., Sias, C. & Köhl, M. Quantum transport through a Tonks–Girardeaugas. Phys. Rev. Lett. 103, 150601 (2009).
Sommer, A., Ku, M., Roati, G. & Zwierlein, M. W. Universal spin transport in a stronglyinteracting Fermi gas. Nature 472, 201–204 (2011).
Krinner, S. et al. Mapping out spin and particle conductances in a quantum point contact.
Proc. Natl. Acad. Sci. USA 113, 8144–8149 (2016).
Vasseur, R. & Moore, J. E. Nonequilibrium quantum dynamics and transport: from
integrability to many-body localization. J. Stat. Mech. 2016, 064010 (2016).
Bertini, B. et al. Finite-temperature transport in one-dimensional quantum lattice models.
Preprint at https://arxiv.org/abs/2003.03334 (2020).
Ljubotina, M., Žnidarič, M. & Prosen, T. Spin diffusion from an inhomogeneous quench in
an integrable system. Nat. Commun. 8, 16117 (2017).
Gopalakrishnan, S. & Vasseur, R. Kinetic theory of spin diffusion and superdiffusion in XXZ
spin chains. Phys. Rev. Lett. 122, 127202 (2019).
Ilievski, E., De Nardis, J., Medenjak, M. & Prosen, T. Superdiffusion in one-dimensional
quantum lattice models. Phys. Rev. Lett. 121, 230602 (2018).
Else, D. V., Monroe, C., Nayak, C. & Yao, N. Y. Discrete time crystals. Annu. Rev. Condens.
Matter Phys. 11, 467–499 (2020).
Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum
materials. Nat. Mater. 16, 1077–1088 (2017).
Langen, T., Gasenzer, T. & Schmiedmayer, J. Prethermalization and universal dynamics innear-integrable quantum systems. J. Stat. Mech. 2016, 064009 (2016).
Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit
quantum simulator. Nature 551, 601–604 (2017).
Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature
551, 579–584 (2017).
Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting
circuit. Nat. Commun. 6, 7654 (2015).
Davis, E. J. et al. Protecting spin coherence in a tunable Heisenberg model. Phys. Rev.
Lett. 125, 060402 (2020).
Signoles, A. et al. Glassy dynamics in a disordered Heisenberg quantum spin system.
Preprint at https://arxiv.org/abs/1909.11959 (2019).
Trotzky, S. et al. Time-resolved observation and control of superexchange interactions
with ultracold atoms in optical lattices. Science 319, 295–299 (2008).
Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science
357, 995–1001 (2017).

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.
© The Author(s), under exclusive licence to Springer Nature Limited 2020

Nature | Vol 588 | 17 December 2020 | 407

Article
MethodsVσσ ′ = gσσ ′ ∫ d3 r w ∗(r − δr)w ∗(r)w(r − δr)w(r),Determination of the Heisenberg parameters
We calibrate the lattice depth using amplitude modulation spectroscopy55. We record the excitation spectrum of a Bose–Einstein condensate in a 1D lattice when the depth of the lattice is modulated by 3%
providing the cloud-averaged lattice depth with a statistical uncertaintyof 0.2%. Owing to an asymmetric excitation profile, we estimate a systematic error of 1%.
The lowest and second-lowest hyperfine states of 7Li realize the|↓⟩ and |↑⟩ states. We use our previous measurements of U⇈ and U⇊(lattice-depth modulation) as well as measurements of U⇈ − U⇅ andU⇅ − U⇊ (interaction spectroscopy)33 to determine the three scattering
lengths a⇈, a⇅ and a⇊ (under inclusion of higher-band corrections) for
several magnetic fields B. The determined anisotropies Δ are shownin Fig. 1b (points). Hyperbolic fits to a⇈(B), a⇅(B) and a⇊(B) are used tointerpolate the values for the anisotropy (solid line). Extended Data
Fig. 1 shows a⇈, a⇅ and a⇊, as well as Jxy, Jz and Δ with (and without) corrections. A recent detailed theoretical analysis56 of the interaction
spectroscopy data also provided precise scattering lengths acrossseveral Feshbach resonances. However, this analysis slightly disagreed
with our lattice-depth modulation data in the range of magnetic fields
studied here, and therefore we relied on the experimental data.
The Gaussian intensity profile of the lattice beams (1/e2 radius of
125 μm) causes inhomogeneity of the lattice depth by up to 1.7% acrossa Mott insulator of 44 lattice sites in diameter. The 1.0% variation of U∼
across the atom cloud can be neglected. The tunnelling coefficient t
within the spin chains is defined by the lattice depth Vz along the chains.
Although it is constant within each spin chain, it varies among the chainsby up to 4.4%, which increases superexchange Jxy by up to 9.0%. The
curvature of the transverse lattice beams (with Vx, Vy = 35ER) causes
1
12
harmonic confinement 2 mωtrapz 2 = 2 κz 2 along the chains with
ωtrap = 2π × 770 Hz or curvature κ = h × 116.6 Hz a−2, where h is the Planck
constant. The curvature leads to a varying energy offset δ between
neighbouring lattice sites, which is largest at the ends z = ±22a of thelongest spin chains. It modifies the energy of the intermediate state in
the superexchange process U ↦ U ± δ (refs. 26,50) and therefore increases
the superexchange rate at the ends of the longest chains by 24% (Δ ≈ 0)
and 11% (Δ ≈ 1), with an average value over all atoms of 4.2% (Δ ≈ 0) and
2.0% (Δ ≈ 1), respectively.
The lattice depth calibration and the experimental determination of
the scattering length a⇅ lead to an uncertainty for the spin-exchange
times ħ/Jxy of about ±10%. The accuracy of the determined anisotropies
Δ is limited by the experimental determination of all scattering lengthsa⇈, a⇅, a⇊. The uncertainty of Δ is estimated to be about ±0.1.
In the experiment (see Fig. 1b), Δ ≈ 0 was realized by tuning the magnetic field to B0 = 882.63 G. Here the measured Hubbard parameters
result in Δ = −0.12 including higher-order corrections (and Δ = −0.02
without corrections). We find that the spin dynamics in this regime is
only weakly dependent on Δ, so we refer to measurements here as
Δ ≈ 0. B′0 = 842.95 G is a second magnetic field value, which also realizes
Δ ≈ 0 (Δ = −0.13 including corrections and Δ = 0.01 without corrections).
We directly compare these two points B0 and B′0 in Extended Data Fig. 5,
and observe quantitative agreement. Arbitrary anisotropies were realized by using the magnetic field region in between: B′0 < B < B0. In particular, for values Δ > 0 (Δ < 0) we used magnetic fields B < 850 G(B > 850 G). The isotropic point Δ ≈ 1 was realized at B1 = 847.30 G (actually Δ = 1.01 and corrections here are negligible).
In the following we give typical absolute values for Hubbard parameters at a lattice depth of Vz = 11ER and Vx, Vy = 35ER. The recoil energy
is ER/h = 25.12 kHz.

where δr is defined as before. One finds that Jz is modified by the addition of 2(V⇈ + V⇊ − 2V⇅). Depending on the signs and magnitudes of
the three interactions, the off-site terms can add to or subtract fromthe two previously discussed corrections to Jz (Extended Data Fig. 1).

XX model. Δ ≈ 0 is realized by the scattering lengths a⇈ = +307a0,a⇅ = −50a0 and a⇊ = −44a0, resulting in the following values for theon-site interactions U ⇈/h = +30.6 kHz, U ⇅/h = −5.8 kHz andU⇊/h = −5.1 kHz. The bare single-particle tunnelling coefficient isExtended Hubbard model
To determine the parameters Jxy and Jz in equation (1)Jxy = −

∼2
∼2  ∼2
∼2
4t
4t
4t
4t 

, Jz =−+U⇅  U⇊
U⇊ 
U⇅



we use measurements of the lattice depths Vx, Vy, Vz and of the threescattering lengths a⇈, a⇅, a⇊. From the calibrated lattice depths, theHubbard parameters t~(0) (in the non-interacting limit) and U(0) (in the
single-band approximation) are calculated as28 ħ 2 d2∼(0)t = − ∫ dzwz∗(z − a)  −+ Vz sin2 (kz) wz(z),2 2m dz

and
(0)
3
4
Uσσ
′ = gσσ ′ ∫ d r |w(r)| ,where gσσ′ = 4πħ2aσσ′/m and σ, σ′ = ↑, ↓ and w(r) = wx(x) wy(y) wz(z) using
the calculated lowest-band Wannier functions wx(x), wy(y) and wz(z)for each lattice depth51.
For the precision needed to compare experiment to theory, threecorrections are applied7,52–54.
Correction 1. Tunnelling is modified by the so-called single-band bondcharge54, which for single-occupancy is∼∼(0)tσσ ′ = t − gσσ ′ ∫ d3 r w ∗(r − δr)w ∗(r)w(r)w(r),where δr = (0, 0, a) is a displacement by one lattice constant a in thetunnelling direction. Through this correction, the tunnelling matrix∼ ∼∼elements t ⇈, t ⇅ and t⇊ are now slightly spin-dependent.
Correction 2. For the on-site interaction, we include admixtures ofhigher bands53,54. The dominant part is captured by a perturbative correction due to the first and second excited bands
2
Uσσ ′ = U (0)σσ ′ − g σσ ′1

∑ E ∫ d3r w∗n (r)w∗n (r)w(r)w(r),n1 , n2bg

12where n1 and n2 are the three-dimensional band indices of the two
atoms and Ebg is the sum of the bandgap energies. Corrections to the∼tunnelling rate t owing to population of higher bands is negligible fora Mott insulator with occupation n = 1, because higher bands are
admixed only through virtual doubly occupied sites. These modifica∼
tions of t and U both contribute to a modification of superexchange
J xy and J z (Extended Data Fig. 1). The relative correction to
∼
(0)
(0)
J (0)
= − 4(t (0)) 2 /U (0)⇅ , given by (Jxy − J xy )/J xy , is almost independent ofxylattice depth (for the range of lattice depths studied here), and in theexperiment Jxy is typically reduced by 10% to 15%. Note that the correction has the opposite sign as in ref. 7, because Jxy > 0 is antiferromagnetic in the present work.
Correction 3. To accurately determine Jz, one must also consider off-site
interactions of the form54

∼given by t (0)/h = 380 Hz , but interactions make tunnelling∼∼∼spin-dependent: t ⇈ /h = 480 Hz, t ⇅ /h = 370 Hz, and t⇊ /h = 370 Hz. That
results in the following Heisenberg parameters: Jxy/h = 93.3 Hz andJz/h = −10.9 Hz.
XXX model. Δ ≈ 1 is realized by the scattering lengths a⇈ = −107a0,a⇅ = −71a0 and a⇊ = −53a0, resulting in U⇈/h = −12.7 kHz, U⇅/h = −8.4 kHz∼(0)∼∼and U⇊/h = −6.1 kHz; and t /h = 380 Hz, t ⇈ /h = 350 Hz, t ⇅ /h = 360 Hz∼
and t⇊ /h = 370 Hz; and Jxy/h = 62.6 Hz and Jz/h = 63.5 Hz.
Superexchange coupling ( J ⊥xy /h ≈ 20 mHz) between chains is negligible, owing to the deep transverse optical lattices (Vx, Vy = 35ER) andis 3 to 4 orders of magnitude smaller than Jxy/h within the chains.

Experimental setup
In the experiment, we prepare 4.5 × 104 7Li atoms in an optical latticewith spacing a = 532 nm in the Mott insulating regime with one atom
per site50. A Bose–Einstein condensate (with barely detectable thermal
fraction Nth/N ≤ 0.05) with all atoms in the |↑⟩ state is loaded into theoptical lattice, with the scattering length set to strong repulsive interactions a⇈ = +307a0. This suppresses doubly occupied sites50, which wecounted directly by interaction spectroscopy33 to be ≤0.5% of the total
atom number. (Even a long spin chain of length L = 44a has then only aprobability ≤20% to contain a doubly occupied site.) The hole fraction isestimated to be between 5% and 10% through comparison of spin dynamics with theory (Fig. 2a). The density degree of freedom is frozen out
after loading into the deep optical lattice, and the on-site interactions
U⇈, U⇅ and U⇊ can then be varied freely without affecting the global atom
distribution, as long as the atoms stay in the Mott insulating regime50.
We then prepare a far-from-equilibrium initial spin state and probe
the spin dynamics in one dimension. The lattice beams in the x andy directions are kept at a large constant depth of Vx, Vy = 35ER separating
the atoms into an array of independent 1D chains, with a typical maximum length of Lmax = 44a (given by the diameter of the Mott insulator),
and with an average length of ⟨L⟩ = 33a (Extended Data Fig. 10). The
depth of 35ER is sufficient to prevent superexchange coupling in thex and y directions (ħ /J ⊥xy ≈ 10 s) on experimental timescales. Initially,
the z-lattice depth is also 35ER. The magnetic field is then ramped to
the value required for a desired anisotropy Δ. Using radio frequency
pulses and a magnetic field gradient, a helical spin pattern is created
where the spin component along the chain winds in the xz plane of the
Bloch sphere with a wavevector Q = 2π/λ, where λ is the wavelength of
the spin helix (see Fig. 1c and Methods section ‘Preparation of the spin
helix’). For λ smaller than the system size, the total magnetization ofthis state is close to zero.
The power of the lattice beam in the z direction controls the superexchange rate within the chains. Time evolution is initiated by ramping
down the z-lattice depth Vz to a value between 9ER and 13ER. The ramp
time is 0.5 ms, fast compared to superexchange ħ/Jxy, but slow compared∼to tunnelling ħ /t . The ensuing coherent dynamics along each chain isgoverned by a 1D Heisenberg XXZ model with anisotropy Δ, equation (1).
This is a quantum quench to a far-from-equilibrium initial state. Aftera variable evolution time t the dynamics is frozen by rapidly increasing
the lattice depth back to 35ER. The atoms are then imaged in the |↑⟩state via state-selective polarization-rotation imaging (see below).
Preparation of the spin helix
A global π/2 pulse of 75 μs rotates the spin |↑⟩i on each site i into thexy plane of the Bloch sphere|φ⟩i = [|↑⟩i + |↓⟩i ]/ 2. An applied magnetic
field gradient in the z direction causes spin precession at rates that
depend linearly on position zi of the spin thus creating a spin helix|φ⟩i = [|↑⟩i + e−iQzi |↓⟩i ]/ 2 where the spin winds in the xy plane7–9.
The strength and duration of the gradient determine the wavevector
Q = 2π/λ where λ is the wavelength of the spin helix. After turning
off the gradient, an additional π/2 pulse rotates the spin helix
into a state where the spin winding occurs in the xz plane|φ⟩i = cos(Qzi /2)|↓⟩i − sin(Qzi /2)|↑⟩i, so that the full many-body xz spinhelix state is |ψ(Q)⟩ = ∏i |φ⟩i . In practice, the phase of the winding, θ,varies from realization to realization, which amounts to replacing
Qzi ↦ Qzi + θ. This is caused by small magnetic bias field drifts on the
10−5 level. The range of λ used in the experiment was limited on
the short side by optical resolution to λ ≥ 5.6a and on the long side by
the length of the chains Lmax = 44a.
When turning off the magnetic field gradient, special care was taken
to cancel any residual gradients to better than |B′| ≤ 0.4 mG cm−1. The
differential magnetic moment is typically Δμ/h = (μ↑ − μ↓)/h ≈ 30 kHz G−1(it varies by approximately 10% depending on the bias field B). Thistranslates to a maximum energy difference of ΔμB′Lmax/h = 0.3 Hz acrossthe chain length, completely negligible compared to the spin-exchange
coupling Jxy/h, which is 2 to 3 orders of magnitude larger.

Imaging
The optical density of the atomic ensemble is too high (>14) to allow forin situ observation of the modulation of ⟨Sz⟩ via absorption imaging.
Instead, we use dispersive imaging, which uses the phase accumulated by the transmitted light to form an image of the atomic density
distribution. When light at frequency ωL is detuned from the atomic
resonance ω0 by many natural linewidths Γ, it picks up an approximate
phase θ ≈ −2Δω/Γ × OD(y, x, Δω), where OD is the optical density at
detuning Δω = ωL − ω0, while absorption is suppressed by a sufficiently
large detuning Δω. To form an image, the phase-shifted light must be
interfered with a reference beam. In this work, we make use of the factthat the optical transition we use for imaging is driven only by a single
polarization component; after passing through the atoms, the shifted
and unshifted components are combined on a polarizer. A judicious
choice of input and output polarizers yields an interference signal I on
the camera57,58 that is I = I0(1 − sinθ)/2.
The optical resolution of our imaging system (with a numerical aperture of NA ≈ 0.2) was determined to have a cut-off at modulation wavelength λ ≈ 3.0 μm = 5.6a (330 line pairs per mm). The reduction of
the modulation transfer function MTF(Q) near the cut-off reduces
the observed contrast C(t) = MTF(Q)c(t) compared to the real contrast
c(t). This does not affect the decay times τ. Assuming that the experimental preparation sequence for the initial spin-helix state achieves
full contrast c(0) = 1 for any wavevector Q (based on careful pulse calibration and characterization), we can use C(0) as a direct measurementof MTF(Q) and determine the real contrast as c(t) = C(t)/C(0).
Constant background contrast
For long evolution times t, the contrast c(t) does not fully decay, butreaches a background value c0, for example c0 = 0.08(1) in Fig. 2a. The
numerical simulations, however, show a decay to zero. Therefore we add
here the fit value for c0 to the simulations for better comparison withexperimental results. The experimental offset is caused by the inhomogeneous density of the atom cloud: only 90% of the atoms are in the
Mott insulator state, which realizes an array of 1D spin chains. A small
fraction of atoms are in dilute spatial wings, separated by holes that are
immobile, owing to the gradient of the trapping potential (which suppresses first-order tunnelling, as shown in our previous work)50. These
atoms preserve an imprinted spin-modulation pattern for long times.
We have checked this mechanism by increasing the amount of thermal
atoms and clearly observe an increase of the background c0 (Extended
Data Fig. 4). Furthermore, a position-sensitive measurement of thecontrast confirms that the main contribution is indeed from atoms
in the spatial wings (Extended Data Fig. 4f–g). In agreement with thismodel, numerical simulations always show a decay to zero for long spin
chains (see Extended Data Figs. 6, 7, 9, 10). Experiments throughout this
Article were performed with the lowest thermal fraction Nth/N ≤ 0.05,
where the contrast decays (almost) uniformly across the whole atom
cloud, and offsets are small, so that it was not necessary to restrict thefits to the central part of the cloud (see Extended Data Fig. 2).

Article
Time-reversal invariance of spin dynamics
For an xz spin-helix initial state and time evolution via the XXZ
Hamiltonian, the contrast is time-reversal symmetric: c(t) = c(−t), which
follows because the state, Hamiltonian and observable (the local magnetization Siz ) can be all expressed as real. This also implies invarianceagainst the overall sign of the Hamiltonian H ↦ −H. The same argument∼
holds for the system with holes evolving under the bosonic t –J model.
The initial dynamics of the contrast (in the ideal scenario) in both cases
is therefore quadratic, c(t) = 1 + Γ2t2 + …, with
Γ2=−

1 2a
∑ cos(Qzi + θ)⟨ψ(Q)|[H , [H , Siz ]]|ψ(Q)⟩.
2ħ L i

Therefore the timescale of the initial quadratic decay |Γ|−1 is the superexchange timescale ħ/Jxy (XXZ model) or the tunnelling timescale∼ ∼ħ /t (t –J model). The fact that, experimentally, we do not observe an
initial quadratic behaviour indicates either (i) the presence of holes, but∼that we are not resolving the fast timescale ħ /t ; or (ii) that the initial state
is not time-reversal invariant (that is, cannot be expressed as real, in thesame basis that the Hamiltonian is written in). The latter could arise frompulse imperfections, or the fact that the ramp-down of the optical lattice
takes place over a finite duration of time, leading to deviations from theideal initial state. Nevertheless, we expect that the overall behaviour ofthe decay of the contrast—for example, the scaling behaviour of dynamics with wavevector Q—is not strongly affected by (i) or (ii).

Power-law scalings in the continuum limit
In the experiments we measured the exponents α of the power lawsτ ∝ Q−α in the short-wavelength regime Q = 2π/(30a) to 2π/(6a) (Fig. 3).
Now we investigate how the exponents depend on the range of wavevectors Q, in particular in the continuum limit Q → 0 (where the wavelength
λ of the modulation becomes large compared to the lattice spacing aand the discreteness of the underlying lattice is no longer relevant).
The short-time (t ≪ ħ/J xy) Taylor expansion of the contrast
c(t) = 1 + Γ2t2 + … enables us to define a ‘time constant’ τ = |Γ|−1, which we
can compute analytically even in the thermodynamic limit L/a → ∞. Thisis because the object ⟨ψ(Q)|[H , [H , Siz ]]|ψ(Q)⟩ is a strictly local quantity
as the spin-helix state |ψ(Q)⟩ is a product state and the commutator of
the Hamiltonian with the local term Siz only produces terms near site i.
Evaluating (with θ = 0 for simplicity), we obtain
Γ2=

Jxy 1
2

ħ 16

[7 − Δ − 8 cos(Qa) + 2 cos(2Qa) + (Δ − 1)cos(3Qa)].

The limiting behaviour of τ as Q → 0 is therefore τ ≈ (Q 2a 2 /4)−1ħ /Jxy for
Δ = 1 and τ ≈ [(3 2 /8)Qa 1 − Δ ]−1ħ /Jxy for Δ < 1.
This implies a sharp crossover from ballistic scaling to diffusive
scaling as Δ approaches 1 from below in the regime of small enoughwavevectors, as shown in Extended Data Fig. 8d, e. In the same figure,
we show the exponents determined for a range of finite wavevectors(Q = 2π/(20a) to 2π/(6a)) as used in experiments and numerics, and finda smooth crossover from a superdiffusive regime to diffusive regime,in agreement with numerical simulations in Fig. 3.
Although this short-time analysis for t ≪ ħ/Jxy predicts the scalings
seen in the numerical simulations, it is not clear how much of thisanalysis is applicable to the intermediate to long times t at which theexperiments were performed. It also remains an open question as towhy numerics predict a superballistic exponent α < 1 for the XX model(Δ = 0), but the experiment measures a ballistic exponent α ≈ 1.
The discrepancy is possibly caused by the presence of holes, as sug∼gested by the t –J model simulation (Fig. 3c).

