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 twodimensional 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, twodimensional fluid relaxes, the system can become more ordered in time. Largescale, coherentlyswirling 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 modelfree 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 twodimensional superfluid dynamics with a meanfield 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 thoughtprovoking 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 eSolutionsResearch 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 TwoDimensional 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 Groundstates 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 Groundstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 [ω0 ]. The eigen functions of the modes labelled ah are shown in Figure 2.3. Angular frequency is given in units of the trapping frequency ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Eigenfunctions of the collective modes shown in Figure 2.3 for the nonrotating interacting harmonic groundstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 The probability density (left) and phase (right) of the harmonicallytrapped nonrotating single positivelycharged vortex groundstate (gN p = 4000). x and y are given in units of the harmonic oscillator lengthscale a0 = ~/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 0  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 doublyquantised 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 [ω0 ]. (b) shows bound vortexantivortex 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) 10vortex 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 10vortex 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/ω0 ]. 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 eg were identified as collective excitations, and d and hl 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 downscaled 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 zerovortex harmonic groundstates subject to a variety trap perturbations. . . 30 4.3 PCA of the singlevortex harmonic groundstate subject to trap displacement, with (b) and without (a) vortex filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 PCA of a 10vortex 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 harmonictrap 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 100vortex configuration in a disktrapped 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 Nv 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/ω0 ]. 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 vortexantivortex 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 vortexantivortex 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 10vortex 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 disktrapped simulation and its PCA. (a) shows the number of surviving vortices Nv 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 disktrapped simulation shown in Figure 7.5 at various stages of evolution. . . 56 v Chapter 1 Introduction Bosons are indistinguishable, integerspin 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 meanfield 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 manyparticle 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 longrange quantum coherence in otherwise classically behaving systems. It allows collections of many quantum particles to be described by a singleparticle wavefunction, where large particle densities evolve like a singleparticle 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 groundstate. 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 Ni among eigenstates with energy i in a system of N bosons at temperature T and thermal equilibrium, as −1 hNi i = e(i −µ)/kB T − 1 , (2.1) where kB 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), hN0 i lim = O(1), (2.2) N →∞ N N/V =c for some state denoted i = 0. That is, a noninfinitesimal fraction of the growing total population remains expected in a particular quantum state, which is typically the groundstate. This is enabled by the divergence of hN0 i (in Equation 2.1) above a critical particle density, or below a critical temperature Tc . 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 singleparticle wavefunction ψ(r, t). In this work, we adopt the normalisation Z ψ2 dr = 1, (2.3) where the manyparticle density relates to the singleparticle 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] ( hN0 i 1 − (T /Tc )3/2 T < Tc N 3 = and λ (Tc ) = ζ(3/2), (2.5) N 0 T ≥ Tc V where ζ is the Euler–Riemann zeta function and λ is the thermal wavelength (roughly the average of each particle’s de Broglie wavelength) s 2π~2 λ(T ) = . (2.6) mkB T 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 Tc0 ) respectively satisfy [18] ( D 1 − (T /Tc )D T < Tc kB Tc0 hN0 i = and N = ζ(D) , (2.7) N 0 T ≥ Tc ~ω 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 1.0 V =0 V = mω 2 r2 /2, D = 3 0.8 V = mω 2 r2 /2, D = 2 hN0 i /N 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T /Tc 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 twoparticle collisions, described by the pseudopotential (between particles at ri and rj ) U (ri , rj ) = g δ(ri − rj ). (2.8) The Born approximation admits the coupling constant g in terms of the lowenergy swave scattering length a, a parameter of classical scattering theory [27], as 4π~2 g= a. (2.9) m Einstein’s condition for condensation in a uniform system is unchanged, though interactions shift the critical temperature in harmonic systems [28] by r 0 mω 1/6 δTc ' −1.33Tc a N , (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 manybody Hamiltonian 2 X ~ X Ĥ = − ∇2i + V (ri ) + ΩL̂z + g δ(ri − rj ) . (2.11) i 2m j<i Having O(N 2 ) terms, the Hamiltonian is impractical to use for analysing systems with experimentally typical populations (N ∼ 106 [20]). 4 2.1.2.1 Mean Field Theory The Hartree–Fock approximation assumes complete condensation, whereby every particle has the same wavefunction [30]: N Y ψ(r1 , ..., rN ) = ψ(ri ). (2.12) i=1 This is valid for dilute, weakly interacting gases at temperatures very close to absolute zero. In this limit, the singleparticle Hamiltonian has a mean field expression (derived in Appendix 7.A) ~2 2 Ĥ = − ∇ + V (r) + ΩL̂z + gN ψ2 , (2.13) 2m as is used to describe the entire system. This presents the Gross–Pitaevskii equation (GPE) [18]; a non linear Schrödinger equation with timedependent form ∂ i~ ψ(r, t) = Ĥψ(r, t). (2.14) ∂t 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 timeindependent GPE ~2 2 2 µψ(r) = − ∇ + V (r) + ΩL̂z + gN ψ(r) ψ(r). (2.15) 2m 2.1.2.2 Thomas–Fermi Approximation The timeindependent GPE remains nonlinear 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 groundstate profile; the latter may then be neglected. This is the Thomas–Fermi approximation (TFA) [31], and is valid in the regime ~ωR3 g (2.16) 2N for a harmonic trapping frequency ω and a condensate radius R. The timeindependent GPE reduces