Development of a new Dy quantum gas experiment

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Development of a new Dy quantum gas experiment

by

William Lunden

B.S., Stony Brook University (2013)

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

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Wolfgang Ketterle

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Development of a new Dy quantum gas experiment by

William Lunden

Submitted to the Department of Physics on January 17, 2020, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

Since the first realization of Bose-Einstein condensation in an atomic gas at the end of the twentieth century, ultracold atomic gases have become a widely adopted platform for the study of various quantum phenomena. In recent years, attention has increas-ingly turned to species with large magnetic dipole moments due to the much stronger long-range interactions that these species exhibit in comparison with the more com-monly studied alkalis. Dysprosium, with a magnetic moment of about 10IB, is the most magnetic atomic species and therefore has become an attractive platform for studying systems in which the long-range (dipole-dipole) interactions compete with or dominate over the contact interactions.

In this thesis I describe the design and optimization of a new dysprosium quantum gas machine. Apart from giving a detailed description of the components of the appa-ratus and their performance, I describe in detail the characterization and optimization of the "angled slowing" technique which is used to enhance the loading rate of our magneto-optical trap (MOT). I also describe in detail the production and detection of the first Bose-Einstein condensates (BECs) produced using the apparatus.

This thesis also contains a detailed description of the development of new control hardware and software which are used in the dysprosium experiment, but can be (and have been) used with other quantum gas experiments. On the hardware side,

I discuss the design of high-performance analog voltage control channels which offer

advantages over commercially available alternatives. On the software side, I discuss a laboratory control and logging database system which I designed, which both expands the capabilities of our control software and simplifies the storage of and accessibility of lab data.

Thesis Supervisor: Wolfgang Ketterle

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Acknowledgments

I have a lot of people to thank for a variety of things over the last 6.5 years. I owe my

advisor Wolfgang more thanks than I can describe here. From my first weeks until my last week he has helped me to develop my physical intuition, hone my sense of prioritization and how to be goal-oriented in the lab, and learn to assess and carefully consider my arguments before making them. He also trusted me to work in not just one, but two of his labs and on a variety of different projects. I've learned so much through both his direct mentorship and the projects I've worked on under him, and

I want to thank him for giving me so many opportunities to do so.

I also want to thank Professor Martin Zwierlein for all of the extremely helpful

advice he's given me over the years. Aside from all of his help with navigating course selections and doctoral exam preparations when I was a younger student, he also helped me to figure out that it was a good idea for me to switch from my first lab (working on lithium) to the dysprosium experiment. I also want to thank him for putting me in touch with Jamil from Vector Atomic, whom I'm excited to start working for when I leave MIT.

Alan Jamison has been a great friend and mentor over the years. He joined the Ketterle group as a postdoc only a year after I arrived, and in my earlier years he was a great person to ask all sorts of random physics questions. When I switched to the dysprosium experiment his leadership and advice were invaluable in helping me to understand this new platform, and how to best go about making the experiment work.

I spent my first 4 years of grad school working in BEC 5 with Jesse, Niki, and

Ivana, finishing the construction of and doing some first experiments with a bosonic lithium machine. All three of them helped me learn about BEC experiments super quickly, and gave me the opportunity to make meaningful contributions even early on, for which I'm very grateful. They've also been great friends, with whom I've gotten the chance to travel to both Italy (for the Varenna summer school) and to Bulgaria (for Ivana's wedding), plus have countless beers with.

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Li, Michael, and Pierre have been awesome labmates in my 2.5 years in Dypole (the dysprosium lab). It's been awesome seeing how quickly all three of them have become experts in various aspects of physics and lab work. And it's been a pleasure to learn about dysprosium alongside such talented people. I'm excited to see what happens next in the lab with such a strong team: Li's work ethic is extremely impressive, and this is something I was very grateful for during the long nights/overnights of debugging and data taking as we were optimizing our MOT and eventually evaporating to BEC. Michael's thorough experience with circuit design and broad problem solving skills are going to be invaluable as the needs of the lab continue to evolve. And Pierre's deep understanding of atomic physics and software savvy will likewise enormously benefit the lab for as long as he is there.

I also want to extend thanks to some of the friends I met in Boston outside of grad school, and with whom I had the chance to work on a variety of musical endeavors. The senses of accomplishment and teamwork that came out of these musical projects made the last several years extremely enjoyable, and also had a cross-pollinating effect, motivating me and giving me new perspective that helped with lab work.

Since 2014 I've been involved in a death metal band called The Beast of Nod. Paul, Brendan, and Nate have become my best friends over the years, probably because of the sheer number of things we've done together: we wrote and recorded three CDs in a variety of studios around the state, toured a large chunk of the U.S., filmed music videos and performance videos, shared the stage with internationally acclaimed metal artists, and developed a genuine online fanbase. So I'm grateful to have met these guys and have had the chance to have all of these great experiences (and more to come!).

In 2016 I joined a progressive metal project called Episodes that was formed at Berklee college of music. I owe a lot of my musical development and understanding of live performance technology to Amaury, OG, Billy, and David, who are all insanely talented and were a genuine pleasure to perform with. I wish we'd been able to tour more together, but it was great fun while it lasted, and I'm excited for the future incarnation of Episodes that's in the works.

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In 2018 I started a new project called Mimesis, with three super talented musicians I'd met in the New England music scene: Jen, Tony, and Kilian. We pulled off an album which combined aspects of modern rock and metal in a way that none of us had heard before, and the reception has been extremely positive. I'm extremely grateful to all three of them for working on this project with me, and for everything I've learned from them about multimedia, promotion, the music industry.

