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.
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.
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.
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.
In this thesis I report on experiments that enter a new regime in the many body physics of ultracold atomic gases. A Bose-Einstein condensate is loaded into a three-dimensional optical lattice potential formed by a standing wave laser light field. In this novel quantum system we have been able to both realize a quantum phase transition from a superfluid to a Mott insulator, and to observe the collapse and revival of a macroscopic matter wave field. Quantum phase transitions are driven by quantum fluctuations and occur, even at zero temperature, as the relative strength of two competing energy terms in the underlying Hamiltonian is varied across a critical value. In the first part of this work I report on the observation of such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. In the superfluid ground state, each atom is spread-out over the entire lattice, whereas in the Mott insulating state, exact numbers of atoms are localized at individual lattice sites. We observed the reversible transition between those states and detected the gap in the excitation spectrum of the Mott insulator. A Bose-Einstein condensate is usually described by a macroscopic matter wave field. However, a quantized field underlies such a "classical" matter wave field of a Bose-Einstein condensate. The striking behavior of ultracold matter due to the field quantization and the nonlinear interactions between the atoms is the focus of the second part of this work. The matter wave field of a Bose-Einstein condensate is observed to undergo a series of collapses and revivals as time evolves. Furthermore, we show that the collisions between individual pairs of atoms lead to a fully coherent collisional phase shift in the corresponding many-particle state, which is a crucial cornerstone of proposed novel quantum computation schemes with neutral atoms. With these experiments we enter a new field of physics with ultracold quantum gases. In this strongly correlated regime, interactions between atoms dominate the behavior of the many-body system such that it can no longer be described by the usual theories for weakly interacting Bose gases. This novel quantum system offers the unique possibility to experimentally address fundamental questions of modern solid state physics, atomic physics, quantum optics, and quantum information.
Cold atoms trapped in optical lattices (crystals of light) provide a pristine platform for exploring quantum many body physics. Motivated by several recent experiments, this thesis examines the equilibrium and non-equilibrium dynamics of a Bose-Einstein condensate (BEC) loaded in a low dimensional optical lattice in order to realize novel quantum phases. There are two main research directions in this thesis. The first one involves the possibility that exotic order spontaneously forms when two-component bosons are trapped in a honeycomb lattice. My studies on this theme is motivated by the observation of a “twisted superfluid” state in Prof. Klaus Sengstock’s group at Hamburg (Soltan-Panahi et al., Nat. Phys. 8, 71 (2012)). A twisted superfluid involves Bose-Einstein condensation into a state whose order parameter has a spatially varying phase. In chapter 3, I study the stability of a Bose-Einstein condensate towards forming a twisted superfluid within the framework of mean field theory. Despite a exhaustive numerical search I do not find a parameter regime with a twisted superfluid. This search involved all experimentally relevant parameter regimes and therefore mean field theory predicted that the experimentalists should not observe a twisted superfluid. I conclude that the experimental observations were either a manifestation of counter superfluidity or due to interactions during time-of-flight. Subsequent experiments showed that the observations were an artifact of the measurement process. The second research direction in this thesis is an exploration of the stability of periodically driven quantum systems (also known as Floquet systems). Floquet systems can be used to realize exotic non-equilibrium quantum phases which do not have a counterpart in static systems. However, the driving can cause these systems to heat up which presents a major obstacle to creating exotic states. To explore this issue in a concrete example, I model an experiment (Parker, Ha, and Chin, Nat. Phys. 9, 769 (2013)) where a Bose-Einstein condensate loaded in an optical lattice is subjected to periodic shaking. I investigate the stability of this Floquet BEC to interactions. This research direction consists of 3 studies. In chapter 4, I first do this analysis for a purely one-dimensional system and identify a large parameter regime where the BEC is stable. In the next two chapters, I go beyond 1D and consider the role of transverse degrees of freedom. This is because the shaken lattice experiments that I model involves a 1D array of pancakes. I find that this geometry leads to much more dissipation than a purely 1D system. This extra dissipation arises because interactions can transfer energy between different directions. In chapter 5, I consider the extreme case where there is no transverse confinement. I find that in the absence of transverse confinement, a one-dimensional Floquet BEC is generically unstable. Finally, in chapter 6, I consider harmonic transverse confinement modeling the crossover between chapters 4 and 5. I find that as the transverse confinement is made stronger, the atom loss rate initially increases, but beyond a critical transverse confinement, the atom loss disappears due to unavailability of phase space for scattering. I also predict that if the transverse confinement is tuned to the vicinity of certain magic values, the heating rate exhibits a sharp drop. I perform similar analyses for a shaken square lattice and find that generically a low-dimensional Floquet BEC can be stabilized by suitably designing the transverse confinement.
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.
In the field of ultracold atoms in optical lattices a plethora of phenomena governed by the hopping energy J and the interaction energy U have been studied in recent years. However, the trapping potential typically present in these systems sets another energy scale and the effects of the corresponding time scale on the quantum dynamics have rarely been considered. Here we study the quantum collapse and revival of a lattice Bose-Einstein condensate (BEC) in an arbitrary spatial potential, focusing on the special case of harmonic confinement. Analyzing the time evolution of the single-particle density matrix, we show that the physics arising at the (temporally) recurrent quantum phase revivals is essentially captured by an effective single-particle theory. This opens the possibility of preparing exotic nonequilibrium condensate states with a large degree of freedom by engineering the underlying spatial lattice potential.