to µψ(r) ' V (r) + gN ψ(r)2 ψ(r) (2.17) and supports an analytic solution ( µ−V (r) 2 gN r<R ψ(r) = (2.18) 0 r≥R where in 2D (derived in Appendix 7.B), r r 1/4 gN 2 gN µ=ω and R= . (2.19) π ω π 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 dissipationfree 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 nondiffusive, wavelike transmission of heat, anomalously high capillarity and zero entropy. Superfluidity can arise in BECs, though the criteria are complicated and systemdependent [26, 34]. In this thesis, we work in the 2D zerotemperature limit and consider only harmonically and homogeneous diskshape trapped BECs of interacting atoms, where a nonzero superfluid fraction exists. Furthermore, we simulate pure entirelysuperfluid 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)eiθ(r) , (2.20) allows us to express the superfluid velocity (derived in Appendix 7.C) as ~ v(r) = ∇θ(r). (2.21) m 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 ψ0 in response to weak excitation can be modelled by a superposition of weak perturbations / quasiparticles δψq . We can write ! X −iµt/~ ψ(r, t) = e ψ0 (r) + δψq (r, t) , (2.23) q where the qth quasiparticle has angular frequency ωq and energy ~ωq , and is expressed as a sum of spatial components uq and vq : δψq (r, t) = e−iωq t uq (r) + eiωq t vq∗ (r). (2.24) Such a formulation allows us to linearise the timedependent Gross–Pitaevskii equation (performed in Appendix 7.D) to the Bogoliubovde Gennes (BdG) equations; coupled differential equations for uq , vq and ωq [12]: gψ02 (r) L̂(r) − µ uq (r) uq (r) = ~ωq , (2.25) −gψ0∗ (r)2 −L̂∗ (r) + µ vq (r) vq (r) where we define ~2 ∇2 L̂(r) = − + V (r) + 2gN ψ(r)2 − ΩL̂z . (2.26) 2m Solutions of the BdG equations represent the possible excitations of the system, with real ωq implying dynamically stable states, as are found around the stable groundstate. Under the normalisation Z uq 2 − vq 2 dr = 1, (2.27) 6 the average angular momentum of an excitation [35] is given by R ∗ uq L̂z uq + vq∗ L̂z vq − ψ0∗ L̂z ψ0 dr lq = R . (2.28) uq 2 + vq 2 dr We note that in axisymmetric systems, such as in nonrotating isotropic harmonic traps, the q energy degenerate excitations can be expressed with a zero angularmomentum basis [35]. A small rotational term can be applied to break the degeneracy between different lq modes in numerical solutions of the BdG equations. Low energy excitations correspond to collective behaviour, whereas uq and vq decouple (with uq  vq ) for high energies and the excitations become singleparticle. 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 freespace of volume V , where the BdG equations can be solved analytically, and give the Bogoliubov dispersion relation [18] s 2N g ~ωq = ~ωq0 ~ωq0 + . (2.29) V For high momenta lq 0, this approaches the energy of a free particle ~2 lq2 ~ωq → ~ωq0 = , (2.30) 2m q Ng though for low momenta, presents a linear regime characteristic of a phonon moving at speed c = mV : ~ωq ∼ c lq . (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: ωq vc = min . = c. (2.32) lq 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 [ω0 ]. The eigenfunctions of the modes labelled ah are shown in Figure 2.3. Angular frequency is given in units of the trapping frequency ω0 . Figure 2.3: Eigenfunctions of the collective modes shown in Figure 2.3 for the nonrotating interacting harmonic groundstate. 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 centerofmass 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 ω0 [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 TwoDimensional 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 twodimensional [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 2πn (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 harmonicallytrapped nonrotating single positivelycharged vortex p groundstate (gN = 4000). x and y are given in units of the harmonic oscillator lengthscale a0 = ~/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 0  is the wavefunction far from the vortex core at r = (x, y) = 0) √ reiatan2(y,x) ψ(r) = N ψ 0  p (2.33) r2 + 1.47ξ 2 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 1.0 0.8 ψ(r)/ψ 0  0.6 0.4 0.2 1 2 3 4 r [ξ] Figure 2.5: Amplitude of a vortex core wavefunction as a function of radius r from the core. ψ 0  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; I θ dr = 2πn. (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 doublyquantised 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 nonzero 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 (− sin θ, cos θ), (2.35) mr 10 with vorticity 2π~ ∇ × v(r) = q δ(r), (2.36) m and implies a quantised circulation I 2π~ v(r) · dr = q . (2.37) m 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 offcenter 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. Multiquantised vortices can decay into many singlyquantised 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 samesigned 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 Nv vortices at (xi , yi ) and with charges qi can be written in the form dx ∂H qi i = , dt ∂yi (2.38) qi dyi = − ∂H , dt ∂xi 11 in terms of the energy integral 1 X qi qj ln (xj − xi )2 + (yj − yi )2 . H=− (2.39) 4π i>j This is Onsager’s pointvortex 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 vortices can relate to the collective and thermodynamic states of the superfluid. For instance, rotating systems can nucleate many samesigned vortices which form a regular triangu lar lattice [16]. An example of an Abrikosov lattice, the subject of the 2003 Nobel Prize in phyiscs, is demonstrated in Figure 2.8a. The lattice supports the Tkachenko mode; an anomalous collective vortex oscillation where the vortex cores move elliptically [35]. Another example is the pairing of oppositely signed vortices at the BerezinskiiKosterlitzThouless (BKT) crossover (illustrated in Figure 2.8b). The pairing accompanies algebraic decay of the coherence in 2D homogenous systems, which otherwise decays exponentially under the Mermin–Wagner theorem [34]. The BKT crossover thus allows superfluidity to emerge in otherwise incoherent systems; vortexantivortex pairing in a 2D homogeneous Bose gas can therefore lead to changes in the scissors excitation. This thesis is partly inspired by the recent experimental evidencing of the BKT crossover, by using statistical techniques to analyse the scissors excitation [5]. We acknowledge the BKT crossover is the subject of the 2016 Nobel Prize in Physics [51]. (a) (b) Figure 2.8: Vortex configurations associated with special thermodynamic states. (a) shows a triangular vortex lattice in a harmonically trapped superfluid rotating at Ω = 0.7 [ω0 ]. (b) shows bound vortex antivortex pairs in a disk trapped superfluid. Blue (red) circles hover over positively (negatively) charged vortices. 