Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differential equations that are nonlinear. Even richer dynamics can be observed with additional long-range spatial coupling. For example, an interesting type of pattern discovered recently can form in systems with nonlocal coupling. The pattern, called chimera states, is composed of both phase coherent and incoherent regions coexisting in the same system. Through numerical studies of oscillatory media with nonlocal diffusive coupling, I show here for the first time that stable chimera knot structures can exist in 3D. Knots were not previously known to be stable in oscillatory media, nor were such non-trivial chimera patterns known to exist in 3D. To realize different nonlinear phenomena in a controlled way in experiments, a flexible physical system is required. Ultracold atomic systems, specifically, Bose-Einstein condensates (BECs), are good candidates because of the high controllability of almost all parameters, including the nonlinearity, in real time. Hence, experimental studies can be carried out for a variety physical systems, including many classical and quantum field equations. In particular, in this thesis I study a setup of BECs with a third-order Kerr nonlinearity to generate Schr\"odinger cat states, which have applications in quantum metrology. I showed that cat states involving hundreds of atoms should be realizable in BECs. This requires careful optimization of the experimental parameters and analysis of the atom loss. Inspired by the previous two projects, it is an interesting question if chimera states can exist in BECs. By analyzing the underlying mechanism of the effective nonlocal diffusive coupling, I establish here a new analogous mechanism to achieve mediated nonlocal spatial hopping for particles in BECs with two interconvertible states. By adiabatically eliminating the fast mediating channel, I obtain the mean-field of Bose-Hubbard model with fully tunable hopping strength, hopping range, and nonlinearity. This is the first known conservative system exhibiting chimera patterns. More importantly, I show that the model should be implementable in BECs with current technology.
Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differential equations that are nonlinear. Even richer dynamics can be observed with additional long-range spatial coupling. For example, an interesting type of pattern discovered recently can form in systems with nonlocal coupling. The pattern, called chimera states, is composed of both phase coherent and incoherent regions coexisting in the same system. Through numerical studies of oscillatory media with nonlocal diffusive coupling, I show here for the first time that stable chimera knot structures can exist in 3D. Knots were not previously known to be stable in oscillatory media, nor were such non-trivial chimera patterns known to exist in 3D. To realize different nonlinear phenomena in a controlled way in experiments, a flexible physical system is required. Ultracold atomic systems, specifically, Bose-Einstein condensates (BECs), are good candidates because of the high controllability of almost all parameters, including the nonlinearity, in real time. Hence, experimental studies can be carried out for a variety physical systems, including many classical and quantum field equations. In particular, in this thesis I study a setup of BECs with a third-order Kerr nonlinearity to generate Schr"odinger cat states, which have applications in quantum metrology. I showed that cat states involving hundreds of atoms should be realizable in BECs. This requires careful optimization of the experimental parameters and analysis of the atom loss. Inspired by the previous two projects, it is an interesting question if chimera states can exist in BECs. By analyzing the underlying mechanism of the effective nonlocal diffusive coupling, I establish here a new analogous mechanism to achieve mediated nonlocal spatial hopping for particles in BECs with two interconvertible states. By adiabatically eliminating the fast mediating channel, I obtain the mean-field of Bose-Hubbard model with fully tunable hopping strength, hopping range, and nonlinearity. This is the first known conservative system exhibiting chimera patterns. More importantly, I show that the model should be implementable in BECs with current technology.
Experimental realizations of chimera patterns, characterized by coexisting regions of phase coherence and incoherence, have so far only been achieved for non-conservative systems with dissipation. Moreover, theoretical studies of chimera patterns have also been limited either to the non-conservative case or to simplified models that describe the dynamics only in terms of a scalar phase field. Here, we show for the first time explicitly that the formation of chimera patterns can also be observed in conservative Hamiltonian systems with nonlocal hopping in which both energy and particle number are conserved, and where the local phase and amplitude are non-separable even in the weak coupling regime. Effective nonlocality can be realized in a physical system with only local coupling if different time scales exist, which we illustrate by a minimal conservative model with an additional mediating channel. Finally, we show that chimera patterns should be observable in ultracold atomic systems: Nonlocal spatial hopping over up to tens of lattice sites with independently tunable hopping strength and on-site nonlinearity can be implemented in a two-component Bose-Einstein condensate with a spin-dependent optical lattice, where the untrapped component serves as the matter-wave mediating field.
