Generation of Counter-Circulating Vortex Lines in a Bose-Einstein Condensate

Transcription

1 Generation of Counter-Circulating Vortex Lines in a Bose-Einstein Condensate Thomas K. Langin Advisor: Professor David S. Hall May 5, 2011 Submitted to the Department of Physics of Amherst College in partial fulfilment of the requirements for the degree of Bachelors of Arts with honors c 2011 Thomas K. Langin

2 Abstract The intriguing properties of superfluids, such as inviscid flow and the quantization of vorticity, have fascinated physicists for nearly a century. Interest has recently centered on the dynamics of interactions between vortex-antivortex pairs in superfluids, since these interactions are central to the physics of quantum turbulence. Dilute-gas Bose-Einstein condensates (BECs) provide a clean system with which we may obtain a better understanding of the dynamics of these interactions. In this thesis, we present observations of dilute-gas BECs containing two, three, and four counter-circulating vortex lines. We also observe possible vortex recombination and/or pair annihilation events. To generate counter-circulating vortices, we make novel use of the ability to generate vortices of known circulation and the ability to radially translate vortices though exchange of angular momentum between the condensate and a rotating thermal cloud. Further exploration of the mechanisms of vortex generation and manipulation will help generate additional counter-circulating states of interest, such as stable vortex dipoles and tripoles, as well as conclusively identifying reconnection and annihilation events.

3 Acknowledgments First thanks go out to my advisor, David Hall. His willingness to let me seek out my own topics for research helped lead me to the fascinating subject of quantum turbulence, a subject which I am considering pursuing further once I leave the confines of Amherst College. Professor Hall s constant enthusiasm for experimental physics is truly inspiring, and I encourage every student considering experimental physics as a possible career path to consider working in his lab for a summer/interterm/semester. There s always something to do, and, even if that something seems like least interesting thing in the world, his enthusiasm will get you excited about it. Also, without his timely and meticulous editing, this thesis would certainly be a lot less readable. So, anyone reading this should thank him too! Second thanks go out to fellow workers in Professor Hall s lab, both past and present. In particular, I d like to thank Daniel Freilich, Emine Altuntas, and Aftaab Dewan. I learned a lot by working with Daniel during interterm my junior year, both about the apparatus and about what being a senior thesis writer was all about (occasionally lots of work...but a great payoff!). Working with Aftaab over this past summer was a real treat. He rewrote the condensate fitting program so that it could fit an arbitrary number of vortices. He also was i

4 (almost) always down for a good game of Halo, which greatly enhanced the Amherst experience. Last but not least, Emine has been a great lab buddy. Her constant focus and determination to get to the bottom of things really helped put me on the right track, both during our work together over the summer and during our more individual work during the academic year. So thanks for putting up with my shenanigans Emine! I ve also got to thank the rest of the class of 2011 physics majors. From the all night/day I-Lab sessions to the fun times at the hbar and everything in between, I cannot imagine going through Amherst with a better group of kids. Special thanks go to Andrew Eddins, whose willingness to have thesis conversations (not the scary kind!) even when he had millions of other things on his plate (i.e., all the time) was truly admirable. Additional thanks go out to the physics faculty at Amherst, the interest that each one of you has in your students future is unbelievable at times. The bros of Taplin 101 were instrumental in the completion of this thesis. Without you guys to occasionally distract me with a game of midnight soccer, strikers, or a late night diner run, I may have gone crazy during the past few months. Most importantly, thank you Mom, Dad, and Kim. Without your unconditional support, I wouldn t have made it as far as I have. You guys mean the world to me. This research has been supported by the National Science Foundation through grant PHY ii

5 Contents 1 Introduction What is a BEC? Quantized Vortices Overview Apparatus An Introduction to 87 Rb The BEC Refrigerator Magneto-Optical Trap Magnetic Trapping Evaporative Cooling Imaging Extraction Imaging Deforming and Rotating the Magnetic Trap Vortex Generation Vortex Generation by Evaporating in a Rotating Frame Vortex Generation Through Quadrupole Mode Excitations: Theory The Quadrupole Mode [1] Vortex Nucleation Process Vortex Generation Through Quadrupole Mode Excitations: Experiment Quadrupole Mode Excitations of Condensates with Zero Vortices Quadrupole Mode Excitations of Condensates with One or More Vortices Vortex Generation by Simultaneously Driving the m = 2 and m = 2 Quadrupole Modes iii

6 4 Vortex Manipulation Radially Translating a Single Vortex: Theory Radially Translating One Vortex: Experiment Stirring a Vortex to the Center Stirring Out a Vortex Stirring In a Vortex from r Radially Translating Multiple Co-Rotating Vortices Observations of Counter-Circulating Vortices Generation and Observation of Vortex-Antivortex Clusters Disappearance of Counter-Circulating Vortices Conclusion 119 A Derivation of Hydrodynamic Equations 122 iv

7 List of Figures 1.1 (color) Bose Statistics in action (color) Plot of Boltzmann factor vs. N for indistinguishable particles (color) 87 Rb hyperfine structure (color) Zeeman Splitting of the ground state of 87 Rb (color) The TOP Trap (color) RF Evaporation (color) Schematic of the imaging process (color) Extraction imaging Example of extraction imaging The Rotating Trap (color) Change in energy of a condensate in a rotating frame (color) Plot of E diff vs. Ω Vortex state generated by rotating trapping potential during evaporation Graph of Vortex Number vs. Ω/2π for condensates produced in a rotating trap Condensates produced in a rotating trap Cartoon of quadrupole mode instability Initially vortex free condensates after driving the quadrupole mode for 1500 ms Initially vortex free condensates after driving the quadrupole mode for 3000 ms Vortex number vs. Ω/ω r for an initially vortex free condensate (color) The Sagnac effect in a rotating condensate Response of one-vortex condensate to driving the co-rotating quadrupole mode vs. Ω/2π Response of one-vortex condensate to driving the counter-rotating quadrupole mode vs. Ω/2π v

8 3.13 One-vortex condensates after driving the co-rotating quadrupole mode One-vortex condensates after driving the counter-rotating quadrupole mode Response of two-vortex condensate to driving the co-rotating quadrupole mode vs. Ω/2π Response of two-vortex condensate to driving the counter-rotating quadrupole mode vs. Ω/2π Two-vortex condensates after driving the co-rotating quadrupole mode Two-vortex condensates after driving the counter-rotating quadrupole mode Two-vortex condensates after driving the co-rotating quadrupole mode for various t stir The stretched trap A vortex-free condensate after stretching the trap One-vortex condensates after stretching the trap (color) The exit of a vortex in a finite temperature condensate (color) Response of a vortex to a change in the angular momentum of the condensate Stirring a vortex to the center Stirring Out a Vortex Plot of r v vs. t + stir (color) Plot of ln r v / ( ω v Ω stir) + vs. tstirout (color) Plot of our data for r v vs. t against the theoretical predictions (Stirring Out) Stirring In a Vortex Plot of r v vs. t stir (color) Plot of our data for r v vs. t against the theoretical predictions (Stirring In) Lattice formation in presence of co-rotating thermal cloud Stirring out two vortices Stirring out three vortices The steps of the counter-circulation generation procedure Condensates containing two counter-circulating vortices Condensates containing three counter-circulating vortices Condensates containing four counter-circulating vortices Disappearing vortices vi

9 Chapter 1 Introduction Over the past decade, the quantity of research on Bose-Einstein condensation has increased tremendously, as a result of the first experimental observations of condensation in dilute gases in 1995 [2]. One such area of research is the study of quantum turbulence (QT), which consists of intersecting and counter-circulating quantized vortex lines (i.e., a vortex tangle), in a dilute gas Bose-Einstein condensate (BEC) [3]. Quantum turbulence has been previously studied in superfluid 4 He [4 6], however, the clean environment provided by BECs allows us to focus more on the dynamics of individual vortex lines, as opposed to the large scale behavior of vortex tangles. Research in QT is particularly fascinating because it is a simpler analog of classical turbulence (CT) which remains, in the words of Richard Feynman, the most important unsolved problem of classical physics. [7] The reason why QT is simpler than CT stems from the quantization of circulation in quantum fluids. In classical fluids, vortices can contain a contin- 1

10 uous spectrum of circulation, allowing for an innumerable quantity of possible vortex-vortex interactions. Moreover, in classical fluids vortices are unstable, and thus will continuously disappear and reappear, further complicating studies of CT [3]. On the other hand, vortices in quantum fluids, like BECs, are topological features which cannot simply disappear and reappear. They also contain a circulation that is quantized, thus limiting the number of possible vortex-vortex interactions. In BECs, each vortex actually contains the same amount of circulation, as we explain later in Section 1.2. This allows us to reduce the problem of turbulence to the problem of interactions between vortices containing the same magnitude of circulation. Once the dynamics of interactions between these vortex lines are well understood, a theory of QT can be built up by using these interactions as building blocks. The development of a theory of QT will hopefully lend insight into the outstanding problem of CT, perhaps leading to its solution. One problem with using BECs to study these building blocks of quantum turbulence stems from small size of the vortex cores, which is on the order of the healing length, ξ, of the condensate. The healing length is typically on the order of a few hundred nanometers, which is smaller than the wavelength of light used for imaging (Sec ). We can get around this problem by imaging after releasing the condensate from the trap, which causes the cores to expand. This precludes the study of vortex dynamics, however, since the dynamics come to an abrupt halt once the condensate is released. One solution to this problem is extraction imaging, which allows us to take multiple images of the same condensate [8, 9]. 2

11 Another problem stems from the need to generate and observe vortex lines which circulate in opposite directions (i.e., clockwise and counter-clockwise). The first method for generating an array of vortex lines which circulate in the same direction was discovered as early as 2000 [10], and many more have been discovered since [11 14]. There are limited methods of generating countercirculating behavior in a BEC, and most of them can only generate relatively few vortex lines [8, 9, 15, 16]. To date, there has only been one static observation of turbulence in a BEC [17, 18]. This thesis is an effort to supply methods by which vortex lines are generated and the methods by which they can be manipulated. We discuss these methods with an emphasis on how they can be used to generate vortexantivortex clusters, i.e., clusters of oppositely circulating vortex lines. We also present two processes which combine methods of vortex generation and manipulation to produce interesting counter-circulating behavior, in which occasionally vortex lines abruptly disappear from the condensates. 1.1 What is a BEC? Thermodynamically, Bose-Einstein condensation is achieved when a macroscopic population of bosons enter the energetic ground state of a confining potential. This phenomenon ultimately derives from the fact that bosons are indistinguishable particles which can share states. The latter requirement is obvious: if particles cannot share the same state, then it s impossible to have a macroscopic population of particles in the ground state. The reason why 3

12 this phenomenon can only occur for indistinguishable particles is a little less obvious; it ultimately has to do with the amount of available states at a given temperature. Consider a system of N particles where there are two possible states, a ground state and an excited state. For both systems, there is only one possible system state in which every particle is in the ground state. Now, let s consider the system state where there is one particle in the excited state. For the system of distinguishable particles, there are N such states, one for each individual particle. In the indistinguishable system, however, there is only one such state, since we cannot tell which one of the particles is the one in the excited state. Figure 1.1 illustrates this phenomenon for 4 particles. In the general case of N particles and Z 1 states the number of accessible energy states (i.e., states with energy of order k B T ) is Z N 1 for the distinguishable system. The relative probability of the system being in a particular state where all of the particles are in excited states with energy of order k B T is given by the Boltzmann factor, e Nk BT/k B T = e N. Therefore, the system state where all particles have energy of order k B T has a Boltzmann factor of Z1 N e N. The system state in which all particles are in the ground state, on the other hand, has a Boltzmann factor of 1. Since typically Z 1 1, we see that the excited system state has a larger Boltzmann factor than the system state in which all particles are in the ground state. This makes condensation impossible for distinguishable particles, since it is incredibly unlikely for the ground state to be macroscopically occupied. For the system of indistinguishable particles, the number of states is [19] 4

13 Ground State Excited State Indistinguishable Distinguishable Figure 1.1: (color) Comparison of distinguishable and indistinguishable particles. We see that there is only one ground state for both types of particles. The gas of distinguishable particles, however, has N states with one particle in the excited state. The gas of indistinguishable particles, on the other hand, has only one state with one particle in the excited state. 5

14 Figure 1.2: (color) Plot of the Boltzmann factor of the excited state vs. N for indistinguishable particles in the case where Z 1 = 100. We see that as N increases, the Boltzmann becomes infinitesimally small. N + Z 1 1 (ez 1/N) N when Z 1 N; N (en/z 1 ) Z 1 when Z 1 N. (1.1) Therefore, in the case where Z 1 N, we find that the Boltzmann factor for the system state where all particles have energy k B T is given by e Z1 N (N/Z 1 ) Z 1. Figure 1.2 shows a plot of the Boltzmann factor of this excited state for Z 1 = 100 vs. N for 100 < N < 150. We observe that the Boltzmann factor becomes incredibly small as N increases. Since the Boltzmann factor for the system state in which all particles are in the ground state remains 1, it becomes a far more likely system state than any excited state in the case where N Z 1. This allows for macroscopic population of the ground state, and thus Bose- Einstein condensation. Insight into the precise conditions required for Bose-Einstein condensation 6