I also want to separately thank Jen for being such a supportive partner and positive influence on my life. Her disciplined approach to life is inspiring when it seems like there is an overwhelming number of things to do. Certain periods of the last year of grad school were particularly draining, and I'm super grateful for her endless patience when I was semi-sleep deprived for weeks at a time.

Finally, I want to thank my family for their endless support, both during grad school and in all the years before it. My mom and step father Lewis gave me a lot of encouragement in the right directions all through high school and college, helping me to get on a good path without forcing me onto it, and I'm very grateful for this. Moving to MA meant being able to see my father more often too, and I'm grateful for all of the life advice and support he gave me during my grad school years. My grandparents have also always been extremely supportive of my academic endeavors, and I want to thank them for that.

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Chapter 1 Introduction

As time goes on, mankind's ability to meaningfully describe the world around us

-that is, to accurately predict the behavior of real physical systems -- is increasingly limited not by the validity of our physical models but by the tractability of the nec-essary calculations. This is especially true when it comes to describing the quantum mechanical behavior of systems of many particles, where the required number of states to keep track of grows exponentially with the system size. Sometimes, we luck out and can apply approximations such a mean-field approach, where we essentially solve a self-consistent single particle problem in place of the true manybody problem. A clas-sic example of this is the description of N identical, contact-interacting bosons in an external potential. Formally, one must solve a problem with a potentially enormous Hamiltonian which acts on the N-body, symmetrized wavefunction [57]:

,Q1

fI 1 V -0 - )(1.1)' )=

H =

EN

2 +ZNV(rN) < rij) (1.2)

Fortunately, much can be learned about the system by instead solving the

single-particle "non-linear Schrodinger equation," also known as the Gross-Pitaevski equa-tion [57, 22j:

-$29247r a.

ih() ( hV + V(r) + 12(r-) 1 () (1.3)

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which is a much more tractable problem.

It is not always possible to construct a meaningful mean-field description of a prob-lem, and so alternative approaches to finding a solution must be considered. Naive numerical methods, such as diagonalization of a manybody Hamiltonian, are usually only computationally tractable for small numbers of particles. More sophisticated ap-proaches such as density-matrix renormalization group (DMRG) have been successful primarily in the case of 1D systems, but most 2D and 3D manybody systems remain computationally intractable [60].

An alternative approach to studying quantum manybody problems is to develop a "quantum simulation" of the problem of interest [19, 24, 26, 1]: a different quantum system which is easy to manipulate and measure is used to implement the Hamilto-nian, and the measured behavior of this system can be used to understand the Hamil-tonian of interest. Quantum degenerate gases - dilute ensembles of atoms which have been cooled to the point that the effects of quantum statistics dominate the behavior of the gas - have become an increasingly popular platform for developing quantum simulations of a wide variety of Hamiltonians. Examples include simulation of gauge fields [20], realization of spin-orbit coupling [8], and the observation of phase tran-sitions in lattice systems [22]. In addition to allowing us to simulate Hamiltonians related to real-world problems of interest, the tunability of quantum gas systems also opens up the consideration of totally novel Hamiltonians. This enables the study of the physics of systems which have no known analog in nature, such as the spin-orbit coupling of high-spin bosons [73].

1.1 Bose-Einstein Condensation

Quantum degenerate atomic gases of bosons1 exhibit a striking behavior called Bose-Einstein condensation. Bose-Bose-Einstein condensation is a phenomenon that occurs when an ensemble of particles obeying bosonic statistics is cooled down to the point that

'Individual atoms have a total composite spin (consisting of the summed spin angular momenta of all nucleons and all electrons) which is either integer or half-integer, meaning that the atoms are respectively composite bosons or composite fermions.

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entropy is maximized when the absolute ground state is macroscopically occupied [46]. The resulting state of matter - the BEC - is essentially made of 103 - 106 copies

of the ground state wavefunction (for the case of the ideal noninteracting BEC) on top of one another. The large particle number allows for the techniques of atomic physics, including various imaging techniques, to be used to manipulate and probe the behavior of the condensate, allowing for an unprecedented level of control over

an inherently quantum system [46, 57].

The temperature (and corresponding density) at which condensation occurs can be understood with a simple physical picture. Let us consider the atoms in the ensemble to be individual wavepackets, with a spatial extent given by their thermal deBroglie wavlengths [46]:

27rh2

AdB = _(1.

mkBT

where m is the atomic mass, kB is the Boltzmann constant, and T is the temperature of the ensemble. The interparticle spacing is given by the inverse of the atomic number density n. If the temperature is lowered, eventually, the deBroglie wavelength will be on the order of the interparticle spacing:

nAB~ 1 (1.5)

and the particles will "overlap" with one another. This corresponds to the notion of the particles losing their distinguishability, and beginning to macroscopically occupy the same state. A more rigorous calculation yields that the critical value of nAdB) the so-called phase space density of the ensemble, for Bose-Einstein condensation is actually [57, 46]

nA 3 -2.612 (1.6)

The Bose-Einstein condensation of dilute atomic gases, first achieved in 1995 [5, 16], opened up a new field of quantum research. The seminal work on achieving BEC soon led to the preparation of quantum-degenerate Fermi gases [47], the loading of ultracold (i.e., quantum degenerate) atoms into optical lattices [22], and a wide

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variety of other milestones. Initially a topic of study in its own right due to its relationship to e.g. superfluid He and superconductivity, the BEC has increasingly become a general platform for the study of quantum mechanical phenomena. This is due to the unprecedented level of control over a quantum system that a BEC offers to experimenters: highly tunable "designer" single-particle and manybody Hamiltonians can be implemented by addressing the BEC with the tools of atomic physics such as Raman transitions, periodic optical potentials, and Feshbach resonances.