12 2.2.3.1 Onsager Clusters In this work, we study clustered vortex configurations; a thermodynamic state entered at negative tem peratures. Coherent, macroscopic, ordered vortex structures can spontaneously emerge from systems with initially disordered vortex configurations [46]. Onsager clustering is a spontaneous spatial cluster ing of likesigned vortices, common in classical, turbulent fluids. In superfluids, where the wavefunction encodes all state information and implies a bounded phase space, Onsager clusters emerge as quasistable equilibrium states at negative temperatures. They are formed by an inverse energy cascade [1], as vortexantivortex annihilations increase the mean energy per vortex [46]. The clusters rotate like rigid bodies and precess through the fluid [52], while oppositely charged clusters orbit. Though free vortices roam practically at random and yield disorganised contributions to the total flow, macroscopic clusters move more regularly and dominate the system motion [1]. This suggests strong, incompressible behaviour should emerge at the formation of clustering [52, 53, 40]. The magnitude of the macroscopic vortex dipole moment X d = d = qi ri (2.40) i can be taken as an order parameter of clustering, where qi and ri are the charge and position of the ith vortex respectively. We can express this nondimensionally in units of κNv R0 [d] = , (2.41) 2 where R0 is the radius of, and Nv the number of vortices in, the system. In these units d ∼ 0 for disordered configurations and d ∼ 1 for clusters with centres separated by the radius of the condensate [2]. Monte Carlo estimates suggest a spontaneous transition from disordered to clustered vortex states occurs at d ∼ 0.5 [46]. Examples of instantaneously clustered and disordered vortex configurations are presented, with their dipole moments, in Figure 2.9; we stress these merely illustrate the vortex configurations and are not necessarily in a stable thermodynamic state. The contrast in orderedness of these states is easily visible in their complex phases. The order parameter is not constant in time, even for systems that do not undergo a phase transition. See Figure 2.10a for the order parameter over the evolution of the systems in Figure 2.9. Variation can reflect system dynamics, like slow undulation to cluster orbital motion [46] or rapid undulation due to compression excitations; these effects can be seen in Figures 2.10a and 2.10b 13 (b) (a) (d) (c) Figure 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 (a) (b) Figure 2.10: The clustering order parameter d over the evolution of 10vortex 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 excited breathing mode. Red lines indicate instantaneous values and blue lines their sliding average over a window of ∆t = 20 [1/ω0 ]. Vor tices temporarily straying too far to the condensate edge can be missed by the vortex detection routine, causing sharp, large spikes in the red curve. Onsager clusters have emerged in GPE simulations with an initial periodic array of vortices [52], random vortex configurations [46, 47] and vortices nucleated from dynamic instabilities [54, 55, 56], in systems with harmonic, annular [53] and disk shaped traps and with uniform periodic domains [40]. It was recently shown that uniformly trapped superfluids have a greater tendency to exhibit Onsager clustering than those harmonically trapped [47]. 2.3 Experimental Methods To motivate our theoretical and numerical investigations, we review the experimental status of 2DQT. As an immensely large and active area of research, we summarise only the progress and limitations relevant to this work. 2.3.1 Achieving Superfluids Superfluids are typically realised through Bose–Einstein condensation of ultracold bosonic gases. Va pors of alkali atoms are loaded into magnetooptical traps and cooled to microkelvin by damping with tuned lasers. The result is transferred to magnetic or optical traps where the most energetic atoms are encouraged to evaporate and the remaining atoms rethermalise at nanokelvin temperatures [57]. BEC has been achieved in a wide variety of trapping geometries, such as harmonic [20] and diskshaped [58], and in regular optical lattices [59]. Effective 2D regimes are accessed by tight confinement in a particular dimension [58, 60]. Vortices [3, 42], collective excitations [37, 61] and superfluidity [62] have long been observed in BECs. 2.3.2 Imaging Superfluids Images of BEC are obtained by absorption imaging. The shadow of the Bose cloud is incident on a chargecoupled device array, where the optical density is proportional to the column density of atoms 15 [20]. This is generally destructive; successive images are obtained by repeatedly and precisely recreating the system and imaging after different suspension times [60]. Collective oscillations can be observed in this way [61, 37]. Absorption imaging is noisy, and is typically subject to stray light, fluctuations of atom number and mechanical vibration. Statistical analysis can identify and isolate these noises however [4]. Not discussed here are methods of imaging the velocity distribution, and nondestructive BEC imaging, such as through dispersive and phasecontrast imaging techniques [63, 64]. 2.3.3 Imaging Vortices Vortex cores are identified in absorption images by a reduced column density. Typically this requires imaging some time after removing the external trapping potential, causing the condensate and vortex cores to expand and become optically resolvable [56]. While the net circulation can be found by indirect measurements of the angular momentum, individual circulations are not easily accessed [52]. Experimental vortexsign detection is currently an area of active research [65]. Bragg spectroscopy was very recently applied to image the velocity field of a 2D turbulent BEC, from which the circulations of the vortices was determined [17]. We note that despite recent efforts [17, 55], Onsager clustering has not yet been experimentally ob served. This may be attributed to experimental conditions in which clustering is not prevalent [47]. We distinguish this from the observation of nonspontaneous clustering [56], where very tight clustering enables identification of a multiply charged dipole in a single density image. 2.4 Principal Component Analysis Principal Component Analysis (PCA) is a generic statistical method for probing the essential patterns in data. It generates a basis of uncorrelated variations present in the data, referred to as the principal components (PCs). Each PC is weighted by an eigenvalue or principal component number (PCN) which reflects the significance of the PC’s associated variation in the data. The sum of the PCs, which we refer to as the principal component number total (PCNT), signifies the total variation in the data. The PCNs are typically normalised by the PCNT to sum to unity. PCA can be applied to a collection of images represented as a timeseries of equallysized 2D arrays [66], where the PCs are images themselves. Being model free, PCA is useful for analysing data with unpre dictable or complicated noise patterns. We caution that PCA shares a name with a related but distinct statistical technique, otherwise referred to as principal factor analysis, singular value decomposition and Karhunen–Loeve expansion [67]. 