15 can be gleaned by considering the quantum properties of the atoms in the gas. Wave-particle duality informs us that each particle of energy k B T has a thermal debroglie wavelength given by λ db = h 2mE = h 2mkB T. (1.2) Through a consideration of Bose statistics, it can be determined that condensation occurs when the λ db is on the order of the interatomic spacing [1]. Equivalently, condensation occurs when the number of atoms contained within a cube whose sides have length L = λ db is 1 (actually for a gas confined by rigid walls, see Ref. [1]). This condition occurs when the phase space density, where n is the density of the gas, is ( ) h 2 3/2 D = n, (1.3) 2mk B T When λ db is on the order of the interatomic spacing, the wavelengths of each atom begin to overlap. At this point, it no longer makes sense to say that each atom has its own wavefunction. We instead consider the motion of the system to be governed by one wavefunction which we call an order parameter. The equation describing the behavior of this function is the Gross-Pitaevskii equation (GPE) h2 2m 2 ψ (r, t) + V (r) ψ (r, t) + U 0 ψ (r, t) 2 ψ (r, t) ψ (r, t) = i h, (1.4) t 7

16 where m is the mass of the atom which composes the gas, V is the confining potential, and U 0 is a parameter characterizing the strength of the interatomic interactions within the gas. Except for the addition of the nonlinear term, U 0 ψ (r, t) 2, this is exactly the same as the Schrödinger equation. The quantization of circulation within a BEC arises as a consequence of Eq. 1.4, as we discuss in the next section. 1.2 Quantized Vortices We can use the GPE to derive the velocity field of a BEC, yielding (Appendix A) v = Substituting ψ = fe iφ into Eq. 1.5 yields h (ψ ψ ψ ψ ) 2mi ψ 2. (1.5) v = h φ (1.6) m In the absence of phase singularities, a velocity field of this form is irrotational ( v = 0), since ( φ) = 0. (1.7) However, rotation can occur around a region containing a phase singularity, since Eq. 1.7 does not apply in a non simply-connected region. This singularity manifests itself as a region of zero density within the condensate (i.e., a vortex 8

17 line). The single-valuedness of the order parameter requires that the change in φ along a closed contour must be a multiple of 2π: φ = φ dl = 2πl, (1.8) where l is an integer. Using Eqs. 1.6 and 1.8, the circulation, Γ, around a closed contour is defined by [1] Γ = v dl = h m 2πl = l h m. (1.9) The circulation around a vortex line, therefore, is quantized in units of h/m. As previously mentioned, each vortex in a condensate contains the same quantized circulation, specifically h/m (i.e., l = 1). This is because the energy of a vortex in a condensate that can accurately be described by the Thomas- Fermi approximation (Section 2.1) is E v = l2 4πn(0) h 2 ( 3 m Z ln r ), (1.10) ξ where n(0) is the density of the condensate at the center of the trap and Z is the height of the condensate. From Eq. 1.10, the energy of a condensate containing a vortex line with l quantized units of circulation is l times greater than the energy of a condensate containing l vortex lines with a single quantum of circulation. Therefore, multiple quantized vortices are energetically unstable and break up into singly quantized vortices. Vortex lines for an oblate condensate trapped in a harmonic oscillator potential are parallel to the strong trap axis (i.e., the one with the largest angu- 9

18 lar frequency ω), which we call the z-axis. We can therefore define countercirculating condensates to be condensates containing one or more vortex lines of each sense of circulation (e.g., clockwise or counter-clockwise) along the z-axis. We can determine the sense of circulation of a vortex because a vortex precesses about the center of the condensate in the same sense as its circulation [20]. This precession is due to a buoyant force arising from the inhomogeneity of the condensate, and has a predicted frequency of [20] ω v = ( ) ( ) 2 hωr ω2 r 2µ ln 8µ (1 r 2 /R 2 ) 5ωz 2 ω r h (1.11) in a stationary trap, where µ is the chemical potential, r is the radius of the vortex, R is the radial extent of the condensate, and ω r (ω z ) is the radial (vertical) trap frequency of the 3-D harmonic oscillator trapping potential. For condensates produced by our apparatus, ω v /2π 4 Hz. Extraction imaging (Sec ) allows us to take images spaced by tens of milliseconds, allowing us to resolve the precession of a vortex and thus determine its sense of circulation. Thus, when we do generate counter-circulating condensates, we are easily able to identify them. 1.3 Overview We begin, in Chapter 2, with a brief discussion of the apparatus used in the experiments discussed in later chapters. We provide references to prior theses when appropriate, in order to guide the reader to more in-depth descriptions of each component of the apparatus. 10

19 Chapters 3 and 4 introduce the methods of vortex generation and manipulation, respectively, which we utilize to generate counter-circulating condensates. The vortex generation methods introduced in Chapter 3 include condensation in a rotating frame and vortex nucleation through dynamical instabilities resulting from an excitation of the l = 2, m = ±2 collective mode (i.e., the quadrupole mode). The techniques used to manipulate vortices discussed in Chapter 4 center on changing the radius of the vortices by using a rotating thermal population to transfer angular momentum to and from the condensate. We conclude with Chapter 5, which demonstrates how we use the methods discussed in the preceding two chapters to generate clusters of vortices and antivortices. We also introduce images of interesting counter-circulating phenomena, including observations of possible pair annihilation and/or vortex recombination events. 11

20 Chapter 2 Apparatus A basic understanding of the atomic properties of 87 Rb and the equipment we use to take advantage of those properties are essential for the chapters ahead. Anyone who is interested additional details any of the techniques of this chapter should examine previous theses [8, 21 32]. 2.1 An Introduction to 87 Rb All of the condensates considered in this thesis are composed of 87 Rb atoms. Rubidium-87 has a number of important features which make it a convenient atom. The two main properties that allow for Bose-condensation of 87 Rb are the fact that it is a composite boson, and the fact that it contains one valence electron. Rubidium-87 is a composite boson because the sum of its electrons (37), protons (37), and neutrons (50) is an even number (124). Since protons, neutrons, and electrons are spin 1/2 particles, this gives the atom an integer spin, making it a composite boson. With its single valence electron, 12

21 87 Rb is well-suited to the laser cooling and trapping techniques that rely on manipulating the electronic state of the atom. A gas of bosonic atoms is Bose-condensed when a macroscopic fraction of the atoms is in its lowest motional state. This occurs when the phase-space density, D, which is the amount of atoms contained within a volume equal to the cube of the thermal de-broglie wavelength, λ T = 2π h 2 /mkt, of the gas in the trap becomes large enough such that [1] ( ) 2π h [ζ (3)] 1/3 3/2 D > n (2.1) ωn 1/3 m where n is the atomic density, N is the number of atoms in the condensate volume, T is the temperature, ω is the geometric mean of the trap frequencies, k B is Boltzmann s constant, h is Planck s constant divided by 2π, and ζ (3) is the Riemann-Zeta function, ζ(α) = n=1 n α, evaluated for α = 3. Rewriting Eq. 2.1 we see that the condensation condition becomes ( ) 3/2 ( ) 3/2 T kb 1 N 1/3 h ω ζ(3) = (2.2) For the sake of comparison, the left hand side of Eq. 2.2 is for a gas containing 10 6 atoms (the size of our largest condensates) at room temperature. For convenience, we want to eliminate N in Eq. 2.2 and replace it with n = N/V, the average atomic density within the volumetric extent of the condensate, V. Our condensates are large enough that they can be considered to be in the Thomas-Fermi limit (see Section 3.1), allowing a few convenient approximations to determine V R 2 Z, where R is the radial extent of the con- 13

22 densate and Z is the vertical extent of the condensate (again, see Section 3.1). The use of these approximations yields the condition ( ) [ ] 1/4 m5 ω k B T h 9 n (15a) 2 1/4 m ω = (2.3) 5/6 h [ζ (3)] 4/3 where m is the mass of one 87 Rb atom. The scattering length, a, characterizes the strength of atomic interactions (the cross section of scattering interactions is σ = 4πa 2 ). For 87 Rb, a = m [1]. For sake of comparison, the quantity on the left hand side of Eq. 2.3 is for a cloud of 87 Rb atoms at standard temperature and pressure. As shown in Eq. 2.3, to reach condensation we need ways to either increase n or decrease T. We can do both by slowing down and confining the atoms through the use of transitions within the hyperfine structures of the ground state and the first excited state of 87 Rb. The ground state (5 2 S 1/2 ) and the first excited state (5 2 P 3/2 ) of the electron are separated by E=hc/λ, where λ = 780 nm. This transition is known as the D 2 line. Each state also has a hyperfine structure associated with the different possible values of the total quantum spin of the atom F = I + (L + S), where I is the nuclear spin, 3/2, L is the orbital angular momentum of the valence electron (0 for the ground state, 1 for the first excited state) and S = 1/2 is the electron spin. The hyperfine splitting stems from a spin-spin coupling term in the Hamiltonian [1], H hf = AI J, (2.4) 14

23 Figure 2.1: (color) The hyperfine structure of 87 Rb. Energy levels (except for the D2 line [34]) are not drawn to scale. The important atomic transitions used for trapping, confining, and imaging condensates are labeled. where A is a constant whose value has been determined in Ref. [33]. Thus, for both the ground state and the first excited state, each possible value of F is associated with a different energy. The ground state splits into two energy levels, corresponding to F {1, 2} while the excited state splits into four levels, corresponding to F {0, 1, 2, 3 }, where prime notation refers to the excited state. The hyperfine structure, along with the three most important transitions for this work, is illustrated in Fig In the presence of a magnetic field, B, each hyperfine level splits into 2F +1 Zeeman levels due to the Zeeman effect [1], 15

24 E = m F g F µ B B (2.5) where m F { F, F + 1,..., 0,..., F 1, F } is the quantum number that determines the projection of the total atomic angular momentum F along the magnetic field vector B, and g F is the Landé g-factor for 87 Rb in its ground state, this quantity is, in the case where we ignore the negligible proton g- factor, given by [1] g F = F 2 + F 3 F 2 + F, (2.6) and is therefore 1/2 for F = 1 and 1/2 for F = 2. Using Eqs. 2.6 and 2.5 we see that, for atoms in the F = 1 manifold, states with negative m f values will have their energies shifted higher in a magnetic field, whereas in the F = 2 manifold, states with positive m f values have their energies shifted higher. The Zeeman sub-levels for the F = 1 and F = 2 manifolds are illustrated in Fig The red lines in Fig. 2.2 represent states in which the energy increases in the presence of a magnetic field. If we were to create a local magnetic field minimum at some point in space, atoms in these states would be attracted to that point in order to minimize their potential energy. This is why atoms in these states are typically described as weak field seeking. We exploit this property to create the magnetic trap (see section 2.2.2). There are three important transitions, labeled in Fig. 2.1, between the hyperfine levels that we use to confine, cool, and image atoms. The most 16

25 Figure 2.2: (color) Zeeman splitting of the ground state of 87 Rb. Energy levels are not drawn to scale. Red lines represent states that are weak field seeking, and can thus be attracted to a local field minimum. 17

26 important of these is labeled the cycling transition, 2, 2 3, 3. We call it the cycling transition because it can be driven repeatedly, which makes it useful for the purposes of laser cooling (see section 2.2.1). We drive the cycling transition by shining σ + circularly polarized light, tuned to resonance with the F = 2 to F = 3 transition, at atoms in the 2, 2 state. This brings the atoms to the 3, 3 state, since m F must increase by 1 whenever an electron absorbs σ + polarized light in order to conserve angular momentum. The atoms then decay from the 3, 3 state. Decays from a higher energy state to a lower one are determined by the selection rules, F = ±1, 0 and m = ±1, 0. Thus, a decay from the 3, 3 state to the ground state necessarily leaves the atom in the 2, 2 state, where the cycling process can begin again. We use the cycling transition for laser cooling (section 2.2.1) and imaging (section 2.2.4). An atom may undergo an off-resonant (and off-polarization ) transition to the F = 2 manifold in the presence of light tuned to the cycling transition. In this event, the atom decays with nonnegligible probability to the F = 1 manifold, where it is no longer resonant with the cycling transition. In order to bring atoms in the F = 1 manifold back to the F = 2 manifold, we expose the atom to light tuned to the repump transition. The repump light transfers atoms from the F = 1 manifold to the F = 2 manifold, from which the atom may decay back into the F = 2 manifold, whence it is again subject to light tuned to the cycling transition. If the atom instead decays back into the F = 1 manifold, we drive the repump transition again to give us another chance to bring it to the F = 2 manifold. The last useful transition is the optical pumping transition, between the 18

27 F = 2 and F = 2 manifolds. We drive this transition with σ + circularly polarized light in order to bring atoms in states where m F < 2 to the 2, 2 state before loading them into the magnetic trap (Sec ). This works because, when the atom absorbs the σ + circularly polarized light, m F increases by one. The atom then decays, at which point m F changes by m F = ±1, 0. The atom can then absorb more σ + polarized light, until eventually it reaches the 2, 2 state, at which point it cannot absorb any more σ + light. 2.2 The BEC Refrigerator In this section we consider the steps required to make and image a BEC. We start by trapping the atoms in a Magneto-optical Trap (Section 2.2.1), which increases the phase-space density to within five orders of magnitude of the required density for Bose-condensation [35]. Then, we transfer the atoms into a magnetic trap (Section 2.2.2). The magnetic trap allows us to further increase the phase-space density before we begin the evaporative cooling process (Section 2.2.3). Evaporative cooling removes the most energetic atoms, which reduces the temperature of the condensate enough to reach the phase-space density required for macroscopic population of the ground state. We then discuss how we take data by imaging the condensates (Section 2.2.4). The section concludes with a discussion of extraction imaging, which we use to study the real-time dynamics of a BEC (Section 2.2.5) [8, 9]. 19

