M ONASH U NIVERSITY H ONOURS T HESIS P HYSICS & A STRONOMY Detecting Vortex Clusters with Principal Component Analysis Using statistical correlations to probe the collective behaviour of turbulent two-dimensional superfluid and the formation of Onsager vortex clusters. Tyson Jones (2413 2756) under the supervision of Dr. Tapio Simula Semester 2, 2016 Abstract When isolated, turbulent, two-dimensional fluid relaxes, the system can become more ordered in time. Large-scale, coherently-swirling vortex clusters can emerge from disordered vortex configurations in bounded, planar fluids [1]. This intriguing phase transition to macroscopic structure, occurring at nega- tive temperatures, results from a bounded phase space, and is extremely common in classical fluids. There has been immense interest in the formation of quantum vortex clusters in superfluids [2]. Although the theoretical framework began with Onsager almost 70 years ago [1], and despite experimental realisa- tion of turbulent superfluids [3], spontaneous clustering has not yet been observed outside of simulation. This is related, in part, to difficulties in verifying the direction in which a quantum vortex swirls, and the thermodynamic state of an imaged superfluid. Principal component analysis (PCA) was recently applied to images of an ultracold gas to unveil its collective behaviour [4]. The model-free data analysis technique was further adapted to evidence su- perfluidity by the emergence of an anomalous collective mode over the Berezinskii–Kosterlitz–Thouless phase transition [5]. In this work, we review superfluidity and its realisation through Bose–Einstein condensation. We simulate two-dimensional superfluid dynamics with a mean-field description and use PCA to investigate superfluid thermodynamics. We adapt PCA to analyse turbulent superfluid images, to evidence both collective and vortex dynamics, and find a signature of clustering therein. We propose extensions to our study to learn more about the collective behaviour of Onsager clusters, and discuss how our existing results can be utilised for experimental cluster detection. Figure 1: Coherent vortex clusters in a harmonically trapped superfluid. Acknowledgements I am extremely grateful to Dr Simula for being an endlessly patient guide and role model, for providing an immensely interesting and rewarding project, and working around my obnoxious sleeping patterns. I owe enormous thanks to Dr Kuopanportti for helpful discussions about the theoretical and computa- tional aspects of the project, and sharing my puzzlement as I debugged my code. I thank my close friend and peer, Shi Qui, for comradeship over many sleepless nights. I also thank Drs Anderson and Turner, and Michael Kewming, for helpful discussions about experimental BEC physics. I thank the School of Physics and Astronomy for financial support through a personally tumultuous year. Finally, I thank Boris Deletic and Thomas Reyment for their engaging, inquisitive, and thought-provoking questions, which repeatedly reminded me of my passion for physics, and were a welcome distraction from my growing workload. This research was supported in part by the Monash eResearch Centre and eSolutions-Research Support Services through the use of the MonARCH HPC Cluster. Contents 1 Introduction 1 2 Theory 2 2.1 Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.1 Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.2 Interacting Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2.1 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2.2 Thomas–Fermi Approximation . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Superfluid Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3.1 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3.2 Collective Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Two-Dimensional Quantum Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Vortex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Vortex Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3.1 Onsager Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Achieving Superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Imaging Superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Imaging Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.3 Application to Imaging Collective Excitations . . . . . . . . . . . . . . . . . . . . . 17 2.4.4 Application to Imaging Quantum Turbulence . . . . . . . . . . . . . . . . . . . . . 