Numerical simulations
In the numerical simulations we consider: (i) a spin-helix quench under
XXZ Hamiltonian (equation (1)) dynamics, and (ii) a spin helix with∼
5% to 10% hole probability evolving under the bosonic t –J model (thatis, assuming no doubly occupied sites), given by
H = ∑ Jxy (Six S xj + Siy S yj ) + Jz Siz S zj  + Hd,


⟨ij ⟩

 t∼2
∼ †
†
nσj aσk
Hd = − ∑ t aσi
aσj − ∑  aσi
Uσ ,⟨ij ⟩σ ,⟨ijk ⟩ 
 ⇅
∼2
∼2

2t †
t † σ
+ aσi
a n a  + h.c.,
S j aσk +Uσσ σi σj σk U⇅where spin σ = ↑, ↓. Here aσi and aσi† are bosonic lowering and raising
operators at site i, respectively, S↑i ≡ Si+ is defined as a↑†i a↓i, and S↓i ≡ Si−
is defined as a↓†i a↑i . We use parameters from experiments and focus
on a lattice depth of 11ER, in which case we have U⇅/U⇈ = 1.206, 1.406,
1.401, 1.398, 1.397 and 1.392, U⇅/U⇊ = −0.188, 0.264, 0.459, 0.575,∼
0.659 and 0.862, and U⇅ /t = −17.94, −24.32, −24.17, −24.08, −24.05 and−23.94 for anisotropies Δ = 0.020, 0.670, 0.860, 0.973, 1.055 and 1.256,
respectively. In the absence of holes, the action of the term Hd vanishes,
and the Hamiltonian reduces to the XXZ Hamiltonian.
In both cases we use the real-time time-evolving block decimation(TEBD) method with matrix product states (MPS) on a spin chain with
open boundary conditions. In case (i): local Hilbert space dimension = 2,
length L = 40a and bond dimensions up to 800; case (ii): local Hilbert
space dimension = 3, L = 40a for Δ = 0, or L = 20a for all other Δ. Simulations are cut-off in simulated time, owing to the rapid increase inentanglement of the state, requiring ever-increasing computationaltimes. For the special case Δ = 0, without holes, we alternatively use
free fermionic methods, considerably speeding up the calculations. For
technical reasons, instead of simulating the full distribution of holes
we simply average over the situations in which there are either exactly
one or two (or four) holes in the chain. For L = 20a (40a), a single holecorresponds to an average of 5% (2.5%) holes.
To obtain the contrast, we determine the local magnetization⟨Siz (t)⟩ = [⟨ni ↑(t)⟩ − ⟨ni ↓(t)⟩]/2 and determine the Fourier component
4a
at wavevector Q via c(t) = L ∑i ⟨Siz (t)⟩cos(Qzi + θ) , assuming that theinitial spin helix has wavevector Q with a given phase θ (see Methods
section ‘Creation of the spin helix’); we then compare this to the experimental contrast. However, as noted in Methods section ‘Time-reversal
invariance of spin dynamics’, the numerically simulated contrast always
has an initial quadratic decay, unlike that seen in the experiments. We
therefore determine the decay timescales and its power-law scaling
with Q using one of the following methods. For positive anisotropies
Δ ≥ 0, we define the decay time constant as the time it takes for thecontrast c(t) to decay from 1 to 0.6, divided by −ln(0.6) to convert to a
‘1/e’ time (see Extended Data Fig. 6). For negative anisotropies Δ < 0,for regime I we take the time to decay from 1 to 0.9, multiplied by 10 toextrapolate to zero, whereas for regime II we fitted a simple exponentialdecay profile to obtain the decay timescales once the curves start ‘peeling off’ when plotted with time units rescaled by λ (see Fig. 4c andExtended Data Fig. 7). The resulting power-law scalings (Extended Data
Fig. 8) and resulting exponents (Fig. 3c) are in reasonable agreementwith the experimental results.

Finite-size effects
Numerical simulations can study the effect of different chain lengths
L and initial phases θ of the spin helix on the dynamics. This is relevant
because in the experiment, the atoms are distributed uniformly over asphere with a typical diameter of Lmax = 44a in three dimensions, leading to an ensemble of 1D chains of varying lengths. The experimentallymeasured contrast c(t) is an average over all chain lengths with a probability distribution shown in Extended Data Fig. 10. Furthermore, owing
to drifts of the applied magnetic field, the initial phase θ of the spin-helix
state varies from shot to shot. Here we study numerically both effects.

We concentrate on the XX model (Δ = 0) without holes. Extended
Data Fig. 9 shows that the strong dependence of the contrast c(t) on theinitial phase θ is due to reflection of magnetization off the boundariesof the chain. This suggests that averaging over various chain lengths
or averaging over initial phases should give similar results, which is
confirmed in Extended Data Fig. 10. It is even sufficient to average over
only two phases, θ = 0 and π/2, to achieve insensitivity to initial andboundary conditions. The distribution of chains, and the simulationof a magnetization profile averaged over this distribution of chains,
are illustrated in Extended Data Fig. 10b–d.
The conclusion is that the experiment is naturally performing an
average over different phases and different chain lengths, washing out
the sensitive dependence of the spin dynamics on initial conditions.
For comparison with simulations, it is sufficient to use a system witha fixed chain length L = 40a, and average over only the two phases
θ = 0 and π/2.

Data availability
The data that support the findings of this study are available from thecorresponding author upon reasonable request.

54. Lühmann, D.-S., Jürgensen, O. & Sengstock, K. Multi-orbital and density-induced
tunneling of bosons in optical lattices. New J. Phys. 14, 033021 (2012).
55. Stöferle, T., Moritz, H., Schori, C., Köhl, M. & Esslinger, T. Transition from a stronglyinteracting 1D superfluid to a Mott insulator. Phys. Rev. Lett. 92, 130403 (2004).
56. Secker, T., Amato-Grill, J., Ketterle, W. & Kokkelmans, S. High-precision analysis of
Feshbach resonances in a Mott insulator. Phys. Rev. A 101, 042703 (2020).
57. Bradley, C. C., Sackett, C. A. & Hulet, R. G. Bose–Einstein condensation of lithium:observation of limited condensate number. Phys. Rev. Lett. 78, 985–989 (1997).
58. Ketterle, W., Durfee, D. S. & Stamper-Kurn, D. M. Making, probing and understanding
Bose–Einstein condensates. In Bose–Einstein Condensation in Atomic Gases (eds
Inguscio, M. et al.) 67–176 (IOS Press, 1999).
Acknowledgements We thank M. D. Lukin, N. Y. Yao and M. Knap for discussions, Y. K. Lee forexperimental assistance, as well as C. M. E. Paus for sharing computing resources, and J. de
Hond for comments on the manuscript. We acknowledge support from the NSF through theCenter for Ultracold Atoms and grant number 1506369, NSF (grant number OAC-193471),
ARO-MURI Non-Equilibrium Many-Body Dynamics (grant number W911NF-14-1-0003),
AFOSR-MURI Photonic Quantum Matter (grant number FA9550-16-1-0323), AFOSR-MURI
Quantum Phases of Matter (grant number FA9550-14-1-0035), ONR (grant number N00014-171-2253), ARO (grant number W911NF-20-1-0163), the Vannevar Bush Faculty Fellowship, and
the Gordon and Betty Moore Foundation EPiQS Initiative (grant number GBMF4306).
Author contributions P.N.J., J.A.-G., I.D. and W.K. conceived the experiment. P.N.J., J.A.-G. and
I.D. developed the experimental setup and took the data. P.N.J. led the data analysis. W.W.H.
and E.D. developed the theoretical analysis. W.W.H. performed the numerical simulations.
All authors discussed the results and contributed to the writing of the manuscript.
Competing interests The authors declare no competing interests.

51.

Greiner, M. Ultracold Quantum Gases in Three-dimensional Optical Lattice Potentials.
PhD thesis, Ludwig-Maximilians-Universität München (2003).
52. Łącki, M., Delande, D. & Zakrzewski, J. Dynamics of cold bosons in optical lattices: effects
of higher Bloch bands. New J. Phys. 15, 013062 (2013).
53. Will, S. et al. Time-resolved observation of coherent multi-body interactions in quantum
phase revivals. Nature 465, 197–201 (2010).

Additional information
Correspondence and requests for materials should be addressed to P.N.J.
Peer review information Nature thanks Jean-Philippe Brantut, Tomaž Prosen and the other,anonymous, reviewer(s) for their contribution to the peer review of this work.
Reprints and permissions information is available at http://www.nature.com/reprints.

Article
Scattering lengths (a0)

a
0.6

0.4

200

b

0

-200

-1

Spin couplings, J/ћ (ms )Relative corrections

0.2

0

-0.2

1
0.5
0
-0.5
-1

Anisotropy, Jz/Jxy

2

-0.4

-0.6

c

d

1
0
-1
-2

-200

0
Scattering length (a0)200

Extended Data Fig. 1 | Determination of the Heisenberg parameters.
∼ ∼∼a, Corrections for tunnelling (t − t (0))/t (0) (green), on-site interactions(0)(0)(U −U )/U (blue), superexchange ( J − J(0))/J(0) (purple) and off-site interactions∼∼−2V/J(0) (orange), where t (0), U(0) and J (0) = 4(t (0)) 2 /U (0) are the uncorrected∼∼2
values and t , U and J = 4t /U include corrections, at a lattice depth of 13ER
(solid line) and 9ER (dotted line). b, As a function of magnetic field B we show the
scattering lengths a⇈ (blue), a⇅ (purple) and a⇊ (orange) measured in our

820

840

860

Magnetic field, B (G)880

previous work (points)33. Here a0 is the Bohr radius. c, d, We interpolate the datain b using hyperbolic fits (solid lines) and calculate values for the transverse
coupling constant J xy (purple), the longitudinal coupling constant J z (yellow)
and the anisotropy Δ = J z /J xy (black), without corrections (dashed line) andincluding corrections (solid line) for a lattice depth of 11ER. The excluded region(|a⇈| > 700a0) is around a Feshbach resonance in the |↑⟩ state near 845.4 G.

c

60

d

Position, y (a)-2bColumn density (a )

-20

a

0

30

0

-1

Linear density (a )

20
1,500

0
-20

0

20

-20

0

-20
20
Position, z (a)Extended Data Fig. 2 | Contrast measurement. a–d, The distribution of
atoms in the |↑⟩ state. Every pixel is a local measurement of the column density(number of atoms per unit area). The y and z axes are displayed in units of
the lattice spacing a = 0.532 μm. The images are projected (integrated) along
the y direction from y = −30a to +30a to obtain the linear density (number ofatoms per unit length). The resulting 1D distributions are fitted withf (z ) = g (z )[1 + C cos(Qz + θ )]/2 (solid line), where g(z) is a Gaussian envelope

0

20

-20

0

20

(dashed line), between z = ±54a. The data in a–d were measured at different
evolution times t = 0 (a), 2.3ħ/J xy (b), 6.3ħ/J xy (c) and 12.0ħ/J xy (d), for anisotropy
Δ ≈ 0 and wavelength λ = 10.4a. The obtained contrast C(t ) is shown in Fig. 2a.
In general, we also normalize by the initial contrast C(0) to correct for finite
optical imaging resolution. This is important for shorter wavelengths λ close to
the optical resolution of 3 μm, where the measured contrast C(t ) is reduced
compared to the real contrast c (t ) = C(t )/C(0).

Article

Extended Data Fig. 3 | Dispersion relations. For all positive anisotropies
Δ ≥ 0, the time evolution of the contrast c(t) shows a damped oscillatorycomponent, in addition to the overall exponential decay. For larger Δ, theoscillations become smaller. a–c, Decay and weak oscillation at the isotropic
point Δ ≈ 1 measured for different wavelengths λ, at three different lattice
depths 9ER (orange), 11ER (blue) and 13ER (yellow). Solid lines are fitsc(t) = [a0 + b0cos(ωt)]e−t/τ + c0 and dashed lines show the overall decaya0e−t/τ + c0, around which the oscillations take place. The oscillations become
more pronounced for short wavelengths λ, because the decay time (τ ∝ λ 2)
decreases with smaller wavelength more strongly than the oscillation period(T ∝ λ). d, The oscillation frequencies follow linear dispersion relations
ω(Q) = vQ shown for Δ = −0.12 (red), 0.35 (orange), 0.78 (yellow), 1.01 (blue)and 1.27 (light blue). e, The obtained velocities v decrease with increasing

anisotropy Δ. For Δ = 1.58 (open symbol), oscillations are small and themeasurement was limited to large values of Q, which precluded recording afull dispersion relation. We note that although the oscillations are difficult todiscern by eye (for example, in a), especially for large anisotropies Δ and small
wavevectors Q, the fitted oscillation frequencies ω all fall very well on linear
dispersion relations, which demonstrates that those barely visible oscillations
are real. The linear scaling ω(Q) = vQ persists even in the superdiffusive,
diffusive and subdiffusive regimes, where the power-law scaling of the decay
time constant τ ∝ Q−α is strongly nonlinear. This small ballistic (oscillatory)component may be related to our initial condition of a spin helix, which in the
mapping to lattice fermions is a 100% density modulation, which reduces
scattering at early times.

0.6

c
0.4

0.2

0

0

5

10

15

20

0.2
0.1

10

0.5

5

0

e

0.6

g 1

6

0.4

4

0.2

2

0

1

0

0

Oscillating fraction

Contrast, c(t)

0.8

f

15

Contrast

0.43
0.37
0.30
0.20
0.10
0.05

d
0.3

0

0.2

Time, t (ћ/Jxy)

Extended Data Fig. 4 | Effect of finite hole concentration. By varying thethermal fraction Nth/N of the Bose–Einstein condensate before it is loaded into
the optical lattice, we vary the energy and entropy of the atoms in the spin
chain, and therefore the concentration of holes. (For our conditions, doubly
occupied sites have higher energies than holes). Measurements are shown here
for Δ ≈ 0 and λ = 10.4a, at a lattice depth of 11ER. a, Decay curves c(t) for varying
hole concentrations ranging from low (blue) to high (orange) thermal fraction.
Solid lines are fits c(t) = [a0 + b0cos(ωt)]e−t/τ + c0. b, The background contrast c0
increases monotonously with thermal fraction Nth/N. A linear fit (solid line)
extrapolates to c0 = 0.01(2), consistent with zero, for Nth/N = 0. This suggeststhat all of the background contrast is due to hole excitations. c, Higher holeconcentrations suppress the oscillating fraction b0/(a0 + b0). d, Holes do notaffect the oscillation period T = 2π/ω. e, Holes decrease the decay time τ, albeit
slightly. b–e show that almost all of our measurements are not sensitive to asmall thermal fraction, which is usually Nth/N ≤ 0.05 throughout this work. The
behaviour shown in c and e is most probably caused by mobile holes in thecentral part of the Mott insulator. Indeed, numerical simulations of the∼t –J model reproduce such effects (Fig. 2a). Note though that for the isotropic

0.4

0Contrast

Nth/N =

Period, T (ћ/Jxy)1

Background, c0

b

a

0.5

0

Thermal fraction, Nth/N

0.2

0.4

0

0

10

20

30

Radius, r (a)case Δ ≈ 1, a previous work7 found a ~50% change in decay time when the holeconcentration changed from 0 to 5%. Our numerical simulations (Extended
Data Fig. 8b) do not show such strong sensitivity (for any anisotropy, even at
Δ = 1), possibly owing to asymmetry in the on-site interactions (U⇈ ≠ U⇅ ≠ U⇊) inour system. On the other hand, a finite background contrast (b) is probably
caused by immobile holes located in the outer parts of the atom distribution
where first-order tunnelling is suppressed by the gradient of the (harmonic)
trapping potential50. Immobile holes disrupt spin transport, and so we expectthat the imprinted spin modulation in these regions will not (or only very
slowly) decay. f, g, The region with immobile holes is visible as a shell of low
atomic density surrounding the Mott insulator in the in situ images for largehole concentration (f) and is absent for low hole concentration (g). The three
curves in both f and g show the local contrast as a function of distance r fromthe centre of the atom cloud for the evolution times t = 0 (top), 2.7ħ/J xy (middle)
and 21.7ħ/J xy (bottom). The two in situ images in both f and g are for t = 0 (top)and 21.7ħ/J x (bottom). The dashed lines indicate contours of constant radius,r = 30a (f) and r = 20a (g).

Article

Extended Data Fig. 5 | Decay behaviour as a function of anisotropy.
a, b, Decay behaviour ranging from negative (a) to positive (b) anisotropy, for a
fixed wavelength λ = 10.4a. Using Δ ≈ 0 as a reference point, we show how the
temporal profile of the decay curve c(t) changes when we introduce positive or
negative interactions. Every data point is an average of two measurements atlattice depths 11ER and 13ER. In a, from bottom to top: Δ = −0.12 (red),−0.59 (pink), −0.81 (yellow), −1.02 (blue), −1.43 (green) and −1.79 (purple). In b,from bottom to top: Δ = −0.13 (red), 0.08 (purple), 0.35 (pink), 0.55 (orange),
0.78 (yellow), 1.01 (blue), 1.27 (light blue) and 1.58 (green). Regardless of sign,for increasing |Δ| the decay always slows down and the revivals damp more
quickly. However, there is a big difference in how this slowdown happens:for increasing positive interactions Δ > 0, the initial rate of decay decreases
continuously (b); by contrast, for all negative interactions Δ < 0, the initial rateof decay stays constant (and is ballistic), coinciding with the Δ ≈ 0 case (a). It isonly after a critical time t0 that the decay suddenly starts slowing down (andbecomes diffusive) for times t > t0. This critical time t0 decreases withincreasing negative interaction strength |Δ|.

Extended Data Fig. 6 | Collapse of decay curves for positive anisotropies. All
decay curves c(t) for wavelengths λ = 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a,
7.2a and 6.7a collapse very well into a single curve for all evolution times t, whentime units are rescaled by λ α, where the exponent α is a function of anisotropy Δ,
both for experiment (points) and theory (solid lines). Experimental points were
measured for lattice depths 9ER (red), 11ER (blue) and 13ER (yellow). a, b, Ballistic
regime (α = 1). c, Superdiffusion (α = 1.5). d, Diffusion (α = 2). e, f, Subdiffusion(α = 2.5, 3 for experiment and α = 3.5, 4.5 for numerical simulations; inf, experiments covered a reduced range λ ≤ 10.4a). For all anisotropies Δ ≥ 0(a–f) the experimentally measured oscillation frequencies ω follow linear
dispersion relations (Extended Data Fig. 3) and have a scaling behaviour
different from the decay rates. However, such oscillations are small outside theballistic regime α ≈ 1, and therefore only lead to a small deviation from thecollapse behaviour. Note also the different timescales in experiments andsimulations for Δ > 1.

Article

Extended Data Fig. 7 | Collapse at short times for negative anisotropies. All
decay curves c(t) for different wavelengths λ collapse into a single curve at earlytimes, when time units are rescaled by λ (indicating ballistic behaviour). For
later times the decay is diffusive with different scaling. a–c, Theory (from top
to bottom: λ = 31.3a, 23.5a, 18.8a, 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a, 7.2a,

6.7a and 6.3a). The dotted lines are exponential fits e−t /τII to the diffusive
regime and the time constants τII are shown in Extended Data Fig. 8a.
d–f, Experiment (from top to bottom: λ = 18.8a, 13.4a, 10.4a, 8.5a, 7.2a and6.3a). Every data point is an average of two measurements at lattice depths 11ER
and 13ER. The black dashed line indicates the ballistic case Δ ≈ 0 (see Fig. 2c).

Extended Data Fig. 8 | Power-law scalings (theory) and diffusion
coefficients. a, b, Decay time constants τ for different anisotropies Δ ranging
from negative (a) to positive (b). Numerical results are shown in a for
Δ = −1 (blue), −1.5 (green) and −2 (purple) and in b for Δ = 0 (red), 0.5 (orange),
0.85 (yellow), 1 (blue) and 1.5 (green). Solid lines are power-law fits (to the filled
symbols). Open symbols are excluded from the fit owing to finite-size effects.
∼
Crossed symbols are results from t –J model simulations including 5% holefraction. Fitted power-law exponents are shown in Fig. 3c. For positive
anisotropies Δ ≥ 0 the decay time τ is defined as τ = τ′/ln(1/0.60) withc(τ′) = 0.60. For negative anisotropies Δ < 0, the decay time τI for short times (I)is defined as τI = 10τ I′ with c (τ I′) = 0.90. For longer times (II), the decay time τII is
obtained from exponential fits e−t /τII to the diffusive long-time tail (see dottedcurves in Extended Data Fig. 7a–c). c, Diffusion coefficients for the diffusive
long-time regime (II) obtained from theory (open symbols) and experiment(filled symbols). For negative anisotropies Δ < 0, values were determined from
quadratic power-law fits 1/τ = DQ2 to the data points in a (theory) and Fig. 3a(experiment) for the diffusive regime (II). Note that for Δ ≥ 0 the system is only
diffusive for Δ = +1, as shown in b (theory) and Fig. 3b (experiment). Fromthe experimental diffusion coefficients, we estimate mean free paths δx using
the velocity v = 0.76(1)vF from the ballistic short-time regime (I), and obtainδx = 3.35(15)a, 1.07(7)a and 0.66(4)a for Δ = −1.02, −1.43 and −1.79, respectively.
d, Short-time (t ≪ ħ/J xy) decay constant τ = |Γ|−1 obtained from Taylor expansionof the contrast c(t) = 1 + Γ2t 2 + … as a function of Q, for Δ = 0 (red), 0.55 (orange),
0.85 (yellow), 0.95 (purple) and 1 (blue). For Δ < 1 all curves in the log–log plot
asymptote to the same slope as Q → 0 (continuum limit), whereas there are
deviations for larger wavevectors Q. For Δ = 1 the slope is instead different. This
indicates that the power-law exponent α in τ ∝ Q −α depends on the range of
wavevectors Q used to determine it. e, Power-law exponents α determined forthe short-wavelength regime between λ = 6a and 20a (filled symbols) as inexperiments and numerics, and for the long-wavelength regime between
λ = 150a and 200a (open symbols) approaching the continuum limit. In theformer case, the exponents show a smooth crossover from superballistic todiffusive as Δ → 1 similar to that in the experiments and numerics, whereas in thelatter case the exponents show a sharp jump from ballistic to diffusive
occurring exactly at Δ = 1.

Article
a

b

1
0.8

1

1
+–
2

c

Lattice site index, i

10Contrast, c(t)

0.6
0.4
0.2
0

20

0

30

-0.2
0

10

20

30

40

0

10
20
Time, t (ћ/Jxy)

Extended Data Fig. 9 | Finite-size effects from the initial phase of the spin
helix. a, The time evolution of the contrast c(t) depends strongly on the initial
phase θ, illustrated here by simulations for Δ = 0 and λ = 10.4a. b, c, The
dynamics of the local magnetization ⟨Siz (t )⟩ for phases θ = 0 (b) and π/2 (c)

30

0

10

20

30

1
-–
2

reveals that this arises owing to the reflection of ballistically propagating
magnetization off the ends of the chain. Depending on the initial phase of thespin helix, the reflected magnetization interferes constructively or
destructively with the pattern of the bulk magnetization.

-20

0

0.5
0

-0.2
0

10
20
Time, t (ћ/Jxy)

30

c

60
40
20
0

3

Contrast, c(t)0.20Atom number (10 )Position, y (a)0.4Number of chains

b 20

a

-0.5

-20

0
Position, z (a)Extended Data Fig. 10 | Finite-size effects from the chain length. a, Contrast
c(t) obtained after a weighted average over all different chain lengths
between L = 0 and 44a (shown in b), for Δ = 0 and λ = 10.4a. The averaged
dynamics (orange, yellow, blue) shows almost no dependence on the phase θ,
in contrast to the dynamics determined from a single chain length L = 40a(Extended Data Fig. 9a). Also overlaid are the contrasts for a fixed chain length(L = 40a) averaged over all initial phases 0 ≤ θ < 2π (black solid line), andaveraged over only the two phases θ = 0 and π/2 (black dashed line). The close
agreement implies that averaging over either chain lengths or phases
suppresses the dependence on initial or boundary conditions. b, A cut through

20

d

3
2
1
0

0

10

20
30
Chain length, L (a)

40

Lmax

the spherical Mott insulator with diameter Lmax = 44a (as in the experiment)
illustrates the distribution of different chain lengths (oriented along thez direction). Averaging the local magnetization S z over the x and y directions
provides a 1D magnetization profile (bottom panel), which is an average over all
chains. c, The number of chains with length L is given by (π/2)(L/a). The total
number of chains is πLmax/(2a)2 ≈ 1,500. d, The number of atoms in chains withlength L is given by (π/2)(L/a)2. The contribution of each chain to the imaging
signal is proportional to the atom number in the chain, and so the relevant
average over chain lengths is weighted by the atom number and is⟨L⟩ = (3/4)Lmax = 33a.