28 2.2.1 Magneto-Optical Trap Magneto-optical traps (MOTs) utilize both magnetic fields and laser light to trap atoms. Atoms are trapped when they are stably confined to a certain volume. The principles behind laser cooling are quite simple [1, 36 38]. We focus three orthogonal pairs of counter-propagating laser beams, which are all slightly red-detuned (i.e. tuned to a lower frequency) from resonance with the cycling transition (Fig. 2.1) on the atoms in the trap. Atoms moving towards a laser see the red-detuned light from that laser blue-shifted (i.e. shifted to a higher frequency) closer to resonance via the Doppler effect, and will absorb a photon from that laser, bringing the atom to the 3, 3 state. The absorption of a photon reduces the component of the velocity of the atom opposite that of the direction of laser propagation by v = h/(mλ). This slowing effect gives this type of laser cooling its nickname, optical molasses. The atom will eventually decay back to the 2, 2 state, where it can continue to be slowed down by further absorption of laser light. Adding a magnetic field gradient provides a means of providing spatial confinement in conjunction with the optical molasses. We generate the field gradient by running current through a pair of anti-helmholtz coils. We can then use a system of coordinated beam polarization to take advantage of the Zeeman shift resulting from the introduction of the magnetic field gradient in a way that confines atoms to a region near the center of the trap. [1, 22, 31]. Our apparatus employs a double MOT system. Rubidium-87 atoms are loaded into the first of our two MOTs, which we call the collection MOT, by 20

29 heating a so-called Rb getter. In order to further cool the atoms we must isolate the 87 Rb atoms from background atoms. We do this by transferring them to the science MOT, which is held under a higher vacuum than the collection MOT. We transfer the atoms by sending brief pulses of laser light, tuned to the cycling transition, from the collection MOT to the science MOT. Photons are repeatedly absorbed by atoms, which provides net momentum kicks in the direction of the science MOT. Sadly, we can t quite reach the phase-space density D c required for condensation using only MOTs, since there is an upper limit on the phase-space density for a cloud of atoms trapped in a MOT which is about 10 5 orders of magnitude above D c [35]. We must continue to reduce T and/or increase n. To do this, we must transfer the atoms to a trap which uses only magnetic fields for confinement. In order to prepare the atoms to be magnetically trapped, we further reddetune the MOT beams and reduce the magnetic field gradient. As a result, atoms scatter less light, which reduces the scattered photon pressure within the gas. This reduction in the scattered photon pressure causes a corresponding reduction in the interatomic collision frequency, which increases the atomic density. This is why we call this stage the compressed MOT stage (CMOT) [39]. In addition to increasing the atomic density, the CMOT also heats the gas, since the efficacy of laser cooling is reduced when the MOT beams are detuned. We then move to optical molasses by re-tuning the trapping beams closer to resonance and reducing the magnetic fields, causing the cloud to expand 21

30 and cool. After a brief period of cooling we prepare the atoms for magnetic trapping by using the optical pumping transition, in the presence of a rotating bias field, to bring them to either the 1, 1 state or the 2, 2 state, both of which can be magnetically trapped (Fig. 2.2) [29, 30] Magnetic Trapping To magnetically trap the atoms we use a pair of anti-helmoltz coils to create a magnetic field given by Eq. 2.7 [1, 27] B = B x ˆx + B y ŷ 2B z ẑ, (2.7) where B is the magnetic field gradient along the x and y axes. Equation 2.7 indicates that the magnetic field vanishes at the origin. This minimum defines the center of the trap. A point in space which is magnetic field minimum is also an energy minimum (see Section 2.1) for the weak-field seeking states of 87 Rb. Therefore, atoms in the 87 Rb gas are attracted to the center of the trap if they are in one of the three weak field seeking states. We typically use atoms in the 1, 1 weak-field seeking state. This magnetic trap rapidly loses atoms as they are cooled. These losses occur when atoms move through the field minimum, undergoing transitions to un-trapped (or anti-trapped) states within the same hyperfine manifold. They then exit the trap. These transitions are called Majorana transitions, and must be avoided if we are to cool the atoms to degeneracy. [40]. We circumvent this problem by using a time-averaged orbiting potential 22

31 (TOP) trap [41]. The TOP trap superposes a rotating bias field on the static field gradient produced by the anti-helmholtz coils. This bias field may be written [1, 27] B bias (t) = (B 0 cos ωt) ˆx + (B 0 sin ωt) ŷ, (2.8) where B 0 is the field magnitude and ω = (2π) 2 khz is the angular frequency at which the field vector rotates. Adding the bias field to the quadrupole field (Eq. 2.7) yields B(t) = (B x + B 0 cos ωt) ˆx + (B y + B 0 sin ωt) ŷ 2B z ẑ. (2.9) The addition of the rotating bias field displaces the minimum of the combined magnetic field from the origin and causes the minimum to rotate in the horizontal plane. For traps in which all of the atoms are located at a radius such that r B 0 /B, the magnitude of this field is given by [1] ] 1/2 B(t) = [(B 0 cos ωt + B x) 2 + (B 0 sin ωt + B y) 2 + 4B 2 z 2 B 0 + B (x cos ωt + y sin ωt) + B 2 2B 0 [ x 2 + y 2 + 4z 2 (x cos ωt + y sin ωt) 2] (2.10) We choose a bias field strong enough to displace magnetic field minimum outside the cloud of atoms, thereby preventing Majorana transitions (see Fig. 2.3). The atoms attempt to chase the rotating minimum, but move 23

32 Figure 2.3: (color) An illustration of the TOP Trap. The location of the magnetic field zero rotates at ω/2π=2 khz outside the cloud of atoms, thereby preventing Majorana transitions. so slowly in comparison to the 2 khz rotation that their displacement from the center of the trap is negligible. The potential energy of an atom in a state s as a function of its position within the trap, according to Eq. 2.5, is E s = C s µ s B(t) (2.11) where C s is a constant and µ s = m f g f µ B is the magnetic moment associated with the internal state of the atom. The relatively low speed of the atoms compared to the rotation frequency of the trap allows us to consider only the time averaged magnitude, B t, when determining the potential energy of the atoms in the magnetic trap. The time averaged magnitude for a field B(t) given by Eq. 2.9 is 24

33 B t = ω 2π 2π/ω 0 B(t) dt B 0 + B 2 4B 0 ( x 2 + y 2 + 8z 2). (2.12) Using Eq. 2.12, we find that the potential of an atom in the magnetic trap is E s = E s (B 0 ) µ s (B 0 ) B 2 4B 0 ( x 2 + y 2 + 8z 2). (2.13) Therefore, we see that the combination of the oscillating bias field and the quadrupole field results in an anisotropic harmonic potential: V (x, y, z) = 1 2 ω2 rr λ2 ω 2 rz 2, (2.14) where ω r = µ s B 2 /2mB 0 and λ = 8. For our trap, ω r /2π is found experimentally to be 36.3 Hz Evaporative Cooling Once the atoms are safely loaded into the TOP trap we can begin the final step of the cooling process, evaporative cooling. During this process, we remove the most energetic atoms in the cloud. This reduces the total energy per particle of atoms within the cloud and, after rethermalization, the temperature of the system. We do this by taking advantage of the spatial dependence of the magnetic field. The magnitude of the magnetic field, and thus the magnitude of the Zeeman shift between trapped and untrapped states, increases with the distance from the center of the trap. We apply an RF field to the atoms that drives transitions between trapped and untrapped states for the atoms with 25

34 the highest potential energy. The RF field removes all atoms past a certain radius r RF, the radius from the trap center at which the RF field is resonant with the frequency associated with the energy gap between Zeeman states, from the trap. This process preferentially removes energetic atoms from the trap, since atoms further from the center of the trap have a higher potential energy. Removing these atoms is important because, when they move towards the center of the trap, their potential energy is converted to kinetic energy, which results in the heating of the condensate. To continue removing the most energetic atoms from the condensate, we further reduce r RF by lowering the frequency of the field. We do this slowly enough for the atoms to continuously rethermalize after the energetic atoms have been removed. This allows us to reduce the energy per particle of the gas until the temperature has reached T c, the critical condition for Bosecondensation defined in Eq An illustration of this process is shown in Fig Once we have successfully produced a condensate, we maintain the RF radiation to shield the cold sample from energetic atoms that have escaped from the cooling process or that arise from inelastic heating. These RF shields provide significantly longer condensate lifetimes Imaging We collect experimental data by taking pictures of the condensates. This is difficult to do in-situ since the radial extent of a BEC is on the order of 10 µm, and our imaging resolution is only about 5 µm; even worse, the characteristic 26

35 Figure 2.4: (color) An illustration of RF Evaporation. The picture on the left is at the start of the evaporation process, when the frequency of the RF radiation field is so high that no atoms wander to the space where the RF radiation is resonant with the 1, 1 2, 0 transition. The picture on the right is at the end of the evaporation process (t = t f ), when enough high energy atoms have been evaporated (and thus removed from the trap) through the process of reducing the RF frequency such that T < T c. 27

36 size of the vortex core is only a few hundred nanometers! Fortunately, the condensate can be expanded prior to imaging. We do this by releasing the BEC from the trap, causing the interatomic interaction energy to be converted to kinetic energy and expanding the cloud. We typically image the atoms with laser beams that are directed either parallel to the axis of symmetry of the condensate (generating a top image of the condensate) or perpendicular to that axis (generating a side image of the condensate) after 23 ms of expansion. The imaging process is illustrated in Fig The absorptive imaging technique works as follows: First, after releasing the atoms from the trap we shine laser light resonant with the cycling transition at the atoms. The atoms absorb the light, and the beam is directed to a chargecoupled device (CCD) camera, which records and rasterizes the beam profile. The beam profile has a visible shadow where atoms scatter light from the laser beam. Next, we take two more frames with the CCD camera, one in which the probe beam is on (the light frame) and one in which the probe beam is off (the dark frame). A computer then processes these images to determine the atomic density of the BEC from the optical depth of the condensate. The computer calculates the atomic density by subtracting the dark frame from the other two frames, and then calculating the light ratio of the condensate frame to the light frame for each pixel [42 44]. Since 1, 1 atoms do not respond to light at the cycling transitions, we need to pump them into the F = 2 manifold. We do this by using microwave radiation to transfer some percentage of atoms from the F = 1 to the F = 2 manifold as the atoms are falling and expanding after being released from the 28

37 To Top Camera BEC Top Camera To Side Camera Side Camera low n high n Figure 2.5: (color) Schematic of the imaging process. We direct a laser beam, tuned to resonance with the cycling transition, at the condensate. The atoms in the condensate absorb some of the photons in the beam. The beam is then directed to either the side or the top camera. The data from the beam profile is sent to the computer, which computes the optical depth of the condensate and then converts the optical depth to atomic density. The computer then generates an image based on the calculation of the atomic density. 29

38 trap [8, 26]. Unsaturated images of 1, 1 condensates can be obtained by imaging only a portion of the atoms in this way. [26] Extraction Imaging Recently we have developed a way to take pictures of the condensate in real time [8, 9]. This allows us to study the real-time vortex dynamics of a BEC, and has opened up many experimental possibilities. We briefly summarize this process here; it is explained in greater detail in Ref. [8]. To image the condensate in real time, we extract portions of the condensate by sending a short, spectrally broad microwave pulse tuned to resonance with the transition from the 1, 1 state to the 2, 0 state. It is important that the pulse be spectrally broad since the resonant frequency of the transition differs depending on the position of the atom in the magnetic field (due to the Zeeman shift). A spectrally broad pulse therefore allows us to extract atoms from all sections of the condensate, providing a representative, and presumably faithful, sample. The source of the microwave pulse is a 20 W Hughes traveling wave amplifier. We can transfer 5% of the atoms to the 2, 0 state by presenting the BEC with a 2 µs pulse [8]. All extraction images presented in this thesis are taken using 5% extraction. Fig. 2.6 provides a visual guide to the extraction imaging process. Atoms in the 2, 0 state are untrapped, so they fall due to gravity. We then image the atoms using the top camera in a way similar to the imaging method we introduced in the last section. During extraction, the trap (and in particular the magnetic field gradient) remains on, thus, the cycling transition 30

39 Figure 2.6: (color) An illustration of the extraction imaging process. The condensate (the red dot) experiences a short, spectrally broad microwave pulse (the green arrows from the microwave horn), causing a small percentage of atoms (the black particles) to fall and expand in the magnetic field gradient (blue arrows). After the extracted atoms have fallen for 23 ms, we image them from either the side or the bottom. Image adapted from Refs. [8] and [9]. 31

40 frequency is affected by the Zeeman shift. This means that we cannot use the same laser frequency to image the extracted atoms, since that frequency is now off resonance. The solution is not difficult: we use a different laser beam that is tuned to the Zeeman shifted resonance frequency of the cycling transition. Another adjustment we make to the usual imaging method arises from the relatively short interval between images required to successfully study vortex dynamics. The precession frequency of a single vortex about the center of a condensate is, for our trap parameters, on the order of a few Hz [8, 9, 20]. To resolve this vortex motion, we require images that are spaced by tens of milliseconds. To do this, we utilize the Fast Kinetics (FK) mode of the camera. In FK mode we mask most of the camera s CCD array, so that no light can reach it. The mask prevents double exposure as the previously exposed portions of the CCD array are shifted behind it. We use the mask to store up to nine pixel images. We take images in FK mode in the same fashion described earlier (see section 2.2.4), but we only expose a section of the pixel CCD array. After an image is exposed, it is shifted behind the mask. This allows us to extract and image eight times. The ninth image is reserved for the remnant condensate, which we image by turning off the trap and dropping the condensate, as in section Only after all of the images have been exposed do we transfer the data to the computer. We then take the light and dark frames in the same fashion. Fig. 2.7 shows an example of a set of extracted images. 32