19 3 Simulation 20 3.1 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 GPE Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.2 Solving Ground-states and Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Imprinting Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.4 Detecting Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.5 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.5.1 Vortex Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.5.2 Collective and Vortex PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Integrity Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Results 27 i 4.1 PCA Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.1 2D Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.2 2D Superfluid Ground-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.3 2D Turbulent Superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 PCA of 2DQT in a Harmonic Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 PCA of 2DQT in a Disk Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.1 Collective Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.2 Vortex Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Discussion 41 5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.1.1 PCA of Reinitialised Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.1.2 Signatures of Clustering in Collective Mode Populations . . . . . . . . . . . . . . . 41 5.1.3 Evolution of Collective Mode Populations . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.4 Study of Vortex PCA to Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.5 Simulating Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Conclusion 43 7 Appendix 44 7.A GPE Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.B TFA Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.C Superfluid Velocity Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.D BdG Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.E PCA of Harmonic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.F PCA of Disk Trapped Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Bibliography 57 ii List of Figures 1 Coherent vortex clusters in a harmonically trapped superfluid. . . . . . . . . . . . . . . . . i 2.1 Condensate fractions of the ideal gas as a function of temperature for different trapping potentials and dimensionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Energy and angular momentum spectrum of the lowest energy modes of a harmonically trapped superfluid of coupling constant gN = 100 , rotating at Ω = 0 01 [ ω 0 ]. The eigen- functions of the modes labelled a - h are shown in Figure 2.3. Angular frequency is given in units of the trapping frequency ω 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Eigenfunctions of the collective modes shown in Figure 2.3 for the non-rotating interacting harmonic ground-state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 The probability density (left) and phase (right) of the harmonically-trapped non-rotating single positively-charged vortex ground-state ( gN = 4000 ). x and y are given in units of the harmonic oscillator length-scale a 0 = √ ~ /mω The profile of the superfluid is emphasised in the phase plots by transparency. . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Amplitude of a vortex core wavefunction as a function of radius r from the core. | ψ ′ | is the wavefunction magnitude far from the vortex core and ξ is the healing length. . . . . . . . 10 2.6 The probability density (left) and phase (right) of a two doubly-quantised vortex state. Blue (plus) and red (minus) circles [markers] indicate a positively (negatively) charged vortex in the probability density [phase]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Trajectories of vortices in a probability density, as indicated by dashed lines, in simulations of harmonically trapped, single and double vortex sates. . . . . . . . . . . . . . . . . . . . 11 2.8 Vortex configurations associated with special thermodynamic states. (a) shows a triangu- lar vortex lattice in a harmonically trapped superfluid rotating at Ω = 0 7 [ ω 0 ]. (b) shows bound vortex-antivortex pairs in a disk trapped superfluid. Blue (red) circles hover over positively (negatively) charged vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.9 (a) and (b) are the probability density and phase respectively of a clustered ( d ∼ 0 9 ) 10-vortex configuration. (c) and (d) are those for a disordered ( d ∼ 0 2 ) configuration. Blue ( + ) and red ( − ) circles (markers) in the probability densities (phases) indicate the location of positively and negatively charged vortices respectively. A white line connects the cluster centres. We note these vortices are precisely imprinted without evolution and are not thermodynamic Onsager cluster states. Evolution of the order parameter in these