1.1.1 The Dipolar Quantum Gas

Many real quantum systems involve interactions long-ranged interactions. The re-alization of quantum gases with long-range, anisotropic interactions opens the door to studying a variety of effects which cannot be studied with the purely contact-interacting alkali species that were first Bose-condensed [34]. The effects range from roton instabilities in anisotropic trap geometries [68] to interacting lattice gases which don't require superexchange [6]. Dipolar quantum gases - both BECs and degenerate Fermi gases - have thus increasingly become a focus of attention within the atomic physics community, with experimental progress being made on producing and study-ing gases of both ultracold polar molecules and of dipolar atoms.

The interaction potential between two dipoles oriented along a guiding magnetic field can be written as [34]

pop 2 1 - 3 cos2 0

Udd = 4o2 1 (1.7)

where p is the magnetic moment of a dipole, 0 is the angle between between the quanti-zation axis and the internuclear separation, and r is the magnitude of the internuclear separation. This interaction is repulsive in the case of side-by-side orientations of the dipoles, attractive in the case of head-to-tail orientations, and continuously variable as a function of 0 between these two cases. The effect of this interaction potential on the behavior of the quantum gas can be quite striking; Figure 1-1 shows some examples of phenomena which have been observed so far in dipolar atomic gases.

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n1 (a.u.)

a

~f 1.0F$ 0A 06k

0.2 ,

1 50 rns bitd 10 00 Ms 20000 0 50 100 150

Nubrof atoms N Atom. per-xe

Figure 1-1: Some recently observed phenomena in ultracold dipolar gases. Top left: the roton-like dispersion of a dipolar gas in an anisotropic trap [14]. Top right: the observation of supersolid-like properties in a dipolar BEC [13]. Bottom: droplet

formation as reported in [30].

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1.2 Why Dysprosium?

The path to quantum degeneracy in all of the stable alkalis (Li, Na, K, Rb, Cs) has been paved for decades, and an enormous number of systems have been realized using these species, ranging from BCS pairing of fermions [70] to lattice-based spin models [52]. With alkalis alone it is possible to create bosonic or fermionic systems (or mixtures thereof) in highly controllable potentials and with tunable interactions

[57]. Alkalis are typically cooled, trapped, and manipulated entirely using light which

addresses the lowest-lying S-to-P transition of the single valence electron (further split into the D1 and D2 lines by fine structure and into further manifolds by the hyperfine interaction, of course). The properties of the transition are similar across all of alkalis, namely that they are all optically accessible and all have linewidths that are suitable for creating magneto-optical traps (MOTs). Although there are virtually endless systems that can be realized with alkalis, these species have their limitations. Namely,

1. The interactions between the atoms are completely dominated by short-range

interactions. At low temperatures, the atoms interact with an effective isotropic contact potential. This precludes the study of, e.g., lattice models with signifi-cant next-nearest neighbor interactions.

2. Unlike alkaline earth and transition metal species, they possess no narrow transitions (loosely speaking, less than 1 MHz), which can be used for lower-temperature Doppler cooling, population shelving, and optical clocks.

3. The optical Raman transitions between hyperfine sublevels, which are often

leveraged for simulating gauge fields, effectively require that the single-photon detuning be the order of the fine structure splitting. This means that the resultant heating due to Rayleigh scattering from each beam limits the lifetime of the cold Raman-coupled system [74]. The ratio of Raman coupling strength to heating rate can be made arbitrarily large by using larger detuning and more laser power in atomic species in which there is already spin-orbit coupling in

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the electronic ground state (see Chapter 2 for more details).

Atomic dysprosium offers a platform for quantum gas experiment which suffers from none of these limitations. As will be described in Chapter 2, Dysprosium has a large magnetic moment of p 0 10pB in its ground state, approximately ten times higher than the corresponding moment of alkali atoms and in fact the highest of any atomic species [49]. As a result, the dipole-dipole interaction between two dysprosium atoms is up to 100 times stronger than the corresponding interaction between alkalis; combined with the tunability of the contact interaction strength by way of Feshbach resonanes (also described in Chapter 2), this means that dysprosium can be studied in regimes where either the contact interaction dominates, the dipolar interaction dominates, or the two interactions are comparable.

Dysprosium also has a wealth of optically accessible, closed (or mostly closed) transitions with different linewidths, allowing for a high level of control over slowing, cooling, and trapping using different lasers.

Finally, the high angular momentum of the Dy ground state means that Raman transition amplitudes among ground state sublevels scale favorably compared to the heating rate as large detunings are used; this alleviates a practical limitation of im-plementing Raman-coupling based synthetic gauge field schemes in alkali species [74].

1.3 Scientific Goals

The scientific goals of our experiment involve the exploration of dipolar-interacting systems which involve more than one spin state (i.e., more than one ground state

my projection). Specifically, we are interested in studying the spin-orbit coupling of

dipolar-interacting atoms, in two and possibly three dimensions. Because the valid-ity of mean-field approaches to the problem of an interacting 2D spin-orbit coupled system of bosons is questionable, it is an interesting goal in its own right to realize and study the properties of such a system's ground state [73]. 2D spin-orbit coupling of dipolar-interacting bosons is also predicted to give rise to exotic quasicrystalline ground states

[211,

and so realizing such a system would constitute a first detection of

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these states. By tuning the dipolar interactions with a tilted or rotating [66] guiding magnetic field, we can also study the effect of dipolar interactions on ID spin-orbit coupled bosons in both the "classic" Raman-coupled scheme [8, 36, 12] and in topo-logically interesting lattice-based spin-orbit coupling schemes.