136

Appendix B
Transverse spin dynamics in theanisotropic Heisenberg model
realized with ultracold atoms
P. N. Jepsen, W. W. Ho, J. Amato-Grill, I. Dimitrova, E. Demler, W. Ketterle
Transverse Spin Dynamics in the Anisotropic Heisenberg Model Realized with Ultracold Atoms
Phys. Rev. X 11, 041054 (2021)

137

PHYSICAL REVIEW X 11, 041054 (2021)

Transverse Spin Dynamics in the Anisotropic Heisenberg Model Realized with Ultracold Atoms
Paul Niklas Jepsen ,1,2,3,* Wen Wei Ho ,3,4,5 Jesse Amato-Grill,1,2,3,6 Ivana Dimitrova ,1,2,3,4
Eugene Demler,3,4,7 and Wolfgang Ketterle 1,2,3
1

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
3
MIT-Harvard Center for Ultracold Atoms, Cambridge, Massachusetts 02139, USA
4
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
5
Department of Physics, Stanford University, Stanford, California 94305, USA
6
QuEra Computing Inc., Boston, Massachusetts 02135, USA
7
Institute for Theoretical Physics, Wolfgang-Pauli-Strasse 27, ETH Zurich, 8093 Zurich, Switzerland(Received 14 March 2021; accepted 23 August 2021; published 17 December 2021)

In Heisenberg models with exchange anisotropy, transverse spin components are not conserved and can
decay not only by transport, but also by dephasing. Here, we utilize ultracold atoms to simulate the dynamics
of 1D Heisenberg spin chains and observe fast, local spin decay controlled by the anisotropy. However, evenfor isotropic interactions, we observe dephasing due to a new effect: an effective magnetic field created bysuperexchange. If spatially uniform, it leads only to uniform spin precession and is, therefore, typicallyignored. However, we show through experimental studies and extensive numerical simulations how thissuperexchange-generated field is relevant and leads to additional dephasing mechanisms over the exchange
anisotropy: There is dephasing due to (i) inhomogeneity of the effective field from variations of lattice depth
between chains; (ii) a twofold reduction of the field at the edges of finite chains; and (iii) fluctuations of theeffective field due to the presence of mobile holes in the system. The latter is a new coupling mechanism
between holes and magnons. All these dephasing mechanisms have not been observed before with ultracold
atoms and illustrate basic properties of the underlying Hubbard model.
DOI: 10.1103/PhysRevX.11.041054

Subject Areas: Atomic and Molecular Physics
Condensed Matter Physics, Magnetism

I. INTRODUCTION
The famous Heisenberg Hamiltonian, also called theHeisenberg–Dirac–van Vleck Hamiltonian [1–3], describes
localized particles on a lattice interacting via spin-exchange
couplings. Despite its apparent simplicity, it serves as aparadigmatic model for a host of emergent phenomena, suchas ferromagnetism (due to Coulomb exchange, also called
potential or direct exchange), antiferromagnetism (due tokinetic exchange from tunneling, also called superexchange)[4], and spin-glass physics [5], as well as exotic states ofmatter like topologically ordered quantum spin liquids [6].
The dynamics of such models is also very rich and multifaceted and is under active, intense investigation. For
example, in one dimension, Heisenberg spin models (withspin quantum number S ¼ 1=2) have the special property of
being integrable, whereby stable quasiparticles exist at all*

jepsen@mit.edu

Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI.

2160-3308=21=11(4)=041054(18)temperatures. This gives rise to a breakdown of simple
hydrodynamics with accompanying varied spin transport
behaviors [7–11]. Understanding this has led to the recent
development of a theory of generalized hydrodynamics[12,13]. In higher dimensions, the interplay of spontaneous
symmetry breaking can lead to long-lived, metastable,
prethermal states in addition to the onset of regular spin
diffusion [14–16] or even turbulent relaxation with universal
scaling of spin-spin correlations [17].
Ultracold atoms in optical lattices form an ideal platform torealize Heisenberg spin models and probe their dynamics in acontrolled fashion [18]. In deep lattices where atoms are
localized and Mott insulators form [19], superexchange
processes via second-order tunneling yield effective
Heisenberg spin models, with potential tunability of thestrength, sign, and anisotropy of the spin-exchange interactions [20–23]. Until very recently, all experimental studies
addressed the special case of an isotropic Heisenberg model[24–31]. However, in Ref. [32], we show how to overcome
this limitation and implement Heisenberg models withtunable anisotropy of the nearest-neighbor spin-spin couplings, by using 7Li and varying the interactions throughFeshbach resonances. We are able to show that the anisotropy
profoundly changes the nature of transport of longitudinal041054-1

Published by the American Physical Society

PAUL NIKLAS JEPSEN et al.

PHYS. REV. X 11, 041054 (2021)(a) SzλSx(b) Sz
z
Sx

a

z

z

x
ImagingyFIG. 1. Geometry of the experiment. The initial state is atransverse (a) or longitudinal (b) spin helix where the spin vector
winds within the Sx -Sy plane (a) or Sz -Sx plane (b). The
transverse helix (a) is a pure phase modulation of spin j↑i andj↓i states, whereas the longitudinal helix (b) also involves
population modulation. Deep optical lattices along the x and y
directions create an array of independent spin chains. The z lattice
is shallower and controls spin dynamics along each chain.

spin components after a quantum quench from a so-calledlongitudinal spin helix [Fig. 1(b)] and observe ballistic,
subdiffusive, diffusive, and superdiffusive behavior in different parameter regimes. These results bear some similarities
with those of spin transport close to equilibrium but strikingly differ in other aspects, prompting the need for further
theoretical investigation.
In this paper, we study the relaxation of transverse spincomponents after quantum quenches from a transverse spin
helix [Fig. 1(a)] and observe even more dramatic effects of
the anisotropy. In the classical limit, any transverse spin
helix for any anisotropy is stationary, since the torques
exerted on a given spin by its neighboring spins cancel
exactly (see the Appendix C 1). Therefore, what we study
here are the effects of quantum fluctuations on theirstability. In contrast to longitudinal spin patterns, which
can decay only by transport, transverse spin components
can decay also by dephasing. We focus here on twoparadigmatic models, which represent complementary spin
physics: the XX model, which has only transverse spin-spin
couplings and can be mapped to a noninteracting system offermions, and the XXX model, which has isotropic spin
couplings. For the XX model, we observe and explain thatthe decay is faster for spin-helix patterns with longer

wavelengths, in strong contrast to spin transport, where
slower dynamics occurs for longer modulations. We also
identify several dephasing mechanisms not discussed
before. For the XXX model, we identify a symmetrybreaking term in the Bose-Hubbard model: an effective
magnetic field caused by different scattering lengths for the
spin j↑i and spin j↓i states. This superexchange-induced
effective magnetic field is often ignored, since a spatially
uniform field can be eliminated in the bulk by going into an
appropriate corotating frame. Here, we show that thepresence of the effective field is actually significant andgives rise to three additional dephasing mechanisms, all ofwhich more or less contribute equally, resulting in drastically different decay behavior of spin-helix patterns withdifferent orientations, i.e., transverse or longitudinal. (Bothwould have naïvely been expected to decay with identicaltimescales for isotropic spin interactions.) One is aninhomogeneous effect where the effective magnetic field
is nonuniform between different chains in our sample. Thiscan be eliminated with a spin-echo technique. The second isdue to dephasing occurring at the ends of finite chains. The
third is due to the presence of mobile holes resulting in afluctuating effective magnetic field in the bulk, i.e., a holemagnon coupling.
Our work shows the limitations of a pure spin model in
capturing spin dynamics realized with ultracold atoms anddemonstrates the need for a theoretical model explicitly
featuring hole-magnon couplings (the so-called bosonic t̃-Jmodel) in order to reach a more complete description ofexperiments. The new insight into hole-magnon couplingshould be important for other systems and materials wheresuch couplings are present, such as high-temperature
superconductors [33–35].
II. EXPERIMENTAL METHODS
The spin models are implemented with a system of twocomponent bosons in an optical lattice, which is welldescribed by the Bose-Hubbard model. These two states(lowest and second-lowest hyperfine states of 7Li), labeledj↓i and j↑i, form a spin-1=2 system. In the idealized scenario
of a Mott insulating regime at unity filling, bosons cannot
tunnel, and the effective Hamiltonian for the remaining spin
degree of freedom is given by the spin-1=2 Heisenberg XXZ
model [20–23]:

X
h
H¼
Jxy ðSxi Sxj þ Syi Syj Þ þ Jz Szi Szj − z ðSzi þ Szj Þ ; ð1Þ
2
hiji
where Sx , Sy , and Sz are the spin-1=2 Pauli operatorsand the sum is over nearest-neighbor pairs of sites hiji. In
leading order, one obtains for the transverse coupling
Jxy ¼ −4t̃2 =U↑↓ and for the longitudinal couplingJz ¼ 4t̃2 =U↑↓ − ð4t̃2 =U↑↑ þ 4t̃2 =U↓↓ Þ, both mediated bysuperexchange. Here, t̃ is the tunneling matrix element
between neighboring sites, while U ↑↑ , U ↑↓ , and U↓↓ are

041054-2

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …

PHYS. REV. X 11, 041054 (2021)

the on-site interaction energies. The effective magnetic fieldstrength is hz P
¼ 4t̃2 =U↑↑ − 4t̃2 =U↓↓ . Note that the totalmagnetization i Szi in the Sz direction is conserved by the
Hamiltonian.
The magnitude of superexchange can be varied over 2
orders of magnitude by changing the lattice depth, whichscales the entire Hamiltonian. The anisotropy Δ ≔ Jz =J xy
is controlled via an applied magnetic field which tunes theinteractions through Feshbach resonances in the lowest two
hyperfine states. In the regime studied here, the transverse
coupling is positive (Jxy > 0). The ability to tune theanisotropy over a wide range of values, both positive
and negative, allows us to explore dynamics beyond
previous experiments [24–28] in which Δ ≈ 1.
One-dimensional (1D) chains are created by two
perpendicular optical lattices whose depths V x ,V y ¼ 35ER
are sufficient to prevent superexchange coupling on experimental timescales. A third orthogonal lattice along the z
direction with adjustable depth V z controls the superexchange rate in the chains (Fig. 1). Here, ER ¼ h2 =ð8ma2 Þdenotes the recoil energy, where a ¼ 0.532 μm is the latticespacing, m the atomic mass, and h Planck’s constant. After
preparing a transverse spin helix [Fig. 1(a); this work] or alongitudinal spin helix [Fig. 1(b); as in our previous work
[32]] with wavelength λ (and wave vector Q ¼ 2π=λ) in eachchain [27,36,37], time evolution is initiated by rapidly
lowering V z . The dynamics following this quench is then(approximately) governed by the 1D XXZ model Eq. (1).
After an evolution time t, the dynamics is frozen by rapidly
increasing V z , and the atoms are imaged in the j↑i state via
state-selective polarization-rotation imaging with an opticalresolution of about six lattice sites. For imaging the transverse spin, we apply a π=2 pulse first, so that we observe themagnetization in the Sx direction. To distinguish homogeneous from inhomogeneous dephasing, we can use a spinecho by applying a π pulse (with a typical duration oftπ ¼ 150 μs ≪ ℏ=J xy ) after half of the evolution time t.
Integrating the images along the direction perpendicular
to the chains yields a 1D spatial profile of the population in
the j↑i state, averaged over all spin chains. As in Fig. 1, thespin helix exhibits a sinusoidal spatial modulation of thedensity of j↑i atoms, observed as a characteristic stripe
pattern with a normalized contrast cðtÞ. During the evolution time t, the 100% contrast of the initial spin helix
decays, and we determine the dependence of cðtÞ on lattice
depth V z , wave vector Q, and anisotropy Δ. Analyzing eachdecay curve cðtÞ yields a decay rate γ, among other fit
parameters (see Fig. 6 in Appendix A for data fitting
methods).
In general, we measure the spin dynamics at two or moredifferent lattice depths V z and verify that the decay curvescðtÞ collapse when time is rescaled by the corresponding
spin-exchange time ℏ=J xy , confirming that the dynamics isdriven by superexchange. These time units are obtained
from the experimentally determined lattice depth using an
extended Hubbard model (detailed in Ref. [32]).

III. RESULTS
A. XX model
We first consider a very anisotropic system by realizing
the Heisenberg model tuned to the noninteracting point(Δ ¼ 0) and study the decay of the transverse spin helix for
different wave vectors Q [Figs. 2(a) and 2(b)]. We find the
decays are quick, all having timescales on the order of a few
spin-exchange times ℏ=J xy , much faster than the decay ofthe longitudinal spin helix which is driven by ballistictransport. Importantly, the transverse decay rate evenincreases for longer wavelengths of the helix, showingthat the decay is not caused by transport where magnetization redistributes in space and equilibrates, for which wewould expect slower rates for longer modulations, but bydephasing where magnetization is locally created or
removed.
Some insight into the fast timescales of transverse decay
is obtained by taking the Q → 0 limit, where the initial state
becomes a uniformly polarized state. This state is obviously
not an eigenstate of the quantum XX model and is,therefore, unstable. Further simplification to a two-site(double-well) system allows us to analytically diagonalize
the Heisenberg Hamiltonian, which gives a level structure
as shown in Fig. 2(c) (for hz ¼ 0). When Δ ¼ 1, thetransverse spin state j →→i ≔ ðj↑i þ j↓iÞðj↑i þ j↓iÞ=2 isan eigenstate of the Hamiltonian as all triplet states aredegenerate and, hence, does not evolve. However, forΔ ≠ 1, the degeneracy is lifted, and the state j →→i showsa beat note at the frequency of the energy splitting
Jxy ð1 − ΔÞ=2. For Δ ¼ 0, this indicates a dephasing time
for transverse spins on the order of a few spin-exchange
times ℏ=J xy , in qualitative agreement with our observations. For many sites, there will be a spectrum of beatfrequencies leading to irreversible dephasing locally.
We can explain the unusual Q dependence of the experimentally observed transverse decay with a semiclassicalanalysis of spin dynamics (see Appendices C 1 and C 2). In
the classical limit, spin-helix states satisfy the LandauLifshitz equations of motion ∂ t S⃗ i ðtÞ¼ð∂H=∂ S⃗ i Þ× S⃗ i ðtÞ¼0
for any wave vector Q and, therefore, do not decay (this is, infact, true for any anisotropy Δ), because the torques exertedon S⃗ i by its neighbors cancel exactly. Here, S⃗ i is a classicalspin vector, which corresponds to the S → ∞ limit of aquantum mechanical spin. For finite S, we can study theeffects of quantum fluctuations with a large spin (1=S)
expansion. We find that the Fourier modes of the fluctuations carrying momentum k have a dispersion relation
ωk ∝ Jxy j cosðQaÞ sinðka=2Þj. As the characteristic energy
scales of all modes are proportional to j cosðQaÞj, thisindicates, in a somewhat surprising fashion, that the dynamics of the spin helix is faster for longer wavelengths thanfor smaller wavelengths. The slowest dynamics occurs at
Qa ¼ π=2 (or λ ¼ 4a), where neighboring spins are at anangle of 90°. This state is robust (at least to leading order)

041054-3

(c)

0

Δ=1

(d)

Δ

1

0.8
0.6

2

Spin echo

Jxy /2Q = 0.032

0

0.6

1
+–
2

Q = 0.160

0.4

0

4
Time t [ /Jxy]

6

8

(b)
0.3

0.4
0.2

No echo

0
0

10

20
Time t [ /Jxy]

30

0

(e)
Jxy /2

Transverse

0.3

Transverse helix

Jxy

1
-–
2

0.2

0.2

0.1

0.1
Longitudinal

0

1
0.8

0.2

Decay rate γ [Jxy / ]

Δ=0

0

z

0.05
0.1
0.15
0.2
Wave vector Q [2π/a]

z

x

x

y

y

H = Jz S1S2 + Jxy (S1S2 + S1S2)

Theory0.25

Decay rate γ [Jxy / ]Δ

Energy [J xy ]Contrast c(t)(a) 1PHYS. REV. X 11, 041054 (2021)

Contrast c(t)PAUL NIKLAS JEPSEN et al.

Longitudinal helix

0

0.05
0.1
Wave vector Q [2 π/a]

0.15

0

FIG. 2. Spin dephasing and spin transport for the XX model [Δ ≈ 0, (a),(b)] and the isotropic XXX model [Δ ≈ 1, (d),(e)]. (a) Spinrelaxation at Δ ≈ 0. Transverse spin-helix contrast cðtÞ for Qa ¼ 2π × 0.032, 0.138, and 0.160 (bright to dark orange). The curves fordifferent lattice depths 11ER (∘) and 13ER (□) collapse when times are rescaled in units of the corresponding spin-exchange timesℏ=J xy ¼ 1.71 (∘) and 4.30 ms (□). The transverse spin decays on a timescale of a few spin-exchange times which increases for smaller
wavelengths λ ¼ 2π=Q. (b) Wave-vector dependence. The initial decay rates [orange points; with values from (a) highlighted] follow acosine dependence γðQÞ ¼ γ 1 cosðQaÞ þ γ 0 (solid orange line) with a constant background rate γ 0 ¼ 0.20ð2ÞJ xy =ℏ (dashed orangeline). This is in strong contrast to the longitudinal spin helix (purple) which shows linear scaling with Q (indicating ballistic transport).
A spin echo (π pulse at time t=2) reduces the background rate to γ 0 ¼ 0.13ð3ÞJ xy =ℏ (dashed blue line). The black solid squares are
numerical results for a single chain and the defect-free XX model (with hz ¼ 0) with a fit γðQÞ ¼ γ 1 cosðQaÞ (solid line) to the points
with Qa ≤ 2π × 0.075. The gray open squares are numerical results for the t̃-J model with 5% hole fraction (see Appendix E 1).
(c) Energy levels of the Heisenberg Hamiltonian for two spins in a double-well potential. For Δ ≠ 1, the triplet states are split. (d) Spinrelaxation at Δ ≈ 1. Transverse spin-helix contrast cðtÞ for Qa ¼ 2π × 0.056 without spin echo (orange) and with spin echo (blue). The
difference shows the presence of inhomogeneous dephasing. The curves taken for different lattice depths 11ER (∘) and 13ER (□)
collapse when times are rescaled in units of the corresponding spin-exchange times ℏ=J xy ¼ 2.55 (∘) and 6.42 ms (□). (e) Wave-vector
dependence. Decay rates for the transverse helix [orange and blue points; with values from (d) highlighted] and longitudinal helix(purple points), for Δ ≈ 1. The orange (no echo) and blue (with echo) solid lines are fits γðQÞ ¼ DQ2 þ γ 0 assuming two contributions
to the decay rate: one quadratic term (indicating diffusive transport) with diffusion constant D (taken from the longitudinal spin
dynamics shown in purple), the other Q-independent γ 0 (shown by the dashed lines). The spin echo reduces the background decay rate
γ 0 by an amount of 0.036J xy =ℏ (e) and 0.069J xy =ℏ (b). These values are consistent with an inhomogeneous dephasing rate proportional
to the effective magnetic field hz , with (predicted) values hz ¼ 0.89J xy (e) and 1.43J xy (b).

against decay induced by quantum fluctuations, because thefluctuations of a spin which points, e.g., in the Sx direction
are in the Sy -Sz plane and, for Δ ¼ 0, do not cause any
precession of their immediate neighboring spins (which
point along the Sy direction).
This prediction is furthermore corroborated by a fully
quantum (S ¼ 1=2) but short-time expansion of the orderparameter of the spin helix (see Appendix C 3). Numericalsimulations, as seen in Figs. 2(b) and 8(a), also verify thisby showing a very good collapse of the decay curves of all
experimentally considered wave vectors Q upon rescaling
time by a factor of cosðQaÞ. This holds even up to

evolution times t longer than would be expected to bevalid for the semiclassical analysis or short-time expansion.
Unsurprisingly, deviations from this relation are seen as thewave vector approaches Q ¼ π=ð2aÞ, for which the simple
approaches would predict a vanishing decay rate. However,note that the transverse spin helix with Q ¼ π=ð2aÞ can beshown to be an exact eigenstate of the XX model withappropriate boundary conditions [38,39].
Experimentally, we find that the decay rate of thetransverse helix as a function of wave vector Q can be
fitted very well as the sum of the predicted cosðQaÞ
dependence together with a constant term, as shown in

041054-4

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …
Fig. 2(b). The constant term represents additional dephasing
mechanisms that go beyond the idealizations of the spinmodel (1), which we discuss below.
B. XXX model
When Δ ¼ 1, we realize the isotropic Heisenberg spinmodel which,Paside from the effective magnetic field hz ,satisfies ½H; i Sαi  ¼ 0 (α ¼ x, y,Pz). The presence of theeffective magnetic field termðhz =2ÞðSzi þ Szj Þ inhijiPz
Eq. (1), which we can rewrite asi hz;i Si , explicitly
breaks this spin-rotational symmetry. Now, a uniform(i.e., site-i-independent) field hz;i can be transformed away
by going into an appropriate rotating frame. In such a case,the transverse and the longitudinal spin helix should show
exactly the same dynamical behavior.
However, as the results in Figs. 2(d) and 2(e) show, there isa dramatic difference: The transverse helix decays much
faster than the longitudinal helix with the same wave vector
Q. In particular, the longitudinal helix exhibits a purely
diffusive scaling with wave vector Q; i.e., its decay rateobeys γðQÞ ¼ DQ2 , where D is a diffusion constant(as shown in Ref. [32]), whereas the transverse helix has
an additional Q-independent decay rate of γ 0 ¼
0.096ð10ÞJxy =ℏ, with the net decay rate fitted well byγðQÞ ¼ DQ2 þ γ 0 . This lends naturally to the interpretationthat the system at this point realizes dynamics close, but notequal, to a model where global transverse magnetization isexactly conserved. In other words, there is not only atransport channel under which the transverse helix decays
by, such that local magnetization profiles redistribute inspace and eventually equilibrate, but also a nontransport(dephasing) channel, whereby transverse magnetization canbe created or removed locally, arising from explicit symmetry-breaking terms. (Note that for the highly anisotropic
case Δ ¼ 0 we cannot separate out such a transport part.)
In the following, we discuss and quantify three plausible
mechanisms leading to such dephasing. First, a careful readof the Hamiltonian Eq. (1) indicates that the effective
magnetic field hz;i is explicitly nonuniform for systems offinite sizes, as encountered in experiments. This stems fromthe fact that all terms in the Hamiltonian arise fromsuperexchange, which requires pairs of sites, and that sites
at the edges of the system have fewer neighbors than sites inthe bulk (discussed further below). Hence, relevant
symmetry-breaking terms (i.e., which cannot be transformed away) are already present in the idealized spinmodel. Second, our experiment consists of an ensemble of
1D systems, and there is a small inhomogeneity in theeffective magnetic field strengths between chains due toslight variations in the lattice depth caused by the Gaussian
shape of the laser beam. This can lead to a loss of measured(ensemble-averaged) contrast due to destructive interference from spins precessing at different rates, leading to a
Q-independent decay rate. Third, a natural deviation of our