41 0 ms Figure 2.7: An example of the extraction imaging process. The first 8 images are images of 5% of the atoms in the BEC. Notice that the vortex is clearly resolved, and the images were taken rapidly enough to show the clockwise precession of the vortex. 33

42 2.3 Deforming and Rotating the Magnetic Trap The vortex generation experiments we discuss in Chapter 3 all rely on the ability to deform and rotate the TOP trap (Sec ). Rotating the trap means that the magnetic field minimum, instead of moving in a circle, moves in the pattern of a rotating ellipse. This motion produces a time-dependent, non-axisymmetric average trapping potential. To rotate the trap we must: Elliptically deform the trap by applying a bias field where the x and y components are unequal; and Modulate the components of the bias field such that the elliptically deformed trap rotates about the z-axis. We perform both of these steps by modulating the current, at angular frequency Ω, in the two pairs of coils that produce the bias field. This causes the trap to become elliptically deformed and to rotate at angular frequency Ω. These variations are illustrated in Fig In the case where the trap is elliptically deformed but does not rotate, the magnetic field produced by the bias coils is B bias (t) = (B 0 + B mod ) cos ωt ˆx + (B 0 B mod ) sin ωt ŷ (2.15) where B mod is the magnitude of the additional field component produced by the modulating current. Since the magnitude of the x component of the bias field increases by roughly the same amount that the y component decreases, the 34

43 a) b) c) ω B0 B0-Bmod ω B0+Bmod Ω ω B0-Bmod B0+Bmod Figure 2.8: An illustration of the rotating trap. (a) In an undeformed, nonrotating trap, the magnetic field minimum travels in a circle at ω/2π = 2 khz, producing an axisymmetric potential. (b) Modulation of the amplitude of the field vector at frequency ω creates a rotating bias field in which the x component is enhanced, and y component diminished, by the amount B mod. The magnetic field minimum traces an elliptical path. (c) Modulation at frequency ω Ω causes the ellipse traced by the magnetic field minimum to rotate at angular frequency Ω. mean radial trap frequency, ω = (ω x ω y ) /2, does not change significantly when the trap is distorted. This provides an advantage when we calculate the critical rotation frequencies that result in vortex nucleation, which often depend on ω (see sections 3.1 and 3.2). The magnitude of the field produced by summing the bias field and the quadrupole field (Eq 2.7) is B(t) = [(B 0 + B mod ) cos ωt + B x] 2 + [(B 0 B mod ) sin ωt + B y] 2 + 4B 2 z 2. (2.16) In order to derive the trap frequencies of the elliptically deformed trap, we need to determine the time-averaged magnetic field magnitude. We cannot do this in the general case because the B mod term in Eq makes it impos- 35

44 sible to derive an analytical expression for the time-averaged magnetic field magnitude in the same way in which we derived Eq In the limit that the ratio between the magnitude of the bias field along the x and y axes, C = (B 0 + B mod ) / (B 0 B mod ), is small, however, the ratio between the x and y trap frequencies is given by [12] ω x = 1 (C 1) + 1. (2.17) ω y 4 In the following chapters we will be concerned with the ellipticity of the trapping potential, ɛ = (ω y/ω x ) 2 1 (ω y /ω x ) = 1 2 [(C/4) + (3/4)] (2.18) In order to rotate the elliptical trap deformation at angular frequency Ω, we make B mod in Eq time dependent. We can determine the x and y components of the bias field at a given time t by applying a rotation matrix to the x and y components of the modulation term in the non-rotating elliptically deformed bias field (Eq. 2.15) B x = B B mod cos Ωt 0 1 sin Ωt B y sin Ωt cos ωt, (2.19) cos Ωt sin ωt producing a trap with a magnetic field minimum that traces a rotating ellipse, as in Fig. 2.8(c). 36

45 Chapter 3 Vortex Generation There are many methods of generating vortex lines in Bose-Einstein condensates (BECs), including rotating the trapping potential during evaporative cooling (Sec )[10, 13], exciting the dynamically unstable quadrupole mode [11, 12, 14], shedding a vortex dipole in response to a moving barrier [16], and spontaneous generation of vortex lines during rapid evaporative cooling [8, 9, 15]. In this chapter we discuss two of these methods in depth: (1) the generation of vortices by quenching from a thermal cloud to a condensate while in a rotating potential (Section 3.1); and (2) the production of vortices through an excitation of the quadrupole mode (Section 3.2.1). We also discuss how we can combine these two processes to create vortex/antivortex clusters. 37

46 3.1 Vortex Generation by Evaporating in a Rotating Frame We can reliably create a condensate containing one or more vortices, all of which are circulating in the same sense as the trap rotation, by rotating and elliptically deforming the axisymmetric trap (see section 2.3) as we Bosecondense the cloud of 87 Rb atoms. This is similar to the experiment in Ref. [10], which used a laser beam instead of a magnetic deformation to create the rotating trap. The effect of rotating the trap at some angular frequency Ω is to change the energy of the condensate. In a frame rotating with the trap, the energy of the condensate becomes E = E L Ω, (3.1) where E is the energy of the condensate in the rotating frame, E is the energy of the condensate in a non-rotating frame, L is the angular momentum, and Ω is the angular velocity vector describing the rotation of the trap. When L is zero (i.e., there are no vortex lines) the energy of the condensate is the same in both reference frames. When L is non-zero, however, the energy of the condensate is affected by the rotation of the trap. Figure 3.1 illustrates how the energy of the condensate changes in a rotating frame. A condensate with a vortex line becomes energetically favorable when E v, the energy of a condensate containing a vortex, is less than E 0, the energy of a vortex-free condensate. Assuming that L and Ω are parallel, Eq

47 Non-Rotating Frame Rotating Frame Ω No Vortex BEC BEC E=E0 E=E0 Vortex Core Ω Vortex Core Vortex BEC BEC E=E v E=E - L Ω v v Figure 3.1: (color) Illustration showing how the energy of the condensate changes in a rotating frame. The energy of the vortex state is indicated by E v while the angular momentum of the condensate is indicated by L v. Condensates with no vortices have no angular momentum (since condensates are irrotational, see Section 1.2), so the energy of a vortex-free condensate is unaffected by rotating the trap. However, condensates containing a vortex (and thus with non-zero angular momentum) have their energy reduced in a rotating frame. 39

48 indicates that the critical rotation frequency, Ω c, at which the vortex state is energetically favorable is given by Ω c = E V 0 E 0, (3.2) L where E V 0 is the energy, in a non-rotating trap, of a condensate that contains one vortex. The difference in energies between a condensate with and without one vortex, in the Thomas-Fermi approximation (which we discuss later in this section) is given by [1] E V 0 E 0 = 4πn (0) h 2 ) ( m Z ln Rξ0, (3.3) where n (0) is the density of the condensate at the center of the trap, m is the mass of a 87 Rb atom, R is the radial extent of the condensate, Z is the vertical extent of the condensate, the factor comes from a numerical integration of the Gross-Pitaevskii equation in a rotating frame [1], and ξ 0 is the healing length, defined by ξ 0 = h 2mµ. (3.4) The angular momentum of a condensate in the Thomas-Fermi regime with a vortex at its center is [1] L = 8π 15 n (0) R2 Z h. (3.5) 40

49 Using Eqs. 3.1, 3.5, and 3.3, we generate the expression for E diff (Ω), the difference between E v and E 0 in a frame rotating at angular frequency Ω parallel to L: E diff (Ω) = 4πn (0) h 2 ) ( m Z ln Rξ0 8πΩ 15 n (0) R2 Z h. (3.6) We also obtain an expression for Ω c by substituting Eqs. 3.3 and 3.5 into Eq. 3.2, which yields Ω c = 5 ) h ( mr ln Rξ0. (3.7) 2 In order to calculate Ω c we need to be able to express R in terms of experimental parameters and fundamental constants. We can do this fairly easily, since our condensates are accurately described by the Thomas-Fermi approximation, valid when Na/ā 1 [1]. For condensates produced in our apparatus, N a/ā > 1000, which easily satisfies this requirement. In the Thomas-Fermi limit, the ratio of kinetic to potential energy is small [1]. We therefore neglect the kinetic energy term in the Gross-Pitaevskii equation, yielding the algebraic equation [ V (r) + U0 ψ (r) 2] ψ (r) = µψ (r) (3.8) which has the solution n (r) = ψ (r) 2 = [µ V (r)] U 0 (3.9) 41

50 in the region where µ > V (r), and has the solution n (r) = 0 otherwise. The condensate, therefore, extends to a radius r such that V (r) = µ. (3.10) For our trap, V (r) is given by: V (x, y, z) = 1 2 m ( ω 2 rr 2 + λ 2 ω 2 rz 2), (3.11) where ω r is the radial trap frequency and λ = ω z /ω r, where ω z is the axial trap frequency. If we substitute Eq into Eq. 3.10, we obtain R 2 = 2µ. (3.12) mωr 2 The chemical potential µ for a harmonically trapped condensate in the Thomas- Fermi limit is [1] µ = 152/5 2 ( ) 2/5 Na h ω, (3.13) where N is the number of atoms in the condensate, a is the scattering length for 87 Rb, ω is the geometric mean of the trap frequencies, and ā is the characteristic length ā ā = Substituting Eq into Eq 3.12, we obtain h m ω. (3.14) 42

51 R 2 = 152/5 h ω mω 2 r ( ) 2/5 Na. (3.15) Finally, we can use Eq 3.15, Eq 3.14, and Eq 3.4 to rewrite the equation for Ω c in terms of fundamental constants and experimental parameters: ā Ω c = 5ω2 r 2 ω ( ) 2/5 h 1 ln ω m ω (15Na m ω 15Na ω r h ) 2/5. (3.16) For our trap, ω r /2π = 36.3 Hz, ω/2π = 50.8 Hz, a = m, and the initial condensate size is typically about atoms. Inserting these values, along with the fundamental constants, into Eq(3.16) yields Ω c /2π = 3.73 Hz. Figure 3.2, which shows E diff vs. Ω (Eq. 3.6) for these conditions, confirms this value. We have been able to generate a condensate containing a single vortex by rotating the trap at Ω/2π = 5 Hz while condensing (Fig. 3.3). We have also observed that, as we increase Ω further above Ω c, the condensate contains more than one vortex line. This is a consequence of Eq In the nonrotating frame, the energy of a condensate containing multiple vortex lines is larger than the energy of a condensate containing a single vortex line. When Ω becomes large enough, however, the L Ω term causes multiple vortex states to be energetically favorable compared to both the single vortex state and the zero vortex state. We indeed observe that as Ω/2π becomes larger, the number of observed vortex lines increases. For Ω c /2π between 28 Hz and 36 Hz, however, the condensate vanishes 43

52 Figure 3.2: (color) Plot of E diff (Eq 3.6) for our trap parameters in the case where N = We see that E v becomes larger than E 0 when Ω = rad/s = 2π 3.73 Hz. Figure 3.3: Images taken of a vortex state created by rotating the trapping potential counter-clockwise at 5 Hz while evaporating a thermal cloud to a condensate 44

53 from the trap. We believe that this phenomenon is due to the ejection of the condensate resulting from rotating the trap at a frequency too close to the radial trap frequency, 2π (36.3 Hz) [45]. At Ω/2π > 36.3 Hz, we observe condensates, albeit with no vortices. Figure 3.4 shows a plot of the number of vortex lines observed versus Ω, while Fig. 3.5 shows images of condensates produced in rotating traps at various Ω. Figure 3.4: Graph of Vortex Number vs. Ω/2π for evaporation in a rotating trap with an elliptical distortion ɛ =

54 0 ms Ω/2π (Hz) Figure 3.5: Images of condensates produced in a rotating trap for various Ω/2π. The elliptical distortion, ɛ, is 0.194, and the evaporation time is 3500 ms. 46

55 3.2 Vortex Generation Through Quadrupole Mode Excitations: Theory We can also generate vortices by driving the quadrupole mode, which is the l = 2, m = ±2 collective mode excitation (Sec ). We drive this mode by weakly distorting the trap and then rotating the distortion in the same manner as we do when we generate vortices during evaporation (Sec. 3.1). This time, however, we rotate the trap after we have already produced a condensate, and the vortex lines arise from a completely different process. Dynamical instabilities associated with the quadrupole mode allow for the generation of a low density region on the outside of the condensate (called the outer cloud ) [11, 12, 46, 47], where vortices are nucleated. These vortices eventually penetrate into the bulk condensate (the inner cloud ), possibly as a result of collisions between condensate fragments in the outer cloud with atoms in the inner cloud. [15] The Quadrupole Mode [1] Collective modes of BECs in a trap are periodic density oscillations that are solutions of the hydrodynamic equations (derived in Appendix A): and n t + (nv) = 0, (3.17) 47