Unfortunately, the rapid dipolar relaxation of spin mixtures of bosons precludes the study of systems with more than one spin state populated unless measures to suppress this relaxation are taken [8]. The first goal of our experiment is thus to attempt to suppress dipolar relaxation by means of geometrical constraints imposed

by optical lattice potentials.

In the section below, spin-orbit coupling and the physical mechanisms that can lead to the suppression of dipolar relaxation are discussed in detail.

1.3.1 Spin-Orbit Coupling Schemes

Spin-orbit coupling is a phenomenon that arises naturally in the description of elec-trons in many real systems and materials. The motion of an electron through an electric potential, such as the crystalline potential of a solid material, causes it to see an effective magnetic field, which in turn gives rise to a spin-dependent energy shift. From special relativity, we know that a particle moving with velocity 6Y through an electric field $ sees an effective magnetic field B given by [51]

-_ 1

B = -E x V (1.8)

c

For an electron with spin ' = A8, there is then a motional- and spin- dependent

energy shift given by

_ geA~B.

soc h uB (1.9)

If we associate the electric field with a potential V and denote the electron's

momen-tum as p-', we generically end up with a term in the Hamiltonian of the form

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In real systems, this phenomenon has a number of effects:

* Spin-orbit coupling contributes to the fine structure splitting in atomic

transi-tions.

* It creates the spin galvanic effect in solid materials: because spin and momentum

are locked in the band structure of the material, preparing a spin density induces a charge current to flow.

* It also creates the inverse spin galvanic effect: inducing a charge current gives

rise to a transverse spin density.

* It is the origin of the spin hall effect in solid materials: spins accumulate on the

outsides of a current-carrying piece of material, analogous to charge accumula-tion in Hall effect

* It gives rise to topological states like 2D and 3D topological insulators

1.3.2 Suppressing Dipolar Relaxation

A consequence of dysprosium's large dipole-dipole interaction strength is that dipolar

relaxation - inelastic two-body collisions which couple internal angular momentum to external angular momentum, converting Zeeman energy into kinetic energy - occurs at a much higher rate than in alkali species. Pairs of atoms in the absolute lowest-energy state (m = -8 in the case of 1 6 2_{Dy) cannot undergo dipolar relaxation because}

there is no lower-energy state to flip into, and so clouds which are spin-polarized into this state are collisionally stable.

Spin-orbit coupling requires that at least the next-highest spin state is populated. As will be discussed in Chapter 2, the rate of dipolar relaxation of fermions is lower than that of bosons, and can be suppressed by increasing the magnitude of the guiding magnetic field. While the suppression of dipolar relaxation in fermions scales as V'IB [25], the enhancement of dipolar relaxation in bosons scales by the same factor. In

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so far, a 1D Raman-coupling scheme, the loss rate of the bosonic spin mixture was prohibitively high and a fermionic spin mixture was utilized instead [8, 9].

A short (< 10ms) lifetime was observed for Bosons even at low magnetic fields [8], suggesting that suppressing dipolar relaxation by working at small controlled

fields2, would not be a feasible approach. We thus are interested in implementing new approaches to suppressing dipolar relaxation by confining the geometry of our system to one or two dimensions.

One approach, which has been demonstrated in Cr atoms [56], is to load the atoms into 1D tubes (i.e., freeze out motion along two directions using a 2D optical lattice) and point the guiding field for the atoms along the unconfining direction. To conserve angular momentum, dipolar relaxation of a spin mixture in this configuration necessitates the creation of transverse relative motion of the particles; however, if the released Zeeman energy falls into the bandgap of the transverse directions, the then the dipolar relaxation process is suppressed. This is illustrated in Figure 1-2. For the lattice depths we can achieve, this approach would require very careful control of a small magnetic field - a technical challenge. If successful, it could allow for investigation of 1D spin-orbit coupled Hamiltonians along the unconfined direction.

1.4 This Thesis

The work described in this thesis constitutes the development of a new experiment to investigate the physics of dipolar Bose gases. A new apparatus for creating Bose-Einstein condensates of dysprosium is described in detail, including the array of cooling and trapping techniques used to create a degenerate gas. This description includes a detailed report of "angled slowing," a new technique for which we have performed the first rigorous characterization and which we believe to be of general

2

This is a technical challenge in its own right. At large magnetic fields, stray transverse compo-nents of the magnetic field are "projected out", and magnetic field control is essentially equivalent to current control in the coils generating the guiding field; at low magnetic fields, precise control of the magnetic field necessitates measurement and stabilization against stray fields. The scale of distinction between "high" and "low" is approximately a few hundred mG - the magnitude of the earth's magnetic field.

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( A1 (a) (r(b) 0.4 . 0.0 008 a IT ~ 0.04-.2- * 0.02-P 0.00._____________ ______ 0 40 80 120 160 20 60100 140

-0.2 0.0 0.2 0.4 Larmor frequency (o,/2xr (kHz) Larrmor frequency jj,i2n (kHz) Angle (rad)

Figure 1-2: Suppression of dipolar relaxation in ID tubes as reported in [56]. Left: the dependence of the temperature of the confined atoms after a fixed hold time on the angle between the guiding magnetic field and the long axis of the tubes. Center: the population of the first excited band of the lattice as a function of the Zeeman energy of the atoms in the guiding field. Right: the temperature of the atoms as a function of the Zeeman energy after a fixed hold time for two different lattice depths (the shaded regions correspond to the widths of the first excited band in each case).

interest to experiments employing narrow transitions for magneto-optical trapping

[45]. In addition to the description of the apparatus, the detailed description of two technology development projects is given. The first is the development of low-noise, high-precision analog control voltage hardware; initially developed to improve upon the limitations of commercially available analog channels for precision work on an-other experiment, this newly developed hardware forms an isolated, low-noise control backbone of the dysprosium apparatus. The second is the development of a database backend and associated applications for our laboratory control system, which has extended the interoperability, flexibility, and automation capabilities of the exist-ing Cicero control system; this new software has been implemented in several labs, including the dysprosium experiment.