PHYS. REV. X 11, 041054 (2021)

experiment from the idealized spin physics governed by thepure Heisenberg Hamiltonian Eq. (1) is the presence of asmall fraction of mobile holes in the spin chains caused by
nonadiabatic preparation of the Mott insulator and nonzero
temperature. Typical hole fractions lie between 5% and 10%
in the central part of the Mott insulator [27,32]. In a simplified
picture, holes cause dephasing, because spins next to holes
experience only half the effective magnetic field. A mobile
hole, therefore, creates a fluctuating effective magnetic field,
causing dephasing of the transverse spin component. Below,
we present experimental studies and extensive numericalsimulations which find that all of these effects contributeroughly equally a Q-independent decay rate.
We note also that, in the final data analysis, a reevaluation of the scattering lengths shows that our data isactually not taken exactly at the isotropic point but at
Δ ¼ 0.93  0.05. This deviation from isotropy is responsible for a Q-independent decay rate of 0.015Jxy =ℏ, or 15%of the observed difference between longitudinal and transverse spin decay (Appendix E 3). When all these effects aretaken together, this accounts approximately for the experimentally measured Q-independent decay rate γ 0 for thetransverse helix.
C. Imaging the effective magnetic field
All the symmetry-breaking mechanisms we discuss above
involve the effective magnetic field. Therefore, we firstpresent how this field can be directly observed and quantified.
If we assume that the ensemble experiences two pronounced values of the effective magnetic field, then the time
evolution of the cloud-averaged contrast cðtÞ will show a beatnote at a frequency which corresponds to the difference of thetwo values of the effective magnetic field. This is the case forour atom clouds, which feature a Mott insulator plateau
surrounded by a dilute shell of individual atoms [32] which
are pinned to their lattice sites by the gradient of the harmonic
trapping potential [40]. Many of these individual atoms do
not have neighbors for spin exchange and, therefore, do not
feel an effective magnetic field, while those in the Mott
insulator plateau do. The observed beat frequencies Ω¼0.90ð1ÞJxy =ℏ (Fig. 3) agree well with the predicted valueof the superexchange-generated effective magnetic fieldhz ¼ 0.89Jxy . As expected, the beat note is more pronounced
by spatially selecting the outer parts of the cloud [Fig. 3(a)]and disappears with a spin echo [Fig. 2(d)]. In Appendix B,
we describe an alternate spectroscopic method to observe theeffective magnetic field as a shift in the spin-flip frequency.
The presence of the effective magnetic field can also be
directly imaged by introducing a sufficiently large gradientin the lattice depth between the chains. This can beachieved by vertically displacing the z lattice relative tothe atom cloud (see Fig. 1), causing a gradient of theeffective magnetic field. As this field sets the “spiraling”
frequencies of the individual spin helices across the cloud(simply arising from the on-site precession of the spins

041054-5

PAUL NIKLAS JEPSEN et al.
(b)

0.5

0.4
0.2

0
0
0

10
20
Time t [ /Jxy]846
848
Magnetic field B [G]FIG. 3. Absolute measurement of the effective magnetic fieldvalue hz as a beat note between the spin precession frequencies in
the inner and outer parts of the cloud. (a) Transverse spin-helix
contrast cðtÞ for λ ¼ 23.5a (all filled symbols) and Δ ≈ 1. For
measurements averaged over the whole atom cloud (purple), they
are performed at 11ER (∘) and 13ER (□). The contrast at thecenter of the atom cloud for radii r ≤ 8.5a (orange filled symbols)
decays slower with less pronounced oscillations, whereas thecontrast in the spatial wings for radii r ≥ 20a (blue filled
symbols, including approximately 7% of the atoms in the cloud)decays faster with more pronounced oscillations. Data points fordiamonds are an average of measurements at 11ER and 13ER .
Open diamond symbols represent data for λ ¼ 10.4a and show
the same oscillation frequency but decay faster due to spintransport. Lines are fits described in Appendix A. Curves areoffset from each other for clarity. The dotted lines indicate theirrespective zeros. (b) The beat frequency Ω varies as a function of
the externally applied magnetic field B. The measured values forℏΩ (points) follow the theoretical prediction (without any
adjustable parameter) for the effective magnetic field hz (solidline) which is tuned by varying the scattering lengths via
Feshbach resonances.

about the Sz axis), this translates to an observable tilt of thewhole stripe pattern [Fig. 4(a)].
The tilt angle grows linearly in time, with a rate
proportional to the effective magnetic field and the gradientof the lattice depth (which we keep fixed). Externallyapplied magnetic fields change the scattering lengths via
broad Feshbach resonances, and so we can tune theeffective magnetic field. The observed rates for the tilt
rotation versus applied magnetic field B are shown in
Figs. 4(b) and 4(c) and agree well with our theoreticalprediction. In particular, near B ¼ 848.1 G, the spin j↑i
and j↓i scattering lengths are identical a↑↑ ¼ a↓↓ and theeffective magnetic field is zero, evinced by the absence ofany tilt in time [Fig. 4(b); yellow data points].
We note that, in principle, such a rotation could also be
caused by an external magnetic field gradient. However, thetilt angle would then not depend on the lattice depth V z and

20

15

10

5

0

-5
0

(b)

2
Time t [ /Jxy]

4

6

8

2

(c) 4

B = 846.37 G

10

15

10

5

0

-5

3

1.5

2

1

1

0.5

0

0

-1

-0.5

Effective magnetic field hz [Jxy]

0.6

25

Rotation rate φ̇ [degrees/( /Jxy)]

1

0.8

(a)

Rotation angle φ [degrees]

Contrast c(t)1Rotation angle φ [degrees]

1.5

Effective magnetic field h z [Jxy](a)PHYS. REV. X 11, 041054 (2021)

B = 848.53 G

04Time t [ /Jxy]

8

845

847
Magnetic field B [G]849

FIG. 4. Direct observation of the effective magnetic field
through spin precession. (a) Rotation angle of the stripe pattern
as a function of evolution time t (filled symbols) for two lattice
depths 11ER (blue) and 13ER (yellow) without (solid line) andwith (dashed line) spin-echo pulse at t ¼ 5ℏ=J xy . (b),(c) Tunability of the effective magnetic field hz . (b) Rotation angle as afunction of evolution time t for different magnetic fields B ¼
846.37 (red), 847.17 (blue), 847.59 (light blue), 848.00 (orange),
848.17 (yellow), 848.34 (purple), and 848.53 G (green). (c) Angular velocities obtained from linear fits in (b) compared to
predicted effective magnetic fields hz (solid line) with the scale
factor between the two y axes as a fitting parameter yielding_ϕℏ=J
xy ¼ 0.037hz =J xy ¼ 2.1°hz =J xy , representing the (uncalibrated) gradient of the lattice depth. Times are normalized by the
spin-exchange time ℏ=J xy for the central part of the atom cloud.
The scaling factor is consistent with a displacement of the zlattice beam (1=e2 radius of 125 μm) by an amount of
29 μm ¼ 55a.

041054-6

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …
the external magnetic field B. In a sufficiently deep lattice(V z ¼ 35ER Þ, for up to at least 40 ms, we do not observe
any discernible rotation, hence ruling out an external field
gradient.
When an echo pulse is added, the direction of the stripe
rotation is reversed, and, at twice the echo time, the stripe
pattern is vertical again, resulting in high contrast for verticallyintegrated images [Fig. 4(a); bottom]. This shows how the spinecho eliminates the effect of inhomogeneous effective fields
across the cloud, a technique we use below to quantify theircontribution to the dephasing of transverse spin patterns.
D. Dephasing mechanisms for the transverse spin helix
We now elaborate on and quantify the effects of threepossible dephasing mechanisms we identify, induced by theeffective magnetic field.
1. Effect of inhomogeneity in the effective magnetic field
between chains
The effect of inhomogeneity in the effective magnetic
field strengths between chains in the ensemble, arising fromslight variations in the lattice depth from the Gaussian
nature of the laser beams, can be eliminated by applying a
spin-echo pulse at half of the evolution time t.
Indeed, when adding spin-echo pulses to the experimental sequence, we find the Q-independent background decayrate γ 0 is reduced from 0.096(10) to 0.060ð3ÞJxy =ℏ [dashedlines in Fig. 2(e)]. This is compatible with an effective
magnetic field distribution over different chains with a full
width at half maximum (FWHM) of 8.2%, corresponding
to variations in the lattice depth V z of 1.6%, compatible
with experimental parameters.
2. Effect of finite chain lengths
As noted before, the effective magnetic field of the ideal
Heisenberg Hamiltonian Eq. (1) is necessarily nonuniform forsystems of finite sizes. To elaborate, the effective magnetic
field arises from superexchange involving nearest-neighborpairs of atoms, indicated already in the Hamiltonian
P Eq. (1),where we deliberately write the magnetic field hiji ðSzi þ
Szj Þ as a sum over pairs hiji of sites to emphasize this fact.
This means, in particular, that, for a 1D chain, the effective
magnetic field is reduced to hz =2 at the ends, half the value inthe bulk. For hz ≠ 0, this nonuniformity hence cannot simply
be transformed away by going into an appropriate corotating
frame. Although this reduced field is localized at the ends ofthe chain, it cannot be eliminated with an echo protocol, since
its effect propagates along the chain via spin dynamics as
illustrated in Fig. 10 (note the echo pulse removes inhomogeneous dephasing only for subsystems, which do not interact
with each other, such as separate chains in the ensemble).
This edge effect results in differences in the relaxation
between the transverse and the longitudinal spin helix: The
spins at the edges dephase rapidly, and this perturbation

PHYS. REV. X 11, 041054 (2021)

then propagates through the entire chain. We perform
numerical simulations which show that, for chain lengthsof 10–20 spins, the edge effect causes a dephasing rate of
approximately 0.02Jxy =ℏ (Appendix E 4). Although thediameter of the Mott insulator plateau we experimentallyrealize is measured to be around 40 sites, we expect that
holes (with an estimated concentration of 5%–10%) [27,32]
in the outer region of the Mott plateau are localized by thegradient of the trapping potential and create effectively
shorter chains in our sample. In contrast to that, in thecentral region of the Mott plateau, where the trapping
potential is flat, holes are mobile, and their effect is
discussed below.
The edge effect can be viewed as an inhomogeneity ofthe effective magnetic field realized within a single chain.
In addition to this effect, we have a harmonic trapping
potential in the experimental system, which along a given
chain creates an additional inhomogeneity, since it creates
an energy offset between neighboring sites and modifies
the superexchange rate by up to approximately 10% (seethe methods section in Ref. [32] for details). We estimatethat this is somewhat less important than the effect of the
50% reduced effective field at the ends of the chain.
3. Effect of mobile holes
We now explore a dephasing mechanism beyond thepure spin model, namely, the presence of mobile holes inthe system. For negligible interactions, the dynamics ofholes can be described by a quantum random walk where
the time-dependent wave function at site i for a hole
initially localized at i ¼ 0 is the Bessel function
Ji (t=ðℏ=2t̃Þ). The square of the Bessel function shows
oscillations at frequencies ω ¼ 4t̃=ℏ or periods of
T ¼ ðπ=2Þℏ=t̃. Now, for a hold time T and with an echopulse at T=2 (described in the previous subsection), thosefluctuations are “rectified” and lead to enhanced dephasing(note that the spin echo removes stationary inhomogeneous
magnetic fields but can enhance the effect of time-dependent fluctuations [41]). This is evidence for hole-magnon
coupling: Holes carry a localized magnetic field which
couples to spin dynamics. Indeed, Fig. 5 shows a feature in
the spin-echoed contrast at early times on the order of thetunneling time ℏ=t̃ in accordance with these predictions,providing experimental evidence that mobile holes are
possibly present. By using a series of echo pulses atfrequency ω, one could map out the frequency spectrumof the effective magnetic field, using concepts fromdynamic decoupling [41].
For times on the order of superexchange scales, the effectof the hole-induced fluctuating effective magnetic field canbe captured by a simple model. From nuclear magnetic
resonance, it is well known that the dephasing time T 2 of a
localized spin at z ¼ 0 is related to the magnetic field
fluctuations hz (measured in units of energy) and theircoherence time τc via 1=T 2 ¼hh2z iτc =ℏ2 ¼Gðz¼0;ω¼0Þ=ℏ2 ,041054-7

PAUL NIKLAS JEPSEN et al.

PHYS. REV. X 11, 041054 (2021)

1

c(t)/cfit(t )1Contrast c(t)

0.8

0.9

0.6
0

5

10

Time t [ /t˜]

0.4

0.2

0

0

2

4

6

8

10

Time t [ /Jxy]
FIG. 5. Tunneling dynamics of mobile holes. Decay curves for
the XX model (Δ ≈ 0) at λ ¼ 10.4a without a spin-echo pulse(orange) and with a spin-echo pulse (blue) after half the evolution
time t, measured for different lattice depths 9ER (△), 11ER (∘),
13ER (□), and 15ER (▽). The time axis has units of thecorresponding spin-exchange times ℏ=J xy ¼ 0.64 (△), 1.71(∘), 4.30 (□), and 10.29 ms (▽). Solid lines are fits (seeAppendix A). The spin echo generally slows down relaxationslightly by eliminating inhomogeneous dephasing, except at earlytimes where relaxation is actually enhanced and the data points
reproducibly deviate from the solid line (data points at t ¼ 0.22
and 0.43ℏ=J xy have to be excluded from fitting). The inset shows
the data points divided by the fit function and plotted with time inunits of the tunneling time ℏ=t̃, showing enhanced relaxation
around 1–2 tunneling times (the solid line is a guide for the eye).
More extensive studies with higher time resolution reproduce
similar features at around the tunneling time, but their shape issensitive to experimental conditions, including the atom number.

where Gðz; tÞ ¼ hhz ðz; tÞhz ð0; 0Þi is the autocorrelation of
the fluctuating magnetic field along the chain [42]. The
dephasing time T 2 is the same for spin patterns with arbitrary
wave vector Q.
For a moving hole, the effective magnetic field has acorrelation function Gðz; tÞ which is identical to the (normalized) density-density correlation function Jðz; tÞ of thehole, multiplied by h2z [here, we neglect the fact that theeffective magnetic field at a given site depends on the holeson the neighboring sites; see Eq. (1)]. For uncorrelated holes
with hole probability p, the variance of the local occupation isp with a coherence time τc ¼ 1.14ðℏ=2t̃Þ, where t̃ is thetunneling matrix element [43]. The associated correlations of
the fluctuating effective magnetic field determine the T 2dephasing time for the spin helix 1=T 2 ¼ 0.57h2z p=t̃ℏ.
Assuming p ¼ 0.1 (10% hole fraction) and using t̃ ¼
6.15J xy (at V z ¼ 11ER ) and hz =J xy ¼ 0.89, this leads toan estimate of 1=T 2 ≈ 0.007Jxy =ℏ.
This simple model of magnon-hole dynamics indicatesalready the non-negligible effect of a small fraction of holes
and strongly suggests that a more generalized model

beyond the pure spin Heisenberg model Eq. (1) should
be considered. To this end and to substantiate the roughestimate of the dephasing rate from the simple model, we
perform simulations of the bosonic t̃-J model which
explicitly take into account the presence of holes in the
Mott insulator near unity filling. This model allows holes tobe present but does not include double occupancies
of bosons, since they are suppressed by the large on-site
repulsion U. See Appendix F for its explicit form andderivation beginning from the Bose-Hubbard model, as
well as an interpretation of its constituent terms in terms of
magnon-hole couplings. There, we also show that a nonzero field strength hz is a necessary and sufficient condition
for the t̃-J model to break spin rotational symmetry,
justifying our identification of the effective magnetic fieldas the agent giving rise to differences in dynamics between
the transverse and the longitudinal spin helix.
Numerical simulations using the bosonic t̃-J model for amobile hole fraction of 10%, with experimentally realized
parameters that would yield the XXX model (Δ ¼ 1) in theideal spin limit, show a Q-independent dephasing at a ratearound 0.026Jxy =ℏ for the experimental conditions in
Fig. 2(e) (see Appendix E 2), supporting the simplified
model of hole-induced dephasing presented above (butproviding a rate 3 times higher). Note that we cannot
experimentally measure the hole fraction and use here thetypical range of values of 5%–10% observed or inferred inother experiments [27,32].
4. Summary of the dephasing mechanisms
At this stage, we account for the isotropy-breaking
dephasing rate of 0.096 (in units of J xy =ℏ) through moreor less equal contributions from the edge effect (0.020),
effective magnetic field inhomogeneity between chains
which can be eliminated via spin echo (0.036), and mobile
holes (0.013–0.026, assuming a hole fraction p between
5% and 10% and linear dependence on p), as well as asmall deviation from isotropy (0.015). These numbers are
summarized in Table I. We regard some of the numbers as
only semiquantitative due to the nonexponential characterof the measured and calculated decay curves, but they
indicate which phenomena have to be accounted for inquantum simulations of spin dynamics using ultracold
atoms in optical lattices.
5. Revisiting the XX model
Our considerations above illustrate the role of edge
effects and a small number of holes in the dephasing oftransverse spin. This should also contribute to the dephasing in the XX model. Hence, we revisit it via simulations of
the bosonic t̃-J model with the appropriate experimental
parameters (see Appendix E 1) and can now account for
two experimental findings. For the Q-independent decayrate, we note that the hz ¼ 1.43Jxy term is 1.6 times larger

041054-8

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …
TABLE I. Q-independent decay rates for different dephasing
mechanisms at Δ ≈ 1, obtained through experimental studies(spin echo for field inhomogeneity) and extensive numericalsimulations (edge effect, mobile holes, and experimental
deviation from isotropy). Numbers should be regarded as semiquantitative but, taken as a whole (second last row), strongly
suggest that they account for the net decay rate experimentallyobserved (last row).
Dephasing mechanism

Decay rate (J xy =ℏ)

Edge effectField inhomogeneity
Mobile holes
Experimental deviation from Δ ¼ 1

0.020
0.036
0.013–0.026
0.015

Total

0.084–0.097

Experimentally measured rate

0.096(10)compared to our experimental realization of the XXX
model, which explains the larger observed Q-independentdephasing rate γ 0 ¼ 0.20ð2ÞJxy =ℏ [Fig. 2(b)] semiquantitatively. Furthermore, the simulations show that the amplitude γ 1 of the cosðQaÞ dependence is a function of theconcentration of holes [see Figs. 2(b) and 8]. We find thatthe experimental data agree best with numerical simulations for 5% holes.
IV. CONCLUSIONS
We have used ultracold atoms to implement the
Heisenberg model with tunable anisotropy. For the relaxation of transverse spin patterns, we have studied for the
first time four decay mechanisms: intrinsic dephasing by
anisotropic spin-exchange couplings, inhomogeneousdephasing through a static superexchange-induced effective
magnetic field, dephasing through the ends of the chain, anddephasing by a fluctuating effective magnetic field due to thepresence of mobile holes. One reason why several of thesemechanisms have not been observed before is that mostprevious studies of spin dynamics in optical lattices haveused either fermions [24], for which the t̃-J model is always
explicitly spin rotationally symmetric and, therefore, hz ¼ 0,
or bosons comprised of 87Rb [25–28], for which thespin j↑i and spin j↓i scattering lengths are almost identical(a↑↑ ¼ 99.0a0 , a↑↓ ¼ 99.0a0 , and a↓↓ ¼ 100.4a0 , with the
Bohr radius a0 ) [44], leading to a value of hz ≈ 0.014J xy ,approximately 100 times smaller than for 7Li.
The experimental and theoretical results presented in thiswork go beyond pure spin physics. They illustrate effects
caused by a small hole fraction that is generally present incold atomic quantum simulators. A more complete description of spin dynamics in such systems, therefore, requires
using the t̃-J model, which features magnon-hole couplings.
This coupling between density and spin is analogous to
the interplay of spin and charge degrees of freedom in

PHYS. REV. X 11, 041054 (2021)

strongly correlated electronic systems, which is important,for example, in understanding emergent many-body phenomena like high-temperature superconductivity in cuprates [33–35]. Therefore, our platform presents an elegant
new setting where such physics can be emulated. Moregenerally, we regard our work as a starting point for
exploring spin dynamics in different dynamical regimes
as well as in generalized Heisenberg models.
Experimentally, the effect of mobile holes can be studied
by varying the lattice depth over a large range, which
affects hole dynamics (tunneling) differently than spin
dynamics (superexchange). A quantum gas microscope
will be able to select chains of a certain length and tomeasure spin and hole dynamics with single-site resolution.
Exciting future directions include the study of spin polaron
dynamics [45], realizing long-lived, metastable prethermal
states in higher dimensions [14–16], and probing the onsetof turbulent spin relaxation utilizing larger spin quantum
numbers [17].
ACKNOWLEDGMENTS
We thank Y. K. Lee and H. Lin for experimental assistance, J. Rodriguez-Nieva for discussions, and J. de Hond forcomments on the manuscript. We acknowledge support fromthe NSF through the Center for Ultracold Atoms and Grant
No. 1506369, ARO-MURI Non-Equilibrium Many-Body
Dynamics (Grant No. W911NF-14-1-0003), AFOSR-MURI
Photonic Quantum Matter (Grant No. FA9550-16-1-0323),
AFOSR-MURI Quantum Phases of Matter (Grant
No. FA9550-14-1-0035), ONR (Grant No. N00014-17-12253), the Vannevar-Bush Faculty Fellowship, and theGordon and Betty Moore Foundation EPiQS Initiative(Grant No. GBMF4306). W. W. H. is supported in part by
the Stanford Institute of Theoretical Physics. Numericalsimulations involving matrix product states were performed
using the TeNPy library [46].
APPENDIX A: DATA ANALYSIS
We determine the contrast C by a fit fðy; zÞ ¼gðy; zÞ · ½1 þ C cosðQz þ θÞ=2 to the two-dimensional
phase-contrast images (see Fig. 6). Here, Q ¼ 2π=λ is thewave vector, θ is a random phase which varies from shot toshot due to small magnetic bias field drifts, and gðy; zÞ is a
two-dimensional envelope function which accounts for the
spatial distribution of all atoms n ¼ n↑ þ n↓ inside a spherepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
of radius R such that gðy; zÞ ¼ A 1 − y2 =R2 − z2 =R2 ·Hð1 − y2 =R2 − z2 =R2 Þ with the Heaviside function HðrÞ.
During the evolution time t, the contrast CðtÞ decays, and westudy the dependence of cðtÞ ¼ CðtÞ=Cð0Þ on lattice depth
V z , wavelength λ, and anisotropy Δ.
For both longitudinal and transverse spin relaxation, we
find the decay curves can be well described by the sum of adecaying part with decay rate γ and a (damped) oscillating
part with frequency ω, resulting in a fitting function

041054-9

PHYS. REV. X 11, 041054 (2021)-1

Linear density [ a ]PAUL NIKLAS JEPSEN et al.
1500

0

60

Position y [a]-2Column density [a ]

-20

0

30

20
-20

0

20
-20
Position z [a]

0

20

0
0
750
-1
Linear density [a ]FIG. 6. Contrast measurement. The central image represents raw data and shows the distribution of atoms in the j↑i state. The left
image is the two-dimensional fit as described in Appendix A. Every pixel is a local measurement of the column density (number of j↑iatoms per unit area). The image is projected (integrated) along both the horizontal z direction (right) and the vertical y direction (top)from y; z ¼ −42a to þ42a to obtain the linear densities (number of j↑i atoms per unit length) for the measured data (points) and the fit(solid line). The dashed line in the y projection shows the (parabolic) envelope function for the spatial density distribution in the cloud.
The y and z axes are displayed in units of the lattice spacing a ¼ 0.532 μm.

cðtÞ ¼ ða0 þ b0 cos ωtÞe−γt þ c0 . Here, a0 , b0 , c0 , ω, and γ
are fitting parameters. These fits are used for Fig. 2(b)(purple), in Fig. 2(d) (blue), and for Fig. 2(e) (purple andblue). Special fitting procedures are used for the XX modeland for the beat note due to the effective magnetic field.
1. XX model
The decay curves cðtÞ for the transverse spin helix[Fig. 2(a)] clearly show a slower decay rate for larger
values of Q. We can use the fitting function describedabove, with the only differenceof adding the constant offsetﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c0 in quadrature cðtÞ ¼ ½ða0 þ b0 cos ωtÞe−γt 2 þ c20[Fig. 2(a)] to reflect that the offset c0 arises due to an
experimental detection noise floor at the 10−2 level. The
actual physical contrast does decay to zero cðtÞ → 0.
Remarkably, the fitted oscillation periods [e.g., T ¼
11.6ð4Þℏ=J xy at λ ¼ 10.4a] agree fairly well with theenergy splitting J xy =2 for the two-spin Heisenberg model[Fig. 2(c)], which implies an oscillation period of
T ¼ 4πℏ=J xy ≈ 12.57ℏ=J xy . With this fit function, theslower decay for large wave vectors Q shows up mainlyin the oscillation frequency ω and not in the decay rate γ.
For a simpler characterization of the decay, we obtain theinitial decay rate by fitting a linear slope cðtÞ ¼ c0 ð1 − γtÞ
to the initial decay in the range cðtÞ ≤ 0.4. Results of suchfits are shown in Fig. 2(b) (orange).
2. XXX modela. Longitudinal spin relaxation [Fig. 2(e), purple]
As in our previous work [32] on spin transport, we use
the fitting function cðtÞ ¼ ða0 þ b0 cos ωtÞe−γt þ c0 .