56 m v ( µ t = + 12 ) mv2, (3.18) where µ = V + nu 0 h2 2m n 2 n (3.19) and U 0 = 4π h 2 a/m. We are interested in solutions of the form n = n 0 + δn, (3.20) where n 0 is the equilibrium density, and δn e iωt. By considering the velocity v, and the incremental density change δn to be small, we can rewrite Eqs and 3.18 as δn t = (n 0 v), (3.21) and m v t = δ µ, (3.22) where δ µ = U 0 δn (we ignore the kinetic energy, 2 n, in the Thomas-Fermi limit). By taking the time derivative of Eq and replacing v/ t with Eq. 3.22, we obtain m 2 δn δt 2 = (n 0 δ µ). (3.23) 48

57 Since we are interested in solutions where δn e iωt, Eq can be rewritten, by using the product rule of vector calculus, as where we have replaced δ µ with U 0 δn. ω 2 δn = U 0 ( n0 δn + n 0 2 δn ), (3.24) m To find n 0, we substitute Eq and Eq into Eq. 3.9, yielding n 0 = µ ( ) 1 r2 U 0 R λ2 z 2, (3.25) 2 R 2 where R is given by Eq Substituting Eq into Eq yields, after some manipulation, ω 2 δn = ω 2 r ( r r + λ2 z ) δn ω2 r z 2 ( R 2 r 2 λ 2 z 2) 2 δn. (3.26) Equation 3.26 can be solved analytically. The class of solutions which are important for discussing the quadrupole mode are those where l = m. These solutions are given by [1] δn r l Y l,±l (θ, φ) e iωt. (3.27) To get ω we substitute Eq into Eq which, since 2 δn = 0, yields ω 2 l = lω 2 r, (3.28) where ω l is the frequency of the density oscillation for a mode with total 49

58 angular momentum l. To resonantly drive the quadrupole mode we do not rotate the trap at the resonant frequency of the mode, ω l. This is because the centrifugal term in the Hamiltonian (Eq. 3.1), ΩL z, shifts the surface mode frequency by lω [48]. Resonantly driving the quadrupole mode therefore requires that we rotate the trap at a frequency Ω such that 2Ω = ω l = 2ω r. (3.29) Thus, Ω c = ω r / Vortex Nucleation Process Simply exciting the quadrupole mode does not generate vortex lines; we need some mechanism for nucleating the vortices. If this weren t the case, then we would see vortices whenever the trap is rotated at Ω/2π > 3.73 Hz, since that is the point at which a vortex state is energetically favorable according to the results from Section 3.1, and we wouldn t need to excite the quadrupole mode at all to generate vortices in a Bose-condensed gas of atoms. Since vortices have been observed after the quadrupole mode is excited [11, 12, 14], the vortex generation mechanism is most likely associated with a quadrupole mode excitation. Vortices nucleated during a quadrupole mode excitation are believed to result from a three step process, which we first briefly summarize; afterwards, we provide a more detailed explanation [46]: 50

59 For certain rotation frequencies and elliptical deformations, rotating the trap creates a dynamical instability within the condensate, causing it to eject material into an outer, low density, cloud where vortices are nucleated [46, 48]. We call these vortices ghost vortices because they are invisible while in the low density outer cloud [49]. Figure 3.6 illustrates this process. The elliptical condensate becomes more and more asymmetric due to fluctuations in the two-fold symmetry (i.e. the condensate becomes less symmetric across its long and short axis) of the condensate. When this asymmetry becomes large enough, modes other than the quadrupole mode can be excited, allowing for more energy and angular momentum to couple into the system (in particular the outer cloud). The outer cloud recombines with the inner cloud. The energy and angular momentum of the outer cloud is transferred to inner cloud, and vortices are nucleated in the inner cloud, likely due to phase dislocations in the merging condensate fragments, as in Ref. [15]. In previous experiments, vortices were observed after rotating the trap at frequencies approximately resonant with the quadrupole mode driving frequency [11, 12], indicating that the dynamical instability manifests when the quadrupole mode is excited. To find out how these excitations of the quadrupole mode can be dynamically unstable, we must first look at the solutions to the Gross-Pitaevskii equation (GPE) in a frame rotating at angular frequency Ω [47] 51

60 a) b) ω L Inner Atoms Ejected Atoms Inner Atoms ω L Outer Atoms c) d) Outer Cloud ω L Inner Cloud ω L Outer Cloud Inner Cloud Ghost Vortices Figure 3.6: Illustration of the generation of ghost vortices after the ejection of atoms due to instability in the quadrupole mode. (a) a condensate before the instability manifests. (b) Ejection of fragments of the condensate at the beginning of the instability. This is followed by (c), where the ejected atoms have formed a low density outer cloud. (d) The formation of ghost vortices. Throughout the instability, the inner cloud undergoes quadrupole oscillations at angular frequency ω L. 52

61 i h ψ ] [ t = h2 2m 2 + V (r, t) + U 0 ψ 2 Ω(t) ˆL z ψ. (3.30) The addition of the Ω(t) ˆL z term in the Hamiltonian forces us to add a rotating term to the hydrodynamic equations, Eqs and 3.18, yielding and n t + (nv) n(ω r) = 0, (3.31) m v ( t = V + nu 0 h2 2 n 2m n + 1 ) 2 mv2 mv [Ω r]. (3.32) Since we are in the Thomas-Fermi limit, we once again ignore the kinetic energy ( 2 n/ n) in Eq We now solve Eq for stationary solutions ( n/ t = v/ t = 0) of n. We can assume that, since we are attempting to drive the quadrupole mode, that the velocity field is an irrotational quadrupolar flow of the form v = α (xy). Using that assumption, we can solve for the stationary solutions of n, which are [50] n 0 = 1 U 0 [µ 1 2 m ( ω x 2 x 2 + ω y 2 y 2 + ω 2 zz 2)], (3.33) where ω x 2 = [ (1 ɛ) + α 2 2αΩ ] ω 2 r, (3.34) and 53

62 ω y 2 = [ (1 + ɛ) + α 2 + 2αΩ ] ω 2 r, (3.35) in the region where µ > m( ω x 2 x 2 + ω y 2 y 2 + ω 2 zz 2 )/2; otherwise n 0 = 0. The modified trap frequencies ω x 2 and ω y 2 can be thought of as the effective trap frequencies induced by rotating the trap. Substituting Eq into Eq yields the solution [51], ( ) 2 2 ωx ω y α = Ω ω 2 2. (3.36) x + ω y Just because a solution is a stationary solution of Eq doesn t mean that it is a stable solution, however. The stability of solutions to Eq can be determined by considering small perturbations δn and δφ of the stationary solutions for the density n 0 and phase φ of the condensate [47, 50]. One can do this by taking variational derivatives of Eq and Eq (since the velocity is related to the phase by the relation v = h φ/m), which yields [47, 50] t δφ = (v Ω r) U 0 /m δφ (3.37) δn (n 0 ) (v Ω r) δn Eigenfunctions of Eq vary in time with e λt, where λ is the eigenvalue associated with the eigenfunction. The solutions to Eq are unstable when one or more of the eigenfunctions blow up as time advances. Therefore, if any eigenvalue of a solution to Eq has a positive real part, the solution 54

63 is unstable [47, 50]. Unstable combinations of ɛ and Ω have been determined by numerically solving Eq [47, 50]. There are three ranges of instability associated with the value of Ω/ω r. The instability that we most likely observe in our experiments is the ripple instability, as described in Ref. [50] and detailed in the following paragraphs. The ripple instability occurs when the trap is rotated at Ω/ω r < 1/ 2 and ɛ is linearly ramped past a critical value that depends on Ω. The start of the instability is characterized by the ejection of atoms on the outside of the condensate, eventually resulting in the formation of a low density outer cloud, which does not undergo quadrupolar oscillations. The numerical simulations also show that ghost vortices form in the outer cloud. This can possibly be explained by the process of phase negotiation between the inner cloud (i.e., the atoms that have not been ejected from the condensate) and the condensate fragments in the outer cloud with which it collides. As the inner cloud continues undergoing quadrupole oscillations, the long axis of the inner cloud collides with atoms in the ejected outer cloud [46]. The order parameters of the inner cloud and the fragments in the outer cloud have different phases. When they collide, a closed contour around the point of collision between the inner cloud and the outer fragments occasionally encloses a 2π phase winding. If so, when the inner cloud and the outer fragments come into contact, a ghost vortex can form in the outer cloud, since a 2*π phase winding around a closed contour is a necessary condition for vortex generation (Sec. 1.2). Then, when ɛ is increased past the critical value, the outer cloud 55

64 becomes so dense (on the order of 10% of n 0 [50]) that the dynamical instability is able to generate shape oscillations that permit ghost vortices to enter the inner cloud. Eventually, small fluctuations in the two-fold symmetry of the condensate cause it to look less and less like an ellipse over time, and it becomes asymmetric with respect to the z-axis. These fluctuations can be caused by interactions between the condensate and the thermal cloud, or fluctuations in the fields that define the trap [46]. Once the asymmetry becomes large enough, the quadrupolar mode is no longer the only mode excited by the rotation. Numerical simulations [46] reveal that the excitation of these additional modes results in the rapid transfer of energy into the outer cloud. Once the condensate becomes so asymmetric that the quadrupole oscillations of the inner cloud begin to break down, the outer cloud merges with the inner cloud, resulting in the transferral of energy to the inner cloud [46]. The merging of the outer cloud with the inner cloud also results in the nucleation of vortices in the inner cloud, which may again be due to the phase dislocations between the inner cloud and the ejected condensate fragments in the outer cloud [15, 52 54]. 3.3 Vortex Generation Through Quadrupole Mode Excitations: Experiment We examine the generation of vortices through excitations of the quadrupole mode for condensates under three initial conditions: condensates that initially have zero vortices, condensates that initially contain one or more vortices with 56

65 circulation in the same direction as the trap rotation, and condensates that initially contain one or more vortices with circulation in the opposite direction as the trap rotation. In the latter case, we observe counter-circulating vortices, as one or more vortices circulating in a direction opposite that of the initial vortex enter the condensate during the excitation. In all cases we excite the quadrupole mode by fixing the stirring frequency Ω, and linearly ramping on the deformation parameter, ɛ, from zero to ɛ f = over 50 ms. Once the deformation parameter reaches ɛ f we keep stirring the trap for time t stir before ramping the deformation back down to zero over 50 ms Quadrupole Mode Excitations of Condensates with Zero Vortices In order to test how weakly rotating the trap affects a condensate containing zero vortices, we first generate a condensate and immediately image it (Section 2.2.5) before we excite the quadrupole mode, in order to ensure the condensate did not initially contain a vortex. This image is referred to as the before stirring image. We then rotate the trap at Ω for a time, t stir. After rotating the trap, we take a series of images of the condensate in order to determine the response of the condensate to the rotating trap. We took two sets of images; one set with t stir = 1500 ms (Fig. 3.7), and the other with t stir = 3000 ms (Fig. 3.8). Both sets of images show that the condensate had the largest response to the rotation somewhere between Ω/ω r = 0.66 and Ω/ω r = 0.68, which is slightly below the resonant driving frequency for 57

66 the quadrupole mode excitation, Ω/ω r Figure 3.9 shows a graph of vortex number vs. Ω/ω r for condensates in which t stir = 3000 ms. The fact that Ω c is below the quadrupole mode excitation frequency indicates that the instability in this case is likely the ripple instability (Section 3.2.2). We also note that we observe these instabilities for trap rotations with ellipticity an order of magnitude lower than in previous experiments [11, 12] Quadrupole Mode Excitations of Condensates with One or More Vortices When there is a vortex present in the condensate, we observe the splitting of the m = 2 and m = 2 quadrupole modes [55, 56]. This mode splitting can be understood in comparison to the Sagnac effect, which was discovered in Georges Sagnac s experiments with rotating interferometers [57]. Sagnac observed that the interference pattern produced by the two light sources in the interferometer shifted while the interferometer was rotated: the rotation causes one beam to traverse a longer distance than the other, since the position of the detector has moved during the time the light takes to move from the light source to the detector. A similar effect is observed in a rotating condensate with one or more vortex lines. Consider a condensate that contains one vortex line of counterclockwise circulation in its center. The atoms in the condensate all rotate counter-clockwise about the center of the condensate. Now, if we rotate the trap clockwise at a frequency Ω in the laboratory frame, we see that in a frame 58

67 before stirring 0 ms Ω/ω r Figure 3.7: Images of initially vortex free condensates after weakly rotating the trap CCW for t stir = 1500 ms. All condensates were stirred with a deformation parameter ɛ f = One early picture was taken before stirring. The strongest reactions are at Ω/ω r = and Ω/ω r =

68 before stirring 0 ms Ω/ω r Figure 3.8: Images of initially vortex free condensates after weakly rotating the trap CCW for t stir = 3000 ms. All condensates were stirred with a deformation parameter ɛ f = One early picture was taken before stirring. The strongest reactions are at Ω/ω r = and Ω/ω r =

69 Figure 3.9: Graph of vortex number vs. Ω/ω r for condensates starting with zero vortices. All condensates were stirred with a deformation parameter, ɛ f, of for t stir = 3000 ms. 61

70 Figure 3.10: (color) An illustration of the Sagnac effect in a rotating condensate. In the lab frame, we see that the condensate observes a trap rotation frequency that differs from Ω. In the co-rotating case, this observed rotation frequency is Ω ω cond, where ω cond is the rotation frequency of the condensate. In the counter-rotating case, the observed rotation frequency is Ω + ω cond. This picture is a good approximation but is not entirely accurate, since the rotation frequency of an atom in the condensate depends on how far that atom is from the condensate center. rotating with an atom in the condensate the trap looks as if it is rotating at a frequency greater than Ω. In this case, a mode excited by rotating the trap at Ω c for a condensate with no vortices will now be excited by stirring at a frequency slightly below Ω c, due to the increase in observed frequency. Figure 3.10 shows an illustration of the Sagnac effect for a rotating condensate. The magnitude of the mode splitting is [56] 62

