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«ABSTRACT Disordered Ultracold Two-Dimensional Title of dissertation: Bose Gases Matthew Charles Beeler Doctor of Philosophy, 2011 Dr. Steven Rolston ...»

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ABSTRACT

Disordered Ultracold Two-Dimensional

Title of dissertation:

Bose Gases

Matthew Charles Beeler

Doctor of Philosophy, 2011

Dr. Steven Rolston

Dissertation directed by:

Department of Physics

Ultracold bose gas systems can perform quantum simulations of high temperature superconductors in certain parameter regimes. Specifically, 2D bose gases at

low temperatures exhibit a superfluid to thermal gas phase transition analogous to the superconductor to insulator transition in certain superconductors. The unbinding of thermally activated vortex pairs drives this phase transition, and disorder is expected to affect vortex motion in this system. In addition, disorder itself can drive phase transitions in superconductors.

We have designed and built a system which produces two 2D ultracold Bose gas systems separated by a few microns. In addition, we have also produced a disordered speckled laser intensity pattern with a grain size of ∼1 µm, small enough to provide a disordered potential for the two systems. We have observed the superfluid phase transition with and without the presence of disorder. The coherence of the system, which is related to superfluidity, is strongly reduced by the presence of disorder, even at small disorder strength, but the effect of the disorder on observed vortices in the system is less clear.

Disordered Ultracold Two-Dimensional Bose Gases by Matthew Charles Beeler Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Advisory Committee:

Dr. Steven Rolston, Chair/Advisor Dr. Mario Dagenais, Dean’s Representative Dr. Luis Orozco Dr. James Porto Dr. Eite Tiesinga c Copyright by Matthew Charles Beeler Dedication To M.H.D.

Je t’aime ii Acknowledgments This thesis has in no way been a solitary endeavor, and I have a long list of people to thank for making this possible. First, I have to thank my advisor, Dr. Steve Rolston, who let us take a newly renovated empty lab and fill it with expensive and complicated toys to play with, and helped guide us in their intended use. Emily Edwards was my partner on this project for almost my entire time in graduate school, and she played a large role in the direction and construction of the apparatus, including the implementation of the incommensurate lattices which led to our first paper. Kevin Teo and Brendan Wyker helped in the initial construction of the apparatus, back when we were still doing plumbing, ordering shelving, and locking lasers. We have since had two postdocs, Zhaoyuan Ma and Tao Hong, whom have left their mark on the experiment. Ilya Arakelyan and Jenn Johnson have been sharing the apparatus with us while pursuing different science, and I must thank them for their help as well as their willingness to let me use the apparatus far more often than was fair. Matt Reed will be taking over the apparatus after I leave, and it has been fun working with him for the last year or so.

I have also benefited from conversations with the large number of students who have worked for the Rolston group over the years. Scott Fletcher, Xianli Zhang, and now Kevin Twedt have worked on the ultracold plasma experiment across the hall, and have been invaluable sources of technical advice and equipment. Tommy Willis was always interested in what I was doing, and was extremely useful as someone who could provide an educated but outside opinion. Alessandro Zenesini spent

–  –  –

insights.

Dr. Luis Orozco and his students have also been here since the beginning, and they have been assets to my growth as a graduate student. Michael Scholten, David Norris, and Jon Sterk have been willingly listening to me ramble about optical lattices and superconductors for the last few years now, even though it is far afield from their specialty, and I even managed to coerce Michael into machining a few parts for me. The formation of the JQI expanded the number of knowledgeable people around, and the opening of Dr. Chris Monroe’s labs down the hall brought an influx of new ideas and equipment. Dr. Trey Porto and Dr. Ian Spielman were extremely helpful when they started spending time at UMD.

My undergraduate advisor at Miami University, Dr. Samir Bali, pushed me onto this path and inspired me to continue doing research on ultracold atoms in graduate school.