Many interesting physical systems, e.g. condensed matter systems, are hard to study. From the experimental point of view, in such systems most parameters are ''frozen" for a given sample. From the theoretical point of view, they often cannot be simply modelled from first principles. In contrast, laser-matter interactions are very well known theoretically, and a very high degree of control of laser beams is available for the experimentalist. In the last years, one learned to ''engineer" quantum hamiltonians using ultracold atoms interacting with electromagnetic fields. For example, laser beams may form an interference pattern that is ''seen" by the atoms as a periodic potential, thus creating an analog of a crystal. However, the analog has great advantages over the original system, as one has a larger control of its properties: One can vary the parameters of the potential by changing the properties of the laser beams, one can make a ''mesoscopic" quantum system by using a Bose-Einstein condensate, one can change it to a degenerate Fermi gas by using a fermionic isotope, one can control atom-atom interactions by using the so-called . Building on seminal ideas of Feynman, the notion of ''quantum simulator" -- a ''simpler" system able to mimic the physics of a complicate one -- recently became a reality. In a variety of situations of experimental interest, weakly interacting degenerate boson gases (Bose-Einstein condensates) are very well described by the Nonlinear Schödinger Equation with a cubic nonlinearity (called, in the present context, the {\it Gross-Pitaevskii equation}), which is much simpler than the underlying many-body problem. This opened a new field for both theoretical and experimental studies: The {\it nonlinear} quantum mechanics, where the familiar techniques like the spectral analysis in terms of eigenvalues and eigenvectors, which derive from the linearity of the usual Schödinger equation, cannot be used. The mathematical framework of this new quantum mechanics is still largely to be invented. Meanwhile, one can resort to numerical simulations -- and to experiments. In this talk, I will discuss two simple yet important examples illustrating these ideas. First, I will show how one can ''quantum simulate" with ultracold atoms the celebrated Anderson model, which describes quantum disordered systems. One can even enrich this system by adding particle-particle interactions in a controlled way. Second, I will show a simple model displaying ''quasiclassical" chaos, that is, chaotic behavior related to sensitivity to the initial conditions, which cannot exist in {\it linear} quantum mechanics. I will show that the observed behavior, although intrinsically quantum, can be described by the KAM theorem.
Experimental realizations of chimera patterns, characterized by coexisting regions of phase coherence and incoherence, have so far been achieved for non-conservative systems with dissipation and exclusively in classical settings. The possibility of observing chimera patterns in quantum systems has rarely been studied and it remains an open question if chimera patterns can exist in closed, or conservative quantum systems. Here, we tackle these challenges by first proposing a conservative Hamiltonian system with nonlocal hopping, where the energy is well-defined and conserved. We show explicitly that such a system can exhibit chimera patterns. Then we propose a physical mechanism for the nonlocal hopping by using an additional mediating channel. This leads us to propose a possible experimentally realizable quantum system based on a two-component Bose-Einstein condensate (BEC) with a spin-dependent optical lattice, where an untrapped component serves as the matter-wave mediating field. In this BEC system, nonlocal spatial hopping over tens of lattice sites can be achieved and simulations suggest that chimera patterns should be observable in certain parameter regimes.