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Chapter 2 Properties of Dysprosium

A fairly unremarkable element outside the field of atomic physics, dysprosium

pos-sesses a number of properties which make it a very attractive platform for studying physics [15, 42, 7, 10, 29, 48, 49, 44, 67, 11] in the ultracold regime. Most of these properties arise from its [Xe]4f1 06s2 electronic structure: a large (P ~ 10pB) mag-netic moment, several closed (or nearly closed) optical transitions with a variety of linewidths, and spin-orbit coupling in the ground state which enables long-lived Ra-man coupling schemes.

2.1 Isotopes of Dysprosium

The four most abundant isotopes of dysprosium are 16 1Dy (18.9% abundance), 162_Dy

(25.5%), 1 63_{Dy(24.9%), and} 164_{Dy (28.3%).} 162_{Dy and}164_{Dy are bosons with no}

nuclear spin (and hence no hyperfine structure), while 161Dy and 1 6 3_{Dy are fermions}

with nuclear spin I == [49, 41]. The work described in this thesis was done using the 162_{Dy isotope (see Figure 2-1).}

This isotope was chosen from between the two abundant bosons out of convenience based on available AOMs and the distance between the dysprosium 626 nm resonance and the available spectroscopic lines in our iodine reference cells. In the future the work may be extended to other isotopes. As has been demonstrated by other groups [40, 48] there is no need for consideration of additional repumping schemes when

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4-4

913 MHz

E

164 162

Dy Dy

Ti:Sapphire Cavity Control Voltage (Arbitrary)

Figure 2-1: Single-beam absorption spectroscopy at 421 nm of the atomic beam, mea-sured transverse to the direction of beam propagation, showing the presence of both abundant bosons and a variety of hyperfine states of the abundant fermions in the atomic beam. The isotopes have been identified by comparison to the spectra reported

in [49].

switching isotopes, and it is a straightforward matter of adjusting the frequencies of the cooling lasers to form a MOT of a different isotope.

2.2 Electronic Structure

of

162

_Dy

The electronic structure of dysprosium is rich, giving rise to many transitions that are optically accessible and which have a useful range of linewidths. The full electronic structure of the ground state is

(1s22 2_2p6_3s2_3p6_3d'0_4s2_4p6_4d'0_5s2_5p6_)4f'0_6s2 _(2.1)

A cartoon depiction of the electronic structure is shown in Figure 2-2. Several of the

strong optical transitions only involve the excitation of one of the two 6s electrons to a 6p state; in this sense, Dy's spectrum has many similarities to the spectra of alkaline earth species. Transitions involving the 4f electrons give rise to additional

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Ground State

5

18

[Xe]4f°6s

2

Figure 2-2: Cartoon depiction of 11 2Dy, showing the electrons which are relevant to the transitions and properties used in quantum gas experiments. The green 4f shell has been unconventionally placed outside of the 6s shell to emphasize the larger spatial extent of the radial wavefunctions of these high-i orbitals.

lines which can be very narrow; for example, the 4f to 5d transition at 1001nm was recently measured to have a linewidth of around 27r x 10 Hz [64].

Although seemingly intractably complicated (see Figure 2-4), the spectrum con-tains a few optically allowed transitions which have been well characterized and pos-sess useful properties for laser cooling. The three transitions of interest to us (421 nm,

626 nm, and 741 nm), along with their electronic configurations [33], are shown in

Fig-ure 2-3. As shown in the figFig-ure, the 421 nm and 626 nm transitions involve the same orbital configuration in the excited state, but different electron spin orientations of the outer (n=6) electrons. The singlet transition is the usual, allowed transition and hence has a larger matrix element and broader linewidth than the triplet transition, which is only allowed as a result of spin-orbit coupling in the atom.

2.3 Angular Momentum and Magnetic Moment

The ground state of Dysprosium has a large magnetic moment of approximately

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421nm 4f'0(1)6s6p(1_Pi) 626 nm 4f"(5 )6s6p(3P1) - 741 nm 4fP(6Ho)5d6s2 sK5 4f'06s2 (si) J=8 J=9

Figure 2-3: The three transitions relevant to our experiment. The 421 nm and 626 nm transitions involve the same electron orbital configurations in the excited states, but correspond to different orientations of the electron spins of the outer 6s electrons: the broad 421nm transition is a spin singlet-to-singlet transition while the narrower

626nm transition is an intercombination line (i.e., a spin singlet-to-triplet transition).

separately considering the orbital and spin angular momentum of all of the valence electrons. In reality, the two angular momentum components couple and the total atomic angular momentum is best described in terms of composite angular momentum

J=L+S.

First we consider the contribution of the orbital angular momentum of the outer

electrons. The 6s electrons possess no angular momentum of course, but the 4f

electrons each carry up to 3h of angular momentum, depending on their m, quantum

number. Only the angular momenta of unpaired electrons contribute to the total atomic angular momentum, and so we must identify which electrons are unpaired. By applying Hund's rules [49], we can conclude that the arrangement of the 10 electrons in the 4f shell should leave the electrons with mi = {3, 2, 1, 0} unoccupied. This is

depicted in Figure 2-5.