2.4.1 Algorithm n What follows is an outline of PCA as applied to an ensemble of N scalar images Iij with w × h pixels [4]. n We first flatten each image into a single rowvector v , where n n Iij = v(i−1)w+j . (2.42) The vectors are centred about zero: N 1 X k vn 7→ vn − v . (2.43) N k=1 16 We next compute the covariance matrix of the centred ensemble, where the diagonal elements contain the variance of each pixel and the offdiagonal quantify correlations between pixels. This is elegantly formulated by first constructing a N × wh matrix n Bnm = vm , (2.44) and computing the wh × wh covariance matrix as 1 S= B T B. (2.45) N −1 The PCs and PCNs are the eigenvectors and eigenvalues of S; since S is real and symmetric, the PCs are real and orthogonal. The PCs have dimensions 1 × wh and are typically restored to w × h for ease of viewing. Though not presented here, iterative methods of diagonalising S are suitable since the PCs of largest PCN are a priori sought, and recommended since S is typically very large. Due to the limited number of sources of variation in a collection of images, not all of the wh PCs are meaningful. Traditional, a threshold is set on the PCNs, below which PCs are excluded [68], though qualitative assessment of the validity of a PC can also be made. We refer to the included PCs as the dominant PCs (DPCs). If the input images are temporal, we can compute the frequencies of oscillation of the variational modes uncovered by PCA [4]. We project each PC onto the original images to find its timedependent weight X wk (n∆t) = PCkij Iij n , (2.46) i,j then apply a sinusoidal fit; we perform this numerically using the discretetime Fourier transform. We refer to wk (t) as the timedependent projection (TDP) of the kth PC. The entire set of PCs forms an exact basis for the centred image vectors, or the original images minus their average. We note that the DPCs can be used for reducedfidelity reconstruction of the input images, or represent the data with reduced dimensionality [66], though we do not perform this. 2.4.2 Requirements The input data should sample at least a couple of periods of the expected variation source of lowest frequency. In general, the PCs are more clearly resolved the greater the number of periods sampled. The temporal resolution should be sufficiently high such that clear oscillatory behaviour can be sampled; the sampling frequency should larger than the greatest frequency of the modes of interest. We caution that regularly sampled images may feature aliasing, and PCA will fail to recover variations beating with the sample rate. We avoid this by sampling at a rate which is an irrational multiple of the expected periods of our variations. 2.4.3 Application to Imaging Collective Excitations PCA can be utilised for quantum state tomography [69]. It was recently applied to images of a quasi2D Bose gas to study its dynamical properties in situ, by Dubessy et al. [4]. Since the collective modes of a superfluid are orthogonal, and present uncorrelated dynamics, they manifest as isolated and separate PCs [68], if adequately sampled. Dubessy et al. dynamically perturbed the trapping potential of an oblate quantum degenerate gas of 87 Rb atoms, to excite several lowlying modes. Temporal gas density profiles were collected by repumping and absorption imaging after regular intervals, to be analysed with PCA. The PCs and their PCNs are displayed 17 in Figure 2.11, and were identified as several collective excitations and noise sources in the system. We remark on their resemblance to the solutions of the Bogoliubov–de Genne equations in Figure 2.3. Specifically, the anisotropic trap minimum was displaced, the trapping frequencies slightly changed and the trap axes was rotated; as expected, this excited dipole, compression and scissors modes. The PCNs reflect the system’s population of the PCassociated excitation. To support their identification, the PCs were projected onto the experimental images, and the projections fit to sinusoids; see Figure 2.12. The frequencies of the TDPs matched those of the collective excitations to which the PCs resembled, supporting that PCA has successfully imaged the collective spectrum. PCA has also been applied to examine the evolution of a system’s collective spectra over a phase tran sition. The same group recently applied PCA to locate the phase boundary of the BerezinskiiKosterlitz Thouless crossover in a homogeneous 2D Bose gas [5]. The transition to a superfluid state was evidenced by the emergence of the scissors excitation. Simultaneous superfluid and thermal populations with dif ferent scissors oscillation frequencies initially inhibited resolution of the scissors excitation by PCA. To combat this, the group developed a novel local correlation analysis, repeating PCA for varying radial trun cations of the experimental images. The BKT crossover was identified by the emergence of two regimes in the scissors frequency as a function of the cutoff radius, attributed to the central superfluid and outer normal populations. This clever adaptation demonstrates the versatility of PCA, and its robustness for imaging collective New J. Phys. behaviour 16 (2014) 122001in superfluids. Figure 2.11: DPCsFigure 2. First (and their principalPCNs normalised components in squareofbrackets) an ensemble of 133 images of experimental (61of×excited images 61 pixels) 87 Rb sampling a time interval of 100 ms. Figure a is the mean image of the data Bose gas [4]. a is the average condensate profile, b  c and eg were identified as collective excitations, set containing and d and hl indicated totalthe averaged particle density number proﬁle, and fluctuation andthe subsequent image images (b through l) are the noise respectively. ﬁrst principal components (sorted by decreasing eigenvalue). The number between brackets is the corresponding eigenvalue, expressed as a fraction of the total variance. The color scale is arbitrary for each image. This analysis of the PCs is supported by the study of the timedependent oscillations of the associated weights, computed by projecting the centered original data set on to the PCs. The result of this computation is displayed in ﬁgure 3 for the dipoles and scissors components. Let us focus ﬁrst on the ﬁrst two weights: they exhibit sinusoidal oscillations at the expected trap frequencies (44 Hz and 33 Hz). This supports 18the fact that PCA has correctly identiﬁed, as independent components, the center of mass motion of the cloud along the trap axes. The scissors component displays a more complex oscillation pattern. We ﬁnd that the best ﬁt to the data is given by a sum of three sinusoids, at frequencies 12 Hz, 55 Hz, and 77 Hz. This is related to the New J. Phys. 16 (2014) 122001 Figure 2.12: Projection Figureof the PCs in Figure 3. Solid 