Since the first experimental observation of the Bose-Einstein condensation of a gas of alkali atoms, six years ago, there has been an increasing interest on this new, highly interdisciplinary, branch of physics. It was realized, from the very beginning that, despite of the diluteness of such systems, the nonlinearity was crucial on the understanding of both the equilibrium and dynamical properties. In this Thesis we point out that there is a further key aspect that must be taken into account when the condensates are trapped in deep optical lattices: discreteness. The interplay between discreteness and nonlinearity raises a new class of deeply non-trivial phenomena which disappear in the continuum limit of the Gross-Pitaevskii equation (GPE). Recognizing and exploiting such interplay is the main topic of this Thesis. We think that the dynamics of a Bose-Einstein condensate (BEC) in optical lattices is going to become of central interest in the next future. Optical potentials provide BEC's waveguides, which prevent the spatial spreading of the BEC wave packets and preserve their coherence over long distances. This has allowed to build high sensitive interferometers, including gyroscopes and gravitometers. From the theoretical point of view, the big challenge is to reconsider the meaning of phenomena which have been well understood in homogeneous systems: superfluidity, for instance, is in the top list. A further example is given by quantum phase transitions. The connection and the interplay with the language and insights of the nonlinear physics community is a second challenge. We believe that the results obtained in the BEC context can provide a new prospective view into old problems. The understanding of the modulational instability in terms of phase randomization and a (classical) dynamical phase transition are among this line, as well as the possibility to generalize the Landau criteria for superfluidity to discrete systems. On the other hand, it is clear that nonlinear physicists can have much to say about BEC. With this Thesis we hope we have provided, together with original results, some links between these different languages and views.
"As the progress of science pushes the frontiers of studying nature to both the extremely large scale and the incredibly small scale, experiments can become increasingly difficult to perform. Therefore, a scale model experiment for these kinds of physics can be useful as long as it is relatively easy to access and control. This is the idea of quantum simulation, where one system exactly mimics another when they are subjected to equivalent physical scenarios. Topological systems, those with a property that is stabilized by their specific configuration, are often quite challenging to experimentally probe. Therefore, test bed systems have been widely and successfully applied to studying topology in an effort to advance the field, which promises enticing applications such as robust quantum technology. Our particular scheme uses a novel type of ""synthetic"" dimension for atoms of a Bose–Einstein condensate, which we call a momentum-space lattice, to emulate one-dimensional lattice models. The idea of a synthetic dimension or synthetic lattices is a useful construct to describe how the particle dynamics can evolve in degrees of freedom that are not directly related to transport in real space. This can relate to population dynamics in an internal degree of freedom, such as spin, or, in our case, in the dynamics of free-particle-like atomic momentum states. We expose a Bose–Einstein condensate to two laser fields that are capable of driving transitions between different momentum states of the condensate atoms through a two-photon Bragg diffraction process. Neighboring states in the free-particle dispersion relation are characterized by a unique momenta and energy spacing such that we can independently control every aspect of their coupling. In this thesis, we report on the design and construction of an apparatus for the production of 87Rb Bose–Einstein condensates as well as the partial construction of an apparatus for the production of 39K Bose–Einstein condensates. We also discuss the experimental realization of, and theoretical background for, our momentum-space lattice. Using the momentum-space lattice, we perform experiments on not only topological systems, but also on driven artificial spins featuring chaotic dynamics, as well as faster-than-adiabatic quantum state control and preparation. Our measurements of topological systems are based in the 1D Su–Schrieffer–Heeger model, where we report on the direct observation of the localized, solitonic edge state created by the topology inherent in the system. Through the addition of exactly tunable tunneling disorder, we observe the destruction of the band topology due to disorder at a random-singlet transition. Furthermore, we drive a trivial system into a topological phase through the addition of disorder, observing a counter-intuitive phenomenon known as the topological Anderson insulator for the first time. We go on to report on our driven, tunable spin experiment, where we probe effective spin ""squeezing"" (in the absence of any entanglement in our noninteracting system), the out-of-time-ordered correlation function, and chaos. And finally, we use our momentum-space lattice to demonstrate a technique which allows a system to be transformed faster than the adiabatic limit with no diabatic transitions in a highly non-trivial regime. Due to the versatility of the momentum-space lattice, the results detailed here have a broad impact. Our observations of topological phenomena in elementary and canonical 1D systems are some of the first experimental verifications of long-standing predictions. Further, our studies of the interplay between classical chaos and quantum systems may help shed light on this complex and rich problem. And from a more ""applied science"" perspective, our demonstrations of generalized counter-diabatic techniques are widely beneficial and applicable to many systems where adiabaticity is challenged."