b. Transverse spin relaxation withspin echo [Figs. 2(d) and 2(e), blue]
The same fitting function yields oscillation frequencies ωand oscillating fractions b0 =ða0 þ b0 Þ which agree fairly wellwith the longitudinal case, especially at large wave vectors Q,
but with much larger error bars at small wave vectors Q,
because the decay rate γ is much faster than the oscillations.
For this reason, we constrain both parameters ω and b0 =ða0 þ
b0 Þ to the values obtained in the longitudinal case.
c. Transverse spin relaxation without
spin echo [Figs. 2(d) and 2(e), orange; Fig. 3]
A beat note between the inner part and outer partof the cloud is visible (Fig. 3), due to the differencein effective magnetic fields. To determine the beatfrequency Ω, we generalize the fitting function to the sumof
two parts which interfere: jc1 ðtÞeiΩt þ c2 ðtÞj ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c1 ðtÞ2 þ 2c1 ðtÞc2 ðtÞ cosðΩtÞ þ c2 ðtÞ2 . Here, c1 ðtÞ ¼
ða0 þ b0 cos ωtÞe−γt is the contrast of the atoms in the innerpart of the cloud, and c2 ðtÞ ¼ c2 is the contrast of the isolated
atoms in the outer part which preserve the contrast for a longtime. We can neglect the background c0 due to the detection
noise. In c1 ðtÞ, we again constrain the two parameters ω andb0 =ða0 þ b0 Þ to the values obtained for longitudinal spin
relaxation.
APPENDIX B: SPECTROSCOPIC OBSERVATION
OF THE EFFECTIVE MAGNETIC FIELD
A spatially uniform effective magnetic field can always betransformed away in a suitable rotating frame. However, eventhen, it can still be observed as a shift of the spin-flip
resonance. We rotate the spins via an adiabatic frequency

041054-10

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …

the XXZ Hamiltonian (assuming the effective magnetic
field is uniform), for any isotropy Δ. The classical limit isreached by taking the spin-quantum number S → ∞ or bytreating the spins as classical vectors S⃗ i ¼ ðSxi ; Syi ; Szi Þ ofarbitrary length S ¼ jS⃗ i j, which we set to 1=2 for comparison to the quantum spin system.
We start with the system initialized at t ¼ 0 in the helixiðQzi þθÞ zstate S; Si ð0Þ ¼ 0, and we untwist thei ð0Þ ¼ Se
helix by using the rotation
0 x1 010 x 1SiTi− sinðQzi Þ 0 cosðQzi Þ
B yC BCB y CðC1Þ
@ Si A ¼ @ cosðQzi Þ 0 sinðQzi Þ A@ T i A

0.8
0.6
0.4
Population

PHYS. REV. X 11, 041054 (2021)

0.2
0
0.8
0.6
0.4
0.2

Szi

0

1

0

T zi

0
-0.6

-0.4

-0.2
0
0.2
Detuning δ/2π [kHz]

0.4

0.6

FIG. 7. Spectroscopic observation of the effective magnetic
field hz as a shift in the spin-flip frequency for hz < 0 (bottom)
compared to hz ≈ 0 (top). Shown is the fraction of atoms in eachstate as a function of the final detuning δ of a 22-ms sweep of therf frequency, starting at δ ¼ þ30 kHz with all atoms in the j↑istate (closed circles) and no atoms in the j↓i state (open squares).
The detuning is relative to the single-particle transition frequency.
The power of the rf drive is also ramped to zero after thefrequency sweep to make the transition sharper. A nonzero
detuning δ for equal spin populations compensates for theeffective magnetic field hz which shifts the curves for a lattice
depth of 11ER (bottom) compared to 35ER (top) where hz ≈ 0.
For the sweep experiment, we choose the second-lowest (closed
circles) and third-lowest (open squares) hyperfine states of 7Li
due to the smaller sensitivity to external magnetic fields(originating from a smaller differential magnetic moment). At
1025 G, the scattering lengths are approximately a↑↑ ≈ a↑↓ ≈−50a0 and a↓↓ ≈ þ350a0 , leading to an estimate for an effective
magnetic field of hz ≈ −1.14J xy ∼ h × ð−100 HzÞ, which isconsistent with our observation.

sweep, where the detuning corresponds to an external Bz
field (in the rotating frame) and the Rabi frequency to a Bx
field, realizing a Heisenberg model with magnetic fields.
Starting from a fully polarized state with all atoms in the j↓istate and large detuning of the rf, we reduce the Bz field
adiabatically and observe the spin imbalance. The spins arebalanced when the detuning compensates for the effective
magnetic field created by superexchange. With this method,
we can observe the effective field for different lattice
depths (Fig. 7).
APPENDIX C: SEMICLASSICAL ANALYSIS AND
SHORT-TIME EXPANSION OF SPIN DYNAMICS
1. The classical transverse spin helix with any wave
vector Q does not evolve for any anisotropy Δ
We show here that, in the classical limit, the transverse
spin helix with any wave vector Q does not evolve under(we ignore the phase θ for simplicity), which gives S⃗ i ↦ T⃗ izwith T i ð0Þ ¼ 0, T i ð0Þ ¼ S. The Hamiltonian Eq. (1) (withhz ¼ 0) then transforms asX
H ↦ HðQÞ ¼ Jxy ½cosðQaÞðT xi T xiþ1 þ T zi T ziþ1 Þiþ sinðQaÞðT zi T xiþ1 − T xi T ziþ1 Þ
Xþ Jz T yi T yiþ1 :

ðC2Þ

i

The Landau-Lifshitz (LL) equations of motion for classicalspins read ∂ t T⃗ i ¼ ∂ T⃗ i HðQÞ × T⃗ i . Upon changing variablesyx
to T 
i ¼ T i  iT i , we have
1
þ
þ
z
−
−
T_þ
i ¼ ifJ z ðT i−1 þ T iþ1 − T i−1 − T iþ1 ÞT i
2
þ
þ
−
−
þ Jxy ½sinðQaÞT þi ðT i−1 − T iþ1 þ T i−1 − T iþ1 Þ
þ 2T zi ðT zi−1 − T ziþ1 Þþzþ cosðQaÞ½−ðT −i−1 þ T −iþ1 þ T þ
i−1 þ T iþ1 ÞT izz
þ 2T þ
i ðT i−1 þ T iþ1 Þg;
T_−i ¼ T_þ
i ;

ðC3Þ
ðC4Þ

1
þ
þ
−
−
T_zi ¼ − ifJ z ðT −i þ T þi ÞðT i−1 þ T iþ1 − T i−1 − T iþ1 Þ
4
þ
þ
−
−
− Jxy ðT −i − T þi Þ½cosðQaÞðT i−1 þ T iþ1 þ T i−1 þ T iþ1 Þ
þ 2 sinðQaÞð−T zi−1 þ T ziþ1 Þg;ðC5Þz
and it is straightforward to verify that T i ðtÞ ¼ 0, T i ðtÞ ¼ S
is a solution to the LL equations, as claimed.

2. Stability of the classical spin-helix state:
Dispersion relation of fluctuations
To understand the stability of the classical spin-helix
states, we linearize the equations of motion about theclassical solution and consider fluctuations. Now, T zi obeys

041054-11

PAUL NIKLAS JEPSEN et al.

PHYS. REV. X 11, 041054 (2021)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ−
the constraint T zi ¼ S2 − T þi T i , so fluctuations about theclassical solution simply entail

δT 
i ¼



2

2
T
i ðtÞ ¼ T̄ i ðtÞ þ δT i ðtÞ þ OððδTÞ Þ ¼ δT i ðtÞ þ OððδTÞ Þ;þ δT þ
iþ1 − 4δT i Þ:ðC6Þ
T zi ðtÞ ¼ S þ OððδTÞ2 Þ:

ðC7ÞTherefore, we get

ωk

δT þ
k
δT −k


¼ SJ xyiSþ½J ðδT −i−1 þ δT −iþ1 − δT þ
i−1 − δT iþ1 Þ
2 z∓ cosðQaÞJxy ðδT −i−1 þ δT −iþ1 þ δT þ
i−1
ðC8Þ

We now expand in Fourier modes δT 
i ¼
P iðkzi þωk tÞ
δT
e
with
momentumkanddispersion
ωk .
kkThis reduces to an eigenvalue problem−½−2 þ cosðkaÞ cosðQaÞ − Δ cosðkaÞ

cosðkaÞ½− cosðQaÞ þ ΔcosðkaÞ½cosðQaÞ − Δ½−2 þ cosðkaÞ cosðQaÞ þ Δ cosðkaÞ

!

δT þ
k



δT −k
ðC9Þ

with solution
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
ωk ¼ 2 2Jxy S cosðQaÞ½−Δ cosðkaÞ þ cosðQaÞ sin2 ðka=2Þ:Reducing to Δ ¼ 0 reproduces the expression quoted in themain text for the XX model in Sec. III A. Note that we
could have equivalently obtained the same dispersion
relations by performing a spin-wave analysis, upon mapping the spins to Holstein-Primakoff bosons (in a large S
expansion) and performing a Bogoliubov transformation todiagonalize the Hamiltonian in second order.

Owing to the factorizable nature of the initial spin-helix
state, we can analytically derive the short-time quantum
dynamics of the state without passing into a semiclassicallimit as done before. The basic object is the Taylor
expansion of a spin operator (in the transverse direction):
1 2 þ
þ
þ
2
hSþ
i ðtÞi ¼ hSi ð0Þi þ h∂ t Si ð0Þit þ h∂ t Si ð0Þit þ    ;2
ðC11Þ
where h·i is Pthe expectation value in the spin statei

Szi Qzia product state, we can easily evaluate the expression forthese terms. We find for general spin S
ih½H; Sþi i ¼ 0;
2 iQzi ½J − J cosðQaÞ2 :−h½H; ½H; Sþzxyi i ¼ −S e

ðC14Þ
ðC15Þ

(The vanishing of the term linear in t follows from timereversal symmetry.) Therefore,

3. Short-time expansion of quantum dynamicsjψðQÞi ¼ e−i
and

ðC10Þ

jþ þ þ   i, where Sxi jþii ¼ Sjþii ,þh∂ t Sþ
i ð0Þi ¼ ih½H; Si i;

ðC12Þ

þ
h∂ 2t Sþi ð0Þi ¼ −h½H; ½H; Si i:

ðC13Þ

Since the Hamiltonian is a sum of strictly local terms andSþ
i is an on-site term, the expressions in the commutators
are comprised of only finite-range terms with support
centered around site i. Using that the state factorizes into

1
iQzi
− S2 eiQzi ½Jz − Jxy cosðQaÞ2 t2 þ    :hSþ
i ðtÞi ¼ Se
2
ðC16ÞExtracting the Fourier component with wave vector Q gives
the normalized contrast
1cðtÞ ¼ 1 − S½Jz − J xy cosðQaÞ2 t2 þ    :
2

ðC17Þ

A characteristic energy rate γ for the initial quadratic decay
can, therefore, be defined as cðtÞ ¼ 1 − γ 2 t2 þ   , yieldingrﬃﬃﬃ
S
jJ ½Δ − cosðQaÞj:
ðC18Þ
γ≔
2 xy
Focusing now on Δ ¼ 0 and S ¼ 1=2, this shows that ahelix of wave vector Q decays with a rate going asγ ∝ j cosðQaÞj. For Δ ¼ 1, we recover that γ ∝ Q2 inthe limit of Q ≪ 1=a (when the wavelength λ is large
compared to the lattice spacing a). More generally, for
arbitrary anisotropy Δ, there is a critical wave vector
ðQc aÞ ¼ arccosðΔÞ where decay is expected to be very

041054-12

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …
slow; seeing such a dependence in experiments would beinteresting. This prediction is actually in line with that the
spin helix with wave vector ðQc aÞ ¼ arccosðΔÞ can beproven to be an exact eigenstate of the XXZ Hamiltonian
with anisotropy Δ with appropriate boundary conditions, as
shown in Refs. [38,39].
APPENDIX D: NUMERICAL SIMULATIONDETAILS
Here, we present general details on the numericalsimulations performed. Unless otherwise specified, we
employ the time-evolving block decimation algorithm onmatrix product states defined on 1D chains of length
L ¼ 40a, with large enough bond dimension to ensure
convergence of local observables to a tolerance 10−4 , via
the TeNPy library [46].
In the absence of holes, we simulate the Heisenberg
model (1), with the initial state the spin helix with wavezvector Q, which reads locallyjψ i ðQÞi ¼ eiðQzi þθÞSi jþii,p
ﬃﬃ
ﬃ
where Sxi jþii ¼ jþii ¼ ð1= 2Þðj↑ii þ j↓ii Þ. Here, θ isthe global initial phase of the spin helix, which varies
from shot to shot in the experiment due to small magnetic
field bias drifts (Appendix A). We measure hSxi ðtÞi and fit
for each time slice a sine function in space with wave vector
Q (allowing its phase to be an independent parameter); theamplitude of the sinusoidal modulation is the numericallydetermined contrast cðtÞ normalized to unity at t ¼ 0. We
also average cðtÞ over θ ¼ 0, π=2 to account for the fact thatthe global phase of the initial state in the experiments shifts
from measurement to measurement (we find that averaging
over these two values suffices to reproduce the full averaging
over θ; also see the methods section in Ref. [32]).
In the presence of holes, we simulate the bosonic t̃-Jmodel [Eq. (F1)]. We assume holes occur independently oneach site with probability p. In order to perform ensemble
averaging over the different hole positions of the initialstate, we employ the following computational trick. Let theon-site Hilbert space be spanned by the states j0i,
j↑ii ≔ a†i↑ j0i, and j↓ii ≔ a†i↓ j0i. We define a pure stateon each site asjΨi ðQÞi ¼ eiφipﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃ
1 − pjψ i ðQÞi þ pj0ii ;Z 2π
0dφ ρi ¼ ð1 − pÞjψ i ðQÞihψ i ðQÞj þ pj0ii h0ji :ðD3ÞThis reproduces the situation where holes occur locally andindependently on each site i with probability p.
In our simulations, we choose a random set of phaseangles ðφ1Q
; …; φN Þ ∈ ½0; 2πÞN and time evolve the globally
pure state i jΨi ðQÞi under the t̃-J Hamiltonian. We repeat
the simulation with different sets of phase angles sampled
randomly uniformly in ½0; 2πÞN and then average theextracted contrast.
We find that, in practice, there are remarkably only verysmall variations between different Q
choices of phase angles[i.e., a given random choice of i jΨi ðQÞi is a typicalconfiguration], allowing us to perform the ensemble average with relatively few repetitions (at most 50 runs for each
Q and global phase θ).
APPENDIX E: NUMERICALSIMULATION RESULTS
1. XX model
For the XX model, we utilize parameters Δ ¼ 0 and hz ¼
1.43J xy (thus, simulating the experimental conditions at
V z ¼ 11ER ), as well as t̃ ¼ 4.11J xy , J ↑↑ =J↑↓ ¼ −0.32, andJ↓↓ =J ↑↓ ¼ 1.15. Here, J σσ 0 ≔ −4t̃2 =Uσσ 0 .
Figure 8 shows decay curves for p ¼ 0, 0.05, and 0.1.
The experimental decay rates γ ¼ 1=τ [in Fig. 2(b)] areobtained from a linear fit cðtÞ ¼ c0 ð1 − t=τÞ to the initial
decay [cðtÞ ≤ 0.4]. We use an equivalent procedure for thetheoretical data and determine the time τ0 where thenumerical simulations show a contrast of cðτ0 Þ ¼ 0.4,
which we then convert to the theoretical decay rate
τ ¼ ð5=3Þτ0 . As shown in Fig. 2(b), the numerical data
verifies the analytically predicted cosðQaÞ dependence ofthe decay rate. This scaling starts to break down as
Q → π=ð2aÞ, because higher-order terms in the expansion(either semiclassical or short-time) become important.
Inclusion of holes washes out the cosðQaÞ dependence.

ðD1Þ
2. XXX model

where φi is some phase with valueQin ½0; 2πÞ. Clearly, in thelimit p → 0, the state jΨðQÞi ≔ i jΨi ðQÞi reduces to thepure-spin helix (i.e., without holes) with wave vector Q.
Consider now the outer product of jΨi ðQÞi with itself whenp ≠ 0:
ρi ¼ ð1 − pÞjψ i ðQÞihψ i ðQÞjpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
þ eiφi 1 − pjψ i ðQÞih0ji þ H:c: þ pj0ii h0ji :

1
ρ̄i ¼
2π

PHYS. REV. X 11, 041054 (2021)

ðD2Þ

If we now average φ over the interval ½0; 2πÞ uniformly, theensemble-averaged reduced density matrix is

For the XXX model, we utilize parameters Δ ¼ 1 andhz ¼ 0.80Jxy (thus, simulating the experimental conditions
at V z ¼ 11ER ), as well as t̃ ¼ 6.13Jxy , J↑↑ =J↑↓ ¼ 0.61,
and J↓↓ =J ↑↓ ¼ 1.40.
Figures 9(a) and 9(b) show the results for a simulation
involving 10% holes, i.e., p ¼ 0.1. We fit the data betweencðtÞ ¼ 0.9 and 0.15 with a straight line (dashed) and obtaina decay rate defined as twice the slope of the fit (this factor
is chosen because the slope of an exponential function at
half decay is reduced by a factor of 2). Because of thenonexponential nature of the decay curves, for both

041054-13

PAUL NIKLAS JEPSEN et al.
(a)PHYS. REV. X 11, 041054 (2021)

1

(b)

(c)

1

0.8

Contrast c(t)

0.5

0.6
0
0

0.4

4

8

0.2

0
0

2

4
Time t [ /Jxy]

6

8

0

2
4
6
Rescaled time t × cos(Qa) [ /Jxy]

8

0

2

4
Time t [ /Jxy]

6

8

FIG. 8. Transverse spin-helix decay for the XX model obtained from numerical simulations, with hole probability p ¼ 0 (a),(b),p ¼ 0.05 (c), and p ¼ 0.1 [(c), inset] with hz ¼ 0 (a),(b) and hz ¼ 1.43J xy (c) as in the experiment. The wavelengths in (a) are (frombottom to top) λ ¼ 31.3a, 23.5a, 18.8a, 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a, 7.2a, 6.7a, 6.3a, 5.9a, 5.6a, 5.3a, 5.0a, 4.8a,
4.6a, 4.4a, and 4.0a. The decay curves for pure spin dynamics (a) show a wave-vector dependence of decay rates of the formγ ∝ cosðQaÞ. Using rescaled time t cosðQaÞ, all decay curves collapse almost perfectly for wavelengths λ ≥ 6.3a, which covers therange studied in the experiment. The presence of holes washes out the cosðQaÞ dependence (c), shown here for λ ≥ 6.3a and p ¼ 0.05.
At sufficiently high hole probability, e.g., p ¼ 0.1 [(c), inset], the wave-vector dependence vanishes almost completely.

numerical simulations and experimental data, there is somearbitrariness in choosing “effective” decay rates which,depending on the parameterization, could differ by up to
50%. By further fitting the decay rates for the six smallest(a) 1

values of Q values to the form γðQÞ ¼ DQ2 þ γ 0 (in orderto focus on the limiting Q → 0 behavior), we obtain a Qindependent decay rate γ 0 ¼ 0.019Jxy =ℏ and a diffusion
constant D ¼ 0.16a2 =ðℏ=J xy Þ.
(c)Contrast c(t)

0.8
0.6
0.4
0.2
0
0

10

20

30

Decay rate γ [Jxy / ]

(b)

40
0
Time t [ /Jxy]

10

20

30

40

(d)

0.2

0.1

0
0

0.05

0.1

0.15
0
Wave vector Q [2π/a]

0.05

0.1

0.15

FIG. 9. Spin relaxation for the XXZ model. (a),(b) Isotropic model Δ ¼ 1 with finite hole concentration (p ¼ 0.1). (c),(d) Slightly
anisotropic Δ ¼ 0.93 with no holes (p ¼ 0). The colored solid lines in (a) and (c) are decay curves cðtÞ for different wavelengths
λ ¼ 31.3a, 23.5a, 18.8a, 15.7a, 13.4a, 11.7a, 10.4a, 9.4a, 8.5a, 7.8a, 7.2a, 6.7a, and 6.3a (from top to bottom). The dotted lines arelinear fits to determine decay rates, which are shown in (b) and (d) with a fit γ ¼ DQ2 þ γ 0 (solid line).

041054-14

TRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …

PHYS. REV. X 11, 041054 (2021)(c) 0.03

Magnetization 〈S 〉Decay rate γ [Jxy / ]z(b)+ –2

Lattice site index i

1

(a) 1

8

16
0

10

20

30

40

0

1
-–
20

10

20

30

Time t [ /Jxy]

40

0.02

0.01
5

10
20
Chain length L [a]3040FIG. 10. Dephasing from the edges of the chains. (a),(b) Dynamics of an Sx -polarized (Q ¼ 0) state for an L ¼ 16a finite chain, withanisotropy Δ ¼ 1 and hz ¼ 0.8J xy reflecting the parameters of the experiment. (a) shows how the initially homogeneous phase of thelocally measured transverse spin hSxi ðtÞi gets distorted at the edges
P and how this perturbation propagates through the chain, leading to aloss of overall contrast of the total transverse magnetization i hSxi ðtÞi (b). A fit (dashed lines) to cosðωt þ ϕÞe−γt =2 determines the
decay rate γ, which is shown as a function of system size L in (c). A log-log plot shows the scaling γ ∝ L−1 for large L.

3. Near-isotropic XXZ model
We also investigate how a small deviation from theisotropic point affects dynamics. One reason is that ourexperimental data for the isotropic model is actually nottaken exactly at the isotropic point but at Δ ¼ 0.93  0.05.
In the two-site model, the triplet splitting of Jxy ð1 − ΔÞ=2
becomes 0.035Jxy =ℏ, but the full simulation presented here
shows that the effect is smaller.
We consider Δ ¼ 0.93 and hz ¼ 0.89Jxy (thus, simulating
the experimental conditions at V z ¼ 11ER ), as well as t̃ ¼
6.15Jxy , J↑↑ =J↑↓ ¼0.53, and J↓↓ =J↑↓ ¼ 1.41. Figures 9(c)and 9(d) show the results for a simulation in the absence ofholes. We use the same method as for the XXX model todetermine decay rates from linear fits. We obtain a Qindependent decay rate γ 0 ¼ 0.015Jxy =ℏ and a diffusion
constant D ¼ 0.15a2 =ðℏ=J xy Þ.
4. Dephasing from edge effects
We also use numerical simulations to study how theinhomogeneity of the effective magnetic field at the endsof finite chains leads to dephasing for transverse spincomponents. We concentrate on the Q → 0 limit, i.e., a state
uniformly polarized in the Sx direction, with anisotropy
Δ ¼ 1.