systems is shown in Figure 2.10a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.10 The clustering order parameter d over the evolution of 10 -vortex configurations in a har- monic trap. (a) shows d for the clustered (top curve) and disordered (bottom curve) systems presented in Figure 2.9. (b) shows d for an arbitrary system with a strongly ex- cited breathing mode. Red lines indicate instantaneous values and blue lines their sliding average over a window of ∆ t = 20 [ 1 /ω 0 ]. Vortices temporarily straying too far to the con- densate edge can be missed by the vortex detection routine, causing sharp, large spikes in the red curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 iii 2.11 DPCs (and their normalised PCNs in square brackets) of experimental images of excited 87 Rb Bose gas [4]. a is the average condensate profile, b - c and e - g were identified as collective excitations, and d and h - l indicated total particle number fluctuation and image noise respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.12 Projection of the PCs in Figure 2.11 back onto the experimental images. Solid blue circles: TDPs of the dipole and scissors PCs. Solid black line: sinusoidal fit to the data. The first 2 PCs oscillate at the anisotropic trapping frequencies, while the fourth PC exhibits more complex behaviour [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 A demonstration of vortex filling (right) on a down-scaled simulation frame (left). We note these vortices have much wider cores than those typically simulated in this work. . . 24 4.1 Illustration of the sampled 2D function f used in PCA testing. . . . . . . . . . . . . . . . . 27 4.2 PCA of the zero-vortex harmonic ground-states subject to a variety trap perturbations. . . 30 4.3 PCA of the single-vortex harmonic ground-state subject to trap displacement, with (b) and without (a) vortex filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 PCA of a 10 -vortex harmonic system (shown in Figure 3.1) with (b) and without (a) vortex filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.5 The clustering order d over time for two harmonic-trap simulations with clustered (top curve) and disordered (bottom curve) vortex configurations. The initial wavefunctions of these simulations are found in Figure 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6 Vortex PCA of the disordered (a) and clustered (b) simulations in Figure 4.5. . . . . . . . . 33 4.7 The probability density (left) and phase (right) of a randomised 100 -vortex configuration in a disk-trapped superfluid. Blue ( + ) and red ( − ) circles (markers) in the probability densities (phases) indicate the location of positively and negatively charged vortices re- spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.8 Total number of vortices N v and the clustering order parameter d over time, for two disk- trapped simulations with (b) and without (a) spontaneous cluster formation. Red lines indicate instantaneous values and blue lines their sliding average over a window of ∆ t = 10 [ 1 /ω 0 ]. Dashed vertical lines indicate times of the instances displayed in Figure 4.9. Spontaneous clustering is gauged by a stable rise in d . . . . . . . . . . . . . . . . . . . . 35 4.9 Instances of the probability density in the disordered (a) and clustered (b) simulations of Figure 4.8, at the times indicated by dashed vertical lines. Blue (red) circles indicate positively (negatively) charged vortices, and a white line connects the cluster centres. Onsager clustering in (b) is seen by the emergence of consistent coherent clusters, while (a) exhibits only brief, unstable instances of clustering. . . . . . . . . . . . . . . . . . . . 35 4.10 Collective PCA of the initial turbulent vortex-antivortex annihilation phase, for the dis- ordered (a) and clustered (b) simulations of Figure 4.8. The projections oscillations are damped as the system relaxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.11 Collective PCA of the late equilibrium phase, for the disordered (a) and clustered (b) simulations of Figure 4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.12 Collective PCA of the entire disordered (a) and clustered (b) simulations of Figure 4.8. An extended plot is provided in Figure 7.2 of Appendix 7.F. . . . . . . . . . . . . . . . . . . . 38 4.13 Vortex PCA of the initial turbulent vortex-antivortex annihilation phase, for the disordered (a) and clustered (b) simulations of Figure 4.8. . . . . . . . . . . . . . . . . . . . . . . . . 