2.11 backtimedependent blue circles: onto the experimental images. weight Solid of the bluedipoles two circles: and the scisso TDPs of the dipole and scissors PCs. Solid black line: sinusoidal fit to the data. The first 2 PCs oscillate components. Solid black line: sinusoidal ﬁt to the data. The vertical scale is arbitrar at the anisotropic trapping frequencies, while the fourth PC exhibits more complex behaviour [4]. and independent for each curve. The ﬁrst principal component can be identiﬁed with th strongest horizontal harmonic trap direction, oscillating at 44 Hz, the second to weake 2.4.4 Application corresponding to ImagingtoQuantum a frequency of 33 Hz. The third component exhibits a mor Turbulence complicated behavior, with oscillations at 12 Hz, 55 Hz, and 77 Hz. We estimate at th To the best of the 1Hz author’s knowledge, level PCA has not of the uncertainty been theapplied to images frequency by the ﬁtting of turbulent superfluids determination with procedure. resolvable vortices. It is unknown how vortex motion will articulate into PCs, or what effect it will have on the PCs of collective behaviour. ﬁgure 2, we This work ﬁnd that is motivated thepotential by the simpleforhydrodynamic models PCA to probe the collective of references behaviour [7,in8]various of superfluids fail to extract thes frequencies present in the oscillation of the density, probably due to the the vortex configurations. In particular, we wonder what PCA can discover of superfluid behaviour over fact that severa Onsager vortex clustering transition. We hope to identify a signature of clustering in the PCs of 2DQT collective which couldmodes areforsimultaneously be utilised excited. the experimental detection In this clusters. of Onsager case it That is really is, PCAessential to use a modelfre of experimental approach to be images could analyze compared the dataresults to our set. to verify the presence of clustering. This would require only successive images of a Bose gas, which are commonplace, in contrast to recent proposals for Onsager cluster detection with complicated experimental setups [65, 17]. 4. Comparison with numerical simulations We pursue our investigation numerically to compare the principal components with norma modes. We use a zero temperature twodimensional meanﬁeld model of our cloud and perform a numerical timedependent simulation which mimics the experimental sequence. We the extract the simulated density proﬁles using a regular time sampling, thus obtaining a data set o 152 computed images. We ﬁnally compare the PCA of this data set with the actual norma modes of the trap, computed using the Bogoliubov–de Gennes equations. Details of th simulations are given in appendix B. Figure 4 displays the result of the simulations. Let us ﬁrst focus on the output of PCA (le panel): the ﬁrst few PCs are like those of ﬁgure 2 except for the atom number ﬂuctuation which are not taken into account in the simulation. In particular, dipole, scissors, monopole like, and quadrupolelike patterns are present 19 (see respectively ﬁgures 4(b) through (f)). This interpretation is supported by the display of the normal modes (right panel), and i particular by the density proﬁles of ﬁgures 4(n)–(q) and (v). To compare these proﬁle Chapter 3 Simulation 3.1 Discretisation We explore system dynamics by numerically propagating the GPE, in 1D and 2D space, and in real and imaginary time, for a variety of harmonic and disc shaped traps. Here we discuss 2D simulations. At each discrete timestep, the wavefunction ψ(r) is numerically represented in a finite L × L space on a discrete npt×npt grid rij of separation ∆x = ∆y = L/(npt−1). A npt2 ×1 column vector representation, where ψij ≈ ψ(rij ) ∈ C, T Ψ = ψ11 ... ψnpt 1 ψ12 ... ψnpt 2 ... ψnpt npt (3.1) allows us to elegantly act differential operators by multiplication with matrices. For example, by employ ing the first order finite difference approximation ∂ψ(rij ) 1 1 ∆x ' − ψ(ri−1,j ) + ψ(ri+1,j ), (3.2) ∂x 2 2 for npt = 5, a first order spatial derivative in x can be applied to Ψ by premultiplication with −1 1 −1 1 D̂x = 1 −1 ⊗I (3.3) 2∆x 1 −1 1 where the Kronecker product has been employed on account of working in 2D. The derivative in y would similarly be −1 1 −1 1 D̂y = I ⊗ 1 −1 . (3.4) 2∆y 1 −1 1 20 Our simulations employ the fourthorder finite difference formulae [70] ∂ψij ψi−4 , j 4ψi−3,j 1 4ψi+3,j ψi+4,j ∆x ' − + (ψi−2,j − 4ψi−1,j + 4ψi+1,j − ψi+2,j ) + − , (3.5) ∂x 280 105 5 105 280 ∂ 2 ψij 205ψi,j 1 8 ∆x2 '− + (8ψi+1,j + 8ψi−1,j − ψi+2,j − ψi−2,j )) + (ψi+3,j + ψi−3,j ) (3.6) ∂x2 72 5 315 1 − (ψi+4,j + ψi−4,j ), (3.7) 560 to construct D̂x and D̂x2 . Scalar operators are trivially represented diagonally. V (r11 ) ψ11 .. .. . . V̂ Ψ = ψnpt 1 , (3.8) V (r npt 1 ) .. .. . . 2 ψ11  ψ11 .. .. ˆ 2Ψ = . . ψ 2 ψnpt 1 . (3.9) ψnpt 1  .. .. . . We choose odd values of npt to give the system a central coordinate and maintain numerically convenient values of ∆x and ∆y. Despite having many elements for typical values of npt ∼ 400, the sparsity of these matrices allows them to be memoryefficiently stored. In this representation, definite integration of the wavefunction is trivially performed by Riemann summation. Z npt X Ψdr = ∆x∆y ψij . (3.10) i,j We express our system quantities in terms of the natural harmonic oscillator length, time and energy scale. r ~ 1 a~2 1 [r] = a0 = , [t] = , [E] = ~ω0 , [g] = , [ψ] = . (3.11) mω ω0 m a0 In these units, the GPE for a harmonic trap becomes dimensionless [27]: ∂ψ(r, t) 1 2 1 2 2 2 i = − ∇ + ω r + gN ψ(r, t) + ΩL̂z ψ(r, t), (3.12) ∂t 2 2 as is easier to handle numerically. Vortex charge becomes integer h [q] = κ = . (3.13) m We create a npt2 × npt2 Hamiltonian matrix by 1 2 ˆ 2 + ΩL̂z . Ĥ(Ψ) = − D̂x + D̂y2 + V̂ + gN ψ (3.14) 2 We construct a general anisotropic harmonic trap with frequencies ωx and ωy , rotated by an angle α from the coordinate axis, by 0 x = x cos α − y sin α, y 0 = x sin α + y cos α, (3.15) V (x, y) = 21 (ωx x0 )2 + (ωy y 0 )2 . 21 Meanwhile, we create disc traps by V (r) = U0 (tanh(0.6(r − 1.12R0 )) + 1) (3.16) where U0 and R0 characterise the potential height and effective radius respectively [46]. 3.2 Algorithms 3.2.1 GPE Evolution We discretise time into steps of duration ∆t, and evolve the wavefunction with the fourthorder Runge– Kutta (RK) method (where Ĥ[Ψ] = Ĥ(Ψ)Ψ) ΨA = −iĤ[Ψ(t)], ΨB = −iĤ[Ψ + ΨA ∆t/2], ΨC = −iĤ[Ψ + ΨB ∆t/2], ΨD = −iĤ[Ψ + ΨC ∆t], Ψ(t + ∆t) = Ψ(t) + (ΨA + 2ΨB + 2ΨC + ΨD )∆t/6, (3.17) ensuring ∆t < ∆x∆y/6 to maintain numerical stability [71]. After every ∆tc > ∆t, the probability density is cached for use in PCA. 3.2.2 Solving Groundstates and Relaxation Groundstates are reached by evolving the GPE with a negative and imaginary timestep: t → −it; this is a Wick rotation, and is frequently employed in GPE simulations [35, 47]. Evolution ends when the change in chemical potential since the previous calculation is less than some threshold ∆µ > 0, otherwise the evolution period is doubled. In this way, a smaller ∆µ means a more precisely converged groundstate. Imaginary evolution over short durations (T ≈ 0.05) is employed to relax the system and remove highenergy excitations. 