We present a flexible scheme to realize exact flat Landau levels on curved spherical geometry in a system of spinful cold atoms. This is achieved by applying the Floquet engineering of a magnetic quadrupole field to create a synthetic monopole field in real space. The system can be exactly mapped to the electron-monopole system on a sphere, thus realizing Haldane's spherical geometry for fractional quantum Hall physics. This method works for either bosons or fermions. We investigate the ground-state vortex pattern for an s-wave interacting atomic condensate by mapping this system to the classical Thompson's problem. The distortion and stability of the vortex pattern are further studied in the presence of dipolar interaction. Our scheme is compatible with the current experimental setup, and may serve as a promising route of investigating quantum Hall physics and exotic spinor vortex matter on curved space.
We present a flexible scheme to realize exact flat Landau levels on curved spherical geometry in a system of spinful cold atoms. This is achieved by applying the Floquet engineering of a magnetic quadrupole field to create a synthetic monopole field in real space. The system can be exactly mapped to the electron-monopole system on a sphere, thus realizing Haldane's spherical geometry for fractional quantum Hall physics. This method works for either bosons or fermions. We investigate the ground-state vortex pattern for an s-wave interacting atomic condensate by mapping this system to the classical Thompson's problem. The distortion and stability of the vortex pattern are further studied in the presence of dipolar interaction. Our scheme is compatible with the current experimental setup, and may serve as a promising route of investigating quantum Hall physics and exotic spinor vortex matter on curved space.
Magnetic monopoles are crucial in explaining the quantization of electric charges and quantum Hall effects, while artificially creating a minimal magnetic monopole in experiments remains a challenge. Here we develop a flexible way to simulate a half-integer-type monopole in Bose gases and investigate the induced vortex dynamics on a sphere. We list the possible experiment parameter settings for different isotopes and discuss their experimental feasibility. With the assumption of a rigid monopole-vortex structure, we analytically predict the vortex trajectory in an external magnetic field. We then confirm the result by numerically solving the Gross-Pitaevskii equation, which employs two gauges simultaneously (the Wu-Yang approach) to prevent singularity in the one-gauge method when a monopole is present. The study offers significant insight into the characteristics of monopoles and vortices, facilitating avenues for experimental validation.
We propose and study systems of coupled atomic wires in a perpendicular synthetic magnetic field as a platform to realize exotic phases of quantum matter. This includes (fractional) quantum Hall states in arrays of many wires inspired by the pioneering work [C. L. Kane et al., Phys. Rev. Lett. 88, 036401 (2002)], as well as Meissner phases and vortex phases in double wires. With one continuous and one discrete spatial dimension, the proposed setup naturally complements recently realized discrete counterparts, i.e., the Harper-Hofstadter model and the two-leg flux ladder, respectively. We present both an in-depth theoretical study and a detailed experimental proposal to make the unique properties of the semicontinuous Harper-Hofstadter model accessible with cold-atom experiments. For the minimal setup of a double wire, we explore how a subwavelength spacing of the wires can be implemented. This construction increases the relevant energy scales by at least an order of magnitude compared to ordinary optical lattices, thus rendering subtle many-body phenomena such as Lifshitz transitions in Fermi gases observable in an experimentally realistic parameter regime. For arrays of many wires, we discuss the emergence of Chern bands with readily tunable flatness of the dispersion and show how fractional quantum Hall states can be stabilized in such systems. Using for the creation of optical potentials Laguerre-Gauss beams that carry orbital angular momentum, we detail how the coupled atomic wire setups can be realized in nonplanar geometries such as cylinders, disks, and tori.
Representative wave functions, which encode the topological properties of the spin polarized fractional quantum Hall states in the lowest Landau level, can be expressed in terms of correlation functions in conformal field theories. Until now, the constructions have been restricted to flat geometries, but in this paper we generalize to the simplest curved geometry, namely that of a sphere. Except for being of interest for numerical studies, that usually are performed on a sphere, the response of the fractional quantum Hall (FQH) liquids to curvature can be used to detect a topological quantity, the shift S which is the average orbital spin of the constituent electrons. We give explicit expressions for representative wave functions on the sphere, for the full Abelian FQH hierarchy, and calculate the corresponding shifts. These microscopic results, based on wave functions, agree with the predictions from the effective Chern-Simons field theory. The methods we develop can also be applied to the planar case. It gives simpler expressions for states with both quasiparticle and quasihole condensates, and allows us to give closed form expressions for a general state in the hierarchy, rather than finding the wave function on a case by case basis.
Artificial monopoles have been engineered in various systems, yet there has been no systematic study of the singular vector potentials associated with the monopole field. We show that the Dirac string, the line singularity of the vector potential, can be engineered, manipulated, and made manifest in a spinor atomic condensate. We elucidate the connection among spin, orbital degrees of freedom, and the artificial gauge, and show that there exists a mapping between the vortex filament and the Dirac string. We also devise a proposal where preparing initial spin states with relevant symmetries can result in different vortex patterns, revealing an underlying correspondence between the internal spin states and the spherical vortex structures. Such a mapping also leads to a new way of constructing spherical Landau levels, and monopole harmonics. Our observation provides insights into the behavior of quantum matter possessing internal symmetries in curved spaces. Published by the American Physical Society 2024
Interplay between matter and fields in physical spaces with nontrivial geometries gives rise to many exotic quantum phenomena. However, their realizations are often impeded by experimental constraints. Here, we realize a Bose-Einstein condensate (BEC) on a synthetic cylindrical surface subject to a net radial synthetic magnetic flux, topologically equivalent to a two-dimensional (2D) Hall ribbon with two edges connected. This cylindrical surface comprises a real spatial dimension and a curved synthetic dimension formed by cyclically-coupled spin states. The BEC on such a Hall cylinder has counterintuitive properties unattainable by its counterparts in 2D planes. We observe Bloch oscillations of the BEC with doubled periodicity of the band structure, analogous to traveling on a Mobius strip, reflecting the BEC's emergent crystalline order with nonsymmorphic symmetry-protected band crossings. We further demonstrate such topological operations as gapping the band crossings and unzipping the cylinder. Our work opens the door to engineering synthetic curved spaces and observing intriguing quantum phenomena inherent to the topology of spaces.
The realization of synthetic gauge fields has attracted a lot of attention recently in relation to periodically driven systems and the Floquet theory. In ultracold atom systems in optical lattices and photonic networks, this allows one to simulate exotic phases of matter such as quantum Hall phases, anomalous quantum Hall phases, and analogs of topological insulators. In this paper, we apply the Floquet theory to engineer anisotropic Haldane models on the honeycomb lattice and two-leg ladder systems. We show that these anisotropic Haldane models still possess a topologically nontrivial band structure associated with chiral edge modes. Focusing on (interacting) boson systems in s-wave bands of the lattice, we show how to engineer through the Floquet theory, a quantum phase transition (QPT) between a uniform superfluid and a Bose-Einstein condensate analog of Fulde-Ferrell-Larkin-Ovchinnikov states, where bosons condense at nonzero wave vectors. We perform a Ginzburg-Landau analysis of the QPT on the graphene lattice, and compute observables such as chiral currents and the momentum distribution. The results are supported by exact diagonalization calculations and compared with those of the isotropic situation. The validity of high-frequency expansion in the Floquet theory is also tested using time-dependent simulations for various parameters of the model. Last, we show that the anisotropic choice for the effective vector potential allows a bosonization approach in equivalent ladder (strip) geometries.