39 4.14 Vortex PCA of the late equilibrium phase, for the disordered (left) and clustered (right) simulations of Figure 4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.15 Vortex PCA of the entire disordered (left) and clustered (right) simulations of Figure 4.8. An extended plot is provided in Figure 7.3 of Appendix 7.F. . . . . . . . . . . . . . . . . . 40 7.1 Collective PCA of 10 -vortex configurations (with vortex filling) in highly excited harmonic superfluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 iv 7.2 Extension of Figure 4.12. Collective PCA of the disordered (left) and clustered (right) systems shown in Figure 4.8, over the entire simulation duration. . . . . . . . . . . . . . . 53 7.3 Extension of Figure 4.15. Vortex PCA of the disordered (left) and clustered (right) systems shown in Figure 4.8, over the entire simulation duration. . . . . . . . . . . . . . . . . . . . 54 7.5 A disk-trapped simulation and its PCA. (a) shows the number of surviving vortices N v and the order parameter d in time. Vertical dashed lines indicate times of the simulation instances presented in (b). (c) is the collective PCA over the entire simulation duration. . . 55 7.6 PCA of the disk-trapped simulation shown in Figure 7.5 at various stages of evolution. . . 56 v Chapter 1 Introduction Bosons are indistinguishable, integer-spin particles and obey Bose–Einstein statistics. They first appeared as Bose’s treatment of photons in his 1924 derivation of Planck’s law [6]. Einstein promptly extended the analysis to composite particles in his treatment of the ideal quantum gas. Einstein then predicted a phenomenon later known as Bose–Einstein condensation, whereby the population of particles in a many- body system occupying a particular quantum state grows macroscopically [7]. This population is referred to as the Bose–Einstein condensate (BEC). The study of BEC was spurred by the investigation of superfluid liquid helium through the 1930s. Ex- perimentalists such as Keesom and Misener probed the anomalously high thermal conductivity and low viscosity of liquid helium [8]. Meanwhile theorists such as London and Tisza sought to explain it by treating liquid helium as a degenerate BEC [9, 10]. Landau objected to a BEC model. His 1941 paper demonstrated liquid helium’s superfluidity was a result of the presence of a linear regime in the spectrum of collective elementary fluid excitations [11]. The matter was settled by Bogoliubov in 1946, with a model of excitations in a BEC of weakly interacting atoms which exhibited Landau’s linear regime [12]. Bogoliubov had demonstrated a weakly interacting BEC can be superfluid. Despite liquid helium being strongly interacting, its thermodynamic conditions for BEC agree with Landau’s criterion for superfluidity. Around this time, Gross and Pitaevskii are attributed with developing a mean-field description of the interacting Bose gas [13, 14]. Attention soon turned to turbulence in superfluids. The behaviour of quantised vortices - swirling regions of vanishing superfluid - caught the interest of physicists like Feynman [15] and Abrikosov [16]. In 1949, Onsager made a tantalising prediction that spontaneous vortex clusters - structures formed by vortices in classical fluids - might be formed by quantum vortices in superfluids [1]. Despite a wealth of theoretical and numerical study, so called Onsager clusters remain undetected [17]. The modern experimental realisation of BECs through ultracold gases has enabled the study of macro- scopic but inherently quantum systems. Bosons are condensed in magnetic and optical traps. Superfluids are excited, quantum vortices are nucleated and the many wonders of quantum turbulence are explored today. This work is inspired by the recent usage of statistical methods to image collective excitations in excited quantum gas, and to evidence phase transitions [4, 5]. We extend these methods to turbulent superfluids, to explore the collective behaviour of Onsager clusters, and in hope of heralding their detection. 1 Chapter 2 Theory 2.1 Bose–Einstein Condensation Bose–Einstein condensation occurs in many-particle systems when a single state becomes macroscopically occupied. A result of the indistinguishability of bosons and the discreteness of quantum states, BEC is the emergence of long-range quantum coherence in otherwise classically behaving systems. It allows collections of many quantum particles to be described by a single-particle wavefunction, where large particle densities evolve like a single-particle probability density [18]. BEC allows description of macroscopic quantum systems with microscopic laws. It underpins our under- standing of superfluidity, and enables the realisation of quantum turbulence [19]. BEC allows fundamental quantum mechanical behaviours to be explored experimentally, such as through ultracold gases, where bosons condense into the ground-state. To date, BEC has been achieved in many species of ultracold elemental gases, molecules, photons and quasiparticles [20, 21, 22, 23]. The thermodynamic conditions to achieve BEC depend on the system, its temperature, dimensionality, the particle species and the nature of their trapping. We review those for both inert and interacting atoms in uniform and harmonic traps. 