Bose-Einstein condensates (BECs), a quantum state of matter formed when bosonic atoms are cooled close to absolute zero, have become the premier platform for investigating fundamental physics with atomic vapours. Experiments on Bose-Einstein condensates now achieve exquisite control over many aspects of the system, including interactions, trapping potential, and dynamics. This has precipitated a new wave of research into many-body quantum phenomena and, in particular, solitons. These structures are fundamental excitations of an interacting non-linear medium, of interest to a multitude of scientific disciplines from non-linear optics to financial markets. The highly controllable environment of BECs form an attractive playground for the study of solitons allowing the non-linearity to be dynamically tuned, facilitating deeper investigations into these structures. Consistently generating and analysing solitons in BEC experiments continues to be problematic. In particular, the non-linear dynamics of BECs, though required for the generation of solitons, produce particularly challenging control and optimization problems. These control problems must be solved before further investigations into the fundamental physics of soliton dynamics can be answered. This thesis makes three important advances in the control and measurement of BECs that will lead to better generation and observation of solitons. (1) a theoretical model for a control scheme capable of highly precise wavefunction engineering, (2) the experimental implementation of a machine learning algorithm for online optimisation, and (3) a continuous non-destructive imaging system capable of directly observing soliton dynamics in real-time. Together, these advances provide a suite of tools for manipulating and exploiting solitons in Bose-Einstein condensates. A novel technique was developed theoretically, offering control of the macroscopic wavefunction of a Bose-Einstein condensate with unprecedented spatial resolution and speed. The ability to control the atomic wavefunction at the fundamental length scale is key to the advancement of many quantum technologies such as quantum simulators. The magnetic resonance control scheme is demonstrated through simulation of a 87Rb condensate with the exemplar model generating a single dark soliton with corresponding π phase kink. The soliton represents a structure at the fundamental length scale of the system, and demonstrates the potential of the scheme for precision state engineering. The scheme is extended to generate higher-order soliton modes which are yet to be experimentally realised. A machine learning algorithm based on Gaussian processes was developed and implemented on the evaporative cooling stage of the production of a 87Rb Bose-Einstein condensate, successfully demonstrating fast optimisation to condensation. The Gaussian process develops a statistical model based on the data that enables the characterisation of the relationship between the experimental controls and resultant quality of the BEC. This relationship is often obfuscated through technical details of the apparatus, frustrating the use of theoretical models to design optimal evaporation ramps. These models often only consider ergodic dynamics with two-body s-wave interactions and no other loss rates with better ramps likely exploiting more complex interactions. The internal model generated from the Gaussian process utilised uncertainty in measured data, making the optimisation more robust to experimental noise than alternate methods. The algorithm is shown to produce high quality Bose-Einstein condensates in 10 times fewer experimental iterations than previously used online optimisation techniques. By exploiting information on the sensitivity of each control, the model can be used to aid experimental design. The convergence of the optimisation is further improved by eliminating a superfluous parameter identified by the model. The general usefulness of machine learning compared with bespoke optimisation algorithms has seen machine learning approach ubiquity. Finally, an experimentally straightforward technique for continuous non-destructive imaging of matter-wave solitons was developed and implemented, facilitating measurements of stochastic phenomena. The technique is readily practicable on any ultracold atom experiment with an existing absorption imaging system, simply requiring the probe laser be far-detuned from resonance. With a signal-to-noise of ∼ 33 at 1.25 GHz detuning, the technique is capable of producing 100 images with no observable heating or atom loss. Coupled with a fast optical phase locked loop, the technique can be used in conjunction with absorption imaging to generate a series of non-destructive images followed by a final high signal-to-noise absorption image solely through moving the laser on resonance for the final image. The high performance and utility of this imaging setup make it a powerful tool for ultra-cold atom experiments.