If the effective magnetic
P field is globally uniform, thetransverse magnetization i hSxi ðtÞi just oscillates in time
without decaying. However, the fact that the edges of thechain feel an effective magnetic field strength which is halfthat of the bulk causes dephasing of spins at the edges.
This disturbance, in turn, propagates into the bulk [seeFig. 10(a)], so that the transverse magnetization decays intime [see Fig. 10(b)]. For long chains, the decay rate
decreases as a function of length L, as the bulk dominates
the edges. This trend starts only for chains with L > 16a.
For smaller L, the trend is reversed due to few-bodydynamics.
We also explore the effect of the effective magnetic fieldin the XX model. Comparison of simulations of the purespin model with and without an effective magnetic field ofhz ¼ 1.43Jxy shows that the edge effect for chains of length
L ¼ 40a give rise to a Q-independent decay rate of
0.04ℏ=J xy , which amounts to shifting the black curve in
Fig. 2(b) (for hz ¼ 0) vertically.
APPENDIX F: DERIVATION
OF THE BOSONIC t̃-J MODEL
We derive here the bosonic t̃-J model


X
hz z
y y
z z
z
x x
Ht̃−J ¼
Jxy ðSi Sj þ Si Sj Þ þ Jz Si Sj − ½Si ðnj↑ þ nj↓ Þ þ ðni↑ þ ni↓ ÞSj  þ cðni↑ þ ni↓ Þðnj↑ þ nj↓ Þ
2
hiji


X t̃2 †
X †2t̃2 †t̃2 † σþ H:c:
a n a þa S a þa n at̃ aiσ ajσ þ H:c: −−U↑↓ iσ jσ̄ kσ U ↑↓ iσ̄ j kσ Uσσ iσ jσ kσhiji;σhijki;σquoted in the main text. Here, spin σ ¼↑; ↓, and aiσ and a†iσ
are bosonic lowering and raising operators, respectively, at
↓
†
þ †
−
site i, such that S↑i ≡ Sþi ≔ ai↑ ai↓ , Si ≡ Si ¼ ðSi Þ , and
þ
1
1
−
y
−
z
Sxi ¼ 12 ðSþi þ Si Þ, S ¼ 2i ðSi − Si Þ, S ¼ 2 ðni↑ − ni↓ Þ
form a representation of the Pauli algebra, upon restricting

ðF1Þ

the on-site Hilbert space to be spanned by three states: an
occupancy of a single boson (either spin j↑i or j↓i) or noboson (hole). Also, c ¼ −ðt̃2 =U↑↑ þ t̃2 =U↓↓ þ t̃2 =U↑↓ Þ.
The above Hamiltonian illustrates Pthat the previously
identified effective magnetic field term hiji ðhz =2ÞðSzi þSzj Þ
in fact stems from a direct interaction term describing

041054-15

PAUL NIKLAS JEPSEN et al.

PHYS. REV. X 11, 041054 (2021)

magnon-density coupling Szi ðnj↑ þ nj↓ Þ þ ði ↔ jÞ, the latterof which reduces to the former upon taking the limit of noholes, i.e., ðni↑ þ ni↓ Þ ¼ 1 for every site i. The terms on thelower line represent dynamics of holes in different flavors:bare tunneling, density-assisted tunneling, and spin-flipassisted tunneling, which are additional magnon-hole couplings. Note that they arise at the same order in perturbation
theory (in t̃=U σσ 0 ) as the pure spin couplings and, thus, inprinciple, cannot be neglected, although they are suppressedby the presence of a small hole probability p. By inspecting theexpressions of Jz , hz , and these extra terms as a function ofU↑↑ , U ↑↓ , and U ↓↓ , we see that at Δ ¼ 1 (isotropic spin
couplings) a nonzero hz is a necessary and sufficient condition
for the t̃-J model to break spin rotational symmetry. (Having
spin rotational symmetry requires all on-site interactionenergies to be equal U↑↑ ¼ U ↑↓ ¼ U↓↓ .) This justifies ouridentification of the effective magnetic field as the agent giving
rise to differences in dynamics between the transverse and thelongitudinal spin helix.
To derive the bosonic t̃-J model, the starting point is theBose-Hubbard Hamiltonian describing cold atoms moving
in a deep optical lattice (so that they are confined to thelowest Bloch band):H¼−

Xt̃ða†iσ ajσ þ H:c:Þ þhiji;σþ U↑↓Xni↑ ni↓ :⃗ ¼ U † ðθÞVU
⃗⃗H0 ðθÞ
0 ðθÞ;0ðF5Þ

⃗ ¼ e−iθ·Γ ;U0 ðθÞ

ðF6Þ⃗ ⃗H0n⃗ ¼

1ð2πÞm

U↓↓ Γ1 ;

⃗

Γ1 ¼

X1

U↑↑ Γ2 ;

Γ2 ¼

X1
i

U↑↓ Γ3 ;Γ3 ¼

2

We assume two components σ ¼↑; ↓ and U σσ 0 ≫ t̃, suchthat there are at most singly occupied sites. We derive theeffective model in this limit. We do not assume U ↑↑ , U↑↓ ,and U ↓↓ are necessarily equal between themselves.
We employ the expansion detailed in Ref. [47], where it isshown how having multiple emergent Uð1Þ charges emerge
in an effective Hamiltonian. The general setup is as such:Let Γ1 ; …; Γm be m mutually commuting operators withinteger eigenvalue spacings, and consider the Hamiltonian

2

ðF8Þ

ni↑ ðni↑ − 1Þ;ðF9ÞXni↑ ni↓and that Γi have integer eigenvalues and mutually commute.
We, therefore, identify ω1 ¼ U↓↓, ω2 ¼ U ↑↑ , andω3 ¼ U↑↓ . We define
ðF11Þ

which gives us⃗ ≔ U 0 ðθÞ⃗ †H0 ðθÞ

X⃗t̃ða†iσ ajσ þ H:c:ÞU 0 ðθÞ:

ðF12Þ

hiji;σWe haveðiθ3 Þ22!
×½ni↑ ni↓ ;½ni↑ ni↓ ;a†i↑ þeiθ3 Γ3 a†i↑ e−iθ3 Γ3 ¼ a†i↑ þðiθ3 Þ½ni↑ ni↓ ;a†i↑ þ¼ a†i↑ þðiθ3 Þa†i↑ ni↓ þ¼ a†i↑ eiθ3 ni↓ :ðiθ3 Þ2 † † 2
a ðn Þ þ
2! i↑ i↓
ðF13Þ

Therefore,

ðF4Þ

n⃗ ≠0

(this turns out to be the so-called van Vleck expansion).
The effective Hamiltonian has emergent symmetries½Heff ; Γi  ¼ 0; i.e., the Hamiltonian is symmetric withrespect to the m Uð1Þ charges Γi . Here, H 0n⃗ is the n⃗ thFourier mode of the “interaction” Hamiltonian defined on
the m-torus T m :

ðF10Þ

i

ðF3Þ

where V need not commute with Γi . In the limit of large j ⃗ωj,
we can derive an effective HamiltonianðF7Þni↓ ðni↓ − 1Þ;⃗ ≔ expð−iθ⃗ · ΓÞ;⃗
U0 ðθÞ
ðF2Þ

X ½H0 ; H0 
n⃗
−⃗n
þ 
⃗
⃗ω
n
·
⃗

⃗
dm θ e−i⃗n·θ H0 ðθÞ:ii

1Heff ¼ ⃗ω · Γ⃗ þ H 0⃗ þ
02Tm

Applying this formalism to the Bose-Hubbard model, we
note that interactions there consist of three kinds:

1X
U n ðn − 1Þ2 i;σ σσ iσ iσH ¼ ⃗ω · Γ⃗ þ V;

Z

Next,

041054-16eiθ3 Γ3 a†i↑ e−iθ3 Γ3 ¼ a†i↑ eiθ3 ni↓ ;ðF14Þeiθ3 Γ3 ai↑ e−iθ3 Γ3 ¼ ai↑ e−iθ3 ni↓ ;ðF15Þ

eiθ3 Γ3 a†i↓ e−iθ3 Γ3 ¼ a†i↓ eiθ3 ni↑ ;ðF16Þeiθ3 Γ3 ai↓ e−iθ3 Γ3 ¼ ai↓ e−iθ3 ni↑ :ðF17ÞTRANSVERSE SPIN DYNAMICS IN THE ANISOTROPIC …

PHYS. REV. X 11, 041054 (2021)






1
ðiθ2 Þ2 1
1††n ðn − 1Þ; ai↑ þn ðn − 1Þ; ni↑ ðni↑ − 1Þ; ai↑ þ    :2 i↑ i↑2
2! 2 i↑ i↑eiθ2 Γ2 a†i↑ e−iθ2 Γ2 ¼ a†i↑ þ ðiθ2 ÞA single commutator yields


1
1
1
ni↑ ðni↑ − 1Þ; a†i↑ ¼ ½ni↑ ; a†i↑ ðni↑ − 1Þ þ ni↑ ½ðni↑ − 1Þ; a†i↑ 
2
2
2
1
1
¼ a†i↑ ðni↑ − 1Þ þ ni↑ a†i↑
22†
¼ ai↑ ni↑

ðF18Þ

ðF19Þ

so the full expression becomeseiθ2 Γ2 a†i↑ e−iθ2 Γ2 ¼ a†i↑ þ ðiθ2 Þa†i↑ ni↑ þ¼ a†i↑ eiθ2 ni↑ :ðiθ2 Þ2 †a ðn Þ2 þ   
2! i↑ i↑
ðF20Þ

Similarly, we haveeiθ2 Γ2 ai↑ e−iθ2 Γ2 ¼ e−iθ2 ni↑ ai↑ :ðF21Þ⃗ † a† aj↑ U0 ðθÞ⃗ ¼ a† eiðθ3 ni↓ þθ2 ni↑ Þ e−iðθ3 nj↓ þθ2 nj↑ Þ aj↑ ;U 0 ðθÞ
i↑
i↑

ðF22Þ

⃗ † a† ai↑ U0 ðθÞ⃗ ¼ a† eiðθ3 nj↓ þθ2 nj↑ Þ e−iðθ3 ni↓ þθ2 ni↑ Þ ai↑ ;U 0 ðθÞ
j↑
j↑

ðF23Þ

⃗ † a† aj↓ U0 ðθÞ⃗ ¼ a† eiðθ3 ni↑ þθ1 ni↓ Þ e−iðθ3 nj↑ þθ1 nj↓ Þ aj↓ ;U 0 ðθÞ
i↓
i↓

ðF24Þ

⃗ † a† ai↓ U0 ðθÞ⃗ ¼ a† eiðθ3 nj↑ þθ1 nj↓ Þ e−iðθ3 ni↑ þθ1 ni↓ Þ ai↓ :U 0 ðθÞ
j↓
j↓

ðF25Þ

⃗
Therefore, there are four terms in H0 ðθÞ:

⃗ enforces projectors ofNow, the n⃗ th Fourier mode of H 0 ðθÞcertain occupation numbers between sites. For example, forthe first term and n1 ¼ 0, n2 ¼ 0, n3 ¼ 1, this isa†i↑ Pni↓ −nj↓ ¼1 Pni↑ −nj↑ ¼0 aj↑ :ðF26Þ

We evaluate Eq. (F4), and the result is Eq. (F1) (ignoring the⃗constant term ⃗ω · Γ).

[1] W. Heisenberg, Mehrkörperproblem und Resonanz in der
Quantenmechanik, Z. Phys. 38, 411 (1926).
[2] P. A. M. Dirac, On the Theory of Quantum Mechanics, Proc.
R. Soc. A 112, 661 (1926).
[3] J. H. V. Vleck, The Theory of Electric and Magnetic
Susceptibilities, International Series of Monographs on
Physics (Oxford University, New York, 1932).
[4] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, Berlin, 1994).

[5] K. Binder and A. P. Young, Spin Glasses: Experimental
Facts, Theoretical Concepts, and Open Questions, Rev.
Mod. Phys. 58, 801 (1986).
[6] L. Savary and L. Balents, Quantum Spin Liquids: A Review,
Rep. Prog. Phys. 80, 016502 (2017).
[7] R. Vasseur and J. E. Moore, Nonequilibrium Quantum
Dynamics and Transport: From Integrability to Many-Body
Localization, J. Stat. Mech. (2016) 064010.
[8] B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R.
Steinigeweg, and M. Žnidarič, Finite-Temperature Transport in One-Dimensional Quantum Lattice Models, Rev.
Mod. Phys. 93, 025003 (2021).
[9] M. Ljubotina, M. Žnidarič, and T. Prosen, Spin Diffusionfrom an Inhomogeneous Quench in an Integrable System,
Nat. Commun. 8, 16117 (2017).
[10] S. Gopalakrishnan and R. Vasseur, Kinetic Theory of Spin
Diffusion and Superdiffusion in XXZ Spin Chains, Phys.
Rev. Lett. 122, 127202 (2019).
[11] E. Ilievski, J. De Nardis, M. Medenjak, and T. Prosen,
Superdiffusion in One-Dimensional Quantum Lattice Models, Phys. Rev. Lett. 121, 230602 (2018).

041054-17

PAUL NIKLAS JEPSEN et al.

PHYS. REV. X 11, 041054 (2021)[12] B. Bertini, M. Collura, J. De Nardis, and M. Fagotti,
Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201(2016).
[13] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura,Emergent Hydrodynamics in Integrable Quantum Systems
out of Equilibrium, Phys. Rev. X 6, 041065 (2016).
[14] M. Babadi, E. Demler, and M. Knap, Far-from-Equilibrium
Field Theory of Many-Body Quantum Spin Systems: Prethermalization and Relaxation of Spin Spiral States in Three
Dimensions, Phys. Rev. X 5, 041005 (2015).
[15] S. Bhattacharyya, J. F. Rodriguez-Nieva, and E. Demler,
Universal Prethermal Dynamics in Heisenberg Ferromagnets, Phys. Rev. Lett. 125, 230601 (2020).
[16] J. F. Rodriguez-Nieva, A. Schuckert, D. Sels, M. Knap, and E.
Demler, Transverse Instability and Universal Decay of Spin
Spiral Order in the Heisenberg Model, arXiv:2011.07058.
[17] J. F. Rodriguez-Nieva, Turbulent Relaxation after a Quench
in the Heisenberg Model, Phys. Rev. B 104, L060302(2021).
[18] C. Gross and I. Bloch, Quantum Simulations with Ultracold
Atoms in Optical Lattices, Science 357, 995 (2017).
[19] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P.
Zoller, Cold Bosonic Atoms in Optical Lattices, Phys. Rev.
Lett. 81, 3108 (1998).
[20] A. B. Kuklov and B. V. Svistunov, Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate
Optical Lattice, Phys. Rev. Lett. 90, 100401 (2003).
[21] L.-M. Duan, E. Demler, and M. D. Lukin, Controlling SpinExchange Interactions of Ultracold Atoms in Optical
Lattices, Phys. Rev. Lett. 91, 090402 (2003).
[22] J. J. García-Ripoll and J. I. Cirac, Spin Dynamics for Bosons
in an Optical Lattice, New J. Phys. 5, 76 (2003).
[23] E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin,
Phase Diagram of Two-Component Bosons on an Optical
Lattice, New J. Phys. 5, 113 (2003).
[24] M. A. Nichols, L. W. Cheuk, M. Okan, T. R. Hartke, E.
Mendez, T. Senthil, E. Khatami, H. Zhang, and M. W.
Zwierlein, Spin Transport in a Mott Insulator of Ultracold
Fermions, Science 363, 383 (2019).
[25] T. Fukuhara et al., Quantum Dynamics of a Mobile Spin
Impurity, Nat. Phys. 9, 235 (2013).
[26] T. Fukuhara, P. Schauß, M. Endres, S. Hild, M. Cheneau, I.
Bloch, and C. Gross, Microscopic Observation of Magnon
Bound States and Their Dynamics, Nature (London) 502, 76(2013).
[27] S. Hild, T. Fukuhara, P. Schauss, J. Zeiher, M. Knap, E.
Demler, I. Bloch, and C. Gross, Far-from-Equilibrium Spin
Transport in Heisenberg Quantum Magnets, Phys. Rev.
Lett. 113, 147205 (2014).
[28] R. C. Brown, R. Wyllie, S. B. Koller, E. A. Goldschmidt, M.
Foss-Feig, and J. V. Porto, Two-Dimensional Superexchange-Mediated Magnetization Dynamics in an Optical
Lattice, Science 348, 540 (2015).
[29] L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H.
Zhang, and M. W. Zwierlein, Observation of 2D Fermionic
Mott Insulators of 40K with Single-Site Resolution, Phys.
Rev. Lett. 116, 235301 (2016).
[30] M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif,and M. Greiner, Site-Resolved Measurement of the Spin-

Correlation Function in the Fermi-Hubbard Model, Science
353, 1253 (2016).
[31] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. KanászNagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M.
Greiner, A Cold-Atom Fermi-Hubbard Antiferromagnet,
Nature (London) 545, 462 (2017).
[32] P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E.
Demler, and W. Ketterle, Spin Transport in a Tunable
Heisenberg Model Realized with Ultracold Atoms, Nature(London) 588, 403 (2020).
[33] J. Spalek, t-J Model Then and Now: A Personal Perspectivefrom the Pioneering Times, Acta Phys. Pol. A 111, 409(2007).
[34] F. C. Zhang and T. M. Rice, Effective Hamiltonian for
the Superconducting Cu Oxides, Phys. Rev. B 37, 3759
(1988).
[35] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N.
Trivedi, and F. C. Zhang, The Physics behind HighTemperature Superconducting Cuprates: The Plain Vanilla
Version of RVB, J. Phys. Condens. Matter 16, R755 (2004).
[36] M. Koschorreck, D. Pertot, E. Vogt, and M. Köhl, Universal
Spin Dynamics in Two-Dimensional Fermi Gases, Nat.
Phys. 9, 405 (2013).
[37] S. Trotzky, S. Beattie, C. Luciuk, S. Smale, A. B. Bardon, T.
Enss, E. Taylor, S. Zhang, and J. H. Thywissen, Observationof the Leggett-Rice Effect in a Unitary Fermi Gas, Phys.
Rev. Lett. 114, 015301 (2015).
[38] V. Popkov, X. Zhang, and A. Klümper, Phantom Bethe
Excitations and Spin Helix Eigenstates in Integrable Periodic and Open Spin Chains, Phys. Rev. B 104, L081410(2021).
[39] X. Zhang, A. Klümper, and V. Popkov, Phantom Bethe
Roots in the Integrable Open Spin-12 XXZ Chain, Phys. Rev.
B 103, 115435 (2021).
[40] I. Dimitrova, N. Jepsen, A. Buyskikh, A. Venegas-Gomez,
J. Amato-Grill, A. Daley, and W. Ketterle, Enhanced
Superexchange in a Tilted Mott Insulator, Phys. Rev. Lett.
124, 043204 (2020).
[41] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum
Sensing, Rev. Mod. Phys. 89, 035002 (2017).
[42] C. P. Slichter, Principles of Magnetic Resonance, Springer
Series in Solid-State Sciences (Springer, Berlin, 1990).
[43] The prefactor 1.14 is somewhat arbitrarily obtained byintegrating the probability for a localized particle to be on
the original site, J 0 ½t=ðℏ=2t̃Þ2 to the first zero of the Bessel
function J 0 .
[44] B. J. Verhaar, E. G. M. van Kempen, and S. J. J. M. F.
Kokkelmans, Predicting Scattering Properties of Ultracold
Atoms: Adiabatic Accumulated Phase Method and Mass
Scaling, Phys. Rev. A 79, 032711 (2009).
[45] G. Martinez and P. Horsch, Spin Polarons in the t-J Model,
Phys. Rev. B 44, 317 (1991).
[46] J. Hauschild and F. Pollmann, Efficient Numerical Simulations with Tensor Networks: Tensor Network Python(TeNPy), SciPost Phys. Lect. Notes 5 (2018). . Codeavailable from https://github.com/tenpy/tenpy.
[47] D. V. Else, W. W. Ho, and P. T. Dumitrescu, Long-Lived
Interacting Phases of Matter Protected by Multiple TimeTranslation Symmetries in Quasiperiodically Driven Systems, Phys. Rev. X 10, 021032 (2020).

041054-18

Appendix C
Catching Bethe phantoms andquantum many-body scars:
Long-lived spin-helix states inHeisenberg magnets
P. N. Jepsen, Y. K. Lee, H. Lin, I. Dimitrova, Y. Margalit, W. W. Ho, W. Ketterle
Catching Bethe phantoms and quantum many-body scars: Long-lived spin-helix
states in Heisenberg magnets
arXiv:2110.12043 (2021)

157

Catching Bethe phantoms and quantum many-body scars:
Long-lived spin-helix states in Heisenberg magnets
Paul Niklas Jepsen,1, 2, 3 Yoo Kyung Lee,1, 2, 3 Hanzhen Lin,1, 2, 3 Ivana
Dimitrova,3, 4 Yair Margalit,1, 2, 3 Wen Wei Ho,4, 5 and Wolfgang Ketterle1, 2, 3
1

arXiv:2110.12043v1 [cond-mat.quant-gas] 22 Oct 20212

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3
MIT-Harvard Center for Ultracold Atoms, Cambridge, MA, USA
4
Department of Physics, Harvard University, Cambridge, MA 02138, USA
5
Department of Physics, Stanford University, Stanford, CA 94305, USA
Exact solutions for quantum many-body systems are rare and provide valuable insight to universal
phenomena. Here we show experimentally in anisotropic Heisenberg chains that special helical spin
patterns can have very long lifetimes. This finding confirms the recent prediction of phantom Bethe
states, exact many-body eigenstates carrying finite momenta yet no energy. We theoretically findanalogous stable spin helices in higher dimensions and in other non-integrable systems, where they
imply non-thermalizing dynamics associated with quantum many-body scars. We use phantom
spin helices to directly measure the interaction anisotropy which has a major contribution fromshort-range off-site interactions that have not been observed before. Phantom helix states open newopportunities for quantum simulations of spin physics and studies of many-body dynamics.