2.1.1 Ideal Bose Gas The Bose–Einstein distribution gives the expected population N i among eigenstates with energy i in a system of N bosons at temperature T and thermal equilibrium, as 〈 N i 〉 = ( e ( i − μ ) /k B T − 1 ) − 1 , (2.1) where k B is the Stefan–Boltzmann constant and μ is the chemical potential. Bose–Einstein condensation occurs when despite a growing total population N → ∞ (with a constant particle density N/V = c ), lim N →∞ N/V = c 〈 N 0 〉 N = O (1) , (2.2) for some state denoted i = 0 That is, a non-infinitesimal fraction of the growing total population remains expected in a particular quantum state, which is typically the ground-state. This is enabled by the divergence of 〈 N 0 〉 (in Equation 2.1) above a critical particle density, or below a critical temperature T c 2 The population not found in i = 0 , not participating in condensation, is referred to as the thermal population. BEC is a saturation of the thermal states, whereby new additional particles are forced into the condensed state of growing population. Achieving global coherence, the condensed population can be described by a single-particle wavefunction ψ ( r , t ) . In this work, we adopt the normalisation ∫ | ψ | 2 d r = 1 , (2.3) where the many-particle density relates to the single-particle wavefunction by n ( r , t ) = N | ψ ( r , t ) | 2 (2.4) and the concept of individual particles is lost. Exact forms of the condensate fraction and the thermodynamic conditions for condensation are system specific. In an ideal gas, BEC is forbidden in 2D uniform systems by the Mermin–Wagner theorem [24]. The condensate fraction and critical temperature in 3D uniform systems respectively are [18] 〈 N 0 〉 N = { 1 − ( T /T c ) 3 / 2 T < T c 0 T ≥ T c and N V λ 3 ( T c ) = ζ (3 / 2) , (2.5) where ζ is the Euler–Riemann zeta function and λ is the thermal wavelength (roughly the average of each particle’s de Broglie wavelength) λ ( T ) = √ 2 π ~ 2 mk B T . (2.6) Referred as Einstein’s condition , this formulation lets one intuit the critical temperature as that required for an ensemble of uniform particles to exhibit quantum behaviour; when their separation approaches their de Broglie wavelengths. BEC can occur in both 2D and 3D harmonically trapped ideal gas. In D ∈ { 1 , 2 } dimensions, the conden- sate fraction and critical temperature (to first order T 0 c ) respectively satisfy [18] 〈 N 0 〉 N = { 1 − ( T /T c ) D T < T c 0 T ≥ T c and N = ζ ( D ) ( k B T 0 c ~ ω ) D , (2.7) where ω is the harmonic trap frequency. The condensate fractions of uniform and harmonically trapped ideal gases are presented in Figure 2.1, which demonstrates the rapid growth of the condensate fraction below the critical temperature. This sudden onset of the condensate fraction is an important feature for detecting BEC [20, 25]. 3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 T /T c 〈 N 0 〉 /N V = 0 V = mω 2 r 2 / 2 , D = 3 V = mω 2 r 2 / 2 , D = 2 Figure 2.1: Condensate fractions of the ideal gas as a function of temperature for different trapping potentials and dimensionality. 2.1.2 Interacting Bose Gas Even very weak interactions between bosons, such as the intermolecular Van der Waals force, can drasti- cally change the conditions and nature of BEC. For instance, interactions are crucial to the phenomenon of superfluidity [26] (to be discussed in the proceeding section). The interactions of dilute, low energy gases can be modelled as two-particle collisions, described by the pseudopotential (between particles at r i and r j ) U ( r i , r j ) = g δ ( r i − r j ) (2.8) The Born approximation admits the coupling constant g in terms of the low-energy s-wave scattering length a , a parameter of classical scattering theory [27], as g = 4 π ~ 2 m a. (2.9) Einstein’s condition for condensation in a uniform system is unchanged, though interactions shift the critical temperature in harmonic systems [28] by δT c ' − 1 33 T 0 c a √ mω ~ N 1 / 6 , (2.10) and generally softens the onset of, and the extent of, the condensate fraction [29]. In an external potential V ( r ) and a frame rotating at Ω , these particles are described by the many-body Hamiltonian ˆ H = ∑ i − ~ 2 2 m ∇ 2 i + V ( r i ) + Ω ˆ L z + g ∑ j<i δ ( r i − r j ) (2.11) Having O ( N 2 ) terms, the Hamiltonian is impractical to use for analysing systems with experimentally typical populations ( N ∼ 10 6 [20]). 