3.2.3 Imprinting Vortices Vortices are imprinted by multiplying the superfluid groundstate with the ansatz in Equation 2.33. The result is simulated in imaginary time for a short duration (t ≈ 0.05) to heal the vortex core and remove unwanted excitations; inaccuracies in the vortex profiles are healed rapidly, and longer times can cause vortices to annihilate [47]. Though we choose the number of positive and negative vortices, their positions are typically randomly chosen within the condensate region. Uniformly random vortices are placed at r where φ = 2πN , 4 √ r = R0 M, 5 r = r(cos φ, sin φ), (3.18) where R0 is the system radius and N and M are uniformly randomly chosen from [0, 1). 22 We generate random cluster centres r1 , r2 at normally random radii with normally random angular separation by R1 , R2 ∼ N (R0 /3, R0 /10) , Ri 0 ≤ Ri ≤ 54 R0 Ri = 0 Ri < 0 , 4 4 5 R0 Ri > 5 R0 θ1 = 2πN1 , ∆θ ∼ N (π, π/10), θ2 = θ1 + ∆θ, ri = Ri (cos θi , sin θi ), (3.19) (3.20) where N (µ, σ 2 ) is the normal distribution with mean µ and variance σ 2 and N1 is uniformly random in [0, 1). Vortices are then randomly positioned around the cluster centres by the same algorithm for generating uniformly random vortex positions around the trap centre. 3.2.4 Detecting Vortices We detect vortices as windings in the wavefunction’s phase around points inside the condensate region. We take the condensate region as where the probability density is above 1% its global maximum; this neglects ghost vortices where the condensate has effectively vanished [47, 72]. 3.2.5 Principal Component Analysis The simulated probably density after every ∆tc is modified before being input to PCA. For large grid sizes (npt > 200), the frames are first scaled down, since the PCA algorithm involves construction of a memoryintense npt2 × npt2 matrix. Scaling is performed with a simple uniform box average; each value in the scaled frame is the average of those in the original frame in a box. Though this may damage localised defects like vortices, averaging preserves the largescale fluctuations, so collective oscillations are still resolvable. We caution that scaling the frames too small reduces the resolution of higher order, high detailed collective modes. We construct the covariance matrix in Equation 2.45 from the centred simulation frames. The largest eigenvalues (PCNs) and their eigenfunctions (PCs) are found iteratively with Lanczos algorithm [73]. The PCs are arbitrarily normalised by their absolute sums, and the PCNs by their sum (the PCNT). Projection weights (TDPs) of each PC are computed with Equation 2.46 and centred. We compute the discrete Fourier transform of the weights using a fast Fourier transform algorithm. Modes are identified by local maxima in the spectral density; weighted summing resolves their frequencies. For long simulations, regimes of dynamics are accessed by performing PCA on contiguous subsets of the simulation frames. Persistent dynamical modes are accessed by PCA of the entire simulation. 3.2.5.1 Vortex Filling We can also fill the vortex cores before scaling and feeding the frames to PCA (motivated in Section 4.1.3). Vortices are located as holes in the condensate region in each frame, and a morphological reconstruc tion algorithm is used to replace each core with the average of the probability density along the core’s boundary [74]. An example of filling atypically large vortices in a frame is shown in Figure 3.1. 23 Keeping the vortex cores small compared to the condensate radius allows us to perform this without damage to the collective dynamics. Contrarily, we demonstrate this can restore otherwise obfuscated col lective behaviour in PCA in Section 4.1.3. We note however that vortex filling does leave small artefacts (circular regions of uniform density) in the frame, occasionally visible in PCA. Figure 3.1: A demonstration of vortex filling (right) on a downscaled simulation frame (left). We note these vortices have much wider cores than those typically simulated in this work. 3.2.5.2 Collective and Vortex PCA We here emphasise an important distinction between the output of PCA on raw simulation frames, and those with filled vortices. PCA on vortexfilled frames images the collective behaviour; we refer to this as collective PCA. PCA on raw frames (no vortex filling) images the vortex behaviour; we refer to this as vortex PCA. The distinction is motivated in Section 4.1.3. 3.3 Parameters When simulating, the wide range of parameters, compiled in Table 3.1, must be carefully chosen. 24 Table 3.1: Parameters of GPE simulation and their typical nondimensional values. Symbol Value Description L 15 Length and height of the simulated spatial domain. npt 401 Number of grid points in each of the x and y directions. ∆t 9e4 Duration of each timestep in wavefunction evolution. ∆tc 0.1 Duration of each timestep between densities used in PCA. T 1500 Total duration of wavefunction evolution. ∆µ 1e8 Threshold of the change in the chemical potential to end groundstate convergence. ωx 1 Harmonic trapping frequency in the x direction. ωy 1 Harmonic trapping frequency in the y direction. α 3◦ Angle of the harmonic trap anisotropy with the coordinate axis. Ω 0.7 Rotational velocity of the simulation frame. U0 30 Height of the disc trap. R0 8.4 Effective radius of the disc trap. gN 4000 Coupling constant. Numerically evolving the GPE by the RK method can lead to a diverging wavefunction if ∆t and ∆x, ∆y are not carefully chosen. The 2D GPE features a Laplacian, which when estimated with a secondorder central finitedifference formula and used in a fourthorder RK method, is stable for [71] √ 3 8 ∆t < ∆x∆y. (3.21) 16 We employ fourthorder central finitedifference formulae with weaker stability requirements, and adopt (for numerical convenience) ∆x∆y ∆t = O . (3.22) 6 When evolving in imaginary time, or in a frame rotating at Ω < ω, the stability requirements are weaker still. The condensate profile must vanish reasonably far from the numerical grid boundaries, which act as in finite potential barriers, and otherwise produce unwanted reflections. Furthermore, numerically solving the BdG equations requires a groundstate wavefunction with plenty of ’empty space’, since some collec tive modes occupy the space beyond the condensate profile. The values of L, gN , Ω, ω and R0 together modulate the relative size of the condensate. In a disc trap, we adopt R0 =≈ 0.6L. In a harmonic trap, the TF approximation is used to estimate the condensate radius RTF before simulation; 1/4 √ gN L = 4RTF = 4 2 . (3.23) π(ω 2 − Ω2 ) Simulating correct vortex dynamics requires spatial resolution of the healing length. We further employ the TF approximation to assume ξ ∼ 1/RTF and choose gN such that npt remains reasonable, since increasing npt is expensive to the simulation running time. We typically employ gN = 4000 and npt = 401, whereby ξ ≈ 5% of the condensate radius and contains ≈ 4 gridpoints. This relative size of healing length and condensate radius allows sufficiently accurate simulation of manyvortex systems (Nv ≈ 100) and is experimentally typical. The vortices are also sufficiently small such that the uncorrelated collective motion can be resolved ’beneath’ and ’between’ them. The rotational velocity Ω should not exceed the trapping frequency ω, else the centrifugal forces ef fectively cancel the trap. When solving the BdG equations, we apply a very small rotation Ω ≈ 0.01 to break the energyangular momentum degeneracy without significantly perturbing the mode energies. Numerical solution of the BdG equations is very sensitive to imperfections in the groundstate. We choose ∆µ ≈ 1e − 8 to ensure we converge very close to the groundstate. 