71 ω + ω = 2 l z (3.38) M r 2, where ω + (ω ) is the quadrupole mode resonance frequency in the case of the co-rotating (counter-rotating) trap, l z = h is the mean angular momentum of the atoms in the condensate for condensates with one vortex in the center ( l z < h if the vortex is not centered), and r 2 = (2/7)R2 is the expectation value of the square of the radial position of an atom in the condensate in the Thomas-Fermi limit. Using Eq to substitute for R the mode splitting becomes, for a single vortex in the center of the condensate, ω + ω = 7ω2 r ω ( ā ) 2/5 (3.39) 15Na with ā is given by Eq After stirring in a vortex by the method in Section 3.1, our condensate typically has N atoms. Substituting into Eq yields (ω + ω )/2π 2.94 Hz. According to Eq. 3.29, driving the quadrupole mode on resonance requires stirring at Ω = ω l /2. Therefore, the difference between the resonant driving frequency of the co-rotating quadrupole mode, Ω co, and the resonant driving frequency of the counterrotating quadrupole mode, Ω counter, should be Ω co Ω counter 2π = ω + ω 2 (2π) 1.47 Hz. (3.40) In order to test the validity of Eq we measured the response of a one-vortex condensate to attempts at driving (ɛ f = ) the m = +2 and m = 2 quadrupole modes as a function of Ω. We generate condensates 63

72 with one counter-clockwise circulating vortex through the process described in section 3.1. The response (determined by the number of additional vortices generated after rotating the trap at Ω) in the co-rotating case is illustrated in Fig. 3.11, while the response in the counter-rotating case is illustrated in Fig Images of the condensates after exciting the co-rotating and counterrotating quadrupole resonances are shown in Figs and Examining Fig closely, it is clear that, in images where there are more than one vortex (the middle three sets of images), the vortex closest to the center is precessing counter-clockwise about the center, while all of the other vortices are precessing clockwise, indicating that they have opposite circulations. Based on the responses, displayed in Figs and 3.12, we determine that Ω co /2π = 25.5 Hz and 23.5 Hz < Ω counter /2π < 24 Hz. Therefore, 1.5 Hz < (Ω co Ω counter ) /2π < 2.0 Hz. This range excludes the 1.47 Hz predicted by Eq The disparity may be explained by the fact that the vortices in our condensates were not always centered, resulting in a lower l z, and reducing the value of the mode splitting predicted by Eq We have demonstrated that the frequencies of the two quadrupole modes do indeed split in the presence of a vortex. Our ability to observe this splitting by counting the additional vortices generated after stirring the trap at Ω helps confirm that the quadrupole mode leads to the dynamical instability that results in vortex nucleation [46, 47]. Equation 3.40 implies that as the angular momentum per atom increases, the splitting between the quadrupole modes should also increase. Therefore, if we start with more than one co-rotating vortex, we should observe a larger 64

73 Figure 3.11: Plot of the response (determined by number of additional vortices generated after rotating the trap at Ω) of a one-vortex condensate to driving the quadrupole mode by rotating the trap in the same direction that the vortex circulates vs. Ω/2π. The elliptical deformation of the trap was ɛ f =0.0027, and t stir = 1500 ms. Based on the plot, we see that Ω co /2π is around 25.5 Hz. 65

74 Figure 3.12: Plot of the response (determined by number of additional vortices generated after rotating the trap at Ω) of a one-vortex condensate to driving the quadrupole mode by rotating the trap in the direction opposite that in which the vortex circulates vs. Ω/2π. The elliptical deformation of the trap was ɛ f =0.0027, and t stir = 1000 ms. Based on the plot, we see that Ω counter /2π is somewhere between 23.5 Hz and 24 Hz. 66

75 before stirring 0 ms Ω/2π (Hz) Figure 3.13: Images of one-vortex condensates after driving the co-rotating quadrupole mode. The first image is always taken before driving the quadrupole mode, and is used to verify that the condensate had a vortex before we excited the quadrupole mode. It is clear that the greatest response occurs at Ω/2π = 25.5 Hz. 67

76 before stirring 0 ms Ω/2π (Hz) Figure 3.14: Images of one-vortex condensates after driving the counterrotating quadrupole mode. The first image is always taken before driving the quadrupole mode, and is used to verify that the condensate had a vortex before we excited the quadrupole mode. It is clear that the greatest responses occur at Ω/2π = 23.5 Hz and Ω/2π = 24 Hz. 68

77 splitting. To test this we perform the same experiment, but stir in two counterclockwise circulating vortices before exciting the quadrupole mode. Figure 3.15 indicates that Ω co /2π is around 25.5 Hz, as in the case with one vortex, while Fig indicates that Ω counter /2π is between 23 Hz and 23.5 Hz. Images of the condensates after exciting the co-rotating and counter-rotating quadrupole resonances are shown in Figs and Figure 3.19 shows images of the condensate after driving the co-rotating quadrupole mode for various t stir. The responses, displayed in Fig and Fig. 3.16, indicate that Ω co /2π = 25.5 Hz and 23 Hz < Ω counter /2π < 23.5 Hz in the case with two vortices. We therefore have 2.0 Hz < (Ω co Ω counter ) /2π < 2.5 Hz. We can see that the upper and lower limits of the splitting have increased in the presence of an additional vortex, as predicted. 3.4 Vortex Generation by Simultaneously Driving the m = 2 and m = 2 Quadrupole Modes. This section is devoted to the following questions: can we generate vortices by driving both the m = 2 and m = 2 quadrupole modes at the same time? Both modes can be driven at the same time by modulating the bias field in a fashion similar to that used to rotate the trap (Sec. 2.3): we add an additional sinusoidal term with frequency Ω 2 to the modulation. To determine the x and y components of the magnetic field at time t we apply a modified rotation 69

78 Figure 3.15: Plot of the response (determined by number of additional vortices generated after rotating the trap at Ω) of a two-vortex condensate to driving the co-rotating quadrupole mode vs. Ω/2π. The elliptical deformation of the trap was ɛ f =0.0027, and t stir = 1500 ms. Based on the plot, we see that Ω co /2π is around 25.5 Hz. 70

79 Figure 3.16: Plot of response (determined by number of additional vortices generated after rotating the trap at Ω) of a two-vortex condensate to driving the counter-rotating quadrupole mode vs. Ω/2π. The elliptical deformation of the trap was ɛ f =0.0027, and t stir = 1000 ms. Based on the plot, we see that Ω counter /2π is somewhere between 23 Hz and 23.5 Hz. 71

80 before stirring 0 ms Ω/2π (Hz) Figure 3.17: Images of two-vortex condensates after driving the co-rotating quadrupole mode. The first image is always taken before driving the quadrupole mode. It is clear that the greatest response occurs at Ω/2π = 25.5 Hz. 72

81 before stirring 0 ms Ω/2π (Hz) Figure 3.18: Images of two-vortex condensates after driving the counterrotating quadrupole mode. The first image is always taken before driving the quadrupole mode. It is clear that the greatest responses occur at Ω/2π = 23 Hz and Ω/2π = 23.5 Hz. 73

82 Figure 3.19: Images of two-vortex condensates after driving the co-rotating quadrupole mode (at Ω/2π = 25.5 Hz) for various t stir. 74

83 matrix to the elliptically deformed field (Eq. 2.15): B x = B y B B mod1 cos Ω 1t sin Ω 1 t 0 1 sin Ω 1 t cos Ω 1 t +B mod2 cos Ω 2t sin Ω 2 t cos ωt, (3.41) sin Ω 2 t cos Ω 2 t sin ωt where B 0 is again the magnitude of the bias field, and B mod1 and B mod2 are the amplitudes of the additional field components produced by the modulating current. The magnitude of the x and y components of the bias field at time t are then given by B x = B 0 cos ωt + B mod1 cos (ω Ω 1 ) t + B mod2 cos (ω Ω 2 ) t, (3.42) and B y = B 0 sin ωt B mod1 sin (ω Ω 1 ) t B mod2 sin (ω Ω 2 ) t, (3.43) respectively. Although we do have the freedom to choose these parameters independently, in the experiments described below we always set Ω 2 = Ω 1 and B mod1 = B mod2. Doing so causes the magnetic field minimum to follow the path illustrated in Fig Due to the periodic extensions of the trap along the x and y axes which occur at angular frequency Ω = Ω 1, we call this process stretching the trap. We define the ellipticity ɛ of the stretched trap to 75

84 be the same as the ellipticity of a rotating trap (see section 2.3) with a ratio C = (B 0 + 2B mod ) / (B 0 2B mod ) between the magnitudes of the magnetic field along the long and short axes. We found that the response of the condensate to stretching the trap with an ellipticity ɛ = for time t stretch depends on how many vortices the condensate had before we stretched the trap. If the condensate contained zero vortices before we stretched the trap at Ω/ω r = (the value of Ω resulting in the largest condensate response) for t stretch = 3.0 s we observed that, though the condensate would become distorted after stretching the trap, no additional vortices were nucleated (Fig. 3.21). We also tried stretching the trap at different values in the ranges < ɛ < and 24.5 Hz < Ω/2π < 28 Hz. We have not yet found a combination of ɛ and Ω that results in the generation of vortices, implying that driving the degenerate m = 2 and m = 2 quadrupole modes at the same time does not produce the dynamical instabilities required for vortex nucleation (section 3.2.2). This makes sense because resonantly driving the m = 2 and m = 2 modes should have the same effect as resonantly driving the sum of those two modes, which has m = 0. The m = 0 mode is not known to produce vortices, so it is not surprising that no vortices are generated by a process which, in effect, drives that mode. In order to test the case where there is already a vortex in the condensate we first stir in a counter-clockwise (CCW) circulating vortex by the method in section 3.1. We then stretch the trap, with an elliptical distortion ɛ, at angular frequency Ω for time t stretch. In this case, we were able to generate additional vortices that could be either co-circulating (CCW) or counter- 76

85 a) t=0 b) t=π/2ω B0+Bmod ω B0 ω B0-Bmod c) t=π/ω d) t=3π/2ω B0-Bmod ω B0 B0+Bmod ω Figure 3.20: Illustration of how we stretch the trap at angular frequency Ω = Ω 1 = Ω 2 in order to excite both the m = 2 and m = 2 quadrupole modes. (a) At t = 0 the trap is circular. (b) At t = T/4 (where T is the period), the trap is stretched along the y axis. (c) At t = T/2 the trap regains its circular shape (d) At t = T/4 the trap is stretched along the x axis. 77

86 Figure 3.21: Image of a vortex-free condensate after stretching the trap at Ω/2π = 24.5 Hz and ɛ = for t stretch = 3.0 s. The condensate clearly becomes distorted, but, no vortices are generated by stretching the trap. circulating (CW) depending on the value of Ω. Figure 3.22 shows images of condensates stretched at various frequencies. We can see that in cases where Ω/2π 24 Hz the vortices on the outside precess CW about the center of the condensate, while the vortex towards the center of the condensate precesses CCW about the center. In cases where Ω/2π 25 Hz we see that all vortices precess CCW about the center of the condensate. The change in the circulation of the vortices generated by stretching the trap is possibly due to the splitting of the two quadrupole modes in the presence of a vortex (Sec ), with the co-circulating mode having a higher resonant driving frequency than the counter-circulating mode. In addition to demonstrating the dependence of Ω on vortex generation, the fourth series of images in Fig shows that we can use this method to generate a vortex dipole configuration, where two vortices of opposite circulation remain (nearly) stationary. Such configurations have been observed previously [9], arising from the evaporation process, but this appears to be the first artificial generation of such a state. 78

87 before stirring ε 0 ms (23,2.5) (23,3.0) (24,4.0) (24,6.0) (25, 2.5) (25.5,2.0) (Ω/2π (Hz), t(s) ) Figure 3.22: Images of one-vortex condensates after stretching the trap at various Ω. One image was taken before stretching the trap in order to confirm that the condensate had a vortex before we stretched it. In cases where Ω/2π 24 Hz, the vortices generated on the outside of the condensate circulate in the direction opposite that of the initial vortex. In cases where Ω/2π 25 Hz, all vortices circulate in the same direction. 79

88 Chapter 4 Vortex Manipulation It can be useful to control the radial positions of the vortex lines in a Bose- Einstein condensate (BEC). For example, we observe that the vortices generated by driving the quadrupole mode (Sec. 3.3) tend to be located in the outer reaches of the condensate. We wish to study the behavior of vortices located at different radial position in order to observe a diverse range of dynamics. In this Chapter we discuss how to radially translate vortices by using frictional interactions between the vortex cores and the surrounding atomic thermal cloud. These are the first experiments, to our knowledge, that examine the interaction between vortex lines and thermal clouds. We begin by first introducing the theory of interactions between a vortex core and a rotating thermal cloud, which indicates that the vortex core moves towards (away from) the center of the condensate when the thermal cloud rotates in the same (opposite) direction that the vortex circulates (Section 4.1). We then present experimental observations of the radial translation of a vortex 80