4 2.1.2.1 Mean Field Theory The Hartree–Fock approximation assumes complete condensation, whereby every particle has the same wavefunction [30]: ψ ( r 1 , ..., r N ) = N ∏ i =1 ψ ( r i ) (2.12) This is valid for dilute, weakly interacting gases at temperatures very close to absolute zero. In this limit, the single-particle Hamiltonian has a mean field expression (derived in Appendix 7.A) ˆ H = − ~ 2 2 m ∇ 2 + V ( r ) + Ω ˆ L z + gN | ψ | 2 , (2.13) as is used to describe the entire system. This presents the Gross–Pitaevskii equation (GPE) [18]; a non- linear Schr ̈ odinger equation with time-dependent form i ~ ∂ ∂t ψ ( r , t ) = ˆ Hψ ( r , t ) (2.14) The GPE is a classical approximation to the equation of motion of the atomic field, and neglects quantum fluctuations. Stationary solutions of the form ψ ( r , t ) = ψ ( r ) e − iμt/ ~ solve the time-independent GPE μψ ( r ) = ( − ~ 2 2 m ∇ 2 + V ( r ) + Ω ˆ L z + gN | ψ ( r ) | 2 ) ψ ( r ) (2.15) 2.1.2.2 Thomas–Fermi Approximation The time-independent GPE remains non-linear and second order, and is difficult to solve analytically. However, in a regime of strong repulsion between harmonically trapped particles, both the interaction and kinetic terms seek to expand the ground-state profile; the latter may then be neglected. This is the Thomas–Fermi approximation (TFA) [31], and is valid in the regime g ~ ωR 3 2 N (2.16) for a harmonic trapping frequency ω and a condensate radius R . The time-independent GPE reduces to μψ ( r ) ' ( V ( r ) + gN | ψ ( r ) | 2 ) ψ ( r ) (2.17) and supports an analytic solution | ψ ( r ) | 2 = { μ − V ( r ) gN r < R 0 r ≥ R (2.18) where in 2D (derived in Appendix 7.B), μ = ω √ gN π and R = √ 2 ω ( gN π ) 1 / 4 (2.19) The TFA is useful for predicting properties of harmonically trapped condensates, such as the effective condensate radius and radii of vortex cores. We note TFAs has been derived for systems of various dimensionalities and trapping potentials [32], and in weaker interaction regimes [33]. 5 2.1.3 Superfluid Bose Gas Superfluids are collections of particles which experience dissipation-free flow. They feature an energy barrier to excitation, and are consequentially undisturbed by very low energy perturbations. Superfluidity is associated with a range of phenomena such as inviscidity, infinite thermal conductivity or non-diffusive, wave-like transmission of heat, anomalously high capillarity and zero entropy. Superfluidity can arise in BECs, though the criteria are complicated and system-dependent [26, 34]. In this thesis, we work in the 2D zero-temperature limit and consider only harmonically and homogeneous disk-shape trapped BECs of interacting atoms, where a non-zero superfluid fraction exists. Furthermore, we simulate pure entirely-superfluid condensates, though note such conditions are not physically achiev- able above absolute zero [26]. Writing the pure condensate wavefunction explicitly in terms of a phase θ ( r ) , ψ ( r ) = | ψ ( r ) | e iθ ( r ) , (2.20) allows us to express the superfluid velocity (derived in Appendix 7.C) as v ( r ) = ~ m ∇ θ ( r ) (2.21) From this, we realise the superfluid flow is irrotational : ∇ × v ( r ) = 0 (2.22) in the absence of singularities in the phase. 2.1.3.1 Linear Response Theory The linear dynamics around a stationary state ψ 0 in response to weak excitation can be modelled by a superposition of weak perturbations / quasiparticles δψ q . We can write ψ ( r , t ) = e − iμt/ ~ ( ψ 0 ( r ) + ∑ q δψ q ( r , t ) ) , (2.23) where the q -th quasiparticle has angular frequency ω q and energy ~ ω q , and is expressed as a sum of spatial components u q and v q : δψ q ( r , t ) = e − iω q t u q ( r ) + e iω q t v ∗ q ( r ) (2.24) Such a formulation allows us to linearise the time-dependent Gross–Pitaevskii equation (performed in Appendix 7.D) to the Bogoliubov-de Gennes (BdG) equations; coupled differential equations for u q , v q and ω q [12]: ( ˆ L ( r ) − μ gψ 2 0 ( r ) − gψ ∗ 0 ( r ) 2 − ˆ L ∗ ( r ) + μ ) ( u q ( r ) v q ( r ) ) = ~ ω q ( u q ( r ) v q ( r ) ) , (2.25) where we define ˆ L ( r ) = − ~ 2 ∇ 2 2 m + V ( r ) + 2 gN | ψ ( r ) | 2 − Ω ˆ L z (2.26) Solutions of the BdG equations represent the possible excitations of the system, with real ω q implying dynamically stable states, as are found around the stable ground-state. Under the normalisation ∫ | u q | 2 − | v q | 2 d r = 1 , (2.27) 6 the average angular momentum of an excitation [35] is given by l q = ∫ u ∗ q ˆ L z u q + v ∗ q ˆ L z v q − ψ ∗ 0 ˆ L z ψ 0 d r ∫ | u q | 2 + | v q | 2 d r (2.28) We note that in axisymmetric systems, such as in non-rotating isotropic harmonic traps, the q energy- degenerate excitations can be expressed with a zero angular-momentum basis [35]. A small rotational term can be applied to break the degeneracy between different l q modes in numerical solutions of the BdG equations. Low energy excitations correspond to collective behaviour, whereas u q and v q decouple (with | u q | | v q | ) for high energies and the excitations become single-particle. The collective nature of low energy excitations is the source of superfluidity. This is most simply demonstrated for the weakly interacting homogeneous Bose gas in free-space of volume V , where the BdG equations can be solved analytically, and give the Bogoliubov dispersion relation [18] ~ ω q = √ ~ ω 0 q ( ~ ω 0 q + 2 N g V ) (2.29) For high momenta l q 0 , this approaches the energy of a free particle ~ ω q → ~ ω 0 q = ~ 2 l 2 q 2 m , (2.30) though for low momenta, presents a linear regime characteristic of a phonon moving at speed c = √ N g mV : ~ ω q ∼ c l q (2.31) As first reasoned by Landau, this linear regime implies superfluidity, whereby classical particles with quadratic spectra moving slower than some critical velocity cannot impart the energy and momentum necessary to excite even the lowest excitation mode. The critical velocity satisfies the Landau criterion [11] and in this case, equals the speed of sound: v c = min ( ω q l q ) = c. (2.32) The Landau criterion also explains the lack of superfluidity in an ideal Bose gas, where c = 0 and there is consequentially no barrier to excitation. For a general inhomogeneous system, the presence of an external potential leads to scattering among the excitation modes and prevents diagonalisation of the BdG equations. Although analytic approximations exist for some systems in hydrodynamic limits [36], the BdG equations are generally solved perturbatively and numerically [35]. 2.1.3.2 Collective Excitations The energy and angular momentum spectrum of the lowest energy excitation modes for a harmonically trapped superfluid are presented in Figure 2.2, as per Equations 2.25 and 2.28. A very small rotation was applied to the system to remove the energy degeneracy between the modes of opposite angular momen- tum, such that the modes in Figure 2.2 are eigenfunctions of both the energy and angular momentum operators. With a few exceptions, the excitation spectrum deviates from the spectrum of an ideal gas, for which the frequencies are multiples of the trapping frequency. The amount of deviation changes adiabatically with the strength of the interaction [37]. The eigenfunctions of some of these modes are plotted in Figure 2.3 7 Figure 2.2: Energy and angular momentum spectrum of the lowest energy modes of a harmonically trapped superfluid of coupling constant gN = 100 , rotating at Ω = 0 01 [ ω 0 ]. The eigenfunctions of the modes labelled a - h are shown in Figure 2.3. Angular frequency is given in units of the trapping frequency ω 0 Figure 2.3: Eigenfunctions of the collective modes shown in Figure 2.3 for the non-rotating interacting harmonic ground-state. We identify some modes in Figures 2.2 and 2.3. The l = 0 modes ( d and h ) are compression or breathing modes, influenced by the compressibility of the condensate. The l = ± 1 modes ( a , f ) are first and higher- order Kohn or dipole modes, corresponding to center-of-mass oscillation of the condensate. b , c and e at l = 1 , l = 2 and l = 3 are the quadrupole, hexapole and octapole surface modes respectively. In general, the energies of these modes depend on the temperature and number of particles, with the exception of the dipole mode which stays close to ω 0 [38]. Though not plotted, we also mention the scissors mode; an irrotational quadrupolar mode which sees a change in energy as a system becomes superfluid [39, 34]. 8 2.2 Two-Dimensional Quantum Turbulence Quantum turbulence concerns the chaotic dynamics of strongly excited quantum fluids. It studies the incompressible fluid behaviour, beyond the compressible collective behaviour. We restrict our study to two dimensional quantum turbulence (2DQT), which although features behaviour very different to three- dimensional systems, is still rich with interesting physics and remains experimentally realisable: strong trapping in a particular dimension can yield very oblate systems with dynamics confined to a plane, which are effectively two-dimensional [40]. 