25 3.4 Integrity Checking We perform a number of tests of our simulation and its numerical operators to ensure their correctness. For example, we diagonalise our Hamiltonian in the noninteracting limit and obtain numerically precise energy eigenfunctions of the quantum harmonic oscillator. We apply our Hamiltonian, angular momen tum and chemical potential operators to a variety of numericallysampled analytic eigenfunctions and accurately compute their eigenvalues. We confirm the integrity of the groundstates Ψ0 with energy E0 converged to by imaginary time propagation, by checking Z (Ĥ − E0 )Ψ0 dr (3.24) is very small. The groundstate is compared to its analytic expression and Thomas–Fermi approximation in the zero and strongly interacting regimes. Our tests assure us that our numerical GPE propagation is physical. 3.5 Method We numerically evolve a wide variety of harmonically and disk trapped systems, which exhibit disordered and clustered vortex configurations. We perform many simulations in parallel with a supercomputer architecture to build a statistical sample. For harmonic systems, we chiefly investigate the emergence of structure in the PCs without vortex filling. We perform short simulations of disordered and preclustered 10vortex and 20vortex states. In this case, the collective behaviour merely reflects the strong excitement caused by our vortex imprinting. We expect that stable vortex clusters, even if artificially created alongside strong excitations, will yield PCs with reflect incompressible fluid behaviour. Longer harmonic simulations are performed for randomised configurations of 100200 vortices, which reach equilibration after vortex annihilation with about 10 remaining vortices, though spontaneous clus ter formation is not observed. This is in line with recent findings that clusters do not tend to form in harmonic traps[47]. For disc systems, we perform very long simulations with randomised configurations of 100 vortices which do spontaneously equilibrate to clusters. We investigate the collective and vortex behaviour, by PCA with and without vortex filling, at various stages of the system evolution. Clustering is quantified by the computation of the order parameter at every timestep, which requires detecting the positions and signs of vortices in the wavefunction. 26 Chapter 4 Results 4.1 PCA Testing The dynamics of turbulent superfluid is generally very complicated. To ensure we correctly interpret the results of PCA on turbulent superfluid simulations, we study its output for simpler systems. Following are our important observations for PCA on very simple image collections and superfluid dynamics. Not presented here, we report the success of PCA in resolving elementary excitations in both 1D and 2D noninteracting quantum harmonic oscillator simulations. 4.1.1 2D Squares f =0 f = A1 sin(ω1 t) f = A2 sin(ω2 t) Figure 4.1: Illustration of the sampled 2D function f used in PCA testing. We first test PCA on a collection of images between which we precisely control the variations. This allows us to study the effect that geometry, frequency and amplitude of collective oscillations in superfluids might have on their PCs and PCNs. Our images feature two additive square oscillators against a zero background. That is, we sample time instances of a function f (x, y, t), which is nonzero in two square regions, where its value oscillates sinusoidally. This is illustrated in Figure 4.1. We vary their spatial sizes, their amplitudes Ai and frequencies ωi of oscillation, their relative positions (overlapping and separated), the total sample period and the time between samples. Depending on the sampling conditions, each square can be resolved as an individual PC in the PCA, as a nonzero square at the oscillator’s positions. Following is a number of observations. 27 The PCN goes linearly with the area of the square, with the square of the amplitudes Ai , and is not affected by the oscillation frequencies ωi . For superfluid PCA, this assures us higherorder modes won’t dominate due to higher energy. It informs us PCN variation will be dominated by population differences in the corresponding collective excitations, rather than the spatial size of the excitations. Spatially orthogonal but equal frequency oscillators articulate to the same PC, and are inseparable by PCA. This assures us PCA can identify collective oscillations with separated lobes, like seen in the dipole mode. When the squares overlapped additively, the PCs contained both square profiles, failing to distin guish them. However, in this case, the squares are not necessarily orthogonal, depending on their relative frequencies and phase. This has no reflection on using PCA for superfluid collective imaging, where the excitations are orthogonal. When the frequency of one oscillator was an integer multiple of the other, each oscillator still appeared as separate PCs This assures us PCA can resolve even noninteracting BEC collective modes. Sampling at irregular times often caused mixing of both squares in single PCs. As the total sampling period is reduced, PCA separates the oscillators into different PCs until the period equals the larger of the oscillator periods. This suggests a single oscillation is sufficient for PCA, in line with observations by Dubessy et al. for a superfluid collective oscillation [4]. For smaller periods, the PCs mix both oscillators. In general, longer sample periods presents better resolved PCs. Sampling at frequencies less than approximately four times an oscillator’s frequency can disable its artic ulation into a PC. We observe however that if the apparent (longer) oscillation period due to aliasing is adequately sampled, an oscillator can still be resolved due to aliasing. This suggests infrequently sampled superfluid images may be used to unveil high energy excitations, provided the sampling is over a large period. 