The dynamics of strongly-interacting, quantum manybody systems is an active frontier of research. It has
broad implications ranging from understanding fundamental phenomena like quantum thermalization or thelack thereof [1–5], to realizing new forms of matter (e.g.
time crystals [6, 7]) and to controlling entanglement forquantum information processing [8, 9].
However, analyzing in full such systems is difficult due
to their complexity. Exactly-solvable models are therefore especially important and desirable since they candirectly reveal the physical mechanisms behind universal phenomena. For example, the transverse field Isingmodel in one spatial dimension can be solved in termsof free fermions and serves as a paradigmatic model for
quantum criticality [10, 11].
The spin-1/2 anisotropic Heisenberg model in one dimension, which is given byX
H = Jxy[Six Sjx + Siy Sjy + ∆Siz Sjz ],
(1)
hiji

is another such example. Here, the transverse and longitudinal spin couplings (between neighboring sites i, j)
are Jxy and Jz := ∆Jxy , where ∆ is the spin-exchange
anisotropy. While the model seems simple and has been
known to be solvable by the Bethe ansatz for almost a
century [12, 13], it gives rise to rich dynamics which arestill not completely understood. Indeed, only recently
was it predicted that the isotropic system (∆ = 1) yields
an exotic Kardar-Parisi-Zhang superdiffusive regime oftransport [14–16] which has been subsequently experimentally confirmed by [17]. Yet another surprise hascome from the recent theoretical discovery of a special
set of degenerate many-body eigenstates in the model
for any anisotropy — so-called phantom Bethe states[18]. These are states composed of quasiparticles whichcarry momentum but contribute zero energy (relative tothe ferromagnetic “vacuum” state), akin to ghost parti-cles, hence the name phantom. Simple patterns of spins
winding in the transverse plane — i.e. spin-helix states— also share these phantom properties if their pitch λ or
wavevector Q := 2π/λ = Qp , where Qp satisfies the phantom condition∆ = cos(Qp a),(2)

with a being the lattice spacing. These states, which wecall phantom helix states, are exact many-body eigenstates and do not decay. Since interactions, even in an integrable model, are expected to cause a system to locallyrelax to a (generalized) Gibbs ensemble [13, 19], sucha long-lived and far-from-equilibrium state represents asurprising exception to (generalized) quantum thermalization.
In this work, we systematically explore the dynamics of spin-helix states using our versatile ultracold atom
quantum simulator platform with tunable anisotropy[20, 21]. Specifically, we study their decay as a function of wavevector Q for different fixed anisotropies ∆,and find a non-monotonic decay rate with a pronounced
minimum near the expected special value Qp . This is
the signature of the phantom spin-helix state, confirming the predictions of [18]. We further theoretically establish generalizations of the phantom spin-helix states
to Heisenberg systems of higher dimensions, with higher
spin quantum numbers, and for non-cubic lattice geometries. While 1D spin-1/2 models are integrable, thesegeneralizations are not; therefore, the existence of stable
far-from-equilibrium helices in such systems leads to genuinely non-thermalizing dynamics associated with quantum many-body scars [22]. We propose an experimental
protocol to realize them with ultracold atoms. Lastly,
we demonstrate how dynamics of phantom helices canbe used as an important tool for quantum simulations ofspin physics. Using the phantom condition (Eq. 2) it is
now possible to experimentally determine the anisotropy

2∆. We find that the anisotropy is strongly affected bynearest-neighbor (i.e. off-site) interactions of the underlying Hubbard model, which have not been observed before
for contact interactions.
Spin-helix states & the phantom condition. In
this work, we study transverse spin-helix statesY|ψ(Q)i =cos(θ/2) |↑in + sin(θ/2)e−iQzn |↓in . (3)nThe polar angle θ determines the local longitudinal spincomponent hSnz i which is constant along the chain, andzn is the position of the n-th spin (see Fig. 1a,b). In theclassical limit, any transverse spin helix is stable for any
anisotropy since the torques exerted on a given spin by
its neighbors cancel exactly [21]. Therefore, the decayof a spin helix is due to quantum fluctuations. However, for wavevectors Qp fulfilling the phantom condition for |∆| ≤1 (Eq. 2), the fluctuations from the twonearest neighbors also cancel exactly, making these special helices particularly long-lived. Intriguingly, one canshow that they are in fact many-body eigenstates of the
Heisenberg model (Eq. 1) for infinite systems, or for finite systems with appropriate boundary conditions [18].
For the finite chains with open boundaries that we prepared in our experiments, the phantom spin helix is only
metastable — defects will propagate from the ends ofthe chain into the bulk, resulting in a nonzero but small
decay rate [21].
For the isotropic system (∆ = 1), the phantom spin
helix has wavevector Qp = 0 and thus reduces to a spinpolarized product state, which is a trivial eigenstateof this model. For ∆ = 0 the phantom condition gives
Qp a = π/2, so that angles between neighboring spinsare 90◦ (assuming all spins lie in the S x -S y plane,i.e. θ = π/2). To explain intuitively how this state ismetastable, consider a spin which points e.g. in the S x
direction, with quantum fluctuations in the S y -S z plane.
For Jz = 0, there is no interaction from the S z component, while the S y component causes no precession on
the neighboring spins, which also point along the ±S y direction. In the following, we will demonstrate the existence of these long-lived phantom helices for general∆, and confirm the predictions of the phantom condition
Eq. 2.
Experimental methods. As in our previous work
[20, 21], the spin model is implemented by loading ultracold lithium-7 atoms in the two lowest hyperfine
states into a three-dimensional optical lattice. This system is well-described by a two-component Bose-Hubbard
model. Two of the three lattice potentials are kept highat 35 ER creating a bundle of isolated 1D-chains. The
lattice depth of the third axis V0 is set to a value between 9 ER and 11 ER , which is deep enough such thatthe system remains in the Mott insulating regime whilestill allowing for spin dynamics. Here ER = h2 /(8ma2 ) is
the recoil energy, m the atomic mass, and h the Planck
constant.
Because particle motion is suppressed in the Mott insu-

Sz

Sy

Qa

a
Sz

θ

Sx

Sy

b
Sx

c
Winding Gradientd
XXZ Interaction

λ = 2π/Qe
Unwinding GradientfMeasurement

Sy

Sz

g

Sx

FIG. 1.
Preparation and observation of spin-helix
states. We prepare a transverse spin helix in the S x -S y plane,i.e. θ = π/2 (a), or with arbitrary polar angle θ (b). An initially spin-polarized state in the S y direction (c) is wound intoa spin helix with variable wavevector Q using a magnetic field
gradient, here illustrated for Qa = π/2 (d). This state evolves
under the XXZ Heisenberg Hamiltonian (e). After unwinding
the remaining spin modulation to a resolvable wavevector (f ),the local S y -magnetization is imaged in-situ (g). Only the S x
and S y components of the spin are shown for c-f.

lator, the dynamics of the remaining degrees of freedom
can be described using a pure spin model. By mapping
the two hyperfine states onto spins |↑i and |↓i, we canrealize the spin-1/2 Heisenberg XXZ model (Eq. 1) inwhich the interactions are mediated by superexchange[23–26]. The transverse and longitudinal spin couplings
are given by
Jz = 4t̃2 /U↑↓ − (4t̃2 /U↑↑ + 4t̃2 /U↓↓ )Jxy = −4t̃2 /U↑↓ ,(4)where t̃ is the tunneling matrix element between neighboring sites and U↑↑ , U↑↓ , U↓↓ are the on-site interaction energies. The spin couplings (Eq. 4) can be varied over two orders of magnitude by changing the lattice
depth V0 , which scales the entire Hamiltonian. We control the anisotropy ∆ := Jz /Jxy via an applied magnetic
field B, which tunes the interactions through Feshbachresonances [20, 21, 27]. In our realization, the transverse
coupling is antiferromagnetic (Jxy > 0).

3
a 1
0.8
0.6

Q=0

0.4
0.2

Contrast c(t)

b 1
0.8
0.6

Q = π/2a

0.4
0.2

c 1
0.8
0.6

Q = π/a

0.4
0.2
0

0

1

2
3
Time t [ћ/Jxy]

4

5

FIG. 2. Decay of spin-helix states. Spin-helix contrast
c(t) measured for ∆ ≈ 0 at two different lattice depths 9 ER(red) and 11 ER (blue), for three wavevectors. a, Qa = 0, allspins aligned. b, Qa = π/2, neighboring spins perpendicular,
and an eigenstate for ∆ = 0. c, Qa = π, all spins anti-aligned.
Decay curves collapse for each wavevector when times are normalized in units of ~/Jxy . The contrast lifetime is significantly
larger for Qa = π/2 compared to Qa = 0 and Qa = π.

The transverse spin helix is created by RF pulses totilt the spins to a finite polar angle θ, followed by magnetic field gradients to wind a helix [21, 28, 29]. Timeevolution is initiated by rapidly lowering V0 . The dynamics following this quench are governed by the 1D XXZ
model (Eq. 1, Fig. 1e) with a selected anisotropy ∆. After a variable evolution time t, the dynamics are frozen
by rapidly increasing V0 ; the sample is then imaged.
Our imaging system limits the direct observation ofspin modulations to a wavelength of λ > 6 a. In orderto image spin helices at any value of Q, we first unwind the remaining spin modulation to a wavelength ofλ ≈ 10 a by applying a π-pulse followed by a magnetic
field gradient as in Fig. 1f. After unwinding, we turnthe transverse helix into a population modulation by applying a π/2-pulse. In the end, we detect the spatial
distribution of spin |↑i atoms in-situ with state-selective
polarization-rotation imaging (Fig. 1g). Compared to ourprevious work [20, 21], this novel unwinding step extends
our observable range of wavevectors Q all the way to
Qa = π, where neighboring spins are anti-aligned. Integrating the images along a direction perpendicular to thechains yields a 1D spatial profile of sinusoidal population
modulation over all spin chains (Fig. 1g). As describedin Refs. [20, 21], we determine the normalized contrast
c(t) which decays during the evolution time t and obtaina decay rate γ by using a linear fit at early times.

Experimental observations of phantom helices.
Fig. 2 illustrates the contrast decay c(t) for spin helices with different wavevectors Q at ∆ ≈ 0. We see thatthe decay for Qa = π/2 is noticeably slower than that of
Qa = 0 or π. This non-monotonic behavior is the signature of phantom spin helices. This feature is emphasized
in Fig. 3 by comparing the contrast decay rates γ at various values of Q. According to the phantom condition
Eq. 2, the wavevector Qp with the smallest decay ratechanges as a function of ∆. Since the superexchange interactions in Eq. 4 depend on the scattering lengths, we
can tune ∆ smoothly by varying the magnetic field. Indeed, in Fig. 3 we see Qp varies accordingly as predicted.
The fit function γ(Q) = γ1 |∆ − cos(Qa)| + γ0 was derived from a short-time expansion of the spin-helix contrast c(t) [21]. Here ∆, γ1 and γ0 are treated as free fit
parameters, where γ0 represents a background decay rateaccounting for effects such as finite chain length, holes inthe spin chains, and inhomogeneous dephasing [21]. The
predicted ∆ based on Eq. 4 and previously determinedscattering lengths [30] agrees qualitatively with the ∆ wefit (see Fig. 4). We also observe phantom helix statesfor various polar angles in Fig. 3b-d. This confirms theprediction [18] that there is a whole family of phantom
helix states for a given value of ∆. We find that a larger
absolute value of the total S z magnetization (i.e. θ close
to 0 or π) leads to an overall slower decay.
For anisotropies |∆| > 1 (Fig. 3a,e; open symbols) there
is no longer a stable spin-helix eigenstate [18]. We see instead that the minimum decay rate is always at Q = 0
for ∆ ≥ 1 and Qa = π for ∆ ≤ −1. Comparing decay
rates across ∆ in this range, we find a Q-independent
increase relative to the |∆| = 1 case which is monotonousin |∆| − 1.
Our improved imaging protocol allows us to accessnew parameter regimes beyond previous work. For
the isotropic system (∆ = 1) we had observed diffusive
spin transport, characterized by γ(Q) ≈ γ1 Q2 a2 /2 + γ0[20, 21, 28]. However, we now see this quadratic behavior
break down for large Q when the wavelength λ becomescomparable to the lattice spacing a. The fastest decay occurs for the Néel state (Qa = π), where neighbouring spins are anti-aligned. This directly demonstrates
that this classical antiferromagnetic state is not the
ground state of the quantum antiferromagnetic Heisenberg Hamiltonian. Nevertheless, the Néel state is an exact highly-excited eigenstate for ∆ = −1 (Fig. 3e).
Extension to higher dimensions. The phantom
spin-helix states were originally discovered in 1D as acoherent superposition of phantom Bethe states, special degenerate solutions to the Bethe ansatz equations[18]. This raises the question if they are a phenomenon
tied exclusively to integrability. We find that they
are not: we can show that stationary phantom helix states exist for any anisotropy, for the anisotropic
Heisenberg model defined on hypercubic lattices in arbitrary spatial dimensions and arbitrary spin quantum numbers. Specifically, we consider the gener-

4aDecay rate γ [Jxy/ћ]

1.2

b

c

Δ ≈ 0.5

Δ≈0

Δ ≈ -0.5

d

e

1.2

0.6
0.8

Δ>1
Δ≈1

0.4
0

0

π/2

0

Δ ≈ -1

0.4

0.2

π

Δ < -1

0.8

0.4

0

π/3

2π/3

π 0

π/2

π 0

π/3

2π/3

π

0

0

π/2

π

Wavevector Q [1/a]FIG. 3. Observation of phantom helix states. Decay rate γ as a function of wavevector Q, shown for fitted anisotropies∆ ranging from positive (a-b) through zero (c) to negative values (d-e) measured at lattice depths of 9 ER (red) and 11 ER(blue). The decay rate minimum occurs at a wavevector Qp which increases smoothly from Qp a = 0 for ∆ ≈ 1 (a) to Qp a = π for∆ ≈ −1 (e). The long-lived spin pattern in the S x -S y plane is illustrated on top of each panel. Fits γ(Q) = γ1 |∆ − cos(Qa)| + γ0(lines) are used to find Qp and the anisotropies ∆ shown in Fig. 4. In addition to the purely transverse helices (polar angle
θ = π/2, circles), panels (b-d) also show the decay of spin helices with polar angles θ = 5π/12 (green triangles) and θ = 2π/3
(orange triangles). The curves above were measured at applied magnetic fields of B = 847.887 G (∆ > 1) and 847.286 G (∆ ≈ 1)(a), 845.760 G (b), 842.905 G (c), 839.376 G (d), 833.004 G (∆ ≈ −1) and 827.287 G (∆ < −1) (e).

alization of the spin-helix states (see Supplementary
Materials) in d dimensions specified by a wavevector
Q = (Q1 , · · · , Qd ), and only require that Q = Qp := Qp x,
where x ∈ {−1, 1}d is a binary vector of −1s and 1s, andQp (0 ≤ Qp a ≤ π) satisfies the phantom condition (Eq. 2).
For example, for the Heisenberg model with ∆ = 1/2
on a 2D square lattice so that Qp a = π/3, there are
four such wavevectors: Qp a = (π/3, π/3), (π/3, −π/3),(−π/3, π/3), (−π/3, −π/3); all such states are phantom
helices (see Fig. S1). In fact, non-trivial phantom helix
states also exist for non-hypercubic lattices, e.g. triangular and kagome lattices (see Fig. S2), but only for the
special value of the anisotropy ∆ = −1/2. We prove thesestatements in the Supplementary Material.
Our experimental protocol can be extended to two or
three dimensions to directly observe quantum scarred dynamics associated with these special helices. An experimental complication is that near Feshbach resonances,
any magnetic field gradient is projected along the direction of the strong bias field. Therefore, to wind a helix
with arbitrary Q = (Qx , Qy , Qz ) vectors, one must load
atoms into a deep 3D lattice, ramp to low fields B whereappropriate gradients in any direction can be created easily, wind the transverse spin helix with arbitrary Q, andthen return to the high fields near Feshbach resonances.
Because of the small scattering lengths of lithium-7 at
low field, a very deep optical lattice is required to stay in
the Mott insulator regime.
Measurement of Anisotropy. Besides being of fundamental interest, the phantom helix states also havepractical applications. We can use the sensitivity ofthe phantom helix states to measure the spin-exchange
anisotropy ∆ := Jz /Jxy precisely as a function of appliedmagnetic field B (shown in Fig. 4). Until now, there has
been no protocol to directly measure ∆; it could only
be derived [20, 21] from measured scattering lengths a↑↑ ,a↑↓ , a↓↓ [27, 30] using Eq. 4. Fig. 4 compares our mea-

sured ∆ to the predictions based on scattering lengths.
They agree quite well far from the Feshbach resonances.
We will first discuss the accuracy of our determination of∆ and possible systematic errors, then comment on the
new physics near Feshbach resonances.
Repeated measurements of ∆ have a reproducibilityof better than 0.1, clearly visible by the small random
scatter of the data points in Fig. 4. In addition, there
are several possible systematic effects. (1) The lineshapesof contrast decay are non-exponential and differ between
short and long decay times (visible in Fig. 2), partly due
to a beat note between the precession of atoms in thespin chains and isolated atoms in the outer part of thecloud [21]. The resulting oscillations average out for long
decay times, but not for shorter decay times. By using
different fit functions, we estimate that |∆| in Fig. 4 could
be overestimated by at most ∼ 0.1.
(2) Our fit function γ(Q) = γ1 |∆ − cos(Qa)| + γ0 was
derived from a short-time quadratic expansion of the decay of contrast [21]. However, experimentally, we canonly observe the contrast at intermediate times where
the decay is more linear (as already discussed in previous work [20, 21, 28]). Numerical simulations reveal thatdifferent methods of extracting γ can lead to a systematic overestimation of |∆| by up to ∼ 0.15. The effect ismaximum around |∆| ≈ 0.5, and falls off for |∆| closer to
0 or 1.
(3) A systematic shift of |∆| would also occur if there
were another Q-dependent decay mechanism due to background decay (e.g. holes) or decay propagating from theends of the chain. Several of these issues can be addressed
experimentally in the future by using a quantum gas microscope and observing dynamics in single isolated spin
chains.
Some of the discrepancies in Fig. 4 can be explained bycorrections to the underlying standard Hubbard model[20, 31, 32] which include density-dependent tunneling5

Anisotropy ∆

1
0.5
0

-0.5
-1

835

840

845

900

1000

1100

Magnetic field B [G]FIG. 4. Tuning the anisotropy with magnetic fields. Measured anisotropies ∆ are compared to the standard model
for superexchange given by Eq. 4 (solid line) using previously measured scattering lengths [30]. The black dashed line includes
the bond-charge correction to tunneling [31] and a small adjustment of the background scattering length of a↑↓ , which was nottightly constrained by previous measurements. Major deviations near the two Feshbach resonances at 845.506 and 893.984 G(vertical dotted lines) are evidence for off-site interactions.

[33], higher-band corrections to U [34, 35], and offsite contact interactions [20]. However, only contributions from off-site interactions get large near a Feshbachresonance. Therefore, the phantom helix states revealthat spin-spin interactions near a Feshbach resonance aredominated by off-site interactions which have never been
observed for contact interactions1 . Off-site interactions
originate from the small overlap of a Wannier function on
one site with those of its neighbours [37] and add a correction term to Jz in Eq. 4 of 2(V↑↑ + V↓↓ − 2V↑↓ ) [20] whereV↑↑ , V↑↓ , V↓↓ are the off-site interaction energies. In aforthcoming publication, we will show that the current
model for off-site interactions [31, 32, 37] is insufficientnear a Feshbach resonance.
Discussion and outlook. Previous studies of the
Heisenberg Hamiltonian have focused on the groundstate [38–40], low-lying elementary excitations including
magnons [41, 42] and Bethe strings [43, 44], or on unstable
dynamics far away from equilibrium [17, 20, 21, 28, 45].
This work captures a new class of excitations for the
Heisenberg model: the phantom spin helices. These arehighly-excited yet long-lived metastable states. Theirstability is not a result of symmetry, but rather due to adelicate cancellation of interactions.
We have theoretically explored spin-helix states in systems in higher dimensions, for different spin quantum
numbers, and in various lattice geometries, which are notintegrable. We find that for the special initial conditionof a phantom helix state, the system does not relax tothermal equilibrium as one would naïvely expect, despite
the presence of strong interactions. Such dynamics in
non-integrable, many-body systems constitute examplesof “weak ergodicity-breaking”, or what are now known
as “quantum many-body scars” [22]. We note that whilevarious toy models hosting exact quantum many-bodyscars have already been discussed in the literature, such

1 Off-site interactions were observed for long-range dipolar interactions [36].

models are primarily theoretical constructs which are difficult to realize experimentally [46–49]. In contrast, wedemonstrate that one of the simplest examples of a manybody system (the XXZ Heisenberg model) can support
quantum many-body scars, a fact which has been overlooked thus far. The simplicity implies that probing suchscarred dynamics experimentally is relatively straightforward, and we propose a way to observe them with anextension of our current experimental setup to higher dimensions.
We expect phantom helix states to have applications
for quantum simulations of spin physics. We have demonstrated the potential of the phantom helix states as a sensitive tool to directly measure the anisotropy ∆. They
have revealed that even short-range interactions can leadto strong off-site interactions in spin models. This can
now be used to realize extended Hubbard models [32, 37]
including the quantum lattice gas (or t-V ) model [50]which supports a supersolid phase [51]. In the future,these long-lived helix states could be an intermediate stepin preparing other many-body quantum states or be used
for robust quantum sensing [52]. An intriguing questionis what will ultimately limit the stability of these states
if periodic boundary conditions are realized with ringshaped atom arrays [53]. Such studies are likely to provide new insight into the rich dynamics of Heisenberg
spin models.
Acknowledgements. We thank J. Rodriguez-Nieva
for helpful discussions, and J. Amato-Grill for important
advice. We thank Jinggang Xiang for experimental assistance, Julius de Hond for comments on the manuscript,
and M. Zwierlein for sharing equipment. Funding: We
acknowledge support from the NSF through the Center for Ultracold Atoms and Grant No. 1506369, theVannevar-Bush Faculty Fellowship, and DARPA (grant
number W911NF2010090). W.W.H. is supported in partby the Stanford Institute of Theoretical Physics. Author contributions: P. N. J., Y. K. L., H. L., W. W. H.
and W. K. conceived the experiment. P. N. J., Y. K. L.,
H. L., I. D., and Y. M. performed the experiment.

6
P. N. J., Y. K. L., and H. L. analyzed the data. W. W. H.
generalized the findings to higher dimensions. All authors discussed the results and contributed to the writingof the manuscript. Competing interests: The authors

declare no competing financial interests. Data and materials availability: The data that support the findingsof this study are available from the corresponding author
upon reasonable request.

[1] J. Berges, S. Borsányi, C. Wetterich, Phys. Rev. Lett. 93,
142002 (2004).
[2] M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, Phys.
Rev. Lett. 98, 050405 (2007).
[3] M. Gring, et al., Science 337, 1318 (2012).
[4] C. Eigen, et al., Nature 563, 221 (2018).
[5] D. A. Abanin, E. Altman, I. Bloch, M. Serbyn, Rev. Mod.
Phys. 91, 021001 (2019).
[6] S. Choi, et al., Nature 543, 221 (2017).
[7] J. Zhang, et al., Nature 543, 217 (2017).
[8] F. Arute, et al., Nature 574, 505 (2019).
[9] F. Flamini, N. Spagnolo, F. Sciarrino, Rep. Prog. Phys.
82, 016001 (2018).
[10] P. Jordan, E. Wigner, Zeitschrift für Physik 47, 631(1928).
[11] S. Sachdev, Quantum Phase Transitions (Cambridge
University Press, 2011), second edn.
[12] H. Bethe, Zeitschrift für Physik 71, 205 (1931).
[13] F. Franchini, An Introduction to Integrable Techniques
for One-Dimensional Quantum Systems (Springer International Publishing, 2017).
[14] M. Žnidarič, Phys. Rev. Lett. 106 (2011).
[15] M. Ljubotina, M. Žnidarič, T. Prosen, Nature Communications 8, 16117 (2017).
[16] M. Ljubotina, M. Žnidarič, T. Prosen, Phys. Rev. Lett.
122, 210602 (2019).
[17] D. Wei, et al., arXiv:2107.00038 ([cond-mat.quant-gas]
30 June 2021).
[18] V. Popkov, X. Zhang, A. Klümper, Phys. Rev. B 104,
L081410 (2021).
[19] A. C. Cassidy, C. W. Clark, M. Rigol, Phys. Rev. Lett.
106, 140405 (2011).
[20] P. N. Jepsen, et al., Nature 588, 403–407 (2020).
[21] P. N. Jepsen, et al., arXiv:2103.07866 [cond-mat.quantgas] (14 March 2021)[22] M. Serbyn, D. A. Abanin, Z. Papić, Nature Physics 17,675 (2021).
[23] A. B. Kuklov, B. V. Svistunov, Phys. Rev. Lett. 90,
100401 (2003).
[24] L.-M. Duan, E. Demler, M. D. Lukin, Phys. Rev. Lett.
91, 090402 (2003).
[25] J. J. García-Ripoll, J. I. Cirac, New Journal of Physics
5, 76 (2003).
[26] E. Altman, W. Hofstetter, E. Demler, M. D. Lukin, New
Journal of Physics 5, 113 (2003).
[27] J. Amato-Grill, N. Jepsen, I. Dimitrova, W. Lunden,
W. Ketterle, Phys. Rev. A 99, 033612 (2019).
[28] S. Hild, et al., Phys. Rev. Lett. 113, 147205 (2014).
[29] S. Trotzky, et al., Phys. Rev. Lett. 114, 015301 (2015).
[30] T. Secker, J. Amato-Grill, W. Ketterle, S. Kokkelmans,
Phys. Rev. A 101, 042703 (2020).
[31] D.-S. Lühmann, O. Jürgensen, K. Sengstock, New Journal of Physics 14, 033021 (2012).
[32] O. Dutta, et al., Reports on Progress in Physics 78,
066001 (2015).