2.2.1 Vortex Structure Strong excitations, rotation and dynamic instabilities in a superfluid can lead to the nucleation of vortices; swirling regions of vanishing fluid density. Vortices accompany a quantised winding of 2 πn ( n ∈ Z ) in the phase and a hole in the fluid density, which can be viewed as a topological defect bringing the density to a new homomorphic class. An example of a single vortex in a harmonically trapped superfluid is present in Figure 2.4. Figure 2.4: The probability density (left) and phase (right) of the harmonically-trapped non-rotating single positively-charged vortex ground-state ( gN = 4000 ). x and y are given in units of the harmonic oscillator length-scale a 0 = √ ~ /mω . The profile of the superfluid is emphasised in the phase plots by transparency. The core shape of a 2D vortex has no analytic expression, though is well approximated by the ansatz [40] (where | ψ ′ | is the wavefunction far from the vortex core at r = ( x, y ) = 0 ) ψ ( r ) = √ N | ψ ′ | re i atan2 ( y,x ) √ r 2 + 1 47 ξ 2 (2.33) and is illustrated in Figure 2.5. The size of the vortex core is on the order of the healing length ξ , which characterises the length scale over which local density perturbations can occur [41]. 9 1 2 3 4 0.2 0.4 0.6 0.8 1.0 r [ ξ ] | ψ ( r ) | / | ψ ′ | Figure 2.5: Amplitude of a vortex core wavefunction as a function of radius r from the core. | ψ ′ | is the wavefunction magnitude far from the vortex core and ξ is the healing length. In two dimensions, vortices swirling clockwise (anticlockwise) are said to have positive (negative) sign , or be positively (negatively) charged or circulating . Negatively charged vortices are commonly referred to as antivortices . The magnitude of the charge reflects the winding number n ∈ Z of the phase winding around the vortex; ∮ θ d r = 2 πn. (2.34) Vortices with unity (greater) charge are said to be singly- (multi-) quantised . An example of two doubly- quantised vortices are presented in Figure 2.6. Figure 2.6: The probability density (left) and phase (right) of a two doubly-quantised vortex state. Blue (plus) and red (minus) circles [markers] indicate a positively (negatively) charged vortex in the probability density [phase]. Vortices allow the superfluid to gain a net non-zero circulation or curl. The velocity field at a position r = r (cos θ, sin θ ) around a charge q vortex core is given by [40] v ( r ) = q ~ mr ( − sin θ, cos θ ) , (2.35) 10 with vorticity ∇ × v ( r ) = q 2 π ~ m δ ( r ) , (2.36) and implies a quantised circulation ∮ v ( r ) · d r = q 2 π ~ m . (2.37) Vortices are experimentally nucleated in many ways, such as through stirring the condensate with lasers [3, 42], rapidly cooling an ultracold gas through the condensate phase transition [43], and rotating the (anisotropic) trapping potential [44]. 2.2.2 Vortex Dynamics Vortices in superfluids exhibit many interesting dynamics. A single off-center vortex will precess around the trapping center. A dynamic instability can nucleate pairs of oppositely charged vortices, and inci- dent oppositely charged vortices can annihilate, exciting phonons. Multi-quantised vortices can decay into many singly-quantised vortices. Proximate vortices interact and move through each other’s velocity fields like rigid bodies. Oppositely charged vortices can become bound and travel across the condensate together, whilst same-signed vortices can orbit one another. Figure 2.7: Trajectories of vortices in a probability density, as indicated by dashed lines, in simulations of harmonically trapped, single and double vortex sates. Some trajectories of vortices in short simulations of harmonically trapped superfluids are presented in Figure 2.7. Generally, the local condensate density and its variation due to collective excitations will affect the motion of a vortex. The dynamics of distant vortices in the absence of collective excitations are approximately Hamiltonian [1]. The equations of motion for N v vortices at ( x i , y i ) and with charges q i can be written in the form q i d x i d t = ∂H ∂y i , q i d y i d t = − ∂H ∂x i , (2.38) 11 in terms of the energy integral H = − 1 4 π ∑ i>j q i q j ln ( ( x j − x i ) 2 + ( y j − y i ) 2 ) (2.39) This is Onsager’s point-vortex model for a 2D fluid, approximating the GPE in the incompressible limit [1, 45], and is frequently employed for simulations of 2DQT [2, 46, 47, 48]. 2.2.3 Vortex Thermodynamics The presence of a vortex in a superfluid affects the superfluid fraction [34], introduces an anomalous collective excitation [49] and perturbs the energies of the existing modes [50]. Furthermore, the distri- bution and dynamics of