89 caused by rotating the thermal cloud (Section 4.2). This chapter concludes with our observations on the effect of rotating the thermal cloud surrounding a condensate with more than one vortex line (Section 4.3). 4.1 Radially Translating a Single Vortex: Theory To determine the effect of frictional interactions between a thermal cloud and a vortex, we derive an analytical formula for the vortex motion of the form: ds dt = ( ) ds + dt 0 ( ) ds dt f (4.1) where s is a three-dimensional curve describing the vortex line, and the 0 and f subscripts indicate the motion of the vortex line in the absence and presence of frictional effects, respectively. Since vortices in an oblate condensate are parallel to the z-axis, we can take s to be the position of the vortex core. Vortex motion without friction can be described by [58, 59] ( ) ds = v s + v i. (4.2) dt 0 The self-induced velocity of the vortex line segment, v i, arises from Magnus forces that involve the inhomogeneity of the condensate (see Sec. 1.2), or the interaction of each line segment with the remainder of the line. The local superfluid velocity at the core of the vortex, v s, arises from bulk motion of the condensate. If there is more than one vortex, we must take into account the 81

90 velocity fields of the other vortices when determining v i. The motion of a vortex that arises as a result of frictional effects has been discussed in Ref. [59], and we follow that derivation here. When there is a net relative velocity between the thermal cloud and the condensate, a frictional force f will be exerted on the fluid surrounding the vortex core. Schwarz uses momentum-conservation arguments to claim that this force f generates an additional motion of the vortex described by [59] ( ) ds dt f = s f ρ s κ (4.3) where s is the unit tangent along the vortex, ρ s is the superfluid density, and κ is the quantized vorticity (κ = 1 for the vortices in a Bose-Einstein condensate; see Sec. 1.2). The frictional force f has been experimentally determined to be [59] ( f = α v n κρ s ( ) ) ( ( )) ds ds α s v n, (4.4) dt 0 dt 0 where v n is the velocity of the thermal cloud, and α and α are parameters that depend on the temperature. Substituting Eq. 4.4 into Eq. 4.3 yields ( ) ds dt f = αs ( v n ( ) ) [ ( ds α s s v n dt 0 ( ) )] ds. (4.5) dt 0 Next we obtain an equation for the vortex motion by substituting Eq. 4.5 and Eq. 4.2 into Eq. 4.1, which yields 82

91 ds dt = v i + v s + αs (v n v i v s ) α s [s (v n v i v s )]. (4.6) The first two terms of Eq. 4.6 indicate that the motion of the vortex at T = 0, where α and α are both zero, is described by v i (since v s = 0). The second two terms indicate that, at T > 0, the thermal cloud exerts a force on the vortex with strength determined by the temperature-dependent parameters α and α. Let us now consider the solution of Eq. 4.6 when the thermal cloud and condensate are both stationary (v s = v n = 0). Assuming that the vortex lines are parallel to the axis of rotation, we see that the self-induced velocity v i results only from the Magnus forces resulting from the inhomogeneity of the condensate. This force causes the vortex to precess about the center of the condensate in the direction of its circulation (Sec. 1.2)) [9, 20, 60]. In cylindrical polar coordinates, v i = v i ˆφ and s = ẑ. Substituting into Eq. 4.6 yields: The azimuthal component of Eq. 4.7 is ds dt = (v i α v i ) ˆφ + αv i ˆρ. (4.7) ( ) ds = v φ = v i α v i. (4.8) dt φ Making the substitution v φ /r v = ω v, where r v is the radial position of the vortex and ω v is the angular frequency at which the vortex precesses about 83

92 the center of the condensate, we obtain ω v = (1 α ) v i r v. (4.9) The radial component of the vortex motion is given by the second term in Eq. 4.7, ( ) ds = αv i. (4.10) dt r The temperature-dependent parameter α in Eq. 4.9 is typically very small and can be ignored, allowing us to substitute Eq. 4.9 into Eq [58]. The result is with solution ds = dr v dt r dt = αω vr v, (4.11) r v = r 0 e αωvt. (4.12) Eq indicates that the radial position of the vortex core increases exponentially until the vortex eventually leaves the condensate, provided we do not rotate the thermal cloud (v s = v n = 0). Figure 4.1 demonstrates the increase in r v as a function of time at various temperatures (parameterized by α). Now let us consider the case where the thermal cloud is rotating about the z-axis, 84

93 a) b) c) d) Figure 4.1: (color) The evolution of r v over time t as a function of α. (a) α = , t = 26.0 s. (b) α = 0.02, t = 4.5 s. (c) α = 0.05, t = 1.85 s. (d) A plot of the radius of the vortex versus time. In all cases the vortex is starting at (0.1, 0.0). This figure is adapted from Ref. [58] 85

94 v n = r v Ω th, (4.13) where Ω th is the angular frequency at which the thermal cloud rotates, taking it to be positive if it is the same sense as ω v. Substituting Eq into Eq. 4.6 yields with azimuthal component ds dt = [v i + α (r v Ω th v i )] ˆφ + [α (v i r v Ω th )] ˆρ, (4.14) ω v = v i α (r v Ω th v i ) r v = (1 α ) v i r v α Ω th, (4.15) and radial component dr v dt = α (v i r v Ω th ). (4.16) As before, we ignore α as a small quantity. Substituting Eq into Eq yields which has solution dr v dt = α (ω v Ω th ) r v, (4.17) r v = r 0 e α(ωv Ω th)t. (4.18) According to Eq. 4.18, rotating the thermal cloud modifies the term in the 86

95 exponential that describes the time evolution of v r. This gives us a parameter that we can adjust in order to cause the vortex to spiral outwards (Ω th < ω v ) or inwards (Ω th > ω v ). For condensates in our trap, ω v 4 Hz. We can understand this process through comparison with the numerical simulations performed in Ref. [61], which demonstrate how the angular momentum per particle, l, of a condensate at T = 0 in a rotating trap increases from 0 to h as a function of r v. First, a vortex approaches from the periphery of the cloud when 0 < l < h. This displaces the center of mass of the condensate, causing it to rotate about the trap center. The orbit of the center of mass then contributes additional angular momentum to the system. As the angular momentum increases, the vortex core moves inward until it reaches the center of the condensate, at which point l = h. At this point, in the absence of a dynamical instability supporting the formation of additional vortices (see Sec ), the condensate cannot gain any more angular momentum regardless of whether or not the condensate and the thermal cloud are in rotational equilibrium. In the case of T > 0, collisions between a thermal cloud rotating in the same direction as the sense of velocity flow in the condensate should transfer angular momentum to the condensate. This transfer of angular momentum causes the vortex core to move inwards, since the angular momentum of a condensate containing a single vortex is maximized when the vortex core is at the center. Likewise, collisions between a condensate and a thermal cloud rotating oppositely should diminish the angular momentum per particle, causing the vortex to move to larger r v. This process is illustrated in Fig

96 a) Stationary vortex b) Rotating ω v Ω > ω v BEC ω v L=L 1 Thermal Cloud c) d) L>L 1 Ω Ω < ω v ω v ω v Ω L=L 1 L<L 1 Figure 4.2: (color) Illustration of the response of a condensate containing a vortex to a change in angular momentum caused by interactions with a rotating thermal cloud. (a,c) A condensate with L = L 1 with a vortex at r 0 precesses about the center of the condensate. (b,d) If we rotate the thermal cloud, angular momentum is transferred from the rotating thermal atoms to the condensate. (b) If the thermal cloud rotates in the same direction as ω v, the angular momentum of the condensate increases, and the vortex moves inwards (blue arrow). (d) If the cloud rotates in the opposite direction, angular momentum leaves the condensate and the vortex moves outward (blue arrow). 88

97 4.2 Radially Translating One Vortex: Experiment We now test the prediction that rotating the thermal cloud at Ω th > ω v in the same direction that the vortex circulates pushes the vortex towards the center. First, we generate condensates containing one vortex using the method described in section 3.1. Next, we take a preliminary image of the condensate (as in Section 2.2.5) in order to measure the initial radial position of the vortex. We then permit a relatively large thermal cloud to form by increasing the frequency of the radiofrequency shields (Section 2.2.3), effectively diminishing its ability to remove atoms heated by inelastic collisions. We then rotate the trap (Section 2.3) at angular frequency Ω and ellipticity ɛ for some time t stir, forcing the entrained thermal cloud into rotation about the condensate. After this rotation, we set ɛ = 0 and restore the shield frequency to bring the temperature of the sample back down. Finally, we take a series of images of the condensate. By comparing the radial position of the vortex in these images to the radial position of the vortex in the first image, we can determine whether the rotating thermal cloud has forced the vortex towards the center of the condensate. We do indeed observe that rotating the thermal cloud at Ω > ω v for a long (10.8 full precessions of the vortex) period of time does force a vortex to move towards the center of the condensate. Figure 4.3 shows a series of images of a condensate stirred at Ω/2π = 16 Hz and ɛ = for 43.2 full rotations of the thermal cloud (t stir = 2.7 s) when the RF Shields were set to 4.85 MHz (as 89

98 opposed to 4.55 MHz that we usually set for a trap with ɛ = 0.102). It is difficult to study the efficacy of this method when the initial radial position of the vortex is different in each experimental run. We can circumvent this difficulty by following the steps below (ɛ = in all cases where we rotate the trap in the steps below): 1. Stir in the vortex: Determine a combination of t center and Ω center that will force a vortex starting at any r 0 to a radial position near the center of the condensate. We then start with a condensate that has a vortex near the center. 2. Stir out the vortex: If we rotate the thermal cloud in the direction opposite to the direction in which the vortex circulates, we can quickly increase the radial position of the vortex. This allows us to choose a particular r v as a starting point for studying the dependence of r v on t + stir for fixed Ω + stir. In particular, for repeated trials conducted on condensates starting with a vortex in the center, if Ω + stir and t+ stir are fixed, the vortex should always appear at the same radial position. 3. Stir in the vortex (again): Then, since we can use steps 1 and 2 to position a vortex at a fixed initial radius, we are able to study the dependence of r v on t stir for fixed Ω stir by re-rotating the thermal cloud in the same direction in which the vortex circulates. 90

99 before stirring 0 ms Figure 4.3: Images of a condensate containing one vortex after rotating the thermal cloud in the same direction as the vortex circulation. The first picture verifies the existence of a single vortex prior to the rotation. The trap was rotated at Ω/2π = 16 Hz for t stir = 2.7 s. 91

100 4.2.1 Stirring a Vortex to the Center To determine the time, t center, required to stir a vortex at any initial r 0 to a region near the center of the condensate, we measure the radial position of the vortex core after stirring at frequency Ω stir for a variable time t center. The condensates initially contain one counter-clockwise circulating vortex line, created by the method outlined in Sec An initial image of the condensate (Section 2.2.5), verifies that it contains a solitary vortex. After the stirring is complete, we take a set of images of the condensate. If, in the first image, the vortex was near the periphery of the condensate, and in the next images it is located near the center, then we conclude that the chosen value of t stir is likely to push a vortex at any radial position to a region near the center, and thus choose it to be t center. Figure 4.3 provides an example: in the image taken before rotating the trap, the vortex is located near the edge of the condensate. In subsequent images the vortex has been translated to the center of the condensate. The parameters adopted in this sequence are Ω center /2π = 16 Hz and t center = 2.7 s Stirring Out a Vortex Using the method in section allows for the generation of a condensate with a vortex near its center. Fixing r 0 in this way allows us to qualitatively observe the dependence of r v on t + stir for a fixed Ω+ stir. Figure 4.4 shows images of condensates that illustrate this dependence in the case where Ω + stir /2π = 15 Hz (negative frequencies indicate a rotation of the trap in the opposite direction to the vortex circulation; See Section 4.1). Figure 4.5 plots the radial position 92

101 of the vortex as determined from the final image of the condensate vs. t + stir. We determine the radial position of the vortex by first fitting the condensate and vortex to a truncated, inverted parabola and an inverted gaussian, respectively. The radius of the vortex is then obtained by comparing the coordinates of center of the gaussian to the center of the parabola [8]. We see from Fig. 4.5 that the results of this stirring do not obviously describe an exponential growth in r v. One explanation is that small changes in r 0 drastically change the expected value of r v (t) expected for given values of α and t + stir. From Eq. 4.18, we see that δr vr, the uncertainty in r v due to uncertainty in r 0 is δr vr (t) r v (t) = δr 0 r 0. (4.19) We found that the mean value for condensates that we declared to be near the center of the condensate was r 0 /R = 0.1 and that the average uncertainty in r 0 /R was Substituting those values into Eq reveals that there is a relative uncertainty of 30% in the expected value of r v due to the uncertainty in r 0. We need to determine α in order to plot the expected exponential growth curve for r v as a function of t + stir. To do so, we rearrange Eq to yield ln r v (ω v Ω th ) = αt + ln r 0 (ω v Ω th ). (4.20) Therefore, α is the slope of a graph of ln r v / (ω v Ω th ) vs. t. Using the data from the first five images in Fig. 4.4 we plot ln r v / ( ) ω v Ω + stir vs. t + stir and 93

102 before stirring 0 ms t + stir Figure 4.4: Images of a condensate containing one vortex after rotating the thermal cloud in the direction opposite that of the vortex circulation. The first picture is taken before the thermal cloud is rotated, in order to confirm the presence of a single vortex. The trap was rotated at Ω stir /2π = 15 Hz. It is clear that increasing t + stir increases the radial position of the vortex core. 94

103 Figure 4.5: Plot of the radial position of the vortex (relative to R, the radial extent of the condensate) as a function of t + stir The trap was rotated at Ω stir /2π = 15 Hz. 95