On a personal level, I couldn’t have done this without my wife, Stephanie, who somehow managed to support me through all of the ups and downs and crazy work hours that grad school entails. Her belief that I could do anything never wavered, and I can’t thank her enough for that. I also have to thank my parents for instilling in me a strong belief in the value of education and following your dreams, although I don’t know if they thought I would take it this far.

Finally, I would like to thank all of the people that have helped me in one way or another that I have forgotten to mention here. It’s been a long journey.

–  –  –

Background

1.1 Introduction The study of ultracold gases has proven to be a very versatile field. The ability to engineer simple designer Hamiltonians has driven innovation in the types of science that can be done in these systems. Within the last few years, the capability to control and investigate the role of strong interactions between particles in these systems has increased their usefulness in understanding science that cannot be analyzed in other systems [25]. The basic building blocks of these systems are not difficult to understand, but an amazing number of Hamiltonians can be built from these few components. Variable parameters in ultracold systems include dimensionality, temperature, atom number, density, quantum states, and strength of interactions. Great physical intuition about the behavior of these systems can be found in the simple Schroedinger equation, and the rudimentary level of complication added by mean field theory for weak interactions goes even farther in aiding understanding. Still, once strong interactions are added to the system, it becomes more difficult to model these systems, and experiments are necessary to determine the physics governing the behavior of the particles.





In fact, because of their versatility and simplicity, ultracold atom systems are ideal candidates to do quantum simulation. Quantum simulation allows us to understand complicated quantum mechanical systems. Richard Feynman recognized that simple quantum systems can often reveal key properties of more complex material systems [60]. Frequently, even the behavior of these simple systems is still theoretically intractable, so the outcome of experiments on analog systems, i.e. quantum simulators, can provide otherwise unavailable insights. Some important quantum systems, such as high-Tc superconductors, are not easily amenable to direct calculation, and may have many important parameters that are not separately adjustable in experiment. Strong interactions between a large number of particles make these systems complex, and their description using Bose condensation of Cooper pairs makes these systems quantum. Ultracold atom systems, through quantum simulation, may be able to shed light on the behavior of complicated solid systems.

The layout of this chapter is to first provide some motivation for studying 2D ultracold Bose gases with disorder, and show that they can be used as quantum simulators. Then, the work that has already been done in 2D systems will be reviewed in Sec. 1.3. Disorder in ultracold gases will be briefly summarized in Sec.

1.4, followed by the small amount of work done with disorder in 2D bosons in Sec.

1.5.

1.2 Motivation: 2D Ultracold Bose Gases with Disorder When designing an experiment on ultracold atoms, it is necessary to restrict the available number of parameters to focus on a specific type of science. We are focusing on the physics of disorder. Disorder is ubiquitous in nature, and we wish to examine its effects in ultracold atom systems. Specifically, this thesis examines the role of disorder in 2D systems.

There are a few reasons for studying 2D systems. Keeping with the theme of quantum simulation, many of the high temperature (high-Tc) superconductors are governed by two-dimensional Bosonic physics [24]. Although it seems odd to model a solid’s electronic properties with a gas of neutral atoms, it turns out that many of the models governing type-II superconductivity have a great deal of similarities to a 2D gas of bosons. First, superconductivity is mediated by Cooper pairs pairs of electrons bound together acting as composite bosons. Superconductivity comes about when these Cooper pairs become phase coherent, and in 2D the pairing temperature and superconducting transition temperature can differ significantly, which is different than in 3D [119]. Bosonic models are applicable in any temperature range when pairing has occurred, and thus the transition to phase coherence can be analogous in ultracold gases. In addition, most of the high-Tc superconductors are stacks of 2D planes weakly coupled together, meaning that much of the physics is 2D. Finally, although electrons interact via the Coulomb interaction, most models of high-Tc superconductivity replicate the fundamental behavior of these materials without including these long-range interaction effects [6].