Thus the sum of the orbital angular momentum projections of the 4f electrons

is L, = Oh + lh + 2h + 3h = 6h, implying a total angular momentum of L = 6.

Considering now the contribution of the spin angular momentum, we again note that only unpaired spins contribute to the total atomic spin angular momentum. Thus only the 4 unpaired 4f electrons contribute their spin angular momentum, with each

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Energy Levels of Dysprosium u

I

U

i

U

i

U U = U U U -- 40000 =35000 30000 25000 - 20000 15000 10000 5000 J=5 J=6 J=7 J=8 J=9 J=100 cm4

Figure 2-4: A portion of the energy level spectrum of Dy from the NIST database

[33], showing the energy in units of wavenumbers for states of various total angular

momentum. The ground state is shown at zero energy with J = 8. Orange lines

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m,=+3

m, = +2

M, = +1

m, =O

mm =

-1

m,

-2

m,=

-3

Figure 2-5: Visualization of the arrangement of the electrons among the orbital an-gular momentum m, levels in the 4f shell in dysprosium. The four unpaired electrons

have orbital angular momenta which align with both each other and with the spin angular momenta, giving rise to the large ground state angular momentum of J = 8.

contributing 1h.

We can now consider the magnetic moment that results from all of these contri-butions to the total atomic angular momentum. Again appealing to Hund's rules, we conclude that because the 4f orbital is more than half-full, that the state with the largest possible total angular momentum J corresponds to the ground state; this is equivalent to the physical picture that all of the constituent orbits and spins contribute angular momentum in the same direction. Noting that the g-factor for electron orbital angular momentum is exactly 1 and that the g-factor for electron spin is approximately 2, we can calculate the total magnetic moment as

3 3

P = -

S

1gimipB - 9s pB ~ O B (2.2)

m1=0 i=0

The proper description of the angular momentum and hence magnetic moment of the atom is given by considering the total angular momentum J = 8, with magnetic

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I

Mo.

J=8

m -8 -7 -6 -5

--Figure 2-6: The angular momentum structure of the J = 8 ground state and any of

the J = 9 excited states in dysprosium. The three arrows indicate transitions from

the mj = -8 state, with the shading of the arrows indicating the relative strengths

of the three transitions.

p = -gj mJpB (2.3)

The experimentally measured value of 9 for bosonic dysprosium is gj = 1.2415867

[49], yielding a moment for the polarized ImI = 8 states of 9_.9 3 2

pB-For i62Dy, the ground state has a multiplicity of 17, corresponding to

integer-spaced projections mj of the total angular momentum J onto the quantization axis running from -8 to

+8.

Since the nuclear spin of both bosons is 0, there is no hyperfine structure in the ground state for i6 2Dy or 16 4Dy. All of the excited states of the

transitions we address in our experiment have J = 9, which has a multiplicity of 19.

Since (for reasons discussed in the following chapters) we are usually addressing atoms that are in the polarized state my = -8, it is worth noting how this large angular

momentum structure gives rise to unusually different Clebsch-Gordon coefficients for transitions to different excited state my levels.

Specifically, if we consider the case of electric dipole transitions from the mj = -8

state to m3 = {-7, -8, -9} by respectively

o-,

7r, .+ light, then we find that the ratio

of the corresponding Clebsch-Gordon coefficients is 1:}: , corresponding to a ratio

of transition strengths of 1:0.11:0.0065. The full ground and excited state sublevel structure, as well as arrows indicating the transitions from the my = -8 state and

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2.4 Scattering: Dipolar Interactions and Feshbach

Resonances

Dysprosium possesses an interesting array of scattering properties. The large mag-netic moment of dysprosium means that the strength of two-body dipole-dipole inter-actions is much higher than is the case in alkali species, leading to both the persistence of higher partial wave scattering at low temperatures and enhanced rates of (inelas-tic) dipolar relaxation. The large number of possible electronic configurations also gives rise to a dense spectrum of Feshbach resonances [49, 7], resulting in a sensitive dependence of both elastic and inelastic scattering on the magnetic field experienced

by the atoms.

2.4.1 Cold Collisions and Interaction Potentials

Many properties of the interactions between dysprosium atoms in an ultracold gas can be described using a combination of short range interactions and dipolar inter-actions. At short ranges, the potential is complicated, containing contributions from the van der Waals (06) interaction, as well higher-order dispersion terms. The short range potential is usually approximated by a pseudopotential which gives a useful description at the low collisional energies that occur at ultracold temperatures; most commonly, a contact pseudopotential is used:

V(r) = gJ(r) (2.4)

where 6(r) is a delta function and

47th

2

g = a (2.5)

m

where m is the mass of a single atom and a is the s-wave scattering length [57]. The scattering length is a notion that arises from the consideration of the scattered wavefunction in a two-body scattering problem; it is the value of the relative radial

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coordinate at which the scattered wavefunction amplitude crosses zero. The scattering length characterizes the behavior of the wavefunction away from the complicated oscillations it exhibits at short ranges (where the scattering potential is relatively complicated). This is specifically the s-wave scattering length because, as will be discussed below, at low temperatures only spherically symmetric s-wave scattering occurs for the short-range interaction.

The dipole-dipole interaction is a longer-ranged interaction which is also respon-sible for many of the collisional properties of dysprosium. For two atoms whose mag-netic moments are aligned along a guiding magmag-netic field, the interaction potential is [34, 57]

po92 1 -3 cos2Og

Udd - P 3 (2.6)

47 r

where p is the dipole moment of an atom, r is the separation, and 0 is the angle between the guiding field and the separation vector. The moment of a polarized dysprosium atom is approximately 10 times as large as the moment of a polarized alkali (whose magnetic moment arises almost entirely from the spin of the single

s-shell valence electron), meaning that the dipolar interaction between two dysprosium atoms is 100 times stronger than alkali-alkali dipolar interactions.