4.1.2 2D Superfluid Groundstate Before applying PCA to simulations of turbulent superfluid, we test its ability to recover the collective excitations of a zerovortex system. We populate some elementary excitations of the groundstate by perturbing the external trap and simulate the resulting dynamics, feeding the simulation frames to PCA. Each PC is projected back onto the simulation frames to product their TDP, and the projections are Fourier analysed to deduce the frequency of the PC’s corresponding excitation. Our trap perturbations mimic those used experimentally by Dubessy et al. [4]. We note the lack of colour bars in the proceeding plots; PC values are arbitrary, and are recognised as collective excitations qualitatively, and by their spectral density. It suffices to know the background colour in each PC corresponds to zero, while blue and red to maximum positive and negative values. We first simulate the groundstate, as computed by imaginary time propagation, subject to no additional excitation. The resulting DPCs are presented in Figure 4.2a, and show profiles and frequencies charac teristic of compression modes. This reflects a small (note the tiny PCNT) error in the normalisation of our groundstate, which can excite axially symmetric modes. Despite the Bogoliubovde Genne modes being orthogonal, the TDPs here feature oscillations at multiple frequencies, likely due to expression of multiple collective excitations in each PC. We then excite several dipole modes by displacing the trap, evident in the DPCs in Figure 4.2b. The PCNT is grows by five orders of magnitude. Isotropically compressing the trap produces PCs corresponding to compression modes (Figure 4.2c), with frequencies of the expected relation to the perturbed trap shape. Meanwhile, anisotropic trap compression yields PCs resembling quadrupole excitations (Figure 4.2d). Displacing and anisotropically compressing the trap still resolves the individual breathing, quadruopolar and dipole modes (see Figure 4.2e). Not shown is a wealth of observed modes, such as higher order dipole modes, compression modes, hexapole and octapole modes. 28 We excited the irrotational scissors mode by rotating an initially anisotropic trapping potential by a small angle (α ≈ 3◦ in Equation 3.15), along with compression, quadropole and higher order modes (Figure 4.2f). The driven TDP oscillations of the quadrupole mode (the second PC) demonstrate growth of the excitation population in time. (a) No perturbation. (b) Trap centre displacement. (d) Anisotropic trap compression: ωx → 1.1 [ω0 ], ωy → (c) Isotropic compression ω → 1.1 [ω0 ]. 0.9 [ω0 ]. 29 (e) Anisotropic trap compression (as above) and trap cen (f) Anisotropic trap rotation by α = 3◦ . tre displacement. Figure 4.2: PCA of the zerovortex harmonic groundstates subject to a variety trap perturbations. In these tests, the PCs resemble the Bogoliubov–de Gennes eigenfunctions and are identified as collective responses of the zerovortex system to trap perturbation. 4.1.3 2D Turbulent Superfluid We now apply PCA to simulations of superfluids containing vortices. A single vortex is imprinted in the centre of the harmonic groundstate which is then subject to the trap perturbations in the preceding section. The vortex core is visible in the PCs of Figure 4.3a. Although it reveals the dipolar vortex oscillation, its presence has obfuscated the collective oscillations. The eigenfunctions of the higher order breathing and dipole modes have very small amplitude in contrast with the anomalous vortex component, and are poorly resolved. In essence, coupling of the vortex and collective motions yields correlated variations that PCA cannot separate; the very steep but local condensate deformation that is the vortex core overpowers the small but long range fluctuations of the collective excitations. This was detrimental for a single, oscillating vortex; we expect the effect for many vortices with chaotic motion to be further troublesome. In order to resolve only the collective behaviour, we fill the vortices in the simulation frames and repeat PCA; restoration of the collective modes are seen in Figure 4.3b. 30 (a) (b) Figure 4.3: PCA of the singlevortex harmonic groundstate subject to trap displacement, with (b) and without (a) vortex filling. We next check this technique will work for harmonic systems of many vortices. An arbitrary disordered 10vortex configuration (shown in Figure 3.1) was simulated. PCA was performed with and without prior vortex filling, as presented in Figure 4.4. We indeed see that vortex filling resolves the otherwise obfuscated collective modes. We remark also that undulation in the projections of the nonvortex filled PCs is on the order of the precession period of vortices around the trap centre. This might suggest that for a clustered system, where the projections are more regular in time, cluster precession might be observed in the projection undulation. We investigate this in the proceeding sections. We henceforth refer to PCA with vortex filling, to image collective behaviour, as collective PCA. PCA without vortex filling, to image incompressible fluid behaviour, is referred to as vortex PCA. We note that in collective PCA (like Figure 4.4), the lowerorder modes have well resolved frequencies, but the projection weights of higher modes show irregularities and largeperiod undulation. This may be due to higher modes having greater spatial detail (e.g. more lobes), making them more adversely affected by the pervading, irregular vortex motions. This nuisance persists despite filling the vortex cores, since filling replaces the vortex core with a uniform region, the motion of which is captured by PCA. 31 (a) (b) Figure 4.4: PCA of a 10vortex harmonic system (shown in Figure 3.1) with (b) and without (a) vortex filling. 4.2 PCA of 2DQT in a Harmonic Trap Although spontaneous cluster formation is inhibited in harmonic traps [47], vortices that begin in coher ent clusters often stay clustered [56]. Here we seek a signature of clustering in vortex PCA. We imprint 5 positive and negative vortices in the harmonic superfluid groundstate at uniformly random positions, and at random positions within clusters, by Equations 3.18 and 3.19. Our imprinting into random clusters is a analogous to the experimental nucleation by Neely et al. [56], though our cluster centres begin separated. Here we present an analysis of two particular simulations, featuring vortices in clustered and disordered configurations. We’ll refer to these as the clustered and disordered simulations respectively. The initial wavefunctions in these simulations are displayed in Figure 2.9, and their order parameters in time in Figure 4.5. Clustering is characterised by an average order parameter above d ∼ 0.5 and by sinusoidal oscillation in the order parameter, due to precession of the clusters [46]. The disordered simulation has an order parameter with irregular variation, and an average below d ∼ 0.5. We remark that undulation at ω ≈ 0.3 [ω0 ] is discernible in d in both simulations. 32
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