It is worth noting that there are models of high-Tc superconductivity which do not make these assumptions, and there is no current agreement about which model is correct. The systems are too complicated to calculate exactly, and clean measurements on solids that reveal microscopic physics are difficult to make. Ultracold gas systems can be used to measure cleaner indicators of the microscopic mechanisms predicted by bosonic models of superconductivity. However, bosonic systems cannot address the many open questions about the pairing mechanisms in superconductors. Our system only simulates the correct physics after the electrons are bound into Cooper pairs. Thus, we can only use a system of 2D bosons to replicate and perhaps constrain bosonic models of high-Tc superconductors.

Predictions about the physics of 2D superconductors with disorder are given by the “dirty boson” model [62, 149]. This model is summarized by the phase diagram given in Fig. 1.1. As a function of temperature, we see that we have the two transitions mentioned earlier. At Tc0, we have the bulk 3D pairing temperature, below which the Bosonic description becomes correct. Tc, meanwhile, is the superconducting transition, where the pairs develop partial long-range phase coherence.

A good starting point on pairing in superconductors is Ref. [131], and our system cannot simulate this type of physics. Therefore, we will examine the microscopic physics of bosonic 2D systems, which our system can simulate.

2D systems exhibit “marginal” behavior. Peierls argued in 1935 that in an infinite uniform 2D system, there can be no long-range order [115], and it has since been rigorously shown [83, 108, 107]. In a 2D fluid, thermal fluctuations at any nonzero temperature will destroy true Bose-Einstein condensation (BEC) [26, 56, 57, 1], when the multiparticle state of the system has all of the particles in the same single-particle ground energy state. BEC is also indicated by the first order Bosonic correlation function being constant in the limit of infinite distance. In an interacting fluid system, at low enough temperature, a superfluid (flow without friction) can still exist without true BEC. In this case, we can define a local superfluid order Figure 1.1: Generic phase diagram of a 2D superconductor with disorder strength ∆ as a function of temperature T. ∆c is the critical disorder strength, Tc is the critical temperature for the superconducting transition, and Tc0 is the pairing temperature for Cooper pairs. Figure from [149].

parameter, and the transport properties of the system exhibit superfluidity, but without all of the particles being in the ground state. In this 2D superfluid, the first order Bosonic correlation function always falls off algebraically, preventing true BEC. This is in contrast to 3D, where BEC and superfluidity happen at the same time. The superfluid transition is often identified with the superconducting phase transition - the Cooper pairs become superfluid.

The mechanism for the transition between this superfluid and a normal fluid is elucidated by the theory of Berezinskii, Kosterlitz, and Thouless (BKT) [95, 94, 16, 17]. This theory says that just above zero temperature, the thermal excitations in the superfluid take two forms - long-wavelength phase fluctuations called phonons and also pairs of bound vortices. A vortex is a spot of zero density in the superfluid with locally circular flow around this spot, and a bound pair of these is two vortices with opposite directions of circulation very close together. Since the derivative of the phase of the superfluid order parameter is the velocity of the fluid, vortices can be thought of as phase defects. If the vortices are closely bound together, they do little to disturb the overall long-range phase coherence of the fluid, since their effects cancel away from the pair. The long-range phase coherence falls off slowly only through phonons. However, as the temperature is raised, the binding of the vortex pairs starts to loosen, and the vortices start to move in the fluid. When the pairs start to unbind and move about the fluid, the phase coherence of the fluid is destroyed, and the superfluid transitions to a normal fluid, or the superconductor becomes an insulator.

BKT theory has been remarkably successful in describing and predicting the physics of 2D superfluids. It is applicable to many other types of systems as well, including Coulomb gases [110], exciton systems [139, 31], polariton systems [91, 4], and spin-polarized hydrogen [133]. Torsional oscillator experiments in thin films of superfluid 4 He showed excellent agreement with the predicted critical temperature [20, 21]. However, in a finite system, especially an inhomogeneous system, the transitions between the different phases become more difficult to discern. In fact, true BEC becomes possible in 2D in a finite system (Sec. 1.3). Identifying both the macroscopic and microscopic indicators of transitions between the various phases is still of experimental interest [75], especially in ultracold gases.



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