2.4.2 Partial Waves

For low-energy scattering we often use the method of a partial wave expansion to determine the scattering cross section. This gives us the usual results that for very low energies, identical bosons only interact through s-wave scattering and thus have scattering properties that are well-described by the notion of a scattering length, and identical fermions do not interact because they do not have sufficient relative momentum to exceed the centrifugal barrier for p-wave interactions (and are forbidden from interacting through s-wave scattering) [57].

The fact that higher-l partial waves become negligible in the limit of low energy can be seen by considering how the scattering phases associated with each partial wave scale with the momentum k. Borrowing a result from general scattering theory (reference

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37 in [34]), we know that for a scattering potential of the form U oc ,

kk21+1

for I < (n-3)

lim J, (k) = 2 (2.7)

k-0 _k"-2 _otherwise

For the van der Waals interaction potential U oc

y

which constitutes the dominant contribution to short-range interactions discussed above, we see that J, oc k, 6p oc k

and 6

d,... oc k4_{. Thus as claimed, the relative contributions of 1 > 0 partial waves}

vanish as k goes to zero.

For the dipole-dipole interaction U oc , we instead see that all phase shifts scale

with the first power of k. Thus, we must include all partial waves even at very low energies, invalidating the treatment of the scattering problem through truncation of the partial wave expansion to only one or two terms. Treatment of scattering due to the dipole-dipole interaction can be carried out in the first Born approximation [25], leading to such results as the scaling of the inelastic collision rate with the magnetic field mentioned in Chapter 1.

2.4.3 Feshbach Resonances

The s-wave scattering length, which characterizes the short range interactions of an ultracold gas, can be experimentally tuned by taking advantage of a phenomenon called Feshbach resonance.

Consider the interaction potential between two free atoms with some specified quantum numbers. The set of quantum numbers is referred to as a channel. The scattering properties between the atoms in the original "entrance" channel depend on the presence of the bound state in the other "closed" channel as long as some coupling between the channels exists. Even a small coupling can give rise to strong modification of the scattering in the entrance channel if the energy of the bound state in the closed channel is tuned into the vicinity of the energy of the scattering particles. This is illustrated in Figure 2-7. In quantum gas experiments, this tunability comes from the fact that the different channels correspond to atoms in different combinations

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>% ... Closed Channel

... ...

Ua -Coupingy. ~ Entrance Channel

Interatomic Separation

Figure 2-7: Schematic depiction of a Feshbach resonance. Two free particles with an energy indicated by the orange arrow interact with the orange potential curve denoted "Entrance Channel." The energy of the free particle pair is nearly-resonant with a bound state in another interaction potential, drawn in blue and denoted "Closed Channel." A coupling between these potentials means that the closed channel bound

state gives rise to a Feshbach resonance.

of my or mf sublevels, which have different total magnetic moments. As a result, the

energy of the closed channel can be shifted up and down relative to the energy of the entrance channel by applying a uniform magnetic field and tuning its strength. The scattering length of a colliding pair then exhibits a dependence on the magnetic field, allowing for the contact interactions to be tuned in value, and made either attractive or repulsive [57, 49].

Compared to alkali species, dysprosium has a large number of possible channels (e.g., 17 magnetic sublevels in the ground state of ie2Dy), and so has many more Feshbach resonances. Many of these resonances are very narrow, and are essentially only detectable by the additional inelastic losses they give rise to. Fortunately, some resonances are sufficiently broad to allow for useful control of the scattering length

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2.4.4 Dipolar Relaxation

To consider the types of processes, both elastic and inelastic, which occur as a result of the dipolar interaction, we can write the potential in terms of in terms of spin

operators [25, 9] F1,2 as

UDDI = O(9FB) F2- 3(F1 3( (2.8)

47rr

Expanding the numerator in terms of angular momentum projection and ladder op-erators, we see that the the interaction potential admits three types of processes

[9]:

F1 -F2- 3(F1 - )(F2-) =FizF2z

1

+ (F1+F2-+ F1 _F2+)

3

-₄(22Fz + iFF₁₊+ +F₁_) - (2iF2z + fF2+ + f+F2-)

where 2 =, =+ - i, and = [25]. The first line includes terms which

give rise to elastic scattering, the second line includes spin-exchanging terms, and the third line includes terms which couple orbital motion and the spin states (giving rise to inelastic scattering). In general we see that the spin projection of either particle can change by 1 or 0, and the orbital angular momentum projection can change by 0 or 2 [9].

The inelastic terms in the third line are the ones that give rise to dipolar relaxation. Because the strength of the dipolar interaction is approximately 100 times higher than in the alkalis, losses due to dipolar relaxation can have a much more significant impact on experiments. For example, evaporation in a magnetic trap, which requires atoms to be in the highest-Zeeman energy (weak field seeking) state is infeasible for dysprosium because of the high rate of dipolar relaxation a gas prepared in such a state would experience.

Since there can be no inelastic dipolar relaxation collisions among atoms spin-polarized in the absolute lowest-energy (strong field seeking) state, the most feasible approach to performing a spin mixture experiment is to use a mixture of the lowest

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and next-lowest energy sublevels. As mentioned in Chapter 1, spin mixtures of Dy bosons (jmj = -8),mj = -7)) have a much higher rate of dipolar relaxation than

the equivalent spin mixture of Dy fermions

(Imj

= -21/2),mj = -19/2)); this has been explored in detail in [8] and [9].