[33] O. Jürgensen, F. Meinert, M. J. Mark, H.-C. Nägerl, D.S. Lühmann, Phys. Rev. Lett. 113, 193003 (2014).
[34] G. K. Campbell, et al., Science 313, 649 (2006).
[35] S. Will, et al., Nature 465, 197 (2010).
[36] S. Baier, et al., Science 352, 201 (2016).
[37] J. C. Amadon, J. E. Hirsch, Phys. Rev. B 54, 6364
(1996).
[38] M. F. Parsons, et al., Science 353, 1253 (2016).
[39] L. W. Cheuk, et al., Science 353, 1260 (2016).
[40] A. Mazurenko, et al., Nature 545, 462 (2017).
[41] T. Fukuhara, et al., Nature Physics 9, 235 (2013).
[42] T. Fukuhara, et al., Nature 502, 76 (2013).
[43] Z. Wang, et al., Nature 554, 219 (2018).
[44] A. K. Bera, et al., Nature Physics 16, 625 (2020).
[45] M. A. Nichols, et al., Science 363, 383 (2019).
[46] N. Shiraishi, et al., Phys. Rev. Lett. 119, 030601 (2017).
[47] D. K. Mark, et al., Phys. Rev. B 101, 195131 (2020).
[48] S. Chattopadhyay, et al., Phys. Rev. B 101, 174308(2020).
[49] S. Moudgalya, et al., Phys. Rev. B 102, 085140 (2020).
[50] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac,
M. Rigol, Rev. Mod. Phys. 83, 1405 (2011).
[51] H. Matsuda, T. Tsuneto, Progress of Theoretical Physics
Supplement 46, 411 (1970).
[52] S. Dooley, PRX Quantum 2, 020330 (2021).
[53] P. Scholl, et al., arXiv:2107.14459 [quant-ph] (30 July2021).

7
SUPPLEMENTARY MATERIALI.

PHANTOM SPIN HELICES IN HIGHER-DIMENSIONS, ARBITRARY SPIN QUANTUM
NUMBERS, AND NON-HYPERCUBIC GEOMETRIESIn this section we extend the phenomenology of stable phantom helices to Heisenberg models in higher dimensions,arbitrary spin quantum numbers, and non-hypercubic geometries. Specifically, we prove that for any given anisotropy
in the easy-plane |∆| ≤ 1, there exist phantom helices which are exact many-body eigenstates of the model, provided(i) the phantom condition (Eq. 2 of the main text) holds; and (ii) appropriate boundary conditions are taken.
A.

Model and spin-helix states

We consider the quantum Heisenberg XXZ model for any spatial dimensionality d, spin-S, and lattice geometry.
The Hamiltonian is given by a sum over pairwise nearest-neighbor interactions (we set Jxy = 1 for simplicity):H(∆) :=X(Six Sjx + Siy Sjy ) + ∆Siz Sjz ,

(5)

hiji

where Siα (α = x, y, z) are spin-S operators.
The generalization of a spin-helix state from the spin-1/2 case (Eq. 3 of the main text) is given byiYhyz|ψ(Q)i =e−iQ·ri Si e−iθSi |Sii .

(6)iHere |Sii is the local maximal spin state satisfying (Si · Si )|Sii = S(S + 1)|Sii and Siz |Sii = S|Sii . Q = (Q1 , · · · , Qd )is a d-dimensional wavevector parameterizing the winding rate and direction of the spiral, and ri is the coordinateof the spin at site i. Therefore, Eq. (6) locally describes a state created by rotation by angle θ from the z-polarizedstate around the S y -axis, before a winding in the S x -S y by a site-dependent angle Q · ri .
B.

Statement of results

For the model and state above, we have the following statements (we set the lattice spacing to a = 1 for brevity):
Theorem 1 (phantom spin helices for the hypercubic lattice). Consider a d-dimensional hypercubic lattice ofvolume L1 × L2 × ... × Ld and the anisotropic spin-S Heisenberg model with anisotropy ∆ defined on it. Let |∆| ≤ 1,
which we parameterize as ∆ = cos(Qp ) (the “phantom condition”) for Qp ∈ [0, π). Define the wavevector Qp := Qp x
where x is a binary vector x = (x1 , · · · , xd ) ∈ {−1, 1}d , and suppose that the linear dimensions Li are such that
Li = 2πmi /Qp for some mi ∈ N. Then, the spin helix with wavevector Qp and any polar angle θ is an exact eigenstateof the model with energy E = S 2 cos(Qp ) × #Links.
Theorem 2 (phantom spin helices for the 2D triangular and kagome lattices). Consider a 2-dimensional
regular triangular or kagome lattice, and the anisotropic spin-S Heisenberg model with anisotropy ∆ = −1/2 definedon it. Define Qp = 2π/3 so that cos(2π/3) = −1/2 = ∆. Then the spin helix with wavevector Qp = (2Qp , 0) or (4Qp , 0)and any polar angle θ, is an exact eigenstate of the model with energy E = S 2 cos(Qp ) × #Links, provided the latticedimensions are such that it is commensurate with the spiral pattern.
Remarks. Figure S1 shows an illustration of the phantom helix for a 2D square lattice, while figure S2 shows
the phantom helix for the 2D triangular and kagome lattices. Note the phantom helix state for the triangular lattice is known as the 120o Néel ordered state in condensed matter physics, which is the ground state of
the classical antiferromagnetic Heisenberg model. We show here that it is an exact eigenstate for√the √quantum
Heisenberg model for ∆ = −1/2. For the kagome lattice, the phantom helix state it is also known as the 3 × 3 state.
Proof. Define the local term of the Hamiltonian
hij = (Six Sjx + Siy Sjy ) + cos(Qp )Siz Sjz

(7)

8
P
so that H(cos(Qp )) = hiji hij . Consider a local part of the spin-helix state with wavevector Qp , specifically on apair of neighboring spins i, j:
z

y

z

y

|ψiij = e−iΦi Si e−iθSi |Sii ⊗ e−iΦj Sj e−iθSj |Sij


y
y
z
z
z
= e−iΦi (Si +Sj ) e−iθSi |Sii ⊗ e−iδΦji Sj e−iθSj |Sij

(8)
(9)

where Φi = Q · ri and δΦji = (Φj − Φi ). A simple but key property is that for the set-ups described in either Theorem,e−iδΦji = e−iQp or eiQp for any nearest-neighbor pair of sites.
for any hypercubic lattice. For the√ (Clearly this is±itrue
4π
3 = e∓iQp . For the kagome lattice the
3/2)
we
have
e
triangle lattice with primitive vectors b1 = (1,
0),
b
=
(1/2,
2
√
primitive vectors are b1 = (2, 0) and b2 = (1, 3) while the lattice vectors within each unit cell are a1 = 12 b1 , a2 = 12 b2
and so the result is the same as the triangular lattice).
With this in mind, we evaluate the action of hij on the state (dropping indices i, j for brevity)
h
i
z
z
y
z
y
e−iΦi (S ⊗I+I⊗S ) (S x ⊗ S x + S y ⊗ S y + cos(Qp )S z ⊗ S z ) e−iθS |Si ⊗ e∓iQp S e−iθS |Si
z

z

z

z

z

z

=e−iΦi (S ⊗I+I⊗S ) (I ⊗ e∓iQp S ) (S x ⊗ (S x cos(Qp ) ∓ S y sin(Qp )) + S y ⊗ (S y cos(Qp ) ± S x sin(Qp )) + cos(Qp )S z ⊗ S z )hiyy× e−iθS |Si ⊗ e−iθS |Si=e−iΦi (S ⊗I+I⊗S ) (I ⊗ e∓iQp S ) (cos(Qp )(S x ⊗ S x + S y ⊗ S y + S z ⊗ S z ) ∓ sin(Qp )(S x ⊗ S y − S y ⊗ S x ))hiyy
× e−iθS |Si ⊗ e−iθS |Si .

(10)

The term proportional to cos(Qp ) is Si · Sj , and the state on the RHS of the last line of the above equation is a
uniformly polarized state, so it evaluates to the original state up to a multiplicative factor
S 2 cos(Qp )|ψiij .

(11)
z

z

z

Now we just have to evaluate the term proportional to sin(Qp ). Ignoring the factor e−iΦi (S ⊗I+I⊗S ) (I ⊗ e∓iQp S ) we
have
h
i
y
y
∓ sin(Qp )(S x ⊗ S y − S y ⊗ S x ) e−iθS |Si ⊗ e−iθS |Si
y

y

y

y

= ∓ sin(Qp )(e−iθS ⊗ e−iθS ) ((S x cos(θ) + S z sin(θ)) ⊗ S y − S y ⊗ (S x cos(θ) + S z sin(θ))) |Si ⊗ |Si= ∓ sin(Qp )(e−iθS ⊗ e−iθS ) (cos(θ)(S x ⊗ S y − S y ⊗ S x ) + sin(θ)S(I ⊗ S y − S y ⊗ I)) |Si ⊗ |Si.

(12)

Now we make use of the fact thatS y |Si = iS x |Si(13)

1(S + − S − )).
(this follows from the definition of |Si as the highest-weight state, S x = 12 (S + + S − ), and S y = 2i
Therefore(S x ⊗ S y − S y ⊗ S x )|Si ⊗ |Si = 0

and Eq. (12) becomes
We add a trivial term

(14)

y

y

(15)

y

y

(16)

∓ sin(Qp )(e−iθS ⊗ e−iθS )i sin(θ)S(I ⊗ S x − S x ⊗ I)|Si ⊗ |Si.
∓ sin(Qp )(e−iθS ⊗ e−iθS )i cos(θ)S(I ⊗ S z − S z ⊗ I)|Si ⊗ |Si

to it, so that Eq. (12) is equal toyy±iS sin(Qp )(I ⊗ S z − S z ⊗ I)(e−iθS ⊗ e−iθS )|Si ⊗ |Si.

z

z

(17)

z

Reinstating the factor e−iΦi (S ⊗I+I⊗S ) (I ⊗ e∓iQp S ) which commutes with (I ⊗ S z − S z ⊗ I), we hence have
hij |ψiij = S 2 cos(Qp )|ψiij ± iS sin(Qp ) Sjz − Siz |ψiij .

(18)This is our final result. The (Siz − Sjz ) term cancels out in the bulk when summed over all sites (it telescopes), so
as claimed. H(cos(Qp ))|ψ(Qp )i = (S 2 cos(Qp ) × #Links)|ψ(Qp )i

(19)

9

FIG. S1.
Phantom spin helix for the square lattice. Each vertex carries a phase, which denotes the state’s local
expectation value hSi+ i = hSix i + ihSiy i. Here, χ := e−iQp . We have assumed the spiral lies fully in the transverse plane (polar
angle θ = π/2), but the latter condition can be straightforwardly lifted to allow for arbitrary polar angles. Thus, the depicted
helix has the particular wavevector Q = (Qp , Qp ), where ∆ = cos(Qp ) (0 ≤ Qp ≤ π). Other valid phantom spin-spirals for thisgeometry are those with wavevectors satisfying Qp = Qp x, where x ∈ {−1, 1}2 .

10

FIG. S2. Phantom spin helices for triangular and kagome lattices. As in the case for the square lattice, we label
each vertex with a phase denoting the local expectation value hSi+ i, defining χ := e−iQp . However, in this case, only Qp = 2π/3
and ∆ = −1/2 defines a valid phantom spin helix. Thus, the spin can only point in one of three directions in the S x -S y plane(denoted by three different colors), with relative angle 2π/3 between them. Note that the collinear neighbors of each red spin
are blue and green spins, and the interactions of the blue and green spins with the red spin cancel for each line. In this way,
one can understand the 2D phantom helix states as arising from simple “stacking” together of phantom helices of 1D chains.

Bibliography[1] Auerbach, A. Interacting Electrons and Quantum Magnetism (Springer-Verlag,
1994).
[2] Zhang, J., Pagano, G., Hess, P. W., Kyprianidis, A., Becker, P., Kaplan, H., Gorshkov, A. V., Gong, Z.-X. & Monroe, C. Observation of a many-body dynamicalphase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).
[3] Bernien, H., Schwartz, S., Keesling, A., Levine, H., Omran, A., Pichler, H., Choi,
S., Zibrov, A. S., Endres, M., Greiner, M., Vuletić, V. & Lukin, M. D. Probing
many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584(2017).
[4] Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nature Communications 6, 7654 (2015).
[5] Davis, E. J., Periwal, A., Cooper, E. S., Bentsen, G., Evered, S. J., Van Kirk,
K. & Schleier-Smith, M. H. Protecting spin coherence in a tunable Heisenberg
model. Phys. Rev. Lett. 125, 060402 (2020).
[6] Signoles, A., Franz, T., Ferracini Alves, R., Gärttner, M., Whitlock, S., Zürn, G.
& Weidemüller, M. Glassy dynamics in a disordered heisenberg quantum spin
system. Phys. Rev. X 11, 011011 (2021).
[7] Trotzky, S., Cheinet, P., Fölling, S., Feld, M., Schnorrberger, U., Rey, A. M.,
Polkovnikov, A., Demler, E. A., Lukin, M. D. & Bloch, I. Time-resolved observation and control of superexchange interactions with ultracold atoms in opticallattices. Science 319, 295–299 (2008).
[8] Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in opticallattices. Science 357, 995–1001 (2017).
[9] Vasseur, R. & Moore, J. E. Nonequilibrium quantum dynamics and transport:
from integrability to many-body localization. Journal of Statistical Mechanics:
Theory and Experiment 2016, 064010 (2016).
[10] Bertini, B., Heidrich-Meisner, F., Karrasch, C., Prosen, T., Steinigeweg, R. &Žnidarič, M. Finite-temperature transport in one-dimensional quantum latticemodels. Rev. Mod. Phys. 93, 025003 (2021).
169

[11] Ljubotina, M., Žnidarič, M. & Prosen, T. Spin diffusion from an inhomogeneous
quench in an integrable system. Nature Communications 8, 16117 (2017).
[12] Gopalakrishnan, S. & Vasseur, R. Kinetic theory of spin diffusion and superdiffusion in 𝑋𝑋𝑍 spin chains. Phys. Rev. Lett. 122, 127202 (2019).
[13] Ilievski, E., De Nardis, J., Medenjak, M. & Prosen, T. Superdiffusion in onedimensional quantum lattice models. Phys. Rev. Lett. 121, 230602 (2018).
[14] Bethe, H. Zur theorie der metalle. Zeitschrift für Physik 71, 205–226 (1931).
[15] Nichols, M. A., Cheuk, L. W., Okan, M., Hartke, T. R., Mendez, E., Senthil, T.,
Khatami, E., Zhang, H. & Zwierlein, M. W. Spin transport in a Mott insulatorof ultracold fermions. Science 363, 383–387 (2019).
[16] Brown, R. C., Wyllie, R., Koller, S. B., Goldschmidt, E. A., Foss-Feig, M. &Porto, J. V. Two-dimensional superexchange-mediated magnetization dynamics
in an optical lattice. Science 348, 540–544 (2015).
[17] Fukuhara, T., Kantian, A., Endres, M., Cheneau, M., Schauß, P., Hild, S.,
Bellem, D., Schollwöck, U., Giamarchi, T., Gross, C., Bloch, I. & Kuhr, S.
Quantum dynamics of a mobile spin impurity. Nature Physics 9, 235–241 (2013).
[18] Fukuhara, T., Schauß, P., Endres, M., Hild, S., Cheneau, M., Bloch, I. & Gross,
C. Microscopic observation of magnon bound states and their dynamics. Nature
502, 76–79 (2013).
[19] Hild, S., Fukuhara, T., Schauß, P., Zeiher, J., Knap, M., Demler, E., Bloch, I. &Gross, C. Far-from-equilibrium spin transport in Heisenberg quantum magnets.
Phys. Rev. Lett. 113, 147205 (2014).
[20] Koschorreck, M., Pertot, D., Vogt, E. & Köhl, M. Universal spin dynamics in
two-dimensional Fermi gases. Nature Physics 9, 405–409 (2013).
[21] Trotzky, S., Beattie, S., Luciuk, C., Smale, S., Bardon, A. B., Enss, T., Taylor,E., Zhang, S. & Thywissen, J. H. Observation of the Leggett-Rice effect in aunitary fermi gas. Phys. Rev. Lett. 114, 015301 (2015).
[22] Palzer, S., Zipkes, C., Sias, C. & Köhl, M. Quantum transport through a TonksGirardeau gas. Phys. Rev. Lett. 103, 150601 (2009).
[23] Sommer, A., Ku, M., Roati, G. & Zwierlein, M. W. Universal spin transport ina strongly interacting Fermi gas. Nature 472, 201–204 (2011).
[24] Krinner, S., Lebrat, M., Husmann, D., Grenier, C., Brantut, J.-P. & Esslinger,
T. Mapping out spin and particle conductances in a quantum point contact.
Proceedings of the National Academy of Sciences 113, 8144–8149 (2016).
170

[25] Cheuk, L. W., Nichols, M. A., Lawrence, K. R., Okan, M., Zhang, H. & Zwierlein, M. W. Observation of 2D fermionic Mott insulators of 40 K with single-site
resolution. Phys. Rev. Lett. 116, 235301 (2016).
[26] Parsons, M. F., Mazurenko, A., Chiu, C. S., Ji, G., Greif, D. & Greiner, M.
Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard
model. Science 353, 1253–1256 (2016).
[27] Mazurenko, A., Chiu, C. S., Ji, G., Parsons, M. F., Kanász-Nagy, M., Schmidt,
R., Grusdt, F., Demler, E., Greif, D. & Greiner, M. A cold-atom Fermi–Hubbard
antiferromagnet. Nature 545, 462–466 (2017).
[28] Jepsen, P. N., Amato-Grill, J., Dimitrova, I., Ho, W. W., Demler, E. & Ketterle,
W. Spin transport in a tunable Heisenberg model realized with ultracold atoms.
Nature 588, 403–407 (2020).
[29] Jepsen, P. N., Ho, W. W., Amato-Grill, J., Dimitrova, I., Demler, E. & Ketterle,
W. Transverse spin dynamics in the anisotropic heisenberg model realized withultracold atoms. Phys. Rev. X 11, 041054 (2021).
[30] Jepsen, P. N., Lee, Y. K., Lin, H., Dimitrova, I., Margalit, Y., Ho, W. W. &Ketterle, W. Catching Bethe phantoms and quantum many-body scars: Longlived spin-helix states in Heisenberg magnets. arXiv e-prints arXiv:2110.12043(2021).
[31] Jordan, P. & Wigner, E. Über das Paulische Äquivalenzverbot. Zeitschrift fur
Physik 47, 631–651 (1928).
[32] Ganahl, M., Rabel, E., Essler, F. H. L. & Evertz, H. G. Observation of complex bound states in the spin-1/2 Heisenberg 𝑋𝑋𝑍 chain using local quantum
quenches. Phys. Rev. Lett. 108, 077206 (2012).
[33] Wei, D., Rubio-Abadal, A., Ye, B., Machado, F., Kemp, J., Srakaew, K.,
Hollerith, S., Rui, J., Gopalakrishnan, S., Yao, N. Y., Bloch, I. & Zeiher, J.
Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion. arXiv e-prints
arXiv:2107.00038 (2021).
[34] Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonicatoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).
[35] Kuklov, A. B. & Svistunov, B. V. Counterflow superfluidity of two-species ultracold atoms in a commensurate optical lattice. Phys. Rev. Lett. 90, 100401(2003).
[36] Duan, L.-M., Demler, E. & Lukin, M. D. Controlling spin exchange interactionsof ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).
[37] García-Ripoll, J. J. & Cirac, J. I. Spin dynamics for bosons in an optical lattice.
New Journal of Physics 5, 76–76 (2003).
171[38] Altman, E., Hofstetter, W., Demler, E. & Lukin, M. D. Phase diagram of twocomponent bosons on an optical lattice. New Journal of Physics 5, 113–113(2003).
[39] Schrieffer, J. R. & Wolff, P. A. Relation between the Anderson and Kondo
Hamiltonians. Phys. Rev. 149, 491–492 (1966).
[40] Luttinger, J. M. & Kohn, W. Motion of electrons and holes in perturbed periodic
fields. Phys. Rev. 97, 869–883 (1955).
[41] Amato-Grill, J., Jepsen, N., Dimitrova, I., Lunden, W. & Ketterle, W. Interaction spectroscopy of a two-component Mott insulator. Phys. Rev. A 99, 033612(2019).
[42] Amadon, J. C. & Hirsch, J. E. Metallic ferromagnetism in a single-band model:
Effect of band filling and coulomb interactions. Phys. Rev. B 54, 6364–6375(1996).
[43] Secker, T., Amato-Grill, J., Ketterle, W. & Kokkelmans, S. High-precision analysis of Feshbach resonances in a Mott insulator. Phys. Rev. A 101, 042703(2020).
[44] Dimitrova, I., Jepsen, N., Buyskikh, A., Venegas-Gomez, A., Amato-Grill, J.,
Daley, A. & Ketterle, W. Enhanced superexchange in a tilted Mott insulator.
Phys. Rev. Lett. 124, 043204 (2020).
[45] Bradley, C. C., Sackett, C. A. & Hulet, R. G. Bose-Einstein condensation oflithium: Observation of limited condensate number. Phys. Rev. Lett. 78, 985–
989 (1997).
[46] Ketterle, W., Durfee, D. S. & Stamper-Kurn, D. M. Making, probing and understanding Bose-Einstein condensates. In Inguscio, M., Stringari, S. & Wieman,
C. E. (eds.) Bose-Einstein Condensation in Atomic Gases, 67–176 (IOS Press,
Amsterdam, 1999).
[47] Popkov, V., Zhang, X. & Klümper, A. Phantom Bethe excitations and spin
helix eigenstates in integrable periodic and open spin chains. Phys. Rev. B 104,
L081410 (2021).
[48] Zhang, X., Klümper, A. & Popkov, V. Phantom Bethe roots in the integrable
open spin- 12 𝑋𝑋𝑍 chain. Phys. Rev. B 103, 115435 (2021).
[49] Busch, T., Englert, B.-G., Rzażewski, K. & Wilkens, M. Two cold atoms in aharmonic trap. Foundations of Physics 28, 549–559 (1998).
[50] Lühmann, D.-S., Jürgensen, O. & Sengstock, K. Multi-orbital and densityinduced tunneling of bosons in optical lattices. New Journal of Physics 14,
033021 (2012).
172

[51] Ljubotina, M., Žnidarič, M. & Prosen, T. A class of states supporting diffusive spin dynamics in the isotropic Heisenberg model. Journal of Physics A:
Mathematical and Theoretical 50, 475002 (2017).
[52] Antal, T., Rácz, Z., Rákos, A. & Schütz, G. M. Transport in the XX chain atzero temperature: Emergence of flat magnetization profiles. Phys. Rev. E 59,
4912–4918 (1999).
[53] Gobert, D., Kollath, C., Schollwöck, U. & Schütz, G. Real-time dynamics in spin1
chains with adaptive time-dependent density matrix renormalization group.
2
Phys. Rev. E 71, 036102 (2005).
[54] Misguich, G., Pavloff, N. & Pasquier, V. Domain wall problem in the quantum
XXZ chain and semiclassical behavior close to the isotropic point. SciPost Phys.
7, 25 (2019).
[55] Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in
integrable quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016).
[56] Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-ofequilibrium 𝑋𝑋𝑍 chains: Exact profiles of charges and currents. Phys. Rev.
Lett. 117, 207201 (2016).
[57] Spałek, J. t-J Model Then and Now: a Personal Perspective from the Pioneering
Times. Acta Physica Polonica A 111, 409 (2007).
[58] Zhang, F. C. & Rice, T. M. Effective Hamiltonian for the superconducting Cu
oxides. Phys. Rev. B 37, 3759–3761 (1988).
[59] Anderson, P. W., Lee, P. A., Randeria, M., Rice, T. M., Trivedi, N. & Zhang,
F. C. The physics behind high-temperature superconducting cuprates: the plain
vanilla version of RVB. Journal of Physics: Condensed Matter 16, R755–R769(2004).
[60] Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys.
89, 035002 (2017).

173