104 Figure 4.6: (color) Plot of ln r v / ( ) ω v Ω + stir vs. t + stir for the first five pictures in Fig The red line is a linear fit to the data with slope α. perform a linear regression to find α. The plot is shown in Fig. 4.6 and the fit yields α = ± Inserting this result into Eq. 4.12, we predict that a vortex starting at r 0 will exit the condensate in 26(8) s if we do not rotate the thermal cloud at all. This prediction is difficult to test directly, since the decay time is considerably longer than a typical condensate lifetime. In order to test the consistency of the data with the response predicted by Eq. 4.18, we plot it on the same graph as r v (t). We also include an upper and lower bound on the predicted r v based on the experimental uncertainty δr v. 96

105 In calculating δr v we must consider δr vα (t), the uncertainty in r v (t) caused by uncertainty in α. From Eq we see that δr vα (t) = [(ω v Ω th ) t th r v (t)] δα. (4.21) Then, to find δr v (t) we sum Eq and Eq in quadrature, yielding δr v (t) = r v (t) δr 2 0 r (ω v Ω th ) 2 t 2 δα 2. (4.22) The resulting plot, is shown in Fig We observe that the data, according to the resulting plot, agrees with the values predicted by Eq when the experimental uncertainties in α and r 0 are taken into account [58]. However, it is true that there is not any one value of α which results in agreement with theory. Although, it is possible that α varies between different sets of images in our experiment. More work needs to be done to determine possible fluctuations in α Stirring In a Vortex from r 0 By using the techniques in sections and we can generate condensates at (roughly) fixed r 0. We do so by first stirring in the vortex to the center. Then, we stir out the vortex for t + stir = 1.2 s and Ω+ stir /2π = 15 Hz. This results in the creation of a condensate with a vortex at r 0 /R = 0.24(3). By then rotating the thermal cloud in the same direction of the vortex precession at Ω stir for time t stir, we can qualitatively observe the dependence of r v on t stir in the case of fixed r 0. 97

106 Figure 4.7: (color) Plot r v vs. t (Fig. 4.5) including the predicted value for r v (t). The points are the experimental data. The blue curve is Eq for our best values of α and r 0. The red (khaki) curve is the upper (lower) bound on r v, which is calculated by summing (taking the difference between) Eqs and The data are consistent with the predictions of Eq

107 In our study of the dependence of r v on t stir for fixed r 0 we set Ω stir /2π = 16 Hz and ran six trials. Figure 4.9 shows a plot of r v vs. t stir for the imaged condensates, which are shown in Fig In Fig we plot our data on the same graph as the prediction of r v (t) (Eq. 4.18), including its error bar given by Eq Once again our data match the values predicted by Eq after experimental uncertainties are taken into account. 4.3 Radially Translating Multiple Co-Rotating Vortices We also observe the effect of a rotating thermal cloud on condensates that contain multiple vortices. As before, the thermal cloud is set into motion by rotating the magnetic trap at Ω stir for a time t stir. The sign of Ω stir specifies trap rotation in the same direction as the vortex circulation, which in these experiments is counterclockwise (CCW) for positive Ω stir. For Ω stir > 0, the principal effect of the rotating thermal cloud is to cause a transition of the vortex cores, from a disordered ensemble to an ordered regular structure near the center of the condensate. This transition is illustrated in Figure 4.11, which shows the final vortex structure for two (line centered at condensate center), three (equilateral triangle centered at condensate center), four (square centered at condensate center), and five (regular pentagon centered at condensate center) vortices. The mechanism behind this response is essentially the same as the mech- 99

108 before stirring 0 ms t - stir Figure 4.8: Images of condensates containing a single vortex after rotating the thermal cloud at Ω stir /2π = 16 Hz for t stir = 2.0 s in the same direction as the vortex circulation. The first picture ensures that there is one vortex in the condensate prior to rotation. It is clear that increasing t stir brings the vortex closer to the center of the condensate. 100

109 Figure 4.9: Plot of the radial position of the vortex (as a fraction of R) as a function of t stir The trap was rotated at Ω/2π = 16 Hz. 101

110 r v Figure 4.10: (color) Plot of our data for r v vs. t (Fig. 4.9) against the predicted value for r v (t). The points are the experimental data. The blue curve is Eq for our best values of α and r 0. The red (khaki) curve is the upper (lower) bound on r v, which is calculated by summing (taking the difference between) Eqs and The data are consistent with the predictions of Eq

111 before stirring 0 ms Figure 4.11: Images of condensates containing multiple vortices after rotating the thermal cloud at Ω stir /2π = 20 Hz and ɛ = for t stir = 2.0 s in the same direction as the vortex circulation. The first picture determines the number of vortices in the condensate prior to rotation. It is clear that rotating the thermal cloud in the same direction as the vortex circulation results in the formation of a lattice. 103

112 anism by which a single vortex moves to the center of the condensate in the presence of a thermal cloud rotating at positive Ω stir (Section 4.1): the collisions increase the angular momentum of the condensate, with a maximum value that depends on the number of pre-existing vortices. The vortex configuration that maximizes angular momentum is a triangular lattice, consistent with our observations [61]. Once the vortices have formed a lattice, the angular momentum cannot be increased without the introduction of additional vortices, which requires the presence of a dynamical instability (Sec ). We next consider Ω stir < 0, i.e., the thermal cloud is rotated in the opposite direction of the vortex circulation. The vortices are initially in a lattice configuration, generated by the process described above. What we observe after rotating the thermal cloud depends sharply on the initial number of vortices. If there are two vortices, we observe that one vortex moves to the center of the condensate while the other exits the condensate within the first 900 ms. Then, the vortex that has moved to the center begins to leave the condensate as well (as in Sec ), and is displaced well towards the edge of the condensate after 1800 ms. Figure 4.12 shows images of condensates taken after rotating for various t stir at Ω stir /2π = 15 Hz. When there are initially three vortices, two vortices exit the condensate at first, while the third moves to the center. The two vortices disappear after 800 ms of rotation. The third vortex then begins to leave the condensate. Figure 4.13 shows images taken after rotating for various t stir at Ω stir /2π = 15 Hz. The process of stirring out multiple vortices may be understood most 104

113 1st Early Picture 2nd Early Picture 0 ms t stir Figure 4.12: Images of condensates containing two vortices after rotating the thermal cloud at Ω stir /2π = 15 Hz and ɛ = in the direction opposite to the vortex circulation. The first picture is taken to determine the number of vortices in the condensate prior rotation. The second picture is taken after the procedure that forces the vortices into a lattice configuration. In the first 0.9 s we observe that one vortex exits the condensate while the other moves towards the center. The remaining vortex then moves outwards. 105

114 1st Early Picture 2nd Early Picture 0 ms t stir Figure 4.13: Images of condensates containing three vortices after rotating the thermal cloud at Ω stir /2π = 15 Hz and ɛ = in the direction opposite to the vortex circulation. The first picture is taken to determine the number of vortices in the condensate prior rotation. The second picture is taken after the procedure that forces the vortices into a lattice configuration. In the first 0.8 s we observe that two vortices exit the condensate while the other moves towards the center. The remaining vortex then moves outwards. 106

115 easily if we compare it to the process of stirring in vortices described earlier in this section. In the latter case, angular momentum is added to the condensate until it reaches a maximum value determined by the initial number of vortices in the condensate. In the former case, angular momentum is removed from the condensate. The minimum angular momentum of the condensate (zero) can be achieved by continuously reducing the angular momentum, independent of the number of initial vortices. The difference is that, while a dynamical instability is required for the entrance of vortices, there is no such requirement for the exit of vortices. In our observations (Figs and 4.13), the first step in this continuous removal of angular momentum is a transition from the state in which angular momentum is maximized for n vortices (a lattice) to the state in which angular momentum is maximized for a single vortex (a single vortex in the center of the condensate). This is accomplished by the coincidence of the exit of n 1 vortices with the movement of one remnant vortex to the center of the condensate. After the first n 1 vortices exeunt, the remnant vortex also exits, and the angular momentum of the condensate ultimately drops to zero. Continued rotation of the thermal cloud, however, is insufficient to induce condensate rotation in the opposite direction. This is, again, a consequence of the inability to introduce vortices to the condensate in absence of a dynamical instability. 107

116 Chapter 5 Observations of Counter-Circulating Vortices The experimental study of counter-circulating vortex clusters is an important step on the path to understanding quantum turbulence (QT). This requires the generation of condensates containing two or more counter-circulating vortices. In this chapter we define this counter-circulating condition in terms of vortices and antivortices, where vortices are vortex lines with one sense of circulation (say, CCW), and antivortices have the opposite sense of circulation (CW). With the notable exception of vortex dipoles, spontaneous generation of vortex-antivortex clusters has not yet been observed in a BEC [8, 9]. We can, however, artificially generate counter-circulating vortices through judicious use of the techniques developed in the previous two chapters. In Section 5.1 we describe how we generate condensates containing two, three, or four counter-circulating vortices. We then present images illustrat- 108

117 ing the resulting behavior. Section 5.2 presents observations of condensates in which vortices abruptly disappear from the condensate. Although the loss mechanism is not yet clear, it is intriguing to imagine that these events correspond to vortex-antivortex reconnection or annihilation events, which are intrinsic to quantum turbulence since these events are required for the untangling of the vortex tangle. 5.1 Generation and Observation of Vortex-Antivortex Clusters The following procedure generates condensates with two or more countercirculating vortices. Throughout, positive Ω corresponds to rotating the trap counter-clockwise (CCW). 1. Generate a CCW vortex Using the method of Sec. 3.1 with ɛ = 0.194, create a vortex with CCW circulation. The RF shield power is then reduced by db, and the frequency increased by 0.3 MHz, which permits the formation of a thermal cloud. The power reduction is essential for reasons that are not well understood. If it is not reduced, however, we obtain a vortex lattice (Fig. 5.1(e)) rather than a cluster of counter-circulating vortices (Fig. 5.1(d)) at the conclusion of the procedure. 2. Stir in the vortex Push the vortex to the center of the condensate by rotating the thermal cloud CCW (ɛ = 0.102, t center = 2.0 s, Ω center /2π = 109

118 15 Hz; Sec ). We push the vortex towards the center so that it is not stirred out (Sec ) during the following steps, which rely on rotating the trap clockwise (CW). The shield frequency is then restored to its usual value to reduce the temperature of the sample. 3. Generate antivortices Generate CW circulating vortices by driving the counter-rotating quadrupole mode (ɛ = 0.010, t quad = 1.5 s, Ω quad /2π = 25 Hz; Sec ). This generates CW vortices at the outer reaches of the condensate. Once again, a thermal cloud is formed by increasing the shield frequency by 0.30 MHz. 4. Stir in the antivortices Push the CW circulating vortices towards the center of the condensate by rotating the trap CW (ɛ = 0.102, t stir = 1.0 s, Ω stir /2π = 15 Hz; Sec. 4.3). This step yields a variety of interesting counter-circulating configurations beyond that of a single CCW vortex at the center and multiple CW vortices near the edge. The trap symmetry is then restored (ɛ = 0.0), as is the shield frequency, which once again reduces the temperature of the sample. 5. Image the condensate Take a series of images of the condensate by repeatedly extracting, releasing, and imaging 5% of the atoms from the trap (Sec ). Figure 5.1 shows image sequences of different condensates taken after each step of this procedure. We can reliably generate condensates containing two, three, and four countercirculating vortices in this way. The three vortex configuration is by far the 110

119 a b c d e Figure 5.1: Images of a condensate after each process in the counter-circulation generation procedure. These images are all of different condensates. (a) We start with a condensate containing a CCW circulating vortex at some r 0. (b) We push that vortex to the center. (c) We generate CW circulating vortices on the outside of the condensate. (d) Pushing the CW circulating vortices towards the center of the condensate generates observable counter-circulating behavior. Notice that the two vortices on the outside precess CW while the one on the inside precesses CCW. (e) If we do not reduce the RF Shields power after evaporation, exciting the quadrupole mode (step 3) generates a condensate with a CW vortex lattice. 111

120 most commonly observed. In the case of four vortices, we have to this date only been able to generate the configuration with three vortices of one sense of circulation and one vortex of the other sense. Figures 5.2, 5.3, and 5.4 show image sequences of two, three, and four counter-circulating vortices, respectively. 5.2 Disappearance of Counter-Circulating Vortices We also have discovered a different, albeit less well understood, process that results in the generation of counter-circulating vortex clusters. The procedure is the same as the one described in the previous section, except in step 1 we stir in more than one CCW vortex by rotating at Ω/2π = 11 Hz and we do not reduce the shield s power. We also replace steps 2 4 with the following two steps: 2. Stir in the vortex: Keeping ɛ = 0.192, we push the vortices to the center by rotating the thermal cloud CCW at Ω center /2π = 11 Hz for t center = 1.0 s. 3. Reverse direction of thermal cloud: We immediately reverse the direction of the thermal cloud rotation by rotating the trap at Ω stir /2π = 12.5 Hz for t stir = 0.2 s. Surprisingly, antivortices are generated during this step by a mechanism that is not yet understood. It is possible that the immediate reversal of direction creates a dynamical instability from which 112

121 1st Early Picture 2nd Early Picture 0 ms Figure 5.2: Images of condensates containing two counter-circulating vortices. Both condensates were produced using our first method of generating countercirculating vortices. The first picture is taken to determine the number of vortices in the condensate immediately after evaporation, while the second picture is taken after exciting the counter-rotating quadrupole mode (step 3). Notice that in both cases the initially generated CCW circulating vortex is closer to the center of condensate than the CW vortex generated by exciting the counter-rotating quadrupole mode. 113