2.5 Properties of the Useful Electronic Transitions

The three transitions we use (or plan to use) in our experiment span four orders of magnitude in linewidth, and as a result have different properties in the context of laser cooling and trapping.

2.5.1

421 nm Transition

The 6s-to-6p 1P1 transition from the dysprosium ground state has a linewidth of

F 2r x 32.2 MHz [49], making it the broadest optical transition from the ground

state. Since the photon scattering rate of an atom is limited to 1 in the limit of₂ large saturation parameter, we have chosen to use this transition for Zeeman slowing and angled slowing. Specifically, the maximal optical force we can generate from this transition is

hkT

Fmax 1.6 x 10-'9 N (2.9)

2

corresponding to an acceleration of over 60,000 times gravity.

The broad linewidth also means that the cross section of the atoms is relatively insensitive to variations in the laser frequency and/or magnetic field, making it a convenient choice of frequency for absorption imaging.

The Doppler temperature of the transition is given by

TD= = 772 pK (2.10)

2kB

which, compared to the Doppler temperature of the 626 nm transition discussed in the following subsection, is very high and thus not the transition we chose for our MOT light.

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2.5.2 626 nm Transition

The 6s-to-6p 3P1 transition at 626 nm has a much narrower linewidth than the 421 nm

transition, with F = 2-r x 136 kHz [49]. This makes it useful for Doppler cooling, with a low Doppler temperature of

TD =3.3 pK (2.11)

2kB

which is similar to the temperatures reached by sub-Doppler cooling in alakli species.

A drawback of utilizing this transition to form a MOT is the correspondingly low

capture velocity of the transition: since the width of this transition is approximately

50 times smaller than the width of typical alkali D-line transitions, the maximum

speed that can be slowed to a stop by a 626nm MOT beam is approximately V55~ 7 times slower than the corresponding maximum speed for an alkali MOT.

As will be described in Chapter 4, we dither the frequency of this transition in order to increase the capture velocity. The angled slowing technique, described in Chapters 3 and 4, can also be thought of as a way of effectively increasing the capture velocity of this narrow-line MOT.

2.6

741 nm Transition

A transition of a 4f electron to the 5d shell gives rise to an exceptionally narrow

transition at 741 nm, with a linewidth of F = 27r x 1.78 kHz. This transition has a

Doppler temperature of

TD =43 nK (2.12)

2kB

which is actually lower than the transition's recoil temperature of

(hk)2

T =215 nK (2.13)

mkB

This transition has been utilized to form a MOT out of atoms initially captured in a MOT formed on the 421 nm tranisition [43], but the final temperature of a compressed

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741 nm MOT is not lower than the temperatures later achieved using 626nm MOTs [48].

The narrow linewidth of this transition (and resulting large tunability with re-spect to the natural linewidth) makes it an ideal candidate for implementing Raman couplings with low heating rates, as will be described in the following section.

2.7 Raman Transitions in Dysprosium

When Raman transitions are induced using pairs of laser beams, there is inevitably some heating of the atoms due to Rayleigh scattering. The photon scattering rate of atoms being excited on a transition of width F by a laser of Rabi frequency Q at a detuning of A scales as [74, 62]

Rs oc (2.14)

'A2

likewise, the two-photon Rabi frequency of a Raman transition induced by two lasers of the same Rabi frequency with a single-photon detuning A scales as

ROC oc (2.15)

A

An ideal Raman-coupling scheme is realized when the ratio of scattering to Raman coupling strength is small; the ratio scales as

Rscat F (2.16)

QR¾

Thus as power is increased and detuning is correspondingly increased while main-taining a fixed Raman coupling strength, the heating rate due to scattering becomes more neglibile. Since this ratio scales as the inverse detuning in units of linewidths, it is technically easier to use AOMs to reach the desired detuning if the linewidth is narrow; this makes the 741 nm transition ai ideal line to use for Raman coupling in

Dy.

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and linewidths in alkalis; however, Raman transition amplitudes do not scale as Q

A

indefinitely in alkali atoms. The physical reason for this is simple: Raman transitions among ground state hyperfine sublevels in an alkali are transitions between states which only differ in the orientation of the electron spin. Because electric dipole transitions do not couple to the electron spin, the spin-orbit coupling of the atom must be used in order to use lasers to move population from one sublevel to another. For detunings much larger than the spin-orbit coupling strength of the atom - i.e., larger than the fine structure splitting AFS of the D-line excited state - the Raman coupling strength instead scales as . 2 [74]. This caps the ratio of scattering to

Raman coupling at

Rscat _F

(2.17)

QR AFS

This result can also be found by explicitly summing up the transition dipole mo-ments of the excited states in the D1 and D2 lines, and finding that they contribute oppositely to the two-photon transition amplitude when the detuning is larger in magnitude than the fine structure splitting.

In dysprosium, the ground state sublevels represent projections of the coupled orbital and spin angular momenta (and nuclear spin angular momenta in the case of the fermionic isotopes). Since there is already spin-orbit coupling in the ground state, arbitrarily large detunings may be used for Raman transitions without abandoning the favorable scaling of the scattering rate with respect to the Raman coupling strength.

2.7.1 Spin-Orbit Coupling Using Raman Transitions

The physics of spin-orbit coupling, discussed in Chapter 1, can be studied in quantum gases by utilizing Raman transitions. In some schemes, one takes advantage of the fact that Raman transitions among ground state sublevels change both the internal state and the momentum of the atoms. Alternatively, in other schemes it is purely the state-selectivity of Raman transitions which is leveraged, creating state-dependent tunneling terms in optical lattice potentials. The following subsection describes each of these types of schemes in detail.

Development of a new Dy quantum gas experiment